**Meet the editor**

Dr. Amimul Ahsan was born in Netrokona, Bangladesh, 1982. He is a Senior Lecturer, Department of Civil Engineering, Faculty of Engineering, University Putra Malaysia, Malaysia. He received a Ph.D in Civil Engineering from the University of Fukui, Japan. He has been involved in collaboration research with many researchers and scientists in the world since 2004 and published

in diverse areas of Civil and Environmental Engineering. He has published 3 books, 6 book chapters, over 19 technical papers, and is a member of the editorial board of numerous international technical journals. He has been serving on the scientific and editorial board of InTech Open Access publisher since 2010.

Contents

**Preface IX** 

**Part 1 Heat Convection 1** 

Chapter 1 **A Mixed Convection Study in Inclined** 

Chapter 2 **Periodically Forced Natural Convection** 

M. A. H. Mamun, T. R. Tanim,

Chapter 4 **Convective Heat Transfer of Unsteady** 

C. Amrouche, M. Batchi and R. Creff

Gregor Kosec and Božidar Šarler

**Capillary Porous Bodies 149** 

Chapter 6 **Hydromagnetic Flow with Thermal Radiation 133**  Cho Young Han and Se-Myong Chang

J. Batina, S. Blancher,

Chapter 5 **Numerical Solution of Natural** 

**Part 2 Heat Conduction 147** 

Nencho Deliiski

Chapter 7 **Transient Heat Conduction in** 

Suvash Chandra Saha

Chapter 3 **Analysis of Mixed Convection** 

**Channels with Discrete Heat Sources 3**  Paulo M. Guimarães and Genésio J. Menon

**in a Lid Driven Trapezoidal Cavity 55** 

M. M. Rahman, R. Saidur and Shuichi Nagata

**Pulsed Flow in Sinusoidal Constricted Tube 83** 

**Convection Problems by a Meshless Method 107** 

**Through the Roof of an Attic-Shaped Building 33** 

### Contents

#### **Preface** XI


Nencho Deliiski

X Contents


### Preface

The convection and conduction heat transfer, thermal conductivity, and phase transformations are significant issues in a design of wide range of industrial processes and devices. This book includes 18 advanced and revised contributions, and it covers mainly (1) heat convection, (2) heat conduction, and (3) heat transfer analysis. The first section introduces mixed convection studies on inclined channels, double diffusive coupling, and on lid driven trapezoidal cavity, forced natural convection through a roof, convection on non-isothermal jet oscillations, unsteady pulsed flow, and hydromagnetic flow with thermal radiation. The second section covers heat conduction in capillary porous bodies and in structures made of functionally graded materials, integral transforms for heat conduction problems, non-linear radiative-conductive heat transfer, thermal conductivity of gas diffusion layers and multi-component natural systems, thermal behavior of the ink, primer and paint, heating in biothermal systems, and RBF finite difference approach in heat conduction. The third section includes heat transfer analysis of reinforced concrete beam, modeling of heat transfer and phase transformations, boundary conditionssurface heat flux and temperature, simulation of phase change materials, and finite element methods of factorial design. The advanced idea and information described here will be fruitful for the readers to find a sustainable solution in an industrialized society.

#### **Acknowledgements**

All praise be to Almighty Allah, the Creator and the Sustainer of the world, the Most Beneficent, Most Benevolent, Most Merciful, and Master of the Day of Judgment. He is Omnipresent and Omnipotent. He is the King of all kings of the world. In His hand is all good. Certainly, over all things Allah hast power.

The editor would like to express appreciation to all who have helped to prepare this book. The editor expresses his gratefulness to Ms. Ivana Lorkovic, Publishing Process Manager, InTech Open Access Publisher for its continued cooperation. In addition, the editor appreciatively remembers the assistance of all authors and reviewers of this book.

#### XII Preface

Gratitude is expressed to Mrs. Ahsan, Ibrahim Bin Ahsan, Mother, Father, Mother-in-Law, Father-in-Law, and Brothers and Sisters for their endless inspirations, mental supports and also necessary help whenever any difficulty.

> **Amimul Ahsan, Ph.D.** Department of Civil Engineering Faculty of Engineering University Putra Malaysia Malaysia

## **Part 1**

**Heat Convection** 

**1** 

*Brazil* 

**A Mixed Convection Study in** 

Paulo M. Guimarães and Genésio J. Menon

*Federal University of Itajubá* 

**Inclined Channels with Discrete Heat Sources** 

In the last two decades, heat transfer study on discrete heat sources has become a subject of increased interest due to advances in the electronics industry. Increased power dissipation is the most significant feature of new generation electronic devices and more significant heat flux densities are obtained as a result of miniaturization. Consequently, the assumption of cooling of electronic devices has increased interest in the analysis of fluid flow and heat transfer in discrete heating situations. Previous works have studied the natural, mixed, and forced convection in inclined channels due to their practical applications such as electronic systems, high performance heat exchangers, chemical process equipments, combustion

An interesting study was reported on the fluid flow and heat transfer characteristics associated with cooling an in-line array of discrete protruding heated blocks in a channel by using a single laminar slot air jet (Arquis et al., 2007). Numerical experiments were carried out for different values of jet Reynolds number, channel height, slot width, spacing between blocks, block height, and block thermal conductivity. The effects of variation of these parameters were detailed to illustrate important fundamental and practical results that are relevant to the thermal management of electronic packages. In general, the effective cooling of blocks was observed to increase with the increase of Reynolds number and the decrease of channel height. Circulation cells that may appear on the top surface of the downstream blocks were shown to decrease the value of Nusselt number for these blocks. The values of surface averaged Nusselt number attained their maximum at the block just underneath the impinging air jet, decreased for the downstream blocks, and approximately reached a

A numerical study (Madhusudhana & Narasimham, 2007) was carried out on conjugate mixed convection arising from protruding heat generating ribs attached to substrates forming a series of vertical parallel plate channels. A channel with periodic boundary conditions in the transverse direction was considered for analysis where identical disposition and heat generation of the ribs on each board were assumed. The governing equations were discretised using a control volume approach on a staggered mesh and a pressure correction method was employed for the pressure–velocity coupling. The solid regions were considered as fluid regions with infinite viscosity; and the thermal coupling between the solid and fluid regions was taken into account by the harmonic thermal

**1. Introduction** 

chambers, environmental control systems and so on.

constant value after the third block.

## **A Mixed Convection Study in Inclined Channels with Discrete Heat Sources**

Paulo M. Guimarães and Genésio J. Menon *Federal University of Itajubá Brazil* 

#### **1. Introduction**

In the last two decades, heat transfer study on discrete heat sources has become a subject of increased interest due to advances in the electronics industry. Increased power dissipation is the most significant feature of new generation electronic devices and more significant heat flux densities are obtained as a result of miniaturization. Consequently, the assumption of cooling of electronic devices has increased interest in the analysis of fluid flow and heat transfer in discrete heating situations. Previous works have studied the natural, mixed, and forced convection in inclined channels due to their practical applications such as electronic systems, high performance heat exchangers, chemical process equipments, combustion chambers, environmental control systems and so on.

An interesting study was reported on the fluid flow and heat transfer characteristics associated with cooling an in-line array of discrete protruding heated blocks in a channel by using a single laminar slot air jet (Arquis et al., 2007). Numerical experiments were carried out for different values of jet Reynolds number, channel height, slot width, spacing between blocks, block height, and block thermal conductivity. The effects of variation of these parameters were detailed to illustrate important fundamental and practical results that are relevant to the thermal management of electronic packages. In general, the effective cooling of blocks was observed to increase with the increase of Reynolds number and the decrease of channel height. Circulation cells that may appear on the top surface of the downstream blocks were shown to decrease the value of Nusselt number for these blocks. The values of surface averaged Nusselt number attained their maximum at the block just underneath the impinging air jet, decreased for the downstream blocks, and approximately reached a constant value after the third block.

A numerical study (Madhusudhana & Narasimham, 2007) was carried out on conjugate mixed convection arising from protruding heat generating ribs attached to substrates forming a series of vertical parallel plate channels. A channel with periodic boundary conditions in the transverse direction was considered for analysis where identical disposition and heat generation of the ribs on each board were assumed. The governing equations were discretised using a control volume approach on a staggered mesh and a pressure correction method was employed for the pressure–velocity coupling. The solid regions were considered as fluid regions with infinite viscosity; and the thermal coupling between the solid and fluid regions was taken into account by the harmonic thermal

A Mixed Convection Study in Inclined Channels with Discrete Heat Sources 5

span influence was important, whenever it was less than 0.45, and eventually, the melting times were shorter and the temperatures on the sources remained equal and moderate

The turbulent convection heat transfer was experimentally investigated in an array of discrete heat sources inside a rectangular channel filled with air (Baskaya et al., 2005). The lower surface of the channel was equipped with 8x4 flush-mounted heat sources subjected to uniform heat flux. The sidewalls and the upper wall were insulated. The experimental parametric study was made for a constant aspect ratio, different Reynolds numbers, and modified Grashof numbers. Results showed that surface temperatures increased with increasing Grashof number and decreased with increasing Reynolds number. However, the increase in the buoyancy forces affected the secondary flow and the onset of instability, and, hence, the temperatures levelled off and even dropped as a result of heat transfer enhancement. This outcome could also be observed from the variation of the row-averaged

A constructal theory was applied to the fundamental problem of how to arrange discrete heat sources on a wall cooled by forced convection (Silva et al., 2004). They aimed to maximize the conductance between the discrete heated wall and the fluid, that is, to minimize the temperature of the hot spot on the wall, when the heat generation rate was specified. The global objective was achieved by the generation of flow configuration, in this case, the distribution of discrete heat sources. Two different analytical approaches were used: (i) large number of small heat sources, and (ii) small number of heat sources with finite length, which were mounted on a flat wall. Both analyses showed that the heat sources should have been placed non-uniformly on the wall, with the smallest distance between them near the tip of the boundary layer. When the Reynolds number was high enough, then, the heat sources should have been mounted flushed against each other, near the entrance of the channel. The analytical results were validated by a numerical study of discrete heat sources that were non-uniformly distributed inside a channel formed by parallel plates. In the present chapter, a heat transfer study in an inclined rectangular channel with heat sources is conducted. The heat source vertical and horizontal positions are also considered. Emphasis is given to the heat transfer distributions on the heat modules, showing their correlation with velocities due to their importance when thermal control in electronic

Figure (1) depicts three heat source lay-outs that will be studied. A mixed convection study is performed in a channel with height H, length L, and inclination γ*.* At the inlet, a constant velocity and temperature profiles, Uo and To, respectively, are imposed. The open boundary conditions (OBC) are arranged in a way that they are calculated, that is, the pressure terms are retrieved in the calculation (Heinrich & Pepper, 1999). Therefore, nothing is directly applied at the open boundary. For more information on this, the reader should refer to (Heinrich & Pepper, 1999). The reference and cooling temperatures To and Tc, respectively, are the same and equal to zero. Initially, the internal fluid domain has velocities and temperatures equal to zero. All surfaces present the no-slip condition. The regime is non-

*One heat source*: One heat source with heat flux q' of finite length B is placed at x1 on the lower surface. Heat transfer will be analyzed according to the variation of the Reynolds

steady, two-dimensional, and laminar. Next, all three situations are described.

throughout the melting process.

equipments is aimed.

**2. Problem description** 

Nusselt number showing an increase towards the exit.

conductivity method. Parametric studies were performed by varying the heat generation based on Grashof number in the range 104–107 and the fan velocity was based on Reynolds number in the range 0–1500, with air as the working fluid. In pure natural convection, the induced mass flow rate varied at 0.44 power of Grashof number. The heat transferred to the working fluid via substrate heat conduction was found to account for 41–47% of the heat removal from the ribs.

The optimum position of a discrete heater was determined by maximizing the conductance and the heat transfer and volume flow rate with the discrete heater at its optimum position in open cavities by using the finite difference-control volume numerical method and considering air (Muftuoglu & Bilgen, 2007). The relevant governing parameters were: the Rayleigh numbers from 106 to 1012, the cavity aspect ratio from 0.5 to 2, the wall thickness from 0.05 to 0.15, the heater size from 0.15 to 0.6, and the conductivity ratio from 1 to 50. They found that the global conductance was an increasing function of the Rayleigh number and the conductivity ratio, and a decreasing function of the wall thickness. The best thermal performance was achieved by positioning the discrete heater eccentrically and slightly closer to the bottom. The Nusselt number and the volume flow rate in and out the open cavity were an increasing function of the Rayleigh number and the wall thickness, and a decreasing function of the conductivity ratio. The Nusselt number was a decreasing function of the cavity aspect ratio and the volume flow rate was an increasing function of it.

Another work conducted a numerical investigation of conjugate convection with surface radiation from horizontal channels with protruding heat sources (Premachandran & Balaji, 2006). The air flow was assumed to be steady, laminar, incompressible, and hydrodynamically and thermally developed. The geometric parameters such as spacing between the channel walls, size of the protruding heat sources, thickness of the substrate and the spacing between the heat sources were fixed. One of the most relevant conclusions was that while carrying out a thermal analysis of a stack of circuit boards with electronic chips (discrete heat sources), the consideration of radiation heat transfer was absolutely essential to accurately predict the non-dimensional maximum temperature.

The mixed convection heat transfer in a top-and-bottom-heated rectangular channel with discrete heat sources using air was experimentally investigated (Dogan et al., 2005). The lower and upper surfaces of the channel were equipped with 8x4 flush-mounted heat sources subjected to uniform heat flux. The lateral and remaining lower and upper walls were insulated. The experimental study was carried out for an aspect ratio equal to 6, Reynolds numbers varying from 955 to 2220 and modified Grashof numbers Gr\*=1.7x107 to 6.7x107. The surface temperature and Nusselt number distributions on the discrete heat sources were obtained. Results showed that the surface temperatures increased with increasing Grashof number. The row-averaged Nusselt numbers decreased with the row number and then, they showed an increase towards the exit as a result of heat transfer enhancement due to the invigoration of buoyancy forces that affected the secondary flow.

A work investigated melting from heat sources that are flush-mounted on discretely heated rectangular vertical enclosures (Binet & Lacroix, 2000). It finds its application in design and operation of thermal energy storage units and the cooling of electronic equipment. The results showed that there were benefits of discrete heating when it comes to optimizing the melting process. The aspect ratio was an important factor that may have led to controlled temperatures on the heat modules. For aspect ratios over 4, controlled temperatures and long melting time were obtained. On the other hand, for aspect ratios up to 4, the source

conductivity method. Parametric studies were performed by varying the heat generation based on Grashof number in the range 104–107 and the fan velocity was based on Reynolds number in the range 0–1500, with air as the working fluid. In pure natural convection, the induced mass flow rate varied at 0.44 power of Grashof number. The heat transferred to the working fluid via substrate heat conduction was found to account for 41–47% of the heat

The optimum position of a discrete heater was determined by maximizing the conductance and the heat transfer and volume flow rate with the discrete heater at its optimum position in open cavities by using the finite difference-control volume numerical method and considering air (Muftuoglu & Bilgen, 2007). The relevant governing parameters were: the Rayleigh numbers from 106 to 1012, the cavity aspect ratio from 0.5 to 2, the wall thickness from 0.05 to 0.15, the heater size from 0.15 to 0.6, and the conductivity ratio from 1 to 50. They found that the global conductance was an increasing function of the Rayleigh number and the conductivity ratio, and a decreasing function of the wall thickness. The best thermal performance was achieved by positioning the discrete heater eccentrically and slightly closer to the bottom. The Nusselt number and the volume flow rate in and out the open cavity were an increasing function of the Rayleigh number and the wall thickness, and a decreasing function of the conductivity ratio. The Nusselt number was a decreasing function

of the cavity aspect ratio and the volume flow rate was an increasing function of it.

accurately predict the non-dimensional maximum temperature.

Another work conducted a numerical investigation of conjugate convection with surface radiation from horizontal channels with protruding heat sources (Premachandran & Balaji, 2006). The air flow was assumed to be steady, laminar, incompressible, and hydrodynamically and thermally developed. The geometric parameters such as spacing between the channel walls, size of the protruding heat sources, thickness of the substrate and the spacing between the heat sources were fixed. One of the most relevant conclusions was that while carrying out a thermal analysis of a stack of circuit boards with electronic chips (discrete heat sources), the consideration of radiation heat transfer was absolutely essential to

The mixed convection heat transfer in a top-and-bottom-heated rectangular channel with discrete heat sources using air was experimentally investigated (Dogan et al., 2005). The lower and upper surfaces of the channel were equipped with 8x4 flush-mounted heat sources subjected to uniform heat flux. The lateral and remaining lower and upper walls were insulated. The experimental study was carried out for an aspect ratio equal to 6, Reynolds numbers varying from 955 to 2220 and modified Grashof numbers Gr\*=1.7x107 to 6.7x107. The surface temperature and Nusselt number distributions on the discrete heat sources were obtained. Results showed that the surface temperatures increased with increasing Grashof number. The row-averaged Nusselt numbers decreased with the row number and then, they showed an increase towards the exit as a result of heat transfer enhancement due to the invigoration of buoyancy forces that affected the secondary flow. A work investigated melting from heat sources that are flush-mounted on discretely heated rectangular vertical enclosures (Binet & Lacroix, 2000). It finds its application in design and operation of thermal energy storage units and the cooling of electronic equipment. The results showed that there were benefits of discrete heating when it comes to optimizing the melting process. The aspect ratio was an important factor that may have led to controlled temperatures on the heat modules. For aspect ratios over 4, controlled temperatures and long melting time were obtained. On the other hand, for aspect ratios up to 4, the source

removal from the ribs.

span influence was important, whenever it was less than 0.45, and eventually, the melting times were shorter and the temperatures on the sources remained equal and moderate throughout the melting process.

The turbulent convection heat transfer was experimentally investigated in an array of discrete heat sources inside a rectangular channel filled with air (Baskaya et al., 2005). The lower surface of the channel was equipped with 8x4 flush-mounted heat sources subjected to uniform heat flux. The sidewalls and the upper wall were insulated. The experimental parametric study was made for a constant aspect ratio, different Reynolds numbers, and modified Grashof numbers. Results showed that surface temperatures increased with increasing Grashof number and decreased with increasing Reynolds number. However, the increase in the buoyancy forces affected the secondary flow and the onset of instability, and, hence, the temperatures levelled off and even dropped as a result of heat transfer enhancement. This outcome could also be observed from the variation of the row-averaged Nusselt number showing an increase towards the exit.

A constructal theory was applied to the fundamental problem of how to arrange discrete heat sources on a wall cooled by forced convection (Silva et al., 2004). They aimed to maximize the conductance between the discrete heated wall and the fluid, that is, to minimize the temperature of the hot spot on the wall, when the heat generation rate was specified. The global objective was achieved by the generation of flow configuration, in this case, the distribution of discrete heat sources. Two different analytical approaches were used: (i) large number of small heat sources, and (ii) small number of heat sources with finite length, which were mounted on a flat wall. Both analyses showed that the heat sources should have been placed non-uniformly on the wall, with the smallest distance between them near the tip of the boundary layer. When the Reynolds number was high enough, then, the heat sources should have been mounted flushed against each other, near the entrance of the channel. The analytical results were validated by a numerical study of discrete heat sources that were non-uniformly distributed inside a channel formed by parallel plates.

In the present chapter, a heat transfer study in an inclined rectangular channel with heat sources is conducted. The heat source vertical and horizontal positions are also considered. Emphasis is given to the heat transfer distributions on the heat modules, showing their correlation with velocities due to their importance when thermal control in electronic equipments is aimed.

#### **2. Problem description**

Figure (1) depicts three heat source lay-outs that will be studied. A mixed convection study is performed in a channel with height H, length L, and inclination γ*.* At the inlet, a constant velocity and temperature profiles, Uo and To, respectively, are imposed. The open boundary conditions (OBC) are arranged in a way that they are calculated, that is, the pressure terms are retrieved in the calculation (Heinrich & Pepper, 1999). Therefore, nothing is directly applied at the open boundary. For more information on this, the reader should refer to (Heinrich & Pepper, 1999). The reference and cooling temperatures To and Tc, respectively, are the same and equal to zero. Initially, the internal fluid domain has velocities and temperatures equal to zero. All surfaces present the no-slip condition. The regime is nonsteady, two-dimensional, and laminar. Next, all three situations are described.

*One heat source*: One heat source with heat flux q' of finite length B is placed at x1 on the lower surface. Heat transfer will be analyzed according to the variation of the Reynolds

A Mixed Convection Study in Inclined Channels with Discrete Heat Sources 7

Tc = constant

L

L

L

B

H

X3

Fig. 1. Geometries and boundary conditions: cases with one, two, and three heat sources

q'3

Tc = constant

q'2

Adiabatic Wall

q'2

B

H

Adiabatic Wall Tc = constant

Exit

OBC

Exit

OBC

H

Adiabatic Wall

B

g

Constant Heat Source (q')

X1 x x

1

2

q'1

> d

X2

g

X1 q'1

> X2

y

Inlet

Inlet Uo = constant To = constant

Uo = constant

To = constant

y

y

Inlet Uo = constant To = constant

γ(+)

x

x

Exit OBC

γ(+)

x

number (Re), the Grashof number (Gr), and the inclination angle (γ). B = H, and L = 15H. *Two heat sources*: Two constant-flux heat sources, q'1and q'2 of finite length B are placed at x1 and x2 on the lower surface. The distance between the heaters is d. Throughout this case, the geometry has x1 = 5H, x2 = 7H, 8H, and 9H, and hence characterizing d = 1, 2, and 3, respectively. Heat transfer will be analyzed according to the variation of Re, Gr, and d. The inclination angle is zero, that is, the channel is in the horizontal position. B = H, and L = 15H. *Three heat sources*: It is an inclined rectangular channel with height H and length L. Three constant heat sources q'1, q'2, and q'3 of length B are placed on the bottom wall at x1, x2, and x3, respectively. The remaining lower wall is isolated. The upper wall in contact with the fluid is constantly cooled at a uniform temperature Tc. At the inlet, constant velocity and temperature profiles, Uo and To, are applied as boundary conditions. Throughout this case, the geometry has x1 = 6.75H, x2 = 14.50H, x3 = 22.25H, B = H = 1, and L = 30H.

#### **3. Problem formulation and the numerical method**

The problem governing equation set is given by the continuity, Navier-Stokes, and energy equations. Variables u and v are, respectively, the velocity components in x and y directions, T is the fluid temperature, t' is the dimensional time, q' is the heat flux, *DT* is the fluid thermal diffusivity, β*T* is the thermal expansion coefficient, ν is the kinematics viscosity, g is the gravity acceleration, and ρ0 is the fluid density and To = Tc and ∆T = Th – Tc. Considering the Boussinesq approximation and the following dimensionless parameters:

$$\begin{aligned} \text{X = \frac{\text{x}}{B}; \quad Y = \frac{\text{y}}{B}; \quad \text{U = \frac{\text{u}}{\text{U}\_o}; \quad V = \frac{\text{v}}{\text{U}\_o}; \quad P = \frac{p}{\rho\_0 \text{U}\_o^2}; \; t = \frac{t^\circ}{\text{(B / } \text{U}\_o)^\circ}; \; \theta = \text{(T - T\_o) / \text{(}\text{q}^\circ\text{B / D}\_T\text{)} \\\ \text{Fr} = \frac{\text{Re}^2}{\text{Gr}} = \frac{\text{U}\_o^2}{\beta\_T \text{g} \Delta \text{TB}}; \quad \text{Pr} = \frac{\nu}{D\_T}; \; \text{Gr} = \frac{\beta\_T \text{g} \Delta \text{TB}^3}{\nu^2}; \; \text{Re} = \frac{\text{U}\_o \rho\_0 \text{B}}{\mu}; \end{aligned} \tag{1}$$

where Fr, Pr, Gr, Re, Uo , and μ are, respectively, the Froude number, the Prandtl number, the Grashof number, the Reynolds number, the average velocity, and the dynamic viscosity, the dimensionless governing equations can be cast into the following form:

$$\frac{\partial \mathcal{U}}{\partial X} + \frac{\partial \mathcal{V}}{\partial Y} = 0 \,\,\,\,\,\tag{2}$$

$$\frac{\partial \mathcal{U}}{\partial t} + \mathcal{U}\frac{\partial \mathcal{U}}{\partial X} + V\frac{\partial \mathcal{U}}{\partial Y} = \cdot \frac{\partial P}{\partial X} + \frac{1}{\text{Re}} \left(\frac{\partial^2 \mathcal{U}}{\partial X^2} + \frac{\partial^2 \mathcal{U}}{\partial Y^2}\right) + \text{sin}(\mathcal{r})\frac{\theta}{Fr} \tag{3}$$

$$\frac{\partial V}{\partial t} + \mathcal{U}\frac{\partial V}{\partial X} + V\frac{\partial V}{\partial Y} = -\frac{\partial P}{\partial Y} + \frac{1}{\text{Re}}\left(\frac{\partial^2 V}{\partial X^2} + \frac{\partial^2 V}{\partial Y^2}\right) + \cos(\gamma)\frac{\theta}{Fr};\tag{4}$$

$$\frac{\partial \mathcal{O}}{\partial t} + \mathcal{U} \frac{\partial \mathcal{O}}{\partial X} + V \frac{\partial \mathcal{O}}{\partial Y} = \frac{1}{\text{Re} \Pr} \left( \frac{\partial^2 \theta}{\partial X^2} + \frac{\partial^2 \theta}{\partial Y^2} \right);\tag{5}$$

The dimensionless boundary conditions are:

UV0 = = (all walls); θ = 0 (inlet and upper walls); U 1 = (inlet) and - 1 Y ∂θ<sup>=</sup> <sup>∂</sup> (heat sources).

number (Re), the Grashof number (Gr), and the inclination angle (γ). B = H, and L = 15H. *Two heat sources*: Two constant-flux heat sources, q'1and q'2 of finite length B are placed at x1 and x2 on the lower surface. The distance between the heaters is d. Throughout this case, the geometry has x1 = 5H, x2 = 7H, 8H, and 9H, and hence characterizing d = 1, 2, and 3, respectively. Heat transfer will be analyzed according to the variation of Re, Gr, and d. The inclination angle is zero, that is, the channel is in the horizontal position. B = H, and L = 15H. *Three heat sources*: It is an inclined rectangular channel with height H and length L. Three constant heat sources q'1, q'2, and q'3 of length B are placed on the bottom wall at x1, x2, and x3, respectively. The remaining lower wall is isolated. The upper wall in contact with the fluid is constantly cooled at a uniform temperature Tc. At the inlet, constant velocity and temperature profiles, Uo and To, are applied as boundary conditions. Throughout this case,

The problem governing equation set is given by the continuity, Navier-Stokes, and energy equations. Variables u and v are, respectively, the velocity components in x and y directions, T is the fluid temperature, t' is the dimensional time, q' is the heat flux, *DT* is the fluid

ν

2

ν



> 2 2 2 2

 θθ⎛ ⎞

'

0 is the fluid density and To = Tc and ∆T = Th – Tc.

( ) ( )( ) <sup>2</sup>

<sup>+</sup> <sup>=</sup> <sup>∂</sup> <sup>∂</sup> ; (2)

γ

γ

cos

θ

are, respectively, the Froude number, the Prandtl number,

0

ρ

θ

θ

Y ∂θ

 μ

is the kinematics viscosity, g

*o T*

; (3)

; (4)

; (5)

<sup>=</sup> <sup>∂</sup> (heat sources).

(1)

the geometry has x1 = 6.75H, x2 = 14.50H, x3 = 22.25H, B = H = 1, and L = 30H.

*T* is the thermal expansion coefficient,

Considering the Boussinesq approximation and the following dimensionless parameters:

0 2 2 3

*U U g TB <sup>B</sup> Fr Gr*

the Grashof number, the Reynolds number, the average velocity, and the dynamic viscosity,

<sup>0</sup> *U V X Y* ∂ ∂

Re

1

Re

1 RePr

= 0 (inlet and upper walls); U 1 = (inlet) and - 1

*t X YY X Y Fr*

*t XY X Y* ∂ ∂ ∂ ∂∂

++= + ⎜ ⎟ ∂∂ ∂ ∂ ∂ ⎝ ⎠

 θ

*t X YX X Y Fr*

*x uv y p <sup>t</sup> XYU V P t T T qB D BBU U <sup>U</sup> B U*

*o o o o*

ν

<sup>Δ</sup> = = = = <sup>=</sup> <sup>Δ</sup>

ρ

;; ; ; ; ; - / ' / / '

Re ; Pr ; ; Re ;

*o o T*

β

ρ

*T T*

the dimensionless governing equations can be cast into the following form:

*U U UP UU U V*

*V V VP VV U V*

*U V*

θ

θ

The dimensionless boundary conditions are:

θ

UV0 = = (all walls);

∂ ∂ ∂∂ ∂∂ ⎛ ⎞ + + =+ + + ⎜ ⎟ ∂ ∂ ∂∂ ∂ ∂ ⎝ ⎠

∂ ∂ ∂∂ ∂∂ ⎛ ⎞ + + =+ + + ⎜ ⎟ ∂ ∂ ∂∂ ∂ ∂ ⎝ ⎠

= = = = = = =

*Gr g TB D*

μ

β

**3. Problem formulation and the numerical method** 

β

is the gravity acceleration, and

where Fr, Pr, Gr, Re, Uo , and

thermal diffusivity,

Fig. 1. Geometries and boundary conditions: cases with one, two, and three heat sources

A Mixed Convection Study in Inclined Channels with Discrete Heat Sources 9

( ) 1 2 <sup>1</sup> *h hh* = +

( ) ( )

; ; 2 2

1 2314 2 2314

*a xxxx a yyyy*

= +−− = +−−

1 1

1 1 ; 2 2

1 3412 2 3412

Vectors **a** and **b** are the vectors that are limited by the middle points of opposite sides and h1 and h2 are the projections of these vectors on the flow direction. The most important characteristic of the Petrov-Galerkin method is that it does not add any transversal diffusion to the flow, that is, an orthogonal diffusion to the flow direction. αj is the so-called artificial diffusion or the equilibrium diffusion added to the convective terms. In fact, it is the optimum diffusion found between the diffusions provided by the traditional Galerkin method and the upwind one, which are respectively, under-diffused and over-diffused methods. This helps one solve some numerical oscillations which arise from the 'battle'

*b xxxx b yyyy*

λ

governing parameters do not change drastically (Bercovier & Engelman, 1979; Carey & Krishnan, 1982). This parameter must have a high value in order to have a 'quasiincompressible' problem. Indeed, the Penalty theory comes from the Stokes viscosity law.

> <sup>2</sup> . <sup>3</sup> *<sup>s</sup> p p* μ μ

where *<sup>s</sup> p* is the static or thermodynamic component of pressure, *p* is the average pressure,

3

. *Ppp* = − =− ∇ *<sup>s</sup>*

and (4). For linear interpolation of velocities, the pressure inside the element can be

μ

λ=

λ

is a very large number, and *P* is a modified pressure, which can be seen in Eqs. (3)

μ= −

is the second viscosity coefficient, and *μ* is the first viscosity.

= +−− = +−−

( ) ( )

**V** (11)

<sup>1</sup> *h* =**a.V** ; 2 *h* =**b.V** (12)

is a problem-independent value, given that the

⎛ ⎞ =− + ∇ ⎜ ⎟ ⎝ ⎠ **<sup>V</sup>** (14)

2 3 μ

represents the fluid bulk viscosity. If the fluid is perfectly

tends to be infinite. The basic idea of the Penalty formulation is to

μ

. It was shown to be true for some gases;

**V** ; (15)

⎛ ⎞ <sup>+</sup> ⎜ ⎟ ⎝ ⎠ is a positive quantity that is

(13)

The element average size *h* in the flow direction is given by :

where h1, h2, a, and b are defined as:

between elliptic and hyperbolic problems.

Stokes hypothesized that *p*<sup>=</sup> *<sup>s</sup> <sup>p</sup>* , hence, <sup>2</sup>

however experiments revealed that with liquids,

μ . λ

λ

Recalling the pressure formula:

μ

much larger than

express the pressure by

incompressible,

λ

expressed by:

where

Applying the Petrov-Galerkin formulation and the Penalty technique to Eqs. (2) to (5), the weak forms of the conservation equations may be written as follows:

$$\begin{aligned} \int\_{\Omega} N\_i \left[ \frac{\partial II}{\partial t} + \frac{1}{\text{Re}} \left( \frac{\partial N\_i}{\partial X} \frac{\partial II}{\partial X} + \frac{\partial N\_i}{\partial Y} \frac{\partial II}{\partial Y} \right) \right] d\Omega + \int\_{\Omega}^{R} \lambda \frac{\partial N\_i}{\partial X} \left( \frac{\partial II}{\partial X} + \frac{\partial V}{\partial Y} \right) d\Omega = \\ \int\_{\Omega} \left[ \left( N\_i + P\_{i1} \right) \left( \mathcal{U} \frac{\partial II}{\partial X} + V \frac{\partial II}{\partial Y} \right) + N\_i \sin \left( \gamma \right) \frac{\theta}{Fr} \right] d\Omega - \int\_{\Gamma\_0} N\_i \, p \, n\_x \, d\Gamma \end{aligned} \tag{6}$$

$$\begin{aligned} \int\_{\Omega} N\_i \left[ \frac{\partial V}{\partial t} + \frac{1}{\text{Re}} \left( \frac{\partial N\_i}{\partial X} \frac{\partial V}{\partial X} + \frac{\partial N\_i}{\partial Y} \frac{\partial V}{\partial Y} \right) \right] d\Omega + \int\_{\Omega}^{R} \lambda \frac{\partial N\_i}{\partial Y} \left( \frac{\partial I}{\partial X} + \frac{\partial V}{\partial Y} \right) d\Omega = \\ \int\_{\Omega} \left[ (N\_i + P\_{i1}) \left( U \frac{\partial V}{\partial X} + V \frac{\partial V}{\partial Y} \right) + N\_i \cos(\boldsymbol{\gamma}) \frac{\theta}{Fr} \right] d\Omega - \int\_{\Gamma\_0} N\_i \, p \, n\_y \, d\Gamma \end{aligned} \tag{7}$$

$$\int\_{\Omega} \left[ N\_i \frac{\partial \theta}{\partial t} + \frac{1}{\text{Re} \Pr} \left( \frac{\partial N\_i}{\partial X} \frac{\partial \theta}{\partial X} + \frac{\partial N\_i}{\partial Y} \frac{\partial \theta}{\partial Y} \right) \right] d\Omega = \int\_{\Omega} \left( N\_i + P\_{i2} \right) \left( M \frac{\partial \theta}{\partial X} + V \frac{\partial \theta}{\partial Y} \right) d\Omega + \int\_{\Gamma\_1} N\_i q \, d\Gamma \tag{8}$$

where the dependent variables are approximated through the finite element method by:

$$\Phi\left(X,Y,t\right) = \sum\_{j} N\_{j}\left(X,Y\right)\Phi\_{j}\left(t\right);\ p\left(X,Y,t\right) = \sum\_{k} M\_{k}\left(X,Y\right)p\_{k}\left(t\right) \tag{9}$$

Ni and Nj are linear shape functions for the quantity Φ , that is, for U, V, andθ . Mk are the pressure piecewise element shape functions. The Petrov-Galerkin perturbations *Pij* , which are only applied to the convective terms, are defined as:

$$P\_{ij} = k\_j \left( \mathbf{U} \frac{\partial N\_i}{\partial X} + V \frac{\partial N\_i}{\partial Y} \right);\ k\_j = \frac{\alpha\_j \overline{h}}{\|V\|};\ \alpha\_j = \coth \frac{\gamma\_j}{2} - \frac{2}{\gamma\_j};\ \gamma\_j = \frac{\|V\| \overline{h}}{\varepsilon\_j};\ j = 1, 2 \tag{10}$$

where γ is the element Péclet number, *V* is the absolute value of the velocity vector that represents the fluid average velocity within the element, *h* is the element average size, 1 2 ε = = 1 /Re, 1 / ε *Pe* , and λ is the Penalty parameter equal to 109. Fig. (2) shows the general linear quadrilateral element with *V* and *h* .

Fig. 2. General bilinear quadrilateral element with vectors a and b and element velocity V

The element average size *h* in the flow direction is given by :

$$\overline{h} = \frac{1}{|\mathbf{V}|} (|h\_1| + |h\_2|) \tag{11}$$

where h1, h2, a, and b are defined as:

8 Convection and Conduction Heat Transfer

Applying the Petrov-Galerkin formulation and the Penalty technique to Eqs. (2) to (5), the

*R*

Ω

*R*

Ω

Ω Γ

∫ ∫

*i i i i N N N d <sup>N</sup> <sup>P</sup> <sup>U</sup> <sup>V</sup> <sup>d</sup> <sup>N</sup> <sup>q</sup> <sup>d</sup>*

> ( , , ( , ) ; , , ( , ) ) *j j* ( ) ( ) *k k* ( ) *j k*

pressure piecewise element shape functions. The Petrov-Galerkin perturbations *Pij* , which

*N N h V h P kU V k <sup>j</sup> XY V* α

α

 is the element Péclet number, *V* is the absolute value of the velocity vector that represents the fluid average velocity within the element, *h* is the element average size,

**V**

2

Fig. 2. General bilinear quadrilateral element with vectors a and b and element velocity V

= + = = −= = ⎜ ⎟

⎡ ⎤ ∂ ∂∂ ⎛ ⎞ ∂ ∂ ⎛ ⎞ ∂ ∂ ⎢ ⎥ <sup>+</sup> ⎜ ⎟ + Ω = + + Ω+ Γ ⎜ ⎟ ⎣ ⎦ ∂ ∂∂ ∂∂ ⎝ ⎠ ⎝ ⎠ ∂ ∂ ∫ ∫∫ (8)

λ

Ω Γ

∫ ∫

λ

( ) ( )

( ) ( )

⎡ ⎤ ⎛ ⎞ ∂ ∂

⎣ ⎦ ⎝ ⎠ ∂ ∂

*i i i*

*t XX YY X Y*

Ni and Nj are linear shape functions for the quantity Φ , that is, for U, V, and

*j j i i ij j j j j*

**a**

1 **b**

4

Ω Ω Γ

where the dependent variables are approximated through the finite element method by:

⎡ ⎤ ∂ ∂∂ ⎛ ⎞ ∂∂ ∂ ⎛ ⎞ ∂ ∂ + +Ω+ +Ω= ⎢ ⎥ ⎜ ⎟ ⎜ ⎟ ⎣ ⎦ ∂ ∂∂ ∂∂ ∂ ∂ ∂ ⎝ ⎠ ⎝ ⎠

*V VV NN N U V N dd t XX YY Y X Y*

⎡ ⎤ ⎛ ⎞ ∂ ∂

⎣ ⎦ ⎝ ⎠ ∂ ∂

*i i i*

⎡ ⎤ ∂ ∂∂ ⎛ ⎞ ∂∂ ∂ ⎛ ⎞ ∂ ∂ + +Ω+ +Ω= ⎢ ⎥ ⎜ ⎟ ⎜ ⎟ ⎣ ⎦ ∂ ∂∂ ∂∂ ∂ ∂ ∂ ⎝ ⎠ ⎝ ⎠

*U UU NN N U V N dd t XX YY X X Y*

<sup>0</sup> <sup>1</sup>

<sup>0</sup> <sup>1</sup>

*i i i i y*

*V V N P U V N d N pn d X Y Fr*

+ ⎜ ⎟ + + Ω− Γ ⎢ ⎥

( )

Φ= Φ = *XYt N XY t pXYt M XYp t* ∑ ∑ (9)

<sup>2</sup> ; ; coth ; ; 1,2 <sup>2</sup>

γ

is the Penalty parameter equal to 109. Fig. (2) shows the

3

 γ

⎝ ⎠ ∂ ∂ (10)

*j j*

x

 ε

 γ

cos

γ

*i i i i x*

*U U N P U V N d N pn d X Y Fr*

+ ⎜ ⎟ + + Ω− Γ ⎢ ⎥

sin

γ

θ

θ

θ

<sup>1</sup> <sup>2</sup>

 θ

θ

. Mk are the

(6)

(7)

weak forms of the conservation equations may be written as follows:

∫ ∫

∫ ∫

*i i*

θθ

are only applied to the convective terms, are defined as:

λ

general linear quadrilateral element with *V* and *h* .

y

⎛ ⎞ ∂ ∂

1 Re

> 1 Re

1 RePr

*Pe* , and

*i*

*i*

θ

where γ

ε

1 2

 = = 1 /Re, 1 / ε

Ω

Ω

$$h\_1 = \mathbf{a}.\mathbf{V} \; ; \; h\_2 = \mathbf{b}.\mathbf{V} \tag{12}$$

$$a\_1 = \frac{1}{2}(\mathbf{x}\_2 + \mathbf{x}\_3 - \mathbf{x}\_1 - \mathbf{x}\_4);\ a\_2 = \frac{1}{2}(y\_2 + y\_3 - y\_1 - y\_4);$$

$$\dots \quad \mathbf{1}.\tag{13}$$

$$b\_1 = \frac{1}{2}(\mathbf{x}\_3 + \mathbf{x}\_4 - \mathbf{x}\_1 - \mathbf{x}\_2);\ b\_2 = \frac{1}{2}(y\_3 + y\_4 - y\_1 - y\_2)$$

Vectors **a** and **b** are the vectors that are limited by the middle points of opposite sides and h1 and h2 are the projections of these vectors on the flow direction. The most important characteristic of the Petrov-Galerkin method is that it does not add any transversal diffusion to the flow, that is, an orthogonal diffusion to the flow direction. αj is the so-called artificial diffusion or the equilibrium diffusion added to the convective terms. In fact, it is the optimum diffusion found between the diffusions provided by the traditional Galerkin method and the upwind one, which are respectively, under-diffused and over-diffused methods. This helps one solve some numerical oscillations which arise from the 'battle' between elliptic and hyperbolic problems. λ is a problem-independent value, given that the governing parameters do not change drastically (Bercovier & Engelman, 1979; Carey & Krishnan, 1982). This parameter must have a high value in order to have a 'quasiincompressible' problem. Indeed, the Penalty theory comes from the Stokes viscosity law. Recalling the pressure formula:

$$p = p\_s \left(\overline{\mu} + \frac{2}{3}\mu\right) \nabla. \mathbf{V} \tag{14}$$

where *<sup>s</sup> p* is the static or thermodynamic component of pressure, *p* is the average pressure, μis the second viscosity coefficient, and *μ* is the first viscosity.

Stokes hypothesized that *p*<sup>=</sup> *<sup>s</sup> <sup>p</sup>* , hence, <sup>2</sup> 3 μ = − μ. It was shown to be true for some gases;

however experiments revealed that with liquids, λ = 2 3 μ μ ⎛ ⎞ <sup>+</sup> ⎜ ⎟ ⎝ ⎠ is a positive quantity that is

much larger than μ . λ represents the fluid bulk viscosity. If the fluid is perfectly incompressible, λ tends to be infinite. The basic idea of the Penalty formulation is to express the pressure by

$$P = p - p\_s = -\mathcal{X} \nabla. \mathbf{V} \; ; \tag{15}$$

where λ is a very large number, and *P* is a modified pressure, which can be seen in Eqs. (3) and (4). For linear interpolation of velocities, the pressure inside the element can be expressed by:

A Mixed Convection Study in Inclined Channels with Discrete Heat Sources 11

The algorithm is validated by comparing the results of the present work with both the ones obtained in experimental and numerical investigations. The first comparison is accomplished not only by using experimental results (Lee & Mateescu, 1998; Armaly et al., 1983), but also by numerical ones (Lee & Mateescu, 1998; Gartling, 1990; Kim & Moin, 1985, and Sohn, 1988). The air flow of the present comparison analysis is taken as two-dimensional, laminar, incompressible, and under the unsteady regime. The domain is a horizontal upstream backward-facing step channel whose inlet has a fully developed velocity profile given by u = 24 y (0.5-y) *U* and v = 0 in which Re = 800. A structured mesh with 6000 quadrilateral linear elements with ∆X = 0.10 e ∆Y = 0.05 and time step of ∆t = 0.05 are used. By comparing the flow separation distance on the upper surface, its reattachment distance on the bottom surface and its reattachment distance on the upper wall, the deviations are in the range from 0.8% to 17.85%. Out of the 14 calculated deviations, just two are over 10%. Therefore, the results of the first comparison agree well with the ones from the literature. The second comparison is performed with the numerical results (Comini et al., 1997). The contrasting study is carried out by considering a problem involving mixed convective heat transfer with the flow being two-dimensional, laminar, and incompressible in the unsteady regime. This is a benchmark problem which involves the Rayleigh-Bérnard convection in a rectangular channel. In this case, some values are chosen such as Re = 10, Pr = 0.67, and Fr = 1/150. The grid has 4000 quadrilateral linear elements with Δ = *x* 0.1 , Δ = *y* 0.15 , Δ = *t* 0.01 and 1000 iterations. After approximately iteration 500, the regime turns to be quasi-periodic with the average Nusselt number on the upper wall oscillating around a mean value of 2.44. This value agrees satisfactorily with 2.34 (Comini et al., 1997), resulting in a deviation of about 4%. It is important to mention that a mesh independency is performed for the most critical situations of the cases that will be studied with the highest values of Re and Gr numbers as well as the study of mass conversation due to the use of the penalty method. Just after reaching reasonable results, the final mesh domain and the penalty parameter are used to obtain the final results. All this study can be found in the doctorate thesis of the present

**4. Validation and mesh independency** 

author (Guimarães, 2007). It is omitted here as a matter of space.

Figure (4) shows the isotherm distributions considering the ranges for the Reynolds number, the Grashof number, and the inclination angle, respectively, as 1 ≤ Re ≤ 10, 103 ≤ Gr ≤ 105, and 0° ≤ γ ≤ 90°. By observing the isotherms from Fig. (4), the orientation of the channel has a more significant effect on the temperature distribution for lower Re, that is, Re = 1, 5, and 10. This effect is even stronger when Gr increases. Later on this chapter, this effect on the Nusselt number will be seen again. On the other hand, for higher Reynolds, over 50 or so (not shown in Fig. (4)), the inclination angle has a weaker effect on the temperature

Figure (5) shows the effect of the source module on the velocity distributions for the horizontal cases (γ = 0°) with Re = 1, 5, and 10, considering Gr = 105. It can be noted that in the natural convection regime, the thermal plume is approximately symmetric about the centerline of the module (Fig. 4). In the thermal plume region, there are two opposite recirculating cells that are almost symmetric. Moreover, as the entrance velocity increases

**5. Results** 

distributions.

**5.1 Case with one heat source** 

$$P^{e} = -\frac{\lambda}{A^{e}} \prod\_{e}^{R} \left(\frac{\partial \mathcal{U}}{\partial X} + \frac{\partial \mathcal{V}}{\partial Y}\right) de\tag{16}$$

where *e* denotes the element restriction and *Ae* the element area.

Let one neglect some terms in the momentum equations and attain to those in the stiffness matrix, just to facilitate understanding of the Penalty parameter study. Substituting Eq. (16) in (3) and (4) with some manipulation and neglection of some terms which are now not important, the final linear system of equations would present this expression.

$$\left[\mu \mathbf{K}\_1 + \lambda \mathbf{K}\_2\right] \mathbf{d} = \mathbf{F} \tag{17}$$

where **F** is the force matrix generated by the boundary conditions, **d** is the velocity vector only. Now, let one suppose that **K**2 is not singular and λ is increased more and more in an attempt to reach incompressibility. Because μ and **K**1 are constant as λ is increased, they can be neglected and the solution to Eq. (17) would be:

$$\mathbf{d} = \frac{1}{\lambda} \mathbf{K}\_2^{-1} \mathbf{F} \tag{18}$$

Being that **K**2 and **F** are also constant, it can be noted that **d** → 0 as λ → ∞ . This is called 'locking' and is strictly related to the incompatibility between the pressure and velocity spaces. Hence, the only solution to this space is the velocity vector **V**=0 . It must be guaranteed that the Penalty matrix **K**2 be singular. This is accomplished by using the selective reduced integration of the Penalty term and the least square quadrature with a degree of precision lower than the necessary to guarantee an optimum convergence rate of matrix **K**<sup>1</sup> . Therefore, Fig. (3) shows a pair of elements used here in order to avoid locking.

Fig. 3. Real (a) and parent (b) elements for full and reduced integration, respectively

The time integration is accomplished by the Euler backward semi-implicit method. Moreover, the convective and viscous terms are, respectively, calculated explicitly and implicitly. Finally, the average Nusselt number along a surface S of a heat source can be written as follows:

$$Nu = \frac{1}{S} \int\_{S} \left[ \frac{1}{\theta} \right] ds \tag{19}$$

#### **4. Validation and mesh independency**

10 Convection and Conduction Heat Transfer

Let one neglect some terms in the momentum equations and attain to those in the stiffness matrix, just to facilitate understanding of the Penalty parameter study. Substituting Eq. (16) in (3) and (4) with some manipulation and neglection of some terms which are now not

> λ

where **F** is the force matrix generated by the boundary conditions, **d** is the velocity vector

μ

'locking' and is strictly related to the incompatibility between the pressure and velocity spaces. Hence, the only solution to this space is the velocity vector **V**=0 . It must be guaranteed that the Penalty matrix **K**2 be singular. This is accomplished by using the selective reduced integration of the Penalty term and the least square quadrature with a degree of precision lower than the necessary to guarantee an optimum convergence rate of matrix **K**<sup>1</sup> . Therefore, Fig. (3) shows a pair of elements used here in order to avoid locking.

2 1

Fig. 3. Real (a) and parent (b) elements for full and reduced integration, respectively

The time integration is accomplished by the Euler backward semi-implicit method. Moreover, the convective and viscous terms are, respectively, calculated explicitly and implicitly. Finally, the average Nusselt number along a surface S of a heat source can be

> 1 1 *S Nu ds S*

θ

ξ

3

Velocity Temperature

1 2 1 λ

4

η

 ⎛ ⎞ ∂ ∂ =− + ⎜ ⎟

⎝ ⎠ ∂ ∂ ∫ (16)

**K KdF** 1 2 + ] = (17)

<sup>−</sup> **d KF** = (18)

is increased more and more in an

is increased, they

→ ∞ . This is called

ξ

λ

λ

2

<sup>⎡</sup> <sup>⎤</sup> <sup>=</sup> <sup>⎢</sup> <sup>⎥</sup> <sup>⎣</sup> <sup>⎦</sup> <sup>∫</sup> (19)

3

Velocity in Penalty terms

λ

and **K**1 are constant as

*R*

*e e U V P de A X Y*

λ

*e*

important, the final linear system of equations would present this expression.

[μ

Being that **K**2 and **F** are also constant, it can be noted that **d** → 0 as

where *e* denotes the element restriction and *Ae* the element area.

only. Now, let one suppose that **K**2 is not singular and

can be neglected and the solution to Eq. (17) would be:

attempt to reach incompressibility. Because

1

written as follows:

4

η

The algorithm is validated by comparing the results of the present work with both the ones obtained in experimental and numerical investigations. The first comparison is accomplished not only by using experimental results (Lee & Mateescu, 1998; Armaly et al., 1983), but also by numerical ones (Lee & Mateescu, 1998; Gartling, 1990; Kim & Moin, 1985, and Sohn, 1988). The air flow of the present comparison analysis is taken as two-dimensional, laminar, incompressible, and under the unsteady regime. The domain is a horizontal upstream backward-facing step channel whose inlet has a fully developed velocity profile given by u = 24 y (0.5-y) *U* and v = 0 in which Re = 800. A structured mesh with 6000 quadrilateral linear elements with ∆X = 0.10 e ∆Y = 0.05 and time step of ∆t = 0.05 are used. By comparing the flow separation distance on the upper surface, its reattachment distance on the bottom surface and its reattachment distance on the upper wall, the deviations are in the range from 0.8% to 17.85%. Out of the 14 calculated deviations, just two are over 10%. Therefore, the results of the first comparison agree well with the ones from the literature. The second comparison is performed with the numerical results (Comini et al., 1997). The contrasting study is carried out by considering a problem involving mixed convective heat transfer with the flow being two-dimensional, laminar, and incompressible in the unsteady regime. This is a benchmark problem which involves the Rayleigh-Bérnard convection in a rectangular channel. In this case, some values are chosen such as Re = 10, Pr = 0.67, and Fr = 1/150. The grid has 4000 quadrilateral linear elements with Δ = *x* 0.1 , Δ = *y* 0.15 , Δ = *t* 0.01 and 1000 iterations. After approximately iteration 500, the regime turns to be quasi-periodic with the average Nusselt number on the upper wall oscillating around a mean value of 2.44. This value agrees satisfactorily with 2.34 (Comini et al., 1997), resulting in a deviation of about 4%.

It is important to mention that a mesh independency is performed for the most critical situations of the cases that will be studied with the highest values of Re and Gr numbers as well as the study of mass conversation due to the use of the penalty method. Just after reaching reasonable results, the final mesh domain and the penalty parameter are used to obtain the final results. All this study can be found in the doctorate thesis of the present author (Guimarães, 2007). It is omitted here as a matter of space.

#### **5. Results**

#### **5.1 Case with one heat source**

Figure (4) shows the isotherm distributions considering the ranges for the Reynolds number, the Grashof number, and the inclination angle, respectively, as 1 ≤ Re ≤ 10, 103 ≤ Gr ≤ 105, and 0° ≤ γ ≤ 90°. By observing the isotherms from Fig. (4), the orientation of the channel has a more significant effect on the temperature distribution for lower Re, that is, Re = 1, 5, and 10. This effect is even stronger when Gr increases. Later on this chapter, this effect on the Nusselt number will be seen again. On the other hand, for higher Reynolds, over 50 or so (not shown in Fig. (4)), the inclination angle has a weaker effect on the temperature distributions.

Figure (5) shows the effect of the source module on the velocity distributions for the horizontal cases (γ = 0°) with Re = 1, 5, and 10, considering Gr = 105. It can be noted that in the natural convection regime, the thermal plume is approximately symmetric about the centerline of the module (Fig. 4). In the thermal plume region, there are two opposite recirculating cells that are almost symmetric. Moreover, as the entrance velocity increases

A Mixed Convection Study in Inclined Channels with Discrete Heat Sources 13

(a)

(b)

(c)

x 5,0 5,2 5,4 5,6 5,8 6,0

Figure (7) depicts the velocities vectors for Gr = 105 and Re = 1, 5, and 10, for γ = 45°. As Re increases, the clockwise recirculating cell tends to vanish, whereas the anticlockwise ones tend to dominate to an extent that reversed flow exists at the exit. The presence of the reversal flow does not necessarily imply an increase of the Nusselt number on the module. It is interesting to observe the formation of secondary recirculating cells in Fig. (7c), and as a

Fig. 6. Module temperature distributions for Gr = 105, γ = 0° and Re = 1, 5, and 10

Re = 1 Re = 5 Re =10

Fig. 5. Velocity distributions for Gr = 105, γ = 0°, and Re = (a) 1, (b) 5, and (c) 10

Gr =10<sup>5</sup> γ = 0°

θ

0,05

0,10

0,15

0,20

0,25

0,30

0,35

Fig. 4. Isotherms: Gr = 103, 104, 105; Re = 1, 5 , 10; and γ = 0°, 15°, 30°, 45°, 60°, 75°, 90°

the plume shifts and stretches towards downstream. This effect features how the natural convection starts to vanish, giving place to the mixed convection. If Re increases more, for instance, Re ≥ 100 (not shown in Fig. (5)), the forced global fluid movement dominates the flow pattern and the heat transfer process. One can note in the temperature distribution along the module in Fig. (6), for Gr = 105, and Re = 1, 5, 10, the transition from the natural convection to the mixed convection. As Re is higher, the maximum temperature on the module decreases. This is due to the fact that when higher velocities are present, the cell recirculation gets further from the module allowing a stronger contact of the cold fluid with the heat module.

60°

Re=1.0; Re=5.0; Re=10.0;

60°

45°

4 Gr=10

0°

15°

15°

30°

0°

90°

90°

75°

75°

15°

90°

75°

30°

0°

Re=1.0; Re=5.0;Gr=10 Re=10.0; <sup>5</sup>

60°

Fig. 4. Isotherms: Gr = 103, 104, 105; Re = 1, 5 , 10; and γ = 0°, 15°, 30°, 45°, 60°, 75°, 90°

the plume shifts and stretches towards downstream. This effect features how the natural convection starts to vanish, giving place to the mixed convection. If Re increases more, for instance, Re ≥ 100 (not shown in Fig. (5)), the forced global fluid movement dominates the flow pattern and the heat transfer process. One can note in the temperature distribution along the module in Fig. (6), for Gr = 105, and Re = 1, 5, 10, the transition from the natural convection to the mixed convection. As Re is higher, the maximum temperature on the module decreases. This is due to the fact that when higher velocities are present, the cell recirculation gets further from the module allowing a stronger contact of the cold fluid with

45°

Gr=105

45°

45°

60°

60°

90°

90°

90°

75°

75°

75°

the heat module.

15°

90°

75°

30°

0°

4 Gr=10

15°

30°

0°

15°

30°

0°

90°

90°

75°

75°

<sup>3</sup> Gr=10

45°

60°

Re=1.0; Re=5.0; Re=10.0;Gr=10<sup>3</sup>

45°

<sup>3</sup> Gr=10

30°

4 Gr=10

15°

30°

0°

<sup>5</sup> Gr=10

15°

30°

0°

15°

30°

45°

45°

45°

60°

60°

60°

0°

Fig. 5. Velocity distributions for Gr = 105, γ = 0°, and Re = (a) 1, (b) 5, and (c) 10

Fig. 6. Module temperature distributions for Gr = 105, γ = 0° and Re = 1, 5, and 10

Figure (7) depicts the velocities vectors for Gr = 105 and Re = 1, 5, and 10, for γ = 45°. As Re increases, the clockwise recirculating cell tends to vanish, whereas the anticlockwise ones tend to dominate to an extent that reversed flow exists at the exit. The presence of the reversal flow does not necessarily imply an increase of the Nusselt number on the module. It is interesting to observe the formation of secondary recirculating cells in Fig. (7c), and as a

A Mixed Convection Study in Inclined Channels with Discrete Heat Sources 15

maximum temperature. Therefore, having these conditions in mind, the increase of the

Figure (9) presents the velocity distributions for Gr = 105, Re = 1, 5, and 10, but now with the channel on the vertical position, γ = 90°. It can be noted that for all the values of Re, there is a reversal flow at the outlet that is stronger than the reversed flow for γ = 45°. For each of these three cases in Fig. (9), there is a secondary recirculating cell. There is not a significant difference in the temperature on the module for the Reynolds values Re = 1, 5, and 10;

inclination angle provides lower values for the maximum temperature.

(a) (b) (c)

diminish as forced fluid global motion dominates.

Fig. 9. Velocity distributions for Gr = 105, γ = 90°, and Re = (a) 1, (b) 5, and (c) 10

Figure (11) makes a comparison of the Nusselt numbers for 1 ≤ Re ≤ 500, 0° ≤ γ ≤ 90° and for the three Grashof numbers Gr =103, 104, and 105. These results are in good agreement with those found in literature (Choi & Ortega, 1993). It was observed according to the 'five percent deviation rule' (Sparrow et al., 1959) that the natural, mixed, and forced convection can be determined. From Figs. (11a-c), the same is found here, where the mixed convection regimes are, approximately, within the ranges 1 < Re < 50 for Gr = 103, 5 < Re < 100 for Gr = 104, and 10 < Re < 500 for Gr = 105. Practically, there is no reduction in the Nusselt number in these cases where the aiding flow is present, which also agrees with the work previously mentioned (Choi & Ortega, 1993). It can be noted that the collapse of the curves at high Re explains that the buoyancy effects and, therefore, the influence of inclination

Figure (12) shows the influence of the orientation angle in the heat transfer for all the cases studied in this work, that is, Re from 1 to 500; γ from 0° to 90°; and Gr = 103, 104, and 105. In a general way, Nu is strongly dependent on the orientation of the channel in the limits of the natural and forced convection. For instance, when Gr = 103, the natural convective cells are not sufficiently strong to influence the flow and the temperature fields (Fig. 4), that means that the heat transfer mode involved is almost conductive along the channel. As Gr increases, the variations in Nu are more significant for low Re. It is said that in the natural

Gr = 105, and γ = 90°(Fig. 10)).

result of this formation, the corresponding isothermal lines are distorted, as shown in Fig. (4). In Fig. (7b), the reversal flow is weak. Comparing Fig. (7) with Fig. (5), there is a small difference in the velocity vectors, hence showing the small effect of the orientation of the channel in these two cases.

Fig. 7. Velocity vectors for Gr = 105, γ = 45°, and (a) Re =1, (b) Re =5, and (c) Re =10; Here, γ = 75° is a better situation when cooling is aimed, with little difference when γ = 90°

Fig. 8. Module temperature distributions or Gr = 105, γ = 45° and Re = 1, 5, and 10

In Figure (8), the temperature distributions along the module are shown for the cases from Fig. (7a-c). In a general way, the maximum temperature on the module decreases for the three cases. It must be remembered that the ideal work conditions in equipments with electronic circuit boards are those in which the module presents the lowest values for the

result of this formation, the corresponding isothermal lines are distorted, as shown in Fig. (4). In Fig. (7b), the reversal flow is weak. Comparing Fig. (7) with Fig. (5), there is a small difference in the velocity vectors, hence showing the small effect of the orientation of the

Fig. 7. Velocity vectors for Gr = 105, γ = 45°, and (a) Re =1, (b) Re =5, and (c) Re =10; Here, γ = 75° is a better situation when cooling is aimed, with little difference when γ = 90°

> x 5,0 5,2 5,4 5,6 5,8 6,0

In Figure (8), the temperature distributions along the module are shown for the cases from Fig. (7a-c). In a general way, the maximum temperature on the module decreases for the three cases. It must be remembered that the ideal work conditions in equipments with electronic circuit boards are those in which the module presents the lowest values for the

Fig. 8. Module temperature distributions or Gr = 105, γ = 45° and Re = 1, 5, and 10

Re = 1 Re =5 Re =10

channel in these two cases.

(a) (b) (c)

Gr =10<sup>5</sup> γ = 45°

θ

0,05

0,10

0,15

0,20

0,25

0,30

0,35

maximum temperature. Therefore, having these conditions in mind, the increase of the inclination angle provides lower values for the maximum temperature.

Figure (9) presents the velocity distributions for Gr = 105, Re = 1, 5, and 10, but now with the channel on the vertical position, γ = 90°. It can be noted that for all the values of Re, there is a reversal flow at the outlet that is stronger than the reversed flow for γ = 45°. For each of these three cases in Fig. (9), there is a secondary recirculating cell. There is not a significant difference in the temperature on the module for the Reynolds values Re = 1, 5, and 10; Gr = 105, and γ = 90°(Fig. 10)).

Fig. 9. Velocity distributions for Gr = 105, γ = 90°, and Re = (a) 1, (b) 5, and (c) 10

Figure (11) makes a comparison of the Nusselt numbers for 1 ≤ Re ≤ 500, 0° ≤ γ ≤ 90° and for the three Grashof numbers Gr =103, 104, and 105. These results are in good agreement with those found in literature (Choi & Ortega, 1993). It was observed according to the 'five percent deviation rule' (Sparrow et al., 1959) that the natural, mixed, and forced convection can be determined. From Figs. (11a-c), the same is found here, where the mixed convection regimes are, approximately, within the ranges 1 < Re < 50 for Gr = 103, 5 < Re < 100 for Gr = 104, and 10 < Re < 500 for Gr = 105. Practically, there is no reduction in the Nusselt number in these cases where the aiding flow is present, which also agrees with the work previously mentioned (Choi & Ortega, 1993). It can be noted that the collapse of the curves at high Re explains that the buoyancy effects and, therefore, the influence of inclination diminish as forced fluid global motion dominates.

Figure (12) shows the influence of the orientation angle in the heat transfer for all the cases studied in this work, that is, Re from 1 to 500; γ from 0° to 90°; and Gr = 103, 104, and 105. In a general way, Nu is strongly dependent on the orientation of the channel in the limits of the natural and forced convection. For instance, when Gr = 103, the natural convective cells are not sufficiently strong to influence the flow and the temperature fields (Fig. 4), that means that the heat transfer mode involved is almost conductive along the channel. As Gr increases, the variations in Nu are more significant for low Re. It is said that in the natural

A Mixed Convection Study in Inclined Channels with Discrete Heat Sources 17

γ = 0° γ = 15° γ = 30° γ = 45° γ = 60° γ = 75° γ = 90°

1 10 100

γ = 0° γ = 15° γ = 30° γ = 45° γ = 60° γ = 75° γ = 90°

(a)

1 10 100

(b)

Gr = 10<sup>3</sup>

Nu

Nu

2

3

4

5

6

7

8

9

1

Gr = 10<sup>4</sup>

2

3

4

5

6

7

and mixed convection, the most suitable situation in the cooling of an electronic module, that is, the lowest Nu, is found when the channel orientation is in the range 45°<γ<90° (considering the convention adopted in the present work), in a general way (Choi and Ortega, 1993). Analyzing Fig. (13), there is still a more favorable inclination within the range proposed by previous literature (Choi & Ortega, 1993), which is around 60° and 75°. The case for γ = 0° should be avoided. A particular case which provides a drastic increase in Nu, within the limits considered in this case, is Gr = 103 and Re = 1 in the range 60° ≤ γ ≤ 75°. This can also be noted in Fig. (5), in the isotherms distributions.

Fig. 10. Module temperature distributions for Gr = 105, γ = 90° and Re = 1, 5, and 10

#### **5.2 Case with two heat sources**

Figure (13) discloses the velocity vector distributions for Gr = 105, Re = 1, 10, and 1000 and d = 1, 2, and 3. In the case where Re = 1, almost symmetric recirculations appear and those are responsible for the thermal plumes mentioned previously. As the source modules are placed further from each other, the recirculation between the sources is powered on and, therefore, higher velocities take part in the flow. For Gr = 105, this does not imply a big difference in temperature on the pair of heat sources. In a general way, as Re is raised, the recirculations tend to cease, making it possible to the cold fluid be more in contact with the module surfaces, hence, invigorating the heat transfer. For high values of Re, the distance d has little influence on the velocity profiles. As mentioned before, for Re = 10, an oscillating behavior of the recirculations brought about by the buoyancy flow near the modules is observed. These oscillations reflect an attempt of the flow to be ruled by a predominant convection, that is, it is a 'fight' between buoyant and forced velocities. The first heat source develops a flow wake that hits the second one, heating it even more. The recirculation originated on the second module becomes more intense and this makes an increase in the

and mixed convection, the most suitable situation in the cooling of an electronic module, that is, the lowest Nu, is found when the channel orientation is in the range 45°<γ<90° (considering the convention adopted in the present work), in a general way (Choi and Ortega, 1993). Analyzing Fig. (13), there is still a more favorable inclination within the range proposed by previous literature (Choi & Ortega, 1993), which is around 60° and 75°. The case for γ = 0° should be avoided. A particular case which provides a drastic increase in Nu, within the limits considered in this case, is Gr = 103 and Re = 1 in the range 60° ≤ γ ≤ 75°.

> x 5,0 5,2 5,4 5,6 5,8 6,0

Figure (13) discloses the velocity vector distributions for Gr = 105, Re = 1, 10, and 1000 and d = 1, 2, and 3. In the case where Re = 1, almost symmetric recirculations appear and those are responsible for the thermal plumes mentioned previously. As the source modules are placed further from each other, the recirculation between the sources is powered on and, therefore, higher velocities take part in the flow. For Gr = 105, this does not imply a big difference in temperature on the pair of heat sources. In a general way, as Re is raised, the recirculations tend to cease, making it possible to the cold fluid be more in contact with the module surfaces, hence, invigorating the heat transfer. For high values of Re, the distance d has little influence on the velocity profiles. As mentioned before, for Re = 10, an oscillating behavior of the recirculations brought about by the buoyancy flow near the modules is observed. These oscillations reflect an attempt of the flow to be ruled by a predominant convection, that is, it is a 'fight' between buoyant and forced velocities. The first heat source develops a flow wake that hits the second one, heating it even more. The recirculation originated on the second module becomes more intense and this makes an increase in the

Fig. 10. Module temperature distributions for Gr = 105, γ = 90° and Re = 1, 5, and 10

Re =1 Re =5 Re =10

This can also be noted in Fig. (5), in the isotherms distributions.

Gr =10<sup>5</sup> γ = 90°

θ

0,05

**5.2 Case with two heat sources** 

0,10

0,15

0,20

0,25

0,30

0,35

A Mixed Convection Study in Inclined Channels with Discrete Heat Sources 19

plays a considerable role on the second module heating. For instance, what has just been

γ (degree) 0 20 40 60 80

(a)

Re=1 Re=5 Re=10 Re=50 Re=100 Re=200

γ (degree) 0 20 40 60 80

(b)

Gr =103

Gr = 10<sup>4</sup>

Re=1 Re=5 Re=10 Re=50 Re=100

said happens in the case where Re = 1000 and d = 1, 2, and 3.

Nu

1

Nu

2

3

4

5

6

7

8

9

2

3

4

5

6

7

Fig. 11. Average Nusselt number vs Re: 0 ≤ γ ≤ 90°, (a) Gr=103, (b) Gr= 04, and (c) Gr=105

fluid pressure coming from backwards. Therefore, it comes to a point that this pressure is strong enough to force that flow barrier downstream. At this moment, the main flow starts to develop and all the process repeats all over again. Consequently, time oscillations in Nu on Heater 2 will appear. This will be seen afterwards.

Figure (14) shows the isotherm distributions for Reynolds numbers Re = 1, 10, 50, 100, and 1000, distances between the heat sources d = 1, 2, and 3 and Grashof number Gr = 105. The isotherms are spaced by Δθ = 0.02. It can be noted when natural convection is concerned, that is, Re = 1, the heat sources in the cases where d = 1 and 2 influence each other whereas for the case where d = 3, it does not happen. For all cases where Re = 1, there are plumes localized in the vicinity of the heaters. For Re = 1 and d = 1, the two plumes are partially glued, making it clear that this distance does make the module temperature fields impact each other. Although this is also true when d = 2, the plumes are further spaced from each other. For Re = 1 and d = 3, the plumes are almost independent. It will be seen that this behavior in Heater 1 and Re = 1 affects smoothly the Nusselt number Nu when Gr = 105. Moreover, for Gr = 103 and 104, the space between the modules does not interfere on heat transfer. On the other hand, when Heater 2 is taken into consideration, the heat source spacing invigorates Nu for Gr = 103, 104, and 105. A perfect symmetry is not present here as long as the flow has a certain forced velocity, Re = 1, in the forced stream. For all Reynolds numbers and Gr = 105, the solutions converge. However, for Re = 10, a periodic flow oscillation in time is brought about for all heat source distances. The temperature distributions (not shown here) where Re = 10 and d = 1, 2, and 3 are plotted in the time shots t = 15, 30, and 50, respectively. It can be perceived in these cases, higher temperature distribution on the second module. This temperature rise tends to weaken as the Reynolds number is increased. The temperature invigoration in the region around Heater 2 is due to the fact that the thermal wake from Heater 1 intensifies the buoyant flow in the localized region in the vicinity of Heater 2. It can be noted that by keeping Re constant, the distance d

γ = 0° γ = 15° γ = 30° γ = 45° γ = 60° γ = 75° γ = 90°

1 10 100

 (c) Fig. 11. Average Nusselt number vs Re: 0 ≤ γ ≤ 90°, (a) Gr=103, (b) Gr= 04, and (c) Gr=105

fluid pressure coming from backwards. Therefore, it comes to a point that this pressure is strong enough to force that flow barrier downstream. At this moment, the main flow starts to develop and all the process repeats all over again. Consequently, time oscillations in Nu

Figure (14) shows the isotherm distributions for Reynolds numbers Re = 1, 10, 50, 100, and 1000, distances between the heat sources d = 1, 2, and 3 and Grashof number Gr = 105. The isotherms are spaced by Δθ = 0.02. It can be noted when natural convection is concerned, that is, Re = 1, the heat sources in the cases where d = 1 and 2 influence each other whereas for the case where d = 3, it does not happen. For all cases where Re = 1, there are plumes localized in the vicinity of the heaters. For Re = 1 and d = 1, the two plumes are partially glued, making it clear that this distance does make the module temperature fields impact each other. Although this is also true when d = 2, the plumes are further spaced from each other. For Re = 1 and d = 3, the plumes are almost independent. It will be seen that this behavior in Heater 1 and Re = 1 affects smoothly the Nusselt number Nu when Gr = 105. Moreover, for Gr = 103 and 104, the space between the modules does not interfere on heat transfer. On the other hand, when Heater 2 is taken into consideration, the heat source spacing invigorates Nu for Gr = 103, 104, and 105. A perfect symmetry is not present here as long as the flow has a certain forced velocity, Re = 1, in the forced stream. For all Reynolds numbers and Gr = 105, the solutions converge. However, for Re = 10, a periodic flow oscillation in time is brought about for all heat source distances. The temperature distributions (not shown here) where Re = 10 and d = 1, 2, and 3 are plotted in the time shots t = 15, 30, and 50, respectively. It can be perceived in these cases, higher temperature distribution on the second module. This temperature rise tends to weaken as the Reynolds number is increased. The temperature invigoration in the region around Heater 2 is due to the fact that the thermal wake from Heater 1 intensifies the buoyant flow in the localized region in the vicinity of Heater 2. It can be noted that by keeping Re constant, the distance d

Gr = 10<sup>5</sup>

Nu

4

on Heater 2 will appear. This will be seen afterwards.

5

6

7

8

9

10

11

12

plays a considerable role on the second module heating. For instance, what has just been said happens in the case where Re = 1000 and d = 1, 2, and 3.

(b)

A Mixed Convection Study in Inclined Channels with Discrete Heat Sources 21

0.020

0.040

0.047

0.061

0.081

0.141

0.061

0.081

0.096

0.065 0.097 0.1<sup>13</sup>

0.161

0.02

0

.094

0.060

0.050

0.021

0.042

0.063

0.147

0.022

0.179

0.201

0

.115

0.03

8

0.045

0.

0.067

089

0.084

0.105 0.105

.189

.2

0.126

0

0

0.084

0.063

0.021

0.089

0.04

0.06

0.04

0.047

0.1

20 0.160

0.071

0.071

0.08

0.08

0.1

0.14

0.18

0.094

0.141 0.165 0.212

0.118 0.141

0.080

0.050

0.021

0.021

0.045

0.020

0.06

7

0.0890. 0.112 134

0.067

0.059

0.049

0.150 0.201 7

0.04

0.024

0.071

0.047

0.020

0.134

0.153

0.191

0.101

0.121

0.141

0.071

0.020

0.024

0.118

0.019

0.016

0.02

0.04

0.048

0.02

0.08

0.1

0.14

0.06

0.08

0.16 0.18

0.08

0.1

0.12

0.24

0.04

0.06

0.024

0.11<sup>8</sup>

0.047

0.071

0.040

0.033

0.008

0.021

0.03845

0.042

0.042

0.022

0.045

0.063

0.084

0.2 10 31

0.157

0.039

0.033

0.008

3

0.05

Fig. 14. Isotherms for Re = 1, 10, 100, 1000, γ = 0° , and Gr = 105, Δθ = 0.02

0.065

0.118

0.100

0.12

0.06

0.040

Re = 1 d = 1

Re = 10 d = 1

Re = 50 d = 1

Re = 1 d = 2

Re = 100 d = 2

Re = 1000 d = 2

Re = 10 d = 3

Re = 50 d = 3

Re = 100 d = 3

Re = 1000 d = 3

Re = 1 d = 3

Re = 50 d = 2

Re = 10 d = 2

Re = 1000 d = 1

Re = 100 d = 1

0.061

0.081

0.095

0.038

0.134

0.17

0.191

2

0.129

0.121 0.121

0.101

0.020

0.024

0.118

0.095

0.071

0.057 0.077

0.007 0.007 0.007 0.015 0.015 0.015 0.022 0.022 0.022 0.0290.029 0.029 0.036 0.029 0.036 0.044 0.073 0.058 0.0580.065 0.0510.044

0.08

0.22

0.1

0.1

0.12 0.12

0.094

0.118

0.024

0.016

0.032

0.047

0.019

0.038

0.02

0.047 0.047

0.10

0.084

0.030

0.042

0.084

0.126

0.045

0.224

0.255

0.067

0.157

0.089

0.063

0.042

0.084 0.084

0.105

0.022

0.179

0.059

0.065 0.082

0.131 0.115

0.109.261

03

.180

0.015

0.157 0.177 0.216

0.246

0

0.024

0.067

0.021

0.067

0.098

0.098

0.024

0.040

0.022

0.079

0.049

0.045

0.016

0.030

0.060

0.04

0.07

0.017

0.11

0.091

0.042

0.105

0.18

0.126

0.147 0.16

9

8

0.063

1

0.06

0.094

0.118

0.040

0.142

0.210

0.166 0.189 0.237

0.142

0.081

0.061

0.162

Fig. 12. Average Nusselt vs γ: 1 ≤ Re ≤ 500 and (a) Gr = 103, (b) Gr = 104, and (c) Gr = 105

Fig. 13. Velocity vector for Gr = 105, Re = 1, 10, 100, and d = 1, 2, 3

Re = 1 Re = 5 Re = 10 Re = 50 Re =100 Re = 200 Re = 500

Nu

4

Fig. 13. Velocity vector for Gr = 105, Re = 1, 10, 100, and d = 1, 2, 3

**Re=1 d=1 Re = 10 d = 1 Re=100 d=1** 

**Re=1 d=2 Re=10 d=2 Re=100 d=2** 

**Re=1 d=3 Re=10 d=3 Re=100 d=3** 

5

6

7

8

9

10

11

12

γ(degree) 0 20 40 60 80

(c) Fig. 12. Average Nusselt vs γ: 1 ≤ Re ≤ 500 and (a) Gr = 103, (b) Gr = 104, and (c) Gr = 105

Gr = 105

Fig. 14. Isotherms for Re = 1, 10, 100, 1000, γ = 0° , and Gr = 105, Δθ = 0.02

A Mixed Convection Study in Inclined Channels with Discrete Heat Sources 23

θ

θ

Fig. 16. Temperature on modules 1 and 2 for d = 1, 2, 3; Re = 100, 1000, and Gr = 105

module is submitted to higher heat transfer since it is constantly been bombarded with cold fluid from the forced convection. On the other hand, it can be seen again that a flow wake from the first source reaches the second one and this is responsible for the bifurcation of the Nusselt number curves. Here, one can note the time spent by the hot fluid coming from the first source and traveling to the second one. For example, for Re = 100 and d = 1, 2, and 3, the time shots are, respectively, around t = 1.4, 3.0, and 4.0. However, the converged values for these last cases are almost the same. As seen earlier, periodic oscillations appear for

The results presented here are obtained using the finite element method (FEM) and a structured mesh with rectangular isoparametric four-node elements in which ΔX = 0.1 and ΔY = 0.05. A mesh sensibility analysis was carried out (Guimaraes, 2008). The temperature distributions for Reynolds numbers Re = 1, 10, 50, and 100, Grashof number Gr = 105, and inclination angles γ = 0° (horizontal), 45°, and 90° (vertical) are available in Fig. (18). For Re = 1 and γ = 0° and 45°, there is a formation of thermal cells which are localized in regions close to the modules. When Re = 1, the flow is predominantly due to natural convection. As Re is increased, these cells are stretched and hence forced convection starts to be characterized. By keeping Re constant, the inclination angle variation plays an important role on the temperature distribution. The effect of γ on temperature is stronger when low velocities are present. For example, when Re = 10 and γ = 0°, 45° and 90°, this behavior is noted, that is, for γ = 0° and Re = 10, a thermal cell is almost present, however, for γ = 45°and Re = 10, those cells vanish. This is more evident when Re =1 and γ = 45° and 90°.

0,06 0,08 0,10 0,12 0,14 0,16 0,18 0,20 0,22 0,24

0,00

0,05

0,10

0,15

0,20

0,25

Re = 100, Gr = 105

Re = 1000, Gr = 105

Heater 2

Heater 2

d = 1 d = 2 d = 3

d = 1 d = 2 d = 3

Re = 100, Gr = 105

Re = 1000, Gr = 105

**5.3 Case with three heat sources** 

θ

θ

0,00

0,00

Re = 10.

0,05

0,10

0,15

0,20

0,25

0,05

0,10

0,15

0,20

0,25

Heater 1

Heater 1

d = 1 d = 2 d = 3

d = 1 d = 2 d = 3

Fig. 15. Nu for Re = 1, 10, 102, 103, d = 1, 2, 3, Gr = 105, 104, and 105 on Heater 1 and 2

Figure (15) depicts the effect of the Reynolds number on heat transfer for Gr = 103, 104, and 105, Re = 1, 10, 100, and 1000, and finally, d = 1, 2, 3, in the pair of heat sources. This picture shows some points already discussed previously such as the module distance effect which is almost negligible on Heater 1 and moderate on Heater 2. It can be clearly seen the balance between forced and natural convections. In a general way, the distance d = 3 is the one which offers better work conditions since the temperatures are lower.

Figure (16) presents the temperature distributions θ on Heater 1 and 2 for Re = 100 and 1000 and d = 1, 2, and 3. The distance between the modules does not affect the temperature on Heater 1 whereas this effect can be distinctively seen on Heater 2. It is interesting noticing in Heater 2, that for Re = 100 and 1000, distances d = 2 and 3 do not present significant changes, but d = 1. Then, there is an optimum distance in which two heat sources can be placed apart to have lower temperatures and this is the case of d = 3 here, although d = 2 does not present a meaningful change in temperature either. This, in a certain way, can lead us to a better layout of the heat sources in an array. Of course, the presence of more heat sources and the geometry of the channel must be taken into account. Anyway, this behavior is food for thought for future studies.

Finally, the time distribution of the Nusselt number along Heater 1 and 2 for Gr = 105, Re = 10, 100, and 1000, d = 1, 2, and 3, is shown in Fig. (17). In all cases, as expected, the first

Nu

NU

NU

Gr=10<sup>4</sup>


G r=10 <sup>5</sup> - H eater 2

Gr=10<sup>3</sup> - Heater 2

1 10 100 1000

1 1 0 100 1000

R e 1 1 0 100 1000

d=1 d=2 d=3

d=1 d=2 d=3

d=1 d=2 d=3

d=1 d=2 d=3

d=1 d=2 d=3

> d=1 d=2 d=3

which offers better work conditions since the temperatures are lower.

Fig. 15. Nu for Re = 1, 10, 102, 103, d = 1, 2, 3, Gr = 105, 104, and 105 on Heater 1 and 2

Figure (15) depicts the effect of the Reynolds number on heat transfer for Gr = 103, 104, and 105, Re = 1, 10, 100, and 1000, and finally, d = 1, 2, 3, in the pair of heat sources. This picture shows some points already discussed previously such as the module distance effect which is almost negligible on Heater 1 and moderate on Heater 2. It can be clearly seen the balance between forced and natural convections. In a general way, the distance d = 3 is the one

Figure (16) presents the temperature distributions θ on Heater 1 and 2 for Re = 100 and 1000 and d = 1, 2, and 3. The distance between the modules does not affect the temperature on Heater 1 whereas this effect can be distinctively seen on Heater 2. It is interesting noticing in Heater 2, that for Re = 100 and 1000, distances d = 2 and 3 do not present significant changes, but d = 1. Then, there is an optimum distance in which two heat sources can be placed apart to have lower temperatures and this is the case of d = 3 here, although d = 2 does not present a meaningful change in temperature either. This, in a certain way, can lead us to a better layout of the heat sources in an array. Of course, the presence of more heat sources and the geometry of the channel must be taken into account. Anyway, this behavior

Finally, the time distribution of the Nusselt number along Heater 1 and 2 for Gr = 105, Re = 10, 100, and 1000, d = 1, 2, and 3, is shown in Fig. (17). In all cases, as expected, the first

Gr=103-Heater1

Gr=10<sup>4</sup> - Heater 1

G r = 10 <sup>5</sup>


1 1 0 100 1000

1 1 0 100 1000

R e 1 1 0 100 1000

is food for thought for future studies.

Fig. 16. Temperature on modules 1 and 2 for d = 1, 2, 3; Re = 100, 1000, and Gr = 105

module is submitted to higher heat transfer since it is constantly been bombarded with cold fluid from the forced convection. On the other hand, it can be seen again that a flow wake from the first source reaches the second one and this is responsible for the bifurcation of the Nusselt number curves. Here, one can note the time spent by the hot fluid coming from the first source and traveling to the second one. For example, for Re = 100 and d = 1, 2, and 3, the time shots are, respectively, around t = 1.4, 3.0, and 4.0. However, the converged values for these last cases are almost the same. As seen earlier, periodic oscillations appear for Re = 10.

#### **5.3 Case with three heat sources**

The results presented here are obtained using the finite element method (FEM) and a structured mesh with rectangular isoparametric four-node elements in which ΔX = 0.1 and ΔY = 0.05. A mesh sensibility analysis was carried out (Guimaraes, 2008). The temperature distributions for Reynolds numbers Re = 1, 10, 50, and 100, Grashof number Gr = 105, and inclination angles γ = 0° (horizontal), 45°, and 90° (vertical) are available in Fig. (18). For Re = 1 and γ = 0° and 45°, there is a formation of thermal cells which are localized in regions close to the modules. When Re = 1, the flow is predominantly due to natural convection. As Re is increased, these cells are stretched and hence forced convection starts to be characterized. By keeping Re constant, the inclination angle variation plays an important role on the temperature distribution. The effect of γ on temperature is stronger when low velocities are present. For example, when Re = 10 and γ = 0°, 45° and 90°, this behavior is noted, that is, for γ = 0° and Re = 10, a thermal cell is almost present, however, for γ = 45°and Re = 10, those cells vanish. This is more evident when Re =1 and γ = 45° and 90°.

A Mixed Convection Study in Inclined Channels with Discrete Heat Sources 25

**0.02**

**0.**

**0**

**0.08**

**0.1**

**.06**

**0**

**0.04**

**0.0**

**4**

**0.06**

**0.08**

**0**

**0.02**

**0.08 0.10 0.10**

**0.08**

**0.06**

**0.04**

**0.06**

**0.04**

**0.0**

**8 0.10**

**0.04 0.06 0.08**

**0.10**

**0**

**.10**

**0.08**

**0.**

**0.08**

**06**

**0.08**

**0.04**

**0.04**

**0.06**

**0.08**

**0.06**

**0.06**

**0.08**

**.06**

**4**

**0.02**

**0.0**

**2**

**0.06**

**0.08**

**0.1**

**0.02 0.04**

**0.04**

**0.04**

**0.08**

**0.06**

**0.04**

**0.08**

**0.06**

**0.08**

**0.06**

**0.04**

**0.06**

**0.08**

**0**

**0.08**

**.06**

**0.02**

**00. 4**

**0.06**

**0.04**

**0**

**0**

**.10**

**0.08**

**0.06 .08**

**0.10 0.12**

**0.10**

**0.08**

**0.06**

**0.10**

**0.08**

**0.02**

**0.02 0.04**

**0.02 0.02** **0.06**

**0**

**0.0**

**6**

**0**

**.08**

**.04**

**0.04**

**0.06**

**0.08**

**0.02**

**0**

**0.06**

**0.04**

**0.02**

**0.06**

**0.04**

**0.02 0.06**

**0.06**

**0.04**

**0.06**

**0.06**

**0.04**

**0.08**

**0.10**

**0.08**

**0.06**

**0.04**

Fig. 18. Isotherms for Gr = 105, Re = 1, 10, 50, 100 and γ = 0°, 45°, 90°

**0.08**

**0.**

**06**

**0.08**

**0.04**

**0.06**

**0.02**

**0.08**

**0.**

**06**

**0.08**

**0.1**

**0.04 .06**

**0.04**

**0.02**

**0.02**

**0.02**

**0.04 0.06**

**0.02**

**0.02 0.04**

**0.02**

**0.04**

**0.02**

**0.02**

**0.0**

**4**

**0.**

**06**

**0.08**

0 0.5 1

Re = 1 γ = 0<sup>0</sup>

Re = 10 γ = 0<sup>0</sup>

Re = 50 γ = 0<sup>0</sup>

Re = 100 γ = 0<sup>0</sup>

Re = 1 γ = 45<sup>0</sup>

Re = 10 γ = 45<sup>0</sup>

Re = 50 γ = 45<sup>0</sup>

Re = 100 γ = 45<sup>0</sup>

Re = 1 γ = 90<sup>0</sup>

Re = 10 γ = 90<sup>0</sup>

Re = 50 γ = 90<sup>0</sup>

Re = 100 γ = 90<sup>0</sup>

0 0.5 1

0 0.5 1

0 0.5 1

0 0.5 1

0 0.5 1

0 0.5 1

0 0.5 1

0 0.5 1

0 0.5 1

0 0.5 1

0 0.5 1

Fig. 17. Average Nusselt number vs Time: Gr = 105, Re = 10, 102, 103, d =1, 2 , 3, Heater 1, 2

NU

<sup>t</sup> 2 4 6 8 10 12 14

NU

t 5 10 15 20

NU

t 5 10 15 20

Fig. 17. Average Nusselt number vs Time: Gr = 105, Re = 10, 102, 103, d =1, 2 , 3, Heater 1, 2

Re = 100, d = 1

Heater 1 Heater 2

Re = 100, d = 2

Heater 1 Heater 2

Re = 100, d = 3

Heater 1 Heater 2

<sup>t</sup> 2 4 6 8 10 12 14

t 5 10 15 20

t 5 10 15 20

Heater 1 Heater 2

Heater 1 Heater 2

> Heater 1 Heater 2

Heater 1 Heater 2

Re = 1000, d = 1

Heater 1 Heater 2

Re = 1000, d = 2

Heater 1 Heater 2

Re = 1000, d = 3

Re = 10, d = 1

Re = 10, d = 2

Re = 10, d = 3

NU

NU

NU

> t 2 4 6 8 10 12 14

> > NU

<sup>t</sup> 5 10 15 20

NU

<sup>t</sup> 5 10 15 20

NU

Fig. 18. Isotherms for Gr = 105, Re = 1, 10, 50, 100 and γ = 0°, 45°, 90°

A Mixed Convection Study in Inclined Channels with Discrete Heat Sources 27

another two secondary recirculations localized just after the sources. Moreover, for these later cases, a reversal fluid flow is present at the outlet. As Re is increased by keeping γ constant, these recirculations get weaker until they disappear for high Re. Clearly, one can note the effect of the inclination on the velocity vectors when Re = 10. The strongest

Figure (20) presents the average Nusselt number distributions on the heat sources, NUH1, NUH2, and NUH3 for Reynolds numbers Re = 1, 10, 50, 100, and 1000, Grashof numbers Gr = 103, 104, and 105, and inclination angles γ = 0°, 45°, and 90°. In general, the average Nusselt number for each source increases as the Reynolds number is increased. By analyzing each graphic separately, it can be observed that NUH1 tends to become more distant from NUH2 and NUH3 as Reynolds number is increased, starting from an initial value for Re = 1 which is almost equal to NUH2 and NUH3. This agreement at the beginning means that the heaters are not affecting one another. Here, it can be better perceived that behavior found in Fig. (13), where a heater is affected by an upstream one. That is the reason why NUH1 shows higher values. The only case in which the heaters show different values for Re = 1 is when Gr = 105 and γ = 90°. Overall, the strongest average Nusselt number variation is between 0° and 45°. Practically in all cases, NUH1, NUH2 ,and NUH3 increase in this angle range, 0° and 45°, while for Gr = 105 and Re = 1000, NUH2 and NUH3 decrease. When electronic circuits are concerned, the ideal case is the one which has the highest Nusselt number. Thus, angles 45° and 90° are the most suitable ones with not so

> Re 100 101 102 103

> Re <sup>100</sup> <sup>101</sup> <sup>102</sup> <sup>103</sup>

Re <sup>100</sup> <sup>101</sup> <sup>102</sup> <sup>103</sup>

Fig. 20. Average Nusselt number vs Reynolds number for Gr = 103, 104, 105, γ = 0°, 45°, 90°

Re 100 101 102 103

Re <sup>100</sup> <sup>101</sup> <sup>102</sup> <sup>103</sup>

Re <sup>100</sup> <sup>101</sup> <sup>102</sup> <sup>103</sup>

NUH1 NUH2 NUH3

NUH1 NUH2 NUH3

NUH1 NUH2 NUH3

, γ = 90°

, γ = 90°

, γ = 90°

Nu Gr=103

Gr=10<sup>4</sup>

NU

Gr=10<sup>5</sup>

NU

NUH! NUH2 NUH3

NUH1 NUH2 NUH3

NUH1 NUH2 NUH3

, γ = 45°

, γ = 45°

, γ = 45°

inclination influence takes place when it is between 0° and 45°.

NU Gr=10<sup>3</sup>

Gr=10<sup>4</sup>

NU

Gr=105

NU

Re 100 101 102 103

Re <sup>100</sup> <sup>101</sup> <sup>102</sup> <sup>103</sup>

Re <sup>100</sup> <sup>101</sup> <sup>102</sup> <sup>103</sup>

NUH1 NUH2 NUH3

NUH1 NUH2 NUH3

NUH1 NUH2 NUH3

NU Gr=10<sup>3</sup> , γ = 0°

> Gr=10<sup>4</sup> , γ = 0°

NU

> Gr=10<sup>5</sup> , γ = 0°

NU

It is worth observing that, the fluid heated in the first heater reaches the second one, and then the third one. Thus, this process of increasing temperature provides undesirable situations when cooling is aimed.

Fig. 19. Velocity vectors for Gr = 105, Re = 10 and 100, and γ = 0°, 45°and 90°

Figure (19) depicts the velocity vectors for Re = 10 and 100 and Gr = 105 for γ = 0°, 45°, and 90°. It can be noted that for Re = 10 and γ = 0°, 45°, and 90°, recirculations are generated by the fluid heated on the sources. For Re = 10 and γ = 0°, three independent recirculations appear. The distance among the heat sources enables the reorganization of the velocity profile until the fluid reaches the next source and then the recirculation process starts all over again. Now, concerning the cases where Re = 10 and γ = 45° and 90°, there are two kinds of recirculations, that is, a primary recirculation along all channel that encompasses

It is worth observing that, the fluid heated in the first heater reaches the second one, and then the third one. Thus, this process of increasing temperature provides undesirable

Fig. 19. Velocity vectors for Gr = 105, Re = 10 and 100, and γ = 0°, 45°and 90°

Figure (19) depicts the velocity vectors for Re = 10 and 100 and Gr = 105 for γ = 0°, 45°, and 90°. It can be noted that for Re = 10 and γ = 0°, 45°, and 90°, recirculations are generated by the fluid heated on the sources. For Re = 10 and γ = 0°, three independent recirculations appear. The distance among the heat sources enables the reorganization of the velocity profile until the fluid reaches the next source and then the recirculation process starts all over again. Now, concerning the cases where Re = 10 and γ = 45° and 90°, there are two kinds of recirculations, that is, a primary recirculation along all channel that encompasses

situations when cooling is aimed.

Re = 10, γ = 0°

Re = 100, γ = 0°

Re = 10, γ = 45°

Re = 100, γ = 45°

Re = 10, γ = 90°

Re = 100, γ = 90°

another two secondary recirculations localized just after the sources. Moreover, for these later cases, a reversal fluid flow is present at the outlet. As Re is increased by keeping γ constant, these recirculations get weaker until they disappear for high Re. Clearly, one can note the effect of the inclination on the velocity vectors when Re = 10. The strongest inclination influence takes place when it is between 0° and 45°.

Figure (20) presents the average Nusselt number distributions on the heat sources, NUH1, NUH2, and NUH3 for Reynolds numbers Re = 1, 10, 50, 100, and 1000, Grashof numbers Gr = 103, 104, and 105, and inclination angles γ = 0°, 45°, and 90°. In general, the average Nusselt number for each source increases as the Reynolds number is increased. By analyzing each graphic separately, it can be observed that NUH1 tends to become more distant from NUH2 and NUH3 as Reynolds number is increased, starting from an initial value for Re = 1 which is almost equal to NUH2 and NUH3. This agreement at the beginning means that the heaters are not affecting one another. Here, it can be better perceived that behavior found in Fig. (13), where a heater is affected by an upstream one. That is the reason why NUH1 shows higher values. The only case in which the heaters show different values for Re = 1 is when Gr = 105 and γ = 90°. Overall, the strongest average Nusselt number variation is between 0° and 45°. Practically in all cases, NUH1, NUH2 ,and NUH3 increase in this angle range, 0° and 45°, while for Gr = 105 and Re = 1000, NUH2 and NUH3 decrease. When electronic circuits are concerned, the ideal case is the one which has the highest Nusselt number. Thus, angles 45° and 90° are the most suitable ones with not so

Fig. 20. Average Nusselt number vs Reynolds number for Gr = 103, 104, 105, γ = 0°, 45°, 90°

A Mixed Convection Study in Inclined Channels with Discrete Heat Sources 29

NUM

2

1 0

NUM

2

1 0

NUM

Fig. 22. Module average Nusselt number in time: Re = 10 e 100 , Gr = 103, 104, 105, γ = 90°

1. The mixed convection was studied in a simple channel considering the effect of the inclination angle and some physical parameters. The ranges performed were as follows: 1 ≤ Re ≤ 500, 103 ≤ Gr ≤ 105, and 0° ≤ γ ≤ 90°. The set of governing equations were discretized and solved using the Galerkin finite element method (FEM) with the Penalty formulation in the pressure terms and the Petrov-Galerkin perturbations in the convective terms in all throughout the chapter. 5980 four-noded elements were used to discretize the spatial domain. Comparisons were performed to validate the computational code. It was observed from the results of the present problem that the effect of the inclination angle on the velocity and temperature distributions plays an important role on the heat transfer for low Re and high Gr. For high Re, the effect of the orientation was negligible. One must understand that when the words 'low' and 'high' were mentioned here, it meant low and high compared to the limits considered in this work. In general, it was also discovered that an inclination angle around 60° and 75° provided a slight most desirable work conditions when cooling is aimed. It was said that this optimal orientation would be 90° (Choi and Ortega, 1993), despite no significant changes were found after γ = 45° (according to the convection adopted here).

2

4

6

8

4

6

8

4

6

8

1 0

Re = 100, Gr = 10 <sup>3</sup>

Re = 100, Gr = 10 <sup>4</sup>

Re = 100, Gr = 10<sup>5</sup>

, γ = 90°

, γ = 90°

, γ = 90°

<sup>t</sup> <sup>0</sup> <sup>5</sup> <sup>10</sup> 1 5 2 0 <sup>25</sup> 3 0

0 10 20 30 40 50 60 70

<sup>t</sup> 0 1 0 20 3 0 4 0 50 6 <sup>0</sup>

H 1 H 2 H 3

H 1 H 2 H 3

> H 1 H 2 H 3

t

H 1 H 2 H 3

> H 1 H 2 H 3

> > H 1 H 2 H 3

Re = 10, Gr = 10<sup>3</sup> ,

Re = 10, Gr = 10 <sup>4</sup>

Re = 10, Gr = 10<sup>5</sup>

NUM

2

1 0

NUM

NUM

2

**6. Conclusions** 

4

6

8

2

1 0

4

6

8

4

6

8

1 0

, γ = 90°

<sup>t</sup> <sup>0</sup> <sup>5</sup> 1 0 <sup>15</sup> 2 0 2 5 <sup>30</sup>

<sup>t</sup> <sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> <sup>25</sup>

, γ = 90°

<sup>t</sup> 0 2 4 6 81 0 12

, γ = 90°

much difference between them. An exception would be the case where Gr = 105, Re = 1000, and γ = 0°.

Figure (21) presents the local dimensionless temperature distributions on the three heat sources for Re = 10, 100, 1000, Gr = 105, γ = 0°, 45°, and 90°. Again, the cases where Re = 10 and 100 show the lowest temperatures when γ = 90°. On the other hand, this does not happen when Re = 1000, where the horizontal position shows the lowest temperatures along the modules. All cases in which γ = 0°, the second and third sources have equal temperatures. However, the first module shows lower temperatures. As mentioned before, this characterizes the fluid being heated by a previous heat source, thus, not contributing to the cooling of an upstream one.

Figure (22) presents the average Nusselt number variation on H1, H2, and H3 against the dimensionless time t considering Re = 10, 100, Gr = 103, 104, 105 and γ = 90°. In the beginning, all three Nusselt numbers on H1, H2, and H3 have the same behavior and value. These numbers tend to converge to different values as time goes on. However, before they do so, they bifurcate at a certain point. This denotes the moment when a heated fluid wake from a previous source reaches a downstream one.

Fig. 21. Module temperatures for Re = 10, 100, 1000; Gr = 105, γ = 0°, 45°, 90°

Fig. 22. Module average Nusselt number in time: Re = 10 e 100 , Gr = 103, 104, 105, γ = 90°

#### **6. Conclusions**

28 Convection and Conduction Heat Transfer

much difference between them. An exception would be the case where Gr = 105, Re = 1000,

Figure (21) presents the local dimensionless temperature distributions on the three heat sources for Re = 10, 100, 1000, Gr = 105, γ = 0°, 45°, and 90°. Again, the cases where Re = 10 and 100 show the lowest temperatures when γ = 90°. On the other hand, this does not happen when Re = 1000, where the horizontal position shows the lowest temperatures along the modules. All cases in which γ = 0°, the second and third sources have equal temperatures. However, the first module shows lower temperatures. As mentioned before, this characterizes the fluid being heated by a previous heat source, thus, not contributing to

Figure (22) presents the average Nusselt number variation on H1, H2, and H3 against the dimensionless time t considering Re = 10, 100, Gr = 103, 104, 105 and γ = 90°. In the beginning, all three Nusselt numbers on H1, H2, and H3 have the same behavior and value. These numbers tend to converge to different values as time goes on. However, before they do so, they bifurcate at a certain point. This denotes the moment when a heated fluid wake

Re=10, Gr=105

Re=100, Gr=105

Re=1000, Gr=105

, γ=45°

, γ=45°

Source

Source

Source

, γ=45°

Source1 Source2 Source3

Source1 Source2 Source3

Source1 Source2 Source3 θ

0,05 0,10 0,15 0,20 0,25 0,30

θ

0,00 0,05 0,10 0,15 0,20 0,25 0,30

θ

0,00 0,02 0,04 0,06 0,08 0,10 0,12 Re=10, Gr=105

Re=100, Gr=105

Re=1000, Gr=105

, γ=90°

, γ=90°

Source

Source

Source

, γ=90°

Source1 Source2 Source3

Source1 Source2 Source3

Source1 Source2 Source3

and γ = 0°.

Re=10, Gr=105 , γ=0°

Re=100, Gr=105

Re=1000, Gr=105

, γ=0°

Source

Source

Source

, γ=0°

θ

0,00 0,05 0,10 0,15 0,20 0,25 0,30

θ

0,00 0,05 0,10 0,15 0,20 0,25 0,30

θ

0,00 0,02 0,04 0,06 0,08 0,10 0,12

the cooling of an upstream one.

from a previous source reaches a downstream one.

Source1 Source2 Source3

> Source1 Source2 Source3

> > Source1 Source2 Source3

θ

0,05 0,10 0,15 0,20 0,25 0,30

θ

0,00 0,05 0,10 0,15 0,20 0,25 0,30

θ

0,00 0,02 0,04 0,06 0,08 0,10 0,12

Fig. 21. Module temperatures for Re = 10, 100, 1000; Gr = 105, γ = 0°, 45°, 90°

1. The mixed convection was studied in a simple channel considering the effect of the inclination angle and some physical parameters. The ranges performed were as follows: 1 ≤ Re ≤ 500, 103 ≤ Gr ≤ 105, and 0° ≤ γ ≤ 90°. The set of governing equations were discretized and solved using the Galerkin finite element method (FEM) with the Penalty formulation in the pressure terms and the Petrov-Galerkin perturbations in the convective terms in all throughout the chapter. 5980 four-noded elements were used to discretize the spatial domain. Comparisons were performed to validate the computational code. It was observed from the results of the present problem that the effect of the inclination angle on the velocity and temperature distributions plays an important role on the heat transfer for low Re and high Gr. For high Re, the effect of the orientation was negligible. One must understand that when the words 'low' and 'high' were mentioned here, it meant low and high compared to the limits considered in this work. In general, it was also discovered that an inclination angle around 60° and 75° provided a slight most desirable work conditions when cooling is aimed. It was said that this optimal orientation would be 90° (Choi and Ortega, 1993), despite no significant changes were found after γ = 45° (according to the convection adopted here).

A Mixed Convection Study in Inclined Channels with Discrete Heat Sources 31

Arquis, E.; Rady M.A.; Nada, S.A. (2007). A numerical investigation and parametric study of

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Facing Step Flow, *J. Fluids and Structures*, Vol. 12, pp. 703-716.

study, *Comput. Methods Appl. Mech Engrg*., Vol. 188, pp. 187-202.

*International Journal of Heat and Fluid Flow*, Vol. 28, pp. 787–805.

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*and Mass Transfer*, Vol. 49, pp. 3568–3582.

cooling an array of multiple protruding heat sources by a laminar slot air jet,

in a vertical channel with heat generating components, *International Journal of Heat* 

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vertical wall, *Int. J. of Numerical Methods for Heat and Fluid Flow, Vol.* 10, pp. 286–306.

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**8. References** 

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3134.

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Some cases presented the reversed flow for low Re and high Gr. The reversal flow did not noticeably influence the heat transfer coefficient on the module, although it did change the velocity and temperature fields.

 In general, the results encourage the use of inclined boards in cabinets. However, some other aspects should be addressed such as the geometrical arrangement of the boards.


The authors found an interesting behaviour in all three cases with one, two or three heat sources: There is a moment in all three cases studied when oscillations in time near the heat sources appear, featuring their initial strength and then reaching its maximum and, eventually, ending up going weaker until the flow does not present these time oscillations anymore. This is a very interesting physical moment that a certain dimensionless number may be created or applied in order to feature this behaviour at its maximum or its appearance length in time. However, this will be left for future works from the same authors or from new authors.

#### **7. Acknowledgment**

The authors thank the following Brazilian support entities: CAPES, CNPq, and FAPEMIG.

#### **8. References**

30 Convection and Conduction Heat Transfer

 In general, the results encourage the use of inclined boards in cabinets. However, some other aspects should be addressed such as the geometrical arrangement of the boards. 2. A mixed convection study in a rectangular channel with two heat sources on the bottom wall was conducted. The upper wall was kept at a constant cold temperature and the remaining part of the lower one was adiabatic. The heat transfer was studied by ranging some physical and geometrical parameters as follows: Re = 1 to 1000; the distances between the modules d = 1, 2, and 3; and Gr = 103, 104, and 105. For Re = 1 and distances d = 1 and 2, the buoyant forces generated plumes that interfered in each other. For higher Reynolds numbers, the heat transfer in Heater 2 was invigorated by a hot wake brought about by Heater 1. In cases where Re = 10 and Gr = 105, the flow oscillations appeared and they strongly affected the flow distributions. For Re = 100 and 1000, the temperature distributions in Heater 1 were not affected by the distances between the modules whereas on Heater 2, they were distinct. An important conclusion is that there was an optimum distance in which two sources could be placed apart from each other, that is, d = 3, although d = 2 did not present a significant change in temperature either. Further investigations are encouraged taking into consideration

3. In this work, mixed convection heat transfer study in an inclined rectangular channel with three heat sources on the lower wall was carried out using the same. Effects on the Nusselt number along the heat sources as well as the velocity vectors in the domain were verified by varying the following parameters: γ = 0°, 45°, 90°, Re = 1, 10, 50, 100, 1000, Gr = 103, 104, 105. In general, the inclination angle had a stronger influence on the flow and heat transfer since lower forced velocities were present, especially when the channel was between 0° and 45°. It could be noted that in some cases some heat sources were reached by a hot wake coming from a previous module, thus, increasing their temperatures. Primary and secondary recirculations and reversal flow were present in some situations such as Re = 10, γ = 45° and 90°. In problems where heat transfer analysis on electronic circuits is aimed, cases with the lowest temperatures, and hence, the highest Nusselt numbers, are the most suitable ones. Therefore, the channel inclination angles 45° and 90° were the best ones with little difference between them. An exception was the case with Gr = 105 and Re = 1000, where γ = 0° was the ideal

The authors found an interesting behaviour in all three cases with one, two or three heat sources: There is a moment in all three cases studied when oscillations in time near the heat sources appear, featuring their initial strength and then reaching its maximum and, eventually, ending up going weaker until the flow does not present these time oscillations anymore. This is a very interesting physical moment that a certain dimensionless number may be created or applied in order to feature this behaviour at its maximum or its appearance length in time. However, this will be left for future works from the same authors

The authors thank the following Brazilian support entities: CAPES, CNPq, and FAPEMIG.

change the velocity and temperature fields.

more heaters and different arrays.

inclination.

or from new authors.

**7. Acknowledgment** 

Some cases presented the reversed flow for low Re and high Gr. The reversal flow did not noticeably influence the heat transfer coefficient on the module, although it did


**0**

**2**

*Australia*

Suvash Chandra Saha

**Periodically Forced Natural Convection Through**

Buoyancy-induced fluid motions in cavities have been discussed widely because of the applications in nature and engineering. A large body of literature exists on the forms of internal and external forcing, various geometry shapes and temporal conditions (steady or unsteady) of the resulting flows. Especially for the classic cases of rectangular, cylindrical or other regular geometries, many authors have investigated imposed temperature or boundary heat fluxes. Reviews of these research can be found in Ostrach (1988) and Hyun (1994). The rectangular cavity is not an adequate model for many geophysical situations where a variable (or sloping) geometry has a significant effect on the system. However, the convective flows in triangular shaped enclosures have received less attention than those in rectangular

Heat transfer through an attic space into or out of buildings is an important issue for attic shaped houses in both hot and cold climates. The heat transfer through attics is mainly governed by a natural convection process, and affected by a number of factors including the geometry, the interior structure and the insulation etc. One of the important objectives for design and construction of houses is to provide thermal comfort for occupants. In the present energy-conscious society, it is also a requirement for houses to be energy efficient, i.e. the energy consumption for heating or air-conditioning of houses must be minimized. A small number of publications are devoted to laminar natural convection in two dimensional

The temperature and flow patterns, local wall heat fluxes and mean heat flux were measured experimentally by Flack (1980; 1979) in isosceles triangular cavities with three different aspect ratios. The cavities, filled with air, were heated/cooled from the base and cooled/heated from the sloping walls covering a wide range of Rayleigh numbers. For the case of heated bottom surface it was found that the flow remained laminar for the low Rayleigh numbers. However, as the Rayleigh number increased, the flow eventually became turbulent. The author also reported the critical Rayleigh numbers of the transition from laminar to turbulent regimes. Kent (2009a) has also investigated the natural convection in an attic space for two different boundary conditions similar to Flack (1980; 1979). The author observed that for top heating and bottom cooling case the flow is dominated by pure conduction and remains stable for higher Rayleigh numbers considered. However, the flow becomes unstable for sufficiently

geometries, even though the topic has been of interest for more than two decades

isosceles triangular cavities in the vast literature on convection heat transfer.

large Rayleigh number for the second case (top cooling and bottom heating).

**1. Introduction**

**the Roof of an Attic-Shaped Building**

*School of Engineering and Physical Sciences, James Cook University*


### **Periodically Forced Natural Convection Through the Roof of an Attic-Shaped Building**

Suvash Chandra Saha

*School of Engineering and Physical Sciences, James Cook University Australia*

#### **1. Introduction**

32 Convection and Conduction Heat Transfer

Gartling, D.K. (1990). A Test Problem for Outflow Boundary Conditions – Flow over a Backward-Facing Step, *Int. J. Num. Meth. in Fluids*, Vol. 11, pp. 953-967. Kim, J. & Moin, P. (1985). Application of a Fractional-Step Method to Incompressible

Sohn, J. (1988). Evaluation of FIDAP on Some Classical Laminar and Turbulent Benchmarks,

Comini, G.; Manzam, M. & Cortella, G.. Open Boundary Conditions for the Streamfunction

Guimarães, P.M. (2007). Phd. Thesis: Heat transfer analysis of mixed convection using the

Guimarães, P. M. (2008). Combined free and forced coonvection in an inclined channel with

–Vorticity Formulation of Unsteady Laminar Convection, *Num. Heat Transfer Part B,* 

discrete heat sources, Int. Communications of Heat and Mass Transfer, Vol. 35, pp.

Navier-Stokes Equations, *J. Comp. Physics*, Vol. 59, pp. 308-323.

*Int. J. Num. Meth. in Fluids*, Vol. 8, pp. 1469-1490.

Petrov-Galerkin technique and the penalty, Brazil.

Vol. 31, pp. 217-234.

1267-1274.

Buoyancy-induced fluid motions in cavities have been discussed widely because of the applications in nature and engineering. A large body of literature exists on the forms of internal and external forcing, various geometry shapes and temporal conditions (steady or unsteady) of the resulting flows. Especially for the classic cases of rectangular, cylindrical or other regular geometries, many authors have investigated imposed temperature or boundary heat fluxes. Reviews of these research can be found in Ostrach (1988) and Hyun (1994). The rectangular cavity is not an adequate model for many geophysical situations where a variable (or sloping) geometry has a significant effect on the system. However, the convective flows in triangular shaped enclosures have received less attention than those in rectangular geometries, even though the topic has been of interest for more than two decades

Heat transfer through an attic space into or out of buildings is an important issue for attic shaped houses in both hot and cold climates. The heat transfer through attics is mainly governed by a natural convection process, and affected by a number of factors including the geometry, the interior structure and the insulation etc. One of the important objectives for design and construction of houses is to provide thermal comfort for occupants. In the present energy-conscious society, it is also a requirement for houses to be energy efficient, i.e. the energy consumption for heating or air-conditioning of houses must be minimized. A small number of publications are devoted to laminar natural convection in two dimensional isosceles triangular cavities in the vast literature on convection heat transfer.

The temperature and flow patterns, local wall heat fluxes and mean heat flux were measured experimentally by Flack (1980; 1979) in isosceles triangular cavities with three different aspect ratios. The cavities, filled with air, were heated/cooled from the base and cooled/heated from the sloping walls covering a wide range of Rayleigh numbers. For the case of heated bottom surface it was found that the flow remained laminar for the low Rayleigh numbers. However, as the Rayleigh number increased, the flow eventually became turbulent. The author also reported the critical Rayleigh numbers of the transition from laminar to turbulent regimes. Kent (2009a) has also investigated the natural convection in an attic space for two different boundary conditions similar to Flack (1980; 1979). The author observed that for top heating and bottom cooling case the flow is dominated by pure conduction and remains stable for higher Rayleigh numbers considered. However, the flow becomes unstable for sufficiently large Rayleigh number for the second case (top cooling and bottom heating).

obtain previously unavailable heat transfer data relevant to air conditioning calculations. This study considered only half of the domain. For the purpose of air conditioning calculations, Asan & Namli (2000) and Kent (2009b) have also reported numerical results for steady, laminar two-dimensional natural convection in a pitched roof of triangular cross-section

Periodically Forced Natural Convection Through the Roof of an Attic-Shaped Building 35

P

Recently, the transient flow development inside the attic space has been analysed by using scaling analysis with numerical verification by (Saha, 2011a;b; Saha et al., 2010a;c; Saha, 2008). The authors considered different types of thermal conditions, such as, sudden heating/cooling and ramp heating/cooling. In real situations, however, the attic space of buildings is subject to alternative heating and cooling over a diurnal cycle as it can be seen in Fig. 1. A very few studies for diurnal heating and cooling effect on the attic space are reported in the literature (Saha et al., 2010b; 2007). The authors discussed a general flow structure and heat transfer due to the effect of periodic thermal forcing. A detailed explanation of choosing the period for the model attic is required as the 24-hour period for the field situation is not

In this study, numerical simulations of natural convection in an attic space subject to diurnal temperature condition on the sloping wall have been carried out. An explanation of choosing the period of periodic thermal effect has been given with help of the scaling analysis which is available in the literature. Moreover, the effects of the aspect ratio and Rayleigh number on the fluid flow and heat transfer have been discussed in details as well as the formation of a

The physical system is sketched in Fig. 2, which is an air-filled isosceles triangular cavity of variable aspect ratios. Here 2*l* is the length of the base or ceiling, *T*<sup>0</sup> is the temperature applied on the base, *TA* is the amplitude of temperature fluctuation on the inclined surfaces, *h* is the

0.00 0.25 0.50 0.75 1.00

Evening

Mid night

under the summer day (day-time) boundary conditions.

Mid day

Fig. 1. Temperature boundary condition on the incline walls of the enclosure

pitchfork bifurcation of the flow at the symmetric line of the enclosure.

height of the enclosure and *P* is the period of the thermal forcing.

T (K)

applicable here.

**2. Formulation of the problem**

290

Morning

295

300

A comparison study is performed by Ridouane et al. (2005) where the authors compare their numerical results produced for two different boundary conditions, (a) cold base and hot inclined walls (b) hot base and cold inclined walls with the experimental results obtained by Flack (1980; 1979). A good agreement has been obtained between the numerical predictions and the experimental measurements of Nusselt number. A numerical study of above mentioned two boundary conditions has also performed by Ridouane et al. (2006). However, the authors cut a significant portion of bottom tips and applied adiabatic boundary condition there. It is revealed from the analysis that the presence of insulated sidewalls, even of very small height, provides a huge gain of energy and helps keep the attic at the desired temperature with a minimum energy.

The attic problem under the night-time conditions was again investigated experimentally by Poulikakos & Bejan (1983a). In their study, the authors modelled the enclosure as a right-angled triangle with an adiabatic vertical wall, which corresponded to the half of the full attic domain. A fundamental study of the fluid dynamics inside an attic-shaped triangular enclosure subject to the night-time conditions was performed by Poulikakos & Bejan (1983b) with an assumption that the flow was symmetric about the centre plane. Del Campo et al. (1988) examined the entire isosceles triangular cavities for seven possible combinations of hot wall, cold wall and insulated wall using the finite element method based on a stream function or vorticity formulation. A two dimensional right triangular cavity filled with air and water with various aspect ratios and Rayleigh numbers are also examined by Salmun (1995a).

The stability of the reported single-cell steady state solution was re-examined by Salmun (1995b) who applied the same procedures developed by Farrow & Patterson (1993) for analysing the stability of a basic flow solution in a wedge-shaped geometry. Later Asan & Namli (2001) carried out an investigation to examine the details of the transition from a single cell to multi cellular structures. Haese & Teubner (2002) investigated the phenomenon for a large-scale triangular enclosure for night-time or winter day conditions with the effect of ventilation.

Holtzmann et al. (2000) modelled the buoyant airflow in isosceles triangular cavities with a heated bottom base and symmetrically cooled top sides for the aspect ratios of 0.2, 0.5, and 1.0 with various Rayleigh numbers. They conducted flow visualization studies with smoke injected into the cavity. The main objective of their research was to validate the existence of the numerical prediction of the symmetry-breaking bifurcation of the heated air currents that arise with gradual increments in Rayleigh number. Ridouane & Campo (2006) has also investigated the numerical prediction of the symmetry-breaking bifurcation. The author reported that as *Ra* is gradually increased, the symmetric plume breaks down and fades away. Thereafter, a subcritical pitchfork bifurcation is created giving rise to an asymmetric plume occurring at a critical Rayleigh number, *Ra* <sup>=</sup> 1.42 <sup>×</sup> 105. The steady state laminar natural convection in right triangular and quarter circular enclosures is investigated by Kent et al. (2007) for the case of winter-day temperature condition. A number of aspect ratios and Rayleigh numbers have been chosen to analyse the flow field and the heat transfer.

Unlike night-time conditions, the attic space problem under day-time (heating from above) conditions has received very limited attention. This may due to the fact that the flow structure in the attics subject to the daytime condition is relatively simple. The flow visualization experiments of Flack (1979) showed that the daytime flow remained stable and laminar for all the tested Rayleigh numbers (up to about 5 <sup>×</sup> 106). Akinsete & Coleman (1982) numerically simulated the attic space with hot upper sloping wall and cooled base. Their aim was to 2 Will-be-set-by-IN-TECH

A comparison study is performed by Ridouane et al. (2005) where the authors compare their numerical results produced for two different boundary conditions, (a) cold base and hot inclined walls (b) hot base and cold inclined walls with the experimental results obtained by Flack (1980; 1979). A good agreement has been obtained between the numerical predictions and the experimental measurements of Nusselt number. A numerical study of above mentioned two boundary conditions has also performed by Ridouane et al. (2006). However, the authors cut a significant portion of bottom tips and applied adiabatic boundary condition there. It is revealed from the analysis that the presence of insulated sidewalls, even of very small height, provides a huge gain of energy and helps keep the attic at the desired

The attic problem under the night-time conditions was again investigated experimentally by Poulikakos & Bejan (1983a). In their study, the authors modelled the enclosure as a right-angled triangle with an adiabatic vertical wall, which corresponded to the half of the full attic domain. A fundamental study of the fluid dynamics inside an attic-shaped triangular enclosure subject to the night-time conditions was performed by Poulikakos & Bejan (1983b) with an assumption that the flow was symmetric about the centre plane. Del Campo et al. (1988) examined the entire isosceles triangular cavities for seven possible combinations of hot wall, cold wall and insulated wall using the finite element method based on a stream function or vorticity formulation. A two dimensional right triangular cavity filled with air and water with various aspect ratios and Rayleigh numbers are also examined by Salmun (1995a). The stability of the reported single-cell steady state solution was re-examined by Salmun (1995b) who applied the same procedures developed by Farrow & Patterson (1993) for analysing the stability of a basic flow solution in a wedge-shaped geometry. Later Asan & Namli (2001) carried out an investigation to examine the details of the transition from a single cell to multi cellular structures. Haese & Teubner (2002) investigated the phenomenon for a large-scale triangular enclosure for night-time or winter day conditions with the effect of

Holtzmann et al. (2000) modelled the buoyant airflow in isosceles triangular cavities with a heated bottom base and symmetrically cooled top sides for the aspect ratios of 0.2, 0.5, and 1.0 with various Rayleigh numbers. They conducted flow visualization studies with smoke injected into the cavity. The main objective of their research was to validate the existence of the numerical prediction of the symmetry-breaking bifurcation of the heated air currents that arise with gradual increments in Rayleigh number. Ridouane & Campo (2006) has also investigated the numerical prediction of the symmetry-breaking bifurcation. The author reported that as *Ra* is gradually increased, the symmetric plume breaks down and fades away. Thereafter, a subcritical pitchfork bifurcation is created giving rise to an asymmetric plume occurring at a critical Rayleigh number, *Ra* <sup>=</sup> 1.42 <sup>×</sup> 105. The steady state laminar natural convection in right triangular and quarter circular enclosures is investigated by Kent et al. (2007) for the case of winter-day temperature condition. A number of aspect ratios and Rayleigh numbers have

Unlike night-time conditions, the attic space problem under day-time (heating from above) conditions has received very limited attention. This may due to the fact that the flow structure in the attics subject to the daytime condition is relatively simple. The flow visualization experiments of Flack (1979) showed that the daytime flow remained stable and laminar for all the tested Rayleigh numbers (up to about 5 <sup>×</sup> 106). Akinsete & Coleman (1982) numerically simulated the attic space with hot upper sloping wall and cooled base. Their aim was to

been chosen to analyse the flow field and the heat transfer.

temperature with a minimum energy.

ventilation.

obtain previously unavailable heat transfer data relevant to air conditioning calculations. This study considered only half of the domain. For the purpose of air conditioning calculations, Asan & Namli (2000) and Kent (2009b) have also reported numerical results for steady, laminar two-dimensional natural convection in a pitched roof of triangular cross-section under the summer day (day-time) boundary conditions.

Fig. 1. Temperature boundary condition on the incline walls of the enclosure

Recently, the transient flow development inside the attic space has been analysed by using scaling analysis with numerical verification by (Saha, 2011a;b; Saha et al., 2010a;c; Saha, 2008). The authors considered different types of thermal conditions, such as, sudden heating/cooling and ramp heating/cooling. In real situations, however, the attic space of buildings is subject to alternative heating and cooling over a diurnal cycle as it can be seen in Fig. 1. A very few studies for diurnal heating and cooling effect on the attic space are reported in the literature (Saha et al., 2010b; 2007). The authors discussed a general flow structure and heat transfer due to the effect of periodic thermal forcing. A detailed explanation of choosing the period for the model attic is required as the 24-hour period for the field situation is not applicable here.

In this study, numerical simulations of natural convection in an attic space subject to diurnal temperature condition on the sloping wall have been carried out. An explanation of choosing the period of periodic thermal effect has been given with help of the scaling analysis which is available in the literature. Moreover, the effects of the aspect ratio and Rayleigh number on the fluid flow and heat transfer have been discussed in details as well as the formation of a pitchfork bifurcation of the flow at the symmetric line of the enclosure.

#### **2. Formulation of the problem**

The physical system is sketched in Fig. 2, which is an air-filled isosceles triangular cavity of variable aspect ratios. Here 2*l* is the length of the base or ceiling, *T*<sup>0</sup> is the temperature applied on the base, *TA* is the amplitude of temperature fluctuation on the inclined surfaces, *h* is the height of the enclosure and *P* is the period of the thermal forcing.

**3. Selection of the physical period**

layer development from Saha et al. (2010c) is

quasi-steady time is (see Saha et al., 2010c)

conditions from Saha et al. (2010a) is

time is given in Saha et al. (2010a) as

*x*<sup>1</sup> ∼ *L*

⎡ ⎣1 −

where *x*<sup>1</sup> is given by

The period is determined in consideration of the scaling predictions of Saha et al. (2010a;b;c) which have demonstrated that the time for the adjustment of the temperature in the thermal boundary layer is by far shorter than the thermal forcing period of 24 hours in field situations. For sudden heating/cooling boundary conditions the steady state time scale for the boundary

Periodically Forced Natural Convection Through the Roof of an Attic-Shaped Building 37

*ARa*1/2*Pr*1/2 � *<sup>h</sup>*<sup>2</sup>

and the heating-up or cooling-down time scale for the enclosure to be filled with hot or cold

On the other hand, the quasi-steady time scale for ramp heating/cooling boundary condition of the boundary layer development for the case when the ramp time is longer than the

*<sup>A</sup>*2/3*Ra*1/3*Pr*1/3 � *tp*

and the heating-up or cooling-down time scale of the enclosure under the same boundary

*<sup>h</sup>*(<sup>1</sup> <sup>+</sup> *<sup>A</sup>*2)1/2 <sup>−</sup> *Ax*<sup>1</sup>

<sup>1</sup> <sup>−</sup> *<sup>κ</sup>A*1/2*Ra*1/4(<sup>1</sup> <sup>+</sup> *<sup>A</sup>*2)1/4

3/7 *p*

However, if the cavity is filled with cold fluid before the ramp is finished then the filling up

Table 1 presents the scaling values of the steady and quasi-steady times for sudden and ramp heating/cooling boundary conditions respectively for different *A* and *Ra*. The highest Rayleigh number considered here for three different aspect ratio is *Ra* <sup>=</sup> 1.5<sup>×</sup> 106. It is noticed that the steady state times for the boundary layer for this Rayleigh number of *A* = 0.2, 0.5 and 1.0 are 8.1*s*, 2.54*s* and 2.26*s* respectively. However, for the lowest Rayleigh number, *Ra* <sup>=</sup> 7.2 <sup>×</sup> 103 the steady state time for *<sup>A</sup>* <sup>=</sup> 0.5 is 35.76*s*. On the other hand, the quasi-steady time for the ramp temperature boundary condition depends on the length of the ramp. If we assume the ramp time to be 1000*s* then the quasi-steady times for these aspect ratios are 40.51*s*, 18.62*<sup>s</sup>* and 17.23*<sup>s</sup>* respectively and for the lowest Rayleigh number, *Ra* <sup>=</sup> 7.2 <sup>×</sup> 103

*κ* �

*κ* �

*h*2/*κ*

�2

*<sup>h</sup>*<sup>2</sup> *tp*

�1/3 � *h*<sup>2</sup>

*κ* �

*<sup>A</sup>*1/2*κRa*1/4(<sup>1</sup> <sup>+</sup> *<sup>A</sup>*2)5/4 , (8)

*<sup>κ</sup>*4/7*Ra*1/7*A*2/7(<sup>1</sup> <sup>+</sup> *<sup>A</sup>*2)1/7 , (10)

�1/2⎤

, (5)

, for *tf* > *ts* (6)

, (7)

⎦ , (9)

*ts* <sup>=</sup> (<sup>1</sup> <sup>+</sup> *Pr*)1/2(<sup>1</sup> <sup>+</sup> *<sup>A</sup>*2)1/2

*<sup>A</sup>*1/2*Ra*1/4*Pr*1/4(<sup>1</sup> <sup>+</sup> *<sup>A</sup>*2)1/4 � *<sup>h</sup>*<sup>2</sup>

fluid under the same boundary conditions as in Saha et al. (2010c) is

*tsr* <sup>=</sup> (<sup>1</sup> <sup>+</sup> *Pr*)1/3(<sup>1</sup> <sup>+</sup> *<sup>A</sup>*2)1/3

�

*tf q* <sup>∼</sup> *<sup>h</sup>*8/7*<sup>t</sup>*

*tf r* =

�

*tf* <sup>=</sup> (<sup>1</sup> <sup>+</sup> *Pr*)1/4

Fig. 2. A schematic of the geometry and boundary conditions of the enclosure

Under the Boussinesq approximations the governing continuity, momentum and energy equations take the following forms.

$$\frac{\partial \mu}{\partial x} + \frac{\partial v}{\partial y} = 0 \tag{1}$$

$$u\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} = -\frac{1}{\rho}\frac{\partial p}{\partial x} + \nu \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\right) \tag{2}$$

$$\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} = -\frac{1}{\rho} \frac{\partial p}{\partial y} + \nu \left( \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} \right) + g\beta \left( T - T\_0 \right) \tag{3}$$

$$\frac{\partial T}{\partial t} + u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} = \kappa \left( \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} \right) \tag{4}$$

where *u* and *v* are the velocity components along *x*− and *y*−directions, *t* is the time, *p* is the pressure, *ν*, *ρ*, *β* and *κ* are kinematic viscosity, density of the fluid, coefficient of thermal expansion and thermal diffusivity respectively, *g* is the acceleration due to gravity and *T* is the fluid temperature.

The boundary conditions for the present numerical simulations are also shown in Fig. 2. Here, the temperature of the bottom wall of the cavity is fixed at *T* = *T*0. A periodic temperature boundary condition is applied to the two inclined walls. The Rayleigh number for the periodic boundary condition has been defined based on the maximum temperature difference between the inclined surface and the bottom over a cycle as

$$Ra = \frac{2g\beta T\_A h^3}{\kappa \nu}$$

Three aspect ratios 0.2, 0.5 and 1.0, four Rayleigh numbers, 1.5 <sup>×</sup> 106, 7.2 <sup>×</sup> 105, 1.5 <sup>×</sup> 104, and 1.5 <sup>×</sup> 103, and a fixed Prandtl number 0.72 are considered in the present investigation. Based on the experimental observations of Flack (1979), which reported the critical Rayleigh number for the flow to become turbulent, we have chosen the maximum Rayleigh number, *Ra* = 1.5 <sup>×</sup> 106 so that the flow stays in the laminar regime. It is understood that in real situations the Rayleigh number may be much higher than this and an appropriate turbulence model should be applied. This is beyond the scope of this study. In order to avoid the singularities at the tips in the numerical simulation, the tips are cut off by 5% and at the cutting points (refer to Fig. 2) rigid non-slip and adiabatic vertical walls are assumed. We anticipate that this modification of the geometry will not alter the overall flow development significantly.

#### **3. Selection of the physical period**

4 Will-be-set-by-IN-TECH

*g* 

*x* 

Under the Boussinesq approximations the governing continuity, momentum and energy

where *u* and *v* are the velocity components along *x*− and *y*−directions, *t* is the time, *p* is the pressure, *ν*, *ρ*, *β* and *κ* are kinematic viscosity, density of the fluid, coefficient of thermal expansion and thermal diffusivity respectively, *g* is the acceleration due to gravity and *T* is

The boundary conditions for the present numerical simulations are also shown in Fig. 2. Here, the temperature of the bottom wall of the cavity is fixed at *T* = *T*0. A periodic temperature boundary condition is applied to the two inclined walls. The Rayleigh number for the periodic boundary condition has been defined based on the maximum temperature difference between

*Ra* <sup>=</sup> <sup>2</sup>*gβTAh*<sup>3</sup> *κν* Three aspect ratios 0.2, 0.5 and 1.0, four Rayleigh numbers, 1.5 <sup>×</sup> 106, 7.2 <sup>×</sup> 105, 1.5 <sup>×</sup> 104, and 1.5 <sup>×</sup> 103, and a fixed Prandtl number 0.72 are considered in the present investigation. Based on the experimental observations of Flack (1979), which reported the critical Rayleigh number for the flow to become turbulent, we have chosen the maximum Rayleigh number, *Ra* = 1.5 <sup>×</sup> 106 so that the flow stays in the laminar regime. It is understood that in real situations the Rayleigh number may be much higher than this and an appropriate turbulence model should be applied. This is beyond the scope of this study. In order to avoid the singularities at the tips in the numerical simulation, the tips are cut off by 5% and at the cutting points (refer to Fig. 2) rigid non-slip and adiabatic vertical walls are assumed. We anticipate that this modification of the geometry will not alter the overall flow development significantly.

 *∂*2*v <sup>∂</sup>x*<sup>2</sup> <sup>+</sup>

> *∂*2*T <sup>∂</sup>x*<sup>2</sup> <sup>+</sup>

*y* 

*<sup>t</sup>* 

 *u v T T TA*

> *∂*2*u <sup>∂</sup>x*<sup>2</sup> <sup>+</sup>

> > *∂*2*v ∂y*<sup>2</sup>

> > > *∂*2*T ∂y*<sup>2</sup>

0

<sup>0</sup> sin 2 / P

S*<sup>t</sup>*

*<sup>∂</sup><sup>y</sup>* <sup>=</sup> <sup>0</sup> (1)

*∂*2*u ∂y*<sup>2</sup>  *H* 

+ *gβ* (*T* − *T*0) (3)

(2)

(4)

 *T* = *T0* , *u* = *v* = 0

2*l*

*∂u ∂x* + *∂v*

Fig. 2. A schematic of the geometry and boundary conditions of the enclosure

 

S

<sup>0</sup> sin 2 / P

*D*

*E*

*∂u <sup>∂</sup><sup>t</sup>* <sup>+</sup> *<sup>u</sup>*

*∂v ∂x* + *v ∂v <sup>∂</sup><sup>y</sup>* <sup>=</sup> <sup>−</sup> <sup>1</sup> *ρ ∂p ∂y* + *ν*

the inclined surface and the bottom over a cycle as

*∂T <sup>∂</sup><sup>t</sup>* <sup>+</sup> *<sup>u</sup>* *∂T ∂x* + *v ∂T <sup>∂</sup><sup>y</sup>* <sup>=</sup> *<sup>κ</sup>*

*∂u ∂x* + *v ∂u <sup>∂</sup><sup>y</sup>* <sup>=</sup> <sup>−</sup> <sup>1</sup> *ρ ∂p ∂x* + *ν*

0

equations take the following forms.

*∂v <sup>∂</sup><sup>t</sup>* <sup>+</sup> *<sup>u</sup>*

the fluid temperature.

*T T TA*

*u v*

The period is determined in consideration of the scaling predictions of Saha et al. (2010a;b;c) which have demonstrated that the time for the adjustment of the temperature in the thermal boundary layer is by far shorter than the thermal forcing period of 24 hours in field situations. For sudden heating/cooling boundary conditions the steady state time scale for the boundary layer development from Saha et al. (2010c) is

$$t\_s = \frac{(1+Pr)^{1/2}(1+A^2)^{1/2}}{ARa^{1/2}Pr^{1/2}} \left(\frac{\hbar^2}{\kappa}\right),\tag{5}$$

and the heating-up or cooling-down time scale for the enclosure to be filled with hot or cold fluid under the same boundary conditions as in Saha et al. (2010c) is

$$t\_f = \frac{(1+Pr)^{1/4}}{A^{1/2}Ra^{1/4}Pr^{1/4}(1+A^2)^{1/4}} \left(\frac{h^2}{\kappa}\right), \quad \text{for} \quad t\_f > t\_s\tag{6}$$

On the other hand, the quasi-steady time scale for ramp heating/cooling boundary condition of the boundary layer development for the case when the ramp time is longer than the quasi-steady time is (see Saha et al., 2010c)

$$t\_{sr} = \frac{(1+Pr)^{1/3}(1+A^2)^{1/3}}{A^{2/3}Ra^{1/3}Pr^{1/3}} \left(\frac{t\_p}{h^2/\kappa}\right)^{1/3} \left(\frac{h^2}{\kappa}\right),\tag{7}$$

and the heating-up or cooling-down time scale of the enclosure under the same boundary conditions from Saha et al. (2010a) is

$$t\_{fr} = \frac{\left[h(1+A^2)^{1/2} - Ax\_1\right]^2}{A^{1/2} \kappa Ra^{1/4} (1+A^2)^{5/4}}\tag{8}$$

where *x*<sup>1</sup> is given by

$$x\_1 \sim L \left[ 1 - \left( 1 - \frac{\kappa A^{1/2} R a^{1/4} (1 + A^2)^{1/4}}{h^2} t\_p \right)^{1/2} \right],\tag{9}$$

However, if the cavity is filled with cold fluid before the ramp is finished then the filling up time is given in Saha et al. (2010a) as

$$\mathbf{t}\_{fq} \sim \frac{\hbar^{8/7} \mathbf{t}\_p^{3/7}}{\kappa^{4/7} \mathrm{R} a^{1/7} A^{2/7} (1 + A^2)^{1/7}},\tag{10}$$

Table 1 presents the scaling values of the steady and quasi-steady times for sudden and ramp heating/cooling boundary conditions respectively for different *A* and *Ra*. The highest Rayleigh number considered here for three different aspect ratio is *Ra* <sup>=</sup> 1.5<sup>×</sup> 106. It is noticed that the steady state times for the boundary layer for this Rayleigh number of *A* = 0.2, 0.5 and 1.0 are 8.1*s*, 2.54*s* and 2.26*s* respectively. However, for the lowest Rayleigh number, *Ra* <sup>=</sup> 7.2 <sup>×</sup> 103 the steady state time for *<sup>A</sup>* <sup>=</sup> 0.5 is 35.76*s*. On the other hand, the quasi-steady time for the ramp temperature boundary condition depends on the length of the ramp. If we assume the ramp time to be 1000*s* then the quasi-steady times for these aspect ratios are 40.51*s*, 18.62*<sup>s</sup>* and 17.23*<sup>s</sup>* respectively and for the lowest Rayleigh number, *Ra* <sup>=</sup> 7.2 <sup>×</sup> 103

Aspect ratio

respectively for different *A* and *Ra*.

Filling-up time (*tf*) for sudden heating/cooling

*Ra* = 1.5u106 *Ra* = 7.2u103 *Ra* = 1.5u106 *Ra* = 7.2u103

Periodically Forced Natural Convection Through the Roof of an Attic-Shaped Building 39

*A*=0.2 83.24s - 213.32s - *A*=0.5 42.39s 159.01s 145.07s 308.77s *A*=1.0 31.61s - 122.67s -

Table 2. Heating-up/cooling-down times for sudden and ramp boundary conditions

**4. Numerical scheme and grid and time step dependence tests**

**5. Flow response to the periodic thermal forcing**

**5.1 General flow response to diurnal heating and cooling**

third cycle are presented.

Rayleigh number considered here is 154.90*s* for *A* = 0.5. These times are very short when compared to the thermal forcing period of 24 hours in field situations. Therefore, the period of the thermal cycle may be considered as 400*s* or more based on the above discussions for the following numerical simulations. However, for a better understanding of the flow at in the quasi-steady mode, we have chosen a thermal forcing period of 2000*s* for all the simulations.

Equations (1) - (4) are solved along with the initial and boundary conditions using the SIMPLE scheme. The finite volume method has been chosen to discretize the governing equations, with the QUICK scheme (see Leonard & Mokhtari, 1990) approximating the advection term. The diffusion terms are discretized using central-differencing with second order accurate. A second order implicit time-marching scheme has also been used for the unsteady term. An extensive mesh and time step dependence tests have been coonducted in Saha et al. (2010a;b;c)

The flow response to the periodic thermal forcing and the heat transfer through the sloping boundary are discussed for the case with *<sup>A</sup>* <sup>=</sup> 0.5, *Pr* <sup>=</sup> 0.72 and *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106 in this section.

Since the initial flow is assumed to be isothermal and motionless, there is a start-up process of the flow response. In order to minimize the start-up effect, three full thermal forcing cycles are calculated in the numerical simulation before consideration of the flow. It is found that the start-up effect for the present case is almost negligible, and the flow response in the third cycle is identical to that in the previous cycle. In the following discussion, the results of the

Fig. 3 shows snapshots of streamlines and the corresponding isotherms and vector field at different stages of the cycle. The flow and temperature structures, shown in Fig. 3 at *t* = 2.00*P*, represent those at the beginning of the daytime heating process in the third thermal forcing cycle. At this time, the inclined surfaces and the bottom surface of the enclosure have the same temperature, but the temperature inside the enclosure is lower than the temperature on the boundaries due to the cooling effect in the previous thermal cycle. The residual temperature structure, which is formed in the previous cooling phase, is still present at *t* = 2.00*P*. The

Filling-up time (*tfr*) for ramp heating/cooling (*tp* = 1000s)


Table 1. Steady state and quasi-steady times for sudden and ramp boundary conditions respectively for different *A* and *Ra*.

the quasi-steady time for the aspect ratio *A* = 0.5 is 108.53*s* which is much shorter than the ramp time (1000*s*). If the ramp time is 200*s* the quasi-steady time of *A* = 0.5 for the lowest Rayleigh number considered here is 63.47*s*. Still the quasi-steady time is about half of the ramp time. Therefore, what happened between the quasi-steady time and the ramp time is, once the quasi-steady state time *tsr* is reached, the boundary layer stops growing according to *κ*1/2*t* 1/2 which is only valid for conductive boundary layers. The thermal boundary layer is in a quasi-steady mode with convection balancing conduction. Further increase of the heat input simply accelerates the flow to maintain the proper thermal balance. The thickness and the velocity scales during this quasi-steady mode is (see Saha et al., 2010c)

$$
\delta\_T \sim \frac{h(1+A^2)^{1/4}}{\text{Ra}^{1/4}A^{1/2}} \left(\frac{t\_p}{t}\right)^{1/4}.\tag{11}
$$

and

$$
\mu \sim \text{Ra}^{1/2} \frac{\kappa}{h} \left( \frac{t}{t\_p} \right)^{1/2}. \tag{12}
$$

respectively. When the hot fluid travels through the boundary layer and reaches the top tip of the cavity then it has no choice but to move downward along the symmetry line of the cavity. In this way the cavity is filled up with hot fluid with a horizontal stratification of the thermal field. However, during the cooling phase, the boundary layer is not stable for the Rayleigh numbers considered here. In that case initially a cold boundary layer develops adjacent to the inclined wall which is potentially unstable to the Rayleigh Bénard instability, which may manifest in a form of sinking plumes. These plumes mix up the cold fluid with the hot fluid inside the cavity until the end of the cooling phase.

Moreover, Table 2 shows the scaling values of the filling-up times for sudden and ramp heating/cooling boundary conditions for different *A* and *Ra*. It is seen that the heating-up or cooling-down times for the sudden heating/cooling boundary condition for *A* = 0.5 and *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106 is 42.39*<sup>s</sup>* and for *Ra* <sup>=</sup> 7.2 <sup>×</sup> 103 and the same aspect ratio is 159.01*s*. For aspect ratios 0.2 and 1.0 the filling-up times are 83.24*<sup>s</sup>* and 31.61*<sup>s</sup>* respectively when *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106. The filling-up times for ramp heating/cooling boundary conditions for *A* = 0.5 are 145.07*s* and 308.77*<sup>s</sup>* when *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106 and 7.2 <sup>×</sup> 103 respectively and *tp* <sup>=</sup> <sup>1000</sup>*s*. For two other aspect ratios, *A* = 0.2 and 1.0, the filling-up times are 213.32*s* and 122.67*s* respectively for *Ra* <sup>=</sup> <sup>1</sup> <sup>×</sup> 106. However, the filling-up time for ramp boundary conditions depends on the length of the ramp time. If the ramp time is 200*s* then the filling-up time for the lowest 6 Will-be-set-by-IN-TECH

*A*=0.2 8.15s - 40.51s - *A*=0.5 2.54s 35.76s 18.62s 108.53s *A*=1.0 2.26s - 17.23s -

Table 1. Steady state and quasi-steady times for sudden and ramp boundary conditions

the quasi-steady time for the aspect ratio *A* = 0.5 is 108.53*s* which is much shorter than the ramp time (1000*s*). If the ramp time is 200*s* the quasi-steady time of *A* = 0.5 for the lowest Rayleigh number considered here is 63.47*s*. Still the quasi-steady time is about half of the ramp time. Therefore, what happened between the quasi-steady time and the ramp time is, once the quasi-steady state time *tsr* is reached, the boundary layer stops growing according

is in a quasi-steady mode with convection balancing conduction. Further increase of the heat input simply accelerates the flow to maintain the proper thermal balance. The thickness and

> *h t tp*

respectively. When the hot fluid travels through the boundary layer and reaches the top tip of the cavity then it has no choice but to move downward along the symmetry line of the cavity. In this way the cavity is filled up with hot fluid with a horizontal stratification of the thermal field. However, during the cooling phase, the boundary layer is not stable for the Rayleigh numbers considered here. In that case initially a cold boundary layer develops adjacent to the inclined wall which is potentially unstable to the Rayleigh Bénard instability, which may manifest in a form of sinking plumes. These plumes mix up the cold fluid with the hot fluid

Moreover, Table 2 shows the scaling values of the filling-up times for sudden and ramp heating/cooling boundary conditions for different *A* and *Ra*. It is seen that the heating-up or cooling-down times for the sudden heating/cooling boundary condition for *A* = 0.5 and *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106 is 42.39*<sup>s</sup>* and for *Ra* <sup>=</sup> 7.2 <sup>×</sup> 103 and the same aspect ratio is 159.01*s*. For aspect ratios 0.2 and 1.0 the filling-up times are 83.24*<sup>s</sup>* and 31.61*<sup>s</sup>* respectively when *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106. The filling-up times for ramp heating/cooling boundary conditions for *A* = 0.5 are 145.07*s* and 308.77*<sup>s</sup>* when *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106 and 7.2 <sup>×</sup> 103 respectively and *tp* <sup>=</sup> <sup>1000</sup>*s*. For two other aspect ratios, *A* = 0.2 and 1.0, the filling-up times are 213.32*s* and 122.67*s* respectively for *Ra* <sup>=</sup> <sup>1</sup> <sup>×</sup> 106. However, the filling-up time for ramp boundary conditions depends on the length of the ramp time. If the ramp time is 200*s* then the filling-up time for the lowest

the velocity scales during this quasi-steady mode is (see Saha et al., 2010c)

inside the cavity until the end of the cooling phase.

*<sup>δ</sup><sup>T</sup>* <sup>∼</sup> *<sup>h</sup>*(<sup>1</sup> <sup>+</sup> *<sup>A</sup>*2)1/4 *Ra*1/4*A*1/2

*<sup>u</sup>* <sup>∼</sup> *Ra*1/2 *<sup>κ</sup>*

1/2 which is only valid for conductive boundary layers. The thermal boundary layer

 *tp t*

1/2

1/4

. (11)

. (12)

*Ra* = 1.5u106 *Ra* = 7.2u103 *Ra* = 1.5u106 *Ra* = 7.2u103

Quasi-steady time (*tsr*) for ramp heating/cooling (*tp* = 1000s)

Steady state time (*ts*) for sudden heating/cooling

Aspect ratio

to *κ*1/2*t*

and

respectively for different *A* and *Ra*.


Table 2. Heating-up/cooling-down times for sudden and ramp boundary conditions respectively for different *A* and *Ra*.

Rayleigh number considered here is 154.90*s* for *A* = 0.5. These times are very short when compared to the thermal forcing period of 24 hours in field situations. Therefore, the period of the thermal cycle may be considered as 400*s* or more based on the above discussions for the following numerical simulations. However, for a better understanding of the flow at in the quasi-steady mode, we have chosen a thermal forcing period of 2000*s* for all the simulations.

#### **4. Numerical scheme and grid and time step dependence tests**

Equations (1) - (4) are solved along with the initial and boundary conditions using the SIMPLE scheme. The finite volume method has been chosen to discretize the governing equations, with the QUICK scheme (see Leonard & Mokhtari, 1990) approximating the advection term. The diffusion terms are discretized using central-differencing with second order accurate. A second order implicit time-marching scheme has also been used for the unsteady term. An extensive mesh and time step dependence tests have been coonducted in Saha et al. (2010a;b;c)

#### **5. Flow response to the periodic thermal forcing**

The flow response to the periodic thermal forcing and the heat transfer through the sloping boundary are discussed for the case with *<sup>A</sup>* <sup>=</sup> 0.5, *Pr* <sup>=</sup> 0.72 and *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106 in this section.

#### **5.1 General flow response to diurnal heating and cooling**

Since the initial flow is assumed to be isothermal and motionless, there is a start-up process of the flow response. In order to minimize the start-up effect, three full thermal forcing cycles are calculated in the numerical simulation before consideration of the flow. It is found that the start-up effect for the present case is almost negligible, and the flow response in the third cycle is identical to that in the previous cycle. In the following discussion, the results of the third cycle are presented.

Fig. 3 shows snapshots of streamlines and the corresponding isotherms and vector field at different stages of the cycle. The flow and temperature structures, shown in Fig. 3 at *t* = 2.00*P*, represent those at the beginning of the daytime heating process in the third thermal forcing cycle. At this time, the inclined surfaces and the bottom surface of the enclosure have the same temperature, but the temperature inside the enclosure is lower than the temperature on the boundaries due to the cooling effect in the previous thermal cycle. The residual temperature structure, which is formed in the previous cooling phase, is still present at *t* = 2.00*P*. The

Fig. 3. A series of snapshots of stream function and temperature contours of the third cycle at

Periodically Forced Natural Convection Through the Roof of an Attic-Shaped Building 41

different times for *<sup>A</sup>* <sup>=</sup> 0.5 and *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106. Left: streamlines; right: isotherms.

corresponding streamline contours at the same time show two circulating cells, and the temperature contours indicate stratification in the upper and lower section of the enclosure with two cold regions in the centre.

As the upper surface temperature increases further, a distinct temperature stratification is established throughout the enclosure by the time *t* = 2.05*P* (see Fig. 3). The streamlines at this stage indicate that the centers of the two circulating cells have shifted closer to the inclined surfaces, indicating a strong conduction effect near those boundaries. This phenomenon has been reported previously in Akinsete & Coleman (1982) and Asan & Namli (2000) for the daytime condition with constant heating at the upper surface or constant cooling at the bottom surface.

At *t* = 2.25*P*, the temperature on the inclined surfaces peaks. Subsequently, the temperature drops, representing a decreasing heating effect. Since the interior flow is stably stratified prior to *t* = 2.25*P*, the decrease of the temperature at the inclined surface results in a cooling event, appearing first at the top corner and expanding downwards as the surface temperature drops further. At *t* = 2.45*P*, two additional circulating cells have formed in the upper region of the enclosure, and the newly formed cells push the existing cells downwards. The corresponding temperature contours show two distinct regions, an expanding upper region responding to the cooling effect, and a shrinking lower region with stratification responding to the decreasing heating effect. By the time *t* = 2.50*P*, the daytime heating ceases; the lower stratified flow region has disappeared completely and the flow in the enclosure is dominated by the cooling effect. At this time, the top and the bottom surfaces again have the same temperature, but the interior temperature is higher than that on the boundaries.

As the upper inclined surface temperature drops below the bottom surface temperature (*t* = 2.70*P*, Fig. 3), the cold-air layer under the inclined surfaces becomes unstable. At the same time, the hot-air layer above the bottom surface also becomes unstable. As a consequence, sinking cold-air plumes and rising hot-air plumes are visible in the isotherm contours and a cellular flow pattern is formed in the corresponding stream function contours. It is also noticeable that the flow is symmetric about the geometric symmetry plane at this time. However, as time increases the flow becomes asymmetric about the symmetric line (see isotherms at *t* = 2.95*P*). The large cell from the right hand side of the centreline, which is still growing, pushes the cell on the left of it towards the left tip. At the same time this large cell also changes its position and attempts to cross the centreline of the cavity and a small cell next to it moves into its position and grows.

At *t* = 2.975*P*, the large cell in the stream lines has crossed the centerline and the cell on the right of it grows and becomes as large as it is after a short time (for brevity figures not included). The flow is also asymmetric at this time. However, it returns to a symmetric flow at the time *t* = 3.00*P* which is the same as that at *t* = 2.00*P*, and similar temperature and flow structures to those at the beginning of the forcing cycle are formed. The above described flow development is repeated in the next cycle.

The horizontal velocity profiles (velocity parallel to the bottom surface) and the corresponding temperature profiles evaluated along the line *DE* shown in Fig. 2 at different time instances of the third thermal forcing cycle are depicted in Fig. 4. At the beginning of the cycle (*t* = 2.00*P*) the velocity is the highest near the roof of the attic (see Fig. 4a), which is the surface driving the flow. At the same time, the body of fluid residing outside the top wall layer moves fast toward the bottom tips to fill up the gap. As time progresses the vertical velocity increases and the horizontal temperature decreases (see *t* = 2.05*P*). A three layer structure in the velocity field 8 Will-be-set-by-IN-TECH

40 Convection and Conduction Heat Transfer

corresponding streamline contours at the same time show two circulating cells, and the temperature contours indicate stratification in the upper and lower section of the enclosure

As the upper surface temperature increases further, a distinct temperature stratification is established throughout the enclosure by the time *t* = 2.05*P* (see Fig. 3). The streamlines at this stage indicate that the centers of the two circulating cells have shifted closer to the inclined surfaces, indicating a strong conduction effect near those boundaries. This phenomenon has been reported previously in Akinsete & Coleman (1982) and Asan & Namli (2000) for the daytime condition with constant heating at the upper surface or constant cooling at the bottom

At *t* = 2.25*P*, the temperature on the inclined surfaces peaks. Subsequently, the temperature drops, representing a decreasing heating effect. Since the interior flow is stably stratified prior to *t* = 2.25*P*, the decrease of the temperature at the inclined surface results in a cooling event, appearing first at the top corner and expanding downwards as the surface temperature drops further. At *t* = 2.45*P*, two additional circulating cells have formed in the upper region of the enclosure, and the newly formed cells push the existing cells downwards. The corresponding temperature contours show two distinct regions, an expanding upper region responding to the cooling effect, and a shrinking lower region with stratification responding to the decreasing heating effect. By the time *t* = 2.50*P*, the daytime heating ceases; the lower stratified flow region has disappeared completely and the flow in the enclosure is dominated by the cooling effect. At this time, the top and the bottom surfaces again have the same

As the upper inclined surface temperature drops below the bottom surface temperature (*t* = 2.70*P*, Fig. 3), the cold-air layer under the inclined surfaces becomes unstable. At the same time, the hot-air layer above the bottom surface also becomes unstable. As a consequence, sinking cold-air plumes and rising hot-air plumes are visible in the isotherm contours and a cellular flow pattern is formed in the corresponding stream function contours. It is also noticeable that the flow is symmetric about the geometric symmetry plane at this time. However, as time increases the flow becomes asymmetric about the symmetric line (see isotherms at *t* = 2.95*P*). The large cell from the right hand side of the centreline, which is still growing, pushes the cell on the left of it towards the left tip. At the same time this large cell also changes its position and attempts to cross the centreline of the cavity and a small cell next

At *t* = 2.975*P*, the large cell in the stream lines has crossed the centerline and the cell on the right of it grows and becomes as large as it is after a short time (for brevity figures not included). The flow is also asymmetric at this time. However, it returns to a symmetric flow at the time *t* = 3.00*P* which is the same as that at *t* = 2.00*P*, and similar temperature and flow structures to those at the beginning of the forcing cycle are formed. The above described flow

The horizontal velocity profiles (velocity parallel to the bottom surface) and the corresponding temperature profiles evaluated along the line *DE* shown in Fig. 2 at different time instances of the third thermal forcing cycle are depicted in Fig. 4. At the beginning of the cycle (*t* = 2.00*P*) the velocity is the highest near the roof of the attic (see Fig. 4a), which is the surface driving the flow. At the same time, the body of fluid residing outside the top wall layer moves fast toward the bottom tips to fill up the gap. As time progresses the vertical velocity increases and the horizontal temperature decreases (see *t* = 2.05*P*). A three layer structure in the velocity field

temperature, but the interior temperature is higher than that on the boundaries.

with two cold regions in the centre.

to it moves into its position and grows.

development is repeated in the next cycle.

surface.

Fig. 3. A series of snapshots of stream function and temperature contours of the third cycle at different times for *<sup>A</sup>* <sup>=</sup> 0.5 and *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106. Left: streamlines; right: isotherms.

remains the same. It is noteworthy that the top surface reaches its peak temperature at *t* = 2.25*P* (for brevity the profile is not included). After this time the top surface temperature starts to decrease which can be seen at time *t* = 2.40*P*. By comparing the temperature profiles at *t* = 2.05*P* and *t* = 2.45*P* shown in Fig. 4(b), it is clear that the temperatures at both the top and bottom surfaces are the same for these two time instances. However, different temperature structures are seen in the interior region. The same phenomenon has been found

Periodically Forced Natural Convection Through the Roof of an Attic-Shaped Building 43

In Fig. 4(c), the velocity profiles at the same location during the night-time cooling phase are displayed. In this phase the flow structure is more complicated. At *t* = 2.55*P* the velocity near the bottom surface is slightly higher than that near the top. Again a three layer structure of the velocity field appeared which is seen at *t* = 2.65*P*, 2.75*P* and 2.85*P*. The maximum velocity near the ceiling occurs at *t* = 2.75*P* when the cooling is at its maximum. After that it decreases and the flow reverses completely at *t* = 3.0*P*. The corresponding temperature profiles for the night-time condition are shown in Fig. 4(d). It is seen that the temperature lines are not as smooth as those observed for the daytime condition. At *t* = 2.55*P*, the temperature near the bottom surface decreases first and then increases slowly with the height and again decreases near the inclined surface. This behaviour near the bottom surface is due to the presence of a rising plume. Similar behaviour has been seen for *t* = 2.75*P* and 2.85*P*. However, at *t* = 2.65*P* it decreases slowly after rapidly decreasing near the bottom surface. At *t* = 3.00*P* again the bottom and top surface temperatures are the same with a lower temperature in the interior

The Nusselt number, which has practical significance, is calculated as follows:

*Nu* <sup>=</sup> *<sup>h</sup>*eff*<sup>h</sup>*

*<sup>h</sup>*eff <sup>=</sup> *<sup>q</sup> TA*

Here *q* is the convective heat flux through a boundary. Since the bottom surface temperature is fixed at 295*K* and the sloping wall surface temperature cycles between 290K and 300K (refer to Figure 1), a zero temperature difference between the surfaces occurs twice in a cycle. Therefore, the amplitude of the temperature fluctuation (*TA*) is chosen for calculating the heat transfer coefficient instead of a changing temperature difference, which would give an

Fig. 5 shows the calculated average Nusselt number on the inclined and bottom surfaces of the cavity. The time histories of the calculated Nusselt number on the inclined surfaces exhibit certain significant features. Firstly, it shows a periodic behaviour in response to the periodic thermal forcing. Secondly within each cycle of the flow response, there is a time period with weak heat transfer and a period with intensive heat transfer. The weak heat transfer corresponds to the daytime condition when the flow is mainly dominated by conduction and the strong heat transfer corresponds to the night-time condition. At night, the boundary layers adjacent to the inclined walls and the bottom are unstable. Therefore, sinking and rising plumes are formed in the inclined and horizontal boundary layers. These plumes dominate

*<sup>k</sup>* (13)

(14)

at the times *t* = 2.50*P* and *t* = 2.00*P*.

**5.2 Heat transfer across the attic**

where the heat transfer coefficient heff is defined by

undefined value of the heat transfer coefficient at particular times.

region.

Fig. 4. Horizontal velocity profile (left) and temperature profile (right) along *DE* for *A* = 0.5 with *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106.

is found at *t* = 2.45*P*. At this time the top portion of the cavity is locally cooled and the bottom portion is still hot (see Fig. 3). After that time the flow completely reverses at *t* = 2.50*P*. It is noted that at this time the horizontal velocity is lower than that at the beginning of the cycle despite that the temperatures on the sloping boundary and the ceiling are the same at both times (see Fig. 4b). This is due to the fact that at the beginning of the cycle the flow is mainly dominated by convection as a result of the cooling effect in the second half of the previous thermal cycle. However, the flow is dominated by conduction at *t* = 2.50*P* as a result of the heating effect in the first half of the current thermal cycle.

As mentioned above, at the beginning of the cycle (*t* = 2.00*P*) the temperatures on the horizontal and inclined surfaces are the same as shown in Fig. 4(b). However the temperature near the mid point of the profile line is lower than that at the surfaces by approximately 0.5*K*, which is consistent with the previous discussion of the flow field. Subsequently the temperature of the top surface increases (*t* = 2.05*P*) while the bottom surface temperature remains the same. It is noteworthy that the top surface reaches its peak temperature at *t* = 2.25*P* (for brevity the profile is not included). After this time the top surface temperature starts to decrease which can be seen at time *t* = 2.40*P*. By comparing the temperature profiles at *t* = 2.05*P* and *t* = 2.45*P* shown in Fig. 4(b), it is clear that the temperatures at both the top and bottom surfaces are the same for these two time instances. However, different temperature structures are seen in the interior region. The same phenomenon has been found at the times *t* = 2.50*P* and *t* = 2.00*P*.

In Fig. 4(c), the velocity profiles at the same location during the night-time cooling phase are displayed. In this phase the flow structure is more complicated. At *t* = 2.55*P* the velocity near the bottom surface is slightly higher than that near the top. Again a three layer structure of the velocity field appeared which is seen at *t* = 2.65*P*, 2.75*P* and 2.85*P*. The maximum velocity near the ceiling occurs at *t* = 2.75*P* when the cooling is at its maximum. After that it decreases and the flow reverses completely at *t* = 3.0*P*. The corresponding temperature profiles for the night-time condition are shown in Fig. 4(d). It is seen that the temperature lines are not as smooth as those observed for the daytime condition. At *t* = 2.55*P*, the temperature near the bottom surface decreases first and then increases slowly with the height and again decreases near the inclined surface. This behaviour near the bottom surface is due to the presence of a rising plume. Similar behaviour has been seen for *t* = 2.75*P* and 2.85*P*. However, at *t* = 2.65*P* it decreases slowly after rapidly decreasing near the bottom surface. At *t* = 3.00*P* again the bottom and top surface temperatures are the same with a lower temperature in the interior region.

#### **5.2 Heat transfer across the attic**

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Fig. 4. Horizontal velocity profile (left) and temperature profile (right) along *DE* for *A* = 0.5

is found at *t* = 2.45*P*. At this time the top portion of the cavity is locally cooled and the bottom portion is still hot (see Fig. 3). After that time the flow completely reverses at *t* = 2.50*P*. It is noted that at this time the horizontal velocity is lower than that at the beginning of the cycle despite that the temperatures on the sloping boundary and the ceiling are the same at both times (see Fig. 4b). This is due to the fact that at the beginning of the cycle the flow is mainly dominated by convection as a result of the cooling effect in the second half of the previous thermal cycle. However, the flow is dominated by conduction at *t* = 2.50*P* as a result of the

As mentioned above, at the beginning of the cycle (*t* = 2.00*P*) the temperatures on the horizontal and inclined surfaces are the same as shown in Fig. 4(b). However the temperature near the mid point of the profile line is lower than that at the surfaces by approximately 0.5*K*, which is consistent with the previous discussion of the flow field. Subsequently the temperature of the top surface increases (*t* = 2.05*P*) while the bottom surface temperature

heating effect in the first half of the current thermal cycle.

with *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106.

The Nusselt number, which has practical significance, is calculated as follows:

$$Nu = \frac{h\_{\rm eff}h}{k} \tag{13}$$

where the heat transfer coefficient heff is defined by

$$h\_{\rm eff} = \frac{q}{T\_A} \tag{14}$$

Here *q* is the convective heat flux through a boundary. Since the bottom surface temperature is fixed at 295*K* and the sloping wall surface temperature cycles between 290K and 300K (refer to Figure 1), a zero temperature difference between the surfaces occurs twice in a cycle. Therefore, the amplitude of the temperature fluctuation (*TA*) is chosen for calculating the heat transfer coefficient instead of a changing temperature difference, which would give an undefined value of the heat transfer coefficient at particular times.

Fig. 5 shows the calculated average Nusselt number on the inclined and bottom surfaces of the cavity. The time histories of the calculated Nusselt number on the inclined surfaces exhibit certain significant features. Firstly, it shows a periodic behaviour in response to the periodic thermal forcing. Secondly within each cycle of the flow response, there is a time period with weak heat transfer and a period with intensive heat transfer. The weak heat transfer corresponds to the daytime condition when the flow is mainly dominated by conduction and the strong heat transfer corresponds to the night-time condition. At night, the boundary layers adjacent to the inclined walls and the bottom are unstable. Therefore, sinking and rising plumes are formed in the inclined and horizontal boundary layers. These plumes dominate

Fig. 6. A series of snapshots of stream function and temperature contours of the third cycle at

Periodically Forced Natural Convection Through the Roof of an Attic-Shaped Building 45

It has been revealed that the flow remains symmetric about the geometrical centreline throughout the cycle for aspect ratio *A* = 0.2, whereas, it is asymmetric during the cooling phase for the other two aspect ratios for *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106. It is also anticipated that the asymmetric solution is one of two possible mirror images of the solutions. Another noticeable variation with different aspect ratios is the formation of a circulation cell near the top of the enclosure. It is seen for *A* = 1.0 that there is an extra vortex (Fig. 6 at *t* = 2.95*P*) on the top of the cavity, which is completely absent for *A* = 0.5 and *A* = 0.2. The flow and temperature fields for the smallest aspect ratio *A* = 0.2 are more complex, with several circulation cells on either side of the central line and many plumes alternately rising and falling throughout the domain, as seen in Fig. 7. These cells and plumes are the result of flow instability described

Fig. 8 illustrates the horizontal velocity and temperature profiles for aspect ratio *A* = 1.0 along the line *DE* as shown in Fig. 2 for *Ra* <sup>=</sup> 1.50 <sup>×</sup> 106. Since the flow is stable and stratified during the day (the heating phase), the structures of the velocity and temperature profiles are

qualitatively the same as those for other aspect ratios.

different times for *<sup>A</sup>* <sup>=</sup> 1.0 and *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106. Left: streamlines; right: isotherms.

earlier.

Fig. 5. Average Nusselt number on the top and bottom surfaces of the cavity for three full cycles where *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106 and *<sup>A</sup>* <sup>=</sup> 0.5

the heat transfer through the sloping walls of the cavity. Finally the calculated maximum Nusselt number on sloping surfaces is 8.72, occurring during the night-time period, whereas the maximum value during the day time is only 3.48, for the selected Rayleigh number and aspect ratio. The corresponding Nusselt number calculated on the bottom surface shows similar behavior as that of top surfaces. Note that the Nusselt number calculated using (13) is based on the total heat flux across the surfaces. Since the surface area of the top surface is larger (0.595*m*2) than the bottom surface (0.532*m*2), therefore the total surface heat flux on the top surfaces will be lower than that of the bottom surface. However, the integral of the heat transfer rate for a cycle on both surfaces has been calculated and it is found that both are the same.

#### **5.3 Effects of the aspect ratio on the flow response**

The flow responses to the periodic thermal forcing for the other two aspect ratios are shown in Figs. 6 and 7, which are compared with the flow response for *A* = 0.5 shown in Fig. 3. It is found that the aspect ratio of the enclosure has a great influence on the flow response as well as heat transfer. The residual effect of the previous cycle on the current cycle has been found similar for all aspect ratios (see at *t* = 2.0*P* in Figs. 3, 6, 7) and the flow and temperature structures during the heating process is qualitatively the same for *A* = 1.0 and *A* = 0.2 as those for *<sup>A</sup>* <sup>=</sup> 0.5 for *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106. However, during the cooling phase there are significant changes of flow and heat transfer among these aspect ratios. For the night-time the high velocity area of these three aspect ratios exists between the two cells where the stream function gradient is higher. Therefore, the buoyancy drives the warm air upwards from the bottom of the geometry and at the same time the gravitational force acts on the cold air downwards from the top. This upward and downward movement can be seen in the temperature contours as a form of rising and sinking plumes.

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Fig. 5. Average Nusselt number on the top and bottom surfaces of the cavity for three full

the heat transfer through the sloping walls of the cavity. Finally the calculated maximum Nusselt number on sloping surfaces is 8.72, occurring during the night-time period, whereas the maximum value during the day time is only 3.48, for the selected Rayleigh number and aspect ratio. The corresponding Nusselt number calculated on the bottom surface shows similar behavior as that of top surfaces. Note that the Nusselt number calculated using (13) is based on the total heat flux across the surfaces. Since the surface area of the top surface is larger (0.595*m*2) than the bottom surface (0.532*m*2), therefore the total surface heat flux on the top surfaces will be lower than that of the bottom surface. However, the integral of the heat transfer rate for a cycle on both surfaces has been calculated and it is found that both are the

The flow responses to the periodic thermal forcing for the other two aspect ratios are shown in Figs. 6 and 7, which are compared with the flow response for *A* = 0.5 shown in Fig. 3. It is found that the aspect ratio of the enclosure has a great influence on the flow response as well as heat transfer. The residual effect of the previous cycle on the current cycle has been found similar for all aspect ratios (see at *t* = 2.0*P* in Figs. 3, 6, 7) and the flow and temperature structures during the heating process is qualitatively the same for *A* = 1.0 and *A* = 0.2 as those for *<sup>A</sup>* <sup>=</sup> 0.5 for *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106. However, during the cooling phase there are significant changes of flow and heat transfer among these aspect ratios. For the night-time the high velocity area of these three aspect ratios exists between the two cells where the stream function gradient is higher. Therefore, the buoyancy drives the warm air upwards from the bottom of the geometry and at the same time the gravitational force acts on the cold air downwards from the top. This upward and downward movement can be seen in the temperature contours as a

cycles where *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106 and *<sup>A</sup>* <sup>=</sup> 0.5

**5.3 Effects of the aspect ratio on the flow response**

form of rising and sinking plumes.

same.

Fig. 6. A series of snapshots of stream function and temperature contours of the third cycle at different times for *<sup>A</sup>* <sup>=</sup> 1.0 and *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106. Left: streamlines; right: isotherms.

It has been revealed that the flow remains symmetric about the geometrical centreline throughout the cycle for aspect ratio *A* = 0.2, whereas, it is asymmetric during the cooling phase for the other two aspect ratios for *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106. It is also anticipated that the asymmetric solution is one of two possible mirror images of the solutions. Another noticeable variation with different aspect ratios is the formation of a circulation cell near the top of the enclosure. It is seen for *A* = 1.0 that there is an extra vortex (Fig. 6 at *t* = 2.95*P*) on the top of the cavity, which is completely absent for *A* = 0.5 and *A* = 0.2. The flow and temperature fields for the smallest aspect ratio *A* = 0.2 are more complex, with several circulation cells on either side of the central line and many plumes alternately rising and falling throughout the domain, as seen in Fig. 7. These cells and plumes are the result of flow instability described earlier.

Fig. 8 illustrates the horizontal velocity and temperature profiles for aspect ratio *A* = 1.0 along the line *DE* as shown in Fig. 2 for *Ra* <sup>=</sup> 1.50 <sup>×</sup> 106. Since the flow is stable and stratified during the day (the heating phase), the structures of the velocity and temperature profiles are qualitatively the same as those for other aspect ratios.

At the night-time the velocity and temperature profiles for *A* = 1.0 are more complicated than that for *A* = 0.5. As seen in Fig. 8, at time *t* = 2.55*P* when the upper surface temperature is lower than the bottom surface, the velocity near the bottom surface is slightly higher than that near the inclined surfaces. After that the velocity increases near both the surfaces until *t* = 2.75*P*. Since a plume-type instability dominates the flow during the cooling phase and the flow has an asymmetric behaviour for a certain period of time, the horizontal velocity is in the same direction near both surfaces and is in an opposite direction in the middle (see *t* = 2.8875*P*). As the flow transits into the next thermal cycle, it becomes very weak. The corresponding temperature contours are plotted in Fig. 8(b). It is seen that the temperature profiles near the bottom surface show a wave shaped for almost the whole cooling phase due to the rising plumes (see Fig. 3). At the time *t* = 2.8875*P*, when three layers of the velocity

Periodically Forced Natural Convection Through the Roof of an Attic-Shaped Building 47

structure is seen, the corresponding temperature profile also shows a wave structure.

Fig. 9. Comparison of the average Nusselt number of three aspect ratios for *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106 Fig. 9 shows the calculated average Nusselt number on the inclined surfaces of the cavity for three different aspect ratios. The time histories of the calculated Nusselt number exhibit certain common features. Within each cycle of the flow response, there is a time period with weak heat transfer and a period with intensive heat transfer for each aspect ratio. The weak heat transfer corresponds to the day time condition when the heat transfer is dominated by conduction and the strong heat transfer corresponds to the night-time condition when convection dominates the flows and the instabilities occur in the form of rising and sinking plumes. During the day time the heat transfer rate is almost the same for all three aspect ratios. However, at the night time the heat transfer rate for *A* = 1.0 is much smaller than that for the other two aspect ratios, and there is a fluctuation of the Nusselt number for a certain period of time. This fluctuation is absent in the other two aspect ratios. This may be due to the fact that less convective cells are present in the streamlines for *A* = 1.0 than those for the other two aspect ratios. Moreover, the movement of the dominating cell for *A* = 1.0 is faster than those for other two. In addition to this, an extra cell appears on the top of the cavity for the aspect ratio *A* = 1.0. It is also noticed that there is not much difference in heat transfer for

Fig. 7. A series of snapshots of stream function and temperature contours of the third cycle at different times for *<sup>A</sup>* <sup>=</sup> 0.2 and *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106. Left: streamlines; right: isotherms.

Fig. 8. Horizontal velocity profile (left) and temperature profile (right) along *DE* for *A* = 1.0 with *Ra* <sup>=</sup> 1.50 <sup>×</sup> 106.

14 Will-be-set-by-IN-TECH

Fig. 7. A series of snapshots of stream function and temperature contours of the third cycle at

Fig. 8. Horizontal velocity profile (left) and temperature profile (right) along *DE* for *A* = 1.0

with *Ra* <sup>=</sup> 1.50 <sup>×</sup> 106.

different times for *<sup>A</sup>* <sup>=</sup> 0.2 and *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106. Left: streamlines; right: isotherms.

At the night-time the velocity and temperature profiles for *A* = 1.0 are more complicated than that for *A* = 0.5. As seen in Fig. 8, at time *t* = 2.55*P* when the upper surface temperature is lower than the bottom surface, the velocity near the bottom surface is slightly higher than that near the inclined surfaces. After that the velocity increases near both the surfaces until *t* = 2.75*P*. Since a plume-type instability dominates the flow during the cooling phase and the flow has an asymmetric behaviour for a certain period of time, the horizontal velocity is in the same direction near both surfaces and is in an opposite direction in the middle (see *t* = 2.8875*P*). As the flow transits into the next thermal cycle, it becomes very weak. The corresponding temperature contours are plotted in Fig. 8(b). It is seen that the temperature profiles near the bottom surface show a wave shaped for almost the whole cooling phase due to the rising plumes (see Fig. 3). At the time *t* = 2.8875*P*, when three layers of the velocity structure is seen, the corresponding temperature profile also shows a wave structure.

Fig. 9. Comparison of the average Nusselt number of three aspect ratios for *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106

Fig. 9 shows the calculated average Nusselt number on the inclined surfaces of the cavity for three different aspect ratios. The time histories of the calculated Nusselt number exhibit certain common features. Within each cycle of the flow response, there is a time period with weak heat transfer and a period with intensive heat transfer for each aspect ratio. The weak heat transfer corresponds to the day time condition when the heat transfer is dominated by conduction and the strong heat transfer corresponds to the night-time condition when convection dominates the flows and the instabilities occur in the form of rising and sinking plumes. During the day time the heat transfer rate is almost the same for all three aspect ratios. However, at the night time the heat transfer rate for *A* = 1.0 is much smaller than that for the other two aspect ratios, and there is a fluctuation of the Nusselt number for a certain period of time. This fluctuation is absent in the other two aspect ratios. This may be due to the fact that less convective cells are present in the streamlines for *A* = 1.0 than those for the other two aspect ratios. Moreover, the movement of the dominating cell for *A* = 1.0 is faster than those for other two. In addition to this, an extra cell appears on the top of the cavity for the aspect ratio *A* = 1.0. It is also noticed that there is not much difference in heat transfer for

Fig. 10. Snapshots of stream function contours (left) and isotherms (right) of the third cycle at

Periodically Forced Natural Convection Through the Roof of an Attic-Shaped Building 49

different times and different Rayleigh numbers with fixed *A* = 0.5.

the aspect ratios *A* = 0.5 and 0.2. The highest average Nusselt numbers for *A* = 1.0, 0.5 and 0.2 are 6.55, 8.72 and 8.76 respectively.

#### **5.4 Effects of Rayleigh number on the flow response and heat transfer**

Fig.10 shows snapshots of stream function and temperature contours for the aspect ratio 0.5 with three different Rayleigh numbers, *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106, 7.2 <sup>×</sup> 104 and 7.2 <sup>×</sup> 103. The contours for *Ra* <sup>=</sup> 7.2 <sup>×</sup> 104 are qualitatively the same as for *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106. It is found that in the heating phase (i.e. when the upper wall temperature is higher than the temperature of the bottom) the flow structures are qualitatively similar for all Rayleigh numbers. However, in the cooling phase the flow behaviour is strongly dependent on the Rayleigh numbers. Stream function and temperature contours are presented at two different times, *t* = 2.70*P* and 2.95*P* for each Rayleigh number in Fig. 10. In the isotherms, rising and sinking plumes are visible for *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106 and 7.2 <sup>×</sup> 104 at both times. A cellular flow pattern is seen in the corresponding stream function contours for *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106. However, only two convective cells are present for *Ra* <sup>=</sup> 7.2 <sup>×</sup> 104. If the Rayleigh number is decreased further (*Ra* <sup>=</sup> 7.2 <sup>×</sup> 103), the flow becomes weaker. Only two cells are seen in the stream function contours and the temperature field is horizontally stratified (see the corresponding isotherms). At *t* = 2.95*P*, the flow seems to be asymmetric along the centre line for *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106. However, for the lower Rayleigh numbers the asymmetric behaviour is not visible.

Fig.11 shows the comparison of the Nusselt number among four Rayleigh numbers for a fixed aspect ratio 0.5. It is seen clearly that during the heating phase the heat transfer rate is weaker, whereas it is much stronger in the cooling phase. With the increase of the Rayleigh number, the Nusselt number increases throughout the thermal cycle, but the rate of increase is much higher in the cooling phase compared to that in the heating phase. The maximum Nusselt number in the cooling phase for *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106 is about 2.5 times of the maximum Nusselt number during the heating phase. It is noticeable that for the lowest Rayleigh number *Ra* = 1.5 <sup>×</sup> 103, the heat transfer rate during the heating and cooling phases are almost the same. The maximum Nusselt number for the four different Rayleigh numbers, *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106, 7.2 <sup>×</sup> 105, 7.2 <sup>×</sup> 104 and 7.2 <sup>×</sup> 103 for the aspect ratio 0.5 are 8.65, 7.34, 4.26 and 3.11 respectively.

#### **5.5 Transition between symmetric and asymmetric flows**

The highest Rayleigh number considered in this study for the three aspect ratios is 1.5 <sup>×</sup> 106. Except for *A* = 0.2, the flow in the cavity for the other two aspect ratios is observed to undergo a supercritical pitchfork bifurcation for this Rayleigh number, in which case one of two possible mirror image asymmetric solutions is obtained. This asymmetric behaviour was first reported numerically and experimentally by Holtzmann et al. (2000) in their study of the case of a sudden cooling boundary condition. If the flow is asymmetric, the horizontal velocity along the midplane of the isosceles triangle would be nonzero. Based on this hypothesis, Fig.12 illustrates the absolute value of maximum horizontal velocity along the geometric center line for *A* = 1.0 and 0.5. It is seen in this figure that, for both aspect ratios, the maximum horizontal velocity is zero up to approximately *t* = 0.70*P* in each cycle, suggesting that the flow is symmetric during this time. However, after this time the maximum horizontal velocity starts to increase, indicating that the flow becomes asymmetric. The asymmetry remains until shortly before the end of each cycle when the flow returns to symmetric again. The same asymmetric behaviour of the flow is seen for the Rayleigh number 7.2 <sup>×</sup> 105 for the aspect ratios 0.5 and 1.0.

16 Will-be-set-by-IN-TECH

48 Convection and Conduction Heat Transfer

the aspect ratios *A* = 0.5 and 0.2. The highest average Nusselt numbers for *A* = 1.0, 0.5 and

Fig.10 shows snapshots of stream function and temperature contours for the aspect ratio 0.5 with three different Rayleigh numbers, *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106, 7.2 <sup>×</sup> 104 and 7.2 <sup>×</sup> 103. The contours for *Ra* <sup>=</sup> 7.2 <sup>×</sup> 104 are qualitatively the same as for *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106. It is found that in the heating phase (i.e. when the upper wall temperature is higher than the temperature of the bottom) the flow structures are qualitatively similar for all Rayleigh numbers. However, in the cooling phase the flow behaviour is strongly dependent on the Rayleigh numbers. Stream function and temperature contours are presented at two different times, *t* = 2.70*P* and 2.95*P* for each Rayleigh number in Fig. 10. In the isotherms, rising and sinking plumes are visible for *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106 and 7.2 <sup>×</sup> 104 at both times. A cellular flow pattern is seen in the corresponding stream function contours for *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106. However, only two convective cells are present for *Ra* <sup>=</sup> 7.2 <sup>×</sup> 104. If the Rayleigh number is decreased further (*Ra* <sup>=</sup> 7.2 <sup>×</sup> 103), the flow becomes weaker. Only two cells are seen in the stream function contours and the temperature field is horizontally stratified (see the corresponding isotherms). At *t* = 2.95*P*, the flow seems to be asymmetric along the centre line for *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106. However, for the lower Rayleigh

Fig.11 shows the comparison of the Nusselt number among four Rayleigh numbers for a fixed aspect ratio 0.5. It is seen clearly that during the heating phase the heat transfer rate is weaker, whereas it is much stronger in the cooling phase. With the increase of the Rayleigh number, the Nusselt number increases throughout the thermal cycle, but the rate of increase is much higher in the cooling phase compared to that in the heating phase. The maximum Nusselt number in the cooling phase for *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106 is about 2.5 times of the maximum Nusselt number during the heating phase. It is noticeable that for the lowest Rayleigh number *Ra* = 1.5 <sup>×</sup> 103, the heat transfer rate during the heating and cooling phases are almost the same. The maximum Nusselt number for the four different Rayleigh numbers, *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106, 7.2 <sup>×</sup> 105, 7.2 <sup>×</sup> 104 and 7.2 <sup>×</sup> 103 for the aspect ratio 0.5 are 8.65, 7.34, 4.26 and 3.11 respectively.

The highest Rayleigh number considered in this study for the three aspect ratios is 1.5 <sup>×</sup> 106. Except for *A* = 0.2, the flow in the cavity for the other two aspect ratios is observed to undergo a supercritical pitchfork bifurcation for this Rayleigh number, in which case one of two possible mirror image asymmetric solutions is obtained. This asymmetric behaviour was first reported numerically and experimentally by Holtzmann et al. (2000) in their study of the case of a sudden cooling boundary condition. If the flow is asymmetric, the horizontal velocity along the midplane of the isosceles triangle would be nonzero. Based on this hypothesis, Fig.12 illustrates the absolute value of maximum horizontal velocity along the geometric center line for *A* = 1.0 and 0.5. It is seen in this figure that, for both aspect ratios, the maximum horizontal velocity is zero up to approximately *t* = 0.70*P* in each cycle, suggesting that the flow is symmetric during this time. However, after this time the maximum horizontal velocity starts to increase, indicating that the flow becomes asymmetric. The asymmetry remains until shortly before the end of each cycle when the flow returns to symmetric again. The same asymmetric behaviour of the flow is seen for the Rayleigh number 7.2 <sup>×</sup> 105 for the aspect

**5.4 Effects of Rayleigh number on the flow response and heat transfer**

0.2 are 6.55, 8.72 and 8.76 respectively.

numbers the asymmetric behaviour is not visible.

**5.5 Transition between symmetric and asymmetric flows**

ratios 0.5 and 1.0.

Fig. 10. Snapshots of stream function contours (left) and isotherms (right) of the third cycle at different times and different Rayleigh numbers with fixed *A* = 0.5.

Fig. 13. Comparison of the average Nusselt number on two inclined surfaces of the enclosure

Periodically Forced Natural Convection Through the Roof of an Attic-Shaped Building 51

Natural convection in an attic-shaped building with the effect of periodic thermal forcing has been carried out in this study based on numerical simulations. An attempt has also been taken to predict the period of the model attic using scaling analysis. Three aspect ratios of *A* = 1.0, 0.5 and 0.2 with four Rayleigh numbers of *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106, 7.2 <sup>×</sup> 105, 7.2 <sup>×</sup> 104 and 7.2 <sup>×</sup> 103 for each aspect ratio have been considered here. Many important features are revealed from the present numerical simulations. It is found that the flow response to the temperature variation on the external surface is fast, and thus the start-up effect is almost negligible. The occurrence of sinking cold-air plumes and rising hot-air plumes in the isotherm contours and the formation of cellular flow patterns in the stream function contours confirm the presence of the Rayleigh-Bénard type instability. It is also observed that the flow undergoes a transition

for (a) *<sup>A</sup>* <sup>=</sup> 1.0 and (b) *<sup>A</sup>* <sup>=</sup> 0.5 with *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106

**6. Conclusions**

Fig. 11. Comparison of the average Nusselt number of four Rayleigh numbers for *A* = 0.5

Fig. 12. The maximum horizontal velocity along the symmetry line for (a) *A* = 1.0 and (b) *<sup>A</sup>* <sup>=</sup> 0.5 with *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106.

The same results have been found when the average Nusselt numbers obtained for both inclined surfaces are compared for the aspect ratios 1.0 and 0.5, which are shown in Fig.13 (a) and (b) respectively. It is seen that at about *t* = 0.70*P*, the calculated Nusselt numbers at the left and right inclined surfaces start to diverge, but later they meet again before the end of each cycle.

Fig. 13. Comparison of the average Nusselt number on two inclined surfaces of the enclosure for (a) *<sup>A</sup>* <sup>=</sup> 1.0 and (b) *<sup>A</sup>* <sup>=</sup> 0.5 with *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106

#### **6. Conclusions**

18 Will-be-set-by-IN-TECH

Fig. 11. Comparison of the average Nusselt number of four Rayleigh numbers for *A* = 0.5

Fig. 12. The maximum horizontal velocity along the symmetry line for (a) *A* = 1.0 and (b)

The same results have been found when the average Nusselt numbers obtained for both inclined surfaces are compared for the aspect ratios 1.0 and 0.5, which are shown in Fig.13 (a) and (b) respectively. It is seen that at about *t* = 0.70*P*, the calculated Nusselt numbers at the left and right inclined surfaces start to diverge, but later they meet again before the end of

*<sup>A</sup>* <sup>=</sup> 0.5 with *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106.

each cycle.

Natural convection in an attic-shaped building with the effect of periodic thermal forcing has been carried out in this study based on numerical simulations. An attempt has also been taken to predict the period of the model attic using scaling analysis. Three aspect ratios of *A* = 1.0, 0.5 and 0.2 with four Rayleigh numbers of *Ra* <sup>=</sup> 1.5 <sup>×</sup> 106, 7.2 <sup>×</sup> 105, 7.2 <sup>×</sup> 104 and 7.2 <sup>×</sup> 103 for each aspect ratio have been considered here. Many important features are revealed from the present numerical simulations. It is found that the flow response to the temperature variation on the external surface is fast, and thus the start-up effect is almost negligible. The occurrence of sinking cold-air plumes and rising hot-air plumes in the isotherm contours and the formation of cellular flow patterns in the stream function contours confirm the presence of the Rayleigh-Bénard type instability. It is also observed that the flow undergoes a transition

Asan, H. & Namli, L. (2000). Laminar natural convection in a pitched roof of triangular cross-section: Summer day boundary conditions. *Energy Buildings*, Vol. 33, 69–73. Del Campo, E. M., Sen, M. & Ramos, E. (1988) Analysis of laminar natural convection in a

Periodically Forced Natural Convection Through the Roof of an Attic-Shaped Building 53

Farrow, D. E. & Patterson, J. C. (1993). On the response of a reservoir sidearm to diurnal

Flack, R. D. (1980) The experimental measurement of natural convection heat transfer in

Flack, R. D. (1979) Velocity measurements in two natural convection air flows using a laser

Haese, P. M. & Teubner, M. D. (2002). Heat exchange in an attic space. *Int. J. Heat Mass*

Holtzmann, G. A., Hill, R. W. & Ball, K. S. (2006). Laminar natural convection in

Hyun, J. M. (1994). Unsteady buoyant convection in an enclosure. *Adv. Heat Transfe*, Vol. 24,

Kent, E. F. (2009a) Numerical analysis of laminar natural convection in isosceles triangular

Kent, E. F. (2009b). Numerical analysis of laminar natural convection in isosceles triangular

Kent, E. F., Asmaz, E. & Ozerbay, S. (2007). Unsteady natural convection in a water-filled

Leonard, B. P. & Mokhtari, S. (1990). ULTRA-SHARP Nonoscillatory Convection Schemes

Ostrach, S. (1988). Natural convection in enclosures. *Trans. ASME: J. Heat Transfer*, Vol. 110,

Poulikakos, D. & Bejan, A. (1983a) Natural convection experiments in a triangular enclosure.

Poulikakos, D. & Bejan, A. (1983b) The fluid dynamics of an attic space. *J. Fluid Mech.*, Vol.

Ridouane, E. H., Campo, A. & Hasnaoui, M. (2006) Benefits derivable from connecting the

Ridouane, E. H. & Campo, A. (2006). Formation of a pitchfork bifurcation in thermal

Ridouane, E. H., Campo, A. & McGarry, M. (2005) Numerical computation of buoyant

Saha, S. C. (2011a). Unsteady natural convection in a triangular enclosure under isothermal

bottom and top walls of attic enclosures with insulated vertical side walls. *Numer.*

convection flow inside an isosceles triangular cavity. *Physics of Fluids*, Vol. 18, 074102.

airflows confined to attic spaces under opposing hot and cold wall conditions. *Int. J.*

triangular enclosures heated or cooled from below, *Trans. ASME: J. Heat Transfer*, Vol.

isosceles triangular enclosures heated from below and symmetrically cooled from

enclosures. *Proceedings of the Institution of Mechanical Engineers Part C - Journal of*

enclosures for cold base and hot inclined walls. *Mechanics Research Communications*,

isosceles triangular enclosure heated from below. *Int. J. Heat Mass Transfer*, Vol. 44,

for High-Speed Steady Multidimensional Flow. *NASA Lewis Research Centre*, NASA

triangular enclosure. *Numer. Heat Transfer.*, Vol. 13, 353–372.

velocimeter. *Trans. ASME: J. Heat Transfer*, Vol. 101, 256–260.

above. *Trans. ASME: J. Heat Transfer*, Vol. 122, 485–491.

*Mechanical Engineering Science*, Vol. 223, 1157–1169.

*Trans. ASME: J. Heat Transfer*, Vol. 105, 652–655.

*Heat Trans. Part A-Applications*, Vol. 49, 175–193.

heating. *Energy Buildings*, Vol. 43, 701–709.

heating and cooling. *J. Fluid Mech.*, Vol. 246, 143–161.

102, 770–772.

277–320.

187–200.

1175–1190.

131, 251–269.

Vol. 36, 497–508.

TM 1-2568 (ICOMP-90-12).

*Therm. Sci.*, Vol. 44, 944–952.

*Transfer*, Vol. 45, 4925–4936.

between symmetry and asymmetry about the geometric symmetry plane over a diurnal cycle for the aspect ratios of *<sup>A</sup>* <sup>=</sup> 1.0 and 0.5 with the Rayleigh numbers 1.5 <sup>×</sup> 106 and 7.2 <sup>×</sup> 105. For all other cases the flow remains symmetric. A three-layer velocity structure has been found along the line at *DE* as shown in Fig. 2 in both the daytime heating phase (due to local cooling effect in the upper sections of the inclined walls) and night-time cooling phase when the flow becomes asymmetric. Furthermore, the flow response in the daytime heating phase is weak, whereas the flow response in the night-time cooling phase, which is dominated by convection, is intensive. At lower Rayleigh numbers the flow becomes weaker for all aspect ratios, and no asymmetric flow behaviour has been noticed.

#### **7. Nomenclature**


#### *Greek letters*


#### **8. References**


20 Will-be-set-by-IN-TECH

between symmetry and asymmetry about the geometric symmetry plane over a diurnal cycle for the aspect ratios of *<sup>A</sup>* <sup>=</sup> 1.0 and 0.5 with the Rayleigh numbers 1.5 <sup>×</sup> 106 and 7.2 <sup>×</sup> 105. For all other cases the flow remains symmetric. A three-layer velocity structure has been found along the line at *DE* as shown in Fig. 2 in both the daytime heating phase (due to local cooling effect in the upper sections of the inclined walls) and night-time cooling phase when the flow becomes asymmetric. Furthermore, the flow response in the daytime heating phase is weak, whereas the flow response in the night-time cooling phase, which is dominated by convection, is intensive. At lower Rayleigh numbers the flow becomes weaker for all aspect ratios, and no

asymmetric flow behaviour has been noticed.

**7. Nomenclature**

*K* Kelvin

t time

*Greek letters*

**8. References**

*A* slope of the attic

*h* height of the attic *h*eff heat transfer coefficient

*Nu* Nusselt number *p* pressure

*P* period of the cycle Pr Prandtl number q convective heat flux Ra Rayleigh number

*T* temperature of the fluid

*x*, *y* cartesian coordinates

*ν* Kinematic viscosity *ρ* Density of the fluid *κ* thermal diffusivity

753–757.

*TA* the amplitude

*T*<sup>0</sup> temperature of the ambient fluid

*β* Volumetric coefficient of thermal expansion

*u*, *v* velocity components along the *x*- and *y*- axes respectively

Δ*T* temperature difference between the surface and the ambient

Akinsete, V. A. & Coleman, T. A. (2000). Heat transfer by steady laminar free convection in

Asan, H. & Namli, L. (2002). Numerical simulation of buoyant flow in a roof of triangular

cross-section under winter day boundary conditions. *Energy Buildings*, Vol. 33,

triangular enclosures. *Int. J. Heat Mass Transfer*, Vol. 25, 991–998.

*k* thermal conductivity

*g* acceleration due to gravity

*l* half horizontal length of the attic


**3** 

*1Japan 2USA 3Malaysia*

**Analysis of Mixed Convection in a** 

**Lid Driven Trapezoidal Cavity** 

M. M. Rahman3, R. Saidur3 and Shuichi Nagata1

*1Institute of Ocean Energy, Saga University 1 Honjo-machi, Saga 840-8502 2Mechanical Engineering Department, Ohio University, Athens, OH-45701* 

Convection is the heat transfer mechanism affected by the flow of fluids. The amount of energy and matter are conveyed by the fluid can be predicted through the convective heat transfer. The convective heat transfer splits into two branches; the natural convection and the forced convection. Forced convection regards the heat transport by induced fluid motion which is forced to happen. This induced flow needs consistent mechanical power. But natural convection differs from the forced convection through the fluid flow driving force which happens naturally. The flows are driven by the buoyancy effect due to the presence of density gradient and gravitational field. As the temperature distribution in the natural convection depends on the intensity of the fluid currents which is dependent on the temperature potential itself, the qualitative and quantitative analysis of natural convection heat transfer is very difficult. Numerical investigation instead of theoretical analysis is more

Two types of natural convection heat transfer phenomena can be observed in the nature. One is that external free convection that is caused by the heat transfer interaction between a single wall and a very large fluid reservoir adjacent to the wall. Another is that internal free convection which befalls within an enclosure. Mathematically, the tendency of a particular

ratio of buoyancy force and viscous force. The parameter *β* is the rate of change of density with respect to the change in temperature (*T*), and *ν* is viscosity. Thus, the Grashof number can be thought of as the ratio of the upwards buoyancy of the heated fluid to the internal friction slowing it down. In very sticky, viscous fluids, the fluid movement is restricted, along with natural convection. In the extreme case of infinite viscosity, the fluid could not

system towards natural convection relies on the Grashof number, (

move and all heat transfer would be through conductive heat transfer.

**1. Introduction** 

needed in this field.

M. A. H. Mamun1, T. R. Tanim2,

*3Department of Mechanical Engineering, University of Malaya, 50603 Kuala Lumpur,* 

> 3 2 *g TL Gr* β

<sup>∇</sup> <sup>=</sup> ), which is a

ν


### **Analysis of Mixed Convection in a Lid Driven Trapezoidal Cavity**

M. A. H. Mamun1, T. R. Tanim2, M. M. Rahman3, R. Saidur3 and Shuichi Nagata1 *1Institute of Ocean Energy, Saga University 1 Honjo-machi, Saga 840-8502 2Mechanical Engineering Department, Ohio University, Athens, OH-45701 3Department of Mechanical Engineering, University of Malaya, 50603 Kuala Lumpur, 1Japan 2USA 3Malaysia*

#### **1. Introduction**

22 Will-be-set-by-IN-TECH

54 Convection and Conduction Heat Transfer

Saha, S. C. (2011b). Scaling of free convection heat transfer in a triangular cavity for Pr > 1.

Saha, S. C., Patterson, J. C. & Lei, C. (2010a). Natural convection in attics subject

Saha, S. C., Patterson, J. C. & Lei, C. (2010b).Natural convection and heat transfer in attics subject to periodic thermal forcing. *Int. J. Therm. Sci.*, Vol. 49, 1899–1910. Saha, S. C., Patterson, J. C. & Lei, C. (2010c). Natural convection in attic-shaped spaces

Saha, S. C. (2008). Natural convection in attics subject to instantaneous and ramp cooling

Saha, S. C., Lei, C. & Patterson, J. C. (2007). Effect of aspect ratio on natural convection in attics subject to periodic thermal forcing. *ANZIAM J.*, Vol. 48, C677–C691. Salmun H. (1995a). Convection patterns in a triangular domain. *Intl. J. Heat Mass Transfer.*,

Salmun H. (1995b). The stability of a single-cell steady-state solution in a triangular enclosure.

to instantaneous and ramp cooling boundary conditions. *Energy Buildings*, Vol. 42,

subject to sudden and ramp heating boundary conditions. *Heat Mass Transfer*, Vol. 46,

boundary conditions. *PhD Thesis*, School of Engineering and Physical Sciences, James

*Energy Buildings*, under review.

1192–1204.

621–638.

Cook University.

Vol. 18, 351–362.

*Intl. J. Heat Mass Transfer.*, Vol. 18, 363–369.

Convection is the heat transfer mechanism affected by the flow of fluids. The amount of energy and matter are conveyed by the fluid can be predicted through the convective heat transfer. The convective heat transfer splits into two branches; the natural convection and the forced convection. Forced convection regards the heat transport by induced fluid motion which is forced to happen. This induced flow needs consistent mechanical power. But natural convection differs from the forced convection through the fluid flow driving force which happens naturally. The flows are driven by the buoyancy effect due to the presence of density gradient and gravitational field. As the temperature distribution in the natural convection depends on the intensity of the fluid currents which is dependent on the temperature potential itself, the qualitative and quantitative analysis of natural convection heat transfer is very difficult. Numerical investigation instead of theoretical analysis is more needed in this field.

Two types of natural convection heat transfer phenomena can be observed in the nature. One is that external free convection that is caused by the heat transfer interaction between a single wall and a very large fluid reservoir adjacent to the wall. Another is that internal free convection which befalls within an enclosure. Mathematically, the tendency of a particular

system towards natural convection relies on the Grashof number, ( 3 2 *g TL Gr* β ν<sup>∇</sup> <sup>=</sup> ), which is a

ratio of buoyancy force and viscous force. The parameter *β* is the rate of change of density with respect to the change in temperature (*T*), and *ν* is viscosity. Thus, the Grashof number can be thought of as the ratio of the upwards buoyancy of the heated fluid to the internal friction slowing it down. In very sticky, viscous fluids, the fluid movement is restricted, along with natural convection. In the extreme case of infinite viscosity, the fluid could not move and all heat transfer would be through conductive heat transfer.

Analysis of Mixed Convection in a Lid Driven Trapezoidal Cavity 57

The flow within an enclosure consisting of two horizontal walls, at different temperatures, is an important circumstance encountered quite frequently in practice. In all the applications having this kind of situation, heat transfer occurs due to the temperature difference across the fluid layer, one horizontal solid surface being at a temperature higher than the other. If the upper plate is the hot surface, then the lower surface has heavier fluid and by virtue of buoyancy the fluid would not come to the lower plate. Because in this case the heat transfer mode is restricted to only conduction. But if the fluid is enclosed between two horizontal surfaces of which the upper surface is at lower temperature, there will be the existence of cellular natural convective currents which are called as Benard cells. For fluids whose density decreases with increasing temperature, this leads to an unstable situation. Benard [1] mentioned this instability as a "top heavy" situation. In that case fluid is completely stationary and heat is transferred across the layer by the conduction mechanism only. Rayleigh [2] recognized that this unstable situation must break down at a certain value of Rayleigh number above which convective motion must be generated. Jeffreys [3] calculated this limiting value of *Ra* to be 1708, when air layer is bounded on both sides by solid walls.

The tilted enclosure geometry has received considerable attention in the heat transfer literature because of mostly growing interest of solar collector technology. The angle of tilt has a dramatic impact on the flow housed by the enclosure. Consider an enclosure heated from below is rotated about a reference axis. When the tilted angle becomes 90º, the flow and thermal fields inside the enclosure experience the heating from side condition. Thereby convective currents may pronounce over the diffusive currents. When the enclosure rotates to 180º, the heat transfer mechanism switches to the diffusion because the top wall is heated.

Flow and heat transfer analysis in lid-driven cavities is one of the most widely studied problems in thermo-fluids area. Numerous investigations have been conducted in the past on lid-driven cavity flow and heat transfer considering various combinations of the imposed temperature gradients and cavity configurations. This is because the driven cavity configuration is encountered in many practical engineering and industrial applications. Such configurations can be idealized by the simple rectangular geometry with regular boundary conditions yielding a well-posed problem. Combined forced-free convection flow in lid-driven cavities or enclosures occurs as a result of two competing mechanisms. The

θ dimensionless temperature ,(TH-TC)/ΔT μ dynamic viscosity of the fluid (Pa s) ν kinematic viscosity of the fluid (m2/s)

ρ density of the fluid (kg/m3) Φ rotational angle of the cavity

v value of cold temperature cH value of hot temperature

**1.1 Flow within enclosure** 

**1.1.1 Tilted enclosure** 

**1.1.2 LID driven enclosure** 

**Subscript** 

a average value

Forced convection is often encountered by engineers designing or analyzing heat exchangers, pipe flow, and flow over flat plate at a different temperature than the stream (the case of a shuttle wing during re-entry, for example). However, in any forced convection situation, some amount of natural convection is always present. When the natural convection is not negligible, such flows are typically referred to as mixed convection.

When analyzing potentially mixed convection, a parameter called the Richardson number (*Ri= Gr/ Re*2) parametizes the relative strength of free and forced convection. The Richardson number is the ratio of Grashof number and the square of the Reynolds number, which represents the ratio of buoyancy force and inertia force, and which stands in for the contribution of natural convection. When *Ri*>>1, natural convection dominates and when *Ri*<<1, forced convection dominates and when *Ri*=1, mixed convection dominates.

 The thermo-fluid fields developed inside the cavity depend on the orientation and the geometry of the cavity. Reviewing the nature and the practical applications, the enclosure phenomena can loosely be organized into two classes. One of these is enclosure heated from the side which is found in solar collectors, double wall insulations, laptop cooling system and air circulation inside the room and the another one is enclosure heated from below which is happened in geophysical system like natural circulation in the atmosphere, the hydrosphere and the molten core of the earth.

#### **Nomenclature**


#### **Greak symbols**


#### **Subscript**

56 Convection and Conduction Heat Transfer

Forced convection is often encountered by engineers designing or analyzing heat exchangers, pipe flow, and flow over flat plate at a different temperature than the stream (the case of a shuttle wing during re-entry, for example). However, in any forced convection situation, some amount of natural convection is always present. When the natural

When analyzing potentially mixed convection, a parameter called the Richardson number (*Ri= Gr/ Re*2) parametizes the relative strength of free and forced convection. The Richardson number is the ratio of Grashof number and the square of the Reynolds number, which represents the ratio of buoyancy force and inertia force, and which stands in for the contribution of natural convection. When *Ri*>>1, natural convection dominates and when

 The thermo-fluid fields developed inside the cavity depend on the orientation and the geometry of the cavity. Reviewing the nature and the practical applications, the enclosure phenomena can loosely be organized into two classes. One of these is enclosure heated from the side which is found in solar collectors, double wall insulations, laptop cooling system and air circulation inside the room and the another one is enclosure heated from below which is happened in geophysical system like natural circulation in the atmosphere, the

convection is not negligible, such flows are typically referred to as mixed convection.

*Ri*<<1, forced convection dominates and when *Ri*=1, mixed convection dominates.

hydrosphere and the molten core of the earth.

h convective heat transfer coefficient (W/m2 K)

CP specific heat at constant pressure (J/kg K)

k thermal conductivity of the fluid (W/m K)

g gravitational acceleration (m/s2)

Gr Grashof number, gβΔTW3/ν<sup>2</sup> Re Reynolds number, U0W/ν Ri Richardson number, Gr/Re2

R length of the inclined sidewalls (m) T temperature of the fluid, (°C)

u velocity component at x-direction (m/s)

v velocity component at y-direction (m/s)

α thermal diffusivity of the fluid (m2/s)

U dimensionless velocity component at X-direction

V dimensionless velocity component at Y-direction

X distance along the non-dimensional x-coordinate Y distance along the non-dimensional y-coordinate

β volumetric coefficient of thermal expansion (K-1) γ inclination angle of the sidewalls of the cavity

**Nomenclature** 

q" Heat Flux (W/m2)

Nu Nusselt number, hW/k Pr Prandtl number, ν/α

A Aspect Ratio, H/W

W length of the cavity, (m) x distance along the x-coordinate

**Greak symbols** 


#### **1.1 Flow within enclosure**

The flow within an enclosure consisting of two horizontal walls, at different temperatures, is an important circumstance encountered quite frequently in practice. In all the applications having this kind of situation, heat transfer occurs due to the temperature difference across the fluid layer, one horizontal solid surface being at a temperature higher than the other. If the upper plate is the hot surface, then the lower surface has heavier fluid and by virtue of buoyancy the fluid would not come to the lower plate. Because in this case the heat transfer mode is restricted to only conduction. But if the fluid is enclosed between two horizontal surfaces of which the upper surface is at lower temperature, there will be the existence of cellular natural convective currents which are called as Benard cells. For fluids whose density decreases with increasing temperature, this leads to an unstable situation. Benard [1] mentioned this instability as a "top heavy" situation. In that case fluid is completely stationary and heat is transferred across the layer by the conduction mechanism only. Rayleigh [2] recognized that this unstable situation must break down at a certain value of Rayleigh number above which convective motion must be generated. Jeffreys [3] calculated this limiting value of *Ra* to be 1708, when air layer is bounded on both sides by solid walls.

#### **1.1.1 Tilted enclosure**

The tilted enclosure geometry has received considerable attention in the heat transfer literature because of mostly growing interest of solar collector technology. The angle of tilt has a dramatic impact on the flow housed by the enclosure. Consider an enclosure heated from below is rotated about a reference axis. When the tilted angle becomes 90º, the flow and thermal fields inside the enclosure experience the heating from side condition. Thereby convective currents may pronounce over the diffusive currents. When the enclosure rotates to 180º, the heat transfer mechanism switches to the diffusion because the top wall is heated.

#### **1.1.2 LID driven enclosure**

Flow and heat transfer analysis in lid-driven cavities is one of the most widely studied problems in thermo-fluids area. Numerous investigations have been conducted in the past on lid-driven cavity flow and heat transfer considering various combinations of the imposed temperature gradients and cavity configurations. This is because the driven cavity configuration is encountered in many practical engineering and industrial applications. Such configurations can be idealized by the simple rectangular geometry with regular boundary conditions yielding a well-posed problem. Combined forced-free convection flow in lid-driven cavities or enclosures occurs as a result of two competing mechanisms. The

Analysis of Mixed Convection in a Lid Driven Trapezoidal Cavity 59

a. To study the variation of average heat transfer in terms of Nusselt number with the variation of Richardson number at different aspect ratios of the rectangular enclosure

b. To find out the optimum configuration by changing the inclination angle of the side

c. To study the variation of average heat transfer in terms of Nusselt number with the

d. To study the variation of average heat transfer in terms of Nusselt number at different

e. To study the variation of average heat transfer in terms of Nusselt number with the variation of Richardson number at different aspect ratios of the optimum trapezoidal enclosure by changing the rotation angle for both aiding and opposing flow conditions.. f. To analyze the flow pattern inside the trapezoidal enclosures in terms of Streamlines

There have been many investigations in the past on mixed convective flow in lid-driven cavities. Many different configurations and combinations of thermal boundary conditions have been considered and analyzed by various investigators. Torrance et al. [9] investigated mixed convection in driven cavities as early as in 1972. Papaniclaou and Jaluria [10-13] carried out a series of numerical studies to investigate the combined forced and natural convective cooling of heat dissipating electronic components, located in rectangular enclosures, and cooled by an external through flow of air. The results indicate that flow patterns generally consists of high of low velocity re-circulating cells because of buoyancy forces induced by the heat source. Koseff and Street [14] studied experimentally as well as numerically the recirculation flow patterns for a wide range of Reynolds (*Re*) and Grashof (*Gr*) numbers. Their results showed that the three dimensional features, such as corner eddies near the end walls, and Taylor- Gortler like longitudinal vortices, have significant effects on the flow patterns for low Reynolds numbers. Khanafer and Chamakha [15] examined numerically mixed convection flow in a lid-driven enclosure filled with a fluidsaturated porous medium and reported on the effects of the Darcy and Richardson numbers

G. A. Holtzman et. al [16] have studied laminar natural convection in isosceles triangular enclosures heated from below and symmetrically cooled from above. This problem is examined over aspect ratios ranging from 0.2 to 1.0 and Grashoff numbers from 103 to 105. Its is found that a pitchfork bifurcation occurs at a critical Grashoff number for each of the aspect ratios considered, above which the symmetric solutions are unstable to finite perturbations and asymmetric solutions are instead obtained. Results are presented detailing the occurance of the pitchfork bifurcation in each of the aspect ratios considered, and the resulting flow patterns are described. A flow visualization study is used to validate the numerical observations. Difference in local values of the Nusselt number between asymmetric and symmetric solutions are found to be more than 500 percent due to the shifting of the buoyancy- driven cells. The phenomenon of natural convection in trapezoidal enclosures where upper and lower walls are not parallel, in particular a triangular geometry, is examined by H. Asan, L. Namli [17] over a parameter domain in which the aspect ratio of the enclosure ranges from 0.1 to 1.0, the Rayleigh number varies between 102

walls of the trapezoidal cavity by analyzing the maximum heat transfer.

variation of Richardson number of the optimum trapezoidal cavity.

and compare it with the established literature.

aspect ratios of the optimum trapezoidal cavity.

on the flow and heat transfer characteristics.

and isotherms.

**2. Literature review** 

first is due to shear flow caused by the movement of one of the walls of the cavity while the second is due to buoyancy flow produced by thermal non homogeneity of the cavity boundaries. Understanding these mechanisms is of great significance from technical and engineering standpoints.

#### **1.2 Application**

Air-cooling is one of the preferred methods for the cooling of computer systems and other electronic equipments, due to its simplicity and low cost. It is very important that such cooling systems should be designed in the most efficient way and the power requirement for the cooling should be minimized. The electronic components are treated as heat sources embedded on flat surfaces. A small fan blows air at low speeds over the heat sources. This gives rise to a situation where the forced convection due to shear driven flow and the natural convection due to buoyancy driven flow are of comparable magnitude and the resulting heat transfer process is categorized as mixed convection. Mixed convection flow and heat transfer also occur frequently in other engineering and natural situations. One important configuration is a lid-driven (or shear- driven) flow in a differentially heated/cooled cavity, which has applications in crystal growth, flow and heat transfer in solar ponds [5], dynamics of lakes [6], thermal-hydraulics of nuclear reactors [7], industrial processes such as food processing, and float glass production [8]. The interaction of the shear driven flow due to the lid motion and natural convective flow due to the buoyancy effect is quite complex and warrants comprehensive analysis to understand the physics of the resulting flow and heat transfer process.

#### **1.3 Motivation behind the selection of problem**

Two dimensional steady, mixed convection heat transfers in a two-dimensional trapezoidal cavity with constant heat flux from heated bottom wall while the isothermal moving top wall has been studied numerically. The present study is based on the configuration of Aydin and Yang [27] where the isothermal heat source at the bottom wall is replaced by a constant flux heat source, which is physically more realistic. The main attribute for choosing the trapezoidal shape cavity is to enhance the heat transfer rate as it could be said intuitionally due to its extended cold top surface. The inclination angle of the sidewalls of the trapezoid has been changed (30°, 45° and 60°) to get the maximum heat transfer in terms of maximum Nusselt number. Then the trapezoid has been rotated (30°, 45° and 60°) and the results have been studied. The tilted position of the enclosure shows a significant influence on the heat transfer. Results are obtained for both the aiding and opposing flow conditions by changing the direction of the lid motion. This study includes additional computations for cavities at various aspect ratios, *A*, ranging from 0.5 to 2 and their effects on the heat transfer process is analyzed in terms of average Nusselt number. Contextually the present study will focus on the computational analysis of the influence of inclination angle of the sidewalls of the cavity, rotational angle of the cavity, Aspect ratio, direction of the lid motion and Richardson number.

#### **1.4 Main objectives of the work**

The investigation is carried out in a two dimensional lid driven trapezoidal enclosure filled with air. The inclined side walls are kept adiabatic and the bottom wall of the cavity is kept at uniform heat flux. The cooled top wall having constant temperature will move with a constant velocity. The specific objectives of the present research work are as follows:


#### **2. Literature review**

58 Convection and Conduction Heat Transfer

first is due to shear flow caused by the movement of one of the walls of the cavity while the second is due to buoyancy flow produced by thermal non homogeneity of the cavity boundaries. Understanding these mechanisms is of great significance from technical and

Air-cooling is one of the preferred methods for the cooling of computer systems and other electronic equipments, due to its simplicity and low cost. It is very important that such cooling systems should be designed in the most efficient way and the power requirement for the cooling should be minimized. The electronic components are treated as heat sources embedded on flat surfaces. A small fan blows air at low speeds over the heat sources. This gives rise to a situation where the forced convection due to shear driven flow and the natural convection due to buoyancy driven flow are of comparable magnitude and the resulting heat transfer process is categorized as mixed convection. Mixed convection flow and heat transfer also occur frequently in other engineering and natural situations. One important configuration is a lid-driven (or shear- driven) flow in a differentially heated/cooled cavity, which has applications in crystal growth, flow and heat transfer in solar ponds [5], dynamics of lakes [6], thermal-hydraulics of nuclear reactors [7], industrial processes such as food processing, and float glass production [8]. The interaction of the shear driven flow due to the lid motion and natural convective flow due to the buoyancy effect is quite complex and warrants comprehensive analysis to understand the physics of

Two dimensional steady, mixed convection heat transfers in a two-dimensional trapezoidal cavity with constant heat flux from heated bottom wall while the isothermal moving top wall has been studied numerically. The present study is based on the configuration of Aydin and Yang [27] where the isothermal heat source at the bottom wall is replaced by a constant flux heat source, which is physically more realistic. The main attribute for choosing the trapezoidal shape cavity is to enhance the heat transfer rate as it could be said intuitionally due to its extended cold top surface. The inclination angle of the sidewalls of the trapezoid has been changed (30°, 45° and 60°) to get the maximum heat transfer in terms of maximum Nusselt number. Then the trapezoid has been rotated (30°, 45° and 60°) and the results have been studied. The tilted position of the enclosure shows a significant influence on the heat transfer. Results are obtained for both the aiding and opposing flow conditions by changing the direction of the lid motion. This study includes additional computations for cavities at various aspect ratios, *A*, ranging from 0.5 to 2 and their effects on the heat transfer process is analyzed in terms of average Nusselt number. Contextually the present study will focus on the computational analysis of the influence of inclination angle of the sidewalls of the cavity, rotational angle of the cavity, Aspect ratio, direction of the lid motion and Richardson number.

The investigation is carried out in a two dimensional lid driven trapezoidal enclosure filled with air. The inclined side walls are kept adiabatic and the bottom wall of the cavity is kept at uniform heat flux. The cooled top wall having constant temperature will move with a

constant velocity. The specific objectives of the present research work are as follows:

engineering standpoints.

the resulting flow and heat transfer process.

**1.4 Main objectives of the work** 

**1.3 Motivation behind the selection of problem** 

**1.2 Application** 

There have been many investigations in the past on mixed convective flow in lid-driven cavities. Many different configurations and combinations of thermal boundary conditions have been considered and analyzed by various investigators. Torrance et al. [9] investigated mixed convection in driven cavities as early as in 1972. Papaniclaou and Jaluria [10-13] carried out a series of numerical studies to investigate the combined forced and natural convective cooling of heat dissipating electronic components, located in rectangular enclosures, and cooled by an external through flow of air. The results indicate that flow patterns generally consists of high of low velocity re-circulating cells because of buoyancy forces induced by the heat source. Koseff and Street [14] studied experimentally as well as numerically the recirculation flow patterns for a wide range of Reynolds (*Re*) and Grashof (*Gr*) numbers. Their results showed that the three dimensional features, such as corner eddies near the end walls, and Taylor- Gortler like longitudinal vortices, have significant effects on the flow patterns for low Reynolds numbers. Khanafer and Chamakha [15] examined numerically mixed convection flow in a lid-driven enclosure filled with a fluidsaturated porous medium and reported on the effects of the Darcy and Richardson numbers on the flow and heat transfer characteristics.

G. A. Holtzman et. al [16] have studied laminar natural convection in isosceles triangular enclosures heated from below and symmetrically cooled from above. This problem is examined over aspect ratios ranging from 0.2 to 1.0 and Grashoff numbers from 103 to 105. Its is found that a pitchfork bifurcation occurs at a critical Grashoff number for each of the aspect ratios considered, above which the symmetric solutions are unstable to finite perturbations and asymmetric solutions are instead obtained. Results are presented detailing the occurance of the pitchfork bifurcation in each of the aspect ratios considered, and the resulting flow patterns are described. A flow visualization study is used to validate the numerical observations. Difference in local values of the Nusselt number between asymmetric and symmetric solutions are found to be more than 500 percent due to the shifting of the buoyancy- driven cells. The phenomenon of natural convection in trapezoidal enclosures where upper and lower walls are not parallel, in particular a triangular geometry, is examined by H. Asan, L. Namli [17] over a parameter domain in which the aspect ratio of the enclosure ranges from 0.1 to 1.0, the Rayleigh number varies between 102

Analysis of Mixed Convection in a Lid Driven Trapezoidal Cavity 61

was adiabatic. A symmetrical isothermal heat source was placed at the otherwise adiabatic bottom wall. They investigated the effects of Richardson number and the length of the heat source on the fluid flow and heat transfer. Shankar et al. [28] presented analytical solution for mixed convection in cavities with very slow lid motion. The convection process has been shown to be governed by an inhomogeneous biharmonic equation for the stream function. Oztop and Dagtekin [29] performed numerical analysis of mixed convection in a square cavity with moving and differentially heated sidewalls. Sharif [30] investigates heat transfer in two-dimensional shallow rectangular driven cavity of aspect ratio 10 and Prandtl number 6.0 with hot moving lid on top and cooled from bottom. They investigated the effect of Richardson number and inclination angle. G. Guo and M. A. R. Sharif [31] studied mixed convection in rectangular cavities at various aspect ratios with moving isothermal sidewalls and constant heat source on the bottom wall. They plotted the streamlines and isotherms for different values of Richardson number and also studied the variation of the average Nu and maximum surface temperature at the heat source with Richardson number with different heat source length. They simulated streamlines and isotherms for asymmetric placements of the heat source and also the effects of asymmetry of the heating elements on the average Nu

The physical model considered here is shown in figure 1 and 2, along with the important geometric parameters. It consists of a trapezoidal cavity filled with air, whose bottom wall and top wall are subjected to hot *TH* and cold *TC* temperatures respectively while the side walls are kept adiabatic. Two cases of thermal boundary conditions for the top moving wall have been considered here. The first case is (figure 1) when the moving cold wall is moving in the positive x direction (opposing flow condition). In that case the shear flow caused by moving top wall opposes the buoyancy driven flow caused by the thermal non-homogeneity of the cavity boundaries. The second case is (figure 2) when the moving cold wall is moving in the negative x direction (aiding flow condition). In that case the shear flow assists the buoyancy flow. The cavity height is H, width of the bottom hot wall is W, is inclined at angle Ф with the horizontal reference axis. γ is the inclination angle of the sidewalls of the cavity. The flow and heat transfer phenomena in the cavity are investigated for a series of

Richardson numbers (*Ri*), aspect ratio (A=H/W), rotation angle of the cavity Ф.

Fig. 1. Schematic diagram of the physical system considering opposing flow condition

and the maximum source length temperature.

**3. Physical model** 

to 105 and Prandtl number correspond to air and water. It is found that the numerical experiments verify the flow features that are known from theoretical asymptotic analysis of this problem (valid for shallow spaces) only over a certain range of the parametric domain.

Moallemi and Jang [18] numerically studied mixed convective flow in a bottom heated square driven cavity and investigated the effect of Prandtl number on the flow and heat transfer process. They found that the effects of buoyancy are more pronounced for higher values of Prandtl number. They also derived a correlation for the average Nusselt number in terms of the Prandtl number, Reynolds number, and Richardson number. Mohammad and Viskanta [19] performed numerical investigation and flow visualization study on two and three-dimensional laminar mixed convection flow in a bottom heated shallow driven cavity filled with water having a Prandtl number of 5.84. They concluded that the lid motion destroys all types of convective cells due to heating from below for finite size cavities. They also implicated that the two-dimensional heat transfer results compare favorably with those based on a three-dimensional model for Gr/Re< 1. Later, Mohammad and Viskanta [20] experimentally and numerically studied mixed convection in shallow rectangular bottom heated cavities filled with liquid Gallium having a low Prandtl number of 0.022. They found that the heat transfer rate is rather insensitive to the lid velocity and an extremely thin shear layer exists along the major portion of the moving lid. The flow structure consists of an elongated secondary circulation that occupies a third of the cavity.

 Mansour and Viskanta [21] studied mixed convective flow in a tall vertical cavity where one of the vertical sidewalls, maintained at a colder temperature than the other, was moving up or downward thus assisting or opposing the buoyancy. They observed that when shear assisted the buoyancy a shear cell developed adjacent to the moving wall while the buoyancy cell filled the rest of the cavity. When shear opposed buoyancy, the heat transfer rate reduced below that for purely natural convection. Iwatsu et al. [22] and Iwatsu and Hyun [23] conducted two-dimensional and three-dimensional numerical simulation of mixed convection in square cavities heated from the top moving wall. Mohammad and Viskanta [24] conducted three-dimensional numerical simulation of mixed convection in a shallow driven cavity filled with a stably stratified fluid heated from the top moving wall and cooled from below for a range of Rayleigh number and Richardson number.

 Prasad and Koseff [25] reported experimental results for mixed convection in deep liddriven cavities heated from below. In a series of experiments which were performed on a cavity filled with water, the heat flux was measured at different locations over the hot cavity floor for a range of Re and Gr. Their results indicated that the overall (i.e. area-averaged) heat transfer rate was a very weak function of Gr for the range of Re examined (2200 < Re < 12000). The data were correlated by Nusselt number vs Reynolds number, as well as Stanton number vs Reynolds number relations.

They observed that the heat transfer is rather insensitive to the Richardson number. Hsu and Wang [26] investigated the mixed convective heat transfer where the heat source was embedded on a board mounted vertically on the bottom wall at the middle in an enclosure. The cooling air flow enters and exits the enclosure through the openings near the top of the vertical sidewalls. The results show that both the thermal field and the average Nusselt number depend strongly on the governing parameters, position of the heat source, as well as the property of the heat-source-embedded board.

Aydin and Yang [27] numerically studied mixed convection heat transfer in a twodimensional square cavity having an aspect ratio of 1. In their configuration the isothermal sidewalls of the cavity were moving downwards with uniform velocity while the top wall was adiabatic. A symmetrical isothermal heat source was placed at the otherwise adiabatic bottom wall. They investigated the effects of Richardson number and the length of the heat source on the fluid flow and heat transfer. Shankar et al. [28] presented analytical solution for mixed convection in cavities with very slow lid motion. The convection process has been shown to be governed by an inhomogeneous biharmonic equation for the stream function. Oztop and Dagtekin [29] performed numerical analysis of mixed convection in a square cavity with moving and differentially heated sidewalls. Sharif [30] investigates heat transfer in two-dimensional shallow rectangular driven cavity of aspect ratio 10 and Prandtl number 6.0 with hot moving lid on top and cooled from bottom. They investigated the effect of Richardson number and inclination angle. G. Guo and M. A. R. Sharif [31] studied mixed convection in rectangular cavities at various aspect ratios with moving isothermal sidewalls and constant heat source on the bottom wall. They plotted the streamlines and isotherms for different values of Richardson number and also studied the variation of the average Nu and maximum surface temperature at the heat source with Richardson number with different heat source length. They simulated streamlines and isotherms for asymmetric placements of the heat source and also the effects of asymmetry of the heating elements on the average Nu and the maximum source length temperature.

#### **3. Physical model**

60 Convection and Conduction Heat Transfer

to 105 and Prandtl number correspond to air and water. It is found that the numerical experiments verify the flow features that are known from theoretical asymptotic analysis of this problem (valid for shallow spaces) only over a certain range of the parametric domain. Moallemi and Jang [18] numerically studied mixed convective flow in a bottom heated square driven cavity and investigated the effect of Prandtl number on the flow and heat transfer process. They found that the effects of buoyancy are more pronounced for higher values of Prandtl number. They also derived a correlation for the average Nusselt number in terms of the Prandtl number, Reynolds number, and Richardson number. Mohammad and Viskanta [19] performed numerical investigation and flow visualization study on two and three-dimensional laminar mixed convection flow in a bottom heated shallow driven cavity filled with water having a Prandtl number of 5.84. They concluded that the lid motion destroys all types of convective cells due to heating from below for finite size cavities. They also implicated that the two-dimensional heat transfer results compare favorably with those based on a three-dimensional model for Gr/Re< 1. Later, Mohammad and Viskanta [20] experimentally and numerically studied mixed convection in shallow rectangular bottom heated cavities filled with liquid Gallium having a low Prandtl number of 0.022. They found that the heat transfer rate is rather insensitive to the lid velocity and an extremely thin shear layer exists along the major portion of the moving lid. The flow structure consists of an

 Mansour and Viskanta [21] studied mixed convective flow in a tall vertical cavity where one of the vertical sidewalls, maintained at a colder temperature than the other, was moving up or downward thus assisting or opposing the buoyancy. They observed that when shear assisted the buoyancy a shear cell developed adjacent to the moving wall while the buoyancy cell filled the rest of the cavity. When shear opposed buoyancy, the heat transfer rate reduced below that for purely natural convection. Iwatsu et al. [22] and Iwatsu and Hyun [23] conducted two-dimensional and three-dimensional numerical simulation of mixed convection in square cavities heated from the top moving wall. Mohammad and Viskanta [24] conducted three-dimensional numerical simulation of mixed convection in a shallow driven cavity filled with a stably stratified fluid heated from the top moving wall

and cooled from below for a range of Rayleigh number and Richardson number.

 Prasad and Koseff [25] reported experimental results for mixed convection in deep liddriven cavities heated from below. In a series of experiments which were performed on a cavity filled with water, the heat flux was measured at different locations over the hot cavity floor for a range of Re and Gr. Their results indicated that the overall (i.e. area-averaged) heat transfer rate was a very weak function of Gr for the range of Re examined (2200 < Re < 12000). The data were correlated by Nusselt number vs Reynolds number, as well as Stanton

They observed that the heat transfer is rather insensitive to the Richardson number. Hsu and Wang [26] investigated the mixed convective heat transfer where the heat source was embedded on a board mounted vertically on the bottom wall at the middle in an enclosure. The cooling air flow enters and exits the enclosure through the openings near the top of the vertical sidewalls. The results show that both the thermal field and the average Nusselt number depend strongly on the governing parameters, position of the heat source, as well

Aydin and Yang [27] numerically studied mixed convection heat transfer in a twodimensional square cavity having an aspect ratio of 1. In their configuration the isothermal sidewalls of the cavity were moving downwards with uniform velocity while the top wall

elongated secondary circulation that occupies a third of the cavity.

number vs Reynolds number relations.

as the property of the heat-source-embedded board.

The physical model considered here is shown in figure 1 and 2, along with the important geometric parameters. It consists of a trapezoidal cavity filled with air, whose bottom wall and top wall are subjected to hot *TH* and cold *TC* temperatures respectively while the side walls are kept adiabatic. Two cases of thermal boundary conditions for the top moving wall have been considered here. The first case is (figure 1) when the moving cold wall is moving in the positive x direction (opposing flow condition). In that case the shear flow caused by moving top wall opposes the buoyancy driven flow caused by the thermal non-homogeneity of the cavity boundaries. The second case is (figure 2) when the moving cold wall is moving in the negative x direction (aiding flow condition). In that case the shear flow assists the buoyancy flow. The cavity height is H, width of the bottom hot wall is W, is inclined at angle Ф with the horizontal reference axis. γ is the inclination angle of the sidewalls of the cavity. The flow and heat transfer phenomena in the cavity are investigated for a series of Richardson numbers (*Ri*), aspect ratio (A=H/W), rotation angle of the cavity Ф.

Fig. 1. Schematic diagram of the physical system considering opposing flow condition

Analysis of Mixed Convection in a Lid Driven Trapezoidal Cavity 63

Firstly the problem is defined as a two dimensional enclosure. Control Volume based finite volume method (FVM) is to be used to discretize the governing differential equations. The pressure- velocity coupling in the governing equations is achieved using the well known SIMPLE method for numerical computations. The set of governing equations are to be solved sequentially. A second order upwind differencing scheme is to be used for the formulation of the coefficients in the finite-volume equations. As the sides of the trapezoidal cavity are not parallel, the present numerical techniques will descretize the computational

In order to obtain the grid independence solution, a grid refinement study is performed for the trapezoidal cavity (A=1) under constant heat flux condition keeping, Re=400, Pr=0.71, Ri= 1.0.n It is found in figure 3 that 5496 regular nodes are sufficient to provide accurate results. This grid resolution is therefore used for all subsequent computations for A≤1. For taller cavities with A>1, a proportionately large number of grids in the vertical direction is

> **1000 2000 3000 4000 5000 6000 7000 Number of Nodes**

The convergence criterion was defined by the required scaled residuals to decrease 10⎯5 for

The computational procedure is validated against the numerical results of Iwatsu et el.[22] for a top heated moving lid and bottom cooled square cavity filled with air (*Pr*=0.71). A 60×60 mesh is used and computations are done for six different *Re* and *Gr* combinations. Comparisons of the average Nusselt number at the hot lid are shown in Table 1. The general

Fig. 3. Grid sensitivity test for trapezoidal cavity at Ri=1.0, Re=400 and A=1

all equations except the energy equations, for which the criterion is 10⎯8.

∂θ∂ = *X* 0

Non-dimensional heat transfer parameter Nusselt number is stated as:

Bottom wall: U=V=0, θ=1 Right and Left wall: U=V=0,

**3.2 Numerical method** 

used.

domain into unstructured triangular elements.

**6**

**7**

**8**

**Nuav**

**9**

Fig. 2. Schematic diagram of the physical system considering aiding flow condition

#### **3.1 Mathematical model**

Using the Boussinesq approximation and neglecting the viscous dissipation effect and compressibility effect the dimensionless governing equations for two dimensional laminar incompressible flows can be written as follows:

$$\frac{\partial U}{\partial X} + \frac{\partial V}{\partial Y} = 0 \tag{1}$$

$$U\frac{\partial U}{\partial X} + V\frac{\partial U}{\partial Y} = -\frac{\partial P}{\partial X} + \frac{1}{\text{Re}} \left( \frac{\partial^2 U}{\partial X^2} + \frac{\partial^2 U}{\partial Y^2} \right) \tag{2}$$

$$U\frac{\partial V}{\partial X} + V\frac{\partial V}{\partial Y} = -\frac{\partial P}{\partial Y} + \frac{1}{\text{Re}} \left( \frac{\partial^2 V}{\partial X^2} + \frac{\partial^2 V}{\partial Y^2} \right) + \frac{Gr}{\text{Re}^2} \theta \tag{3}$$

$$U\frac{\partial\theta}{\partial X} + V\frac{\partial\theta}{\partial Y} = \frac{1}{\text{Re}\Pr} \left( \frac{\partial^2\theta}{\partial X^2} + \frac{\partial^2\theta}{\partial Y^2} \right) \tag{4}$$

The dimensionless variables are as follows:

$$\mathbf{X} \mathbf{\overline{x}} = \mathbf{\overline{x}}/\mathbf{W}, \text{ Y=y/W}, \text{ } \mathbf{\overline{0}=(T\_{\mathbf{H}^{\mathbf{u}}} T\_{\mathbf{C}})/\Delta T, \text{ } \Delta \mathbf{T} = \mathbf{q} \text{"} \mathbf{W}/\mathbf{k}, \text{ U=u/U} \mathbf{0}, \text{ } \mathbf{V} \mathbf{\overline{e}} \text{ } \mathbf{y}/\mathbf{U}\_0, \text{ } \mathbf{P} \mathbf{\overline{e}} \text{ } \mathbf{y}/\mathbf{p} \mathbf{U}\_0 \text{ } \mathbf{P}$$

The dimensionless parameters, appearing in Eqs. (1)-(4) are Reynolds number *Re= U0W/ν* , the Prandtl number *Pr=ν/α*, the Grashof number 3 2 *g TL Gr* β ν <sup>∇</sup> <sup>=</sup> . The ratio of *Gr/Re2* is the mixed convection parameter and is called Richardson number *Ri* and is a measure of the relative strength of the natural convection and forced convection for a particular problem. If *Ri*<<1 the forced convection is dominant while if *Ri*>> 1, then natural convection is dominant. For problems with *Ri*~1 then the natural convection effects are comparable to the forced convection effects.

The boundary conditions for the present problem are specified as follows: Top wall: U=U0 , V=0, θ=0

Bottom wall: U=V=0, θ=1 Right and Left wall: U=V=0,

62 Convection and Conduction Heat Transfer

Fig. 2. Schematic diagram of the physical system considering aiding flow condition

Using the Boussinesq approximation and neglecting the viscous dissipation effect and compressibility effect the dimensionless governing equations for two dimensional laminar

> ∂ ∂ + ∂ ∂

> > *X P*

<sup>∂</sup> <sup>=</sup> <sup>∂</sup>

RePr 1 *Y X Y*

X=x/W, Y=y/W, θ=(TH-TC)/ΔT, ΔT=q"W/k, U=u/U0 , V=v/ U0, P=p/ρUo2 The dimensionless parameters, appearing in Eqs. (1)-(4) are Reynolds number *Re= U0W/ν* ,

mixed convection parameter and is called Richardson number *Ri* and is a measure of the relative strength of the natural convection and forced convection for a particular problem. If *Ri*<<1 the forced convection is dominant while if *Ri*>> 1, then natural convection is dominant. For problems with *Ri*~1 then the natural convection effects are comparable to the

*X*

*Y P*

∂ + ∂ ∂

θ

*V X U* θ

The boundary conditions for the present problem are specified as follows:

*<sup>V</sup> <sup>U</sup>* +⎟

+ ∂ <sup>∂</sup> <sup>=</sup> <sup>−</sup> <sup>∂</sup> ∂ + ∂ ∂

*Y <sup>U</sup> <sup>V</sup>*

*X*

*X*

The dimensionless variables are as follows:

forced convection effects.

Top wall: U=U0 , V=0, θ=0

the Prandtl number *Pr=ν/α*, the Grashof number

*Y <sup>V</sup> <sup>V</sup>*

*Y V*

+ ∂ <sup>∂</sup> <sup>=</sup> <sup>−</sup> <sup>∂</sup> ∂ + ∂ ∂

= 0

Re 1

> 2 2

<sup>+</sup> <sup>⎜</sup> ⎜ ⎝ ⎛ ∂

2 2

θ

*X V*

<sup>+</sup> <sup>⎜</sup> ⎜ ⎝ ⎛ ∂ ∂

*<sup>U</sup>* (1)

⎟ ⎟ ⎠ ⎞

θ

(3)

(4)

<sup>∇</sup> <sup>=</sup> . The ratio of *Gr/Re2* is the

2 2

2 2

⎟ ⎟ ⎠ ⎞

3 2 *g TL Gr* β

ν

2 2

θ

∂ ∂

> ⎟ ⎠ ⎞

*<sup>U</sup> <sup>U</sup>* (2)

*Y U*

<sup>+</sup> <sup>⎜</sup> ⎜ ⎝ ⎛ ∂ ∂

*X U*

Re Re 1 *Gr Y V*

∂ ∂

2 2

2

∂ ∂

**3.1 Mathematical model** 

incompressible flows can be written as follows:

∂θ∂ = *X* 0

Non-dimensional heat transfer parameter Nusselt number is stated as:

#### **3.2 Numerical method**

Firstly the problem is defined as a two dimensional enclosure. Control Volume based finite volume method (FVM) is to be used to discretize the governing differential equations. The pressure- velocity coupling in the governing equations is achieved using the well known SIMPLE method for numerical computations. The set of governing equations are to be solved sequentially. A second order upwind differencing scheme is to be used for the formulation of the coefficients in the finite-volume equations. As the sides of the trapezoidal cavity are not parallel, the present numerical techniques will descretize the computational domain into unstructured triangular elements.

In order to obtain the grid independence solution, a grid refinement study is performed for the trapezoidal cavity (A=1) under constant heat flux condition keeping, Re=400, Pr=0.71, Ri= 1.0.n It is found in figure 3 that 5496 regular nodes are sufficient to provide accurate results. This grid resolution is therefore used for all subsequent computations for A≤1. For taller cavities with A>1, a proportionately large number of grids in the vertical direction is used.

Fig. 3. Grid sensitivity test for trapezoidal cavity at Ri=1.0, Re=400 and A=1

The convergence criterion was defined by the required scaled residuals to decrease 10⎯5 for all equations except the energy equations, for which the criterion is 10⎯8.

The computational procedure is validated against the numerical results of Iwatsu et el.[22] for a top heated moving lid and bottom cooled square cavity filled with air (*Pr*=0.71). A 60×60 mesh is used and computations are done for six different *Re* and *Gr* combinations. Comparisons of the average Nusselt number at the hot lid are shown in Table 1. The general

Analysis of Mixed Convection in a Lid Driven Trapezoidal Cavity 65

In this study the effect of inclination angle of the adiabatic sidewalls has been observed first. The inclination angle of the side walls has been changed to 30º, 45 º and 60º. The Richardson number has been changed from 0.1 to 10. The optimum inclination angle has been selected based on the average Nusselt number which is a non dimensional parameter that indicates the rate of heat transfer between the hot and cold walls. The results are obtained both for

Figure 5-7 reveals the impact of varying inclination angles of the sidewalls of the trapezoidal cavity. These figures show the contours of streamlines and isotherms at different Richardson numbers. For small values of *Ri* number, it can be seen that the shear effect due to the movement of the top wall is dominant. The fluid flow is characterized by a primary recirculating eddy of the size of the cavity generated by the movement of the top lid. The isothermal contour maps are clustered near the bottom and top walls resulting in steep temperature gradient there. In each case as the Richardson number increases the convection current becomes more dominant resulting in stronger flow field. Again at *γ*=45º (figure 6), the flow field is stronger than the *γ*=30º and *γ*=60º (figure 5 and figure 7), which is an indication of better heat transfer. The isothermal plots also complies with the flow field,

From figure 8-9 the average value of the Nusselt number with respect to the Richardson number has been plotted. Here it can be seen that Nusselt number at *γ*=45º dominates the other two cases i.e. *γ*=30º and *γ*=60º, showing better heat transfer. So it is clearly visible that trapezoid having the inclination angle *γ*=45º gives better heat transfer and consequently it

Next the effect of rotational angle, Φ has been studied. When studying the effect of rotational angel, two distinct cases have been taken into consideration. They are aiding and opposing flow condition. The first one is when the shear driven flow opposes the convective flow and in that case the top moving lid is moving in the positive direction at a specified rotational angle [figure 1]. The second condition is the aiding flow condition where the shear driven flow aids the natural convective flow and the moving top lid moves in the opposite direction unlike the first case [figure 2]. Both these cases have been studied for a rotational angle for Φ=30º, 45º and 60º and their heat transfer characteristics has been

Unlike Φ=0º, when the buoyancy is acting only in the y direction, as the rotational angle Φ changes, the flow field changes significantly. In opposing flow condition the shear driven flow opposes the natural convective flow, At low Richardson number (*Ri*<1)the forced convection is dominating, creating a single circulation at the right corner of the top moving

As the Richardson number increases (*Ri*>1), natural convection becomes dominating creating a large circulation at the bottom of the cavity. This large circulation causing by natural convection goes bigger and stronger as *Ri* number increases as well as squeezes the upper circulation, resulting an opposing effect. If we observe the isothermal plots, it changes accordingly with streamlines. As *Ri* number increases, the isothermal lines changes significantly indicating that the convection is the dominating heat transfer for the specified case. The shear driven circulation at the upper right side becomes smaller and smaller as the

showing minimum value of the maximum isotherms at *γ*=45º.

can be taken as the optimum inclination angle.

studied in terms of streamlines and isothermal plots.

*Ri* number increases because of dominating natural convection.

**4.2 Effect of rotational angle, Φ**

lid [figure 10-12].

**4.1 Effect of inclination angle** 

*Re*=400 and *Re*=600.


agreement between the present computation and that of Iwatsu et al. [22] is seen to be very well with a maximum discrepancy of about 3.9%.

Table 1. Comparison of the computed average Nusselt number at the hot plate

The computational procedure has also been validated against the numerical results of Guo and Sharif [31] shown in the figure 4.

Figure 4, reveals that the Average Nusselt numbers in the present study have excellent agreement with those obtained by Guo and Sharif [31] having a maximum discrepancy of about 2.3% Therefore, it can be concluded that the numerical code used in this analysis can solve the present problem with reasonable agreement.

#### **4. Results and disscussion**

Numerical results are presented in order to determine the effects of the inclination angle of the side walls, Richardson number *Ri*, Reynolds number *Re*, Aspect ratio *A*, the rotational angle of the cavity *Ф* on mixed convection flow in trapezoidal enclosure. The inclination angle of the sidewalls of the trapezoidal enclosure has been changed from 30º to 60º with an interval of 15º. The values of Richardson number varies from 0.1 to 10, Aspect ratio, *A* changes from 0.5 to 2.0 taking Rotational angle 30º, 45 º, 60º for two different Reynolds numbers 400 and 600.

#### **4.1 Effect of inclination angle**

64 Convection and Conduction Heat Transfer

agreement between the present computation and that of Iwatsu et al. [22] is seen to be very

Present Iwatsu

The computational procedure has also been validated against the numerical results of Guo

Fig. 4. Variation of the average Nusselt number with different aspect ratio at Ri=10, Re=100

**0.5 1 1.5 2**

G. Guo at Ri=10

G. Guo at Ri=0.1

Present computation at Ri=10

Present computation at Ri=0.1

A

Figure 4, reveals that the Average Nusselt numbers in the present study have excellent agreement with those obtained by Guo and Sharif [31] having a maximum discrepancy of about 2.3% Therefore, it can be concluded that the numerical code used in this analysis can

Numerical results are presented in order to determine the effects of the inclination angle of the side walls, Richardson number *Ri*, Reynolds number *Re*, Aspect ratio *A*, the rotational angle of the cavity *Ф* on mixed convection flow in trapezoidal enclosure. The inclination angle of the sidewalls of the trapezoidal enclosure has been changed from 30º to 60º with an interval of 15º. The values of Richardson number varies from 0.1 to 10, Aspect ratio, *A* changes from 0.5 to 2.0 taking Rotational angle 30º, 45 º, 60º for two different Reynolds

solve the present problem with reasonable agreement.

Nu

**4. Results and disscussion** 

numbers 400 and 600.

400 3.97 3.84 3.3 3.75 3.62 3.5 1.18 1.22 3.2 1000 6.25 6.33 1.2 6.32 6.29 0.47 1.70 1.77 3.9

et al.

Diff. % Present Iwatsu

et al.

Diff. %

*Re Gr*= 102 *Gr*= 104 *Gr*= 106

Table 1. Comparison of the computed average Nusselt number at the hot plate

Diff. %

well with a maximum discrepancy of about 3.9%.

et al.

and Sharif [31] shown in the figure 4.

Present Iwatsu

and ε =0.6

In this study the effect of inclination angle of the adiabatic sidewalls has been observed first. The inclination angle of the side walls has been changed to 30º, 45 º and 60º. The Richardson number has been changed from 0.1 to 10. The optimum inclination angle has been selected based on the average Nusselt number which is a non dimensional parameter that indicates the rate of heat transfer between the hot and cold walls. The results are obtained both for *Re*=400 and *Re*=600.

Figure 5-7 reveals the impact of varying inclination angles of the sidewalls of the trapezoidal cavity. These figures show the contours of streamlines and isotherms at different Richardson numbers. For small values of *Ri* number, it can be seen that the shear effect due to the movement of the top wall is dominant. The fluid flow is characterized by a primary recirculating eddy of the size of the cavity generated by the movement of the top lid. The isothermal contour maps are clustered near the bottom and top walls resulting in steep temperature gradient there. In each case as the Richardson number increases the convection current becomes more dominant resulting in stronger flow field. Again at *γ*=45º (figure 6), the flow field is stronger than the *γ*=30º and *γ*=60º (figure 5 and figure 7), which is an indication of better heat transfer. The isothermal plots also complies with the flow field, showing minimum value of the maximum isotherms at *γ*=45º.

From figure 8-9 the average value of the Nusselt number with respect to the Richardson number has been plotted. Here it can be seen that Nusselt number at *γ*=45º dominates the other two cases i.e. *γ*=30º and *γ*=60º, showing better heat transfer. So it is clearly visible that trapezoid having the inclination angle *γ*=45º gives better heat transfer and consequently it can be taken as the optimum inclination angle.

#### **4.2 Effect of rotational angle, Φ**

Next the effect of rotational angle, Φ has been studied. When studying the effect of rotational angel, two distinct cases have been taken into consideration. They are aiding and opposing flow condition. The first one is when the shear driven flow opposes the convective flow and in that case the top moving lid is moving in the positive direction at a specified rotational angle [figure 1]. The second condition is the aiding flow condition where the shear driven flow aids the natural convective flow and the moving top lid moves in the opposite direction unlike the first case [figure 2]. Both these cases have been studied for a rotational angle for Φ=30º, 45º and 60º and their heat transfer characteristics has been studied in terms of streamlines and isothermal plots.

Unlike Φ=0º, when the buoyancy is acting only in the y direction, as the rotational angle Φ changes, the flow field changes significantly. In opposing flow condition the shear driven flow opposes the natural convective flow, At low Richardson number (*Ri*<1)the forced convection is dominating, creating a single circulation at the right corner of the top moving lid [figure 10-12].

As the Richardson number increases (*Ri*>1), natural convection becomes dominating creating a large circulation at the bottom of the cavity. This large circulation causing by natural convection goes bigger and stronger as *Ri* number increases as well as squeezes the upper circulation, resulting an opposing effect. If we observe the isothermal plots, it changes accordingly with streamlines. As *Ri* number increases, the isothermal lines changes significantly indicating that the convection is the dominating heat transfer for the specified case. The shear driven circulation at the upper right side becomes smaller and smaller as the *Ri* number increases because of dominating natural convection.

Analysis of Mixed Convection in a Lid Driven Trapezoidal Cavity 67

 *Ri* = 0.1

 *Ri* = 5

 *Ri* = 10

Fig. 6. Contours of streamlines and isotherms at Re=400, A=1.0 and γ=45°

*Ri* = 10

Fig. 5. Contours of streamlines and isotherms at Re=400, A=1.0 and γ=30°

 *Ri* = 0.1

> *Ri* = 1

> *Ri* = 5

 *Ri* = 10

Fig. 5. Contours of streamlines and isotherms at Re=400, A=1.0 and γ=30°

*Ri* = 10

Fig. 6. Contours of streamlines and isotherms at Re=400, A=1.0 and γ=45°

Analysis of Mixed Convection in a Lid Driven Trapezoidal Cavity 69

Re=400

0 2 4 6 8 10

Re=600

0 2 4 6 8 10

Ri

Fig. 9. Average Nusselt number, Nuav vs Richardson number at Re=600, A=1

**γ=30º**

**γ=45º**

**γ=60º**

γ = 30º

γ = 45º

γ = 60º

Ri

Fig. 8. Average Nusselt number, Nuav vs Richardson number at Re=400, A=1

1 2

1

3

5

7

Nuav

9

11

13

3

4

5 6

Nuav

7

8

9

10

Fig. 7. Contours of streamlines and isotherms at Re=400, A=1.0 and γ=60°

 *Ri* = 0.1

> *Ri* = 1

> *Ri* = 5

 *Ri* = 10

Fig. 7. Contours of streamlines and isotherms at Re=400, A=1.0 and γ=60°

Fig. 8. Average Nusselt number, Nuav vs Richardson number at Re=400, A=1

Fig. 9. Average Nusselt number, Nuav vs Richardson number at Re=600, A=1

Analysis of Mixed Convection in a Lid Driven Trapezoidal Cavity 71

 *Ri* = 0.1

> *Ri* = 1

> *Ri* = 5

 *Ri* = 10 Fig. 11. Contours of streamlines and isotherms at Re=400, A=1 and Ф=30°, opposing flow

*Ri* = 10

Fig. 10. Contours of streamlines and isotherms at Re=400, A=0.5 and Ф=30°, opposing flow

 *Ri* = 0.1

> *Ri* = 1

> *Ri* = 5

 *Ri* = 10 Fig. 10. Contours of streamlines and isotherms at Re=400, A=0.5 and Ф=30°, opposing flow

 *Ri* = 10

Fig. 11. Contours of streamlines and isotherms at Re=400, A=1 and Ф=30°, opposing flow

Analysis of Mixed Convection in a Lid Driven Trapezoidal Cavity 73

 *Ri* = 0.1

> *Ri* = 1

> *Ri* = 5

 *Ri* = 10

Fig. 13. Contours of streamlines and isotherms at Re=400, A=1 and Ф=45°, aiding flow

Fig. 12. Contours of streamlines and isotherms at Re=400, A=1.5 and Ф=30°, opposing flow

 *Ri* = 1

 *Ri* = 1

 *Ri* = 1

 *Ri* = 1 Fig. 12. Contours of streamlines and isotherms at Re=400, A=1.5 and Ф=30°, opposing flow

 *Ri* = 10

Fig. 13. Contours of streamlines and isotherms at Re=400, A=1 and Ф=45°, aiding flow

Analysis of Mixed Convection in a Lid Driven Trapezoidal Cavity 75

**Φ=30º (Op) Φ=45º (Op) Φ=60º (Op) Φ=30º (Aid) Φ=45º (Aid) Φ=60º (Aid)** 

Φ=30º (Op)

Φ=45º (Op)

Φ=60º (Op)

Φ=30º (Aid)

Φ=45º (Aid)

Φ=60º (Aid)

Re=400, A=1

0 2 4 6 8 10

Re=400, A=1.5

Ri

0 5 10

Ri

2

Fig. 16. Variation of Nuav with Ri at A=1.5, Re=400

3

4

5

6

Nuav

7

8

9

10

11

Fig. 15. Variation of Nuav with Ri at A=1, Re=400

Nuav

In the case of aiding flow, condition when the forced convection aids the natural convection a different scenario has been observed [e.g. figure 13]. In all the cases, a single circulation of the size of the cavity has been observed. Unlike the opposing flow condition, in that case the natural convection aids the shear driven flow from the smaller value of *Ri* number, resulting a much stronger convective current. As the *Ri* number increases, the convection flow fields become more and more stronger resulting better and better heat transfer. The isotherms changes significantly as the Richardson number increases and gives the minimum value at higher *Ri* number.

As the aspect ratio, *A* increases the convective flow fields become more and more stronger. As cavity volume increases with aspect ratio and more volume of cooling air is involved in cooling the heat source leading to better cooling effect. The effect of aspect ratio at different rotational angle has also been studied. In all the cases, it can be observed that *Nuav* increases with increasing aspect ratio for all rotational angles, leading to better heat transfer.

Fig. 14-18 shows a comparative analysis of aiding and opposing flow conditions. There it can be seen that, the aiding flow condition always dominates the opposing flow condition in terms of *Nuav* , which indicates better heat transfer at all rotational angle. The aiding flow condition provides stronger convective currents, which has been visible in the study, as the natural convection aids the shear driven flow. As a result the maximum value of the isotherms is lower in case of aiding flow condition, indicating lower temperature. But in opposing condition the natural convection opposes the shear driven flow, providing weak convective currents.

Fig. 14. Variation of Nuav with Ri at A=0.5, Re=400

In the case of aiding flow, condition when the forced convection aids the natural convection a different scenario has been observed [e.g. figure 13]. In all the cases, a single circulation of the size of the cavity has been observed. Unlike the opposing flow condition, in that case the natural convection aids the shear driven flow from the smaller value of *Ri* number, resulting a much stronger convective current. As the *Ri* number increases, the convection flow fields become more and more stronger resulting better and better heat transfer. The isotherms changes significantly as the Richardson number increases and gives the minimum value at

As the aspect ratio, *A* increases the convective flow fields become more and more stronger. As cavity volume increases with aspect ratio and more volume of cooling air is involved in cooling the heat source leading to better cooling effect. The effect of aspect ratio at different rotational angle has also been studied. In all the cases, it can be observed that *Nuav* increases

Fig. 14-18 shows a comparative analysis of aiding and opposing flow conditions. There it can be seen that, the aiding flow condition always dominates the opposing flow condition in terms of *Nuav* , which indicates better heat transfer at all rotational angle. The aiding flow condition provides stronger convective currents, which has been visible in the study, as the natural convection aids the shear driven flow. As a result the maximum value of the isotherms is lower in case of aiding flow condition, indicating lower temperature. But in opposing condition the natural convection opposes the shear driven flow, providing weak

> **Φ=30º (Op) Φ=45º (Op) Φ=60º (Op) Φ=30º (Aid) Φ=45º (Aid) Φ=60º (Aid)**

with increasing aspect ratio for all rotational angles, leading to better heat transfer.

Re=400, A=0.5

0 2 4 6 8 10

Ri

Fig. 14. Variation of Nuav with Ri at A=0.5, Re=400

higher *Ri* number.

convective currents.

1

2

3

4

5

Nuav

6

7

8

9

10

Fig. 15. Variation of Nuav with Ri at A=1, Re=400

Fig. 16. Variation of Nuav with Ri at A=1.5, Re=400

Analysis of Mixed Convection in a Lid Driven Trapezoidal Cavity 77

The *Nuav* is also sensitive to rotational angle, figure 14-18. At Re=400 it can be seen that, Nusselt number decreases as the rotational angle, *Φ* increases. *Nuav* increases marginally at Φ=30º from Φ=45º but at *Φ*=60º, *Nuav* drops significantly for all the aspect ratios, figure 14- 17.The flow fields also changes accordingly. At *Re*=600, the maximum heat transfer has been obtained at *Φ*=45º, in terms of average Nusselt number, figure 18. *Nuav* increases marginally at Φ=45º from Φ=30º but drops significantly at Φ=60º, indicating poor heat transfer at Φ=60º.

The value of the Richardson number, *Ri=Gr/Re2* provides a measure of the importance of buoyancy driven natural convection relative to the lid driven forced convection. When the Buoyancy effects are relatively small, *Ri*<1, the gross flow features are similar to those of a conventional non-stratified fluid at comparable values of Re. The main circulation fills the entire cavity of the size of the cavity generated by the movement of the top wall. Minor cells may be visible near the bottom corners. The streamlines and isotherms indicated that the hydrodynamic and thermal boundary layers are not developed fully at low Richardson number. The isothermal lines are mostly undistorted and horizontal lines except the large recirculation area inside the cavity at low Richardson number. In the large recirculation zone temperature gradients are very weak. This implies that, due to the vigorous actions of the mechanically driven circulations, fluids are well mixed; consequently, temperature

When *Ri*>1, natural convection begins to dominate the forced convection. The Buoyancy assists the core flow and thus the convection current becomes more and more strong with increasing Richardson number. As Richardson number increases, the main circulation occupies the whole cavity and it become more symmetrical inside the cavity. If we see the isothermal plots, we can see that as the Richardson number increases the isothermal lines becomes more and more denser at the upper cold lid. The crowded streamlines and isothermal lines indicate that the hydrodynamic and thermal boundary layers have been developed along the hot wall and cold wall, respectively, reflecting rigorous heat transfer rate occurred.

Consequently the maximum temperature reduces due to this large heat transfer rate.

The average Nusselt number as a function of Richardson number has been plotted in figure 19-20 for different Reynolds number. It can be observed that as the Richardson number increases the average Nusselt number increases accordingly for all the aspect ratios. When *Ri*<1, *Nuav* grows only slightly with increasing *Ri*. After *Ri* is more than 1, *Nuav* is found to increase more rapidly. Since *Re* is kept constant the forced convection effect remains invariant as *Ri* increases for a particular case. When *Ri*>1, the natural convection aids more and more in the heat transfer process in addition to the forced convection which results in

Changing the aspect ratio, A (A=H/W) causes a change in heat transfer characteristics. In order to investigate the convection heat transfer at different aspect ratios, computations has been done for cavities at aspect ratios of 0.5, 1, 1.5 and 2.0. Keeping Reynolds number fixed at 400 and 600 the Richardson number has been changed from 0.1 to 10. If we compare the flow fields at different aspect ratios from 0.5 to 2.0, it can be revealed that in the convection region adjacent to the heat source, the isotherms become thinner and denser producing higher temperature gradients with increasing aspect ratio. The streamlines becomes stronger as the aspect ratio increases. This is due to the fact that the cavity volume increases with

**4.3 Effect of Richardson number,** *Ri* 

more rapid increase of *Nuav.*

**4.4 Effect of aspect ratio,** *A*

differences in much of this interior region are very small.

Fig. 17. Variation of Nuav with Ri at A=2, Re=400

Fig. 18. Variation of Nuav with Ri at A=1, Re=600

The *Nuav* is also sensitive to rotational angle, figure 14-18. At Re=400 it can be seen that, Nusselt number decreases as the rotational angle, *Φ* increases. *Nuav* increases marginally at Φ=30º from Φ=45º but at *Φ*=60º, *Nuav* drops significantly for all the aspect ratios, figure 14- 17.The flow fields also changes accordingly. At *Re*=600, the maximum heat transfer has been obtained at *Φ*=45º, in terms of average Nusselt number, figure 18. *Nuav* increases marginally at Φ=45º from Φ=30º but drops significantly at Φ=60º, indicating poor heat transfer at Φ=60º.

#### **4.3 Effect of Richardson number,** *Ri*

76 Convection and Conduction Heat Transfer

Φ=30º (Op)

Φ=45º (Op)

Φ=60º (Op)

Φ=30º (Aid)

Φ=45º (Aid)

Φ=60º (Aid)

Φ=30º (Op)

Φ=45º (Op)

Φ=60º (Op)

Φ=30º (Aid)

Φ=45º (Aid)

Φ=60º (Aid)

Re=400, A=2

0 5 10

Re=600, A=1

0 5 10

Ri

Ri

2

3

Fig. 18. Variation of Nuav with Ri at A=1, Re=600

4

5

6

7

8

Nuav

9

10

11

12

13

Fig. 17. Variation of Nuav with Ri at A=2, Re=400

4

6

8

Nuav

10

12

14

The value of the Richardson number, *Ri=Gr/Re2* provides a measure of the importance of buoyancy driven natural convection relative to the lid driven forced convection. When the Buoyancy effects are relatively small, *Ri*<1, the gross flow features are similar to those of a conventional non-stratified fluid at comparable values of Re. The main circulation fills the entire cavity of the size of the cavity generated by the movement of the top wall. Minor cells may be visible near the bottom corners. The streamlines and isotherms indicated that the hydrodynamic and thermal boundary layers are not developed fully at low Richardson number. The isothermal lines are mostly undistorted and horizontal lines except the large recirculation area inside the cavity at low Richardson number. In the large recirculation zone temperature gradients are very weak. This implies that, due to the vigorous actions of the mechanically driven circulations, fluids are well mixed; consequently, temperature differences in much of this interior region are very small.

When *Ri*>1, natural convection begins to dominate the forced convection. The Buoyancy assists the core flow and thus the convection current becomes more and more strong with increasing Richardson number. As Richardson number increases, the main circulation occupies the whole cavity and it become more symmetrical inside the cavity. If we see the isothermal plots, we can see that as the Richardson number increases the isothermal lines becomes more and more denser at the upper cold lid. The crowded streamlines and isothermal lines indicate that the hydrodynamic and thermal boundary layers have been developed along the hot wall and cold wall, respectively, reflecting rigorous heat transfer rate occurred. Consequently the maximum temperature reduces due to this large heat transfer rate.

The average Nusselt number as a function of Richardson number has been plotted in figure 19-20 for different Reynolds number. It can be observed that as the Richardson number increases the average Nusselt number increases accordingly for all the aspect ratios. When *Ri*<1, *Nuav* grows only slightly with increasing *Ri*. After *Ri* is more than 1, *Nuav* is found to increase more rapidly. Since *Re* is kept constant the forced convection effect remains invariant as *Ri* increases for a particular case. When *Ri*>1, the natural convection aids more and more in the heat transfer process in addition to the forced convection which results in more rapid increase of *Nuav.*

#### **4.4 Effect of aspect ratio,** *A*

Changing the aspect ratio, A (A=H/W) causes a change in heat transfer characteristics. In order to investigate the convection heat transfer at different aspect ratios, computations has been done for cavities at aspect ratios of 0.5, 1, 1.5 and 2.0. Keeping Reynolds number fixed at 400 and 600 the Richardson number has been changed from 0.1 to 10. If we compare the flow fields at different aspect ratios from 0.5 to 2.0, it can be revealed that in the convection region adjacent to the heat source, the isotherms become thinner and denser producing higher temperature gradients with increasing aspect ratio. The streamlines becomes stronger as the aspect ratio increases. This is due to the fact that the cavity volume increases with

Analysis of Mixed Convection in a Lid Driven Trapezoidal Cavity 79

**Re=400**

0.5 1 1.5 2 **Aspect Ratio,** *A*

**Re=600**

**0.5 1 1.5 2 Aspect Ratio,** *A*

The average Nusselt number at the heat source surface has been plotted in figure 21-22 for a range of *Ri* and aspect ratios. For a particular aspect ratio, the *Nuav* increases with increasing

**Ri=0.1 Ri=0.2 Ri=0.5 Ri=1 Ri=2 Ri=5 Ri=10**

> **Ri=0.1 Ri=0.2 Ri=0.5 Ri=1 Ri=2 Ri=5 Ri=10**

0

**0**

Fig. 22. Variation of Nuav with A at Re=600 and Ф=0º

**2**

**4**

**6**

**8**

**Nuav**

**10**

**12**

**14**

Fig. 21. Variation of Nuav with A at Re=400 and Ф=0º

2

4

6

**Nuav**

8

10

12

aspect ratio and more volume of cooling air is involved in cooling the heat source leading to better cooling effect.

Fig. 19. Variation of Nuav with Ri at Re=400 and Ф=0º

Fig. 20. Variation of Nuav with Ri at Re=600 and Ф=0º

aspect ratio and more volume of cooling air is involved in cooling the heat source leading to

RE=400

0 2 4 6 8 10

Ri

Re=600

0 2 4 6 8 10

Ri

A=2 A=1.5 A=1 A=0.5

> A=2 1.5 A=1 A=0.5

better cooling effect.

0

2

Fig. 20. Variation of Nuav with Ri at Re=600 and Ф=0º

4

6

8

Nuav

10

12

Fig. 19. Variation of Nuav with Ri at Re=400 and Ф=0º

2

4

6

Nuav

8

10

12

Fig. 21. Variation of Nuav with A at Re=400 and Ф=0º

Fig. 22. Variation of Nuav with A at Re=600 and Ф=0º

The average Nusselt number at the heat source surface has been plotted in figure 21-22 for a range of *Ri* and aspect ratios. For a particular aspect ratio, the *Nuav* increases with increasing

Analysis of Mixed Convection in a Lid Driven Trapezoidal Cavity 81

1. Numerical investigation can be carried out by incorporating different physics like

2. Double diffusive natural convection can be analyzed through including the governing

3. Investigation can be performed by using magnetic fluid or electrically conducting fluid within the trapezoidal cavity and changing the boundary conditions of the cavity's wall. 4. Investigation can be performed by moving the other lids of the enclosure and see the

5. Investigation can be carried out by changing the Prandtl number of the fluid inside the

6. Investigation can be carried out by using a porous media inside the trapezoidal cavity

[1] H. Benard, "Fouration de centers de gyration a L'arriere d'cen obstacle en movement",

[2] L. Rayleigh, "On convection currents in a horizontal layer of fluid when the higher temperature is on the underside", Philos. Mag., vol. 6, no. 32, pp. 529-546, 1916. [3] H. Jeffreys, "Some cases of instabilities in fluid motion", Proc. R. Soc. Ser.A, vol. 118, pp.

[4] F.P. Incropera, Convection heat transfer in electronic equipment cooling, J.Heat Transfer

[5] C. K. Cha and Y. Jaluria, Recirculating mixed convection flow for energy extraction, Int. j.

[6] J. Imberger'and P. F. Hamblin, Dynamics of lakes, reservoirs, and cooling ponds, A. Rev.

[7] F. J. K. Ideriah, Prediction of turbulent cavity flow driven by buoyancy and shear, J.

[8] L. A. B. Pilkington, Review lecture: The float glass process, Proc. R. Sot. Lond., IA 314, 1-

[9] K. Torrance, R. Davis, K. Eike, P. Gill, D. Gutman, A. Hsui, S. Lyons, H. Zien, Cavity flows driven by buoyancy and shear, J. Fluid Mech. 51 (1972) 221–231. [10] E. Papanicolaou, Y. Jaluria, Mixed convection from and isolated heat source in a rectangular enclosure, Numer. Heat Transfer, Part A 18 (1990) 427-461 [11] E. Papanicolaou, Y. Jaluria, Transition to a periodic regime in mixed convection in a

[12] E. Papanicolaou, Y. Jaluria, Mixed convection from a localized heat source in a cavity

[13] E. Papanicolaou, Y. Jaluria, Mixed convection from simulated electronic components at varying relative positions in a cavity J. Heat Transfer, 116 (1994) 960-970 [14] J. R. Kosef and R. L. Street, The Lid-Driven Cavity Flow: A Synthesis of Quantitative and Qualitative Observations, ASME J. Fluids Eng., 106(1984) 390-398. [15] K. Khanafer and A. J. Chamkha, Mixed convection flow in a lid-driven enclosure filled

with conducting walls: A numerical study, Numer. Heat Transfer, Part A 23 (1993)

with a fluid saturated porous medium, Int. J. Heat Mass Transfer, 36 (1993) 1601-

radiation effects, internal heat generation/ absorption, capillary effects.

equation of concentration conservation.

Compt. Rend, vol. 147, pp. 416-418, 1900.

Heat Mass Transfer 27.1801-1810 11984).

FIuid Mech. 14, 153-187 (1982).

Mech. Engng Sci. 22, 287-295 (1980).

square cavity, J. Fluid Mech. 239 (1992) 489-509

heat transfer effect.

instead of air.

**7. References** 

trapezoidal enclosure.

195-208, 1928.

25 (1969).

463-484

1608.

110 (1988) 1097–1111.

*Ri*. As a result, the maximum temperature decreases monotonously which can be recognized from the isothermal plots. As the aspect ratio increases from 0.5 to 1 the *Nuav* increases for a particular *Ri*.

At higher Reynolds number i.e. Re=600, with increasing aspect ratio some secondary eddy at the bottom surface of the cavity has been observed. This is of frictional losses and stagnation pressure. As the *Ri* increases, natural convection dominates more and the bottom secondary eddies blends into the main primary flow. For A>1.5 the variation is almost flat indicating that the aspect ratio does not play a dominant role on the heat transfer process at that range.

#### **4.5 Effect of Reynolds number,** *Re*

This study has been done at two different Reynolds numbers. They are Re=400 and Re=600. With a particular case keeping *Ri* and *A* constant, as the Reynolds number increases the convective current becomes more and more stronger and the maximum value of the isotherms reduces. As we know *Ri=Gr/Re2*. *Gr* is square proportional of *Re* for a fixed *Ri*. So slight change of Re and Ri causes huge change of Gr. Gr increases the buoyancy force. As buoyancy force is increased then heat transfer rate is tremendously high. So changes are very visible to the change of *Re*. From figure 19-20, it can be observed that as the *Re* increases the average Nusselt number also increases for all the aspect ratios.

#### **5. Conclusion**

Two dimensional steady, mixed convection heat transfer in a two-dimensional trapezoidal cavity with constant heat flux from heated bottom wall while the isothermal moving top wall in the horizontal direction has been studied numerically for a range of Richardson number, Aspect ratio, the inclination angle of the side walls and the rotational angle of the cavity. A number of conclusions can be drawn form the investigations:


#### **6. Further recommandations**

The following recommendation can be put forward for the further work on this present research.


#### **7. References**

80 Convection and Conduction Heat Transfer

*Ri*. As a result, the maximum temperature decreases monotonously which can be recognized from the isothermal plots. As the aspect ratio increases from 0.5 to 1 the *Nuav* increases for a

At higher Reynolds number i.e. Re=600, with increasing aspect ratio some secondary eddy at the bottom surface of the cavity has been observed. This is of frictional losses and stagnation pressure. As the *Ri* increases, natural convection dominates more and the bottom secondary eddies blends into the main primary flow. For A>1.5 the variation is almost flat indicating that the aspect ratio does not play a dominant role on the heat transfer process at

This study has been done at two different Reynolds numbers. They are Re=400 and Re=600. With a particular case keeping *Ri* and *A* constant, as the Reynolds number increases the convective current becomes more and more stronger and the maximum value of the isotherms reduces. As we know *Ri=Gr/Re2*. *Gr* is square proportional of *Re* for a fixed *Ri*. So slight change of Re and Ri causes huge change of Gr. Gr increases the buoyancy force. As buoyancy force is increased then heat transfer rate is tremendously high. So changes are very visible to the change of *Re*. From figure 19-20, it can be observed that as the *Re* increases the average

Two dimensional steady, mixed convection heat transfer in a two-dimensional trapezoidal cavity with constant heat flux from heated bottom wall while the isothermal moving top wall in the horizontal direction has been studied numerically for a range of Richardson number, Aspect ratio, the inclination angle of the side walls and the rotational angle of the

• The optimum configuration of the trapezoidal enclosure has been obtained at γ=45º, as

• As the Richardson number increases the Nuav increases accordingly at all Aspect ratios, because at higher Richardson number natural convection dominates the forced

• As Aspect Ratio increases from 0.5 to 2.0, the heat transfer rate increases. This is due to the fact that the cavity volume increases with aspect ratio and more volume of cooling

• The direction of the motion of the lid also affects the heat transfer phenomena. Aiding flow condition always gives better heat transfer rate than opposing flow condition. Because at aiding flow condition, the shear driven flow aids the natural convective flow, resulting a much stronger convective current that leads to better heat transfer. • The *Nuav* is also sensitive to rotational angle Ф. At Re=400 it can be seen that, Nusselt number decreases as the rotational angle, *Φ* increases. *Nuav* increases marginally at

Φ=30 from Φ=45º but at *Φ*=60º, *Nuav* drops significantly for all the aspect ratios.

The following recommendation can be put forward for the further work on this present

particular *Ri*.

that range.

**5. Conclusion** 

convection.

**6. Further recommandations** 

research.

**4.5 Effect of Reynolds number,** *Re*

Nusselt number also increases for all the aspect ratios.

cavity. A number of conclusions can be drawn form the investigations:

at this configuration the Nuav was maximum at all Richardson number.

air is involved in cooling the heat source leading to better cooling effect.


**4** 

*France* 

**Convective Heat Transfer of Unsteady** 

J. Batina1, S. Blancher1, C. Amrouche2, M. Batchi2 and R. Creff1 *1Laboratoire des Sciences de l'Ingénieur Appliquées à la Mécanique et l'Electricité Université de Pau et des Pays de l'Adour, Avenue de l'Université – 64000 Pau;* 

*Université de Pau et des Pays de l'Adour, Avenue de l'Université – 64000 Pau;* 

In many industrial engineering and other technological processes, it is crucial to characterise

Particularly, in the inlet region of internal pulsed flows, unsteady dynamic and thermal effects can present large amplitudes. These effects are mainly located in the wall region. This suggests the existence of intense unsteady stresses at the wall (shear, friction or thermal stresses). Our studies (André et al., 1987; Batina, 1995; Creff et al., 1985) show that there could exist an 'adequacy' of different parameters such as Reynolds or Prandtl numbers, leading to large amplitudes for the unsteady velocity and temperature in the entry zone if compared to those encountered downstream in the fully developed region. Consequently, in order to obtain convective heat transfer enhancement, most of the studies are linked to: - Firstly, the search for optimal geometries (undulated or grooved channels, tube with periodic sections, etc.) : among those geometrical studies, one can quote the investigations of Blancher, 1991; Ghaddar et al., 1986, for the wavy or grooved plane geometries, in order to highlight the influence of the forced or natural disturbances on


The main objective of this study is to analyse the special case of convective heat transfer of an unsteady pulsed, laminar, incompressible flow in axisymmetric tubes with periodic sections. The flow is supposed to be developing dynamically and thermally from the duct

One of the originality of this study is the choice of Chebyshev polynomials basis in both axial and radial directions for spectral methods, the use of spectral collocation method and the introduction of a shift operator to satisfy non homogeneous boundary conditions for spectral Galerkin formulation. A comparison of results obtained by the two spectral

heat and mass transfer, for example to avoid thermo mechanical damages.

**1. Introduction** 

heat transfer.

carry on time periodic deformable walls.

inlet. The wall is heated at constant and uniform temperature.

methods is given. A Crank - Nicolson scheme permits the resolution in time.

*2Laboratoire de Mathématiques Appliquées- CNRS UMR 5142* 

**Pulsed Flow in Sinusoidal Constricted Tube** 


### **Convective Heat Transfer of Unsteady Pulsed Flow in Sinusoidal Constricted Tube**

J. Batina1, S. Blancher1, C. Amrouche2, M. Batchi2 and R. Creff1 *1Laboratoire des Sciences de l'Ingénieur Appliquées à la Mécanique et l'Electricité Université de Pau et des Pays de l'Adour, Avenue de l'Université – 64000 Pau; 2Laboratoire de Mathématiques Appliquées- CNRS UMR 5142 Université de Pau et des Pays de l'Adour, Avenue de l'Université – 64000 Pau; France* 

#### **1. Introduction**

82 Convection and Conduction Heat Transfer

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[17] H. Asan, L. Namli, The laminar natural convection in a pitched roof of triangular cross-

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[20] A.A.Mohammad, R.Viskanta,Flow structures and heat transfer in a lid-driven cavity

[21] R.B. Mansour, R. Viskanta, Shear-opposed mixed-convection flow heat transfer in a

[22] R. Iwatsu, J.M. Hyun, K. Kuwahara, Mixed convection in a driven cavity with a stable vertical temperature gradient, Int. J. Heat Mass Transfer 36 (1993) 1601–1608. [23] R. Iwatsu, J.M. Hyun, Three-dimensional driven cavity flows with a vertical temperature gradient, Int. J. Heat Mass Transfer 38 (1995) 3319–3328. [24] A. A. Mohammad, R. Viskanta, Flow and heat transfer in a lid-driven cavity filled with

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moving lid on top and cooled from bottom, Applied Thermal Engineering 27 (2007)

with moving isothermal sidewalls and constant flux heat source on the bottom

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sources, Numer. Heat Transfer, Part A 38 (2000) 627–652.

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wall, Int. J. Thermal Sciences 43 (2004) 465–475.

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In many industrial engineering and other technological processes, it is crucial to characterise heat and mass transfer, for example to avoid thermo mechanical damages.

Particularly, in the inlet region of internal pulsed flows, unsteady dynamic and thermal effects can present large amplitudes. These effects are mainly located in the wall region. This suggests the existence of intense unsteady stresses at the wall (shear, friction or thermal stresses). Our studies (André et al., 1987; Batina, 1995; Creff et al., 1985) show that there could exist an 'adequacy' of different parameters such as Reynolds or Prandtl numbers, leading to large amplitudes for the unsteady velocity and temperature in the entry zone if compared to those encountered downstream in the fully developed region. Consequently, in order to obtain convective heat transfer enhancement, most of the studies are linked to:


The main objective of this study is to analyse the special case of convective heat transfer of an unsteady pulsed, laminar, incompressible flow in axisymmetric tubes with periodic sections. The flow is supposed to be developing dynamically and thermally from the duct inlet. The wall is heated at constant and uniform temperature.

One of the originality of this study is the choice of Chebyshev polynomials basis in both axial and radial directions for spectral methods, the use of spectral collocation method and the introduction of a shift operator to satisfy non homogeneous boundary conditions for spectral Galerkin formulation. A comparison of results obtained by the two spectral methods is given. A Crank - Nicolson scheme permits the resolution in time.

Convective Heat Transfer of Unsteady Pulsed Flow in Sinusoidal Constricted Tube 85

the upstream ambient temperature. Physical constants are supposed to be independent of the temperature, which involves that the motion and energy equations are uncoupled.

Navier-Stokes equations in which the incompressibility condition is automatically satisfied. In fact, the essential advantage of this formulation compared to the primitive variables (velocity-pressure formulation) is the reduction of the number of unknown functions and the non-used of the pressure. On the other hand, Navier-Stokes equations become a fourth

ˆˆˆ 11 2 ˆˆ ˆ <sup>1</sup> ˆ ˆ *t rzr rr z z r rz r r*

ω ν

*r z r r* ψψ

⎛ ⎞ ∂∂ ∂ <sup>=</sup> = − + − = −Δ ⎜ ⎟ ∂ ∂ ∂ ⎝ ⎠

 ψ

2 2 2 2

and

⎜ ⎜ ⎝ ⎛

*a*

The present problem is unsteady. This unsteadiness is generated at the initial instant *t=0*, and is sustained during all the time by a source of upstream pulsations. For both steady and

• *Entry*: for the thermal problem, the inlet fluid temperature is equal to the upstream

<sup>0</sup> *u T r r*

• *Wall*: no slip condition is imposed and the wall is heated at constant temperature:

*z z*

• *Exit*: the flow velocity is normal to the exit section and verifies the classical condition:

*T* 1

∂ <sup>∂</sup> <sup>⎟</sup> <sup>=</sup> ⎠

*r T*

2 2

<sup>1</sup> <sup>ˆ</sup> *<sup>r</sup>*

∂ ∂∂ ∂∂ ∂ ∂ ∂ ∂ ⎛ ⎞ <sup>−</sup> + + = + − =Δ ⎜ ⎟ ∂ ∂∂ ∂∂ ∂ ∂ ∂ <sup>∂</sup> ⎝ ⎠

2 2

 ω

> ψ

 ω

*v* (3)

⎟ ⎟ ⎠ ⎞

<sup>∂</sup> <sup>∂</sup> <sup>=</sup> <sup>=</sup> <sup>∂</sup> <sup>∂</sup> . (5)

<sup>∂</sup> <sup>∂</sup> <sup>=</sup> = = <sup>∂</sup> <sup>∂</sup> *<sup>v</sup>* . (6)

∂ ∂ +

*r T*

*v* (4)

 ν ω

ψ

(2)

 ω

 ψ

∂*zr* <sup>∂</sup> −= <sup>1</sup> ψ

> ∂ ∂ +

*z r T*

2 2

2 22

ω,ψ

) for the

(1)

. The vorticity

With the 2D hypothesis, we use the vorticity-stream function formulation (

 ψω

by the relation:

ω

<sup>⎞</sup> <sup>⎜</sup>

*z T u t*

It is important to note that we have only one unknown function, i.e.:

1 *u*

*r r* ∂ψ<sup>=</sup> <sup>∂</sup>

∂ ∂ + ∂ ∂ + ∂ ∂

*r T*

unsteady flow, the following boundary conditions are available for any time 0 *t* ≥ :

order Partial Differential Equations whose expressions in cylindrical coordinates are:

**2.2 Governing equations** 

ω

is linked to

Velocity components are given by:

function

ω

The energy equation is:

**3. Boundary conditions** 

ambient temperature: *T T*= <sup>∞</sup> .

 ψω

ψ

⎝ ⎛

<sup>0</sup> *<sup>v</sup>* <sup>=</sup> and 0 *T u*

• *Axis*: the flow preserves at each time an axial symmetry:

ω


#### **1.1 Nomenclature**

#### **1.2 Suggested keywords**

Convective heat transfer – sinusoidal constricted tube – axisymmetric geometry – pulsed laminar, incompressible flow – spectral collocation method – Chebyshev-Gauss-Lobatto mesh – spectral Galerkin formulation – shift operator method – Crank - Nicolson resolution in time.

### **2. General hypothesis and governing equations**

#### **2.1 General hypothesis**

We consider a Newtonian incompressible fluid flow developing inside an axisymmetric cylindrical duct with periodic sinusoidal radius. The unsteadiness imposed to the flow corresponds to a source of periodic pulsations generating plane waves. This flow is described in terms of an unsteady pulsed flow superimposed on a steady one, without reverse flow at the entry and the exit sections. With regard to the thermal problem, the wall is heated at constant and uniform temperature, and the fluid inlet temperature is equal to the upstream ambient temperature. Physical constants are supposed to be independent of the temperature, which involves that the motion and energy equations are uncoupled.

#### **2.2 Governing equations**

84 Convection and Conduction Heat Transfer

λ

θ

θ

μ

ν = μ ρ

ρ

τ

ω

ψ

θ

*<sup>W</sup>* wall heat flux *W m*<sup>2</sup> <sup>⎡</sup> <sup>⎤</sup> <sup>⎣</sup> <sup>⎦</sup> *Subscripts: 0* steady flow; W*:* wall

Convective heat transfer – sinusoidal constricted tube – axisymmetric geometry – pulsed laminar, incompressible flow – spectral collocation method – Chebyshev-Gauss-Lobatto mesh – spectral Galerkin formulation – shift operator method – Crank - Nicolson resolution

We consider a Newtonian incompressible fluid flow developing inside an axisymmetric cylindrical duct with periodic sinusoidal radius. The unsteadiness imposed to the flow corresponds to a source of periodic pulsations generating plane waves. This flow is described in terms of an unsteady pulsed flow superimposed on a steady one, without reverse flow at the entry and the exit sections. With regard to the thermal problem, the wall is heated at constant and uniform temperature, and the fluid inlet temperature is equal to

*u*0 mean bulk velocity [*m s*] *Re* Reynolds number: *Re = Ru*<sup>0</sup> *ν*

*v* radial velocity [*m s*] *Pr* Prandtl number: *Pr a* =

*t* time [*s* ] Ω pulsation [*rad s*]

*z* axial co-ordinate [*m*] *Nu* Nusselt number

**2. General hypothesis and governing equations** 

*u* axial velocity [*m s*] *Dimensionless numbers* 

dimensionless total wavelength

dimensionless temperature:

dynamic viscosity <sup>2</sup> <sup>⎡</sup>*Ns m* <sup>⎤</sup> <sup>⎣</sup> <sup>⎦</sup>

fluid density <sup>3</sup> <sup>⎡</sup>*Kg m* <sup>⎤</sup> <sup>⎣</sup> <sup>⎦</sup>

modulation flow rate

vorticity function [1 *s*]

stream function <sup>3</sup> <sup>⎡</sup>*m s*<sup>⎤</sup> <sup>⎣</sup> <sup>⎦</sup>

0m( ) *x* averaged bulk temperature

kinematic viscosity: <sup>2</sup> <sup>⎡</sup>*m s*<sup>⎤</sup> <sup>⎣</sup> <sup>⎦</sup>

ν

=− − (*TT T T* <sup>∞</sup> ) ( *<sup>W</sup>* <sup>∞</sup> )

**1.1 Nomenclature** 

*<sup>a</sup>*thermal diffusity <sup>2</sup> <sup>⎡</sup>*m s*<sup>⎤</sup> <sup>⎣</sup> <sup>⎦</sup>

H periodic sinusoidal radius [*m*]

*L* geometric half-length tube [*m*]

*R* tube radius at the constriction [*m* ]

e reduced amplitude

*r* radial co-ordinate [*m*]

*T* fluid temperature [*K*]

*Greek symbols*

**1.2 Suggested keywords** 

**2.1 General hypothesis** 

Φ

in time.

*T*∞ duct inlet temperature [*K* ]

*h* wall function

With the 2D hypothesis, we use the vorticity-stream function formulation (ω,ψ ) for the Navier-Stokes equations in which the incompressibility condition is automatically satisfied. In fact, the essential advantage of this formulation compared to the primitive variables (velocity-pressure formulation) is the reduction of the number of unknown functions and the non-used of the pressure. On the other hand, Navier-Stokes equations become a fourth order Partial Differential Equations whose expressions in cylindrical coordinates are:

$$\frac{\partial \hat{\partial} \hat{\partial}}{\partial t} - \frac{1}{r} \frac{\partial \boldsymbol{\nu}}{\partial z} \frac{\partial \hat{\partial}}{\partial r} + \frac{1}{r} \frac{\partial \boldsymbol{\nu}}{\partial r} \frac{\partial \hat{\partial}}{\partial z} + \frac{2}{r^2} \frac{\partial \boldsymbol{\nu}}{\partial z} \hat{\partial} = \nu \left( \frac{\partial^2 \hat{\partial}}{\partial r^2} + \frac{\partial^2 \hat{\partial}}{\partial z^2} - \frac{1}{r} \frac{\partial \hat{\partial}}{\partial r} \right) = \nu \Lambda \hat{\partial} \tag{1}$$

It is important to note that we have only one unknown function, i.e.: ψ . The vorticity function ω is linked to ψby the relation:

$$\hat{\alpha} = r\alpha = -\left(\frac{\hat{\sigma}^2 \mathcal{V}}{\hat{\sigma}r^2} + \frac{\hat{\sigma}^2 \mathcal{V}}{\hat{\sigma}z^2} - \frac{1}{r}\frac{\partial \mathcal{V}}{\partial r}\right) = -\Delta \mathcal{V} \tag{2}$$

Velocity components are given by:

$$u = \frac{1}{r}\frac{\partial \,\nu}{\partial r} \text{ and } \,v = -\frac{1}{r}\frac{\partial \,\nu}{\partial z} \tag{3}$$

The energy equation is:

$$a\left(\frac{\partial T}{\partial t} + \mu \frac{\partial T}{\partial z} + v \frac{\partial T}{\partial r}\right) = a\left(\frac{\partial^2 T}{\partial r^2} + \frac{\partial^2 T}{\partial z^2} + \frac{1}{r} \frac{\partial T}{\partial r}\right) \tag{4}$$

#### **3. Boundary conditions**

The present problem is unsteady. This unsteadiness is generated at the initial instant *t=0*, and is sustained during all the time by a source of upstream pulsations. For both steady and unsteady flow, the following boundary conditions are available for any time 0 *t* ≥ :


$$v = 0 \quad \text{and} \quad \frac{\partial T}{\partial z} = \frac{\partial u}{\partial z} = 0 \,\text{}\,\tag{5}$$

• *Axis*: the flow preserves at each time an axial symmetry:

$$\frac{\partial \mu}{\partial r} = \nu = \frac{\partial T}{\partial r} = 0 \,\,\,\,\tag{6}$$

• *Wall*: no slip condition is imposed and the wall is heated at constant temperature:

Convective Heat Transfer of Unsteady Pulsed Flow in Sinusoidal Constricted Tube 87

( ) 1 1 cos . ( ) 1 1 cos . 1 ( ) ( ) 2 2 *O O ez e Hz R n hx n x L*

Finally, the study domain is transformed into a rectangle 1 1 − ≤ ≤ *x* and 0 1 ≤ ≤

Considering the transformation of variables defined before, the new stream – vorticity

2 12 1 <sup>2</sup>

22 2 2

( ) ( ) ( )

ρρρ

,,,

2 2 2 2 2 2

*g gg*

*Dx Ex Fx x x*

, ; , ; , 6 ;

22 2 2

*ggg*

<sup>⎪</sup> <sup>⎡</sup> ∂∂∂ <sup>⎤</sup> <sup>⎪</sup> <sup>⎢</sup> + + <sup>⎥</sup> <sup>⎨</sup> ∂ ∂ <sup>∂</sup> <sup>⎢</sup> <sup>⎥</sup> ⎪Δ =−⎢ <sup>⎥</sup> <sup>⎪</sup> ∂ ∂ ⎢+ ++ <sup>⎥</sup> <sup>⎪</sup> <sup>⎢</sup> ∂ ∂ <sup>∂</sup> <sup>⎥</sup> <sup>⎩</sup> <sup>⎣</sup> <sup>⎦</sup>

*Ax Bx Cx*

( ) ( ) ( )

ρ

, ,,

ρ

<sup>⎨</sup> ∂ ∂∂ ∂∂ ∂ ∂ ⎛ ⎞⎛ ⎞ ′ <sup>⎪</sup> <sup>+</sup> ⎜ ⎟⎜ ⎟ − + − =Δ

 ψ

> ρ

> > ω

 ω

 ρ

> ρ

λ

ρ

(21)

ρ

ρ

(20)

( )

ρ

(17)

⎧ ⎫ ⎪ ⎪ ⎡ ⎤ ⎛ ⎞ = + − ⇔ =+ − + <sup>⎡</sup> <sup>⎤</sup> ⎨ ⎬ ⎢ ⎥ ⎜ ⎟ <sup>⎣</sup> <sup>⎦</sup> ⎪ ⎪ ⎩ ⎭ ⎣ ⎦ ⎝ ⎠ (15)

 π

Re

 ψ

ρ

ω

( )

 ω

 ρ ω

ρ

ρ

> ρ

*g*

 λ

> ρ ρ

ψ

(18)

(19)

2

λ

ρ

ψ

2

 λ

> ρ ρ

 ω ρ

(16)

π

representing the half - space of the square:[− ×− 1,1 1,1 ] [ ] .

 ψω

ρρ

*x x*

2 2

*h*

 ω

 ρ

**4.1.2 New system of unsteady dynamic governing equations** 

*<sup>h</sup> <sup>h</sup> t xx x h*

 ρ

2 2 2 2 2 2 2 2 2 *<sup>f</sup> h hh h h hh*

 λρ

*x*

ρ

*gg g*

*x x*

 ρ  ρ

*g gg*

ρ ω

ω

ρ

() () ( ) ( )

ρ

<sup>⎪</sup> <sup>=</sup> <sup>=</sup> <sup>+</sup> ′ ′ = −− ′′ <sup>⎪</sup>

*Ax h Bx h C x h hh*

 ρ

2 2 2 2 2 2 2 2 2 *<sup>f</sup> h hh h h hh*

 λρ

The dynamic steady problem corresponding to problem (16) is written as follows:

λ

, 2 ; , 4 ; , 2 3

*D x hh E x hh F x h hh*

<sup>2</sup> Re Re Re = = λ

ψψ

⎧ ⎫ ⎪ ⎪ ∂∂ ∂ ⎡ ⎤ <sup>∂</sup> Δ= − + + + − − ′ ′′ ⎡ ⎤ ′′ ⎨ ⎬ ⎢ ⎥ ⎣ ⎦ ∂ ∂ ∂ ∂ ⎢ ⎥ <sup>∂</sup> ⎪ ⎪ ⎩ ⎭ ⎣ ⎦

<sup>⎪</sup> = − ′ = − ′ ′ = − ′′ ⎪⎩

2 2 2 2 2

( ) ( ) ( ) ( )

ρ

ψψ

⎧ ⎫ ⎪ ⎪ ∂∂ ∂ ⎡ ⎤ <sup>∂</sup> Δ= − + + + − − ′ ′′ ⎡ ⎤ ′′ ⎨ ⎬ ⎢ ⎥ ⎣ ⎦ ∂ ∂ ∂ ∂ ⎢ ⎥ <sup>∂</sup> ⎪ ⎪ ⎩ ⎭ ⎣ ⎦

⎩ ∂ ∂∂ ∂∂ ∂ ∂ ⎝ ⎠⎝ ⎠

 ψω

and (see equation 73)

formulation of this problem is:

ψ

where:

and

with:

2

ω

⎧ = −Δ ⎪

*f*

ψ

*g*

ρ

⎧

⎨

ψ

ρρ

ψ

**4.1.3 The dynamic steady problem formulation** 

ω

ωλ

<sup>⎧</sup> <sup>=</sup> <sup>⎪</sup>

ω

 ψ

$$
\mu = \nu = 0 \quad \text{; } T = T\_W \text{ .} \tag{7}
$$

For dynamic conditions at the entry section, we impose:


$$
\mu\left(z=0,r\right) = 2\overline{u}\_0 \left(1 - \left(\frac{r}{R}\right)^2\right) \tag{8}
$$


$$f(z=0,r,t) = f\_0(z=0,r)\left(1 + \sum\_{n=1}^{N\_F} \tau^n .sin\left(n\Omega t\right)\right) \tag{9}$$

where *f* represents *u* or ψ . At this section, to avoid reverse flow, we impose: 1 τ< .

#### **4. New formulation and resolution of the dynamic and thermal problem**

#### **4.1 New formulation of the dynamic problem**

#### **4.1.1 Dimensionless quantities and variables transformations**

One chooses for dimensionless variables:

$$\tilde{r} = \frac{r}{R} \; ; \; \tilde{z} = \frac{z}{R} \; ; \; \tilde{t} = \frac{t}{\overline{t}\_0} \; ; \; \tilde{\alpha} = \frac{\hat{\alpha}}{\overline{\alpha}\_0} \; ; \; \tilde{\nu} = \frac{\nu}{\overline{\nu}\_0} \; ; \; \tilde{\mu} = \frac{u}{\overline{u}\_0} \; ; \; \tilde{\mathbf{v}} = \frac{v}{u\_o} \tag{10}$$

with

$$\overline{\overline{\mathbf{u}}}\_0 = \frac{\mathbf{L}}{\overline{\mathbf{u}}\_0} \; ; \; \overline{\alpha}\_0 = \frac{\overline{\overline{u}}\_0}{R} \; ; \; \overline{\nu}\_0 = \overline{\mu}\_0 R^2 \tag{11}$$

The Reynolds number *Re* is based on the radius at the duct constriction:

$$\text{Re} = R\overline{u}\_0/\nu \tag{12}$$

In order to obtain a computational square domain permitting the use of two dimensional Chebyshev polynomials, we proceed to a space variables transformation. This one is inspired by Sobey, 1980, and modified by Blancher, 1991. It has been adapted to the axisymmetric geometry used in this study. Afterwards, we note by *H z*( ) the duct periodic radius. Then we define:

$$
\rho = \frac{\tilde{r}}{h(\alpha)} \quad \text{:} \quad \alpha = \frac{\tilde{z}}{\lambda} \quad -1 \tag{13}
$$

with

$$\mathcal{A} = \frac{L}{R};\ h(\mathbf{x}) = \frac{1}{R}H\left[ (\mathbf{x} + \mathbf{1})L \right] \tag{14}$$

and (see equation 73)

86 Convection and Conduction Heat Transfer

<sup>0</sup> 0, 2 1 *<sup>r</sup>*

• *Entry*: the source of imposes a periodic pressure gradient modulation. Then the velocity

ψ

= == + Ω ⎜ ⎟

*f z rt f z r n t*

**4. New formulation and resolution of the dynamic and thermal problem** 

⎛ ⎞ ⎛ ⎞ = = − ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠

• *Entry*: for the dynamic problem, Poiseuille profile boundary condition is chosen

( )

( 0, , ) ( 0, ) 1

**4.1.1 Dimensionless quantities and variables transformations** 

*uz r u*

For dynamic conditions at the entry section, we impose:

axial component and the stream function

ψ

**4.1 New formulation of the dynamic problem** 

One chooses for dimensionless variables:

*- Steady flow (t=0 time step)* 

*- Unsteady flow (t>0)* 

where *f* represents *u* or

radius. Then we define:

with

with

*u* = *v* = 0 ; *T T*= *<sup>W</sup>* . (7)

have a Fourier series expansion in time:

⎝ ⎠ <sup>∑</sup> *<sup>n</sup> <sup>τ</sup> .sin* (9)

*o*

*<sup>0</sup>*=*u R*0 (11)

<sup>−</sup> (13)

Re *= Ru*<sup>0</sup> *ν* (12)

<sup>=</sup> = ⎡+ ⎤ <sup>⎣</sup> <sup>⎦</sup> (14)

(8)

τ< .

(10)

2

*R*

<sup>0</sup> ( ) 1

*n*

=

. At this section, to avoid reverse flow, we impose: 1

0 0 0

*0* ; <sup>2</sup> ψ

 ψ ψ

 ψ

<sup>ˆ</sup> = ; = ; = ; = ; ; ; v=

ω

 = = *0*

ω

0 L = u *t*

The Reynolds number *Re* is based on the radius at the duct constriction:

*<sup>0</sup>* ; <sup>0</sup> <sup>=</sup> *<sup>u</sup>*

( ) <sup>=</sup> *<sup>r</sup> h x* ρ

*R R*

λ

ω

*r zt u v r zt <sup>u</sup> RRt u u* ω

*R*

In order to obtain a computational square domain permitting the use of two dimensional Chebyshev polynomials, we proceed to a space variables transformation. This one is inspired by Sobey, 1980, and modified by Blancher, 1991. It has been adapted to the axisymmetric geometry used in this study. Afterwards, we note by *H z*( ) the duct periodic

> ; = 1 *<sup>z</sup> x* λ

( ) ( ) <sup>1</sup> ; 1 . *<sup>L</sup> hx H x L*

*NF*

⎛ ⎞

$$H(z) = R\left\{1 + \frac{\varepsilon}{2} \left[1 - \cos\left(\pi \pi\_{\rm O} \frac{z}{L}\right)\right] \right\} \Leftrightarrow h(x) = 1 + \frac{\varepsilon}{2} \left[1 - \cos\left(\pi \pi\_{\rm O} (x + 1)\right)\right] \tag{15}$$

Finally, the study domain is transformed into a rectangle 1 1 − ≤ ≤ *x* and 0 1 ≤ ≤ ρ representing the half - space of the square:[− ×− 1,1 1,1 ] [ ] .

#### **4.1.2 New system of unsteady dynamic governing equations**

Considering the transformation of variables defined before, the new stream – vorticity formulation of this problem is:

$$\begin{cases} \bar{o}\bar{o} = -\Lambda\_f \tilde{\nu} \\ \hbar^2 \frac{\partial \bar{o}}{\partial \tilde{t}} + \frac{1}{\rho} \bigg( \frac{\partial \tilde{\nu}}{\partial \rho} \frac{\partial \tilde{o}}{\partial \mathbf{x}} - \frac{\partial \tilde{\nu}}{\partial \mathbf{x}} \frac{\partial \tilde{o}}{\partial \rho} \bigg) + \frac{2}{\rho^2} \bigg( \frac{\partial \tilde{\nu}}{\partial \mathbf{x}} - 2\rho \frac{\hbar'}{\hbar} \frac{\partial \tilde{\nu}}{\partial \rho} \bigg) \bar{o} = \frac{1}{\tilde{\text{Re}}} \Lambda\_g \bar{o} \end{cases} \tag{16}$$

where:

$$\Delta\_f \tilde{\boldsymbol{\varphi}} = \left\{ \hbar^2 \frac{\hat{\boldsymbol{\varphi}}^2 \tilde{\boldsymbol{\varphi}}}{\hat{\boldsymbol{\alpha}}^2} - 2 \, \rho \hbar^\prime \hbar \frac{\hat{\boldsymbol{\alpha}}^2 \tilde{\boldsymbol{\varphi}}}{\hat{\boldsymbol{\alpha}} \hat{\boldsymbol{\alpha}} \hat{\boldsymbol{\varphi}}} + \left[ \hat{\boldsymbol{\lambda}}^2 + \rho^2 \hbar^2 \right] \frac{\hat{\boldsymbol{\alpha}}^2 \tilde{\boldsymbol{\varphi}}}{\hat{\boldsymbol{\alpha}} \rho^2} + \left[ \rho \left( 2 \hbar^2 - \hbar \mathbf{h}^\* \right) - \frac{\hat{\boldsymbol{\lambda}}^2}{\rho} \left| \frac{\partial \tilde{\boldsymbol{\varphi}}}{\partial \rho} \right. \right] \tag{17}$$

and

$$\begin{cases} \bar{\boldsymbol{\rho}} = \lambda^2 \boldsymbol{h}^2 \bar{\boldsymbol{\alpha}} \\\\ \Lambda\_{\boldsymbol{\mathcal{S}}} \bar{\boldsymbol{\rho}} = -\begin{bmatrix} A\_{\mathcal{S}} \left( \mathbf{x}, \boldsymbol{\rho} \right) \frac{\hat{\boldsymbol{\sigma}}^2 \bar{\boldsymbol{\alpha}}}{\partial \mathbf{x}^2} + B\_{\mathcal{S}} \left( \mathbf{x}, \boldsymbol{\rho} \right) \frac{\hat{\boldsymbol{\sigma}}^2 \bar{\boldsymbol{\alpha}}}{\partial \boldsymbol{\rho}^2} + C\_{\mathcal{S}} \left( \mathbf{x}, \boldsymbol{\rho} \right) \frac{\hat{\boldsymbol{\sigma}} \bar{\boldsymbol{\alpha}}}{\partial \boldsymbol{\rho}} \\\\ + D\_{\mathcal{S}} \left( \mathbf{x}, \boldsymbol{\rho} \right) \frac{\hat{\boldsymbol{\sigma}}^2 \bar{\boldsymbol{\alpha}}}{\partial \boldsymbol{\alpha} \partial \boldsymbol{\rho}} + E\_{\mathcal{S}} \left( \mathbf{x}, \boldsymbol{\rho} \right) \frac{\hat{\boldsymbol{\sigma}}^2 \bar{\boldsymbol{\alpha}}}{\partial \mathbf{x}} + F\_{\mathcal{S}} \left( \mathbf{x}, \boldsymbol{\rho} \right) \bar{\boldsymbol{\alpha}} \end{cases} \tag{18}$$

with:

$$\begin{cases} A\_{\mathcal{g}}\left(\mathbf{x},\boldsymbol{\rho}\right) = \boldsymbol{h}^{2} \quad ; \; B\_{\mathcal{g}}\left(\mathbf{x},\boldsymbol{\rho}\right) = \boldsymbol{\lambda}^{2} + \boldsymbol{\rho}^{2}\boldsymbol{h}^{\prime 2} \; ; \; \mathcal{C}\_{\mathcal{g}}\left(\mathbf{x},\boldsymbol{\rho}\right) = \boldsymbol{\rho}\left(6\mathbf{h}^{\prime 2} - \mathbf{h}\mathbf{h}^{\prime}\right) - \frac{\boldsymbol{\lambda}^{2}}{\boldsymbol{\rho}} \; ; \; \mathcal{C}\_{\mathcal{g}}\left(\mathbf{x},\boldsymbol{\rho}\right) = -\frac{\boldsymbol{\lambda}^{2}}{\boldsymbol{\rho}} \; ; \; \mathcal{C}\_{\mathcal{g}}\left(\mathbf{x},\boldsymbol{\rho}\right) = -\frac{\boldsymbol{\lambda}^{2}}{\boldsymbol{\rho}} \; ; \; \mathcal{C}\_{\mathcal{g}}\left(\mathbf{x},\boldsymbol{\rho}\right) = 2\left(3\mathbf{h}^{\prime} - \mathbf{h}\mathbf{h}^{\prime}\right) \end{cases} \tag{19}$$

$$
\hat{\mathbf{Re}} = \tilde{\mathbf{Re}} \mathcal{A}^2 = \mathbf{Re} \mathcal{A} \tag{20}
$$

$$\Delta\_f \tilde{\boldsymbol{\varphi}} = \left\{ h^2 \frac{\partial^2 \tilde{\boldsymbol{\varphi}}}{\partial \mathbf{x}^2} - 2 \rho h' h \frac{\partial^2 \tilde{\boldsymbol{\varphi}}}{\partial \mathbf{x} \partial \boldsymbol{\rho}} + \left[ \boldsymbol{\mathcal{k}}^2 + \rho^2 h'^2 \right] \frac{\partial^2 \tilde{\boldsymbol{\varphi}}}{\partial \boldsymbol{\rho}^2} + \left[ \rho \left( 2h'^2 - hh'' \right) - \frac{\boldsymbol{\mathcal{k}}^2}{\rho} \left| \frac{\partial \tilde{\boldsymbol{\varphi}}}{\partial \boldsymbol{\rho}} \right| \right] \tag{21}$$

#### **4.1.3 The dynamic steady problem formulation**

The dynamic steady problem corresponding to problem (16) is written as follows:

Convective Heat Transfer of Unsteady Pulsed Flow in Sinusoidal Constricted Tube 89

It is necessary to study the influence of the physical parameters such as the Reynolds number to remain in 2D hypothesis. From a numerical point of view we will show the

Because no symmetry condition is imposed at the boundaries of our half-domain of study, we choose basis functions constructed from Chebyshev polynomials (Bernardi & Maday, 1992; Canuto et al., 1988) instead of trigonometric trial functions. Then, *P r <sup>l</sup>*( ) and *Q x <sup>k</sup>* ( ) are written as linear combination of Chebyshev polynomials. Their expressions depend on the

Generally, with Galerkin method, Dirichlet or Neuman boundary conditions imposed to trial functions must be homogeneous, but it is not necessary for collocation method (see

The basis *P r <sup>l</sup>*( ) and *Q x <sup>k</sup>* ( ) are written as a linear combination of Chebyshev polynomials

() () () ( ) ( ) ( ) 1 1

where *n* (respectively *m*) is the number of boundary conditions according to the radial direction *r* (respectively the axial direction *x*), and *T x <sup>k</sup>* ( ) is the Chebyshev polynomial of

Spectral methods are used successfully in many problems of physics, mainly those involving periodic physical phenomena in space and / or in time. Its main advantage is its high degree of accuracy, compared with some methods such as finite differences, finite elements or finite volumes (Bernardi & Maday, 1992; Canuto et al., 1988; Gelfgat, 2004; Shen, 1994, 1995, 1997). Spectral methods are particularly suitable to study instabilities phenomena, selfmaintained or forced, occuring in Computational Fluid Dynamics. However, spectral methods are limited to simple geometries. For complicated study domains, an alternative way may be using spectral finite elements. The second disadvantage of these methods is their cost of implementing and their high CPU calculations. The matrices obtained are

usually full and strategies for solving linear or nonlinear systems remain limited.

**6. Numerical resolution of the dynamic and thermal problem using spectral** 

The steady dynamic problem is given by the equation (22). Generally, this problem is written with classical homogeneous boundary conditions. One of the originalities of this study is the use of a relevment function allowing the introduction of non homogeneous

ϕ

ψψ

0 0

 ϕ

= =

+ +

= +∑ ∑ = + (28)

 β

ψ

( )*r* corresponding to the Poiseuille velocity

(,) (,) () *xr xr r* = + (29)

(,) *x r* is written by

 and *n m l l lli k k kki i i Pr Tr T r Q x T x T x*

α

**5.2.1 Advantages and limitations of spectral methods** 

**6.1 Numerical resolution of the dynamic steady problem** 

mean of the Poiseuille stream function 0

boundary conditions. For this reason, the unknown stream function

boundary conditions and the spectral method used (Galerkin or collocation method).

influence of the polynomials degrees particularly for the thermal problem.

**5.2 The choice of basis functions** 

Galerkin and collocation methods below).

degree *k*.

**galerkin formulation** 

imposed at the duct entry as:

such as (Gelfgat, 2004; Shen, 1994, 1995, 1997):

$$\begin{cases} \ddot{o} = -\Delta\_f \tilde{\nu} \\ \frac{1}{\rho} \left( \frac{\partial \tilde{\nu}}{\partial \rho} \frac{\partial \bar{\alpha}}{\partial \mathbf{x}} - \frac{\partial \tilde{\nu}}{\partial \mathbf{x}} \frac{\partial \bar{\alpha}}{\partial \rho} \right) + \frac{2}{\rho^2} \left( \frac{\partial \tilde{\nu}}{\partial \mathbf{x}} - 2\rho \frac{h'}{h} \frac{\partial \tilde{\nu}}{\partial r} \right) \bar{o} = \frac{1}{\tilde{\mathbf{R}} \mathbf{e}} \Delta\_{\mathcal{g}} \bar{o} \end{cases} \tag{22}$$

*Important:* for reason of convenience, the radius ρwill be noted *r* .

#### **4.2 New formulation of the thermal problem**

For the thermal problem, the temperature θ is made dimensionless in a classic way:

$$\tilde{\theta} = \frac{T - T\_{\infty}}{T\_{\text{W}} - T\_{\infty}} \tag{23}$$

#### **4.2.1 The thermal unsteady problem formulation**

Using (1) and (10)-(15), the dimensionless energy equation can be written as follows:

$$\hbar^2 \frac{\partial \tilde{\theta}}{\partial \tilde{t}} + \hbar^2 \tilde{\mu} \frac{\partial \tilde{\theta}}{\partial \mathbf{x}} + \hbar (\tilde{\mathbf{v}} \lambda - \tilde{u}r\hbar') \frac{\partial \tilde{\theta}}{\partial r} = \frac{1}{\hat{\mathbf{R}} \mathbf{e} \, \mathbf{Pr}} \Lambda\_f \tilde{\theta} \tag{24}$$

with:

$$
\Delta\_f \tilde{\theta} = h^2 \frac{\hat{\sigma}^2 \tilde{\theta}}{\hat{\sigma} \hat{x}^2} - 2r h' h \frac{\hat{\sigma}^2 \tilde{\theta}}{\hat{\sigma} \hat{x} \hat{\sigma} r} + \left[ \hat{\mathcal{L}}^2 + r^2 h'^2 \right] \frac{\hat{\sigma}^2 \tilde{\theta}}{\hat{\sigma} r^2} + \left[ r \left( 2h'^2 - h h'' \right) + \frac{\hat{\mathcal{L}}^2}{r} \right] \frac{\hat{\sigma} \tilde{\theta}}{\hat{\sigma} r} \tag{25}
$$

#### **4.2.2 The thermal steady problem formulation**

The dimensionless steady state energy problem related to the equation (24) is:

$$\hbar^2 \tilde{u} \frac{\partial \bar{\tilde{\theta}}}{\partial \mathbf{x}} + \hbar (\tilde{\mathbf{v}} \mathcal{X} - \tilde{u} r l r') \frac{\partial \bar{\tilde{\theta}}}{\partial r} = \frac{1}{\hat{\mathbf{R}} \mathbf{e} \mathbf{Pr}} \Delta\_f \tilde{\theta} \tag{26}$$

#### **5. Numerical resolution using spectral methods**

#### **5.1 Trial functions and development orders**

The spectral methods consist in projecting any unknown function *f* (*xrt* , , ) on trial functions as follows:

$$f(\mathbf{x}, r, t) = \sum\_{k=0}^{N\_x} \sum\_{l=0}^{N\_r} f\_{kl}(t) P\_l(r) Q\_k(\mathbf{x}) \tag{27}$$

where *Nx* and *Nr* are the development orders according to the axis *x* and *r* respectively. The basis functions *P r <sup>l</sup>*( ) and *Q x <sup>k</sup>* ( ) are generally trigonometric or polynomial functions (Chebyshev, Legendre, etc.) according to different boundary conditions situations. The time dependant coefficients *fkl*(*t*) are the unknowns of the problem. For our problem, the function *f* representsω , ψ orθ . For a steady problem, the coefficients *fkl*( )*t* are time independent.

It is necessary to study the influence of the physical parameters such as the Reynolds number to remain in 2D hypothesis. From a numerical point of view we will show the influence of the polynomials degrees particularly for the thermal problem.

#### **5.2 The choice of basis functions**

88 Convection and Conduction Heat Transfer

 ψ

ρ

*h*

ρ

 ψ

will be noted *r* .

(22)

ω

is made dimensionless in a classic way:

(23)

θ

( )

θ

<sup>=</sup> ∑ ∑ (27)

. For a steady problem, the coefficients *fkl*( )*t* are time

<sup>v</sup> (26)

() ( )

*kl l k*

 λ θ

<sup>v</sup> (24)

2 12 1 <sup>2</sup>

<sup>⎨</sup> ⎛ ⎞ ∂∂ ∂∂ ∂ ∂ ⎛ ⎞ ′ <sup>⎪</sup> ⎜ ⎟ − + − =Δ ⎜ ⎟ <sup>⎩</sup> ⎝ ⎠ ∂∂ ∂∂ ∂ ∂ ⎝ ⎠

*x x x hr*

ρ

θ

θ∞

Using (1) and (10)-(15), the dimensionless energy equation can be written as follows:

( ) 2 2 <sup>1</sup> RePr *<sup>f</sup> h h u h urh tx r*

2 2 2 2 2 2 2 2 2 *<sup>f</sup> h rh h r h r h hh*

 λ

The dimensionless steady state energy problem related to the equation (24) is:

θ

**5. Numerical resolution using spectral methods** 

λ

θθ

Δ= − + + + − + ′ ′′ ⎡ ⎤ ′′ ⎢ ⎥ ⎣ ⎦ ∂ ∂ <sup>∂</sup> ∂ ∂ ⎣ ⎦

( ) <sup>2</sup> <sup>1</sup> RePr *<sup>f</sup> h u h urh x r*

λ

 ∂∂ ∂ <sup>+</sup> +− = Δ ′ ∂∂ ∂

22 2 2

<sup>∂</sup> ∂∂ ∂ ⎡ ⎤

*x r x r r r*

 θ

 ∂ ∂ <sup>+</sup> − =Δ ′ ∂ ∂

The spectral methods consist in projecting any unknown function *f* (*xrt* , , ) on trial

0 l0 ( , ,) () *N N x r*

*f xrt f tP rQ x* = =

where *Nx* and *Nr* are the development orders according to the axis *x* and *r* respectively. The basis functions *P r <sup>l</sup>*( ) and *Q x <sup>k</sup>* ( ) are generally trigonometric or polynomial functions (Chebyshev, Legendre, etc.) according to different boundary conditions situations. The time dependant coefficients *fkl*(*t*) are the unknowns of the problem. For our problem, the

*k*

*W T T T T*

<sup>−</sup> <sup>=</sup> <sup>−</sup>

∞

 θ

(25)

*f*

ψω

 ψω

θθ

 ρ

 ψ

ω

⎧ = −Δ ⎪

ρρ

*Important:* for reason of convenience, the radius

**4.2 New formulation of the thermal problem**  For the thermal problem, the temperature

**4.2.1 The thermal unsteady problem formulation** 

θ

**4.2.2 The thermal steady problem formulation** 

**5.1 Trial functions and development orders** 

ω

 , ψ orθ

θ

functions as follows:

function *f* represents

independent.

with:

Re

*g*

 ω

> Because no symmetry condition is imposed at the boundaries of our half-domain of study, we choose basis functions constructed from Chebyshev polynomials (Bernardi & Maday, 1992; Canuto et al., 1988) instead of trigonometric trial functions. Then, *P r <sup>l</sup>*( ) and *Q x <sup>k</sup>* ( ) are written as linear combination of Chebyshev polynomials. Their expressions depend on the boundary conditions and the spectral method used (Galerkin or collocation method).

> Generally, with Galerkin method, Dirichlet or Neuman boundary conditions imposed to trial functions must be homogeneous, but it is not necessary for collocation method (see Galerkin and collocation methods below).

> The basis *P r <sup>l</sup>*( ) and *Q x <sup>k</sup>* ( ) are written as a linear combination of Chebyshev polynomials such as (Gelfgat, 2004; Shen, 1994, 1995, 1997):

$$P\_l(r) = T\_l(r) + \sum\_{i=1}^{n} \alpha\_l T\_{l+i}(r) \quad \text{and} \quad Q\_k(\mathbf{x}) = T\_k(\mathbf{x}) + \sum\_{i=1}^{m} \beta\_k T\_{k+i}(\mathbf{x}) \tag{28}$$

where *n* (respectively *m*) is the number of boundary conditions according to the radial direction *r* (respectively the axial direction *x*), and *T x <sup>k</sup>* ( ) is the Chebyshev polynomial of degree *k*.

#### **5.2.1 Advantages and limitations of spectral methods**

Spectral methods are used successfully in many problems of physics, mainly those involving periodic physical phenomena in space and / or in time. Its main advantage is its high degree of accuracy, compared with some methods such as finite differences, finite elements or finite volumes (Bernardi & Maday, 1992; Canuto et al., 1988; Gelfgat, 2004; Shen, 1994, 1995, 1997). Spectral methods are particularly suitable to study instabilities phenomena, selfmaintained or forced, occuring in Computational Fluid Dynamics. However, spectral methods are limited to simple geometries. For complicated study domains, an alternative way may be using spectral finite elements. The second disadvantage of these methods is their cost of implementing and their high CPU calculations. The matrices obtained are usually full and strategies for solving linear or nonlinear systems remain limited.

#### **6. Numerical resolution of the dynamic and thermal problem using spectral galerkin formulation**

#### **6.1 Numerical resolution of the dynamic steady problem**

The steady dynamic problem is given by the equation (22). Generally, this problem is written with classical homogeneous boundary conditions. One of the originalities of this study is the use of a relevment function allowing the introduction of non homogeneous boundary conditions. For this reason, the unknown stream function ψ(,) *x r* is written by mean of the Poiseuille stream function 0 ϕ ( )*r* corresponding to the Poiseuille velocity imposed at the duct entry as:

$$
\tilde{\boldsymbol{\varphi}}(\mathbf{x}, r) = \boldsymbol{\varphi}\_0(\mathbf{x}, r) + \boldsymbol{\varphi}\_0(r) \tag{29}
$$

Convective Heat Transfer of Unsteady Pulsed Flow in Sinusoidal Constricted Tube 91

( ) ( ) 1 () () *m k k ki k i i Q x Tx T x*

where 3 *m* = here (see bellow). The velocity boundary conditions imply that the stream

function must satisfy the corresponding homogeneous boundary conditions as:

( 1) 0 *Qk*

(1) 0 *Qk*

 ψ

<sup>Δ</sup> <sup>=</sup>

equation (30) and each test function ( ) ( ) *Q xP r i j* <sup>2</sup> , by writing:

*r x r r r r*

( ) ()

*i j*

**6.2 Numerical resolution of the dynamic unsteady problem** 

Φ

*Q xP r*

2

0 00

 βω

() () ( ) ( )

Let us define the Chebyshev scalar product as:

where Δ is the square: Δ= − × − [ 1,1 1,1 ] [ ] .

φ

0

ϕ

unknowns, solved by Newton algorithm.

ψψ

 α

Φ

 Φ

 β

⎛ ⎞ <sup>∂</sup> = − ⎜ ⎟ ⎝ ⎠ <sup>∂</sup>

From equation (16), introducing the unknown

α ω α

1 1 , Re

αω

Φ

γ

depend now on time:

*L xrt*

ψ *r r*

ψφ

Finally we obtain:

Taking as test function:

=

β+

( )( ) ( ) ( ) ( ) ( ) ( )

++ + ++

*kk k kk* ++ +

1 1 *xr xr dxdr*

*k k k k k k k <sup>k</sup> k k Qx Tx Tx Tx T x*

1 1 ( , ) ( , ) ( , )( . )

the Galerkin spectral method consist to make scalar products between the non linear

2 2

 β

 Φ

<sup>⎛</sup> ∂∂ ∂ <sup>⎞</sup> <sup>+</sup> + ++ − <sup>=</sup> <sup>⎜</sup> <sup>⎟</sup> <sup>⎝</sup> ∂∂∂ <sup>⎠</sup>

Finally, we obtain a system of *NN N xr x r* = ( + + 1 1 )( ) non linear equations with *Nxr*

ψ

and using the equations (46), we define the operator in which the unknown coefficients

*r x r r r r r r*

⎛ ⎞ ∂∂∂ ∂ = − ++ ++ − + − ⎜ ⎟ ⎝ ⎠ ∂∂∂ ∂ (48)

 Φ

 β

ψψϕ

( ) ( ) ( ) ( ) ( ) ( ) <sup>2</sup> 1 1 1 11 <sup>1</sup> ( ,,) Re Re

ψψϕ

 βω

1 1 11 , Re

( ) ( ) ( ) ( ) ( ) ( ) ( ) ()

 ϕ

 βω

function such as:

 βω

(*xrt xrt r At* ,, ,, ) = + ( ) ( ) ( ) (47)

 γω

 φ

+ + + + = − − +

2 22 12 3 3 1 3 1 22 2 22

2 2

(,) *xr Q x P r* = *k l* ( ) 2 xr ( ), for 0 k N , 0 l N ≤ ≤ ≤ ≤ (45)

− − ∫∫ (44)

 γω

*x r*

2 2 2

= +∑ (39)

′ − = at x -1 = ( 0 *v* = at 1 *x* = − ) (40)

′ = at x 1 = ( 0 *v* = at 1 *x* = ) (42)

( )( ) ( )

*i j*

*Q xP r*

Φ

 γ

Φ

 β

 ϕ

(43)

(46)

( 1) 0 *Qk* − = at x -1 = (Poiseuille profile 1 *x* = − ) (41)

where the stream function 0 ψ (,) *x r* verifies homogeneous boundary conditions in both directions *x* and *r* .

The equation (22) becomes:

$$\frac{1}{r^2}\frac{\partial \boldsymbol{\nu}\_0}{\partial \mathbf{x}}(\boldsymbol{a}(\boldsymbol{\nu}) + \boldsymbol{a}\_{\boldsymbol{\Phi}}) + \frac{1}{r}\frac{\partial \boldsymbol{\nu}\_0}{\partial r}(\boldsymbol{\beta}(\boldsymbol{\nu}) + \boldsymbol{\beta}\_{\boldsymbol{\Phi}}) + \frac{1}{r}\frac{\partial \boldsymbol{\rho}\_0}{\partial r}\boldsymbol{\beta}(\boldsymbol{\nu}) - \frac{1}{\widehat{\mathbf{R}}}\boldsymbol{\gamma}(\boldsymbol{\nu}) = \frac{1}{\widehat{\mathbf{R}}}\boldsymbol{\gamma}\_{\boldsymbol{\Phi}} - \frac{1}{r}\frac{\partial \boldsymbol{\rho}\_0}{\partial r}\boldsymbol{\beta}\_{\boldsymbol{\Phi}} \tag{30}$$

with:

$$\begin{aligned} \alpha(\alpha) &= 2\alpha - r \frac{\partial \alpha}{\partial r}; \alpha\_{\Phi} = 2\Phi - r \frac{\partial \Phi}{\partial r}; \beta(\alpha) = \frac{\partial \alpha}{\partial x} - 4 \frac{h'}{h} \alpha; \beta\_{\Phi} = \frac{\partial \Phi}{\partial x} - 4 \frac{h'}{h} \Phi; \\ \gamma(\alpha) &= \Delta\_{\g} \alpha; \gamma\_{\Phi} = \Delta\_{\g} \Phi; \end{aligned} \tag{31}$$

$$
\Phi(\mathbf{x}, r) = -\Delta\_f \phi\_0(r) \,. \tag{32}
$$

The corresponding Galerkin method consists in projecting the discretized equations on a Chebyshev polynomials basis, taking into account the whole boundary conditions (Canuto et al., 1988). Then, according to the general formulation of spectral methods, the streamfunction ψ0 is projected on trial functions as follows:

$$\Psi\_0(\mathbf{x}, r) = \sum\_{k=0}^{N\_x} \sum\_{l=0}^{N\_r} \varphi\_{kl} P\_{2l}(r) Q\_k(\mathbf{x}) \tag{33}$$

Because of the symmetry property on the whole axisymmetric domain of the problem, *P r* <sup>2</sup>*l*( ) will be an even function. To construct the basis *P r* <sup>2</sup>*l*( ) , we choose a linear combination of Chebyshev polynomials such as (Gelfgat, 2004; Shen, 1994, 1995, 1997):

$$P\_{2l}(r) = T\_{2l}(r) + \sum\_{i=1}^{n} a\_{li} T\_{2\{l+i\}}(r) \tag{34}$$

where *n* is the number of boundary conditions according to radial direction *r* (3 *n* = here, see bellow).

The coefficients α*li* are determined so that *P r* <sup>2</sup>*l*( ) satisfies the corresponding homogeneous boundary conditions:

$$\frac{\partial P\_{2l}}{\partial r} = 0 \quad \text{at} \ r = \pm 1 \quad \text{(because } \frac{\partial \tilde{\nu}}{\partial r} = 0 \text{ at} \quad r = \pm 1\text{)}\tag{35}$$

$$P\_{2l}\left(r\right) = 0 \quad \text{at } r = \pm 1 \quad \text{(flow-rate condition at } r = \pm 1\text{)}\tag{36}$$

$$P\_{2l}(r) = 0 \quad \text{at } r = 0 \quad \text{(axial symmetry)}\tag{37}$$

So, one can determine all coefficients α*li* . Finally we have:

$$P\_{2l}(r) = T\_{2l}(r) - \frac{l+1}{l+2} T\_{2(l+1)}(r) - T\_{2(l+2)}(r) + \frac{l+1}{l+2} T\_{2(l+3)}(r) \tag{38}$$

A similar analysis is available for the choice of *Q x <sup>k</sup>* ( ) basis functions:

$$Q\_k(\mathbf{x}) = T\_k(\mathbf{x}) + \sum\_{i=1}^{m} \mathcal{J}\_{ki} T\_{(k+i)}(\mathbf{x}) \tag{39}$$

where 3 *m* = here (see bellow). The velocity boundary conditions imply that the stream function must satisfy the corresponding homogeneous boundary conditions as:

$$Q'\_k(-1) = 0 \quad \text{at } \ge -1 \text{ (} \tilde{\upsilon} = 0 \quad \text{at } \ge -1\text{)}\tag{40}$$

$$Q\_k(-1) = 0 \quad \text{at } \ge -1 \quad \text{(Poiseuille profile } \ge -1\text{)}\tag{41}$$

$$Q'\_k(1) = 0 \text{ at } \ge -1 \quad \left(\tilde{v} = 0 \quad \text{at } \ge -1\right) \tag{42}$$

Finally we obtain:

90 Convection and Conduction Heat Transfer

( ) ( ) ( ) ( ) ( ) ( ) 0 00 <sup>0</sup>

 ϕ

2 ; 2 ; 4; 4 ;

ϕ

Φω

∂ ∂∂ ∂ ′ ′ =− = − = − = − ∂ ∂∂ ∂

<sup>0</sup> (,) () *<sup>f</sup>*

The corresponding Galerkin method consists in projecting the discretized equations on a Chebyshev polynomials basis, taking into account the whole boundary conditions (Canuto et al., 1988). Then, according to the general formulation of spectral methods, the stream-

> 0 2 ( ) () ( ) 0 l0

 ψ *x r P rQ x* = =

Because of the symmetry property on the whole axisymmetric domain of the problem, *P r* <sup>2</sup>*l*( ) will be an even function. To construct the basis *P r* <sup>2</sup>*l*( ) , we choose a linear combination of Chebyshev polynomials such as (Gelfgat, 2004; Shen, 1994, 1995, 1997):

> 2 2 ( ) 2( ) 1 ( ) ( ) *n l l li l i i Pr Tr T r*

where *n* is the number of boundary conditions according to radial direction *r* (3 *n* = here,

at 1 *r* = ± ( because 0

*li* . Finally we have:

+ +

*l l* ++ + + +

() () ( ) ( ) ( ) ( ) ( ) 2 2 ( ) 21 22 2 3 1 1

2 2 *l l ll l l l Pr Tr T r T r T r*

=− − +

α

A similar analysis is available for the choice of *Q x <sup>k</sup>* ( ) basis functions:

α+

*li* are determined so that *P r* <sup>2</sup>*l*( ) satisfies the corresponding homogeneous

*r* ∂ψ<sup>=</sup> <sup>∂</sup>

<sup>2</sup> ( ) 0 *P r <sup>l</sup>* = at 1 *r* = ± (flow-rate condition at 1 *r* = ± ) (36)

<sup>2</sup> ( ) 0 *P r <sup>l</sup>* = at 0 *r* = (axial symmetry) (37)

=

*kl l k*

*N N x r*

*k*

*xr r* = −Δ

*r r xh xh*

 βω

1 1 1 11 1 *r x r r r r* Re Re *r r*

∂∂ ∂ ∂ ++ ++ − = − ∂ ∂∂ ∂ (30)

> βω

 β

(,) *x r* verifies homogeneous boundary conditions in both

 γω

Φ Φ Φ

 Φ

<sup>=</sup> ∑ ∑ (33)

= + ∑ (34)

at 1 *<sup>r</sup>* <sup>=</sup> <sup>±</sup> ) (35)

(38)

 ωβ

*h h*

 Φ

. (32)

 ϕ

Φ

 β

(31)

 γ

where the stream function 0

α ω α

( )

α ω

γ ω Φ

ψψ

 ω

> ωγ

directions *x* and *r* . The equation (22) becomes:

2

with:

function

see bellow). The coefficients

boundary conditions:

α

So, one can determine all coefficients

ψ

ψ

 βω

( ) ( )

Φ

 Φ

0 is projected on trial functions as follows:

ψ

<sup>2</sup> <sup>0</sup> *<sup>P</sup> <sup>l</sup> r* <sup>∂</sup> <sup>=</sup> <sup>∂</sup>

,

 α

*r r*

 Φ

Φ

; ; *g g*

ω

Φ

=Δ =Δ

$$Q\_k(\mathbf{x}) = T\_k(\mathbf{x}) - \frac{\left(k+3\right)^2 \left(k+1\right)}{\left(k+2\right)^2 \left(k+2\right)} T\_{k+1}(\mathbf{x}) - \frac{k^2}{\left(k+2\right)^2} T\_{k+2}\left(\mathbf{x}\right) + \frac{\left(k+3\right)^2 \left(k+1\right)}{\left(k+2\right)^2 \left(k+2\right)} T\_{k+3}\left(\mathbf{x}\right) \tag{43}$$

Let us define the Chebyshev scalar product as:

$$\phi(\boldsymbol{\nu}, \phi) = \iint\_{\Delta} \boldsymbol{\nu}(\mathbf{x}, r) \phi(\mathbf{x}, r) (\frac{1}{\sqrt{1 - \mathbf{x}^2}}, \frac{1}{\sqrt{1 - r^2}}) d\mathbf{x} dr \tag{44}$$

where Δ is the square: Δ= − × − [ 1,1 1,1 ] [ ] . Taking as test function:

$$\phi(\mathbf{x}, r) = Q\_k(\mathbf{x}) P\_{2l}(r)\_{\prime} \text{ for } 0 \le \mathbf{k} \le \mathbf{N}\_{\mathbf{x}^{\prime}} \ 0 \le \mathbf{l} \le \mathbf{N}\_{\mathbf{r}} \tag{45}$$

the Galerkin spectral method consist to make scalar products between the non linear equation (30) and each test function ( ) ( ) *Q xP r i j* <sup>2</sup> , by writing:

$$\begin{split} & \left( \frac{1}{r^2} \frac{\partial \boldsymbol{\nu}\_0}{\partial \mathbf{x}} (\boldsymbol{a}(\boldsymbol{\alpha}) + \boldsymbol{a}\_{\boldsymbol{\Phi}}) + \frac{1}{r} \frac{\partial \boldsymbol{\nu}\_0}{\partial r} (\boldsymbol{\beta}(\boldsymbol{\alpha}) + \boldsymbol{\beta}\_{\boldsymbol{\Phi}}) + \frac{1}{r} \frac{\partial \boldsymbol{\rho}\_0}{\partial r} \boldsymbol{\beta}(\boldsymbol{\omega}) - \frac{1}{\overline{\mathbf{R}}} \boldsymbol{\gamma}(\boldsymbol{\alpha}), \boldsymbol{Q}\_i(\boldsymbol{\alpha}) \boldsymbol{P}\_{2j}(r) \right) = \\ & = \left( \frac{1}{\overline{\mathbf{R}}} \boldsymbol{\gamma}\_{\boldsymbol{\Phi}} - \frac{1}{r} \frac{\partial \boldsymbol{\rho}\_0}{\partial r} \boldsymbol{\rho}\_{\boldsymbol{\Phi}}, \boldsymbol{Q}\_i(\boldsymbol{\alpha}) \boldsymbol{P}\_{2j}(r) \right) \end{split} \tag{46}$$

Finally, we obtain a system of *NN N xr x r* = ( + + 1 1 )( ) non linear equations with *Nxr* unknowns, solved by Newton algorithm.

#### **6.2 Numerical resolution of the dynamic unsteady problem**

From equation (16), introducing the unknown ψfunction such as:

$$
\tilde{\psi}\left(\mathbf{x}, r, t\right) = \psi\left(\mathbf{x}, r, t\right) + \phi\left(r\right) A\left(t\right) \tag{47}
$$

and using the equations (46), we define the operator in which the unknown coefficients depend now on time:

$$\mathcal{L}\_{\boldsymbol{\nu}}(\mathbf{x},\boldsymbol{r},t) = -\left(\frac{1}{r^{2}}\frac{\partial\boldsymbol{\nu}}{\partial\mathbf{x}}(\boldsymbol{a}(\boldsymbol{\alpha}) + \boldsymbol{\alpha}\_{\boldsymbol{\Phi}}) + \frac{1}{r}\frac{\partial\boldsymbol{\nu}}{\partial\boldsymbol{r}}(\boldsymbol{\beta}(\boldsymbol{\alpha}) + \boldsymbol{\beta}\_{\boldsymbol{\Phi}}) + \frac{1}{r}\frac{\partial\boldsymbol{\rho}}{\partial\boldsymbol{r}}\boldsymbol{\beta}(\boldsymbol{\alpha}) - \frac{1}{\hat{\mathbf{R}}}\boldsymbol{\gamma}(\boldsymbol{\alpha})\right) + \frac{1}{\hat{\mathbf{R}}}\boldsymbol{\gamma}\_{\boldsymbol{\Phi}} - \frac{1}{r}\frac{\partial\boldsymbol{\rho}}{\partial\boldsymbol{r}}\boldsymbol{\beta}\_{\boldsymbol{\Phi}}(48)$$

Convective Heat Transfer of Unsteady Pulsed Flow in Sinusoidal Constricted Tube 93

1 1 <sup>2</sup> 11 1 . . 2(1 ) . RePr RePr

∂∂ ∂∂ ∂ ∂ ∂ − + − − Δ= + Δ

This problem is discretized by Galerkin spectral method explained above. The linear system

11 1 1 1 11 1 . . . RePr RePr

<sup>0</sup> ( ) *x r*, is the steady thermal problem solution. The equation (56) is numerically

<sup>1</sup> ( )( ) ( )

 ε

> ϕ

 θ

Δ (57)

 θ

*n n* ,, 1 ,,

*f R*

θ

⎛ ⎞ <sup>∂</sup> ∂ ∂∂ ∂ ∂ = − ⎜ ⎟ − + − Δ ⎝ ⎠ ∂∂ ∂∂ ∂ ∂

*L xrt L xrt*

+

11 1 1 <sup>1</sup> , , . . () RePr

 ψθ

*h rr x rx r rr x*

By projecting (57) in the Galerkin basis ( *i j* () () <sup>2</sup> )*ij <sup>q</sup> <sup>x</sup> <sup>p</sup> <sup>r</sup>* , one obtains at each time step a system

One can notice that the use of Chebyshev polynomials in both axial and radial directions is

**7. Numerical resolution of the dynamic and energy problem using spectral** 

For reasons of simplicity, we describe explicitly only the resolution of the steady dynamic problem. For the unsteady problem, we use Crank-Nicolson method for time integration as

The main interest of collocation method compared with Galerkin formulation is its simplicity: it is not necessary to build a relevment function to take into account non homogeneous boundary conditions. We introduce these conditions directly in the matrix of the system and/or in the basis trial functions. For this reason, it is easy to compute

in (50); the unsteady problem resolution in space is identical to the steady case.

collocation procedure. Let us explain this method for the steady dynamic problem.

*f f <sup>R</sup> t rr x rx r rrx h h rx r*

∂ ∂ ∂ ∂ ∂ ∂∂ ⎛ ⎞ <sup>⎛</sup> ∂ ∂ <sup>⎞</sup> = − ⎜ ⎟ − + − Δ+ <sup>⎜</sup> + Δ <sup>⎟</sup> ∂ ∂ ∂ ∂ ∂ ∂∂ ⎝ ⎠ <sup>⎝</sup> ∂ ∂ <sup>⎠</sup>

 θ

*rr x rx r x rx r*

2 2

 ϕθ

integrated in time by using the second order Crank-Nicolson scheme ( <sup>1</sup>

θ

11 1 . RePr

*L xrt A t*

θ

ψ

> ψ θ

*R*

⎛ ⎞ ∂ ∂ + +Δ ⎜ ⎟ ⎝ ⎠ ∂ ∂

<sup>+</sup> <sup>−</sup> = +−

ε

*xr xr r* = + , the steady thermal problem is written as follows:

*R*

ψ θ

*R*

(56)

θ

θ

= ) which is

(58)

2 ε

*f*

θ

*f f <sup>R</sup> r*

θ

> θ

∂∂ ∂∂ ∂ ∂ ∂ (55)

ψ θ

**6.3.2 Resolution of the steady energy equation** 

 ψθ

obtained is solved by a Gauss type classical method.

**6.3.3 Resolution of the unsteady energy equation** 

 ψ θ

1

2

of linear equations solved by the classical Gauss method.

**7.1 Numerical resolution of the dynamic problem** 

*h rx r*

not obvious, and contribute to emphasize this numerical method.

θ

( ) <sup>2</sup>

*n n*

 θ

*t*

The unsteady problem is written as follows:

θθθ

With (24) and ( , ) ( , ) ( ) *<sup>R</sup>*

ψθ

0

θ

**collocation method** 

 ψ θ

( , ,0) ( , )

 θ

*xr xr*

=

θ

θ

formulated as follows:

θ

where

where

Then the previous problem (16) can take the following form:

$$\text{ch}^2 \frac{\partial \bar{\phi}}{\partial t} = \text{L}\_{\psi} \left( \mathbf{x}\_{\prime} r\_{\prime} t \right) \text{ where } \bar{\phi} = \phi + \phi\_{\Phi} \tag{49}$$

The operator *L xrt* ψ ( , , ) is nonlinear. Notice that ωΦ is the contribution coming from Poiseuille extension. The temporal discretization of (49) is made by using the ε –method, reduced here to Crank - Nicolson method. The advantage of this method is to be unconditionally stable. It leads to the equation below with 1 /2 ε = , which corresponds to a two order scheme:

$$\begin{cases} h^2 \frac{\alpha^{n+1} - \alpha^n}{\Delta t} + h^2 \frac{\partial \alpha\_{\Phi}(\mathbf{x}, r, t)}{\partial t} = \varepsilon \mathbf{L}\_{\boldsymbol{\psi}^{n+1}} \left( \mathbf{x}, r, t \right) + (1 - \varepsilon) \mathbf{L}\_{\boldsymbol{\psi}^n} \left( \mathbf{x}, r, t \right) \\ \alpha^n = -\Delta\_f \boldsymbol{\psi}^n, \forall n \end{cases} \tag{50}$$

where the initial condition is given by the solution of the steady problem.

The unknowns ψ *kl*(*t*) are obtained by solving with Newton algorithm, at each time step, the non linear system obtained with scalar products between relation (50) and test functions *Q xP r i j* ( ) <sup>2</sup> ( ) , as in equation (46).

### **6.3 Numerical resolution of the thermal unsteady problem**

#### **6.3.1 Choices of the basis functions**

The dimensionless energy equation is given by (25) and (25). The choice of the temperature basis functions is made in the same way as in the dynamic problem. In order to apply the Galerkin method, we consider the boundary conditions (heading 3) for the temperature θ . Let us set:

$$
\tilde{\theta}(\mathbf{x}, r, t) = \theta(\mathbf{x}, r, t) + \theta\_{\mathbb{R}}(r) \tag{51}
$$

where θ is the solution satisfying the homogeneous boundary conditions and ( ) *<sup>R</sup>*θ *r* is a smoothed gap temperature imposed at the entry. The homogeneous temperatureθ , truncated at development orders *Mx* according to the axis *x* and *Mr* according to the radius *r*, is projected on the trial functions as follows:

$$\theta(\mathbf{x}, r, t) = \sum\_{k=0}^{M\_x} \sum\_{l=0}^{M\_r} \theta\_{kl}(t) q\_k(\mathbf{x}) p\_{2l}(r) \tag{52}$$

where *p*2*l*( ) *r* and *qk* (*x*) are built from Chebyshev polynomials as in heading 5. According to temperature boundary conditions (heading 3), we obtain, at last:

$$q\_k(\mathbf{x}) = T\_k(\mathbf{x}) + \frac{4(k+1)}{\left(k+1\right)^2 + \left(k+2\right)^2} T\_{k+1}(\mathbf{x}) - \frac{\left(k+1\right)^2 + k^2}{\left(k+1\right)^2 + \left(k+2\right)^2} T\_{k+2}(\mathbf{x}), \quad \text{if } 0 \le k \le M\_\chi \tag{53}$$

The polynomial *p*2*l*(*r*) is given by:

$$p\_{2l}(r) = T\_{2l}(r) - T\_{2(l+1)}(r), \qquad \text{if } 0 \le l \le M\_r \tag{54}$$

#### **6.3.2 Resolution of the steady energy equation**

With (24) and ( , ) ( , ) ( ) *<sup>R</sup>* θθθ*xr xr r* = + , the steady thermal problem is written as follows:

$$\frac{1}{r}\frac{\partial\boldsymbol{\nu}}{\partial r}\cdot\frac{\partial\boldsymbol{\theta}}{\partial\mathbf{x}}-\frac{1}{r}\frac{\partial\boldsymbol{\nu}}{\partial\mathbf{x}}\cdot\frac{\partial\boldsymbol{\theta}}{\partial r}+2(1-r^{2})\frac{\partial\boldsymbol{\theta}}{\partial\mathbf{x}}-\frac{1}{\hat{\mathbf{R}}\mathbf{e}\mathbf{Pr}}\boldsymbol{\Delta}\_{f}\boldsymbol{\theta} = \frac{1}{r}\frac{\partial\boldsymbol{\nu}}{\partial\mathbf{x}}\cdot\frac{\partial\boldsymbol{\theta}\_{\mathbb{R}}}{\partial r}+\frac{1}{\hat{\mathbf{R}}\mathbf{ePr}}\boldsymbol{\Delta}\_{f}\boldsymbol{\theta}\_{\mathbb{R}}\tag{55}$$

This problem is discretized by Galerkin spectral method explained above. The linear system obtained is solved by a Gauss type classical method.

#### **6.3.3 Resolution of the unsteady energy equation**

The unsteady problem is written as follows:

$$\begin{split} \frac{\partial \boldsymbol{\hat{\theta}}}{\partial t} &= -\frac{1}{h^{2}} \Big( \frac{1}{r} \frac{\partial \boldsymbol{\nu}}{\partial r} \cdot \frac{\partial \boldsymbol{\theta}}{\partial \mathbf{x}} - \frac{1}{r} \frac{\partial \boldsymbol{\nu}}{\partial \mathbf{x}} \cdot \frac{\partial \boldsymbol{\theta}}{\partial r} + \frac{1}{r} \frac{\partial \boldsymbol{\rho}}{\partial r} \frac{\partial \boldsymbol{\theta}}{\partial \mathbf{x}} - \frac{1}{\text{Re} \text{Pr}} \boldsymbol{\Delta\_{f}} \boldsymbol{\theta} \Big) + \frac{1}{h^{2}} \Big( \frac{1}{r} \frac{\partial \boldsymbol{\nu}}{\partial \mathbf{x}} \cdot \frac{\partial \boldsymbol{\theta}\_{\mathbf{R}}}{\partial r} + \frac{1}{\text{Re} \text{Pr}} \boldsymbol{\Delta\_{f}} \boldsymbol{\theta}\_{\mathbf{R}} \Big) \\ \boldsymbol{\theta}(\mathbf{x}, r, \mathbf{0}) &= \boldsymbol{\theta}\_{0}(\mathbf{x}, r) \end{split} (56)$$

where θ<sup>0</sup> ( ) *x r*, is the steady thermal problem solution. The equation (56) is numerically integrated in time by using the second order Crank-Nicolson scheme ( <sup>1</sup> 2 ε = ) which is formulated as follows:

$$\frac{\partial^{n+1} - \theta^{n}}{\Delta t} = \varepsilon L\_{\theta^{n+1}} \left( \mathbf{x}, r, t \right) + (1 - \varepsilon) L\_{\theta^{n}} \left( \mathbf{x}, r, t \right) \tag{57}$$

where

92 Convection and Conduction Heat Transfer

reduced here to Crank - Nicolson method. The advantage of this method is to be

2 2 ( ,,) ,, 1 ,,

non linear system obtained with scalar products between relation (50) and test

The dimensionless energy equation is given by (25) and (25). The choice of the temperature basis functions is made in the same way as in the dynamic problem. In order to apply the Galerkin method, we consider the boundary conditions (heading 3) for the temperature

θθθ

smoothed gap temperature imposed at the entry. The homogeneous temperature

0 l0

= =

+ ++ + ++

*k*

*M M x r*

truncated at development orders *Mx* according to the axis *x* and *Mr* according to the

 θ*xrt t q x p r*

where *p*2*l*( ) *r* and *qk* (*x*) are built from Chebyshev polynomials as in heading 5. According

22 22 1 2 4( 1) ( 1) ( ) ( ) ( ), if 0 ( 1) ( 2) ( 1) ( 2) *k k <sup>k</sup> <sup>k</sup> <sup>x</sup> k k k q x Tx T x T x kM kk kk* + + + + + = + − ≤ ≤

is the solution satisfying the homogeneous boundary conditions and ( ) *<sup>R</sup>*

() ( ) ( ) <sup>2</sup>

2 2

2 2 2( 1) ( ) ( ) ( ), if 0 *lll <sup>r</sup> p r Tr T r lM* = − <sup>+</sup> ≤ ≤ (54)

*kl k l*

ε

ψ

+

*kl*(*t*) are obtained by solving with Newton algorithm, at each time step, the

*xrt h h Lx <sup>r</sup> t Lx <sup>r</sup> <sup>t</sup>*

where

ω = ω ω+ Φ

ωΦ

ε

*n n*

<sup>1</sup> ( )( ) ( )

*xrt xrt r* = + (51)

<sup>=</sup> ∑ ∑ (52)

 ε

> ψ

(49)

is the contribution coming from

= , which corresponds to a

ε

–method,

(50)

θ.

θ,

(53)

θ*r* is a

Then the previous problem (16) can take the following form:

The operator *L xrt*

two order scheme:

The unknowns

Let us set:

where

θ

( )

The polynomial *p*2*l*(*r*) is given by:

ψ

ω

ψ

1

ωω

*n n f*

<sup>⎪</sup> = −Δ ∀ <sup>⎩</sup>

functions *Q xP r i j* ( ) <sup>2</sup> ( ) , as in equation (46).

**6.3.1 Choices of the basis functions** 

*n n*

 ψ ,

*t t n*

**6.3 Numerical resolution of the thermal unsteady problem** 

( ,,) ( ,,) () *<sup>R</sup>*

(,,)

to temperature boundary conditions (heading 3), we obtain, at last:

θ

radius *r*, is projected on the trial functions as follows:

∂ω <sup>=</sup> <sup>∂</sup> 

( ) <sup>2</sup> *h L xrt* , , *<sup>t</sup>* ψ

Poiseuille extension. The temporal discretization of (49) is made by using the

( , , ) is nonlinear. Notice that

unconditionally stable. It leads to the equation below with 1 /2

Φ

where the initial condition is given by the solution of the steady problem.

<sup>+</sup> <sup>⎧</sup> − ∂ <sup>⎪</sup> + =+ <sup>−</sup> <sup>⎨</sup> Δ ∂

 ω

$$\begin{split} L\_{\theta}(\mathbf{x}, r, t) &= -\frac{1}{h^{2}} \Big( \frac{1}{r} \frac{\partial \boldsymbol{\nu}}{\partial r} \cdot \frac{\partial \boldsymbol{\theta}}{\partial \mathbf{x}} - \frac{1}{r} \frac{\partial \boldsymbol{\nu}}{\partial \mathbf{x}} \cdot \frac{\partial \boldsymbol{\theta}}{\partial r} + \frac{1}{r} \frac{\partial \boldsymbol{\rho}}{\partial r} A(t) \frac{\partial \boldsymbol{\theta}}{\partial \mathbf{x}} - \frac{1}{\text{Re} \, \text{Pr}} \boldsymbol{\Lambda}\_{f} \boldsymbol{\theta} \Big) \\ &+ \frac{1}{h^{2}} \Big( \frac{1}{r} \frac{\partial \boldsymbol{\nu}}{\partial \mathbf{x}} \cdot \frac{\partial \boldsymbol{\theta}\_{\mathsf{R}}}{\partial r} + \frac{1}{\text{Re} \, \text{Pr}} \boldsymbol{\Lambda}\_{f} \boldsymbol{\theta}\_{\mathsf{R}} \Big) \end{split} \tag{58}$$

By projecting (57) in the Galerkin basis ( *i j* () () <sup>2</sup> )*ij <sup>q</sup> <sup>x</sup> <sup>p</sup> <sup>r</sup>* , one obtains at each time step a system of linear equations solved by the classical Gauss method.

One can notice that the use of Chebyshev polynomials in both axial and radial directions is not obvious, and contribute to emphasize this numerical method.

#### **7. Numerical resolution of the dynamic and energy problem using spectral collocation method**

#### **7.1 Numerical resolution of the dynamic problem**

For reasons of simplicity, we describe explicitly only the resolution of the steady dynamic problem. For the unsteady problem, we use Crank-Nicolson method for time integration as in (50); the unsteady problem resolution in space is identical to the steady case.

The main interest of collocation method compared with Galerkin formulation is its simplicity: it is not necessary to build a relevment function to take into account non homogeneous boundary conditions. We introduce these conditions directly in the matrix of the system and/or in the basis trial functions. For this reason, it is easy to compute collocation procedure. Let us explain this method for the steady dynamic problem.

Convective Heat Transfer of Unsteady Pulsed Flow in Sinusoidal Constricted Tube 95

The collocation method consists to write the above equation on specific points

We recall that homogeneous boundary conditions are imposed on trial functions *P2l(r)* and *Qk*(*x*). The non homogeneous boundary conditions, like Poiseuille profile at the duct entry or

Steady and unsteady energy equations are solved in the same manner as the dynamic problem, using spectral collocation method and Crank-Nicolson time-solver method

Concerning trial functions for steady and unsteady thermal problems, we use directly

( ) , *<sup>W</sup>*

where Δ*Tref* is a typical difference temperature reference. That one depends on the wall boundary conditions hypotheses. The main difficulty with convective unsteady heat transfer

( ) ( )( )

1 0

1 0

= ∫

,, . *<sup>m</sup>*

∫

( )

,, . ,, .

*u x r t T x r t rdr*

*u x r t rdr*

*ref*

*T* Φ

*T*

lies in the temperature reference choice. After several tests, we have chosen:

,

*T xt*

*h xt*

= = of Δ , called collocation points. We chose the collocation points of

) and *r jN <sup>j</sup>* = −cos(

and ( ) ( ) <sup>2</sup> , , *kl*

*A xr*

<sup>∂</sup> <sup>=</sup> ∂ ∂ (65)

*<sup>r</sup>* ) (66)

α β

π

*pl l* (*r Tr* ) = ( ) and *qx Tx k k* ( ) = ( ) (67)

( , , ) ( ) Δ =− *T xt T T xt ref W m* (69)

<sup>=</sup> <sup>Δ</sup> (68)

(70)

*kl*

*A xr* αβ

represents the space variable *x* or *r* . The relation (65) is a function of local

( ) ( , ) , *kl*

<sup>∂</sup> <sup>=</sup> <sup>∂</sup>

*x iN i x* = −cos( π

flow-rate condition at the wall, are directly imposed in the matrix system . The nonlinear system obtained is solved by the Newton algorithm.

*A x r*

α

*kl*

*A xr* α

point (*x r*, )∈ Δ , where Δ is the square [− ×− 1,1 1,1 ] [ ] .

**7.2 Numerical resolution of the thermal problem** 

All boundary conditions are imposed in the matrix system.

where *Tm*(*x*,*t*) is the mean bulk temperature given by:

The local convective heat transfer coefficient *hT* is written as follows:

with:

where α or β

( )<sup>0</sup>

*x xr r* ≤ ≤

described above.

Chebyshev polynomials:

**8. Convective heat transfer** 

0 , *<sup>r</sup>*

*i j j N*

*x*

Chebyshev-Gauss-Lobatto [5], defined by:

*j N*

≤ ≤

with 0 *<sup>x</sup>* ≤ ≤*i N* and 0 *<sup>r</sup>* ≤ *j N*≤ .

According to the general formulation of spectral methods (27), the stream-function ψ is projected on trial functions in the same manner as equation (33):

$$\tilde{\varphi}(\mathbf{x},r) = \sum\_{k=0}^{N\_x} \sum\_{l=0}^{N\_r} \varphi\_{kl} P\_{2l}(r) Q\_k(\mathbf{x}) \tag{59}$$

We can apply the same approach used in Galerkin method to determine trial functions *P r* <sup>2</sup>*l*( ) and *Q x <sup>k</sup>* ( ) . All conditions given by (35-42) are available, except the second condition (36) for *P r* <sup>2</sup>*l*( ) and the second condition (41) for *Q x <sup>k</sup>* ( ) . Then, with the method given by (34), we obtain:

$$P\_{2l}\left(r\right) = T\_{2l}\left(r\right) + \frac{\left(l+2\right)^2 - l^2}{\left(l+2\right)^2 + \left(l+1\right)^2} T\_{2\left(l+1\right)}\left(r\right) - \frac{\left(l+1\right)^2 + l^2}{\left(l+2\right)^2 + \left(l+1\right)^2} T\_{2\left(l+2\right)}\left(r\right) \tag{60}$$

and

$$Q\_k(\mathbf{x}) = T\_k(\mathbf{x}) - \frac{k^2}{\left(k+2\right)^2} T\_{k+2}(\mathbf{x}) \tag{61}$$

The vorticity function can be written as follows:

$$
\bar{\rho} = \sum\_{k=0}^{N\_x} \sum\_{l=0}^{N\_r} A\_{kl} \left( \mathbf{x}\_r r \right) \nu\_{kl} \tag{62}
$$

where:

$$\begin{cases} A\_{kl}(\mathbf{x},r) = A\_f P\_l(r) Q\_k''(\mathbf{x}) + B\_f P\_l''(r) Q\_k(\mathbf{x}) + \mathbf{C}\_f P\_l'(r) Q\_k(\mathbf{x}) + D\_f P\_l'(r) Q\_k'(\mathbf{x}) \\\\ A\_f = h^2 \quad ; \quad B\_f = r^2 + r^2 h'^2 \quad ; \quad \mathbf{C}\_f = \rho \left( 2h'^2 - hh'' \right) - \frac{\lambda^2}{r} \quad ; \quad D\_f = -2rhh' \end{cases} \tag{63}$$

Then, substituting ψ andω by their expressions (59), (62) in the steady dynamic equation (22), we obtain the following discretized dynamic equation:

() ( ) ( ) () ( ) ( ) () ( ) 0 l0 0 l0 0 l0 0 l0 2 0 l0 0 <sup>1</sup> . , . , 2 *x r x r x r x r x r N N N N x l k kl kl kl k k N N N N r l k kl kl kl k k N N N l k kl k k P rQ x A xr r P rQ x A xr <sup>h</sup> P rQ x r r h* ψ ψ ψ ψ ψ = = = = = = = = = = = ⎡⎛ ⎞ ⎛ ⎞ ⎢⎜ ⎟ ′ ⎜ ⎟ ⎢⎣⎝ ⎠ ⎝ ⎠ ⎛ ⎞ ⎛ ⎞⎤ <sup>−</sup> ⎜ ⎟ ′ ⎜ ⎟⎥ ⎥ ⎝ ⎠ ⎝ ⎠⎦ ⎛ ⎞ ′ + − ⎜ ⎟ ′ ⎝ ⎠ ∑∑ ∑∑ ∑∑ ∑∑ ∑ ∑ () ( ) ( ) () () () () ()() () () () () ()() l 0 0 l0 0 l0 , , , ,, ,, 1 Re , , ,, ,, *x r x r x r N N N l k kl kl kl k xx rr r g kl g kl g kl N N kl k xr rr g kl g kl g kl P rQ x A xr A xr A xr B xr A xr C xr A xr D xr A xr E xr A xr F xr A xr* ψ ψ ψ = = = = = ⎡ ⎤ <sup>⎛</sup> ⎞ ⎛ <sup>⎞</sup> ⎢ ⎥ <sup>⎜</sup> ′ ⎟ ⎜ <sup>⎟</sup> <sup>⎜</sup> ⎟ ⎜ <sup>⎟</sup> ⎢ ⎥ ⎣ ⎦ <sup>⎝</sup> ⎠ ⎝ <sup>⎠</sup> ⎡ ⎤ + + ⎢ ⎥ <sup>=</sup> +++ ⎣ ⎦ ∑∑ ∑∑ ∑ ∑ (64)

with:

ψis

(60)

(63)

(64)

(61)

,

 ψ

*kl*

ψ

94 Convection and Conduction Heat Transfer

<sup>2</sup> () ( )

<sup>=</sup> ∑ ∑ (59)

( )( ) ( ) ( )

<sup>=</sup> ∑ ∑ (62)

2

*r* λ

> ψ

*l k kl kl kl k*

*P rQ x A xr*

by their expressions (59), (62) in the steady dynamic equation

*r*

l 0 0 l0

= = =

ψ

*x r x r*

∑∑ ∑∑

*N N N*

*kl l k*

2 2 2 2

*ll ll* + + + − + +

2

( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

; ; 2 ; 2

ρ

() ( ) ( )

∑ ∑ () ( ) ( )

⎡ ⎤ <sup>⎛</sup> ⎞ ⎛ <sup>⎞</sup> ⎢ ⎥ <sup>⎜</sup> ′ ⎟ ⎜ <sup>⎟</sup> <sup>⎜</sup> ⎟ ⎜ <sup>⎟</sup> ⎢ ⎥ ⎣ ⎦ <sup>⎝</sup> ⎠ ⎝ <sup>⎠</sup>

() () () () ()()

*A xr A xr B xr A xr C xr A xr*

, , ,, ,,

*xx rr r*

() () () () ()()

*g kl g kl g kl*

*D xr A xr E xr A xr F xr A xr*

⎣ ⎦

*P rQ x A xr*

⎛ ⎞ ⎛ ⎞⎤ <sup>−</sup> ⎜ ⎟ ′ ⎜ ⎟⎥ ⎥ ⎝ ⎠ ⎝ ⎠⎦

*l k kl kl kl*

 ψ

*kl fl k fl k fl k fl k*

⎧= + + + ′′ ′′ ′ ′ ′ <sup>⎪</sup>

*A h B r r h C h hh D rhh*

<sup>⎪</sup> = = + = − − =− ′ ′ ′′ ′ <sup>⎩</sup>

*x*

*A xr AP rQ x BP rQ x C P rQ x DP rQ x*

,

*kl kl*

ψ

( )

( ) ( )

2 2 2

*ll ll Pr Tr T r T r*

According to the general formulation of spectral methods (27), the stream-function

0 l0

 ψ *x r P rQ x* = =

We can apply the same approach used in Galerkin method to determine trial functions *P r* <sup>2</sup>*l*( ) and *Q x <sup>k</sup>* ( ) . All conditions given by (35-42) are available, except the second condition (36) for *P r* <sup>2</sup>*l*( ) and the second condition (41) for *Q x <sup>k</sup>* ( ) . Then, with the method given by

( )( ) ( ) ( ) ( )

2 2 2 2 2 1 2 2 2 2 2 1 21 21

*l l l l*

*kk k <sup>k</sup> Qx Tx T x <sup>k</sup>* = − <sup>+</sup> +

0 l0

= =

*ff f f*

0 l0 0 l0

Re , , ,, ,,

+++

*g kl g kl g kl N N*

⎡ ⎤ + + ⎢ ⎥ <sup>=</sup>

= = = =

∑∑ ∑∑

ψ

*x r x r*

*N N N N*

*A xr*

*k* ω

2 2 22 2

() ( ) ( )

*P rQ x A xr*

⎡⎛ ⎞ ⎛ ⎞ ⎢⎜ ⎟ ′ ⎜ ⎟ ⎢⎣⎝ ⎠ ⎝ ⎠

*l k kl kl kl*

. ,

*k k*

(22), we obtain the following discretized dynamic equation:

() ( )

*r h*

⎛ ⎞ ′ + − ⎜ ⎟ ′ ⎝ ⎠

*N N N l k kl k k*

0 l0 0

= = =

*<sup>h</sup> P rQ x r*

ψ

*k xr rr*

0 l0 0 l0

= = = =

∑∑ ∑∑

<sup>1</sup> . ,

ψ

*x r x r*

*N N N N*

*k k*

*N N x r*

= + <sup>−</sup> + ++ + ++

() ()

*N N x r*

*k*

projected on trial functions in the same manner as equation (33):

ψ

() () ( )

The vorticity function can be written as follows:

,

ψ andω

*x r*

0 l0

= =

∑ ∑

*x r*

⎨

Then, substituting

*r*

2

1

2

(34), we obtain:

and

where:

(,)

$$A\_{kl}^{\alpha}(\mathbf{x},r) = \frac{\partial A\_{kl}(\mathbf{x},r)}{\partial \alpha} \text{ and } \ A\_{kl}^{\alpha\beta}(\mathbf{x},r) = \frac{\partial^2 A\_{kl}(\mathbf{x},r)}{\partial \alpha \partial \beta} \tag{65}$$

where α or β represents the space variable *x* or *r* . The relation (65) is a function of local point (*x r*, )∈ Δ , where Δ is the square [− ×− 1,1 1,1 ] [ ] .

The collocation method consists to write the above equation on specific points ( )<sup>0</sup> 0 , *<sup>r</sup> x i j j N j N x xr r* ≤ ≤ ≤ ≤ = = of Δ , called collocation points. We chose the collocation points of

Chebyshev-Gauss-Lobatto [5], defined by:

$$\mathbf{x}\_i = -\cos\left(i\pi/N\_x\right) \text{ and } \mathbf{r}\_j = -\cos\left(j\pi/N\_r\right) \tag{66}$$

with 0 *<sup>x</sup>* ≤ ≤*i N* and 0 *<sup>r</sup>* ≤ *j N*≤ .

We recall that homogeneous boundary conditions are imposed on trial functions *P2l(r)* and *Qk*(*x*). The non homogeneous boundary conditions, like Poiseuille profile at the duct entry or flow-rate condition at the wall, are directly imposed in the matrix system .

The nonlinear system obtained is solved by the Newton algorithm.

#### **7.2 Numerical resolution of the thermal problem**

Steady and unsteady energy equations are solved in the same manner as the dynamic problem, using spectral collocation method and Crank-Nicolson time-solver method described above.

Concerning trial functions for steady and unsteady thermal problems, we use directly Chebyshev polynomials:

$$p\_l(r) = T\_l(r) \text{ and } q\_k(\mathbf{x}) = T\_k(\mathbf{x}) \tag{67}$$

All boundary conditions are imposed in the matrix system.

#### **8. Convective heat transfer**

The local convective heat transfer coefficient *hT* is written as follows:

$$h\_T\left(\mathbf{x}, t\right) = \frac{\Phi\_W}{\Delta T\_{ref}}\tag{68}$$

where Δ*Tref* is a typical difference temperature reference. That one depends on the wall boundary conditions hypotheses. The main difficulty with convective unsteady heat transfer lies in the temperature reference choice. After several tests, we have chosen:

$$
\Delta T\_{ref} \left( \mathbf{x}\_{\prime} \mathbf{t} \right) = T\_{\rm IV} - T\_{\rm m} \left( \mathbf{x}\_{\prime} \mathbf{t} \right) \tag{69}
$$

where *Tm*(*x*,*t*) is the mean bulk temperature given by:

$$T\_m(\mathbf{x}, t) = \frac{\int\_0^1 u(\mathbf{x}, r, t) \, T(\mathbf{x}, r, t) \, r dr}{\int\_0^1 u(\mathbf{x}, r, t) \, r dr} \tag{70}$$

Convective Heat Transfer of Unsteady Pulsed Flow in Sinusoidal Constricted Tube 97

, max *kl kl k l e f f* α

on the parameter *Nx*. This means that the increase in the number of polynomials in the radial direction does not improve the convergence of the results. Secondly, figure 2.a shows

For a given value of *Mr*, we observe in figure 2.b a good convergence of the temperature coefficients when *Mx* increases. But, unlike dynamic field, for the range of *Mr* values between 5 and 9, the analysis of the thermal field leads to slightly different conclusions. Indeed, probably due to the temperature conditions imposed on the entry section, the thermal field is more sensitive to the parameter *Mr* than dynamic field. For a fixed value of *Mx*, the temperature truncature errors increase with *Mr*. Then, optimal convergence is obtained for *Mr* = 5. For this value, the truncature error is less than <sup>8</sup> 10<sup>−</sup> when 56 *Mx* > . In conclusion, we have selected for the dynamic problem: *Nx* = 30 and *Nr* = 5, and for the

αβ

For the dynamic point of view, we note first that the truncature errors *e*

αβ

thermal problem, we have chosen: *Mx* =120 and *Mr* = 5.

**Nr** <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>0</sup>

development of the temperature function

**9.2.1 Study of the dynamic field** 

**Legend**

, for the truncature order

αβ

**Nx = 11 Nx = 12 Nx = 13 Nx = 14 Nx = 25 Nx = 26 Nx = 27 Nx = 28 Nx = 29 Nx = 30**

(a) (b)

5.0E-05

(steady flow, Re = 30). b) Maximum truncature error in the Chebyshev basis

(steady flow, Re=30, Pr = 0.73)

Fig. 2. a) Maximum truncature error in the Chebyshev basis development of the stream

θ

In order to study the dynamic behaviour of the flow according to the flow-rate, we varied the Reynolds number from 1 to 50. Figure 3 shows that the flow remains "with parallel lines", i.e. of crawling type, until Re 10 = . From this value, a vortex initially appears in the

1.0E-04

1.5E-04

2.0E-04

2.5E-04

**Error**

of the stream function coefficients

*kl* ) relative to the steady flow. The expression of *e*

β

α.

are negligible, about <sup>4</sup> 2.10<sup>−</sup> .

ψ

= − (74)

decrease when the values of *Nx* increase. With *Nr* fixed

**Mx** 44 46 48 50 52 54 56

αβis:

αβ

*kl* (respectively the

depend mainly

**Mr = 9 Mr = 8 Mr = 7 Mr = 5**

**Legend**

truncature orders

α

where *f*

0.002

function

ψ

**9.2 The steady flow** 

0.004

0.006

**Error**

temperature coefficients

represents

that, globally, the amplitudes of *e*

to 5 and 30 *Nx* ≥ , the truncature errors *e*

αand β

θ

ψ or θ

The instantaneous convective heat transfer in unsteady flows can formally be defined by the local Nusselt number *Nu x t* (,) , given by the relation:

$$\text{Nu}\left(\mathbf{x},t\right) = \frac{\text{Rh}\_T\left(\mathbf{x},t\right)}{k} \tag{71}$$

With the variables transform (heading 4), the Nusselt number can be written as follows:

$$Nu(\mathbf{x}, t) = \frac{1}{h\lambda} \sqrt{\lambda^2 + h'^2} \frac{\left(\frac{\partial \theta(\mathbf{x}, t)}{\partial r}\right)\_W}{1 - \theta\_m(\mathbf{x}, t)}\tag{72}$$

where *h*′ is the derivative oh the function *h* .

#### **9. Numerical results**

#### **9.1 Definition of geometrical, physical and numerical parameters**

All results have been computed with Galerkin spectral method, except those used to make the comparison between Galerkin and collocation method (headings 6 and 7). The source of pulsations is located at the inlet section.The studied fluid is air, under normal conditions of temperature and pressure. The fluid flow is submitted to a pure sinusoidal pulsation. The previous studies [1, 2] showed that the numerical results are in the more stable mode if the ratio *R L* is small, compared to the unit. Consequently, the basic geometry parameters are:

$$R = 0,02\ m \text{ : } L = 0,08\ m \text{ : } e = E/R = 2 \text{ : } R\_V = 3R = 0,06\ m \text{ : } L$$

The sinusoidal surface of the wall is represented by the function *h*:

$$h(\mathbf{x}) = 1 + \frac{e}{2} \left[ 1 - \cos \left( \pi . n\_O(\mathbf{x} + 1) \right) \right] \tag{73}$$

where *nO* indicates the number of geometrical periods chosen here equals to 3.

Fig. 1. Geometry of the study domain showing the projection of the Chebyshev -Gauss-Lobatto mesh grid on the physical geometry (Nx=30 , Nr=5)

#### **9.1.1 Choice for the orders of truncature**

To ensure the accuracy of our results from the numerical point of view, we try several orders of truncature in the Chebyshev basis developments (Batchi, 2005) for details. When the orders of truncature increase, let *e*αβbe the error calculated between two consecutive truncature orders α and β of the stream function coefficients ψ *kl* (respectively the temperature coefficientsθ *kl* ) relative to the steady flow. The expression of *e*αβis:

$$e\_{\alpha\beta} = \max\_{k,l} \left| f\_{kl}^{\alpha} - f\_{kl}^{\beta} \right| \tag{74}$$

where *f* α represents ψ or θ , for the truncature order α.

96 Convection and Conduction Heat Transfer

The instantaneous convective heat transfer in unsteady flows can formally be defined by the

( ) ( , ) , *Rh x t <sup>T</sup> Nu x t*

2 2

*h x t*

<sup>1</sup> , 1 ,

All results have been computed with Galerkin spectral method, except those used to make the comparison between Galerkin and collocation method (headings 6 and 7). The source of pulsations is located at the inlet section.The studied fluid is air, under normal conditions of temperature and pressure. The fluid flow is submitted to a pure sinusoidal pulsation. The previous studies [1, 2] showed that the numerical results are in the more stable mode if the ratio *R L* is small, compared to the unit. Consequently, the basic geometry parameters are:

*R =* 0,02 *m* ; *L m* = 0,08 ; *e ER* = = 2 , 3 0,06 *RR m <sup>V</sup>* = = .

( ) 1 1 cos . 1 ( ) ( ) <sup>2</sup> *<sup>O</sup> <sup>e</sup> h x* =+ − + <sup>⎡</sup>

**INLET OUTLET**

*Total Length = 2L*

**AXIS**

To ensure the accuracy of our results from the numerical point of view, we try several orders of truncature in the Chebyshev basis developments (Batchi, 2005) for details. When

**WALL**

*E H(z)*

where *nO* indicates the number of geometrical periods chosen here equals to 3.

0 5 10 15 20

αβ

Fig. 1. Geometry of the study domain showing the projection of the Chebyshev -Gauss-

π

With the variables transform (heading 4), the Nusselt number can be written as follows:

λ

( )

**9.1 Definition of geometrical, physical and numerical parameters** 

The sinusoidal surface of the wall is represented by the function *h*:

Lobatto mesh grid on the physical geometry (Nx=30 , Nr=5)

**9.1.1 Choice for the orders of truncature** 

the orders of truncature increase, let *e*

*Nu x t h*

λ

*<sup>k</sup>* <sup>=</sup> (71)

( )

*x t r*

⎛∂ ⎞

θ

*m*

 θ ,

( )

*W*

⎜ ⎟ <sup>∂</sup> ⎝ ⎠ = + ′ <sup>−</sup> (72)

*n x* <sup>⎤</sup> <sup>⎣</sup> <sup>⎦</sup> (73)

be the error calculated between two consecutive

local Nusselt number *Nu x t* (,) , given by the relation:

where *h*′ is the derivative oh the function *h* .

**9. Numerical results** 

*R*

For the dynamic point of view, we note first that the truncature errors *e*αβ depend mainly on the parameter *Nx*. This means that the increase in the number of polynomials in the radial direction does not improve the convergence of the results. Secondly, figure 2.a shows that, globally, the amplitudes of *e*αβ decrease when the values of *Nx* increase. With *Nr* fixed to 5 and 30 *Nx* ≥ , the truncature errors *e*αβare negligible, about <sup>4</sup> 2.10<sup>−</sup> .

For a given value of *Mr*, we observe in figure 2.b a good convergence of the temperature coefficients when *Mx* increases. But, unlike dynamic field, for the range of *Mr* values between 5 and 9, the analysis of the thermal field leads to slightly different conclusions. Indeed, probably due to the temperature conditions imposed on the entry section, the thermal field is more sensitive to the parameter *Mr* than dynamic field. For a fixed value of *Mx*, the temperature truncature errors increase with *Mr*. Then, optimal convergence is obtained for *Mr* = 5. For this value, the truncature error is less than <sup>8</sup> 10<sup>−</sup> when 56 *Mx* > . In conclusion, we have selected for the dynamic problem: *Nx* = 30 and *Nr* = 5, and for the thermal problem, we have chosen: *Mx* =120 and *Mr* = 5.

Fig. 2. a) Maximum truncature error in the Chebyshev basis development of the stream function ψ (steady flow, Re = 30). b) Maximum truncature error in the Chebyshev basis development of the temperature function θ(steady flow, Re=30, Pr = 0.73)

#### **9.2 The steady flow 9.2.1 Study of the dynamic field**

In order to study the dynamic behaviour of the flow according to the flow-rate, we varied the Reynolds number from 1 to 50. Figure 3 shows that the flow remains "with parallel lines", i.e. of crawling type, until Re 10 = . From this value, a vortex initially appears in the

Convective Heat Transfer of Unsteady Pulsed Flow in Sinusoidal Constricted Tube 99

by period is equal to 24. The Reynolds number is fixed to 30 corresponding to a total filling of the furrows. The corresponding steady regime is taken as initial condition for the

To understand better the fluid dynamic behaviour in pulsed regime, figure 5 shows the detail of the streamlines for one period *T*. We note that the vortices quickly disappear during the first instants, from *t=0* to *t=2T/8*. This interval of time corresponds to the phase of the flow acceleration, with a maximum reached for *t=2T/8*. After that, a phase of deceleration appears, with a passage to zero for *t=T/2*. The size of the vortex is maximal for

central zone, the flow moves in positive direction, and close to the wall, the flow moves in opposite direction. After this, the fluid moves more closer to the wall. For the acceleration phase which follows, the flow tends to take its initial aspect again. However, with *t=T*, we

t = 6T/8

π

τ= ). In the

*t=6T/8*. This stage corresponds to the maximum of the flow deceleration ( Ω*t* = 3 /2

unsteady mode (instant *t=0*).

t = 4T/8

region.

approximately find the form of the flow for *t=0*.

t=0 t = 2T/8

t = 15T/16 t=T

**9.3.1 Temporal evolution of the unsteady temperature field** 

**Point 2 Point 1** **Point 3**

Fig. 5. Time history streamlines during one period, Ω=0.3, Re=30, Pr=0.73, 0.7

**Point 4**

Let us locate first particular control points in the duct (figure 6). Each point is chosen because we expect significant results on dynamical and thermal phenomena close to this

**Point 5**

**Point 6**

In order to have a global vision of the dynamic and thermal unsteady phenomena, we carried out a spectral analysis with the FFT method, for the velocity and temperature fields, on three temporal periods (t >10). The figures 7.a and 7.b show that the most significant

Fig. 6. Localization of control points for the description of the time-history phenomena

**Point 7**

**Point 8**

**Point 10 Point 9**

first geometrical period, with a center shifted upstream and close to the wall. Then, when Re increases, a less bulky vortex appears in the two other geometrical periods. The center of each vortex moves towards the downstream while moving away from the wall more and more gradually. These results perfectly agree with those previously shown by Blancher, 1991; Batina et al., 2004, 2009.

Fig. 3. Streamlines parametric study versus Reynolds number (steady flow)

#### **9.2.2 Thermal study**

Figure 4 shows a comparative study of the convective heat transfer by means of the Nusselt number, in stationary regime. One can clearly see that the vortex has a negative influence on the heat transfer on almost the totality of the duct, except for the entry. Locally, we observe a light heat transfer enhancement at the constriction which increases with the amplitude of geometry.

Fig. 4. Heat transfer comparison (steady case): parametric study according to the reduced amplitude e of the geometry

#### **9.3 The unsteady flow**

In order to maintain the bidimensional hypothesis, the flow is submitted to low frequencies ( 0 5 ≤ ≤ *f Hz* ) and the amplitude of pulsation τdo not exceed 0.7. The number of time steps

first geometrical period, with a center shifted upstream and close to the wall. Then, when Re increases, a less bulky vortex appears in the two other geometrical periods. The center of each vortex moves towards the downstream while moving away from the wall more and more gradually. These results perfectly agree with those previously shown by Blancher,

Fig. 3. Streamlines parametric study versus Reynolds number (steady flow)

Figure 4 shows a comparative study of the convective heat transfer by means of the Nusselt number, in stationary regime. One can clearly see that the vortex has a negative influence on the heat transfer on almost the totality of the duct, except for the entry. Locally, we observe a light heat transfer enhancement at the constriction which increases with the amplitude of

Legend

z/R

In order to maintain the bidimensional hypothesis, the flow is submitted to low frequencies

τ

0 5 10 15 20

Fig. 4. Heat transfer comparison (steady case): parametric study according to the reduced

Re = 10

Re = 50

e=0 e=1 e=2 e=3

do not exceed 0.7. The number of time steps

1991; Batina et al., 2004, 2009.

Re = 5

Re = 30

Nu

**0**

0

( 0 5 ≤ ≤ *f Hz* ) and the amplitude of pulsation

amplitude e of the geometry

**9.3 The unsteady flow** 

2

4

6

8

10

**9.2.2 Thermal study** 

geometry.

by period is equal to 24. The Reynolds number is fixed to 30 corresponding to a total filling of the furrows. The corresponding steady regime is taken as initial condition for the unsteady mode (instant *t=0*).

To understand better the fluid dynamic behaviour in pulsed regime, figure 5 shows the detail of the streamlines for one period *T*. We note that the vortices quickly disappear during the first instants, from *t=0* to *t=2T/8*. This interval of time corresponds to the phase of the flow acceleration, with a maximum reached for *t=2T/8*. After that, a phase of deceleration appears, with a passage to zero for *t=T/2*. The size of the vortex is maximal for *t=6T/8*. This stage corresponds to the maximum of the flow deceleration ( Ω*t* = 3 /2 π ). In the central zone, the flow moves in positive direction, and close to the wall, the flow moves in opposite direction. After this, the fluid moves more closer to the wall. For the acceleration phase which follows, the flow tends to take its initial aspect again. However, with *t=T*, we approximately find the form of the flow for *t=0*.

Fig. 5. Time history streamlines during one period, Ω=0.3, Re=30, Pr=0.73, 0.7 τ=

#### **9.3.1 Temporal evolution of the unsteady temperature field**

Let us locate first particular control points in the duct (figure 6). Each point is chosen because we expect significant results on dynamical and thermal phenomena close to this region.

Fig. 6. Localization of control points for the description of the time-history phenomena

In order to have a global vision of the dynamic and thermal unsteady phenomena, we carried out a spectral analysis with the FFT method, for the velocity and temperature fields, on three temporal periods (t >10). The figures 7.a and 7.b show that the most significant

Convective Heat Transfer of Unsteady Pulsed Flow in Sinusoidal Constricted Tube 101

On figure 8, we study, on the control point 4, the Nusselt number evolution versus the pulsation frequency Ω . This amplitude analysis is obtained by the FFT method realised on the instantaneous Nusselt number defined by equation (71). We observe the decrease of the

The instantaneous heat transfer does not correspond to a measurable physical reality. Thus

() ( ) ( )

To evaluate the contribution of the pulsation on the heat transfer, we compare *Nu x*( ) with the Nusselt number *Nu*<sup>0</sup> *(x)* obtained in steady flow. We confirm in figure 9, a very significant increase of the heat transfer located at the constriction, and conversely a high

Ω

Ω=5 Ω = 10

z/R

τ= )

0 5 10 15 20

Fig. 9. Heat transfer comparison in steady and unsteady flow(Ω=10, Ω=5, 0.7

Fig. 8. Evolution of Nusselt number (FFT method) versus the pulsation frequency on the

6 Steady case

1 2 3 4 5

π

*Nu x Nu x t Nu x t dt*

2 <sup>0</sup> , , <sup>2</sup>

π

ω ω

= = ∫ (75)

it is necessary to consider the time averaged Nusselt number. So, we define:

**9.4 Unsteady convective heat transfer** 

Nusselt number amplitudes when Ω increases.

reduction at maximum radius areas (zones of dead fluid).

Nu

4.25

Nu

\_\_

1

2

3

4

5

control point 4 (Re=30; Pr=0.73; τ=0.7)

4.3

4.35

4.4

dynamic fluctuations are located at each constriction of the tube for axial velocities and downstream the constriction for radial velocities.

Fig. 7. Amplitudes fluctuations of the axial velocity (a), the radial velocity (b) and the temperature (c). Re=30, Pr=0.73, 0.7 τ=

One can thus expect a substantial modification of the thermal convective heat transfer in these privileged areas, due to the thermal boundary modifications corresponding to the entry section duct, and in the minimum sections as shown in figure 7.c.

(c)

#### **9.4 Unsteady convective heat transfer**

100 Convection and Conduction Heat Transfer

dynamic fluctuations are located at each constriction of the tube for axial velocities and

0.082 0.165 0.247 0.329 0.412 0.494 0.576 0.659 0.741 0.824 0.906 0.988 1.071 1.153 1.235 (a)

0.018 0.035 0.053 0.071 0.088 0.106 0.124 0.141 0.159 0.177 0.194 0.212 0.230 0.247 0.265 (b)

0.004 0.011 0.019 0.026 0.034 0.042 0.049 0.057 0.064 0.072 0.080 0.087 0.095 0.102 0.110

(c)

One can thus expect a substantial modification of the thermal convective heat transfer in these privileged areas, due to the thermal boundary modifications corresponding to the

Fig. 7. Amplitudes fluctuations of the axial velocity (a), the radial velocity (b) and the

τ=

entry section duct, and in the minimum sections as shown in figure 7.c.

temperature (c). Re=30, Pr=0.73, 0.7

downstream the constriction for radial velocities.

On figure 8, we study, on the control point 4, the Nusselt number evolution versus the pulsation frequency Ω . This amplitude analysis is obtained by the FFT method realised on the instantaneous Nusselt number defined by equation (71). We observe the decrease of the Nusselt number amplitudes when Ω increases.

The instantaneous heat transfer does not correspond to a measurable physical reality. Thus it is necessary to consider the time averaged Nusselt number. So, we define:

$$\overline{Nu}\left(\mathbf{x}\right) = \left\langle \mathrm{Nu}\left(\mathbf{x}, t\right) \right\rangle = \frac{\alpha}{2\pi} \int\_0^{2\pi} \overline{\mathrm{Nu}}\left(\mathbf{x}, t\right) dt\tag{75}$$

To evaluate the contribution of the pulsation on the heat transfer, we compare *Nu x*( ) with the Nusselt number *Nu*<sup>0</sup> *(x)* obtained in steady flow. We confirm in figure 9, a very significant increase of the heat transfer located at the constriction, and conversely a high reduction at maximum radius areas (zones of dead fluid).

Fig. 8. Evolution of Nusselt number (FFT method) versus the pulsation frequency on the control point 4 (Re=30; Pr=0.73; τ=0.7)

Fig. 9. Heat transfer comparison in steady and unsteady flow(Ω=10, Ω=5, 0.7 τ= )

Convective Heat Transfer of Unsteady Pulsed Flow in Sinusoidal Constricted Tube 103

not exceed two or three percent. In terms of unsteady heat transfer, comparison between the two methods shows that the average Nusselt number *Nu x*( ) given by equation (75) presents slight differences similar to those observed in steady state regime (figure 11.b.).

*Nu*

*0*

5

10

15

20

25

30

*z/R*

**Legend:**

*Galerkin Collocation*

0 5 10 15 20

*z/R*

case; (b): unsteady case (Ω=0.3, Re=30, Pr=0.73, 0.7

**Legend:**

*Galerkin Collocation*

(a) (b)

**9.5.2 Comparison of performances and speed computations** 

with Galerkin and collocation method respectively.

*CPU CPU*

The table below summarizes their performances. Let us define before:

1 *G C*

Then, we obtain the following results:, performed on the same computer:

Fig. 11. Heat transfer comparison between Galerkin and collocation methods. (a): steady

τ= )

We have shown that Galerkin and collocation spectral methods, developed to the same truncature orders, give results with similar accuracy. The essential difference between these methods lies in their performances, rapidity and simplicity of computational implementation.

• On the one hand: *CPU*0*<sup>G</sup>* and *CPU*0*<sup>C</sup>* the CPU time to obtain the steady regime with Galerkin and collocation method respectively; *CPU*1*<sup>G</sup>* and *CPU*1*<sup>C</sup>* the CPU time to compute one period of the unsteady regime with Galerkin and collocation method

• On the other hand: *Newton*0*<sup>G</sup>* and *Newton*0*<sup>C</sup>* the number of Newton iterations to compute the steady regime with Galerkin and collocation method respectively; *Newton*1*<sup>G</sup>* and *Newton*1*<sup>C</sup>* the number of Newton iterations to compute one time step

> *CPU* <sup>=</sup> ; <sup>0</sup> 0

*CPU*<sup>0</sup> *CPU*<sup>1</sup> *Newton*<sup>0</sup> *Newton*<sup>1</sup> 2.4 5.2 3.2 3.2

Table 1. Comparison of performances between Galerkin and collocation spectral methods

*Newton Newton*

0 *G C*

*Newton* <sup>=</sup> ; <sup>1</sup>

1

*Newton Newton*

1 *G C*

*Newton* <sup>=</sup> (76)

0 5 10 15 20

*Nu*

*\_\_\_*

5

respectively;

At last, we define the ratios:

0

*CPU CPU*

0

*CPU* <sup>=</sup> ; <sup>1</sup> 1

0 *G C*

10

15

20

25

30

#### **9.5 Comparison between Galerkin and collocation spectral methods 9.5.1 Dynamic and thermal results comparison**

In order to make comparison between Galerkin and collocation spectral methods, classical parameters are chosen: Re=30, Nx = 30 and Nr = 5 for both methods.

When the flow is pulsed, we chose to study the dynamic and thermal fields at points 1, 7, 8 and 9 of figure 6. Apart the periodicity previously highlighted, these curves show once again that there is nearly no difference between the two methods as shown in figures 10.

Fig. 10. Comparison between Galerkin and collocation methods during 3 periods, for control points: (a): point 1; (b): point 7; (c): point 8; (d): point 9. (Ω=0.3, Re=30, Pr=0.73, 0.7 τ= )

In steady flow, the longitudinal evolution of heat transfer characterized by the Nusselt number shows a slight difference between the two models, located particularly in the vicinity of geometry furrows and constrictions (figure 11.a.). However, this difference does not exceed two or three percent. In terms of unsteady heat transfer, comparison between the two methods shows that the average Nusselt number *Nu x*( ) given by equation (75) presents slight differences similar to those observed in steady state regime (figure 11.b.).

Fig. 11. Heat transfer comparison between Galerkin and collocation methods. (a): steady case; (b): unsteady case (Ω=0.3, Re=30, Pr=0.73, 0.7 τ= )

#### **9.5.2 Comparison of performances and speed computations**

We have shown that Galerkin and collocation spectral methods, developed to the same truncature orders, give results with similar accuracy. The essential difference between these methods lies in their performances, rapidity and simplicity of computational implementation. The table below summarizes their performances. Let us define before:


At last, we define the ratios:

102 Convection and Conduction Heat Transfer

In order to make comparison between Galerkin and collocation spectral methods, classical

When the flow is pulsed, we chose to study the dynamic and thermal fields at points 1, 7, 8 and 9 of figure 6. Apart the periodicity previously highlighted, these curves show once again that there is nearly no difference between the two methods as shown in figures 10.

**9.5 Comparison between Galerkin and collocation spectral methods** 

Collocation *u* ψ θ

(a) (b)

(c) (d)

θ ψ

> *u, ,*

ψ

θ

0

*u, ,*

Fig. 10. Comparison between Galerkin and collocation methods during 3 periods, for control points: (a): point 1; (b): point 7; (c): point 8; (d): point 9. (Ω=0.3, Re=30, Pr=0.73, 0.7

In steady flow, the longitudinal evolution of heat transfer characterized by the Nusselt number shows a slight difference between the two models, located particularly in the vicinity of geometry furrows and constrictions (figure 11.a.). However, this difference does

ψ

θ

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0.5

1

1.5

2

*Dimensionless time*

*Dimensionless time*

0 0.2 0.4 0.6 0.8

τ= )

0 0.2 0.4 0.6 0.8

Galerkin *u*

parameters are chosen: Re=30, Nx = 30 and Nr = 5 for both methods.

**9.5.1 Dynamic and thermal results comparison** 

**Legends for figures (a), (b), (c) and (d):**

*Dimensionless time*

*Dimensionless time*

0 0.2 0.4 0.6 0.8

0 0.2 0.4 0.6 0.8

*u, ,*

*u, ,*

ψ

θ

0

0.2

0.4

0.6

0.8

1

ψ

θ

0

0.5

1

1.5

2

2.5

$$\text{CPUL}\_{0} = \frac{\text{CPUL}\_{0\text{G}}}{\text{CPUL}\_{0\text{C}}}; \text{ CPU}\_{1} = \frac{\text{CPUL}\_{1\text{G}}}{\text{CPUL}\_{1\text{C}}}; \text{ Newton}\_{0} = \frac{\text{Newton}\_{0\text{G}}}{\text{Newton}\_{0\text{C}}}; \text{ Newton}\_{1} = \frac{\text{Newton}\_{1\text{G}}}{\text{Newton}\_{1\text{C}}} \tag{76}$$

Then, we obtain the following results:, performed on the same computer:


Table 1. Comparison of performances between Galerkin and collocation spectral methods

Convective Heat Transfer of Unsteady Pulsed Flow in Sinusoidal Constricted Tube 105

Actually, our model is extended to non linear coupled problems of unsteady Navier Stokes and Energy equations, and the results are very encouraging. Moreover, our model treats

In a final analysis, our numerical method based on a suitable spectral method is of a good accuracy. One of its originality is the choice of Chebyshev polynomials basis in both axial and radial directions, and the use of a shift operator technique allow the introduction of non homogeneous boundary conditions in Galerkin formulation. The automatic construction of these polynomials basis is of a great interest. These particular mathematical and numerical tools have permitted the resolution of this non obvious problem which consists on pulsated unsteady flows associated to simultaneous developments dynamic and thermal fields.

André, P.; Creff, R. & Batina, J. (1981). Etude des conditions particulières de fréquence

Batchi, M. (2005). Etude Mathématique et Numérique des Phénomènes de Transferts

Batina, J.; Creff, R.; André, P. & Blancher, S. (1989). Numerical model for dynamic and

Batina, J.; André, P.; Creff, R. & Blancher, S. (1991). Dynamic and thermal developments of a

Batina, J. (1995). Etude numérique des écoulements instationnaires pulsés en canalisation

Batina, J.; Creff, R. & Batchi, M. (2004). Etude thermique convective d'un écoulement interne en géométrie axisymétrique sinusoïdale, *Actes du congrès SFT 04*, pp. 987-992. Batina, J.; Batchi, M.; Blancher, S.; Creff, R. & Amrouche, C. (2009). Convective heat transfer

Blancher, S. (1991). Transfert convectif stationnaire et stabilité hydrodynamique en géométrie

Canuto, C.; Hussaini, M.Y.; Quarteroni, A. & Zang, T.A. (1988). *Spectral methods in fluids* 

Chakravarty, S. & Sannigrahi A.K. (1999). A nonlinear mathematical model of blood flow in

Creff, R.; André, P. & Batina, J. (1985). Dynamic and Convective Results for a Developping

a constricted artery experiencing body acceleration, *Mathematical and Computer* 

Laminar Unsteady Flow, *International Journal of Numerical Methods in Fluids*, Vol. 5,

conference., Montreal, Canada, Pineridge Press, pp. 149-154, 1987.

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cylindrique, *Thèse de Doctorat Physique*, Université de Pau.

périodique, *Thèse de Doctorat Physique*, Université de Pau.

n°9, pp. 50-57, Bochum, Germany.

Pau, France.

Springer-Verlag.

pp. 745-760.

*dynamics,* Springer-Verlag.

*Modeling*, vol. 29, pp. 9-25.

favorisant les transferts thermiques en écoulements pulsés en canalisation cylindrique, *International Journal of Heat and Mass Transfer*, Vol. 24, pp. 1211-1219. André, P.; Creff, R. & Batina, J. (1987). Study of thermal fluid for pulsed flow with

compressible fluids, *Numerical Methods in Thermal Problems*., Fifth international

Thermiques liés aux Ecoulements Instationnaires en Géométrie Axisymétrique,

thermal developments of a turbulent pulsed ducted flow*, Proceedings Eurotherm*,

pulsed laminar compressible ducted flow, *Proceedigs Eurotherm*, n°25, pp. 227-234,

augmentation through vortex shedding in sinusoidal constricted tube., *International Journal of Numerical Methods for Heat and Fluid Flow*, Vol.19, N°.3/4, pp. 374-395. Bernardi, C. & Maday, Y. (1992). *Approximations spectrales de problèmes aux limites elliptiques*,

successfully the problem of heat and mass transfer in natural convection.

**11. References** 

### **10. Conclusions**

In this paper, numerical studies have been carried out on pulsating flows through axisymmetric sinusoidal ducts. Thus, the study emphasizes on the heat transfer modifications in this particular flows with rates modulation close to the unit. The results obtained have shown that the flow is influenced by many parameters including Reynolds numbers, rate modulation and amplitude of geometry. We observe that the results are encouraging and offer good perspectives in pulsed internal flows cases. From numerical point of view, the results obtained confirm the previous general conclusions in axisymmetric geometries (André et al., 1987, 1981; Batina et al., 1989, 1991, 2009), i.e.:


The numerical solution was performed using Galerkin and collocation spectral methods whose main features were recalled. The numerical results obtained show that:


However, the collocation method is simpler to use and its computer calculations are faster.

Out of these general conclusions, this study focusing on sinusoidal geometries induce especially zones of dead fluid that locally have a negative influence on heat transfer, particularly for the steady flow. The spatial periodicity of the steady flow in this type of geometry is acquired only at the end of the second, even third geometrical period. The transient phenomena are therefore relatively short in time. Thus, the dynamic and thermal flow behaviours become periodic.

Compared to models based on classical methods such as asymptotic developments, finite differences, finite volumes, finite elements, etc, our numerical method leads to the following remarks:

	- the non linear coupled unsteady terms in Navier Stokes and Energy equations are not taken into account in the present model
	- this problem require smoother geometries
	- the order of polynomials developments increases strongly the computing time
	- when the modulation flow-rate approaches or exceeds the unity, we must choose carefully some data to assume the algorithms convergence. For example, when 100% τ > , convergence conditions impose small time steps, and the CPU time on classical computers can make our code prohibitive.

Actually, our model is extended to non linear coupled problems of unsteady Navier Stokes and Energy equations, and the results are very encouraging. Moreover, our model treats successfully the problem of heat and mass transfer in natural convection.

In a final analysis, our numerical method based on a suitable spectral method is of a good accuracy. One of its originality is the choice of Chebyshev polynomials basis in both axial and radial directions, and the use of a shift operator technique allow the introduction of non homogeneous boundary conditions in Galerkin formulation. The automatic construction of these polynomials basis is of a great interest. These particular mathematical and numerical tools have permitted the resolution of this non obvious problem which consists on pulsated unsteady flows associated to simultaneous developments dynamic and thermal fields.

#### **11. References**

104 Convection and Conduction Heat Transfer

In this paper, numerical studies have been carried out on pulsating flows through axisymmetric sinusoidal ducts. Thus, the study emphasizes on the heat transfer modifications in this particular flows with rates modulation close to the unit. The results obtained have shown that the flow is influenced by many parameters including Reynolds numbers, rate modulation and amplitude of geometry. We observe that the results are encouraging and offer good perspectives in pulsed internal flows cases. From numerical point of view, the results obtained confirm the previous general conclusions in axisymmetric

• For the unsteady regime, dynamic and thermal fields show an important heat transfer enhancement in the entry zone. A dynamic and thermal shock occur nearly this area. Mechanical tube behaviour can be modified in this region and the shear stress occurring during the pulsation can induce some damage if the tube is connected to a big tank. This phenomenon is of great interest in industrial structures. Nevertheless, convective

The numerical solution was performed using Galerkin and collocation spectral methods

• The unsteady dynamic and thermal phenomena are periodic in time, with a frequency

However, the collocation method is simpler to use and its computer calculations are faster. Out of these general conclusions, this study focusing on sinusoidal geometries induce especially zones of dead fluid that locally have a negative influence on heat transfer, particularly for the steady flow. The spatial periodicity of the steady flow in this type of geometry is acquired only at the end of the second, even third geometrical period. The transient phenomena are therefore relatively short in time. Thus, the dynamic and thermal

Compared to models based on classical methods such as asymptotic developments, finite differences, finite volumes, finite elements, etc, our numerical method leads to the following

• The present computational code is easier to build compared to finite elements one, for

• If we consider the CPU time, the present model needs few minutes to compute the numerical equations. This result traduces the efficiency of our model which is easier and more adapted to solve this particular problem. Nevertheless, compared to some


> , convergence conditions impose small time steps, and the CPU time on


industrial codes, our model have some disadvantages, such as:

not taken into account in the present model - this problem require smoother geometries

classical computers can make our code prohibitive.

geometries (André et al., 1987, 1981; Batina et al., 1989, 1991, 2009), i.e.:

heat transfer decrease when the fluid moves forward in the tube.

equal to the frequency of pulsations.

• The accuracy of the present model is high.

flow behaviours become periodic.

remarks:

example.

 100% τ

whose main features were recalled. The numerical results obtained show that: • Steady and unsteady regimes do not depend on the spectral method used;

• For the steady regime: all classical results are obtained with high precision.

**10. Conclusions** 


**5** 

*Slovenia* 

**Numerical Solution of Natural** 

Gregor Kosec and Božidar Šarler

*University of Nova Gorica* 

**Convection Problems by a Meshless Method** 

Natural convection is a phenomenon where fluid motion is generated by density changes due to the temperature or concentration variations in a gravity field. The computational modelling of systems with natural convection (Bejan, 2004) has become a highly popular research subject due to its pronounced influence in better understanding of nature as well as in the development of the advanced technologies. Melting of the polar ice caps, the global oceans dynamics, various weather systems, water transport, soil erosion and denudation, magma transport and manufacturing of nano-materials, improving casting processes, energetic studies, exploitation of natural resources, welding, casting and advanced solidifications are two typical contemporary example groups where natural convection plays an important role. This chapter deals with the numerical approach towards solution of

The main part of the solution procedure is focused on the general transport equation treatment and the pressure velocity coupling strategy. The transport phenomena are solved by a local meshless method and explicit time stepping. The local variant of Radial Basis Function Collocation Method (LRBFCM) has been previously developed for diffusion problems (Šarler and Vertnik, 2006), convection-diffusion solid-liquid phase change problems (Vertnik and Šarler, 2006) and subsequently successfully applied in industrial

The fluid flow, which is generally a global problem, is treated by the proposed local iterative method. Instead of solving the pressure Poisson equation or/and pressure correction Poisson equation (Divo and Kassab, 2007) a more simplified local pressure-velocity coupling (LPVC) (Kosec and Šarler, 2008a) algorithm is proposed where the pressure-correction is predicted from the local mass continuity violation similar to the SOLA algorithm (Hong,

The presented solution procedure represents a variant of already developed global approach (Šarler*, et al.*, 2004a, Šarler, 2005). In this chapter, such a local solution procedure is tested with the standard free fluid flow benchmark test (de Vahl Davis natural convection test (de Vahl Davis, 1983)). The test is especially convenient for benchmarking purposes as there are several numerical solutions published in the literature (Divo and Kassab, 2007, Hortmann*, et al.*, 1990, Manzari, 1999, Prax*, et al.*, 1996, Sadat and Couturier, 2000, Šarler,

**1. Introduction** 

2004).

2005, Wan*, et al.*, 2001).

this type of problems by a meshless technique.

process of direct chill casting (Vertnik*, et al.*, 2006).


## **Numerical Solution of Natural Convection Problems by a Meshless Method**

Gregor Kosec and Božidar Šarler *University of Nova Gorica Slovenia* 

#### **1. Introduction**

106 Convection and Conduction Heat Transfer

Ghaddar, N.K.; Magen, M.; Mikic, B.B. & Patera, A. (1986). Numerical investigation of

Gelfgat, A.Y. (2004). Stability and slightly supercritical oscillatory regimes of natural

Lee, B.S.; Kang, L.S. & Lim, H.C. (1999). Chaotic mixing and mass transfer enhancement by

Moschandreau, T. & Zamir, M. (1997). Heat transfer in a tube with pulsating flow and

Shen, J. (1994). Efficient spectral-Galerkin method I: direct solvers for the second and fourth

Shen, J. (1995). Efficient spectral-Galerkin methods II: direct solvers of second and fourth

Shen, J. (1997). Efficient spectral-Galerkin methods III: polar and cylindrical geometries,

Sobey, I.J. (1980). On flow through channels. Part 1: Calculated flow patterns, *Journal of Fluid* 

Young Kim, S.; Ha Kang, B. & Min Hyun, J. (1998). Forced convection heat transfer from two

enhancement, *Journal of Fluid Mechanics*, vol.168, pp. 541-567.

*Transfer*, vol. 45, pp. 1767-178.

2461-2466.

1489-1505.

pp. 74-87.

*Heat and Mass Transfer*, Vol. 42, pp. 2571-2581.

*SIAM J. Sci. Comput*, vol.18, n°6, 1583-1604.

*Mechanics*, vol. 96, pp. 1-26.

*Transfer*, vol. 41, n°3, pp. 625-634.

incompressible flow in grooved channels, Resonance and oscillatory heat transfer

convection in a 8:1 cavity: solution of the benchmark problem by a global Galerkin method, *International Journal of Numerical Methods in Fluids*, vol. 44, pp. 135-146. Hemida, H.N.; Sabry, M.N.; Abdel-Rahim, A. & Mansour, H. (2002). Theoretical analysis of

heat transfer in laminar pulsating flow, *International Journal of Heat and Mass* 

pulsatile laminar flow in an axisymmetric wavy channel, *International Journal of* 

constant heat flux, *International Journal of Heat and Mass Transfer*, vol.. 40, n°10, pp.

order equations using Legendre polynomials, *SIAM J. Sci. Comput*, vol.15, n°6, pp.

order equations by using Chebyshev polynomials, *SIAM J. Sci. Comput*, vol.16, n°8,

heated blocks in pulsating channel flow, *International Journal of Heat and Mass* 

Natural convection is a phenomenon where fluid motion is generated by density changes due to the temperature or concentration variations in a gravity field. The computational modelling of systems with natural convection (Bejan, 2004) has become a highly popular research subject due to its pronounced influence in better understanding of nature as well as in the development of the advanced technologies. Melting of the polar ice caps, the global oceans dynamics, various weather systems, water transport, soil erosion and denudation, magma transport and manufacturing of nano-materials, improving casting processes, energetic studies, exploitation of natural resources, welding, casting and advanced solidifications are two typical contemporary example groups where natural convection plays an important role. This chapter deals with the numerical approach towards solution of this type of problems by a meshless technique.

The main part of the solution procedure is focused on the general transport equation treatment and the pressure velocity coupling strategy. The transport phenomena are solved by a local meshless method and explicit time stepping. The local variant of Radial Basis Function Collocation Method (LRBFCM) has been previously developed for diffusion problems (Šarler and Vertnik, 2006), convection-diffusion solid-liquid phase change problems (Vertnik and Šarler, 2006) and subsequently successfully applied in industrial process of direct chill casting (Vertnik*, et al.*, 2006).

The fluid flow, which is generally a global problem, is treated by the proposed local iterative method. Instead of solving the pressure Poisson equation or/and pressure correction Poisson equation (Divo and Kassab, 2007) a more simplified local pressure-velocity coupling (LPVC) (Kosec and Šarler, 2008a) algorithm is proposed where the pressure-correction is predicted from the local mass continuity violation similar to the SOLA algorithm (Hong, 2004).

The presented solution procedure represents a variant of already developed global approach (Šarler*, et al.*, 2004a, Šarler, 2005). In this chapter, such a local solution procedure is tested with the standard free fluid flow benchmark test (de Vahl Davis natural convection test (de Vahl Davis, 1983)). The test is especially convenient for benchmarking purposes as there are several numerical solutions published in the literature (Divo and Kassab, 2007, Hortmann*, et al.*, 1990, Manzari, 1999, Prax*, et al.*, 1996, Sadat and Couturier, 2000, Šarler, 2005, Wan*, et al.*, 2001).

Numerical Solution of Natural Convection Problems by a Meshless Method 109

where Ω and Γ stand for interior and boundary nodes indexes, respectively.

North wall

*TH*

*y p*

Fig. 1. The natural convection benchmark test schematics

West wall

*x p*

South wall

The introduced dimensionless quantities (embellishment ~) are defined as

*x y*

*v v* ρ

λ

Prandtl number (Pr and the domain aspect ratio ) (A*<sup>R</sup>* ) defined as

*T*

Ω*H*

 *<sup>y</sup> <sup>x</sup> x y W H p p*

*x W p y H p*

*C H C*

2 *p H t t c* λ

( ) 3 2

*W H*

 *TT c* λμ

 ρ

*v c vc*

*T T <sup>T</sup> T T* <sup>−</sup> <sup>=</sup> <sup>−</sup>

> ρ<sup>=</sup> <sup>Ω</sup>

where Ω*W* and Ω*H* stand for domain width and height, respectively. The problem is characterized by three dimensionless numbers; the thermal Rayleigh number ( ) Ra*<sup>T</sup>* , the

Ra = *TH C H <sup>p</sup>*

Pr *<sup>p</sup>* μ*c* λ

AR

**g** − Ω

β

*T t* (**p**<sup>Ω</sup> , 0 0.5 = =) , (10)

*TC*

East wall

*p p* <sup>=</sup> <sup>=</sup> <sup>Ω</sup> <sup>Ω</sup> , (11)

, (13)

, (14)

<sup>=</sup> , (16)

Ω= <sup>Ω</sup> , (17)

, (15)

ρ

Ω Ω <sup>=</sup> <sup>=</sup> , (12)

 λ

<sup>0</sup> *dT <sup>d</sup>* <sup>=</sup> **<sup>n</sup>**

Ω*W*

<sup>0</sup> *dT <sup>d</sup>* <sup>=</sup> **<sup>n</sup>**

In addition to the basic test, the proposed local solution procedure is tested on the tall cavity, natural convection in the porous media and melting of a pure material (Gobin - Le Quéré test) driven by a natural convection tests. Numerous analyses and comparisons with the published data are performed in order to assess the characteristics of the proposed numerical approach in details. The comprehensive verification procedure shows excellent agreements with the previously published data, based on different numerical methods.

#### **2. Governing equations**

#### **2.1 De Vahl Davis test**

The de Vahl Davis natural convection benchmark problem is described by three coupled partial differential equations (PDEs). The PDEs are mass, momentum and energy conservation equations where all material properties are considered to be constant. The Boussinesq approximation is used for density hypothesis and the phenomena is thus described by the following system of equations

$$\nabla \cdot \mathbf{v} = 0 \,, \tag{1}$$

$$
\rho \frac{\partial \mathbf{v}}{\partial t} + \rho \nabla \cdot (\mathbf{v} \mathbf{v}) = -\nabla P + \nabla \cdot (\mu \nabla \mathbf{v}) + \mathbf{b} \tag{2}
$$

$$c\_p \rho \frac{\partial T}{\partial t} + c\_p \rho \nabla \cdot (T\mathbf{v}) = \nabla \cdot \left(\mathcal{A} \nabla T\right)\_{\mathcal{I}} \tag{3}$$

$$\mathbf{b} = \rho \left[ 1 - \beta\_T (T - T\_{\text{ref}}) \right] \mathbf{g} \,\,\, \,\tag{4}$$

where , , , , , , , , , and ( *x y* ) *T ref t vv P T* **vbg** ρ μ λβ stand for time, velocity, density, pressure, viscosity, body force, thermal conductivity, thermal expansion coefficient, reference temperature and gravitational acceleration, respectively. The south and the north sides of the domain are thermally insulated while the east and the west sides are set to different temperatures. Due to the consideration of the viscous fluid within enclosure, the no-slip and impermeable velocity boundary conditions are used (Figure 1). The boundary conditions are formulated as follows

$$\tilde{\mathbf{v}}\left(\tilde{\mathbf{p}}\_{\Gamma'}\tilde{t}\right) = 0 \;/\tag{5}$$

$$
\tilde{T}\left(\tilde{p}\_{\text{x}} = 0, \tilde{t}\right) = 1 \quad , \tag{6}
$$

$$\tilde{T}\left(\tilde{p}\_{\pm} = 1, \tilde{t}\right) = 0 \,, \tag{7}$$

$$\frac{\partial}{\partial \tilde{p}\_y} \tilde{T}\left(\tilde{p}\_y = 0, \tilde{t}\right) = \frac{\partial}{\partial \tilde{p}\_y} \tilde{T}\left(\tilde{p}\_y = 1, \tilde{t}\right) = 0 \tag{8}$$

and the initial conditions as follows

$$\tilde{\mathbf{v}}\left(\tilde{\mathbf{p}}\_{\Omega'}\tilde{t}=0\right)=\mathbf{0}\,,\tag{9}$$

In addition to the basic test, the proposed local solution procedure is tested on the tall cavity, natural convection in the porous media and melting of a pure material (Gobin - Le Quéré test) driven by a natural convection tests. Numerous analyses and comparisons with the published data are performed in order to assess the characteristics of the proposed numerical approach in details. The comprehensive verification procedure shows excellent agreements with the previously published data, based on different numerical methods.

The de Vahl Davis natural convection benchmark problem is described by three coupled partial differential equations (PDEs). The PDEs are mass, momentum and energy conservation equations where all material properties are considered to be constant. The Boussinesq approximation is used for density hypothesis and the phenomena is thus

( ) *<sup>P</sup>* ( ) *<sup>t</sup>*

( ) ( ) *p p <sup>T</sup> c cT T*

**b g** =− −

viscosity, body force, thermal conductivity, thermal expansion coefficient, reference temperature and gravitational acceleration, respectively. The south and the north sides of the domain are thermally insulated while the east and the west sides are set to different temperatures. Due to the consideration of the viscous fluid within enclosure, the no-slip and impermeable velocity boundary conditions are used (Figure 1). The boundary conditions are

**v p** ( <sup>Γ</sup> , 0 *t*

( ) 0, ( ) 1, 0 *y y*

*Tp t Tp t*

**v p** ( <sup>Ω</sup> ,00 *t*

∂ ∂ <sup>=</sup> = = = ∂ ∂

*y y*

*p p*

 β

<sup>∂</sup> + ∇⋅ =∇⋅ ∇

+ ∇ ⋅ = −∇ + ∇ ⋅ ∇ +

μ

 λ

∇ ⋅ = **v** 0 , (1)

**vv v b** , (2)

[1( ) *<sup>T</sup> T T*ref ] , (4)

**v** , (3)

stand for time, velocity, density, pressure,

) = , (5)

*Tp t* ( *<sup>x</sup>* <sup>=</sup> 0, 1 ) <sup>=</sup> , (6)

*Tp t* ( *<sup>x</sup>* = 1, 0 ) = , (7)

= ) = , (9)

, (8)

**2. Governing equations** 

described by the following system of equations

where , , , , , , , , , and ( *x y* ) *T ref t vv P T* **vbg** ρ

 μ λβ

ρ

∂

∂ **v**

 ρ

*t* ρρ

ρ

∂

**2.1 De Vahl Davis test** 

formulated as follows

and the initial conditions as follows

$$
\tilde{T}\left(\tilde{\mathbf{p}}\_{\Omega\prime}\tilde{t}=0\right)=0.5\tag{10}
$$

where Ω and Γ stand for interior and boundary nodes indexes, respectively.

Fig. 1. The natural convection benchmark test schematics

The introduced dimensionless quantities (embellishment ~) are defined as

$$
\tilde{p}\_x = \frac{p\_x}{\Omega\_W} \qquad \qquad \tilde{p}\_y = \frac{p\_y}{\Omega\_H} \, \, \, \tag{11}
$$

$$
\tilde{\upsilon}\_x = \frac{\upsilon\_x \Omega\_W \rho \upsilon\_p}{\varkappa} \qquad \qquad \tilde{\upsilon}\_y = \frac{\upsilon\_y \Omega\_H \rho \upsilon\_p}{\varkappa} \,, \tag{12}
$$

$$
\tilde{T} = \frac{T - T\_{\odot}}{T\_H - T\_{\odot}} \, \text{} \tag{13}
$$

$$\tilde{\mathbf{f}} = \frac{\tilde{\lambda}}{\rho \mathbf{c}\_p \Omega\_H^2} \mathbf{t}\_{\prime} \tag{14}$$

where Ω*W* and Ω*H* stand for domain width and height, respectively. The problem is characterized by three dimensionless numbers; the thermal Rayleigh number ( ) Ra*<sup>T</sup>* , the Prandtl number (Pr and the domain aspect ratio ) (A*<sup>R</sup>* ) defined as

$$\mathbf{Ra}\_{T} = \frac{|\mathbf{g}| \, \beta\_{\Gamma} \left(T\_{H} - T\_{C}\right) \Omega\_{H}{}^{3} \rho^{2} c\_{p}}{\lambda \mu} \,, \tag{15}$$

$$\mathbf{Pr} = \frac{\mu \mathbf{c}\_p}{\lambda},\tag{16}$$

$$\mathbf{A}\_{\rm R} = \frac{\boldsymbol{\Omega}\_{\rm W}}{\boldsymbol{\Omega}\_{\rm H}} \; \tag{17}$$

Numerical Solution of Natural Convection Problems by a Meshless Method 111

( ) ( ) *<sup>h</sup> h T*

() *hT cT* = + *<sup>p</sup> f <sup>L</sup> L* , (23)

*F L*

<sup>⎪</sup> <sup>−</sup> = + >> <sup>⎨</sup>

δ

*TT T*

*T T fT T T TT <sup>T</sup>*

*T T*

*F*

The phase change of the pure material occurs exactly at the melting temperature which produces discontinues in the enthalpy field due to the latent heat release. The constitutive relation (24) incorporates a smoothing interval near the phase change in order to avoid

**v p** ( <sup>Γ</sup> , 0 *t*

( *y y* 0, ) ( 1, 0 )

*Tp t Tp t*

**v p** ( <sup>Ω</sup> ,00 *t*

<sup>∂</sup> <sup>∂</sup> <sup>=</sup> = = = ∂ ∂

*y y*

*p p*

<sup>∂</sup> + ∇⋅ =∇⋅ ∇

( ) ;

δ

*L FL F*

<sup>⎪</sup> <sup>≤</sup> <sup>⎩</sup>

⎪

<sup>⎧</sup> ≥ + <sup>⎪</sup>

 λ

; 1

*F*

*L*

δ

; 0

**v** , (22)

) = , (25)

= ) = , (29)

*Tp t* ( *<sup>x</sup>* <sup>=</sup> 0, 1 ) <sup>=</sup> , (26)

*Tp t T* ( *x F* <sup>=</sup> 1, 0 ) = = , (27)

, (28)

. (24)

*t* ρρ

∂

with

numerical instabilities.

Fig. 2. The pure phase change test schematics

The boundary conditions are set to

and initial state to

where the ratio between the Prandtl and the Rayleigh number is known as the Grashof number

$$\text{Gr} = \frac{\text{Ra}\_T}{\text{Pr}} \,. \tag{18}$$

The de Vahl Davis benchmark is limited to the natural convection of the air in a rectangular cavity with aspect ratio A 1 <sup>R</sup> = and Pr 0.71. = In this work additional tests are done for lower Prandtl number and higher aspect ratio in order to test the method in regimes similar to those in the early stages of phase change simulations of metal like materials where the oscillatory "steady-state" develops.

#### **2.2 Porous media natural convection**

A variant of the test, where instead of the free fluid, the domain is filled with porous media, is considered in the next test. Similar to the de Vahl Davis benchmark test, the porous natural convection case is also well known in the literature (Chan*, et al.*, 1994, Jecl*, et al.*, 2001, Ni and Beckermann, 1991, Prasad and Kulacky, 1984, Prax*, et al.*, 1996, Raghavan and Ozkan, 1994, Šarler*, et al.*, 2000, Šarler*, et al.*, 2004a, Šarler*, et al.*, 2004b) and therefore a good quantitative comparison is possible.

The only difference from de Vahl Davis case is in the momentum equation, and the consecutive velocity boundary conditions. Instead of the Navier-Stokes the Darcy momentum equation is used to describe the fluid flow in the porous media

$$
\rho \frac{\partial \mathbf{v}}{\partial t} + \rho \nabla \cdot (\mathbf{v} \mathbf{v}) = -\nabla P - \frac{\mu}{K} \mathbf{v} + \mathbf{b} \tag{19}
$$

where *K* stands for permeability. The main difference in the momentum equation is in its order. The Navier-Stokes equation is of the second order while the Darcy equation is of the first order and therefore different boundary conditions for the velocity apply. Instead of the no-slip boundary condition for velocity, the slip and impermeable velocity boundary conditions are used. This is formulated as

$$\mathbf{v}(\mathbf{p}\_{\Gamma},t) \cdot \mathbf{n} = 0 \,. \tag{20}$$

Instead of the thermal Rayleigh and Prandtl numbers, the filtration Rayleigh number defines ( ) Ra*T* the problem

$$\mathbf{Ra}\_{F} = \frac{|\mathbf{g}| \, \mathbb{K} \beta\_{T} \left(T\_{H} - T\_{\mathbb{C}}\right) \Omega\_{H} \, \rho^{2} c\_{p}}{\lambda \mu} \,. \tag{21}$$

#### **2.3 Phase change driven by natural convection**

The benchmark test is similar to the previous cases with an additional phase change phenomenon added. The solid and the liquid thermo-physical properties are assumed to be equal. In this case the energy transport is modelled through enthalpy (*h*) formulation. The concept is adopted in order to formulate a one domain approach. The phase change phenomenon is incorporated within the enthalpy formulation with introduction of liquid fraction (*fL*). The problem is thus defined with equations (1), (2), (4) and

$$
\rho \frac{\partial \mathbf{l}}{\partial t} + \rho \nabla \cdot (\mathbf{l} \mathbf{v}) = \nabla \cdot (\lambda \nabla T) \,' \,' \,' \tag{22}
$$

$$h(T) = c\_p T + f\_{\perp} L \,, \tag{23}$$

with

110 Convection and Conduction Heat Transfer

where the ratio between the Prandtl and the Rayleigh number is known as the Grashof

Ra Gr= Pr

The de Vahl Davis benchmark is limited to the natural convection of the air in a rectangular cavity with aspect ratio A 1 <sup>R</sup> = and Pr 0.71. = In this work additional tests are done for lower Prandtl number and higher aspect ratio in order to test the method in regimes similar to those in the early stages of phase change simulations of metal like materials where the

A variant of the test, where instead of the free fluid, the domain is filled with porous media, is considered in the next test. Similar to the de Vahl Davis benchmark test, the porous natural convection case is also well known in the literature (Chan*, et al.*, 1994, Jecl*, et al.*, 2001, Ni and Beckermann, 1991, Prasad and Kulacky, 1984, Prax*, et al.*, 1996, Raghavan and Ozkan, 1994, Šarler*, et al.*, 2000, Šarler*, et al.*, 2004a, Šarler*, et al.*, 2004b) and therefore a good

The only difference from de Vahl Davis case is in the momentum equation, and the consecutive velocity boundary conditions. Instead of the Navier-Stokes the Darcy

> ( ) *P t K*

where *K* stands for permeability. The main difference in the momentum equation is in its order. The Navier-Stokes equation is of the second order while the Darcy equation is of the first order and therefore different boundary conditions for the velocity apply. Instead of the no-slip boundary condition for velocity, the slip and impermeable velocity boundary

Instead of the thermal Rayleigh and Prandtl numbers, the filtration Rayleigh number

Ra = *TH C H <sup>p</sup>*

The benchmark test is similar to the previous cases with an additional phase change phenomenon added. The solid and the liquid thermo-physical properties are assumed to be equal. In this case the energy transport is modelled through enthalpy (*h*) formulation. The concept is adopted in order to formulate a one domain approach. The phase change phenomenon is incorporated within the enthalpy formulation with introduction of liquid

**g** − Ω

β

( ) <sup>2</sup>

 ρ

*KTT c*

λμ

+ ∇ ⋅ = −∇ − +

μ

**vv v b** , (19)

**vp n** ( <sup>Γ</sup> , 0 *t*)⋅ = . (20)

. (21)

momentum equation is used to describe the fluid flow in the porous media

 ρ

ρ

∂

∂ **v**

*F*

fraction (*fL*). The problem is thus defined with equations (1), (2), (4) and

**2.3 Phase change driven by natural convection** 

*<sup>T</sup>* . (18)

number

oscillatory "steady-state" develops.

quantitative comparison is possible.

conditions are used. This is formulated as

defines ( ) Ra*T* the problem

**2.2 Porous media natural convection** 

$$f\_L(T) = \begin{cases} \begin{array}{c} T \ge T\_F + \delta T\_L; \\ T\_F + \delta T\_L > T > T\_F; \end{array} \frac{T - T\_F}{\delta T\_L} \,. \end{cases} \tag{24}$$

The phase change of the pure material occurs exactly at the melting temperature which produces discontinues in the enthalpy field due to the latent heat release. The constitutive relation (24) incorporates a smoothing interval near the phase change in order to avoid numerical instabilities.

Fig. 2. The pure phase change test schematics

The boundary conditions are set to

$$\tilde{\mathbf{v}}\left(\tilde{\mathbf{p}}\_{\Gamma'}\tilde{t}\right) = 0 \;/\tag{25}$$

$$\tilde{T}\left(\tilde{p}\_x = 0, \tilde{t}\right) = 1\tag{26}$$

$$
\tilde{T}\left(\vec{p}\_x = 1, \tilde{t}\right) = \tilde{T}\_{\tilde{F}} = 0 \; \text{s} \tag{27}
$$

$$\frac{\partial}{\partial \tilde{p}\_y} \tilde{T}\left(\tilde{p}\_y = 0, \tilde{t}\right) = \frac{\partial}{\partial \tilde{p}\_y} \tilde{T}\left(\tilde{p}\_y = 1, \tilde{t}\right) = 0 \tag{28}$$

and initial state to

$$\tilde{\mathbf{v}}\left(\tilde{\mathbf{p}}\_{\Omega'}\tilde{\mathbf{t}}=\mathbf{0}\right)=\mathbf{0}\,,\tag{29}$$

Numerical Solution of Natural Convection Problems by a Meshless Method 113

1 () () *NBasis*

functions, the approximation coefficients and the basis functions, respectively. The basis could be selected arbitrarily, however in this chapter only Hardy's Multiquadrics (MQs)

( ) ( ) ( ) <sup>2</sup> / 1 *n n* Ψ*n C* **p pp pp** = − ⋅− +

with σC standing for the free shape parameter of the basis function, are used. By taking into account all support domain nodes and equation (32), the approximation system is obtained. In this chapter the simplest possible case is considered, where the number of support domain nodes is exactly the same as the number of basis functions. In such a case the approximation simplifies to collocation. With the constructed collocation function an

1

In this work only five node support domains are used and therefore a basis of five MQs is

The implementation of the Dirichlet boundary condition is straightforward. In order to implement Neumann and Robin boundary conditions, however, a special case of interpolation is needed. In these boundary nodes the function directional derivative instead of the function value is known and therefore the equation in the interpolation system changes to

1

*BC n n n*

⎛ ⎞ <sup>∂</sup> Θ = ⎜ ⎟ Ψ +Ψ ⎝ ⎠ <sup>∂</sup> <sup>∑</sup> **p p n**

*a b*

<sup>∂</sup> ∑ **<sup>p</sup> n**

= <sup>∂</sup> Θ= Ψ

1

=

*NBasis*

*n* α

*NBasis BC n n n* α

( )

() ()

=

 α

*NBasis*

*n L L*

=

 α

*n*

θ

( )

θ

arbitrary spatial differential operator (*L*) can be computed

where

θ

used as well.

 α

Fig. 3. The local meshless principle

in the Neumann boundary nodes and to

in the Robin boundary nodes.

*n n*

, , and *NBasis n n* Ψ stand for the interpolation function, the number of basis

σ

( )

*n n*

**p p** = Ψ ∑ , (32)

**p p** = Ψ ∑ . (34)

, (33)

, (35)

, (36)

$$
\bar{T}\left(\tilde{\mathbf{p}}\_{\Omega\prime}\tilde{t}=0\right)=0\,.\tag{30}
$$

Velocity in the solid state is forced to zero by multiplying it with the liquid fraction. This approach introduces additional smoothing in the artificial "mushy" zone. This smoothing produces an error of the same magnitude as smoothing of the enthalpy jump at the phase change temperature. The problem is schematically presented in Figure 2.

Additional dimensionless number to characterize the ratio between the sensible and latent heat, the Stefan number, is introduced

$$\text{Stte} = \frac{c\_p \left(T\_H - T\_C\right)}{L}.\tag{31}$$

#### **3. Solution procedure**

There exist several meshless methods such as the Element free Galerkin method, the Meshless Petrov-Galerkin method, the point interpolation method, the point assembly method, the finite point method, the smoothed particle hydrodynamics method, the reproducing kernel particle method, the Kansa method (Atluri and Shen, 2002a, Atluri and Shen, 2002b, Atluri, 2004, Chen, 2002, Gu, 2005, Kansa, 1990a, Kansa, 1990b, Liu, 2003), etc. However, this chapter is focused on one of the simplest classes of meshless methods in development today, the Radial Basis Function (Buhmann, 2000) Collocation Methods (RBFCM) (Šarler, 2007). The meshless RBFCM was used for the solution of flow in Darcy porous media for the first time in (Šarler*, et al.*, 2004a). A substantial breakthrough in the development of the RBFCM was its local formulation, LRBFCM. Lee at al. (Lee, 2003) demonstrated that the local formulation does not substantially degrade the accuracy with respect to the global one. On the other hand, it is much less sensitive to the choice of the RBF shape and node distribution. The local RBFCM has been previously developed for diffusion problems (Šarler and Vertnik, 2006), convection-diffusion solid-liquid phase change problems (Vertnik and Šarler, 2006) and subsequently successfully applied in industrial process of direct chill casting (Vertnik*, et al.*, 2006).

In this chapter a completely local numerical approach is used. The LRBFCM spatial discretization, combined with local pressure-correction and explicit time discretization, enables the consideration of each node separately from other parts of computational domain. Such an approach has already been successfully applied to several thermo-fluid problems (Kosec and Šarler, 2008a, Kosec and Šarler, 2008b, Kosec and Šarler, 2008c, Kosec and Šarler, 2008d, Kosec and Šarler, 2009) and it shows several advantages like ease of implementation, straightforward parallelization, simple consideration of complex physical models and CPU effectiveness.

An Euler explicit time stepping scheme is used for time discretization and the spatial discretization is performed by the local meshfree method. The general idea behind the local meshless numerical approach is the use of a local influence domain for the approximation of an arbitrary field in order to evaluate the differential operators needed to solve the partial differential equations. The principle is represented in Figure 3.

Each node uses its own support domain for spatial differential operations; the domain is therefore discretized with overlapping support domains. The approximation function is introduced as

Velocity in the solid state is forced to zero by multiplying it with the liquid fraction. This approach introduces additional smoothing in the artificial "mushy" zone. This smoothing produces an error of the same magnitude as smoothing of the enthalpy jump at the phase

Additional dimensionless number to characterize the ratio between the sensible and latent

( ) Ste= *<sup>p</sup> H C cT T L*

There exist several meshless methods such as the Element free Galerkin method, the Meshless Petrov-Galerkin method, the point interpolation method, the point assembly method, the finite point method, the smoothed particle hydrodynamics method, the reproducing kernel particle method, the Kansa method (Atluri and Shen, 2002a, Atluri and Shen, 2002b, Atluri, 2004, Chen, 2002, Gu, 2005, Kansa, 1990a, Kansa, 1990b, Liu, 2003), etc. However, this chapter is focused on one of the simplest classes of meshless methods in development today, the Radial Basis Function (Buhmann, 2000) Collocation Methods (RBFCM) (Šarler, 2007). The meshless RBFCM was used for the solution of flow in Darcy porous media for the first time in (Šarler*, et al.*, 2004a). A substantial breakthrough in the development of the RBFCM was its local formulation, LRBFCM. Lee at al. (Lee, 2003) demonstrated that the local formulation does not substantially degrade the accuracy with respect to the global one. On the other hand, it is much less sensitive to the choice of the RBF shape and node distribution. The local RBFCM has been previously developed for diffusion problems (Šarler and Vertnik, 2006), convection-diffusion solid-liquid phase change problems (Vertnik and Šarler, 2006) and subsequently successfully applied in industrial

In this chapter a completely local numerical approach is used. The LRBFCM spatial discretization, combined with local pressure-correction and explicit time discretization, enables the consideration of each node separately from other parts of computational domain. Such an approach has already been successfully applied to several thermo-fluid problems (Kosec and Šarler, 2008a, Kosec and Šarler, 2008b, Kosec and Šarler, 2008c, Kosec and Šarler, 2008d, Kosec and Šarler, 2009) and it shows several advantages like ease of implementation, straightforward parallelization, simple consideration of complex physical

An Euler explicit time stepping scheme is used for time discretization and the spatial discretization is performed by the local meshfree method. The general idea behind the local meshless numerical approach is the use of a local influence domain for the approximation of an arbitrary field in order to evaluate the differential operators needed to solve the partial

Each node uses its own support domain for spatial differential operations; the domain is therefore discretized with overlapping support domains. The approximation function is

change temperature. The problem is schematically presented in Figure 2.

heat, the Stefan number, is introduced

process of direct chill casting (Vertnik*, et al.*, 2006).

differential equations. The principle is represented in Figure 3.

models and CPU effectiveness.

introduced as

**3. Solution procedure** 

*T t* (**p**<sup>Ω</sup> ,00 = ) = . (30)

<sup>−</sup> . (31)

$$\theta(\mathbf{p}) = \sum\_{n=1}^{N\_{\text{Bn}^\*}} \alpha\_n \Psi\_n(\mathbf{p}) \, \, \, \, \, \tag{32}$$

where θ α , , and *NBasis n n* Ψ stand for the interpolation function, the number of basis functions, the approximation coefficients and the basis functions, respectively. The basis could be selected arbitrarily, however in this chapter only Hardy's Multiquadrics (MQs)

$$\Psi\_n(\mathbf{p}) = \sqrt{\left(\mathbf{p} - \mathbf{p}^n\right) \cdot \left(\mathbf{p} - \mathbf{p}^n\right) / \sigma\_\odot^2 + 1} \,, \tag{33}$$

with σC standing for the free shape parameter of the basis function, are used. By taking into account all support domain nodes and equation (32), the approximation system is obtained. In this chapter the simplest possible case is considered, where the number of support domain nodes is exactly the same as the number of basis functions. In such a case the approximation simplifies to collocation. With the constructed collocation function an arbitrary spatial differential operator (*L*) can be computed

$$L\theta(\mathbf{p}) = \sum\_{n=1}^{N\_{\text{Ras}}} \alpha\_n L\Psi\_n(\mathbf{p}) \,. \tag{34}$$

In this work only five node support domains are used and therefore a basis of five MQs is used as well.

Fig. 3. The local meshless principle

The implementation of the Dirichlet boundary condition is straightforward. In order to implement Neumann and Robin boundary conditions, however, a special case of interpolation is needed. In these boundary nodes the function directional derivative instead of the function value is known and therefore the equation in the interpolation system changes to

$$\Theta\_{\rm BC} = \sum\_{n=1}^{N\_{\rm Bus}} \alpha\_n \frac{\partial}{\partial \mathbf{n}} \Psi\_n(\mathbf{p}) \, \prime \tag{35}$$

in the Neumann boundary nodes and to

$$\Theta\_{BC} = \sum\_{n=1}^{N\_{\text{Rig}}} \alpha\_n \left( a \frac{\partial}{\partial \mathbf{n}} \Psi\_n(\mathbf{p}) + b \Psi\_n(\mathbf{p}) \right) \tag{36}$$

in the Robin boundary nodes.

Numerical Solution of Natural Convection Problems by a Meshless Method 115

( ) Nu( ) ()() *<sup>x</sup> x*

The Nusselt number is computed locally on five nodded influence domains, while the streamfunction is computed on one dimensional influence domains each representing an *x* row, where all the nodes in the row are used as an influence domain. The streamfunction is

ψ

The de Vahl Davis test represents the first benchmark test in the series and therefore some additional assessments regarding the numerical performance as well as the computational effectiveness are done. One of the tests is focused on the global mass continuity conservation, which indicates the pressure-velocity coupling algorithm effectiveness. The

<sup>1</sup> ( ) () ; ( 0)

*tt t t t N*

Δ= − Δ

dimensionless time-step in all cases. The reference values in the Boussinesq approximation

The classical de Vahl Davis benchmark test is defined for the natural convection of air ( Pr 0.71 = ) in the square closed cavity ( A 1 <sup>R</sup> = ). The only physical free parameter of the test remains the thermal Rayleigh number. In the original paper (de Vahl Davis, 1983) de Vahl Davis tested the problem up to the Rayleigh number <sup>6</sup> 10 , however in the latter publications, the results of more intense simulations were presented with the Rayleigh number up to <sup>8</sup> 10 . Lage and Bejan (Lage and Bejan, 1991) showed that the laminar domain of the closed cavity natural convection problem is roughly below <sup>9</sup> Gr<10 . It was reported (Janssen and Henkes, 1993, Nobile, 1996) that the natural convection becomes unsteady for <sup>8</sup> Ra 2 10 = ⋅ . This section deals with the steady state solution and therefore regarding to the published analysis, a

A comparison of the present numerical results with the published data is stated in Table 1

streamfunction, average Nusselt number and maximum mid-plane velocities, respectively. The results of the present work are compared to the (de Vahl Davis, 1983) (a), (Sadat and Couturier, 2000) (b), (Wan*, et al.*, 2001) (c) and (Šarler, 2005) (d). The specifications of the

The temperature contours (yellow-red continuous plot) and the streamlines are plotted in Figure 4 with the streamline contour plot step 0.05 for <sup>3</sup> Ra=10 , 0.2 for <sup>4</sup> Ra=10 , 0.5 for <sup>5</sup> Ra=10 , 1 for <sup>6</sup> Ra=10 , 1.5 for <sup>7</sup> Ra=10 and 2.5 for <sup>8</sup> Ra=10 . The Nusselt number time

Due to the completely symmetric problem formulation (*T t* (**p**<sup>Ω</sup> , 0 0.5 = =) ) the cold side and the hot side average Nusselt numbers should be the same at all times and therefore the

development is plotted in Figure 5 in order to characterize the system dynamics.

<sup>=</sup> (0.5,0.5) , avg Nu , max(0.5, ) *<sup>x</sup> <sup>y</sup> <sup>v</sup> p* and max( ,0.5) *x y v p* stand for mid-point

*D n*

avg avg avg

ρρ

ρ ρρ (0,0 0 ) = .

1 avg

= + Δ = +Δ ∇⋅ = =

*ND*

( )

*N t*

*t*

∑ **<sup>v</sup>**

*n*

 ρ

stand for average density, number domain nodes and density

ζ

 ρ

is set to the same value as the

, (45)

*p* ∂ =− + ∂

set to zero in south west corner of the domain

global mass leakage is analysed by implementing

ρ

The pressure-velocity coupling relaxation parameter

ρ

, and *ND* Δ

are set to the initial values.

maximum <sup>8</sup> Ra 10 *<sup>T</sup>* = case is tested.

 ψ

simulations are stated in Table 2.

**4.1 De Vahl Davis test** 

where the mid ψ

where avg ρ

change.

*<sup>T</sup> v T*

**<sup>p</sup> p p <sup>p</sup>** . (44)

With the defined time and spatial discretization schemes, the general transport equation under the model assumptions can be written as

$$\frac{\partial\_1 - \theta\_0}{\Delta t} = D\nabla^2 \theta\_0 - \nabla \theta\_0 \cdot \left(\rho \mathbf{v}\_0\right) + \mathbf{S}\_0 \,. \tag{37}$$

where 0,1 <sup>0</sup> θ , , and *Dt S* Δ stand for the field value at current and next time step, general diffusion coefficient, time step and for source term, respectively.

To couple the mass and momentum conservation equations a special treatment is required. The intermediate velocity ( **v**ˆ ) is computed by

$$
\hat{\mathbf{v}} = \mathbf{v}\_0 + \frac{\Delta t}{\rho} \Big( -\nabla P\_0 + \nabla \cdot \left( \mu \nabla \mathbf{v}\_0 \right) + \mathbf{b}\_0 - \nabla \cdot \left( \rho \mathbf{v}\_0 \mathbf{v}\_0 \right) \Big). \tag{38}
$$

The equation (38) did not take in account the mass continuity. The pressure and the velocity corrections are added

$$
\hat{\mathbf{v}}^{m+1} = \hat{\mathbf{v}}^{m} + \hat{\mathbf{v}} \quad \hat{P}^{m+1} = \hat{P}^{m} + \hat{P}\_{\ \prime} \tag{39}
$$

where , and *mv P* stand for pressure velocity iteration index, velocity correction and pressure correction, respectively. By combining the momentum and mass continuity equations the pressure correction Poisson equation emerges

$$
\nabla \cdot \hat{\mathbf{v}}^m = \frac{\Delta t}{\rho} \nabla^2 \hat{P} \,. \tag{40}
$$

Instead of solving the global Poisson equation problem, the pressure correction is directly related to the divergence of the intermediate velocity

$$
\hat{P} = \ell^2 \frac{\mathcal{P}}{\Delta t} \nabla \cdot \hat{\mathbf{v}}^m \,, \tag{41}
$$

where A stands for characteristic length. The proposed assumption enables direct solving of the pressure velocity coupling iteration and thus is very fast, since there is only one step needed in each node to evaluate the new iteration pressure and the velocity correction. With the computed pressure correction the pressure and the velocity can be corrected as

$$
\hat{\mathbf{v}}^{m+1} = \hat{\mathbf{v}}^{m} - \boldsymbol{\zeta} \frac{\Delta t}{\rho} \nabla \hat{P} \quad \text{and} \quad \hat{P}^{m+1} = \hat{P}^{m} + \boldsymbol{\zeta} \hat{P} \, \text{.}\tag{42}
$$

where ζ stands for relaxation parameter. The iteration is performed until the criterion · ˆ *<sup>V</sup>* ∇ < **v** εis met in all computational nodes.

#### **4. Results**

The results of the benchmark tests are assessed in terms of streamfunction (Ψ) , cavity Nusselt number (Nu and mid-plane velocity components. )

$$\tilde{\boldsymbol{\varphi}}(\tilde{\mathbf{p}}) = \int\_0^1 \tilde{\boldsymbol{v}}\_x \left( \tilde{\mathbf{p}} \right) d\tilde{\boldsymbol{p}}\_y \tag{43}$$

$$\text{Nu}(\tilde{\mathbf{p}}) = -\frac{\partial \tilde{T}(\tilde{\mathbf{p}})}{\partial \tilde{p}\_x} + \tilde{v}\_x(\tilde{\mathbf{p}}) \tilde{T}(\tilde{\mathbf{p}}) \,. \tag{44}$$

The Nusselt number is computed locally on five nodded influence domains, while the streamfunction is computed on one dimensional influence domains each representing an *x* row, where all the nodes in the row are used as an influence domain. The streamfunction is set to zero in south west corner of the domain ψ(0,0 0 ) = .

The de Vahl Davis test represents the first benchmark test in the series and therefore some additional assessments regarding the numerical performance as well as the computational effectiveness are done. One of the tests is focused on the global mass continuity conservation, which indicates the pressure-velocity coupling algorithm effectiveness. The global mass leakage is analysed by implementing

$$\begin{aligned} \boldsymbol{\rho}^{\text{avg}} \{ \mathbf{t} + \boldsymbol{\Delta t} \} &= \boldsymbol{\rho}^{\text{avg}} \{ \mathbf{t} \} + \boldsymbol{\Delta t} \boldsymbol{\rho} \frac{1}{N\_D} \sum\_{n=1}^{N\_D} \nabla \cdot \mathbf{v}\_n \ & \quad \text{\(\boldsymbol{\rho} = 0\)} = \boldsymbol{\rho} \\ \boldsymbol{\Delta \boldsymbol{\rho}} &= \left| \boldsymbol{\rho} - \boldsymbol{\rho}^{\text{avg}} \{ \mathbf{N}\_t \boldsymbol{\Delta t} \} \right| \end{aligned} \tag{45}$$

where avg ρ , and *ND* Δρ stand for average density, number domain nodes and density change.

The pressure-velocity coupling relaxation parameter ζ is set to the same value as the dimensionless time-step in all cases. The reference values in the Boussinesq approximation are set to the initial values.

#### **4.1 De Vahl Davis test**

114 Convection and Conduction Heat Transfer

With the defined time and spatial discretization schemes, the general transport equation

( ) 1 0 <sup>2</sup> *D S* 0 0 00 *<sup>t</sup>*

To couple the mass and momentum conservation equations a special treatment is required.

0 0 00 0 ( ) ( ) <sup>0</sup> <sup>ˆ</sup> ( ) *<sup>t</sup> <sup>P</sup>* μ

The equation (38) did not take in account the mass continuity. The pressure and the velocity

<sup>1</sup> ˆ ˆ *m m* <sup>+</sup> **v vv** <sup>=</sup> <sup>+</sup> ˆ ˆ *m m* <sup>1</sup> *P PP* <sup>+</sup> <sup>=</sup> <sup>+</sup>

where , and *mv P* stand for pressure velocity iteration index, velocity correction and pressure correction, respectively. By combining the momentum and mass continuity

> <sup>2</sup> <sup>ˆ</sup> *<sup>m</sup> <sup>t</sup> <sup>P</sup>* ρ

Instead of solving the global Poisson equation problem, the pressure correction is directly

<sup>2</sup> ˆ *<sup>m</sup> P t* ρ = ∇ ⋅ Δ

where A stands for characteristic length. The proposed assumption enables direct solving of the pressure velocity coupling iteration and thus is very fast, since there is only one step needed in each node to evaluate the new iteration pressure and the velocity correction. With

The results of the benchmark tests are assessed in terms of streamfunction (Ψ) , cavity

1

0 ( ) *<sup>x</sup> <sup>y</sup>*

( )

*P P PP*

stands for relaxation parameter. The iteration is performed until the criterion

the computed pressure correction the pressure and the velocity can be corrected as

+ + Δ

ρ

ζ

ψ

1 1 ˆ ˆ ˆ ˆ and *m m <sup>t</sup> m m*

<sup>−</sup> <sup>=</sup> ∇ −∇ ⋅ +

 θρ

, , and *Dt S* Δ stand for the field value at current and next time step, general

 ρ

<sup>Δ</sup> ∇⋅ = ∇ **<sup>v</sup>** . (40)

**<sup>v</sup>** <sup>A</sup> , (41)

ζ

<sup>=</sup> *v dp* ∫ **p p** , (43)

**v v** =− ∇ =+ , (42)

**v v** = + −∇ + ∇ ⋅ ∇ + − ∇ ⋅ **v b vv** . (38)

**v** , (37)

, (39)

θ

under the model assumptions can be written as

The intermediate velocity ( **v**ˆ ) is computed by

where 0,1 <sup>0</sup> θ

corrections are added

where

· ˆ *<sup>V</sup>* ∇ < **v** ε

**4. Results** 

ζ

θ θ

Δ

diffusion coefficient, time step and for source term, respectively.

ρ

equations the pressure correction Poisson equation emerges

related to the divergence of the intermediate velocity

is met in all computational nodes.

Nusselt number (Nu and mid-plane velocity components. )

Δ

The classical de Vahl Davis benchmark test is defined for the natural convection of air ( Pr 0.71 = ) in the square closed cavity ( A 1 <sup>R</sup> = ). The only physical free parameter of the test remains the thermal Rayleigh number. In the original paper (de Vahl Davis, 1983) de Vahl Davis tested the problem up to the Rayleigh number <sup>6</sup> 10 , however in the latter publications, the results of more intense simulations were presented with the Rayleigh number up to <sup>8</sup> 10 . Lage and Bejan (Lage and Bejan, 1991) showed that the laminar domain of the closed cavity natural convection problem is roughly below <sup>9</sup> Gr<10 . It was reported (Janssen and Henkes, 1993, Nobile, 1996) that the natural convection becomes unsteady for <sup>8</sup> Ra 2 10 = ⋅ . This section deals with the steady state solution and therefore regarding to the published analysis, a maximum <sup>8</sup> Ra 10 *<sup>T</sup>* = case is tested.

A comparison of the present numerical results with the published data is stated in Table 1 where the mid ψ ψ <sup>=</sup> (0.5,0.5) , avg Nu , max(0.5, ) *<sup>x</sup> <sup>y</sup> <sup>v</sup> p* and max( ,0.5) *x y v p* stand for mid-point streamfunction, average Nusselt number and maximum mid-plane velocities, respectively. The results of the present work are compared to the (de Vahl Davis, 1983) (a), (Sadat and Couturier, 2000) (b), (Wan*, et al.*, 2001) (c) and (Šarler, 2005) (d). The specifications of the simulations are stated in Table 2.

The temperature contours (yellow-red continuous plot) and the streamlines are plotted in Figure 4 with the streamline contour plot step 0.05 for <sup>3</sup> Ra=10 , 0.2 for <sup>4</sup> Ra=10 , 0.5 for <sup>5</sup> Ra=10 , 1 for <sup>6</sup> Ra=10 , 1.5 for <sup>7</sup> Ra=10 and 2.5 for <sup>8</sup> Ra=10 . The Nusselt number time development is plotted in Figure 5 in order to characterize the system dynamics.

Due to the completely symmetric problem formulation (*T t* (**p**<sup>Ω</sup> , 0 0.5 = =) ) the cold side and the hot side average Nusselt numbers should be the same at all times and therefore the

Numerical Solution of Natural Convection Problems by a Meshless Method 117

*<sup>y</sup> v <sup>x</sup> p* avg Nu mid

3.679 0.179 3.634 0.813 1.116 1.174 (a) 3.686 0.188 3.489 0.813 1.117 (c) 3.566 3.544 1.165 (d) 3.991 0.170 3.931 0.825 1.101 1.298 1677 3.699 0.177 3.653 0.812 1.098 1.194 6557

3.695 0.179 3.645 0.820 1.089 1.196 10197

<sup>4</sup> 10 19.51 0.120 16.24 0.823 2.234 5.098 (a) 19.79 0.120 16,17 0.823 2.243 (c) 19.04 15.80 4.971 (d) 19.81 0.120 16.24 0.825 2.075 5.155 1677 19.83 0.120 16.27 0.825 2.120 5.167 6557 20.03 0.120 16.45 0.830 2.258 5.240 10197

<sup>5</sup> 10 68.22 0.066 34.81 0.855 4.510 9.142 (a) 68.52 0.064 34.63 0.852 4.534 9.092 (b) 70.63 0.072 33.39 0.835 4.520 (c) 67.59 32.51 8.907 (d) 67.65 0.070 33.67 0.850 4.624 8.896 1677 68.98 0.062 34.60 0.850 4.813 9.135 6557 69.69 0.069 35.03 0.860 4.511 9.278 10197

<sup>6</sup> 10 216.75 0.038 65.33 0.851 8.798 16.53 (a) 219.41 0.038 64.43 0.852 8.832 16.29 (b) 227.11 0.04 65.4 0.86 8.8 (c) 211.67 61.55 15.91 (d) 195.98 0.045 63.73 0.850 6.1 15.15 1677 219.48 0.038 64.87 0.851 7.67 16.13 6557 221.37 0.039 65.91 0.860 8.97 16.51 10197

<sup>7</sup> 10 687.43 0.023 145.68 0.888 16.59 28.23 (b) 714.48 0.022 143.56 0.922 16.65 (c) 632.60 0.020 127.70 0.925 10.43 24.93 1677 654.803 0.035 143.55 0.902 14.70 27.51 6557 687.20 0.021 149.61 0.900 16.92 28.61 10197

<sup>8</sup> 10 2180.1 0.011 319.19 0.943 30.94 50.81 (b) 2259.08 0.012 296.71 0.93 31.48 (c) 2060.86 0.010 264.96 0.939 29.33 44.85 6557 2095.23 0.009 278.49 0.930 32.12 47.12 10197

ψ

reference /*ND*

Ra max

Table 1. A comparison of the results

<sup>3</sup> 10

*<sup>x</sup> <sup>v</sup> <sup>y</sup> <sup>p</sup>* max

difference between the two can be understood as a numerical error of the solution procedure. A simple relative error measure is introduced as

$$\mathbf{E} = \left| \frac{\mathbf{N} \mathbf{u}^{\text{avg}} \left( \tilde{p}\_x = \mathbf{1}, \tilde{p}\_y \right) - \mathbf{N} \mathbf{u}^{\text{avg}} \left( \tilde{p}\_x = \mathbf{0}, \tilde{p}\_y \right)}{\mathbf{N} \mathbf{u}^{\text{max}} \left( \tilde{\mathbf{p}} \right)} \right| \tag{46}$$

where avg max Nu and Nu stand for average and maximum Nusselt number. The Nusselt number as a function of time is presented in Figure 5. The hot-cold side errors (E are ) plotted in Figure 6 and the mid-plane velocities are presented in Figure 7.

Fig. 4. Temperature and streamline contour plots for de Vahl Davis benchmark test

difference between the two can be understood as a numerical error of the solution

max Nu 1, Nu 0,

where avg max Nu and Nu stand for average and maximum Nusselt number. The Nusselt number as a function of time is presented in Figure 5. The hot-cold side errors (E are )

avg avg

Nu

Fig. 4. Temperature and streamline contour plots for de Vahl Davis benchmark test

plotted in Figure 6 and the mid-plane velocities are presented in Figure 7.

( ) ( ) ( )

, (46)

*<sup>x</sup> <sup>y</sup> <sup>x</sup> <sup>y</sup> p* =− = *p pp* **p** 

procedure. A simple relative error measure is introduced as

E=


Table 1. A comparison of the results

Numerical Solution of Natural Convection Problems by a Meshless Method 119

Fig. 6. The Nusselt number hot-cold side error as a function of time

Fig. 5. The Nusselt number as a function of time. The red plot stands for the domain average and the blue for the cold side average Nusselt number

Fig. 5. The Nusselt number as a function of time. The red plot stands for the domain average

and the blue for the cold side average Nusselt number

Fig. 6. The Nusselt number hot-cold side error as a function of time

Numerical Solution of Natural Convection Problems by a Meshless Method 121

The next test is merely a generalization of de Vahl Davis with the same governing equations, initial state and boundary conditions, where the cold side of the domain is set to the initial temperature. The tall cavity with aspect ratio A 1/4 <sup>R</sup> = is filled with metal like (Al-4.5%Cu) low Prandtl fluid Pr 0.0137 = at the Rayleigh number <sup>5</sup> Ra 2.81 10 = ⋅ is considered. The case is especially interesting due to its oscillating »steady-state« which is a result of a balance between the buoyancy and the shear forces. This case is also relevant for fluid flow behaviour at initial stages of melting of low Prandtl materials (metals), since it has a similar geometrical arrangement. The test case has been already computed by two different numerical methods (Založnik*, et al.*, 2005) (spectral FEM and FVM) with good mutual agreement. Respectively, these solutions have been used for assessment of the present

The temperature contours and the streamlines are plotted as continuous red to yellow fill

Fig. 8. The early stage time development and "steady-state" oscillations of a tall cavity

A comparison with the already published data is done on the analysis of the hot side Nusselt number time development ( ) avg Nu 0, *<sup>x</sup> <sup>y</sup> p p* <sup>=</sup> . To confirm the agreement of the results, the hot side Nusselt number frequency domains are compared, where the early stages of signal development are omitted. From Figure 9 one can see that the agreement with reference results is excellent. In Figure 9 the frequency domains for different node distributions are compared, as well. The Nu *freq* stands for Nusselt number transformation

 stands for dimensionless frequency. The presented case is highly sensitive; even the smallest changes in the case setup affect the results dramatically, for example, changing the aspect ratio for less than 1 % results in completely different flow structure. Instead of two there are three major oscillating vortices. On the other hand, the presented results are computed with the completely different numerical approach in comparison with the reference solutions (meshless spatial

natural convection - streamline and temperature contour plots

to the frequency domain and *f*

and dotted lines with a contour step 0.2, respectively, in Figure 8.

**4.2 Low Prandtl number - tall cavity test** 

method as well.


Table 2. Numerical specifications with time and density loss analysis

Fig. 7. Velocity mid-plane profiles

#### **4.2 Low Prandtl number - tall cavity test**

120 Convection and Conduction Heat Transfer

 10 10e-4 1e-04 6 3208 1e-4 26 26837 5e-05 65 3662 3.37e-7 10 10e-3 1e-04 5 3154 1e-4 15 3259 5e-05 51 3706 8.44e-6 10 10e-2 1e-te04 5 1590 1e-4 14 1244 5e-05 43 4090 1.06e-5 10 1 1e-04 4 5527 1e-4 14 5608 1e-05 283 144089 1.38e-4 10 5 1e-05 6 18250 1e-5 85 71340 5e-06 270 184697 3.01e-4 10 25 5e-6 192 193708 5e-06 387 219885 5.90e-4

Ra *<sup>v</sup>* ε

Δ*t*

Fig. 7. Velocity mid-plane profiles

[s] *ct NPv* Δ*t*

Table 2. Numerical specifications with time and density loss analysis

1677 nodes 6557 nodes 10197 nodes

[s] *ct NPv* Δ*t*

[s] *ct NPv*

Δρ / ρ The next test is merely a generalization of de Vahl Davis with the same governing equations, initial state and boundary conditions, where the cold side of the domain is set to the initial temperature. The tall cavity with aspect ratio A 1/4 <sup>R</sup> = is filled with metal like (Al-4.5%Cu) low Prandtl fluid Pr 0.0137 = at the Rayleigh number <sup>5</sup> Ra 2.81 10 = ⋅ is considered. The case is especially interesting due to its oscillating »steady-state« which is a result of a balance between the buoyancy and the shear forces. This case is also relevant for fluid flow behaviour at initial stages of melting of low Prandtl materials (metals), since it has a similar geometrical arrangement. The test case has been already computed by two different numerical methods (Založnik*, et al.*, 2005) (spectral FEM and FVM) with good mutual agreement. Respectively, these solutions have been used for assessment of the present method as well.

The temperature contours and the streamlines are plotted as continuous red to yellow fill and dotted lines with a contour step 0.2, respectively, in Figure 8.

Fig. 8. The early stage time development and "steady-state" oscillations of a tall cavity natural convection - streamline and temperature contour plots

A comparison with the already published data is done on the analysis of the hot side Nusselt number time development ( ) avg Nu 0, *<sup>x</sup> <sup>y</sup> p p* <sup>=</sup> . To confirm the agreement of the results, the hot side Nusselt number frequency domains are compared, where the early stages of signal development are omitted. From Figure 9 one can see that the agreement with reference results is excellent. In Figure 9 the frequency domains for different node distributions are compared, as well. The Nu *freq* stands for Nusselt number transformation to the frequency domain and *f* stands for dimensionless frequency.

The presented case is highly sensitive; even the smallest changes in the case setup affect the results dramatically, for example, changing the aspect ratio for less than 1 % results in completely different flow structure. Instead of two there are three major oscillating vortices. On the other hand, the presented results are computed with the completely different numerical approach in comparison with the reference solutions (meshless spatial

Numerical Solution of Natural Convection Problems by a Meshless Method 123

1.979 2.863 (a)

17.380 35.889 3.101 4.375 (a) 3.103 (b) 3.010 (c)

13.42 (b)

287.375 4880.108 44.295 49.235 40397 5 5e-06 4 10 1

<sup>2</sup> 10 0.5 27.109 52.136 3.720 3.509 20297 1 1e-05

<sup>3</sup> 10 0.5 120.724 732.806 22.452 15.928 20297 1 1e-05

<sup>2</sup> 10 2 10.779 23.283 11.944 4.630 20297 1 1e-05

<sup>3</sup> 10 2 45.111 241.218 7.250 19.576 20297 1 1e-05

Three different aspect ratios are tested AR = [0.5,1,2] for filtration Rayleigh numbers up to 104. The results are compared against (a) (Šarler*, et al.*, 2000), (b) (Ni and Beckermann, 1991), and (c) (Prax*, et al.*, 1996) with good agreement achieved (Table 3). In addition to the previously treated cases in quoted works, results for RaF = 103 and RaF = 104 are newly represented in this work. The pressure-velocity coupling algorithm was tested for up to 160797 uniformly distributed nodes and it behaves convergent. The temperature and the

2.135 2.148 (a) 50 0.5

1.386 2.639 (a) 50 2

Table 3. A comparison of the results and numerical parameters

ψ

11.174 18.112 1.941 2.860 10197 0.01 1e-04 11.214 17.928 1.962 2.853 40397 0.01 1e-05

11.236 17.845 1.975 2.848 160797 0.01 1e-05

17.500 37.454 3.040 4.510 10197 0.1 1e-04 17.562 37.055 3.071 4.607 40397 0.1 1e-05

17.603 36.873 3.086 4.655 160797 0.1 1e-05

72.560 441.771 13.044 17.062 10197 1 1e-04 74.741 435.107 13.455 18.791 40397 1 1e-05

75.577 432.335 13.529 19.658 160797 1 1e-05 257.394 4946.968 36.720 35.436 10197 1 5e-05

287.375 4880.197 44.295 49.310 160797 5 1e-06

16.562 23.402 2.130 2.090 20297 0.1 1e-04

7.039 11.710 1.367 2.608 20297 0.1 1e-05

reference /*ND <sup>v</sup>*

ε

Δ*t*

*<sup>y</sup> v* avg Nu mid

RaF AR max

50 1

<sup>2</sup> 10 1

<sup>3</sup> 10 1

*<sup>x</sup> <sup>v</sup>* max

discretization against FVM and Spectral method, explicit against implicit time discretization scheme and LPVC against SIMPLE pressure-velocity coupling) and still the comparison shows a high level of agreement. The presented comparison infers on a high level of confidence in the present novel meshless method and the local solution approach.

Fig. 9. The hot side average Nusselt number time development comparison

Fig. 10. The hot side average Nusselt number in the frequency domain

#### **4.3 Porous media test**

The next test is focused on the assessments of the solution procedure behaviour when working with fluid flow in the Darcy porous media. Again, a symmetrical differentially heated rectangular cavity is considered with impermeable velocity boundary condition.

discretization against FVM and Spectral method, explicit against implicit time discretization scheme and LPVC against SIMPLE pressure-velocity coupling) and still the comparison shows a high level of agreement. The presented comparison infers on a high level of

The next test is focused on the assessments of the solution procedure behaviour when working with fluid flow in the Darcy porous media. Again, a symmetrical differentially heated rectangular cavity is considered with impermeable velocity boundary condition.

Fig. 10. The hot side average Nusselt number in the frequency domain

**4.3 Porous media test** 

Fig. 9. The hot side average Nusselt number time development comparison

confidence in the present novel meshless method and the local solution approach.



Three different aspect ratios are tested AR = [0.5,1,2] for filtration Rayleigh numbers up to 104. The results are compared against (a) (Šarler*, et al.*, 2000), (b) (Ni and Beckermann, 1991), and (c) (Prax*, et al.*, 1996) with good agreement achieved (Table 3). In addition to the previously treated cases in quoted works, results for RaF = 103 and RaF = 104 are newly represented in this work. The pressure-velocity coupling algorithm was tested for up to 160797 uniformly distributed nodes and it behaves convergent. The temperature and the

Numerical Solution of Natural Convection Problems by a Meshless Method 125

Fig. 13. Temperature and streamline contour plots for the test with A 1 <sup>R</sup> =

streamfunction contours are presented in Figure 11 for tall cavity, Figure 12 for low cavity and Figure 13 for square cavity with streamline steps 2, 5, 15 and 40 for RaF = 50, RaF = 102, RaF = 103 and RaF = 104, respectively. Additional comparison of the results with reference Finite Volume Method (FVM) solution, previously used in (Šarler*, et al.*, 2000) for mid-plane velocities, hot side Nusselt number, mid-plane and top temperature profiles is done for case with AR = 1 and filtration Rayleigh number RaF = 100 (Figure 14). The comparison shows good agreement with the generally accepted solution.

Fig. 11. Temperature and streamline contour plots for the test with A 1/2 <sup>R</sup> =

Fig. 12. Temperature and streamline contour plots for the test with, A 2 <sup>R</sup> =

streamfunction contours are presented in Figure 11 for tall cavity, Figure 12 for low cavity and Figure 13 for square cavity with streamline steps 2, 5, 15 and 40 for RaF = 50, RaF = 102, RaF = 103 and RaF = 104, respectively. Additional comparison of the results with reference Finite Volume Method (FVM) solution, previously used in (Šarler*, et al.*, 2000) for mid-plane velocities, hot side Nusselt number, mid-plane and top temperature profiles is done for case with AR = 1 and filtration Rayleigh number RaF = 100 (Figure 14). The comparison shows

good agreement with the generally accepted solution.

Fig. 11. Temperature and streamline contour plots for the test with A 1/2 <sup>R</sup> =

Fig. 12. Temperature and streamline contour plots for the test with, A 2 <sup>R</sup> =

Fig. 13. Temperature and streamline contour plots for the test with A 1 <sup>R</sup> =

Numerical Solution of Natural Convection Problems by a Meshless Method 127

All computations are performed on the 10197 uniformly distributed nodes with the pressurevelocity relaxation parameter set to the same numerical value as the time-step. The organiclike (high Prandtl Number) and metal-like (low Prandtl number) materials are subjected to the melting simulation in order to assess the method. The benchmark cases definitions together with the global mass continuity check and time-steps are presented in Table 4. The reference values in the Boussinesq approximation are set to the initial values. A comparison of the results or demonstration for all four cases is done at specific dimensionless time *Ct*

stated in Table 4, unless stated otherwise. The streamfunction and temperature contour plots are shown in Figure 15 with streamline steps: 0.1, 0.5 and 4.0 for *Cases* 1,2 and 3,4,

results are presented for phase change front and mid-plane temperature profile in Figure 19. For mid-plane velocity profiles the results are shown in Figure 20. In Figure 21 the average

> *Case* 1 0.02 0.01 2.5e4 1.0e-5 1.74e-6 10 *Case* 2 0.02 0.01 2.5e5 5.0e-6 8.02e-6 10 *Case* 3 50 0.1 1.0e7 1.0e-4 2.58e-5 0.1 *Case* 4 50 0.1 1.0e8 1.0e-5 1.31e-8 0.1

computational nodes shown. The results confirm convergent behaviour in all analyses.

*Lf* ) time development for all four cases are presented in

 / *Ct* <sup>Δ</sup>ρ ρ *Ct* 

*Lf* as a function of the number of

respectively. The phase change front comparison is demonstrated in Figure 17.

Pr Ste Ra Δ*t*

Figure 16 and the hot side average Nusselt number ( ) avg Nu ( 0, ) *x y <sup>p</sup>* <sup>=</sup> *<sup>p</sup>* in Figure 18. Additional node distribution convergence analysis is done for the *Case* 1 at time *t*

,

= 10 . The

Fig. 16. The average liquid fraction as a function of time

cavity temperature ( ) avg *T* and liquid fractions ( ) avg

**4.4 Melting** 

The average liquid fraction ( avg

Table 4. The benchmark test definition

Fig. 14. A comparison of cross-sections quantities. The mid-plane velocities, the mid-plane and top temperatures and hot side Nusselt number, respectively

Fig. 15. Temperature and streamline contour plots for pure melting benchmark test

Fig. 16. The average liquid fraction as a function of time

#### **4.4 Melting**

126 Convection and Conduction Heat Transfer

 Fig. 14. A comparison of cross-sections quantities. The mid-plane velocities, the mid-plane

Fig. 15. Temperature and streamline contour plots for pure melting benchmark test

and top temperatures and hot side Nusselt number, respectively

All computations are performed on the 10197 uniformly distributed nodes with the pressurevelocity relaxation parameter set to the same numerical value as the time-step. The organiclike (high Prandtl Number) and metal-like (low Prandtl number) materials are subjected to the melting simulation in order to assess the method. The benchmark cases definitions together with the global mass continuity check and time-steps are presented in Table 4. The reference values in the Boussinesq approximation are set to the initial values. A comparison of the results or demonstration for all four cases is done at specific dimensionless time *Ct* , stated in Table 4, unless stated otherwise. The streamfunction and temperature contour plots are shown in Figure 15 with streamline steps: 0.1, 0.5 and 4.0 for *Cases* 1,2 and 3,4, respectively. The phase change front comparison is demonstrated in Figure 17.

The average liquid fraction ( avg *Lf* ) time development for all four cases are presented in Figure 16 and the hot side average Nusselt number ( ) avg Nu ( 0, ) *x y <sup>p</sup>* <sup>=</sup> *<sup>p</sup>* in Figure 18.

Additional node distribution convergence analysis is done for the *Case* 1 at time *t* = 10 . The results are presented for phase change front and mid-plane temperature profile in Figure 19. For mid-plane velocity profiles the results are shown in Figure 20. In Figure 21 the average cavity temperature ( ) avg *T* and liquid fractions ( ) avg *Lf* as a function of the number of computational nodes shown. The results confirm convergent behaviour in all analyses.


Table 4. The benchmark test definition

Numerical Solution of Natural Convection Problems by a Meshless Method 129

Fig. 19. The phase change front position (left) and mid-plane temperature (right) profile for

Fig. 20. The mid-plane velocities profiles for *Case* 1 as a function of different discretizations

Fig. 21. The cavity average temperature (left) and cavity average liquid fraction (right) as a

=

*Case* 1 as a function of different discretizations at 10 *t*

at *t* = 10

function of domain nodes count

Fig. 17. The phase change front position comparison

Fig. 18. The hot side average Nusselt number as a function of time

Fig. 17. The phase change front position comparison

Fig. 18. The hot side average Nusselt number as a function of time

Fig. 19. The phase change front position (left) and mid-plane temperature (right) profile for *Case* 1 as a function of different discretizations at 10 *t* =

Fig. 20. The mid-plane velocities profiles for *Case* 1 as a function of different discretizations at *t* = 10

Fig. 21. The cavity average temperature (left) and cavity average liquid fraction (right) as a function of domain nodes count

Numerical Solution of Natural Convection Problems by a Meshless Method 131

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*Sciences* Vol. 198, pp. 289-306.

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Physics Publishing Bristol*.*

Vol. B24, pp. 191-207.

*Applications*, Vol. 19, pp. 127-145.

*Applications*, Vol. 19, pp. 147-161.

*Flow*, Vol. 18, pp. 868-882.

Vol. 35, pp. 35-59.

Springer*,* Dordrecht*.*

#### **5. Conclusions**

In this chapter it is shown that the proposed numerical method as well as the proposed solution procedure performs well for the natural convection problems at different flow constitutive relations and regimes.

A detailed analysis of the de Vahl Davis test has been performed in order to assess the method in details. The mass leakage test and the cold-hot side Nusselt number comparison both confirmed that the method is accurate when such type of problems are considered. Furthermore, a comparison with already published data shows good agreement as well. The test involving a tall cavity, where the method is compared to the two completely different approaches, gives excellent agreement in case of the oscillatory flow regimes. The time average hot side Nusselt number development shows good quantitative comparison with the benchmark data. To complete the tests, the natural convection in a Darcy momentum regime is performed. Again, good agreement with the published data is achieved.

The final test is the phase change driven by a natural convection. The present results show good agreement with other approaches in terms of interphase boundary dynamics and complicated flow structures despite the simplest LRBFCM implementation. The Nusselt number oscillations in the *Case* 2 were already reported by Mencinger (Mencinger, 2003). The oscillations are a result of an unstable flow regime in the low Prandtl fluid, similar as in the tall cavity natural convection case, where the periodic solutions occur. The potential instabilities can occur in the natural convection in liquid metals, due to their low Prandtl number (Založnik*, et al.*, 2005, Založnik and Šarler, 2006). Complex flow patterns and fast transients can occur already in laminar regimes at relatively low Rayleigh numbers. With another words; lowering the Prandtl number increases the nonlinearity of the natural convection. Generally, the main sources of instabilities for all presented cases stems from nonlinearities due to the complex liquid flow pattern and enthalpy behaviour at the phase change interphase. The enthalpy jump problem is partly resolved by introducing numerical smoothing of the phase change, at least enough to get stable results, but at the price of physical model accuracy. A detailed discussion on the parameter range with appearance of the flow physics based oscillations can be found in the work (Hannoun, et al., 2003). However, the results are in good agreement with the already known solutions (Gobin and Le Quéré, 2000). There is a bit higher deviance in the *Case* 4, still the deviations of other authors for that case is very high and it is difficult to conclude which solution is more credible.

#### **6. Acknowledgment**

The authors would like to express their gratitude to Slovenian Research Agency for support in the framework of the projects Young Researcher Programme 1000-06-310232 (G. K.) and project P2-0379 (B. Š.).

#### **7. References**

Atluri, S. N. & Shen, S. (2002a), *The Meshless Method*, Tech Science Press*,* Encino*.*


Bejan, A. (2004), *Convection Heat Transfer* John Wiley & Sons*,* New Jersey*.*

In this chapter it is shown that the proposed numerical method as well as the proposed solution procedure performs well for the natural convection problems at different flow

A detailed analysis of the de Vahl Davis test has been performed in order to assess the method in details. The mass leakage test and the cold-hot side Nusselt number comparison both confirmed that the method is accurate when such type of problems are considered. Furthermore, a comparison with already published data shows good agreement as well. The test involving a tall cavity, where the method is compared to the two completely different approaches, gives excellent agreement in case of the oscillatory flow regimes. The time average hot side Nusselt number development shows good quantitative comparison with the benchmark data. To complete the tests, the natural convection in a Darcy momentum

The final test is the phase change driven by a natural convection. The present results show good agreement with other approaches in terms of interphase boundary dynamics and complicated flow structures despite the simplest LRBFCM implementation. The Nusselt number oscillations in the *Case* 2 were already reported by Mencinger (Mencinger, 2003). The oscillations are a result of an unstable flow regime in the low Prandtl fluid, similar as in the tall cavity natural convection case, where the periodic solutions occur. The potential instabilities can occur in the natural convection in liquid metals, due to their low Prandtl number (Založnik*, et al.*, 2005, Založnik and Šarler, 2006). Complex flow patterns and fast transients can occur already in laminar regimes at relatively low Rayleigh numbers. With another words; lowering the Prandtl number increases the nonlinearity of the natural convection. Generally, the main sources of instabilities for all presented cases stems from nonlinearities due to the complex liquid flow pattern and enthalpy behaviour at the phase change interphase. The enthalpy jump problem is partly resolved by introducing numerical smoothing of the phase change, at least enough to get stable results, but at the price of physical model accuracy. A detailed discussion on the parameter range with appearance of the flow physics based oscillations can be found in the work (Hannoun, et al., 2003). However, the results are in good agreement with the already known solutions (Gobin and Le Quéré, 2000). There is a bit higher deviance in the *Case* 4, still the deviations of other authors for

regime is performed. Again, good agreement with the published data is achieved.

that case is very high and it is difficult to conclude which solution is more credible.

Atluri, S. N. & Shen, S. (2002a), *The Meshless Method*, Tech Science Press*,* Encino*.*

Bejan, A. (2004), *Convection Heat Transfer* John Wiley & Sons*,* New Jersey*.*

The authors would like to express their gratitude to Slovenian Research Agency for support in the framework of the projects Young Researcher Programme 1000-06-310232 (G. K.) and

Atluri, S. N. & Shen, S. (2002b), The meshless local Petrov-Galerkin (MLPG) method: a

simple & less-costly alternative to the finite element and boundary element methods, *CMES: Computer Modeling in Engineering & Sciences*, Vol. 3, pp. 11-52. Atluri, S. N. (2004), *The Meshless Method (MLPG) for Domain and BIE Discretization*, Tech

**5. Conclusions** 

**6. Acknowledgment** 

project P2-0379 (B. Š.).

Science Press*,* Forsyth*.*

**7. References** 

constitutive relations and regimes.

Buhmann, M. D. (2000), *Radial Basis Functions*, Cambridge University Press*,* Cambridge*.*


**6** 

**Hydromagnetic Flow with Thermal Radiation** 

Magnetohydrodynamic (MHD) flows, which is the simplest plasma model, has been the subject of a great number of empirical and theoretical investigations in many industrial fields. Especially the MHD flows associating with heat transfer have received considerable attention so far, as their applications reside in many industrial fields such as electric propulsion for space exploration, crystal growth in liquids, cooling of nuclear reactors,

The most common type of body force, which acts on fluid, is attributed to gravity so that the body force vector can be deduced from the gravitational acceleration. On the other hand, when an electrically conducting fluid is subjected to a magnetic field, the fluid motion induces an electric current such that the fluid velocity is reduced on account of interaction between the electric current and the fluid motion. Therefore, in case of free convection of an electrically conducting fluid in the presence of a magnetic field, there should be two body forces, i.e., a buoyancy force and a Lorentz force. They interact with each other, and in turn

Among various studies for MHD free flows, rather small amount of studies have been accomplished for the confined enclosures. Seki et al. (1979) studied the laminar natural convection of mercury subjected to a magnetic field parallel to gravity in a rectangular enclosure. Numerical results were obtained and compared to their experiment in the consideration of a partially heated vertical wall by a uniform heat generator. Rudraiah et al. (1995) performed a numerical simulation about natural convection in a two-dimensional cavity filled with an electrically conducting fluid in the presence of a magnetic field aligned to gravity. They selected the Grashof and Hartmann numbers as controlling parameters to examine the effect of a magnetic field on free convection and associated heat transfer. The three-dimensional free convective flow in a cubical enclosure in the presence of a transverse

For the free convection in an inclined enclosure under a magnetic field, the following representative works have been conducted. Bian et al. (1996) have studied the effect of a transverse magnetic field on buoyancy-driven convection in an inclined rectangular porous cavity, saturated with an electrically conducting fluid. Recently Wang et al. (2007) investigated numerically the natural convection in an inclined enclosure filled with porous media when a strong magnetic field was applied. They modelled the cubic enclosure, such that the direction of an applied magnetic field is varied in accordance with the inclination

**1. Introduction** 

electronic packages, micro electronic devices, etc.

influence the transport phenomena of heat and mass.

magnetic field was analysed by Kolsi et al. (2007) numerically.

Cho Young Han1 and Se-Myong Chang2

*1Korea Aerospace Research Institute, 2Kunsan National University,* 

*Republic of Korea* 

Liu, G.R. (2003), *Mesh Free Methods*, CRC Press*,* Boca Raton*.*


### **Hydromagnetic Flow with Thermal Radiation**

Cho Young Han1 and Se-Myong Chang2

*1Korea Aerospace Research Institute, 2Kunsan National University, Republic of Korea* 

#### **1. Introduction**

132 Convection and Conduction Heat Transfer

Manzari, M. T. (1999), An explicit finite element algorithm for convection heat transfer problems, *International Journal of Numerical Methods for Heat and Fluid Flow*, Vol. 9, pp. 860-877. Mencinger, J. (2003), Numerical simulation of melting in two-dimensional cavity using adaptive grid, *Journal of Computational Physics*, Vol. 198, pp. 243-264. Ni, J. & Beckermann, C. (1991), Natural convection in a vertical enclosure filled with anisotropic porous media, *ASME Journal of Heat Transfer*, Vol. 113, pp. 1033-1037. Nobile, E. (1996), Simulation of time-dependent flow in cavities with the additive-correction multigrid method, part II: Apllications, *Numerical Heat Transfer, Part B*, Vol. 30, pp. Prasad, V. & Kulacky, F. A. (1984), Convective heat transfer in a rectangular porous cavity.

Effect of aspect ratio on flow structure and heat transfer, *ASME Journal of Heat* 

media - dual reciprocity boundary element method solution of the Darcy model,

of natural convection in porous Media, *International Journal of Numerical Methods for* 

element method solution of natural convection in Darcy-Brinkman porous media,

method for diffusion problems, *Computers and Mathematics with Applications*, Vol.

convective-diffusive solid-liquid phase change problems, *International Journal of* 

billet casting problem with simultaneous material and interphase moving boundaries by a meshless method, *Engineering Analysis with Boundary Elements*, Vol.

buoyancy-driven cavity by discrete singular convolution, *Numerical Heat Transfer*,

macrosegregation in direct chill casting, *International Journal of Numerical Methods* 

Prax, C.; Sadat, H. & Salagnac, P. (1996), Diffuse approximation method for solving natural convection in porous Media, *Transport in Porous Media*, Vol. 22, pp. 215-223. Raghavan, R. & Ozkan, E. A. (1994), *Method for computing unsteady flows in porous media*, John

Sadat, H. & Couturier, S. (2000), Performance and accuracy of a meshless method for laminar natural convection, *Numerical Heat Transfer*, Vol. B37, pp. 455-467. Šarler, B.; Gobin, D.; Goyeau, B.; Perko, J. & Power, H. (2000), Natural convection in porous

Šarler, B.; Perko, J.; Gobin, D.; Goyeau, B. & Power, H. (2004b), Dual reciprocity boundary

Šarler, B. (2005), A radial basis function collocation approach in computational fluid dynamics, *CMES: Computer Modeling in Engineering & Sciences*, Vol. 7, pp. 185-193. Šarler, B. & Vertnik, R. (2006), Meshfree explicit local radial basis function collocation

Šarler, B. (2007), *From global to local radial basis function collocation method for transport* 

Vertnik, R. & Šarler, B. (2006), Meshless local radial basis function collocation method for

Vertnik, R.; Založnik, M. & Šarler, B. (2006), Solution of transient direct-chill aluminium

Wan, D. C.; Patnaik, B. S. V. & Wei, G. W. (2001), A new benchmark quality solution for the

Založnik, M.; Xin, S. & Šarler, B. (2005), Verification of a numerical model of

Založnik, M. & Šarler, B. (2006), "Thermosolutal flow in metals and implications for DC casting", *Modeling of Casting, Welding and Advanced Solidification Processes XI.*

*International Journal of Numerical Methods in Fluids*, Vol. 33, pp. 279-312. Šarler, B.; Perko, J. & Chen, C. S. (2004a), Radial basis function collocation method solution

*Engineering Analysis with Boundary Elements*, Vol. 28, pp. 23-41.

*Numerical Methods for Heat and Fluid Flow*, Vol. 16, pp. 617-640.

Liu, G.R. (2003), *Mesh Free Methods*, CRC Press*,* Boca Raton*.*

*Transfer*, Vol. 106, pp. 158-65.

Wiley and Sond inc.*,* Essex*.*

51, pp. 1269-1282.

30, pp. 847-855.

Vol. B40, pp. 199-228.

*Heat & Fluid Flow*, Vol. 14, pp. 187-212.

*phenomena*, Springer*,* Berlin, pp. 257-282.

*for Heat & Fluid Flow*, Vol. 18, pp. 308-324.

Magnetohydrodynamic (MHD) flows, which is the simplest plasma model, has been the subject of a great number of empirical and theoretical investigations in many industrial fields. Especially the MHD flows associating with heat transfer have received considerable attention so far, as their applications reside in many industrial fields such as electric propulsion for space exploration, crystal growth in liquids, cooling of nuclear reactors, electronic packages, micro electronic devices, etc.

The most common type of body force, which acts on fluid, is attributed to gravity so that the body force vector can be deduced from the gravitational acceleration. On the other hand, when an electrically conducting fluid is subjected to a magnetic field, the fluid motion induces an electric current such that the fluid velocity is reduced on account of interaction between the electric current and the fluid motion. Therefore, in case of free convection of an electrically conducting fluid in the presence of a magnetic field, there should be two body forces, i.e., a buoyancy force and a Lorentz force. They interact with each other, and in turn influence the transport phenomena of heat and mass.

Among various studies for MHD free flows, rather small amount of studies have been accomplished for the confined enclosures. Seki et al. (1979) studied the laminar natural convection of mercury subjected to a magnetic field parallel to gravity in a rectangular enclosure. Numerical results were obtained and compared to their experiment in the consideration of a partially heated vertical wall by a uniform heat generator. Rudraiah et al. (1995) performed a numerical simulation about natural convection in a two-dimensional cavity filled with an electrically conducting fluid in the presence of a magnetic field aligned to gravity. They selected the Grashof and Hartmann numbers as controlling parameters to examine the effect of a magnetic field on free convection and associated heat transfer. The three-dimensional free convective flow in a cubical enclosure in the presence of a transverse magnetic field was analysed by Kolsi et al. (2007) numerically.

For the free convection in an inclined enclosure under a magnetic field, the following representative works have been conducted. Bian et al. (1996) have studied the effect of a transverse magnetic field on buoyancy-driven convection in an inclined rectangular porous cavity, saturated with an electrically conducting fluid. Recently Wang et al. (2007) investigated numerically the natural convection in an inclined enclosure filled with porous media when a strong magnetic field was applied. They modelled the cubic enclosure, such that the direction of an applied magnetic field is varied in accordance with the inclination

Hydromagnetic Flow with Thermal Radiation 135

conducting fluid is negligible compared to the external magnetic field applied. Based on this assumption the electromagnetic retarding force and the buoyancy force terms are appeared in the relevant momentum equations, respectively, such that the governing equations are not amenable to the boundary layer type of analysis. In this simulation the SIMPLER algorithm (Patankar, 1980) is involved to estimate the flow field, which is numerically stable and being widely used. The resultant solution of the governing equations is to be proposed in association with the finite-volume method (Chai et al., 1994), which is compatible with assessing the radiant heat exchange between enclosure walls. The effect of controlling parameters pertaining to fluid flow, heat transfer characteristics and radiation involvement

A schematic of the two-dimensional rectangular enclosure with width L and height H is shown in Fig. 1. It is filled with the electrically conducting fluid that is viscous and incompressible. The left- and right-hand-side walls are kept at TC and TH, respectively. The

**adiabatic**

λ

**TH**

**C + QH**

**R**

**QH**

γ

ceiling and floor are assumed to be insulated for both conduction and radiation.

**H**

**adiabatic**

**g**

The medium within the enclosure does not participate in radiation. Such nonparticipating medium example is intrinsically important in the analysis of nongray gas effects because there are wavelength regions where the medium is essentially transparent. All four walls of the enclosure are radiatively active surfaces, which are black and diffuse, so that the

The fluid is permeated by the uniform magnetic field Bo. The enclosure is tilted at an angle of γ with respect to the horizontal plane, and the orientation of an external magnetic field could vary from 0 to 2π radians corresponding to an angle of λ. In addition the induced electric current does not distort considerably the magnetic field applied. The fluid properties including the electrical conductivity are assumed to be constant except for the density, so that the Boussinesq approximation is used. Neglecting viscous and ohmic dissipations, the governing equations for mass, momentum and energy of the steady laminar flow are as

**L**

**Bo**

**x**

**TC**

**R**

**y**

**0**

radiative interaction between walls is taken into account.

Fig. 1. Schematic diagram of the enclosure

follows.

**QC**

**C + QC**

is evaluated numerically.

**2.1 Governing equations** 

angle. Differentially heated two side walls were kept to be vertical regardless of an inclination angle in their analysis.

Investigations considering the variation in the orientation of an external magnetic field applied were carried out too. The effect of direction of an external magnetic field on a low Prandtl number fluid in a cubical enclosure was numerically examined by Ozoe & Okada (1989). Hua & Walker (1995) examined the three-dimensional liquid metal MHD flow in rectangular ducts with thin conducting walls and an inclined transverse magnetic field against the principal axes. On the other hand, Bessaih et al. (1999) studied the buoyancyinduced flow of gallium in cavities simulating the apparatus for crystal growth. The combined effect on the flow structure of wall electrical conductivity and magnetic field orientation were numerically investigated in their work. Sivasankaran & Ho (2008) conducted the numerical analysis for natural convection of water near its density maximum in the presence of a magnetic field in a cavity with temperature dependent properties. They observed the effect of the direction of an external magnetic field on the flow field and accompanying heat transfer when varying it from 0 to π/2 radians.

Moreover many researchers have taken an interest in the effect of thermal radiation on the hydromagnetic flow and heat transfer problems, by reason of its great importance in diverse industrial fields. Radiation effect can be quite significant at high operating temperature. Many processes in engineering areas occur at high temperatures and the knowledge of radiation heat transfer becomes very important for the design of pertinent equipments. The effects of thermal radiation on hydromagnetic boundary layer flows were studied by Chamkha (2000), Seddeek (2002), Ghaly (2002) and Raptis et al. (2004). However they converted the complex radiation problem to the simpler conduction one accounting for the radiation conductivity by introducing the similarity transformation as a sort of the onedimensional analysis. This analytical approach is not suitable for the assessment of the radiant heat exchange between surfaces of an enclosure operating under the high temperature environment, as the multi-dimensional analysis should be required in this case. Mahmud & Fraser (2002) examined analytically radiation effects on mixed convection through a vertical channel in the presence of a transverse magnetic field, but the underlying drawbacks are similar to those for studies aforementioned.

The aim of present chapter is to investigate in detail the effect of a magnetic field as well as thermal radiation on free convection associating with heat transfer in an enclosure filled with an electrically conducting fluid. Basically the full two-dimensional analysis is performed rather than the boundary layer type of analysis, even when considering the thermal radiation. It is motivated by a desire to find any effects of the controlling parameters on the thermally driven hydromagnetic flows found in many engineering applications. In this context this chapter can be classified into three subjects. In the first place the changes in the buoyant flow patterns and temperature distributions due to the tilting of the enclosure are examined, neglecting thermal radiation. Secondly the flow and thermal field variation is investigated in terms of the orientation of an external magnetic field. Finally the effect of combined radiation and a magnetic field on the convective flow and heat transfer characteristics of an electrically conducting fluid is analysed and discussed in detail.

#### **2. Analysis model**

For many electrically conducting fluids used in laboratories, the electrical conductivity is usually small. Subsequently the magnetic Reynolds number should be very small. Therefore it is reasonable to assume that the induced magnetic field by the motion of the electrically

angle. Differentially heated two side walls were kept to be vertical regardless of an

Investigations considering the variation in the orientation of an external magnetic field applied were carried out too. The effect of direction of an external magnetic field on a low Prandtl number fluid in a cubical enclosure was numerically examined by Ozoe & Okada (1989). Hua & Walker (1995) examined the three-dimensional liquid metal MHD flow in rectangular ducts with thin conducting walls and an inclined transverse magnetic field against the principal axes. On the other hand, Bessaih et al. (1999) studied the buoyancyinduced flow of gallium in cavities simulating the apparatus for crystal growth. The combined effect on the flow structure of wall electrical conductivity and magnetic field orientation were numerically investigated in their work. Sivasankaran & Ho (2008) conducted the numerical analysis for natural convection of water near its density maximum in the presence of a magnetic field in a cavity with temperature dependent properties. They observed the effect of the direction of an external magnetic field on the flow field and

Moreover many researchers have taken an interest in the effect of thermal radiation on the hydromagnetic flow and heat transfer problems, by reason of its great importance in diverse industrial fields. Radiation effect can be quite significant at high operating temperature. Many processes in engineering areas occur at high temperatures and the knowledge of radiation heat transfer becomes very important for the design of pertinent equipments. The effects of thermal radiation on hydromagnetic boundary layer flows were studied by Chamkha (2000), Seddeek (2002), Ghaly (2002) and Raptis et al. (2004). However they converted the complex radiation problem to the simpler conduction one accounting for the radiation conductivity by introducing the similarity transformation as a sort of the onedimensional analysis. This analytical approach is not suitable for the assessment of the radiant heat exchange between surfaces of an enclosure operating under the high temperature environment, as the multi-dimensional analysis should be required in this case. Mahmud & Fraser (2002) examined analytically radiation effects on mixed convection through a vertical channel in the presence of a transverse magnetic field, but the underlying

The aim of present chapter is to investigate in detail the effect of a magnetic field as well as thermal radiation on free convection associating with heat transfer in an enclosure filled with an electrically conducting fluid. Basically the full two-dimensional analysis is performed rather than the boundary layer type of analysis, even when considering the thermal radiation. It is motivated by a desire to find any effects of the controlling parameters on the thermally driven hydromagnetic flows found in many engineering applications. In this context this chapter can be classified into three subjects. In the first place the changes in the buoyant flow patterns and temperature distributions due to the tilting of the enclosure are examined, neglecting thermal radiation. Secondly the flow and thermal field variation is investigated in terms of the orientation of an external magnetic field. Finally the effect of combined radiation and a magnetic field on the convective flow and heat transfer

characteristics of an electrically conducting fluid is analysed and discussed in detail.

For many electrically conducting fluids used in laboratories, the electrical conductivity is usually small. Subsequently the magnetic Reynolds number should be very small. Therefore it is reasonable to assume that the induced magnetic field by the motion of the electrically

accompanying heat transfer when varying it from 0 to π/2 radians.

drawbacks are similar to those for studies aforementioned.

**2. Analysis model** 

inclination angle in their analysis.

conducting fluid is negligible compared to the external magnetic field applied. Based on this assumption the electromagnetic retarding force and the buoyancy force terms are appeared in the relevant momentum equations, respectively, such that the governing equations are not amenable to the boundary layer type of analysis. In this simulation the SIMPLER algorithm (Patankar, 1980) is involved to estimate the flow field, which is numerically stable and being widely used. The resultant solution of the governing equations is to be proposed in association with the finite-volume method (Chai et al., 1994), which is compatible with assessing the radiant heat exchange between enclosure walls. The effect of controlling parameters pertaining to fluid flow, heat transfer characteristics and radiation involvement is evaluated numerically.

#### **2.1 Governing equations**

A schematic of the two-dimensional rectangular enclosure with width L and height H is shown in Fig. 1. It is filled with the electrically conducting fluid that is viscous and incompressible. The left- and right-hand-side walls are kept at TC and TH, respectively. The ceiling and floor are assumed to be insulated for both conduction and radiation.

Fig. 1. Schematic diagram of the enclosure

The medium within the enclosure does not participate in radiation. Such nonparticipating medium example is intrinsically important in the analysis of nongray gas effects because there are wavelength regions where the medium is essentially transparent. All four walls of the enclosure are radiatively active surfaces, which are black and diffuse, so that the radiative interaction between walls is taken into account.

The fluid is permeated by the uniform magnetic field Bo. The enclosure is tilted at an angle of γ with respect to the horizontal plane, and the orientation of an external magnetic field could vary from 0 to 2π radians corresponding to an angle of λ. In addition the induced electric current does not distort considerably the magnetic field applied. The fluid properties including the electrical conductivity are assumed to be constant except for the density, so that the Boussinesq approximation is used. Neglecting viscous and ohmic dissipations, the governing equations for mass, momentum and energy of the steady laminar flow are as follows.

Hydromagnetic Flow with Thermal Radiation 137

<sup>∂</sup> (10)

<sup>∂</sup> (11)

( <sup>v</sup> \*T \*) \* <sup>T</sup> \* <sup>y</sup> \* (u \*T \*)

Non-slip conditions of the velocities are assigned on the ceiling and floor. The thermal boundary condition at adiabatic ceiling and floor is found from the following energy balancing equation, in which the net radiation into the wall surface is balanced by

∂ ∂ +

2 2 2 2 2

Pr \* <sup>u</sup> \* GrPr <sup>T</sup> \* cos PrHa ( <sup>u</sup> \* sin <sup>v</sup> \* cos sin ) <sup>x</sup> \*

<sup>2</sup> <sup>2</sup> <sup>2</sup> <sup>2</sup> <sup>2</sup> + ∇ + γ + λ λ − λ

0

= δ

⎦

w

R w \*

\* R \* w w

Nuw is the total average Nusselt number, which denotes the total

<sup>1</sup> Nu (17)

<sup>1</sup> Nu (16)

Nuw =Nu + Nu (18)

Nuw and <sup>R</sup>

Nuw ,

4Pl q \*

y \* T \* <sup>R</sup> w

∂

C

4Pl

R

average Nusselt numbers at both hot and cold walls are same.

w

+

To estimate the heat transfer rates at two isothermal walls, various types of average Nusselt

∫ <sup>⎥</sup>

<sup>w</sup> dy x \* T \*

<sup>w</sup> sign(nˆ ) <sup>q</sup> dy <sup>A</sup>

C w

From the above equations, A means the aspect ratio, i.e., A = H/L. The conductive and

amount of heat transfer at the wall. Since the steady state is assumed in this study, the total

∂

⎣ ⎡

∫ <sup>⋅</sup> <sup>δ</sup> <sup>=</sup> <sup>A</sup> 0

<sup>∂</sup> <sup>=</sup> <sup>−</sup> <sup>A</sup> 0

A

1

T

radiative average Nusselt numbers at the walls are represented by <sup>C</sup>

<sup>⎤</sup> <sup>⎢</sup>

+ ∇ − γ + − λ+ λ λ

Pr \* v \* GrPr T \* sin PrHa (u \* cos sin v \* cos )

2 =∇

<sup>∂</sup> (12)

u\* = v\* = 0, T\* = -0.5 at x\* = 0 (13)

u\* = v\* = 0, T\* = 0.5 at x\* = 1 (14)

<sup>∂</sup> <sup>−</sup> (15)

*X-momentum* 

*Y-momentum* 

x \*

∂

*Energy* 

∂

<sup>y</sup> \* (u \* ) <sup>x</sup> \*

<sup>y</sup> \* (u \* v\*)

∂ +

> ∂ +

<sup>p</sup> \* (u \* v\*)

<sup>p</sup> \* (v \* )

<sup>∂</sup> =− <sup>∂</sup>

**2.2 Dimensionless boundary conditions** 

conductive heat loss from the wall:

**2.3 Heat transfer rates** 

numbers are defined as follows:

respectively. Their sum, <sup>T</sup>

<sup>∂</sup> =− <sup>∂</sup>

∂

y \*

x \*

The boundary conditions for two isothermal walls are as follows:

∂

∂

*Continuity* 

$$\frac{\partial \mathbf{u}}{\partial \mathbf{x}} + \frac{\partial \mathbf{v}}{\partial \mathbf{y}} = \mathbf{0} \tag{1}$$

*X-momentum* 

$$\frac{\partial}{\partial \mathbf{x}} (\mathbf{u}^{\mathrm{i}}) + \frac{\partial}{\partial \mathbf{y}} (\mathbf{u} \cdot \mathbf{v}) = -\frac{1}{\rho} \frac{\partial \mathbf{p}}{\partial \mathbf{x}} + \mathbf{v} \nabla^{\mathrm{i}} \mathbf{u} - \mathbf{g} \, \mathbb{B} (\mathbf{T} - \mathbf{T}\_{\circ}) \cos \gamma + \frac{\sigma \mathbf{B}\_{\circ}^{\mathrm{i}}}{\rho} (-\mathbf{u} \sin^{\mathrm{i}} \lambda + \mathbf{v} \cos \lambda \sin \lambda) \tag{2}$$

*Y-momentum* 

$$\frac{\partial}{\partial \mathbf{x}} (\mathbf{u} \cdot \mathbf{v}) + \frac{\partial}{\partial \mathbf{y}} (\mathbf{v}^{\circ}) = -\frac{1}{\rho} \frac{\partial \mathbf{p}}{\partial \mathbf{y}} + \mathbf{v} \nabla^{\mathbf{z}} \mathbf{v} + \mathbf{g} \otimes (\mathbf{T} - \mathbf{T}\_{\circ}) \sin \gamma + \frac{\sigma \mathbf{B}\_{\circ}^{\ddagger}}{\rho} (\mathbf{u} \cos \lambda \sin \lambda - \mathbf{v} \cos^{\sharp} \lambda) \tag{3}$$

*Energy* 

$$\frac{\partial}{\partial \mathbf{x}} (\mathbf{u} \, \mathbf{T}) + \frac{\partial}{\partial \mathbf{y}} (\mathbf{v} \, \mathbf{T}) = a \, \nabla^{\mathbf{z}} \mathbf{T} \tag{4}$$

In the above equations, u and v are the velocity component in the x and y directions. The density, pressure and temperature are denoted by ρ, p and T, respectively. The relevant fluid properties are kinematic viscosity ν, thermal diffusivity α, electrical conductivity σ and volumetric expansion coefficient β. The governing equations are nondimensionalised using the following variables:

$$\mathbf{x}^\* = \mathbf{x} / \mathbf{L} \text{ , } \mathbf{y}^\* = \mathbf{y} / \mathbf{L} \text{ , } \mathbf{u}^\* = \mathbf{u} / \mathbf{u}\_{\circ \circ \prime} \text{ } \mathbf{v}^\* = \mathbf{v} / \mathbf{u}\_{\circ} \tag{5}$$

$$\mathbf{p}^\* = \frac{\mathbf{p}}{\rho\_\circ \mathbf{u}\_\circ^2}, \quad \mathbf{T}^\* = \frac{\mathbf{T} - \mathbf{T\_\circ}}{\mathbf{T\_m} - \mathbf{T\_c}} \tag{6}$$

$$\text{Pr} = \text{v} / \text{a} \,, \text{Ra} = \frac{\text{g} \, \text{f} \, \Delta \text{T} \, \text{L}^{\circ}}{\text{v} \, \text{a}} \,, \text{Gr} = \text{Ra} / \text{Pr} \, \text{a} \,, \text{Ha} = \text{B}\_{\text{o}} \, \text{L} \, \sqrt{\frac{\sigma}{\mu}} \tag{7}$$

$$\mathbf{G} = \frac{\mathbf{T}\_{\text{in}} - \mathbf{T}\_{\text{c}}}{\mathbf{T}\_{\text{o}}}, \text{ Pl} = \frac{\mathbf{k} / \text{L}}{4\overline{\sigma}\,\text{T}\_{\text{o}}^{\text{v}}}, \text{ q}^{\text{v}} \,\text{\*} = \frac{\mathbf{q}^{\text{v}}}{\overline{\sigma}\,\text{T}\_{\text{o}}^{\text{v}}} \tag{8}$$

The reference velocity is defined as uo = α / L. The overheat ratio is expressed by δ, while the reference temperature is defined as the arithmetic mean of the two isothermal wall temperatures, i.e., To = (TH + TC) / 2. The Prandtl, Rayleigh, Grashof, Hartmann and Planck numbers are represented by Pr, Ra, Gr, Ha and Pl, respectively, where μ and k are dynamic viscosity and thermal conductivity. Radiative heat flux qR is nondimensionalised by the reference emissive power, i.e., 4 σTo , in which σ is Stefan-Boltzmann constant. Based on the presumptions above, the dimensionless governing equations can be shown as follows.

*Continuity* 

$$\frac{\partial \mathbf{u}^\*}{\partial \mathbf{x}^\*} + \frac{\partial \mathbf{v}^\*}{\partial \mathbf{y}^\*} = \mathbf{0} \tag{9}$$

*X-momentum* 

136 Convection and Conduction Heat Transfer

u g ( T T ) cos

v g ( T T )sin

+ ν∇ + β − γ +

y

In the above equations, u and v are the velocity component in the x and y directions. The density, pressure and temperature are denoted by ρ, p and T, respectively. The relevant fluid properties are kinematic viscosity ν, thermal diffusivity α, electrical conductivity σ and volumetric expansion coefficient β. The governing equations are nondimensionalised using

∂ ∂ +

2 ouo <sup>p</sup> <sup>p</sup> \* <sup>ρ</sup> <sup>=</sup> ,

3

( u T ) x

+ ν∇ − β − γ +

x <sup>u</sup> <sup>=</sup> <sup>∂</sup> ∂ + ∂

x

y

∂

Pr =ν/<sup>α</sup> , να

βΔ <sup>=</sup>

o H C T

<sup>T</sup> <sup>−</sup><sup>T</sup> δ= , 3

2 2

∂ ∂

∂ ∂

<sup>1</sup> <sup>p</sup> ( <sup>u</sup> <sup>v</sup> )

<sup>1</sup> <sup>p</sup> ( <sup>v</sup> )

2 2

<sup>ρ</sup> <sup>=</sup> <sup>−</sup> <sup>∂</sup>

<sup>ρ</sup> <sup>=</sup> <sup>−</sup> <sup>∂</sup>

y

y

∂ +

∂ +

0 y v

o

o

( v T ) T

2 =α ∇

x \* = x/L , y \* = y /L , u/uo u \* = , v/uo v \* = (5)

o

σTo , in which σ is Stefan-Boltzmann constant. Based on the

<sup>∂</sup> (9)

R

T

H C o T T <sup>T</sup> <sup>T</sup> <sup>T</sup> \* <sup>−</sup>

<sup>g</sup> TL Ra , Gr <sup>=</sup>Ra/Pr , <sup>μ</sup>

<sup>σ</sup> <sup>=</sup> , 4

0 y \* v \*

R

To 4 k/L Pl

The reference velocity is defined as uo = α / L. The overheat ratio is expressed by δ, while the reference temperature is defined as the arithmetic mean of the two isothermal wall temperatures, i.e., To = (TH + TC) / 2. The Prandtl, Rayleigh, Grashof, Hartmann and Planck numbers are represented by Pr, Ra, Gr, Ha and Pl, respectively, where μ and k are dynamic viscosity and thermal conductivity. Radiative heat flux qR is nondimensionalised by the

presumptions above, the dimensionless governing equations can be shown as follows.

x \* <sup>u</sup> \* <sup>=</sup> <sup>∂</sup> ∂ +

∂

<sup>∂</sup> (3)

<sup>∂</sup> (2)

<sup>∂</sup> (1)

2 o

> 2 o

<sup>∂</sup> (4)

ρ σ

ρ σ

( usin <sup>v</sup> cos sin ) <sup>B</sup>

( ucos sin <sup>v</sup> cos ) <sup>B</sup>

λ λ − λ

2

− λ + λ λ

2

<sup>−</sup> <sup>=</sup> (6)

<sup>σ</sup> Ha <sup>=</sup>Bo L (7)

<sup>q</sup> <sup>q</sup> \* <sup>σ</sup> <sup>=</sup> (8)

*Continuity* 

*X-momentum* 

∂

*Y-momentum* 

∂

*Energy* 

( u ) x

> ( u v ) x

the following variables:

reference emissive power, i.e., 4

*Continuity* 

$$\frac{\partial}{\partial \mathbf{x}^{\star}} \langle \mathbf{u}^{\star \star} \rangle + \frac{\partial}{\partial \mathbf{y}^{\star}} \langle \mathbf{u}^{\star} \mathbf{v}^{\star} \rangle = -\frac{\partial \mathbf{p}^{\star}}{\partial \mathbf{x}^{\star}} + \text{Pr} \nabla^{\star \star} \mathbf{u}^{\star} - \text{Gr} \text{Pr}^{\star} \mathbf{T}^{\star} \cos \gamma + \text{Pr} \text{H} \mathbf{a}^{\star} \langle -\mathbf{u}^{\star} \sin^{\gamma} \lambda + \mathbf{v}^{\star} \cos \lambda \sin \lambda \rangle \text{ (10)}$$

*Y-momentum* 

$$\frac{\partial}{\partial \mathbf{x}^\*} \text{(}\mathbf{u}^\* \text{v}^\*\text{)} + \frac{\partial}{\partial \mathbf{y}^\*} \text{(}\mathbf{v}^{\*\text{e}}\text{)} = -\frac{\partial \mathbf{p}^\*}{\partial \mathbf{y}^\*} + \text{Pr} \mathbf{V}^{\*\text{e}} \text{ v}^\* + \text{Gr} \text{Pr}^\circ \text{T}^\* \sin \gamma + \text{Pr} \text{H} \text{\dot{z}} \text{\(}\mathbf{u}^\* \cos \lambda \sin \lambda - \mathbf{v}^\* \cos^\circ \lambda\) \text{(11)}$$

*Energy* 

$$\frac{\partial}{\partial \mathbf{x}^\*} (\mathbf{u}^\* \mathbf{T}^\*) + \frac{\partial}{\partial \mathbf{y}^\*} (\mathbf{v}^\* \mathbf{T}^\*) = \nabla^{\ast \varepsilon} \mathbf{T}^\* \tag{12}$$

#### **2.2 Dimensionless boundary conditions**

The boundary conditions for two isothermal walls are as follows:

$$\mathbf{u}^\* = \mathbf{v}^\* = \mathbf{0}, \text{ T}^\* = \text{-0.5 at } \mathbf{x}^\* = \mathbf{0} \tag{13}$$

$$\mathbf{u}^\* = \mathbf{v}^\* = \mathbf{0}, \mathbf{T}^\* = \mathbf{0}.\\\mathbf{5} \text{ at } \mathbf{x}^\* = \mathbf{1} \tag{14}$$

Non-slip conditions of the velocities are assigned on the ceiling and floor. The thermal boundary condition at adiabatic ceiling and floor is found from the following energy balancing equation, in which the net radiation into the wall surface is balanced by conductive heat loss from the wall:

$$-\frac{\left.\partial \operatorname{Tr}^\* \right|}{\left.\partial \operatorname{y}^\* \right|\_\*} \Big|\_\* + \frac{\left.\operatorname{q}\_{\text{w}}^{\text{v}} \ast \right|\_\*} = 0 \tag{15}$$

#### **2.3 Heat transfer rates**

To estimate the heat transfer rates at two isothermal walls, various types of average Nusselt numbers are defined as follows:

$$\overline{\mathbf{Nu}^{c}\_{\bullet}} = \frac{1}{\mathbf{A}} \int\_{\circ}^{\wedge} \left[ -\frac{\partial \mathbf{T}^{\ast}}{\partial \mathbf{x}^{\ast}} \right]\_{\circ} \, \mathbf{dy}^{\cdot} \tag{16}$$

$$\overline{\rm Nu}\_{\text{v}}^{\ast} = \frac{1}{4 \text{Pl} \,\delta \,\text{A}} \frac{1}{\text{A}} \int\_{0}^{\wedge} \text{sign}(\hat{\mathbf{n}}\_{\text{v}}) \cdot \mathbf{q}\_{\text{v}}^{\ast \cdot} \,\text{dy}^{\ast} \tag{17}$$

$$\overline{\mathbf{Nu}^{\mathrm{r}}\_{\ast}} = \overline{\mathbf{Nu}^{\mathrm{c}}\_{\ast}} + \overline{\mathbf{Nu}^{\mathrm{s}}\_{\ast}} \tag{18}$$

From the above equations, A means the aspect ratio, i.e., A = H/L. The conductive and radiative average Nusselt numbers at the walls are represented by <sup>C</sup> Nuw and <sup>R</sup> Nuw , respectively. Their sum, <sup>T</sup> Nuw is the total average Nusselt number, which denotes the total amount of heat transfer at the wall. Since the steady state is assumed in this study, the total average Nusselt numbers at both hot and cold walls are same.

Hydromagnetic Flow with Thermal Radiation 139

(a)

(b)

(c)

(d)

(e)

= π/3, (d) γ = 5π/12 and (e) γ = π/2 radians

Fig. 2. Isotherm and streamline contours for Gr = 106 and Ha = 0: (a) γ = π/6, (b) γ = π/4, (c) γ

#### **3. Numerical analysis**

A numerical analysis of thermo-fluid dynamics characteristics is conducted by adopting the SIMPLER algorithm developed by Patankar (1980). While the convection term is discretised using the QUICK finite-difference scheme (Thakur & Shyy, 1993), the central difference method is chosen for the diffusion term. In order to compute the radiative heat fluxes on enclosure walls, the finite-volume method (FVM) for radiation heat transfer is implemented here as proposed by Chai et al. (1994). After some preliminary calculations for checking convergence and accuracy, the spatial domain is discretised into 51 × 51 non-uniform control volumes in the x and y directions. In the case of radiation, 48 control angles are employed to estimate the radiative wall heat fluxes accordingly.

Computations are proceeded by the following procedures. To begin with, the radiative wall heat fluxes are determined by FVM. The velocity field is estimated from the momentum equation. After the updated temperature is estimated from the energy equation based on the interior point temperature at the previous iteration, the wall conductive heat fluxes are estimated. Then, these values of the wall conductive heat flux and the radiative wall heat flux are used to update the non-prescribed wall surface temperatures by solving the energy balance equation (15). In computation, to reduce the error in the wall conductive heat flux, the grid system should be clustered near the walls. Since the calculation of radiative heat fluxes is not necessarily required at every iteration to produce a reliable steady-state result, the radiative heat flux is updated at every tenth iteration. However, the wall surface temperature is recalculated and changed at each iteration with an estimated wall conductive heat flux.

Finally, the convergence criteria for main variables are checked if the steady state is reached. Computations are terminated when the difference in total average Nusselt numbers for the hot and cold walls is within less than 10-3 tolerance to meet the overall energy balance in the enclosure.

$$\left| \overline{\mathbf{Nu}^{\mathrm{r}}} - \overline{\mathbf{Nu}^{\mathrm{r}}} \right| \le 10^{-\mathfrak{r}} \tag{19}$$

#### **4. Results and discussion**

A numerical investigation is presented for free convection in a two-dimensional enclosure filled with an electrically conducting fluid in the presence of an external magnetic field. The enclosure is such that the two opposing side walls are differentially heated with a temperature difference specified, while the top and bottom walls are insulated. Those four walls are radiatively active surfaces, and the enclosure is tilted as well as the orientation of an external magnetic field can be arbitrary. As a whole computations are carried out for the Grashof numbers ranging from 1 × 104 to 2 × 106, and Hartmann numbers from 0 to 100. The assumption of two-dimensional laminar flow is valid for above values of the Grashof numbers (Larson & Viskanta, 1976).

It is difficult to study the influence of all parameters involved in the present problem on the flow and thermal field. Therefore a selected set of parameters is accounted for this numerical investigation. The aspect ratio, A = H/L, of the enclosure is set to be of 1, and the Prandtl number of 0.733 is utilised. The overheat ratio is taken to be δ = 2/3, i.e., TH = 2 TC. The constant fluid property and Boussinesq approximations are reasonable for above values (Fusegi & Farouk, 1989). The Planck number is assigned to be 0.02.

A numerical analysis of thermo-fluid dynamics characteristics is conducted by adopting the SIMPLER algorithm developed by Patankar (1980). While the convection term is discretised using the QUICK finite-difference scheme (Thakur & Shyy, 1993), the central difference method is chosen for the diffusion term. In order to compute the radiative heat fluxes on enclosure walls, the finite-volume method (FVM) for radiation heat transfer is implemented here as proposed by Chai et al. (1994). After some preliminary calculations for checking convergence and accuracy, the spatial domain is discretised into 51 × 51 non-uniform control volumes in the x and y directions. In the case of radiation, 48 control angles are

Computations are proceeded by the following procedures. To begin with, the radiative wall heat fluxes are determined by FVM. The velocity field is estimated from the momentum equation. After the updated temperature is estimated from the energy equation based on the interior point temperature at the previous iteration, the wall conductive heat fluxes are estimated. Then, these values of the wall conductive heat flux and the radiative wall heat flux are used to update the non-prescribed wall surface temperatures by solving the energy balance equation (15). In computation, to reduce the error in the wall conductive heat flux, the grid system should be clustered near the walls. Since the calculation of radiative heat fluxes is not necessarily required at every iteration to produce a reliable steady-state result, the radiative heat flux is updated at every tenth iteration. However, the wall surface temperature is recalculated and changed at each iteration with an estimated wall conductive

Finally, the convergence criteria for main variables are checked if the steady state is reached. Computations are terminated when the difference in total average Nusselt numbers for the hot and cold walls is within less than 10-3 tolerance to meet the overall energy balance in the

A numerical investigation is presented for free convection in a two-dimensional enclosure filled with an electrically conducting fluid in the presence of an external magnetic field. The enclosure is such that the two opposing side walls are differentially heated with a temperature difference specified, while the top and bottom walls are insulated. Those four walls are radiatively active surfaces, and the enclosure is tilted as well as the orientation of an external magnetic field can be arbitrary. As a whole computations are carried out for the Grashof numbers ranging from 1 × 104 to 2 × 106, and Hartmann numbers from 0 to 100. The assumption of two-dimensional laminar flow is valid for above values of the Grashof

It is difficult to study the influence of all parameters involved in the present problem on the flow and thermal field. Therefore a selected set of parameters is accounted for this numerical investigation. The aspect ratio, A = H/L, of the enclosure is set to be of 1, and the Prandtl number of 0.733 is utilised. The overheat ratio is taken to be δ = 2/3, i.e., TH = 2 TC. The constant fluid property and Boussinesq approximations are reasonable for above values

(Fusegi & Farouk, 1989). The Planck number is assigned to be 0.02.

T NuH Nu 10<sup>−</sup>

<sup>3</sup> <sup>T</sup> C

− ≤ (19)

employed to estimate the radiative wall heat fluxes accordingly.

**3. Numerical analysis** 

heat flux.

enclosure.

**4. Results and discussion** 

numbers (Larson & Viskanta, 1976).

Fig. 2. Isotherm and streamline contours for Gr = 106 and Ha = 0: (a) γ = π/6, (b) γ = π/4, (c) γ = π/3, (d) γ = 5π/12 and (e) γ = π/2 radians

Hydromagnetic Flow with Thermal Radiation 141

Isotherm and streamline plots will be reported for different values of controlling parameters. The contour lines of isotherm plots correspond to equally-spaced values of the dimensionless temperature T\*, i.e., ΔT\* = 0.1, in the range between -0.5 and +0.5. On the other hand the dimensionless stream function is obtained from the velocity field solution by

The contour lines of the streamline plots are correspondent to equally-spaced values of the

A numerical investigation is presented for natural convection of an electrically conducting fluid in a tilted square cavity in the presence of a vertical magnetic field aligned to the

In the present study, the Grashof number is fixed as Gr = 106. Computations are carried out for tilted angles ranging from 0 to π/2 radians, and the thermal radiation is neglected. Figure 2 shows the isotherm and streamline contours for natural convection in inclined cavities in the absence of a magnetic field. The multi-cellular inner core consists of a central roll (designated by "+" in the figures) sandwiched between two rolls. As the tilting angle decreases, the fluid motion becomes progressively intensive. The temperature is stratified at the core region in case of γ = π/2 rad. When the tilting angle decreases, this trend is maintained until γ = π/4 rad. The stratification of the temperature field in the interior begins to diminish as the inclination angle reaches π/6 rad due to the increasing buoyant action. The results depicted in Fig. 3 demonstrate the influence of the magnetic field on the fluid flow and the temperature distributions along with the tilting angle. For relatively strong Hartmann number (Ha =50), the temperature stratification in the core tends to diminish, and the thermal boundary layers at the two side walls disappear, together with the decrease in inclination angle. Also, the streamlines are elongated, and the core region becomes broadly stagnated. Furthermore, the axes of the streamlines are changed, which is due to the retarding effect of the Lorentz force. In addition, the flow strength displays maximum at γ = π/4 rad in this case, then, it decreases when γ reaches π/6 rad. This phenomenon is different from the previous result for pure free convection; hence, a considerable interaction between the buoyant and the magnetic forces is evidently caused by the tilting, as the magnitude of

the Lorentz force in the x and y directions is subjected to the inclination angle.

Hydromagnetic flow in a horizontal enclosure (γ = π/2 rad) under a uniform magnetic field is studied. The changes in the flow and thermal field based on the orientation of an external magnetic field, which varies from 0 to 2π radians, are investigated in the absence of the

The source terms caused by the Lorentz force in Eqs. (10) & (11) are such that they are function of sin2λ and cosλsinλ as well as cos2λ, which have the common period of π radians. Thus the numerical simulation is conducted with directional variation of a magnetic field

In Fig. 4, thermo-fluidic behaviour in an enclosure is displayed as to the slanted angle of a magnetic field when Ha = 50. The flow intensity varies in accordance with the change of λ and it becomes strongest as λ = 3π/4 rad. This phenomenon can be explained from the flow

**4.2 Effect of the orientation of a magnetic field without radiation** 

thermal radiation. Assuming constant buoyant action, Gr is fixed as 106.

applied from λ = 0 to π rad on account of the phase difference of π radians.

\* along constant x\* lines, setting u \*dy \* Ψ\* = 0 at x\* = y\* = 0.

integrating the integral ∫ <sup>Ψ</sup> <sup>=</sup> <sup>1</sup>

gravity, i.e., λ = - γ.

0

dimensionless stream function, unless otherwise specified.

**4.1 Influence of the tilting of an enclosure without radiation** 

Fig. 3. Isotherm and streamline contours for Gr = 106 and Ha = 50: (a) γ = π/6, (b) γ = π/4, (c) γ = π/3, (d) γ = 5π/12 and (e) γ = π/2 radians

(a)

(b)

(c)

(d)

(e)

γ = π/3, (d) γ = 5π/12 and (e) γ = π/2 radians

Fig. 3. Isotherm and streamline contours for Gr = 106 and Ha = 50: (a) γ = π/6, (b) γ = π/4, (c)

Isotherm and streamline plots will be reported for different values of controlling parameters. The contour lines of isotherm plots correspond to equally-spaced values of the dimensionless temperature T\*, i.e., ΔT\* = 0.1, in the range between -0.5 and +0.5. On the other hand the dimensionless stream function is obtained from the velocity field solution by integrating the integral ∫ <sup>Ψ</sup> <sup>=</sup> <sup>1</sup> 0 \* along constant x\* lines, setting u \*dy \* Ψ\* = 0 at x\* = y\* = 0.

The contour lines of the streamline plots are correspondent to equally-spaced values of the dimensionless stream function, unless otherwise specified.

#### **4.1 Influence of the tilting of an enclosure without radiation**

A numerical investigation is presented for natural convection of an electrically conducting fluid in a tilted square cavity in the presence of a vertical magnetic field aligned to the gravity, i.e., λ = - γ.

In the present study, the Grashof number is fixed as Gr = 106. Computations are carried out for tilted angles ranging from 0 to π/2 radians, and the thermal radiation is neglected.

Figure 2 shows the isotherm and streamline contours for natural convection in inclined cavities in the absence of a magnetic field. The multi-cellular inner core consists of a central roll (designated by "+" in the figures) sandwiched between two rolls. As the tilting angle decreases, the fluid motion becomes progressively intensive. The temperature is stratified at the core region in case of γ = π/2 rad. When the tilting angle decreases, this trend is maintained until γ = π/4 rad. The stratification of the temperature field in the interior begins to diminish as the inclination angle reaches π/6 rad due to the increasing buoyant action.

The results depicted in Fig. 3 demonstrate the influence of the magnetic field on the fluid flow and the temperature distributions along with the tilting angle. For relatively strong Hartmann number (Ha =50), the temperature stratification in the core tends to diminish, and the thermal boundary layers at the two side walls disappear, together with the decrease in inclination angle. Also, the streamlines are elongated, and the core region becomes broadly stagnated. Furthermore, the axes of the streamlines are changed, which is due to the retarding effect of the Lorentz force. In addition, the flow strength displays maximum at γ = π/4 rad in this case, then, it decreases when γ reaches π/6 rad. This phenomenon is different from the previous result for pure free convection; hence, a considerable interaction between the buoyant and the magnetic forces is evidently caused by the tilting, as the magnitude of the Lorentz force in the x and y directions is subjected to the inclination angle.

#### **4.2 Effect of the orientation of a magnetic field without radiation**

Hydromagnetic flow in a horizontal enclosure (γ = π/2 rad) under a uniform magnetic field is studied. The changes in the flow and thermal field based on the orientation of an external magnetic field, which varies from 0 to 2π radians, are investigated in the absence of the thermal radiation. Assuming constant buoyant action, Gr is fixed as 106.

The source terms caused by the Lorentz force in Eqs. (10) & (11) are such that they are function of sin2λ and cosλsinλ as well as cos2λ, which have the common period of π radians. Thus the numerical simulation is conducted with directional variation of a magnetic field applied from λ = 0 to π rad on account of the phase difference of π radians.

In Fig. 4, thermo-fluidic behaviour in an enclosure is displayed as to the slanted angle of a magnetic field when Ha = 50. The flow intensity varies in accordance with the change of λ and it becomes strongest as λ = 3π/4 rad. This phenomenon can be explained from the flow

Hydromagnetic Flow with Thermal Radiation 143

The changes in flow and thermal fields together with λ are illustrated in Fig. 5, in the case of a strong magnetic field, i.e., Ha = 100. The tendency in the variation of flow and thermal fields influenced by λ, seems to be similar to that for the prior case. A multi-cellular core structure, however, start to appear at the later stage comparing with the case of Ha = 50; in contrast a uni-cellular core structure is recovered at the earlier stage. It is inferred that stronger magnetic field plays a role to suppress the transition of the inner core structure as λ varies 0 to π/2 radians. Inclination of isotherms is obvious than Fig. 4. With a vertically

Computation is carried out for free convection of an electrically conducting fluid in a square enclosure encompassed with radiatively active walls in the presence of a vertically assigned magnetic field parallel to the gravity. In that case, γ is fixed as π/2 rad so that λ is - π/2 rad. Radiation-affected temperature and buoyant flow fields in a square enclosure are demonstrated with Gr = 2 × 106, in the absence of an external magnetic field, i.e., Ha = 0, as presented in Fig. 6 (a). The radiative interaction between the hot and cold walls is significant so that the colder region is extended further into the mid-region. The temperature gradients at the adiabatic walls are steeper owing to the increased interaction by means of the surface radiation. The flow field displays a multi-cellular structure, and the inner core consists of

permeated magnetic field, the inclination of isotherms is most conspicuous.

(a) (b) (c) (d)

and (d) Ha = 100

Ψ\* =

T\* =

Fig. 6. Isotherm and streamline contours with Gr = 2 × 106: (a) Ha = 0, (b) Ha = 10, (c) Ha = 50

It is seen that for a weak magnetic field (Ha = 10), as shown in Fig. 6 (b), the isotherms and streamlines are almost similar to those in the absence of an external magnetic field, i.e., Ha = 0. The flow field becomes less intensive a little bit than that corresponding to the streamline plot in Fig. 6 (a). As a relatively strong magnetic field is applied, i.e., Ha = 50, the thermal and flow fields are considerably changed as depicted in Fig. 6 (c). The streamlines are elongated laterally and the axis of the streamline is slanted. The former convective roll at

**4.3 Effect of combined radiation and a magnetic field** 

two convective rolls in upper and lower halves, respectively.

retardation induced by direct interaction between the magnetic field and the velocity component perpendicular to the direction of the magnetic field. As for streamlines, the orientation of a magnetic field affects the elongation of streamlines. A uni-cellular inner core is formed along with a transverse magnetic field. Following the change in λ, the inner core gets a multi-cellular structure accompanying the elongation of streamlines at the central region. In terms of the thermal field, the tilting of isotherms is most severe with a vertical magnetic field.

Fig. 4. Streamlines and isotherms for Gr = 106 and Ha = 50: (a) λ = 0, π and 2π; (b) λ = π/4 and 5π/4; (c) λ = π/2 and 3π/2; (d) λ = 3π/4 and 7π/4 radians

Fig. 5. Streamlines and isotherms for Gr = 106 and Ha = 100: (a) λ = 0, π and 2π; (b) λ = π/4 and 5π/4; (c) λ = π/2 and 3π/2; (d) λ = 3π/4 and 7π/4 radians

retardation induced by direct interaction between the magnetic field and the velocity component perpendicular to the direction of the magnetic field. As for streamlines, the orientation of a magnetic field affects the elongation of streamlines. A uni-cellular inner core is formed along with a transverse magnetic field. Following the change in λ, the inner core gets a multi-cellular structure accompanying the elongation of streamlines at the central region. In terms of the thermal field, the tilting of isotherms is most severe with a vertical

(a) (b) (c) (d)

(a) (b) (c) (d)

and 5π/4; (c) λ = π/2 and 3π/2; (d) λ = 3π/4 and 7π/4 radians

Fig. 5. Streamlines and isotherms for Gr = 106 and Ha = 100: (a) λ = 0, π and 2π; (b) λ = π/4

and 5π/4; (c) λ = π/2 and 3π/2; (d) λ = 3π/4 and 7π/4 radians

Fig. 4. Streamlines and isotherms for Gr = 106 and Ha = 50: (a) λ = 0, π and 2π; (b) λ = π/4

magnetic field.

The changes in flow and thermal fields together with λ are illustrated in Fig. 5, in the case of a strong magnetic field, i.e., Ha = 100. The tendency in the variation of flow and thermal fields influenced by λ, seems to be similar to that for the prior case. A multi-cellular core structure, however, start to appear at the later stage comparing with the case of Ha = 50; in contrast a uni-cellular core structure is recovered at the earlier stage. It is inferred that stronger magnetic field plays a role to suppress the transition of the inner core structure as λ varies 0 to π/2 radians. Inclination of isotherms is obvious than Fig. 4. With a vertically permeated magnetic field, the inclination of isotherms is most conspicuous.

#### **4.3 Effect of combined radiation and a magnetic field**

Computation is carried out for free convection of an electrically conducting fluid in a square enclosure encompassed with radiatively active walls in the presence of a vertically assigned magnetic field parallel to the gravity. In that case, γ is fixed as π/2 rad so that λ is - π/2 rad. Radiation-affected temperature and buoyant flow fields in a square enclosure are demonstrated with Gr = 2 × 106, in the absence of an external magnetic field, i.e., Ha = 0, as presented in Fig. 6 (a). The radiative interaction between the hot and cold walls is significant so that the colder region is extended further into the mid-region. The temperature gradients at the adiabatic walls are steeper owing to the increased interaction by means of the surface radiation. The flow field displays a multi-cellular structure, and the inner core consists of two convective rolls in upper and lower halves, respectively.

Fig. 6. Isotherm and streamline contours with Gr = 2 × 106: (a) Ha = 0, (b) Ha = 10, (c) Ha = 50 and (d) Ha = 100

It is seen that for a weak magnetic field (Ha = 10), as shown in Fig. 6 (b), the isotherms and streamlines are almost similar to those in the absence of an external magnetic field, i.e., Ha = 0. The flow field becomes less intensive a little bit than that corresponding to the streamline plot in Fig. 6 (a). As a relatively strong magnetic field is applied, i.e., Ha = 50, the thermal and flow fields are considerably changed as depicted in Fig. 6 (c). The streamlines are elongated laterally and the axis of the streamline is slanted. The former convective roll at

Hydromagnetic Flow with Thermal Radiation 145

Free convection in a two-dimensional enclosure filled with an electrically conducting fluid in the presence of an external magnetic field was investigated numerically. The effects of the controlling parameters on the thermally driven hydromagnetic flows have been scrutinised. In the first place the changes in the buoyant flow patterns and temperature distributions due to the tilting of the enclosure were examined neglecting thermal radiation. In general terms, the effect of the tilting angle on the flow patterns and associated heat transfer was found to be considerable. The variation of flow strength was affected by the orientation of the cavity with imposition of the magnetic field because the effective electromagnetic retarding force in each flow direction was subjected closely to the inclination angle. The flow structure and the temperature field were enormously affected by the strength of the magnetic field,

Secondly the flow and thermal field variation was investigated in terms of the orientation of an external magnetic field. The flow intensity and structure varied in accordance with the change of the direction of an external magnetic field. The flow retardation appeared by direct interaction between the magnetic field and the velocity component perpendicular to the direction of the magnetic field. In terms of the thermal field, the tilting of isotherms was

Finally the effects of combined radiation and a magnetic field on the convective flow and heat transfer characteristics of an electrically conducting fluid were investigated. It was concluded that the radiation was the dominant mode of heat transfer and surpassed convective heat transfer so that it played an important role in developing the hydromagnetic

As a consequence, all the numerical analyses so far have been subjected to the rectangular enclosure. Hence the future studies are supposed to be related to the general geometries containing an electrically conducting fluid with the permeation of an external magnetic field

This work is partly supported by KETEP (Korea Institute of Energy Technology Evaluation and Planning) under the Ministry of Knowledge Economy, Korea (2008-E-AP-HM-P-19-

Bessaih, R.; Kadja, M. & Marty, Ph. (1999). Effect of wall electrical conductivity and magnetic

Bian, W.; Vasseur, P., Bilgen, E. & Meng, F. (1996). Effect of an electromagnetic field on

Chai, J. C.; Lee, H. S. & Patankar, S. V. (1994). Finite-volume method for radiation heat

transfer. *J. Thermophysics and Heat Transfer*, Vol.8, pp. 419-425

field orientation on liquid metal flow in a geometry similar to the horizontal Bridgman configuration for crystal growth. *Int. J. Heat Mass Transfer*, Vol.42, pp.

natural convection in an inclined porous layer. *Int. J. Heat and Fluid Flow*, Vol.17, pp.

**5. Conclusions** 

regardless of the tilting angle.

free convective flow in a differentially heated enclosure.

as well as the participation in radiation.

**6. Acknowledgment** 

4345-4362

36-44

observed.

0000).

**7. References** 

the lower left part of the enclosure moves upward. On the contrary the convective roll which was at the upper right region moves downward as to increase in the strength of a magnetic field applied. In the case of the thermal field, severe temperature gradients caused by the surface radiation are maintained at adiabatic top and bottom walls. In mid-region the tilting of isotherms coincides with steeper temperature gradient observed by in-between distance of isotherms getting narrower. These tendencies are preserved until Ha reaches 100, as illustrated in Fig. 6 (d). Besides such typical influence of a magnetic field as the tilting of isotherms and streamlines, appears to be emphasised with the suppression of convection in an enclosure.


Table 1. Nusselt numbers estimated

The rate of heat transfer across the enclosure is attained by evaluating the conductive, radiative, and total average Nusselt numbers, i.e., <sup>C</sup> Nu , <sup>R</sup> Nu , and <sup>T</sup> Nu , respectively, at the hot and cold walls, and tabulated in Table 1 for various combinations of parameters. From this table it can be demonstrated that the introduction of a magnetic field suppresses the convection in the enclosure. With the thermal radiation getting involved in, the radiative contribution to the combined heat transfer is predominant at both hot and cold walls. In addition the convective contribution to the combined heat transfer at the cold wall is always larger than that at the hot wall disregarding the Grashof number and the radiation effect.

#### **5. Conclusions**

144 Convection and Conduction Heat Transfer

the lower left part of the enclosure moves upward. On the contrary the convective roll which was at the upper right region moves downward as to increase in the strength of a magnetic field applied. In the case of the thermal field, severe temperature gradients caused by the surface radiation are maintained at adiabatic top and bottom walls. In mid-region the tilting of isotherms coincides with steeper temperature gradient observed by in-between distance of isotherms getting narrower. These tendencies are preserved until Ha reaches 100, as illustrated in Fig. 6 (d). Besides such typical influence of a magnetic field as the tilting of isotherms and streamlines, appears to be emphasised with the suppression of convection in

Left cold wall Right hot wall Gr Radiation Ha C Nu <sup>R</sup>

Nu <sup>C</sup>

50 1.118 0.000 1.118 0.000 1.118 Without

50 2.997 0.000 2.997 0.000 2.997 Without

50 8.891 0.000 8.891 0.000 8.891 Without

The rate of heat transfer across the enclosure is attained by evaluating the conductive,

hot and cold walls, and tabulated in Table 1 for various combinations of parameters. From this table it can be demonstrated that the introduction of a magnetic field suppresses the convection in the enclosure. With the thermal radiation getting involved in, the radiative contribution to the combined heat transfer is predominant at both hot and cold walls. In addition the convective contribution to the combined heat transfer at the cold wall is always larger than that at the hot wall disregarding the Grashof number and the radiation effect.

0 2.523 0.000 2.523 0.000 2.523 10 2.220 0.000 2.220 0.000 2.220

100 1.116 0.000 1.116 0.000 1.116 0 4.049 36.733 2.105 38.678 40.783 10 3.754 36.759 1.874 38.641 40.513 50 3.021 36.841 1.368 38.494 39.862

100 2.997 36.846 1.357 38.487 39.843 0 5.090 0.000 5.090 0.000 5.090 10 4.983 0.000 4.983 0.000 4.983

100 1.454 0.000 1.454 0.000 1.454 0 6.138 36.486 3.639 38.987 42.624 10 5.986 36.513 3.530 38.970 42.499 50 4.083 36.704 2.068 38.721 40.787

100 3.174 36.808 1.446 38.537 39.982 0 9.904 0.000 9.904 0.000 9.904 10 9.863 0.000 9.863 0.000 9.863

100 6.640 0.000 6.640 0.000 6.640 0 10.413 36.047 6.946 39.514 46.460 10 10.339 36.073 6.914 39.499 46.412 50 9.025 36.313 6.050 39.289 45.338

100 6.699 36.531 4.178 39.054 43.230

Nu , <sup>R</sup>

Nu , and <sup>T</sup>

Nu , respectively, at the

Nu <sup>R</sup>

Nu

T Nu

an enclosure.

2 × 104

2 × 105

2 × 106

With

With

With

radiative, and total average Nusselt numbers, i.e., <sup>C</sup>

Table 1. Nusselt numbers estimated

Free convection in a two-dimensional enclosure filled with an electrically conducting fluid in the presence of an external magnetic field was investigated numerically. The effects of the controlling parameters on the thermally driven hydromagnetic flows have been scrutinised.

In the first place the changes in the buoyant flow patterns and temperature distributions due to the tilting of the enclosure were examined neglecting thermal radiation. In general terms, the effect of the tilting angle on the flow patterns and associated heat transfer was found to be considerable. The variation of flow strength was affected by the orientation of the cavity with imposition of the magnetic field because the effective electromagnetic retarding force in each flow direction was subjected closely to the inclination angle. The flow structure and the temperature field were enormously affected by the strength of the magnetic field, regardless of the tilting angle.

Secondly the flow and thermal field variation was investigated in terms of the orientation of an external magnetic field. The flow intensity and structure varied in accordance with the change of the direction of an external magnetic field. The flow retardation appeared by direct interaction between the magnetic field and the velocity component perpendicular to the direction of the magnetic field. In terms of the thermal field, the tilting of isotherms was observed.

Finally the effects of combined radiation and a magnetic field on the convective flow and heat transfer characteristics of an electrically conducting fluid were investigated. It was concluded that the radiation was the dominant mode of heat transfer and surpassed convective heat transfer so that it played an important role in developing the hydromagnetic free convective flow in a differentially heated enclosure.

As a consequence, all the numerical analyses so far have been subjected to the rectangular enclosure. Hence the future studies are supposed to be related to the general geometries containing an electrically conducting fluid with the permeation of an external magnetic field as well as the participation in radiation.

#### **6. Acknowledgment**

This work is partly supported by KETEP (Korea Institute of Energy Technology Evaluation and Planning) under the Ministry of Knowledge Economy, Korea (2008-E-AP-HM-P-19- 0000).

#### **7. References**


**Part 2** 

**Heat Conduction** 


## **Part 2**

**Heat Conduction** 

146 Convection and Conduction Heat Transfer

Chamkha, A. J. (2000). Thermal radiation and buoyancy effects on hydromagnetic flow over

Fusegi, T. & Farouk, B. (1989). Laminar and turbulent natural convection-radiation

Ghaly, A. Y. (2002). Radiation effects on a certain MHD free-convection flow. *Chaos, Solitons* 

Hua, T. Q. & Walker, J. S. (1995). MHD flow in rectangular ducts with inclined non-uniform transverse magnetic field. *Fusion Engineering and Design*, Vol.27, pp. 703-710 Kolsi, L.; Abidi, A., Borjini, M. N., Daous, N. & Aissia, H. B. (2007). Effect of an external

Larson, D. W. & Viskanta, R. (1976). Transient combined laminar free convection and radiation in a rectangular enclosure. *J. Fluid Mech.*, Vol.78, pp. 65-85 Mahmud, S. & Fraser, R. A. (2002). Analysis of mixed convection-radiation interaction in a vertical channel: entropy generation. *Exergy, an Internal Journal*, Vol.2, pp. 330-339 Ozoe, H. & Okada, K. (1989). The effect of the direction of the external magnetic field on the

Patankar, S. V. (1980). *Numerical Heat Transfer and Fluid Flow*, Hemisphere, McGraw-Hill,

Raptis, A.; Perdikis, C. & Takhar, H. S. (2004). Effect of thermal radiation on MHD flow.

Rudraiah, N.; Barron, R. M., Venkatachalappa, M. & Subbaraya, C. K. (1995). Effect of a

Seddeek, M. A. (2002). Effects of radiation and variable viscosity on a MHD free convection

Seki, M.; Kawamura, H. & Sanokawa, K. (1979). Natural convection of mercury in a magnetic field parallel to the gravity. *J. Heat Transfer*, Vol.101, pp. 227-232 Sivasankaran, S. & Ho, C. J. (2008). Effect of temperature dependent properties on MHD

Thakur, S. & Shyy, W. (1993). Some implementational issues of convection schemes for finite-volume formulations. *Numer. Heat Transfer B*, Vol.24, pp. 31-55 Wang, Q. W.; Zeng, M., Huang, Z. P., Wang, G. & Ozoe, H. (2007). Numerical investigation

*Applied Mathematics and Computation*, Vol.153, pp. 645-649

unsteady flow. *Int. J. Heat Mass Transfer*, Vol.45, pp. 931-935

magnetic field. *Int. J. Heat Mass Transfer*, Vol.50, pp. 3684-3689

pp. 1699-1712

Vol.15, pp. 303-322

*and Fractals*, Vol.13, pp. 1843-1850

*Transfer*, Vol.32, pp. 1939-1954

Washington, DC

Vol.33, pp. 1075-1084

Vol.47, pp. 1184-1194

*Numer. Heat Transfer A*, Vol.51, pp. 1003-1021

an accelerating permeable surface with heat source or sink. *Int. J. Engng Sci.*, Vol.38,

interactions in a square enclosure filled with a nongray gas. *Numer. Heat Transfer A*,

magnetic field on the 3-D unsteady natural convection in a cubical enclosure.

three-dimensional natural convection in a cubical enclosure. *Int. J. Heat Mass* 

magnetic field on free convection in a rectangular enclosure. *Int. J. Engng Sci*.,

flow past a semi-infinite flat plate with an aligned magnetic field in the case of

convection of water near its density maximum in a square cavity. *Int. J. Therm. Sci.*,

of natural convection in an inclined enclosure filled with porous medium under

**7** 

Nencho Deliiski *University of Forestry* 

*Bulgaria* 

**Transient Heat Conduction in** 

Capillarity is a well known phenomenon in physics and engineering. Porous materials such as soil, sand, rocks, mineral building elements (cement stone or concrete, gypsum stone or plasterboards, bricks, mortar, etc.), biological products (wood, grains, fruit, etc.) have microscopic capillaries and pores which cause a mixture of transfer mechanisms to occur

In the most general case each capillary porous material is a peculiar system characterized by the extremely close contact of three intermixed phases: gas (air), liquid (water) and solid. Water may appear in them as physically bounded water and capillary water (Chudinov, 1968, Twardowski, Richinski & Traple, 2006). Both the bounded water and the capillary

Physically bounded water co-operates with the surface of a solid phase of the materials and has different properties than the free water. The maximum amount of bounded water in porous materials corresponds to the maximal hygroscopicity, i.e. moisture absorbed by the material at the 100% relative vapour pressure. The maximum hygroscopicity of the biological capillary porous bodies is known as fibre saturation point. Capillary water fills the capillary tube vessels, small pores or sharp, narrow indentions of bigger pores. It is not bound physically and is called free water. Free water is not in the same thermodynamic state as liquid water: energy is required to overcome the capillary forces, which arise

For the optimization of the heating and/or cooling processes in the capillary porous bodies, it is required that the distribution of the temperature and moisture fields in the bodies and the consumed energy for their heating at every moment of the process are known. The intensity of heating or cooling and the consumption of energy depend on the dimensions and the initial temperature and moisture content of the bodies, on the texture and microstructural features of the porous materials, on their anisotropy and on the content and aggregate condition of the water in them, on the law of change and the values of the temperature and humidity of the heating or cooling medium, etc. (Deliiski, 2004, 2009). The correct and effective control of the heating and cooling processes is possible only when its physics and the weight of the influence of each of the mentioned above as well as of many other specific factors for the concrete capillary porous body are well understood. The summary of the influence of a few dozen factors on the heating or cooling processes of the

**1. Introduction** 

simultaneously when subjected to heating or cooling.

water can be found in liquid or hard aggregate condition.

between the free water and the solid phase of the materials.

**Capillary Porous Bodies** 

Nencho Deliiski *University of Forestry Bulgaria* 

#### **1. Introduction**

Capillarity is a well known phenomenon in physics and engineering. Porous materials such as soil, sand, rocks, mineral building elements (cement stone or concrete, gypsum stone or plasterboards, bricks, mortar, etc.), biological products (wood, grains, fruit, etc.) have microscopic capillaries and pores which cause a mixture of transfer mechanisms to occur simultaneously when subjected to heating or cooling.

In the most general case each capillary porous material is a peculiar system characterized by the extremely close contact of three intermixed phases: gas (air), liquid (water) and solid. Water may appear in them as physically bounded water and capillary water (Chudinov, 1968, Twardowski, Richinski & Traple, 2006). Both the bounded water and the capillary water can be found in liquid or hard aggregate condition.

Physically bounded water co-operates with the surface of a solid phase of the materials and has different properties than the free water. The maximum amount of bounded water in porous materials corresponds to the maximal hygroscopicity, i.e. moisture absorbed by the material at the 100% relative vapour pressure. The maximum hygroscopicity of the biological capillary porous bodies is known as fibre saturation point. Capillary water fills the capillary tube vessels, small pores or sharp, narrow indentions of bigger pores. It is not bound physically and is called free water. Free water is not in the same thermodynamic state as liquid water: energy is required to overcome the capillary forces, which arise between the free water and the solid phase of the materials.

For the optimization of the heating and/or cooling processes in the capillary porous bodies, it is required that the distribution of the temperature and moisture fields in the bodies and the consumed energy for their heating at every moment of the process are known. The intensity of heating or cooling and the consumption of energy depend on the dimensions and the initial temperature and moisture content of the bodies, on the texture and microstructural features of the porous materials, on their anisotropy and on the content and aggregate condition of the water in them, on the law of change and the values of the temperature and humidity of the heating or cooling medium, etc. (Deliiski, 2004, 2009).

The correct and effective control of the heating and cooling processes is possible only when its physics and the weight of the influence of each of the mentioned above as well as of many other specific factors for the concrete capillary porous body are well understood. The summary of the influence of a few dozen factors on the heating or cooling processes of the

needed for the melting of the ice, which has been formed from the freezing of the hygroscopically bounded water in the wood, although the specific heat capacity of that ice is comparable by value to the capacity of the frozen wood itself (Chudinov, 1966), has not been taken into account. These models assume that the fibre saturation point is identical for all wood species and that the melting of the ice, formed by the free water in the wood, occurs at 0ºC. However, it is known that there are significant differences between the fibre saturation point of the separate wood species and that the dependent on this point quantity of ice, formed from the free water in the wood, thaws at a temperature in the range between -2°C

This paper presents the creation and numerical solutions of the 3D, 2D, and 1D mathematical models for the transient non-linear heat conduction in anisotropic frozen and non-frozen prismatic and cylindrical capillary porous bodies, where the physics of the processes of heating and cooling of bodies is taken into account to a maximum degree and the indicated complications and incompleteness in existing analogous models have been overcome. The solutions include the non-stationary temperature distribution in the volume of the bodies for each *u* ≥ 0 kg.kg-1 at every moment of their heating and cooling at prescribed surface temperature, equal to the temperature of the processing medium or

and -1°C (Chudinov, 1968).

**2. Nomenclature** 

*d* = thickness (m) *D* = diameter (m) *L* = length (m)

*b* = width (m)

*R* = radius (m)

*r* = radial coordinate: *0* 

*T* = temperature (К)

(W.m-2.K-1)

ρ = density (kg.m-3)

τ = time (s)

during the time of convective thermal processing.

*а* = temperature conductivity (m2.s-1)

*c* = specific heat capacity (W.kg-1.K-1)

*q* = internal source of heat, W.m-3.s-1

*t* = temperature (°C): *t* = *T* – 271,15 *u* = moisture content (kg.kg-1 = %/100) *x* = coordinate on the thickness: 0

λ = thermal conductivity (W.m-1.K-1)

Δτ = interval between time levels (s)

*y* = coordinate on the width: 0

*z* = longitudinal coordinate: 0

φ = angular coordinate (rad)

≤ *r* ≤ *R* (m)

β = coefficients in the equations for determining of λ γ = coefficients in the equations for determining of λ

≤ *x* ≤ *d/*2 (m)

Δ*r* = distance between mesh points in space coordinates for the cylinders (m) Δ*x* = distance between mesh points in space coordinates for the prisms (m)

≤ *y* ≤ *b/*2 (m)

≤ *z* ≤ *L/*2 (m) α = heat transfer coefficient between the body and the processing medium

capillary porous bodies is a difficult task and its solution is possible only with the assistance of adequate for these processes mathematical models.

There are many publications dedicated to the modelling and computation of distribution of the temperature and moisture content in the subjected to drying capillary porous bodies at different initial and boundary conditions. Drying is fundamentally a problem of simultaneous heat and mass transfer under transient conditions. Luikov (1968) and later Whitaker (1977) defined a coupled system of non-linear partial differential equations for heat and mass transfer in porous bodies. Practically all drying models of capillary porous bodies are based on these equations and include a description of the specific initial and boundary conditions, as well as of thermo- and mass physical characteristics of the subjected to drying bodies (Ben Nasrallah & Perre, 1988, Doe, Oliver & Booker, 1994, Ferguson & Lewis, 1991, Kulasiri & Woodhead 2005, Murugesan et al., 2001, Zhang, Yang & Liu 1999, etc.).

As a rule the non-defective drying is a very long continuous process, which depending on the dimensions of the bodies and their initial moisture content can last many hours, days or even months. However there are a lot of cases in the practice where the capillary porous bodies are subjected to a relatively short heating and/or cooling, as a result of which a significant change in their temperature and a relatively small change in their moisture content occurs. A typical example for this is the change in temperature in building elements under the influence of the changing surrounding temperature during the day and night or during fire.

Widely used during the production of veneer, plywood or furniture parts technological processes of thermal treatment of wood materials (logs, lumber, etc.) with the aim of plasticizing or ennoblement of the wood are characterized by a controlled change in the temperature in the volume of the processed bodies, without it being accompanied by significant changes in their moisture content. In these as well as in other analogous cases of heating and/or cooling of capillary porous bodies the calculation of the non-stationary change in the temperature field in the bodies can be carried out with the assistance of models, which do not take into account the change in moisture content in the bodies. The number of such published models is very limited, especially in comparison to the existing large variety of drying models.

Axenenko (1995) presents and uses a 1D model for the computation of the temperature change in the exposed to fire gypsum plasterboards. According to the author, after obtaining the temperature fields inside the plasterboard, changes in the material properties and temperature deformations can be calculated and used as initial data for the study of the structural behaviour of the entire plasterboard assembly.

Considerate contribution to the calculation of the non-stationary distribution of the temperature in frozen and non-frozen logs and to the duration of their heating has been made by H. P. Steinhagen. For this purpose, he, alone, (Steinhagen, 1986, 1991) or with coauthoring (Steinhagen, Lee & Loehnertz, 1987), (Steinhagen & Lee, 1988) has created and solved a 1-dimensional, and later a 2-dimensional (Khattabi & Steinhagen, 1992, 1993) mathematical model, whose application is limited only for *u* ≥ 0,3 kg.kg-1. The development of these models is dominated by the usage of the method of enthalpy, which is rather more complicated than its competing temperature method.

The models contain two systems of equations, one of which is used for the calculation of the change in temperature at the axis of the log, and the other – for the calculation of the temperature distribution in the remaining points of its volume. The heat energy, which is needed for the melting of the ice, which has been formed from the freezing of the hygroscopically bounded water in the wood, although the specific heat capacity of that ice is comparable by value to the capacity of the frozen wood itself (Chudinov, 1966), has not been taken into account. These models assume that the fibre saturation point is identical for all wood species and that the melting of the ice, formed by the free water in the wood, occurs at 0ºC. However, it is known that there are significant differences between the fibre saturation point of the separate wood species and that the dependent on this point quantity of ice, formed from the free water in the wood, thaws at a temperature in the range between -2°C and -1°C (Chudinov, 1968).

This paper presents the creation and numerical solutions of the 3D, 2D, and 1D mathematical models for the transient non-linear heat conduction in anisotropic frozen and non-frozen prismatic and cylindrical capillary porous bodies, where the physics of the processes of heating and cooling of bodies is taken into account to a maximum degree and the indicated complications and incompleteness in existing analogous models have been overcome. The solutions include the non-stationary temperature distribution in the volume of the bodies for each *u* ≥ 0 kg.kg-1 at every moment of their heating and cooling at prescribed surface temperature, equal to the temperature of the processing medium or during the time of convective thermal processing.

#### **2. Nomenclature**


150 Convection and Conduction Heat Transfer

capillary porous bodies is a difficult task and its solution is possible only with the assistance

There are many publications dedicated to the modelling and computation of distribution of the temperature and moisture content in the subjected to drying capillary porous bodies at different initial and boundary conditions. Drying is fundamentally a problem of simultaneous heat and mass transfer under transient conditions. Luikov (1968) and later Whitaker (1977) defined a coupled system of non-linear partial differential equations for heat and mass transfer in porous bodies. Practically all drying models of capillary porous bodies are based on these equations and include a description of the specific initial and boundary conditions, as well as of thermo- and mass physical characteristics of the subjected to drying bodies (Ben Nasrallah & Perre, 1988, Doe, Oliver & Booker, 1994, Ferguson & Lewis, 1991,

As a rule the non-defective drying is a very long continuous process, which depending on the dimensions of the bodies and their initial moisture content can last many hours, days or even months. However there are a lot of cases in the practice where the capillary porous bodies are subjected to a relatively short heating and/or cooling, as a result of which a significant change in their temperature and a relatively small change in their moisture content occurs. A typical example for this is the change in temperature in building elements under the influence of the changing surrounding temperature during the day and night or

Widely used during the production of veneer, plywood or furniture parts technological processes of thermal treatment of wood materials (logs, lumber, etc.) with the aim of plasticizing or ennoblement of the wood are characterized by a controlled change in the temperature in the volume of the processed bodies, without it being accompanied by significant changes in their moisture content. In these as well as in other analogous cases of heating and/or cooling of capillary porous bodies the calculation of the non-stationary change in the temperature field in the bodies can be carried out with the assistance of models, which do not take into account the change in moisture content in the bodies. The number of such published models is very limited, especially in comparison to the existing

Axenenko (1995) presents and uses a 1D model for the computation of the temperature change in the exposed to fire gypsum plasterboards. According to the author, after obtaining the temperature fields inside the plasterboard, changes in the material properties and temperature deformations can be calculated and used as initial data for the study of the

Considerate contribution to the calculation of the non-stationary distribution of the temperature in frozen and non-frozen logs and to the duration of their heating has been made by H. P. Steinhagen. For this purpose, he, alone, (Steinhagen, 1986, 1991) or with coauthoring (Steinhagen, Lee & Loehnertz, 1987), (Steinhagen & Lee, 1988) has created and solved a 1-dimensional, and later a 2-dimensional (Khattabi & Steinhagen, 1992, 1993) mathematical model, whose application is limited only for *u* ≥ 0,3 kg.kg-1. The development of these models is dominated by the usage of the method of enthalpy, which is rather more

The models contain two systems of equations, one of which is used for the calculation of the change in temperature at the axis of the log, and the other – for the calculation of the temperature distribution in the remaining points of its volume. The heat energy, which is

Kulasiri & Woodhead 2005, Murugesan et al., 2001, Zhang, Yang & Liu 1999, etc.).

of adequate for these processes mathematical models.

during fire.

large variety of drying models.

structural behaviour of the entire plasterboard assembly.

complicated than its competing temperature method.


 λ

<sup>∂</sup> <sup>=</sup> − ±

This form holds for each coordinate system and for each processing medium – both for

In the most general case *с*, ρ and λ of the capillary porous bodies depend on *Т* and *u,* i.e. in

As it was described in the introduction, the water contained in these bodies can be found in liquid or hard aggregate condition. It is known that the specific heat capacity of the liquid (non-frozen) water at 0°С is equal to 4237 J.kg-1.K-1, and the specific heat capacity of ice is 2261 J.kg-1.K-1, i.e. almost two times smaller (Chudinov, 1966, 1984). Because of this, the frozen water in capillary porous bodies causes smaller values of *c* in comparison to the case,

The ice in capillary porous bodies can be formed from the freezing of higroscopically bounded water or of the free water in them. It is widely accepted that the phase transition of water into ice and vice versa to be expressed with the help of the so-called "latent heat" in the ice of the frozen body. When solving problems, connected to transient heat conduction in frozen bodies, it makes sense to include the latent heat in the so-called effective specific heat capacity *c*e (Chudinov, 1966), which is equal to the sum of the own specific heat capacity of the body *с* and the specific heat capacity of the ice, formed in them from the

When solving the problems, it must be taken into consideration that the formation and thawing of both types of ice in these bodies takes place at different temperature ranges. Because of this for each of the diapasons in equation (2) the sum of *с* with *c*bw and/or *c*fw (shown below as an example) participates. When modeling processes of transient heat conduction in anisotropic capillary porous bodies it is also necessary to take into consideration that the thermal conductivity of these bodies λ apart from *Т* and *u* depends additionally on the direction of the influencing heat flux towards the anatomic directions of

**4. Mathematical models for transient heat conduction in prismatic bodies** 

following form of this equation in the Cartesian coordinate system is obtained:

*y z*

τ

If it is assumed, that the anatomical directions of a prismatic capillary porous body coincide with the coordinate axes, in the absence of an internal source of heat *q* in equation (1), the

( ) ( ) ( ) ( )

*T xyz T xyz c Tu Tu T u*

τ

*y yz z*

τ

thermal processing) of the capillary porous bodies with prismatic form is obtained:

,,, ,,, , (,) ,

( ) ( ) ( ) ( )

*T xyz T xyz T u T u*

After the differentiation of the right side of equation (3) on the spatial coordinates *x, y,* and *z*, excluding the arguments in the brackets for shortening of the record, the following mathematical model of the process of non-stationary heating or cooling (further calling

∂ ∂ ⎡ ∂ ∂ ⎤ ⎡ ⎤ + + <sup>⎢</sup> ⎥ ⎢ <sup>⎥</sup> ∂ ∂∂ ∂ <sup>⎣</sup> ⎦ ⎣ <sup>⎦</sup>

,,, ,,, , ,

*x*

∂ ∂ <sup>∂</sup> <sup>⎡</sup> <sup>⎤</sup> <sup>=</sup> <sup>⎢</sup> <sup>⎥</sup> <sup>+</sup> ∂∂ ∂ ⎣ ⎦

 λ

 λ

*x x*

 τ

. (3)

 τ

*q*

<sup>∂</sup> . (1)

*c cc c* e bw fw = + + . (2)

div( grad ) *<sup>T</sup> c T*

ρ

immobile and mobile.

when the water in them is completely liquid.

τ

equation (1) the functional dependencies participate *с*(*Т*,*u*), ρ(*Т*,*u*) and λ(*Т*,*u*).

freezing of the hygroscopically bounded water and of the free water, i.e.

the body – radial, tangential and longitudinal to the fibers.

e

ρ

λ

#### **Subscripts:**


or 1, 2, 3, …, [*L*/(2Δ*x*)]+1 for the prisms

m = medium


#### **Superscripts:**

```
n = time level 0, 1, 2, … 
20 = 20°С
```
#### **3. Mechanism of heat distribution in capillary porous bodies**

During the heating or cooling of the capillary porous materials along with the purely thermal processes, a mass-exchange occurs between the processing medium and the materials. The values of the mass diffusion in these materials are usually hundreds of times smaller than the values of their temperature conductivity. These facts determine a not so big change in defunding mass in the materials, which lags significantly from the distribution of heat in them during the heating or cooling. This allows to disregarding the exchange of mass between the materials and the processing medium and the change in temperature in them to be viewed as a result of a purely thermo-exchange process, where the heat in them is distributed only through thermo-conductivity.

Because of this the mechanism of heat distribution in capillary porous bodies can be described by the equation of heat conduction (also known as the equation of Fourier-Kirchhoff). Its most compact form is as follows:

152 Convection and Conduction Heat Transfer

b = basic (for density, based on dry mass divided to green volume)

*i* = nodal point in radial direction for the cylinders: 1, 2, 3, …, (R/Δ*r*)+1

**3. Mechanism of heat distribution in capillary porous bodies** 

During the heating or cooling of the capillary porous materials along with the purely thermal processes, a mass-exchange occurs between the processing medium and the materials. The values of the mass diffusion in these materials are usually hundreds of times smaller than the values of their temperature conductivity. These facts determine a not so big change in defunding mass in the materials, which lags significantly from the distribution of heat in them during the heating or cooling. This allows to disregarding the exchange of mass between the materials and the processing medium and the change in temperature in them to be viewed as a result of a purely thermo-exchange process, where the heat in them

Because of this the mechanism of heat distribution in capillary porous bodies can be described by the equation of heat conduction (also known as the equation of Fourier-

or in the direction along the thickness for the prisms: 1, 2, 3, …, [*d*/(2Δ*x*)]+1 *j* = nodal point in the direction along the prisms' width: 1, 2, 3, …, [*b*/(2Δ*x*)]+1 *k* = nodal point in longitudinal direction: 1, 2, 3, …, [*L*/(2Δ*r*)]+1 for the cylinders

**Subscripts:** 

a = anatomical direction

fsp = fiber saturation point

e = effective (for specific heat capacity)

or 1, 2, 3, …, [*L*/(2Δ*x*)]+1 for the prisms

p/cr = parallel to the cross sectional

*x* = direction along the thickness *y* = direction along the width *z* = longitudinal direction

r = radial direction (radial to the fibers)

t = tangential direction (tangential to the fibers)

is distributed only through thermo-conductivity.

Kirchhoff). Its most compact form is as follows:

bw = bound water c = center (of the body) cr = cross sectional to the fibers

d = dimension

fw = free water

m = medium

w = wood

**Superscripts:** 

20 = 20°С

nfw = non-frozen water 0 = initial (at 0°C for λ) p = parallel to the fibers

p/r = parallel to the radial

*n* = time level 0, 1, 2, …

$$
\varepsilon \rho \frac{\partial T}{\partial \tau} = -\text{div}(\lambda \text{grad} T) \pm q \,\tag{1}
$$

This form holds for each coordinate system and for each processing medium – both for immobile and mobile.

In the most general case *с*, ρ and λ of the capillary porous bodies depend on *Т* and *u,* i.e. in equation (1) the functional dependencies participate *с*(*Т*,*u*), ρ(*Т*,*u*) and λ(*Т*,*u*).

As it was described in the introduction, the water contained in these bodies can be found in liquid or hard aggregate condition. It is known that the specific heat capacity of the liquid (non-frozen) water at 0°С is equal to 4237 J.kg-1.K-1, and the specific heat capacity of ice is 2261 J.kg-1.K-1, i.e. almost two times smaller (Chudinov, 1966, 1984). Because of this, the frozen water in capillary porous bodies causes smaller values of *c* in comparison to the case, when the water in them is completely liquid.

The ice in capillary porous bodies can be formed from the freezing of higroscopically bounded water or of the free water in them. It is widely accepted that the phase transition of water into ice and vice versa to be expressed with the help of the so-called "latent heat" in the ice of the frozen body. When solving problems, connected to transient heat conduction in frozen bodies, it makes sense to include the latent heat in the so-called effective specific heat capacity *c*e (Chudinov, 1966), which is equal to the sum of the own specific heat capacity of the body *с* and the specific heat capacity of the ice, formed in them from the freezing of the hygroscopically bounded water and of the free water, i.e.

$$
\mathfrak{c}\_{\mathbf{e}} = \mathfrak{c} + \mathfrak{c}\_{\mathbf{bw}} + \mathfrak{c}\_{\mathbf{fw}} \,. \tag{2}
$$

When solving the problems, it must be taken into consideration that the formation and thawing of both types of ice in these bodies takes place at different temperature ranges. Because of this for each of the diapasons in equation (2) the sum of *с* with *c*bw and/or *c*fw (shown below as an example) participates. When modeling processes of transient heat conduction in anisotropic capillary porous bodies it is also necessary to take into consideration that the thermal conductivity of these bodies λ apart from *Т* and *u* depends additionally on the direction of the influencing heat flux towards the anatomic directions of the body – radial, tangential and longitudinal to the fibers.

#### **4. Mathematical models for transient heat conduction in prismatic bodies**

If it is assumed, that the anatomical directions of a prismatic capillary porous body coincide with the coordinate axes, in the absence of an internal source of heat *q* in equation (1), the following form of this equation in the Cartesian coordinate system is obtained:

$$\begin{split} c\_{\varepsilon}(T,\mu)\,\rho(T,\mu)\frac{\partial T\left(\mathbf{x},\mathbf{y},z,\tau\right)}{\partial\tau} &= \frac{\partial}{\partial\mathbf{x}}\bigg[\mathcal{A}\_{\mathbf{x}}\big(T,\mu\big)\frac{\partial T\left(\mathbf{x},\mathbf{y},z,\tau\right)}{\partial\mathbf{x}}\bigg] + \\ + \frac{\partial}{\partial\mathbf{y}}\bigg[\mathcal{A}\_{\mathbf{y}}\big(T,\mu\big)\frac{\partial T\left(\mathbf{x},\mathbf{y},z,\tau\right)}{\partial\mathbf{y}}\bigg] + \frac{\partial}{\partial\mathbf{z}}\bigg[\mathcal{A}\_{\mathbf{z}}\big(T,\mu\big)\frac{\partial T\left(\mathbf{x},\mathbf{y},z,\tau\right)}{\partial\mathbf{z}}\bigg] \end{split} \tag{3}$$

After the differentiation of the right side of equation (3) on the spatial coordinates *x, y,* and *z*, excluding the arguments in the brackets for shortening of the record, the following mathematical model of the process of non-stationary heating or cooling (further calling thermal processing) of the capillary porous bodies with prismatic form is obtained:

• for thermal processing of the prisms at their prescribed surface temperature:

r

t

λ

λ

α

*x y* τα

*y x* τ

from the frontal sides, can be calculated using the following 1D model:

ρ

τ

*c*

*x* τ

from rectangular to cylindrical coordinates (Deliiski, 1979)

 λ

τ

τ

e r 2 22

• for convective thermal processing of the prisms:

r

λ

• for convective thermal processing of the prisms:

λ

2 2,5

length by at least ( ) <sup>p</sup>

with an initial condition

and boundary conditions:

ρ

λτ

*T y Tx T* (0, , ,0, τ

 ) = = ( τ

(0, , ) (0, , ) (0, , ) ( ) (0, , ) *Ty y Ty T*

( ,0, ) ( ,0, ) ( ,0, ) ( ) ( ,0, ) *T x <sup>x</sup> Tx T*

 τ

 τ

When the thickness of the body is smaller than its width by at least 2 ÷ 3 times, and than its

the radial direction *x* of the body, coinciding with its thickness in the section, equally distant

<sup>r</sup> e r <sup>2</sup> *TT T*

 λ

• for thermal processing of the prisms at their prescribed surface temperature:

*T T* (0,τ ) = <sup>m</sup> (τ

r

**5. Mathematical models for transient heat conduction in cylindrical bodies**  The mechanism of the heat distribution in the volume of cylindrical capillary porous bodies during their thermal processing can be described by the following non-linear differential equation with partial derivatives, which is obtained from the equation (3) after passing in it

λ τ

 ατ

( )( ) ( ) ( ) ( ) ( ) ( )

, , , , 1 1 ,, ,, , , ,

r p

φ

 τ

*c Tu Tu T u r r r r*

(0, ) (0, ) (0, ) ( ) (0, ) *<sup>T</sup> T T*

( ) ( ) ( ) ( ) ( ) ( ) ( )

, ,, <sup>1</sup> , , , , , , , ,

λ

*T u*

2 2 p

*T r r z T z*

*Tu Trz Trz Trz T u Trz*

∂ ⎡∂ ⎤ ⎡∂ ⎤ ⎧ ⎫ ⎪ ⎪ <sup>∂</sup> <sup>∂</sup> ⎡∂ ⎤ + ++ + ⎨ ⎬ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ∂∂ ∂ <sup>∂</sup> ∂ ∂ ⎪ ⎪ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎩ ⎭

2 2 2 2

φ

 τ

ττ

λ

 τ . (19)

2 2

∂ ∂ ∂∂ ⎡ ⎤ = + ++ ⎢ ⎥ ∂ ∂ ∂ ∂ ⎣ ⎦

*Trz Trz Trz Trz*

 τ

 τ

 τ  τ

[ ] <sup>r</sup> <sup>m</sup>

[ ] <sup>t</sup> <sup>m</sup>

2 2

[ ] <sup>r</sup> <sup>m</sup>

τ τ

<sup>∂</sup> <sup>=</sup> − − <sup>∂</sup> . (18)

*x T x* λ

τ

τ

<sup>÷</sup> times, then the non-stationary change in *T*, for example along

<sup>∂</sup> <sup>=</sup> − − <sup>∂</sup> , (13)

<sup>∂</sup> <sup>=</sup> − − <sup>∂</sup> . (14)

τ

τ

∂ ∂ ∂∂⎛ ⎞ = + ⎜ ⎟ ∂ ∂∂ <sup>∂</sup> ⎝ ⎠ (15)

( ) <sup>0</sup> *Tx T* ,0 = (16)

) , (17)

) <sup>m</sup> ( ) , (12)

$$\begin{split} c\_{\text{e}} \rho \frac{\partial T}{\partial \tau} &= \lambda\_{\text{e}} \frac{\partial^2 T}{\partial \mathbf{x}^2} + \frac{\partial \lambda\_{\text{e}}}{\partial T} \left( \frac{\partial T}{\partial \mathbf{x}} \right)^2 + \\ &+ \lambda\_{\text{e}} \frac{\partial^2 T}{\partial \mathbf{y}^2} + \frac{\partial \lambda\_{\text{e}}}{\partial T} \left( \frac{\partial T}{\partial \mathbf{y}} \right)^2 + \lambda\_{\text{p}} \frac{\partial^2 T}{\partial \mathbf{z}^2} + \frac{\partial \lambda\_{\text{p}}}{\partial T} \left( \frac{\partial T}{\partial \mathbf{z}} \right)^2 \end{split} \tag{4}$$

with an initial condition

$$T\left(\mathbf{x}, y, z, \mathbf{0}\right) = T\_0 \tag{5}$$

and boundary conditions:

• during the time of thermal processing of the prisms at their prescribed surface temperature, equal to the temperature of the processing medium:

$$T\left(0, y, z, \tau\right) = T\left(x, 0, z, \tau\right) = T\left(x, y, 0, \tau\right) = T\_{\text{m}}\left(\tau\right),\tag{6}$$

• during the time of convective thermal processing of the prisms:

$$\frac{\partial T(0, y, z, \tau)}{\partial \mathbf{x}} = -\frac{\alpha\_{\mathbf{r}}(0, y, z, \tau)}{\lambda\_{\mathbf{r}}(0, y, z, \tau)} [T(0, y, z, \tau) - T\_{\mathbf{m}}(\tau)] \, \tag{7}$$

$$\frac{\partial T(\mathbf{x},0,z,\tau)}{\partial y} = -\frac{a\_{\mathbf{t}}(\mathbf{x},0,z,\tau)}{\lambda\_{\mathbf{t}}(\mathbf{x},0,z,\tau)}[T(\mathbf{x},0,z,\tau) - T\_{\mathbf{m}}(\tau)],\tag{8}$$

$$\frac{\partial T(\mathbf{x}, \mathbf{y}, 0, \tau)}{\partial z} = -\frac{\alpha\_{\mathbf{p}}(\mathbf{x}, \mathbf{y}, 0, \tau)}{\lambda\_{\mathbf{p}}(\mathbf{x}, \mathbf{y}, 0, \tau)} [T(\mathbf{x}, \mathbf{y}, 0, \tau) - T\_{\mathbf{m}}(\tau)]\,. \tag{9}$$

The system of equations (4) ÷ (9) presents a 3D mathematical model, which describes the change in temperature in the volume of capillary porous bodies with prismatic form during the time of their thermal processing at corresponding initial and boundary conditions.

When the length of the subjected to thermal processing body exceeds its thickness by at least ( ) <sup>p</sup> 2 2 2,5 λ λλ<sup>÷</sup> <sup>+</sup> times, then the heat transfer through the frontal sides of the body can be

neglected, because it does not influence the change in temperature in the cross-section, which is equally distant from the frontal sides. In these cases for the calculation of the change in *T* in this section (i.e. only along the coordinates *x* and *y*) the following 2D model can be used:

$$\mathcal{L}\_{\text{e}}\rho\frac{\partial T}{\partial \tau} = \mathcal{\lambda}\_{\text{r}}\frac{\partial^{2}T}{\partial \mathbf{x}^{2}} + \frac{\partial \mathcal{\lambda}\_{\text{r}}}{\partial T} \left(\frac{\partial T}{\partial \mathbf{x}}\right)^{2} + \mathcal{\lambda}\_{\text{t}}\frac{\partial^{2}T}{\partial y^{2}} + \frac{\partial \mathcal{\lambda}\_{\text{t}}}{\partial T} \left(\frac{\partial T}{\partial y}\right)^{2} \tag{10}$$

with an initial condition

r t

$$T\left(\mathbf{x}, \mathbf{y}, \mathbf{0}\right) = T\_0 \tag{11}$$

and boundary conditions:

2 2

λ

*x T x*

<sup>2</sup> 2 2 <sup>2</sup> t p

λ

 ττ

τ

τ

> τ

<sup>∂</sup> <sup>=</sup> − − <sup>∂</sup> . (9)

<sup>∂</sup> <sup>=</sup> − − <sup>∂</sup> , (8)

<sup>∂</sup> <sup>=</sup> − − <sup>∂</sup> , (7)

[ ] <sup>r</sup> <sup>m</sup>

[ ] <sup>t</sup> <sup>m</sup>

[ ] <sup>p</sup>

( ) <sup>0</sup> *T x*, , ,0 *y z T*= (5)

) = ( ) = <sup>m</sup> ( ) , (6)

τ

τ

m

τ

(4)

*TT T T y z T y T z*

 λ

∂ ∂∂ ∂ <sup>∂</sup> ⎛ ⎞ <sup>∂</sup> ⎛ ⎞ ++ + + ⎜ ⎟ ⎜ ⎟ ∂ ∂ ∂ ∂ ⎝ ⎠ ∂ ∂ ⎝ ⎠

• during the time of thermal processing of the prisms at their prescribed surface

*T yz T x z T xy T* (0, , , ,0, , , ,0,

r

t

p

α

λ

λ

λ

α

*x y z* τα

*y xz*

*z xy*

τ

(0, , , ) (0, , , ) (0, , , ) ( ) (0, , , ) *T yz yz T yz T*

 τ

 τ

( ,0, , ) ( ,0, , ) ( ,0, , ) ( ) ( ,0, , ) *Tx z x z Tx z T*

 τ

 τ

( , ,0, ) ( , ,0, ) ( , ,0, ) ( ) ( , ,0, ) *Txy x y Txy T*

The system of equations (4) ÷ (9) presents a 3D mathematical model, which describes the change in temperature in the volume of capillary porous bodies with prismatic form during the time of their thermal processing at corresponding initial and boundary conditions. When the length of the subjected to thermal processing body exceeds its thickness by at least

 τ

 τ

<sup>÷</sup> <sup>+</sup> times, then the heat transfer through the frontal sides of the body can be

<sup>2</sup> 2 2 <sup>2</sup>

λ

λ

( ) <sup>0</sup> *T x*, ,0 *y* = *T* (11)

(10)

*x y Tx T y*

neglected, because it does not influence the change in temperature in the cross-section, which is equally distant from the frontal sides. In these cases for the calculation of the change in *T* in this section (i.e. only along the coordinates *x* and *y*) the following 2D model

> ∂ ∂ ∂∂ ∂ ∂ ⎛ ⎞ ∂ ⎛ ⎞ =+ ++ ⎜ ⎟ ⎜ ⎟ ∂ ∂∂ ∂∂ ∂ ∂ ⎝ ⎠ ⎝ ⎠

<sup>r</sup> <sup>t</sup> e r 2 2 <sup>t</sup> *TT T T T*

λ

<sup>r</sup> e r <sup>2</sup>

λ

temperature, equal to the temperature of the processing medium:

τ) = (

• during the time of convective thermal processing of the prisms:

τ

τ  λ

*c*

with an initial condition

and boundary conditions:

( ) <sup>p</sup>

can be used:

2 2,5

r t

with an initial condition

and boundary conditions:

λ

*c*

ρ

τ

 λ

λ

2

λ ρ

λ

τ

t p 2 2

*TT T*

∂ ∂ ∂∂⎛ ⎞ =+ + ⎜ ⎟ ∂ ∂∂ <sup>∂</sup> ⎝ ⎠

• for thermal processing of the prisms at their prescribed surface temperature:

$$T\left(0, y, \tau\right) = T\left(\infty, 0, \tau\right) = T\_{\text{m}}\left(\tau\right), \tag{12}$$

• for convective thermal processing of the prisms:

$$\frac{\partial T(0, y, \tau)}{\partial \mathbf{x}} = -\frac{\alpha\_{\mathbf{r}}(0, y, \tau)}{\lambda\_{\mathbf{r}}(0, y, \tau)} [T(0, y, \tau) - T\_{\mathbf{m}}(\tau)] \, \tag{13}$$

$$\frac{\partial T(\mathbf{x},0,\tau)}{\partial y} = -\frac{\alpha\_{\mathbf{t}}(\mathbf{x},0,\tau)}{\lambda\_{\mathbf{t}}(\mathbf{x},0,\tau)}[T(\mathbf{x},0,\tau) - T\_{\mathbf{m}}(\tau)].\tag{14}$$

When the thickness of the body is smaller than its width by at least 2 ÷ 3 times, and than its length by at least ( ) <sup>p</sup> r 2 2,5 λ λ <sup>÷</sup> times, then the non-stationary change in *T*, for example along the radial direction *x* of the body, coinciding with its thickness in the section, equally distant from the frontal sides, can be calculated using the following 1D model:

$$
\omega\_{\rm e} \rho \frac{\partial T}{\partial \tau} = \lambda\_{\rm r} \frac{\partial^2 T}{\partial \mathbf{x}^2} + \frac{\partial \lambda\_{\rm r}}{\partial T} \left(\frac{\partial T}{\partial \mathbf{x}}\right)^2 \tag{15}
$$

with an initial condition

$$T\left(\mathbf{x},0\right) = T\_0 \tag{16}$$

and boundary conditions:

• for thermal processing of the prisms at their prescribed surface temperature:

$$T\left(0,\tau\right) = T\_{\rm m}\left(\tau\right),\tag{17}$$

• for convective thermal processing of the prisms:

$$\frac{\partial T(0,\tau)}{\partial \mathbf{x}} = -\frac{a\_{\mathbf{r}}(0,\tau)}{\lambda\_{\mathbf{r}}(0,\tau)} [T(0,\tau) - T\_{\mathbf{m}}(\tau)] \,. \tag{18}$$

#### **5. Mathematical models for transient heat conduction in cylindrical bodies**

The mechanism of the heat distribution in the volume of cylindrical capillary porous bodies during their thermal processing can be described by the following non-linear differential equation with partial derivatives, which is obtained from the equation (3) after passing in it from rectangular to cylindrical coordinates (Deliiski, 1979)

$$c\_{\rm e}(T,\mu)\rho(T,\mu)\frac{\partial T(r,z,\tau)}{\partial z} = \lambda\_{\rm r}(T,\mu)\left[\frac{\hat{\sigma}^{2}T(r,z,\tau)}{\hat{\sigma}^{2}} + \frac{1}{r^{2}}\frac{\hat{\sigma}^{2}T(r,z,\tau)}{\hat{\sigma}\phi^{2}} + \frac{1}{r}\frac{\hat{\sigma}T(r,z,\tau)}{\hat{\sigma}r}\right] + \tag{19}$$

$$+\frac{\partial\lambda\_{\rm r}(T,\mu)}{\partial T}\left[\left[\frac{\hat{\sigma}T(r,z,\tau)}{\hat{\sigma}r}\right]^{2} + \frac{1}{r^{2}}\left[\frac{\hat{\sigma}T(r,z,\tau)}{\hat{\sigma}\phi}\right]^{2}\right] + \lambda\_{\rm p}(T,\mu)\frac{\hat{\sigma}^{2}T(r,z,\tau)}{\hat{\sigma}z^{2}} + \frac{\partial\lambda\_{\rm p}(T,\mu)}{\hat{\sigma}T}\left[\frac{\hat{\sigma}T(r,z,\tau)}{\hat{\sigma}z}\right]^{2} \tag{10}$$

Analytical solution of mathematical models, which contain non-linear differential equations with partial derivatives in the form of (4) and (20), is practically impossible without

For the numerical solution of models with such equations the methods of finite differences or of the finite elements can be used. When the bodies have a correct shape – prismatic or cylindrical, the method of finite differences is preferred, because its implementation requires

For the numerical solution of the above presented models for transient heat conduction it makes sense to use the explicit form of the finite-difference method, which allows for the exclusion of any simplifications. The large calculation resources of the contemporary computers eliminate the inconvenience, which creates the limitation for the value of the step

According to the main idea of the finite-difference method, the temperature, which is a uninterrupted function of space and time, is presented using a grid vector, and the

along the spatial and time coordinates through their finite-difference (discrete) analogues. For this purpose the subjected to thermal processing body, or 1/8 of it in the presence of mirror symmetry towards the other 7/8, is "pierced" by a system of mutually perpendicular lines, which are parallel to the three spatial coordinate axes. The distances between the lines, also called as the step of differentiation, is constant for each coordinate direction (Fig. 1). The knots from 0 to 6 with temperatures accordingly from *T*0 to *T*<sup>6</sup> are centers for the presented and neighboring it volume elements. The length of the sides of the volume element Δ*x* , Δ*y* and Δ*z* are steps of differentiation, the size of which determines the distance between the separate knots of the mesh. The entire time of thermal processing of the body is also

element of a subjected to thermal processing body used for the solution of equation (4) is shown on Fig. 1, together with its belonging part from the rectangular calculation mesh.

Fig. 1. A volume element of the body with a built on it calculation mesh for the solution of

equation (4) using the explicit form of the finite-difference method

separated into definite number *n* intervals (steps) with equal duration Δ

by using the explicit form (refer to equation (47)).

<sup>∂</sup> are approximated using the built computational mesh

τ

. A volume

**6. Transformation of the models for transient heat conduction in suitable** 

significant simplifications of these equations and of their boundary conditions.

**form for programming** 

along the time coordinate Δ

*x* ∂ <sup>∂</sup> *, <sup>T</sup> y* ∂ <sup>∂</sup> *, <sup>T</sup> z* ∂ ∂

derivatives *<sup>T</sup>*

less computational resources from the computers.

τ

 and *<sup>T</sup>* τ

∂

If heating or cooling cylindrical bodies of material, which is homogenous in their cross section, the distribution of *T* in their volume does not depend on φ, but only depends on *r* and *z*. Consequently, when excluding the participants in the equation (19), containing φ and when also omitting the arguments in the brackets for the shortening of the record, the following 2D mathematical model is obtained, which describes the change of temperature in the volume of capillary porous bodies with cylindrical form:

$$\begin{split} c\_{\rm e} \rho \frac{\partial T}{\partial \tau} &= \lambda\_{\rm r} \left( \frac{\partial^2 T}{\partial r^2} + \frac{1}{r} \frac{\partial T}{\partial r} \right) + \frac{\partial \lambda\_{\rm r}}{\partial T} \left( \frac{\partial T}{\partial r} \right)^2 + \\ &+ \lambda\_{\rm p} \frac{\partial^2 T}{\partial z^2} + \frac{\partial \lambda\_{\rm p}}{\partial T} \left( \frac{\partial T}{\partial z} \right)^2 \end{split} \tag{20}$$

with an initial condition

$$T\left(r, z, 0\right) = T\_0 \tag{21}$$

and boundary conditions:

• for thermal processing of the bodies at their prescribed surface temperature:

$$T\left(0, z, \tau\right) = T\left(r, 0, \tau\right) = T\_{\text{m}}\left(\tau\right), \tag{22}$$

• for convective thermal processing of the bodies:

$$\frac{\partial T(0, z, \tau)}{\partial r} = -\frac{\alpha\_{\text{r}}(0, z, \tau)}{\lambda\_{\text{r}}(0, z, \tau)} [T(0, z, \tau) - T\_{\text{m}}(\tau)] \, \tag{23}$$

$$\frac{\partial T(r,0,\tau)}{\partial z} = -\frac{\alpha\_{\rm p}(r,0,\tau)}{\lambda\_{\rm p}(r,0,\tau)}[T(r,0,\tau) - T\_{\rm m}(\tau)].\tag{24}$$

When the length of the body exceeds its diameter by at least ( ) <sup>p</sup> r 2 2,5 λ λ<sup>÷</sup> times, then the

heat transfer through the frontal sides of the body can be neglected, because it does not influence the change in temperature of its cross section, which is equally distant from the frontal sides. In such cases, for the calculation of the change in *T* only along the coordinate *r* of this section, the following 1D model can be used:

$$
\rho c\_{\rm e} \rho \frac{\partial T}{\partial \tau} = \lambda\_{\rm r} \left( \frac{\partial^2 T}{\partial r^2} + \frac{1}{r} \frac{\partial T}{\partial r} \right) + \frac{\partial \lambda\_{\rm r}}{\partial T} \left( \frac{\partial T}{\partial r} \right)^2 \tag{25}
$$

with an initial condition

$$T\left(r,0\right) = T\_0 \tag{26}$$

and with boundary conditions, which are identical to the ones in equations (17) and (18), but with derivative of *T* along *r* instead of along *x* in (18).

*x*

*y*

*z*

τ

∂

156 Convection and Conduction Heat Transfer

If heating or cooling cylindrical bodies of material, which is homogenous in their cross section, the distribution of *T* in their volume does not depend on φ, but only depends on *r* and *z*. Consequently, when excluding the participants in the equation (19), containing φ and when also omitting the arguments in the brackets for the shortening of the record, the following 2D mathematical model is obtained, which describes the change of temperature in

*T TT T* 1

∂ ∂ ∂ ∂∂ ⎛ ⎞ ⎛ ⎞ <sup>=</sup> ⎜ ⎟ ++ + ⎜ ⎟ <sup>∂</sup> <sup>∂</sup> ∂ ∂∂ ⎝ ⎠ ⎝ ⎠

<sup>r</sup> e r <sup>2</sup>

2 2 p

• for thermal processing of the bodies at their prescribed surface temperature:

r

p

λ

α

λ

*r z* τα

*T r r*

*z r* τ

When the length of the body exceeds its diameter by at least ( ) <sup>p</sup>

 λ

*T z Tr T* (0, , ,0, τ

 ) = = ( τ

(0, , ) (0, , ) (0, , ) ( ) (0, , ) *Tz z Tz T*

( ,0, ) ( ,0, ) ( ,0, ) ( ) ( ,0, )

 τ

 τ

heat transfer through the frontal sides of the body can be neglected, because it does not influence the change in temperature of its cross section, which is equally distant from the frontal sides. In such cases, for the calculation of the change in *T* only along the coordinate *r*

> <sup>r</sup> e r <sup>2</sup> *T TT T* 1

∂ ∂ ∂ ∂∂ ⎛ ⎞ ⎛ ⎞ = ++ ⎜ ⎟ ⎜ ⎟ <sup>∂</sup> <sup>∂</sup> ∂ ∂∂ ⎝ ⎠ ⎝ ⎠

and with boundary conditions, which are identical to the ones in equations (17) and (18), but

 τ

 τ

*T T z T z*

λ

 λ

∂ ∂ <sup>∂</sup> ⎛ ⎞ + + ⎜ ⎟ <sup>∂</sup> ∂ ∂ ⎝ ⎠

2 2

λ

> τ

[ ] <sup>r</sup> <sup>m</sup>

[ ] <sup>p</sup>

2 2

λ

*r rr T r*

*Tr T*

τ

τ

<sup>∂</sup> <sup>=</sup> − − <sup>∂</sup> , (23)

<sup>∂</sup> <sup>=</sup> − − <sup>∂</sup> . (24)

m

τ

τ

( ) <sup>0</sup> *Trz T* , ,0 = (21)

) <sup>m</sup> ( ) , (22)

r

<sup>÷</sup> times, then the

(25)

λ

λ

2 2,5

( ) <sup>0</sup> *Tr T* ,0 = (26)

(20)

*r rr T r*

the volume of capillary porous bodies with cylindrical form:

ρ

λ

*c*

• for convective thermal processing of the bodies:

of this section, the following 1D model can be used:

*c*

with derivative of *T* along *r* instead of along *x* in (18).

ρ

τ

with an initial condition

and boundary conditions:

with an initial condition

p 2

τ

#### **6. Transformation of the models for transient heat conduction in suitable form for programming**

Analytical solution of mathematical models, which contain non-linear differential equations with partial derivatives in the form of (4) and (20), is practically impossible without significant simplifications of these equations and of their boundary conditions.

For the numerical solution of models with such equations the methods of finite differences or of the finite elements can be used. When the bodies have a correct shape – prismatic or cylindrical, the method of finite differences is preferred, because its implementation requires less computational resources from the computers.

For the numerical solution of the above presented models for transient heat conduction it makes sense to use the explicit form of the finite-difference method, which allows for the exclusion of any simplifications. The large calculation resources of the contemporary computers eliminate the inconvenience, which creates the limitation for the value of the step along the time coordinate Δτby using the explicit form (refer to equation (47)).

According to the main idea of the finite-difference method, the temperature, which is a uninterrupted function of space and time, is presented using a grid vector, and the derivatives *<sup>T</sup>* ∂ <sup>∂</sup> *, <sup>T</sup>* ∂ <sup>∂</sup> *, <sup>T</sup>* ∂ and *<sup>T</sup>* ∂ <sup>∂</sup> are approximated using the built computational mesh

along the spatial and time coordinates through their finite-difference (discrete) analogues.

For this purpose the subjected to thermal processing body, or 1/8 of it in the presence of mirror symmetry towards the other 7/8, is "pierced" by a system of mutually perpendicular lines, which are parallel to the three spatial coordinate axes. The distances between the lines, also called as the step of differentiation, is constant for each coordinate direction (Fig. 1). The knots from 0 to 6 with temperatures accordingly from *T*0 to *T*<sup>6</sup> are centers for the presented and neighboring it volume elements. The length of the sides of the volume element Δ*x* , Δ*y* and Δ*z* are steps of differentiation, the size of which determines the distance between the separate knots of the mesh. The entire time of thermal processing of the body is also separated into definite number *n* intervals (steps) with equal duration Δτ . A volume element of a subjected to thermal processing body used for the solution of equation (4) is shown on Fig. 1, together with its belonging part from the rectangular calculation mesh.

Fig. 1. A volume element of the body with a built on it calculation mesh for the solution of equation (4) using the explicit form of the finite-difference method

 β

Then after carrying out the differentiation of λ along *Т* in equation (4) and substituting in it with finite differences of the first derivative along the time and the first and second derivatives along the spatial coordinates, the following system of equations is obtained:

1 2 ,, ,, 1, , 1, , , , , , 1, , e 0r , , 2 2

*n n n n n nn ijk ijk n i jk i jk ijk ijk i jk*

*T T T T T TT*

0t , , 2 2

0p , , 2 2

After transformation of the equation (28) it is obtained, that the value of the temperature in whichever knot of the built in the volume of a subjected to thermal processing body 3D

<sup>⎧</sup> +− − <sup>⎫</sup> <sup>⎪</sup> <sup>⎪</sup> + +− <sup>⎨</sup>⎡ ⎤ <sup>+</sup> <sup>⎬</sup> ⎣ ⎦ ⎪⎩ ⎭ Δ Δ <sup>⎪</sup>

τ

( ) ( ) ( )

*n n n n nn ijk i jk i jk ijk ijk i jk*

<sup>2</sup> 0r 2 , , 1, , 1, , , , , , 1, ,

<sup>Δ</sup> ⎡ ⎤ ++ + − + − + − + ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ <sup>Δ</sup> ⎣ ⎦

The boundary conditions (6) ÷ (9) get the following easy for programming form: • for thermal processing of the prisms at their prescribed surface temperature:

> 1 1, ,

1

+

1 ,1,

*n i k*

where

<sup>+</sup> <sup>+</sup> <sup>+</sup> <sup>+</sup> = = = *<sup>n</sup> <sup>n</sup> i j*

⎡ ⎤ ⎡ ⎤ + − +− + − + ⎢ ⎥ ⎣ ⎦ <sup>Δ</sup> ⎣ ⎦

<sup>2</sup> 0t 2 , , , 1, , 1, , , , , , 1,

<sup>⎧</sup> <sup>⎫</sup> <sup>⎪</sup> <sup>⎪</sup> <sup>⎪</sup> <sup>⎪</sup> <sup>⎪</sup> <sup>⎪</sup> <sup>⎪</sup> <sup>⎪</sup> <sup>⎨</sup> <sup>⎬</sup> <sup>⎪</sup> <sup>⎪</sup> <sup>⎪</sup> <sup>⎪</sup> ⎡ ⎤ <sup>⎪</sup> ⎡ ⎤ − +− + − <sup>⎪</sup> ⎪⎩ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎪⎭

<sup>Δ</sup> ( ) ( ) ( )

The initial condition (5) in the 3D mathematical model is presented using the following

*T T T T TT*

( ) ( ) ( )

, , ,, 1 ,, 1 ,, ,, ,, 1 273,15 2 *<sup>n</sup> n n n nn T T T T TT ijk ijk ijk ijk ijk ijk* + −

0

*n n n n nn ijk ij k ij k ijk ijk ij k*

*T T T T TT*

<sup>−</sup> <sup>⎧</sup> +− − <sup>⎫</sup> <sup>⎪</sup> <sup>⎪</sup> <sup>=</sup> <sup>⎨</sup>⎡ ⎤ + − <sup>+</sup> <sup>⎬</sup> <sup>+</sup> <sup>Δ</sup> ⎪⎩ ⎭ ⎣ ⎦ Δ Δ <sup>⎪</sup> ⎧ +− − ++− ⎡ ⎤ <sup>+</sup> ⎣ ⎦ Δ Δ

2( ) 1 273,15

2( ) 1 ( 273,15)

<sup>⎫</sup> <sup>⎪</sup> <sup>⎪</sup> <sup>⎨</sup> <sup>⎬</sup> <sup>+</sup> ⎪⎩ ⎭⎪

2( ) 1 ( 273,15)

*n ijk ijk ijk ijk ijk*

*n ij k ij k ijk ijk ij k*

<sup>p</sup> (*T*) are linear both for containing ice, as well as for not

= +− <sup>0</sup> [1 ( 273,15) *T* ]. (27)

, 1, , 1, , , , , , 1.

,, 1 ,, 1 ,, ,, ,, 1

+ − −

*x x*

β

β

β+ − <sup>−</sup>

using the following system of finite

*y y*

*z z*

*n n n nn*

*n n n nn*

is determined depending on the already computed values

 β

+ − −

1 m 1 , ,1

*<sup>x</sup> <sup>G</sup>*

λγ

*<sup>n</sup> T <sup>j</sup> <sup>k</sup> T T T* , (31)

( ) <sup>r</sup> 1, , 0r 1, , [1 273,15 ] *j k <sup>n</sup> j k*

 β

<sup>Δ</sup> <sup>=</sup> + −

*T* α

+ − −

 β

β

<sup>0</sup> *Ti*, *<sup>j</sup>*,*<sup>k</sup>* = *T* . (30)

−

*T T T TT*

+ − −

*T T T TT*

2

(28)

. (29)

, (32)

2

2

the functions

ρ λ<sup>r</sup> ( ) *T* ,

*c T*

mesh at the moment ( 1) *n* + Δ

differences equations:

= +

*x*

λ

λ

> 0p 2

+ +

λ

*z*

finite differences equation:

1, ,

*T*

*j k*

1

β

β

β

• for convective thermal processing of the prisms:

*T GT*

1 2, , 1, , m

*n jk jk*

<sup>+</sup> <sup>+</sup> <sup>=</sup> <sup>+</sup>

1, , 1

*G*

*j k*

*n n*

1 ,, ,,

+

*T T*

*n n ijk ijk*

e

ρ

γ τ

*c y*

τ

+

λγ

λγ

λγ

λ

<sup>t</sup> (*T*) and

λ

containing ice bodies and are described with an equation of the kind:

λ λγ

( )

*ijk*

*ijk*

*ijk*

*T*

*T*

τ

1 273,15 2

1 273,15 2

for the temperature in the preceding moment *n*Δ

 β

 β

> β

As a result of the described procedures, the process for the solution of the non-linear differential equation with partial derivatives (4) is transformed to the solution of the equivalent to it system of linear finite-difference equations, requiring the carrying out of numerous one-type, but not complicated algebraic operations. In the built in the body 3D rectangular mesh, the heat transfer is taken into consideration only along its gradient lines. For each cross point of three mesh lines, called a knot, a finite-difference equation is derived, using which the temperature of this knot is calculated.

Finite-difference equations are obtained by substituting derivatives in the differential equation (4), with their approximate expressions, which are differences between the values of the function in selected knots of the calculation mesh. The temperature in knot 0 with coordinates ( *<sup>i</sup> x , j y , k z , n* τ ) on Fig. 1 is designated as *<sup>n</sup> T*<sup>0</sup> = *Ti*, *<sup>j</sup>*,*<sup>k</sup>* , and the temperature in neighboring knots accordingly as 1 1, , *<sup>n</sup> T T*= *<sup>i</sup>*<sup>+</sup> *<sup>j</sup> <sup>k</sup>* (knot 1), 2 1, , *<sup>n</sup> T T*= *<sup>i</sup>*<sup>−</sup> *<sup>j</sup> <sup>k</sup>* (knot 2), 3 , 1, *<sup>n</sup> T T*= *<sup>i</sup> <sup>j</sup>*<sup>+</sup> *<sup>k</sup>* (knot 3), 4 , 1, *<sup>n</sup> T T*= *<sup>i</sup> <sup>j</sup>*<sup>−</sup> *<sup>k</sup>* (knot 4), 5 ,, 1 *<sup>n</sup> T T*= *ijk*<sup>+</sup> (knot 5), and 6 ,, 1 *<sup>n</sup> T T*= *ijk*<sup>−</sup> (knot 6). For the calculation of temperature *T*0 in each following moment of time ( 1) *n* + Δτ the values of the temperatures from *T*1 to *T*6 in the preceding moment *n*Δτneed to be known.

#### **6.1 Discrete analogues of models for transient heat conduction in prismatic bodies**

The transformation of the non-linear differential equation with partial derivatives (4) in its discrete analogue with the help of the explicit form of the finite-difference method is carried out using the shown on Fig. 2 coordinate system for the positioning of the knots of the calculation mesh, in which the distribution of the temperature in a subjected to thermal processing capillary porous body with prismatic form is computed.

Fig. 2. Positioning of the knots in the calculation mesh on 1/8 of the volume of a subjected to thermal processing prismatic capillary porous body

For the carrying out of the differentiation of λ along *Т* in equation (4), it is necessary to have the function λ ( ) *T* in the separate anatomical directions of the capillary porous body. For the purpose of determination of subsequent transformations of the models, we accept that

As a result of the described procedures, the process for the solution of the non-linear differential equation with partial derivatives (4) is transformed to the solution of the equivalent to it system of linear finite-difference equations, requiring the carrying out of numerous one-type, but not complicated algebraic operations. In the built in the body 3D rectangular mesh, the heat transfer is taken into consideration only along its gradient lines. For each cross point of three mesh lines, called a knot, a finite-difference equation is derived,

Finite-difference equations are obtained by substituting derivatives in the differential equation (4), with their approximate expressions, which are differences between the values of the function in selected knots of the calculation mesh. The temperature in knot 0 with

**6.1 Discrete analogues of models for transient heat conduction in prismatic bodies**  The transformation of the non-linear differential equation with partial derivatives (4) in its discrete analogue with the help of the explicit form of the finite-difference method is carried out using the shown on Fig. 2 coordinate system for the positioning of the knots of the calculation mesh, in which the distribution of the temperature in a subjected to thermal

Fig. 2. Positioning of the knots in the calculation mesh on 1/8 of the volume of a subjected to

For the carrying out of the differentiation of λ along *Т* in equation (4), it is necessary to have

the purpose of determination of subsequent transformations of the models, we accept that

( ) *T* in the separate anatomical directions of the capillary porous body. For

) on Fig. 1 is designated as *<sup>n</sup> T*<sup>0</sup> = *Ti*, *<sup>j</sup>*,*<sup>k</sup>* , and the temperature in

τ

*<sup>n</sup> T T*= *<sup>i</sup>*<sup>−</sup> *<sup>j</sup> <sup>k</sup>* (knot 2), 3 , 1,

τ

need to be known.

*<sup>n</sup> T T*= *ijk*<sup>−</sup> (knot 6). For the

*<sup>n</sup> T T*= *<sup>i</sup> <sup>j</sup>*<sup>+</sup> *<sup>k</sup>*

the values of the

*<sup>n</sup> T T*= *<sup>i</sup>*<sup>+</sup> *<sup>j</sup> <sup>k</sup>* (knot 1), 2 1, ,

*<sup>n</sup> T T*= *ijk*<sup>+</sup> (knot 5), and 6 ,, 1

using which the temperature of this knot is calculated.

τ

*<sup>n</sup> T T*= *<sup>i</sup> <sup>j</sup>*<sup>−</sup> *<sup>k</sup>* (knot 4), 5 ,, 1

thermal processing prismatic capillary porous body

the function

λ

temperatures from *T*1 to *T*6 in the preceding moment *n*Δ

calculation of temperature *T*0 in each following moment of time ( 1) *n* + Δ

processing capillary porous body with prismatic form is computed.

neighboring knots accordingly as 1 1, ,

coordinates ( *<sup>i</sup> x , j y , k z , n*

(knot 3), 4 , 1,

the functions λ<sup>r</sup> ( ) *T* , λ<sup>t</sup> (*T*) and λ<sup>p</sup> (*T*) are linear both for containing ice, as well as for not containing ice bodies and are described with an equation of the kind:

$$
\lambda = \lambda\_0 \gamma \left[ 1 + \beta(T - 273, 15) \right]. \tag{27}
$$

Then after carrying out the differentiation of λ along *Т* in equation (4) and substituting in it with finite differences of the first derivative along the time and the first and second derivatives along the spatial coordinates, the following system of equations is obtained:

$$\begin{split} c\_{\sigma} \rho \frac{T\_{i,j,k}^{n+1} - T\_{i,j,k}^{n}}{\Delta x} &= \lambda\_{01} \mathcal{V} \left[ \left[ 1 + \mathcal{J} \left( T\_{i,j,k}^{n} - 273, 15 \right) \right] \frac{T\_{i+1,j,k}^{n} + T\_{i-1,j,k}^{n} - 27 T\_{i,j,k}^{n}}{\Delta x^{2}} + \beta \frac{\left( T\_{i,j,k}^{n} - T\_{i-1,j,k}^{n} \right)^{2}}{\Delta x^{2}} \right] + \\ &+ \lambda\_{01} \mathcal{V} \left\{ \left[ 1 + \beta (T\_{i,j,k}^{n} - 273, 15) \right] \frac{T\_{i,j+1,k}^{n} + T\_{i,j-1,k}^{n} - 27 T\_{i,j,k}^{n}}{\Delta y^{2}} + \beta \frac{\left( T\_{i,j,k}^{n} - T\_{i,j-1,k}^{n} \right)^{2}}{\Delta y^{2}} \right] + \\ &+ \lambda\_{0p} \mathcal{V} \left\{ \left[ 1 + \beta (T\_{i,j,k}^{n} - 273, 15) \right] \frac{T\_{i,j,k+1}^{n} + T\_{i,j,k-1}^{n} - 2T\_{i,j,k}^{n}}{\Delta z^{2}} + \beta \frac{\left( T\_{i,j,k}^{n} - T\_{i,j,k-1}^{n} \right)^{2}}{\Delta z^{2}} \right] \end{split} \tag{28}$$

After transformation of the equation (28) it is obtained, that the value of the temperature in whichever knot of the built in the volume of a subjected to thermal processing body 3D mesh at the moment ( 1) *n* + Δτ is determined depending on the already computed values for the temperature in the preceding moment *n*Δτ using the following system of finite differences equations:

$$\begin{split} T\_{i,j,k}^{n+1} &= T\_{i,j,k}^{n+1} + \\ & \left[ \frac{\dot{\lambda}\_{0\tau}}{\Delta x^{2}} \bigg[ \left[ 1 + \mathcal{J}\left( T\_{i,j,k}^{n} - 273, 15 \right) \right] \Big( T\_{i+1,j,k}^{n} + T\_{i-1,j,k}^{n} - 2T\_{i,j,k}^{n} \right) + \mathcal{J}\left( T\_{i,j,k}^{n} - T\_{i-1,j,k}^{n} \right)^{2} \Bigg] + \\ & + \frac{\dot{\lambda}\_{0\tau}}{c\_{e}\rho} \Bigg[ \left[ 1 + \mathcal{J}\left( T\_{i,j,k}^{n} - 273, 15 \right) \right] \Big( T\_{i,j+1,k}^{n} + T\_{i,j-1,k}^{n} - 2T\_{i,j,k}^{n} \Big) + \mathcal{J}\left( T\_{i,j,k}^{n} - T\_{i,j-1,k}^{n} \right)^{2} \Bigg] + \\ & + \frac{\dot{\lambda}\_{0\tau}}{\Delta x^{2}} \Bigg[ \left[ 1 + \mathcal{J}\left( T\_{i,j,k}^{n} - 273, 15 \right) \right] \Big( T\_{i,j,k+1}^{n} + T\_{i,j,k-1}^{n} - 2T\_{i,j,k}^{n} \Big) + \mathcal{J}\left( T\_{i,j,k}^{n} - T\_{i,j,k-1}^{n} \right)^{2} \Bigg] \end{split} \tag{29}$$

The initial condition (5) in the 3D mathematical model is presented using the following finite differences equation:

$$T\_{i,j,k}^0 = T\_0 \,. \tag{30}$$

The boundary conditions (6) ÷ (9) get the following easy for programming form:

• for thermal processing of the prisms at their prescribed surface temperature:

$$T\_{1,j,k}^{n+1} = T\_{i,1,k}^{n+1} = T\_{i,j,1}^{n+1} = T\_{\mathbf{m}}^{n+1} \; \prime \tag{31}$$

• for convective thermal processing of the prisms:

$$T\_{1,j,k}^{n+1} = \frac{T\_{2,j,k}^{n} + G\_{1,j,k} T\_{\text{m}}^{n+1}}{1 + G\_{1,j,k}} \quad \text{where} \quad G\_{1,j,k} = \frac{a\_{\text{r}} \Delta \mathbf{x}}{\lambda\_{0\text{r}} \gamma [1 + \beta \left(T\_{1,j,k}^{n} - 273, 15\right)]} \tag{32}$$

<sup>+</sup> <sup>+</sup> <sup>+</sup> = = *<sup>n</sup> <sup>n</sup> i*

1, 0cr 1,

,1 0cr ,1

<sup>2</sup> <sup>1</sup> 0cr

0

1 m 1 1

λγ

1 0cr 1 where 1 [1 273,15 ]

*G T*

Using the system of equations (37) and (30) ÷ (36) of the 3D case, (38) ÷ (42) of the 2D case and (43) ÷ (46) of the 1D case, the integration of the differential equation (4) and of its reduced along the spatial coordinates analogues, is transformed to a consequent determination of the values of *T* in the knots of the calculation mesh, built in the subjected to thermal processing prismatic material, where during the computations the distribution of *T* 

corresponding system of equations. It must not exceed the smaller of the two values

e min min e max max d min d max , ( ,) , ( ,) and , , *cT u T ux cT u T ux KT u KT u*

 τ

*G T*

The discrete analogue of the 1D model, in which equations (15) ÷ (18) participate, becomes

1 273,15 2 *n n <sup>n</sup> n n n nn T T i i T T T T TT <sup>i</sup> i i i ii*

=+ + − + − + − ⎡ ⎤

*G T*

1 [1 273,15 ]

1 [1 273,15 ]

λγ

λγ

{ ( ) ( ) ( ) }

<sup>Δ</sup> ⎣ ⎦ (43)

2 1 1 1

• for thermal processing of the prisms at their prescribed surface temperature:

1 1,

<sup>1</sup> 2, 1, m cr 1, 1,

where

<sup>1</sup> ,2 ,1 m cr ,1 ,1

*i i <sup>n</sup> <sup>i</sup> <sup>i</sup> T GT <sup>x</sup> T G*

• for thermal processing of the prisms at their prescribed surface temperature:

<sup>1</sup> 2 1m cr 1 1

*T GT <sup>x</sup> T G*

<sup>+</sup> <sup>+</sup> <sup>+</sup> <sup>Δ</sup> <sup>=</sup> <sup>=</sup> <sup>+</sup> + −

where

*j j <sup>n</sup> <sup>j</sup> <sup>j</sup> T GT <sup>x</sup> T G*

<sup>+</sup> <sup>+</sup> <sup>Δ</sup> <sup>=</sup> <sup>=</sup> <sup>+</sup> + −

<sup>+</sup> <sup>+</sup> <sup>Δ</sup> <sup>=</sup> <sup>=</sup> <sup>+</sup> + −

1

1

+

β

1

is used in the preceding moment, distanced from the current by Δ

( )

ρ

λ

+

• for convective thermal processing of the prisms:

*n j j*

*n i i*

e

*c x* λ γ τ

Δ

• for convective thermal processing of the prisms:

*n n <sup>n</sup>*

τ

( )

ρ

*n n*

*n n*

0

1 m 1 ,1

<sup>0</sup> *Ti*, *<sup>j</sup>* = *T* (39)

*<sup>n</sup> T <sup>j</sup> T T* , (40)

α

α

 β

 β ( )

( )

 β

<sup>0</sup> *Ti* = *T* (44)

<sup>+</sup> <sup>+</sup> = *<sup>n</sup> <sup>n</sup> T T* , (45)

( )

τ.

( )

ρ

*n*

α

 β

is determined by the requirement for stability of the solution of the

( )

 λ

2 2

Δ Δ Δ = Δ = , (47)

+ − −

, (41)

. (42)

. (46)

with an initial condition

and boundary conditions:

+

with an initial condition

and boundary conditions:

The value of the step Δ

obtained from the equations

τ

$$T\_{i,1,k}^{n+1} = \frac{T\_{i,2,k}^n + G\_{i,1,k} T\_{\text{m}}^{n+1}}{1 + G\_{i,1,k}} \quad \text{where} \quad G\_{i,1,k} = \frac{a\_t \Delta \mathbf{x}}{\lambda\_{0t} \gamma [1 + \beta \left(T\_{i,1,k}^n - 273, 15\right)]},\tag{33}$$

$$T\_{i,j,1}^{n+1} = \frac{T\_{i,j,2}^{n} + G\_{i,j,1} T\_{\text{m}}^{n+1}}{1 + G\_{i,j,1}} \quad \text{where} \quad G\_{i,j,1} = \frac{a\_{\text{p}} \Delta \chi}{\lambda\_{\text{0p}} \chi \left[1 + \beta \left(T\_{i,j,1}^{n} - 273, 15\right)\right]}. \tag{34}$$

In the boundary conditions (31) ÷ (34) is reflected the requirement for the used for the solution of the models computation environment of FORTRAN, that the knots of the mesh, which are positioned on the corresponding surface of the subjected to thermal processing prism, to be designated with number 1, i.e. 1 ≤ *i* ≤ M; 1 ≤ *j* ≤ N; 1 ≤ *k* ≤ KD (Dorn & McCracken 1972). Using the system of equations (29) ÷ (34) the distribution of the temperature in subjected to thermal processing prismatic materials can be calculated, whose radial, tangential and parallel to the fibers direction coincide with its coordinate axes *х*, *у* and *z*. Since the subjected to thermal processing in the practice prismatic materials usually do not have a clear radial or clear tangential sides, but are with partially radial or partially tangential, then in equation (29) instead of the coefficients *λ*0 in the observed two anatomical directions their average arithmetic value can be substituted, which determines the thermal

$$
\mathcal{A}\_{0\text{cr}} = \frac{\mathcal{A}\_{0\text{r}} + \mathcal{A}\_{0\text{t}}}{2} \,. \tag{35}
$$

Also the thermal conductivity at 0°С in the direction parallel to the fibers *λ*0p can be expressed through *λ*0cr using the equation

conductivity at 0°С cross sectional to the body's fibers *λ*0cr :

$$\mathcal{A}\_{\text{0p}} = \mathcal{K}\_{\text{p}/\text{cr}} \mathcal{A}\_{\text{0cr}} \,\prime \,\tag{36}$$

where the coefficient 0p p/cr 0cr *K* λ λ <sup>=</sup> depends on the type of the capillary porous material. For the unification of the calculations it makes sense to use one such step of the calculation mesh along the spatial coordinates ∆*x* = ∆*y* = ∆*z* (refer to Fig. 2). Taking into consideration this condition, and also of equations (35) and (36), the system of equations (29) becomes

$$\begin{aligned} &T\_{i,j,k}^{n+1} = T\_{i,j,k}^{n} + \\ &+ \frac{\lambda\_{0\pi}\gamma\mu\Delta\tau}{c\_{e}\rho\Delta\textbf{r}^{2}} \Bigg[ \begin{bmatrix} 1 + \beta(T\_{i,j,k}^{n} - 273, 15) \\ \left[T\_{i+1,j,k}^{n} + T\_{i-1,j,k}^{n} + T\_{i,j+1,k}^{n} + T\_{i,j-1,k}^{n} + K\_{\text{p/cr}}(T\_{i,j,k+1}^{n} + T\_{i,j,k-1}^{n}) - (\mathbf{4} + 2K\_{\text{p/cr}})T\_{i,j,k}^{n} \right] \\ + \beta\left[\left(T\_{i,j,k}^{n} - T\_{i-1,j,k}^{n}\right)^{2} + \left(T\_{i,j,k}^{n} - T\_{i,j-1,k}^{n}\right)^{2} + K\_{\text{p/cr}}(T\_{i,j,k}^{n} - T\_{i,j,k-1}^{n})^{2} \right] \end{aligned} \tag{37}$$

In this case in equations (32) and (33) instead of α<sup>r</sup> and αt , αcr must be used, and instead of λ0r and λ0t , λ0cr must be used.

The discrete analogue of the 2D model, in which equations (10) ÷ (14) participate, becomes

$$T\_{i,j}^{n+1} = T\_{i,j}^{n} + \frac{\lambda\_{0 \text{cr}} \rho \Delta \tau}{c\_{\text{e}} \rho \Delta \mathbf{x}^{2}} \begin{bmatrix} \left[1 + \beta \left(T\_{i,j}^{n} - 273, 15\right)\right] \left(T\_{i+1,j}^{n} + T\_{i-1,j}^{n} + T\_{i,j+1}^{n} + T\_{i,j-1}^{n} - 4T\_{i,j}^{n}\right) + \\ + \beta \left[\left(T\_{i,j}^{n} - T\_{i-1,j}^{n}\right)^{2} + \left(T\_{i,j}^{n} - T\_{i,j-1}^{n}\right)^{2}\right] \end{bmatrix} \tag{38}$$

with an initial condition

160 Convection and Conduction Heat Transfer

,1, 0t ,1,

, ,1 0p , ,1

0r 0t

λ

<sup>=</sup> depends on the type of the capillary porous material.

0cr 2 λ

Also the thermal conductivity at 0°С in the direction parallel to the fibers *λ*0p can be

0p p/cr 0cr = *K* λ

For the unification of the calculations it makes sense to use one such step of the calculation mesh along the spatial coordinates ∆*x* = ∆*y* = ∆*z* (refer to Fig. 2). Taking into consideration this condition, and also of equations (35) and (36), the system of equations (29) becomes

2 1, , 1, , , 1, , 1, p/cr , , 1 , , 1 p/cr , ,

The discrete analogue of the 2D model, in which equations (10) ÷ (14) participate, becomes

( )( )

− −

= + <sup>⎨</sup> <sup>⎬</sup> <sup>Δ</sup> ⎡ ⎤ ⎪+ − +− <sup>⎪</sup> ⎢ ⎥ ⎩ ⎭ ⎣ ⎦

*ij i j ij ij*

*i j i j i j ij ij ij n n*

+− +− <sup>+</sup>

+ + + + + + −+ + ⎡ ⎤ ⎣ ⎦

*T T T T KT T KT*

*nnnn nn n i jk i jk ij k ij k ijk ijk ijk*

<sup>⎧</sup> <sup>⎫</sup> <sup>⎪</sup> <sup>⎪</sup> <sup>⎪</sup> <sup>⎪</sup> <sup>⎨</sup> <sup>⎬</sup> <sup>⎪</sup> <sup>⎪</sup> <sup>⎪</sup> ⎡ ⎤ <sup>−</sup> <sup>⎪</sup> <sup>⎩</sup> ⎣ ⎦ <sup>⎭</sup>

α<sup>r</sup> and αt , α

, , 1, , , , , 1, p/cr , ,

*ijk i jk ijk ij k ijk*

e , 1, , , 1

*c x TT TT*

+− +− +−

+ − +− + <sup>2</sup>

( )( ) (

− −

<sup>1</sup> 0cr , , <sup>2</sup> 2 2

*ij ij nn nn*

β

β

*T T T T KT*

λ

λ

*G T*

*G T*

In the boundary conditions (31) ÷ (34) is reflected the requirement for the used for the solution of the models computation environment of FORTRAN, that the knots of the mesh, which are positioned on the corresponding surface of the subjected to thermal processing prism, to be designated with number 1, i.e. 1 ≤ *i* ≤ M; 1 ≤ *j* ≤ N; 1 ≤ *k* ≤ KD (Dorn & McCracken 1972). Using the system of equations (29) ÷ (34) the distribution of the temperature in subjected to thermal processing prismatic materials can be calculated, whose radial, tangential and parallel to the fibers direction coincide with its coordinate axes *х*, *у* and *z*. Since the subjected to thermal processing in the practice prismatic materials usually do not have a clear radial or clear tangential sides, but are with partially radial or partially tangential, then in equation (29) instead of the coefficients *λ*0 in the observed two anatomical directions their average arithmetic value can be substituted, which determines the thermal

1 [1 273,15 ]

1 [1 273,15 ]

λγ

λγ

1

1

+

, ,1 , ,1

*T G*

conductivity at 0°С cross sectional to the body's fibers *λ*0cr :

expressed through *λ*0cr using the equation

p/cr

*K*

, ,

*n ijk*

*T*

β

β

1 ( 273,15)

⎡ ⎤ + − ⎣ ⎦

<sup>e</sup> 2 2

In this case in equations (32) and (33) instead of

0cr must be used.

λ γτ

> ρ

*nn nn*

0cr

λ

λ

where the coefficient 0p

1 ,, ,,

+

*T T*

*n n ijk ijk*

= +

*T T*

0cr

λ γΔτ

λ0r and λ0t , λ

*c x*

ρΔ

+

*n n*

*n n*

*n ik ik*

*n ij ij*

<sup>1</sup> ,2, ,1, m <sup>t</sup> ,1, ,1,

where

<sup>+</sup> <sup>+</sup> <sup>Δ</sup> <sup>=</sup> <sup>=</sup> <sup>+</sup> + −

*T GT x*

<sup>+</sup> <sup>+</sup> <sup>Δ</sup> <sup>=</sup> <sup>=</sup> <sup>+</sup> + −

*i k i k <sup>n</sup> i k i k T GT <sup>x</sup> T G*

1 , ,2 , ,1 m p

where

*i j i j <sup>n</sup> i j i j*

( )

( )

, (33)

. (34)

. (37)

(38)

α

α

<sup>+</sup> <sup>=</sup> . (35)

( ) (4 2 )

,, 1 ) *n n Tijk*<sup>−</sup>

( ) ( )

*T TTTT T*

1 273,15 4 *n nnnn n*

⎧⎡ ⎤ ⎫ + − +++− + Δ ⎪⎣ ⎦ ⎪

, 1, 1, , 1 , 1 ,

, (36)

cr must be used, and instead of

 β

 β

$$T\_{i,j}^0 = T\_0 \tag{39}$$

and boundary conditions:

• for thermal processing of the prisms at their prescribed surface temperature:

$$T\_{1,j}^{n+1} = T\_{i,1}^{n+1} = T\_{\mathbf{m}}^{n+1} \; \prime \tag{40}$$

• for convective thermal processing of the prisms:

$$T\_{1,j}^{n+1} = \frac{T\_{2,j}^n + G\_{1,j} T\_{\text{m}}^{n+1}}{1 + G\_{1,j}} \quad \text{where} \quad G\_{1,j} = \frac{a\_{\text{cr}} \Delta \mathbf{x}}{\lambda\_{0 \text{cr}} \gamma [1 + \beta \left( T\_{1,j}^n - 273, 15 \right)]},\tag{41}$$

$$T\_{i,1}^{n+1} = \frac{T\_{i,2}^{n} + G\_{i,1} T\_{\text{m}}^{n+1}}{1 + G\_{i,1}} \quad \text{where} \quad G\_{i,1} = \frac{a\_{\text{cr}} \Delta \mathbf{x}}{\lambda\_{0 \text{cr}} \gamma [1 + \beta \left( T\_{i,1}^{\n \quad \pi} - 273, 15 \right)]} \tag{42}$$

The discrete analogue of the 1D model, in which equations (15) ÷ (18) participate, becomes

$$T\_i^{n+1} = T\_i^n + \frac{\mathcal{J}\_{0x}\eta\Delta\tau}{c\_e\rho\Delta x^2} \left[ \left( 1 + \mathcal{J}\left(T\_i^n - 273, 15\right) \right) \left( T\_{i+1}^n + T\_{i-1}^n - 2T\_i^n \right) + \mathcal{J}\left(T\_i^n - T\_{i-1}^n\right)^2 \right] \tag{43}$$

with an initial condition

$$T\_i^0 = T\_0 \tag{44}$$

and boundary conditions:

• for thermal processing of the prisms at their prescribed surface temperature:

$$T\_1^{n+1} = T\_{\text{m}}^{n+1} \, \prime \tag{45}$$

• for convective thermal processing of the prisms:

$$T\_1^{n+1} = \frac{T\_2^n + G\_1 T\_m^{n+1}}{1 + G\_1} \quad \text{where} \quad G\_1 = \frac{a\_{\text{cr}} \Delta \mathbf{x}}{\lambda\_{0 \text{cr}} \gamma [1 + \beta \left(T\_1^n - 273, 15\right)]} \cdot \tag{46}$$

Using the system of equations (37) and (30) ÷ (36) of the 3D case, (38) ÷ (42) of the 2D case and (43) ÷ (46) of the 1D case, the integration of the differential equation (4) and of its reduced along the spatial coordinates analogues, is transformed to a consequent determination of the values of *T* in the knots of the calculation mesh, built in the subjected to thermal processing prismatic material, where during the computations the distribution of *T*  is used in the preceding moment, distanced from the current by Δτ.

The value of the step Δτ is determined by the requirement for stability of the solution of the corresponding system of equations. It must not exceed the smaller of the two values obtained from the equations

$$
\Delta \tau = \frac{c\_{\rm e} \left( T\_{\rm min}, \mu \right) \rho \left( T\_{\rm min}, \mu \right) \Delta \mathbf{x}^2}{K\_{\rm d} \mathcal{L} \left( T\_{\rm min}, \mu \right)} \text{ and } \Delta \tau = \frac{c\_{\rm e} \left( T\_{\rm max}, \mu \right) \rho \left( T\_{\rm max}, \mu \right) \Delta \mathbf{x}^2}{K\_{\rm d} \mathcal{L} \left( T\_{\rm max}, \mu \right)}, \tag{47}
$$

<sup>+</sup> <sup>+</sup> <sup>+</sup> = = *<sup>n</sup> <sup>n</sup> i*

1, 0r 1, where 1 [1 273,15 ]

,1 0p ,1 where 1 [1 273,15 ]

e 1 1 1 1

<sup>Δ</sup> ⎪⎣ ⎦ <sup>⎪</sup> <sup>⎪</sup> <sup>⎪</sup> = + <sup>⎨</sup> <sup>⎬</sup> <sup>Δ</sup> ⎡ ⎤ <sup>⎪</sup> +−+ − + − <sup>⎪</sup> ⎢ ⎥ ⎪⎩ ⎭ ⎣ ⎦ <sup>−</sup> <sup>⎪</sup>

*c r TT T TT TT i*

<sup>1</sup> <sup>2</sup>

with an initial condition presented through equation (44) and boundary conditions,

the presented models, is determined by the condition for stability, described by equation (47). When solving the 2D model d p/r *K K* = 4 , and when solving the 1D model d *K* = 2 in

The mathematical description of the thermo-physical characteristics of the capillary porous bodies consists of the deduction of summarized dependencies as a function of the factors which have an influence on them, which with maximum precision correspond to the their experimentally determined values in the interesting for the practice sufficiently wide ranges for the change in the factors. It is necessary to have such a description when solving the above presented mathematical models of the transient heat conduction in capillary porous

The following thermo-physical characteristics are present in these models: specific heat

The mathematical description of the above mentioned characteristics of the wood, which is a

capacity, thermal conductivity and density of the capillary porous bodies.

typical representative of the studied bodies, is shown as an example below.

**7. Mathematical description of the thermo-physical characteristics of the** 

*G T*

The discrete analogue of the 1D model, in which equations (25), (26), (17) and (18)

*G T*

1 m 1 ,1

λγ

λγ

2 2

*ii i ii ii*

⎧⎡ ⎤ ⎫ + −

1

− + − −

• for heating or cooling of the bodies at their prescribed surface temperature:

• for convective heating or cooling of the bodies in the processing medium:

<sup>1</sup> 2, 1, m <sup>r</sup> 1, 1,

*k k <sup>n</sup> <sup>k</sup> <sup>k</sup> T GT <sup>x</sup> T G*

1 ,2 ,1 m p

<sup>+</sup> <sup>+</sup> <sup>Δ</sup> <sup>=</sup> <sup>=</sup> <sup>+</sup> + −

( )

*i i nn n nn nn*

1 273,15

*n i*

*T*

β

presented through equations (45) and (46), where in (46) instead of

τ

*T GT x*

*i i <sup>n</sup> <sup>i</sup> <sup>i</sup>*

<sup>+</sup> <sup>+</sup> <sup>Δ</sup> <sup>=</sup> <sup>=</sup> <sup>+</sup> + −

1

1

+

,1 ,1

*T G*

+

*n n*

*n n*

*n k k*

*n i i*

1 0r

λγτ

ρ

*n n*

0r are used. The value of the Δ

**capillary porous bodies** 

*T T*

+

λ

(47).

bodies.

1 1,

<sup>0</sup> *T T i k*, 0 = (49)

*<sup>n</sup> T <sup>k</sup> T T* , (50)

α

α

 β

 β ( )

( )

( )( )

, which guarantees the obtaining of robust solutions for

β

αcr and

, (51)

. (52)

(53)

λ0cr , αr and

with an initial condition

and boundary conditions:

participate, takes the form

in which *T*min and *T*max are correspondingly the smallest and biggest of all values of the temperatures, encountered in the initial and boundary conditions of the heat transfer when solving the mathematical model, and *K*<sup>d</sup> is a coefficient, reflecting the dimensioning of the heat flux when calculating the step Δτ : for a 3D heat flux d p/cr *K K* = 6 ; for a 2D heat flux <sup>d</sup> *K* = 4 and for a 1D heat flux d *K* = 2 .

#### **6.2 Discrete analogues of models for transient heat conduction in cylindrical bodies**

For the obtaining of a discrete analogues of equations (20) ÷ (24) the explicit form of the finite-difference method has been used in a way comparable to the reviewed above case of 2D thermal processing of prismatic bodies. The transformation of the non-linear differential equation with partial derivatives (20) in its discrete analogue is carried out using the shown on Fig. 3 coordinate system for the positioning of the knots from the calculation mesh, in which the distribution of temperature in the longitudinal section of subjected to thermal processing cylindrical body is calculated. The calculation mesh for the solution of the 2D model with the help of the finite-differences method is built on ¼ of the longitudinal section of the cylindrical body due to the circumstance that this ¼ is mirror symmetrical towards the remaining ¾ of the same section.

Fig. 3. Positioning of the knots of the calculation mesh on ¼ of the longitudinal section of a subjected to thermal processing cylindrical body

Taking into consideration equations (2) and (27) and using the coefficient 0p p/r 0r *K* λ λ <sup>=</sup> , after applying the explicit form of the finite-differences method to equations (20) ÷ (24), they are transformed into the following system of equations:

$$T\_{i,k}^{n+1} = T\_{i,k}^{n} + \frac{\mathcal{I}\_{0\gamma}\rho\Delta\tau}{c\_{e}\rho\Delta\tau^{2}} \begin{bmatrix} \left[1 + \mathcal{J}\left(T\_{i,k}^{n} - 273.15\right)\right] \\\\ \left[T\_{i-1,k}^{n} + T\_{i+1,k}^{n} + K\_{\mathrm{p}/\tau}\left(T\_{i,k-1}^{n} + T\_{i,k+1}^{n}\right) - \right] \\\\ \left(2 + 2K\_{\mathrm{p}/\tau}\right)T\_{i,k}^{n} + \frac{1}{i-1}\left(T\_{i-1,k}^{n} - T\_{i,k}^{n}\right) \\\\ \mathcal{J}\left[\left(T\_{i-1,k}^{n} - T\_{i,k}^{n}\right)^{2} + K\_{\mathrm{p}/\tau}\left(T\_{i,k-1}^{n} - T\_{i,k}^{n}\right)^{2}\right] \end{bmatrix} + \begin{bmatrix} \left\{\mathrm{s}\_{i}^{\mathrm{s}}\right\} & \mathrm{s}\_{i,k-1}^{\mathrm{s}} \\\\ \left[\mathrm{s}\_{i,k-1}^{\mathrm{s}} - \mathrm{s}\_{i,k-1}^{\mathrm{s}}\right] & \mathrm{s}\_{i,k-1}^{\mathrm{s}} \end{bmatrix} \end{bmatrix} \tag{48}$$

with an initial condition

162 Convection and Conduction Heat Transfer

in which *T*min and *T*max are correspondingly the smallest and biggest of all values of the temperatures, encountered in the initial and boundary conditions of the heat transfer when solving the mathematical model, and *K*<sup>d</sup> is a coefficient, reflecting the dimensioning of the

**6.2 Discrete analogues of models for transient heat conduction in cylindrical bodies**  For the obtaining of a discrete analogues of equations (20) ÷ (24) the explicit form of the finite-difference method has been used in a way comparable to the reviewed above case of 2D thermal processing of prismatic bodies. The transformation of the non-linear differential equation with partial derivatives (20) in its discrete analogue is carried out using the shown on Fig. 3 coordinate system for the positioning of the knots from the calculation mesh, in which the distribution of temperature in the longitudinal section of subjected to thermal processing cylindrical body is calculated. The calculation mesh for the solution of the 2D model with the help of the finite-differences method is built on ¼ of the longitudinal section of the cylindrical body due to the circumstance that this ¼ is mirror symmetrical towards

Fig. 3. Positioning of the knots of the calculation mesh on ¼ of the longitudinal section of a

applying the explicit form of the finite-differences method to equations (20) ÷ (24), they are

,

*T*

*n i k*

1 273,15

1, 1, p/r , 1 , 1 <sup>1</sup> 0r , , <sup>2</sup>

−+ −+ <sup>+</sup>

*ik ik ik ik n n ik ik n nn*

β

β

( )

e p/r , 1, ,

<sup>Δ</sup> ⎪⎢ <sup>⎥</sup> <sup>⎪</sup> <sup>=</sup> + + ⎨⎢ <sup>⎥</sup> <sup>⎬</sup> <sup>Δ</sup> ⎪⎢ ++ − <sup>⎥</sup> <sup>⎪</sup> ⎪⎣ <sup>−</sup> <sup>⎦</sup> <sup>⎪</sup> <sup>⎪</sup> <sup>⎪</sup> <sup>⎡</sup> <sup>⎤</sup> <sup>⎪</sup> −+ − <sup>⎪</sup> <sup>⎢</sup> <sup>⎥</sup> ⎩ ⎭ <sup>⎣</sup> <sup>⎦</sup>

<sup>1</sup> 2 2

*c r KT T T*

( ) ( )

− −

*i T T KT T*

*nn nn*

*T T KT T*

( )( )

*nn nn i k ik ik ik*

<sup>⎧</sup> <sup>⎫</sup> <sup>⎪</sup>⎡ ⎤ + − <sup>⎪</sup> ⎣ ⎦ <sup>⎪</sup> <sup>⎪</sup> ⎪⎡ ++ + −⎤ <sup>⎪</sup>

1, , p/r , 1 ,

1

*i k i k ik*

( )

2 2

−

Taking into consideration equations (2) and (27) and using the coefficient 0p

: for a 3D heat flux d p/cr *K K* = 6 ; for a 2D heat flux

p/r

*K*

0r

(48)

λ

λ<sup>=</sup> , after

τ

heat flux when calculating the step Δ

<sup>d</sup> *K* = 4 and for a 1D heat flux d *K* = 2 .

the remaining ¾ of the same section.

subjected to thermal processing cylindrical body

transformed into the following system of equations:

λγτ

ρ

*T T*

$$T\_{i,k}^{0} = T\_0 \tag{49}$$

and boundary conditions:

• for heating or cooling of the bodies at their prescribed surface temperature:

$$T\_{1,k}^{n+1} = T\_{i,1}^{n+1} = T\_{\mathbf{m}}^{n+1} \; \prime \tag{50}$$

• for convective heating or cooling of the bodies in the processing medium:

$$T\_{1,k}^{n+1} = \frac{T\_{2,k}^n + G\_{1,k} T\_{\text{m}}^{n+1}}{1 + G\_{1,k}} \quad \text{where} \quad G\_{1,k} = \frac{a\_{\text{r}} \Delta \mathbf{x}}{\lambda\_{0\text{r}} \gamma [1 + \beta \left(T\_{1,k}^n - 273, 15\right)]} \tag{51}$$

$$T\_{i,1}^{n+1} = \frac{T\_{i,2}^n + G\_{i,1} T\_{\text{m}}^{n+1}}{1 + G\_{i,1}} \quad \text{where} \quad G\_{i,1} = \frac{a\_{\text{p}} \Delta x}{\lambda\_{\text{0p}} \gamma [1 + \beta \left(T\_{i,1}^n - 273, 15\right)]} \,. \tag{52}$$

The discrete analogue of the 1D model, in which equations (25), (26), (17) and (18) participate, takes the form

$$T\_{i}^{n+1} = T\_{i}^{n} + \frac{\lambda\_{0r}\rho\Delta\tau}{c\_{e}\rho\Delta r^{2}} \begin{bmatrix} \left[1 + \mathcal{J}\left(T\_{i}^{n} - 273, 15\right)\right] \\\\ \left[T\_{i-1}^{n} + T\_{i+1}^{n} - 2T\_{i}^{n} + \frac{1}{i-1}\left(T\_{i-1}^{n} - T\_{i}^{n}\right)\right] + \mathcal{J}\left(T\_{i-1}^{n} - T\_{i}^{n}\right)^{2} \end{bmatrix} \tag{53}$$

with an initial condition presented through equation (44) and boundary conditions, presented through equations (45) and (46), where in (46) instead of αcr and λ0cr , αr and λ0r are used. The value of the Δτ , which guarantees the obtaining of robust solutions for the presented models, is determined by the condition for stability, described by equation (47). When solving the 2D model d p/r *K K* = 4 , and when solving the 1D model d *K* = 2 in (47).

#### **7. Mathematical description of the thermo-physical characteristics of the capillary porous bodies**

The mathematical description of the thermo-physical characteristics of the capillary porous bodies consists of the deduction of summarized dependencies as a function of the factors which have an influence on them, which with maximum precision correspond to the their experimentally determined values in the interesting for the practice sufficiently wide ranges for the change in the factors. It is necessary to have such a description when solving the above presented mathematical models of the transient heat conduction in capillary porous bodies.

The following thermo-physical characteristics are present in these models: specific heat capacity, thermal conductivity and density of the capillary porous bodies.

The mathematical description of the above mentioned characteristics of the wood, which is a typical representative of the studied bodies, is shown as an example below.

The following equations for fsp *cT uu* (,, ) have been derived, which are valid for all wood species for the intervals 0 ≤ *u* ≤ 1,2 kg.kg-1 and 223 ≤ *Т* ≤ 373 К (Deliiski 1990, 2003b, 2004): • When *Т* > 271,15 К, or when *Т* ≤ 271,15 К and simultaneously with this *u u* ≤ nfw :

> 1 9,92 2,55

> 1 5,49 2,95

<sup>2862</sup> <sup>555</sup> *<sup>T</sup> <sup>u</sup> <sup>T</sup> <sup>u</sup> u*

+ +

*<sup>T</sup> <sup>T</sup> <sup>u</sup> <sup>u</sup> <sup>c</sup> Kc* <sup>+</sup> <sup>+</sup> <sup>+</sup> <sup>+</sup> <sup>+</sup> <sup>=</sup> <sup>1</sup>

*<sup>u</sup> <sup>c</sup>* <sup>+</sup>

*<sup>u</sup> <sup>c</sup>* <sup>+</sup>

+

+

*K u*

where the content of non-frozen water in the wood *u*nfw is equals to

<sup>2097</sup> <sup>826</sup> *<sup>T</sup> <sup>u</sup> <sup>T</sup> <sup>u</sup> u*

+ +

1

1

+

+

*u*

*u*

c

fsp *u* = 0,3 kg. kg-1 and fsp *u* = 0,4 kg. kg-1 is shown on Fig. 5.

0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4

**Non-frozen water** 

*u*

**nfw, kg.kg-1**

• When *Т* ≤ 271,15 К and simultaneously with this *u u* > nfw :

2

2

( )

<sup>−</sup> =+ + , (57)

nfw

nfw ( ) ( ) fsp *uu T* <sup>=</sup> 0,12 0,12 exp 0,0567 271,15 +− ⎡ − ⎤ <sup>⎣</sup> <sup>⎦</sup> . (58)

*T*

*u*

, (56)

Ufsp=0,2

Ufsp=0,3

. (59)

W A T E R Ufsp=0,4 kg/kg

<sup>+</sup> <sup>=</sup> , @ *u u* <sup>&</sup>lt; fsp (54)

<sup>+</sup> <sup>=</sup> , @ *u u* <sup>≥</sup> fsp (55)

1 0,0002

1 0,0036

+

+

*u*

526 2,95 0,0022 2261 1976 nfw 2

0,00075 271,15 1,06 0,04

The calculated according to equation (58) change in *u*nfw depending on *t* at fsp *u* = 0,2 kg. kg-1,

I C E


for the calculation of *c*fw (Deliiski, 2003b, 2004, 2009, Dzurenda & Deliiski, 2010):

5 fsp

+

*u*

*u u*

**0 C**

Fig. 5. Change in the content of non-frozen water in the wood *u*nfw depending on *t* and *u*fsp Taking into consideration the latent heat in the phase transition (crystallization) of the water, equal to <sup>5</sup> 3,34.10 J.kg-1 (Chudinov, 1966) the following equation has been obtained

fw fsp 3,34.10 @ 271,15 272,15 & 1

<sup>−</sup> <sup>=</sup> <≤ >

*c T uu*

#### **7.1 Mathematical description of the specific heat capacity of the wood**

The mathematical description of e *c* is done according to equation (2) and is based on an analysis of the physics of the wood frosting and defrosting process. The description reveals the facts determined by Chudinov (1966, 1984), which state that the thawing of the ice from the situated in the wood free water occurs in the temperature range between 271,15 and 272,15 K, and the thawing of the ice from the situated in the wood hygroscopically bounded water depends on the temperature, which is *Т* < 271,15 К.

Besides this, at *Т* < 271,15, a certain portion, *u*nfw , from the hygroscopically bounded water is found in a non-frozen state, where the value of *u*nfw is calculated by equation (58).

The own specific heat capacity of the wood *c* is described mathematically using the shown on Fig. 4 experimentally determined in the dissertations of Kanter (1955) and Chudinov (1966) data for its change as a function of *t* and *u*. In the description apart from *t* and *u*, the independent parameter *u*fsp has been input, which is different for the separate wood species and reflects the influence of the anatomic characteristics of the wood on *c*.

From the analysis on Fig. 4 it can be seen that at *t* = -2°C and all values for the content of water *u* > 0,3 kg.kg-1 during the thawing of wood a jump in the values of *c* is observed. This jump is explained by the phase transition of the frozen free water in the wood at the given values for *t* and *u*, when the influence on *c* of a significant difference in the specific heat capacity of the water in a liquid and hard aggregate condition is observed.

On Fig. 4 the jump of *c* for *u* = 0,2 kg.kg-1 occurs at a temperature around –17°С. It is characterized by the phase transition of the frozen part of the hygroscopically bounded water in the wood at these values for *t* and *u*. Both at 0 *u* = , and at 0,1 *u* = kg.kg-1 there is a lack of a jump in the values for *c*, because even at –40°С there is no ice in the wood.

Fig. 4. Change in the specific heat capacity of the wood *c* depending on *t* and *u*

The mathematical description of e *c* is done according to equation (2) and is based on an analysis of the physics of the wood frosting and defrosting process. The description reveals the facts determined by Chudinov (1966, 1984), which state that the thawing of the ice from the situated in the wood free water occurs in the temperature range between 271,15 and 272,15 K, and the thawing of the ice from the situated in the wood hygroscopically bounded

Besides this, at *Т* < 271,15, a certain portion, *u*nfw , from the hygroscopically bounded water

The own specific heat capacity of the wood *c* is described mathematically using the shown on Fig. 4 experimentally determined in the dissertations of Kanter (1955) and Chudinov (1966) data for its change as a function of *t* and *u*. In the description apart from *t* and *u*, the independent parameter *u*fsp has been input, which is different for the separate wood species

From the analysis on Fig. 4 it can be seen that at *t* = -2°C and all values for the content of water *u* > 0,3 kg.kg-1 during the thawing of wood a jump in the values of *c* is observed. This jump is explained by the phase transition of the frozen free water in the wood at the given values for *t* and *u*, when the influence on *c* of a significant difference in the specific heat

On Fig. 4 the jump of *c* for *u* = 0,2 kg.kg-1 occurs at a temperature around –17°С. It is characterized by the phase transition of the frozen part of the hygroscopically bounded water in the wood at these values for *t* and *u*. Both at 0 *u* = , and at 0,1 *u* = kg.kg-1 there is a

u = 0

u = 0,1

u = 0,2

u = 0,3

u = 0,4

u = 0,6

u = 0,8

u = 1,0

u = 1,2 kg/kg

lack of a jump in the values for *c*, because even at –40°С there is no ice in the wood.


Fig. 4. Change in the specific heat capacity of the wood *c* depending on *t* and *u*

**0 C**

is found in a non-frozen state, where the value of *u*nfw is calculated by equation (58).

**7.1 Mathematical description of the specific heat capacity of the wood** 

and reflects the influence of the anatomic characteristics of the wood on *c*.

capacity of the water in a liquid and hard aggregate condition is observed.

water depends on the temperature, which is *Т* < 271,15 К.

**Specific heat capacity** *c*

**, J.kg-1.K-1**

The following equations for fsp *cT uu* (,, ) have been derived, which are valid for all wood species for the intervals 0 ≤ *u* ≤ 1,2 kg.kg-1 and 223 ≤ *Т* ≤ 373 К (Deliiski 1990, 2003b, 2004):

• When *Т* > 271,15 К, or when *Т* ≤ 271,15 К and simultaneously with this *u u* ≤ nfw :

$$c = \frac{2097u + 826}{1+u} + \frac{9,92u + 2,55}{1+u}T + \frac{0,0002}{1+u}T^2 \quad \text{ @ } \ u < u\_{\text{fsp}} \tag{54}$$

$$\mathcal{L} = \frac{2862u + 555}{1+u} + \frac{5,49u + 2,95}{1+u}T + \frac{0,0036}{1+u}T^2 \quad \text{ @} \quad u \ge \mathfrak{u}\_{\text{fsp}} \tag{55}$$

• When *Т* ≤ 271,15 К and simultaneously with this *u u* > nfw :

$$\mathbf{c} = K\_c \frac{526 + 2.95T + 0.0022T^2 + 2261u + 1976u\_{\mathrm{nfw}}}{1 + u} \,\mathrm{}\,\mathrm{}\tag{56}$$

$$K\_c = 1,06 + 0,04\mu + \frac{0,00075(T - 271, 15)}{\mu\_{\text{rfw}}},\tag{57}$$

where the content of non-frozen water in the wood *u*nfw is equals to

$$
\mu\_{\rm rfrw} = 0,12 + \left(\mu\_{\rm fsp} - 0,12\right) \exp\left[0,0567(T - 271, 15)\right].\tag{58}
$$

The calculated according to equation (58) change in *u*nfw depending on *t* at fsp *u* = 0,2 kg. kg-1, fsp *u* = 0,3 kg. kg-1 and fsp *u* = 0,4 kg. kg-1 is shown on Fig. 5.

Fig. 5. Change in the content of non-frozen water in the wood *u*nfw depending on *t* and *u*fsp Taking into consideration the latent heat in the phase transition (crystallization) of the water, equal to <sup>5</sup> 3,34.10 J.kg-1 (Chudinov, 1966) the following equation has been obtained for the calculation of *c*fw (Deliiski, 2003b, 2004, 2009, Dzurenda & Deliiski, 2010):

$$u\_{\rm fw} = 3,34.10^5 \frac{u - u\_{\rm fsp}}{1 + u} \quad \text{@} \quad 271,15 < T \le 272,15 \quad \& \quad u > u\_{\rm fsp} \,. \tag{59}$$

For the calculation of *c*bw the following equation (Deliiski 2003b, 2009) has been obtained:

( ) ( ) <sup>4</sup>

The change in the calculated according to equation (60) values for *c*bw depending on *T* and *u* is shown on Fig. 7 for the comparatively more often subjected to thermal processing beech

When *Т* and *u* are outside of the indicated in equations (59) and (60) ranges, the values of *c*fw and *c*bw have been assumed to be equal to zero during the solution of the models.

A mathematical description of the thermal conductivity of the wood λ has been done using the shown on Fig. 8 experimentally determined in the dissertation by Kanter (1955) and Chudinov (1966) data for its change as a function of *t* and *u*. The graphs of this figure relate

in both the European (Shubin, 1990, Trebula, 1996, Videlov, 2003, etc.) and the American specialized literature (Steinhagen, 1986, 1991, etc.) when calculating various processes of thermal processing of wood. From the analysis on Fig. 8 it can be seen that at *u u* > fsp the temperature *t* = -2°C for the defrosting of the wood is critical, because at this temperature a

explained by a significant difference in the thermal conductivity of the water in liquid and in


**0 C**

**Temperature** *t***,** 

Fig. 8. Change in the thermal conductivity cross sectional to the fibers

= 515 kg.m-3 and fsp *u* = 0,30 kg.kg-1 depending on *t* and *u*

*T*

*u*

(, ) *t u* shown on Fig. 8, as well as the data for *ctu* (, ) on Fig. 4 find a wide use

cr is observed. This jump increases with the increase in *u*. It is

@ *u u* > nfw . (60)

λ

cr of birch wood,

u = 0

u = 0,1

u = 0,2 u = 0,3 u = 0,4

u = 0,6 u = 0,8 u = 1,0

cr of birch wood

λ

u = 1,2 kg/kg

exp 0,0567 271,15 1,8938.10 0,12 <sup>1</sup>

**7.2 Mathematical description of the thermal conductivity of the wood** 

to the change in the thermal conductivity cross sectional to the fibers

<sup>b</sup> = 515 kg.m-3 and fsp *u* = 0,3 kg.kg-1.

λ

<sup>⎡</sup> − ⎤ <sup>⎣</sup> <sup>⎦</sup> <sup>=</sup> <sup>−</sup> <sup>+</sup>

bw fsp

*c u*

wood with fsp *u* = 0,31 kg.kg-1.

having

ρ

λ

jump in the values of

hard aggregate condition.

The data for cr

0,1

0,2

0,3

0,4

**Thermal conductivity** 

with b ρ

λ**cr, W.m-1.K-1**

0,5

0,6

0,7

0,8

0,9

The change in the calculated according to equation (59) values for *c*fw depending on *u* and *u*fsp of the wood is shown on Fig. 6. It can be seen that the decrease in *u*fsp at a given *u* causes a proportional increase of the capacity *c*fw due to the increase in the numerator in equation (59). The increase of *u* causes an increase of *c*fw , which is caused by the increase of the quantity of ice in the more humid wood, which was formed from the free water.

Fig. 6. Change in the specific heat capacity *c*fw depending on *u* and *u*fsp

Fig. 7. Change in *c*bw of beech wood depending on *T* and *u* at fsp *u* = 0,31 kg.kg-1

The change in the calculated according to equation (59) values for *c*fw depending on *u* and *u*fsp of the wood is shown on Fig. 6. It can be seen that the decrease in *u*fsp at a given *u* causes a proportional increase of the capacity *c*fw due to the increase in the numerator in equation (59). The increase of *u* causes an increase of *c*fw , which is caused by the increase of

Ufsp = 0,2

Ufsp = 0,3

Ufsp = 0,4 kg/kg

u < 0,2

u = 0,2

u = 0,3

u = 0,4

u = 0,6

u = 0,8

u = 1,0

u = 1,2 kg/kg

the quantity of ice in the more humid wood, which was formed from the free water.

0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 1,1 1,2

**Wood moisture content** *u* **, kg.kg-1**


Fig. 7. Change in *c*bw of beech wood depending on *T* and *u* at fsp *u* = 0,31 kg.kg-1

**0 C**

Fig. 6. Change in the specific heat capacity *c*fw depending on *u* and *u*fsp

0

20000

0

500

1000

**Specific heat capacity** *c*

**bw, J.kg-1.K-1**

1500

2000

2500

3000

40000

60000

**Specific heat capacity** *c*

**fw , J.kg-1.K-1**

80000

100000

120000

140000

160000

For the calculation of *c*bw the following equation (Deliiski 2003b, 2009) has been obtained:

$$\omega\_{\rm bw} = 1,8938.10^4 \left( u\_{\rm fsp} - 0.12 \right) \frac{\exp \left| \begin{matrix} 0,0567 \left( T - 271,15 \right) \right|}{1+u} \right\rangle \quad \text{@} \quad u > u\_{\rm rfw} \,\,\,\,\tag{60}$$

The change in the calculated according to equation (60) values for *c*bw depending on *T* and *u* is shown on Fig. 7 for the comparatively more often subjected to thermal processing beech wood with fsp *u* = 0,31 kg.kg-1.

When *Т* and *u* are outside of the indicated in equations (59) and (60) ranges, the values of *c*fw and *c*bw have been assumed to be equal to zero during the solution of the models.

#### **7.2 Mathematical description of the thermal conductivity of the wood**

A mathematical description of the thermal conductivity of the wood λ has been done using the shown on Fig. 8 experimentally determined in the dissertation by Kanter (1955) and Chudinov (1966) data for its change as a function of *t* and *u*. The graphs of this figure relate to the change in the thermal conductivity cross sectional to the fibers λcr of birch wood, having ρ<sup>b</sup> = 515 kg.m-3 and fsp *u* = 0,3 kg.kg-1.

The data for cr λ (, ) *t u* shown on Fig. 8, as well as the data for *ctu* (, ) on Fig. 4 find a wide use in both the European (Shubin, 1990, Trebula, 1996, Videlov, 2003, etc.) and the American specialized literature (Steinhagen, 1986, 1991, etc.) when calculating various processes of thermal processing of wood. From the analysis on Fig. 8 it can be seen that at *u u* > fsp the temperature *t* = -2°C for the defrosting of the wood is critical, because at this temperature a jump in the values of λcr is observed. This jump increases with the increase in *u*. It is explained by a significant difference in the thermal conductivity of the water in liquid and in hard aggregate condition.

Fig. 8. Change in the thermal conductivity cross sectional to the fibers λcr of birch wood with b ρ= 515 kg.m-3 and fsp *u* = 0,30 kg.kg-1 depending on *t* and *u*

Equation (70), together with equation (27), make the calculation of λ applicable for all temperatures larger than –50°С (i.e. 223,15 *T* = К), because only at 223,15 *T* = К and the minimum possible for the wood species fsp *u* = 0,2 kg.kg-1, the value of *u*nfw is *u*nfw = 0,125

The mathematical description of the wood density ρ depending on the moisture content *u*

experimental values set by (Sergovski, 1975) and shown in (Deliiski, 1977, 2003b). The

b fsp

*u* − ρ − +

The values for <sup>20</sup> *u*fsp are given in the specialized literature (Siau, 1984, Videlov, 2003, etc.).

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 **Wood moisture content** *u* **, kg.kg-1**

**8. Computer solution of the models for transient heat conduction in the** 

For the numerical solution of the above presented models for transient heat conduction in prismatic and cylindrical capillary porous bodies a software package has been prepared in

Fig. 9. Change in ρ of all wood species depending on *u* and

( ) *u u*

following equations, which are applicable to all wood species, have been derived:

20

ρ = ρ <sup>−</sup>

4 b 1 9,3.10

<sup>b</sup> which influence it has been carried out using the shown on Fig. 9

<sup>1</sup> @ *u u* <sup>≤</sup> fsp , (71)

ROb = 200 ROb = 300 ROb = 400 ROb = 500 ROb = 600 ROb = 700 ROb = 800

ROb = 900 kg/m3

(1 ) ρ = ρ<sup>b</sup> + *u* @ *u u* > fsp , (72)

fsp fsp *uu T* =− − 0,001( 293,15) . (73)

ρb

kg.kg-1 (Fig. 5).

and basic wood density

where according to (Siau 1984)

200

**bodies** 

400

600

800

**Wood density** ρ**, kg.m-3**

1000

1200

**7.3 Mathematical description of the wood density** 

ρ

In the description of λcr , as well as in that of <sup>e</sup> *c* , *u*fsp has been introduced as an independent variable for different wood species, instead of the ordinarily used averaged value of fsp *u* = 0,30 kg.kg-1 for all wood species. The following equations for the calculation of the coefficients of λ in equation (27) for the ranges 0 1,2 ≤ *u* ≤ kg.kg-1 and 223 373 ≤ *T* ≤ К have been derived (Deliiski, 1994, 2003b, Dzurenda & Deliiski, 2010):

$$\mathcal{R}\_0 = K\_\text{a} \nu \left[ 0, 165 + (1, 39 + 3, 8u) \left( 3, 3.10^{-7} \rho\_\text{b}^2 + 1, 015.10^{-3} \rho\_\text{b} \right) \right] \tag{61}$$

$$\mathbf{v} = \mathbf{0}, \mathbf{1}\mathbf{5} - \mathbf{0}, \mathbf{0}7\boldsymbol{\mu} \quad \text{@} \quad \boldsymbol{\mu} \le \boldsymbol{\mu}\_{\text{fsp}} + \mathbf{0}, \mathbf{1}\,\mathbf{k}\,\mathbf{g}.\,\text{kg}\cdot\text{s}\,\text{}\,\text{s}\,\tag{62}$$

$$\nu = 0, 1284-0, 013u \text{ @ } \mu > \mu\_{\text{fsp}} + 0, 1 \text{ kg.} \text{kg} \text{-} 1. \tag{63}$$

The equations, which have been suggested by Chudinov (1966, 1968) and shown in (Deliiski, 1977) can be used for the determining of the values of the coefficient *K*a in equation (61), which takes into account the influence on λ0 of the heat flux towards the separate anatomic directions of the wood, i.e. for the determining of λr , λt and λ<sup>p</sup> . The coefficients γ and β in equation (27) are calculated by the following equations:

• When *Т* > 271,15 К or when *Т* ≤ 271,15 К and simultaneously with this *u u* < nfw :

γ = 1,0 , (64)

$$\mathfrak{F} = \left( 2, 0.5 + 4u \left( \frac{579}{\rho\_y} - 0, 124 \right) 10^{-3} \quad \text{ @} \quad u \le u\_{\text{fsp}} + 0, 1 \text{ kg.} \text{kg} \text{m}^{-1} \right) \tag{65}$$

$$\beta = 3,65 \left( \frac{579}{\rho\_Y} - 0, 124 \right) 10^{-3} \text{ @ } \mu > u\_{\text{fsp}} + 0, 1 \text{ kg.kg}^{-1}. \tag{66}$$

• When *Т* ≤ 271,15 К and simultaneously with this *u u* ≥ fsp or when *T T*≤ nfw and simultaneously with this fsp 0,125 ≤ *u u* < :

$$\gamma = 1 + 0, \Im4\{1, 1\Im\{\mu - \mu\_{\rm fsp}\}\}\_{\prime} \tag{67}$$

$$\beta = 0,002(\mu - \mu\_{\rm fsp}) - 0,0038\left(\frac{\rm 579}{\rho\_{\rm b}} - 0,124\right). \tag{68}$$

$$T\_{\rm nfw} = 271.15 + \frac{\ln \frac{u\_{\rm nfw} - 0.12}{u\_{\rm fsp} - 0.12}}{0,0567} \,\text{.}\tag{69}$$

For the calculation of λ in the cases, when *T T*≤ nfw and simultaneously with this fsp 0,125 ≤ < *u u* , the value of λnfw is initially determined by substituting *T T*= nfw in equation (27) as well as the values of γ from equation (64) and of β from equation (65). After this, with the usage of the coefficient β from equation (68), the values of λ are calculated with the following equation:

$$
\lambda = \lambda\_{\rm nfw} \left[ 1 + \Re \{ T - T\_{\rm nfw} \} \right] \text{ @ } T \le T\_{\rm nfw} \text{ } \&\ \mathbf{0}, 125 \le \mu \le \nu\_{\rm isp} \,\, . \tag{70}
$$

variable for different wood species, instead of the ordinarily used averaged value of fsp *u* = 0,30 kg.kg-1 for all wood species. The following equations for the calculation of the coefficients of λ in equation (27) for the ranges 0 1,2 ≤ *u* ≤ kg.kg-1 and 223 373 ≤ *T* ≤ К have

0 a b b

− − = ++ <sup>⎡</sup> <sup>+</sup> <sup>⎤</sup> <sup>⎣</sup> <sup>⎦</sup>

*v* = 0,15 − 0,07*u* @ fsp *u u* ≤ + 0,1 kg.kg-1, (62)

*v* = 0,1284 − 0,013*u* @ fsp *u u* > + 0,1 kg.kg-1. (63)

The equations, which have been suggested by Chudinov (1966, 1968) and shown in (Deliiski, 1977) can be used for the determining of the values of the coefficient *K*a in equation (61),

> λr , λt and λ

λ

• When *Т* > 271,15 К or when *Т* ≤ 271,15 К and simultaneously with this *u u* < nfw :

⎟ ⎟ ⎠

• When *Т* ≤ 271,15 К and simultaneously with this *u u* ≥ fsp or when *T T*≤ nfw and

<sup>β</sup> <sup>=</sup> <sup>−</sup> <sup>−</sup> 0,124 <sup>579</sup> 0,002( ) 0,0038

271,15 fsp

For the calculation of λ in the cases, when *T T*≤ nfw and simultaneously with this

equation (27) as well as the values of γ from equation (64) and of β from equation (65). After this, with the usage of the coefficient β from equation (68), the values of λ are calculated

= +

⎜ ⎜ ⎝ <sup>⎛</sup> <sup>−</sup> <sup>ρ</sup>

nfw

*u u*

0,0567

− − b

0,12 0,12 ln

[1 ( )] λ = λnfw + β *T* −*T*nfw @ *TT uu* ≤ nfw & 0,125 ≤ ≤ fsp . (70)

⎟ ⎟ ⎠

⎞

⎞

( ) <sup>3</sup> 0,124 .10 <sup>579</sup> 2,05 <sup>4</sup> <sup>−</sup>

*y*

<sup>3</sup> 0,124 .10 <sup>579</sup> 3,65 <sup>−</sup>

nfw

λ

⎜ ⎜ ⎝ <sup>⎛</sup> <sup>−</sup> <sup>ρ</sup>

*y*

⎜ ⎜ ⎝ <sup>⎛</sup> <sup>−</sup> <sup>ρ</sup>

*K u* 0,165 1,39 3,8 3,3.10 1,015.10

cr , as well as in that of <sup>e</sup> *c* , *u*fsp has been introduced as an independent

ρ

 ρ

0 of the heat flux towards the separate anatomic

γ = 1,0 , (64)

@ fsp *u u* > + 0,1 kg.kg-1. (66)

*u* @ fsp *u u* ≤ + 0,1 kg.kg-1, (65)

1 0,34[1,15( )] fsp γ = + *u* − *u* , (67)

⎟ ⎟ ⎠ ⎞

*T* . (69)

nfw is initially determined by substituting *T T*= nfw in

fsp *u u* , (68)

<sup>p</sup> . The coefficients γ and β in

, (61)

In the description of

λ

λ

 ν

which takes into account the influence on

directions of the wood, i.e. for the determining of

β = +

β =

fsp 0,125 ≤ < *u u* , the value of

with the following equation:

simultaneously with this fsp 0,125 ≤ *u u* < :

equation (27) are calculated by the following equations:

been derived (Deliiski, 1994, 2003b, Dzurenda & Deliiski, 2010):

( )( ) 7 2 <sup>3</sup>

Equation (70), together with equation (27), make the calculation of λ applicable for all temperatures larger than –50°С (i.e. 223,15 *T* = К), because only at 223,15 *T* = К and the minimum possible for the wood species fsp *u* = 0,2 kg.kg-1, the value of *u*nfw is *u*nfw = 0,125 kg.kg-1 (Fig. 5).

#### **7.3 Mathematical description of the wood density**

The mathematical description of the wood density ρ depending on the moisture content *u* and basic wood density ρ <sup>b</sup> which influence it has been carried out using the shown on Fig. 9 experimental values set by (Sergovski, 1975) and shown in (Deliiski, 1977, 2003b). The following equations, which are applicable to all wood species, have been derived:

$$\rho = \rho\_{\rm b} \frac{1+\mu}{1-9.3.10^{-4}\,\rho\_{\rm b}\left(\mu\_{\rm fsp}-\mu\right)} \quad \text{ @} \quad \mu \le \mu\_{\rm fsp} \,\,\,\tag{71}$$

$$
\mathfrak{p} = \mathfrak{p}\_{\mathfrak{b}} (\mathfrak{l} + \mathfrak{u}) \quad \oplus \ \mathfrak{u} > \mathfrak{u}\_{\text{fsp}}.\tag{72}
$$

where according to (Siau 1984)

$$
\mu\_{\rm fsp} = \mu\_{\rm fsp}^{20} - 0.001(T - 293, 15) \,\text{.}\tag{73}
$$

The values for <sup>20</sup> *u*fsp are given in the specialized literature (Siau, 1984, Videlov, 2003, etc.).

Fig. 9. Change in ρ of all wood species depending on *u* and ρb

#### **8. Computer solution of the models for transient heat conduction in the bodies**

For the numerical solution of the above presented models for transient heat conduction in prismatic and cylindrical capillary porous bodies a software package has been prepared in

reaching of the minimally required for cutting of veneer temperature in their centre, equal to

During the time of cooling of the heated prisms a redistribution and equalization of *t* in their cross section takes place, which is especially appropriate for the obtaining of quality veneer. The change in *t*<sup>m</sup> and *t* is shown on Fig. 11 in 5 characteristic points from the cross section of

With the help of the 2D model the change in the *t* in the longitudinal section of non-frozen

time of thermal processing during 20 hours at a prescribed surface temperature <sup>0</sup> *t*<sup>m</sup> = 80 C . The change in *t*<sup>m</sup> and *t* is shown on Fig. 12 in 4 characteristic points of the longitudinal

<sup>0</sup>*t* = 0 C and frozen beech prisms with <sup>0</sup>

section of the logs with coordinates, which are given in the legend of the graphs.

R/2, L/4

R/2, L/2

R, L/4

R, L/2

<sup>b</sup> = 560 kg.kg-1, *u* = 0,6 kg.kg-1 and <sup>20</sup> *<sup>u</sup>*fsp = 0,31 kg.kg-1

Fig. 12. 2D heating at <sup>0</sup> *t*<sup>m</sup> = 80 C of frozen (left) and non-frozen (right) beech logs with

**Temperature** *t***,** 

**0**

**C**

tm

Fig. 11. 2D heating at <sup>0</sup> *t*<sup>m</sup> = 60 C and consequent cooling at <sup>0</sup> *t*<sup>m</sup> = 20 C of frozen (left) and

**Temperature** *t,*

 **0**

**C**

0 4 8 12 16 20 **Time** τ**, h**

fsp *u* = 0,31 kg.kg-1 has been calculated during the

0 4 8 12 16 20 **Time** τ**, h**

<sup>b</sup> = 670 kg.kg-1 and *u* = 0,6

<sup>0</sup>*t* = −2 C with *D* = 0,4 m, *L* = 0,8

tm

R/2, L/4

R/2, L/2

R, L/4

R, L/2

ρ

fsp *u* = 0,29 kg.kg-1. The heating of the prisms continues until the

ρ<sup>b</sup> = 670

 tm d=0, b=0 d/8, b/8 d/4, b/4 d/2, b/4 d/2, b/2

The prisms have the following characteristics: *d* = 0,25 m, *b* = 0,40 m, *L* > 1,0 m,

the prisms with coordinates, which are given in the legend of the graphs.

 tm d=0, b=0 d/8, b/8 d/4, b/4 d/2, b/4 d/2, b/2

kg.kg-1, *u* = 0,6 kg.kg-1 and <sup>20</sup>

0 4 8 12 16 20 24 28 **Time** τ**, h**

= 560 kg.kg-1, *u* = 0,6 kg.kg-1 and <sup>20</sup>

0 4 8 12 16 20 **Time** τ**, h**

ρ

non-frozen (right) oak prisms with cross section 0,25 x 0,40 m,

**8.2 Non-stationary thermal processing of cylindrical wood materials** 

0 <sup>c</sup>*t* = 50 C .

**Temperature** *t,*

kg.kg-1

m, b ρ

R=0,2 m, L=0,8 m,

**Temperature** *t***,** 

**0**

**C**

beech prisms with <sup>0</sup>

 **0**

**C**

FORTRAN (Dorn & McCracken, 1972), which has been input in the developed by Microsoft calculation environment of VISUAL FORTRAN PROFESSIONAL (Deliiski 2003b).

The software package can be used for the calculation and colour visualization (either as animation of the whole process or as 3D, 2D, 1D graphs of each desired moment of the process) of the non-stationary distribution of the temperature fields in the materials containing or not containing ice during their thermal processing. The computation of the change in the temperature field in the volume of materials containing ice in the beginning of their thermal processing is interconnected for the periods of the melting of the ice and after that, taking into account the flexible spatial boundary of the melting ice.

The computation of the temperature fields is done interconnectedly and for the processes of heating and consequent cooling of the materials, i.e. the calculation of the non-stationary change in temperatures in the volume of the materials during the time of their cooling begins from the already reached during the time of calculations distribution of temperature in the end of the heating. Based on the calculations it can be determined when the moment of reaching in the entire volume of the heated wood has occurred for the necessary optimal temperatures needed for bending of the parts or for cutting the veneer, as well as the stage of the ennoblement of the wood desired by the clients.

#### **8.1 Non-stationary thermal processing of prismatic wood materials**

With the help of the 3D model the change in *t* in the volume of non-frozen beech prisms with <sup>0</sup> <sup>0</sup>*t C* = 0 and frozen beech prisms with <sup>0</sup> <sup>0</sup>*t C* = −10 with *d* = 0,4 m, *b* = 0,4 m, *L* = 0,8 m, b ρ = 560 kg.kg-1, *u* = 0,6 kg.kg-1 and <sup>20</sup> fsp *u* = 0,31 kg.kg-1 has been calculated during the time of thermal processing during 20 hours at a prescribed surface temperature <sup>0</sup> *t*<sup>m</sup> = 80 C .

The change in *t*<sup>m</sup> and *t* is shown on Fig. 10 in 6 characteristic points of the volume of the prisms with coordinates, which are given in the legend of the graphs. The increase in *t*<sup>m</sup> from <sup>0</sup> m0 *t* = 0 C to <sup>0</sup> *t*<sup>m</sup> = 80 C is done exponentially with a time constant equal to 1800 s.

Fig. 10. 3D heating at <sup>0</sup> *t*<sup>m</sup> = 80 C of frozen (left) and non-frozen (right) beech prisms with dimensions 0,4 x 0,4 x 0,8 m, b ρ = 560 kg.kg-1, *u* = 0,6 kg.kg-1 and <sup>20</sup> fsp *u* = 0,31 kg.kg-1

With the help of the 2D model the change in temperature in 5 characteristic points of cross section of non-frozen oak prisms with <sup>0</sup> <sup>0</sup>*<sup>t</sup>* = 0 C and frozen oak prisms with <sup>0</sup> <sup>0</sup>*t* = −10 C has been calculated during the time of their thermal processing with prescribed surface temperature <sup>0</sup> *<sup>t</sup>*<sup>m</sup> = 60 C and during the time of the consequent cooling with surface convection at <sup>0</sup> *<sup>t</sup>*<sup>m</sup> = 20 C .

FORTRAN (Dorn & McCracken, 1972), which has been input in the developed by Microsoft

The software package can be used for the calculation and colour visualization (either as animation of the whole process or as 3D, 2D, 1D graphs of each desired moment of the process) of the non-stationary distribution of the temperature fields in the materials containing or not containing ice during their thermal processing. The computation of the change in the temperature field in the volume of materials containing ice in the beginning of their thermal processing is interconnected for the periods of the melting of the ice and after

The computation of the temperature fields is done interconnectedly and for the processes of heating and consequent cooling of the materials, i.e. the calculation of the non-stationary change in temperatures in the volume of the materials during the time of their cooling begins from the already reached during the time of calculations distribution of temperature in the end of the heating. Based on the calculations it can be determined when the moment of reaching in the entire volume of the heated wood has occurred for the necessary optimal temperatures needed for bending of the parts or for cutting the veneer, as well as the stage

With the help of the 3D model the change in *t* in the volume of non-frozen beech prisms

m0 *t* = 0 C to <sup>0</sup> *t*<sup>m</sup> = 80 C is done exponentially with a time constant equal to 1800 s.

**Temperature** *t*

 **, 0**

**C**

of thermal processing during 20 hours at a prescribed surface temperature <sup>0</sup> *t*<sup>m</sup> = 80 C . The change in *t*<sup>m</sup> and *t* is shown on Fig. 10 in 6 characteristic points of the volume of the prisms with coordinates, which are given in the legend of the graphs. The increase in

tm

d/4, b/4, L/8 d/2, b/2. L/8 d/4, b/4, L/4 d/2, b/2, L/4 d/4, b/4, L/2 d/2, b/2, L/2

With the help of the 2D model the change in temperature in 5 characteristic points of cross

been calculated during the time of their thermal processing with prescribed surface temperature <sup>0</sup> *<sup>t</sup>*<sup>m</sup> = 60 C and during the time of the consequent cooling with surface

= 560 kg.kg-1, *u* = 0,6 kg.kg-1 and <sup>20</sup>

Fig. 10. 3D heating at <sup>0</sup> *t*<sup>m</sup> = 80 C of frozen (left) and non-frozen (right) beech prisms with

<sup>0</sup>*t C* = −10 with *d* = 0,4 m, *b* = 0,4 m, *L* = 0,8 m,

tm

d/4, b/4, L/8 d/2, b/2. L/8 d/4, b/4, L/4 d/2, b/2, L/4 d/4, b/4, L/2 d/2, b/2, L/2

<sup>0</sup>*t* = −10 C has

0 4 8 12 16 20 **Time** τ**, h**

<sup>0</sup>*<sup>t</sup>* = 0 C and frozen oak prisms with <sup>0</sup>

fsp *u* = 0,31 kg.kg-1

fsp *u* = 0,31 kg.kg-1 has been calculated during the time

calculation environment of VISUAL FORTRAN PROFESSIONAL (Deliiski 2003b).

that, taking into account the flexible spatial boundary of the melting ice.

**8.1 Non-stationary thermal processing of prismatic wood materials** 

of the ennoblement of the wood desired by the clients.

<sup>0</sup>*t C* = 0 and frozen beech prisms with <sup>0</sup>

= 560 kg.kg-1, *u* = 0,6 kg.kg-1 and <sup>20</sup>

0 4 8 12 16 20 **Time** τ**, h**

section of non-frozen oak prisms with <sup>0</sup>

ρ

dimensions 0,4 x 0,4 x 0,8 m, b

convection at <sup>0</sup> *<sup>t</sup>*<sup>m</sup> = 20 C .

with <sup>0</sup>

*t*<sup>m</sup> from <sup>0</sup>

**Temperature** *t***,** 

**0**

**C**

b ρ

The prisms have the following characteristics: *d* = 0,25 m, *b* = 0,40 m, *L* > 1,0 m, ρ <sup>b</sup> = 670 kg.kg-1, *u* = 0,6 kg.kg-1 and <sup>20</sup> fsp *u* = 0,29 kg.kg-1. The heating of the prisms continues until the reaching of the minimally required for cutting of veneer temperature in their centre, equal to 0 <sup>c</sup>*t* = 50 C .

During the time of cooling of the heated prisms a redistribution and equalization of *t* in their cross section takes place, which is especially appropriate for the obtaining of quality veneer. The change in *t*<sup>m</sup> and *t* is shown on Fig. 11 in 5 characteristic points from the cross section of the prisms with coordinates, which are given in the legend of the graphs.

Fig. 11. 2D heating at <sup>0</sup> *t*<sup>m</sup> = 60 C and consequent cooling at <sup>0</sup> *t*<sup>m</sup> = 20 C of frozen (left) and non-frozen (right) oak prisms with cross section 0,25 x 0,40 m, ρ <sup>b</sup> = 670 kg.kg-1 and *u* = 0,6 kg.kg-1

#### **8.2 Non-stationary thermal processing of cylindrical wood materials**

With the help of the 2D model the change in the *t* in the longitudinal section of non-frozen beech prisms with <sup>0</sup> <sup>0</sup>*t* = 0 C and frozen beech prisms with <sup>0</sup> <sup>0</sup>*t* = −2 C with *D* = 0,4 m, *L* = 0,8 m, b ρ = 560 kg.kg-1, *u* = 0,6 kg.kg-1 and <sup>20</sup> fsp *u* = 0,31 kg.kg-1 has been calculated during the time of thermal processing during 20 hours at a prescribed surface temperature <sup>0</sup> *t*<sup>m</sup> = 80 C . The change in *t*<sup>m</sup> and *t* is shown on Fig. 12 in 4 characteristic points of the longitudinal section of the logs with coordinates, which are given in the legend of the graphs.

Fig. 12. 2D heating at <sup>0</sup> *t*<sup>m</sup> = 80 C of frozen (left) and non-frozen (right) beech logs with R=0,2 m, L=0,8 m, ρ<sup>b</sup> = 560 kg.kg-1, *u* = 0,6 kg.kg-1 and <sup>20</sup> *<sup>u</sup>*fsp = 0,31 kg.kg-1

Fig. 14. 3D graphs and 2D contour plots for the temperature distribution with time in ¼ of longitudinal section of beech log with *R* = 0,2 m, *L* = 0,8 m, *u* = 0,6 kg⋅kg−1 and *t*0 = -2°C

The change in *t* in the longitudinal section of non-frozen beech logs with <sup>0</sup> <sup>0</sup>*t* = 0 C and in frozen beech logs with <sup>0</sup> <sup>0</sup>*t* = −10 C has been also calculated with the given above parameters during the time of a 3-stage high temperature thermal processing in autoclave and during the time of the consequent cooling with surface convection at <sup>0</sup> *<sup>t</sup>*<sup>m</sup> = 20 C (Fig. 13).

Fig. 13. 2D high temperature heating in autoclave and consequent cooling of frozen (left) and non-frozen (right) beech logs with R=0,2 m, L=0,8 m, ρ<sup>b</sup> = 560 kg.kg-1 and *u* = 0,6 kg.kg-1

Using 3D graphs and 2D diagrams a part of the results is shown on Fig. 14 from the simulation studies on the heat transfer in the radial and longitudinal direction of the frozen beech logs with *t*0 = -2°C, whose temperature field is shown on the left side on Fig. 12. The non-stationary temperature distribution during specific time intervals of the thermal processing is clearly observed from the 3D graphs (left columns on Fig. 14). The 2D diagrams which show in more detail the results from the simulations can be used rather for qualitative than quantitative analysis of the thermal processing of the materials (right columns on Fig. 14).

On the left parts of Fig. 10, Fig. 11, Fig. 12 and Fig. 13 the characteristic non-linear parts can be seen well, which show a slowing down in the change in *t* in the range from -2°С to - 1°С, in which the melting of the ice takes place, which was formed in the wood from the freezing of the free water in it. This signifies the good quality and quantity adequacy of the mathematical models towards the real process of heating of ice-containing wood materials. The calculated with the help of the models results correspond with high accuracy to wide experimental data for the non-stationary change in *t* in the volume of the containing and not containing ice wood logs, which have been derived in the publications by (Schteinhagen, 1986, 1991) and (Khattabi & Steinhagen, 1992, 1993).

The results presented on the figures show that the procedures for calculation of nonstationary change in *t* in prepared software package, realizing the solution of the mathematical models according to the finite-differences method, functions well for the cases of heating and cooling both for frozen and non-frozen materials at various initial and boundary conditions of the heat transfer during the thermal processing of the materials.

The good adequacy and precision of the models towards the results from numerous own and foreign experimental studies allows for the carrying out of various calculations with the models, which are connected to the non-stationary distribution of *t* in prismatic and cylindrical materials from various wood species and also to the heat energy consumption by the wood at random encountered in the practice conditions for thermal processing.

during the time of a 3-stage high temperature thermal processing in autoclave and during

Fig. 13. 2D high temperature heating in autoclave and consequent cooling of frozen (left)

Using 3D graphs and 2D diagrams a part of the results is shown on Fig. 14 from the simulation studies on the heat transfer in the radial and longitudinal direction of the frozen beech logs with *t*0 = -2°C, whose temperature field is shown on the left side on Fig. 12. The non-stationary temperature distribution during specific time intervals of the thermal processing is clearly observed from the 3D graphs (left columns on Fig. 14). The 2D diagrams which show in more detail the results from the simulations can be used rather for qualitative than quantitative analysis of the thermal processing of the materials (right

On the left parts of Fig. 10, Fig. 11, Fig. 12 and Fig. 13 the characteristic non-linear parts can be seen well, which show a slowing down in the change in *t* in the range from -2°С to - 1°С, in which the melting of the ice takes place, which was formed in the wood from the freezing of the free water in it. This signifies the good quality and quantity adequacy of the mathematical models towards the real process of heating of ice-containing wood materials. The calculated with the help of the models results correspond with high accuracy to wide experimental data for the non-stationary change in *t* in the volume of the containing and not containing ice wood logs, which have been derived in the publications by (Schteinhagen,

The results presented on the figures show that the procedures for calculation of nonstationary change in *t* in prepared software package, realizing the solution of the mathematical models according to the finite-differences method, functions well for the cases of heating and cooling both for frozen and non-frozen materials at various initial and boundary conditions of the heat transfer during the thermal processing of the materials. The good adequacy and precision of the models towards the results from numerous own and foreign experimental studies allows for the carrying out of various calculations with the models, which are connected to the non-stationary distribution of *t* in prismatic and cylindrical materials from various wood species and also to the heat energy consumption by

the wood at random encountered in the practice conditions for thermal processing.

ρ

**Temperature** *t*

 **, C**

> 0 4 8 12 **Time** τ**, h**

<sup>b</sup> = 560 kg.kg-1 and *u* = 0,6 kg.kg-1

<sup>0</sup>*t* = −10 C has been also calculated with the given above parameters

<sup>0</sup>*t* = 0 C and in

tm R=0, L=0 R/2, L/4 R/2, L/2 R, L/4 R, L/2

The change in *t* in the longitudinal section of non-frozen beech logs with <sup>0</sup>

the time of the consequent cooling with surface convection at <sup>0</sup> *<sup>t</sup>*<sup>m</sup> = 20 C (Fig. 13).

tm

R=0, L=0 R/2, L/4 R/2, L/2 R, L/4 R, L/2

frozen beech logs with <sup>0</sup>

0 4 8 12 16 **Time** τ**, h**

and non-frozen (right) beech logs with R=0,2 m, L=0,8 m,

1986, 1991) and (Khattabi & Steinhagen, 1992, 1993).

columns on Fig. 14).

**Temperature** *t***, 0**

**С**

Fig. 14. 3D graphs and 2D contour plots for the temperature distribution with time in ¼ of longitudinal section of beech log with *R* = 0,2 m, *L* = 0,8 m, *u* = 0,6 kg⋅kg−1 and *t*0 = -2°C

Axenenko, O. (1995). Structural Failure of Plasterboard Assembles in Fires. In: *Australian* 

Ben Nasrallah, S., Perre, P. (1988). Detailed Study of a Model of Heat and Mass Transfer

Chudinov, B. S. (1966). *Theoretical Research of Thermo-physical Properties and Thermal Treatment of Wood,* Dissertation for Dr.Sc., SibLTI, Krasnoyarsk, USSR (in Russian) Chudinov, B. S. (1968). *Theory of Thermal Treatment of Wood*, Publishing Company "Nauka",

Chudinov, B. S. (1984). *Water in Wood,* Publishing Company "Nauka", Moscow, USSR (in

Deliiski, N. (1977). Berechnung der instationären Temperaturverteilung im Holz bei der Er-

Deliiski, N. (1979). Mathematical Modeling of the Process of Heating of Cylindrical Wood

Deliiski, N. (1994). Mathematical Description of the Thermal Conductivity Coefficient of

Deliiski, N. (2003a). Microprocessor System for Automatic Control of Logs' Steaming

Deliiski, N. (2003b). *Modelling and Technologies for Steaming Wood Materials in Autoclaves.*

Deliiski, N. (2004). Modelling and Automatic Control of Heat Energy Consumption Required for Thermal Treatment of Logs. *Drvna Industria*, Volume 55, № 4, pp. 181-199 Deliiski, N. (2009). Computation of the 2-dimensional Transient Temperature Distribution

Doe, P. D., Oliver, A. R., Booker, J. D. (1994). A Non-linear Strain and Moisture Content

Dzurenda, L., Deliiski, N. (2010). *Thermal Processes in the Technologies for Wood Processing*, TU

Ferguson, W. J., Lewis, R. W. (1991). A Comparison of a Fully Non-linear and a Partially

Kanter, K. R. (1955). *Investigation of the Thermal Properties of Wood.* Dissertation, MLTI,

in Zvolen, ISBN 978-80-228-2169-8, Zvolen, Slovakia (in Slovakian)

*International Wood Drying Conference,* Rotorua, New Zealand, pp. 203-210 Dorn, W. S., McCracken, D. D. (1972). *Numerical Methods with FORTRAN IV: Case Studies*,

Dissertation for Dr.Sc., University of Forestry, Sofia (in Bulgarian)

Institute in Sofia, Volume XXV- MTD, 1979, pp. 21-26 (in Bulgarian) Deliiski, N. (1990). Mathematische Beschreibung der spezifischen Wärmekapazität des

*on Fundamental Research of Wood*. Warsaw, Poland, pp. 229-233

*Structure and Properties '94*, Zvolen, Slovakia, pp. 127-134

Process. *Drvna Industria,* Volume 53, № 4, pp. 191-198.

*Mathematical Society Gazette*, Available from: http://www.austms.org.au/Gazette/

During Convective Drying of Porous Media, *International Journal of Heat and Mass* 

wärmung durch Wärmeleitung. Teil I.: Mathematisches Modell für die Erwärmung des Holzes durch Wärmeleitung. *Holz Roh- Werkstoff*, Volume 35, № 4, pp. 141−145

Materials by Thermal Conductivity. Scientific Works of the Higher Forest-technical

aufgetauten und gefrorenen Holzes, *Proceedings of the VIIIth International Symposium* 

Non-frozen and Frozen Wood. *Proceedings of the 2nd International Symposium on Wood* 

and Heat Energy Consumption of Frozen and Non-frozen Logs. *Wood Research*,

Model of Variable Hardwood Drying Schedules. *Proceedings of the 4th IUFRO* 

Non-linear Heat and Mass Transfer of a Timber Drying Problem, *Proceedings of the 7*th *Conference on Numerical methods in Thermal Problems*, Vol. VII, Part 2, pp. 973-984

**11. References** 

1995/Jun95/struct.html

Moscow, USSR (in Russian)

Volume 54, № 3, pp. 67−78

Moscow, USSR (in Russian)

John Willej & Sons, Inc., New York

Russian)

*Transfer*, Volume 31, № 5, pp. 297-310

#### **9. Conclusion**

This paper describes the creation and solution of non-linear mathematical models for the transient heat conduction in anisotropic frozen and non-frozen capillary porous bodies with prismatic and cylindrical shape and at any *u* ≥ 0 kg.kg-1. The mechanism of the heat distribution in the entire volume of the bodies is described only by one partial differential equation of heat conduction. For the first time the own specific heat capacity of the bodies and the specific heat capacity of the ice, formed in them from the freezing of the hygroscopically bounded water and of the free water are taken into account in the models.

The models take into account the physics of the described processes and allow the 3D, 2D and 1D calculation of the temperature distribution in the volume of subjected to heating and/or cooling anisotropic or isotropic bodies in the cases, when the change in their moisture content during the thermal processing is relatively small. For the solution of the models an explicit form of the finite-difference method is used, which allows for the exclusion of any simplifications in the models.

For the usage of the models it is required to have the knowledge and mathematical description of several properties of the subjected to thermal processing frozen and nonfrozen capillary porous materials. In this paper the approaches for mathematical description of thermo-physical characteristics of materials from different wood species, which are typical representatives of anisotropic capillary porous bodies, widely subjected to thermal treatment in the practice are shown as examples.

For the numerical solution of the models a software package has been prepared in FORTRAN, which has been input in the developed by Microsoft calculation environment of Visual Fortran Professional. The software allows for the computations to be done for heating and cooling of the bodies at prescribed surface temperature, equal to the temperature of the processing medium or during the time of convective thermal processing. The computation of the change in the temperature field in the volume of materials containing ice in the beginning of their thermal processing is interconnected for the periods of the melting of the ice and after that, taking into account the flexible boundary of the melting ice. The computation of the temperature fields is done interconnectedly and for the processes of heating and consequent cooling of the materials, i.e. the calculation of the change in temperatures in the volume of the materials during the time of their cooling begins from the already reached during the time of calculations distribution of temperature in the end of the heating. It is shown how based on the calculations it can be determined when the moment of reaching in the entire volume of the heated and after that cooling body has occurred for the necessary optimal temperatures needed, for example, for bending of wood parts or for cutting the veneer from plasticised wooden prisms or logs.

The models can be used for the calculation and colour visualization (either as animation of the whole process or as 3D, 2D, 1D graphs of each desired moment of the process) of the distribution of the temperature fields in the bodies during their thermal processing. The development of the models and algorithms and software for their solution is consistent with the possibility for their usage in automatic systems with a model based (Deliiski 2003a, 2003b, 2009) or model predicting control of different processes for thermal treatment.

#### **10. Acknowledgement**

This work was supported by the Scientific Research Sector of the University of Forestry, Sofia, Bulgaria.

#### **11. References**

174 Convection and Conduction Heat Transfer

This paper describes the creation and solution of non-linear mathematical models for the transient heat conduction in anisotropic frozen and non-frozen capillary porous bodies with prismatic and cylindrical shape and at any *u* ≥ 0 kg.kg-1. The mechanism of the heat distribution in the entire volume of the bodies is described only by one partial differential equation of heat conduction. For the first time the own specific heat capacity of the bodies and the specific heat capacity of the ice, formed in them from the freezing of the hygroscopically bounded water and of the free water are taken into account in the models. The models take into account the physics of the described processes and allow the 3D, 2D and 1D calculation of the temperature distribution in the volume of subjected to heating and/or cooling anisotropic or isotropic bodies in the cases, when the change in their moisture content during the thermal processing is relatively small. For the solution of the models an explicit form of the finite-difference method is used, which allows for the

For the usage of the models it is required to have the knowledge and mathematical description of several properties of the subjected to thermal processing frozen and nonfrozen capillary porous materials. In this paper the approaches for mathematical description of thermo-physical characteristics of materials from different wood species, which are typical representatives of anisotropic capillary porous bodies, widely subjected to thermal

For the numerical solution of the models a software package has been prepared in FORTRAN, which has been input in the developed by Microsoft calculation environment of Visual Fortran Professional. The software allows for the computations to be done for heating and cooling of the bodies at prescribed surface temperature, equal to the temperature of the processing medium or during the time of convective thermal processing. The computation of the change in the temperature field in the volume of materials containing ice in the beginning of their thermal processing is interconnected for the periods of the melting of the ice and after that, taking into account the flexible boundary of the melting ice. The computation of the temperature fields is done interconnectedly and for the processes of heating and consequent cooling of the materials, i.e. the calculation of the change in temperatures in the volume of the materials during the time of their cooling begins from the already reached during the time of calculations distribution of temperature in the end of the heating. It is shown how based on the calculations it can be determined when the moment of reaching in the entire volume of the heated and after that cooling body has occurred for the necessary optimal temperatures needed, for example, for bending of wood parts or for

The models can be used for the calculation and colour visualization (either as animation of the whole process or as 3D, 2D, 1D graphs of each desired moment of the process) of the distribution of the temperature fields in the bodies during their thermal processing. The development of the models and algorithms and software for their solution is consistent with the possibility for their usage in automatic systems with a model based (Deliiski 2003a,

This work was supported by the Scientific Research Sector of the University of Forestry,

2003b, 2009) or model predicting control of different processes for thermal treatment.

**9. Conclusion** 

exclusion of any simplifications in the models.

treatment in the practice are shown as examples.

cutting the veneer from plasticised wooden prisms or logs.

**10. Acknowledgement** 

Sofia, Bulgaria.


**0**

**8**

*Brazil*

**Non-Linear Radiative-Conductive Heat Transfer**

Marco T.M.B. de Vilhena, Bardo E.J. Bodmann and Cynthia F. Segatto

Radiative transfer considers problems that involve the physical phenomenon of energy transfer by radiation in media. These phenomena occur in a variety of realms (Ahmad & Deering, 1992; Tsai & Ozi¸sik, 1989; Wilson & Sen, 1986; Yi et al., 1996) including optics (Liu et al., 2006), astrophysics (Pinte et al., 2009), atmospheric science (Thomas & Stamnes, 2002), remote sensing (Shabanov et al., 2007) and engineering applications like heat transport by radiation (Brewster, 1992) for instance or radiative transfer laser applications (Kim & Guo, 2004). Furthermore, applications to other media such as biological tissue, powders, paints among others may be found in the literature (see ref. (Yang & Kruse, 2004) and references therein). Although radiation in its basic form is understood as a photon flux that requires a stochastic approach taking into account local microscopic interactions of a photon ensemble with some target particles like atoms, molecules, or effective micro-particles such as impurities, this scenario may be conveniently modelled by a radiation field, i.e. a radiation intensity, in a continuous medium where a microscopic structure is hidden in effective model parameters, to be specified later. The propagation of radiation through a homogeneous or heterogeneous medium suffers changes by several isotropic or non-isotropic processes like absorption, emission and scattering, respectively, that enter the mathematical approach in form of a non-linear radiative transfer equation. The non-linearity of the equation originates from a local thermal description using the Stefan-Boltzmann law that is related to heat transport by radiation which in turn is related to the radiation intensity and renders the radiative transfer problem a radiative-conductive one (Ozisik, 1973; Pomraning, 2005). Here, local thermal description means, that the domain where a temperature is attributed to, is sufficiently large in order to allow for the definition of a temperature, i.e. a local radiative

The principal quantity of interest is the intensity *I*, that describes the radiation energy flow through an infinitesimal oriented area *d*Σˆ = *nd*ˆ Σ with outward normal vector *n*ˆ into the solid angle *d*Ωˆ = Ωˆ *d*Ω, where Ωˆ represents the direction of the flow considered, with angle *θ* of the normal vector and the flow direction *<sup>n</sup>*<sup>ˆ</sup> · <sup>Ω</sup><sup>ˆ</sup> <sup>=</sup> cos *<sup>θ</sup>* <sup>=</sup> *<sup>μ</sup>*. In the present case we focus on the non-linearity of the radiative-conductive transfer problem and therefore introduce the simplification of an integrated spectral intensity over all wavelengths or equivalently all frequencies that contribute to the radiation flow and further ignore possible effects due to polarization. Also possible effects that need in the formalism properties such as coherence

**1. Introduction**

equilibrium.

**in a Heterogeneous Gray Plane-Parallel**

**Participating Medium**

*Universidade Federal do Rio Grande do Sul*


### **Non-Linear Radiative-Conductive Heat Transfer in a Heterogeneous Gray Plane-Parallel Participating Medium**

Marco T.M.B. de Vilhena, Bardo E.J. Bodmann and Cynthia F. Segatto *Universidade Federal do Rio Grande do Sul Brazil*

#### **1. Introduction**

176 Convection and Conduction Heat Transfer

Khattabi, A., Steinhagen, H. P. (1992). Numerical Solution to Two-dimensional Heating of

Khattabi, A., Steinhagen, H. P. (1993). Analysis of Transient Non-linear Heat Conduction in

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Radiative transfer considers problems that involve the physical phenomenon of energy transfer by radiation in media. These phenomena occur in a variety of realms (Ahmad & Deering, 1992; Tsai & Ozi¸sik, 1989; Wilson & Sen, 1986; Yi et al., 1996) including optics (Liu et al., 2006), astrophysics (Pinte et al., 2009), atmospheric science (Thomas & Stamnes, 2002), remote sensing (Shabanov et al., 2007) and engineering applications like heat transport by radiation (Brewster, 1992) for instance or radiative transfer laser applications (Kim & Guo, 2004). Furthermore, applications to other media such as biological tissue, powders, paints among others may be found in the literature (see ref. (Yang & Kruse, 2004) and references therein). Although radiation in its basic form is understood as a photon flux that requires a stochastic approach taking into account local microscopic interactions of a photon ensemble with some target particles like atoms, molecules, or effective micro-particles such as impurities, this scenario may be conveniently modelled by a radiation field, i.e. a radiation intensity, in a continuous medium where a microscopic structure is hidden in effective model parameters, to be specified later. The propagation of radiation through a homogeneous or heterogeneous medium suffers changes by several isotropic or non-isotropic processes like absorption, emission and scattering, respectively, that enter the mathematical approach in form of a non-linear radiative transfer equation. The non-linearity of the equation originates from a local thermal description using the Stefan-Boltzmann law that is related to heat transport by radiation which in turn is related to the radiation intensity and renders the radiative transfer problem a radiative-conductive one (Ozisik, 1973; Pomraning, 2005). Here, local thermal description means, that the domain where a temperature is attributed to, is sufficiently large in order to allow for the definition of a temperature, i.e. a local radiative equilibrium.

The principal quantity of interest is the intensity *I*, that describes the radiation energy flow through an infinitesimal oriented area *d*Σˆ = *nd*ˆ Σ with outward normal vector *n*ˆ into the solid angle *d*Ωˆ = Ωˆ *d*Ω, where Ωˆ represents the direction of the flow considered, with angle *θ* of the normal vector and the flow direction *<sup>n</sup>*<sup>ˆ</sup> · <sup>Ω</sup><sup>ˆ</sup> <sup>=</sup> cos *<sup>θ</sup>* <sup>=</sup> *<sup>μ</sup>*. In the present case we focus on the non-linearity of the radiative-conductive transfer problem and therefore introduce the simplification of an integrated spectral intensity over all wavelengths or equivalently all frequencies that contribute to the radiation flow and further ignore possible effects due to polarization. Also possible effects that need in the formalism properties such as coherence

coefficient *κ* and the geometrical projection on the direction perpendicular to that plane, say along the *z*-axis, so that *dτ* = −*κρdz*. Further the temperature is measured in multiples of a

Based on the photon number balance and in the spirit of a Boltzmann type equation one arrives at the radiative transfer equation in a volume that shall be chosen in a way so that no boundaries that separate media with different physical properties cross the control volume. To this end, five photon number changing contributions shall be taken into account which may be condensed into the four terms that follow. The first term describes the net rate of streaming of photons through the bounding surface of an infinitesimal control volume, the second term combines absorption and out-scattering from *μ* to all possible directions *μ*� in the control volume. The third term contemplates in-scattering from all directions *μ*� into the direction *μ*, and last not least a black-body like emission term according to the temperature

reference temperature *T*(*τ*) = Θ(*τ*)*Tr*, typically taken at *τ* = 0.

in a Heterogeneous Gray Plane-Parallel Participating Medium

dependence of Stefan-Boltzmann's law for the control volume.

*<sup>I</sup>*(*τ*, *<sup>μ</sup>*) = *<sup>ω</sup>*(*τ*)

2

P(*μ*, *μ*�

*n* ∑ *m*=1

) *dμ*� =

) + 2

)*I*(*τ*, *μ*�

and write the integral on the right hand side of side of equation (1) as

2*μ*

 1 −1 P(*μ*�

P(*μ*) *dμ* = 1.

) = ∞ ∑ =0

Here, *ω* is the single scattering albedo and P(*μ*) signifies the differential scattering coefficient or also called the phase function, that accounts for the rate at which photons are scattered into an angle *dμ*� and with inclination *μ* with respect to the normal vector of the sheet. Note, that

Upon simplifying the phase function in plane geometry one may expand the angular

with *β<sup>n</sup>* the expansion coefficients that follow from orthogonality. Further one may employ the addition formula for Legendre polynomials using azimuthal symmetry (hence the zero

> *Pm <sup>n</sup>* (*μ*)*P<sup>m</sup> <sup>n</sup>* (*μ*� ) 2*π* 0

(*n* − *m*)! (*n* + *m*)!

> ∞ ∑ =0 *β* 1 −1

where the summation index refers to the degree of anisotropy. For practical applications only a limited number of terms indexed with have to be taken into account in order to characterise qualitatively and quantitatively the anisotropic contributions to the problem. Also higher terms oscillate more significantly and thus suppress the integral's significance in the solution. The degree of anisotropy may be indicated truncating the sum by an upper limit *L*. The integro-differential equation (1) together with the afore mentioned manipulations may be cast into an approximation known as the *SN* equation upon reducing the continuous angle cosine to a discrete set of *N* angles. This procedure opens a pathway to apply standard vector algebra techniques to obtain a solution from the equation system, discussed in detail in section 3.

)*I*(*τ*, *μ*�

*βnPn*(*μ*� − *μ*) ,

*P*(*μ*)*P*(*μ*�

)*I*(*τ*, *μ*�

) *dμ*� +

1 − *ω*(*τ*) *μ*

cos(*m*(*φ* − *φ*�

 =0

) *dμ*� ,

)) *dφ*�

,

Θ4(*τ*) (1)

179

*dI*(*τ*, *μ*) *dτ*

the phase function is normalized <sup>1</sup>

*P*(*μ*� − *μ*) = *P*(*μ*)*P*(*μ*�

 1 −1

P(*μ*, *μ*�

integral)

+ 1 *μ*

Non-Linear Radiative-Conductive Heat Transfer

dependence in Legendre Polynomials *Pn*(*μ*),

and diffraction are not taken into account. In general the Radiative-Conductive Transfer Equation is difficult to solve without introducing some approximations, like linearisation or a reduction to a diffusion like equation, that facilitate the construction of a solution but at the cost of predictive power in comparison to experimental findings, or more sophisticated approaches. The present approach is not different in the sense that approximations shall be introduced, nevertheless the non-linearity that represents the crucial ingredient in the problem is solved without resorting to linearisation or perturbation like procedures and to the best of our knowledge is the first approach of its kind. The solution of the modified or approximate problem can be given in closed analytical form, that permits to calculate numerical results in principle to any desired precision. Moreover, the influence of the non-linearity can be analysed in an analytical fashion directly from the formal solution. Solutions found in the literature are typically linearised and of numerical nature (see for instance (Asllanaj et al., 2001; 2002; Attia, 2000; Krishnapraka et al., 2001; Menguc & Viskanta, 1983; Muresan et al., 2004; Siewert & Thomas, 1991; Spuckler & Siegel, 1996) and references therein). To the best of our knowledge no analytical approach for heterogeneous media and considering the non-linearity exists so far, that are certainly closer to realistic scenarios in natural or technological sciences. A possible reason for considering a simplified problem (homogeneous and linearised) is that such a procedure turns the determination of a solution viable. It is worth mentioning that a general solution from an analytical approach for this type of problems exists only in the discrete ordinate approximation and for homogeneous media as reported in reference (Segatto et al., 2010).

Various of the initially mentioned applications allow to segment the medium in plane parallel sheets, where the radiation field is invariant under translation in directions parallel to that sheet. In other words the only spatial coordinate of interest is the one perpendicular to the sheet that indicates the penetration depth of the radiation in the medium. Frequently, it is justified to assume the medium to have an isotropic structure which reduces the angular degrees of freedom of the radiation intensity to the azimuthal angle *θ* or equivalently to its cosine *μ*. Further simplifications may be applied which are coherent with measurement procedures. One the one hand measurements are conducted in finite time intervals where the problem may be considered (quasi-)stationary, which implies that explicit time dependence may be neglected in the transfer equation. On the other hand, detectors have a finite dimension (extension) with a specific acceptance angle for measuring radiation and thus set some angular resolution for experimental data. Such an uncertainty justifies to segment the continuous angle into a set of discrete angles (or their cosines), which renders the original equation with angular degrees of freedom a set of equations known as the *SN* approximation to be introduced in detail in section 3.

Our chapter is organised as follows: in the next section we motivate the radiative-conductive transfer problem. Sections 3 and 4 are dedicated to the hierarchical construction procedure of analytical solutions for the heterogeneous radiative-conductive transfer problem from its reduction to the homogeneous case, using two distinct philosophies. In section 4.3 we apply the method to specific cases and present results. Last, we close the chapter with some remarks and conclusions.

#### **2. The radiative conductive transfer problem**

In problems of radiative transfer in plane parallel media it is convenient to measure linear distances normal to the plane of stratification using the concept of optical thickness *τ* which is measured from the boundary inward and is related through the density *ρ*, an attenuation 2 Will-be-set-by-IN-TECH

and diffraction are not taken into account. In general the Radiative-Conductive Transfer Equation is difficult to solve without introducing some approximations, like linearisation or a reduction to a diffusion like equation, that facilitate the construction of a solution but at the cost of predictive power in comparison to experimental findings, or more sophisticated approaches. The present approach is not different in the sense that approximations shall be introduced, nevertheless the non-linearity that represents the crucial ingredient in the problem is solved without resorting to linearisation or perturbation like procedures and to the best of our knowledge is the first approach of its kind. The solution of the modified or approximate problem can be given in closed analytical form, that permits to calculate numerical results in principle to any desired precision. Moreover, the influence of the non-linearity can be analysed in an analytical fashion directly from the formal solution. Solutions found in the literature are typically linearised and of numerical nature (see for instance (Asllanaj et al., 2001; 2002; Attia, 2000; Krishnapraka et al., 2001; Menguc & Viskanta, 1983; Muresan et al., 2004; Siewert & Thomas, 1991; Spuckler & Siegel, 1996) and references therein). To the best of our knowledge no analytical approach for heterogeneous media and considering the non-linearity exists so far, that are certainly closer to realistic scenarios in natural or technological sciences. A possible reason for considering a simplified problem (homogeneous and linearised) is that such a procedure turns the determination of a solution viable. It is worth mentioning that a general solution from an analytical approach for this type of problems exists only in the discrete ordinate approximation and for homogeneous media as reported in reference (Segatto

Various of the initially mentioned applications allow to segment the medium in plane parallel sheets, where the radiation field is invariant under translation in directions parallel to that sheet. In other words the only spatial coordinate of interest is the one perpendicular to the sheet that indicates the penetration depth of the radiation in the medium. Frequently, it is justified to assume the medium to have an isotropic structure which reduces the angular degrees of freedom of the radiation intensity to the azimuthal angle *θ* or equivalently to its cosine *μ*. Further simplifications may be applied which are coherent with measurement procedures. One the one hand measurements are conducted in finite time intervals where the problem may be considered (quasi-)stationary, which implies that explicit time dependence may be neglected in the transfer equation. On the other hand, detectors have a finite dimension (extension) with a specific acceptance angle for measuring radiation and thus set some angular resolution for experimental data. Such an uncertainty justifies to segment the continuous angle into a set of discrete angles (or their cosines), which renders the original equation with angular degrees of freedom a set of equations known as the *SN* approximation

Our chapter is organised as follows: in the next section we motivate the radiative-conductive transfer problem. Sections 3 and 4 are dedicated to the hierarchical construction procedure of analytical solutions for the heterogeneous radiative-conductive transfer problem from its reduction to the homogeneous case, using two distinct philosophies. In section 4.3 we apply the method to specific cases and present results. Last, we close the chapter with some remarks

In problems of radiative transfer in plane parallel media it is convenient to measure linear distances normal to the plane of stratification using the concept of optical thickness *τ* which is measured from the boundary inward and is related through the density *ρ*, an attenuation

et al., 2010).

to be introduced in detail in section 3.

**2. The radiative conductive transfer problem**

and conclusions.

coefficient *κ* and the geometrical projection on the direction perpendicular to that plane, say along the *z*-axis, so that *dτ* = −*κρdz*. Further the temperature is measured in multiples of a reference temperature *T*(*τ*) = Θ(*τ*)*Tr*, typically taken at *τ* = 0.

Based on the photon number balance and in the spirit of a Boltzmann type equation one arrives at the radiative transfer equation in a volume that shall be chosen in a way so that no boundaries that separate media with different physical properties cross the control volume. To this end, five photon number changing contributions shall be taken into account which may be condensed into the four terms that follow. The first term describes the net rate of streaming of photons through the bounding surface of an infinitesimal control volume, the second term combines absorption and out-scattering from *μ* to all possible directions *μ*� in the control volume. The third term contemplates in-scattering from all directions *μ*� into the direction *μ*, and last not least a black-body like emission term according to the temperature dependence of Stefan-Boltzmann's law for the control volume.

$$\frac{dI(\boldsymbol{\tau},\boldsymbol{\mu})}{d\boldsymbol{\tau}} + \frac{1}{\mu}I(\boldsymbol{\tau},\boldsymbol{\mu}) = \frac{\omega(\boldsymbol{\tau})}{2\mu} \int\_{-1}^{1} \mathcal{P}(\mu')I(\boldsymbol{\tau},\mu') \, d\mu' + \frac{1-\omega(\boldsymbol{\tau})}{\mu} \Theta^4(\boldsymbol{\tau}) \tag{1}$$

Here, *ω* is the single scattering albedo and P(*μ*) signifies the differential scattering coefficient or also called the phase function, that accounts for the rate at which photons are scattered into an angle *dμ*� and with inclination *μ* with respect to the normal vector of the sheet. Note, that the phase function is normalized <sup>1</sup> 2 P(*μ*) *dμ* = 1.

Upon simplifying the phase function in plane geometry one may expand the angular dependence in Legendre Polynomials *Pn*(*μ*),

$$\mathcal{P}(\mu,\mu') = \sum\_{\ell=0}^{\infty} \beta\_{\mathbb{N}} P\_{\mathbb{N}}(\mu'-\mu) \dots$$

with *β<sup>n</sup>* the expansion coefficients that follow from orthogonality. Further one may employ the addition formula for Legendre polynomials using azimuthal symmetry (hence the zero integral)

$$P\_{\ell}(\mu'-\mu) = P\_{\ell}(\mu)P\_{\ell}(\mu') + 2\sum\_{m=1}^{n} \frac{(n-m)!}{(n+m)!}P\_{n}^{m}(\mu)P\_{n}^{m}(\mu') \underbrace{\int\_{0}^{2\pi} \cos(m(\phi-\phi')) \,d\phi'}\_{=0} \,\lambda$$

and write the integral on the right hand side of side of equation (1) as

$$\int\_{-1}^{1} \mathcal{P}(\mu, \mu') I(\tau, \mu') \, d\mu' = \sum\_{\ell=0}^{\infty} \beta\_{\ell} \int\_{-1}^{1} P\_{\ell}(\mu) P\_{\ell}(\mu') I(\tau, \mu') \, d\mu' \,, \mu'$$

where the summation index refers to the degree of anisotropy. For practical applications only a limited number of terms indexed with have to be taken into account in order to characterise qualitatively and quantitatively the anisotropic contributions to the problem. Also higher terms oscillate more significantly and thus suppress the integral's significance in the solution. The degree of anisotropy may be indicated truncating the sum by an upper limit *L*. The integro-differential equation (1) together with the afore mentioned manipulations may be cast into an approximation known as the *SN* equation upon reducing the continuous angle cosine to a discrete set of *N* angles. This procedure opens a pathway to apply standard vector algebra techniques to obtain a solution from the equation system, discussed in detail in section 3.

radiative convective transfer problem in the *SN* approximation

2*μ<sup>n</sup>*

*L* ∑ �=0

*N* ∑ *k*=1

for *n* = 1, . . . , *N* and are subject to the following boundary conditions.

*β*�*P*�(*μn*)

*N* ∑ *k*=1

(0)*IN*−*n*+1(0) + <sup>2</sup>*ρd*(0)

(*τ*0)*In*(*τ*0) + 2*ρd*(*τ*0)

Note, that the integrals over the angular variables are replaced by a Gaussian quadrature

For convenience we introduce a shorthand notation in matrix operator form, where the

**<sup>Φ</sup>**(*τ*)=(**I**, <sup>Θ</sup>(*τ*))*<sup>T</sup>* = (*I*1(*τ*), ··· , *IN*(*τ*), <sup>Θ</sup>(*τ*))*<sup>T</sup>* combines the anisotropic intensities and the isotropic temperature function, the non-linear terms and boundary terms from integration (i.e. the temperature gradient and the conduction

Θ4(*τ*),

*d*Θ

*ω*(*τ*) 2*μ<sup>n</sup>*

*L* ∑ �=0

*<sup>d</sup><sup>τ</sup>* (0) <sup>−</sup> <sup>1</sup>

2*Nc*

**Φ** − L*M***Φ** = **Ψ** (10)

1 for *x* > 0 0 else

+ *fnk* for *n*, *k* = 1, . . . , *N* + 1 (11)

*β*�*P*�(*μn*)*wkP*�(*μk*)

. (12)

*N* ∑ *k*=1

*wk Ik*(0)*μ<sup>k</sup>*

*<sup>T</sup>*

scheme with weight factors *wk*, where *k* refers to one of the discrete directions *μk*.

1 − *ω*(*τ*) *μn*

*μn*

0 else , *<sup>θ</sup>H*(*x*) =

*μk* 2*Nc*

*d dτ* *wkP*�(*μk*)*Ik*(*τ*) + <sup>1</sup> <sup>−</sup> *<sup>ω</sup>*(*τ*)

*wk* (*Ik*(*τ*) − *Ik*(0)) *μ<sup>k</sup>* , (8)

*N* 2 ∑ *k*=1

*N* 2 ∑ *k*=1 *μn*

*wk IN*−*k*+1(0)*μ<sup>k</sup>*

*wk Ik*(*τ*0)*μ<sup>k</sup>* (9)

Θ4(*τ*) , (7)

181

*In*(*τ*) = *<sup>ω</sup>*(*τ*)

in a Heterogeneous Gray Plane-Parallel Participating Medium

<sup>=</sup> <sup>1</sup> 2*Nc*

*In*(0) = *�*(0)Θ4(0) + *ρ<sup>s</sup>*

radiation intensity at *τ* = 0) are absorbed in an inhomogeneity

<sup>Θ</sup>4(*τ*), ··· ,

(L*M*)*nk* <sup>=</sup> *<sup>δ</sup>nk*(<sup>1</sup> <sup>−</sup> *<sup>δ</sup>n*,*N*+1) <sup>1</sup>

Here, *δij* is the Kronecker delta, *θ<sup>H</sup>* the Heaviside functional

1 for *i* = *j*

*fnk* = *θH*(*N* − *n* + 1/2)*θH*(*N* − *k* + 1/2)

+(1 − *δk*,*N*+1)*δn*,*N*+<sup>1</sup>

which allows to cast the equation system (7) and (8) in compact form

*IN*−*n*+1(*τ*0) = *�*(*τ*0)Θ4(*τ*0) + *<sup>ρ</sup><sup>s</sup>*

**3.1 The** *SN* **approach in matrix representation**

1 − *ω*(*τ*) *μ*1

where L*<sup>M</sup>* has the following elements.

*δij* =

 *τ*=0

Non-Linear Radiative-Conductive Heat Transfer

*dIn*(*τ*) *dτ*

*d*Θ(*τ*)

column vector

**Ψ** = 

and the factors *fnk* are

+ 1 *μn*

*<sup>d</sup><sup>τ</sup>* <sup>−</sup> *<sup>d</sup>*Θ(*τ*) *dτ*

In order to define boundary conditions we have to specify in more details the scenario in consideration. In the further we analyse non-linear radiative-conductive transfer in a grey plane-parallel participating medium with opaque walls, where specular (mirror like) as well as diffuse reflections occur besides thermal photon emission according to the Stefan-Boltzmann law (see (Elghazaly, 2009) and references therein). If one thinks the medium being subdivided into sheets of thickness Δ*τ* with sufficiently small depth so that for each sheet a homogeneous medium applies, than for each face or interface the condition for the top boundary (at *τ* = *τi*) is

$$I(\tau,\mu) = \varepsilon(\tau)\Theta^4(\tau) + \rho^s(\tau)I(\tau,-\mu) + 2\rho^d(\tau)\int\_0^1 I(\tau,-\mu')\mu' \,d\mu' \,, \tag{2}$$

with *ρ<sup>s</sup>* and *ρ<sup>d</sup>* the specular and diffuse reflections at the boundary, which are related to the emissivity *�* by *�* + *ρ<sup>s</sup>* + *ρ<sup>d</sup>* = 1. For the limiting bottom boundary (*τ* = *τ<sup>i</sup>* + Δ*τ*) *μ* and *μ*� change their sign in the argument of *I*(*τ*, *μ*(�)) in equation (2). Suppose we have *NS* sheets and *NS* + 1 boundaries, one might think that for a first order differential equation (1) in *τ* the supply of *NS* + 1 boundary conditions results in an ill-posed problem with no solutions at all. However, we still have to set up an equation that uniquely defines the non-linearity in terms of the radiation intensity.

The relation may be established in two steps, first recognizing that the dimensionless radiative flux is expressed in terms of the intensity by

$$q\_{\nu}^{\*} = 2\pi \int\_{-1}^{1} I(\tau, \mu)\mu \, d\mu \,, \tag{3}$$

and the energy equation for the temperature that connects the radiative flux to a temperature gradient is

$$\frac{d^2}{d\tau^2}\Theta(\tau) = \frac{1}{4\pi N\_\mathrm{c}}\frac{d}{d\tau}q\_r^\*(\tau) = \frac{1}{4\pi N\_\mathrm{c}}\frac{d}{d\tau}\left(2\pi \int\_{-1}^1 I(\tau,\mu)\mu \,d\mu\right). \tag{4}$$

Here*Nc* is the conduction-radiation parameter, defined as

$$N\_{\mathcal{L}} = \frac{k \mathcal{B}\_{\text{ext}}}{4 \sigma \mathbf{n}^2 T\_r^3} \,'\text{.}\tag{5}$$

with *k* the thermal conductivity, *βext* the extinction coefficient, *σ* the Stefan-Boltzmann constant and n the refractive index. Note that the radiative flux results from the integration of the intensity over angular variables, so that the thermal conductivity is considered here isotropic. Equation (4) is subject to prescribed temperatures at the top- and bottommost boundary

$$
\Theta(0) = \Theta\_T \qquad \text{and} \qquad \Theta(\tau\_0) = \Theta\_B \; . \tag{6}
$$

#### **3. The** *SN* **approximation for the heterogeneous problem**

The set of equations (1) and (4), that are continuous in the angle cosine, may be simplified using an enumerable set of discrete angles following the collocation method, that defines the 4 Will-be-set-by-IN-TECH

In order to define boundary conditions we have to specify in more details the scenario in consideration. In the further we analyse non-linear radiative-conductive transfer in a grey plane-parallel participating medium with opaque walls, where specular (mirror like) as well as diffuse reflections occur besides thermal photon emission according to the Stefan-Boltzmann law (see (Elghazaly, 2009) and references therein). If one thinks the medium being subdivided into sheets of thickness Δ*τ* with sufficiently small depth so that for each sheet a homogeneous medium applies, than for each face or interface the condition for the top

(*τ*)*I*(*τ*, <sup>−</sup>*μ*) + <sup>2</sup>*ρd*(*τ*)

with *ρ<sup>s</sup>* and *ρ<sup>d</sup>* the specular and diffuse reflections at the boundary, which are related to the emissivity *�* by *�* + *ρ<sup>s</sup>* + *ρ<sup>d</sup>* = 1. For the limiting bottom boundary (*τ* = *τ<sup>i</sup>* + Δ*τ*) *μ* and *μ*� change their sign in the argument of *I*(*τ*, *μ*(�)) in equation (2). Suppose we have *NS* sheets and *NS* + 1 boundaries, one might think that for a first order differential equation (1) in *τ* the supply of *NS* + 1 boundary conditions results in an ill-posed problem with no solutions at all. However, we still have to set up an equation that uniquely defines the non-linearity in terms

The relation may be established in two steps, first recognizing that the dimensionless radiative

and the energy equation for the temperature that connects the radiative flux to a temperature

4*πNc*

*d dτ* 2*π* 1 −1

Θ(0) = Θ*<sup>T</sup>* and Θ(*τ*0) = Θ*<sup>B</sup>* . (6)

 1 −1

*<sup>r</sup>* (*τ*) = <sup>1</sup>

*Nc* <sup>=</sup> *<sup>k</sup>βext* 4*σ*n2*T*<sup>3</sup> *r*

with *k* the thermal conductivity, *βext* the extinction coefficient, *σ* the Stefan-Boltzmann constant and n the refractive index. Note that the radiative flux results from the integration of the intensity over angular variables, so that the thermal conductivity is considered here isotropic. Equation (4) is subject to prescribed temperatures at the top- and bottommost

The set of equations (1) and (4), that are continuous in the angle cosine, may be simplified using an enumerable set of discrete angles following the collocation method, that defines the

*q*∗ *<sup>r</sup>* = 2*π*

*d <sup>d</sup><sup>τ</sup> <sup>q</sup>*<sup>∗</sup>  1 0

*I*(*τ*, −*μ*�

*I*(*τ*, *μ*)*μ dμ* , (3)

*I*(*τ*, *μ*)*μ dμ*

, (5)

. (4)

)*μ*� *dμ*� , (2)

boundary (at *τ* = *τi*) is

of the radiation intensity.

gradient is

boundary

*I*(*τ*, *μ*) = *�*(*τ*)Θ4(*τ*) + *ρ<sup>s</sup>*

flux is expressed in terms of the intensity by

*<sup>d</sup>τ*<sup>2</sup> <sup>Θ</sup>(*τ*) = <sup>1</sup>

4*πNc*

**3. The** *SN* **approximation for the heterogeneous problem**

Here*Nc* is the conduction-radiation parameter, defined as

*d*2

radiative convective transfer problem in the *SN* approximation

$$\frac{dI\_{\boldsymbol{n}}(\boldsymbol{\tau})}{d\boldsymbol{\tau}} + \frac{1}{\mu\_{\boldsymbol{n}}}I\_{\boldsymbol{n}}(\boldsymbol{\tau}) = \frac{\omega(\boldsymbol{\tau})}{2\mu\_{\boldsymbol{n}}} \sum\_{\ell=0}^{L} \beta\_{\ell} P\_{\ell}(\mu\_{\boldsymbol{n}}) \sum\_{k=1}^{N} w\_{k} P\_{\ell}(\mu\_{k}) I\_{k}(\boldsymbol{\tau}) + \frac{1 - \omega(\boldsymbol{\tau})}{\mu\_{\boldsymbol{n}}} \Theta^{4}(\boldsymbol{\tau}) \,, \tag{7}$$

$$\left. \frac{d\Theta(\tau)}{d\tau} - \left. \frac{d\Theta(\tau)}{d\tau} \right|\_{\tau=0} = \frac{1}{2N\_c} \sum\_{k=1}^{N} w\_k \left( I\_k(\tau) - I\_k(0) \right) \mu\_k \,\,\,\tag{8}$$

for *n* = 1, . . . , *N* and are subject to the following boundary conditions.

$$I\_{\boldsymbol{\eta}}(0) = \boldsymbol{\epsilon}(0)\Theta^{4}(0) + \rho^{\boldsymbol{s}}(0)I\_{N-\boldsymbol{\eta}+1}(0) + 2\rho^{\boldsymbol{d}}(0)\sum\_{k=1}^{\frac{N}{2}}w\_{k}I\_{N-k+1}(0)\mu\_{k}$$

$$I\_{N-\boldsymbol{\eta}+1}(\tau\_{0}) = \boldsymbol{\epsilon}(\tau\_{0})\Theta^{4}(\tau\_{0}) + \rho^{\boldsymbol{s}}(\tau\_{0})I\_{\boldsymbol{\eta}}(\tau\_{0}) + 2\rho^{\boldsymbol{d}}(\tau\_{0})\sum\_{k=1}^{\frac{N}{2}}w\_{k}I\_{k}(\tau\_{0})\mu\_{k} \tag{9}$$

Note, that the integrals over the angular variables are replaced by a Gaussian quadrature scheme with weight factors *wk*, where *k* refers to one of the discrete directions *μk*.

#### **3.1 The** *SN* **approach in matrix representation**

For convenience we introduce a shorthand notation in matrix operator form, where the column vector

$$\Phi(\tau) = (\mathbf{I}, \Theta(\tau))^T = (I\_1(\tau), \dots, I\_N(\tau), \Theta(\tau))^T$$

combines the anisotropic intensities and the isotropic temperature function, the non-linear terms and boundary terms from integration (i.e. the temperature gradient and the conduction radiation intensity at *τ* = 0) are absorbed in an inhomogeneity

$$\mathbf{Y} = \left( \frac{1 - \omega(\tau)}{\mu\_1} \Theta^4(\tau), \dots, \frac{1 - \omega(\tau)}{\mu\_n} \Theta^4(\tau), \frac{d\Theta}{d\tau}(0) - \frac{1}{2\mathcal{N}\_c} \sum\_{k=1}^N w\_k I\_k(0)\mu\_k \right)^T$$

which allows to cast the equation system (7) and (8) in compact form

$$\frac{d}{d\tau}\Phi - \mathcal{L}\_M \Phi = \Psi \tag{10}$$

where L*<sup>M</sup>* has the following elements.

$$(\mathcal{L}\_M)\_{nk} = \delta\_{nk} (1 - \delta\_{nN+1}) \frac{1}{\mu\_n} + f\_{nk} \qquad \text{for} \quad n, k = 1, \dots, N+1 \tag{11}$$

Here, *δij* is the Kronecker delta, *θ<sup>H</sup>* the Heaviside functional

$$\delta\_{ij} = \begin{cases} 1 \text{ for } i = j \\ 0 \text{ else} \end{cases}, \qquad \qquad \theta\_H(\mathbf{x}) = \begin{cases} 1 \text{ for } \mathbf{x} > \mathbf{0} \\ 0 \text{ else} \end{cases}$$

and the factors *fnk* are

$$f\_{nk} = \theta\_H (N - n + 1/2) \theta\_H (N - k + 1/2) \frac{\omega(\tau)}{2\mu\_n} \sum\_{\ell=0}^L \beta\_\ell P\_\ell(\mu\_n) w\_k P\_\ell(\mu\_k)$$

$$+ (1 - \delta\_{k, N+1}) \delta\_{n, N+1} \frac{\mu\_k}{2N\_c} \cdot \tag{12}$$

0

183

Θ*<sup>T</sup>*

Non-Linear Radiative-Conductive Heat Transfer

in a Heterogeneous Gray Plane-Parallel Participating Medium

*ω*1, *ρ<sup>s</sup>* <sup>1</sup>, *<sup>ρ</sup><sup>s</sup>* <sup>1</sup>, *�*<sup>1</sup>

*ω*2, *ρ<sup>s</sup>* <sup>2</sup>, *<sup>ρ</sup><sup>s</sup>* <sup>2</sup>, *�*<sup>2</sup>

*ωi*, *ρ<sup>s</sup> <sup>i</sup>* , *<sup>ρ</sup><sup>s</sup> <sup>i</sup>* , *�<sup>i</sup>*

. . .

. . .

*ωNS* , *ρ<sup>s</sup> NS* , *ρ<sup>s</sup> NS* , *�NS*

heterogeneous medium in form of a multi-layer slab (see figure 1). For each of the layers the problem reduces to a homogeneous problem but with the same number of boundary conditions as the original problem. The procedure that determines the solution for each slab is presented in detail in section 4. In order to solve the unknown boundary values of the intensities and the temperatures at the interfaces between the slabs, matching these quantities using the bottom boundary values of the upper slab and the top boundary values of the lower

In the second approach we introduce a new procedure to work the heterogeneity. To begin

 *τ*<sup>0</sup> 0

and rewrite the problem as a homogeneous problem plus an inhomogeneous correction. Note that L*<sup>M</sup>* as well as Ψ depend on the local albedo coefficient *ω*(*τ*). Since the terms containing the coefficient are linear in *ω* permits to separate an average factor *ω*¯ and the difference *ω*(*τ*) −

Now, following the idea of the Decomposition method proposed originally by Adomian (Adomian, 1988), to solve non-linear problems without linearisation, we handle equation

formal decomposition and the non-linearity is written in terms of the so-called Adomian

as in a homogeneous slab, and the influence of the heterogeneity is governed by the source term. The homogeneous problem is explicitly solved in section 4 so that we concentrate here

(17), constructing the following recursive system of equations. Here, **Ψ** = ∑<sup>∞</sup>

**Φ** − L*M*(*ω*¯)**Φ** = **Ψ**(*ω*¯) + L*M*(*ω*(*τ*) − *ω*¯)**Φ** + **Ψ**(*ω*(*τ*) − *ω*¯) (17)

*<sup>m</sup>*=<sup>0</sup> *<sup>A</sup>*ˆ*m*(*τ*). The first equation of the recursive system is the same

*<sup>ω</sup>*¯ <sup>=</sup> <sup>1</sup> *τ*0

with, we take the averaged value for the albedo coefficient *ω*(*τ*),

Fig. 1. Schematic illustration of a heterogeneous medium in form of a multi-layer slab.

Θ<sup>1</sup>

Θ<sup>2</sup>

<sup>Θ</sup>*i*−<sup>1</sup>

Θ*NS*−<sup>1</sup>

Θ*<sup>B</sup>*

slab eliminates these incognitos.

*dτ*

polynomials Θ4(*τ*) = ∑<sup>∞</sup>

*<sup>ω</sup>*¯ . *<sup>d</sup>*

Θ*i*

*τ*1

*τ*2

*τi*−<sup>1</sup>

*τNS*−<sup>1</sup>

*τ*0

*ω*(*τ*) *dτ* (16)

*<sup>m</sup>*=<sup>0</sup> **Ψ***<sup>m</sup>* is a

*τi*

Note, that the increment 1/2 in the Heaviside functional was introduced merely to make the argument positive definite in the range of interest which otherwise could lead to conflicts with possible definitions for *θH*(*x*) at *x* = 0.

The boundary conditions are combined accordingly, except for the limiting temperatures (equation (6)) that are kept separately for simplicity because they would add only an additional diagonal block leading to a reducible representation and thus this does not bring any advantage.

$$
\mathcal{B}\_D \mathbf{I} - \mathcal{B}\_M \mathbf{I} = \Gamma \tag{13}
$$

Equation (13) has a block form where one block represents forward angle contributions *μ* > 0 and the other one backward terms *μ* < 0 originating from the top and bottom boundary, respectively. Here, B*<sup>D</sup>* is the *N* × *N* unit matrix, and

$$\mathcal{B}\_M = \begin{pmatrix} 0 & \rho^s \mathbb{C}\_{N/2} + 2\rho^d \mathbb{G}\_{N/2}^-\\ \rho^s \mathbb{C}\_{N/2} + 2\rho^d \mathbb{G}\_{N/2}^+ & 0 \end{pmatrix} \tag{14}$$

with C*N*/2 an *N*/2 × *N*/2 matrix which results from column reversion in the unit matrix, i.e. after mapping column position *k* to position *N*/2 − *k* + 1. The remaining matrices that control the diffuse forward and backward reflection (G± *<sup>N</sup>*/2), respectively have the elements

$$
\begin{split}
\left(\mathcal{G}\_{N/2}^{+}\right)\_{nk} &= \theta\_{H}(N/2 - n + 1/2)\theta\_{H}(k - N/2 - 1/2)\mu\_{N-k+1}w\_{N-k+1} \\
\left(\mathcal{G}\_{N/2}^{-}\right)\_{nk} &= \theta\_{H}(n - N/2 - n - 1/2)\theta\_{H}(N/2 - k + 1/2)\mu\_{k}w\_{k} \end{split} \tag{15}
$$

In these expressions the Heaviside functions restrict the non-zero elements to the off-diagonal blocks with row indices *n* ∈ {1, . . . , *N*/2} and column indices *k* ∈ {*N*/2 + 1, . . . , *N*} and with row indices *n* ∈ {*N*/2 + 1, . . . , *N*} and column indices *k* ∈ {1, . . . , *N*/2}, respectively. The vector representation for the intensity is

$$\mathbf{I} = (\mathbf{I}\_+, \mathbf{I}\_-)^T \quad \text{with} \quad \mathbf{I}\_+ = (I\_\mathbf{I}(\tau), \dots, I\_{\mathbf{N}/2}(\tau)) \quad \text{and} \quad \mathbf{I}\_- = (I\_{\mathbf{N}/2+1}(\tau), \dots, I\_{\mathbf{N}}(\tau)) \; .$$

The inhomogeneity **Γ** has the same emission term in each component.

$$
\Gamma\_n = \epsilon(\tau) \Theta^4(\tau) \qquad \forall n
$$

#### **3.2 Constructing the solution by the decomposition method**

The principal difficulty in constructing a solution for the radiative conductive transfer problem in the *SN* approximation (10) subject to the boundary conditions (13) and (6) is due to the fact that the single scattering albedo *ω*(*τ*), the emissivity *�*(*τ*) and the specular and diffuse reflection (*ρs*(*τ*) and *ρd*(*τ*)) have an explicit dependence on the optical depth *τ*, that is the heterogeneity of the medium in consideration. It is worth mentioning that the proposed methodology is quite general in the sense that it can be applied to other approximations of equation (1) that make use of spectral methods, as for instance the spherical harmonic *PN*-, the Chebychev *ChN*- and the Walsi *WN*-approximation (Vilhena & Segatto, 1999; Vilhena et al., 1999), among others.

In the sequel we report on two approaches to solve the heterogeneous problem (equations. (10), (13), (6)). The principal idea of this techniques relies on the reduction of the Radiative Conductive transfer problem in heterogeneous media to a set of problems in domains of homogeneous media. In the first approach we consider the standard approximation of the 6 Will-be-set-by-IN-TECH

Note, that the increment 1/2 in the Heaviside functional was introduced merely to make the argument positive definite in the range of interest which otherwise could lead to conflicts with

The boundary conditions are combined accordingly, except for the limiting temperatures (equation (6)) that are kept separately for simplicity because they would add only an additional diagonal block leading to a reducible representation and thus this does not bring

Equation (13) has a block form where one block represents forward angle contributions *μ* > 0 and the other one backward terms *μ* < 0 originating from the top and bottom boundary,

<sup>0</sup> *<sup>ρ</sup>s*C*N*/2 <sup>+</sup> <sup>2</sup>*ρd*G<sup>−</sup>

with C*N*/2 an *N*/2 × *N*/2 matrix which results from column reversion in the unit matrix, i.e. after mapping column position *k* to position *N*/2 − *k* + 1. The remaining matrices that control

In these expressions the Heaviside functions restrict the non-zero elements to the off-diagonal blocks with row indices *n* ∈ {1, . . . , *N*/2} and column indices *k* ∈ {*N*/2 + 1, . . . , *N*} and with row indices *n* ∈ {*N*/2 + 1, . . . , *N*} and column indices *k* ∈ {1, . . . , *N*/2}, respectively. The

**<sup>I</sup>** = (**I**+,**I**−)*<sup>T</sup>* with **<sup>I</sup>**<sup>+</sup> = (*I*1(*τ*),..., *IN*/2(*τ*)) and **<sup>I</sup>**<sup>−</sup> = (*IN*/2+1(*τ*),..., *IN*(*τ*)) .

<sup>Γ</sup>*<sup>n</sup>* <sup>=</sup> *�*(*τ*)Θ4(*τ*) <sup>∀</sup>*<sup>n</sup>*

The principal difficulty in constructing a solution for the radiative conductive transfer problem in the *SN* approximation (10) subject to the boundary conditions (13) and (6) is due to the fact that the single scattering albedo *ω*(*τ*), the emissivity *�*(*τ*) and the specular and diffuse reflection (*ρs*(*τ*) and *ρd*(*τ*)) have an explicit dependence on the optical depth *τ*, that is the heterogeneity of the medium in consideration. It is worth mentioning that the proposed methodology is quite general in the sense that it can be applied to other approximations of equation (1) that make use of spectral methods, as for instance the spherical harmonic *PN*-, the Chebychev *ChN*- and the Walsi *WN*-approximation (Vilhena & Segatto, 1999; Vilhena et

In the sequel we report on two approaches to solve the heterogeneous problem (equations. (10), (13), (6)). The principal idea of this techniques relies on the reduction of the Radiative Conductive transfer problem in heterogeneous media to a set of problems in domains of homogeneous media. In the first approach we consider the standard approximation of the

The inhomogeneity **Γ** has the same emission term in each component.

**3.2 Constructing the solution by the decomposition method**

*<sup>N</sup>*/2 0

*nk* <sup>=</sup> *<sup>θ</sup>H*(*N*/2 <sup>−</sup> *<sup>n</sup>* <sup>+</sup> 1/2)*θH*(*<sup>k</sup>* <sup>−</sup> *<sup>N</sup>*/2 <sup>−</sup> 1/2)*μN*−*k*+1*wN*−*k*+<sup>1</sup>

*nk* <sup>=</sup> *<sup>θ</sup>H*(*<sup>n</sup>* <sup>−</sup> *<sup>N</sup>*/2 <sup>−</sup> *<sup>n</sup>* <sup>−</sup> 1/2)*θH*(*N*/2 <sup>−</sup> *<sup>k</sup>* <sup>+</sup> 1/2)*μkwk* . (15)

*<sup>ρ</sup>s*C*N*/2 + <sup>2</sup>*ρd*G<sup>+</sup>

B*D***I** − B*M***I** = **Γ** (13)

*N*/2

*<sup>N</sup>*/2), respectively have the elements

(14)

possible definitions for *θH*(*x*) at *x* = 0.

respectively. Here, B*<sup>D</sup>* is the *N* × *N* unit matrix, and

B*<sup>M</sup>* =

the diffuse forward and backward reflection (G±

vector representation for the intensity is

 G<sup>+</sup> *N*/2 

 G− *N*/2 

al., 1999), among others.

any advantage.

Fig. 1. Schematic illustration of a heterogeneous medium in form of a multi-layer slab.

heterogeneous medium in form of a multi-layer slab (see figure 1). For each of the layers the problem reduces to a homogeneous problem but with the same number of boundary conditions as the original problem. The procedure that determines the solution for each slab is presented in detail in section 4. In order to solve the unknown boundary values of the intensities and the temperatures at the interfaces between the slabs, matching these quantities using the bottom boundary values of the upper slab and the top boundary values of the lower slab eliminates these incognitos.

In the second approach we introduce a new procedure to work the heterogeneity. To begin with, we take the averaged value for the albedo coefficient *ω*(*τ*),

$$
\omega = \frac{1}{\tau\_0} \int\_0^{\tau\_0} \omega(\tau) \, d\tau \tag{16}
$$

and rewrite the problem as a homogeneous problem plus an inhomogeneous correction. Note that L*<sup>M</sup>* as well as Ψ depend on the local albedo coefficient *ω*(*τ*). Since the terms containing the coefficient are linear in *ω* permits to separate an average factor *ω*¯ and the difference *ω*(*τ*) − *<sup>ω</sup>*¯ . *<sup>d</sup>*

$$\frac{d}{d\tau}\Phi - \mathcal{L}\_M(\bar{\omega})\Phi = \mathbf{Y}(\bar{\omega}) + \mathcal{L}\_M(\omega(\tau) - \bar{\omega})\Phi + \mathbf{Y}(\omega(\tau) - \bar{\omega})\tag{17}$$

Now, following the idea of the Decomposition method proposed originally by Adomian (Adomian, 1988), to solve non-linear problems without linearisation, we handle equation (17), constructing the following recursive system of equations. Here, **Ψ** = ∑<sup>∞</sup> *<sup>m</sup>*=<sup>0</sup> **Ψ***<sup>m</sup>* is a formal decomposition and the non-linearity is written in terms of the so-called Adomian polynomials Θ4(*τ*) = ∑<sup>∞</sup> *<sup>m</sup>*=<sup>0</sup> *<sup>A</sup>*ˆ*m*(*τ*). The first equation of the recursive system is the same as in a homogeneous slab, and the influence of the heterogeneity is governed by the source term. The homogeneous problem is explicitly solved in section 4 so that we concentrate here

1973), the index *N* signifies here the number of the discrete directions of the angular space. More specifically, we circumvent limitations that arose in the discussion of the same problem in ref. (Vargas & Vilhena, 1999). Furthermore, differently than some iterative schemes found in the literature (Abulwafa, 1999; Ozisik, 1973; Siewert & Thomas, 1991), we construct an analytical solution sequence which in the limit of the truncation parameter M → ∞ converges

For any arbitrary truncation and using Laplace transform (LT) the original *SN* problem may be cast into *LTSN* form, which allows for matrix orthogonalisation and thus opens the advantage of handling *SN* problems with *N* as large as for instance ∼ 1500; for further details see references. (Segatto et al., 1999) and (Goncalves et al., 2000). Once the *A*ˆ*<sup>m</sup>* polynomials are

The hybrid *LTSN* Adomian approach is not new, see references. (Vargas & Vilhena, 1999) and (Brancher et al., 1999), but in the present discussion we present a procedure based on the same reasoning but in a novel and optimised form. This progress is partially due to the more effective handling of the boundary conditions. In a previous attempt (Vargas et al., 2003) the boundary conditions entered in every step of recursion which posed limitations on the solutions so that it was only possible to resolve angles with *N* = 30 and truncate the expansion after the first term. As the following discussion will show, the present approach henceforth denoted the *D*M*LTSN* approach circumvents these shortcomings (here *D*M*LTSN*

The dimensionless non-linear *SN* radiative transfer equation in a grey plane-parallel homogeneous medium results from equation (7) upon substitution of the albedo coefficient

*βP*(*μn*)

for *n* = 1, . . . , *N* and subject to the boundary conditions with constant emissivity and

<sup>1</sup> *IN*−*n*+1(0) + <sup>2</sup>*ρ<sup>d</sup>*

2 *N*/2 ∑ *k*=1

<sup>2</sup> *In*(*τ*0) + <sup>2</sup>*ρ<sup>d</sup>*

The equation for the temperature (8) may be solved by integrating twice from the boundary

*τ τ*0

Recalling, that equation (8) relates the intensity to the temperature, equation (23) shows the connection between the temperature and the radiative flux that permits to cast the problem

 *τ*<sup>0</sup> 0 *q*∗ *<sup>r</sup>* (*τ*�

*N* ∑ *k*=1

> 1 *N*/2 ∑ *k*=1

> > )*dτ*� +

1 4*πNc*  *τ* 0 *q*∗ *<sup>r</sup>* (*τ*�

*<sup>P</sup>*(*μk*)*Ik*(*τ*) + <sup>1</sup> <sup>−</sup> *<sup>ω</sup>*¯

*μn*

*wkμ<sup>k</sup> IN*−*k*+1(0) , (21)

*wkμ<sup>k</sup> Ik*(*τ*0) . (22)

Θ4(*τ*) (20)

185

)*dτ*� (23)

to the exact solution of the equation that characterises the *SN* problem.

Non-Linear Radiative-Conductive Heat Transfer

in a Heterogeneous Gray Plane-Parallel Participating Medium

stands for Decomposition Laplace Transform *SN* approach).

*In*(*τ*) = *<sup>ω</sup>*¯

*In*(0) = *�*1Θ<sup>4</sup>

*IN*−*n*+1(*τ*0) = *�*2Θ<sup>4</sup>

<sup>Θ</sup>(*τ*) = <sup>Θ</sup><sup>1</sup> + (Θ<sup>2</sup> <sup>−</sup> <sup>Θ</sup>1) *<sup>τ</sup>*

2*μ<sup>n</sup>*

<sup>1</sup> <sup>+</sup> *<sup>ρ</sup><sup>s</sup>*

<sup>2</sup> + *<sup>ρ</sup><sup>s</sup>*

*τ*0

into a form that depends only on the directional intensity **I**.

Note, that *β* are the expansion coefficients explicitly given in case study 4.3.1.

<sup>−</sup> <sup>1</sup> 4*πNc*

*L* ∑ =0

**4.1 The** *LTSN* **formalism**

*d dτ*

*τ* = 0 to any *τ* ∈ [0, *τ*0].

*In*(*τ*) + <sup>1</sup>

*μn*

by its average value.

reflectivity.

known up to M the *LTSN* provides the sum up to M of the expanded solution.

on the inhomogeneity.

$$\begin{aligned} \frac{d}{d\tau}\Phi\_0 - \mathcal{L}\_M(\bar{\omega})\Phi\_0 &= \mathbf{Y}\_0(\bar{\omega})\\ \frac{d}{d\tau}\Phi\_i - \mathcal{L}\_M(\bar{\omega})\Phi\_i &= \mathbf{Y}\_i(\bar{\omega}) + \mathcal{L}\_M(\omega(\tau) - \bar{\omega})\Phi\_{i-1} + \mathbf{Y}\_{i-1}(\omega(\tau) - \bar{\omega}) \quad \text{for} \quad i \le 1 \end{aligned} \tag{18}$$
 
$$\mathbf{Y}\_{i-1}(\omega(\tau) - \bar{\omega}) = (\bar{\omega} - \omega(\tau))A\_m(\tau)(\mu\_1^{-1}, \dots, \mu\_1^{-N}, 0)^T \tag{18}$$

with **<sup>Ψ</sup>***i*−1(*ω*(*τ*) <sup>−</sup> *<sup>ω</sup>*¯)=(*ω*¯ <sup>−</sup> *<sup>ω</sup>*(*τ*))*Am*(*τ*)(*μ*−<sup>1</sup>

Note, that the *N* + 1-th component of **Ψ**0(*ω*¯) contains the inhomogeneous term of the temperature equation.

$$(\mathbf{Y}\_0(\omega))\_{N+1} = \mathbf{Y}\_{N+1} = \frac{d\Theta}{d\tau}(0) - \frac{1}{2\mathcal{N}\_\mathbf{c}} \sum\_{k=1}^N w\_k I\_k(0)\mu\_k \tag{19}$$

The determination of the Adomian polynomials *Am*(*τ*) in equation (18) in terms of the temperature is shown in section 4.

To complete our analysis considering the boundary conditions, the first equation of the recursive system satisfies the boundary condition, whereas the remaining equations satisfy homogeneous boundary conditions. By this procedure we guarantee that the solution **Φ** determined from the recursive scheme and truncated at a convenient limit M satisfies the boundary conditions of the problem (13) and (6). Therefore we are now in a position to construct a solution with a prescribed accuracy by controlling the number of terms in the series solution given by equation (18). From the previous discussion it becomes apparent that it is possible by the proposed procedure to obtain a solution of the heterogeneous problem by a reduction to a set of homogeneous problems. To complete the construction of a solution for the heterogeneous problem in the next section we present the derivation of the solution of the *SN* radiative-conductive transfer problem in a homogeneous slab.

#### **4. The solution for the homogeneous radiative conductive heat transfer problem**

In this section we consider the non-linear radiative-conductive transfer problem in a grey plane-parallel participating medium with combined specular and diffuse reflection (Siewert & Thomas, 1991) and its solution in an analytical form using a composite method by Laplace transform and the decomposition method (Adomian, 1988). Before the advent of the decomposition method analytical solutions were restricted to a few special problems like the Bernoulli and Ricatti equations, to mention only two. The basic idea of the decomposition technique understands the following steps: The non-linear problem is interpreted as an operator equation (as already introduced in section 3) and split into a sum of linear and non-linear terms. Next, one expands the solution (in the present discussion the intensity *I*) and the non-linear term (here the quartic dimensionless temperature term Θ4), respectively, as a series *I*(*τ*) = ∑<sup>∞</sup> *<sup>m</sup>*=<sup>1</sup> **<sup>U</sup>***<sup>m</sup>* and <sup>Θ</sup><sup>4</sup> <sup>=</sup> <sup>∑</sup><sup>∞</sup> *<sup>m</sup>*=<sup>1</sup> *<sup>A</sup>*ˆ*m*, where *<sup>A</sup>*ˆ*<sup>m</sup>* are to be determined self consistently according to Adomian's procedure (Adomian, 1988). Upon insertion of these expansions in the split equation, one may construct a set of linear recursive problems that can be solved by classical methods for linear problems.

Although the method is designed for general non-linear problems, it is not straight forward to apply it to any given problem and to any desired precision. One specific equation system which we solve in the sequel considers the *SN* problem equation (10) for non-linear radiative-conductive heat transfer in plane parallel geometry as introduced in ref. (Ozisik, 1973), the index *N* signifies here the number of the discrete directions of the angular space. More specifically, we circumvent limitations that arose in the discussion of the same problem in ref. (Vargas & Vilhena, 1999). Furthermore, differently than some iterative schemes found in the literature (Abulwafa, 1999; Ozisik, 1973; Siewert & Thomas, 1991), we construct an analytical solution sequence which in the limit of the truncation parameter M → ∞ converges to the exact solution of the equation that characterises the *SN* problem.

For any arbitrary truncation and using Laplace transform (LT) the original *SN* problem may be cast into *LTSN* form, which allows for matrix orthogonalisation and thus opens the advantage of handling *SN* problems with *N* as large as for instance ∼ 1500; for further details see references. (Segatto et al., 1999) and (Goncalves et al., 2000). Once the *A*ˆ*<sup>m</sup>* polynomials are known up to M the *LTSN* provides the sum up to M of the expanded solution.

The hybrid *LTSN* Adomian approach is not new, see references. (Vargas & Vilhena, 1999) and (Brancher et al., 1999), but in the present discussion we present a procedure based on the same reasoning but in a novel and optimised form. This progress is partially due to the more effective handling of the boundary conditions. In a previous attempt (Vargas et al., 2003) the boundary conditions entered in every step of recursion which posed limitations on the solutions so that it was only possible to resolve angles with *N* = 30 and truncate the expansion after the first term. As the following discussion will show, the present approach henceforth denoted the *D*M*LTSN* approach circumvents these shortcomings (here *D*M*LTSN* stands for Decomposition Laplace Transform *SN* approach).

#### **4.1 The** *LTSN* **formalism**

8 Will-be-set-by-IN-TECH

Note, that the *N* + 1-th component of **Ψ**0(*ω*¯) contains the inhomogeneous term of the

The determination of the Adomian polynomials *Am*(*τ*) in equation (18) in terms of the

To complete our analysis considering the boundary conditions, the first equation of the recursive system satisfies the boundary condition, whereas the remaining equations satisfy homogeneous boundary conditions. By this procedure we guarantee that the solution **Φ** determined from the recursive scheme and truncated at a convenient limit M satisfies the boundary conditions of the problem (13) and (6). Therefore we are now in a position to construct a solution with a prescribed accuracy by controlling the number of terms in the series solution given by equation (18). From the previous discussion it becomes apparent that it is possible by the proposed procedure to obtain a solution of the heterogeneous problem by a reduction to a set of homogeneous problems. To complete the construction of a solution for the heterogeneous problem in the next section we present the derivation of the solution of the

**4. The solution for the homogeneous radiative conductive heat transfer problem** In this section we consider the non-linear radiative-conductive transfer problem in a grey plane-parallel participating medium with combined specular and diffuse reflection (Siewert & Thomas, 1991) and its solution in an analytical form using a composite method by Laplace transform and the decomposition method (Adomian, 1988). Before the advent of the decomposition method analytical solutions were restricted to a few special problems like the Bernoulli and Ricatti equations, to mention only two. The basic idea of the decomposition technique understands the following steps: The non-linear problem is interpreted as an operator equation (as already introduced in section 3) and split into a sum of linear and non-linear terms. Next, one expands the solution (in the present discussion the intensity *I*) and the non-linear term (here the quartic dimensionless temperature term Θ4), respectively, as

according to Adomian's procedure (Adomian, 1988). Upon insertion of these expansions in the split equation, one may construct a set of linear recursive problems that can be solved by

Although the method is designed for general non-linear problems, it is not straight forward to apply it to any given problem and to any desired precision. One specific equation system which we solve in the sequel considers the *SN* problem equation (10) for non-linear radiative-conductive heat transfer in plane parallel geometry as introduced in ref. (Ozisik,

**<sup>Φ</sup>***<sup>i</sup>* − <sup>L</sup>*M*(*ω*¯)**Φ***<sup>i</sup>* = **<sup>Ψ</sup>***i*(*ω*¯) + <sup>L</sup>*M*(*ω*(*τ*) − *<sup>ω</sup>*¯)**Φ***i*−<sup>1</sup> + **<sup>Ψ</sup>***i*−1(*ω*(*τ*) − *<sup>ω</sup>*¯) for *<sup>i</sup>* ≤ <sup>1</sup>

*<sup>d</sup><sup>τ</sup>* (0) <sup>−</sup> <sup>1</sup>

<sup>1</sup> ,..., *<sup>μ</sup>*−*<sup>N</sup>*

2*Nc*

*N* ∑ *k*=1

*<sup>m</sup>*=<sup>1</sup> *<sup>A</sup>*ˆ*m*, where *<sup>A</sup>*ˆ*<sup>m</sup>* are to be determined self consistently

<sup>1</sup> , 0)*<sup>T</sup>* (18)

*wk Ik*(0)*μ<sup>k</sup>* (19)

on the inhomogeneity.

**Φ**<sup>0</sup> − L*M*(*ω*¯)**Φ**<sup>0</sup> = **Ψ**0(*ω*¯)

with **<sup>Ψ</sup>***i*−1(*ω*(*τ*) <sup>−</sup> *<sup>ω</sup>*¯)=(*ω*¯ <sup>−</sup> *<sup>ω</sup>*(*τ*))*Am*(*τ*)(*μ*−<sup>1</sup>

(**Ψ**0(*ω*¯))*N*+<sup>1</sup> <sup>=</sup> **<sup>Ψ</sup>***N*+<sup>1</sup> <sup>=</sup> *<sup>d</sup>*<sup>Θ</sup>

*SN* radiative-conductive transfer problem in a homogeneous slab.

*<sup>m</sup>*=<sup>1</sup> **<sup>U</sup>***<sup>m</sup>* and <sup>Θ</sup><sup>4</sup> <sup>=</sup> <sup>∑</sup><sup>∞</sup>

classical methods for linear problems.

*d dτ*

> *d dτ*

temperature equation.

a series *I*(*τ*) = ∑<sup>∞</sup>

temperature is shown in section 4.

The dimensionless non-linear *SN* radiative transfer equation in a grey plane-parallel homogeneous medium results from equation (7) upon substitution of the albedo coefficient by its average value.

$$\frac{d}{d\tau}I\_{\mathbb{H}}(\tau) + \frac{1}{\mu\_{\text{n}}}I\_{\mathbb{H}}(\tau) = \frac{\omega}{2\mu\_{\text{n}}} \sum\_{\ell=0}^{L} \mathcal{J}\_{\ell} P\_{\ell}(\mu\_{\text{n}}) \sum\_{k=1}^{N} P\_{\ell}(\mu\_{k}) I\_{k}(\tau) + \frac{1-\omega}{\mu\_{\text{n}}} \Theta^{4}(\tau) \tag{20}$$

for *n* = 1, . . . , *N* and subject to the boundary conditions with constant emissivity and reflectivity.

$$I\_n(0) = \varepsilon\_1 \Theta\_1^4 + \rho\_1^s I\_{N-n+1}(0) + 2\rho\_1^d \sum\_{k=1}^{N/2} w\_k \mu\_k I\_{N-k+1}(0) \tag{21}$$

$$I\_{N-n+1}(\tau\_0) = \varepsilon\_2 \Theta\_2^4 + \rho\_2^s I\_{\mathcal{H}}(\tau\_0) + 2\rho\_2^d \sum\_{k=1}^{N/2} w\_k \mu\_k I\_k(\tau\_0) \,. \tag{22}$$

Note, that *β* are the expansion coefficients explicitly given in case study 4.3.1. The equation for the temperature (8) may be solved by integrating twice from the boundary *τ* = 0 to any *τ* ∈ [0, *τ*0].

$$\Theta(\tau) = \Theta\_1 + (\Theta\_2 - \Theta\_1)\frac{\tau}{\tau\_0} - \frac{1}{4\pi N\_c} \frac{\tau}{\tau\_0} \int\_0^{\tau\_0} q\_r^\*(\tau') d\tau' + \frac{1}{4\pi N\_c} \int\_0^{\tau} q\_r^\*(\tau') d\tau' \tag{23}$$

Recalling, that equation (8) relates the intensity to the temperature, equation (23) shows the connection between the temperature and the radiative flux that permits to cast the problem into a form that depends only on the directional intensity **I**.

where **D** and **X** are respectively the matrices of eigenvalues and eigenfunctions resulting from the spectral decomposition of the matrix **A**. The components of the diagonal matrix **E**(**D***τ*)

Note, that the matrix expression **D**, **E** and eigenvectors **X** are from the solution of the Laplace transformed problem equations (30) and (31). In equation (33) *dii* are entries of the eigenvalue

for *<sup>m</sup>* <sup>=</sup> 1, . . . , <sup>M</sup> and (∗) denotes the convolution operator. The constant vectors **<sup>V</sup>**(*m*) are

on the left hand side of equation (34). The effectiveness of this recursive scheme is due to the fact that the boundary condition for the problem (20) is already absorbed in the first recursion instruction whereas the remaining problems satisfy homogeneous boundary conditions only. To complete the construction of the analytical solution of problem (20) by the decomposition method, we present in the next section, a convergent scheme to generate the Adomian

The role of the Adomian polynomials is to constitute the non-linear term in equation (20), i.e. the dimensionless non-linear temperature term Θ4. Using a finite functional expansion in

where one of the possible identifications of the *A*ˆ*<sup>m</sup>* is to group together terms with *Ti* in the right hand side of the equation (35) in a way, such that the index *i* of *Ti* ranges from 0 to

or generically, *A*ˆ*<sup>m</sup>* = *A*ˆ*m*(*T*0,..., *Tm*). Note, that the significance of the *Tm* becomes clear further down in equation (39) and is used here merely as a term of a functional expansion. The

*edii<sup>τ</sup>* if *dii* < 0 *edii*(*τ*−*τ*0) if *dii* > 0

**<sup>U</sup>***m*(*τ*) = **XE**(**D***τ*)**V**(*m*) <sup>+</sup> **XeD***τ***X**−1∗*A*ˆ*m*−1(*τ*)**<sup>M</sup>** (34)

for *m* = 0

for *m* = 1, . . . , M

*<sup>m</sup>*=<sup>0</sup> *Tm*(*τ*) implies

, (35)

<sup>1</sup> on *T*0, *T*1,

12*T*<sup>2</sup> 0 2! � <sup>M</sup> ∑ *i*=1 *Ti* �<sup>2</sup> (33)

187

**E**(**D***τ*) =

⎧ ⎨ ⎩

matrix **D**. Further the general solution for the remaining problems are given by

determined from the application of the inhomogeneous boundary conditions

**U**0(0) = **I**(0) **U**0(*τ*0) = **I**(*τ*0)

**U***m*(0) = 0 **U***m*(*τ*0) = 0

polynomials *<sup>A</sup>*ˆ*<sup>m</sup>* for *<sup>m</sup>* ∈ {1, . . . , <sup>M</sup>}, for any generic <sup>M</sup>.

*Tm*(*τ*) for the dimensionless temperature Θ(*τ*) = ∑<sup>M</sup>

M ∑ *m*=0

*A*ˆ*<sup>m</sup>* = *T*<sup>4</sup>

<sup>0</sup> <sup>+</sup> <sup>4</sup>*T*<sup>3</sup> 0 M ∑ *i*=1 *Ti* +

> + 24*T*<sup>0</sup> 3! � <sup>M</sup> ∑ *i*=1 *Ti* �<sup>3</sup> + 24 4! � <sup>M</sup> ∑ *i*=1 *Ti* �<sup>4</sup>

*m*. This can be seen explicitly in equation (36) where *A*ˆ <sup>0</sup> depends on *T*<sup>0</sup> only, *A*ˆ

**4.2 The determination of the** *A*ˆ*<sup>m</sup>* **polynomials**

Θ<sup>4</sup> =

and the homogeneous boundary conditions

Non-Linear Radiative-Conductive Heat Transfer

in a Heterogeneous Gray Plane-Parallel Participating Medium

are:

In order to apply the decomposition method to the problem (20) and (23), we expand the non-linear source term into a series of Adomian polynomials Adomian (1988), which are determined in the next section.

$$\Theta^4(\tau) = \sum\_{m=0}^{\infty} \hat{A}\_m(\tau) \tag{24}$$

Upon inserting this ansatz in equation (20) yields a first order matrix differential equation:

$$\frac{d}{d\tau}\mathbf{I}(\tau) - \mathbf{A}\mathbf{I}(\tau) = \sum\_{m=0}^{\infty} \hat{A}\_m(\tau)\mathbf{M}.\tag{25}$$

Here **<sup>I</sup>**(*τ*)=(**I**+(*τ*),**I**−(*τ*))*<sup>T</sup>* is the intensity radiation vector, where the sub-vectors **<sup>I</sup>**+(*τ*) and **I**−(*τ*) are the intensity radiation for the positive (0 < *μ* < 1) and negative (−1 < *μ* < 0) directions, respectively, and of order *N*/2 each. Further, **M** is a vector of order *N* with entries:

$$\mathbf{M} = (1 - \omega) \left(\frac{1}{\mu\_1}, \dots, \frac{1}{\mu\_N}\right)^T \tag{26}$$

Finally, the components of matrix **A** have the form:

$$A\_{ij} = -\frac{1}{\mu\_i} \delta\_{ij} + \frac{\omega}{2\mu\_i} \sum\_{\ell=0}^{L} \beta\_\ell P\_\ell(\mu\_i) P\_\ell(\mu\_j) \,. \tag{27}$$

where *δij* is the Kronecker symbol. The radiation intensity can formally be written as a series:

$$\mathbf{I}(\tau) = \sum\_{m=0}^{\infty} \mathbf{U}\_m(\tau) \tag{28}$$

which upon substitution in equation (25) results in:

$$\sum\_{m=0}^{\infty} \left( \frac{d}{d\tau} \mathbf{U}\_m(\tau) - \mathbf{A} \mathbf{U}\_m(\tau) \right) = \sum\_{m=0}^{\infty} \hat{A}\_m(\tau) \mathbf{M} \tag{29}$$

One possibility of solving the equation system (29) starts with the initialisation

$$\frac{d}{d\tau}\mathbf{U}\_0(\tau) - \mathbf{A}\mathbf{U}\_0(\tau) = 0\tag{30}$$

$$\frac{d}{d\tau}\mathbf{U}\_m(\tau) - \mathbf{A}\mathbf{U}\_m(\tau) = \hat{A}\_{m-1}(\tau)\mathbf{M} \,, \qquad m = 1, 2, \ldots, \infty \tag{31}$$

which is then solved by the Laplace transform procedure (i.e. the *LTSN* method) for any arbitrary but finite *m* ≤ M. Here, M is a truncation of the series which has to be chosen such that the remaining dropped terms are only a small correction to the approximate solution. Details of the method may be found in references. (Segatto et al., 1999) and (Goncalves et al., 2000). In the further we make use of the results of the Laplace transformed equations (30) and (31) and write **U***<sup>m</sup>* in form of a Laplace inversion. The Adomian polynomials are given explicitly in equation (36).

So far the *LTSN* solution to the first problem of the recursive system has the form:

$$\mathbf{U}\_0(\tau) = \mathbf{X} \mathbf{E}(\mathbf{D}\tau) \mathbf{V}^{(0)} \tag{32}$$

10 Will-be-set-by-IN-TECH

In order to apply the decomposition method to the problem (20) and (23), we expand the non-linear source term into a series of Adomian polynomials Adomian (1988), which are

> ∞ ∑ *m*=0

> > ∞ ∑ *m*=0

> > > ,..., <sup>1</sup> *μ<sup>N</sup>*

*<sup>T</sup>*

Upon inserting this ansatz in equation (20) yields a first order matrix differential equation:

Here **<sup>I</sup>**(*τ*)=(**I**+(*τ*),**I**−(*τ*))*<sup>T</sup>* is the intensity radiation vector, where the sub-vectors **<sup>I</sup>**+(*τ*) and **I**−(*τ*) are the intensity radiation for the positive (0 < *μ* < 1) and negative (−1 < *μ* < 0) directions, respectively, and of order *N*/2 each. Further, **M** is a vector of order *N* with entries:

> 1 *μ*1

> > *L* ∑ �=0

*ω* 2*μ<sup>i</sup>*

**I**(*τ*) =

**U***m*(*τ*) − **AU***m*(*τ*)

One possibility of solving the equation system (29) starts with the initialisation

So far the *LTSN* solution to the first problem of the recursive system has the form:

where *δij* is the Kronecker symbol. The radiation intensity can formally be written as a series:

∞ ∑ *m*=0

which is then solved by the Laplace transform procedure (i.e. the *LTSN* method) for any arbitrary but finite *m* ≤ M. Here, M is a truncation of the series which has to be chosen such that the remaining dropped terms are only a small correction to the approximate solution. Details of the method may be found in references. (Segatto et al., 1999) and (Goncalves et al., 2000). In the further we make use of the results of the Laplace transformed equations (30) and (31) and write **U***<sup>m</sup>* in form of a Laplace inversion. The Adomian polynomials are given

 = ∞ ∑ *m*=0

**U**0(*τ*) − **AU**0(*τ*) = 0 (30)

**<sup>U</sup>***m*(*τ*) <sup>−</sup> **AU***m*(*τ*) = *<sup>A</sup>*ˆ*m*−1(*τ*)**<sup>M</sup>** , *<sup>m</sup>* <sup>=</sup> 1, 2, ..., <sup>∞</sup> (31)

**U**0(*τ*) = **XE**(**D***τ*)**V**(0) (32)

*A*ˆ*m*(*τ*) (24)

*A*ˆ*m*(*τ*)**M**. (25)

*β*�*P*�(*μi*)*P*�(*μj*) , (27)

**U***m*(*τ*) (28)

*A*ˆ*m*(*τ*)**M** (29)

(26)

Θ4(*τ*) =

**I**(*τ*) − **AI**(*τ*) =

**M** = (1 − *ω*)

*d dτ*

*Aij* <sup>=</sup> <sup>−</sup> <sup>1</sup>

*μi δij* +

Finally, the components of matrix **A** have the form:

which upon substitution in equation (25) results in: ∞ ∑ *m*=0

> *d dτ*

*d dτ*

explicitly in equation (36).

 *d dτ*

determined in the next section.

where **D** and **X** are respectively the matrices of eigenvalues and eigenfunctions resulting from the spectral decomposition of the matrix **A**. The components of the diagonal matrix **E**(**D***τ*) are:

$$\mathbf{E}(\mathbf{D}\tau) = \begin{cases} e^{d\_{li}\tau} & \text{if } \qquad d\_{li} < 0 \\ e^{d\_{li}(\tau - \tau\_0)} & \text{if } \qquad d\_{li} > 0 \end{cases} \tag{33}$$

Note, that the matrix expression **D**, **E** and eigenvectors **X** are from the solution of the Laplace transformed problem equations (30) and (31). In equation (33) *dii* are entries of the eigenvalue matrix **D**. Further the general solution for the remaining problems are given by

$$\mathbf{U}\_{m}(\tau) = \mathbf{X}\mathbf{E}(\mathbf{D}\tau)\mathbf{V}^{(m)} + \mathbf{X}\mathbf{e}^{\mathbf{D}\tau}\mathbf{X}^{-1} \ast \hat{A}\_{m-1}(\tau)\mathbf{M} \tag{34}$$

for *<sup>m</sup>* <sup>=</sup> 1, . . . , <sup>M</sup> and (∗) denotes the convolution operator. The constant vectors **<sup>V</sup>**(*m*) are determined from the application of the inhomogeneous boundary conditions

$$\begin{aligned} \mathbf{U}\_0(0) &= \mathbf{I}(0) \\ \mathbf{U}\_0(\tau\_0) &= \mathbf{I}(\tau\_0) \end{aligned} \quad \text{for} \quad \quad m = 0$$

and the homogeneous boundary conditions

$$\begin{aligned} \mathbf{U}\_m(0) &= 0 \\ \mathbf{U}\_m(\pi\_0) &= 0 \end{aligned} \quad \text{for} \quad \begin{aligned} \boldsymbol{m} &= 1, \dots, \mathcal{M} \\ \boldsymbol{0} &= \boldsymbol{1}, \dots, \mathcal{M} \end{aligned}$$

on the left hand side of equation (34). The effectiveness of this recursive scheme is due to the fact that the boundary condition for the problem (20) is already absorbed in the first recursion instruction whereas the remaining problems satisfy homogeneous boundary conditions only. To complete the construction of the analytical solution of problem (20) by the decomposition method, we present in the next section, a convergent scheme to generate the Adomian polynomials *<sup>A</sup>*ˆ*<sup>m</sup>* for *<sup>m</sup>* ∈ {1, . . . , <sup>M</sup>}, for any generic <sup>M</sup>.

#### **4.2 The determination of the** *A*ˆ*<sup>m</sup>* **polynomials**

The role of the Adomian polynomials is to constitute the non-linear term in equation (20), i.e. the dimensionless non-linear temperature term Θ4. Using a finite functional expansion in *Tm*(*τ*) for the dimensionless temperature Θ(*τ*) = ∑<sup>M</sup> *<sup>m</sup>*=<sup>0</sup> *Tm*(*τ*) implies

$$\begin{split} \Theta^{4} = \sum\_{m=0}^{\mathcal{M}} \hat{A}\_{m} &= T\_{0}^{4} + 4T\_{0}^{3} \sum\_{i=1}^{\mathcal{M}} T\_{i} + \frac{12T\_{0}^{2}}{2!} \left( \sum\_{i=1}^{\mathcal{M}} T\_{i} \right)^{2} \\ &+ \frac{24T\_{0}}{3!} \left( \sum\_{i=1}^{\mathcal{M}} T\_{i} \right)^{3} + \frac{24}{4!} \left( \sum\_{i=1}^{\mathcal{M}} T\_{i} \right)^{4} . \end{split} \tag{35}$$

where one of the possible identifications of the *A*ˆ*<sup>m</sup>* is to group together terms with *Ti* in the right hand side of the equation (35) in a way, such that the index *i* of *Ti* ranges from 0 to *m*. This can be seen explicitly in equation (36) where *A*ˆ <sup>0</sup> depends on *T*<sup>0</sup> only, *A*ˆ <sup>1</sup> on *T*0, *T*1, or generically, *A*ˆ*<sup>m</sup>* = *A*ˆ*m*(*T*0,..., *Tm*). Note, that the significance of the *Tm* becomes clear further down in equation (39) and is used here merely as a term of a functional expansion. The

*�*<sup>1</sup> *�*<sup>2</sup> *ρ<sup>s</sup>*

in a Heterogeneous Gray Plane-Parallel Participating Medium

Non-Linear Radiative-Conductive Heat Transfer

Table 1. Parameters of case 1.

<sup>1</sup> *<sup>ρ</sup><sup>s</sup>* <sup>2</sup> *<sup>ρ</sup><sup>d</sup>* <sup>1</sup> *<sup>ρ</sup><sup>d</sup>*

M Θ(*τ*) *Qc*(*τ*) *Qr*(*τ*) *Q*(*τ*)

Table 2. The *D*M*LTS*<sup>300</sup> results for M ranging from 0 to 200, assuming *τ*/*τ*<sup>0</sup> = 0.5.

adopted to show the smooth convergence with increasing M in the three cases for *N*.

M Θ(*τ*) *Qc*(*τ*) *Qr*(*τ*) *Q*(*τ*)

Table 3. The *D*M*LTS*<sup>350</sup> results for M ranging from 0 to 200, assuming *τ*/*τ*<sup>0</sup> = 0.5.

Table 4. The *D*M*LTS*<sup>400</sup> results for M ranging from 0 to 200, assuming *τ*/*τ*<sup>0</sup> = 0.5.

 0.8177175726732399 0.5016457333417628 1.5158272165190925 2.0174729498608555 0.7698083350919457 0.4588373357092782 1.5859090000727045 2.0447463357819826 0.7775833211907642 0.4652872662455577 1.5788010270489736 2.0440882932945312 0.7775903655034493 0.4652925826127813 1.5787958707789687 2.0440884533917498 0.7775903643992450 0.4652925817800941 1.5787958715859125 2.0440884533660064 0.7775903643992450 0.4652925817800941 1.5787958715859125 2.0440884533660064 0.7775903643992450 0.4652925817800941 1.5787958715859125 2.0440884533660064 0.7775903643992450 0.4652925817800941 1.5787958715859125 2.0440884533660064

M Θ(*τ*) *Qc*(*τ*) *Qr*(*τ*) *Q*(*τ*)

<sup>2</sup> Θ<sup>1</sup> Θ<sup>2</sup> *ω τ*<sup>0</sup> *Nc L*

189

0.6 0.4 0.1 0.2 0.3 0.4 1.0 0.5 0.95 1.0 0.05 299

 0.8177177955027018 0.5016457699309158 1.5158278540320405 2.0174736239629563 0.7698084721160902 0.4588373285860757 1.5859093685802235 2.0447466971662993 0.7775834780564881 0.4652872714487863 1.5788014097534921 2.0440886812022785 0.7775905224102305 0.4652925878442414 1.5787962534790447 2.0440888413232861 0.7775905213060152 0.4652925870115464 1.5787962542859926 2.0440888412975391 0.7775905213060152 0.4652925870115464 1.5787962542859926 2.0440888412975391 0.7775905213060152 0.4652925870115464 1.5787962542859926 2.0440888412975391 0.7775905213060152 0.4652925870115464 1.5787962542859926 2.0440888412975391

The numerical results for Θ, *Qr*(*τ*), *Qc*(*τ*) and *Q*(*τ*) are shown in table 2,3 and 4. The stability and convergence of the method was tested for *τ*/*τ*<sup>0</sup> = 0.5, varying M from 0 to 200, and using for *N* the values 300, 350 and 400, respectively. The displayed precision with 16 digits was

 0.8177176602853717 0.5016457476711904 1.5158274669152312 2.0174732145864214 0.7698083890454525 0.4588373328783077 1.5859091448625833 2.0447464777408908 0.7775833829280683 0.4652872682514292 1.5788011774222819 2.0440884456737112 0.7775904272568637 0.4652925846297308 1.5787960211505467 2.0440886057802774 0.7775904261526551 0.4652925837970406 1.5787960219574921 2.0440886057545327 0.7775904261526551 0.4652925837970406 1.5787960219574921 2.0440886057545327 0.7775904261526551 0.4652925837970406 1.5787960219574921 2.0440886057545327 0.7775904261526551 0.4652925837970406 1.5787960219574921 2.0440886057545327 0.7775904261526551 0.4652925837970406 1.5787960219574921 2.0440886057545327

resulting scheme for the Adomain polynomials reads then, which for later use we indicate in factorized form:

$$\begin{aligned} \dot{A}\_0 &= T\_0^4 = T\_0 T\_0 T\_0^2 \\ \dot{A}\_1 &= 4T\_0^3 T\_1 + 6T\_0^2 T\_1^2 + 4T\_0 T\_1^3 + T\_1^4 = T\_1 (2T\_0 + T\_1)(2T\_0^2 + 2T\_0 T\_1 + T\_1^2) \\ \dot{A}\_2 &= 4T\_0^3 T\_2 + 12T\_0^2 T\_1 T\_2 + 12T\_0 T\_1^2 T\_2 + 4T\_1^3 T\_2 + 6T\_0^2 T\_2^2 + 12T\_0 T\_1 T\_2^2 \\ &\quad + 6T\_1^2 T\_2^2 + 4T\_0 T\_2^3 + 4T\_1 T\_2^3 + T\_2^4 \\ &= T\_2 (2T\_0 + 2T\_1 + T\_2)(2T\_0^2 + 4T\_0 T\_1 + 2T\_1^2 + 2T\_0 T\_2 + 2T\_1 T\_2 + T\_2^2) \\ &\vdots \end{aligned} \tag{36}$$

In shorthand notation the recursive scheme for the Adomian polynomials may be written as

$$
\hat{A}\_m = T\_m S\_m R\_m \tag{37}
$$

where *Sm* and *Rm* are determined by the formulas for *m* = 1, . . . , M.

$$S\_m = S\_{m-1} + T\_m + T\_{m-1} \qquad \text{and} \qquad R\_m = R\_{m-1} + S\_{m-1}T\_{m-1} + S\_mT\_m \tag{38}$$

The recursive procedure according to equation (36) starts with *S*<sup>0</sup> = *T*<sup>0</sup> and *R*<sup>0</sup> = *T*<sup>2</sup> <sup>0</sup> . From equation (23), we construct then the recursive formulation for the temperature.

$$T\_0(\tau) = \Theta\_1 + (\Theta\_2 - \Theta\_1)\frac{\tau}{\tau\_0} \tag{39}$$

$$T\_{m+1}(\tau) = -\frac{1}{2\mathcal{N}\_c}\frac{\tau}{\tau\_0} \left\langle \mathbf{W}\_\prime \int\_0^{\tau\_0} \mathbf{U}\_m(\tau') d\tau' \right\rangle + \frac{1}{2\mathcal{N}\_c} \left\langle \mathbf{W}\_\prime \int\_0^{\tau} \mathbf{U}\_m(\tau') d\tau' \right\rangle$$

Here *m* = 0, . . . , M and the column vector **W** = (*w*1*μ*1, ...., *wNμN*)*<sup>T</sup>* contains as components the discrete directions *μ<sup>i</sup>* and the Gaussian quadrature weights *wi*. The bracket signifies the vector inner product. Note, that equation (39) establishes the Adomian polynomials in terms of the temperature at the boundaries and the expansion terms of the intensity, which in principle could be determined until infinity.

#### **4.3 Numerical results**

In this section we present three cases that show the robustness and quantitative coincidence of the *D*M*LTSN* approach with solutions of the *SN* radiative-conductive problem in a slab in the literature. As results we evaluate the normalised temperature, conductive, radiative and total heat fluxes.

$$Q\_r(\tau) = \frac{1}{4\pi N\_\circ} q\_r^\*(\tau) \quad Q\_\circ(\tau) = -\frac{d}{d\tau} \Theta(\tau) \quad \text{and} \quad Q(\tau) = Q\_r(\tau) + Q\_\circ(\tau)$$

#### **4.3.1 Case 1**

In this case we determine the numerical values for M and *N* in order to get results with a considerable accuracy. The numerical values of the parameters used in cases 1 to 2 are given in table 1. The coefficient *β* is defined considering a binomial scattering law which also permits a comparison with the results of (Siewert & Thomas, 1991).

$$
\beta\_{\ell} = \left(\frac{2\ell + 1}{2\ell - 1}\right) \left(\frac{L + 1 - \ell}{L + 1 + \ell}\right) \beta\_{\ell - 1} \qquad 0 \le \ell \le L \qquad \text{and} \qquad \beta\_0 = 1.
$$


Table 1. Parameters of case 1.

12 Will-be-set-by-IN-TECH

resulting scheme for the Adomain polynomials reads then, which for later use we indicate in

<sup>1</sup> *<sup>T</sup>*<sup>2</sup> <sup>+</sup> <sup>4</sup>*T*<sup>3</sup>

<sup>0</sup> <sup>+</sup> <sup>4</sup>*T*0*T*<sup>1</sup> <sup>+</sup> <sup>2</sup>*T*<sup>2</sup>

In shorthand notation the recursive scheme for the Adomian polynomials may be written as

<sup>2</sup> <sup>+</sup> <sup>4</sup>*T*1*T*<sup>3</sup>

*Sm* = *Sm*−<sup>1</sup> + *Tm* + *Tm*−<sup>1</sup> and *Rm* = *Rm*−<sup>1</sup> + *Sm*−<sup>1</sup>*Tm*−<sup>1</sup> + *SmTm* (38)

<sup>1</sup> <sup>=</sup> *<sup>T</sup>*1(2*T*<sup>0</sup> <sup>+</sup> *<sup>T</sup>*1)(2*T*<sup>2</sup>

<sup>1</sup> *<sup>T</sup>*<sup>2</sup> <sup>+</sup> <sup>6</sup>*T*<sup>2</sup>

<sup>2</sup> <sup>+</sup> *<sup>T</sup>*<sup>4</sup>

<sup>0</sup> *<sup>T</sup>*<sup>2</sup>

<sup>1</sup> <sup>+</sup> <sup>2</sup>*T*0*T*<sup>2</sup> <sup>+</sup> <sup>2</sup>*T*1*T*<sup>2</sup> <sup>+</sup> *<sup>T</sup>*<sup>2</sup>

*A*ˆ*<sup>m</sup>* = *TmSmRm* (37)

 **W**, *τ* 0

Θ(*τ*) and *Q*(*τ*) = *Qr*(*τ*) + *Qc*(*τ*)

*<sup>β</sup>*<sup>−</sup><sup>1</sup> <sup>0</sup> ≤ ≤ *<sup>L</sup>* and *<sup>β</sup>*<sup>0</sup> = <sup>1</sup>

**U***m*(*τ*�

)*dτ*� 

<sup>0</sup> <sup>+</sup> <sup>2</sup>*T*0*T*<sup>1</sup> <sup>+</sup> *<sup>T</sup>*<sup>2</sup>

2

<sup>2</sup> (36)

2 )

<sup>2</sup> <sup>+</sup> <sup>12</sup>*T*0*T*1*T*<sup>2</sup>

1 )

<sup>0</sup> . From

(39)

<sup>1</sup> <sup>+</sup> *<sup>T</sup>*<sup>4</sup>

<sup>2</sup> <sup>+</sup> <sup>4</sup>*T*0*T*<sup>3</sup>

The recursive procedure according to equation (36) starts with *S*<sup>0</sup> = *T*<sup>0</sup> and *R*<sup>0</sup> = *T*<sup>2</sup>

**U***m*(*τ*�

Here *m* = 0, . . . , M and the column vector **W** = (*w*1*μ*1, ...., *wNμN*)*<sup>T</sup>* contains as components the discrete directions *μ<sup>i</sup>* and the Gaussian quadrature weights *wi*. The bracket signifies the vector inner product. Note, that equation (39) establishes the Adomian polynomials in terms of the temperature at the boundaries and the expansion terms of the intensity, which in

In this section we present three cases that show the robustness and quantitative coincidence of the *D*M*LTSN* approach with solutions of the *SN* radiative-conductive problem in a slab in the literature. As results we evaluate the normalised temperature, conductive, radiative and

*dτ*

In this case we determine the numerical values for M and *N* in order to get results with a considerable accuracy. The numerical values of the parameters used in cases 1 to 2 are given in table 1. The coefficient *β* is defined considering a binomial scattering law which also

)*dτ*� + 1 2*Nc*

equation (23), we construct then the recursive formulation for the temperature.

*τ*0

*<sup>r</sup>* (*τ*) *Qc*(*τ*) = <sup>−</sup> *<sup>d</sup>*

permits a comparison with the results of (Siewert & Thomas, 1991).

 *<sup>L</sup>* <sup>+</sup> <sup>1</sup> <sup>−</sup> *L* + 1 + factorized form:

*A*ˆ <sup>0</sup> = *T*<sup>4</sup>

<sup>1</sup> = 4*T*<sup>3</sup>

*A*ˆ2 = 4*T*<sup>3</sup>

. . .

*A*ˆ

<sup>0</sup> <sup>=</sup> *<sup>T</sup>*0*T*0*T*<sup>2</sup>

<sup>0</sup> *<sup>T</sup>*<sup>1</sup> <sup>+</sup> <sup>6</sup>*T*<sup>2</sup>

<sup>0</sup> *<sup>T</sup>*<sup>2</sup> <sup>+</sup> <sup>12</sup>*T*<sup>2</sup>

= *T*2(2*T*<sup>0</sup> + 2*T*<sup>1</sup> + *T*2)(2*T*<sup>2</sup>

*<sup>T</sup>*0(*τ*) = <sup>Θ</sup><sup>1</sup> + (Θ<sup>2</sup> <sup>−</sup> <sup>Θ</sup>1) *<sup>τ</sup>*

*τ τ*0 **W**, *τ*<sup>0</sup> 0

2*Nc*

*Tm*+1(*τ*) = <sup>−</sup> <sup>1</sup>

principle could be determined until infinity.

4*πNc q*∗

2 + 1 2 − 1

**4.3 Numerical results**

*Qr*(*τ*) = <sup>1</sup>

*β* =

total heat fluxes.

**4.3.1 Case 1**

0

<sup>0</sup> *<sup>T</sup>*<sup>2</sup>

<sup>1</sup> <sup>+</sup> <sup>4</sup>*T*0*T*<sup>3</sup>

+6*T*<sup>2</sup> <sup>1</sup> *<sup>T</sup>*<sup>2</sup>

where *Sm* and *Rm* are determined by the formulas for *m* = 1, . . . , M.

<sup>0</sup> *<sup>T</sup>*1*T*<sup>2</sup> <sup>+</sup> <sup>12</sup>*T*0*T*<sup>2</sup>


Table 2. The *D*M*LTS*<sup>300</sup> results for M ranging from 0 to 200, assuming *τ*/*τ*<sup>0</sup> = 0.5.

The numerical results for Θ, *Qr*(*τ*), *Qc*(*τ*) and *Q*(*τ*) are shown in table 2,3 and 4. The stability and convergence of the method was tested for *τ*/*τ*<sup>0</sup> = 0.5, varying M from 0 to 200, and using for *N* the values 300, 350 and 400, respectively. The displayed precision with 16 digits was adopted to show the smooth convergence with increasing M in the three cases for *N*.


Table 3. The *D*M*LTS*<sup>350</sup> results for M ranging from 0 to 200, assuming *τ*/*τ*<sup>0</sup> = 0.5.


Table 4. The *D*M*LTS*<sup>400</sup> results for M ranging from 0 to 200, assuming *τ*/*τ*<sup>0</sup> = 0.5.

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

Θ

Θ

Θ

Θ

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

heat flux *Q* (right) against the relative optical depth *<sup>τ</sup>*

behaviour in the considered range (see also figure 2).

τ τ0

τ τ0

τ τ0

0 0.5 1 1.5 2 2.5

0 0.5 1 1.5 2 2.5

0 0.5 1 1.5 2 2.5

0 0.5 1 1.5 2 2.5

Q

Fig. 3. Numerical comparisons of the *D*M*LTSN* (solid line) and Abulwafa's results (dotted line) for the parameter combinations (*ω*, *Nc*) = {(0.5, 0.1),(0.5, 0.5),(0.9, 0.1),(0.9, 0.5)} from top to bottom. The temperature profile Θ (left) and the conductive *Qc*, radiative *Qr* and total

curves, the conductive (radiative) heat fluxes show predominantly convex (concave)

Q

Q

Q

0 0.2 0.4 0.6 0.8 1

191

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

τ τ0

0 0.2 0.4 0.6 0.8 1

*<sup>τ</sup>*<sup>0</sup> . The total heat fluxes are the constant

τ τ0 τ τ0

τ τ0

τ τ0

Non-Linear Radiative-Conductive Heat Transfer

in a Heterogeneous Gray Plane-Parallel Participating Medium

Fig. 2. Numerical results for *D*10*LTS*<sup>350</sup> that exactly reproduce the results of reference (Siewert & Thomas, 1991) within the adopted precision. The temperature profile Θ (left), the conductive *Qc*, radiative *Qr* and total heat flux *Q* (right) against the relative optical depth *<sup>τ</sup> τ*0 .

Comparing the corresponding lines in tables 2 to 4 for different *N* one observes that an analytical expression with M = 10 and *N* = 300 is already close to the solution with M as large as 200 or ideally in the limit M → ∞. However, in the subsequent problems we set M = 10 and *N* = 350 in order to reproduce within the adopted precision the numerical results of ref. (Siewert & Thomas, 1991) where the *PN* method was applied to the same problem.

#### **4.3.2 Case 2**

The numerical evaluation in case 1 may now be refined varying the optical depth which was maintained fixed previously. To this end we determine Θ(*τ*), *Qc*(*τ*), *Qr*(*τ*) and *Q*(*τ*) for *τ*/*τ*<sup>0</sup> ranging from 0 to 1. The numerical results are shown in figure 2 which coincide with the findings in reference (Siewert & Thomas, 1991) beyond a six digit precision. In their work Siewert and Thomas left open the question of convergence of their applied method, which by virtue of numerical coincidence with the present approach may be positively answered. Although not presented here with mathematical rigour, convergence of the decomposition method is formally guaranteed (see references (Adomian, 1988; Cherruault, 1989; Pazos & Vilhena, 1999a;b)) by the manifest exact solution in the limit M → ∞.

#### **4.3.3 Case 3**

A third comparison is elaborated making contact to a work by (Abulwafa, 1999), considering a conductive radiative problem in a slab assuming isotropy (*L* = 0) and with thickness *τ*0, which also serves as a unit length. The parameter set is with either *ω* = 0.9 or *ω* = 0.5, with *�*<sup>1</sup> = *�*<sup>2</sup> = Θ<sup>1</sup> = 1 and *ρ<sup>d</sup>* <sup>1</sup> <sup>=</sup> *<sup>ρ</sup><sup>d</sup>* <sup>2</sup> = *<sup>ρ</sup><sup>s</sup>* <sup>1</sup> <sup>=</sup> *<sup>ρ</sup><sup>s</sup>* <sup>2</sup> = 0. In this article the author uses a variational technique to solve the radiative problem, while an iterative method is implemented to include the non-linearity effect of the temperature distribution of the medium from the conductive energy equation. Figure 3 shows the numerical findings of *D*10*LTS*<sup>350</sup> in comparison to results from (Abulwafa, 1999) for two conduction-radiation parameters *Nc* = 0.5 and *Nc* = 0.1, respectively.

The comparison of the *D*M*LTSN* results with the ones of reference (Abulwafa, 1999) shows a fairly good agreement between the methods. It seems that a decrease in *Nc* opens slightly the difference between the two solutions, whereas increase in *ω* closes the difference between the solutions. Moreover, the larger *Nc* the closer one gets to a linear temperature profile. The difference is probably due to the fact that the approach in reference (Abulwafa, 1999) makes 14 Will-be-set-by-IN-TECH

Q

Fig. 2. Numerical results for *D*10*LTS*<sup>350</sup> that exactly reproduce the results of reference (Siewert & Thomas, 1991) within the adopted precision. The temperature profile Θ (left), the conductive *Qc*, radiative *Qr* and total heat flux *Q* (right) against the relative optical depth *<sup>τ</sup>*

Comparing the corresponding lines in tables 2 to 4 for different *N* one observes that an analytical expression with M = 10 and *N* = 300 is already close to the solution with M as large as 200 or ideally in the limit M → ∞. However, in the subsequent problems we set M = 10 and *N* = 350 in order to reproduce within the adopted precision the numerical results of ref. (Siewert & Thomas, 1991) where the *PN* method was applied to the same problem.

The numerical evaluation in case 1 may now be refined varying the optical depth which was maintained fixed previously. To this end we determine Θ(*τ*), *Qc*(*τ*), *Qr*(*τ*) and *Q*(*τ*) for *τ*/*τ*<sup>0</sup> ranging from 0 to 1. The numerical results are shown in figure 2 which coincide with the findings in reference (Siewert & Thomas, 1991) beyond a six digit precision. In their work Siewert and Thomas left open the question of convergence of their applied method, which by virtue of numerical coincidence with the present approach may be positively answered. Although not presented here with mathematical rigour, convergence of the decomposition method is formally guaranteed (see references (Adomian, 1988; Cherruault, 1989; Pazos &

A third comparison is elaborated making contact to a work by (Abulwafa, 1999), considering a conductive radiative problem in a slab assuming isotropy (*L* = 0) and with thickness *τ*0, which also serves as a unit length. The parameter set is with either *ω* = 0.9 or *ω* = 0.5, with

technique to solve the radiative problem, while an iterative method is implemented to include the non-linearity effect of the temperature distribution of the medium from the conductive energy equation. Figure 3 shows the numerical findings of *D*10*LTS*<sup>350</sup> in comparison to results from (Abulwafa, 1999) for two conduction-radiation parameters *Nc* = 0.5 and *Nc* = 0.1,

The comparison of the *D*M*LTSN* results with the ones of reference (Abulwafa, 1999) shows a fairly good agreement between the methods. It seems that a decrease in *Nc* opens slightly the difference between the two solutions, whereas increase in *ω* closes the difference between the solutions. Moreover, the larger *Nc* the closer one gets to a linear temperature profile. The difference is probably due to the fact that the approach in reference (Abulwafa, 1999) makes

Vilhena, 1999a;b)) by the manifest exact solution in the limit M → ∞.

<sup>2</sup> = *<sup>ρ</sup><sup>s</sup>*

<sup>1</sup> <sup>=</sup> *<sup>ρ</sup><sup>s</sup>*

<sup>1</sup> <sup>=</sup> *<sup>ρ</sup><sup>d</sup>*

0 0.5 1 1.5 2 2.5

Q

Q<sup>r</sup>

Q<sup>c</sup>

0 0.2 0.4 0.6 0.8 1

*τ*0 .

τ τ0

<sup>2</sup> = 0. In this article the author uses a variational

0 0.2 0.4 0.6 0.8 1

**4.3.2 Case 2**

**4.3.3 Case 3**

respectively.

*�*<sup>1</sup> = *�*<sup>2</sup> = Θ<sup>1</sup> = 1 and *ρ<sup>d</sup>*

Θ

0 0.2 0.4 0.6 0.8 1

τ τ0

Fig. 3. Numerical comparisons of the *D*M*LTSN* (solid line) and Abulwafa's results (dotted line) for the parameter combinations (*ω*, *Nc*) = {(0.5, 0.1),(0.5, 0.5),(0.9, 0.1),(0.9, 0.5)} from top to bottom. The temperature profile Θ (left) and the conductive *Qc*, radiative *Qr* and total heat flux *Q* (right) against the relative optical depth *<sup>τ</sup> <sup>τ</sup>*<sup>0</sup> . The total heat fluxes are the constant curves, the conductive (radiative) heat fluxes show predominantly convex (concave) behaviour in the considered range (see also figure 2).

Nevertheless, we consider our work as an essential step for implementation of problems considering heterogeneous media. Furthermore, for thick media were the total optical depth lies outside the convergence radius a multi-slab treatment shall be used. In section 3 we showed explicitly how dependencies of the Albedo, the emissivity and the reflectivity on the optical depth are handled reducing partially the problem to a homogeneous one and including corrections in form of source terms. As an additional task due to this particular procedure matching of the partial solutions at the interfaces of the slabs has to be performed in addition, as a consequence of the local character of the physical parameter in each slab. So far, our findings show that for the *SN* problem the series truncated with M = 10 yields already a fairly good solution in form of an analytical expression. One advantage in comparison to numerical approaches lies in the fact that the dependence of the solution on the physical parameter may

193

In section 4.3 we solved a selection of cases that may constitute a partial problem in a more complex medium and showed systematically, how a reliable solution may be obtained following the construction steps of 4.1 and 4.2. The application given in case 1 indicates the limits for M and *N*, and showed in case 2 that for a sequence of optical depths the same numerical results appear as given in ref. (Siewert & Thomas, 1991). Since convergence in the present approach is guaranteed one may elaborate a genuine convergence criterion depending on a desired precision. As a third test we compared our results for an isotropic problem to ref. (Abulwafa, 1999), where also agreement between the findings was verified. Since the proposed method reproduces the exact analytical solution in the limit M → ∞, approximate analytical expressions with finite M gain the character of benchmark results, which are of

Abulwafa, E.M. (1999). Conductive-radiative heat transfer in an inhomogeneous slab with

Adomian, G., 1988. A review of the Decomposition method in applied-mathematics. *Journal of Mathematical Analysis and Applications*, Vol. 135, No. 2, (November 1988), 501-544. Ahmad, S. & Deering, D. (1992). A Simple Analytical Function for Bidirectional Reflectance.

Asllanaj, F., Jeandel, G., Roche, J.R. (2001). Numerical Solution of Radiative Transfer

Attia, M.T. (2000). On the Exact Solution of a Generalized Equation of Radiative Transfer in

directional reflecting boundaries. *Journal of Physics D*, Vol. 32, No. 14, (July 1999),

*Journal of Geophysical Research D*, Vol. 97, No. 17 (April 1992), 18867–18886,

Equation Coupled with Non-linear Heat Conduction Equation. *International Journal of Numerical Methods for Heat and Fluid Flow*, Vol. 11, No. 5 (July 2001), 449-473. Asllanaj, F., Milandri, A., Jeandel, G., Roche, J.R. (2002). A Finite Difference Solution

of Non-linear Systems of RadiativeâA ¸ ˘ SConductive Heat Transfer Equations. *International Journal for Numerical Methods in Engineering*, Vol. 54, No. 11, (August

a Two-region Inhomogeneous Slab. *Journal of Quantitative Spectroscopy & Radiative*

be analytically explored from the resulting expressions.

Non-Linear Radiative-Conductive Heat Transfer

in a Heterogeneous Gray Plane-Parallel Participating Medium

special interest in applications considering heterogeneous media.

*Transfer*, Vol. 66, No. 6, (September 2000), 529-538.

**6. References**

1626-1632.

18867-18886.

2002), 1649-1668.

use of a trial function. In general in such type of approaches convergence depends crucially on how close the trial function is to the true solution. The comparison of cases 2 and 3 and the quantitative agreement with two different approaches shows the quality of the present method, especially because of the fact that it reproduces the exact analytical solution in the limit M → ∞ and thus allows to implement computationally a genuine convergence criterion. Some further information concerning the computational issue, the three cases were calculated on a Notebook computer with 64-bit Athlon 3200+ processor (1Ghz, 512kb Cache) and 1GB RAM. All calculations terminated with less than a minute execution time (some examples returned the result within seconds) and used typically between 10 and 20 iterations.

#### **5. Conclusion**

In the present work we discussed and compared an analytical approach to the non-linear *SN* radiative-conductive transfer problem in plane-parallel geometry and a heterogeneous medium using a composite method by the Laplace transform and the Adomian decomposition (Adomian, 1988). We showed by two options how the heterogeneous problem may be cast into a set of homogeneous problems, so that the general solution may be obtained by a hierarchical algorithm. The Laplace technique opens pathways to resort to classical methods for linear problems, whereas the decomposition procedure allows to disentangle the non-linear contribution of the problem, that permits to solve the equations by a recursion scheme. It is worth mentioning two limiting cases, i.e. with single scattering albedo either *ω* = 0 or *ω* = 1. The latter case turns Adomian obsolete, because the non-linear term vanishes, whereas *ω* = 0 diagonalizes the equation system and thus turns Laplace obsolete, since the solution may be obtained directly by integration.

The decomposition method as originally introduced is designed for general non-linear problems, but several ways are possible to construct a solution (Cardona et al., 2009; Segatto et al., 2008). The present study may be considered a guideline on how to distribute the influence of the boundary conditions and the non-linearity in order to solve the given problem. The boundary condition is absorbed in the part of the solution that belongs to the inversion of the differential operator without the non-linear contribution and the non-linear part simplifies to a problem for homogeneous boundary conditions only. Since existence and uniqueness of the solution for radiative-conductive transfer problems was discussed in references (Kelley, 1996; Thompson et al., 2004; 2008) the only critical issue of the recursive scheme is convergence. According to (Adomian, 1988; Cherruault, 1989; Pazos & Vilhena, 1999a;b) the resulting scheme is manifest exact and converges in the limit M → ∞ to the exact analytical solution. A genuine control of errors opens thus the possibility of model validation in comparison to experimental findings. In numerical approaches it is not straight forward to distinguish between model and numerical uncertainties, especially when non-linearities are present in the problem. Moreover, in general error analysis in numerical procedures is based on a heuristic basis, whereas the present approach permits a mathematical proof of convergence. This may not be that crucial for homogeneous problems, or those that permit linear approximations, but in the case of a heterogeneous problem in form of a multi-slab medium the question of convergence certainly plays a major role, especially because of the matching of solutions at the slab interfaces.

We are completely aware of the fact that the present procedure is limited by the convergence relation between the optical depth and the convergence radius of the Adomian approximation. Nevertheless, we consider our work as an essential step for implementation of problems considering heterogeneous media. Furthermore, for thick media were the total optical depth lies outside the convergence radius a multi-slab treatment shall be used. In section 3 we showed explicitly how dependencies of the Albedo, the emissivity and the reflectivity on the optical depth are handled reducing partially the problem to a homogeneous one and including corrections in form of source terms. As an additional task due to this particular procedure matching of the partial solutions at the interfaces of the slabs has to be performed in addition, as a consequence of the local character of the physical parameter in each slab. So far, our findings show that for the *SN* problem the series truncated with M = 10 yields already a fairly good solution in form of an analytical expression. One advantage in comparison to numerical approaches lies in the fact that the dependence of the solution on the physical parameter may be analytically explored from the resulting expressions.

In section 4.3 we solved a selection of cases that may constitute a partial problem in a more complex medium and showed systematically, how a reliable solution may be obtained following the construction steps of 4.1 and 4.2. The application given in case 1 indicates the limits for M and *N*, and showed in case 2 that for a sequence of optical depths the same numerical results appear as given in ref. (Siewert & Thomas, 1991). Since convergence in the present approach is guaranteed one may elaborate a genuine convergence criterion depending on a desired precision. As a third test we compared our results for an isotropic problem to ref. (Abulwafa, 1999), where also agreement between the findings was verified. Since the proposed method reproduces the exact analytical solution in the limit M → ∞, approximate analytical expressions with finite M gain the character of benchmark results, which are of special interest in applications considering heterogeneous media.

#### **6. References**

16 Will-be-set-by-IN-TECH

use of a trial function. In general in such type of approaches convergence depends crucially on how close the trial function is to the true solution. The comparison of cases 2 and 3 and the quantitative agreement with two different approaches shows the quality of the present method, especially because of the fact that it reproduces the exact analytical solution in the limit M → ∞ and thus allows to implement computationally a genuine convergence criterion. Some further information concerning the computational issue, the three cases were calculated on a Notebook computer with 64-bit Athlon 3200+ processor (1Ghz, 512kb Cache) and 1GB RAM. All calculations terminated with less than a minute execution time (some examples

returned the result within seconds) and used typically between 10 and 20 iterations.

In the present work we discussed and compared an analytical approach to the non-linear *SN* radiative-conductive transfer problem in plane-parallel geometry and a heterogeneous medium using a composite method by the Laplace transform and the Adomian decomposition (Adomian, 1988). We showed by two options how the heterogeneous problem may be cast into a set of homogeneous problems, so that the general solution may be obtained by a hierarchical algorithm. The Laplace technique opens pathways to resort to classical methods for linear problems, whereas the decomposition procedure allows to disentangle the non-linear contribution of the problem, that permits to solve the equations by a recursion scheme. It is worth mentioning two limiting cases, i.e. with single scattering albedo either *ω* = 0 or *ω* = 1. The latter case turns Adomian obsolete, because the non-linear term vanishes, whereas *ω* = 0 diagonalizes the equation system and thus turns Laplace obsolete, since the

The decomposition method as originally introduced is designed for general non-linear problems, but several ways are possible to construct a solution (Cardona et al., 2009; Segatto et al., 2008). The present study may be considered a guideline on how to distribute the influence of the boundary conditions and the non-linearity in order to solve the given problem. The boundary condition is absorbed in the part of the solution that belongs to the inversion of the differential operator without the non-linear contribution and the non-linear part simplifies to a problem for homogeneous boundary conditions only. Since existence and uniqueness of the solution for radiative-conductive transfer problems was discussed in references (Kelley, 1996; Thompson et al., 2004; 2008) the only critical issue of the recursive scheme is convergence. According to (Adomian, 1988; Cherruault, 1989; Pazos & Vilhena, 1999a;b) the resulting scheme is manifest exact and converges in the limit M → ∞ to the exact analytical solution. A genuine control of errors opens thus the possibility of model validation in comparison to experimental findings. In numerical approaches it is not straight forward to distinguish between model and numerical uncertainties, especially when non-linearities are present in the problem. Moreover, in general error analysis in numerical procedures is based on a heuristic basis, whereas the present approach permits a mathematical proof of convergence. This may not be that crucial for homogeneous problems, or those that permit linear approximations, but in the case of a heterogeneous problem in form of a multi-slab medium the question of convergence certainly plays a major role, especially because of the matching of solutions at

We are completely aware of the fact that the present procedure is limited by the convergence relation between the optical depth and the convergence radius of the Adomian approximation.

**5. Conclusion**

the slab interfaces.

solution may be obtained directly by integration.


Pomraning, G.C. (2005). *The Equations of Radiation Hydrodynamics*, Dover Publ. Inc., ISBN:

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**9** 

*Poland*

**Optimization of the Effective** 

Hubert Jopek and Tomasz Strek

**Thermal Conductivity of a Composite** 

*Poznan University of Technology, Institute Of Applied Mechanics* 

Composite materials by definition are a combination of two or more materials. Although the idea of combining two or more components to produce materials with controlled properties has been known and used from time immemorial, modern composites were developed only several decades ago and have found by now intensive application in different fields of

These materials are used in various design to improve the characteristic of various construction and reduce their weight. The properties of these materials and the problems of obtaining structural elements based upon them have been studied by researchers and engineers all over the world. The fields of composite applications are diversified (Freger et al., 2004). They include structural elements of flying vehicles, their casinos, wings, fuselages, tails and nose cones, jet engine stators, panels form various purposes, main rotors of helicopters, heat – proofing components, construction elements such as panels, racks,

Any property of a composite which is made of two (or more) materials has the value which is the resultant of a few factors. Obviously, the most important are the values of a certain property of each constituent material. However, one of the factor that also influences the resultant value of a property of a composite as a whole is its geometrical structure. Such resultant properties are commonly called effective properties of a composite. Temperature is the most important of all environmental factors affecting the behaviour of composite materials, mainly because composites are rather sensitive to temperature and have relatively low effective thermal conductivity. For instance, advanced composites for engineering applications are characterized with low density providing high specific strength and stiffness, low thermal conductivity resulting in high heat insulation, and negative thermal expansion coefficient allowing us to construct hybrid composite elements that do not change

Because experimental evaluation of effective properties (e.g. thermal conductivity) of composites is expensive and time consuming, computational methods have been found to provide efficient alternatives for predicting the best parameters of composites, especially those having complex geometries. To achieve a reliable prediction, one needs to work on two aspects: a good description of the structural details of fibrous materials, and an efficient numerical method for the solution of energy equations through the fibrous structures (Wang et al. 2009). The need to determine the thermal conductivity of fibres for design purposes has been the motivation of work (Al-Sulaiman et al., 2006). Authors developed four

**1. Introduction**

engineering (Vasiliev&Morozov, 2001).

shields, banking elements, etc.

their dimensions under heating (Vasiliev & Morozov, 2001).


### **Optimization of the Effective Thermal Conductivity of a Composite**

Hubert Jopek and Tomasz Strek

*Poznan University of Technology, Institute Of Applied Mechanics Poland*

#### **1. Introduction**

20 Will-be-set-by-IN-TECH

196 Convection and Conduction Heat Transfer

Wilson, S.J. & Sen, K.K. (1986). Generalized Eddington Approximation Method for Radiative

*Transfer*, Vol. 35, No. 6, (June 1986), 467-472.

Yang, L. & Kruse, B. (2004). Revised KubelkaâA ¸

Vol. 55, No. 1 (January 1996), 1-5.

Transfer Problems in Slab Medium. *Journal of Quantitative Spectroscopy & Radiative*

*Journal of the Optical Society of America A*, Vol. 21, No. 10 (October 2004) 1933-1941. Yi, W., Yingkui, M., Peizhu, M. (1996). A Linear Spline Approximation for Radiative Transfer

Problems in Slab Medium. *Journal of Quantitative Spectroscopy & Radiative Transfer*,

˘ SMunk theory. I. Theory and application.

Composite materials by definition are a combination of two or more materials. Although the idea of combining two or more components to produce materials with controlled properties has been known and used from time immemorial, modern composites were developed only several decades ago and have found by now intensive application in different fields of engineering (Vasiliev&Morozov, 2001).

These materials are used in various design to improve the characteristic of various construction and reduce their weight. The properties of these materials and the problems of obtaining structural elements based upon them have been studied by researchers and engineers all over the world. The fields of composite applications are diversified (Freger et al., 2004). They include structural elements of flying vehicles, their casinos, wings, fuselages, tails and nose cones, jet engine stators, panels form various purposes, main rotors of helicopters, heat – proofing components, construction elements such as panels, racks, shields, banking elements, etc.

Any property of a composite which is made of two (or more) materials has the value which is the resultant of a few factors. Obviously, the most important are the values of a certain property of each constituent material. However, one of the factor that also influences the resultant value of a property of a composite as a whole is its geometrical structure. Such resultant properties are commonly called effective properties of a composite. Temperature is the most important of all environmental factors affecting the behaviour of composite materials, mainly because composites are rather sensitive to temperature and have relatively low effective thermal conductivity. For instance, advanced composites for engineering applications are characterized with low density providing high specific strength and stiffness, low thermal conductivity resulting in high heat insulation, and negative thermal expansion coefficient allowing us to construct hybrid composite elements that do not change their dimensions under heating (Vasiliev & Morozov, 2001).

Because experimental evaluation of effective properties (e.g. thermal conductivity) of composites is expensive and time consuming, computational methods have been found to provide efficient alternatives for predicting the best parameters of composites, especially those having complex geometries. To achieve a reliable prediction, one needs to work on two aspects: a good description of the structural details of fibrous materials, and an efficient numerical method for the solution of energy equations through the fibrous structures (Wang et al. 2009). The need to determine the thermal conductivity of fibres for design purposes has been the motivation of work (Al-Sulaiman et al., 2006). Authors developed four

Optimization of the Effective Thermal Conductivity of a Composite 199

In the present paper, a composite material consisting of two materials is analysed. It is a fibrous material with unidirectional fibres. The material of the matrix is homogenous and its thermal conductivity is constant. Fibres are also homogenous, however, they may differ

Composite materials typically consist of stiff and strong material phase, often as fibres, held together by a binder of matrix material, often an organic polymer. Matrix is soft and weak, and its direct load bearing is negligible. In order to achieve particular properties in preferred directions, continuous fibres are usually employed in structures having essentially two

Applying the fundamental definition of thermal conductivity to a unit cell of unidirectional fibre reinforced composite with air voids, one can deduce simple empirical formula to predict the thermal conductivity of the composite material with estimated air void volume percent (Al-Sulaiman et al., 2006). The ability to accurately predict the thermal conductivity of composite has several practical applications. The most basic thermal-conductivity models

���� � ∑ ���� �

���� � �∑ ��

composite thermal conductivity of a unidirectional continuous fibre composite.

consolidated within the relationship given by McCullough (McCullough, 1985).

where *λeff* is the effective thermal conductivity, *λi, Vi* - thermal conductivity and volume

The composite thermal conductivity in the filler direction is estimated by the rule of mixtures. The rule of mixtures is the weighted average of filler and matrix thermal conductivities. This model is typically used to predict the thermal conductivity of a unidirectional composite with continuous fibres. In the direction perpendicular to the fillers (through plane direction), the series model (inverse mixing rule) is used to estimate

Another model similar to the two standard-mixing rule models is the geometric model (Ott,

���� � ∑ ��

Numerous existing relationships are obtained as special cases of above equations. Filler shapes ranging from platelet, particulate, and short-fibre, to continuous fibre are

The effective thermal conductivity for a composite solid depends, however, on the geometry assumed for the problem. In general, to calculate the effective thermal conductivity of fibrous materials, we have to solve the energy transport equations for the temperature and heat flux fields. For a steady pure thermal conduction with no phase change, no convection and no contact thermal resistance, the equations to be solved are a series of Poisson equations subject to temperature and heat flux continuity constraints at the phase interfaces.

� ��

�� � ��� � ��

��� (1)

��� (3)

, (2)

from each other when it comes to radius or thermal conductivity.

(McCullough, 1985) start with the standard mixture rule

fraction of *i-th* composite constituents (e.g. resin, fibre, void).

**2. Fibrous composite material** 

**2.1 Effective thermal conductivity**

dimensional characteristics.

and inverse mixture rule

1981)

empirical formulas to predict the thermal conductivities of fibre reinforced composite laminates and their constituents. In the paper (Boguszewski et al., 2008) the analysis of structure and features of phases composite was considered in order to study heat transfer phenomena. Models of three phases composite matrix, filler and interface with discontinuities were analyzed. Distance between particles was also considered. The paper (Kidalov & Shakhov, 2009) presents results in studying the possibility of developing composites in diamond-containing systems with a view of obtaining materials with a high thermal conductivity. The main objectives of project (Weber, 2001) were to develop a model to predict thermal conductivity of the carbon filled polymer composites and to determine if synergism between the fillers exists. Thermally conductive polymer composites can replace metals in many applications. The article (Zhou&Li, 2008) presents a numerical procedure to design two-phase periodic microstructural composites with tailored thermal conductivities, which is generalized as a topology optimization problem. The objective function is formulated in a least-square of the difference between the target and effective conductivities. Various microstructures both in 2- and 3-dimensions are presented to demonstrate such a systematic procedure of conductive material design. The effective thermal conductivity enhancement of carbon fibre composites was investigated in contribution (Wang et al., 2009) using a threedimensional numerical method. The authors of the paper (Wang&Pan, 2008) have developed a random generation-growth method to reproduce the microstructures of opencell foam materials via computer modelling, and then solve the energy transport equations through the complex structure by using a high-efficiency lattice Boltzmann method. The effective thermal conductivities of open-cell foam materials are thus numerically calculated and the predictions are compared with the existing experimental data. In the paper (Karkri, 2010) thermal properties of composites are investigated numerically and experimentally. In the numerical study, finite elements method is used for modelling heat transfer and to calculate the effective thermal conductivity of the composite for three elementary cells, such as simple cubic, body centered cubic and face centered cubic. The effect of the filler concentrations, the ratio of thermal conductivities of filler to matrix material and the Kapitza resistance of the contact inclusion/matrix on the effective conductivity was investigated. In the paper (Brucker&Majdalani, 2005), several analytical expressions are derived for an effective thermal conductivity. These explicit solutions embody many possible heat pathways and base plate geometries that arise in microelectronic packages. From a physical stand point, the effective thermal conductivity represents a figure-of-merit that assumes an intermediate value greater than that of the coolant, and smaller than that of the metal.

The objective of this contribution is to investigate the effective hybrid numerical method to predict effective thermal conductivity of composite material with fibres distributed in matrix phase. This method is combination of finite element method and genetic algorithm (FEM-GA). FEM-GA was used to find distribution of fibre in composite domain giving maximum, minimum or required value of effective thermal conductivity. The Algorithm is implemented in Comsol Multiphysics environment using Comsol Script language (Comsol, 2007). Comsol solver uses finite element method which today has been widely employed in solving field problems arising in modern industrial practices (Zienkiewicz & Taylor, 2000).

It is assumed that both the matrix and fibres of the considered composite are homogenous, isotropic and their thermal conductivities are constant. The fibres are cylindrical, arranged parallel, continuous with circular cross-section. The fibre diameter is relatively small in comparison to their length, thus fibres can be treated as infinitely long. Fibres can be different in size and thermal properties (thermal conductivity).

#### **2. Fibrous composite material**

198 Convection and Conduction Heat Transfer

empirical formulas to predict the thermal conductivities of fibre reinforced composite laminates and their constituents. In the paper (Boguszewski et al., 2008) the analysis of structure and features of phases composite was considered in order to study heat transfer phenomena. Models of three phases composite matrix, filler and interface with discontinuities were analyzed. Distance between particles was also considered. The paper (Kidalov & Shakhov, 2009) presents results in studying the possibility of developing composites in diamond-containing systems with a view of obtaining materials with a high thermal conductivity. The main objectives of project (Weber, 2001) were to develop a model to predict thermal conductivity of the carbon filled polymer composites and to determine if synergism between the fillers exists. Thermally conductive polymer composites can replace metals in many applications. The article (Zhou&Li, 2008) presents a numerical procedure to design two-phase periodic microstructural composites with tailored thermal conductivities, which is generalized as a topology optimization problem. The objective function is formulated in a least-square of the difference between the target and effective conductivities. Various microstructures both in 2- and 3-dimensions are presented to demonstrate such a systematic procedure of conductive material design. The effective thermal conductivity enhancement of carbon fibre composites was investigated in contribution (Wang et al., 2009) using a threedimensional numerical method. The authors of the paper (Wang&Pan, 2008) have developed a random generation-growth method to reproduce the microstructures of opencell foam materials via computer modelling, and then solve the energy transport equations through the complex structure by using a high-efficiency lattice Boltzmann method. The effective thermal conductivities of open-cell foam materials are thus numerically calculated and the predictions are compared with the existing experimental data. In the paper (Karkri, 2010) thermal properties of composites are investigated numerically and experimentally. In the numerical study, finite elements method is used for modelling heat transfer and to calculate the effective thermal conductivity of the composite for three elementary cells, such as simple cubic, body centered cubic and face centered cubic. The effect of the filler concentrations, the ratio of thermal conductivities of filler to matrix material and the Kapitza resistance of the contact inclusion/matrix on the effective conductivity was investigated. In the paper (Brucker&Majdalani, 2005), several analytical expressions are derived for an effective thermal conductivity. These explicit solutions embody many possible heat pathways and base plate geometries that arise in microelectronic packages. From a physical stand point, the effective thermal conductivity represents a figure-of-merit that assumes an intermediate value greater than that of the coolant, and smaller than that of the metal. The objective of this contribution is to investigate the effective hybrid numerical method to predict effective thermal conductivity of composite material with fibres distributed in matrix phase. This method is combination of finite element method and genetic algorithm (FEM-GA). FEM-GA was used to find distribution of fibre in composite domain giving maximum, minimum or required value of effective thermal conductivity. The Algorithm is implemented in Comsol Multiphysics environment using Comsol Script language (Comsol, 2007). Comsol solver uses finite element method which today has been widely employed in solving field problems arising in modern industrial practices (Zienkiewicz & Taylor, 2000). It is assumed that both the matrix and fibres of the considered composite are homogenous, isotropic and their thermal conductivities are constant. The fibres are cylindrical, arranged parallel, continuous with circular cross-section. The fibre diameter is relatively small in comparison to their length, thus fibres can be treated as infinitely long. Fibres can be

different in size and thermal properties (thermal conductivity).

In the present paper, a composite material consisting of two materials is analysed. It is a fibrous material with unidirectional fibres. The material of the matrix is homogenous and its thermal conductivity is constant. Fibres are also homogenous, however, they may differ from each other when it comes to radius or thermal conductivity.

#### **2.1 Effective thermal conductivity**

Composite materials typically consist of stiff and strong material phase, often as fibres, held together by a binder of matrix material, often an organic polymer. Matrix is soft and weak, and its direct load bearing is negligible. In order to achieve particular properties in preferred directions, continuous fibres are usually employed in structures having essentially two dimensional characteristics.

Applying the fundamental definition of thermal conductivity to a unit cell of unidirectional fibre reinforced composite with air voids, one can deduce simple empirical formula to predict the thermal conductivity of the composite material with estimated air void volume percent (Al-Sulaiman et al., 2006). The ability to accurately predict the thermal conductivity of composite has several practical applications. The most basic thermal-conductivity models (McCullough, 1985) start with the standard mixture rule

$$
\lambda\_{eff} = \Sigma\_{l=1}^n \lambda\_l V\_l \tag{1}
$$

and inverse mixture rule

$$
\lambda\_{eff} = \left(\Sigma\_{l=1}^{n} \frac{\nu\_l}{\lambda\_l}\right)^{-1},
\tag{2}
$$

where *λeff* is the effective thermal conductivity, *λi, Vi* - thermal conductivity and volume fraction of *i-th* composite constituents (e.g. resin, fibre, void).

The composite thermal conductivity in the filler direction is estimated by the rule of mixtures. The rule of mixtures is the weighted average of filler and matrix thermal conductivities. This model is typically used to predict the thermal conductivity of a unidirectional composite with continuous fibres. In the direction perpendicular to the fillers (through plane direction), the series model (inverse mixing rule) is used to estimate composite thermal conductivity of a unidirectional continuous fibre composite.

Another model similar to the two standard-mixing rule models is the geometric model (Ott, 1981)

$$
\lambda\_{eff} = \Sigma\_{l=1}^{n} \lambda\_l^{\nu\_l} \tag{3}
$$

Numerous existing relationships are obtained as special cases of above equations. Filler shapes ranging from platelet, particulate, and short-fibre, to continuous fibre are consolidated within the relationship given by McCullough (McCullough, 1985).

The effective thermal conductivity for a composite solid depends, however, on the geometry assumed for the problem. In general, to calculate the effective thermal conductivity of fibrous materials, we have to solve the energy transport equations for the temperature and heat flux fields. For a steady pure thermal conduction with no phase change, no convection and no contact thermal resistance, the equations to be solved are a series of Poisson equations subject to temperature and heat flux continuity constraints at the phase interfaces.

$$
\lambda\_{eff} = \frac{L \int q dA}{\Delta \Gamma \int d\mathcal{A}'} \tag{4}
$$

$$q = -\lambda \frac{\partial \overline{r}}{\partial n} \tag{5}$$

$$
\nabla^2 \mathbf{T} = \mathbf{0}.\tag{6}
$$

$$\frac{\partial \mathbf{r}\_{\mathbf{M}}}{\partial \mathbf{x}} = \mathbf{0} \quad for \, \mathbf{x} = \mathbf{0} \, and \, \mathbf{x} = \mathbf{1}, \tag{7}$$

$$\mathbf{T\_{C}} = \mathbf{290K} \text{ for } \begin{array}{c} \text{y} = \mathbf{0}, \end{array} \tag{8}$$

$$\text{T}\_{\text{H}} = 300 \text{K} \quad \text{for} \quad \text{y} = 1,\tag{9}$$

$$\mathbf{T\_{M}} = \mathbf{T\_{F}} \cdot for \prime \quad \partial \mathbf{F},\tag{10}$$

$$
\lambda\_{\mathbf{M}} \frac{\partial \mathbf{T\_M}}{\partial \mathbf{n}} = \lambda\_{\mathbf{F}} \frac{\partial \mathbf{T\_F}}{\partial \mathbf{n}} \quad \text{for} \ \partial \mathbf{F}. \tag{11}
$$

$$
\frac{
\partial
}{
\partial x
}
\left(
\lambda \frac{
\partial T
}{
\partial x
}
\right)
+
\frac{
\partial
}{
\partial y
}
\left(
\lambda \frac{
\partial T
}{
\partial y
}
\right)
+
\dot{Q}
=
0,
\tag{12}
$$

$$T(\mathbf{x}, \mathbf{y}) = T\_1 \tag{13}$$

$$\left(\lambda \frac{\partial \boldsymbol{T}}{\partial \boldsymbol{x}} \boldsymbol{n}\_{\boldsymbol{x}}\right) + \left(\lambda \frac{\partial \boldsymbol{T}}{\partial \boldsymbol{y}} \boldsymbol{n}\_{\boldsymbol{y}}\right) = \dot{\boldsymbol{Q}}\_{2} \tag{14}$$

$$
\left(\lambda \frac{\partial \tau}{\partial x} n\_{\chi}\right) + \left(\lambda \frac{\partial \tau}{\partial \chi} n\_{\mathcal{Y}}\right) = \alpha (T\_m - T) \tag{15}
$$

$$T^e(\mathbf{x}, \mathbf{y}) = \Sigma\_{f=1}^M T\_f^e \cdot N\_f^e(\mathbf{x}, \mathbf{y}) = [N] \{ T \} \tag{16}$$

$$\int\_{\Omega^{\varepsilon}} \left( \frac{\partial}{\partial x} \left( \lambda \frac{\partial T^{\varepsilon}}{\partial x} \right) + \frac{\partial}{\partial y} \left( \lambda \frac{\partial T^{\varepsilon}}{\partial y} \right) + \dot{Q} \right) N\_{l}^{\varepsilon} (\mathfrak{x}, \mathfrak{y}) dxd\mathfrak{y} = 0. \tag{17}$$

$$\int\_{\Omega^{\varepsilon}} \left( \frac{\partial}{\partial \mathbf{x}} \left( \lambda \frac{\partial T^{e}}{\partial \mathbf{x}} N\_{l}^{\varepsilon} \right) + \frac{\partial}{\partial \mathbf{y}} \left( \lambda \frac{\partial T^{e}}{\partial \mathbf{y}} N\_{l}^{\varepsilon} \right) \right) d\mathbf{x} d\mathbf{y} = \int\_{\Gamma^{\varepsilon}} \left( \lambda \frac{\partial T^{e}}{\partial \mathbf{x}} d\mathbf{y} - \lambda \frac{\partial T^{e}}{\partial \mathbf{y}} d\mathbf{x} \right) N\_{l}^{\varepsilon} \tag{18}$$

$$\begin{split} \int\_{\Omega^{e}} \left( \frac{\partial}{\partial \mathbf{x}} \left( \lambda \frac{\partial T^{e}}{\partial \mathbf{x}} \right) + \frac{\partial}{\partial \mathbf{y}} \left( \lambda \frac{\partial T^{e}}{\partial \mathbf{y}} \right) \right) N\_{l}^{e} \, d\mathbf{x} dy &= \\ = - \int\_{\Omega^{e}} \left( \lambda \frac{\partial T^{e}}{\partial \mathbf{x}} \frac{\partial N\_{l}^{e}}{\partial \mathbf{x}} + \lambda \frac{\partial T^{e}}{\partial \mathbf{y}} \frac{\partial N\_{l}^{e}}{\partial \mathbf{y}} \right) d\mathbf{x} dy + \int\_{\Gamma^{e}} \left( \lambda \frac{\partial T^{e}}{\partial \mathbf{x}} \, dy - \lambda \frac{\partial T^{e}}{\partial \mathbf{y}} \, d\mathbf{x} \right) N\_{l}^{e} . \end{split} \tag{19}$$

$$Q = \left(\lambda \frac{\partial T}{\partial x} n\_x\right) + \left(\lambda \frac{\partial T}{\partial y} n\_y\right) \tag{20}$$

$$\int\_{\Omega^{\varepsilon}} \left( \left( \lambda \frac{\partial \mathcal{T}^{\varepsilon}}{\partial x} \frac{\partial N\_{l}^{\varepsilon}}{\partial x} \right) + \left( \lambda \frac{\partial \mathcal{T}^{\varepsilon}}{\partial y} \frac{\partial N\_{l}^{\varepsilon}}{\partial y} \right) \right) d\mathbf{x} d\mathbf{y} = \int\_{\Omega^{\varepsilon}} Q N\_{l}^{\varepsilon} d\mathbf{x} d\mathbf{y} + \int\_{\Gamma^{0}} \left( \lambda \frac{\partial \mathcal{T}^{\varepsilon}}{\partial x} d\mathbf{y} - \lambda \frac{\partial \mathcal{T}^{\varepsilon}}{\partial y} d\mathbf{x} \right) N\_{l}^{\varepsilon}. \tag{21}$$

$$\int\_{\Omega^{e}} \left( \left( \lambda \frac{\partial T^{e}}{\partial x} \frac{\partial N\_{l}^{e}}{\partial x} \right) + \left( \lambda \frac{\partial T^{e}}{\partial y} \frac{\partial N\_{l}^{e}}{\partial y} \right) \right) dxdy = \tag{22}$$

$$= \int\_{\Omega^{e}} \dot{Q} N\_{l}^{e} dxdy + \int\_{\Gamma\_{2}^{e}} \dot{Q}\_{2} N\_{l}^{e} ds + \int\_{\Gamma\_{3}^{e}} a \left( T\_{m} - T\_{1} \right) N\_{l}^{e} ds,$$

$$Qds = \left(\lambda \frac{\partial T}{\partial x} n\_x ds\right) + \left(\lambda \frac{\partial T}{\partial y} n\_y ds\right) \tag{23}$$

$$\begin{split} \int\_{\mathcal{B}^{\varepsilon}} \left( \left( \lambda \frac{\partial N\_{l}^{\varepsilon}}{\partial x} \sum\_{j=1}^{M} \left( T\_{j}^{\varepsilon} \frac{\partial N\_{j}^{\varepsilon}}{\partial x} \right) \right) + \left( \lambda \frac{\partial N\_{l}^{\varepsilon}}{\partial y} \sum\_{j=1}^{M} \left( T\_{j}^{\varepsilon} \frac{\partial N\_{j}^{\varepsilon}}{\partial y} \right) \right) \right) dx dy = \\ = \int\_{\mathcal{B}^{\varepsilon}} \mathcal{Q} N\_{l}^{\varepsilon} dx dy + \int\_{I\_{2}^{\varepsilon}} \mathcal{Q}\_{2} N\_{l}^{\varepsilon} ds - \int\_{I\_{3}^{\varepsilon}} a \sum\_{j=1}^{M} \left( T\_{j}^{\varepsilon} N\_{l}^{\varepsilon} \right) N\_{l}^{\varepsilon} ds + \int\_{I\_{3}^{\varepsilon}} a T\_{m} N\_{l}^{\varepsilon} ds. \end{split} \tag{24}$$

$$
\mathbf{K}\mathbf{a} = f \tag{25}
$$

$$K = K\_c^e + K\_{\Gamma\_v}^e \tag{26}$$

$$f = f\_q^e + f\_{\Gamma\_2}^e + f\_{\Gamma\_3}^e \tag{27}$$

$$K\_{c,lj}^{e} = \int\_{\Omega^e} \left( \left( \lambda \frac{\partial N\_l^e}{\partial x} \frac{\partial N\_f^e}{\partial x} \right) + \left( \lambda \frac{\partial N\_l^e}{\partial y} \frac{\partial N\_f^e}{\partial y} \right) \right) d\mathbf{x} d\mathbf{y}, \tag{28}$$

$$K\_{\Gamma\_3,lj}^{\varepsilon} = \int\_{\Gamma\_3^{\varepsilon}} a N\_l^{\varepsilon} N\_l^{\varepsilon} \,\mathrm{d}s,\tag{29}$$

$$f\_{q,l}^{e} = \int\_{\Omega^e} \dot{Q} N\_l^e dx dy,\tag{30}$$

$$f\_{\Gamma\_2,l}^e = \int\_{\Gamma\_2^e} \dot{Q}\_2 N\_l^e ds,\tag{31}$$

$$f\_{\Gamma\_3,l}^{\mathfrak{e}} = \int\_{\Gamma\_\tau^{\mathfrak{e}}} a T\_m \, N\_l^{\mathfrak{e}} \, \text{ds} \tag{32}$$

$$\mathbf{f}(\hat{\mathbf{x}}) \ge \mathbf{f}(\mathbf{x}), \mathbf{x} \in \mathbf{D}, \tag{33}$$

Optimization of the Effective Thermal Conductivity of a Composite 207

Modification of the roulette wheel selection that we introduced is caused by the fact that we needed to perform constrained optimization. The constrains are the result of the fact that fibres cannot overlap with each other. There are some possible options to handle this problem, one of which would to use penalty function. During calculations, however, it turned out that this approach is less effective than the other one based on elitist selection. We decided that in case of chromosome representing arrangement of overlapping fibres

Cross-over operation requires two chromosomes (parents) which are cut in one, randomly chosen point (locus) and since this point the binary code is swapped between the

Mutation procedure in case of binary representation of solution is an operation of bit inversion at randomly chosen position Fig6. The following purpose of this procedure is to introduce some diversity into population and so to avoid premature convergence to

All optimization problems considered in this chapter are governed by Eq. 6 for each constituent of the composite with appropriate boundary conditions (7-11). In our calculations we assumed the same sizes of the unit cell i.e. 1x1cm ( Fig1.). Temperatures on the lower and upper boundaries were: TC=290K (upper), TH=300K (lower) respectively. We analysed several cases in which the number of fibres Nf and fibres radii R were changed, also thermal conductivity of the matrix λM and fibres λF were also changed. Finite element calculation were made using second order triangular Lagrange elements. The stationary problem of heat transfer was solved using direct UMFPACK linear system solver. The mesh structure depends on the number and positions of fibres and so the number of mesh

We performed three types of optimization in terms of effective thermal conductivity: minimization, maximization and determination of arrangement which gives desired value of effective thermal conductivity. In the latter case we defined the objective function as the minimization of the deviation from the expected value. The results of optimization are

such chromosome should be replaced with the best one.

chromosomes creating two, new chromosomes, as it is shown at Fig. 5.

**3.2.3 Genetic operators** 

Fig. 5. Crossover procedure scheme

Fig. 6. Mutation procedure scheme

elements was not larger than 5000.

presented at Figs 7-9.

**4. Numerical results** 

local maximum.

Fig. 4. Genetic algorithm scheme

to optimize *2N* variables ��. Furthermore, it is assumed that each coordinate is determined with finite precision �� and limited to a certain range �� � ���� ��� - a, b denoting the lower and upper limit of the range respectively. It means that each domain �� needs to be divided into ��� � ������� sub-domains. Hence we can calculate �� – number of bits required to encode variables:

$$(\mathbf{b}\_{\mathbf{l}} - \mathbf{a}\_{\mathbf{l}})10^{\mathbf{p}\_{\mathbf{l}}} \le 2^{\mathbf{h}\_{\mathbf{l}}} - 1. \tag{34}$$

Consequently, we can calculate the number of bits � required to encode a chromosome:

$$\mathbf{H} = \sum\_{\mathbf{l}=\mathbf{1}}^{2\mathbf{N}} \mathbf{h}\_{\mathbf{l}} \tag{35}$$

In our calculation we assume three significant digits precision which means we need 2�� bits to encode each variable.

#### **3.2.2 Fitness and selection**

Selection is a procedure in which parents for the new generation are chosen using the fitness function. There are many procedures possible to select chromosomes which will create another population. The most common are: roulette wheel selection, tournament selection, rank selection, elitists selection.

In our case, modified fitness proportionate selection also called roulette wheel selection is used. Based on values assigned to each solution by fitness function������, the probability ����� of being selected is calculated for every individual chromosome. Consequently, the candidate solution whose fitness is low will be less likely selected as a parent whereas it is more probable for candidates with higher fitness to become a parent. The probability of selection is determined as follows:

$$P(\mathbf{x}\_l) = \frac{\mathbf{f}(\mathbf{x}\_l)}{\sum\_{k=1}^{\mathbf{f}} \mathbf{f}(\mathbf{x}\_k)} \tag{36}$$

where *S* is the number of chromosomes in population.

Modification of the roulette wheel selection that we introduced is caused by the fact that we needed to perform constrained optimization. The constrains are the result of the fact that fibres cannot overlap with each other. There are some possible options to handle this problem, one of which would to use penalty function. During calculations, however, it turned out that this approach is less effective than the other one based on elitist selection. We decided that in case of chromosome representing arrangement of overlapping fibres such chromosome should be replaced with the best one.

#### **3.2.3 Genetic operators**

206 Convection and Conduction Heat Transfer

to optimize *2N* variables ��. Furthermore, it is assumed that each coordinate is determined with finite precision �� and limited to a certain range �� � ���� ��� - a, b denoting the lower and upper limit of the range respectively. It means that each domain �� needs to be divided into ��� � ������� sub-domains. Hence we can calculate �� – number of bits required to

Consequently, we can calculate the number of bits � required to encode a chromosome:

� � ∑ h� ��

In our calculation we assume three significant digits precision which means we need 2�� bits

Selection is a procedure in which parents for the new generation are chosen using the fitness function. There are many procedures possible to select chromosomes which will create another population. The most common are: roulette wheel selection, tournament selection,

In our case, modified fitness proportionate selection also called roulette wheel selection is used. Based on values assigned to each solution by fitness function������, the probability ����� of being selected is calculated for every individual chromosome. Consequently, the candidate solution whose fitness is low will be less likely selected as a parent whereas it is more probable for candidates with higher fitness to become a parent. The probability of

P�x�� � �����

∑ ����� <sup>S</sup> ���

��� � ������� � 2�� � �*.* (34)

��� (35)

(36)

Fig. 4. Genetic algorithm scheme

encode variables:

to encode each variable.

**3.2.2 Fitness and selection** 

rank selection, elitists selection.

selection is determined as follows:

where *S* is the number of chromosomes in population.

Cross-over operation requires two chromosomes (parents) which are cut in one, randomly chosen point (locus) and since this point the binary code is swapped between the chromosomes creating two, new chromosomes, as it is shown at Fig. 5.

Mutation procedure in case of binary representation of solution is an operation of bit inversion at randomly chosen position Fig6. The following purpose of this procedure is to introduce some diversity into population and so to avoid premature convergence to local maximum.

$$\{\begin{smallmatrix} \textsf{110} \textsf{|01001\rangle} \\ \textsf{(101)} \textsf{|11101\rangle} \end{smallmatrix} \} \xrightarrow[\mbox{\bf{\bf{0}} \textsf{0} \textsf{\bf \bf{1}} \textsf{0} \textsf{1} \textsf{0} \textsf{1} \textsf{0} \textsf{1} \textsf{0} \textsf{1} \textsf{0} \textsf{1} \textsf{0} \textsf{1} \textsf{0} \textsf{1} \textsf{0} \textsf{1} \textsf{0} \textsf{1} \textsf{0} \textsf{1} \textsf{0} \textsf{1} \textsf{0} \textsf{1} \textsf{0} \textsf{1} \textsf{0} \textsf{1} \textsf{0} \textsf{1} \textsf{0} \textsf{1} \textsf{0} \textsf{1} \textsf{0} \textsf{1} \textsf{0} \textsf{1} \textsf{0} \textsf{1} \textsf{0} \textsf{1} \textsf{0} \textsf{1} \textsf{0} \textsf{1} \textsf{0} \textsf{1} \textsf{0} \textsf{1} \textsf{0} \textsf{1} \textsf{0} \textsf{1} \textsf{0} \textsf{1} \textsf{0} \textsf{1} \textsf{0} \textsf{1} \textsf{0} \textsf{1} \textsf{0} \textsf{1} \textsf{0} \textsf{1} \textsf{0} \textsf{1} \textsf{0} \textsf{1} \textsf{0} \textsf{1} \textsf{0} \textsf{1} \textsf{0} \textsf{1} \textsf{0} \textsf{0} \textsf{1} \textsf{0} \textsf{1} \textsf{0} \textsf{1} \textsf{0} \textsf{1} \textsf{0} \textsf{0} \textsf{$$

Fig. 5. Crossover procedure scheme

$$\left(\text{1\u000001}\right) \xrightarrow[\text{locus=5}]{\text{mutation}} \left(\text{1\u000001}\right)$$

Fig. 6. Mutation procedure scheme

#### **4. Numerical results**

All optimization problems considered in this chapter are governed by Eq. 6 for each constituent of the composite with appropriate boundary conditions (7-11). In our calculations we assumed the same sizes of the unit cell i.e. 1x1cm ( Fig1.). Temperatures on the lower and upper boundaries were: TC=290K (upper), TH=300K (lower) respectively. We analysed several cases in which the number of fibres Nf and fibres radii R were changed, also thermal conductivity of the matrix λM and fibres λF were also changed. Finite element calculation were made using second order triangular Lagrange elements. The stationary problem of heat transfer was solved using direct UMFPACK linear system solver. The mesh structure depends on the number and positions of fibres and so the number of mesh elements was not larger than 5000.

We performed three types of optimization in terms of effective thermal conductivity: minimization, maximization and determination of arrangement which gives desired value of effective thermal conductivity. In the latter case we defined the objective function as the minimization of the deviation from the expected value. The results of optimization are presented at Figs 7-9.

Optimization of the Effective Thermal Conductivity of a Composite 209

A B

C D

E F

Fig. 8. Resultant arrangement for five and six fibres

Fig. 7. Resultant arrangement for three and four fibres

A B

C D

E F

Fig. 7. Resultant arrangement for three and four fibres

Fig. 8. Resultant arrangement for five and six fibres

Optimization of the Effective Thermal Conductivity of a Composite 211

thermal conductivity at Figs. 8C, 8D. One can notice that the arrangement of fibres is also close to horizontal in case of minimization and close to vertical in case of maximization, although fibres are not localised next to each other and initialization of the second row in case of six fibres can be observed. In general, however, we may not assume that fibres are always aligned in rows in case of minimum and maximum values of effective thermal conductivity. The situation changes when the thermal conductivity of fibres is not the same

 NF R λ<sup>F</sup> λM Opt. **λeff** Fig 8A 5 0.1 0.1 2.0 Min **1,0**  Fig 8B 5 0.1 0.1 2.0 Max **1,61**  Fig 8C 5 0.1 0.1 2.0 1.5 **1.5**  Fig 8D 6 0.1 2.0 0.1 0.15 **0.15**  Fig 8E 6 0.1 2.0 0.1 Min **0.13**  Fig 8F 6 0.1 2.0 0.1 Max **0.19**  Table 2. The values assigned for calculations and the resultant λeff for five and six fibres

**4.3 Optimization of four and five fibres arrangement with different radii and thermal** 

Apart from the simplest case in which the composite consisted of identical fibres we also analysed the case in which fibres differ from each other. We used two sizes of fibres with different values of thermal conductivities. All parameters used in calculations were presented in Table 3. The symbol NR denotes the number of fibres having the same dimension and

> NF NR R λ<sup>F</sup> λM Opt. **λeff** Fig 9A 4 2 0.12 0.1 2.0 Min **1.68** 2 0.15 10

> Fig 9B 4 2 0.12 0.1 2.0 Max **2.39** 2 0.15 10

Fig 9C 4 2 0.12 0.1 2.0 2.0 **2,0** 2 0.15 10

Fig 9D 5 4 0.075 0.1 0.1 1.85 **1.85** 1 0.15 10

Fig 9E 5 4 0.075 0.1 0.1 Min **1.65** 1 0.15 10

Fig 9F 5 4 0.075 20.1 0.1 Max **2.08** 1 0.15 10

Table 3. The values assigned for calculations and the resultant λeff for four and five fibres of

We performed the optimization of the arrangement of four and five fibres in a composite cell. The minimization results were presented at Figs 9A, 9E while maximization at Figs 9B, 9F. The arrangements obtained for the assumed values of effective thermal conductivity for four and five fibres were presented at Figs 9C,9D respectively. It is remarkable, that in these

**conductivity of fibres** 

different radii and thermal conductivities

properties.

in each fibre. The result for such situation was presented in the next section.

#### **4.1 Optimization of three and four fibres arrangement**

In the beginning we assumed the same sizes of the fibres, as well as the same value of thermal conductivity for each fibre. Numerical values of parameters used in calculations, and the resultant effective thermal conductivity was shown in Table 1. The 'Opt.' column refers to optimization criteria i.e. minimum, maximum or expected value of λeff The column entitled **λeff** contains obtained results. Not surprisingly did minimization and maximization results agree with results presented in section 2.3. Figures 7A and 7E present the arrangement obtained during minimization. All fibres are aligned horizontally perpendicularly to heat flux direction, next to each other. In case of maximization (Figs 7B, 7F) fibres are aligned vertically – along with heat flux direction.

 However, there are many possible ways of arrangement of intermediate values of effective thermal conductivity – fibres do not have to be aligned anymore as it was assumed at Fig 2C. We also presented one of possible arrangements that result in a composite with effective thermal conductivity equal to the one expected for each number of fibres: (Figs 7C, 7D). If one would like to achieve certain value of effective thermal conductivity with respect to some geometrical assumptions (for instance minimum/maximum distance between fibres) it is also possible to perform such optimization, however penalty function should be implemented or objective function modified to include such conditions.


Table 1. The values assigned for calculations and the resultant λeff for three and four fibres

#### **4.2 Optimization of five and six fibres arrangement**

Calculation performed for five and six fibres were similar to those presented above for three and four fibres. However, the more fibres the more complex problem. As it was mentioned in section 3.2.1 each fibre is described by two variables changing within the range [0,1] with the 10-3 precision which means 210 bits. Consequently, by adding one fibre we enlarge the search space by 220 elements. So, the search space dimension for three fibres arrangement optimization equals 260, while for six fibres it equals 2120. The size of search space has a direct impact on calculation time and so it takes far more time to find optimal solution.

The terminating condition of GA was set to 2000 iterations for three and four fibres. It resulted in almost perfect arrangement in case of three fibres whereas the arrangement for four fibres was not equally well. While increasing the number of fibres to five and six fibres, we also increased the number of iteration to 10000.

Another important aspect of the considered problem was that in case of five and six fibres of assumed radii (Table 2) it was not possible to align them in one row so the relation presented in section 2.3 could not be applied anymore.

The minimization results for five and six fibres were presented at Figs 8A and 8E, the maximization results at Figs 8B and 8F and the arrangement for expected value of effective

In the beginning we assumed the same sizes of the fibres, as well as the same value of thermal conductivity for each fibre. Numerical values of parameters used in calculations, and the resultant effective thermal conductivity was shown in Table 1. The 'Opt.' column refers to optimization criteria i.e. minimum, maximum or expected value of λeff The column entitled **λeff** contains obtained results. Not surprisingly did minimization and maximization results agree with results presented in section 2.3. Figures 7A and 7E present the arrangement obtained during minimization. All fibres are aligned horizontally perpendicularly to heat flux direction, next to each other. In case of maximization (Figs 7B,

 However, there are many possible ways of arrangement of intermediate values of effective thermal conductivity – fibres do not have to be aligned anymore as it was assumed at Fig 2C. We also presented one of possible arrangements that result in a composite with effective thermal conductivity equal to the one expected for each number of fibres: (Figs 7C, 7D). If one would like to achieve certain value of effective thermal conductivity with respect to some geometrical assumptions (for instance minimum/maximum distance between fibres) it is also possible to perform such optimization, however penalty function should be

Figure's number NF R λ<sup>F</sup> λM Opt. **λeff** Fig 7A 3 0.15 2.0 0.1 Min **0.13**  Fig 7B 3 0.15 2.0 0.1 Max **0.23**  Fig 7C 3 0.15 2.0 0.1 0.15 **0.15**  Fig 7D 4 0.12 0.1 2.0 1.35 **1.35**  Fig 7E 4 0.12 0.1 2.0 Min **1.1**  Fig 7F 4 0.12 0.1 2.0 Max **1.56**  Table 1. The values assigned for calculations and the resultant λeff for three and four fibres

Calculation performed for five and six fibres were similar to those presented above for three and four fibres. However, the more fibres the more complex problem. As it was mentioned in section 3.2.1 each fibre is described by two variables changing within the range [0,1] with the 10-3 precision which means 210 bits. Consequently, by adding one fibre we enlarge the search space by 220 elements. So, the search space dimension for three fibres arrangement optimization equals 260, while for six fibres it equals 2120. The size of search space has a direct impact on calculation time and so it takes far more time to find optimal solution. The terminating condition of GA was set to 2000 iterations for three and four fibres. It resulted in almost perfect arrangement in case of three fibres whereas the arrangement for four fibres was not equally well. While increasing the number of fibres to five and six fibres,

Another important aspect of the considered problem was that in case of five and six fibres of assumed radii (Table 2) it was not possible to align them in one row so the relation

The minimization results for five and six fibres were presented at Figs 8A and 8E, the maximization results at Figs 8B and 8F and the arrangement for expected value of effective

**4.1 Optimization of three and four fibres arrangement** 

7F) fibres are aligned vertically – along with heat flux direction.

implemented or objective function modified to include such conditions.

**4.2 Optimization of five and six fibres arrangement** 

we also increased the number of iteration to 10000.

presented in section 2.3 could not be applied anymore.

thermal conductivity at Figs. 8C, 8D. One can notice that the arrangement of fibres is also close to horizontal in case of minimization and close to vertical in case of maximization, although fibres are not localised next to each other and initialization of the second row in case of six fibres can be observed. In general, however, we may not assume that fibres are always aligned in rows in case of minimum and maximum values of effective thermal conductivity. The situation changes when the thermal conductivity of fibres is not the same in each fibre. The result for such situation was presented in the next section.


Table 2. The values assigned for calculations and the resultant λeff for five and six fibres

#### **4.3 Optimization of four and five fibres arrangement with different radii and thermal conductivity of fibres**

Apart from the simplest case in which the composite consisted of identical fibres we also analysed the case in which fibres differ from each other. We used two sizes of fibres with different values of thermal conductivities. All parameters used in calculations were presented in Table 3. The symbol NR denotes the number of fibres having the same dimension and properties.



We performed the optimization of the arrangement of four and five fibres in a composite cell. The minimization results were presented at Figs 9A, 9E while maximization at Figs 9B, 9F. The arrangements obtained for the assumed values of effective thermal conductivity for four and five fibres were presented at Figs 9C,9D respectively. It is remarkable, that in these

Optimization of the Effective Thermal Conductivity of a Composite 213

cases the optimal arrangement of fibres is no longer that predictable. Fibres are not aligned in a row, although there was enough space. However, fibres still tend to be close to each

This study has examined the effect of multi fibres filler in composite on thermal conductivity. Three types of optimization were performed in terms of effective thermal conductivity: minimization, maximization and determination of arrangement which gives expected value of effective thermal conductivity. Hybrid method combining optimization with genetic algorithm and differential equation solver by finite element method were used to find optimal arrangement of fibres position in composite matrix was used in this work.

It was proved that the geometrical structure of the composite (matrix and filler arrangement) may have a great impact on the resultant effective conductivity of the composite. In many research works it is assumed that fibres are arranged in various

Through this study, some areas were found that need to be investigated further. Composite constituents can be anisotropic, and with temperature dependent thermal conductivity of

Al-Sulaiman FA, Mokheimer EM, Al-Nassar YN (2006). Prediction of the thermal

Al-Sulaiman FA, Al-Nassar YN and Mokheimer EM (2006). Composite Laminates: Voids

Boguszewski T., Ciupiński Ł., Kurzydłowski K.J. (2008). Numerical Calculation of The

Brucker Kyle A., Majdalani Joseph (2005). Effective thermal conductivity of common geometric shapes, *International Journal of Heat and Mass Transfer*, 48, pp.4779–4796. Comsol Multiphysics User's Guide (2007). *Modeling Guide and Model Library*, Documentation

Duc N.D., Boi L.V., Dac N.T. (2008). Determining thermal expansion coefficients of three-

Freger G.E., Kestelman V.N., Freger D.G. (2004). *Spirally Anisotropic Composites*, Springer-

Holland J. H. (1975). *Adaptation in natural and artificial systems*, The University Michigan

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phase fiber composite material reinforced by spherical particles, VNU Journal of

Proposed algorithm was implemented in Comsol Multiphysics environment.

geometrical arrays or they are distributed randomly in the cross-section.

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other but spatial configuration is changed.

constituents (e.g. resin, fibre, void).

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**5. Conclusion** 

**6. References** 

Fig. 9. Resultant arrangements for fibres of different sizes and thermal conductivities

cases the optimal arrangement of fibres is no longer that predictable. Fibres are not aligned in a row, although there was enough space. However, fibres still tend to be close to each other but spatial configuration is changed.

#### **5. Conclusion**

212 Convection and Conduction Heat Transfer

A B

C D

E F

Fig. 9. Resultant arrangements for fibres of different sizes and thermal conductivities

This study has examined the effect of multi fibres filler in composite on thermal conductivity. Three types of optimization were performed in terms of effective thermal conductivity: minimization, maximization and determination of arrangement which gives expected value of effective thermal conductivity. Hybrid method combining optimization with genetic algorithm and differential equation solver by finite element method were used to find optimal arrangement of fibres position in composite matrix was used in this work. Proposed algorithm was implemented in Comsol Multiphysics environment.

It was proved that the geometrical structure of the composite (matrix and filler arrangement) may have a great impact on the resultant effective conductivity of the composite. In many research works it is assumed that fibres are arranged in various geometrical arrays or they are distributed randomly in the cross-section.

Through this study, some areas were found that need to be investigated further. Composite constituents can be anisotropic, and with temperature dependent thermal conductivity of constituents (e.g. resin, fibre, void).

#### **6. References**


**10** 

**Computation of Thermal Conductivity of** 

**Gas Diffusion Layers of PEM Fuel Cells** 

While fuel cells in general are expected to play a major role in the future energy supply, proton exchange membrane (PEM) fuel cells are considered especially interesting for automotive applications due to their relatively low operating temperature which allows for fast start-up and flexibility in power output. Other promising applications of PEM fuel cells are back-up power units, small portable power supplies, micro combined heat and power

> Proton conducting membrane

H2 O2 H O2

Fig. 1 illustrates the principle of a PEM fuel cell. At the anode (left hand side) protons are produced from hydrogen and have to move through the proton-conducting (but not electron-conducting) membrane to the cathode side (right hand side). Electrons will be transported via the electrical load outside the fuel cell to the cathode side where water is

Fig. 1. Sketch of a PEM fuel cell (not to scale). A PEM fuel cell contains two gas diffusion

Cathode catalyst layer

 Gas diffusion layer

installations, but also large scale stationary PEM fuel cell plants.

 Gas diffusion layer

layers, one on the anode and one on the cathode side

 Anode catalyst layer

**1. Introduction** 

produced as 'waste'.

Andreas Pfrang, Damien Veyret and Georgios Tsotridis *European Commission, Joint Research Centre, Institute for Energy* 

*P.O. Box 2, NL-1755 ZG Petten,* 

*The Netherlands* 


### **Computation of Thermal Conductivity of Gas Diffusion Layers of PEM Fuel Cells**

Andreas Pfrang, Damien Veyret and Georgios Tsotridis *European Commission, Joint Research Centre, Institute for Energy P.O. Box 2, NL-1755 ZG Petten, The Netherlands* 

#### **1. Introduction**

214 Convection and Conduction Heat Transfer

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Wang M., Pan N. (2008). Modeling and prediction of the effective thermal conductivity of

Wang M., Kang Q., Pan N. (2009). Thermal conductivity enhancement of carbon fiber

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Zhou S., Qing Li (2008). Computational Design of Microstructural Composites with Tailored Thermal Conductivity, *Numerical Heat Transfer*, Part A, 54, pp.686–708. Zienkiewicz O.C., Taylor R.L. (2000). *The Finite Element Method*, Vol. 1-3: The Basis, Solid Mechanics, Fluid Dynamics (5th ed.), Butterworth-Heinemann, Oxford.

random open-cell porous foams, *International Journal of Heat and Mass Transfer*, 51,

study, *Proceedings of the COMSOL Conference 2010*, Paris.

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*Selection*, MIT Press, Cambridge.

Elsevier.

pp. 1325–1331.

*and Applications*, Vol. 1, 1981, pp. 9-24.

While fuel cells in general are expected to play a major role in the future energy supply, proton exchange membrane (PEM) fuel cells are considered especially interesting for automotive applications due to their relatively low operating temperature which allows for fast start-up and flexibility in power output. Other promising applications of PEM fuel cells are back-up power units, small portable power supplies, micro combined heat and power installations, but also large scale stationary PEM fuel cell plants.

Fig. 1. Sketch of a PEM fuel cell (not to scale). A PEM fuel cell contains two gas diffusion layers, one on the anode and one on the cathode side

Fig. 1 illustrates the principle of a PEM fuel cell. At the anode (left hand side) protons are produced from hydrogen and have to move through the proton-conducting (but not electron-conducting) membrane to the cathode side (right hand side). Electrons will be transported via the electrical load outside the fuel cell to the cathode side where water is produced as 'waste'.

Computation of Thermal Conductivity of Gas Diffusion Layers of PEM Fuel Cells 217

dimensional X-ray images taken around a single axis of rotation (see Fig. 2). The 3D image of the object consists of voxels with a certain gray value. Each voxel is then assigned to one material that is present in the object e.g. by considering its gray value. This assignment is

Fig. 2. Principle of X-ray computed tomography (CT). A carbon cloth is shown as sample

X-ray computed tomography (Ostadi et al., 2008; Pfrang et al., 2010) as well as synchrotron based tomography (Becker et al., 2008; Becker et al., 2009) have been used for imaging of gas

Sample

Step by step rotation

2D detector

Also membranes and membrane electrode assemblies (Garzon et al., 2007; Pfrang et al., 2011) have been imaged by X-ray computed tomography and even functioning fuel cells have been imaged by synchrotron-based methods and soft X-ray radiography e.g. for imaging of liquid water in the GDL (Sinha et al., 2006; Bazylak, 2009; Sasabe et al., 2010;

**Gas diffusion layer PTFE / wt% Thickness / mm Porosity**  E-Tek, EC-CC1-060T 30 0.33 0.75 E-Tek, EC-TP1-060T 30 0.19 0.72

(with microporous layer) 5 0.325 0.80

Table 1. Properties of gas diffusion layers investigated by X-ray computed tomography

Here, CT data from three different commercially available gas diffusion layers will be discussed which were imaged by a nanotom X-ray computed tomography system (GE Sensing and Inspection Technologies, phoenix X-ray, Wunstorf, Germany) at a resolution below 1 µm. Table 1 shows PTFE content, thickness and porosity of the investigated gas diffusion layers. Segmentation into solid material (i.e. carbon and PTFE) and air was carried out based on a gray level threshold. Further details can be found in (Pfrang et al., 2010).

The second approach is the random generation of three-dimensional fiber structures using the FiberGeo module of the Geodict software package (Fraunhofer ITWM, 2011). Geometric

(Toray Industries, 2005b; SGL Group, 2009; Pfrang et al., 2010)

**2.1.2 Random generation of 3D structures** 

denoted as 'segmentation'.

diffusion layers at resolutions below 1 µm.

X-ray source

Tsushima & Hirai, 2011).

SGL Carbon, Sigracet 35 BC

The two gas diffusion layers (GDL) have multiple functions in a PEM fuel cell: provide gas access to the catalyst layers, allow removal of product water on the cathode side while also keeping the membrane and electrode layers humidified when gas conditions are subsaturated, mechanically stabilize the membrane-electrode assembly while compensating for thickness variations of the membrane, and providing electrical and thermal conductivity.

A GDL has typically a thickness of 200 to 400 μm and consists of carbon fiber papers or carbon fiber felts which are impregnated with polytetrafluoroethylene (PTFE) to achieve a partial hydrophobization of the surfaces (Mathias et al., 2003). Carbon binder can be added for a mechanical joining of neighbouring fibers. Furthermore, a microporous layer (MPL, typical pore sizes around 100 nm) consisting of a mixture of carbon black and PTFE is often applied with a thickness of a few 10 µm on the side facing the catalyst layer for a further optimization of the water management (Paganin et al., 1996; Giorgi et al., 1998; Mathias et al., 2003).

An operating PEM fuel cell is not isothermal, mainly because heat is generated within the membrane electrode assembly and at the same time this assembly can be considered 'insulated' by the gas diffusion layers (Burheim et al., 2011) leading to temperature gradients within the fuel cell. A detailed knowledge of the temperature distribution and therefore of thermal conductivity of the GDLs is essential for a proper understanding and the optimization of not only heat transfer in the PEM fuel cell, but also for water management and optimization of cell performance and durability. A direct measurement of thermal conductivity is possible and has been performed mainly for the through–plane direction (see Table 2). Nevertheless, the direct measurement is for several reasons nontrivial: due to the anisotropy of the GDLs, the thermal conductivity is expected to be anisotropic as well. Furthermore, the through-plane thermal conductivity as well as the contact resistance change with a compression of the GDL (Burheim et al., 2010; Sadeghi et al., 2011a). Recently, the measurement of in-plane thermal conductivities has been reported (Sadeghi et al., 2011b; Teertstra et al., 2011).

Alternatively, the anisotropic thermal conductivity of gas diffusion layers can be calculated based on the 3D microstructure of the GDL and the knowledge of thermal conductivity of the different materials which are present in the GDL. This approach is presented in the following using X-ray computed tomography structure data of gas diffusion layers as well as randomly computer-generated 3D structures based on structural models of gas diffusion layers.

### **2. Materials and methods**

#### **2.1 3D structure of gas diffusion layers**

The computation of anisotropic thermal conductivity requires the knowledge of the 3D structure of the gas diffusion layer, i.e. also the 3D distribution of the different materials that are present in the gas diffusion layer. This is especially important as the thermal conductivities of these different materials differ considerably: air 0.026 W m-1 K-1 (Taine & Petit, 1989) , PTFE 0.25 W m-1 K-1 (Marotta & Fletcher, 1996) and a typical value for PANbased carbon fibers with relatively high strength and at the same time relatively high modulus is 120 W m-1 K-1 (Toray Industries, 2005a).

#### **2.1.1 Characterization of 3D structures by X-ray computed tomography**

The first approach presented here is the application of X-ray computed tomography (CT) where a 3D image of an object is determined by digital processing of a large series of two-

The two gas diffusion layers (GDL) have multiple functions in a PEM fuel cell: provide gas access to the catalyst layers, allow removal of product water on the cathode side while also keeping the membrane and electrode layers humidified when gas conditions are subsaturated, mechanically stabilize the membrane-electrode assembly while compensating for thickness variations of the membrane, and providing electrical and thermal conductivity. A GDL has typically a thickness of 200 to 400 μm and consists of carbon fiber papers or carbon fiber felts which are impregnated with polytetrafluoroethylene (PTFE) to achieve a partial hydrophobization of the surfaces (Mathias et al., 2003). Carbon binder can be added for a mechanical joining of neighbouring fibers. Furthermore, a microporous layer (MPL, typical pore sizes around 100 nm) consisting of a mixture of carbon black and PTFE is often applied with a thickness of a few 10 µm on the side facing the catalyst layer for a further optimization of the water management (Paganin et al., 1996; Giorgi et al., 1998; Mathias et

An operating PEM fuel cell is not isothermal, mainly because heat is generated within the membrane electrode assembly and at the same time this assembly can be considered 'insulated' by the gas diffusion layers (Burheim et al., 2011) leading to temperature gradients within the fuel cell. A detailed knowledge of the temperature distribution and therefore of thermal conductivity of the GDLs is essential for a proper understanding and the optimization of not only heat transfer in the PEM fuel cell, but also for water management and optimization of cell performance and durability. A direct measurement of thermal conductivity is possible and has been performed mainly for the through–plane direction (see Table 2). Nevertheless, the direct measurement is for several reasons nontrivial: due to the anisotropy of the GDLs, the thermal conductivity is expected to be anisotropic as well. Furthermore, the through-plane thermal conductivity as well as the contact resistance change with a compression of the GDL (Burheim et al., 2010; Sadeghi et al., 2011a). Recently, the measurement of in-plane thermal conductivities has been reported

Alternatively, the anisotropic thermal conductivity of gas diffusion layers can be calculated based on the 3D microstructure of the GDL and the knowledge of thermal conductivity of the different materials which are present in the GDL. This approach is presented in the following using X-ray computed tomography structure data of gas diffusion layers as well as randomly computer-generated 3D structures based on structural models of gas diffusion

The computation of anisotropic thermal conductivity requires the knowledge of the 3D structure of the gas diffusion layer, i.e. also the 3D distribution of the different materials that are present in the gas diffusion layer. This is especially important as the thermal conductivities of these different materials differ considerably: air 0.026 W m-1 K-1 (Taine & Petit, 1989) , PTFE 0.25 W m-1 K-1 (Marotta & Fletcher, 1996) and a typical value for PANbased carbon fibers with relatively high strength and at the same time relatively high

The first approach presented here is the application of X-ray computed tomography (CT) where a 3D image of an object is determined by digital processing of a large series of two-

**2.1.1 Characterization of 3D structures by X-ray computed tomography** 

al., 2003).

layers.

(Sadeghi et al., 2011b; Teertstra et al., 2011).

**2.1 3D structure of gas diffusion layers** 

modulus is 120 W m-1 K-1 (Toray Industries, 2005a).

**2. Materials and methods** 

dimensional X-ray images taken around a single axis of rotation (see Fig. 2). The 3D image of the object consists of voxels with a certain gray value. Each voxel is then assigned to one material that is present in the object e.g. by considering its gray value. This assignment is denoted as 'segmentation'.

Fig. 2. Principle of X-ray computed tomography (CT). A carbon cloth is shown as sample

X-ray computed tomography (Ostadi et al., 2008; Pfrang et al., 2010) as well as synchrotron based tomography (Becker et al., 2008; Becker et al., 2009) have been used for imaging of gas diffusion layers at resolutions below 1 µm.

Also membranes and membrane electrode assemblies (Garzon et al., 2007; Pfrang et al., 2011) have been imaged by X-ray computed tomography and even functioning fuel cells have been imaged by synchrotron-based methods and soft X-ray radiography e.g. for imaging of liquid water in the GDL (Sinha et al., 2006; Bazylak, 2009; Sasabe et al., 2010; Tsushima & Hirai, 2011).


Table 1. Properties of gas diffusion layers investigated by X-ray computed tomography (Toray Industries, 2005b; SGL Group, 2009; Pfrang et al., 2010)

Here, CT data from three different commercially available gas diffusion layers will be discussed which were imaged by a nanotom X-ray computed tomography system (GE Sensing and Inspection Technologies, phoenix X-ray, Wunstorf, Germany) at a resolution below 1 µm. Table 1 shows PTFE content, thickness and porosity of the investigated gas diffusion layers. Segmentation into solid material (i.e. carbon and PTFE) and air was carried out based on a gray level threshold. Further details can be found in (Pfrang et al., 2010).

#### **2.1.2 Random generation of 3D structures**

The second approach is the random generation of three-dimensional fiber structures using the FiberGeo module of the Geodict software package (Fraunhofer ITWM, 2011). Geometric

Computation of Thermal Conductivity of Gas Diffusion Layers of PEM Fuel Cells 219

Whereas in the randomly generated 3D structures the distribution of PTFE and carbon is well known, these two materials could not be distinguished in the CT data. As a rough approximation, all solid voxels in the CT datasets were assumed to have a thermal conductivity that was calculated as the weighted average of the thermal conductivities of the carbon fibers and the thermal conductivity of PTFE, even though these two materials do

Several analytical models for the estimation of thermal conductivity of heterogeneous materials exist (Progelhof et al., 1976; Carson et al., 2005; Wang et al., 2006) and can be applied to gas diffusion layers. In the following, the most fundamental models – parallel model, series model, Maxwell Eucken model, effective medium theory model and cocontinous model – are presented. Furthermore it is possible to use combinations of two or more of these fundamental models for the estimation of thermal conductivity (Krischer,

If conduction is the only or the dominating heat transfer mechanism, it may be assumed that thermal conductivity of a porous material will lie between the parallel and series model values. Equation (2) describes the result using the parallel model which considers the thermal resistances to be in parallel, i.e. heat can flow through both materials in parallel. The parallel model gives the upper bound of effective thermal conductivity of the heterogeneous

kh is the thermal conductivity of the heterogeneous material; the second subscript denotes the model used for its estimation (e.g. p for parallel). f is the filling factor i.e. the volume fraction of the solid phase, ks is the thermal conductivity of the solid phase and kair the

In the series model (see equation (3)), the thermal resistances are considered to be in series with respect to the heat flux and kh,s gives the lower bound of effective thermal conductivity.

f /k (1 f) /k <sup>=</sup> + −

The effective medium theory (EMT) model (see equation (4)) assumes a random, mutual

( )( ) ( )( ) <sup>2</sup>

Equation (5) shows the result of the co-continuous model (Wang et al., 2008) where both

( ) , kh, C-C , . 18 / 1 <sup>2</sup>

<sup>1</sup> k 3 1 23 3 1 23 8

*h s*

s air

<sup>4</sup> *<sup>s</sup> air <sup>s</sup> air air s <sup>f</sup> k fk f k fk k k* ⎛ ⎞ = − +− + − +− + ⎡ ⎤ ⎜ ⎟ ⎣ ⎦ ⎝ ⎠ (4)

*h p hs*

*<sup>k</sup>* <sup>=</sup> + − *k k* (5)

k fk (1 f)k h, p s = + − air (2)

(3)

not intermix. The remaining, non-solid voxels were assumed to be filled with air.

**3.1 Estimation of thermal conductivity of heterogeneous materials** 

thermal conductivity of air. The volume fraction of air is 1-f.

dispersion of two components (Carson et al., 2005).

h, EMT

phases are assumed to be continuous.

h, s

<sup>1</sup> <sup>k</sup>

**3. Results and discussion** 

1963; Wang et al., 2006).

material.

parameters such as fiber diameter and length, fiber volume fraction were specified as well as the size of the grid as specified by the number of voxels in x, y and z-direction, nx, ny and nz.

The degree of orientation anisotropy was characterized by the anisotropy parameter β. Using spherical coordinates, β characterizes the directional distribution of fibers. The density of the directional distribution is given by Equation (1), (Schladitz et al., 2006):

$$\mathbf{P}\left(\theta,\phi\right) = \frac{1}{4\pi} \frac{\beta\sin\theta}{\left(1 + \left(\beta^2 - 1\right)\cos^2\theta\right)}\tag{1}$$

with the inclination θ ∈ ⎡0,π ) ⎣ and the azimuth ϕ ∈ ⎡0,2π ) ⎣ . The density is thus independent of φ, i.e. the density exhibits rotational symmetry with respect to the z-axis. The case β = 1 describes the isotropic system. For β →∞, the cylinders tend to be more and more parallel to the xy-plane. For β → 0 the cylinders tend to be more and more parallel to the z-axis.

For the randomly generated structures presented here, the selected structure size was 200 x 200 x 271 voxels with a voxel length of 0.7 µm. The fiber volume fraction was 21 % and an anisotropy factor ß of 1000 was chosen. The fibers had a diameter of 10 voxels and were assumed to have infinite length. For further details of the random generation of 3D structures, see also (Veyret & Tsotridis, 2010).

Additionally, model structures consisting of layers of equidistant parallel fibers were generated for the examination of the influence of PTFE distribution on thermal conductivity. For each model structure (see e.g. Fig. 6, before and after the addition of PTFE), the distance between adjacent fiber layers was fixed, but model structures were generated for 6 different layer distances. As the filling factor of the carbon fibers was kept constant at 22 %, the lateral distance between parallel fibers within a layer was adjusted accordingly.

The addition of PTFE to the fiber structures – the randomly generated structures as well as the model structures – was implemented by using the 'add binder'-function in GeoDict, where pores are filled starting from the smallest pores and then continuing to bigger pores until the desired binder volume fraction is reached. The algorithm used here to determine the size of a pore does not distinguish between through pores, closed pores and blind pores and is in this sense purely geometrical. A pore radius is determined by fitting spheres into the pore volume, i.e a point belongs to a pore of radius larger than r, if it is inside any sphere of radius r, which can be fitted into the pore space (Fraunhofer ITWM, 2011).

#### **2.2 Numerical method for the computation of effective thermal conductivity**

For the computation of the effective thermal conductivity of fibrous materials, the steady, purely diffusive, three-dimensional heat transfer equation has to be solved. In the case of large three-dimensional geometries (e.g. large data sets from CT imaging, see section 2.1.1 or generated randomly, see section 2.1.2), partial differential equation solvers are not efficient. (Wiegmann & Zemitis, 2006) use a different approach where the energy equation is solved by harmonic averaging. Fast Fourier transform and bi-conjugate gradient stabilized (BiCGStab) methods are then used to solve the Schur-complement formulation. This method – where convection and radiation transport, as well as thermal contact resistance and phase changes are not taken into account – is implemented in the GeoDict software which was also used for the random generation of 3D structures. Further details can be found in (Veyret & Tsotridis, 2010).

Whereas in the randomly generated 3D structures the distribution of PTFE and carbon is well known, these two materials could not be distinguished in the CT data. As a rough approximation, all solid voxels in the CT datasets were assumed to have a thermal conductivity that was calculated as the weighted average of the thermal conductivities of the carbon fibers and the thermal conductivity of PTFE, even though these two materials do not intermix. The remaining, non-solid voxels were assumed to be filled with air.

#### **3. Results and discussion**

218 Convection and Conduction Heat Transfer

parameters such as fiber diameter and length, fiber volume fraction were specified as well as the size of the grid as specified by the number of voxels in x, y and z-direction, nx, ny and

The degree of orientation anisotropy was characterized by the anisotropy parameter β. Using spherical coordinates, β characterizes the directional distribution of fibers. The

( ) ( ) 2 2

θ

(1)

) ⎣ . The density is thus independent

β θ

β

ϕ ∈ ⎡0,2π

of φ, i.e. the density exhibits rotational symmetry with respect to the z-axis. The case β = 1 describes the isotropic system. For β →∞, the cylinders tend to be more and more parallel to

For the randomly generated structures presented here, the selected structure size was 200 x 200 x 271 voxels with a voxel length of 0.7 µm. The fiber volume fraction was 21 % and an anisotropy factor ß of 1000 was chosen. The fibers had a diameter of 10 voxels and were assumed to have infinite length. For further details of the random generation of 3D

Additionally, model structures consisting of layers of equidistant parallel fibers were generated for the examination of the influence of PTFE distribution on thermal conductivity. For each model structure (see e.g. Fig. 6, before and after the addition of PTFE), the distance between adjacent fiber layers was fixed, but model structures were generated for 6 different layer distances. As the filling factor of the carbon fibers was kept constant at 22 %, the lateral

The addition of PTFE to the fiber structures – the randomly generated structures as well as the model structures – was implemented by using the 'add binder'-function in GeoDict, where pores are filled starting from the smallest pores and then continuing to bigger pores until the desired binder volume fraction is reached. The algorithm used here to determine the size of a pore does not distinguish between through pores, closed pores and blind pores and is in this sense purely geometrical. A pore radius is determined by fitting spheres into the pore volume, i.e a point belongs to a pore of radius larger than r, if it is inside any sphere

For the computation of the effective thermal conductivity of fibrous materials, the steady, purely diffusive, three-dimensional heat transfer equation has to be solved. In the case of large three-dimensional geometries (e.g. large data sets from CT imaging, see section 2.1.1 or generated randomly, see section 2.1.2), partial differential equation solvers are not efficient. (Wiegmann & Zemitis, 2006) use a different approach where the energy equation is solved by harmonic averaging. Fast Fourier transform and bi-conjugate gradient stabilized (BiCGStab) methods are then used to solve the Schur-complement formulation. This method – where convection and radiation transport, as well as thermal contact resistance and phase changes are not taken into account – is implemented in the GeoDict software which was also used for the random generation of 3D structures. Further details can be found in (Veyret &

+ −

density of the directional distribution is given by Equation (1), (Schladitz et al., 2006):

π

the xy-plane. For β → 0 the cylinders tend to be more and more parallel to the z-axis.

=

1 sin P , <sup>4</sup> 1 1 cos

( )

) ⎣ and the azimuth

distance between parallel fibers within a layer was adjusted accordingly.

of radius r, which can be fitted into the pore space (Fraunhofer ITWM, 2011).

**2.2 Numerical method for the computation of effective thermal conductivity** 

θ φ

nz.

with the inclination

Tsotridis, 2010).

θ ∈ ⎡0,π

structures, see also (Veyret & Tsotridis, 2010).

#### **3.1 Estimation of thermal conductivity of heterogeneous materials**

Several analytical models for the estimation of thermal conductivity of heterogeneous materials exist (Progelhof et al., 1976; Carson et al., 2005; Wang et al., 2006) and can be applied to gas diffusion layers. In the following, the most fundamental models – parallel model, series model, Maxwell Eucken model, effective medium theory model and cocontinous model – are presented. Furthermore it is possible to use combinations of two or more of these fundamental models for the estimation of thermal conductivity (Krischer, 1963; Wang et al., 2006).

If conduction is the only or the dominating heat transfer mechanism, it may be assumed that thermal conductivity of a porous material will lie between the parallel and series model values. Equation (2) describes the result using the parallel model which considers the thermal resistances to be in parallel, i.e. heat can flow through both materials in parallel. The parallel model gives the upper bound of effective thermal conductivity of the heterogeneous material.

$$\mathbf{k}\_{\mathbf{h},\mathbf{p}} = \mathbf{f}\mathbf{k}\_{\mathbf{s}} + (\mathbf{1} - \mathbf{f})\mathbf{k}\_{\text{air}} \tag{2}$$

kh is the thermal conductivity of the heterogeneous material; the second subscript denotes the model used for its estimation (e.g. p for parallel). f is the filling factor i.e. the volume fraction of the solid phase, ks is the thermal conductivity of the solid phase and kair the thermal conductivity of air. The volume fraction of air is 1-f.

In the series model (see equation (3)), the thermal resistances are considered to be in series with respect to the heat flux and kh,s gives the lower bound of effective thermal conductivity.

$$\mathbf{k}\_{\mathbf{h},s} = \frac{1}{\mathbf{f} \;/\; \mathbf{k}\_s + (1 - \mathbf{f}) \;/\; \mathbf{k}\_{\text{air}}} \tag{3}$$

The effective medium theory (EMT) model (see equation (4)) assumes a random, mutual dispersion of two components (Carson et al., 2005).

$$\mathbf{k}\_{\rm h,EMI} = \frac{1}{4} \Big( (\Im f - 1)k\_s + (2 - \Im f)k\_{\rm air} + \sqrt{\left[ (\Im f - 1)k\_s + \left( 2 - \Im f \right)k\_{\rm air} \right]^2 + 8k\_{\rm air}k\_s \Big)} \tag{4}$$

Equation (5) shows the result of the co-continuous model (Wang et al., 2008) where both phases are assumed to be continuous.

$$\mathbf{k}\_{\rm h,C-C} = \frac{k\_{h,s}}{2} \left( \sqrt{1 + 8k\_{h,p} \left/ k\_{h,s} - 1} \right) \tag{5}$$

Computation of Thermal Conductivity of Gas Diffusion Layers of PEM Fuel Cells 221

Nevertheless, the models can be applied to gas diffusion layers and Fig. 3 shows normalized thermal conductivity estimated using the models mentioned above assuming a ratio of thermal conductivities ks/kair of 120 Wm-1K-1 / 0.026 Wm-1K-1 which is an estimate for gas diffusion layers consisting of carbon fibers in air. As an example of a GDL, the in-plane and through-plane thermal conductivities of Toray carbon paper without the addition of PTFE as given by the manufacturer (Toray Industries, 2005b) are included assuming a ks of 120

Overall, the presented models allow estimating the order of magnitude of the thermal conductivity of gas diffusion layers, but – also due to the anisotropic microstructure of a

As more accurate thermal conductivity data is required, one further approach is the computation based on 3D structure data (Becker et al., 2008; Pfrang et al., 2010; Veyret & Tsotridis, 2010; Zamel et al., 2010). Fig. 4 illustrates the two approaches applied here: the characterization of GDL 3D structure by X-ray computed tomography (section 2.1.1), left and the random generation of 3D models of the GDL structure (section 2.1.2), right. Both approaches have certain advantages and drawbacks: While the randomly generated structures allow an accurate definition of the distribution of each material, in CT it was not possible to discriminate carbon from PTFE due to similar X-ray adsorption. CT, on the other hand, provides the realistic 3D structure; whereas there are deviations from the real structure after random structure generation (e.g. straight fibers are assumed). In both approaches there are limitations with respect to spatial resolution; resolution of X-ray CT is limited while essentially computer hardware and computing time limit the number of

Even though PTFE cannot be discriminated from carbon fibers by the contrast in the CT datasets, the carbon fibers are clearly visible in the 3D structure (see top left of Fig. 4) and the micro porous layer (MPL) is clearly visible in the cross-section (see bottom left of Fig. 4). For one of the samples investigated by CT – EC-TP1-060T – a corresponding randomly generated structural model was generated. This sample was selected as it contains relatively straight fibers (as compared to the carbon cloth) and does not contain a micro porous layer. The parameters for the structure generation were chosen according to the manufacturer's datasheet (Toray Industries, 2005b): 22 % filling factor of carbon; these 22 % were distributed into 21 % carbon fiber and 1 % carbon binder. After fiber generation and addition of the

The results are given in Table 2 with gray background together with a selection of thermal conductivity data published in the literature. For in-plane thermal conductivity, two values are given if the thermal conductivity was determined independently for two orthogonal inplane directions, and one value is given when the average in-plane conductivity was

In an earlier study (Ihonen et al., 2004), thermal conductivity for several types of GDLs was estimated to 0.2-0.4 W/m K, which was considered unrealistically low in the study. Nevertheless, this estimated range is in agreement with more recent measurements. Finally, a study on the computation of thermal conductivity based on model structures with different geometries was published by (Zamel et al., 2010) where the calculated values are given relative to the thermal conductivity of the solid phase and therefore the values are not given in Table 2. It is reasonable to give this ratio, because the thermal conductivity of carbon fibers is often not known as it depends on the type of carbon fiber and can vary by

typical GDL – a more precise a-priori estimation seems impossible.

**3.2 Computation of thermal conductivity of gas diffusion layers** 

voxels of randomly generated structures.

carbon binder, 30 wt. % of PTFE were added.

determined.

Wm-1K-1.

Even though this model is independent of parallel and series model, the result kh, C-C can be expressed as function of kh. p and kh, s, which are the thermal conductivities calculated for the parallel and series model (see equations (2) and (3)).

Whereas all four models mentioned so far are symmetric with respect to exchange of the two phases, the Maxwell-Euken model (Eucken, 1940) is not, as one phase is assumed to be dispersed in a second, continuous phase. The heterogeneous conductivity calculated following the Maxwell-Euken model kh, M-E is given in (6) where the index 'cont' refers to the continuous phase and the index 'dis' to the dispersed phase.

$$\mathbf{k}\_{\rm h,M-E} = \frac{k\_{\rm cont} f\_{\rm cont} + k\_{\rm dis} f\_{\rm dis} \frac{\mathfrak{D} k\_{\rm cont}}{\mathfrak{D} k\_{\rm cont} + k\_{\rm dis}}}{\mathbf{f}\_{\rm cont} + \mathbf{f}\_{\rm dis} \frac{\mathfrak{D} k\_{\rm cont}}{\mathfrak{D} k\_{\rm cont} + k\_{\rm dis}}} \tag{6}$$

Obviously, each of the models assumes a certain geometry which does not reflect exactly the microstructure of a GDL. Gas diffusion layers typically exhibit anisotropy of the microstructure as carbon fibers are preferentially oriented in–plane. Furthermore, carbon fibers are expected to exhibit an anisotropic thermal conductivity as e.g. pyrolytic graphite (Wen & Huang, 2008) – the degree of anisotropy depending on the type of fiber – due to their anisotropic, partly graphite-like structure. Both, the anisotropy in microstructure and the anisotropy of thermal conductivity in carbon fibers is not into account in any of the presented models.

Fig. 3. Estimated thermal conductivity of the heterogeneous material kh normalized with respect to the thermal conductivity of the solid phase ks dependent on porosity. Different models were used for the estimation using a ratio of ks/kair of 120 Wm-1K-1 / 0.026 Wm-1K-1. For some models, the assumed structure is shown as insert; the small arrows indicate the direction of heat flux where appropriate. Additionally, the thermal conductivity of Toray carbon paper as given by the manufacturer (Toray Industries, 2005b) is shown

Even though this model is independent of parallel and series model, the result kh, C-C can be expressed as function of kh. p and kh, s, which are the thermal conductivities calculated for the

Whereas all four models mentioned so far are symmetric with respect to exchange of the two phases, the Maxwell-Euken model (Eucken, 1940) is not, as one phase is assumed to be dispersed in a second, continuous phase. The heterogeneous conductivity calculated following the Maxwell-Euken model kh, M-E is given in (6) where the index 'cont' refers to the

cont dis

Obviously, each of the models assumes a certain geometry which does not reflect exactly the microstructure of a GDL. Gas diffusion layers typically exhibit anisotropy of the microstructure as carbon fibers are preferentially oriented in–plane. Furthermore, carbon fibers are expected to exhibit an anisotropic thermal conductivity as e.g. pyrolytic graphite (Wen & Huang, 2008) – the degree of anisotropy depending on the type of fiber – due to their anisotropic, partly graphite-like structure. Both, the anisotropy in microstructure and the anisotropy of thermal conductivity in carbon fibers is not into account in any of the

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Porosity

Fig. 3. Estimated thermal conductivity of the heterogeneous material kh normalized with respect to the thermal conductivity of the solid phase ks dependent on porosity. Different models were used for the estimation using a ratio of ks/kair of 120 Wm-1K-1 / 0.026 Wm-1K-1. For some models, the assumed structure is shown as insert; the small arrows indicate the direction of heat flux where appropriate. Additionally, the thermal conductivity of Toray

carbon paper as given by the manufacturer (Toray Industries, 2005b) is shown

+ <sup>+</sup> <sup>=</sup> +

2

Maxwell

(gas phase dispersed)

*cont cont cont dis dis*

*<sup>k</sup> k f kf k k*

<sup>2</sup> <sup>k</sup> <sup>3</sup> f f

Parallel

EMT

Maxwell

(solid phase dispersed)

3

*cont cont dis*

+

*k k k*

*cont dis*

(6)

Toray carbon paper in-plane through-plane

parallel and series model (see equations (2) and (3)).

presented models.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Series

Co-continuous

kh / ks

continuous phase and the index 'dis' to the dispersed phase.

h, M-E

Nevertheless, the models can be applied to gas diffusion layers and Fig. 3 shows normalized thermal conductivity estimated using the models mentioned above assuming a ratio of thermal conductivities ks/kair of 120 Wm-1K-1 / 0.026 Wm-1K-1 which is an estimate for gas diffusion layers consisting of carbon fibers in air. As an example of a GDL, the in-plane and through-plane thermal conductivities of Toray carbon paper without the addition of PTFE as given by the manufacturer (Toray Industries, 2005b) are included assuming a ks of 120 Wm-1K-1.

Overall, the presented models allow estimating the order of magnitude of the thermal conductivity of gas diffusion layers, but – also due to the anisotropic microstructure of a typical GDL – a more precise a-priori estimation seems impossible.

#### **3.2 Computation of thermal conductivity of gas diffusion layers**

As more accurate thermal conductivity data is required, one further approach is the computation based on 3D structure data (Becker et al., 2008; Pfrang et al., 2010; Veyret & Tsotridis, 2010; Zamel et al., 2010). Fig. 4 illustrates the two approaches applied here: the characterization of GDL 3D structure by X-ray computed tomography (section 2.1.1), left and the random generation of 3D models of the GDL structure (section 2.1.2), right. Both approaches have certain advantages and drawbacks: While the randomly generated structures allow an accurate definition of the distribution of each material, in CT it was not possible to discriminate carbon from PTFE due to similar X-ray adsorption. CT, on the other hand, provides the realistic 3D structure; whereas there are deviations from the real structure after random structure generation (e.g. straight fibers are assumed). In both approaches there are limitations with respect to spatial resolution; resolution of X-ray CT is limited while essentially computer hardware and computing time limit the number of voxels of randomly generated structures.

Even though PTFE cannot be discriminated from carbon fibers by the contrast in the CT datasets, the carbon fibers are clearly visible in the 3D structure (see top left of Fig. 4) and the micro porous layer (MPL) is clearly visible in the cross-section (see bottom left of Fig. 4).

For one of the samples investigated by CT – EC-TP1-060T – a corresponding randomly generated structural model was generated. This sample was selected as it contains relatively straight fibers (as compared to the carbon cloth) and does not contain a micro porous layer. The parameters for the structure generation were chosen according to the manufacturer's datasheet (Toray Industries, 2005b): 22 % filling factor of carbon; these 22 % were distributed into 21 % carbon fiber and 1 % carbon binder. After fiber generation and addition of the carbon binder, 30 wt. % of PTFE were added.

The results are given in Table 2 with gray background together with a selection of thermal conductivity data published in the literature. For in-plane thermal conductivity, two values are given if the thermal conductivity was determined independently for two orthogonal inplane directions, and one value is given when the average in-plane conductivity was determined.

In an earlier study (Ihonen et al., 2004), thermal conductivity for several types of GDLs was estimated to 0.2-0.4 W/m K, which was considered unrealistically low in the study. Nevertheless, this estimated range is in agreement with more recent measurements. Finally, a study on the computation of thermal conductivity based on model structures with different geometries was published by (Zamel et al., 2010) where the calculated values are given relative to the thermal conductivity of the solid phase and therefore the values are not given in Table 2. It is reasonable to give this ratio, because the thermal conductivity of carbon fibers is often not known as it depends on the type of carbon fiber and can vary by

Computation of Thermal Conductivity of Gas Diffusion Layers of PEM Fuel Cells 223

When considering all data from Table 2, the range of thermal conductivity values for the through-plane direction is 0.13-2.8 W / m K, as compared to a range of 1.75-21 W / m K for the in-plane direction. As carbon fibers have a higher thermal conductivity than PTFE or air, this can be explained by the preferred orientation of the carbon fibers in-plane, i.e. heat can be transported mainly along the fibers in-plane, whereas for through-plane heat transfer

In a PEM fuel cell, the GDL is subject to compression and consequently in most experimental setups used for the determination of thermal conductivity the GDL is investigated under compression. As a general trend, an increase of thermal conductivity and a reduction of thermal resistance were found with increasing compression. More recently, also the effect of load cycling – i.e. cycles of increasing compression up to a maximum value and releasing compression to 0 – was investigated and steady-state (in the investigated case

Further, it was found that residual water in the GDL leads to a significant increase of through-plane thermal conductivity (Burheim et al., 2010; Burheim et al., 2011). The

In the following paragraphs, our data on thermal conductivity will be compared with literature data. For the computation for E-Tek EC-CC1-060T, only one measurement on a similar sample – but without PTFE – is available and gives a clearly lower thermal conductivity. One explanation could be that the addition of PTFE leads to an increase of thermal conductivity, but this is not in accordance with the trend of a decrease of thermal conductivity observed for other samples. Because PTFE cannot be discriminated from carbon fibers in the CT dataset, it seems reasonable to assume that the thermal contact between fibers is overestimated in our computation (as an arithmetic average of thermal conductivity between carbon and PTFE was used in the computation for all solid voxels)

For Sigracet 35 BC, the MPL was not considered in the computation of thermal conductivity, i.e. it was replaced by air. This obviously leads to an overall underestimation of thermal

One way forward would be the clear identification of MPL material (maybe applying advanced segmentation techniques or using improved CT imaging techniques) and its

When comparing our results computed from the CT data of EC-TP1-060T and the randomly generated model (based on EC-TP1-060T), through-plane thermal conductivities agree well - 1.7 vs. 1.65 W / m K – whereas in-plane thermal conductivities are significantly larger for the randomly generated model. The in-plane heat flux is expected to flow mainly along the fibers. Therefore the different thermal conductivities assumed for solid voxels – 120 W / m K for the carbon fibers in the randomly generated model vs. 93 W / m K as weighted average between carbon and PTFE for the CT dataset – could explain this difference. Nevertheless, the computed thermal conductivities lie well within the range of values available in the literature for Toray carbon paper based materials for through-plane as well

Experimental results have shown that an increase of PTFE loading leads to a reduction of through-plane thermal conductivity (Khandelwal & Mench, 2006; Burheim et al., 2011) in

reached after 5 cycles) properties were determined (Sadeghi et al., 2010).

which results in an overestimation of thermal conductivity.

several, but not all types of gas diffusion layers (see Table 2).

conductivity as observed.

as in-plane direction.

**3.3 Influence of PTFE distribution** 

inclusion into the computation.

influence of PTFE distribution on thermal conductivity is discussed in section 3.3.

from fiber to fiber is required to a larger extent.

orders of magnitude (Blanco et al., 2002) depending e.g. on the heat treatment of the fibers. This may explain the apparent discrepancy between the results for structural models of carbon paper by (Becker et al., 2008) and (Veyret & Tsotridis, 2010) where thermal conductivity values of 17 W/ m K and 130 W/ m K were used, respectively.

Fig. 4. 3D structure and cross-section of a GDL as determined by X-ray computed tomography (left) and randomly generated (right). The micro porous layer (MPL) is clearly visible in the CT cross section. In the 3D structure of the randomly generated structure only the carbon fibers are shown (not carbon binder and PTFE), while in the cross section, the fibers (red) can be clearly discriminated from carbon binder (dark gray) and PTFE (light gray)

The focus of earlier work was on through-plane thermal conductivity of the GDL, as the heat flows predominantly through-plane in a PEM fuel cell. Nevertheless, for a detailed understanding of the heat flux, also in-plane thermal conductivity is relevant, e.g. because thermal contact between bipolar plate and GDL is not homogeneous due to the gas flow channels in the bipolar plate. Only recently measurements of in-plane thermal conductivity were published (Sadeghi et al., 2011b; Teertstra et al., 2011).

orders of magnitude (Blanco et al., 2002) depending e.g. on the heat treatment of the fibers. This may explain the apparent discrepancy between the results for structural models of carbon paper by (Becker et al., 2008) and (Veyret & Tsotridis, 2010) where thermal

Cross-section Cross-section

Fig. 4. 3D structure and cross-section of a GDL as determined by X-ray computed

tomography (left) and randomly generated (right). The micro porous layer (MPL) is clearly visible in the CT cross section. In the 3D structure of the randomly generated structure only the carbon fibers are shown (not carbon binder and PTFE), while in the cross section, the fibers (red) can be clearly discriminated from carbon binder (dark gray) and PTFE (light gray) The focus of earlier work was on through-plane thermal conductivity of the GDL, as the heat flows predominantly through-plane in a PEM fuel cell. Nevertheless, for a detailed understanding of the heat flux, also in-plane thermal conductivity is relevant, e.g. because thermal contact between bipolar plate and GDL is not homogeneous due to the gas flow channels in the bipolar plate. Only recently measurements of in-plane thermal conductivity

**Randomly generated fiber structure** 

3D structure

conductivity values of 17 W/ m K and 130 W/ m K were used, respectively.

**X-ray computed tomography of SGL GDL 35 BC** 

3D structure

MPL **100 µm** 

were published (Sadeghi et al., 2011b; Teertstra et al., 2011).

When considering all data from Table 2, the range of thermal conductivity values for the through-plane direction is 0.13-2.8 W / m K, as compared to a range of 1.75-21 W / m K for the in-plane direction. As carbon fibers have a higher thermal conductivity than PTFE or air, this can be explained by the preferred orientation of the carbon fibers in-plane, i.e. heat can be transported mainly along the fibers in-plane, whereas for through-plane heat transfer from fiber to fiber is required to a larger extent.

In a PEM fuel cell, the GDL is subject to compression and consequently in most experimental setups used for the determination of thermal conductivity the GDL is investigated under compression. As a general trend, an increase of thermal conductivity and a reduction of thermal resistance were found with increasing compression. More recently, also the effect of load cycling – i.e. cycles of increasing compression up to a maximum value and releasing compression to 0 – was investigated and steady-state (in the investigated case reached after 5 cycles) properties were determined (Sadeghi et al., 2010).

Further, it was found that residual water in the GDL leads to a significant increase of through-plane thermal conductivity (Burheim et al., 2010; Burheim et al., 2011). The influence of PTFE distribution on thermal conductivity is discussed in section 3.3.

In the following paragraphs, our data on thermal conductivity will be compared with literature data. For the computation for E-Tek EC-CC1-060T, only one measurement on a similar sample – but without PTFE – is available and gives a clearly lower thermal conductivity. One explanation could be that the addition of PTFE leads to an increase of thermal conductivity, but this is not in accordance with the trend of a decrease of thermal conductivity observed for other samples. Because PTFE cannot be discriminated from carbon fibers in the CT dataset, it seems reasonable to assume that the thermal contact between fibers is overestimated in our computation (as an arithmetic average of thermal conductivity between carbon and PTFE was used in the computation for all solid voxels) which results in an overestimation of thermal conductivity.

For Sigracet 35 BC, the MPL was not considered in the computation of thermal conductivity, i.e. it was replaced by air. This obviously leads to an overall underestimation of thermal conductivity as observed.

One way forward would be the clear identification of MPL material (maybe applying advanced segmentation techniques or using improved CT imaging techniques) and its inclusion into the computation.

When comparing our results computed from the CT data of EC-TP1-060T and the randomly generated model (based on EC-TP1-060T), through-plane thermal conductivities agree well - 1.7 vs. 1.65 W / m K – whereas in-plane thermal conductivities are significantly larger for the randomly generated model. The in-plane heat flux is expected to flow mainly along the fibers. Therefore the different thermal conductivities assumed for solid voxels – 120 W / m K for the carbon fibers in the randomly generated model vs. 93 W / m K as weighted average between carbon and PTFE for the CT dataset – could explain this difference. Nevertheless, the computed thermal conductivities lie well within the range of values available in the literature for Toray carbon paper based materials for through-plane as well as in-plane direction.

#### **3.3 Influence of PTFE distribution**

Experimental results have shown that an increase of PTFE loading leads to a reduction of through-plane thermal conductivity (Khandelwal & Mench, 2006; Burheim et al., 2011) in several, but not all types of gas diffusion layers (see Table 2).


Table 2. (continues) Published values of thermal conductivity of gas diffusion layers

Computation of Thermal Conductivity of Gas Diffusion Layers of PEM Fuel Cells 225

**Thermal conductivity / W m-1 K-1**

**Gas diffusion layer**

**PTFE / % through-**

0 2.8 0 1.8 -1.2 0.41 -0.66

5 1.6

dry GDL Toray carbon paper TGP-H-060

E-Tek EC-TP1-060T

based on Toray TGP

30 1.7

8.9, 8.0

based on x


based on structural model computation EJ


estimate based on experiment

21 datasheet

ex situ unidirectional heat flux and measuring temperature difference (Ramousse et al., 2008)

not available (Toray Industries, 2005b)

(Radhakrishnan, 2009)


 -H-060 Carbon paper EC-TP1-060T

(structural model) Toray carbon paper, Quintech, thickness 230 μm

Table 2. (continued) (results by the authors marked by gray background)

Toray carbon paper TGP-H-060 / -090 and -120

Toray carbon paper TGP-H-060 and -120

5 1.4 -2.1

1.4

 -0.73

Toray carbon paper TGP-H-090

10 0.48 -0.69 60 0.28 -0.32 0 0.33 -0.59 0.62 -0.89

estimate based on experiment

4.6 dry GDL 5 1.4

4.6


residual water ex situ unidirectional heat flux and measuring temperature difference (Burheim et al., 2011)

at quasi steady state, load cycling 0.25-1.5 MPa, in vacuum ex situ unidirectional heat flux and measuring temperature difference (Sadeghi et al., 2010)

in vacuum ex situ thermal resistance measurements using GDL samples of different lengths (Sadeghi et al., 2011b)

 ments using the parallel thermal conductance (PTC) technique


in vacuum ex situ thermal resistance measure

5 1.55 -2.1

Toray carbon paper TGP-H-120 5-30 17.3

30 15.1

0.296

based on synchrotron 3D structure data Carbon paper (structural model) ≤ 20

0.388

0 1.86

0-25 1.86

Carbon paper (structural model)

9.79, 9.82



2.55

compressed by 20%, based on synchrotron 3D structure data computation EJ

1.75, 2.05

 -17.8


ex situ unidirectional heat flux and measuring temperature difference (Ramousse et al., 2008)

(Teertstra et al., 2011)

Toray carbon paper, Quintech, thickness 280 μm


residual water 5 0.5

4.6


dry GDL ex situ unidirectional heat flux and measuring temperature difference (Burheim et al., 2011)


0 1.7 0 1.6 -1.9

0.04

0.2



ex situ guarded hot plate, two similar samples with different thicknesses used

ex situ unidirectional heat flux and measuring temperature difference (Sadeghi et al., 2011a)

yes 0.20 -0.30

30 1.65

0 0.36 -1.36

13.8

compression 1.9 MPa

temperature 26 - 73 °C

4.6 4.6


residual water ex situ unidirectional heat flux and measuring temperature difference (Burheim et al., 2011)


**plane in-plane Remarks**

**Type of data Technique Reference**

in situ estimated from temperature differences in running fuel cell (Kawase et al., 2009)

ex situ unidirectional heat flux and measuring temperature difference (Khandelwal & Mench, 2006)


#### Computation of Thermal Conductivity of Gas Diffusion Layers of PEM Fuel Cells 225

224 Convection and Conduction Heat Transfer

**Thermal conductivity / W m-1 K-1**

**Gas diffusion layer**

E-Tek EC-CC1-060

E-Tek EC-CC1-060T

E-Tek ELAT carbon cloth LT1200


E-Tek ELAT carbon cloth, thickness 180 μm E-Tek ELAT carbon cloth, thickness 410 Freudenberg FCCT H2315

Freudenberg FCCT H2315 T10A 10 0.15

 -0.16 0.30 -0.42

dry GDL SGL Sigracet 10 AA

0 0.7 -0.87

 -0.33

SGL Sigracet 10 BA

Table 2. (continues) Published values of thermal conductivity of gas diffusion layers

SGL Sigracet 24 AA

SGL Si gracet 24 BA

SGL Sigracet 24 DA

SGL Sigracet 25BC and 35BC

SGL Sigracet 35 BC

SGL Carbon, thickness 420 μm

yes 0.26 -0.34

30 0.25 -0.52 30 3.2

0.27 -0.40

dry GDL SolviCore carbon paper not av. 0.45 -0.57

0 0.26 -0.7 12, 19, 29 0.28 -0.6

SpectraCarb carbon paper

0 12.8, 13.8

6, 11, 19 9.78

 -14.2

 -3.87 4.6 4.6


0.7 - 13.8 bar compression

ex situ ASTM Standard D-5470

in vacuum ex situ thermal resistance measurements using the parallel thermal conductance (PTC) technique

(Teertstra et al., 2011)

 -06 with modifications; unidirectional heat flux

(Karimi et al., 2010)

residual water ex situ unidirectional heat flux and measuring temperature difference (Burheim et al., 2010)


5 0.16

1.9, 2.0

based on x -ray CT data (without MPL) estimate based on experiment

0.7 - 13.8 bar compression

ex situ ASTM Standard D-5470

in vacuum ex situ thermal resistance measurements using the parallel thermal conductance (PTC) technique


(Teertstra et al., 2011)

5 0.6 -0.9

0.04


computation EJ

ex situ guarded hot plate, two similar samples with different thicknesses used


ex situ unidirectional heat flux and measuring temperature difference (Ramousse et al., 2008)

0 0.48

5 0.31

20 0.22

5 1.18

conductivity almost independent of compression up to 5.5 MPa

4.6 4.6 4.6



) 5 0.26

residual water ex situ unidirectional heat flux and measuring temperature difference (Burheim et al., 2011

dry GDL ex situ unidirectional heat flux and measuring temperature difference (Burheim et al., 2011)

ex situ unidirectional heat flux and measuring temperature difference (Nitta et al., 2008)

ex situ unidirectional heat flux and measuring temperature difference (Khandelwal & Mench, 2006)

(Radhakrishnan, 2009)


0 0.14 -0.15

4.6


dry GDL ex situ unidirectional heat flux and measuring temperature difference (Burheim et al., 2011)

μm not av. 0.13

 -0.19

0 0.28 -0.32

30 1.4

0 0.22

0 0.2

5.6, 6.2

based on x


combined value for catalyst surface + GDL in situ calculated from temperature profiles in running fuel cell


ex situ unidirectional heat flux and measuring temperature difference (Khandelwal & Mench, 2006)

**PTFE / % through-**

**plane in-plane Remarks**

4.6


dry GDL ex situ unidirectional heat flux and measuring temperature difference (Burheim et al., 2011)

**Type of data Technique Reference**

in situ estimated from temperature differences in running fuel cell (Burford & Mench, 2004)

(Vie & Kjelstrup, 2004)

Table 2. (continued) (results by the authors marked by gray background)

Computation of Thermal Conductivity of Gas Diffusion Layers of PEM Fuel Cells 227

Fig. 6. Detail of model structures of layers of equidistant parallel fibers. The filling factor of the carbon fibers is 22 %. Without PTFE (left) and with 30 wt. % of PTFE shown in gray

> With PTFE No PTFE Air (0.026 W/Km) PTFE (0.25 W/Km)

01234 **Distance between fiber layers / µm**

Fig. 7. Computed through-plane thermal conductivity dependent on the distance between fiber layers (based on model structures as illustrated in Fig. 6). For comparison, the thermal

In both cases, with and without PTFE, the conductivity drops sharply from distance 0, i.e. contact between the fibers layers, to a distance of 0.175 µm, the smallest considered distance. For the model structures without PTFE, the conductivity drops to values close to the thermal conductivity of air. These cases are of course not realistic for a GDL as fibers will always be in contact due to the compression of the GDL, but it is illustrated that heat transfer between

conductivities of pure air and pure PTFE are included into the graph

(right)

0.0

non-touching fibers is strongly inhibited.

0.5

1.0

1.5

**Through plane thermal conductivity / W K-1m-1**

2.0

2.5

**No PTFE 30 wt. % of PTFE** 

Assuming that the structure of the GDL remains unchanged, one would expect that the addition of material that has a higher thermal conductivity than air – like PTFE – should increase the overall thermal conductivity.

Fig. 5. Scanning electron micrographs of the cross section of carbon cloth EC-CC1-060T: the top image gives an overview of the whole thickness of the cross section, the bottom two images show carbon fibers with PTFE in between

However, since the thermal conductivity of PTFE is orders of magnitude smaller than that of carbon, PTFE potentially can insulate carbon fibers from each other, i.e. the contact region between carbon fibers can have a big impact on the thermal conductivity of the GDL. This is especially relevant for carbon papers where the carbon fibers are oriented preferably inplane, which means that through-plane heat transfer requires heat transfer from fiber to fiber. Fig. 5 shows scanning electron micrographs of the cross section of carbon cloth EC-CC1-060T. In the selected region, PTFE separates two neighbouring carbon fibers. For a further illustration of this possible insulation effect, thermal conductivity of model structures of layers of parallel fibers with a filling factor of 22 % were randomly generated using GeoDict (see Fig. 6). Model structures with different distances between the fiber layers were investigated and thermal conductivity of each of these structures was evaluated without PTFE and after the addition of 30 wt. % of PTFE. These model structures are surely not a realistic description of any GDL, but they will be used to examine different possible geometries of the contact region between fibers. As these fibers are oriented exactly in-plane, the model structures represent an extreme case with respect to through-plane conductivity in the sense that heat transfer along the fibers cannot contribute significantly.

The computed thermal conductivities of these model structures are summarized in Fig. 7. For the case of touching fibers (distance 0 µm), there is only a small increase in thermal conductivity due to the addition of PTFE around the contact region between fibers.

Assuming that the structure of the GDL remains unchanged, one would expect that the addition of material that has a higher thermal conductivity than air – like PTFE – should

PTFE

C fiber

Fig. 5. Scanning electron micrographs of the cross section of carbon cloth EC-CC1-060T: the top image gives an overview of the whole thickness of the cross section, the bottom two

C fiber

However, since the thermal conductivity of PTFE is orders of magnitude smaller than that of carbon, PTFE potentially can insulate carbon fibers from each other, i.e. the contact region between carbon fibers can have a big impact on the thermal conductivity of the GDL. This is especially relevant for carbon papers where the carbon fibers are oriented preferably inplane, which means that through-plane heat transfer requires heat transfer from fiber to fiber. Fig. 5 shows scanning electron micrographs of the cross section of carbon cloth EC-CC1-060T. In the selected region, PTFE separates two neighbouring carbon fibers. For a further illustration of this possible insulation effect, thermal conductivity of model structures of layers of parallel fibers with a filling factor of 22 % were randomly generated using GeoDict (see Fig. 6). Model structures with different distances between the fiber layers were investigated and thermal conductivity of each of these structures was evaluated without PTFE and after the addition of 30 wt. % of PTFE. These model structures are surely not a realistic description of any GDL, but they will be used to examine different possible geometries of the contact region between fibers. As these fibers are oriented exactly in-plane, the model structures represent an extreme case with respect to through-plane conductivity

in the sense that heat transfer along the fibers cannot contribute significantly.

conductivity due to the addition of PTFE around the contact region between fibers.

The computed thermal conductivities of these model structures are summarized in Fig. 7. For the case of touching fibers (distance 0 µm), there is only a small increase in thermal

increase the overall thermal conductivity.

images show carbon fibers with PTFE in between

**2 µm 500 nm** 

Fig. 6. Detail of model structures of layers of equidistant parallel fibers. The filling factor of the carbon fibers is 22 %. Without PTFE (left) and with 30 wt. % of PTFE shown in gray (right)

Fig. 7. Computed through-plane thermal conductivity dependent on the distance between fiber layers (based on model structures as illustrated in Fig. 6). For comparison, the thermal conductivities of pure air and pure PTFE are included into the graph

In both cases, with and without PTFE, the conductivity drops sharply from distance 0, i.e. contact between the fibers layers, to a distance of 0.175 µm, the smallest considered distance. For the model structures without PTFE, the conductivity drops to values close to the thermal conductivity of air. These cases are of course not realistic for a GDL as fibers will always be in contact due to the compression of the GDL, but it is illustrated that heat transfer between non-touching fibers is strongly inhibited.

Computation of Thermal Conductivity of Gas Diffusion Layers of PEM Fuel Cells 229

distribution of PTFE in the contact region between carbon fibers on the macroscopic thermal conductivity is considerable. Therefore an improved characterisation of the contact region between carbon fibers could be a worthwhile objective for future similar studies. This

Overall, it was shown that the computation of thermal conductivity of gas diffusion layers based on 3D structure can lead to an improved understanding of the thermal properties of a GDL. Both approaches presented here have advantages: CT measurements guarantee a realistic 3D structure, while the random generation of 3D structures allows the precise definition of the distribution of all components which can be especially relevant for virtual

This work has been carried out within the multiannual programme of the European Commission's Joint Research Centre, as part of the FCPOINT activities. The authors thank Marc Steen (Institute for Energy, European Commission) for critical reading of the manuscript, Gaby Janssen (Energy Research Centre of the Netherlands, ECN) for the provision of gas diffusion layers and GE sensing and inspection technologies/phoenix X-ray

Bazylak, A. (2009). Liquid water visualization in PEM fuel cells: A review. *International Journal of Hydrogen Energy* Vol. 34, No. 9, pp. 3845-3857, ISSN 03603199 Becker, J., Flückiger, R., Reum, M., Büchi, F. N., Marone, F. & Stampanoni, M. (2009).

*Electrochemical Society* Vol. 156, No. 10, pp. B1175-B1181, ISSN 00134651 Becker, J., Schulz, V. & Wiegmann, A. (2008). Numerical determination of two-phase

Determination of material properties of gas diffusion layers: Experiments and simulations using phase contrast tomographic microscopy. *Journal of the* 

material parameters of a gas diffusion layer using tomography images. *Journal of Fuel Cell Science and Technology* Vol. 5, No. 2, pp. art. no. 021006, ISSN 1550624X Blanco, C., Appleyard, S. P. & Rand, B. (2002). Study of carbon fibres and carbon-carbon

composites by scanning thermal microscopy. *Journal of Microscopy* Vol. 205, No. 1,

thermal conductivity of PEMFC porous transport layers. *Journal of Fuel Cell Science* 

through-plane thermal conductivities in a polymer electrolyte fuel cell. *Journal of* 

bounds for isotropic, porous materials. *International Journal of Heat and Mass Transfer*

Burford, D. J. & Mench, M. M. (2004). *Heat transport and temperature distribution in PEFCS*. American Society of Mechanical Engineers, Heat Transfer Division, Anaheim, CA. Burheim, O. S., Pharoah, J. G., Lampert, H., Vie, P. J. S. & Kjelstrup, S. (2011). Through-plane

Burheim, O. S., Vie, P. J. S., Pharoah, J. G. & Kjelstrup, S. (2010). Ex situ measurements of

Carson, J. K., Lovatt, S. J., Tanner, D. J. & Cleland, A. C. (2005). Thermal conductivity

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GDL design.

**6. References** 

**5. Acknowledgment** 

For the model structures with PTFE, the conductivity drops to values below the thermal conductivity of PTFE, which is in the same order of magnitude as thermal conductivities measured for realistic gas diffusion layers (compare Table 2).

It has to be mentioned that in the model structures the contact area between touching fibers (distance 0 µm) was defined by the voxel size of the model structures. It is clear that this contact area will have a huge impact on heat flux through the contact. In this context, the presence of carbon binder – which is used for the mechanical stabilization of the GDL – can play an important role.

Overall, it is obvious from the results presented in Fig. 7 that the contact region between fibers is important for the thermal conductivity of a GDL. When looking back to the models estimating thermal conductivity of heterogeneous materials (section 2.1) and considering the contact area between fibers consisting of PTFE and carbon, both extreme cases – parallel heat flux through carbon and PTFE and heat flux in series through the two components – seem to be potentially realistic for the description of a specific single contact between two fibers in a gas diffusion layer. The macroscopic thermal conductivity of a GDL will depend strongly on the statistical distribution of the different contact geometries.

Considering the drop of thermal conductivity with the increase of PTFE loading observed experimentally, it might be speculated that PTFE can be introduced between fibers thereby reducing the direct contact area between fibers which would explain the observed drop in thermal conductivity. Of course compression as well as mechanical cycling potentially can lead to a modification of contacts between fibers, i.e. the statistical distribution of contact geometries and thereby thermal conductivity could change. Reversible as well as irreversible modifications of contact geometries – and consequently thermal conductivity could occur under mechanical loading.

Also the observation that the presence of residual liquid water leads to an increase of through-plane thermal conductivity (Burheim et al., 2011) can possibly be explained; the water – with a thermal conductivity of 0.6 W/ m K (Taine & Petit, 1989) – could either improve heat flux at already existing contacts between fibers or even provide new heat flux paths that were 'insulated' by air before the addition of water.

#### **4. Conclusion**

Three commercially available gas diffusion layers of PEM fuel cells were investigated by 3D X-ray computed tomography (CT). The 3D structure of carbon fiber based gas diffusion layers was clearly resolved, but carbon fibers and PTFE could not be discriminated. Further, 3D fiber structures were randomly generated using GeoDict and model structures of layers of equidistant parallel fibers were generated. Binder material – this could be PTFE only or carbon and PTFE – was added to these fiber structures.

The application of analytical models for the estimation of thermal conductivity of heterogeneous materials was discussed. Based on the measured and generated 3D structures, the macroscopic, anisotropic effective thermal conductivities of the gas diffusion layers were computed. The average in-plane thermal conductivity for all structures is by about a factor 4 to 12 larger than the average through-plane thermal conductivity. These results were compared with the available literature data on thermal conductivity of gas diffusion layers and a good agreement was found.

Finally, it was shown that – due to the big difference between the thermal conductivities of PTFE and carbon – the impact of the contact area between carbon fibers and the spatial distribution of PTFE in the contact region between carbon fibers on the macroscopic thermal conductivity is considerable. Therefore an improved characterisation of the contact region between carbon fibers could be a worthwhile objective for future similar studies. This includes also the discrimination of PTFE and carbon in CT datasets.

Overall, it was shown that the computation of thermal conductivity of gas diffusion layers based on 3D structure can lead to an improved understanding of the thermal properties of a GDL. Both approaches presented here have advantages: CT measurements guarantee a realistic 3D structure, while the random generation of 3D structures allows the precise definition of the distribution of all components which can be especially relevant for virtual GDL design.

#### **5. Acknowledgment**

228 Convection and Conduction Heat Transfer

For the model structures with PTFE, the conductivity drops to values below the thermal conductivity of PTFE, which is in the same order of magnitude as thermal conductivities

It has to be mentioned that in the model structures the contact area between touching fibers (distance 0 µm) was defined by the voxel size of the model structures. It is clear that this contact area will have a huge impact on heat flux through the contact. In this context, the presence of carbon binder – which is used for the mechanical stabilization of the GDL – can

Overall, it is obvious from the results presented in Fig. 7 that the contact region between fibers is important for the thermal conductivity of a GDL. When looking back to the models estimating thermal conductivity of heterogeneous materials (section 2.1) and considering the contact area between fibers consisting of PTFE and carbon, both extreme cases – parallel heat flux through carbon and PTFE and heat flux in series through the two components – seem to be potentially realistic for the description of a specific single contact between two fibers in a gas diffusion layer. The macroscopic thermal conductivity of a GDL will depend

Considering the drop of thermal conductivity with the increase of PTFE loading observed experimentally, it might be speculated that PTFE can be introduced between fibers thereby reducing the direct contact area between fibers which would explain the observed drop in thermal conductivity. Of course compression as well as mechanical cycling potentially can lead to a modification of contacts between fibers, i.e. the statistical distribution of contact geometries and thereby thermal conductivity could change. Reversible as well as irreversible modifications of contact geometries – and consequently thermal conductivity

Also the observation that the presence of residual liquid water leads to an increase of through-plane thermal conductivity (Burheim et al., 2011) can possibly be explained; the water – with a thermal conductivity of 0.6 W/ m K (Taine & Petit, 1989) – could either improve heat flux at already existing contacts between fibers or even provide new heat flux

Three commercially available gas diffusion layers of PEM fuel cells were investigated by 3D X-ray computed tomography (CT). The 3D structure of carbon fiber based gas diffusion layers was clearly resolved, but carbon fibers and PTFE could not be discriminated. Further, 3D fiber structures were randomly generated using GeoDict and model structures of layers of equidistant parallel fibers were generated. Binder material – this could be PTFE only or

The application of analytical models for the estimation of thermal conductivity of heterogeneous materials was discussed. Based on the measured and generated 3D structures, the macroscopic, anisotropic effective thermal conductivities of the gas diffusion layers were computed. The average in-plane thermal conductivity for all structures is by about a factor 4 to 12 larger than the average through-plane thermal conductivity. These results were compared with the available literature data on thermal conductivity of gas

Finally, it was shown that – due to the big difference between the thermal conductivities of PTFE and carbon – the impact of the contact area between carbon fibers and the spatial

measured for realistic gas diffusion layers (compare Table 2).

strongly on the statistical distribution of the different contact geometries.

paths that were 'insulated' by air before the addition of water.

carbon and PTFE – was added to these fiber structures.

diffusion layers and a good agreement was found.

play an important role.

could occur under mechanical loading.

**4. Conclusion** 

This work has been carried out within the multiannual programme of the European Commission's Joint Research Centre, as part of the FCPOINT activities. The authors thank Marc Steen (Institute for Energy, European Commission) for critical reading of the manuscript, Gaby Janssen (Energy Research Centre of the Netherlands, ECN) for the provision of gas diffusion layers and GE sensing and inspection technologies/phoenix X-ray (Wunstorf, Germany) for X-ray computed tomography data.

#### **6. References**


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**11** 

**Analytical Methods for** 

**in Permafrost Areas** 

*Melnikov Permafrost Institute SB RAS* 

Rev I. Gavriliev

(1)

*Russia* 

**Estimating Thermal Conductivity** 

**of Multi-Component Natural Systems** 

Frozen soils consist of soil solids, ice, unfrozen water, and gas (vapour). The solid particles vary in size and composition and may be composed of one or more minerals or of organic material. Based on particle size, soils are classified into soil types which vary between the many classification systems in use throughout the world. The classification which is most generally used in Russia is that of V.V. Okhotin (Sergeev, 1971), with the basic soil types being sand, sand-silt, silt-clay, and clay which are further subdivided into a large number of subtypes. Soils that have been subject to repeated cycles of freezing and thawing generally

The bound water is structurally and energetically heterogeneous. Water bonding to the mineral particles is provided predominantly by the active centres on the surface and the exchange cations. The most important active centres for water adsorption in the crystalline lattice of clay minerals are hydroxyl groups and coordinately unsaturated atoms of oxygen,

In quantitative terms, it is an undeniable fact that the pore water freezes over a range of negative temperatures rather than at a single temperature, depending on soil moisture content and solute concentration. This is due to distortion of the bound water structure by the active centres on the particle surfaces and dissolved ions, resulting in a kinetic barrier

The phase composition of water (or solution) changes with temperature following the dynamic equilibrium state principle established by Tsytovich (1945) and experimentally confirmed by Nersesova (1953). This principle states that the amount of unfrozen water for a given soil type (non-saline) is a function of the temperature below 0°C and is virtually independent of the total soil moisture content. It is quantitatively described by the equation

> uw 0 2 <sup>1</sup> W W A' 1 , 1 a' t b' t

= + <sup>⎢</sup> <sup>−</sup> <sup>⎥</sup> ⎢⎣ + Δ+ Δ ⎥⎦

⎡ ⎤

**1. Introduction** 

have higher silt contents.

(Ivanov, 1962):

silicon, aluminium and other elements.

which makes water crystallization difficult.


## **Analytical Methods for Estimating Thermal Conductivity of Multi-Component Natural Systems in Permafrost Areas**

Rev I. Gavriliev *Melnikov Permafrost Institute SB RAS Russia* 

#### **1. Introduction**

232 Convection and Conduction Heat Transfer

Veyret, D. & Tsotridis, G. (2010). Numerical determination of the effective thermal

Vie, P. J. S. & Kjelstrup, S. (2004). Thermal conductivities from temperature profiles in the

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*of Heat and Mass Transfer* Vol. 49, No. 17-18, pp. 3075-3083, ISSN 00179310 Wang, J., Carson, J. K., North, M. F. & Cleland, D. J. (2008). A new structural model of

Wen, C. Y. & Huang, G. W. (2008). Application of a thermally conductive pyrolytic graphite

Wiegmann, A. & Zemitis, A. (2006). EJ-HEAT: A fast explicit jump harmonic averaging

Zamel, N., Li, X., Shen, J., Becker, J. & Wiegmann, A. (2010). Estimating effective thermal

03787753

00134686

Vol. 94

2397, ISSN 00179310

No. 1, pp. 132-140, ISSN 03787753

No. 13, pp. 3994-4006, ISSN 00092509

conductivity of fibrous materials. Application to proton exchange membrane fuel cell gas diffusion layers. *Journal of Power Sources* Vol. 195, No. 5, pp. 1302-1307, ISSN

polymer electrolyte fuel cell. *Electrochimica Acta* Vol. 49, No. 7, pp. 1069-1077, ISSN

the effective thermal conductivity of heterogeneous materials. *International Journal* 

effective thermal conductivity for heterogeneous materials with co-continuous phases. *International Journal of Heat and Mass Transfer* Vol. 51, No. 9-10, pp. 2389-

sheet to thermal management of a PEM fuel cell. *Journal of Power Sources* Vol. 178,

solver for the effective heat conductivity of composite materials. *Fraunhofer ITWM*

conductivity in carbon paper diffusion media. *Chemical Engineering Science* Vol. 65,

Frozen soils consist of soil solids, ice, unfrozen water, and gas (vapour). The solid particles vary in size and composition and may be composed of one or more minerals or of organic material. Based on particle size, soils are classified into soil types which vary between the many classification systems in use throughout the world. The classification which is most generally used in Russia is that of V.V. Okhotin (Sergeev, 1971), with the basic soil types being sand, sand-silt, silt-clay, and clay which are further subdivided into a large number of subtypes. Soils that have been subject to repeated cycles of freezing and thawing generally have higher silt contents.

The bound water is structurally and energetically heterogeneous. Water bonding to the mineral particles is provided predominantly by the active centres on the surface and the exchange cations. The most important active centres for water adsorption in the crystalline lattice of clay minerals are hydroxyl groups and coordinately unsaturated atoms of oxygen, silicon, aluminium and other elements.

In quantitative terms, it is an undeniable fact that the pore water freezes over a range of negative temperatures rather than at a single temperature, depending on soil moisture content and solute concentration. This is due to distortion of the bound water structure by the active centres on the particle surfaces and dissolved ions, resulting in a kinetic barrier which makes water crystallization difficult.

The phase composition of water (or solution) changes with temperature following the dynamic equilibrium state principle established by Tsytovich (1945) and experimentally confirmed by Nersesova (1953). This principle states that the amount of unfrozen water for a given soil type (non-saline) is a function of the temperature below 0°C and is virtually independent of the total soil moisture content. It is quantitatively described by the equation (Ivanov, 1962):

$$\mathbf{W}\_{\text{uw}} = \mathbf{W}\_0 + \mathbf{A}' \left[ \frac{1}{1 + \mathbf{a}' |\Delta \mathbf{t}| + \mathbf{b}' \Delta \mathbf{t}^2} - 1 \right] \tag{1}$$

Analytical Methods for Estimating Thermal Conductivity

the porosity m2 (dry density γs):

heat transfer across the contact spot, W/(m•K):

and the fill (air, water and ice), respectively.

ad 2

at δ ≤ 1

1996).

of Multi-Component Natural Systems in Permafrost Areas 235

number is constant and equal to 6; the relation between the thermal conductivity and porosity is realized by changing the particle shape at various size ratios of the ellipsoids of revolution. At a/R ≥ 1, a contact spot appears automatically in the model, which represents rigid bonding between the particles that provides hard, monolithic rock structure (Gavriliev,

Fig. 1. Particle shapes in the soil thermal conductivity model at different semi-axes ratios of ellipsoids δ = a/R: 1 – faceted (δ > 1); 2 – spherical (δ = 1); 3 – worn (δ < 1); 4 – cruciate (δ < 1) All calculations are made in terms of the parameter δ = a/R, which is a unique function of

where λmod is the resulting thermal conductivity of the model and ϕsc is the correction for

mod ad 2 2

λ =λ + <sup>⎢</sup> ⎜ ⎟ − π <sup>⎥</sup> <sup>⎢</sup> ⎝ ⎠ + ϑ− ϑ <sup>⎥</sup> <sup>⎣</sup> <sup>⎦</sup>

where 2 1 ϑ=λ λ ≤ϑ≤ ;0 1; λad is the thermal conductivity of the system where the elementary cell is divided by adiabatic planes; the subscripts "1" and "2" refer to the particle

The thermal conductivity of the model, λad, is given by the following equations:

4 1 1

( ) ( )

1 1 K 4arcsin / X 1 ln 1 1 XK X 2 1

2 1 1 1

1 1 1 H arcsin X 1 1 ln 2 KK 1 K

<sup>λ</sup> πδ <sup>π</sup> <sup>⎡</sup> ⎛ ⎞ <sup>=</sup> − −δ +δ δ + −δ − <sup>⎢</sup> ⎜ ⎟ <sup>−</sup> λ δ <sup>⎢</sup> <sup>δ</sup> <sup>−</sup> <sup>δ</sup> <sup>⎣</sup> ⎝ ⎠

⎛ ⎞ δ δ <sup>⎤</sup> ⎡ ⎤ <sup>−</sup> πδ − −δ + − × − ⎜ ⎟ <sup>⎥</sup> ⎢ ⎥ δ π <sup>⎥</sup> <sup>−</sup> <sup>δ</sup> ⎝ ⎠ <sup>⎦</sup> ⎣ ⎦

1.3 1 1 sin <sup>m</sup> , 1 0.5 0.26 ⎡ ⎛ ⎞ ⎤

( )

λ =λ +ϕ mod sc , (2)

(3)

, (4)

where Δt = t – tf; tf is the initial freezing temperature of water; W0 is the equilibrium moisture content at tf; and A', a' and b' are the characteristic soil parameters. For a narrow range of freezing temperatures (|Δt| ≤ 10°C), Eq. (1) can be simplified by assuming b' = 0.

The thermodynamic instability of the phase composition of water in frozen soils causes their properties to be highly dynamic at subzero temperatures. The presence of unfrozen water below 0°C provides conditions for water migration during freezing. This results in the formation of cryostructures and cryotextures that, in turn, cause the anisotropy of soil thermal and other properties. All cryostructural types can be grouped into three board classes: massive, layered, and reticulate (Everdingen, 2002).

Model calculations generally consider heat conduction in frozen soils. It is characterized by an effective value of the heat flux transferred by the solid particles and interstitial medium (ice, water and vapour) and through the contacts. It depends on multiple variables which reflect the origin and history of the soil, including moisture content, temperature, dry density, grain size distribution, mineralogical composition, salinity, structure, and texture.

A large number of theoretical models and methods were developed for estimating the thermal conductivity of various particulate materials. However, most of them do not address the structural transformations and their validity is limited to a narrow range of material's density. In permafrost investigations, it is essential that properties of snow, soils and rocks be studied in relation to the history of sediment formation through geologic time. Therefore, a universal theoretical model with changing particle shapes was proposed by the present author to describe the processes of rock formation, snow compaction and glacierization with account for diagenetic and post-diagenetic structural modifications, as well the processes of rock weathering and soil formation. A detailed description of the model was given in earlier publications (Gavril'ev, 1992, Gavriliev, 1996, Gavriliev, 1998). Since then, the model has been amended and improved. We therefore find it necessary to present a brief description of the geometric models and the final predictive equations.

#### **2. Theoretical model accounting for structural transformations of sediments**

#### **2.1 Soils and sedimentary rocks**

A model for estimating the thermal conductivity of soils and sedimentary rocks should take into account the changes in particle shape over the entire range of porosity from 0 to 1 in order to consider the entire cycle of sediment changes since its deposition. In developing such a model, it should be kept in mind that mineral rock particles undergo some kind of plastic deformation through geologic time, gradually filling the entire space. Particles bind together at the contacts ("the contact spot") and rigid crystal bindings develop between the particles.

Following the real picture of rock weathering and particle shape changes through diagenesis, the author has proposed a model, which presents the solid component in a cubic cell by three intersecting ellipsoids of revolution (Fig. 1) (Gavril'ev, 1992).

In this scheme, depending on the semi-axes ratio of the ellipsoids a/R, the porosity of the system varies from 0 to 1 and the particle attains a variety of shapes, such as cubical, faceted, spherical, worn, and cruciate. This logically represents the real changes in particle shape through the sedimentary history, i.e., the key requirement to the model - adequate representation of the real system – is met. In this scheme, the particles always maintain contacts with each other and the system remains stable and isotropic. The coordinate

where Δt = t – tf; tf is the initial freezing temperature of water; W0 is the equilibrium moisture content at tf; and A', a' and b' are the characteristic soil parameters. For a narrow range of freezing temperatures (|Δt| ≤ 10°C), Eq. (1) can be simplified by assuming b' = 0. The thermodynamic instability of the phase composition of water in frozen soils causes their properties to be highly dynamic at subzero temperatures. The presence of unfrozen water below 0°C provides conditions for water migration during freezing. This results in the formation of cryostructures and cryotextures that, in turn, cause the anisotropy of soil thermal and other properties. All cryostructural types can be grouped into three board

Model calculations generally consider heat conduction in frozen soils. It is characterized by an effective value of the heat flux transferred by the solid particles and interstitial medium (ice, water and vapour) and through the contacts. It depends on multiple variables which reflect the origin and history of the soil, including moisture content, temperature, dry density, grain size distribution, mineralogical composition, salinity, structure, and texture. A large number of theoretical models and methods were developed for estimating the thermal conductivity of various particulate materials. However, most of them do not address the structural transformations and their validity is limited to a narrow range of material's density. In permafrost investigations, it is essential that properties of snow, soils and rocks be studied in relation to the history of sediment formation through geologic time. Therefore, a universal theoretical model with changing particle shapes was proposed by the present author to describe the processes of rock formation, snow compaction and glacierization with account for diagenetic and post-diagenetic structural modifications, as well the processes of rock weathering and soil formation. A detailed description of the model was given in earlier publications (Gavril'ev, 1992, Gavriliev, 1996, Gavriliev, 1998). Since then, the model has been amended and improved. We therefore find it necessary to present a brief description of the geometric models and the final predictive equations.

**2. Theoretical model accounting for structural transformations of sediments**

A model for estimating the thermal conductivity of soils and sedimentary rocks should take into account the changes in particle shape over the entire range of porosity from 0 to 1 in order to consider the entire cycle of sediment changes since its deposition. In developing such a model, it should be kept in mind that mineral rock particles undergo some kind of plastic deformation through geologic time, gradually filling the entire space. Particles bind together at the contacts ("the contact spot") and rigid crystal bindings develop between the

Following the real picture of rock weathering and particle shape changes through diagenesis, the author has proposed a model, which presents the solid component in a cubic

In this scheme, depending on the semi-axes ratio of the ellipsoids a/R, the porosity of the system varies from 0 to 1 and the particle attains a variety of shapes, such as cubical, faceted, spherical, worn, and cruciate. This logically represents the real changes in particle shape through the sedimentary history, i.e., the key requirement to the model - adequate representation of the real system – is met. In this scheme, the particles always maintain contacts with each other and the system remains stable and isotropic. The coordinate

cell by three intersecting ellipsoids of revolution (Fig. 1) (Gavril'ev, 1992).

classes: massive, layered, and reticulate (Everdingen, 2002).

**2.1 Soils and sedimentary rocks** 

particles.

number is constant and equal to 6; the relation between the thermal conductivity and porosity is realized by changing the particle shape at various size ratios of the ellipsoids of revolution. At a/R ≥ 1, a contact spot appears automatically in the model, which represents rigid bonding between the particles that provides hard, monolithic rock structure (Gavriliev, 1996).

Fig. 1. Particle shapes in the soil thermal conductivity model at different semi-axes ratios of ellipsoids δ = a/R: 1 – faceted (δ > 1); 2 – spherical (δ = 1); 3 – worn (δ < 1); 4 – cruciate (δ < 1)

All calculations are made in terms of the parameter δ = a/R, which is a unique function of the porosity m2 (dry density γs):

λ =λ +ϕ mod sc , (2)

where λmod is the resulting thermal conductivity of the model and ϕsc is the correction for heat transfer across the contact spot, W/(m•K):

$$
\lambda\_{\rm mod} = \lambda\_{\rm ad} \left[ 1 + \left( \frac{1.3}{1 + 0.5 \,\text{s} - 0.26 \,\text{g}^2} - 1 \right) \sin \pi \text{m}\_2 \right], \tag{3}
$$

where 2 1 ϑ=λ λ ≤ϑ≤ ;0 1; λad is the thermal conductivity of the system where the elementary cell is divided by adiabatic planes; the subscripts "1" and "2" refer to the particle and the fill (air, water and ice), respectively.

The thermal conductivity of the model, λad, is given by the following equations: at δ ≤ 1

$$\begin{split} \frac{\lambda\_{\text{ad}}}{\lambda\_{2}} &= 1 - \frac{\pi \delta}{2} - \delta^{2} \mathcal{H} + \delta \arcsin \delta \beta \mathcal{X} + \frac{\pi}{\delta \mathcal{K}\_{1}} \bigg[ (1 - \delta) \Big( 1 - \frac{1}{\delta \mathcal{K}\_{1}} \ln \left| \frac{1}{1 - \delta \mathcal{K}\_{1}} \right| \right] - \\ &- \left( 1 - \delta^{4} \right) \left( \frac{1}{\mathcal{X}} + \frac{1}{\delta \mathcal{K}\_{1}} \ln \left| 1 - \frac{\delta \mathcal{K}\_{1}}{\mathcal{X}} \right| \right) \Bigg[ \times \left[ 1 - \frac{4 \arcsin \delta \,/\mathcal{X} - \pi \delta}{2 \pi (1 - \delta)} \right] \end{split} \tag{4}$$

Analytical Methods for Estimating Thermal Conductivity

( ) <sup>2</sup> <sup>2</sup> <sup>2</sup>

<sup>1</sup> <sup>2</sup> <sup>2</sup>

at δ ≥ 1

The dry density of the soil is

where ρs is the solids unit weight.

statistical particle distribution.

(Gavriliev, 1989):

**2.2 Snow** 

the volume fraction ms and dry density γs by

of Multi-Component Natural Systems in Permafrost Areas 237

1 2 1 11 <sup>1</sup> m 1 4 arcsin . 6X X 16X 2 X πδ ⎛ ⎞ δ − δ +δ ⎛ ⎞ ⎛ ⎞ =−+ ⎜ ⎟ ⎜ ⎟ + × δ −π ⎜ ⎟ δ δ ⎝ ⎠ ⎝ ⎠ + δ ⎝ ⎠

> 22 2 cc c 2 sc 22 2 2 rr r m 2 11 . 4 RR R <sup>⎡</sup> ⎛ ⎞ <sup>⎤</sup> <sup>π</sup> ⎛ ⎞ = − δ− − − <sup>⎢</sup> ⎜ ⎟ <sup>⎥</sup> ⎜ ⎟ <sup>⎢</sup> <sup>δ</sup> <sup>⎥</sup> ⎝ ⎠ <sup>⎣</sup> ⎝ ⎠ <sup>⎦</sup>

The above equations can be used to calculate the thermal conductivity of soils and sedimentary rocks in the saturated frozen and unfrozen states, as well as in the air-dry state in relation to the porosity m2 and the thermal conductivity λ1 of the solid particles (a twocomponent system). The predictions obtained are presented as nomograms in Fig. 2. It should be noted that in this case, the porosity m2 refers to the entire volume fraction of the soil or rock which is completely filled either with ice, water, or air. This porosity is related to

The model assumes that the material consists of mineral particles of the same composition. However, naturally occurring soils always contain particles of various compositions and they can be treated in modelling as multi-component heterogeneous systems with a

In computations based on the universal model, the average thermal conductivity of soil solid particles may be used, which is approximately estimated in terms of the thermal conductivity and volume fraction of constituent minerals according to the equation

> <sup>1</sup> 0.5 m , <sup>=</sup> m

> > j 1 j

=

∑

n <sup>1</sup> j j <sup>n</sup> <sup>j</sup> j 1

rocks characterized by the plane contacts between mineral aggregates.

<sup>⎡</sup> <sup>⎤</sup> <sup>⎢</sup> <sup>⎥</sup> <sup>⎢</sup> <sup>⎥</sup> λ= λ + <sup>⎢</sup> <sup>⎥</sup> <sup>⎢</sup> <sup>⎥</sup> <sup>⎢</sup> <sup>λ</sup> <sup>⎥</sup> <sup>⎣</sup> <sup>⎦</sup> ∑

where λj and mj are the thermal conductivity and volume fraction of the j-th mineral of the soil, respectively. This equation can also be used for calculating the thermal conductivity of

In snowpack, the structural changes of ice crystals occur continuously throughout the winter. The thermodynamic processes in snowpack result in a multi-branch openwork structure of

The increase in the volume fraction of the solids due to the contact spot is expressed by

(10)

(11)

(14)

γd 1 sc s =+ ρ (mm , ) (12)

m 1m 1m m 1 . 2 s 1 sc s s = − = − − = −γ ρ (13)

at δ ≥ 1

$$\begin{split} \frac{\lambda\_{\text{ad}}}{\lambda\_{2}} &= 1 - \frac{\pi}{4} - \delta^{2} \mathcal{H} + \frac{\pi}{2\mathcal{K}\_{1}} \left( \delta^{2} - \frac{1}{\delta^{2}} \right) \times \left( \frac{1}{\mathcal{K}\_{1}} \ln \left| 1 - \frac{\delta \mathcal{K}\_{1}}{\mathcal{K}} \right| + \frac{\delta}{\mathcal{X}} \right) + \\ &+ \frac{\delta \mathcal{K}\_{1}}{\mathcal{X} - \delta \mathcal{K}\_{1}} \left( \delta \arcsin \mathbf{1} / \mathcal{X} - \frac{\pi}{4} \right) \end{split} \tag{5}$$

where δ = a/R; <sup>2</sup> X1 ; = +δ <sup>1</sup> 1 1 <sup>1</sup> H ln 1 K 1 ; 2K K π ⎛ ⎞ = −+ ⎜ ⎟ ⎝ ⎠ 2 1 1 K1 . <sup>λ</sup> = − <sup>λ</sup>

The correction factor ϕsc is given by

$$\mathbf{p}\_{\rm sc} = \frac{\pi \lambda\_1 \mathcal{S}^2}{2} \left[ \frac{\mathbf{r}\_c^2}{2\mathbf{R}^2 \mathcal{S}^2} + \frac{9}{\mathbf{K}\_1} \left( 1 - \sqrt{1 - \frac{\mathbf{r}\_c^2}{\mathbf{R}^2 \mathcal{S}^2}} \right) + \frac{9}{\mathbf{K}\_1^2} \ln \frac{1 - \mathbf{K}\_1}{1 - \mathbf{K}\_1 \sqrt{1 - \frac{\mathbf{r}\_c^2}{\mathbf{R}^2 \mathcal{S}^2}}} \right],\tag{6}$$

where rc is the radius of the contact spot between the particles.

It is assumed in Eq. (6) that the spot contact between particles is formed of the same material as the particle by its flattening at high pressure or by its squeezing (solution and crystallization) due to selective growth of cement in sandstones (quartz cement grows on quartz particles and feldspar on feldspar particles). In a general case however, the contact spot may consist of a foreign material resulting, for example, from precipitation of salts from solution at the particle contacts. In this case, the correction factor ϕsc is given by

$$\begin{split} \boldsymbol{\phi}\_{\text{sc}} &= \frac{\pi \mathbf{a}}{2\,\text{R}^2} \Bigg[ \frac{\lambda\_3}{\mathbf{K}\_1^2} \ln \left| \frac{1 - \mathbf{K}\_1 \sqrt{1 - \mathbf{r}\_c^2/\mathbf{a}}}{\left(1 - \mathbf{K}\_1\right)} \right| - \frac{\lambda\_2}{\mathbf{K}\_2^2} \ln \left| \frac{1 - \mathbf{K}\_2 \sqrt{1 - \mathbf{r}\_c^2/\mathbf{a}^2}}{\left(1 - \mathbf{K}\_2\right)} \right| + \\ &+ \left( \frac{\lambda\_2}{\mathbf{K}\_2} - \frac{\lambda\_3}{\mathbf{K}\_1} \right) \left(1 - \sqrt{1 - \mathbf{r}\_c^2/\mathbf{a}^2} \right) \Bigg] \end{split} \tag{7}$$

where K2 = 1 - λ3/λ1; λ1, λ2 and λ3 are the thermal conductivities of the solid, medium and contact spot (contact cement), respectively.

The relative size of the contact spot is expressed in terms of the system's porosity as:

$$\frac{\mathbf{r}\_c}{\mathbf{R}} = 1.69 \sqrt{1 - \sqrt[3]{\frac{\pi}{6 \left(1 - \mathbf{m}\_2\right)}}}\ . \tag{8}$$

The soil porosity m2 or the volume fraction of the mineral particle m1 is a unique function of the parameter δ and is given by the following equations: at δ ≤ 1

$$\mathbf{m}\_1 = \frac{\pi \delta^2}{6} \left[ 1 - \frac{1 - \delta^2}{\delta} + \frac{1 - \delta}{\chi} \left( \frac{1 + \delta}{\delta} - \frac{1}{\chi^2} + 3 \right) \right],\tag{9}$$

#### at δ ≥ 1

236 Convection and Conduction Heat Transfer

2

4 2K K XX

, (5)

, (6)

, (7)

(9)

2

1

R

1

K1 . <sup>λ</sup> = − <sup>λ</sup>

ad 2 2 1

1 1 <sup>1</sup> H ln 1 K 1 ; 2K K π ⎛ ⎞ = −+ ⎜ ⎟ ⎝ ⎠

1 1 <sup>K</sup> 1 H ln 1

<sup>λ</sup> π π ⎛ ⎞ ⎛ ⎞ <sup>δ</sup> <sup>δ</sup> <sup>=</sup> − −δ + δ − × − + + ⎜ ⎟ ⎜ ⎟ <sup>λ</sup> ⎝ ⎠ <sup>δ</sup> ⎝ ⎠

1 1 c c sc 2 2 22 2 <sup>2</sup> <sup>1</sup> <sup>1</sup> <sup>c</sup> <sup>1</sup> 2 2

It is assumed in Eq. (6) that the spot contact between particles is formed of the same material as the particle by its flattening at high pressure or by its squeezing (solution and crystallization) due to selective growth of cement in sandstones (quartz cement grows on quartz particles and feldspar on feldspar particles). In a general case however, the contact spot may consist of a foreign material resulting, for example, from precipitation of salts from

<sup>⎡</sup> <sup>⎤</sup> <sup>⎢</sup> <sup>⎥</sup> ⎛ ⎞ πλ δ <sup>⎢</sup> <sup>ϑ</sup> ϑ − <sup>⎥</sup> ϕ = + −− + ⎜ ⎟ <sup>⎢</sup> ⎜ ⎟ <sup>⎥</sup> δ δ <sup>⎢</sup> ⎝ ⎠ − − <sup>⎥</sup> <sup>⎢</sup> <sup>δ</sup> <sup>⎥</sup> <sup>⎣</sup> <sup>⎦</sup>

r r 1 K 1 1 ln 2 K 2R R K <sup>r</sup> 1K 1

( ) ( )

2

<sup>π</sup> = − <sup>−</sup> (8)

3 1 c 2 2 c

<sup>a</sup> 1K 1ra 1K 1r a ln ln 2R K 1 K K 1 K

where K2 = 1 - λ3/λ1; λ1, λ2 and λ3 are the thermal conductivities of the solid, medium and

( ) <sup>c</sup> <sup>3</sup>

<sup>r</sup> 1.69 1 . R 6 <sup>1</sup> <sup>m</sup>

The soil porosity m2 or the volume fraction of the mineral particle m1 is a unique function of

1 2 1 11 1 m 1 3 , 6 X <sup>X</sup> πδ <sup>⎡</sup> −δ −δ +δ <sup>⎤</sup> ⎛ ⎞ = − + −+ <sup>⎢</sup> ⎜ ⎟ <sup>⎥</sup> ⎢⎣ δ δ ⎝ ⎠ ⎥⎦

<sup>⎡</sup> <sup>π</sup> <sup>λ</sup> −− −− <sup>λ</sup> ϕ = <sup>⎢</sup> <sup>−</sup> <sup>+</sup> <sup>⎢</sup> − − <sup>⎣</sup>

1 2 1 2

2 2 2

2 11

<sup>K</sup> arcsin 1 / X X K <sup>4</sup>

2 2 2

solution at the particle contacts. In this case, the correction factor ϕsc is given by

( )

1 1r a

2 2

c

The relative size of the contact spot is expressed in terms of the system's porosity as:

sc 2 2 2

2 3 2 2

⎛ ⎞ λ λ ⎤ + − −− ⎜ ⎟ <sup>⎥</sup> ⎝ ⎠ ⎥⎦

2 1

the parameter δ and is given by the following equations:

K K

contact spot (contact cement), respectively.

at δ ≤ 1

where rc is the radius of the contact spot between the particles.

<sup>δ</sup> ⎛ ⎞ <sup>π</sup> +δ − ⎜ ⎟ − δ ⎝ ⎠

1 1

where δ = a/R; <sup>2</sup> X1 ; = +δ <sup>1</sup>

The correction factor ϕsc is given by

at δ ≥ 1

$$\mathbf{m}\_{1} = \frac{\pi \delta^{2}}{6} \left( 1 - \frac{\delta^{2} - 1}{\delta \mathcal{X}} \right) + \frac{\delta \left( 1 + 2 \delta^{2} \right)}{16 \mathcal{X}^{2}} \left( \frac{1}{\mathcal{X}} + \frac{1}{\sqrt{2 + \delta^{2}}} \right) \times \left( 4 \delta \arcsin \frac{1}{\delta \mathcal{X}} - \pi \right). \tag{10}$$

The increase in the volume fraction of the solids due to the contact spot is expressed by

$$\mathbf{m}\_{\rm sc} = \frac{\pi}{4} \left[ \frac{\mathbf{r}\_{\rm c}^2}{\mathbf{R}^2} - 2 \left( \boldsymbol{\delta}^2 - \frac{\mathbf{r}\_{\rm c}^2}{\mathbf{R}^2} \right) \left( 1 - \sqrt{1 - \frac{\mathbf{r}\_{\rm c}^2}{\mathbf{R}^2 \boldsymbol{\delta}^2}} \right) \right]. \tag{11}$$

The dry density of the soil is

$$\gamma\_{\rm d} = \left(\mathbf{m}\_1 + \mathbf{m}\_{\rm sc}\right) \rho\_{\rm s} \,\,\,\,\,\tag{12}$$

where ρs is the solids unit weight.

The above equations can be used to calculate the thermal conductivity of soils and sedimentary rocks in the saturated frozen and unfrozen states, as well as in the air-dry state in relation to the porosity m2 and the thermal conductivity λ1 of the solid particles (a twocomponent system). The predictions obtained are presented as nomograms in Fig. 2. It should be noted that in this case, the porosity m2 refers to the entire volume fraction of the soil or rock which is completely filled either with ice, water, or air. This porosity is related to the volume fraction ms and dry density γs by

$$\mathbf{m}\_2 = \mathbf{1} - \mathbf{m}\_s = \mathbf{1} - \mathbf{m}\_1 - \mathbf{m}\_{sc} = \mathbf{1} - \boldsymbol{\gamma}\_s / \mathbf{p}\_s \; . \tag{13}$$

The model assumes that the material consists of mineral particles of the same composition. However, naturally occurring soils always contain particles of various compositions and they can be treated in modelling as multi-component heterogeneous systems with a statistical particle distribution.

In computations based on the universal model, the average thermal conductivity of soil solid particles may be used, which is approximately estimated in terms of the thermal conductivity and volume fraction of constituent minerals according to the equation (Gavriliev, 1989):

$$\overline{\lambda\_1} = 0.5 \left[ \sum\_{j=1}^n \lambda\_j \mathbf{m}\_j + \frac{1}{\sum\_{j=1}^n \frac{\mathbf{m}\_j}{\lambda\_j}} \right],\tag{14}$$

where λj and mj are the thermal conductivity and volume fraction of the j-th mineral of the soil, respectively. This equation can also be used for calculating the thermal conductivity of rocks characterized by the plane contacts between mineral aggregates.

#### **2.2 Snow**

In snowpack, the structural changes of ice crystals occur continuously throughout the winter. The thermodynamic processes in snowpack result in a multi-branch openwork structure of

Analytical Methods for Estimating Thermal Conductivity

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7

9 – 4.5; 10 – 5.0; 11 – 6.0; 12 – 7.0

thermal conductivity of calm air.

given by Vargaftik (1963):

**Thermal conductivity (**

**λ), W/(m•K)**

of Multi-Component Natural Systems in Permafrost Areas 239

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

**Porosity (m2)**

(c)

These changes in snow structure through the whole cycle from deposition to glacier formation can be fairly well represented by the same model shown in Fig. 1 (Gavrilyev, 1996a). But the calculations should take into account the heat convection by vapour diffusion due to a temperature gradient in the snow. This can be done by substituting in Eqs. (3) - (6) the effective thermal conductivity of air in snow for its thermal conductivity (λa) which is given by

( <sup>0</sup> ) s 0 ae a <sup>2</sup>

⎛ ⎞ <sup>⎡</sup> <sup>−</sup> <sup>⎤</sup> λ =λ + ⎜ ⎟ − × <sup>⎢</sup> <sup>⎥</sup> ⎝ ⎠ <sup>⎣</sup> <sup>⎦</sup>

where e0 = 6.1⋅102 Pa is the saturation vapour pressure at 0°С (T0 = 273 K); Rv = 4.6⋅102 J/(kg•K) is the gas constant of water vapour; T is the absolute temperature, K; L is the latent heat of ice sublimation; Ds is the diffusion coefficient of water vapour in snow; and λa is the

The thermal conductivity of air in relation to temperature may be calculated by an equation

0 a a

⎛ ⎞ λ =λ ⎜ ⎟ ⎝ ⎠

0.82

0 <sup>T</sup> , <sup>T</sup>

v v v 0 LD e L LT T 1 exp , R T R T RTT

(15)

(16)

Fig. 2. Nomograms for calculating the thermal conductivity of soils and rocks in dry (a), saturated unfrozen (b) and frozen (c) states in terms of total porosity m2 and solids thermal conductivity λ1 (W/(m•K)): 1 – 0.5; 2 – 1.0; 3 – 1.5; 4 – 2.0; 5 – 2.5; 6 – 3.0; 7 – 3.5; 8 – 4.0;

contacting ice crystals with shapes that continuously change throughout the period of snow existence.

**Porosity (m2)** (b)

contacting ice crystals with shapes that continuously change throughout the period of snow

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 **Porosity (m2)**

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 **Porosity (m2)**

(b)

(a)

existence.

0

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7

**Thermal conductivity (λ), W/(m•K)**

0.5

1

1.5

2

2.5

**Thermal conductivity (λ), W/(m•K)**

3

3.5

4

4.5

5

Fig. 2. Nomograms for calculating the thermal conductivity of soils and rocks in dry (a), saturated unfrozen (b) and frozen (c) states in terms of total porosity m2 and solids thermal conductivity λ1 (W/(m•K)): 1 – 0.5; 2 – 1.0; 3 – 1.5; 4 – 2.0; 5 – 2.5; 6 – 3.0; 7 – 3.5; 8 – 4.0; 9 – 4.5; 10 – 5.0; 11 – 6.0; 12 – 7.0

These changes in snow structure through the whole cycle from deposition to glacier formation can be fairly well represented by the same model shown in Fig. 1 (Gavrilyev, 1996a). But the calculations should take into account the heat convection by vapour diffusion due to a temperature gradient in the snow. This can be done by substituting in Eqs. (3) - (6) the effective thermal conductivity of air in snow for its thermal conductivity (λa) which is given by

$$
\lambda\_{\rm ae} = \lambda\_{\rm a} + \frac{\rm LD\_{s}}{R\_{\rm v}T^{2}} \left(\frac{\rm L}{R\_{\rm v}T} - 1\right) \times \exp\left[\frac{\rm L\left(T - T\_{0}\right)}{R\_{\rm v}T\_{0}T}\right] \tag{15}
$$

where e0 = 6.1⋅102 Pa is the saturation vapour pressure at 0°С (T0 = 273 K); Rv = 4.6⋅102 J/(kg•K) is the gas constant of water vapour; T is the absolute temperature, K; L is the latent heat of ice sublimation; Ds is the diffusion coefficient of water vapour in snow; and λa is the thermal conductivity of calm air.

The thermal conductivity of air in relation to temperature may be calculated by an equation given by Vargaftik (1963):

$$
\lambda\_{\mathbf{a}} = \lambda\_{\mathbf{a}}^{0} \left(\frac{\mathbf{T}}{\mathbf{T}\_{0}}\right)^{0.82},
\tag{16}
$$

Analytical Methods for Estimating Thermal Conductivity

**on thermal conductivity of frozen soils** 

cube) developed a similar equation of the type:

range of inclusion contents: 0 ≤ m1 ≤ 1.

2

cobble inclusions may affect the accuracy of calculations.

2

and ( ) ( ) 22 22 p 1ba 1ca =− − – is the modulus of the integral.

where Kf is the shape factor of particles or inclusions.

following generalized form (Gavriliev, 1986):

In Eq. (20), the inclusion shape factor, Kf, is

form (Ovchinnikov, 1971)

consider two examples.

of Multi-Component Natural Systems in Permafrost Areas 241

**3. Effects of coarse inclusions and the layered and reticulate cryostructures** 

For the thermal conductivity of media containing spherical and cubic inclusions with no contacts (or with point contacts), Maxwell (1873) (for a sphere) and Odelevsky (1951) (for a

( )

where (as before) the subscripts "1" and "2" refer to the inclusions (particles) and the medium, respectively. For the cubical particle shape, Eq. (19) is formally valid across the

The advantage of Eq. (19) is its simplicity. In some cases, Eq. (19) is applicable to permafrost problems, for example, for estimating the thermal conductivity of soils with a cryostructure or of soils containing gravel- or cobble-size inclusions. However, at large differences between the λ1 and λ2 values, such as in air-dry soils, the degree of roundness of gravel and

For a more general formulation of the problem, an ellipsoidal particle shape may be considered in Eq. (19), since with the change in the ratio of semi-axes the particles transform into other figures, such as a sphere, plate, or cylinder. Eq. (19) may be presented in the

( )

where a, b, and c are the semi-axes of the ellipsoids (a > b > c); and C(0) is the integral of the

2 2 3 2 <sup>2</sup> 1 p sin tg E( ,p) C(0) a 1p <sup>⎡</sup> <sup>⎤</sup> − ψ ψ− ψ <sup>=</sup> <sup>⎢</sup> <sup>⎥</sup> <sup>⎢</sup> <sup>−</sup> <sup>⎥</sup> <sup>⎣</sup> <sup>⎦</sup>

E(ψ, p) is the elliptic integral of the second kind, 2 2 ψ= − arcsin 1 c a – is the amplitude

The elliptic integral E(ψ, p) is tabulated, and the shape factor of inclusions can be readily found from the ratio of the particle dimensions a, b, and c. For practical purposes, calculations can be limited to the more simple case of ellipsoids of revolution. Then, the integral C(0) is expressed in terms of elementary functions (Carslaw & Jaeger, 1959). Let us

2 f 112

1 , K1m ⎡ λ −λ ⎤ λ=λ + <sup>⎢</sup> <sup>⎥</sup> ⎢⎣ λ + − λ −λ ⎥⎦

( )( ) 121

K abcC(0), <sup>f</sup> = (21)

m

2 1 1 2

1 , 0.33 1 m ⎡ λ −λ ⎤ λ=λ + <sup>⎢</sup> <sup>⎥</sup> ⎢⎣ λ + − λ −λ ⎥⎦

( )( ) 121

(19)

(20)

. (22)

m

where <sup>0</sup> λ<sup>a</sup> = 0.0244 W/(m•K) is the thermal conductivity of air at temperature T0. It is convenient for practical calculations to express the radius of a contact spot directly in terms of the parameter δ = a/R, although this relationship is indirectly reflected in Eq. (8) in terms of porosity. The following correlations have been derived (Gavriliev, 1998): at a/R ≤ 1

$$\frac{\mathbf{r}\_c}{\mathbf{R}} = 0.25 \left( \frac{\mathbf{a}}{\mathbf{R}} \right)^{2.5} \text{ \,\,\,\,\tag{17}$$

at a/R ≥ 1

$$\frac{\mathbf{r}\_c}{\mathbf{R}} = 1.25 - \exp\left[-0.6\left(\frac{\mathbf{a}}{\mathbf{R}} - 1\right)\right].\tag{18}$$

Fig. 3 presents a nomogram which can be used to find the thermal conductivity of snow from its temperature and porosity. This nomogram has been developed based on the above theoretical model which takes into account the heat transfer by thermal diffusion of water vapour. In the computations, the diffusion coefficient of water vapour in snow, Ds, is taken to be 0.66 cm2/s, which is the average of the experimental values reported in the literature ranging from 0.40 cm2/s (Sulakvelidze & Okudzhava, 1959) to 0.90 cm2/s (Pavlov, 1962).

Fig. 3. Nomogram for the calculation of thermal conductivity of snowcover from its density and temperature, °C: (1) -0; (2) -5; (3) -10; (4) -20; (5) -30

It is convenient for practical calculations to express the radius of a contact spot directly in terms of the parameter δ = a/R, although this relationship is indirectly reflected in Eq. (8) in

2.5

⎝ ⎠ , (17)

<sup>⎡</sup> ⎛ ⎞ <sup>⎤</sup> =− − − <sup>⎢</sup> ⎜ ⎟ <sup>⎥</sup> <sup>⎣</sup> ⎝ ⎠ <sup>⎦</sup> (18)

where <sup>0</sup> λ<sup>a</sup> = 0.0244 W/(m•K) is the thermal conductivity of air at temperature T0.

terms of porosity. The following correlations have been derived (Gavriliev, 1998):

<sup>c</sup><sup>r</sup> <sup>a</sup> 0.25 R R ⎛ ⎞ <sup>=</sup> ⎜ ⎟

<sup>c</sup><sup>r</sup> <sup>a</sup> 1.25 exp 0.6 1 . R R

Fig. 3 presents a nomogram which can be used to find the thermal conductivity of snow from its temperature and porosity. This nomogram has been developed based on the above theoretical model which takes into account the heat transfer by thermal diffusion of water vapour. In the computations, the diffusion coefficient of water vapour in snow, Ds, is taken to be 0.66 cm2/s, which is the average of the experimental values reported in the literature ranging from 0.40 cm2/s (Sulakvelidze & Okudzhava, 1959) to 0.90 cm2/s

0 0.2 0.4 0.6

**Density (γ), g/cm3**

Fig. 3. Nomogram for the calculation of thermal conductivity of snowcover from its density

at a/R ≤ 1

at a/R ≥ 1

(Pavlov, 1962).

0

and temperature, °C: (1) -0; (2) -5; (3) -10; (4) -20; (5) -30

0.2

0.4

0.6

**Thermal conductivity (λ), W/(m•K)**

0.8

1

1.2

#### **3. Effects of coarse inclusions and the layered and reticulate cryostructures on thermal conductivity of frozen soils**

For the thermal conductivity of media containing spherical and cubic inclusions with no contacts (or with point contacts), Maxwell (1873) (for a sphere) and Odelevsky (1951) (for a cube) developed a similar equation of the type:

$$\lambda = \lambda\_2 \left[ 1 + \frac{\left(\lambda\_1 - \lambda\_2\right) \mathbf{m}\_1}{\lambda\_2 + 0.33 \left(1 - \mathbf{m}\_1\right) \left(\lambda\_1 - \lambda\_2\right)} \right],\tag{19}$$

where (as before) the subscripts "1" and "2" refer to the inclusions (particles) and the medium, respectively. For the cubical particle shape, Eq. (19) is formally valid across the range of inclusion contents: 0 ≤ m1 ≤ 1.

The advantage of Eq. (19) is its simplicity. In some cases, Eq. (19) is applicable to permafrost problems, for example, for estimating the thermal conductivity of soils with a cryostructure or of soils containing gravel- or cobble-size inclusions. However, at large differences between the λ1 and λ2 values, such as in air-dry soils, the degree of roundness of gravel and cobble inclusions may affect the accuracy of calculations.

For a more general formulation of the problem, an ellipsoidal particle shape may be considered in Eq. (19), since with the change in the ratio of semi-axes the particles transform into other figures, such as a sphere, plate, or cylinder. Eq. (19) may be presented in the following generalized form (Gavriliev, 1986):

$$
\lambda = \lambda\_2 \left[ 1 + \frac{\left(\lambda\_1 - \lambda\_2\right) \mathbf{m}\_1}{\lambda\_2 + \mathbf{K}\_\mathbf{f} \left(1 - \mathbf{m}\_1\right) \left(\lambda\_1 - \lambda\_2\right)} \right] \tag{20}
$$

where Kf is the shape factor of particles or inclusions. In Eq. (20), the inclusion shape factor, Kf, is

$$\mathbf{K}\_{\mathbf{f}} = \mathbf{a}\mathbf{b}\mathbf{c}\mathbf{C}(\mathbf{0}),\tag{21}$$

where a, b, and c are the semi-axes of the ellipsoids (a > b > c); and C(0) is the integral of the form (Ovchinnikov, 1971)

$$\mathbf{C(0)} = \frac{2}{\mathbf{a}^3} \left[ \frac{\sqrt{1 - \mathbf{p}^2 \sin^2 \psi} \mathbf{t} \mathbf{g} \mathbf{y} - \mathbf{E(w, p)}}{1 - \mathbf{p}^2} \right]. \tag{22}$$

E(ψ, p) is the elliptic integral of the second kind, 2 2 ψ= − arcsin 1 c a – is the amplitude and ( ) ( ) 22 22 p 1ba 1ca =− − – is the modulus of the integral. The elliptic integral E(ψ, p) is tabulated, and the shape factor of inclusions can be readily found from the ratio of the particle dimensions a, b, and c. For practical purposes,

calculations can be limited to the more simple case of ellipsoids of revolution. Then, the integral C(0) is expressed in terms of elementary functions (Carslaw & Jaeger, 1959). Let us consider two examples.

Analytical Methods for Estimating Thermal Conductivity

cryostructure; q – heat-flow vector

of Multi-Component Natural Systems in Permafrost Areas 243

Soils with a reticulate cryostructure are also anisotropic. The degree of anisotropy depends on the geometry of a reticulate ice network and the direction of the heat-flow vector (Fig. 5 d).

As an example, we will consider frozen soils with cryostructures in more detail below. Soils with a layered cryostructure exhibit the highest anisotropy of thermophysical properties. In thermal terms, it makes sense to identify the following categories of layered cryostructure: vertical layered, cross layered, and horizontal layered. These cryostructural categories are equivalent to the three main directions of the heat-flow vector relative to the

orientation of ice layers: perpendicular, parallel, and intermediate (Fig. 5 a-c).

(a) (b) (c) (d)

Fig. 5. Schematic representation of frozen soils with layered and reticulate cryostructures at different directions of heat-flow vector. Layered cryostructure for normal (a), parallel (b) and intermediate (c) directions of heat-flow vector relative ice orientation; d – reticulate

The mechanism by which cryostructures develop in sediment is not as yet clearly understood, but the underlying effect is known to be the movement of water to the freezing front. Growing ice lenses dissect the homogeneous (massive) frozen soil into bands or blocks, i.e., the soil elements in the cryostructure are approximately similar in composition and thermal properties. In the reticulate structure, ice is the matrix material and the enclosed soil blocks are commonly rectangular in shape. For estimating the thermal conductivity of soils containing a cryostructure, Ivanov & Gavriliev (1965) considered series and parallel heat flows separately for the layered cryostructure and in combination for the reticulate cryostructure. In the latter case, difficulty arose in practice with how to account for the thickness of ice layers separately along and across the heat flow. A more simple way of taking into account the cryostructure in frozen soils can be found from the theory of generalized conductivity of media containing foreign inclusions. For generality, let us consider the inclusions of ellipsoidal shape, because any type of cryostructure can be obtained by changing the ratio of ellipsoid's semi-axes. For ellipsoids of revolution, for example, the layered cryostructure is obtained by flattening the ellipsoids: с/а → 0 (с and а are the semi-minor and semi-major axes of the ellipsoid, respectively), when they change into plane layers. In the case of prolate ellipsoids of revolution with radius с, at с/а → 0 the soil inclusions in the cryostructure become cylindrical. Any other values of the с/а ratio give reticulate cryostructures with one or other degree of elongation or flattening of the soil inclusions. At с/а = 1 the inclusions attain a spherical shape (an analogue of a cubic shape).

1. The particles have a shape of an oblate ellipsoid of revolution (a = b > c). Then, along the semi-axes we have

$$\mathbf{K}\_{\rm fc} = \frac{1}{\beta} \left( 1 - \frac{\mathbf{c}}{\mathbf{a}\beta} \arcsin \sqrt{\beta} \right) \tag{23}$$

$$\mathbf{K}\_{\rm fa} = \mathbf{K}\_{\rm fb} = \frac{\mathbf{c}}{2\mathbf{a}\beta} \left( \frac{\mathbf{c}}{\sqrt{\beta}} \arcsin \sqrt{\beta} - \frac{\mathbf{c}}{\mathbf{a}} \right) \tag{24}$$

where 2 2 β= −1ca .

2. The inclusions have a shape of a prolate ellipsoid of revolution (b = c < a)

$$\mathbf{K}\_{\rm fc} = \mathbf{K}\_{\rm fb} = \frac{1}{2\mathfrak{B}} \left( 1 - \frac{\mathbf{c}^2}{2\mathbf{a}^2 \overline{\mathfrak{B}}} \ln \frac{1 + \sqrt{\mathfrak{B}}}{1 - \sqrt{\mathfrak{B}}} \right) \tag{25}$$

$$\mathbf{K}\_{\rm fa} = \frac{\mathbf{c}^2}{\mathbf{a}^2 \mathfrak{B}} \left( \frac{1}{2\sqrt{\mathfrak{B}}} \ln \frac{1 + \sqrt{\mathfrak{B}}}{1 - \sqrt{\mathfrak{B}}} \right). \tag{26}$$

Fig. 4 shows graphically the shape factors Kf for oblate and prolate ellipsoids of revolution calculated with Eqs. (23) - (26) in relation to the ratio of the ellipse's semi-minor (c) and semi-major (a) axes at different directions. In the case of a layered cryostructure (c/a = 0), we have Kf = 1 (curve 1) for the ice-soil layers oriented across the flow, and Kf = 0 (curve 1′) for the orientation along the flow. In the case of cylindrical inclusions (c/a = 0), it follows that perpendicular to the heat flow Kf = 1/2 (curve 2′) and parallel to the flow Kf = 0 (curve 2).

Fig. 4. Shape factor Kf of soil particles in the form of oblate (1 and 1') and prolate (2 and 2') ellipsoids of revolution versus parameter с/а for different directions: 1 and 2 – along the axis of revolution; 1' and 2' – perpendicular to the axis of revolution

1. The particles have a shape of an oblate ellipsoid of revolution (a = b > c). Then, along

1 c K 1 arcsin , a ⎛ ⎞ <sup>=</sup> ⎜ ⎟ − β β β ⎝ ⎠

cc c K K arcsin , 2a <sup>a</sup> ⎛ ⎞ = = ⎜ ⎟ β − <sup>β</sup> <sup>β</sup> ⎝ ⎠

1 c <sup>1</sup> K K 1 ln , <sup>2</sup> 2a 1 ⎛ ⎞ <sup>+</sup> <sup>β</sup> = = − ⎜ ⎟ <sup>β</sup> <sup>β</sup> − β ⎝ ⎠

> c 1 <sup>1</sup> K ln . a2 1 ⎛ ⎞ <sup>+</sup> <sup>β</sup> <sup>=</sup> ⎜ ⎟ β β −β ⎝ ⎠

Fig. 4 shows graphically the shape factors Kf for oblate and prolate ellipsoids of revolution calculated with Eqs. (23) - (26) in relation to the ratio of the ellipse's semi-minor (c) and semi-major (a) axes at different directions. In the case of a layered cryostructure (c/a = 0), we have Kf = 1 (curve 1) for the ice-soil layers oriented across the flow, and Kf = 0 (curve 1′) for the orientation along the flow. In the case of cylindrical inclusions (c/a = 0), it follows that perpendicular to the heat flow Kf = 1/2 (curve 2′) and parallel to the flow Kf = 0 (curve 2).

> 0 0.2 0.4 0.6 0.8 1 **Parameter c/a**

Fig. 4. Shape factor Kf of soil particles in the form of oblate (1 and 1') and prolate (2 and 2') ellipsoids of revolution versus parameter с/а for different directions: 1 and 2 – along the

axis of revolution; 1' and 2' – perpendicular to the axis of revolution

1'

2

2' 1

2

fa fb

2. The inclusions have a shape of a prolate ellipsoid of revolution (b = c < a)

fc fb 2

2 fa 2

(23)

(24)

(25)

(26)

the semi-axes we have

where 2 2 β= −1ca .

fc

0

0.2

0.4

**Shape factor (Kf)**

0.6

0.8

1

As an example, we will consider frozen soils with cryostructures in more detail below. Soils with a layered cryostructure exhibit the highest anisotropy of thermophysical properties. In thermal terms, it makes sense to identify the following categories of layered cryostructure: vertical layered, cross layered, and horizontal layered. These cryostructural categories are equivalent to the three main directions of the heat-flow vector relative to the orientation of ice layers: perpendicular, parallel, and intermediate (Fig. 5 a-c).

Soils with a reticulate cryostructure are also anisotropic. The degree of anisotropy depends on the geometry of a reticulate ice network and the direction of the heat-flow vector (Fig. 5 d).

Fig. 5. Schematic representation of frozen soils with layered and reticulate cryostructures at different directions of heat-flow vector. Layered cryostructure for normal (a), parallel (b) and intermediate (c) directions of heat-flow vector relative ice orientation; d – reticulate cryostructure; q – heat-flow vector

The mechanism by which cryostructures develop in sediment is not as yet clearly understood, but the underlying effect is known to be the movement of water to the freezing front. Growing ice lenses dissect the homogeneous (massive) frozen soil into bands or blocks, i.e., the soil elements in the cryostructure are approximately similar in composition and thermal properties. In the reticulate structure, ice is the matrix material and the enclosed soil blocks are commonly rectangular in shape. For estimating the thermal conductivity of soils containing a cryostructure, Ivanov & Gavriliev (1965) considered series and parallel heat flows separately for the layered cryostructure and in combination for the reticulate cryostructure. In the latter case, difficulty arose in practice with how to account for the thickness of ice layers separately along and across the heat flow. A more simple way of taking into account the cryostructure in frozen soils can be found from the theory of generalized conductivity of media containing foreign inclusions. For generality, let us consider the inclusions of ellipsoidal shape, because any type of cryostructure can be obtained by changing the ratio of ellipsoid's semi-axes. For ellipsoids of revolution, for example, the layered cryostructure is obtained by flattening the ellipsoids: с/а → 0 (с and а are the semi-minor and semi-major axes of the ellipsoid, respectively), when they change into plane layers. In the case of prolate ellipsoids of revolution with radius с, at с/а → 0 the soil inclusions in the cryostructure become cylindrical. Any other values of the с/а ratio give reticulate cryostructures with one or other degree of elongation or flattening of the soil inclusions. At с/а = 1 the inclusions attain a spherical shape (an analogue of a cubic shape).

Analytical Methods for Estimating Thermal Conductivity

moisture content will be discussed in the next section.

0

0

1

2

**Thermal conductivity (λ║), W/(m•K)**

3

4

5

1

2

**Thermal conductivity (λ┴), W/(m•K)**

3

4

5

of Multi-Component Natural Systems in Permafrost Areas 245

thermal conductivity from 0.5 to 5.0 W/(m•K). It is assumed that the soils comprising the intermediate layers and inclusions are perennially frozen; their thermal properties in relation to natural moisture content have been fairly well studied (Gavriliev, 1989, 1998; Gavriliev & Eliseev, 1970). It is known, for example, that the thermal conductivity of peat in its naturally frozen state is independent of moisture content and is approximately equal to 1.27 W/(m•K). The thermal conductivity of perennially frozen soils in relation to natural

> 0 0.2 0.4 0.6 0.8 1 **Volume fraction of ice (mi)**

0 0.2 0.4 0.6 0.8 1 **Volume fraction ice (mi)**

(b)

(a)

Let us consider the cryostructure as an ice matrix with soil inclusions in the form of ellipsoids. We assume in the general case that the soil inclusions are non-uniform in composition. Then, for the effective thermal conductivity of frozen soil, λ⊥,||,+, we can use the equation derived earlier by the author (Gavrilyev, 1996b) for very coarse soils with particles of different mineralogical compositions

$$
\lambda\_{\perp,\parallel} \parallel\_{\perp} = \lambda\_{\mathbf{i}} \left( \mathbf{1} + \frac{\mathbf{1}}{\mathbf{B} - \mathbf{K}\_{\mathbf{f}}} \right) \tag{27}
$$

where

$$\mathbf{B} = 1 \Bigg/ \sum\_{\mathbf{j} = \mathbf{I}}^{\mathbf{n}} \frac{\left(\lambda\_{\mathbf{j}} - \lambda\_{\mathbf{i}}\right) \mathbf{m}\_{\mathbf{j}}}{\left(\lambda\_{\mathbf{j}} - \lambda\_{\mathbf{i}}\right) / \mathbf{K}\_{\mathbf{f}} + \lambda\_{\mathbf{i}}} \Bigg/ \tag{28}$$

Kf is the shape factor of soil inclusions or layers; λi is the thermal conductivity of the ice matrix; λj and mj are the thermal conductivity and the volumetric content of the j-th soil inclusion.

The volume fraction of ice, mi, is

$$\mathbf{m}\_{\mathbf{i}} = \mathbf{1} - \sum\_{\mathbf{j}=1}^{n} \mathbf{m}\_{\mathbf{j}} \ . \tag{29}$$

It is assumed in Eqs. (27) - (29) that the soil inclusions have a massive structure and are fully saturated (see Section 3 for permafrost soils with a massive cryostructure).

When the ice lenses occur in the soil at angle α to the direction of heat flow, the thermal conductivity of the frozen soil mass is given by

$$
\lambda\_{\alpha} = \sqrt{\lambda\_{\perp}^{2} \sin^{2} \alpha + \lambda\_{\text{II}}^{2} \cos^{2} \alpha} \tag{30}
$$

where λ⊥ and λ|| are the thermal conductivities of the soil with a layered cryostructure, defined by Eq. (27) at corresponding Kf values, for the ice-soil layers perpendicular and parallel to heat flow direction.

The change of rectangular soil inclusions for ellipsoidal ones does not detract from the accuracy of calculations, as it is known from the theoretical predictions of thermal parameters that in the case of inclusions dispersed in a medium, the shape of inclusions has no significant effect on the final calculation results. In case of the uniform composition of frozen soil inclusions, Eq. (27) simplifies to Eq. (20).

Computations of the thermal conductivity of frozen soils with layered and reticulate cryostructures were performed for the ice layers parallel, perpendicular and at 45° angle to the heat flow vector, and for spherical or cubical (c/a = 1 and Kf = 0.33) soil inclusions in the reticulate cryostructure. The thermal conductivity of the soil containing a cryostructure depends on the size (volume fraction) and orientation of the ice and soil layers relative heat flow direction, as well as the thermal conductivity of these layers.

In the cryostructures, the intermediate layers or inclusions are made of a macroscopically isotropic (massive structure) frozen mass of mineral or organic soils which can vary in

Let us consider the cryostructure as an ice matrix with soil inclusions in the form of ellipsoids. We assume in the general case that the soil inclusions are non-uniform in composition. Then, for the effective thermal conductivity of frozen soil, λ⊥,||,+, we can use the equation derived earlier by the author (Gavrilyev, 1996b) for very coarse soils with particles

<sup>1</sup> 1 , ,||, <sup>i</sup> B Kf

( )

Kf is the shape factor of soil inclusions or layers; λi is the thermal conductivity of the ice matrix; λj and mj are the thermal conductivity and the volumetric content of the j-th soil

> n i j j 1 m 1 m. =

It is assumed in Eqs. (27) - (29) that the soil inclusions have a massive structure and are fully

When the ice lenses occur in the soil at angle α to the direction of heat flow, the thermal

22 2 2 sin cos , λα ⊥ = λ α+λ α ΙΙ (30)

where λ⊥ and λ|| are the thermal conductivities of the soil with a layered cryostructure, defined by Eq. (27) at corresponding Kf values, for the ice-soil layers perpendicular and

The change of rectangular soil inclusions for ellipsoidal ones does not detract from the accuracy of calculations, as it is known from the theoretical predictions of thermal parameters that in the case of inclusions dispersed in a medium, the shape of inclusions has no significant effect on the final calculation results. In case of the uniform composition of

Computations of the thermal conductivity of frozen soils with layered and reticulate cryostructures were performed for the ice layers parallel, perpendicular and at 45° angle to the heat flow vector, and for spherical or cubical (c/a = 1 and Kf = 0.33) soil inclusions in the reticulate cryostructure. The thermal conductivity of the soil containing a cryostructure depends on the size (volume fraction) and orientation of the ice and soil layers relative heat

In the cryostructures, the intermediate layers or inclusions are made of a macroscopically isotropic (massive structure) frozen mass of mineral or organic soils which can vary in

<sup>n</sup> jij <sup>m</sup> B 1 ; j 1 /K <sup>j</sup> i i <sup>f</sup>

λ −λ

= λ −λ +λ

λ =λ + ⎜ ⎟ ⊥ + ⎜ ⎟ <sup>−</sup> ⎝ ⎠

⎛ ⎞

(27)

(28)

= −∑ (29)

of different mineralogical compositions

The volume fraction of ice, mi, is

parallel to heat flow direction.

conductivity of the frozen soil mass is given by

frozen soil inclusions, Eq. (27) simplifies to Eq. (20).

flow direction, as well as the thermal conductivity of these layers.

( )

= ∑

saturated (see Section 3 for permafrost soils with a massive cryostructure).

where

inclusion.

thermal conductivity from 0.5 to 5.0 W/(m•K). It is assumed that the soils comprising the intermediate layers and inclusions are perennially frozen; their thermal properties in relation to natural moisture content have been fairly well studied (Gavriliev, 1989, 1998; Gavriliev & Eliseev, 1970). It is known, for example, that the thermal conductivity of peat in its naturally frozen state is independent of moisture content and is approximately equal to 1.27 W/(m•K). The thermal conductivity of perennially frozen soils in relation to natural moisture content will be discussed in the next section.

Analytical Methods for Estimating Thermal Conductivity

even in the field.

considerable variation.

of Multi-Component Natural Systems in Permafrost Areas 247

It is sufficient for analysis of the thermal behaviour of permafrost to only consider one of these parameters. Natural moisture content is preferably chosen, since it is easily measured

The total moisture content of frozen soils, especially fine-grained soils, varies over a wide range due to moisture migration. For frozen alluvial deposits in Yakutia, for example, the typical values range from 0.07 to 0.30 for sands and from 0.20 to 0.60 for sand-silts and siltclays (Votyakov, 1975). Correspondingly, the thermal conductivity of soils may exhibit

Fig. 7 shows the experimental results for thermal conductivity, λ, of frozen Yakutian alluvial soils in the wide range of saturation moisture contents Wsat. It should be noted that full saturation was assumed in the experiments as a model of the natural state of permafrost soils. As is seen, the dependence of λ on Wsat differs between course- and fine-grained soils. With increasing Wsat, the thermal conductivity of the frozen sand at the point of full saturation decreases, while that of the silt-clay increases tending to the thermal conductivity

The observed differences in λ (Wsat) can be explained by the differences in the unfrozen water content and in the mineralogical composition of the soils. In the fine soils, the unfrozen water content is quite high (about 0.1) at the measurement temperatures (about - 10°C). At low moisture contents, silt-clay can thus be considered as an unfrozen soil. The effect of ice inclusions on overall heat conduction increases with increasing water content, resulting in higher soil thermal conductivity. At high moisture contents, the thermal conductivity of the silt-clay tends to that of ice. In the sands, the unfrozen water content is low and the mineral particles are in direct contact with ice. As the mineral particles have a higher thermal conductivity than ice, the thermal conductivity of the sand decreases with increasing water (ice) content. The same is true for the unfrozen soils. The effect of the unfrozen water film coating the mineral particles is less in the sand-silt compared to the siltclay. The thermal conductivity of the "mineral particle + unfrozen water" system is likely to have the same values as for ice. The frozen saturated sand-silt has therefore a nearly constant thermal conductivity over the entire range of saturation moisture contents. The mineralogical composition has also an effect, resulting in an increase in the thermal

The above features of permafrost thermal conductivity can be estimated based on the analytical theory of thermal conductivity of composite materials. The possible structural models of soil follow from the mechanism of water binding by mineral particles. Soil particles possess excess surface energy which depends on their size and mineral composition. When water enters the ground, it interacts with the mineral particles under the influence of molecular forces and surrounds them in concentric layers until the excess of surface energy is removed. The particles interact through the bound water layer, forming a stable system with dispersed particles. The remaining part of the soil pores is filled with free water. As the soil temperature decreases, primarily near 0°C, the free water begins to freeze. Then more of the bound water freezes with a further decrease in temperature. The strongly bound water remains unfrozen down to about -20°C. When frozen, the system of dispersed particles is cemented by ice, becoming even more stable. Hence, the thermal conductivity of fine-grained permafrost soils at different subzero temperatures can be estimated considering a three-component shell system (mineral particle + unfrozen water + ice) as shown in Fig. 8.

of ice. The sand-silts are intermediate between these two soil types.

Mineral particles in this scheme are assumed to be spherical in shape.

conductivity of solids from finer to coarser soils.

Fig. 6. Thermal conductivities λ⊥ (a), λ||(b) and λ+ (c) of frozen soils with a cryostructure as a function of volume fraction of ice layers (mi) for various thermal conductivities of intermediate layers or inclusions comprised of frozen organic and mineral soils (λfl, W/(m•K)). λfl values: 1-0.5; 2-1.0; 3-1.5; 4-2.0; 5-2.5; 6-3.0; 7-3.5; 8-4.0; 9-4.5; 10-5.0

Based on the calculated thermal conductivity values for the frozen soils with layered and reticulate cryostructures as a function of the volume fraction of ice layers mi at different thermal conductivities of intermediate layers λfl (here the subscript "fl" refers to frozen soil), nomograms were developed shown in Fig. 6. For the layered cryostructure, the volume fraction of ice layers mi is equal to their relative thickness li.

At the same values of mi and λfl, the thermal conductivity of frozen soils is highest for a layered cryostructure with the soil and ice layers parallel to heat flow and lowest for that with the heat flow direction normal to the ice and soil layers. The soils containing reticulate and layered cryostructures with the ice and soil layers at 45о to heat flow direction have intermediate thermal conductivity values.

#### **4. Permafrost soils with a massive cryostructure**

In engineering practice, thermal properties of a given soil type are usually examined in relation to moisture content and dry density. For permafrost soils, there is a unique relationship between these parameters, because naturally occurring soils are near saturation and the air porosity comprises only 2-3% of the total soil volume. The density of frozen soil is then given by Votyakov's equation (1975):

$$\gamma\_{\rm f} = \frac{2.4(1+\rm W)}{2.7\rm W + 0.9} \,\tag{31}$$

where W is the gravimetric moisture content of the frozen soil expressed as a fraction.

0 0.2 0.4 0.6 0.8 1 **Volume fraction of ice (mi)**

(c) Fig. 6. Thermal conductivities λ⊥ (a), λ||(b) and λ+ (c) of frozen soils with a cryostructure as a function of volume fraction of ice layers (mi) for various thermal conductivities of intermediate layers or inclusions comprised of frozen organic and mineral soils (λfl, W/(m•K)). λfl values:

Based on the calculated thermal conductivity values for the frozen soils with layered and reticulate cryostructures as a function of the volume fraction of ice layers mi at different thermal conductivities of intermediate layers λfl (here the subscript "fl" refers to frozen soil), nomograms were developed shown in Fig. 6. For the layered cryostructure, the volume

At the same values of mi and λfl, the thermal conductivity of frozen soils is highest for a layered cryostructure with the soil and ice layers parallel to heat flow and lowest for that with the heat flow direction normal to the ice and soil layers. The soils containing reticulate and layered cryostructures with the ice and soil layers at 45о to heat flow direction have

In engineering practice, thermal properties of a given soil type are usually examined in relation to moisture content and dry density. For permafrost soils, there is a unique relationship between these parameters, because naturally occurring soils are near saturation and the air porosity comprises only 2-3% of the total soil volume. The density of frozen soil

> 2.4(1 W) , 2.7W 0.9 +

+

(31)

f

γ =

where W is the gravimetric moisture content of the frozen soil expressed as a fraction.

0

intermediate thermal conductivity values.

is then given by Votyakov's equation (1975):

1

1-0.5; 2-1.0; 3-1.5; 4-2.0; 5-2.5; 6-3.0; 7-3.5; 8-4.0; 9-4.5; 10-5.0

fraction of ice layers mi is equal to their relative thickness li.

**4. Permafrost soils with a massive cryostructure** 

2

**Thermal condyctivity (λ+), W/(m•K)**

3

4

5

It is sufficient for analysis of the thermal behaviour of permafrost to only consider one of these parameters. Natural moisture content is preferably chosen, since it is easily measured even in the field.

The total moisture content of frozen soils, especially fine-grained soils, varies over a wide range due to moisture migration. For frozen alluvial deposits in Yakutia, for example, the typical values range from 0.07 to 0.30 for sands and from 0.20 to 0.60 for sand-silts and siltclays (Votyakov, 1975). Correspondingly, the thermal conductivity of soils may exhibit considerable variation.

Fig. 7 shows the experimental results for thermal conductivity, λ, of frozen Yakutian alluvial soils in the wide range of saturation moisture contents Wsat. It should be noted that full saturation was assumed in the experiments as a model of the natural state of permafrost soils. As is seen, the dependence of λ on Wsat differs between course- and fine-grained soils. With increasing Wsat, the thermal conductivity of the frozen sand at the point of full saturation decreases, while that of the silt-clay increases tending to the thermal conductivity of ice. The sand-silts are intermediate between these two soil types.

The observed differences in λ (Wsat) can be explained by the differences in the unfrozen water content and in the mineralogical composition of the soils. In the fine soils, the unfrozen water content is quite high (about 0.1) at the measurement temperatures (about - 10°C). At low moisture contents, silt-clay can thus be considered as an unfrozen soil. The effect of ice inclusions on overall heat conduction increases with increasing water content, resulting in higher soil thermal conductivity. At high moisture contents, the thermal conductivity of the silt-clay tends to that of ice. In the sands, the unfrozen water content is low and the mineral particles are in direct contact with ice. As the mineral particles have a higher thermal conductivity than ice, the thermal conductivity of the sand decreases with increasing water (ice) content. The same is true for the unfrozen soils. The effect of the unfrozen water film coating the mineral particles is less in the sand-silt compared to the siltclay. The thermal conductivity of the "mineral particle + unfrozen water" system is likely to have the same values as for ice. The frozen saturated sand-silt has therefore a nearly constant thermal conductivity over the entire range of saturation moisture contents. The mineralogical composition has also an effect, resulting in an increase in the thermal conductivity of solids from finer to coarser soils.

The above features of permafrost thermal conductivity can be estimated based on the analytical theory of thermal conductivity of composite materials. The possible structural models of soil follow from the mechanism of water binding by mineral particles. Soil particles possess excess surface energy which depends on their size and mineral composition. When water enters the ground, it interacts with the mineral particles under the influence of molecular forces and surrounds them in concentric layers until the excess of surface energy is removed. The particles interact through the bound water layer, forming a stable system with dispersed particles. The remaining part of the soil pores is filled with free water. As the soil temperature decreases, primarily near 0°C, the free water begins to freeze. Then more of the bound water freezes with a further decrease in temperature. The strongly bound water remains unfrozen down to about -20°C. When frozen, the system of dispersed particles is cemented by ice, becoming even more stable. Hence, the thermal conductivity of fine-grained permafrost soils at different subzero temperatures can be estimated considering a three-component shell system (mineral particle + unfrozen water + ice) as shown in Fig. 8. Mineral particles in this scheme are assumed to be spherical in shape.

Analytical Methods for Estimating Thermal Conductivity

the mineral solids, ice and unfrozen water, respectively.

ε =

<sup>s</sup> <sup>s</sup>

1 W <sup>ρ</sup> <sup>=</sup> <sup>+</sup> <sup>ρ</sup>

m

conductivity λ of a saturated frozen soil (Gavriliev, 1989):

i

3 3 s uw s 4R 4R / 3 3

solids ms and unfrozen water muw

water content muw

where

instead of λi.

be expressed as:

where

of Multi-Component Natural Systems in Permafrost Areas 249

component system consisting of mineral solids and unfrozen water; σ is the volume fraction of the mineral solids and unfrozen water in the soil; the subscripts "s", "i" and "uw" refer to

Parameters ε and υ in Eq. (32) can be expressed in terms of the volume fractions of soil

Considering the relation of ms and muw to the saturation moisture content Wsat and unfrozen

( ρs is the solids density), we finally obtain the following expression for the thermal

In Eqs. (33) - (35), all limiting conditions are satisfied. At Wsat→∞, λ=λi. If Wsat=0 and Wuw=0, then λ=λs. When Wuw=0, the well-known Maxwell-Odolevsky equation for a twocomponent medium is obtained, which can be expressed in terms of moisture content as

1W 2 2

<sup>+</sup> ρ λ +λ + λ −λ λ=λ

Eq. (36) is also applicable to unfrozen soils, if the thermal conductivity of water λw is used

The presence of entrapped air reduces the thermal conductivity of frozen soils, and this can

( ) ( ) s

s sat 2 1W 2 3W W λ+ρ λ = +ρ −

1W 2

where W is the actual moisture content of the soil which should vary in the range

( ) ( ) ( ) ( )( ) ( ) sat s i s s i

sat s i s s i

s s uw m m m

sat s

N= ( ) ( uw s i uw ) ( )

sat s i uw

i

1 W

<sup>2</sup> 1W 2

 M= ( )( ) ( uw s uw i ) ( ) uw s uw i

+

π π <sup>+</sup> ε = is the volume fraction of the mineral soil solids in the two-

and σ= + m m s uw

and s uw uw

m

N 2М N М

1W 2 <sup>⎡</sup> λ −λ λ −λ <sup>⎤</sup> + ρ λ +λ + ⋅ <sup>⎢</sup> <sup>⎥</sup> + ρ λ +λ <sup>⎣</sup> <sup>⎦</sup>

sat s

<sup>+</sup> λ=λ <sup>−</sup> , (33)

, (34)

W

1 W <sup>ρ</sup> <sup>=</sup> <sup>+</sup> <sup>ρ</sup> ,

uw s uw s

λ −λ λ +λ + ρ λ −λ − λ +λ . (35)

2

uw s 2

<sup>+</sup> ρ λ +λ − λ −λ . (36)

, (37)

Fig. 7. Thermal conductivity vs. saturation moisture content for alluvial sediments in frozen state: 1 - sand; 2 – sand-silt; 3 – silt-clay; 4 - experimental curves; 5 – predicted curves

Fig. 8. Three-component shell medium: 1 – soil mineral particle; 2 – unfrozen water; 3 – ice

The effective conductivity λ of such a shell system can be predicted using the Maxwell method based on the solution of Laplace's equation for a medium with a constant temperature gradient at a distance from the spherical particle with a shell. The equation has the form (Belskaya, 1981)

$$\frac{\lambda - \lambda\_{\rm i}}{\lambda - 2\lambda\_{\rm i}} = \frac{\sigma \left| \lambda\_{\rm uw} - \lambda\_{\rm i} - \frac{\varepsilon \left( \lambda\_{\rm uw} - \lambda\_{\rm s} \right) \left( 2\lambda\_{\rm uw} + \lambda\_{\rm i} \right)}{2\lambda\_{\rm uw} + \lambda\_{\rm s}} \right|}{2\lambda\_{\rm i} + \lambda\_{\rm uw} + \frac{2\varepsilon \left( \lambda\_{\rm uw} - \lambda\_{\rm s} \right) \left( \lambda\_{\rm i} - \lambda\_{\rm uw} \right)}{2\lambda\_{\rm uw} + \lambda\_{\rm s}}}\tag{32}$$

where 3 3 s uw s 4R 4R / 3 3 π π <sup>+</sup> ε = is the volume fraction of the mineral soil solids in the two-

component system consisting of mineral solids and unfrozen water; σ is the volume fraction of the mineral solids and unfrozen water in the soil; the subscripts "s", "i" and "uw" refer to the mineral solids, ice and unfrozen water, respectively.

Parameters ε and υ in Eq. (32) can be expressed in terms of the volume fractions of soil solids ms and unfrozen water muw

$$\varepsilon = \frac{\mathbf{m}\_s}{\mathbf{m}\_s + \mathbf{m}\_{\text{uw}}} \text{ and } \quad \sigma = \mathbf{m}\_s + \mathbf{m}\_{\text{uw}}$$

Considering the relation of ms and muw to the saturation moisture content Wsat and unfrozen water content muw

$$\mathbf{m}\_{\rm s} = \frac{\rho\_{\rm s}}{1 + \mathbf{W}\_{\rm sat} \rho\_{\rm s}} \quad \text{and} \quad \mathbf{m}\_{\rm uw} = \frac{\rho\_{\rm s} \mathbf{W}\_{\rm uw}}{1 + \mathbf{W}\_{\rm sat} \rho\_{\rm s}} \; \prime$$

( ρs is the solids density), we finally obtain the following expression for the thermal conductivity λ of a saturated frozen soil (Gavriliev, 1989):

$$
\lambda = \lambda\_i \frac{\mathbf{N} + 2\mathbf{M}}{\mathbf{N} - \mathbf{M}} \,' \,. \tag{33}
$$

where

248 Convection and Conduction Heat Transfer

Fig. 7. Thermal conductivity vs. saturation moisture content for alluvial sediments in frozen state: 1 - sand; 2 – sand-silt; 3 – silt-clay; 4 - experimental curves; 5 – predicted curves

Fig. 8. Three-component shell medium: 1 – soil mineral particle; 2 – unfrozen water; 3 – ice The effective conductivity λ of such a shell system can be predicted using the Maxwell method based on the solution of Laplace's equation for a medium with a constant temperature gradient at a distance from the spherical particle with a shell. The equation has

> i uw s i uw s i uw

<sup>⎡</sup> ε λ −λ λ +λ <sup>⎤</sup> σ λ −λ − <sup>⎢</sup> <sup>⎥</sup> λ−λ λ +λ <sup>⎣</sup> <sup>⎦</sup> <sup>=</sup> λ− λ ε λ −λ λ −λ λ +λ +

uw i

i uw

2 2 2

( )( )

2

2

uw s uw i

2

( )( )

, (32)

uw s

λ +λ

the form (Belskaya, 1981)

$$\mathbf{N} = \left(\mathbf{1} + \mathbf{W\_{sat}}\rho\_s\right) \left[2\lambda\_\mathrm{i} + \lambda\_\mathrm{uw} + \frac{\mathbf{2}}{1 + \mathbf{W\_{uw}}\rho\_s} \cdot \frac{\left(\lambda\_\mathrm{uw} - \lambda\_s\right)\left(\lambda\_\mathrm{i} - \lambda\_\mathrm{uw}\right)}{2\lambda\_\mathrm{uw} + \lambda\_s}\right],\tag{34}$$

$$\mathbf{M} = \left(\mathbf{1} + \mathbf{W}\_{\text{uw}} \boldsymbol{\rho}\_{\text{s}}\right) \left(\boldsymbol{\lambda}\_{\text{uw}} - \boldsymbol{\lambda}\_{\text{i}}\right) - \frac{\left(\boldsymbol{\lambda}\_{\text{uw}} - \boldsymbol{\lambda}\_{\text{s}}\right) \left(2\boldsymbol{\lambda}\_{\text{uw}} + \boldsymbol{\lambda}\_{\text{i}}\right)}{2\boldsymbol{\lambda}\_{\text{uw}} + \boldsymbol{\lambda}\_{\text{s}}}.\tag{35}$$

In Eqs. (33) - (35), all limiting conditions are satisfied. At Wsat→∞, λ=λi. If Wsat=0 and Wuw=0, then λ=λs. When Wuw=0, the well-known Maxwell-Odolevsky equation for a twocomponent medium is obtained, which can be expressed in terms of moisture content as

$$\lambda = \lambda\_i \frac{\left(1 + \mathcal{W}\_{\rm sat} \rho\_s\right)\left(2\lambda\_i + \lambda\_s\right) + \mathcal{Z}\left(\lambda\_s - \lambda\_i\right)}{\left(1 + \mathcal{W}\_{\rm sat} \rho\_s\right)\left(2\lambda\_i + \lambda\_s\right) - \left(\lambda\_s - \lambda\_i\right)}.\tag{36}$$

Eq. (36) is also applicable to unfrozen soils, if the thermal conductivity of water λw is used instead of λi.

The presence of entrapped air reduces the thermal conductivity of frozen soils, and this can be expressed as:

$$\lambda = \frac{2\lambda \left(1 + W\rho\_s\right)}{2 + \rho\_s \left(\Im W\_{\rm sat} - W\right)},\tag{37}$$

where W is the actual moisture content of the soil which should vary in the range

Analytical Methods for Estimating Thermal Conductivity

effect on the soil thermal properties.

virtually no experimental data available.

2 – colloid aggregate; 3 – unfrozen water; 4 - ice

where

of Multi-Component Natural Systems in Permafrost Areas 251

microorganisms), climate, and human activity. The presence of organic matter has a strong

Humic substances of the soil are specific high-molecular compounds. They play a significant role in creating the soil structure (Tsyganov, 1958). The organic substances exist in the form of very fine particles smaller than 0.2 μ referred to as colloids. Colloids have a very large specific surface area that provides strong bonding of water in soil and the presence of large amounts of unfrozen water at temperatures below freezing. Colloidal particles occur as sols and gels. Sols are smaller particles which can aggregate into gels by coagulation. When bound with water, organic particles form colloidal micelles having a core of an electrically neutral mineral particle surrounded by ionic layers of adsorbed molecules of colloid aggregate matter (hydrates: SiO2, Al2O3, MnO2, etc.) and electrolyte (water). At temperatures below 0°C, the diffuse layer of water freezes, while the bound water remains unfrozen. At the present stage of research, the problem of organic content effect on the thermal conductivity of soils can only be approached using analytical methods, since there are

Fig. 9. Schematic representation of the saturated frozen organic soil: 1 – mineral particle;

f 4

Let us consider the saturated soils. Based on the above consideration, the saturated soil containing organic matter can be represented as a four-component shell system within a cubic cell (Fig. 9) consisting of a mineral particle (1), organic matter (humus) (2), unfrozen water (3), and ice (4). The thermal conductivity of this system (frozen organic soil) calculated with the successive use of Maxwell's method is described by the equation (Gavrilyev, 2001)

<sup>⎡</sup> <sup>−</sup> <sup>⎤</sup> λ =λ + <sup>⎢</sup> <sup>⎥</sup> λ + <sup>⎣</sup> <sup>⎦</sup>

uw uw org i uw uw i uw i uw org uw 2 m D Q 3 1 m 2m Z , D31 m m m Q λ +λ ⎡ − − ⎤ ⎣ ⎦ <sup>=</sup> λ −λ λ ⎡ − − ⎤+λ ⎣ ⎦

( )

( <sup>i</sup>)

(39)

(40)

i i 3Z 1 m 1 , 3 Zm

( )

D m 3m m , =λ +λ + s org org s org ( ) (41)

$$\mathbf{W} \gtrsim \left( 1 - \frac{0.4}{\rho\_{\rm moi}} \right) \mathbf{W}\_{\rm sat} - \frac{0.4}{\rho\_s} \,\mathrm{}^{\prime}\tag{38}$$

where ρmoi is the parameter dependent on the soil condition which has a value of 1 above 0°C and 0.92 below 0°C.

The relation (38) may be particularly useful for estimating the thermal conductivity of the thawed soils where any excess water escapes on thawing (if the thawing layer is not underlain by frozen soil) and only part of the moisture is retained due to surface tension.

For actual computations, the values for thermal conductivity of soil constituents should be specified in Eqs. (34) and (35). For pure ice at t = 0°C λ = 2.25 W/(m•K). The thermal conductivity of unfrozen water can be taken approximately equal to that of free, i.e., λ = 0.58 W/(m•K), since all anomalies in the properties of bound water are related to its strongly adsorbed portion which is insignificant in amount. The thermal conductivity of mineral soil solids depends on the mineral composition of particles and may be approximately estimated by Eq. (14).

The distribution of minerals in soils is influenced by sedimentary conditions which vary widely in nature. The amount of minerals in a soil can be estimated approximately based on the relationship between particle mineralogy and size distribution. The three particle sizes used for soil classification are clay (< 0.002 mm), silt (0.002-0.05 mm), and sand (0.05-2.0 mm). In practice, it is assumed that the content of clay minerals, such as kaolinite, is equal to the amount of clay-sized particles and 50% of silt-sized particles (Kokshenov, 1957). The remainder of the soil consists predominantly of quartz and feldspar, and their relative proportions vary widely depending on the soil origin. If no appropriate data are available, the ratio of quartz to feldspar may be taken as 0.6:0.4 (Kokshenov, 1957).

In computations, the following values for λj may be used [W/(m•K)]: 6-7 for quartz, 1.9 for feldspars, and 1.2 for kaolinite.

The distribution of particle sizes in a soil strongly depends on sedimentation conditions. If no granulometric data are available, the values given in Table 1 may be used for approximate estimations.


Table 1. Relative proportions of particles sizes in soils

Comparison of the predicted and experimental data (see Fig. 7) shows that Eqs. (33) - (35) provide satisfactory results. The following values were used in the computations: for λ<sup>s</sup> [W/(m•K)]:3.50 for sand, 2.70 for sand-silt, 2.30 for silt-clay; for Wuw:0 for sand, 0.03 for sand-silt and 0.10 for silt-clay.

#### **5. Effect of organic matter on soil thermal conductivity**

The unconsolidated soil layer on the immediate surface of the earth is enriched with organic remains in the form of humus due to the effects of vegetation, animals (mainly

⎜ ⎟ − − <sup>ρ</sup> <sup>ρ</sup> ⎝ ⎠

where ρmoi is the parameter dependent on the soil condition which has a value of 1 above

The relation (38) may be particularly useful for estimating the thermal conductivity of the thawed soils where any excess water escapes on thawing (if the thawing layer is not underlain by frozen soil) and only part of the moisture is retained due to surface tension. For actual computations, the values for thermal conductivity of soil constituents should be specified in Eqs. (34) and (35). For pure ice at t = 0°C λ = 2.25 W/(m•K). The thermal conductivity of unfrozen water can be taken approximately equal to that of free, i.e., λ = 0.58 W/(m•K), since all anomalies in the properties of bound water are related to its strongly adsorbed portion which is insignificant in amount. The thermal conductivity of mineral soil solids depends on the mineral composition of particles and may be approximately estimated

The distribution of minerals in soils is influenced by sedimentary conditions which vary widely in nature. The amount of minerals in a soil can be estimated approximately based on the relationship between particle mineralogy and size distribution. The three particle sizes used for soil classification are clay (< 0.002 mm), silt (0.002-0.05 mm), and sand (0.05-2.0 mm). In practice, it is assumed that the content of clay minerals, such as kaolinite, is equal to the amount of clay-sized particles and 50% of silt-sized particles (Kokshenov, 1957). The remainder of the soil consists predominantly of quartz and feldspar, and their relative proportions vary widely depending on the soil origin. If no appropriate data are available,

In computations, the following values for λj may be used [W/(m•K)]: 6-7 for quartz, 1.9 for

The distribution of particle sizes in a soil strongly depends on sedimentation conditions. If no granulometric data are available, the values given in Table 1 may be used for

> Particle size Soil type Clay Silt Sand Sand 0.02 0.10 0.88 Sand-silt 0.06 0.30 0.64 Silt-clay 0.20 0.37 0.43

Comparison of the predicted and experimental data (see Fig. 7) shows that Eqs. (33) - (35) provide satisfactory results. The following values were used in the computations: for λ<sup>s</sup> [W/(m•K)]:3.50 for sand, 2.70 for sand-silt, 2.30 for silt-clay; for Wuw:0 for sand, 0.03 for

The unconsolidated soil layer on the immediate surface of the earth is enriched with organic remains in the form of humus due to the effects of vegetation, animals (mainly

the ratio of quartz to feldspar may be taken as 0.6:0.4 (Kokshenov, 1957).

Table 1. Relative proportions of particles sizes in soils

**5. Effect of organic matter on soil thermal conductivity** 

⎛ ⎞

moi s 0.4 0.4 1 W

, (38)

W≥ sat

0°C and 0.92 below 0°C.

feldspars, and 1.2 for kaolinite.

approximate estimations.

sand-silt and 0.10 for silt-clay.

by Eq. (14).

microorganisms), climate, and human activity. The presence of organic matter has a strong effect on the soil thermal properties.

Humic substances of the soil are specific high-molecular compounds. They play a significant role in creating the soil structure (Tsyganov, 1958). The organic substances exist in the form of very fine particles smaller than 0.2 μ referred to as colloids. Colloids have a very large specific surface area that provides strong bonding of water in soil and the presence of large amounts of unfrozen water at temperatures below freezing. Colloidal particles occur as sols and gels. Sols are smaller particles which can aggregate into gels by coagulation. When bound with water, organic particles form colloidal micelles having a core of an electrically neutral mineral particle surrounded by ionic layers of adsorbed molecules of colloid aggregate matter (hydrates: SiO2, Al2O3, MnO2, etc.) and electrolyte (water). At temperatures below 0°C, the diffuse layer of water freezes, while the bound water remains unfrozen.

At the present stage of research, the problem of organic content effect on the thermal conductivity of soils can only be approached using analytical methods, since there are virtually no experimental data available.

Fig. 9. Schematic representation of the saturated frozen organic soil: 1 – mineral particle; 2 – colloid aggregate; 3 – unfrozen water; 4 - ice

Let us consider the saturated soils. Based on the above consideration, the saturated soil containing organic matter can be represented as a four-component shell system within a cubic cell (Fig. 9) consisting of a mineral particle (1), organic matter (humus) (2), unfrozen water (3), and ice (4). The thermal conductivity of this system (frozen organic soil) calculated with the successive use of Maxwell's method is described by the equation (Gavrilyev, 2001)

$$
\lambda\_{\rm f} = \lambda\_4 \left[ 1 + \frac{3Z(1 - \text{m}\_i)}{3\lambda\_i + \text{Zm}\_i} \right] \tag{39}
$$

where

$$\mathbf{Z} = \boldsymbol{\lambda}\_{\text{uw}} \frac{\mathbf{Z} \boldsymbol{\lambda}\_{\text{uw}} \mathbf{m}\_{\text{uw}} \mathbf{D} + \boldsymbol{\lambda}\_{\text{org}} \mathbf{Q} \Big[\mathbf{3} \left(\mathbf{1} - \mathbf{m}\_{\text{i}}\right) - \mathbf{2} \mathbf{m}\_{\text{uw}}\Big]}{\boldsymbol{\lambda}\_{\text{uw}} \mathbf{D} \Big[\mathbf{3} \left(\mathbf{1} - \mathbf{m}\_{\text{i}}\right) - \mathbf{m}\_{\text{uw}}\Big] + \boldsymbol{\lambda}\_{\text{org}} \mathbf{m}\_{\text{uw}} \mathbf{Q}} - \boldsymbol{\lambda}\_{\text{i}} \Big] \tag{40}$$

$$\mathbf{D} = \boldsymbol{\lambda}\_{\rm s} \mathbf{m}\_{\rm org} + \boldsymbol{\lambda}\_{\rm org} \left( \mathbf{3} \,\mathbf{m}\_{\rm s} + \mathbf{m}\_{\rm org} \right) \,\prime \tag{41}$$

Analytical Methods for Estimating Thermal Conductivity

at low humus contents (norg < 0.1).

(unit fraction): 1 - 0; 2 – 0.1; 3 – 0.2; 4 – 0.3

component (morg=0).

**6. Summary** 

distribution.

view.

of Multi-Component Natural Systems in Permafrost Areas 253

The λj values of minerals are available in reference books (Birch, 1942; Clark, 1966; Kobranova, 1962; Missenard, 1965; Smyslov et al., 1979). The unit weight of organic matter

Fig. 10 shows the relationship between the thermal conductivity of a silt-clay [λs=2.50 W/(m•K)] and the saturation moisture content at different organic contents. It is seen that reduction in the soil thermal conductivity due to the presence of organic matter is strongest

(ρorg) by analogy with peat (Gavriliev & Eliseev, 1970) can be taken as 1.48⋅103 kg/m3.

Fig. 10. Thermal conductivity of saturated silt-clay in frozen (solid lines) and thawed (dashed lines) states vs saturation moisture content Wsat at different organic contents norg

We note in conclusion that Eq. (39) can lead to Eq. (33) for permafrost with no second

Frozen soils are complex multi-component and multi-phase systems consisting of mineral and organic particles, ice, unfrozen water, and gas (vapour). The specific conditions of sedimentation and subsequent diagenesis in permafrost environments result in sediments of permafrost-type with a very complex composition, structure and statistical particle

A deep understanding of the thermal properties of frozen soils can only be gained through an integral combination of experimental and analytical methods. Experimental methods have limitations in quantitative terms. Having obtained some basic information with experimental techniques, further in-depth study can be made using analytical methods. The development of theoretical approach is needed to understand heat transfer processes and to analyze experimental data on thermal properties of soils and rocks from a common point of

$$\mathbf{Q} = \mathcal{Z}\boldsymbol{\lambda}\_{\mathrm{org}}\mathbf{m}\_{\mathrm{org}} + \boldsymbol{\lambda}\_{\mathrm{s}} \left( \mathbf{3}\,\mathrm{m}\_{\mathrm{s}} + \mathbf{m}\_{\mathrm{org}} \right) \prime \tag{42}$$

λ and m are the thermal conductivity and volume fractions of the components, respectively. In the saturated unfrozen organic soil, the ice content mi is zero, and from Eq. (39) we obtain the following equation for the thermal conductivity λ<sup>t</sup>

$$
\lambda\_{\rm t} = \lambda\_{\rm uw} \frac{2\lambda\_{\rm uw}\mathbf{m}\_{\rm uw}\mathbf{D} + \lambda\_{\rm og}\mathbf{Q}(3 - 2\,\mathbf{m}\_{\rm uw})}{\lambda\_{\rm uw}\mathbf{D}(3 - \mathbf{m}\_{\rm uw}) + \lambda\_{\rm og}\mathbf{m}\_{\rm uw}\mathbf{Q}} \,. \tag{43}
$$

The volume fractions of the components of the organic soil in the saturated state, ms, morg, muw and mi, can be found using the following equations:

$$\mathbf{m}\_s = \frac{1}{1 + \mathbf{W}\_{\rm sat}^0 \boldsymbol{\rho}\_s} \,'\,\tag{44}$$

$$\mathbf{m}\_{\rm org} = \frac{\mathbf{n}\_{\rm org} \rho\_{\rm s} \mathbf{m}\_{\rm s}}{\rho\_{\rm org} \left(1 - \mathbf{n}\_{\rm org}\right)}\tag{45}$$

$$\mathbf{m}\_{\text{uw}} = \mathbf{W}\_{\text{uw}} \boldsymbol{\gamma}\_{\text{d}}.\tag{46}$$

$$\mathbf{m}\_{\rm i} = \left(\mathbf{W}\_{\rm sat} - \mathbf{W}\_{\rm uw}\right) \mathbf{\gamma}\_{\rm d} \tag{47}$$

$$
\gamma\_{\rm d} = \rho\_{\rm s} \mathbf{m}\_{\rm s} + \rho\_{\rm org} \mathbf{m}\_{\rm org},\tag{48}
$$

$$\mathbf{m}\_s + \mathbf{m}\_{\text{org}} + \mathbf{m}\_{\text{uw}} + \mathbf{m}\_i = \mathbf{1},\tag{49}$$

where norg = Porg/(Ps+Porg) is the relative weight of organic matter; Рs and Рorg are the weights of the soil mineral particles and organic matter in the dry state; ρ is the unit weight of the components; <sup>0</sup> W is the saturation moisture content of the soil containing no organic sat matter (fraction).

The saturation moisture content of the organic soil Wsat is related to that of the soil containing no organics <sup>0</sup> W by the relationship: sat

$$\mathbf{W}\_{\rm sat} = \mathbf{W}\_{\rm sat}^{0} \left( 1 - \mathbf{n}\_{\rm org} \right) - \frac{\mathbf{n}\_{\rm org} \rho\_{\rm ww}}{\rho\_{\rm org}} \,. \tag{50}$$

For the saturated organic soil, the following relationship is valid:

$$\gamma\_{\rm d} = \frac{\rho\_{\rm s}^{\prime}}{1 + \mathcal{W}\_{\rm sat} \rho\_{\rm s}^{\prime}} \; \; \; \tag{51}$$

where ( ) ( ) ' ρ = ρ +ρ s s s or m m mm <sup>g</sup> org s or + g is the unit weight of the organic soil.

In computations of the thermal conductivity of organic soils using Eqs. (39) - (43), the following λ values (W/(m•K)) can be taken for components: λorg=0.26 (Farouki 1986), λuw=0.58 and λi=2.25. The value of λs is a function of the soil type of the C horizon and can be estimated from the mineral composition of the particles by Eq. (14).

λ and m are the thermal conductivity and volume fractions of the components, respectively. In the saturated unfrozen organic soil, the ice content mi is zero, and from Eq. (39) we obtain

( )

The volume fractions of the components of the organic soil in the saturated state, ms, morg,

org

uw uw org uw

sat s

( ) org s s

org org

where norg = Porg/(Ps+Porg) is the relative weight of organic matter; Рs and Рorg are the weights of the soil mineral particles and organic matter in the dry state; ρ is the unit weight of the components; <sup>0</sup> W is the saturation moisture content of the soil containing no organic sat

The saturation moisture content of the organic soil Wsat is related to that of the soil

sat sat org

<sup>s</sup> <sup>d</sup>

In computations of the thermal conductivity of organic soils using Eqs. (39) - (43), the following λ values (W/(m•K)) can be taken for components: λorg=0.26 (Farouki 1986), λuw=0.58 and λi=2.25. The value of λs is a function of the soil type of the C horizon and can

ρ′ γ =

where ( ) ( ) ' ρ = ρ +ρ s s s or m m mm <sup>g</sup> org s or + g is the unit weight of the organic soil.

be estimated from the mineral composition of the particles by Eq. (14).

sat s , 1 W

n m m , 1 n

<sup>ρ</sup> <sup>=</sup> ρ −

1 m , 1 W <sup>=</sup> <sup>+</sup> <sup>ρ</sup>

uw uw org uw 2 m D Q 3 2m . D3 m m Q λ +λ −

the following equation for the thermal conductivity λ<sup>t</sup>

t uw

λ =λ

muw and mi, can be found using the following equations:

s <sup>0</sup>

containing no organics <sup>0</sup> W by the relationship: sat

( ) <sup>0</sup> org uw

For the saturated organic soil, the following relationship is valid:

matter (fraction).

Q 2 m 3m m , = λ +λ + org org s s org ( ) (42)

( )

λ − +λ (43)

m W, uw uw d = γ (46)

mWW , i sat uw d = ( − γ) (47)

d s s or m m, <sup>g</sup> org γ =ρ +ρ (48)

m m m m 1, s org uw i + + += (49)

org

W W 1n . <sup>ρ</sup> = −− <sup>ρ</sup> (50)

<sup>+</sup> ρ′ (51)

n

(44)

(45)

The λj values of minerals are available in reference books (Birch, 1942; Clark, 1966; Kobranova, 1962; Missenard, 1965; Smyslov et al., 1979). The unit weight of organic matter (ρorg) by analogy with peat (Gavriliev & Eliseev, 1970) can be taken as 1.48⋅103 kg/m3.

Fig. 10 shows the relationship between the thermal conductivity of a silt-clay [λs=2.50 W/(m•K)] and the saturation moisture content at different organic contents. It is seen that reduction in the soil thermal conductivity due to the presence of organic matter is strongest at low humus contents (norg < 0.1).

Fig. 10. Thermal conductivity of saturated silt-clay in frozen (solid lines) and thawed (dashed lines) states vs saturation moisture content Wsat at different organic contents norg (unit fraction): 1 - 0; 2 – 0.1; 3 – 0.2; 4 – 0.3

We note in conclusion that Eq. (39) can lead to Eq. (33) for permafrost with no second component (morg=0).

#### **6. Summary**

Frozen soils are complex multi-component and multi-phase systems consisting of mineral and organic particles, ice, unfrozen water, and gas (vapour). The specific conditions of sedimentation and subsequent diagenesis in permafrost environments result in sediments of permafrost-type with a very complex composition, structure and statistical particle distribution.

A deep understanding of the thermal properties of frozen soils can only be gained through an integral combination of experimental and analytical methods. Experimental methods have limitations in quantitative terms. Having obtained some basic information with experimental techniques, further in-depth study can be made using analytical methods. The development of theoretical approach is needed to understand heat transfer processes and to analyze experimental data on thermal properties of soils and rocks from a common point of view.

Analytical Methods for Estimating Thermal Conductivity

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186-189, Harbin, China, September 11-14, 1996

Maxwell, J. C. (1873). *Treatise on Electricity and Magnetism,* Oxford

June 3-5, 1996

2003

Russia

1996

Moscow, USSR

ISSN 0044-4642

SSSR, Moscow, USSR

Ugletekhizdat, Moscow, USSR

Editions Eyrolles, Paris, France

Izd-vo AN SSSR, Moscow, USSR

USSR

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Frozen Soils. Permafrost. *Proceedings of the Eighth International Conference on Permafrost,* Vol. 1, pp. 283-287, ISBN 90 5809 584 3, Zurich, Switzerland, July 21-25,

*Permafrost Zone* (in Russian), Izdatelstvo SO RAN, ISBN 5-7692-0551-2, Novosibirsk,

Properties of Frozen Soils (in Russian). *Kriosfera Zemli,* Vol. 7, No.2, (April-June,

Subject to Metamorphism, *Proceedings of 5th International Symposium on Thermal Engineering and Sciences for Cold Regions,* pp. 516-521, Ottawa, Canada, May 19-22,

at the Non-uniform Composition of Gruss-crushed Stone and Gravel-pebble Inclusions. *Proceedings of International Symposium on Cold Regions Engineering,* pp.

*Proceedings of 7th International Symposium on Thermal Engineering and Sciences for Cold* 

Considering the history of sediment formation over a geologic time, a universal model with changing particle shapes is proposed which describes the processes of rock formation from sediments, snow compaction and glacierization with account for diagenetic and postdiagenetic structural modifications, as well the processes of rock weathering and soil formation.

Based on the Maxwell's model with dispersed particles in the main medium, methods of estimating the thermal conductivity are given for:


#### **7. References**


Considering the history of sediment formation over a geologic time, a universal model with changing particle shapes is proposed which describes the processes of rock formation from sediments, snow compaction and glacierization with account for diagenetic and postdiagenetic structural modifications, as well the processes of rock weathering and soil

Based on the Maxwell's model with dispersed particles in the main medium, methods of

1. Soils containing coarse inclusions and frozen soils of layered or reticulate cryostructure with account for variation in composition (a two-component system: coarse inclusions + fine fill, layers of frozen soil and ice for a case of layered cryostructure, and soil

2. Permafrost with a massive cryostructure (a three-component spherical shell system:

3. Organic soils (a four-component spherical shell system: mineral particle + humus +

Belskaya, E., Postnikov, V., Vasiliev, L. & Khusid, B. (1981). Thermal Conductivity of

Birch, F., Schairer, J. & Spicer, H. (Eds.). (1942*)*. *Handbook of Physical Constants,* Spec. Pap. 36,

Carslaw, H. & Jaeger, J. (1959*). Conduction of Heat in Solids* (2nd edition), Oxford University

Clark, S. (Ed.). (1966). *Handbook of Physical Constants* (rev. ed.), Geological Society of

Everdingen, R. van (Ed.). (1998, revised January, 2002). *Multi-Language Glossary of Permafrost* 

Farouki, O. (1986). *Thermal Properties of Soils,* Trans Tech Publications, ISBN 0-87849-055-8,

Gavril'ev, R. I. (1989). Thermal Conductivity of Permafrost Soils in Relation to Natural

Gavril'ev, R. I. (1992). Model for Calculating the Thermal Conductivity of Soils with Their

Gavriliev, R. & Eliseev, S. (1970). Thermal Properties of Moss and Peat (in Russian), In:

Gavriliev, R. (1986). Investigating the Thermal Properties of Soils with Coarse Inclusions (in

Gavriliev, R. (1996). A Model for Estimating the Thermal Conductivity of Sediments (in

*and Related Ground-Ice Terms*, National Snow and Ice Data Center/World Data Center for Glaciology, Boulder, Colorado, Available from

Moisture Content. *Journal of Engineering Physics and Thermophysics,* Vol.56, No.6,

Genesis Taken into Account*. Journal of Engineering Physics and Thermophysics,*

*Methods for Determining the Thermal Properties of Soils and Rocks,* N. Ivanov, (Ed.),

Russian). *Inzhenernaya Geologiya,* No. 5 (September-October, 1986), pp. 60-71, ISSN

Russian), *Proceedings of 1st Conference of Russian Geocryologists,* Book 2, Part 1,

Materials with Shell-Covered Dispersed Fillers (in Russian). *Vestsi AN BSSR Seryia fizika-matematicheskikh navuk,* No.1 (January-March 1981), pp. 91-95, ISSN 0374-4760

inclusions in the ice matrix for a case of reticulate cryostructure);

Geological Society of America, Boulder, Colorado, USA

Press, ISBN 0-19-85-3303-9, Oxford, UK

http://nsidc.org/fgdc/glossary/

(June, 1989), pp. 701-706, ISSN 1062-0125

Vol.62, No.1, (January, 1992), pp. 68-76, ISSN 1062-0125

Clausthal-Zellerfeld, Germany

139-153, Nauka, Moscow, USSR

0208-0292

America, ISBN 0813710979, New York, USA

formation.

estimating the thermal conductivity are given for:

mineral particle + unfrozen water + ice);

unfrozen water + ice).

**7. References** 

Physics, Chemistry and Mechanics of Frozen Ground, pp. 101-111, Moscow, Russia, June 3-5, 1996


**12** 

*Taiwan* 

**Heating in Biothermal Systems** 

Huang, Huang-Wen1 and Liauh, Chihng-Tsung2

*2Department of Mechanical Engineering, Kun-Shan University* 

*1Department of Innovative Information and Technology, Tamkang University* 

Heat therapy is a traditional healing method. The first written reference to hyperthermia occurred in an Egyptian papyrus scroll 5000 years ago (Hall & Cox, 2003). In this chapter, we planned to introduce promising medical treatments which applied thermal energy in living tissues to raise tissue temperatures to therapeutic temperatures. The treatments, based on duration of time, were classified into low-temperature level in the range of 41~45°C (hyperthermia or thermal therapy) and much higher temperatures (thermal ablation

Heating in the biothermal systems involves two primary heat transport modes: thermal conduction and convection. Extreme complicating living vasculatures and organs make heating the target volume and raising temperature to therapeutic temperature at the target

In this chapter, we will introduce thermal models which were used to describe biothermal systems. One of well-known models, Pennes bio-heat transfer equation (PBHTE) was initially introduced in 1949. Up until now, it still is used widely by research groups around the globe, to deal with temperature or heat that associated with human or living creatures. As well as other models which also described the impact of thermally significant blood vessel (or vascular system) and blood perfusion during treatments will also be introduced in

The heating methods that deposited the thermal energy in transport medium (i.e. solid tumor, tissues and liquid so forth) will also be addressed. For example, high-intensity focused ultrasound (HIFU) is one of popular non-invasive method which could deposit heat

Lastly, we addressed promising future in combining other treating modalities (such as drug treatment) with hyperthermia. It has shown significant results for medical treatments in

Thermal transport modes in bio-thermal systems involve three typical modes: conduction, convection and radiation. Limitation and restrictions of therapeutic temperatures on heating (or freezing) subjects are first, to remove tumorous tissues and at the same time without damaging the normal tissues. Table 1 shows significance of thermal transport modes in

**1. Introduction** 

this chapter.

treatments) than previous one.

into deep seated tissue region.

curing patients with cancers.

**2. Heat transport** 

volume difficult and, thus, a challenging task.


## **Heating in Biothermal Systems**

Huang, Huang-Wen1 and Liauh, Chihng-Tsung2

*1Department of Innovative Information and Technology, Tamkang University 2Department of Mechanical Engineering, Kun-Shan University Taiwan* 

### **1. Introduction**

256 Convection and Conduction Heat Transfer

Sergeev, E., Golodkovskaya, G., Osipov, V. & Trofimov, V. (1971). *Soil and Rock Engineering*

Smyslov, A., Moiseenko, U. & Chadovich, T. (1979). *Thermal Regime and Radioactivity of the* 

Sulakvelidze, G.K. & Okudzhava, A.M. (1959). Snow and Its Properties (in Russian),

Tsytovich, N. (1945). On the Theory of the Equilibrium State of Water in Frozen Soils (in

Vargaftik, N. (Ed.). (1963). *Handbook of Thermophysical Properties of Gases and Fluids* (in

Votyakov, I. (1975). *Physical and Mechanical Properties of Frozen and Thawing Soils in Yakutia*

*Transactions of the Elbrus Alpine Expedition,* Vol.1, No.4 (October-December, 1959),

Russian). *Izvestia AN SSSR,* Seria Geograficheskaya i Geofizicheskaya, Vol. 9, No. 5-

(in Russian), Izdatelstvo MGU, Moscow, USSR

Tsyganov, M. (1958). *Soil Science* (in Russian). Selkhozgiz, Moscow, USSR

*Earth* (in Russian), Nedra, Leningrad, USSR

6 (November-December, 1945), pp. 493-502

Russian), Fizmatgiz, Moscow, USSR

(in Russian), Nauka, Novosibirsk, USSR

рр. 9-49

Heat therapy is a traditional healing method. The first written reference to hyperthermia occurred in an Egyptian papyrus scroll 5000 years ago (Hall & Cox, 2003). In this chapter, we planned to introduce promising medical treatments which applied thermal energy in living tissues to raise tissue temperatures to therapeutic temperatures. The treatments, based on duration of time, were classified into low-temperature level in the range of 41~45°C (hyperthermia or thermal therapy) and much higher temperatures (thermal ablation treatments) than previous one.

Heating in the biothermal systems involves two primary heat transport modes: thermal conduction and convection. Extreme complicating living vasculatures and organs make heating the target volume and raising temperature to therapeutic temperature at the target volume difficult and, thus, a challenging task.

In this chapter, we will introduce thermal models which were used to describe biothermal systems. One of well-known models, Pennes bio-heat transfer equation (PBHTE) was initially introduced in 1949. Up until now, it still is used widely by research groups around the globe, to deal with temperature or heat that associated with human or living creatures. As well as other models which also described the impact of thermally significant blood vessel (or vascular system) and blood perfusion during treatments will also be introduced in this chapter.

The heating methods that deposited the thermal energy in transport medium (i.e. solid tumor, tissues and liquid so forth) will also be addressed. For example, high-intensity focused ultrasound (HIFU) is one of popular non-invasive method which could deposit heat into deep seated tissue region.

Lastly, we addressed promising future in combining other treating modalities (such as drug treatment) with hyperthermia. It has shown significant results for medical treatments in curing patients with cancers.

#### **2. Heat transport**

Thermal transport modes in bio-thermal systems involve three typical modes: conduction, convection and radiation. Limitation and restrictions of therapeutic temperatures on heating (or freezing) subjects are first, to remove tumorous tissues and at the same time without damaging the normal tissues. Table 1 shows significance of thermal transport modes in

Heating in Biothermal Systems 259

Nevertheless, the equation does have some limitations. It does not, nor was it ever intended to, handle several physical effects. The most significant problem is that it does not consider the effect of the directionality of blood flow, and hence does not describe any convective

Several investigators have developed alternative formulations to predict temperatures in living tissues. In 1980, Chen and Holmes (Chen and Holmes, 1980) derived the one and it

> ( ) ( ) *p p m b a bb <sup>p</sup> <sup>T</sup> k k T q q Wc T T c u T c <sup>t</sup>*

Comparing this equation with Pennes' equation, two extra terms have been added. The term

In 1985, Weinbaum and Jiji (Weinbaum and Jiji, 1985) proposed an alternative mathematical formulation of the bio-heat transfer equation. Their formulation is based on their observations from vascular network of rabbit thighs that blood vessels which are significant for heat transfer in tissues always occur in counter-current pairs. Hence, the major heat transfer mechanism between blood and tissues is the "incomplete counter-current heat exchanger" between thermally significant arteries and veins (with diameters about 50-

<sup>2</sup> ( ) <sup>4</sup>

⎡ ⎤ ∂∂ ∂ <sup>∂</sup> <sup>∂</sup> − = ⎢ ⎥ − + ∂∂ ∂ ⎢ ⎥ ∂ ∂ ⎣ ⎦

product, a is the local blood vessel radius, σ is a shape factor for the thermal conduction resistance between adjacent counter-current vessels, n is the number density of blood vessels of size a, kb is the blood thermal conductivity, Pe is the local Peclet number

<sup>2</sup> () ( ) <sup>4</sup>

*ij eff ij i j na k k k Pe l l*

π δ

*i j j j na k l c k Pe l Q t x x k kx*

*<sup>b</sup>* cnau/kb), us is average blood flow velocity in the vessels and li is the direction cosine of the ith pair of countercurrent vessels. ( ) *ij eff k* is the effective conductivity tenor element,

2 22

*k*

σ

*b*

2

π

2 22

σ

*ij eff j m*

*b i*

 θ

*c* is the volume average tissue density and specific heat

= + (4)

(3)

500μm). Their formulation uses tensor notation and it can be written as

θθ

ρ

 *cu T* i is the convective heat transfer term, which accounts for the thermal interactions between blood vessels and tissues. The term *<sup>p</sup>* ∇i*k T*∇ accounts for the enhanced tissue conductive heat transfer due to blood perfusion term in tissues, where kp is called the perfusion conductivity, and is a function of blood perfusion rate. The blood perfusion term ( ) − − *Wc T T b a* shown in CH model, accounts for the effects of the large number of capillary structures whose individual dimensions are small relative to the macroscopic phenomenon under their study. Relatively CH model has a more solid physical basis than Pennes model. However, it requires knowledge of the details of the vascular anatomy and flow pattern to

<sup>∂</sup> ∇ + ∇+ + − − − ∇=

ρ

<sup>∂</sup> i i (2)

 ρ

has a very strong physical and physiological basis. The equation can be written as,

heat transfer mechanism.

*b b* − ∇ ρ

**3.2 The Chen and Holmes (CH) model** 

solve it, and that was an extreme complicating task.

**3.3 The Weinbaum and Jiji (WJ) model** 

ρ

where *θ* is the local temperature,

(=2 ρ

which is given by,

typical components of bio-thermal systems, as our subject of discussion refers to cancer treatments using heat. Thermal diffusion plays a dominant transport mode in tissues, and convection is less significant as blood perfuses in solid tissues at capillary level vessels which are small in size and blood within are in slow motion.


Table 1. Significance of thermal transport modes in typical components of vio-thermal systems

#### **3. Mathematical modelings**

To accurately describe physical phenomena in living tissues, we have to rely upon mathematical models (or bio-heat transfer formulation). There are many significant models appeared in history during the development of heating in tissues. Those models help us in analyzing temperature management in treatment planning with accuracy. We introduced them here:

#### **3.1 Pennes bio-heat transfer equation (PBHTE)**

In 1948, Pennes (Pennes, 1948) performed a series of experiments which measured temperatures on human forearms of volunteers, and derived a thermal energy conservation equation: the well known bio-heat transfer equation (BHTE) or the traditional bio-heat transfer equation. It is written as

$$\nabla \cdot \mathbf{k} \nabla T + q\_p + q\_m - \mathcal{V} \mathbf{c}\_b (T - T\_a) = \rho \mathbf{c}\_p \, \frac{\partial T}{\partial t} \tag{1}$$

where T(°>°C) is the local tissue temperature; Ta (°C) is the arterial temperature; cb (J/kg/°C) is the blood specific heat; cp (J/kg/°C) is the tissue specific heat; W(kg/m3/s) is the local tissue blood perfusion rate; k(w/m/°C) is the tissue thermal conductivity; ρ(kg/m3) is the tissue density; qp(w/m3) is the energy deposition rate; and qm (w/m3) is the metabolism, which is usually very small compared to the external power deposition term qp (Roemer et al, 1988). The term Wcb(T-Ta), which accounts for the effects of blood perfusion, can be the dominant form of energy removal when considering heating process. It assumes that the blood enters the control volume at some arterial temperature Ta, and then comes to equilibrium at the tissue temperature. Thus, the blood leaves the control volume it carries away the energy, and hence acts as an energy sink in hyperthermia treatment.

Although Pennes' equation is an approximation equation and it does not have a physically consistent theoretical basis, it is surprisingly that this simple mathematical formulation predicted temperature fields well in many application situations. The reasons why PBHTE has been widely used in hyperthermia modeling field are two important factors: one is its mathematical simplicity and the other is its ability to predict the temperature field reasonably in applied fields.

typical components of bio-thermal systems, as our subject of discussion refers to cancer treatments using heat. Thermal diffusion plays a dominant transport mode in tissues, and convection is less significant as blood perfuses in solid tissues at capillary level vessels

Tissues Significant Less Significant Insignificant Bones Significant Insignificant Insignificant Blood vessels Less Significant Significant Insignificant Skins Insignificant Significant Significant Table 1. Significance of thermal transport modes in typical components of vio-thermal

To accurately describe physical phenomena in living tissues, we have to rely upon mathematical models (or bio-heat transfer formulation). There are many significant models appeared in history during the development of heating in tissues. Those models help us in analyzing temperature management in treatment planning with accuracy. We introduced

In 1948, Pennes (Pennes, 1948) performed a series of experiments which measured temperatures on human forearms of volunteers, and derived a thermal energy conservation equation: the well known bio-heat transfer equation (BHTE) or the traditional bio-heat

<sup>∂</sup> ∇∇+ + − − =

where T(°>°C) is the local tissue temperature; Ta (°C) is the arterial temperature; cb (J/kg/°C) is the blood specific heat; cp (J/kg/°C) is the tissue specific heat; W(kg/m3/s) is the local tissue blood perfusion rate; k(w/m/°C) is the tissue thermal conductivity; ρ(kg/m3) is the tissue density; qp(w/m3) is the energy deposition rate; and qm (w/m3) is the metabolism, which is usually very small compared to the external power deposition term qp (Roemer et al, 1988). The term Wcb(T-Ta), which accounts for the effects of blood perfusion, can be the dominant form of energy removal when considering heating process. It assumes that the blood enters the control volume at some arterial temperature Ta, and then comes to equilibrium at the tissue temperature. Thus, the blood leaves the control volume it carries

Although Pennes' equation is an approximation equation and it does not have a physically consistent theoretical basis, it is surprisingly that this simple mathematical formulation predicted temperature fields well in many application situations. The reasons why PBHTE has been widely used in hyperthermia modeling field are two important factors: one is its mathematical simplicity and the other is its ability to predict the temperature field

away the energy, and hence acts as an energy sink in hyperthermia treatment.

( ) *pm b a p <sup>T</sup> k T q q Wc T T c <sup>t</sup>*

ρ

<sup>∂</sup> <sup>i</sup> (1)

Conduction Convection Radiation

which are small in size and blood within are in slow motion.

systems

them here:

**3. Mathematical modelings** 

transfer equation. It is written as

reasonably in applied fields.

**3.1 Pennes bio-heat transfer equation (PBHTE)** 

Nevertheless, the equation does have some limitations. It does not, nor was it ever intended to, handle several physical effects. The most significant problem is that it does not consider the effect of the directionality of blood flow, and hence does not describe any convective heat transfer mechanism.

#### **3.2 The Chen and Holmes (CH) model**

Several investigators have developed alternative formulations to predict temperatures in living tissues. In 1980, Chen and Holmes (Chen and Holmes, 1980) derived the one and it has a very strong physical and physiological basis. The equation can be written as,

$$\nabla \bullet (k + k\_p) \nabla T + q\_p + q\_m - \mathcal{Wc}\_b (T - T\_a) - \rho\_b c\_b \mu \bullet \nabla T = \rho c\_p \frac{\partial T}{\partial t} \tag{2}$$

Comparing this equation with Pennes' equation, two extra terms have been added. The term *b b* − ∇ ρ *cu T* i is the convective heat transfer term, which accounts for the thermal interactions between blood vessels and tissues. The term *<sup>p</sup>* ∇i*k T*∇ accounts for the enhanced tissue conductive heat transfer due to blood perfusion term in tissues, where kp is called the perfusion conductivity, and is a function of blood perfusion rate. The blood perfusion term ( ) − − *Wc T T b a* shown in CH model, accounts for the effects of the large number of capillary structures whose individual dimensions are small relative to the macroscopic phenomenon under their study. Relatively CH model has a more solid physical basis than Pennes model. However, it requires knowledge of the details of the vascular anatomy and flow pattern to solve it, and that was an extreme complicating task.

#### **3.3 The Weinbaum and Jiji (WJ) model**

In 1985, Weinbaum and Jiji (Weinbaum and Jiji, 1985) proposed an alternative mathematical formulation of the bio-heat transfer equation. Their formulation is based on their observations from vascular network of rabbit thighs that blood vessels which are significant for heat transfer in tissues always occur in counter-current pairs. Hence, the major heat transfer mechanism between blood and tissues is the "incomplete counter-current heat exchanger" between thermally significant arteries and veins (with diameters about 50- 500μm). Their formulation uses tensor notation and it can be written as

$$
\rho c \frac{\partial \theta}{\partial t} - \frac{\partial}{\partial \mathbf{x}\_i} \left[ (k\_{ij})\_{\text{eff}} \frac{\partial \theta}{\partial \mathbf{x}\_j} \right] = -\frac{\pi^2 n a^2 k\_b^2}{4 \sigma k} P e^2 l\_j \frac{\partial l\_i}{\partial \mathbf{k}\_j} \frac{\partial \theta}{\partial \mathbf{x}\_j} + Q\_m \tag{3}
$$

where *θ* is the local temperature, ρ*c* is the volume average tissue density and specific heat product, a is the local blood vessel radius, σ is a shape factor for the thermal conduction resistance between adjacent counter-current vessels, n is the number density of blood vessels of size a, kb is the blood thermal conductivity, Pe is the local Peclet number (=2 ρ*<sup>b</sup>* cnau/kb), us is average blood flow velocity in the vessels and li is the direction cosine of the ith pair of countercurrent vessels. ( ) *ij eff k* is the effective conductivity tenor element, which is given by,

$$(k\_{ij})\_{eff} = k(\delta\_{ij} + \frac{\pi^2 n a^2 k\_b^2}{4\sigma k^2} Pe^2 l\_i l\_j) \tag{4}$$

Heating in Biothermal Systems 261

The blood leaving these terminal arterial vessels at any cross-section is assumed to perfuse the tissue at a constant rate. The detailed description is shown in Huang (Huang et al, 1996). As to venous thermal model, for all of veins except the smallest terminal veins, the above equation (5) holds. For the smallest veins, the Tb replaced by the venous return temperature, Tvr(xi). In the presented study this temperature is taken to be average temperature of four

> 4 , 1

For tissue matrix thermal equations, they can be explained most succinctly by considering

Here, k is the thermal conductivity of the tissue matrix, T(x,y,z) is the tissue temperature, *W* is the "perfusion" value and Ta is the arterial blood temperature at some reference

A few studies (Leeuwen et al, 2000; Devashish and Roemer, 2006; Baish, 1994) have modeled the effect of collections of a large number of parallel vessels or of networks of vessels on the resulting temperature distributions. Those were developed in attempt to describe the impact of blood vessels and to properly predict heat transfer processes in bio-thermal systems in a

As mentioned above the mathematical models for actual thermal problems of interest in hyperthermia or thermal ablation are too complicated to be conveniently solved with exact formulas. The majority of unsolved problems in medical fields is governed by non-linear partial differential equations. In most cases, one thereby reduces the problems to rather simplified models which can be exactly analyzed, for example, analytical solution of the 3D Pennes equation presented by Liu (Liu, 2001; Liu and Deng, 2002) using multidimensional Green function, and 1D transient Pennes equation by Shih et. al. (Shih et al, 2007) using the Laplace transform. But occasionally such an approach does not suffice. Consequently, specialists have recently devoted increasing attention to numerical, as opposed to analytical, techniques. Nowadays one of the major challenges for thermal ablation and hyperthermia simulation is the incorporation of the very detailed information coming from biophysical models into the numerical simulations. Thanks to advanced imaging techniques, accurate tumor static models including detailed description of all vascular matrix objects are currently available. Unfortunately, most of the discretization methods commonly used in computer simulation, mainly based on structured grids, are not capable to represent the detailed geometry of such treatment regions or other complicated entities such as microvascular matrix, horizontal wells, and uniformity, etc. The complexity of multidimensional heat transfer problems in hyperthermia suggests the application of numerical techniques. Several numerical methods have been used in engineering and

<sup>=</sup> ∑ (7)

<sup>2</sup> ( ) *b aap* −∇ + − = *k T Wc T T Q* (8)

1 4 *vr i adj i T T* =

tissue nodes adjacent to the terminal vein in the plane perpendicular to that vein,

the Pennes Bio-Heat Transfer Equation as the most general formulation,

location.

**3.5 Others** 

more accurate way.

**4. Numerical modelings** 

where *δij* is the kronecker delta function, and k is the tissue thermal conductivity. Clearly, this equation represents one of the most significant contributions to the bio-heat transfer formulation. But, in practical situations, this equation needs detailed knowledge of the sizes, orientations, and blood flow velocities in the countercurrent vessels to solve it and that presents a formidable task. Furthermore, there are several issues related to the WJ model. First, thoroughly comparison for both predicted temperatures and macroscopic experiments are required. Secondly, the formulation was developed for superficial normal tissues in which the counter-current heat transfer occurs. In tumors, the vascular anatomy is different from the superficial normal tissues, and therefore a new model should be derived for tumors. Some (Wissler, 1987) has questioned the two basic assumptions of WJ model: first, that the arithmetic mean of the arteriole and venule blood temperature can be approximated by the mean tissue temperature; and second, that there is negligible heat transfer between the thermally significant arteriole-venule pairs and surrounding tissue.

#### **3.4 Thermally significant blood vessel model**

As CH and WJ models presented, many investigators (Baish et al, 1986; Charny and Levin, 1990) during late 1980, questioned mostly on blood perfusion term or how to estimate blood temperature and local tissue temperatures where blood vessels (counter-current vessels) are involved. As arterial and veinous capillary vessels are small, their thermal contributions to local tissue temperatures are insignificant. However, some larger vessel sizes than the capillaries do have thermally significant impacts on tissue temperatures in either cooling or heating processes. Several investigators (Chato, 1980; Lagendijk, 1982; Huang et al, 1994) examined the effect of large blood vessels on temperature distribution using theoretical studies. Huang et al (Huang et al, 1996) in 1996 presented a more fundamental approach to model temperatures in tissues than do the generally used approximate equations such as the Pennes' BHTE or effective thermal conductivity equations. As such, this type of model can be used to study many important questions at a more basic level. For example, in the particular hyperthermia application studied herein, a simple vessel network model predicts that the role of counter current veins is minimal and that their presence does not significantly affect the tissue temperature profiles: the arteries, however, removed a significant fraction of the power deposited in the tissue. The Huang's model used a simple convective energy balance equation to calculate the blood temperature as a function of position,

$$
\dot{M}\_i \mathfrak{C}\_b \frac{dT\_b}{d\mathfrak{x}\_i} = \dot{Q}\_{ap} - h\_i A\_i (T\_b - T\_w) \tag{5}
$$

Here, *Mi* is the mass flow rate of blood in artery i, cb is the specific heat of blood, Tb(xi) is the average blood temperature at position xi, xi indicates the direction along the vessel I (either x, y or z depending on the vessel level). *Qap* is the applied power deposition xi, hi is the heat transfer coefficient between the blood and the tissue, Ai is the perimeter of blood vessel i, and Tw(xi) is the temperature of the tissue at the vessel wall. For the smallest, terminal arterial vessels a decreasing blood flow rate is present giving the energy balance equation,

$$
\dot{M}\_i \mathbf{c}\_b \frac{dT\_b}{d\mathbf{x}\_i} = \dot{Q}\_{ap} - h\_i A\_i (T\_b - T\_w) - \frac{d\dot{M}\_i}{d\mathbf{x}\_i} \mathbf{c}\_b T\_b \tag{6}
$$

where *δij* is the kronecker delta function, and k is the tissue thermal conductivity. Clearly, this equation represents one of the most significant contributions to the bio-heat transfer formulation. But, in practical situations, this equation needs detailed knowledge of the sizes, orientations, and blood flow velocities in the countercurrent vessels to solve it and that presents a formidable task. Furthermore, there are several issues related to the WJ model. First, thoroughly comparison for both predicted temperatures and macroscopic experiments are required. Secondly, the formulation was developed for superficial normal tissues in which the counter-current heat transfer occurs. In tumors, the vascular anatomy is different from the superficial normal tissues, and therefore a new model should be derived for tumors. Some (Wissler, 1987) has questioned the two basic assumptions of WJ model: first, that the arithmetic mean of the arteriole and venule blood temperature can be approximated by the mean tissue temperature; and second, that there is negligible heat transfer between

As CH and WJ models presented, many investigators (Baish et al, 1986; Charny and Levin, 1990) during late 1980, questioned mostly on blood perfusion term or how to estimate blood temperature and local tissue temperatures where blood vessels (counter-current vessels) are involved. As arterial and veinous capillary vessels are small, their thermal contributions to local tissue temperatures are insignificant. However, some larger vessel sizes than the capillaries do have thermally significant impacts on tissue temperatures in either cooling or heating processes. Several investigators (Chato, 1980; Lagendijk, 1982; Huang et al, 1994) examined the effect of large blood vessels on temperature distribution using theoretical studies. Huang et al (Huang et al, 1996) in 1996 presented a more fundamental approach to model temperatures in tissues than do the generally used approximate equations such as the Pennes' BHTE or effective thermal conductivity equations. As such, this type of model can be used to study many important questions at a more basic level. For example, in the particular hyperthermia application studied herein, a simple vessel network model predicts that the role of counter current veins is minimal and that their presence does not significantly affect the tissue temperature profiles: the arteries, however, removed a significant fraction of the power deposited in the tissue. The Huang's model used a simple convective energy balance equation to calculate the blood temperature as a function of

> ( ) *<sup>b</sup> i b ap i i b w*

Here, *Mi* is the mass flow rate of blood in artery i, cb is the specific heat of blood, Tb(xi) is the average blood temperature at position xi, xi indicates the direction along the vessel I (either x, y or z depending on the vessel level). *Qap* is the applied power deposition xi, hi is the heat transfer coefficient between the blood and the tissue, Ai is the perimeter of blood vessel i, and Tw(xi) is the temperature of the tissue at the vessel wall. For the smallest, terminal arterial vessels a decreasing blood flow rate is present giving the energy balance

> ( ) *b i ib ap i i b w bb i i*

*dT Mc Q hA T T dx* =− − (5)

*dT dM <sup>M</sup> c Q hA T T cT dx dx* =− −− (6)

*i*

the thermally significant arteriole-venule pairs and surrounding tissue.

**3.4 Thermally significant blood vessel model** 

position,

equation,

The blood leaving these terminal arterial vessels at any cross-section is assumed to perfuse the tissue at a constant rate. The detailed description is shown in Huang (Huang et al, 1996). As to venous thermal model, for all of veins except the smallest terminal veins, the above equation (5) holds. For the smallest veins, the Tb replaced by the venous return temperature, Tvr(xi). In the presented study this temperature is taken to be average temperature of four tissue nodes adjacent to the terminal vein in the plane perpendicular to that vein,

$$T\_{vr} = \frac{1}{4} \sum\_{i=1}^{4} T\_{i,adj} \tag{7}$$

For tissue matrix thermal equations, they can be explained most succinctly by considering the Pennes Bio-Heat Transfer Equation as the most general formulation,

$$-k\nabla^2 T + \dot{V}\dot{\mathcal{L}}\mathbf{c}\_b (T - T\_a) = \dot{\mathcal{Q}}\_{ap} \tag{8}$$

Here, k is the thermal conductivity of the tissue matrix, T(x,y,z) is the tissue temperature, *W* is the "perfusion" value and Ta is the arterial blood temperature at some reference location.

#### **3.5 Others**

A few studies (Leeuwen et al, 2000; Devashish and Roemer, 2006; Baish, 1994) have modeled the effect of collections of a large number of parallel vessels or of networks of vessels on the resulting temperature distributions. Those were developed in attempt to describe the impact of blood vessels and to properly predict heat transfer processes in bio-thermal systems in a more accurate way.

#### **4. Numerical modelings**

As mentioned above the mathematical models for actual thermal problems of interest in hyperthermia or thermal ablation are too complicated to be conveniently solved with exact formulas. The majority of unsolved problems in medical fields is governed by non-linear partial differential equations. In most cases, one thereby reduces the problems to rather simplified models which can be exactly analyzed, for example, analytical solution of the 3D Pennes equation presented by Liu (Liu, 2001; Liu and Deng, 2002) using multidimensional Green function, and 1D transient Pennes equation by Shih et. al. (Shih et al, 2007) using the Laplace transform. But occasionally such an approach does not suffice. Consequently, specialists have recently devoted increasing attention to numerical, as opposed to analytical, techniques. Nowadays one of the major challenges for thermal ablation and hyperthermia simulation is the incorporation of the very detailed information coming from biophysical models into the numerical simulations. Thanks to advanced imaging techniques, accurate tumor static models including detailed description of all vascular matrix objects are currently available. Unfortunately, most of the discretization methods commonly used in computer simulation, mainly based on structured grids, are not capable to represent the detailed geometry of such treatment regions or other complicated entities such as microvascular matrix, horizontal wells, and uniformity, etc. The complexity of multidimensional heat transfer problems in hyperthermia suggests the application of numerical techniques. Several numerical methods have been used in engineering and

Heating in Biothermal Systems 263

( ) *<sup>p</sup>* Π ≈Γ + *T T Oh*

where *h* = max{∆*x,* ∆*y,* ∆*z*}. Then the semi-discrete equation corresponding to Equation (11)

2 2 *n n n nn*

where ∆*t* is the time step size and T*<sup>n</sup>* is the discrete solution vector at time *tn* = *n*∆*t*. This numerical scheme is known as the Crank–Nicolson scheme (Crank and Nicolson, 1947). It yields a truncation error at the *n*th time-level: ( ) <sup>2</sup> *<sup>p</sup> Error O t h* = Δ+ . In the matrix form we

> <sup>1</sup> ( ) 2 2 *n n n*

<sup>1</sup> ( ) 2 2

<sup>+</sup> ⎛ ⎞⎛ ⎞ Δ ΔΔ <sup>⎡</sup> <sup>⎤</sup> =− Π + Π + + ⎜ ⎟⎜ ⎟ <sup>⎢</sup> <sup>⎥</sup> ⎝ ⎠⎝ ⎠ <sup>⎣</sup> <sup>⎦</sup>

Obviously other standard techniques for numerical discretization in time have also been used. For instance the unconditionally stable Alternating Direction Implicit (ADI) finite difference method (Peaceman and Rachford, 1955) was successfully used in the solution of

When an analysis is performed in complex geometries, the finite element method (Dennis et al, 2003; Hinton and Owen, 1974) usually handles those geometries better than finite difference. In the finite element method the domain where the solution is sought is divided into a finite number of mesh elements. (for example, a pyramid mesh, as shown in Figure 3). Applying the method of weighted residual to Pennes equation with a weight function, ω,

*<sup>T</sup> c k T wc T T Q d <sup>t</sup>*

A large but finite number of known functions are proposed as the representation of the temperature. The (shape) functions are constructed from simple interpolation functions within each element into which the domain is divided. The value of the function everywhere inside the element is determined by values at the nodes of that element. The

⎡ ⎤ <sup>∂</sup> <sup>−</sup> ∇• ∇ + − − Λ = ⎢ ⎥ ⎣ ⎦ <sup>∂</sup>

*c cc*

ρρ

ρρ

⎝ ⎠⎝ ⎠ (13)

() 0

v∫ (15)

*bb a e*

*t tt I T I T Q wcT*

*t tt T I I T Q wcT*

 *c cc* <sup>+</sup> ⎛ ⎞⎛ ⎞ Δ ΔΔ ⎜ ⎟⎜ ⎟ − Π =+ Π + +

*n n n*

1

−

ρ

the bio-heat equation in (Qi and Wissler, 1992; Yuan et al, 1995).

*T T c TT <sup>Q</sup> <sup>w</sup> <sup>c</sup> <sup>T</sup>*

<sup>1</sup> 1 1 <sup>1</sup>

*t*

ρ

ρ

That is at time *tn*+1 the discrete solution is given by:

<sup>∂</sup> . To integrate in time, one can use the two-level implicit time-

reads *bb a <sup>T</sup> c T Q wcT <sup>t</sup>*

<sup>∂</sup> =Π + +

ρ

stepping scheme:

can represent (2) as:

**4.2 Finite element method** 

over a single element, Λ*e* results in:

temperature can be expressed by,

*e*

Λ

ω ρ <sup>2</sup> Γ =∇ − *T k T wcT b b* (11)

*bb a*

*bb a*

*bb a*

(14)

<sup>+</sup> <sup>−</sup> ⎛ ⎞ <sup>+</sup> =Π + + + ⎜ ⎟ <sup>Δ</sup> ⎝ ⎠ (12)

science fields; finite difference method, finite element method, finite volume approach, etc. (Morton and Mayers, 2005; Derziger, Peric, 2001; Thomas, 1995; Minkowycz et al, 1988; Anderson et al, 1984).

#### **4.1 Finite difference method**

Several mathematical models were discussed above to describe the continuum models of heat transfer in living biological tissue, with blood flow and metabolism. The general form of these equations is given by:

$$
\rho c \frac{DT}{Dt} = \rho c \left(\frac{\partial T}{\partial t} + V \bullet \nabla T\right) = \nabla \bullet k \nabla T - w\_b c\_b (T - T\_a) + Q \tag{9}
$$

The partial differential equations for thermal ablation or hyperthermia are discretized at the grid point by using the finite difference approximation using Pennes equation.

$$
\rho c \frac{\partial T}{\partial t} = k \nabla^2 T - w\_b c\_b (T - T\_a) + Q \tag{10}
$$

The Pennes equation is solved with the finite difference formulation when the exact geometry is not particularly important or when the fundamental behavior of a bio-thermal system is analyzed, in particular with heterogeneous and at times anisotropic thermal properties. Define an *Nx* x *Ny* x *Nz* lattice in the (x, y, z) plane that spans our region of interest in 3D with dimension of L*x* x L*y* x L*z* as shown in Figure 2. Let *Nx*, *Ny* and *Nz* be the numbers of equally spaced grid points in the *x*-, *y*-, and *z*-directions, respectively, and {*xijk* := (*i*∆*x*, *j*∆*y*, *k*∆*z*)} the grid points in the computational domain, where ∆*x* = Lx*/Nx*, ∆*y* = Ly*/Ny*, and ∆*z* = Lz*/Nz*.

Fig. 1. Schematic representation of the grid system using a finite difference scheme

In a typical numerical treatment, the dependent variables are described by their values at discrete points (a lattice) of the independent variables (e.g. space and/or time), and the partial differential equation is reduced to a large set of difference equations. It would be useful to revise our description of difference equations. Let Γ be the elliptic operator and Π a finite difference approximation of Γ with *p*th order accuracy, i.e.,

science fields; finite difference method, finite element method, finite volume approach, etc. (Morton and Mayers, 2005; Derziger, Peric, 2001; Thomas, 1995; Minkowycz et al, 1988;

Several mathematical models were discussed above to describe the continuum models of heat transfer in living biological tissue, with blood flow and metabolism. The general form

*c c V T k T wc T T Q*

The partial differential equations for thermal ablation or hyperthermia are discretized at the

<sup>∂</sup> <sup>=</sup> ∇− − +

The Pennes equation is solved with the finite difference formulation when the exact geometry is not particularly important or when the fundamental behavior of a bio-thermal system is analyzed, in particular with heterogeneous and at times anisotropic thermal properties. Define an *Nx* x *Ny* x *Nz* lattice in the (x, y, z) plane that spans our region of interest in 3D with dimension of L*x* x L*y* x L*z* as shown in Figure 2. Let *Nx*, *Ny* and *Nz* be the numbers of equally spaced grid points in the *x*-, *y*-, and *z*-directions, respectively, and {*xijk* := (*i*∆*x*, *j*∆*y*, *k*∆*z*)} the grid points in the computational domain, where ∆*x* = Lx*/Nx*, ∆*y* = Ly*/Ny*,

grid point by using the finite difference approximation using Pennes equation.

⎛ ⎞ <sup>∂</sup> <sup>=</sup> + •∇ =∇• ∇ − − + ⎜ ⎟

<sup>2</sup> ( ) *bb a <sup>T</sup> c k T wc T T Q <sup>t</sup>*

( ) *bb a*

<sup>∂</sup> (10)

(i,j,k+1)

(i,j,k-1)

(i,j-1,k) (i-1,j,k)

(i+1,j,k) (i,j,k)

⎝ ⎠ <sup>∂</sup> (9)

Anderson et al, 1984).

and ∆*z* = Lz*/Nz*.

nz

Z

X ny

Y nx

Fig. 1. Schematic representation of the grid system using a finite difference scheme

a finite difference approximation of Γ with *p*th order accuracy, i.e.,

In a typical numerical treatment, the dependent variables are described by their values at discrete points (a lattice) of the independent variables (e.g. space and/or time), and the partial differential equation is reduced to a large set of difference equations. It would be useful to revise our description of difference equations. Let Γ be the elliptic operator and Π

Liver

**4.1 Finite difference method** 

of these equations is given by:

ρ

*DT T*

*Dt t*

ρ

 ρ

$$
\Gamma T = k \nabla^2 T - w\_b c\_b T \tag{11}
$$

( ) *<sup>p</sup>* Π ≈Γ + *T T Oh*

where *h* = max{∆*x,* ∆*y,* ∆*z*}. Then the semi-discrete equation corresponding to Equation (11) reads *bb a <sup>T</sup> c T Q wcT <sup>t</sup>* ρ <sup>∂</sup> =Π + + <sup>∂</sup> . To integrate in time, one can use the two-level implicit timestepping scheme:

$$
\rho c \frac{T^{n+1} - T^n}{\Delta t} = \Pi \left( \frac{1}{2} T^{n+1} + \frac{1}{2} T^n \right) + Q^n + w\_b c\_b T\_a \tag{12}
$$

where ∆*t* is the time step size and T*<sup>n</sup>* is the discrete solution vector at time *tn* = *n*∆*t*. This numerical scheme is known as the Crank–Nicolson scheme (Crank and Nicolson, 1947). It yields a truncation error at the *n*th time-level: ( ) <sup>2</sup> *<sup>p</sup> Error O t h* = Δ+ . In the matrix form we can represent (2) as:

$$\left(I - \frac{\Delta t}{2\rho c} \Pi \right) T^{n+1} = \left(I + \frac{\Delta t}{2\rho c} \Pi \right) T^n + \frac{\Delta t}{\rho c} (Q^n + w\_b c\_b T\_a) \tag{13}$$

That is at time *tn*+1 the discrete solution is given by:

$$T^{n+1} = \left(I - \frac{\Delta t}{2\rho \infty} \Pi\right)^{-1} \left[ \left(I + \frac{\Delta t}{2\rho \infty} \Pi\right) T^n + \frac{\Delta t}{\rho \infty} (Q^n + w\_b c\_b T\_a) \right] \tag{14}$$

Obviously other standard techniques for numerical discretization in time have also been used. For instance the unconditionally stable Alternating Direction Implicit (ADI) finite difference method (Peaceman and Rachford, 1955) was successfully used in the solution of the bio-heat equation in (Qi and Wissler, 1992; Yuan et al, 1995).

#### **4.2 Finite element method**

When an analysis is performed in complex geometries, the finite element method (Dennis et al, 2003; Hinton and Owen, 1974) usually handles those geometries better than finite difference. In the finite element method the domain where the solution is sought is divided into a finite number of mesh elements. (for example, a pyramid mesh, as shown in Figure 3). Applying the method of weighted residual to Pennes equation with a weight function, ω, over a single element, Λ*e* results in:

$$\oint\_{\Lambda\_e} \rho \left[ \rho c \frac{\partial T}{\partial t} - \nabla \bullet k \nabla T + w\_b c\_b (T - T\_a) - Q \right] d\Lambda\_e = 0 \tag{15}$$

A large but finite number of known functions are proposed as the representation of the temperature. The (shape) functions are constructed from simple interpolation functions within each element into which the domain is divided. The value of the function everywhere inside the element is determined by values at the nodes of that element. The temperature can be expressed by,

Heating in Biothermal Systems 265

[ ] [ ] { } [ ] [ ] { } ( ) { }{ } 11 11 1 1 1

where the superscript n+1 denotes the current time step and the superscript n, the previous

Finite volume methods are based on an integral form instead of a differential equation and the domains of interest are broken into a number of volumes, or grid cells, rather than pointwise approximations at grid points. Some of the important features of the finite volume method are thus similar to those of the finite element method (Oden, 1991). The basic idea of using finite volume method is to eliminate the divergence terms by applying the Gaussian divergence theorem. As a result an integral formulation of the fluxes over the boundary of the control volume is then obtained. Furthermore they allow for arbitrary geometries, using structured or unstructured meshing cells. An additional feature is that the numerical flux is conserved from one discretization cell to its neighbor. This characteristic makes the finite volume method quite attractive when modeling problems for which the flux is of importance, such as in fluid dynamics, heat transfer, acoustics and electromagnetic

Since finite volume methods are especially designed for equations incorporating divergence terms, they are a good choice for the numerical treatment of the bio-heat-transfer-equation. The computational domain is discretized into an assembly of grid cells as shown in Figure 4.

3D volume

*n n <sup>n</sup> <sup>n</sup> A MT A MT B B*

⎛ ⎞⎛ ⎞ + + + =+ + + ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠ Δ Δ (18)

2 22

*t t*

time step.

simulations, etc.

node *i* as

Liver

Z

Y

Fig. 3. Schematic representation of the grid cell system using a finite volume scheme

Then the governing equation is applied over each control volume in the mesh. So the volume integrals of Pennes equation can be evaluated over the control volume surrounding

*<sup>T</sup> c k T wc T T Q d <sup>t</sup>*

⎡ ⎤ <sup>∂</sup> <sup>−</sup> ∇• ∇ + − − Ω= ⎢ ⎥ ⎣ ⎦ <sup>∂</sup>

() 0 *bb a*

v∫ (19)

X

*i*

Ω

ρ

**4.3 Finite volume method** 

$$T^{(\epsilon)}\left(\mathbf{x}, y, z, t\right) = \sum\_{i=1}^{Nr} N\_i\left(\mathbf{x}, y, z\right) T\_i\left(t\right) \tag{16}$$

Or in a matrix form, ( ) ( )() ,,, ,, { ( )} *<sup>e</sup> T x <sup>y</sup> zt N x* <sup>=</sup> ⎡ ⎤ *<sup>y</sup> z Tt* ⎣ ⎦

Fig. 2. Schematic representation of the mesh element system using a finite element scheme

In Eq. (16), i , is an element local node number, Nr is the total number of element nodes and N(x,y,z) is the shape function associated with node *i*. Applying integration by parts into Eq. (15) one can obtain

$$\oint\_{\Lambda\_{\varepsilon}} \phi \left[ \rho \mathbf{c} \frac{\partial \Gamma}{\partial t} + w\_{b} c\_{b} (\Gamma - T\_{a}) - Q \right] d\Lambda\_{\varepsilon} + \int\_{\Lambda\_{\varepsilon}} \phi k \nabla T \bullet \nabla N\_{i} d\Lambda\_{\varepsilon} - \int\_{\Gamma\_{\varepsilon}} \phi (k \nabla T \bullet \overset{\wedge}{\bullet} n) N\_{i} d\Gamma\_{\varepsilon} = 0 \tag{17}$$

Here, Γ*e* is the surface element. Using the Galerkin method, the weight function, ω, is chosen to be the same as the interpolation function for T. Evaluation of each element and then assembling into the global system of linear equations for each node in the domain yields

$$
\begin{aligned}
\begin{bmatrix} M \end{bmatrix} \begin{Bmatrix} \bullet \\ T \end{Bmatrix} + \begin{bmatrix} K \end{bmatrix} \begin{Bmatrix} T \end{Bmatrix} + \begin{bmatrix} \mathcal{W} \end{Bmatrix} \begin{Bmatrix} T \end{Bmatrix} = \begin{Bmatrix} R \end{Bmatrix} + \begin{Bmatrix} P \end{Bmatrix} \\\\ \text{or } \begin{bmatrix} M \end{bmatrix} \begin{Bmatrix} T \\ T \end{Bmatrix} + \begin{bmatrix} A \end{bmatrix} \begin{Bmatrix} T \end{Bmatrix} = \begin{Bmatrix} B \end{Bmatrix}
\end{aligned}
$$

where

$$\begin{aligned} \mathcal{M}\_{\vec{\boldsymbol{\eta}}} &= \int\_{\boldsymbol{\Lambda}\_{\boldsymbol{\varepsilon}}} \boldsymbol{\alpha} \rho \mathbf{c} \mathbf{N}\_{i} \mathbf{N}\_{j} d\boldsymbol{\Lambda}\_{\boldsymbol{\varepsilon}} \; \prime \; \mathcal{K}\_{\vec{\boldsymbol{\eta}}} = \int\_{\boldsymbol{\Lambda}\_{\boldsymbol{\varepsilon}}} \boldsymbol{\alpha} \mathbf{k} \{\nabla \mathbf{N}\_{i} \bullet \nabla \mathbf{N}\_{j}\} d\boldsymbol{\Lambda}\_{\boldsymbol{\varepsilon}} \; \; \mathcal{W}\_{\vec{\boldsymbol{\eta}}} &= \int\_{\boldsymbol{\Lambda}\_{\boldsymbol{\varepsilon}}} \boldsymbol{\alpha} \mathbf{W}\_{b} \mathbf{c}\_{b} \mathbf{N}\_{i} \mathbf{N}\_{j} d\boldsymbol{\Lambda}\_{\boldsymbol{\varepsilon}} \; \prime \\\ P\_{i} &= \int\_{\boldsymbol{\Gamma\_{\varepsilon}}} \boldsymbol{\alpha} \mathbf{k} \{\nabla \sum\_{j=1}^{Nr} T\_{j} \mathbf{N}\_{j} \bullet \mathbf{n}\} \mathbf{N}\_{i} d\boldsymbol{\Gamma}\_{\boldsymbol{\varepsilon}} \; \; \mathcal{R}\_{i} = \int\_{\boldsymbol{\Lambda\_{\varepsilon}}} \boldsymbol{\alpha} \mathbf{Q}\_{i} \mathbf{N}\_{i} d\boldsymbol{\Lambda}\_{\boldsymbol{\varepsilon}} \; \; \mathcal{A}\_{\vec{\boldsymbol{\eta}}} = \mathbf{K}\_{\vec{\boldsymbol{\eta}}} + \mathbf{V} \boldsymbol{\delta\_{ij}} \; \; \mathcal{B}\_{i} = \mathbf{R}\_{i} + \mathbf{P}\_{i} \end{aligned}$$

This set of equations cane be solved with any kind on numerical integration in time to obtain the approximate temperature distribution in the domain. For instance one can use the Crank-Nicolson algorithm,

( ) ( ) () ( ) 1 ,,, ,,

*i T xyzt N xyzT t* =

Fig. 2. Schematic representation of the mesh element system using a finite element scheme In Eq. (16), i , is an element local node number, Nr is the total number of element nodes and N(x,y,z) is the shape function associated with node *i*. Applying integration by parts into Eq.

*<sup>T</sup> c wc T T Q d k T Nd k T nNd <sup>t</sup>*

Here, Γ*e* is the surface element. Using the Galerkin method, the weight function, ω, is chosen to be the same as the interpolation function for T. Evaluation of each element and then assembling into the global system of linear equations for each node in the domain yields

[ ] [] *<sup>M</sup> T KT WT R P* { } [ ]{} {} {} • <sup>⎧</sup> <sup>⎫</sup> ⎨ ⎬ + + =+

or [ ] [] *<sup>M</sup> T AT B* {} {} • <sup>⎧</sup> <sup>⎫</sup> ⎨ ⎬ + = ⎩ ⎭

This set of equations cane be solved with any kind on numerical integration in time to obtain the approximate temperature distribution in the domain. For instance one can use the

+ − − Λ + ∇ •∇ Λ − ∇ • Γ = ⎢ ⎥

ω

*e e e*

Λ Λ Γ

⎩ ⎭

*K k N Nd ij* ω

*i j e*

<sup>=</sup> ∇ •∇ Λ ∫ ,

*e R QNd i ii e* ω

Λ

( ) () 0

*e W WcNNd ij* ω

Λ

<sup>=</sup> <sup>Λ</sup> ∫ , *Aij* <sup>=</sup> *K W ij* <sup>+</sup> *ij* , *BRP i ii* <sup>=</sup> <sup>+</sup>

*bb a e i e i e*

v∫ ∫∫ (17)

*i i*

<sup>=</sup> ∑ (16)

3D element

^

ω

,

*bb i j e*

<sup>=</sup> <sup>Λ</sup> ∫ ,

*Nr <sup>e</sup>*

( )() ,,, ,, { ( )} *<sup>e</sup> T x <sup>y</sup> zt N x* <sup>=</sup> ⎡ ⎤ *<sup>y</sup> z Tt* ⎣ ⎦

Liver

Or in a matrix form, ( )

Z

X

(15) one can obtain

where

*e Mij <sup>i</sup> <sup>j</sup> <sup>e</sup>* ωρ

ω

Γ =

Λ

*e*

*cN N d*

1 ( )

Crank-Nicolson algorithm,

=∇ •Γ ∫ ∑ ,

*Nr i j j i e j P k TN nNd*

ω ρ Y

⎡ ⎤ ∂

⎣ ⎦ ∂

<sup>=</sup> <sup>Λ</sup> ∫ , ( )

^

*e*

Λ

$$
\left(\frac{1}{2}[A] + \frac{1}{\Delta t}[M]\right)\left|T^{n+1}\right\rangle = \left(\frac{1}{2}[A] + \frac{1}{\Delta t}[M]\right)\left|T^{n}\right\rangle + \frac{1}{2}\left(\left\{B^{n+1}\right\} + \left\{B^{n}\right\}\right) \tag{18}
$$

where the superscript n+1 denotes the current time step and the superscript n, the previous time step.

#### **4.3 Finite volume method**

Finite volume methods are based on an integral form instead of a differential equation and the domains of interest are broken into a number of volumes, or grid cells, rather than pointwise approximations at grid points. Some of the important features of the finite volume method are thus similar to those of the finite element method (Oden, 1991). The basic idea of using finite volume method is to eliminate the divergence terms by applying the Gaussian divergence theorem. As a result an integral formulation of the fluxes over the boundary of the control volume is then obtained. Furthermore they allow for arbitrary geometries, using structured or unstructured meshing cells. An additional feature is that the numerical flux is conserved from one discretization cell to its neighbor. This characteristic makes the finite volume method quite attractive when modeling problems for which the flux is of importance, such as in fluid dynamics, heat transfer, acoustics and electromagnetic simulations, etc.

Since finite volume methods are especially designed for equations incorporating divergence terms, they are a good choice for the numerical treatment of the bio-heat-transfer-equation. The computational domain is discretized into an assembly of grid cells as shown in Figure 4.

Fig. 3. Schematic representation of the grid cell system using a finite volume scheme

Then the governing equation is applied over each control volume in the mesh. So the volume integrals of Pennes equation can be evaluated over the control volume surrounding node *i* as

$$\oint\_{\Omega} \oint\_{} \rho \mathbf{c} \frac{\partial T}{\partial t} - \nabla \bullet k \nabla T + w\_b c\_b (T - T\_a) - Q \left[ d\Omega = 0 \tag{19}$$

Heating in Biothermal Systems 267

method (Lyra, 1994; Zienkiewicz & Morgan, 1983), multi-stage Runge-Kutta scheme (Lyra,

Other classes of methods have also been applied to the partial differential equations, such as boundary element method (Wrobel and Aliabadi, 2002), spectral method (Canuto et al,

Heating in bio-thermal systems that have many forms, they can be appeared in different power deposition calculations in PBHTE. They can be classified into three types which are invasive, minimal invasive and non-invasive methods. We introduced most clinical methods

Hyperthermia is a heat treatment, and traditionally refers to raise tissue temperatures to therapeutic temperatures in the range of 41~45°C (significantly higher than the usual bodytemperature) by external means. In history, the first known, more than 5000 years old, written medical report from the ancient Egypt mentions hyperthermia (Smith, 2002). Also, an ancient tradition in China, "Palm Healing", has used the healing properties of far infrared rays for 3000 years. As our bodies radiate far infrared energy through the skin at 3 to 50 microns, with a peak around 9.4 microns, these natural healers emit energy and heat radiating from their hands to heal. It could be applied in several various treatments: cure of common cold (Tyrrell et al, 1989), help in the rheumatic diseases (Robinson et al, 2002; Brosseau et al, 2003) or application in cosmetics (Narins & Narins, 2003) and for numerous other indicators.

The differentiation between thermal ablation and hyperthermia relates to the treatment temperature and times. Thermal ablation usually refers to heat treatments delivered at temperatures above 55°C for short periods of time (i.e. few seconds to 1 min.). Hyperthermia usually refers to treatments delivered at temperatures around 41-45°C for 30~60 minutes. The goal of thermal ablation is to destroy entire tumors, killing the malignant cells using heat with only minimal damage to surrounding normal tissues. The principle of operation of the thermal ablation techniques is that to produces a concentrated thermal energy (heating or freezing), creating a hyperthermic/hypothermic injury, for example, by a needle-like applicator placed directly into the tumor or using focused ultrasound beams. Thermal ablation comprises several distinct techniques as shown in Figure 1: radiofrequency (RF) ablation, microwave ablation, laser ablation, cryoablation, and high-intensity focused ultrasound ablation. To have a good treatment, it is also crucial to destroy a thin layer of tissue surrounding the tumor because of the uncertainty of tumor margin and the possibility

When it is not applicable for patients to surgery, one of alternative therapies for malignant tumors is thermal ablation. It is a technique that provides clinicians and patients a repeatable, effective, low cost, and safe treatment to effectively alleviate, and in some cases cure, both primary and metastatic malignancies. However, the common procedures for each thermal ablation technique are not yet clearly defined because the decision to use ablation,

1994) can be implemented if higher-order time accuracy is required.

2006), multigrid method (Briggs et al, 2000) ect.

**4.4 Others** 

here.

**5. Heating methods** 

**5.1 Hyperthermia** 

**5.2 Thermal ablation** 

of microscopic disease (Dodd et al, 2000).

By the use of the divergent theorem,

$$\oint\_{\Omega} \left[ -\nabla \bullet k \nabla T \right] d\Omega = -\int\_{\Gamma\_i} \left( k \nabla T \bullet \hat{n} \right) d\Gamma = \int\_{\Gamma\_i} q\_j n\_j d\Gamma \tag{20}$$

where the heat flux *q kT* = − ∇ and

$$\oint\_{\Omega} \oint\_{\Omega} \rho \mathbf{c} \frac{\partial T}{\partial t} \Big|\_{} d\Omega \cong \rho \mathbf{c} \frac{\partial T\_i}{\partial t} \cap\_i \oint\_{\Omega} w\_b c\_b T d\Omega \cong w\_b c\_b T\_i \cap\_i \,, \oint\_{\Omega} [Q + w\_b c\_b T\_a] d\Omega \cong \left(Q\_i + w\_b c\_b T\_a\right) \cap\_i \mathbf{c}$$

where ∩*i* is the volume of the control volume, *Ti* and *Qi* represent the numerical calculated temperature and source term at node *i*, respectively. The boundary integral presented in equation (a) is computed over the boundary of the control volume, Ω*<sup>i</sup>* , that surrounds node *i* using an edge-based representation of the mesh, i.e.

$$\int\_{\Gamma\_i} q\_j n\_j d\Gamma \cong \sum\_{\text{all edges}} G\_{ij} q\_j + \sum\_{\text{all edges}} H\_{ij} q\_j \tag{21}$$

where *Gij* denotes the coefficients that must be applied to the edge value of the flux *<sup>j</sup> q* in the *xj* direction to obtain the contribution made by the edge to node *i* and *Hij* represents the boundary edges coefficients that relate to the boundary edge flux *<sup>j</sup> q* when the edge lies on the boundary, where *Hij* =0 on all edges except on the domain boundaries. The approximation of *<sup>j</sup> q* on edge is evaluated by different schemes based on the temperatures between nodes. For example,

$$q\_j = \frac{T\_j - T\_i}{d\_{\bar{\eta}}}$$

where *ij d* is the distance between the center of the cells *i* and *j*. The semi-discrete form of the transient bioheat heat transfer equation represents a coupled system of first order differential equations, which can be rewritten in a compact matrix notation as

$$P\frac{\partial \widetilde{T}}{\partial t} + RT = S \tag{22}$$

with an initial condition. In equation (22), *P* represents the heat capacity matrix which is a diagonal matrix. *R* is the conductivity matrix including the contributions from the surface integral and perfusion terms. The vector *S* is formed by the independent terms, which arises from the thermal loads and boundary conditions. *T* is the vector of the nodal unknowns. Equation (22) can be further discretized in time to produce a system of algebraic equations. With the objective of validating the finite volume formulation described, one can use the simplest two-level explicit time step and rewrite equation (22) as the following expression

$$P\frac{T^{n+1} - T^n}{\Delta t} + RT^n = S^n \tag{23}$$

where *n n* <sup>1</sup> *tt t* Δ= − <sup>+</sup> is the length of the time interval and the superscripts represent the time levels. Such scheme is just first order accurate in time and the Δ*t* must be chosen according to a stability condition (Lyra, 1994). Other alternatives, such as the generalized trapezoidal method (Lyra, 1994; Zienkiewicz & Morgan, 1983), multi-stage Runge-Kutta scheme (Lyra, 1994) can be implemented if higher-order time accuracy is required.

#### **4.4 Others**

266 Convection and Conduction Heat Transfer

*i i*

Ω ≅ ∫ <sup>∩</sup> � , [ *bb a i bb a i* ] ( ) *i*

*k T d k T n d qnd*

Ω

where ∩*i* is the volume of the control volume, *Ti* and *Qi* represent the numerical calculated temperature and source term at node *i*, respectively. The boundary integral presented in equation (a) is computed over the boundary of the control volume, Ω*<sup>i</sup>* , that

> *j j ij j ij j all edges all edges qnd Gq H q*

where *Gij* denotes the coefficients that must be applied to the edge value of the flux *<sup>j</sup> q* in the *xj* direction to obtain the contribution made by the edge to node *i* and *Hij* represents the boundary edges coefficients that relate to the boundary edge flux *<sup>j</sup> q* when the edge lies on the boundary, where *Hij* =0 on all edges except on the domain boundaries. The approximation of *<sup>j</sup> q* on edge is evaluated by different schemes based on the temperatures

*j i*

*ij T T <sup>q</sup> <sup>d</sup>* <sup>−</sup> <sup>=</sup>

The semi-discrete form of the transient bioheat heat transfer equation represents a coupled system of first order differential equations, which can be rewritten in a compact matrix

*<sup>T</sup> P RT S <sup>t</sup>*

with an initial condition. In equation (22), *P* represents the heat capacity matrix which is a diagonal matrix. *R* is the conductivity matrix including the contributions from the surface integral and perfusion terms. The vector *S* is formed by the independent terms, which arises from the thermal loads and boundary conditions. *T* is the vector of the nodal unknowns. Equation (22) can be further discretized in time to produce a system of algebraic equations. With the objective of validating the finite volume formulation described, one can use the simplest two-level explicit time step and rewrite equation (22) as the following expression

*n n* <sup>1</sup> *T T n n P RT S <sup>t</sup>*

where *n n* <sup>1</sup> *tt t* Δ= − <sup>+</sup> is the length of the time interval and the superscripts represent the time levels. Such scheme is just first order accurate in time and the Δ*t* must be chosen according to a stability condition (Lyra, 1994). Other alternatives, such as the generalized trapezoidal

*j*

*j j*

*Q wcT d Q wcT*

∫ Γ≅ + ∑ ∑ (21)

+ Ω≅ + ∫ <sup>∩</sup> �

<sup>∂</sup> <sup>+</sup> <sup>=</sup> <sup>∂</sup> (22)

<sup>+</sup> <sup>−</sup> <sup>+</sup> <sup>=</sup> <sup>Δ</sup> (23)

−∇• ∇ Ω=− ∇ • Γ= Γ �∫ ∫∫ (20)

[ ] ( ˆ)

*w c Td w c T*

surrounds node *i* using an edge-based representation of the mesh, i.e.

*i*

where *ij d* is the distance between the center of the cells *i* and *j*.

Γ

Ω ΓΓ

By the use of the divergent theorem,

where the heat flux *q kT* = − ∇ and

 ρ

between nodes. For example,

notation as

*<sup>T</sup> <sup>T</sup> cd c t t*

⎡ ⎤ <sup>∂</sup> <sup>∂</sup> Ω ≅ ⎢ ⎥

*i*

Ω

ρ

*i i*

⎣ ⎦ ∂ ∂ ∫ <sup>∩</sup> � , *bb bbi i i*

*i*

Ω

Other classes of methods have also been applied to the partial differential equations, such as boundary element method (Wrobel and Aliabadi, 2002), spectral method (Canuto et al, 2006), multigrid method (Briggs et al, 2000) ect.

#### **5. Heating methods**

Heating in bio-thermal systems that have many forms, they can be appeared in different power deposition calculations in PBHTE. They can be classified into three types which are invasive, minimal invasive and non-invasive methods. We introduced most clinical methods here.

#### **5.1 Hyperthermia**

Hyperthermia is a heat treatment, and traditionally refers to raise tissue temperatures to therapeutic temperatures in the range of 41~45°C (significantly higher than the usual bodytemperature) by external means. In history, the first known, more than 5000 years old, written medical report from the ancient Egypt mentions hyperthermia (Smith, 2002). Also, an ancient tradition in China, "Palm Healing", has used the healing properties of far infrared rays for 3000 years. As our bodies radiate far infrared energy through the skin at 3 to 50 microns, with a peak around 9.4 microns, these natural healers emit energy and heat radiating from their hands to heal. It could be applied in several various treatments: cure of common cold (Tyrrell et al, 1989), help in the rheumatic diseases (Robinson et al, 2002; Brosseau et al, 2003) or application in cosmetics (Narins & Narins, 2003) and for numerous other indicators.

#### **5.2 Thermal ablation**

The differentiation between thermal ablation and hyperthermia relates to the treatment temperature and times. Thermal ablation usually refers to heat treatments delivered at temperatures above 55°C for short periods of time (i.e. few seconds to 1 min.). Hyperthermia usually refers to treatments delivered at temperatures around 41-45°C for 30~60 minutes. The goal of thermal ablation is to destroy entire tumors, killing the malignant cells using heat with only minimal damage to surrounding normal tissues. The principle of operation of the thermal ablation techniques is that to produces a concentrated thermal energy (heating or freezing), creating a hyperthermic/hypothermic injury, for example, by a needle-like applicator placed directly into the tumor or using focused ultrasound beams. Thermal ablation comprises several distinct techniques as shown in Figure 1: radiofrequency (RF) ablation, microwave ablation, laser ablation, cryoablation, and high-intensity focused ultrasound ablation. To have a good treatment, it is also crucial to destroy a thin layer of tissue surrounding the tumor because of the uncertainty of tumor margin and the possibility of microscopic disease (Dodd et al, 2000).

When it is not applicable for patients to surgery, one of alternative therapies for malignant tumors is thermal ablation. It is a technique that provides clinicians and patients a repeatable, effective, low cost, and safe treatment to effectively alleviate, and in some cases cure, both primary and metastatic malignancies. However, the common procedures for each thermal ablation technique are not yet clearly defined because the decision to use ablation,

Heating in Biothermal Systems 269

Cryoablation

Cryoprobe

Argon (gas) or N2 (liquid)

Liver

ablation (c) Laser ablation (d) Cryoablation (e) HIFU ablation

fever, discomfort, and anorexia.

**5.2.1 High-intensity focused ultrasound (HIFU)** 

Liver

(d)

(e)

Patients referred for thermal ablation are initially evaluated in a clinic setting where the patient's history and pertinent imaging information are reviewed. Meanwhile, the applicability of ablation and the risks and benefits of the procedure are also discussed. Prior to ablation, the evaluation is very similar to a surgical evaluation that any possible risks of bleeding or serious cardiopulmonary issues are considered. Side effects from thermal ablation are also discussed, including postablation syndrome—for example, a short term

HIFU is a non-invasive power deposition method via mechanical oscillation motion of object molecules. One of important features in the heating methods is non-invasiveness and it reduces external surgical operations on body object. Thus this method has become a promising tool for localized tumor therapy. Compared to hyperthermia which lasts long period of treating time, HIFU referred as thermal surgery, could heat the target region elevated temperature up to 50~55°C within a short period of time (i.e. few seconds to 1 min.). Another important feature is that this comparably higher temperature during treatment could cause thermal coagulation and thermal lesion. Therefore, precise location management and monitoring are required during clinic HIFU treatment to prevent

irreversible heating process on tissues. Figure 1.e illustrated the method.

Fig. 4. Schematics of different thermal ablation techniques. (a) RF ablation (b) Microwave

HIFU Ablation, Freq=0.5~5MHz Transducer (Focused/Phased array)

and which ablation technique to use, depends on several factors. In practice, the decision of whether to use thermal ablation depends on the training and preference of the physician in charge and the equipment resources available at his/her medical center. Moreover, physical characteristics of the treatment zone using ablation are also needed to concern, including the zone shape, uniformity, and its location. Up to now clinical results have been indicated that the different techniques of thermal ablation have roughly equivalent effectiveness for treating various tumors.

(c)

and which ablation technique to use, depends on several factors. In practice, the decision of whether to use thermal ablation depends on the training and preference of the physician in charge and the equipment resources available at his/her medical center. Moreover, physical characteristics of the treatment zone using ablation are also needed to concern, including the zone shape, uniformity, and its location. Up to now clinical results have been indicated that the different techniques of thermal ablation have roughly equivalent effectiveness for

> Radiofrequency Ablation Freq=460~500 kHz Needle electrode

Microwave Ablation Freq ~2450MHz, Bipolar

antenna needle

Laser Ablation

Optical fibe

Nd-YAG Laser*ν*=1064nm

Liver

(a)

Liver

(b)

Liver

(c)

treating various tumors.

Fig. 4. Schematics of different thermal ablation techniques. (a) RF ablation (b) Microwave ablation (c) Laser ablation (d) Cryoablation (e) HIFU ablation

Patients referred for thermal ablation are initially evaluated in a clinic setting where the patient's history and pertinent imaging information are reviewed. Meanwhile, the applicability of ablation and the risks and benefits of the procedure are also discussed. Prior to ablation, the evaluation is very similar to a surgical evaluation that any possible risks of bleeding or serious cardiopulmonary issues are considered. Side effects from thermal ablation are also discussed, including postablation syndrome—for example, a short term fever, discomfort, and anorexia.

#### **5.2.1 High-intensity focused ultrasound (HIFU)**

HIFU is a non-invasive power deposition method via mechanical oscillation motion of object molecules. One of important features in the heating methods is non-invasiveness and it reduces external surgical operations on body object. Thus this method has become a promising tool for localized tumor therapy. Compared to hyperthermia which lasts long period of treating time, HIFU referred as thermal surgery, could heat the target region elevated temperature up to 50~55°C within a short period of time (i.e. few seconds to 1 min.). Another important feature is that this comparably higher temperature during treatment could cause thermal coagulation and thermal lesion. Therefore, precise location management and monitoring are required during clinic HIFU treatment to prevent irreversible heating process on tissues. Figure 1.e illustrated the method.

Heating in Biothermal Systems 271

chemotherapy or other therapies with hyperthermia is that the available armamenterium for tumor heating has been substantially improved. The potential to control power distributions in clinic has been significantly improved lately by the development of advanced imaging techniques (particularly, online magnetic resonance tomography), planning systems and

The efficacy of standard radiotherapy for patients with different tumor sites, for example, cervical, gastrointestinal, and genitourinary tumors, might become poor because the localcontrol rates after locoregional treatment are disappointingly low. To compensate this defect the combinations of radiotherapy with other therapies have been used. It has been known that hyperthermia probably is the strongest radiosensitizer known, with an enhancement factor of up to 5 (Kampinga and Dikomey, 2001). Although the exact mechanism why heat can cause cells more sensitive to radiation is not known, clinical results reveal that heat primarily interferes with the cells' ability to deal with radiation-induced DNA damage

Clinical studies have shown that the combination of radiotherapy with hyperthermia increases cytotoxic effects and higher locoregional control rates. In the Netherlands 358 patients with tumors of the bladder, cervix, or rectum were treated with radiotherapy alone (n=176) or radiotherapy plus hyperthermia (n=182) from 1990 to 1996. Results showed the complete-response rates were 39% after radiotherapy and 55% after radiotherapy plus hyperthermia (Van der Zee et al, 2000). Radiotherapy plus hyperthermia was superior to

Moreover, other clinical results of the combination of radiotherapy with hyperthermia are summarized in some recent studies (Wust et al, 2002; Van der Zee, 2002). The supplementary values of this combined therapy are from 41% to 61% in 3-year local control rate and from 27% to 51% in 3-year overall survival in cervix cancer, from 24% to 69% in 5 year local control rate in lymph nodes of head-and-neck tumors, and from 24% to 42% in 3 year overall survival in esophageal cancer. The differences reported for the other radiosensitizing agents (Horsman et al, 2006), insofar as there are clinical results, are in the range of 10% to 20%. Significant improvements in clinical outcomes by additional treatment with hyperthermia were also shown for cancer of the breast, brain, rectum, bladder, and

Whether the combination of radiation and heat is given in a simultaneous or sequential schedule, the thermal enhancement will be dependent on the heating time and temperature

Besides, hyperthermia has a direct cell-killing effect, specifically in insufficiently perfused parts of the tumor. Several randomized clinical trials have shown that the beneficial effect of hyperthermia, when added to radiotherapy, can be substantial, even while the temperature of 43°C that was thought to be necessary was not achieved in the whole tumor volume. The improvements in clinical outcomes, despite the inadequacy to heat the whole tumor volume to temperatures of 43°C, can be explained by the more recent findings that hyperthermia has more effects than just that of direct cell kill and radiosensitization. Several additional effects that become apparent at different temperatures between 39° and 45°C have been described: vascular damage resulting in secondary cell death; improvement f perfusion and oxygenation, which results in a better effect of radiation; and stimulation of

of both tumors and normal tissues (Horsman and Overgaard, 2002 & 2007).

other modeling tools.

lung, and for melanoma.

(Kampinga and Dikomey, 2001; Roti, 2004).

radiotherapy alone and improved tumor response and survival.

**6.1 Radiotherapy** 

#### **5.2.2 Radiofrequency (RF) ablation**

Radiofrequency ablation is a "minimally invasive" treatment method mostly for primary and metastatic liver tumors. It is becoming a promising treating method to replace surgical resection. A study (Solbiati et al, 2001) in 2001 of RF ablation in 117 patients has shown 1-, 2- , and 3-year survival rates of 93%, 62%, and 41%, respectively. As compared to traditionally only low or 10-20 % of patients, those will have disease amenable to surgical resection due to limited hepatic reserve, high surgical risk, or unfavorable tumor location.

The mechanics of RF ablation uses the electromagnetic energy which is converted to heat by ionic friction. Tissue damage can occur at temperatures above 43°C with long heating times of several hours (Sapateto and Dewey, 1984). Elevated tissue temperature to 50°C near the probe required 3-min of heating time. Traditional and commercial design of the probe uses 17-gauge needles with active tip exposures of 1, 2 and 3 cm and the remainder of the needle is electrically insulated. Within the probe, water is circulated during the ablation procedure to cool tissue next to the probe and prevent tissue charring. Figure 1.a illustrated the method.

#### **5.2.3 Microwave (MW) ablation**

Microwave tumor ablation is also a "minimally invasive" treatment method. In contrast, while RF employs radio-frequency current to generate heat, MW ablation produces an electromagnetic wave that is emitted from a 14.5 gauge (standard) microwave antenna placed directly within the treatment site. The electromagnetic wave produces 60 W of power at a frequency ranging from 900 to 2450 MHz, which generate frictional heat from the agitation of polar water molecules (McTaggart and Dupuy, 2007; Liang and Wang, 2007; Simon et al, 2005). In principle the electromagnetic wave passes through the tissues, it causes polar water molecules to rapidly change their orientation in accordance with the magnetic field. Additionally, the design of MW antenna contributes significantly to the efficiency of MW therapy. Figure 1.b illustrated the method.

#### **5.2.4 Others**

Another interesting method to kill tumor cells is cryo-ablation, as shown in Figure 1.d. In contrast with other methods, cryo-ablation use lower temperature to ablate tumors. The procedure can be performed either by a laparoscopic or percutaneous approach under MRI, US or CT guidance. Cryoablation involves a number of freeze-thaw-freeze cycles with argon and helium gas (McTaggart and Dupuy, 2007). Gases are used to remove heat and induce thawing. It is used to treat lesions of the prostate, kidney, liver, lung, bone, and breast (Hayek et al, 2008; Orlacchio et al, 2005). As the tissue freezes, osmolarity increases and causes an imbalance of solutes between the intracellular and extracellular environments. Cellular death initially occurs through cellular dehydration and protein denaturization.

#### **6. Adjuvant to other tumor treatment modalities**

Although the effectiveness of hyperthermia alone as a cancer treatment may be not so promising, significant improvements in clinical trials using combined therapies with hyperthermia are observed. Recently, hyperthermia has been applied as an adjunctive therapy with various established cancer treatments such as radiotherapy, chemotherapy, and nano-particle drug treatments, etc. The combination therapies seem to be safe and effective approaches even when other treatments have failed. The rationale of combining

Radiofrequency ablation is a "minimally invasive" treatment method mostly for primary and metastatic liver tumors. It is becoming a promising treating method to replace surgical resection. A study (Solbiati et al, 2001) in 2001 of RF ablation in 117 patients has shown 1-, 2- , and 3-year survival rates of 93%, 62%, and 41%, respectively. As compared to traditionally only low or 10-20 % of patients, those will have disease amenable to surgical resection due

The mechanics of RF ablation uses the electromagnetic energy which is converted to heat by ionic friction. Tissue damage can occur at temperatures above 43°C with long heating times of several hours (Sapateto and Dewey, 1984). Elevated tissue temperature to 50°C near the probe required 3-min of heating time. Traditional and commercial design of the probe uses 17-gauge needles with active tip exposures of 1, 2 and 3 cm and the remainder of the needle is electrically insulated. Within the probe, water is circulated during the ablation procedure to cool tissue next to the probe and prevent tissue charring. Figure 1.a illustrated the

Microwave tumor ablation is also a "minimally invasive" treatment method. In contrast, while RF employs radio-frequency current to generate heat, MW ablation produces an electromagnetic wave that is emitted from a 14.5 gauge (standard) microwave antenna placed directly within the treatment site. The electromagnetic wave produces 60 W of power at a frequency ranging from 900 to 2450 MHz, which generate frictional heat from the agitation of polar water molecules (McTaggart and Dupuy, 2007; Liang and Wang, 2007; Simon et al, 2005). In principle the electromagnetic wave passes through the tissues, it causes polar water molecules to rapidly change their orientation in accordance with the magnetic field. Additionally, the design of MW antenna contributes significantly to the

Another interesting method to kill tumor cells is cryo-ablation, as shown in Figure 1.d. In contrast with other methods, cryo-ablation use lower temperature to ablate tumors. The procedure can be performed either by a laparoscopic or percutaneous approach under MRI, US or CT guidance. Cryoablation involves a number of freeze-thaw-freeze cycles with argon and helium gas (McTaggart and Dupuy, 2007). Gases are used to remove heat and induce thawing. It is used to treat lesions of the prostate, kidney, liver, lung, bone, and breast (Hayek et al, 2008; Orlacchio et al, 2005). As the tissue freezes, osmolarity increases and causes an imbalance of solutes between the intracellular and extracellular environments. Cellular death initially occurs through cellular dehydration and protein denaturization.

Although the effectiveness of hyperthermia alone as a cancer treatment may be not so promising, significant improvements in clinical trials using combined therapies with hyperthermia are observed. Recently, hyperthermia has been applied as an adjunctive therapy with various established cancer treatments such as radiotherapy, chemotherapy, and nano-particle drug treatments, etc. The combination therapies seem to be safe and effective approaches even when other treatments have failed. The rationale of combining

to limited hepatic reserve, high surgical risk, or unfavorable tumor location.

**5.2.2 Radiofrequency (RF) ablation** 

**5.2.3 Microwave (MW) ablation** 

efficiency of MW therapy. Figure 1.b illustrated the method.

**6. Adjuvant to other tumor treatment modalities** 

method.

**5.2.4 Others** 

chemotherapy or other therapies with hyperthermia is that the available armamenterium for tumor heating has been substantially improved. The potential to control power distributions in clinic has been significantly improved lately by the development of advanced imaging techniques (particularly, online magnetic resonance tomography), planning systems and other modeling tools.

#### **6.1 Radiotherapy**

The efficacy of standard radiotherapy for patients with different tumor sites, for example, cervical, gastrointestinal, and genitourinary tumors, might become poor because the localcontrol rates after locoregional treatment are disappointingly low. To compensate this defect the combinations of radiotherapy with other therapies have been used. It has been known that hyperthermia probably is the strongest radiosensitizer known, with an enhancement factor of up to 5 (Kampinga and Dikomey, 2001). Although the exact mechanism why heat can cause cells more sensitive to radiation is not known, clinical results reveal that heat primarily interferes with the cells' ability to deal with radiation-induced DNA damage (Kampinga and Dikomey, 2001; Roti, 2004).

Clinical studies have shown that the combination of radiotherapy with hyperthermia increases cytotoxic effects and higher locoregional control rates. In the Netherlands 358 patients with tumors of the bladder, cervix, or rectum were treated with radiotherapy alone (n=176) or radiotherapy plus hyperthermia (n=182) from 1990 to 1996. Results showed the complete-response rates were 39% after radiotherapy and 55% after radiotherapy plus hyperthermia (Van der Zee et al, 2000). Radiotherapy plus hyperthermia was superior to radiotherapy alone and improved tumor response and survival.

Moreover, other clinical results of the combination of radiotherapy with hyperthermia are summarized in some recent studies (Wust et al, 2002; Van der Zee, 2002). The supplementary values of this combined therapy are from 41% to 61% in 3-year local control rate and from 27% to 51% in 3-year overall survival in cervix cancer, from 24% to 69% in 5 year local control rate in lymph nodes of head-and-neck tumors, and from 24% to 42% in 3 year overall survival in esophageal cancer. The differences reported for the other radiosensitizing agents (Horsman et al, 2006), insofar as there are clinical results, are in the range of 10% to 20%. Significant improvements in clinical outcomes by additional treatment with hyperthermia were also shown for cancer of the breast, brain, rectum, bladder, and lung, and for melanoma.

Whether the combination of radiation and heat is given in a simultaneous or sequential schedule, the thermal enhancement will be dependent on the heating time and temperature of both tumors and normal tissues (Horsman and Overgaard, 2002 & 2007).

Besides, hyperthermia has a direct cell-killing effect, specifically in insufficiently perfused parts of the tumor. Several randomized clinical trials have shown that the beneficial effect of hyperthermia, when added to radiotherapy, can be substantial, even while the temperature of 43°C that was thought to be necessary was not achieved in the whole tumor volume.

The improvements in clinical outcomes, despite the inadequacy to heat the whole tumor volume to temperatures of 43°C, can be explained by the more recent findings that hyperthermia has more effects than just that of direct cell kill and radiosensitization. Several additional effects that become apparent at different temperatures between 39° and 45°C have been described: vascular damage resulting in secondary cell death; improvement f perfusion and oxygenation, which results in a better effect of radiation; and stimulation of

Heating in Biothermal Systems 273

The nanoparticles have been applied to facilitate drug delivery and to overcome some of the problems of drug delivery for cancer treatment. In the past, cancer therapies using anticancer drugs were dissatisfied and had major side effects. Because of their multifunctional character the nanoparticles can deliver larger and more effective doses of chemotherapeutic agents or therapeutic genes into malignant cells, minimize toxic effects on healthy tissues and then alleviate patients suffering from the side-effects of chemotherapy. Nanoparticles can be used to deliver hydrophilic drugs, hydrophobic drugs, proteins, vaccines, biological macromolecules, etc. Several nanoscale delivery devices, such as ceramic nanoparticles, virus, dendrimers (spherical, branched polymers), silica-coated micelles, cross linked liposomes, and carbon nanotubes (Portney and Ozkan, 2006) have been used to improve delivery of anticancer agents to tumor cells (Brigger el al, 2002). Some of the challenges in effectively delivering anticancer drugs have to be solved: how to ensure therapeutic drug molecules reach the targeted tumor, how to release them slowly over a

longer period, and how to avoid the human immune system destroying them.

and eventually to reduce the side effects caused by most chemotherapy drugs.

Normally, the structure of a nanoparticle drug carrier has four elements. The first of them is the targeted chemotherapy drug, for example, docetaxel or Taxotere. The second is a matrix made of a biodegradable polymer (polylactic acid), which contains the anticancer drug and breaks down slowly so that the drug is released gradually over several days. The third element is a coating layer of polyethylene glycol, which is used to prevent from attacking by antibodies and macrophage cells of the human immune system. The final element is a targeted tag, in the form of special enzymes attached to the outer coating, which can form electrostatic or covalent bonds with positively charged agents and biomolecules. This tag allows the nanoparticles to bind directly to desired tumor cells but to bypass healthy tissues

Several different drug delivery methods (Jain, 2005) have been shown their feasibility to treat human cancers. Lipid-based cationic nanoparticles (Cavalcanti et al, 2005), one of new promising tumor therapies, by loading suitable cytotoxic compounds can cause strong human immune responses and result in the destruction of tumor. Magnetic nanoparticles as the carrier have been used in cancer treatment avoiding side effects of conventional chemotherapy (Alexiou et al, 2006). Recent progress has been made in the application of nanoparticles to cancer treatment, including their use as delivery systems for potent anticancer drugs or genes, as well as agents for more advanced cancer treatment modalities, such as the combination treatments of radiotherapy, chemotherapy, and gene therapy with

Major factors determining the effectiveness of gene therapy are the method of gene delivery and the details of the therapeutic gene expression in the targeted tissue. Some researchers have reported that heat can not only enhance the immunogenicity of tumor cells (Kubista et al, 2002), but also regulate the heat-sensitive promoters in the region of interest (Ito et al, 2003; Ito et al, 2006; Ito et al, 2004; Todryk et al, 2003). Heat shock proteins (HSPs) as sensitive promoters are recognized as significant participants in immune reactions (Kubista et al, 2002). Animal studies showed that the hyperthermia-regulated gene therapy using

**6.3 Nano-particle drug therapy** 

hyperthermia (Kong et al, 2000).

**Hyperthermia-regulated gene therapy** 

**6.4 Others** 

the immune system (Dewhirst et al, 2007). All these effects may contribute o the desired eventual effect, which (certainly when combined with RT) is he achievement of local control. Several phase III trials comparing radiotherapy alone or with hyperthermia have shown a beneficial effect of hyperthermia (with existing standard equipment) in terms of local control (eg, recurrent breast cancer and malignant melanoma) and survival (eg, head and neck lymph-node metastases, glioblastoma, cervical carcinoma). Therefore, further development of existing technology and elucidation of molecular mechanisms are justified.

#### **6.2 Chemotherapy**

The combination of hyperthermia and chemotherapy has demonstrated several advantages over chemotherapy alone. The architecture of the vasculature in solid tumors is often insufficient due to the rapid growth of tumor tissue compared to normal tissue and/or chaotic, resulting in regions with hypoxia and low pH levels, which is not found in normal tissue. When using a mild hyperthermia (temperatures < 42 C), heat results in vasodilatation which improves the oxygenation of tissue (Iwata et al, 1996). Results reveal that the changes in tumor oxygenation are temperature dependent. This relationship could possibly influence treatment outcome of thermo-chemotherapy when the activity of chemotherapeutic agents is known to be oxygen dependent. This improvement of the blood supply can increase the cell metabolism which allows a greater effect of the chemotherapeutic agent on the tumor cells. Besides, heat also improves the cellular permeability which leads to the increase of the drug uptake by the tumor cells and intracellular spaces, the reaction of chemotherapy with DNA, and the prevention of DNA repair (Herman et al, 1988).

Moreover, the pathologic studies have shown that the enhanced drug cytotoxicity by heating induces both apoptosis and necrosis above a certain threshold temperature (Harmon et al, 1990; Yonezawa et al, 1996). In addition, several studies have also shown that some of the advantages of combining chemotherapy with hyperthermia are not only treating the primary cancers, but also reducing the risk of treatment-induced secondary cancers ( Kampinga and Dikomey, 2001; Hurt et al, 2004; Hunt et al, 2007). These factors make cells more sensitive to heat especially in low perfused tissues. Therefore, in addition to direct cytotoxicity, hyperthermia leads in vivo to a selective destruction of tumor cells in hypoxic and, consequently, acidic environment within parts of malignant tumors (Vaupel et al, 1989; Vaupel, 2004).

More recent in vivo studies have demonstrated that the thermal enhancement of cytotoxicity of many chemotherapeutic agents is maximized with heat (Hahn, 1979; Marmor, 1979; Engelhardt, 1987; Dahl, 1988; Bull, 1984; Hildebrandt et al, 2002; Urano et al, 1999). The positive results of thermo-chemotherapy are observed that the rate at which cells are killed by the drug increases with temperatures. Besides, the efficacy of thermochemotherapy also depends on the treatment planning. In general, promising results indicate that patients need to take chemotherapeutic agents immediately before hyperthermia. However, some of agents like the antimetabolite gemcitabine, are taken prior to hyperthermia at least 24 h to achieve a synergistic effect in vitro and in vivo (Haveman et al, 1995; Van Bree et al, 1999).

Although the working mechanism of thermo-chemotherapy is not fully understood, with the promising results of clinical trials and the thermal enhancement of drug cytotoxicity from pathologic studies, hyperthermia combined with chemotherapy has demonstrated as one of effective modalities in the present cancer treatment.

the immune system (Dewhirst et al, 2007). All these effects may contribute o the desired eventual effect, which (certainly when combined with RT) is he achievement of local control. Several phase III trials comparing radiotherapy alone or with hyperthermia have shown a beneficial effect of hyperthermia (with existing standard equipment) in terms of local control (eg, recurrent breast cancer and malignant melanoma) and survival (eg, head and neck lymph-node metastases, glioblastoma, cervical carcinoma). Therefore, further development of existing technology and elucidation of molecular mechanisms are justified.

The combination of hyperthermia and chemotherapy has demonstrated several advantages over chemotherapy alone. The architecture of the vasculature in solid tumors is often insufficient due to the rapid growth of tumor tissue compared to normal tissue and/or chaotic, resulting in regions with hypoxia and low pH levels, which is not found in normal tissue. When using a mild hyperthermia (temperatures < 42 C), heat results in vasodilatation which improves the oxygenation of tissue (Iwata et al, 1996). Results reveal that the changes in tumor oxygenation are temperature dependent. This relationship could possibly influence treatment outcome of thermo-chemotherapy when the activity of chemotherapeutic agents is known to be oxygen dependent. This improvement of the blood supply can increase the cell metabolism which allows a greater effect of the chemotherapeutic agent on the tumor cells. Besides, heat also improves the cellular permeability which leads to the increase of the drug uptake by the tumor cells and intracellular spaces, the reaction of chemotherapy with

Moreover, the pathologic studies have shown that the enhanced drug cytotoxicity by heating induces both apoptosis and necrosis above a certain threshold temperature (Harmon et al, 1990; Yonezawa et al, 1996). In addition, several studies have also shown that some of the advantages of combining chemotherapy with hyperthermia are not only treating the primary cancers, but also reducing the risk of treatment-induced secondary cancers ( Kampinga and Dikomey, 2001; Hurt et al, 2004; Hunt et al, 2007). These factors make cells more sensitive to heat especially in low perfused tissues. Therefore, in addition to direct cytotoxicity, hyperthermia leads in vivo to a selective destruction of tumor cells in hypoxic and, consequently, acidic environment within parts of malignant tumors (Vaupel et

More recent in vivo studies have demonstrated that the thermal enhancement of cytotoxicity of many chemotherapeutic agents is maximized with heat (Hahn, 1979; Marmor, 1979; Engelhardt, 1987; Dahl, 1988; Bull, 1984; Hildebrandt et al, 2002; Urano et al, 1999). The positive results of thermo-chemotherapy are observed that the rate at which cells are killed by the drug increases with temperatures. Besides, the efficacy of thermochemotherapy also depends on the treatment planning. In general, promising results indicate that patients need to take chemotherapeutic agents immediately before hyperthermia. However, some of agents like the antimetabolite gemcitabine, are taken prior to hyperthermia at least 24 h to achieve a synergistic effect in vitro and in vivo (Haveman et

Although the working mechanism of thermo-chemotherapy is not fully understood, with the promising results of clinical trials and the thermal enhancement of drug cytotoxicity from pathologic studies, hyperthermia combined with chemotherapy has demonstrated as

DNA, and the prevention of DNA repair (Herman et al, 1988).

one of effective modalities in the present cancer treatment.

**6.2 Chemotherapy** 

al, 1989; Vaupel, 2004).

al, 1995; Van Bree et al, 1999).

#### **6.3 Nano-particle drug therapy**

The nanoparticles have been applied to facilitate drug delivery and to overcome some of the problems of drug delivery for cancer treatment. In the past, cancer therapies using anticancer drugs were dissatisfied and had major side effects. Because of their multifunctional character the nanoparticles can deliver larger and more effective doses of chemotherapeutic agents or therapeutic genes into malignant cells, minimize toxic effects on healthy tissues and then alleviate patients suffering from the side-effects of chemotherapy. Nanoparticles can be used to deliver hydrophilic drugs, hydrophobic drugs, proteins, vaccines, biological macromolecules, etc. Several nanoscale delivery devices, such as ceramic nanoparticles, virus, dendrimers (spherical, branched polymers), silica-coated micelles, cross linked liposomes, and carbon nanotubes (Portney and Ozkan, 2006) have been used to improve delivery of anticancer agents to tumor cells (Brigger el al, 2002). Some of the challenges in effectively delivering anticancer drugs have to be solved: how to ensure therapeutic drug molecules reach the targeted tumor, how to release them slowly over a longer period, and how to avoid the human immune system destroying them.

Normally, the structure of a nanoparticle drug carrier has four elements. The first of them is the targeted chemotherapy drug, for example, docetaxel or Taxotere. The second is a matrix made of a biodegradable polymer (polylactic acid), which contains the anticancer drug and breaks down slowly so that the drug is released gradually over several days. The third element is a coating layer of polyethylene glycol, which is used to prevent from attacking by antibodies and macrophage cells of the human immune system. The final element is a targeted tag, in the form of special enzymes attached to the outer coating, which can form electrostatic or covalent bonds with positively charged agents and biomolecules. This tag allows the nanoparticles to bind directly to desired tumor cells but to bypass healthy tissues and eventually to reduce the side effects caused by most chemotherapy drugs.

Several different drug delivery methods (Jain, 2005) have been shown their feasibility to treat human cancers. Lipid-based cationic nanoparticles (Cavalcanti et al, 2005), one of new promising tumor therapies, by loading suitable cytotoxic compounds can cause strong human immune responses and result in the destruction of tumor. Magnetic nanoparticles as the carrier have been used in cancer treatment avoiding side effects of conventional chemotherapy (Alexiou et al, 2006). Recent progress has been made in the application of nanoparticles to cancer treatment, including their use as delivery systems for potent anticancer drugs or genes, as well as agents for more advanced cancer treatment modalities, such as the combination treatments of radiotherapy, chemotherapy, and gene therapy with hyperthermia (Kong et al, 2000).

#### **6.4 Others**

#### **Hyperthermia-regulated gene therapy**

Major factors determining the effectiveness of gene therapy are the method of gene delivery and the details of the therapeutic gene expression in the targeted tissue. Some researchers have reported that heat can not only enhance the immunogenicity of tumor cells (Kubista et al, 2002), but also regulate the heat-sensitive promoters in the region of interest (Ito et al, 2003; Ito et al, 2006; Ito et al, 2004; Todryk et al, 2003). Heat shock proteins (HSPs) as sensitive promoters are recognized as significant participants in immune reactions (Kubista et al, 2002). Animal studies showed that the hyperthermia-regulated gene therapy using

Heating in Biothermal Systems 275

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hsp70 obtained strong prevention of tumor growth, complete regression of tumors, and the induction of systemic antitumor immunity in the cured mice (Ito et al, 2006; Ito et al, 2004; Todryk et al, 2003).

These studies suggest that the combination gene therapy with hyperthermia using hsp70 have great potential in cancer treatment. Nevertheless, results also suggested that inappropriate immune reactivity to hsp70 might lead to pro-inflammatory responses and the development of autoimmune disease. Moreover, endotoxin contamination has been reported to be responsible for the human hsp70 preparation (Gao & Tsan, 2003; Bausinger et al, 2002) and uncontrolled promoter activation by other than heat stressors for the HSP70B promoter system was found (Siddiqui et al, 2008). To have a safe and controllable gene therapy the unintentional activation of heat-responsive promoters should be avoided. Although some effects of heat shock proteins on the immunogenicity of tumor cells have been studied, more work is still needed before the hyperthermia-regulated gene therapy using hsp70 can applied into clinical cancer treatments.

Recently, a combination of gene therapy with magnetic resonance imaging (MRI), highintensity focused ultrasound (HIFU) and a temperature-sensitive promoter is being evaluated in the cancer field (Moonen, 2007; Silcox et al, 2005; Plathow et al, 2005; Frenkel, 2006; Rome et al, 2005; Walther & Stein, 2009). With the help of advanced imaging techniques one can noninvasively monitor the temperature field induced by a high-intensity focused ultrasound system and simultaneously regulate the gene expression in the treatment region. Results indicated that although the application of MRI-guided HIFU in gene therapy is promising, further technical requirements of the heating and monitoring systems for precise control are still needed.

In recent molecular and biological investigations there have been novel applications such as gene therapy or immunotherapy (vaccination) with temperature acting as an enhancer, to trigger or to switch mechanisms on and off. However, for every particular temperaturedependent interaction exploited for clinical purposes, sophisticated control of temperature, spatially as well as temporally, in deep body regions will further improve the potential.

#### **7. Conclusion**

Thermal transport in bio-thermal systems signifies that temperature management in living systems can help us in curing and treatment of ill conditions. In analytic perspective, efforts on mathematical and numerical modelings have showed great progrssing in calculating temperature distributions. Advance in computer technology is one of critical contributing factors. In clinical perspective, many thermal energy related experiments and tests that are adjuvant to other tumor treatment modalities have identified effectiveness in treatments. In this regard, the well knowledge of heat transfer process revealed significant in optimal controling temperature in bio-thermal systems. Thus it shows us another great research and career opportunities in this field.

#### **8. Acknowledgment**

Authors would like to express special thanks to Professor Win-li Lin for his valuable suggestions and discussion. The authors would like to thank the National Science Council of Taiwan for partially supporting this research under no. NSC 99-2221-E-168 -026 and NSC 100-2221-E-032 -013.

#### **9. References**

274 Convection and Conduction Heat Transfer

hsp70 obtained strong prevention of tumor growth, complete regression of tumors, and the induction of systemic antitumor immunity in the cured mice (Ito et al, 2006; Ito et al, 2004;

These studies suggest that the combination gene therapy with hyperthermia using hsp70 have great potential in cancer treatment. Nevertheless, results also suggested that inappropriate immune reactivity to hsp70 might lead to pro-inflammatory responses and the development of autoimmune disease. Moreover, endotoxin contamination has been reported to be responsible for the human hsp70 preparation (Gao & Tsan, 2003; Bausinger et al, 2002) and uncontrolled promoter activation by other than heat stressors for the HSP70B promoter system was found (Siddiqui et al, 2008). To have a safe and controllable gene therapy the unintentional activation of heat-responsive promoters should be avoided. Although some effects of heat shock proteins on the immunogenicity of tumor cells have been studied, more work is still needed before the hyperthermia-regulated gene therapy

Recently, a combination of gene therapy with magnetic resonance imaging (MRI), highintensity focused ultrasound (HIFU) and a temperature-sensitive promoter is being evaluated in the cancer field (Moonen, 2007; Silcox et al, 2005; Plathow et al, 2005; Frenkel, 2006; Rome et al, 2005; Walther & Stein, 2009). With the help of advanced imaging techniques one can noninvasively monitor the temperature field induced by a high-intensity focused ultrasound system and simultaneously regulate the gene expression in the treatment region. Results indicated that although the application of MRI-guided HIFU in gene therapy is promising, further technical requirements of the heating and monitoring

In recent molecular and biological investigations there have been novel applications such as gene therapy or immunotherapy (vaccination) with temperature acting as an enhancer, to trigger or to switch mechanisms on and off. However, for every particular temperaturedependent interaction exploited for clinical purposes, sophisticated control of temperature, spatially as well as temporally, in deep body regions will further improve the potential.

Thermal transport in bio-thermal systems signifies that temperature management in living systems can help us in curing and treatment of ill conditions. In analytic perspective, efforts on mathematical and numerical modelings have showed great progrssing in calculating temperature distributions. Advance in computer technology is one of critical contributing factors. In clinical perspective, many thermal energy related experiments and tests that are adjuvant to other tumor treatment modalities have identified effectiveness in treatments. In this regard, the well knowledge of heat transfer process revealed significant in optimal controling temperature in bio-thermal systems. Thus it shows us another great research and

Authors would like to express special thanks to Professor Win-li Lin for his valuable suggestions and discussion. The authors would like to thank the National Science Council of Taiwan for partially supporting this research under no. NSC 99-2221-E-168 -026 and NSC

Todryk et al, 2003).

**7. Conclusion** 

career opportunities in this field.

**8. Acknowledgment** 

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D. Stevens1, A. LaRocca1, H. Power1 and V. LaRocca2 <sup>1</sup>*Faculty of Engineering, Division of Energy and Sustainability,*

**on Unstructured Datasets** 

<sup>2</sup>*Dipartimento di Ricerche Energetiche ed Ambientali, University of Palermo, Sicily*

**A Generalised RBF Finite Difference Approach** 

**to Solve Nonlinear Heat Conduction Problems** 

Radial Basis Functions have traditionally been used to provide a continuous interpolation of scattered data sets. However, this interpolation also allows for the reconstruction of partial derivatives throughout the solution field, which can then be used to drive the solution of a partial differential equation. Since the interpolation takes place on a scattered dataset with no local connectivity, the solution is essentially meshless. RBF-based methods have been

Such full-domain RBF methods are highly flexible and can exhibit spectral convergence rates Madych & Nelson (1990). However, in their traditional implementation the fully-populated matrix systems which are produced lead to computational complexities of at least order-*N*<sup>2</sup> with datasets of size *N*. In addition, they suffer from increasingly poor numerical conditioning as the size of the dataset grows, and also with increasingly flat interpolating functions. This is a consequence of ill-conditioning in the determination of RBF weighting coefficients (as demonstrated in Driscoll & Fornberg (2002)), and is described by Robert Schaback Schaback (1995) as the uncertainty relation; better conditioning is associated with worse accuracy, and worse conditioning is associated with improved accuracy. Many techniques have been developed to reduce the effect of the uncertainty relation in the traditional RBF formulation, such as RBF-specific preconditioners Baxter (2002); Beatson et al. (1999); Brown (2005); Ling & Kansa (2005), or adaptive selection of data centres Ling et al. (2006); Ling & Schaback (2004). However, at present the only reliable methods of controlling numerical ill-conditioning and computational cost as problem size increases are domain decomposition Hernandez Rosales & Power (2007); Wong et al. (1999); Zhang (2007); Zhou et al. (2003), or the use of locally supported basis functions Fasshauer (1999); Schaback (1997); Wendland (1995); Wu (1995). In this work the domain decomposition principle is applied, forming a large number of heavily overlapping systems that cover the solution domain. A small RBF collocation system is formed around each global data centre, with each collocation system used to approximate the governing PDE at its centrepoint, in terms of the solution value at surrounding collocation points. This leads to a sparse global linear system which may be solved using a variety

successfully used to solve a wide variety of PDEs in this fashion.

**1. Introduction**

*University of Nottingham*

<sup>1</sup>*UK* <sup>2</sup>*Italy*

**13**


### **A Generalised RBF Finite Difference Approach to Solve Nonlinear Heat Conduction Problems on Unstructured Datasets**

D. Stevens1, A. LaRocca1, H. Power1 and V. LaRocca2 <sup>1</sup>*Faculty of Engineering, Division of Energy and Sustainability, University of Nottingham* <sup>2</sup>*Dipartimento di Ricerche Energetiche ed Ambientali, University of Palermo, Sicily* <sup>1</sup>*UK* <sup>2</sup>*Italy*

#### **1. Introduction**

280 Convection and Conduction Heat Transfer

Wust, P; Hildebrandt, B; Sreenivasa, G, *et al.* (2002) Hyperthermia in combined treatment of

Yonezawa, M; Otsuka, T; Matsui, N; et al. (1996) Hyperthermia induces apoptosis in

Yuan, D.Y.; et al. (1995) Advances in Heat and Mass Transfer in Biotechnology. in *ASME* 

malignant fibrous histiocytoma cell *in vitro*. *Int J Cancer* 66:347–51.

cancer. *Lancet Oncol* 3:487– 497.

*Winter Annual Meeting*.

Radial Basis Functions have traditionally been used to provide a continuous interpolation of scattered data sets. However, this interpolation also allows for the reconstruction of partial derivatives throughout the solution field, which can then be used to drive the solution of a partial differential equation. Since the interpolation takes place on a scattered dataset with no local connectivity, the solution is essentially meshless. RBF-based methods have been successfully used to solve a wide variety of PDEs in this fashion.

Such full-domain RBF methods are highly flexible and can exhibit spectral convergence rates Madych & Nelson (1990). However, in their traditional implementation the fully-populated matrix systems which are produced lead to computational complexities of at least order-*N*<sup>2</sup> with datasets of size *N*. In addition, they suffer from increasingly poor numerical conditioning as the size of the dataset grows, and also with increasingly flat interpolating functions. This is a consequence of ill-conditioning in the determination of RBF weighting coefficients (as demonstrated in Driscoll & Fornberg (2002)), and is described by Robert Schaback Schaback (1995) as the uncertainty relation; better conditioning is associated with worse accuracy, and worse conditioning is associated with improved accuracy. Many techniques have been developed to reduce the effect of the uncertainty relation in the traditional RBF formulation, such as RBF-specific preconditioners Baxter (2002); Beatson et al. (1999); Brown (2005); Ling & Kansa (2005), or adaptive selection of data centres Ling et al. (2006); Ling & Schaback (2004). However, at present the only reliable methods of controlling numerical ill-conditioning and computational cost as problem size increases are domain decomposition Hernandez Rosales & Power (2007); Wong et al. (1999); Zhang (2007); Zhou et al. (2003), or the use of locally supported basis functions Fasshauer (1999); Schaback (1997); Wendland (1995); Wu (1995). In this work the domain decomposition principle is applied, forming a large number of heavily overlapping systems that cover the solution domain. A small RBF collocation system is formed around each global data centre, with each collocation system used to approximate the governing PDE at its centrepoint, in terms of the solution value at surrounding collocation

points. This leads to a sparse global linear system which may be solved using a variety

4), and the ability of the formulation to handle strongly nonlinear problems is demonstrated

<sup>283</sup> A Generalised RBF Finite Difference Approach

The Hermitian RBF collocation method operates on a domain which is covered by a series of *N* scattered data points, including a distribution of points over all domain boundaries. The solution is constructed using *N* distinct basis functions, which are composed of a partial differential operator applied to a radial basis function Ψ*<sup>j</sup>* which is centred on the data point *j*. The partial differential operator that is applied to each basis function is the PDE boundary operator for points lying on the domain boundary, and the governing PDE for points lying within the domain (see equation 3). A polynomial term is required to complete the underlying vector space. In the present work, the multiquadric radial basis function is used throughout,

> � *m* 2

The multiquadric RBF is a conditionally positive definite function of order *m*, which requires the addition of a polynomial term of order *m* − 1, together with a homogeneous constraint condition, in order to obtain an invertible interpolation matrix. For the *m* = 1 case used in this work, the polynomial is simply a constant term. The '*c*' term is known as a 'shape parameter', and describes the relative width of the RBF functions about their centres. The tuning of the shape parameter can have a significant effect upon the accuracy of the solution

*L* [*u*] = *S* (*x*) *on* Ω

where the operators *L* [] and *B* [] are linear partial differential operators on the domain Ω and on the boundary *∂*Ω, describing the governing equation and boundary conditions respectively. Data points *ξ<sup>j</sup>* are distributed over the boundary and inside the domain, and the solution is constructed from basis functions centred around the *ξj*. At data points lying on *∂*Ω the boundary operator is applied to the RBF, in order to form the basis function, while the

> *N* ∑ *j*=*NB*+1

*BxL<sup>ξ</sup>* � <sup>Ψ</sup>*ij*� *Bx* � *Pi m*−1 �

*LxL<sup>ξ</sup>* � <sup>Ψ</sup>*ij*� *Lx* � *Pi m*−1 �

The RBF formula (3) is then collocated at each of the data points *ξ<sup>j</sup>* applying the PDE boundary equation, *B* [*u*] = *g*, at points on the domain boundary, and the PDE governing equation, *L* [*u*] = *f* , at points within the domain. This leads to a symmetric collocation system, as represented by equation (4), which can be solved to obtain the RBF weighting coefficients *λi*.

*λjLξ*Ψ

�� � � *x* − *ξ<sup>j</sup>* � � � � + *NP* ∑ *j*=1

0

⎤ ⎥ ⎥ ⎥ ⎦ *λ<sup>i</sup>* =

⎡ ⎣ *gi Si* 0

⎤

*<sup>m</sup>* <sup>∈</sup> <sup>2</sup>**Z**<sup>+</sup> <sup>−</sup> <sup>1</sup> (1)

*<sup>λ</sup>j*+*NP<sup>j</sup>*

*<sup>m</sup>*−<sup>1</sup> (*x*) (3)

⎦ (4)

*B* [*u*] = *g* (*x*) *on ∂*Ω (2)

in the solution of a food freezing problem (see section 5).

to Solve Nonlinear Heat Conduction Problems on Unstructured Datasets

Ψ (*r*) =

PDE governing operator is applied at those points inside the domain:

and the conditioning of the numerical system. Consider a typical linear boundary value problem

� *r*<sup>2</sup> + *c*<sup>2</sup>

**2. RBF method formulation**

with *m* = 1.

*u* (*x*) =

*NB* ∑ *j*=1

*λjBξ*Ψ

⎡ ⎢ ⎢ ⎢ ⎣ *BxB<sup>ξ</sup>* � <sup>Ψ</sup>*ij*�

*LxB<sup>ξ</sup>* � <sup>Ψ</sup>*ij*�

*Bξ* � *Pj m*−1 �*T Lξ* � *Pj m*−1 �*T*

�� � � *x* − *ξ<sup>j</sup>* � � � � +

of standard solvers. In this way, the proposed formulation emulates a finite difference method, with the RBF collocation systems replacing the polynomial interpolation functions used in traditional finite difference methods. However, unlike such polynomial functions RBF collocation is well suited to scattered data, and the method may be applied to both structured and unstructured datasets without modification.

The method is applied here to solve the nonlinear heat conduction equation. In order to reduce the nonlinearity in the governing equation the Kirchhoff integral transformation is applied, and the transformed equation is solved using a Picard iterative process. The application of the Kirchhoff transform necessitates that the thermal property functions be transformed to Kirchhoff space also. If the thermal properties are a known and integrable function of temperature then the transformation may be performed analytically. Otherwise, an integration-interpolation procedure can be performed using 1D radial basis functions, as described in Stevens & Power (2010).

In recent years a number of local RBF collocation techniques have been proposed, and applied a wide variety of problems (for example; Divo & Kassab (2007); Lee et al. (2003); Sarler & Vertnik (2006); Wright & Fornberg (2006)). A more comprehensive review of such methods is given in Stevens et al. (2009). Unlike most local RBF collocation methods that are used in the literature, the technique described here utilises the Hermitian RBF collocation formulation (see section 2 for more details), and allows both the PDE-boundary and PDE-governing operators to be included within in the local collocation systems. This inclusion of the governing PDE within the basis functions is shown in Stevens et al. (2009) to significantly improve the accuracy and stability of solutions obtained for linear transport problems. Additionally, the incorporation of information about the convective velocity field into the basis functions was shown to have a stabilising effect, similar to traditional upwinding methods but without the requirement to alter the stencil configuration based on the local convective field.

The standard approach to the solution of linear and nonlinear heat conduction problems is the use of finite difference and finite volume methods with simple polynomial interpolants Bejan (1993); Holman (2002); Kreith & Bohn (2000). Due to the dominance of diffusion in most cases, central differencing techniques are commonly used to compute the heat fluxes. However, limiter methods (such as the unconditionally stable TVD schemes) may be used for nonlinear heat conduction problems where the effective convection term, which results from the non-zero variation of thermal conductivity with temperature, can be expected to approach the magnitude of the diffusive term (see, for example, Shen & Han (2002)). Full-domain RBF methods have also been examined for use with nonlinear heat conduction problems (see Chantasiriwan (2007)), however such methods are restricted to small dataset sizes, due to the computational cost and numerical conditioning experienced by full-domain RBF techniques on large datasets.

The present work demonstrates how local RBF collocation may be used as an alternative to traditional finite difference and finite volume methods, for nonlinear heat conduction problems. The described method retains freedom from a volumetric mesh, while allowing solution over unstructured datasets. A central stencil configuration is used in each case, and the solution is stabilised via the inclusion of the governing and boundary PDEs within the local collocation systems ("implicit upwinding"), rather than by adjusting the stencil configuration based on the local solution field ("traditional upwinding"). The method is validated using a transient numerical example with a known analytical solution (see section 4), and the ability of the formulation to handle strongly nonlinear problems is demonstrated in the solution of a food freezing problem (see section 5).

#### **2. RBF method formulation**

2 Heat Transfer Book 2

of standard solvers. In this way, the proposed formulation emulates a finite difference method, with the RBF collocation systems replacing the polynomial interpolation functions used in traditional finite difference methods. However, unlike such polynomial functions RBF collocation is well suited to scattered data, and the method may be applied to both structured

The method is applied here to solve the nonlinear heat conduction equation. In order to reduce the nonlinearity in the governing equation the Kirchhoff integral transformation is applied, and the transformed equation is solved using a Picard iterative process. The application of the Kirchhoff transform necessitates that the thermal property functions be transformed to Kirchhoff space also. If the thermal properties are a known and integrable function of temperature then the transformation may be performed analytically. Otherwise, an integration-interpolation procedure can be performed using 1D radial basis functions, as

In recent years a number of local RBF collocation techniques have been proposed, and applied a wide variety of problems (for example; Divo & Kassab (2007); Lee et al. (2003); Sarler & Vertnik (2006); Wright & Fornberg (2006)). A more comprehensive review of such methods is given in Stevens et al. (2009). Unlike most local RBF collocation methods that are used in the literature, the technique described here utilises the Hermitian RBF collocation formulation (see section 2 for more details), and allows both the PDE-boundary and PDE-governing operators to be included within in the local collocation systems. This inclusion of the governing PDE within the basis functions is shown in Stevens et al. (2009) to significantly improve the accuracy and stability of solutions obtained for linear transport problems. Additionally, the incorporation of information about the convective velocity field into the basis functions was shown to have a stabilising effect, similar to traditional upwinding methods but without the requirement to alter the stencil configuration based on the local

The standard approach to the solution of linear and nonlinear heat conduction problems is the use of finite difference and finite volume methods with simple polynomial interpolants Bejan (1993); Holman (2002); Kreith & Bohn (2000). Due to the dominance of diffusion in most cases, central differencing techniques are commonly used to compute the heat fluxes. However, limiter methods (such as the unconditionally stable TVD schemes) may be used for nonlinear heat conduction problems where the effective convection term, which results from the non-zero variation of thermal conductivity with temperature, can be expected to approach the magnitude of the diffusive term (see, for example, Shen & Han (2002)). Full-domain RBF methods have also been examined for use with nonlinear heat conduction problems (see Chantasiriwan (2007)), however such methods are restricted to small dataset sizes, due to the computational cost and numerical conditioning experienced by full-domain RBF techniques

The present work demonstrates how local RBF collocation may be used as an alternative to traditional finite difference and finite volume methods, for nonlinear heat conduction problems. The described method retains freedom from a volumetric mesh, while allowing solution over unstructured datasets. A central stencil configuration is used in each case, and the solution is stabilised via the inclusion of the governing and boundary PDEs within the local collocation systems ("implicit upwinding"), rather than by adjusting the stencil configuration based on the local solution field ("traditional upwinding"). The method is validated using a transient numerical example with a known analytical solution (see section

and unstructured datasets without modification.

described in Stevens & Power (2010).

convective field.

on large datasets.

The Hermitian RBF collocation method operates on a domain which is covered by a series of *N* scattered data points, including a distribution of points over all domain boundaries. The solution is constructed using *N* distinct basis functions, which are composed of a partial differential operator applied to a radial basis function Ψ*<sup>j</sup>* which is centred on the data point *j*. The partial differential operator that is applied to each basis function is the PDE boundary operator for points lying on the domain boundary, and the governing PDE for points lying within the domain (see equation 3). A polynomial term is required to complete the underlying vector space. In the present work, the multiquadric radial basis function is used throughout, with *m* = 1.

$$\Psi'(r) = \left(r^2 + c^2\right)^{\frac{m}{2}} \quad m \in 2\mathbb{Z}^+ - 1 \tag{1}$$

The multiquadric RBF is a conditionally positive definite function of order *m*, which requires the addition of a polynomial term of order *m* − 1, together with a homogeneous constraint condition, in order to obtain an invertible interpolation matrix. For the *m* = 1 case used in this work, the polynomial is simply a constant term. The '*c*' term is known as a 'shape parameter', and describes the relative width of the RBF functions about their centres. The tuning of the shape parameter can have a significant effect upon the accuracy of the solution and the conditioning of the numerical system.

Consider a typical linear boundary value problem

$$L\begin{bmatrix}u\end{bmatrix} = \mathcal{S}\begin{pmatrix}\chi\end{pmatrix}\quad on\ \Omega$$

$$\begin{aligned}B\begin{bmatrix}u\end{bmatrix} = \mathcal{g}\begin{pmatrix}\chi\end{pmatrix}\quad on\ \partial\Omega\end{aligned}\tag{2}$$

where the operators *L* [] and *B* [] are linear partial differential operators on the domain Ω and on the boundary *∂*Ω, describing the governing equation and boundary conditions respectively. Data points *ξ<sup>j</sup>* are distributed over the boundary and inside the domain, and the solution is constructed from basis functions centred around the *ξj*. At data points lying on *∂*Ω the boundary operator is applied to the RBF, in order to form the basis function, while the PDE governing operator is applied at those points inside the domain:

$$\mu\left(\mathbf{x}\right) = \sum\_{j=1}^{NB} \lambda\_j B\_{\tilde{\xi}} \mathbf{Y} \left( \left\| \left\| \mathbf{x} - \tilde{\xi}\_j \right\| \right\| \right) + \sum\_{j=NB+1}^{N} \lambda\_j L\_{\tilde{\xi}} \mathbf{Y} \left( \left\| \left\| \mathbf{x} - \tilde{\xi}\_j \right\| \right\| \right) + \sum\_{j=1}^{NP} \lambda\_{j+N} P\_{m-1}^j \left( \mathbf{x} \right) \tag{3}$$

The RBF formula (3) is then collocated at each of the data points *ξ<sup>j</sup>* applying the PDE boundary equation, *B* [*u*] = *g*, at points on the domain boundary, and the PDE governing equation, *L* [*u*] = *f* , at points within the domain. This leads to a symmetric collocation system, as represented by equation (4), which can be solved to obtain the RBF weighting coefficients *λi*.

$$
\begin{bmatrix} B\_X B\_\xi \begin{bmatrix} \Psi\_{ij} \\ \Psi\_{ij} \end{bmatrix} & B\_X L\_\xi \begin{bmatrix} \Psi\_{ij} \\ \Psi\_{ij} \end{bmatrix} & B\_X \begin{bmatrix} P\_{m-1}^i \\ \Psi\_{ij} \end{bmatrix} \\ L\_X B\_\xi \begin{bmatrix} P\_{m-1}^j \\ P\_{m-1}^j \end{bmatrix}^T L\_\xi \begin{bmatrix} P\_{ij}^j \\ P\_{m-1}^j \end{bmatrix}^T & 0 \end{bmatrix} \lambda\_i = \begin{bmatrix} \mathcal{G}\_i \\ \mathcal{S}\_i \\ 0 \end{bmatrix} \tag{4}
$$

where the "reconstruction vector" *H* (*x*)

(*k*) = Ψ *x* − *ξ<sup>j</sup>* , *B<sup>ξ</sup>* Ψ *x* − *ξ<sup>j</sup>*

= *Lx* Ψ *x*−*ξ<sup>j</sup>*

governing operator can be given by

the values of *ui* within the local system

*L u*(*k*) (*x*) = *L H*(*k*) (*x*) *α*(*k*)

number of finite difference time advancement schemes.

By inverting the matrix system (6), i.e. *α*(*k*) =

*H* (*x*)

*L u*(*k*) (*x*) = *L H* (*x*) (*k*) *α*(*k*)

where *W*(*k*)

(*k*) is given by

<sup>285</sup> A Generalised RBF Finite Difference Approach

 *A*(*k*) −<sup>1</sup>

variable *u* at any point within the stencil of local system *k* in terms of the data vector *d*(*k*), i.e.

(*k*) *A*(*k*) −<sup>1</sup>

Similarly, the PDE governing operator, *L*, can be applied to the reconstruction vector *H*(*k*) in

As before, by expressing the vector *α*(*k*) in terms of the data vector *d*(*k*), the value of the PDE

*<sup>L</sup>* (*x*) *<sup>d</sup>*(*k*)

reconstructing the PDE operator at the system centrepoint, a relation can be obtained linking

= *<sup>W</sup>*(*k*)

Applying the above reconstruction to each local system *k*, a series of *N* simultaneous equations are produced for *ui*, *i* = 1, . . . , *N*, where *N* is the global number of solution centres. In the resulting global linear system the corresponding boundary conditions of the problem have already been imposed, at the local interpolation, in those stencils containing boundary points. The resulting linear system is sparse, with the number of non-zero entries in each row equal to the number of solution centres in the corresponding collocation system. Therefore, if the local system size is kept constant, the method may be scaled efficiently to very large datasets. The method of solution for time dependent problems requires the creation of a modified PDE-operator via a finite difference approximation of the time derivative. The procedure is illustrated here using a Θ-weighted Crank-Nicolson approach, but is easily extensible to any

*<sup>L</sup>* is a *stencil weights* vector for the differential operator *L* at local system *k*. By

*u*(*k*) (*xcentre*)

*u*(*k*) = *H* (*x*)

order to reconstruct its value at any location within the valid domain:

to Solve Nonlinear Heat Conduction Problems on Unstructured Datasets

, *LxB<sup>ξ</sup>*

= *L H*(*k*) (*x*)

= *W*(*k*)

*S* (*xcentre*) = *L*

 Ψ *x*−*ξ<sup>j</sup>*

 , *<sup>L</sup><sup>ξ</sup>* Ψ *x* − *ξ<sup>j</sup>*

, *LxL<sup>ξ</sup>*

*<sup>A</sup>*(*k*)

 Ψ *x*−*ξ<sup>j</sup>*

−<sup>1</sup> 

*<sup>L</sup>* (*xcentre*) *<sup>d</sup>*(*k*) (13)

, *Pm*−<sup>1</sup>

*d*(*k*), it is possible to express the field

*d*(*k*) (10)

, *Lx* [*Pm*−1]

*d*(*k*) (12)

*α*(*k*) (11)

(9)

Fig. 1. Local system representation

The basic Hermitian RBF collocation method, as outlined above, is highly accurate and offers excellent convergence rates. However due to the presence of a fully-populated collocation matrix (4) solving even moderately-sized problems becomes computationally expensive. The poor numerical conditioning of this system also places limits upon the size of the dataset that can be solved. To address these issues of computational cost and numerical conditioning, the Hermitian method is applied here in "finite difference mode". In this way a large number of small and highly overlapping collocation systems are formed, leading to a numerical method that is of order-N computational complexity, while maintaining many of the flexibilities present in the global Hermitian RBF collocation method.

The Hermitian RBF method in finite difference mode takes a scattered dataset of internal ("solution") and boundary centres, and forms a local stencil around each non-boundary node (see Figure 1). Additionally, within each stencil a number of PDE centres are distributed. At the boundary centres the boundary operator is enforced within the Hermitian collocation system, at the solution centres the unknown solution value is collocated, and at the PDE centres the PDE governing operator is enforced. In this way a series of small symmetric collocation systems are formed, which can be solved independently to interpolate the solution over the domain in a piecewise fashion.

A series of local RBF collocation systems

$$A^{(k)}a^{(k)} = d^{(k)}\tag{5}$$

are constructed for the interpolation coefficients *α*(*k*), where

$$A^{(k)} = \begin{bmatrix} \Psi\_{ij} & B\_{\xi} \begin{bmatrix} \mathbf{V}\_{ij} \end{bmatrix} & L\_{\xi} \begin{bmatrix} \mathbf{V}\_{ij} \end{bmatrix} & P\_{m-1} \\ B\_{X} \begin{bmatrix} \mathbf{V}\_{ij} \end{bmatrix} & B\_{X} B\_{\xi} \begin{bmatrix} \mathbf{V}\_{ij} \end{bmatrix} & B\_{x} L\_{\xi}^{\tau} \begin{bmatrix} \mathbf{V}\_{ij} \end{bmatrix} & B\_{x} \begin{bmatrix} P\_{m-1} \end{bmatrix} \\ L\_{\chi} \begin{bmatrix} \mathbf{V}\_{ij} \end{bmatrix} & L\_{X} B\_{\xi} \begin{bmatrix} \mathbf{V}\_{ij} \end{bmatrix} & L\_{X} L\_{\xi} \begin{bmatrix} \mathbf{V}\_{ij} \end{bmatrix} & L\_{x} \begin{bmatrix} P\_{m-1} \end{bmatrix} \\ P\_{m-1}^{\top} & B\_{\xi} \begin{bmatrix} P\_{m-1} \end{bmatrix}^{\top} & L\_{\xi} \begin{bmatrix} P\_{m-1} \end{bmatrix}^{\top} & 0 \end{bmatrix} \end{pmatrix} \quad \text{and} \quad d^{(k)} = \begin{bmatrix} u\_{i} \\ \mathcal{S}\_{i} \\ 0 \end{bmatrix} \tag{6}$$

Here the *ui* are the unknown values of the solution field *u*, at the solution centres contained within the local system *k*. The shape parameter, *c*, in the multiquadric RBF (1) is kept constant within the local system, but may be allowed to vary from system to system. By defining the shape parameter in terms of the local system support domain size, *R*, a dimensionless shape parameter may be described (7) which is consistent over varying stencil sizes.

$$c^\* = \frac{c}{R} \tag{7}$$

The value of *u* within the bounds of the local system may be reconstructed as follows

$$
\mu^{(k)}\left(\mathbf{x}\right) = H\left(\mathbf{x}\right)^{(k)}\boldsymbol{a}^{(k)}\tag{8}
$$

where the "reconstruction vector" *H* (*x*) (*k*) is given by

4 Heat Transfer Book 2

The basic Hermitian RBF collocation method, as outlined above, is highly accurate and offers excellent convergence rates. However due to the presence of a fully-populated collocation matrix (4) solving even moderately-sized problems becomes computationally expensive. The poor numerical conditioning of this system also places limits upon the size of the dataset that can be solved. To address these issues of computational cost and numerical conditioning, the Hermitian method is applied here in "finite difference mode". In this way a large number of small and highly overlapping collocation systems are formed, leading to a numerical method that is of order-N computational complexity, while maintaining many of the flexibilities

The Hermitian RBF method in finite difference mode takes a scattered dataset of internal ("solution") and boundary centres, and forms a local stencil around each non-boundary node (see Figure 1). Additionally, within each stencil a number of PDE centres are distributed. At the boundary centres the boundary operator is enforced within the Hermitian collocation system, at the solution centres the unknown solution value is collocated, and at the PDE centres the PDE governing operator is enforced. In this way a series of small symmetric collocation systems are formed, which can be solved independently to interpolate the solution

*A*(*k*)

*Lξ* � Ψ*ij* �

*<sup>T</sup> <sup>L</sup><sup>ξ</sup>* [*Pm*−1]

parameter may be described (7) which is consistent over varying stencil sizes.

Here the *ui* are the unknown values of the solution field *u*, at the solution centres contained within the local system *k*. The shape parameter, *c*, in the multiquadric RBF (1) is kept constant within the local system, but may be allowed to vary from system to system. By defining the shape parameter in terms of the local system support domain size, *R*, a dimensionless shape

*<sup>c</sup>*<sup>∗</sup> <sup>=</sup> *<sup>c</sup>*

The value of *u* within the bounds of the local system may be reconstructed as follows

*u*(*k*) (*x*) = *H* (*x*)

*Pm*−<sup>1</sup>

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

*Bx* [*Pm*−1]

*Lx* [*Pm*−1]

*<sup>T</sup>* 0

*α*(*k*) = *d*(*k*) (5)

and *d*(*k*) =

*<sup>R</sup>* (7)

(*k*) *α*(*k*) (8)

⎡ ⎢ ⎢ ⎣

*ui gi Si* 0

⎤ ⎥ ⎥ ⎦

(6)

Fig. 1. Local system representation

over the domain in a piecewise fashion. A series of local RBF collocation systems

*A*(*k*)=

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

*Bx* � Ψ*ij* � *BxB<sup>ξ</sup>* � Ψ*ij* � *BxL<sup>ξ</sup>* � Ψ*ij* �

*Lx* � Ψ*ij* � *LxB<sup>ξ</sup>* � Ψ*ij* � *LxL<sup>ξ</sup>* � Ψ*ij* �

*PT*

present in the global Hermitian RBF collocation method.

are constructed for the interpolation coefficients *α*(*k*), where

*<sup>m</sup>*−<sup>1</sup> *<sup>B</sup><sup>ξ</sup>* [*Pm*−1]

� Ψ*ij* �

Ψ*ij B<sup>ξ</sup>*

$$H(\mathbf{x})^{(k)} = \left[\Psi\left(\mathbf{x} - \boldsymbol{\xi}\_{j}\right), B\_{\tilde{\xi}}\left[\Psi\left(\mathbf{x} - \boldsymbol{\xi}\_{j}\right)\right], L\_{\tilde{\xi}}\left[\Psi\left(\mathbf{x} - \boldsymbol{\xi}\_{j}\right)\right], P\_{m-1}\right] \tag{9}$$

By inverting the matrix system (6), i.e. *α*(*k*) = *A*(*k*) −<sup>1</sup> *d*(*k*), it is possible to express the field variable *u* at any point within the stencil of local system *k* in terms of the data vector *d*(*k*), i.e.

$$u^{(k)} = H\left(\mathbf{x}\right)^{(k)} \left[A^{(k)}\right]^{-1} d^{(k)}\tag{10}$$

Similarly, the PDE governing operator, *L*, can be applied to the reconstruction vector *H*(*k*) in order to reconstruct its value at any location within the valid domain:

$$\begin{split} L\left[u^{(k)}\left(\mathbf{x}\right)\right] &= L\left[H\left(\mathbf{x}\right)^{(k)}\right]a^{(k)}\\ &= \left[L\_{\mathbf{x}}\left[\Psi\left(\mathbf{x}-\tilde{\xi}\_{j}\right)\right], L\_{\mathbf{x}}B\_{\tilde{\xi}}\left[\Psi\left(\mathbf{x}-\tilde{\xi}\_{j}\right)\right], L\_{\mathbf{x}}L\_{\tilde{\xi}}\left[\Psi\left(\mathbf{x}-\tilde{\xi}\_{j}\right)\right], L\_{\mathbf{x}}\left[P\_{m-1}\right]\right]a^{(k)} \end{split} \tag{11}$$

As before, by expressing the vector *α*(*k*) in terms of the data vector *d*(*k*), the value of the PDE governing operator can be given by

$$\begin{aligned} \left[\boldsymbol{\mu}^{(k)}\left(\mathbf{x}\right)\right] &= \boldsymbol{L}\left[\boldsymbol{H}^{(k)}\left(\mathbf{x}\right)\right] \boldsymbol{a}^{(k)}\\ &= \left(\boldsymbol{L}\left[\boldsymbol{H}^{(k)}\left(\mathbf{x}\right)\right] \left[\boldsymbol{A}^{(k)}\right]^{-1}\right) \boldsymbol{d}^{(k)}\\ &= \boldsymbol{W}\_{\boldsymbol{L}}^{(k)}\left(\mathbf{x}\right) \boldsymbol{d}^{(k)} \end{aligned} \tag{12}$$

where *W*(*k*) *<sup>L</sup>* is a *stencil weights* vector for the differential operator *L* at local system *k*. By reconstructing the PDE operator at the system centrepoint, a relation can be obtained linking the values of *ui* within the local system

$$\begin{split} S\left(\mathfrak{x}\_{centre}\right) &= L\left[\mathfrak{u}^{(k)}\left(\mathfrak{x}\_{centre}\right)\right] \\ &= \mathcal{W}\_{L}^{(k)}\left(\mathfrak{x}\_{centre}\right)d^{(k)}\end{split} \tag{13}$$

Applying the above reconstruction to each local system *k*, a series of *N* simultaneous equations are produced for *ui*, *i* = 1, . . . , *N*, where *N* is the global number of solution centres. In the resulting global linear system the corresponding boundary conditions of the problem have already been imposed, at the local interpolation, in those stencils containing boundary points. The resulting linear system is sparse, with the number of non-zero entries in each row equal to the number of solution centres in the corresponding collocation system. Therefore, if the local system size is kept constant, the method may be scaled efficiently to very large datasets. The method of solution for time dependent problems requires the creation of a modified PDE-operator via a finite difference approximation of the time derivative. The procedure is illustrated here using a Θ-weighted Crank-Nicolson approach, but is easily extensible to any number of finite difference time advancement schemes.

where

rewritten as:

derivatives in the diffusive term:

*∂ ∂xi*

sampled in pointwise fashion.

*<sup>k</sup>* (*T*) *<sup>∂</sup><sup>T</sup> ∂xi* 

used for applied heat-flux boundary conditions:

Neumann condition in Kirchhoff-space (see equation (23)).

1 *cv* (*T*)

*T* is the temperature *ρ* is the material density *Cp* is the specific heat capacity *k* is the thermal conductivity For convenience, the density and specific heat capacity terms may be combined into the

in the governing equation (19). The Kirchhoff transformation is taken as:

to Solve Nonlinear Heat Conduction Problems on Unstructured Datasets

*∂ψ*

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> *<sup>k</sup>* (*ψ*) *cv* (*ψ*)

<sup>=</sup> <sup>1</sup> *cv* (*T*)

<sup>=</sup> <sup>1</sup> *cv* (*T*)

*ψ* (*T*) =

The Kirchhoff integral transformation is applied, in order to reduce the degree of nonlinearity

<sup>287</sup> A Generalised RBF Finite Difference Approach

 *T T*0

By applying the Kirchhoff transformation to the temperature field, equation (19) can be

 *∂*2*ψ ∂x*<sup>2</sup> *i*

The transformation of the nonlinear diffusion term into Kirchhoff space reduces the equation from a strongly nonlinear to a weakly nonlinear form, by removing the multiplication of first

> *k ∂*2*T ∂x*<sup>2</sup> *i* + *dk dT <sup>∂</sup><sup>T</sup> ∂xi*

*∂*2*ψ ∂x*<sup>2</sup> *i*

In order to complete the transformation it is necessary to obtain a closed-form representation of *ψ* (*T*) and its inverse function *T* (*ψ*), as well as for the two functions *k* (*ψ*) and *cv* (*ψ*). If *k* (*T*) and *cv* (*T*) are known analytically and are integrable, it is possible to describe these functions exactly. In other cases the functions must be computed numerically . A procedure for this is suggested in Stevens & Power (2010), which can be applied to functions of *k* and *cv* that are

The surface heat flux *qs* (*x*, *t*) has a corresponding value in Kirchhoff space, which must be

*qs* (*x*, *<sup>t</sup>*) <sup>=</sup> <sup>−</sup>*ni <sup>k</sup>* (*T*) *<sup>∂</sup><sup>T</sup>*

= −*ni*

In this case the transformation to Kirchhoff space removes the nonlinearity in the boundary condition; a nonlinear heat-flux condition in temperature-space is reduced to a linear

The solution to the nonlinear equation (21) requires a process of nonlinear iterations. The solution procedure performs Picard iterations on the Kirchhoff transform variable and functions thereof, in order to iteratively approach the solution at each timestep. An approximation is taken to the Kirchhoff transform variable *ψ*, written henceforth as *ψ*. From

*∂ψ ∂xi* *∂xi*

+ *s*(*xi*, *t*)

<sup>2</sup> 

*k* (*χ*) *dχ* (20)

Strongly nonlinear

Weakly nonlinear

(21)

(22)

(23)

volumetric heat capacity; *cv* (*T*) = *ρ* (*T*) *Cp* (*T*).

For a general initial-boundary value problem

$$\frac{\partial u\left(\mathbf{x},t\right)}{\partial t} = L\left[u\left(\mathbf{x},t\right)\right] + S\_2\left(\mathbf{x},t\right) \quad \text{in } \Omega\\ u\left(\mathbf{x},0\right) \quad = f\left(\mathbf{x}\right) \quad \qquad \text{in } \Omega\end{aligned} \tag{14}$$

$$\begin{aligned} \mu \left( \mathfrak{x}, 0 \right) &= f \left( \mathfrak{x} \right) \\ B \left[ \mu \left( \mathfrak{x}, t \right) \right] &= \mathfrak{g} \left( \mathfrak{x}, t \right) \end{aligned} \qquad \begin{aligned} \text{in } \Omega\\ \text{on } \Gamma \end{aligned}$$

a finite difference approximation is made to the time-derivative

$$\frac{\mu^n - \mu^{n-1}}{\Delta t} = \Theta L \left[ \mu^n \right] + (1 - \Theta) L \left[ \mu^{n-1} \right] + \mathcal{S}\_2 \left( \mathbf{x}, t^n \right) \tag{15}$$

From this approximation, a modified PDE operator is obtained

$$\overline{L}\left[u^{n}\right] = \hat{L}\left[u^{n-1}\right] + \mathcal{S}\_{2}\left(\mathfrak{x}, t^{n}\right) \tag{16}$$

where

$$\begin{aligned} \hat{L} &= 1 - \Theta \Delta t \, L \\ \hat{L} &= 1 + (1 - \Theta) \, \Delta t \, L \end{aligned} \tag{17}$$

The time stepping algorithm implies that the original initial boundary value problem reduces at each time step to the solution of a boundary value problem defined by the non-homogeneous partial differential equation (16), with the non-homogeneous term given in terms of the solution of the problem at the previous time step. As such the solution procedure at each time step is performed in the same way as for the steady problem, using the modified operator, *L*, and the corresponding non-homogeneous term, *L*ˆ *un*−<sup>1</sup> + *S*<sup>2</sup> (*x*, *tn*).

After solution of the global linear system, and using the current collocation systems and updated data-vectors, a reconstruction of *L*ˆ [*un*] at the solution and PDE centres must be made, ready for the next time step. This is done via the creation of further reconstruction arrays

$$\begin{split} \mathcal{L}\left[\boldsymbol{u}^{n}\left(\mathbf{x}\_{i}\right)\right] &= \mathcal{L}\left[\boldsymbol{H}^{(k)}\left(\mathbf{x}\_{i}\right)\right] \boldsymbol{a}^{(k)} \\ &= \mathcal{W}\_{\hat{L}}^{(k)}\left(\mathbf{x}\_{i}\right) \boldsymbol{d}^{(k)} \end{split} \tag{18}$$

for every *xi* at which the reconstruction is required, within local system *k*.

The initial time step is always performed using the Θ = 1 first-order implicit time stepping formulation. Were a value of Θ < 1 to be used, *L u*0 would be required in order to calculate the non-homogeneous term *L*ˆ *u*0 . This quantity is unknown at the initial configuration, and in order to be calculated would require an interpolation of the initial solution field, using a different interpolation system which does not rely on a non-existent previous time step. At subsequent time steps any value of Θ can be chosen without altering the interpolation system.

#### **3. Solution procedure**

The nonlinear heat conduction equation is solved, with an arbitrary body source:

$$\rho\left(T\right)\mathbb{C}\_{p}\left(T\right)\frac{\partial T}{\partial t} = \frac{\partial}{\partial \mathbf{x}\_{l}}\left(k\left(T\right)\frac{\partial T}{\partial \mathbf{x}\_{l}}\right) + \mathbf{s}\left(\mathbf{x}\_{l}, t\right) \tag{19}$$

where

(14)

*<sup>n</sup>*) (15)

*<sup>n</sup>*) (16)

+ *S*<sup>2</sup> (*x*, *tn*).

6 Heat Transfer Book 2

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> *<sup>L</sup>* [*<sup>u</sup>* (*x*, *<sup>t</sup>*)] <sup>+</sup> *<sup>S</sup>*<sup>2</sup> (*x*, *<sup>t</sup>*) in <sup>Ω</sup> *u* (*x*, 0) = *f* (*x*) in Ω *B* [*u* (*x*, *t*)] = *g* (*x*, *t*) on Γ

> *un*−<sup>1</sup>

+ *S*<sup>2</sup> (*x*, *t*

+ *S*<sup>2</sup> (*x*, *t*

*L* = 1 + (1 − Θ) Δ*t L* (17)

*un*−<sup>1</sup> 

. This quantity is unknown at the initial configuration, and

*<sup>L</sup>*<sup>ˆ</sup> (*xi*) *<sup>d</sup>*(*k*) (18)

would be required in order to calculate

+ *s* (*xi*, *t*) (19)

For a general initial-boundary value problem

*∂u* (*x*, *t*)

a finite difference approximation is made to the time-derivative

From this approximation, a modified PDE operator is obtained

operator, *L*, and the corresponding non-homogeneous term, *L*ˆ

formulation. Were a value of Θ < 1 to be used, *L*

the non-homogeneous term *L*ˆ

**3. Solution procedure**

<sup>Δ</sup>*<sup>t</sup>* <sup>=</sup> <sup>Θ</sup>*<sup>L</sup>* [*un*] <sup>+</sup> (<sup>1</sup> <sup>−</sup> <sup>Θ</sup>) *<sup>L</sup>*

 *un*−<sup>1</sup> 

*L* = 1 − ΘΔ*t L*

The time stepping algorithm implies that the original initial boundary value problem reduces at each time step to the solution of a boundary value problem defined by the non-homogeneous partial differential equation (16), with the non-homogeneous term given in terms of the solution of the problem at the previous time step. As such the solution procedure at each time step is performed in the same way as for the steady problem, using the modified

After solution of the global linear system, and using the current collocation systems and updated data-vectors, a reconstruction of *L*ˆ [*un*] at the solution and PDE centres must be made, ready for the next time step. This is done via the creation of further reconstruction arrays

= *W*(*k*)

The initial time step is always performed using the Θ = 1 first-order implicit time stepping

in order to be calculated would require an interpolation of the initial solution field, using a different interpolation system which does not rely on a non-existent previous time step. At subsequent time steps any value of Θ can be chosen without altering the interpolation system.

*<sup>k</sup>* (*T*) *<sup>∂</sup><sup>T</sup> ∂xi* 

 *u*0

*H*(*k*) (*xi*)

 *α*(*k*)

*L* [*un*] = *L*ˆ

*L*ˆ [*u<sup>n</sup>* (*xi*)] = *L*ˆ

The nonlinear heat conduction equation is solved, with an arbitrary body source:

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> *<sup>∂</sup> ∂xi*

for every *xi* at which the reconstruction is required, within local system *k*.

*u*0 

*<sup>ρ</sup>* (*T*) *Cp* (*T*) *<sup>∂</sup><sup>T</sup>*

*<sup>u</sup><sup>n</sup>* <sup>−</sup> *<sup>u</sup>n*−<sup>1</sup>

where

*T* is the temperature *ρ* is the material density *Cp* is the specific heat capacity *k* is the thermal conductivity

For convenience, the density and specific heat capacity terms may be combined into the volumetric heat capacity; *cv* (*T*) = *ρ* (*T*) *Cp* (*T*).

The Kirchhoff integral transformation is applied, in order to reduce the degree of nonlinearity in the governing equation (19). The Kirchhoff transformation is taken as:

$$
\psi\left(T\right) = \int\_{T\_0}^{T} k\left(\chi\right)d\chi\tag{20}
$$

By applying the Kirchhoff transformation to the temperature field, equation (19) can be rewritten as:

$$\frac{\partial \psi}{\partial t} = \frac{k \left(\psi\right)}{c\_{\upsilon} \left(\psi\right)} \left(\frac{\partial^2 \psi}{\partial x\_i^2} + s \left(x\_i, t\right)\right) \tag{21}$$

The transformation of the nonlinear diffusion term into Kirchhoff space reduces the equation from a strongly nonlinear to a weakly nonlinear form, by removing the multiplication of first derivatives in the diffusive term:

$$\begin{split} \frac{1}{\mathbf{c}\_{\upsilon}(T)} \frac{\partial}{\partial \mathbf{x}\_{i}} \left( k(T) \frac{\partial T}{\partial \mathbf{x}\_{i}} \right) &= \frac{1}{\mathbf{c}\_{\upsilon}(T)} \left[ k \frac{\partial^{2} T}{\partial \mathbf{x}\_{i}^{2}} + \frac{dk}{dT} \left( \frac{\partial T}{\partial \mathbf{x}\_{i}} \right)^{2} \right] & \text{Strongly nonlinear} \\ &= \frac{1}{\mathbf{c}\_{\upsilon}(T)} \frac{\partial^{2} \psi}{\partial \mathbf{x}\_{i}^{2}} & \text{Weakly nonlinear} \end{split} \tag{22}$$

In order to complete the transformation it is necessary to obtain a closed-form representation of *ψ* (*T*) and its inverse function *T* (*ψ*), as well as for the two functions *k* (*ψ*) and *cv* (*ψ*). If *k* (*T*) and *cv* (*T*) are known analytically and are integrable, it is possible to describe these functions exactly. In other cases the functions must be computed numerically . A procedure for this is suggested in Stevens & Power (2010), which can be applied to functions of *k* and *cv* that are sampled in pointwise fashion.

The surface heat flux *qs* (*x*, *t*) has a corresponding value in Kirchhoff space, which must be used for applied heat-flux boundary conditions:

$$q\_s \left( \mathbf{x}\_\prime \, t \right) = -n\_i k \left( T \right) \frac{\partial T}{\partial \mathbf{x}\_i}$$

$$= -n\_i \frac{\partial \Psi}{\partial \mathbf{x}\_i} \tag{23}$$

In this case the transformation to Kirchhoff space removes the nonlinearity in the boundary condition; a nonlinear heat-flux condition in temperature-space is reduced to a linear Neumann condition in Kirchhoff-space (see equation (23)).

The solution to the nonlinear equation (21) requires a process of nonlinear iterations. The solution procedure performs Picard iterations on the Kirchhoff transform variable and functions thereof, in order to iteratively approach the solution at each timestep. An approximation is taken to the Kirchhoff transform variable *ψ*, written henceforth as *ψ*. From

with *α* and *λ* as arbitrary scalar parameters, the governing equation (19) becomes

*T* (*x*, *t*) = *r*

 1 + *αe* −*λt* 

<sup>289</sup> A Generalised RBF Finite Difference Approach

At the origin, the value of the thermal conductivity becomes zero, and as such the solution

The computational domain is considered to be a cuboid, represented by *x* ∈ [*xmin*, *xmax*], *y* ∈ [*ymin*, *ymax*], *z* ∈ [*zmin*, *zmax*]. The analytical solution field (27) is applied at the *xmax*, *ymax* and *zmax* boundaries as a Dirichlet boundary condition. The gradient of the analytical

where *ni* represents the surface normal at the boundary. At the locations where the Neumann boundaries converge, each of the converging temperature gradient conditions is imposed simultaneously, taking advantage of the double collocation property of the RBF Hermitian method (see LaRocca & Power (2007)). The initial condition is obtained by applying the

Two separate domains are examined. The first domain takes *xmin* = *ymin* = *zmin* = 0, and *xmax* = *ymax* = *zmax* = 2. As such, the singularity at the origin is included within each of the three Neumann boundaries of the problem. However, the normal temperature gradient at each of these three Neumann surfaces is considered to be zero at the origin, where the singularity lies. This can be considered valid, since the temperature gradient is analytically zero over each of these three surfaces in the limit *r* → 0. The second domain takes *xmin* = *ymin* = *zmin* = 1, and *xmax* = *ymax* = *zmax* = 3, leading to a non singular solution throughout the domain, and a non-zero temperature gradient over each of the Neumann boundaries. The two domains described above will be henceforth referred to as the *singular* and the *translated*

The solution domain is discretised using (11 × 11 × 11) uniformly distributed solution and boundary centres. Local systems are formed by connecting each of the solution centres to the solution or boundary centre within a single Cartesian index of itself, leading to a stencil of 27 solution or boundary centres. PDE centres are placed at every Cartesian half-index, leading to 8 PDE centres within each local stencil. A dimensionless shape parameter of value *c*∗ = 5 is used throughout. The value of the parameters *α* and *λ* are both taken to be 0.5. For the transient case, local systems are reformed after every timestep. For the steady case local systems are reformed after every nonlinear iteration. In both cases, the convergence parameter is taken to be *�NL* = 10−<sup>6</sup> in the *L*<sup>∞</sup> norm. For the transient case a timestep of size Δ*t* = 0.01

Figure 3 shows the variation of the *L*<sup>2</sup> solution error with time, for the singular case, using three different time advancement schemes. In each case the profile of the error appears similar. The error rises rapidly from the initial condition, reaching its maximum value at around *t* = 0.6. From here the error decreases towards a minimum value at around *t* = 5, before rising again towards a steady value. The second-order Θ = 0.5 time advancement scheme offers the

<sup>=</sup> *ni xi r* 1 + *αe* −*λt* 

<sup>−</sup> <sup>3</sup> <sup>−</sup> (<sup>6</sup> <sup>+</sup> *<sup>λ</sup>r*) *<sup>e</sup>*−*λ<sup>t</sup>* <sup>−</sup> <sup>3</sup>*e*−2*λ<sup>t</sup>* (26)

(27)

(28)

 *<sup>T</sup> <sup>∂</sup><sup>T</sup> ∂xi* 

to Solve Nonlinear Heat Conduction Problems on Unstructured Datasets

temperature field is imposed at the *xmin*, *ymin* and *zmin* boundaries, i.e.

*ni ∂T ∂xi*

analytical solution (27) over the interior of the solution domain, at *t* = 0.

is used, with a variety of time advancement schemes.

*∂T <sup>∂</sup><sup>t</sup>* <sup>=</sup> *<sup>∂</sup> ∂xi*

which has an analytical solution given by:

becomes singular at this location.

domains respectively.

Fig. 2. Schematic description of the nonlinear solution procedure

this, the current guess-values of *k* = *k ψ* and *<sup>c</sup><sup>v</sup>* <sup>=</sup> *cv ψ* can be obtained. Within the nonlinear iteration these quantities are considered constant, leading to the linearised equation:

$$\frac{\partial \widetilde{\psi}}{\partial t} = \frac{\widetilde{k}}{\widetilde{c}\_{\upsilon}} \left( \frac{\partial^2 \widetilde{\psi}}{\partial \mathbf{x}\_i \partial \mathbf{x}\_j} + s \left( \mathbf{x}\_{i\prime}, t \right) \right) \tag{24}$$

This equation can be solved directly, using the procedure outlined in section 2. Convergence is examined by comparing the maximum change in *ψ* over the nonlinear iteration against a user specified convergence parameter *�NL*. Once convergence has been reached, the *L*ˆ *ψn* quantity (see equation (16)) can be reconstructed ready for the next time step. This procedure is summarised in Figure 2. The procedure for steady problems is similar, except for the absence of the time-advancement loop and the *L*ˆ *ψn* reconstruction.

#### **4. Validation test case**

To validate the implementation of the HRBF-FD method, the unsteady nonlinear heat conduction equation (19) is solved in 3D over a domain, using hypothetical values for the thermal properties and source term. Taking

$$\begin{aligned} k\left(T\right) &= T\\ c\_{\upsilon}\left(T\right) &= 1\\ s\left(\mathbf{x}, t\right) &= -\mathbf{3} - \left(\mathbf{6} + \lambda r\right)e^{-\lambda t} - \mathbf{3}e^{-2\lambda t} \end{aligned} \tag{25}$$

with *α* and *λ* as arbitrary scalar parameters, the governing equation (19) becomes

$$\frac{\partial T}{\partial t} = \frac{\partial}{\partial \mathbf{x}\_l} \left( T \frac{\partial T}{\partial \mathbf{x}\_l} \right) - 3 - \left( 6 + \lambda r \right) e^{-\lambda t} - 3e^{-2\lambda t} \tag{26}$$

which has an analytical solution given by:

8 Heat Transfer Book 2

Obtain *k*

<sup>~</sup> and *<sup>v</sup> <sup>c</sup>* ~ at

required locations Update boundary operator values, if required

*n* = *n* +1 *<sup>n</sup> t* = *t* + Δ*t*

Solve linearised equation for ψ

~

2 <sup>2</sup> ψ ψ

> ~ converged to within *NL* ε ?

> > Yes

Reconstruct [ ]*<sup>n</sup> <sup>L</sup>*ˆψ at solution and PDE centres

and *<sup>c</sup><sup>v</sup>* <sup>=</sup> *cv*

+ *s* (*xi*, *t*)

nonlinear iteration these quantities are considered constant, leading to the linearised equation:

This equation can be solved directly, using the procedure outlined in section 2. Convergence is examined by comparing the maximum change in *ψ* over the nonlinear iteration against a user specified convergence parameter *�NL*. Once convergence has been reached, the *L*ˆ

quantity (see equation (16)) can be reconstructed ready for the next time step. This procedure is summarised in Figure 2. The procedure for steady problems is similar, except for the

To validate the implementation of the HRBF-FD method, the unsteady nonlinear heat conduction equation (19) is solved in 3D over a domain, using hypothetical values for the

*<sup>s</sup>*(*x*, *<sup>t</sup>*) <sup>=</sup> <sup>−</sup><sup>3</sup> <sup>−</sup> (<sup>6</sup> <sup>+</sup> *<sup>λ</sup>r*) *<sup>e</sup>*−*λ<sup>t</sup>* <sup>−</sup> <sup>3</sup>*<sup>e</sup>*

*ψn*

 *ψ*

reconstruction.

*cv* (*T*) = 1 (25)

−2*λt*

can be obtained. Within the

(24)

*ψn*

Set ψ ψ ~ =*<sup>n</sup>*

⎣ <sup>⎡</sup> <sup>+</sup> <sup>∂</sup> <sup>∂</sup> <sup>=</sup> <sup>∂</sup> <sup>∂</sup> *<sup>s</sup> <sup>x</sup> <sup>t</sup> <sup>c</sup> <sup>x</sup> k t <sup>i</sup> v i*

~ ~ ~

Has ψ

No

 *ψ*

> *<sup>∂</sup>*2*<sup>ψ</sup> ∂xi∂xj*

Fig. 2. Schematic description of the nonlinear solution procedure

*∂ψ <sup>∂</sup><sup>t</sup>* <sup>=</sup> *<sup>k</sup> cv*

*k* (*T*) = *T*

this, the current guess-values of *k* = *k*

absence of the time-advancement loop and the *L*ˆ

thermal properties and source term. Taking

**4. Validation test case**

~ :

( )⎥ ⎦

,

<sup>⎤</sup> <sup>⎢</sup>

Transform to ( ) <sup>0</sup> <sup>0</sup> ψ =ψ *<sup>T</sup>* and set ~ <sup>0</sup> ψ =ψ

Provide initial *T* - data ( *n* = 0 )

$$T\left(\mathbf{x},t\right) = r\left(1+\alpha e^{-\lambda t}\right) \tag{27}$$

At the origin, the value of the thermal conductivity becomes zero, and as such the solution becomes singular at this location.

The computational domain is considered to be a cuboid, represented by *x* ∈ [*xmin*, *xmax*], *y* ∈ [*ymin*, *ymax*], *z* ∈ [*zmin*, *zmax*]. The analytical solution field (27) is applied at the *xmax*, *ymax* and *zmax* boundaries as a Dirichlet boundary condition. The gradient of the analytical temperature field is imposed at the *xmin*, *ymin* and *zmin* boundaries, i.e.

$$m\_{\bar{l}} \frac{\partial T}{\partial \mathbf{x}\_{\bar{l}}} = \frac{n\_{\bar{l}} \cdot \mathbf{x}\_{\bar{l}}}{r} \left(1 + \alpha e^{-\lambda t}\right) \tag{28}$$

where *ni* represents the surface normal at the boundary. At the locations where the Neumann boundaries converge, each of the converging temperature gradient conditions is imposed simultaneously, taking advantage of the double collocation property of the RBF Hermitian method (see LaRocca & Power (2007)). The initial condition is obtained by applying the analytical solution (27) over the interior of the solution domain, at *t* = 0.

Two separate domains are examined. The first domain takes *xmin* = *ymin* = *zmin* = 0, and *xmax* = *ymax* = *zmax* = 2. As such, the singularity at the origin is included within each of the three Neumann boundaries of the problem. However, the normal temperature gradient at each of these three Neumann surfaces is considered to be zero at the origin, where the singularity lies. This can be considered valid, since the temperature gradient is analytically zero over each of these three surfaces in the limit *r* → 0. The second domain takes *xmin* = *ymin* = *zmin* = 1, and *xmax* = *ymax* = *zmax* = 3, leading to a non singular solution throughout the domain, and a non-zero temperature gradient over each of the Neumann boundaries. The two domains described above will be henceforth referred to as the *singular* and the *translated* domains respectively.

The solution domain is discretised using (11 × 11 × 11) uniformly distributed solution and boundary centres. Local systems are formed by connecting each of the solution centres to the solution or boundary centre within a single Cartesian index of itself, leading to a stencil of 27 solution or boundary centres. PDE centres are placed at every Cartesian half-index, leading to 8 PDE centres within each local stencil. A dimensionless shape parameter of value *c*∗ = 5 is used throughout. The value of the parameters *α* and *λ* are both taken to be 0.5. For the transient case, local systems are reformed after every timestep. For the steady case local systems are reformed after every nonlinear iteration. In both cases, the convergence parameter is taken to be *�NL* = 10−<sup>6</sup> in the *L*<sup>∞</sup> norm. For the transient case a timestep of size Δ*t* = 0.01 is used, with a variety of time advancement schemes.

Figure 3 shows the variation of the *L*<sup>2</sup> solution error with time, for the singular case, using three different time advancement schemes. In each case the profile of the error appears similar. The error rises rapidly from the initial condition, reaching its maximum value at around *t* = 0.6. From here the error decreases towards a minimum value at around *t* = 5, before rising again towards a steady value. The second-order Θ = 0.5 time advancement scheme offers the

best accuracy overall, with the first-order Θ = 1 scheme providing the least accurate solution for most runtimes. The Θ = 0.65 mixed scheme offers accuracy intermediate to the other two

<sup>291</sup> A Generalised RBF Finite Difference Approach

to Solve Nonlinear Heat Conduction Problems on Unstructured Datasets

Figure 4 shows the equivalent error variations for the translated case. Here the accuracy is significantly improved for each of the time advancement schemes, in comparison to the singular case. This is most likely a consequence of the singularity not being present within the solution domain; with the singular case the maximum error is always found at the solution centre closest to the singularity, whereas with the translated case the maximum error location may change as the solution progresses. Once again, the Θ = 0.5 case provides the most accurate solution and the Θ = 1 scheme the least accurate. The error profile appears similar to the singular case, with the main difference being that the peak error is achieved at a much earlier runtime (around *t* = 0.2). In both the singular and translated cases, the solution is

replicated to a high degree of accuracy throughout the time advancement procedure.

solver begins with an initial guess of *T* (*x*) = 0.

optimisation procedure are given in Stevens et al. (2011).

**5. Phase change example**

processes challenging.

When the steady solution is obtained directly, using the steady solution procedure, the *L*<sup>2</sup> error at the solution centres is 1.49 <sup>×</sup> <sup>10</sup>−<sup>3</sup> for the singular case, and 3.44 <sup>×</sup> <sup>10</sup>−<sup>4</sup> for the translated case. Therefore it appears that approaching the steady solution using any of the transient solution schemes offers a higher degree of accuracy than can be achieved by using the steady solution procedure, when a consistent shape parameter value is used. This is likely a consequence of providing an accurate initial condition to the transient solver. The steady

To demonstrate the capability of the method to handle rapid changes in thermal properties, the freezing of mashed potato is considered. The functions for heat capacity and thermal conductivity typically vary rapidly during phase-change, which leads to strong nonlinearity in the PDE governing equation. In this case, a piecewise-linear approximation is taken to the

The thermal properties for different foodstuffs may vary significantly, however they all share common features (see Figure 5). As their temperature is reduced they go from an unfrozen "liquid" state, through a transitional state, to a fully frozen state at some temperature several degrees below zero. During the transition zone the thermal conductivity changes significantly, and a large spike is observed in the heat capacity, representing the latent heat of fusion. The rapid change in the magnitude of the heat capacity makes the accurate simulation of freezing

Experiments performed at the University of Palermo, Dipartimento di Ricerche Energetiche ed Ambientali, provided data for the freezing of a hemispherical sample of mashed potato. The experiment was then replicated numerically, adjusting the functions for *k* and *cv* using the piecewise linear approximations described above in order to better represent the experimental data. More detail on the experimental setup, the functional parameterisation, and the

To model the freezing process, a 3D hemispherical dataset was created. The dataset is represented in Figure 6, and consists of an unstructured, though fairly regular, distribution of 3380 nodes in total. The base surface of the hemisphere consists of 367 nodes, and at these locations a zero heat-flux boundary condition is applied, representing contact with the insulating material beneath the sample. The upper surface of the hemisphere consists of 1164 nodes, over which a time-varying temperature profile is enforced, as obtained from the (smoothed) experimental results. Additionally, 66 nodes are present along the base edge,

thermal properties in order to facilitate their tuning to experimental results.

schemes.

Fig. 3. Variation of *L*<sup>2</sup> error norm with runtime : Singular case

Fig. 4. Variation of *L*<sup>2</sup> error with runtime : Translated case

best accuracy overall, with the first-order Θ = 1 scheme providing the least accurate solution for most runtimes. The Θ = 0.65 mixed scheme offers accuracy intermediate to the other two schemes.

Figure 4 shows the equivalent error variations for the translated case. Here the accuracy is significantly improved for each of the time advancement schemes, in comparison to the singular case. This is most likely a consequence of the singularity not being present within the solution domain; with the singular case the maximum error is always found at the solution centre closest to the singularity, whereas with the translated case the maximum error location may change as the solution progresses. Once again, the Θ = 0.5 case provides the most accurate solution and the Θ = 1 scheme the least accurate. The error profile appears similar to the singular case, with the main difference being that the peak error is achieved at a much earlier runtime (around *t* = 0.2). In both the singular and translated cases, the solution is replicated to a high degree of accuracy throughout the time advancement procedure.

When the steady solution is obtained directly, using the steady solution procedure, the *L*<sup>2</sup> error at the solution centres is 1.49 <sup>×</sup> <sup>10</sup>−<sup>3</sup> for the singular case, and 3.44 <sup>×</sup> <sup>10</sup>−<sup>4</sup> for the translated case. Therefore it appears that approaching the steady solution using any of the transient solution schemes offers a higher degree of accuracy than can be achieved by using the steady solution procedure, when a consistent shape parameter value is used. This is likely a consequence of providing an accurate initial condition to the transient solver. The steady solver begins with an initial guess of *T* (*x*) = 0.

#### **5. Phase change example**

10 Heat Transfer Book 2

0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

t

θ = 1.0 θ = 0.65 θ = 0.5

θ = 1.0 θ = 0.65 θ = 0.5

t

1.0E-05

1.0E-05

1.0E-04

1.0E-03

L2 Error Norm

1.0E-02

Fig. 3. Variation of *L*<sup>2</sup> error norm with runtime : Singular case

Fig. 4. Variation of *L*<sup>2</sup> error with runtime : Translated case

1.0E-04

1.0E-03

L2 Error Norm

1.0E-02

To demonstrate the capability of the method to handle rapid changes in thermal properties, the freezing of mashed potato is considered. The functions for heat capacity and thermal conductivity typically vary rapidly during phase-change, which leads to strong nonlinearity in the PDE governing equation. In this case, a piecewise-linear approximation is taken to the thermal properties in order to facilitate their tuning to experimental results.

The thermal properties for different foodstuffs may vary significantly, however they all share common features (see Figure 5). As their temperature is reduced they go from an unfrozen "liquid" state, through a transitional state, to a fully frozen state at some temperature several degrees below zero. During the transition zone the thermal conductivity changes significantly, and a large spike is observed in the heat capacity, representing the latent heat of fusion. The rapid change in the magnitude of the heat capacity makes the accurate simulation of freezing processes challenging.

Experiments performed at the University of Palermo, Dipartimento di Ricerche Energetiche ed Ambientali, provided data for the freezing of a hemispherical sample of mashed potato. The experiment was then replicated numerically, adjusting the functions for *k* and *cv* using the piecewise linear approximations described above in order to better represent the experimental data. More detail on the experimental setup, the functional parameterisation, and the optimisation procedure are given in Stevens et al. (2011).

To model the freezing process, a 3D hemispherical dataset was created. The dataset is represented in Figure 6, and consists of an unstructured, though fairly regular, distribution of 3380 nodes in total. The base surface of the hemisphere consists of 367 nodes, and at these locations a zero heat-flux boundary condition is applied, representing contact with the insulating material beneath the sample. The upper surface of the hemisphere consists of 1164 nodes, over which a time-varying temperature profile is enforced, as obtained from the (smoothed) experimental results. Additionally, 66 nodes are present along the base edge,

Fig. 6. Computational dataset; boundary and solution centres

to Solve Nonlinear Heat Conduction Problems on Unstructured Datasets

0 5000 10000 15000 20000 25000 30000

<sup>293</sup> A Generalised RBF Finite Difference Approach

Experimental Numerical

Time (seconds)

lower shape parameters. In this case, a shape parameter of *c*<sup>∗</sup> ≥ 2 can lead to instability in

By adjusting eight parameters defining the thermal property functions, it is possible to achieve a good representation of the experimental data. Figure (7) represents the predicted temperature profile at the centre of the base of the hemisphere, compared with the experimental data. The agreement between computational and experimental results is excellent, until around *t* = 18000. At this point, the experimental results show a relatively

Fig. 7. Comparison of numerical and experimental temperature profiles at the core

some configurations of the thermal properties shown in Figure 5.




00.0

Temperature (C)

10.0

20.0

30.0

(c) Variation of *α*

Fig. 5. Typical variation of thermal properties with temperature (food freezing case)

where the top surface meets the bottom surface. Over these nodes, both boundary conditions are enforced simultaneously, taking advantage of the double collocation property of the local Hermite collocation method.

The local system size varies slightly, however the modal number of boundary and solution centres present in each local system is 14. Additionally, PDE centres are added to each local system. A tetrahedralisation is performed on each local system, using the boundary and solution centres as nodes, with PDE centres placed at the centre of each resulting tetrahedron. The modal number of PDE centres present in each local system is 24. It is important to note that the tetrahedralisation is performed only to provide suitable staggered locations for the PDE centres, and plays no part in the actual solution procedure, which is entirely meshless. Since the tetrahedralisation is local, it may be performed very cheaply. It is also possible to collocate the PDE centres with the solution and boundary centres, however previous research (see Stevens et al. (2009)) indicates that a staggered placement leads to the most accurate results in the majority of cases.

The simulation is performed using a second-order Crank-Nicholson implicit time advancement scheme, and a timestep of size 50 seconds. The nonlinear convergence parameter is set to *�NL* = 10−5. The shape parameter is taken as *c*<sup>∗</sup> = 1.0; significantly lower than in the validation example of section 4. It is typical among RBF methods that cases involving irregular datasets and rapid variations in governing properties will tend to favour

12 Heat Transfer Book 2

0


(c) Variation of *α*

where the top surface meets the bottom surface. Over these nodes, both boundary conditions are enforced simultaneously, taking advantage of the double collocation property of the local

The local system size varies slightly, however the modal number of boundary and solution centres present in each local system is 14. Additionally, PDE centres are added to each local system. A tetrahedralisation is performed on each local system, using the boundary and solution centres as nodes, with PDE centres placed at the centre of each resulting tetrahedron. The modal number of PDE centres present in each local system is 24. It is important to note that the tetrahedralisation is performed only to provide suitable staggered locations for the PDE centres, and plays no part in the actual solution procedure, which is entirely meshless. Since the tetrahedralisation is local, it may be performed very cheaply. It is also possible to collocate the PDE centres with the solution and boundary centres, however previous research (see Stevens et al. (2009)) indicates that a staggered placement leads to the most accurate

The simulation is performed using a second-order Crank-Nicholson implicit time advancement scheme, and a timestep of size 50 seconds. The nonlinear convergence parameter is set to *�NL* = 10−5. The shape parameter is taken as *c*<sup>∗</sup> = 1.0; significantly lower than in the validation example of section 4. It is typical among RBF methods that cases involving irregular datasets and rapid variations in governing properties will tend to favour

Fig. 5. Typical variation of thermal properties with temperature (food freezing case)

Temperature (Celcius)


Temperature (Celcius) (b) Variation of *Cp*

20

40

60

Specific heat Capacity (KJ / Kg.K

80

100

0

Hermite collocation method.

results in the majority of cases.


(a) Variation of *k*

Temperature (Celcius)

Thermal diffusivity (m2/s)

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012

0.5

1

Thermal Conductivity (W/m.K

1.5

2

Fig. 6. Computational dataset; boundary and solution centres

Fig. 7. Comparison of numerical and experimental temperature profiles at the core

lower shape parameters. In this case, a shape parameter of *c*<sup>∗</sup> ≥ 2 can lead to instability in some configurations of the thermal properties shown in Figure 5.

By adjusting eight parameters defining the thermal property functions, it is possible to achieve a good representation of the experimental data. Figure (7) represents the predicted temperature profile at the centre of the base of the hemisphere, compared with the experimental data. The agreement between computational and experimental results is excellent, until around *t* = 18000. At this point, the experimental results show a relatively

**7. References**

11: 253–270.

33: 127–159.

55: 1950–1960.

Holman, J. (2002). *Heat Transfer*, McGraw-Hill.

*Transfer* 129: 109–113.

Bejan, A. (1993). *Heat Transfer*, John Wiley and Sons.

to Solve Nonlinear Heat Conduction Problems on Unstructured Datasets

Baxter, B. (2002). Preconditioned conjugate gradients, radial basis functions, and toeplitz

<sup>295</sup> A Generalised RBF Finite Difference Approach

Beatson, R., Cherrie, J. & Mouat, C. (1999). Fast fitting of radial basis functions: Methods

Brown, D. (2005). On approximate cardinal preconditioning methods for solving pdes with radial basis functions, *Engineering Analysis with Boundary Elements* 29: 343–353. Chantasiriwan, S. (2007). Multiquadric collocation method for time-dependent heat

Divo, E. & Kassab, A. (2007). An efficient localised radial basis function meshless method for fluid flow and conjugate heat transfer, *Journal of heat transfer* 129: 124–136. Driscoll, T. & Fornberg, B. (2002). Interpolation in the limit of increasingly flat radial basis

Fasshauer, G. (1999). Solving differential equations with radial basis functions: multilevel methods and smoothing, *Advances in Computational Mathematics* 11: 139–159. Hernandez Rosales, A. & Power, H. (2007). Non-overlapping domain decomposition

LaRocca, A. & Power, H. (2007). A double boundary collocation Hermitian approach for

Lee, C., Liu, X. & Fan, S. (2003). Local multiquadric approximation for solving boundary value

Ling, L. & Kansa, E. (2005). A least-squares preconditioner for radial basis functions collocation methods, *Advances in Computational Mathematics* 23: 31–54. Ling, L., Opfer, R. & Schaback, R. (2006). Results on meshless collocation techniques,

Ling, L. & Schaback, R. (2004). On adaptive unsymmetric meshless collocation, *in* S. Atluri

Madych, W. R. & Nelson, S, A. (1990). Multivariate interpolation and conditionally positive

Sarler, B. & Vertnik, R. (2006). Meshless explicit local radial basis function collocation methods

Schaback, R. (1995). Multivariate interpolation and approximation by translates of a basis

Schaback, R. (1997). On the efficiency of interpolation by radial basis functions, *in*

Shen, W. & Han, S. (2002). An explicit TVD scheme for hyperbolic heat conduction in complex

geometry, *Numerical Heat Transfer, Part B: Fundamentals* 41: 565–590.

A. LeMehaute, C.Rabut & L.L.Schumaker (eds), *Surface fitting and multiresolution*

& A. Tadeu (eds), *Proceedings of the 2004 international conference on computational and*

functions, *Computers and mathematics with applications* 43: 413–422.

Kreith, F. & Bohn, M. (2000). *Principles of Heat Transfer (sixth edition)*, Brookes/Cole.

problems, *Computational Mechanics* 30: 396–409.

*Engineering Analysis with Boundary Elements* 30: 247–253.

*experimental engineering and sciences*, Tech Science Press.

definite functions ii, *Mathematics of Computation* 54: 211–230.

for diffusion problems, *Comput. Math. Appl.* 51: 1269–1282.

*methods*, Vanderbilt University Press, pp. 309–318.

function, *Approximation Theory VIII*.

based on preconditioned gmres iteration, *Advances in Computational Mathematics*

conduction problems with temperature-dependent thermal properties, *Journal of Heat*

algorithm for the hermite radial basis function meshless collocation approach: applications to convection diffusion problems, *Journal of Algorithms and Technology*

the solution of steady state convection diffusion problems, *Comput. Math. Appl.*

matrices, *Computers and mathematics with applications* 43: 305–318.

gradual drop in temperature between *Tmax* <sup>=</sup> <sup>−</sup>1.5*oC* and *Ts* <sup>=</sup> <sup>−</sup>4.5*oC*, occurring between *t* = 18000 and *t* = 18500. In contrast, the numerical results predict a near-instantaneous drop in temperature, from *Tmax* down to well below *Ts*, at a slightly later time. This "sudden dropoff" behaviour was replicated across a wide range of thermal parameters, and could represent a limitation in the piecewise-linear approximation to the thermal properties.

The local Hermitian method was able to produce stable results using a wide range of thermal parameters, and convergence at each timestep was typically relatively fast. The size and intensity of the spike in the function for *cp* (see Figure 5(b)) is the feature that has most impact upon numerical stability. By increasing the height of the spike sufficiently, it is possible to find configurations where the method is unstable at any shape parameter. This is not unexpected, as an increasingly sharp spike will represent increasingly strong nonlinearities in the governing equation (19), within the phase transition zone. Tests were performed using stencil configurations without PDE centres, i.e. without the "implicit upwinding" feature. However, it was not found to be possible to obtain a stable solution for spikes of intensity close to that which was required to match the experimental results. That inclusion of PDE centres provides a stabilising effect has previously been demonstrated for convection-diffusion problems Stevens et al. (2009), and the stabilising effect appears to be present here also.

#### **6. Discussion**

The use of local radial basis function methods in finite difference mode (HRBF-FD) appears to be a viable option for the simulation of nonlinear heat conduction processes, particularly when irregular datasets are required. Traditional polynomial-based finite difference methods are difficult to implement on irregular datasets, and RBF collocation allows a natural generalisation of the principle to irregular data. The inclusion of arbitrary boundary operators within the local collocation systems allows the flexibility to enforce a wide variety of boundary conditions, and the double-collocation property of the Hermitian RBF formulation allows multiple boundary operators to be enforced at a single location where required (such as on converging boundaries).

The inclusion of the governing PDE operator within the local collocation systems is optional, but when present introduces an "implicit upwinding" effect, which stabilises the solution and improves accuracy, at the expense of larger local systems and hence higher computational cost (discussed further in Stevens et al. (2009)). The stabilisation effect is similar to that of stencil-based upwinding, but operates on a centrally defined stencil. Therefore, the HRBF-FD method may be of benefit to problems which may otherwise require upwinding schemes, in particular with unstructured datasets, where the selection of appropriate upwinding stencils may be particularly challenging.

The application of the Kirchhoff transformation greatly simplifies the PDE governing equation and linearises heat-flux boundary conditions, at the cost of requiring thermal property functions to be transformed to Kirchhoff space. Using this Kirchhoff formulation the HRBF-FD method is able to solve a benchmark heat transfer problem to a high degree of accuracy, using both steady and transient solution procedures. Additionally, the method was able to produce stable results for a phase change model involving the freezing of food, in the presence of strongly varying thermal properties. By tuning the thermal properties it was possible to replicate the experimental data to a good degree of accuracy, potentially allowing the calibrated thermal properties to be used in further numerical simulations.

#### **7. References**

14 Heat Transfer Book 2

gradual drop in temperature between *Tmax* <sup>=</sup> <sup>−</sup>1.5*oC* and *Ts* <sup>=</sup> <sup>−</sup>4.5*oC*, occurring between *t* = 18000 and *t* = 18500. In contrast, the numerical results predict a near-instantaneous drop in temperature, from *Tmax* down to well below *Ts*, at a slightly later time. This "sudden dropoff" behaviour was replicated across a wide range of thermal parameters, and could represent a limitation in the piecewise-linear approximation to the thermal properties. The local Hermitian method was able to produce stable results using a wide range of thermal parameters, and convergence at each timestep was typically relatively fast. The size and intensity of the spike in the function for *cp* (see Figure 5(b)) is the feature that has most impact upon numerical stability. By increasing the height of the spike sufficiently, it is possible to find configurations where the method is unstable at any shape parameter. This is not unexpected, as an increasingly sharp spike will represent increasingly strong nonlinearities in the governing equation (19), within the phase transition zone. Tests were performed using stencil configurations without PDE centres, i.e. without the "implicit upwinding" feature. However, it was not found to be possible to obtain a stable solution for spikes of intensity close to that which was required to match the experimental results. That inclusion of PDE centres provides a stabilising effect has previously been demonstrated for convection-diffusion problems Stevens et al. (2009), and the stabilising effect appears to be

The use of local radial basis function methods in finite difference mode (HRBF-FD) appears to be a viable option for the simulation of nonlinear heat conduction processes, particularly when irregular datasets are required. Traditional polynomial-based finite difference methods are difficult to implement on irregular datasets, and RBF collocation allows a natural generalisation of the principle to irregular data. The inclusion of arbitrary boundary operators within the local collocation systems allows the flexibility to enforce a wide variety of boundary conditions, and the double-collocation property of the Hermitian RBF formulation allows multiple boundary operators to be enforced at a single location where required (such as on

The inclusion of the governing PDE operator within the local collocation systems is optional, but when present introduces an "implicit upwinding" effect, which stabilises the solution and improves accuracy, at the expense of larger local systems and hence higher computational cost (discussed further in Stevens et al. (2009)). The stabilisation effect is similar to that of stencil-based upwinding, but operates on a centrally defined stencil. Therefore, the HRBF-FD method may be of benefit to problems which may otherwise require upwinding schemes, in particular with unstructured datasets, where the selection of appropriate upwinding stencils

The application of the Kirchhoff transformation greatly simplifies the PDE governing equation and linearises heat-flux boundary conditions, at the cost of requiring thermal property functions to be transformed to Kirchhoff space. Using this Kirchhoff formulation the HRBF-FD method is able to solve a benchmark heat transfer problem to a high degree of accuracy, using both steady and transient solution procedures. Additionally, the method was able to produce stable results for a phase change model involving the freezing of food, in the presence of strongly varying thermal properties. By tuning the thermal properties it was possible to replicate the experimental data to a good degree of accuracy, potentially allowing

the calibrated thermal properties to be used in further numerical simulations.

present here also.

**6. Discussion**

converging boundaries).

may be particularly challenging.


**Part 3** 

**Heat Transfer Analysis** 


## **Part 3**

**Heat Transfer Analysis** 

16 Heat Transfer Book 2

296 Convection and Conduction Heat Transfer

Stevens, D., LaRocca, A., Power, H. & LaRocca, V. (2011). Estimation of temperature evolution

Stevens, D. & Power, H. (2010). A scalable meshless formulation based on RBF Hermitian

Stevens, D., Power, H., Lees, M. & Morvan, H. (2009). The use of PDE centres in the

Wendland, H. (1995). Piecewise polynomial, positive definite and compactly supported radial

Wong, A., Hon, Y., Li, T., S.L.Chung & Kansa, E. (1999). Multizone decomposition for

Wright, G. & Fornberg, B. (2006). Scattered node compact finite difference-type formulas generated from radial basis functions, *Journal of Computational Physics* 212: 99–123. Wu, Z. (1995). Compactly supported positive definite radial basis functions, *Adv. Comput.*

Zhang, Y. (2007). Reconstruct multiscale functions using different RBFs in different

Zhou, X., Hon, Y. & Li, J. (2003). Overlapping domain decomposition method by radial basis

subdomains, *Applied Mathematics and Computation* 189: 893–901.

functions, *Applied Numerical Mathematics* 44: 241–255.

*Journal of Food Engineering* -: (submitted).

*and Mathematics with Applications* 37: 23–43.

*engineering and sciences* 55: 111–145.

228: 4606–4624.

*Math.* 4: 75–97.

during the freezing of foodstuffs using a three-dimensional meshless approach,

interpolation for 3D nonlinear heat conduction problems, *Computer modelling in*

local RBF Hermitian method for 3D Convective-Diffusion problems, *J. Comput. Phys.*

basis functions of minimal degree, *Advances in Computational Mathematics* 4: 389–396.

simulation of time-dependent problems using the multiquadric scheme, *Computers*

**14** 

Rami A. Hawileh

*United Arab Emirates* 

*American University of Sharjah* 

**Heat Transfer Analysis of Reinforced** 

**Concrete Beams Reinforced with GFRP Bars** 

Corrosion of steel reinforcement has been identified as a key factor of deterioration and structural deficiency (Masoudi et al., 2011) in reinforced concrete (RC) structural members. The corrosion state of current RC bridges and high-rise buildings has been a source of concern to designers and engineers. In addition, such structures have been invulnerable to harsh environmental exposures, with little or no maintenance. Furthermore, such structures are experiencing larger amount of loads than their original capacities due to the increase number of users over the years (Bisby, 2003). Several different solutions were proposed to retrofit deteriorated structural members (Masoudi et al., 2011; Hawelih et al., 2011; Al-Tamimi et al., 2011) by replacing cracked concrete, using epoxy injected supplements, and

The use of embedded FRP bar reinforcement seems to be a promising solution (Masoudi et al., 2011; Bisby, 2003; Abbasi & Hogg, 2005; Abbasi & Hogg, 2006; Qu et al., 2009; Aiello & Ombres, 2002) to strengthen structural RC members in flexure and shear. Compared to the conventional reinforcing steel bars, the FRP bars seem to have a high strength to weight ratio, moderate modulus of elasticity and resistance to chemical and electrical corrosion. Although FRP materials were shown to have a brittle failure, due to their natural composition, still if designed properly they can show considerable amount of ductility (Rasheed et al, 2010; De Lorenzis & Teng, 2007). One of the draw backs of using FRP embedded bars is their low glass temperature and tendency to change state; from solid to liquid at elevated temperatures. Hence, the performance of FRP reinforced structural members under elevated temperatures draws many doubts and concerns and warrants further investigation. Few experimental tests have been conducted in the previous years on the fire performance of RC beams reinforced with FRP bars due to the high costs of such tests, tremendous amount of preparation, and shortage of specialized facilities (Franssen et

Sadek et al. (Sadek et al., 2006) conducted a full scale experimental program on the fire resistance of RC beams reinforced with steel and Glass Fibre Reinforced Polymer (GFRP) bars. The test matrix composed of different reinforcing rebars used along with different concrete compressive strengths. The testing took place in a special testing facility and the beams were loaded statically at 60% of their ultimate load capacity during the course of the fire test. The tests followed the ASTM E119 (ASTM E119, 2002) standard and fire curve.

**1. Introduction** 

al., 2009).

FRP externally bonded systems.

### **Heat Transfer Analysis of Reinforced Concrete Beams Reinforced with GFRP Bars**

#### Rami A. Hawileh

*American University of Sharjah United Arab Emirates* 

#### **1. Introduction**

Corrosion of steel reinforcement has been identified as a key factor of deterioration and structural deficiency (Masoudi et al., 2011) in reinforced concrete (RC) structural members. The corrosion state of current RC bridges and high-rise buildings has been a source of concern to designers and engineers. In addition, such structures have been invulnerable to harsh environmental exposures, with little or no maintenance. Furthermore, such structures are experiencing larger amount of loads than their original capacities due to the increase number of users over the years (Bisby, 2003). Several different solutions were proposed to retrofit deteriorated structural members (Masoudi et al., 2011; Hawelih et al., 2011; Al-Tamimi et al., 2011) by replacing cracked concrete, using epoxy injected supplements, and FRP externally bonded systems.

The use of embedded FRP bar reinforcement seems to be a promising solution (Masoudi et al., 2011; Bisby, 2003; Abbasi & Hogg, 2005; Abbasi & Hogg, 2006; Qu et al., 2009; Aiello & Ombres, 2002) to strengthen structural RC members in flexure and shear. Compared to the conventional reinforcing steel bars, the FRP bars seem to have a high strength to weight ratio, moderate modulus of elasticity and resistance to chemical and electrical corrosion. Although FRP materials were shown to have a brittle failure, due to their natural composition, still if designed properly they can show considerable amount of ductility (Rasheed et al, 2010; De Lorenzis & Teng, 2007). One of the draw backs of using FRP embedded bars is their low glass temperature and tendency to change state; from solid to liquid at elevated temperatures. Hence, the performance of FRP reinforced structural members under elevated temperatures draws many doubts and concerns and warrants further investigation. Few experimental tests have been conducted in the previous years on the fire performance of RC beams reinforced with FRP bars due to the high costs of such tests, tremendous amount of preparation, and shortage of specialized facilities (Franssen et al., 2009).

Sadek et al. (Sadek et al., 2006) conducted a full scale experimental program on the fire resistance of RC beams reinforced with steel and Glass Fibre Reinforced Polymer (GFRP) bars. The test matrix composed of different reinforcing rebars used along with different concrete compressive strengths. The testing took place in a special testing facility and the beams were loaded statically at 60% of their ultimate load capacity during the course of the fire test. The tests followed the ASTM E119 (ASTM E119, 2002) standard and fire curve.

Heat Transfer Analysis of Reinforced Concrete Beams Reinforced with GFRP Bars 301

and atoms. On the other hand, convection transfers the heat from the source to the RC beam via cycles of heating and cooling of the surrounding fluids. Radiation is the transfer of heat by electromagnetic waves. The basic one dimensional steady state governing equations for

4

*r te q*′′ =Φε σ

*Φ* is a configuration or view factor depends on the area (A) of the emitting surface and

Furthermore, the three dimensional transient governing heat transfer equation as a function of time is given by Eq. 4. Equation 4 is derived from the Law of Conservation of Energy which states that the total inflow of heat in a unit time across a certain body must be equal to the total outflow per unit time for the same body. It should be noted that Eq. 4 can be solved giving both initial and boundary conditions on a division or all the boundary of the body in question (domain). The initial conditions define the temperature distribution over the domain at the initiation of the heat transfer (i.e. at *t* = 0). The initial and boundary

> 222 222 *TTTT ck k k S t xyz*

> > *c S* ( )( ) *<sup>f</sup> r S <sup>f</sup>*

<sup>∂</sup> ∂∂∂ (4)

*T x*( , , ,0 , , ,0 *y z Tx* ) = <sup>0</sup> ( *y z* ) (5)

∂ (6)

∂∂ ∂ ∂ <sup>=</sup> +++

*<sup>T</sup> k hT T hT T*

*S* is the internally generated heat on unit volume per unit time; *T* is the temperature gradient

*TS* is the temperature of solid surface; *Tf* is the temperature of fluid; *hr* is the radiation heat

*t* is time; *u* is the direction of heat; *hc* is the heat transfer coefficient of solid surface

− = −+ −

*h* is the convective heat transfer coefficient in (W/m2K), typical vale is 25W/m2K *∆T* is the temperature difference between the solid surface and fluid in (°C or K)

*<sup>k</sup> q k dT dx* ′′ = (1)

*<sup>h</sup> q hT* ′′ = Δ (2)

*T* (3)

conduction, convection and radiation are presented in Eq. 1-3, respectively.

where,

 is the density *c* is the specific heat *k* is the conductivity

ρ

where,

transfer coefficient given by Eq. 7

*<sup>k</sup> q*′′ is the heat flux due to conduction *<sup>h</sup> q*′′ is the heat flux due to convection *<sup>r</sup> q*′′ is the heat flux due to radiation

distance (r) to the receiving surface. (Φ = A/πr2) *ε<sup>t</sup>* is the emissivity factor, ranged from 0-1.0

conditions can be given by Eqs. 5 and 6, respectively:

ρ

*u* ∂

*σ* is the Stefan-Boltzmann constant taken as (5.67×10-8 W/m2K4) *Te* is the absolute temperature of the emitting surface (K)

Because of forming of flexure and shear cracks, fire was able to penetrate through the crosssection of the tested beams. The beams with low and normal strength concrete achieved a 30 and 45min fire endurance, respectively. On the other hand, the steel reinforced concrete beam achieved 90min fire endurance. The short fire endurance observed was mainly due to the small concrete cover used to protect the flexural reinforcements.

Abbasi and Hogg (Abbasi & Hogg, 2006) conducted two full scale fire tests on RC beams reinforced with GFRP bars as the main reinforcement having a concrete cover of 75mm. The beams were fully loaded up to 40kN and subjected to the ISO 834 (ISO, 1975) fire standard curve. Eurocode 2 (Eurocode, 1992) and ACI-440 (ACI, 2008) procedures were used to design the beams. The beam reinforced with the steel stirrups achieved a 128min fire endurance while the beam reinforced with GFRP stirrups achieved a 94min fire endurance. Both RC beams limited the mid-span deflection to less than L/20; the deflection limit used in the load bearing capacity of BS 476: Part 20. In addition, the RC beams showed that they can pass the building regulations for fire safety by withstanding the fire test more than 90 min.

Hawileh et al. (Hawileh et al., 2009, 2011) developed FE models that predicted the performance of RC beams strengthened with insulated carbon CFRP plates subjected to bottom and top fire loading. The models predicted with reasonable accuracy the experimental results of Williams et al. (Williams et al., 2008). It was concluded the developed models can serve as a valid alternative tool to expensive experimental testing especially in design oriented parametric studies, to capture the response of such beams when subjected to thermal loading.

Different building codes recommend conducting further experimental and analytical research studies to investigate the thermal effect on RC members strengthened or reinforced with FRP sheets, plates or bars. Such studies would lead to a reduction on the tough restrictions and requirements set by the current codes of practice on the use of FRP materials in building and other types of structures. In addition, such studies would draw a better understanding on the behavior of FRP materials under fire actions that would enhance the available documentation and literature that in turn would encourage designers and engineers to use FRP bars more frequently to reinforce RC structural members.

This chapter aims to develop a 3D nonlinear FE model that can accurately predict the temperature distribution at any location with RC beams reinforced with GFRP bars when exposed to the standard fire curve, ISO 834. The model is validated by comparing the predicted average temperature in the GFRP bars with the measured experimental data obtained by Abbasi and Hogg (Abbasi & Hogg, 2006). The developed FE model incorporates the different thermal nonlinear temperature dependant material properties associated with each material including density, specific heat, and thermal conductivity. Transient thermal analysis was carried out using the available FE code, ANSYS (ANSYS, 2007). The results of the developed FE model showed a good matching with the experimental results at all stages of fire loading. Several other observations and conclusion were drawn based on the results of the developed model.

#### **2. Heat transfer equations**

Heat transfers via the following three methods: Conduction, Convection, and Radiation. They can occur together or individually depending on the heat source exposure and environment. Conduction transfers heat within the RC beam by movement or vibrations of free electrons and atoms. On the other hand, convection transfers the heat from the source to the RC beam via cycles of heating and cooling of the surrounding fluids. Radiation is the transfer of heat by electromagnetic waves. The basic one dimensional steady state governing equations for conduction, convection and radiation are presented in Eq. 1-3, respectively.

$$
\sigma\_k'' = k \, dT \not\!f \, d\mathbf{x} \tag{1}
$$

$$
\sigma\_{\text{lt}}'' = \text{lt}\Delta T \tag{2}
$$

$$q\_r'' = \mathcal{Q} \varepsilon\_t \sigma T\_e^4 \tag{3}$$

where,

300 Convection and Conduction Heat Transfer

Because of forming of flexure and shear cracks, fire was able to penetrate through the crosssection of the tested beams. The beams with low and normal strength concrete achieved a 30 and 45min fire endurance, respectively. On the other hand, the steel reinforced concrete beam achieved 90min fire endurance. The short fire endurance observed was mainly due to

Abbasi and Hogg (Abbasi & Hogg, 2006) conducted two full scale fire tests on RC beams reinforced with GFRP bars as the main reinforcement having a concrete cover of 75mm. The beams were fully loaded up to 40kN and subjected to the ISO 834 (ISO, 1975) fire standard curve. Eurocode 2 (Eurocode, 1992) and ACI-440 (ACI, 2008) procedures were used to design the beams. The beam reinforced with the steel stirrups achieved a 128min fire endurance while the beam reinforced with GFRP stirrups achieved a 94min fire endurance. Both RC beams limited the mid-span deflection to less than L/20; the deflection limit used in the load bearing capacity of BS 476: Part 20. In addition, the RC beams showed that they can pass the building regulations for fire safety by withstanding the fire test more than 90

Hawileh et al. (Hawileh et al., 2009, 2011) developed FE models that predicted the performance of RC beams strengthened with insulated carbon CFRP plates subjected to bottom and top fire loading. The models predicted with reasonable accuracy the experimental results of Williams et al. (Williams et al., 2008). It was concluded the developed models can serve as a valid alternative tool to expensive experimental testing especially in design oriented parametric studies, to capture the response of such beams when subjected to thermal

Different building codes recommend conducting further experimental and analytical research studies to investigate the thermal effect on RC members strengthened or reinforced with FRP sheets, plates or bars. Such studies would lead to a reduction on the tough restrictions and requirements set by the current codes of practice on the use of FRP materials in building and other types of structures. In addition, such studies would draw a better understanding on the behavior of FRP materials under fire actions that would enhance the available documentation and literature that in turn would encourage designers and engineers

This chapter aims to develop a 3D nonlinear FE model that can accurately predict the temperature distribution at any location with RC beams reinforced with GFRP bars when exposed to the standard fire curve, ISO 834. The model is validated by comparing the predicted average temperature in the GFRP bars with the measured experimental data obtained by Abbasi and Hogg (Abbasi & Hogg, 2006). The developed FE model incorporates the different thermal nonlinear temperature dependant material properties associated with each material including density, specific heat, and thermal conductivity. Transient thermal analysis was carried out using the available FE code, ANSYS (ANSYS, 2007). The results of the developed FE model showed a good matching with the experimental results at all stages of fire loading. Several other observations and conclusion were drawn based on the results

Heat transfers via the following three methods: Conduction, Convection, and Radiation. They can occur together or individually depending on the heat source exposure and environment. Conduction transfers heat within the RC beam by movement or vibrations of free electrons

the small concrete cover used to protect the flexural reinforcements.

to use FRP bars more frequently to reinforce RC structural members.

min.

loading.

of the developed model.

**2. Heat transfer equations** 

*<sup>k</sup> q*′′ is the heat flux due to conduction

*<sup>h</sup> q*′′ is the heat flux due to convection

*<sup>r</sup> q*′′ is the heat flux due to radiation

ρis the density

*c* is the specific heat

*k* is the conductivity

*h* is the convective heat transfer coefficient in (W/m2K), typical vale is 25W/m2K

*∆T* is the temperature difference between the solid surface and fluid in (°C or K)

*Φ* is a configuration or view factor depends on the area (A) of the emitting surface and distance (r) to the receiving surface. (Φ = A/πr2)

*ε<sup>t</sup>* is the emissivity factor, ranged from 0-1.0

*σ* is the Stefan-Boltzmann constant taken as (5.67×10-8 W/m2K4)

*Te* is the absolute temperature of the emitting surface (K)

Furthermore, the three dimensional transient governing heat transfer equation as a function of time is given by Eq. 4. Equation 4 is derived from the Law of Conservation of Energy which states that the total inflow of heat in a unit time across a certain body must be equal to the total outflow per unit time for the same body. It should be noted that Eq. 4 can be solved giving both initial and boundary conditions on a division or all the boundary of the body in question (domain). The initial conditions define the temperature distribution over the domain at the initiation of the heat transfer (i.e. at *t* = 0). The initial and boundary conditions can be given by Eqs. 5 and 6, respectively:

$$
\rho c \frac{\partial T}{\partial t} = k \frac{\partial^2 T}{\partial \mathbf{x}^2} + k \frac{\partial^2 T}{\partial y^2} + k \frac{\partial^2 T}{\partial z^2} + \mathbf{S} \tag{4}
$$

$$T\left(\mathbf{x}, y, z, \mathbf{0}\right) = T\_0\left(\mathbf{x}, y, z, \mathbf{0}\right) \tag{5}$$

$$-k\frac{\partial T}{\partial u} = h\_c \left(T\_S - T\_f\right) + h\_r \left(T\_S - T\_f\right) \tag{6}$$

where,

*S* is the internally generated heat on unit volume per unit time; *T* is the temperature gradient *t* is time; *u* is the direction of heat; *hc* is the heat transfer coefficient of solid surface *TS* is the temperature of solid surface; *Tf* is the temperature of fluid; *hr* is the radiation heat transfer coefficient given by Eq. 7

Heat Transfer Analysis of Reinforced Concrete Beams Reinforced with GFRP Bars 303

The fire testing was conducted at the building research establishment (Abbasi & Hogg, 2006). The internal dimensions of the furnace were 4000mm wide, 4000mm long and 2000mm deep. Each side of the furnace contained 10 burners lined in parallel to each other. The top side of the furnace is closed with either the test specimen, or lined with steel cover slabs. On the other hand, the furnace is lined with 1400 grade insulating brick to comply

The developed FE model has the same geometry, material properties, and loading as the tested GFRP-RC beam by (Abbasi & Hogg, 2006). The FE model was developed and simulated using the commercial FE code, ANSYS 11.0 (ANSYS, 2007). Figure 2 shows a detailed view of the developed FE model. To take advantage of the symmetrical nature of the geometry, material properties and heat transfer actions, only one-quarter of the RC beam was modeled. The development of a one-quarter model will still yields the same accuracy as

In order to simulate such complex behavior, an analytical procedure must be determined. Firstly, the different material properties and corresponding constitutive laws were collected from the open literature. Then, the development of the geometry and simulation enviroment was conducted using ANSYS (ANSYS, 2007) were different element types, meshing and simulation techniques were incorporated to simulate the concrete and reinforcing GFRP bars elements. Finally, a 3D transient thermal analysis is conducted to simulate the applied

Symmetry Axis no. 2

Symmetry Axis no. 1

with British Standard and ISO 834 requirements.

the full scale model and saves a lot of computational time.

Concrete

GFRP rebars

(a) Isoperimetric view

**4. Finite element model development** 

ISO 834 fire curve.

$$\mathcal{H}\_r = \sigma \varepsilon\_S \left( T\_s^2 + T\_f^2 \right) \left( T\_S + T\_f \right) \tag{7}$$

where,

*S* εis the emissivity of the surface in question

σis the Stefan-Boltzmann constant 5.669 × 10–8 W/m2K4 (0.1714× 10–8 BTU/hr ft2 R4)

#### **3. Experimental program**

The experimental program of Abbasi and Hogg (Abbasi & Hogg, 2006) is used as a benchmark in this study to validate the accuracy of the developed model. The experimental program (Abbasi & Hogg, 2006) consisted of three RC beams reinforced with GFRP bars. The RC beams were casted using marine siliceous gravel coarse aggregates. Figure 1 shows the cross-section detailing of the tested RC beams. The beams had a height and width of 400 mm and 350 mm, respectively and effective depth of 325 mm. The concrete cover from the beam's soffit to the GFRP flexural reinforcement was 75 mm. The total length of the beam specimens was 4400 mm having an exposed span length of 4250 mm. The beams were reinforced with nine Φ12.7 mm (Area= 1303.6mm2) GFRP bars, seven were placed in two layers at the tension side and two serving as compression reinforcement. In addition, Φ9 mm stirrups were used as shear reinforcements spaced at 160mm center to center. The concrete compressive strength was 42MPa. The first beam specimen was tested under monotonic loading at ambient temperature conditions to serve as a control beam. The other two beams were tested under sustained static and transient fire loading defined according to ISO 834.

Fig. 1. Details of the tested RC beams (Abbasi & Hogg, 2006)

The fire testing was conducted at the building research establishment (Abbasi & Hogg, 2006). The internal dimensions of the furnace were 4000mm wide, 4000mm long and 2000mm deep. Each side of the furnace contained 10 burners lined in parallel to each other. The top side of the furnace is closed with either the test specimen, or lined with steel cover slabs. On the other hand, the furnace is lined with 1400 grade insulating brick to comply with British Standard and ISO 834 requirements.

### **4. Finite element model development**

302 Convection and Conduction Heat Transfer

is the Stefan-Boltzmann constant 5.669 × 10–8 W/m2K4 (0.1714× 10–8 BTU/hr ft2 R4)

b = 350mm

50mm

12.7mm 50mm

main rebars

9mm stirrups

The experimental program of Abbasi and Hogg (Abbasi & Hogg, 2006) is used as a benchmark in this study to validate the accuracy of the developed model. The experimental program (Abbasi & Hogg, 2006) consisted of three RC beams reinforced with GFRP bars. The RC beams were casted using marine siliceous gravel coarse aggregates. Figure 1 shows the cross-section detailing of the tested RC beams. The beams had a height and width of 400 mm and 350 mm, respectively and effective depth of 325 mm. The concrete cover from the beam's soffit to the GFRP flexural reinforcement was 75 mm. The total length of the beam specimens was 4400 mm having an exposed span length of 4250 mm. The beams were reinforced with nine Φ12.7 mm (Area= 1303.6mm2) GFRP bars, seven were placed in two layers at the tension side and two serving as compression reinforcement. In addition, Φ9 mm stirrups were used as shear reinforcements spaced at 160mm center to center. The concrete compressive strength was 42MPa. The first beam specimen was tested under monotonic loading at ambient temperature conditions to serve as a control beam. The other two beams were tested under sustained static and transient fire loading defined according to ISO 834.

σε

is the emissivity of the surface in question

**3. Experimental program** 

h = 400mm

concrete

Fig. 1. Details of the tested RC beams (Abbasi & Hogg, 2006)

where, *S* ε

σ

( )( ) 2 2 *S f rS S <sup>f</sup> h T TTT* = ++

(7)

d = 325mm

The developed FE model has the same geometry, material properties, and loading as the tested GFRP-RC beam by (Abbasi & Hogg, 2006). The FE model was developed and simulated using the commercial FE code, ANSYS 11.0 (ANSYS, 2007). Figure 2 shows a detailed view of the developed FE model. To take advantage of the symmetrical nature of the geometry, material properties and heat transfer actions, only one-quarter of the RC beam was modeled. The development of a one-quarter model will still yields the same accuracy as the full scale model and saves a lot of computational time.

In order to simulate such complex behavior, an analytical procedure must be determined. Firstly, the different material properties and corresponding constitutive laws were collected from the open literature. Then, the development of the geometry and simulation enviroment was conducted using ANSYS (ANSYS, 2007) were different element types, meshing and simulation techniques were incorporated to simulate the concrete and reinforcing GFRP bars elements. Finally, a 3D transient thermal analysis is conducted to simulate the applied ISO 834 fire curve.

(a) Isoperimetric view

Heat Transfer Analysis of Reinforced Concrete Beams Reinforced with GFRP Bars 305

J

(a) SOLID70

(b) LINK33

The thermal material properties are required as inputs to the developed FE model. Table 1 tabulates the thermal properties for the concrete and GFRP bars materials used in this study at room temperatures. Figure 4 shows the thermal concrete material temperature-dependent material properties including the conductivity and normalized density and specific heat.

> Material *Ko (W/mm K) Co (J/kg K) ρ (kg/mm3)*  Concrete 2.7×10-3 722.8 2.32×10-6 GFRP 4.0×10-5 1310 1.60×10-6

1

3

4

N

L

6

P

I

Y

I

Table 1. Thermal material properties at ambient room temperature

Y

X

X

**4.1 Material constitutive models** 

where,

Fig. 3. 3-D Thermal elements (ANSYS, 2007)

*Ko* is the thermal conductivity (W/mm K)

*Co* is the heat capacity (J/kg K) *ρ* is the density (kg/mm3)

Z

Z

M

2

5

M O,P

N

J

Prism Option

J Tetrahedral Option

M.N.O.P

<sup>I</sup> <sup>K</sup> J

Pyramid Option

L

K,L

M.N.O.P

K,L

J

I

I

K

O

(c) Cross-sectional view

Fig. 2. Developed FE model

Different element types were selected from the ANSYS element type selection library. The thermal elements implemented to model the concrete material and reinforcing GFRP rebars were SOLID70 and LINK33, respectively. SOLID70 has eight nodes with a single degree of freedom at each node, defined as temperature as well as 3-D thermal conduction capability. SOLID70 has 2×2×2 integration scheme for both conductivity and specific heat matrices. On the other hand, GFRP rebars were modeled using LINK33. LINK33 is a thermal uniaxial element with the ability to conduct heat between its two nodes. The element has a single degree of freedom SDF, temperature, at each node. In addition, both elements are applicable to conduct 3-D, steady-state and/or transient thermal analysis (ANSYS, 2007). Figure 3 shows the different element types used in this study.

(b) LINK33

Fig. 3. 3-D Thermal elements (ANSYS, 2007)

#### **4.1 Material constitutive models**

The thermal material properties are required as inputs to the developed FE model. Table 1 tabulates the thermal properties for the concrete and GFRP bars materials used in this study at room temperatures. Figure 4 shows the thermal concrete material temperature-dependent material properties including the conductivity and normalized density and specific heat.


Table 1. Thermal material properties at ambient room temperature

#### where,

304 Convection and Conduction Heat Transfer

(b) Side view

(c) Cross-sectional view

Different element types were selected from the ANSYS element type selection library. The thermal elements implemented to model the concrete material and reinforcing GFRP rebars were SOLID70 and LINK33, respectively. SOLID70 has eight nodes with a single degree of freedom at each node, defined as temperature as well as 3-D thermal conduction capability. SOLID70 has 2×2×2 integration scheme for both conductivity and specific heat matrices. On the other hand, GFRP rebars were modeled using LINK33. LINK33 is a thermal uniaxial element with the ability to conduct heat between its two nodes. The element has a single degree of freedom SDF, temperature, at each node. In addition, both elements are applicable to conduct 3-D, steady-state and/or transient thermal analysis (ANSYS, 2007). Figure 3

GFRP Bars

shows the different element types used in this study.

Fig. 2. Developed FE model

Symmetry Axis no. 2

Symmetry Axis no. 2

*Ko* is the thermal conductivity (W/mm K) *Co* is the heat capacity (J/kg K) *ρ* is the density (kg/mm3)

Heat Transfer Analysis of Reinforced Concrete Beams Reinforced with GFRP Bars 307

Fig. 5. The applied ISO 834 fire curve

Fig. 6. Location of applied nodal temperatures

Exposed surfaces to the applied fire loading

Fig. 4. Thermal temperature-dependent material properties of concrete material

Few limited studies were conducted on the mechanical temperature-dependant properties of GFRP bars (Abbasi & Hogg, 2005). Abbasi and Hogg (Abbasi & Hogg, 2005) recommended empirical equations to predict the mechanical properties (stiffness, tensile strength, etc.) as a function of increasing temperature. To the top of the author's knowledge, there is no available data in the open literature on the thermal material-dependant properties of GFRP bars. Thus, the thermal material properties of the GFRP bars at ambient room temperature are used in this study.

Fig. 5. The applied ISO 834 fire curve

Fig. 4. Thermal temperature-dependent material properties of concrete material

room temperature are used in this study.

Few limited studies were conducted on the mechanical temperature-dependant properties of GFRP bars (Abbasi & Hogg, 2005). Abbasi and Hogg (Abbasi & Hogg, 2005) recommended empirical equations to predict the mechanical properties (stiffness, tensile strength, etc.) as a function of increasing temperature. To the top of the author's knowledge, there is no available data in the open literature on the thermal material-dependant properties of GFRP bars. Thus, the thermal material properties of the GFRP bars at ambient

Fig. 6. Location of applied nodal temperatures

Heat Transfer Analysis of Reinforced Concrete Beams Reinforced with GFRP Bars 309

bars. In this study, a critical temperature of 462°C in the GFRP is defined as the failure criteria. Thus, the time to failure (fire endurance) of the RC beam specimen is defined when the temperature in the GFRP bars reached 462°C during fire exposure. The same criteria is

The predicted FE and experimental results were compared in order to validate the accuracy of the developed model presented in this study. Figure 7 shows a comparison between the predicted and measured average temperature of the GFRP bars during the course of fire loading. The predicted average temperature in the GFRP bars is obtained every 0.5 second time increment. It is clear from Fig. 7 that there is a good correlation between the predicted and measured results throughout the entire thermal fire exposure. It should be noted from Fig. 7 that the predicted average temperature results overestimates the temperature after 25 minutes of fire exposure. This slight deviation could be related to the lack of temperaturedependant material properties of the GFRP bars. The tested RC beam failed approximately after 128min of fire exposure, when the average temperature in the GFRP bars reached 462°C. Similarly, the predicted time to failure is 130 minutes. The percentage difference between the FE model and experimental testing time to failure was 1.5%. Thus, the developed model could serve as a valid numerical tool to predict the temperature distribution of RC

used in the experimental program of Abbasi and Hogg (Abbasi & Hogg, 2006).

beams strengthened with GFRP bars when exposed to transient thermal loading.

as an alternative to the expensive experimental testing.

The experimental programs are restricted to limited number of instrumentations (thermocouples and strain gauges) due to their high cost and complex preparation. On the contrary, full fields of temperature distribution at any location (node) within the beam could predicted from the FE simulation of the validated model. Having a viable FE model, further results can be extracted and used. For instance, Fig. 8 shows the temperature evolution across the beam's cross section at different time periods of fire exposure. Figure 8 could be used to determine the nodal temperature at any point (node) within the beam especially at the GFRP bar reinforcement. As expected, the temperature evolution starts at the edges, then propagate within the beam. It seems that the available concrete cover is sufficient to delay the increase of temperature at the GFRP reinforcement level up to 130min. Figure 9 shows the temperature distribution along the RC beam specimen at the end of fire. The FE simulation can thus provide a wide range of results and the developed model could be used

The experimental program lacked data on the increase of temperature in the individual GFRP bars during the fire exposure. However, the FE model can predict the increase of temperature at any specific location and time. Figures 10 and 11 shows the location of the GFRP bars and temperature evolution in the individual bars at mid-span during fire exposure. It is clear from Fig. 11 that the temperature rise in GFRP Bar 2 is the highest. This is due to the application of the fire curve to the soffit and vertical side of the RC beam specimen. Being located at the corner, Bar 2 thus has the shortest distance to the exposed surfaces. Although Bars 1 and 3 seem to have the same perpendicular distance from the exposed edge, the increase in temperature of Bar 1 seems to be slightly lower than that of

**5. Results and discussions** 

**5.1 Model validation** 

**5.2 Model behavior** 

#### **4.2 Loading & boundary conditions**

The developed FE model was exposed to thermal transient temperature-time curve, ISO834 (ISO, 1975). The applied ISO 834 fire curve is shown in Fig. 5. The ISO 834 curve was applied as nodal temperature loading versus time at the soffit and vertical sides of the RC beam specimen. The locations of the applied nodal transient temperatures are shown in Fig 6. It must be noted that the applied nodal temperatures in the transient analysis domain started at 200mm away from the edge of the RC beam to simulate the furnace boundary conditions.

Since the fire nozzles in the furnace are very close to the tested RC beams, the author applied the average furnace temperature directly to the soffit and sides of the developed FE model (Hawileh et al., 2009, 2011). Thus, heat is transferred mainly by conduction in the developed model. This approach resulted in good matching with the temperature results recorded in the experimental program by Abbasi and Hogg (Abbasi & Hogg, 2006) and will be discussed in the subsequent section.

The average furnace temperature was applied in terms of small time incremental steps. Each time step is composed of several smaller sub-steps that are solved using Newton-Raphson's technique. In this study, automatic time stepping option is turned on to predict and control time step sizes. At the end of each time (temperature) step, convergence is achieved by Newton-Raphson equilibrium iterations when the temperature difference at each node from each iteration to another is less than one degree.

Fig. 7. Comparison between measured and predicted average temperature in the GFRP bars

#### **4.3 Failure criteria**

The failure criterion adopted in this study was based upon reaching a critical temperature in the GFRP bars. There has been a lot of debate on defining a critical temperature for FRP bars. In this study, a critical temperature of 462°C in the GFRP is defined as the failure criteria. Thus, the time to failure (fire endurance) of the RC beam specimen is defined when the temperature in the GFRP bars reached 462°C during fire exposure. The same criteria is used in the experimental program of Abbasi and Hogg (Abbasi & Hogg, 2006).

#### **5. Results and discussions**

#### **5.1 Model validation**

308 Convection and Conduction Heat Transfer

The developed FE model was exposed to thermal transient temperature-time curve, ISO834 (ISO, 1975). The applied ISO 834 fire curve is shown in Fig. 5. The ISO 834 curve was applied as nodal temperature loading versus time at the soffit and vertical sides of the RC beam specimen. The locations of the applied nodal transient temperatures are shown in Fig 6. It must be noted that the applied nodal temperatures in the transient analysis domain started at 200mm away from the edge of the RC beam to simulate the furnace boundary

Since the fire nozzles in the furnace are very close to the tested RC beams, the author applied the average furnace temperature directly to the soffit and sides of the developed FE model (Hawileh et al., 2009, 2011). Thus, heat is transferred mainly by conduction in the developed model. This approach resulted in good matching with the temperature results recorded in the experimental program by Abbasi and Hogg (Abbasi & Hogg, 2006) and will

The average furnace temperature was applied in terms of small time incremental steps. Each time step is composed of several smaller sub-steps that are solved using Newton-Raphson's technique. In this study, automatic time stepping option is turned on to predict and control time step sizes. At the end of each time (temperature) step, convergence is achieved by Newton-Raphson equilibrium iterations when the temperature difference at each node from

Fig. 7. Comparison between measured and predicted average temperature in the GFRP bars

The failure criterion adopted in this study was based upon reaching a critical temperature in the GFRP bars. There has been a lot of debate on defining a critical temperature for FRP

**4.2 Loading & boundary conditions** 

be discussed in the subsequent section.

each iteration to another is less than one degree.

conditions.

**4.3 Failure criteria** 

The predicted FE and experimental results were compared in order to validate the accuracy of the developed model presented in this study. Figure 7 shows a comparison between the predicted and measured average temperature of the GFRP bars during the course of fire loading. The predicted average temperature in the GFRP bars is obtained every 0.5 second time increment. It is clear from Fig. 7 that there is a good correlation between the predicted and measured results throughout the entire thermal fire exposure. It should be noted from Fig. 7 that the predicted average temperature results overestimates the temperature after 25 minutes of fire exposure. This slight deviation could be related to the lack of temperaturedependant material properties of the GFRP bars. The tested RC beam failed approximately after 128min of fire exposure, when the average temperature in the GFRP bars reached 462°C. Similarly, the predicted time to failure is 130 minutes. The percentage difference between the FE model and experimental testing time to failure was 1.5%. Thus, the developed model could serve as a valid numerical tool to predict the temperature distribution of RC beams strengthened with GFRP bars when exposed to transient thermal loading.

#### **5.2 Model behavior**

The experimental programs are restricted to limited number of instrumentations (thermocouples and strain gauges) due to their high cost and complex preparation. On the contrary, full fields of temperature distribution at any location (node) within the beam could predicted from the FE simulation of the validated model. Having a viable FE model, further results can be extracted and used. For instance, Fig. 8 shows the temperature evolution across the beam's cross section at different time periods of fire exposure. Figure 8 could be used to determine the nodal temperature at any point (node) within the beam especially at the GFRP bar reinforcement. As expected, the temperature evolution starts at the edges, then propagate within the beam. It seems that the available concrete cover is sufficient to delay the increase of temperature at the GFRP reinforcement level up to 130min. Figure 9 shows the temperature distribution along the RC beam specimen at the end of fire. The FE simulation can thus provide a wide range of results and the developed model could be used as an alternative to the expensive experimental testing.

The experimental program lacked data on the increase of temperature in the individual GFRP bars during the fire exposure. However, the FE model can predict the increase of temperature at any specific location and time. Figures 10 and 11 shows the location of the GFRP bars and temperature evolution in the individual bars at mid-span during fire exposure. It is clear from Fig. 11 that the temperature rise in GFRP Bar 2 is the highest. This is due to the application of the fire curve to the soffit and vertical side of the RC beam specimen. Being located at the corner, Bar 2 thus has the shortest distance to the exposed surfaces. Although Bars 1 and 3 seem to have the same perpendicular distance from the exposed edge, the increase in temperature of Bar 1 seems to be slightly lower than that of

Heat Transfer Analysis of Reinforced Concrete Beams Reinforced with GFRP Bars 311

115.56 222.22 328.88 435.54 542.2 648.86 755.52 862.18 968.84

Bar

3

r 4

Cente

rline

Ba

Fig. 9. Temperature distribution at the end of thermal loading

Bar

1

2

Fig. 10. Location of GFRP bars within Beam's cross-section

Bar

Bar 3. This could be related to the small bar spacing Bar 2 and Bar 4 that would increase of temperature in the closely spaced GFRP bars. Furthermore, upon comparing the increase of temperature in Bars 3 and 4, Bar 3 seems to experience higher temperatures because it is closer to the exposed faces than that of Bar 4. Figure 12 shows the temperature distribution along the GFRP bars after 130min of fire exposure. Figure 12 also shows that the edge GFRP bar (Bar 2) experienced the highest increase of temperature during fire exposure.

#### **6. Summary and conclusions**

A nonlinear 3D FE model was developed in this study and validated against the experimental program conducted by Abbasi and Hogg (Abbasi & Hogg, 2006). Good agreement between the measured experimental and predicted FE simulation was obtained for the average temperature in the GFRP bars at all stages of fire exposure. Although the UK Building Regulations (Building Regulations, 2000) for fire safety recommends the minimum periods of the fire resistance for the most structural elements to be of 90 min, the fire tests and FE simulation results showed that concrete beams reinforced with GFRP bars can achieve a fire endurance of about 130 minutes. Thus, using GFRP bars as concrete reinforcement seems to meet the fire design requirements.. Upon the validation of the measured data, the FE modeling could provide full field of results, in terms of 3D temperature distribution. It could be concluded that the developed FE model is a great tool to aid designers and researchers to predict numerically the temperature distribution of RC beams reinforced with GFRP bars. Thus, the validated model could be used as a valid tool in lieu of experimental testing especially in design oriented parametric studies. Furthermore, the developed and verified FE model in this study could be used as a tool for further investigation of the fire performance of RC beams reinforced with GFRP bars under different applied fire curves and boundary conditions.

Fig. 8. Temperature evolutions at different time of fire exposures

Bar 3. This could be related to the small bar spacing Bar 2 and Bar 4 that would increase of temperature in the closely spaced GFRP bars. Furthermore, upon comparing the increase of temperature in Bars 3 and 4, Bar 3 seems to experience higher temperatures because it is closer to the exposed faces than that of Bar 4. Figure 12 shows the temperature distribution along the GFRP bars after 130min of fire exposure. Figure 12 also shows that the edge GFRP

A nonlinear 3D FE model was developed in this study and validated against the experimental program conducted by Abbasi and Hogg (Abbasi & Hogg, 2006). Good agreement between the measured experimental and predicted FE simulation was obtained for the average temperature in the GFRP bars at all stages of fire exposure. Although the UK Building Regulations (Building Regulations, 2000) for fire safety recommends the minimum periods of the fire resistance for the most structural elements to be of 90 min, the fire tests and FE simulation results showed that concrete beams reinforced with GFRP bars can achieve a fire endurance of about 130 minutes. Thus, using GFRP bars as concrete reinforcement seems to meet the fire design requirements.. Upon the validation of the measured data, the FE modeling could provide full field of results, in terms of 3D temperature distribution. It could be concluded that the developed FE model is a great tool to aid designers and researchers to predict numerically the temperature distribution of RC beams reinforced with GFRP bars. Thus, the validated model could be used as a valid tool in lieu of experimental testing especially in design oriented parametric studies. Furthermore, the developed and verified FE model in this study could be used as a tool for further investigation of the fire performance of RC beams reinforced with GFRP bars under

30min 60min 90min 120min 130min

Fig. 8. Temperature evolutions at different time of fire exposures

25.529 142.248 258.967 375.686 492.405 609.124 725.843 842.562 959.281

bar (Bar 2) experienced the highest increase of temperature during fire exposure.

**6. Summary and conclusions** 

different applied fire curves and boundary conditions.

Fig. 9. Temperature distribution at the end of thermal loading

Fig. 10. Location of GFRP bars within Beam's cross-section

Heat Transfer Analysis of Reinforced Concrete Beams Reinforced with GFRP Bars 313

• Extend the current analysis into a 3D nonlinear thermal-stress analysis that can capture

• Conduct several parametric studies on the developed model to investigate the

• Investigate the temperature distribution when the beams get exposed to different fire

Abbasi, A. & Hogg, PJ. (2005). Temperature and environmental effects on glass fibre rebar:

Abbasi, A. & Hogg, PJ. (2005). Prediction of the Failure Time of Glass Fiber Reinforced

Abbasi, A. & Hogg, PJ. (2006). Fire testing of concrete beams with fibre reinforced plastic

ACI Committee, Guide for the design and construction of concrete reinforced with FRP bars.

Aiello, M. & Ombres, L. (2002). Structural Performances of Concrete Beams with Hybrid

Al-Tamimi, A., Hawileh, R., Abdalla, J. & Rasheed, H. (2011). Effects of Ratio of CFRP Plate

ANSYS – Release Version 11. A Finite Element Computer Software and User Manual for

ASTM E119. Standard test methods for fire tests of building construction and materials.

Bisby L. Fire behavior of fiber-reinforced polymer reinforced or confined concrete. Ph.D.

Building Regulations 2000. (2000). Fire Safety, Amendments 2002 to Approved Document B

De Lorenzis L. & Teng JG. (2007). Near-surface mounted FRP reinforcement: an emerging

Franssen, J., Kodur, V. & Zaharia, R. (2009). *Designing Steel Structures for Fire Safety*. Taylor &

Hawileh, R. Naser, M. Zaidan, W. & Rasheed, H. (2009). Modeling of insulated CFRP-

Hawileh, R., Tamimi, A., Abdalla, J.A. & Wehbi, M. (2011). Retrofitting Pre-cracked RC

Nonlinear Structural Analysis, ANSYS 2007; Inc. Canonsburg, PA.

West Conshohocken, PA: ASTM international; 2002. 22 pp

(Fire safety) published by TSO (The Stationary Office).

*Conference On Composite Science* and *Technology (ICCST/8)*.

Eurocode 2, Design of concrete structures, ENV EC2 Part 1.2; 1992.

Francis Group, London, UK.

(2009), pp. 3072-79.

modulus, strength and interfacial bond strength with concrete. *Composites: part B*,

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(Fiber-Reinforced Polymer-Steel) Reinforcements. *Journal of Composites for* 

Length to Shear Span and End Anchorage on Flexural Behavior of SCC R/C Beams.

thesis. Kingston (Canada): Department of Civil Engineering, Queen's University;

technique for strengthening structures. *J Compos, Part B*, Vol.38, (2007), pp. 119–143.

strengthened reinforced concrete T-beam exposed to fire. *Eng Struct,* Vol.31, No.12,

Beams Using CFRP and Epoxy Injections. *In Proceeding of the Eight International* 

the response of RC beams reinforced with GFRP reinforcement.

local exposures.

2003. 371 pp.

Vol.36, No.5, (2005), pp. 394-404

*Construction*, Vol.9, No.5, (2005), pp. 450-457

rebar. *Composites: part A*, Vol.37, (2006), pp. 1142–1150

Reported by ACI Committee 440; January 5, 2001.

*Construction*, Vol.13, No.5, (2002), pp. 133-140

*Journal of Composites for Construction. Accepted.*

**8. References** 

temperature distribution under different applied fire exposure curves.

Fig. 11. Predicted temperature evolution in the GFRP bars

Fig. 12. Temperature distribution along the GFRP bars after 130min of fire exposure

#### **7. Future research**

• Experimental studies on the temperature-dependent thermal properties of GFRP bars are warranted.


#### **8. References**

312 Convection and Conduction Heat Transfer

271.79 306.395 341 375.605 410.21 444.815 479.42 514.024 548.629

• Experimental studies on the temperature-dependent thermal properties of GFRP bars

Fig. 12. Temperature distribution along the GFRP bars after 130min of fire exposure

Fig. 11. Predicted temperature evolution in the GFRP bars

**7. Future research** 

are warranted.


**0**

**15**

**Modelling of Heat Transfer and Phase**

**Titanium Components**

António Crespo

*Portugal*

*Instituto Superior Técnico*

**Transformations in the Rapid Manufacturing of**

Over the last thirty years a new generation of manufacturing processes have been developed whereby fabrication takes place by the sequential addition of material at specific locations on a layer-by-layer basis to produce near-net-shape three-dimensional parts. These processes are generically known as solid freeform fabrication or rapid manufacturing, and their most prominent features are fast delivery times, material wastage reduction and the ability to produce parts directly from a CAD file in a single fabrication step without the need for hard tooling. These characteristics lead to lower costs and faster production cycles than those of hard-tooling based methods, and have made solid freeform fabrication an attractive choice for the production of customised one-of-a-kind parts, prototyping and short-run production. For the manufacture of metallic components these techniques involve the use of a localised energy source, typically a laser beam, to consolidate layers of metallic materials supplied in the form a pre-placed powder bed, blown powder stream or wire feeding. By scanning the interaction zone of the energy source and the material over a substrate and according to a previously defined trajectory, a track of resolidified material is produced and the overlapping of tracks enables the manufacture of near-net-shape components, as shown schematically in

One important feature of rapid manufacturing techniques is that, as a result of layer overlap during part buildup, the deposited material undergoes consecutive thermal cycles with a duration and amplitude which depend on the processing parameters and on the shape and dimensions of the part being manufactured. These thermal cycles induce phase transformations in the material leading to a progressive modification of its microstructure and properties. Since the thermal history varies from point to point in the part, the material will in general present complex distributions of microstructure and properties. To optimise the manufacturing process to obtain parts fulfilling specific requirements, the influence of the processing conditions and build-up strategy on the final microstructure and properties of the material must be assessed. The complexity of the thermal cycles taking place during the manufacture of components and the impossibility of directly monitoring the thermal and microstructural evolution of the material during the fabrication process make empirical or experimental build-up strategy optimisation impractical. However, the optimisation of the process can be carried out efficiently using a computational approach based on a model

**1. Introduction**

Figure 1.


### **Modelling of Heat Transfer and Phase Transformations in the Rapid Manufacturing of Titanium Components**

António Crespo *Instituto Superior Técnico Portugal*

#### **1. Introduction**

314 Convection and Conduction Heat Transfer

Hawileh, R., Naser, M. & Rasheed, H. (2011). Thermal-Stress Finite Element Analysis of

ISO-834, Fire resistance tests, Elements of building construction. International Standards

Kodur, V. &Ahmed, A. (2010). A Numerical Model for Tracing the Response of FRP-

Masoudi, R., Masoudi, A., Ouezdou, B. & Daoud, A. (2011). Long-term bond performance of

Rasheed, H.A., Harrison, R.R., Peterman, R.J. & Alkhrdaji, T. (2010). Ductile Strengthening

Sadek, A., El-Hawary, M. & El-Deeb, A. (2006). Fire Resistance Testing of Concrete Beams

Qu, W., Zhang, X., & Huang, H. (2009). Flexural Behavior of Concrete Beams Reinforced

Williams, B., Kodur, V.K.R., Green, M.F. & Bisby, L. (2008). Fire Endurance of Fiber-

*Advanced Materials and Structures*, Vol.18, No.3, (2011). Loading

Organisation, Geneva; 1975.

(2006), pp. 190-200

(2008), pp. 60–67.

No.2, (2009), pp. 350-359.

*Construction*, Vol.14, No.6, 2010, pp. 730-742

*Structures,* Vol.92, No.10, (2009), pp. 2379-2390

*and Building Materials,* Vol.25, No.2, (2011), pp. 486-493

CFRP Strengthened Concrete Beam Exposed to Top Surface Fire. *Mechanics of* 

Strengthened Reinforced Concrete Beams Exposed to Fire. *Journal of Composites for* 

GFRP bars in concrete under temperature ranging from 20°C to 80°C. *Construction* 

Using Externally Bonded and Near Surface Mounted Composite Systems. *Composite* 

Reinforced by GFRP Rebars. *European Journal of Scientific Research,* Vol.15, No.2,

with Hybrid (GFRP and Steel) Bars. *Journal of Composites for Construction*, Vol.6,

Reinforced Polymer Strengthened Concrete T-Beams. *ACI Struct. J.*, Vol.105, No.1,

Over the last thirty years a new generation of manufacturing processes have been developed whereby fabrication takes place by the sequential addition of material at specific locations on a layer-by-layer basis to produce near-net-shape three-dimensional parts. These processes are generically known as solid freeform fabrication or rapid manufacturing, and their most prominent features are fast delivery times, material wastage reduction and the ability to produce parts directly from a CAD file in a single fabrication step without the need for hard tooling. These characteristics lead to lower costs and faster production cycles than those of hard-tooling based methods, and have made solid freeform fabrication an attractive choice for the production of customised one-of-a-kind parts, prototyping and short-run production. For the manufacture of metallic components these techniques involve the use of a localised energy source, typically a laser beam, to consolidate layers of metallic materials supplied in the form a pre-placed powder bed, blown powder stream or wire feeding. By scanning the interaction zone of the energy source and the material over a substrate and according to a previously defined trajectory, a track of resolidified material is produced and the overlapping of tracks enables the manufacture of near-net-shape components, as shown schematically in Figure 1.

One important feature of rapid manufacturing techniques is that, as a result of layer overlap during part buildup, the deposited material undergoes consecutive thermal cycles with a duration and amplitude which depend on the processing parameters and on the shape and dimensions of the part being manufactured. These thermal cycles induce phase transformations in the material leading to a progressive modification of its microstructure and properties. Since the thermal history varies from point to point in the part, the material will in general present complex distributions of microstructure and properties. To optimise the manufacturing process to obtain parts fulfilling specific requirements, the influence of the processing conditions and build-up strategy on the final microstructure and properties of the material must be assessed. The complexity of the thermal cycles taking place during the manufacture of components and the impossibility of directly monitoring the thermal and microstructural evolution of the material during the fabrication process make empirical or experimental build-up strategy optimisation impractical. However, the optimisation of the process can be carried out efficiently using a computational approach based on a model

conducive to mechanically sound components (Crespo et al., 2008; Deus & Mazumder, 2006; Ghosh & Choi, 2005; Kahlen & Kar, 2001; Labudovich et al., 2003). Although the study of residual stresses formation during manufacture is out of the scope of the present work, the model presented in this chapter has been extended to study these phenomena and a detailed description of the methodology can be found in the work of A. Crespo (Crespo, 2010). This chapter reports the development of a finite element thermo-kinetic model that couples heat transfer calculations with phase transformation kinetics theory to investigate the influence of the build-up strategy and processing parameters on the distributions of microstructure and properties of titanium parts manufactured by rapid manufacturing processes. The model was validated by comparing its predictions for the phase constitution, Young's modulus and hardness distributions with experimental results measured on parts fabricated by a laser based rapid manufacturing technique, and used to construct processing maps relating the deposition parameters to the microstructure and properties of the parts.

<sup>317</sup> Modelling of Heat Transfer

Rapid manufacturing processes are characterised by the continuous addition of material on a layer-by-layer basis to produce a final component. In order to process the different layers into a consolidated part an energy source is typically used to heat up and melt the feedstock material before incorporation into the part (Figure 1). This type of manufacturing subjects the processed material to consecutive thermal cycles which induce a sequence of phase transformations that alter the microstructure and properties of the material, and which, if not properly controlled, may render the final parts useless. Therefore a critical aspect of additive manufacturing is the control of the different heat transfer phenomena that take place during

The phase transformations during rapid manufacturing are determined by the thermal history of the material, which is the result of energy absorption by the material, heat conduction within the workpiece being built and heat losses by convection and radiation to the environment. The temperature field evolution within the material can be calculated by solving the three-dimensional heat conduction equation. Thermal conduction in an isotropic solid region Ω is a time dependent 3-D problem described by the following partial differential

where *T*(*r*, *t*) is the temperature (◦*C*), *ρ* the density of the solid (*kgm*−3), *k* the thermal conductivity (*W*◦*C*−1*m*−1), *Cp* the specific heat (*J*◦*C*−1*Kg*−1) and *Q* a generic internal heat generation rate (*Wm*−3) which can account, for example, for phenomena associated with latent heats of transformation. The thermal properties of materials are usually temperature dependant, that is: *ρ* = *ρ*(*T*), *k* = *k*(*T*) and *Cp* = *Cp*(*T*). To apply the heat conduction equation to time-dependants problems, such as simulating a fabrication process, an initial condition specifying the initial temperature distribution in the part must also be provided:

*dt* <sup>=</sup> <sup>∇</sup> [*k*∇*T*(*<sup>r</sup>*, *<sup>t</sup>*)] <sup>+</sup> *<sup>Q</sup>*, (1)

*T*(*r*, *t* = 0) = *T*0(*r*), *r* ∈ Ω. (2)

**2. Modelling of heat transfer in rapid manufacturing processes**

and Phase Transformations in the Rapid Manufacturing of Titanium Components

build-up and which determine the properties and quality of the final parts.

*dT*(*r*, *t*)

*ρCp*

**2.1 Heat transfer equations**

equation:

Fig. 1. Rapid manufacturing using a laser beam to consolidate blown powder particles into a final part

that simulates the microstructure formation mechanisms in the material, in particular the phase transformations caused by the heat and mass transfer phenomena that take place during fabrication. Such a model should use as input the processing parameters and the part shape and dimensions to predict the final microstructure and properties distributions in the manufactured part. This type of approach provides a cheap and rapid way of optimising the manufacturing process and obtaining parts fulfilling unique and specific properties requirements. Several researchers such as L. Costa (Costa et al., 2005), S. M. Kelly (Kelly & Kampe, 2004b), Toyserkani et al. (Toyserkani et al., 2004), have successfully used computational approaches to analyse the influence of the processing parameters on several aspects of the process, such as the heat transfer mechanisms, the microstructure formation processes or the evolution of the properties of the material, among others (Crespo et al., 2006; Labudovich et al., 2003; Vasinonta et al., 2001; Wang et al., 2008).

In addition to microstructural transformations, the consecutive thermal cycles imposed on the material are also responsible for the accumulation of internal stresses in parts produced by rapid manufacturing techniques. During the fabrication process, the thermal cycles experienced by the material will give rise to thermal expansion and contraction. Since the deposited material is cyclically subjected to steep temperature variations, the thermal expansion gradients ocurring in the material give rise to the development of internal stresses in the workpiece. These stresses can be high enough to cause yield in some regions or lead to complex residual stresses patterns in the final parts. The study of the development of residual stresses during rapid manufacturing processes has been the subject of several investigations aimed at acquiring a thorough insight into this phenomenon and at establishing processing parameters sets and build-up strategies conducive to mechanically sound components (Crespo et al., 2008; Deus & Mazumder, 2006; Ghosh & Choi, 2005; Kahlen & Kar, 2001; Labudovich et al., 2003). Although the study of residual stresses formation during manufacture is out of the scope of the present work, the model presented in this chapter has been extended to study these phenomena and a detailed description of the methodology can be found in the work of A. Crespo (Crespo, 2010).

This chapter reports the development of a finite element thermo-kinetic model that couples heat transfer calculations with phase transformation kinetics theory to investigate the influence of the build-up strategy and processing parameters on the distributions of microstructure and properties of titanium parts manufactured by rapid manufacturing processes. The model was validated by comparing its predictions for the phase constitution, Young's modulus and hardness distributions with experimental results measured on parts fabricated by a laser based rapid manufacturing technique, and used to construct processing maps relating the deposition parameters to the microstructure and properties of the parts.

#### **2. Modelling of heat transfer in rapid manufacturing processes**

Rapid manufacturing processes are characterised by the continuous addition of material on a layer-by-layer basis to produce a final component. In order to process the different layers into a consolidated part an energy source is typically used to heat up and melt the feedstock material before incorporation into the part (Figure 1). This type of manufacturing subjects the processed material to consecutive thermal cycles which induce a sequence of phase transformations that alter the microstructure and properties of the material, and which, if not properly controlled, may render the final parts useless. Therefore a critical aspect of additive manufacturing is the control of the different heat transfer phenomena that take place during build-up and which determine the properties and quality of the final parts.

#### **2.1 Heat transfer equations**

2 Will-be-set-by-IN-TECH

Fig. 1. Rapid manufacturing using a laser beam to consolidate blown powder particles into a

that simulates the microstructure formation mechanisms in the material, in particular the phase transformations caused by the heat and mass transfer phenomena that take place during fabrication. Such a model should use as input the processing parameters and the part shape and dimensions to predict the final microstructure and properties distributions in the manufactured part. This type of approach provides a cheap and rapid way of optimising the manufacturing process and obtaining parts fulfilling unique and specific properties requirements. Several researchers such as L. Costa (Costa et al., 2005), S. M. Kelly (Kelly & Kampe, 2004b), Toyserkani et al. (Toyserkani et al., 2004), have successfully used computational approaches to analyse the influence of the processing parameters on several aspects of the process, such as the heat transfer mechanisms, the microstructure formation processes or the evolution of the properties of the material, among others (Crespo et al., 2006;

In addition to microstructural transformations, the consecutive thermal cycles imposed on the material are also responsible for the accumulation of internal stresses in parts produced by rapid manufacturing techniques. During the fabrication process, the thermal cycles experienced by the material will give rise to thermal expansion and contraction. Since the deposited material is cyclically subjected to steep temperature variations, the thermal expansion gradients ocurring in the material give rise to the development of internal stresses in the workpiece. These stresses can be high enough to cause yield in some regions or lead to complex residual stresses patterns in the final parts. The study of the development of residual stresses during rapid manufacturing processes has been the subject of several investigations aimed at acquiring a thorough insight into this phenomenon and at establishing processing parameters sets and build-up strategies

Labudovich et al., 2003; Vasinonta et al., 2001; Wang et al., 2008).

final part

The phase transformations during rapid manufacturing are determined by the thermal history of the material, which is the result of energy absorption by the material, heat conduction within the workpiece being built and heat losses by convection and radiation to the environment. The temperature field evolution within the material can be calculated by solving the three-dimensional heat conduction equation. Thermal conduction in an isotropic solid region Ω is a time dependent 3-D problem described by the following partial differential equation:

$$
\rho \mathbb{C}\_p \frac{dT(\vec{r}, t)}{dt} = \nabla \left[ k \nabla T(\vec{r}, t) \right] + Q\_{\prime} \tag{1}
$$

where *T*(*r*, *t*) is the temperature (◦*C*), *ρ* the density of the solid (*kgm*−3), *k* the thermal conductivity (*W*◦*C*−1*m*−1), *Cp* the specific heat (*J*◦*C*−1*Kg*−1) and *Q* a generic internal heat generation rate (*Wm*−3) which can account, for example, for phenomena associated with latent heats of transformation. The thermal properties of materials are usually temperature dependant, that is: *ρ* = *ρ*(*T*), *k* = *k*(*T*) and *Cp* = *Cp*(*T*). To apply the heat conduction equation to time-dependants problems, such as simulating a fabrication process, an initial condition specifying the initial temperature distribution in the part must also be provided:

$$T(\vec{r}, t=0) = T\_0(\vec{r}), \ \vec{r} \in \Omega. \tag{2}$$

Calculating the solution to a problem using the finite element method is accomplished by using several mathematical tools and can be summarised by the following sequence of steps

<sup>319</sup> Modelling of Heat Transfer

1. the physical domain of the problem being solved is partitioned into a collection of simple

2. in each element, the solution is approximated by a linear combination of adequately chosen interpolation functions and time dependant temperature values at the nodes of

used to establish algebraic relations between the unknown nodal temperatures *Tj*(*t*); 3. the algebraic equations developed for the different elements are assembled by requiring the continuity of *T*(*r*, *t*) and the balance of certain relevant physical quantities at the interface between neighbouring elements. This produces a system of equations which can be solved to find the coefficients *Tj*(*t*) and produce an approximate solution to the

The approximate solution differs from the exact solution due to the approximations made in each of the three steps described above: in the first step errors are introduced because a continuous physical domain is represented by a finite element mesh that does not match it exactly; in the second step errors arise from approximating the exact solution over each element by a finite sum of functions; the solution of the system of assembled equations in the third step usually requires numerical integration, which is another source of error. The overall error in the computed solution can be minimised, for example, by refining the mesh to ensure a more accurate representation of the physical domain (first step), by using a better approximation to the local solution during the development of the algebraic equations in each element (second step) and by defining stricter convergence criteria in the numerical integration (third step). However, error reduction is achieved at the expense of computation time and usually a compromise has to be accepted between the accuracy of the solution and

In the finite element method the equations governing heat transfer in the elements are expressed in terms of a variational formulation, typically the weak form of the heat conduction equation. Although the weak form of the heat conduction equation will not be derived here,

where the superscript *e* is an identification label for each element in the mesh and *n* is the

*k*∇*ψi*.∇*ψjdV*

where the dependencies of *T* on*r* and *t* have been omitted for simplicity. *Q*(*r*, *t*) represents the heat flux across the common surfaces of neighbouring elements and the equations of the

*qlaser* − *h* (*T* − *T*∞) − *�σ<sup>s</sup>*

it produces the following set of equations for each element (Reddy, 2006):

*ijT<sup>e</sup> <sup>e</sup>* (*r*, *t*) <sup>−</sup> *<sup>Q</sup><sup>e</sup> <sup>i</sup>* <sup>−</sup> *<sup>q</sup><sup>e</sup>*

order of the interpolation functions used to approximate the solution. *M<sup>e</sup>*

*ij* = Ω*<sup>e</sup>*

*<sup>i</sup>* = Γ *ψi* 

*ρCψiψjdV*, *K<sup>e</sup>*

*ψiQ*(*r*, *t*)*dV*, *q<sup>e</sup>*

*<sup>j</sup>*=<sup>1</sup> *Tj*(*t*)*ψj*(*r*), with the governing equations of the problem being

*<sup>i</sup>* = 0, *i* = 1, ..., *n*, (7)

*ij*, *<sup>K</sup><sup>e</sup> ij*, *<sup>Q</sup><sup>e</sup>*

*<sup>T</sup>*<sup>4</sup> <sup>−</sup> *<sup>T</sup>*<sup>4</sup> ∞ *dS*,

*<sup>i</sup>* and *<sup>q</sup><sup>e</sup>*

*<sup>i</sup>* are

(8)

(Desai & Abel, 1972; Reddy, 2006; Zienkiewicz & Taylor, 2000):

and Phase Transformations in the Rapid Manufacturing of Titanium Components

geometrical sub-domains called finite elements;

the elements: *<sup>T</sup>*(*<sup>r</sup>*, *<sup>t</sup>*) <sup>≈</sup> <sup>∑</sup>*<sup>n</sup>*

problem.

the time to compute it .

defined by:

*M<sup>e</sup> ij* = Ω*<sup>e</sup>*

*Qe <sup>i</sup>* = Ω*<sup>e</sup>*

*n* ∑ *j*=1  *M<sup>e</sup> ij dT<sup>e</sup> <sup>j</sup>* (*r*, *t*) *dt* <sup>+</sup> *<sup>K</sup><sup>e</sup>*

Additionally, the influence of the energy source used to process the material has to be taken into consideration. The best way to model the interaction between the heat source and the material depends on the particular characteristics of the energy source being used such as its power, power density, shape and the way it is absorbed or transferred into the material. Probably the most common way of processing metallic materials for rapid manufacturing purposes is to utilise a laser beam. Assuming, for example, a gaussian distribution of the laser beam power , the energy input into the material can be described by:

$$q\_{laser} = a \frac{2.P}{\pi r\_l^2} exp[-\frac{2.r^2}{r\_l^2}] \,\text{\AA} \tag{3}$$

where *P* is the laser beam power, *α* the absorptivity of the material to the radiation being used, *rl* the spot radius of the laser beam and *r* the distance to its centre, respectively. Energy losses to the environment by convection and radiation are described by:

$$Q\_{\text{convection}} = h \left[ T(\vec{r}, t) - T\_{\infty} \right] \tag{4}$$

$$Q\_{radiation} = \epsilon \sigma\_{\rm s} \left[ T^4(\vec{r}, t) - T\_{\infty}^4 \right] \tag{5}$$

where *h*, *�*, *σs* and *T*∞ are the convective heat transfer coefficient, emissivity, Stefan-Boltzmann constant and temperature of the surrounding environment (room temperature), respectively. On the surface of the workpiece, the balance between the heat lost to the environment by convection and radiation and the energy absorbed from the laser radiation can be written as:

$$-k\nabla T\vec{n} = h\left[T(\vec{r},t) - T\_{\infty}\right] + \epsilon\sigma\_{\rm s}\left[T^{4}(\vec{r},t) - T\_{\infty}^{4}\right] - q\_{laser\prime} \quad \vec{r} \in \Gamma\_{\prime} \tag{6}$$

where Γ represents the surface (or boundary) of the solid region Ω and *n* is the unitary vector normal to the surface Γ.

#### **2.2 The finite element method**

Although the mathematical description of heat transfer during rapid manufacturing processes is easy to formulate, its solution can rarely be analytically determined. In particular, rapid manufacturing processes involve the continuous incorporation of material into the parts being built, which means that the shape of the part is continuously changing and an adequate mathematical description of this type of process must also account for an evolving geometry. The finite element method has been successfully used by previous authors (Costa et al., 2005; Deus & Mazumder, 2006; Kelly & Kampe, 2004b; Toyserkani et al., 2004) to study additive manufacturing processes. The finite element method is a numerical technique for calculating approximate solutions to algebraic, differential and integral equations for which analytical solutions cannot be found. It is frequently applied to physical and engineering problems in areas such as structural mechanics, dynamics, fluid mechanics and heat transfer. This method can efficiently handle complicated geometries and account for non-uniform properties and boundary conditions, and is easily implemented in computer programs to solve complex simulation problems, taking advantage of the large computational power presently available, therefore making this technique a prime choice to study heat transfer during rapid manufacturing processes.

4 Will-be-set-by-IN-TECH

Additionally, the influence of the energy source used to process the material has to be taken into consideration. The best way to model the interaction between the heat source and the material depends on the particular characteristics of the energy source being used such as its power, power density, shape and the way it is absorbed or transferred into the material. Probably the most common way of processing metallic materials for rapid manufacturing purposes is to utilise a laser beam. Assuming, for example, a gaussian distribution of the laser

> 2.*P π*.*r*<sup>2</sup> *l*

where *P* is the laser beam power, *α* the absorptivity of the material to the radiation being used, *rl* the spot radius of the laser beam and *r* the distance to its centre, respectively. Energy losses

where *h*, *�*, *σs* and *T*∞ are the convective heat transfer coefficient, emissivity, Stefan-Boltzmann constant and temperature of the surrounding environment (room temperature), respectively. On the surface of the workpiece, the balance between the heat lost to the environment by convection and radiation and the energy absorbed from the laser radiation can be written as:

where Γ represents the surface (or boundary) of the solid region Ω and *n* is the unitary vector

Although the mathematical description of heat transfer during rapid manufacturing processes is easy to formulate, its solution can rarely be analytically determined. In particular, rapid manufacturing processes involve the continuous incorporation of material into the parts being built, which means that the shape of the part is continuously changing and an adequate mathematical description of this type of process must also account for an evolving geometry. The finite element method has been successfully used by previous authors (Costa et al., 2005; Deus & Mazumder, 2006; Kelly & Kampe, 2004b; Toyserkani et al., 2004) to study additive manufacturing processes. The finite element method is a numerical technique for calculating approximate solutions to algebraic, differential and integral equations for which analytical solutions cannot be found. It is frequently applied to physical and engineering problems in areas such as structural mechanics, dynamics, fluid mechanics and heat transfer. This method can efficiently handle complicated geometries and account for non-uniform properties and boundary conditions, and is easily implemented in computer programs to solve complex simulation problems, taking advantage of the large computational power presently available, therefore making this technique a prime choice to study heat

*exp*[−2.*r*<sup>2</sup> *r*2 *l*

*<sup>T</sup>*4(*<sup>r</sup>*, *<sup>t</sup>*) <sup>−</sup> *<sup>T</sup>*<sup>4</sup>

*<sup>T</sup>*4(*<sup>r</sup>*, *<sup>t</sup>*) <sup>−</sup> *<sup>T</sup>*<sup>4</sup>

*Qconvection* = *h* [*T*(*r*, *t*) − *T*∞] (4)

∞ 

> ∞

], (3)

, (5)

− *qlaser*, *r* ∈ Γ, (6)

beam power , the energy input into the material can be described by:

to the environment by convection and radiation are described by:

− *k*∇*Tn* = *h* [*T*(*r*, *t*) − *T*∞] + *�σ<sup>s</sup>*

normal to the surface Γ.

**2.2 The finite element method**

transfer during rapid manufacturing processes.

*qlaser* = *α*

*Qradiation* = *�σ<sup>s</sup>*

Calculating the solution to a problem using the finite element method is accomplished by using several mathematical tools and can be summarised by the following sequence of steps (Desai & Abel, 1972; Reddy, 2006; Zienkiewicz & Taylor, 2000):


The approximate solution differs from the exact solution due to the approximations made in each of the three steps described above: in the first step errors are introduced because a continuous physical domain is represented by a finite element mesh that does not match it exactly; in the second step errors arise from approximating the exact solution over each element by a finite sum of functions; the solution of the system of assembled equations in the third step usually requires numerical integration, which is another source of error. The overall error in the computed solution can be minimised, for example, by refining the mesh to ensure a more accurate representation of the physical domain (first step), by using a better approximation to the local solution during the development of the algebraic equations in each element (second step) and by defining stricter convergence criteria in the numerical integration (third step). However, error reduction is achieved at the expense of computation time and usually a compromise has to be accepted between the accuracy of the solution and the time to compute it .

In the finite element method the equations governing heat transfer in the elements are expressed in terms of a variational formulation, typically the weak form of the heat conduction equation. Although the weak form of the heat conduction equation will not be derived here, it produces the following set of equations for each element (Reddy, 2006):

$$\sum\_{j=1}^{n} \left( M\_{ij}^{\varepsilon} \frac{d T\_j^{\varepsilon}(\vec{r}, t)}{dt} + K\_{ij}^{\varepsilon} T\_\varepsilon^{\varepsilon}(\vec{r}, t) \right) - Q\_i^{\varepsilon} - q\_i^{\varepsilon} = 0, \qquad \qquad i = 1, \dots, n,\tag{7}$$

where the superscript *e* is an identification label for each element in the mesh and *n* is the order of the interpolation functions used to approximate the solution. *M<sup>e</sup> ij*, *<sup>K</sup><sup>e</sup> ij*, *<sup>Q</sup><sup>e</sup> <sup>i</sup>* and *<sup>q</sup><sup>e</sup> <sup>i</sup>* are defined by:

$$\begin{aligned} M\_{\vec{l}\vec{j}}^{\varepsilon} &= \int\_{\Omega^{\varepsilon}} \rho \mathbb{C} \boldsymbol{\psi}\_{i} \boldsymbol{\psi}\_{\vec{l}} dV, \quad K\_{\vec{l}\vec{j}}^{\varepsilon} = \int\_{\Omega^{\varepsilon}} k \nabla \boldsymbol{\psi}\_{i} \cdot \nabla \boldsymbol{\psi}\_{\vec{l}} dV \\ Q\_{\vec{i}}^{\varepsilon} &= \int\_{\Omega^{\varepsilon}} \boldsymbol{\psi}\_{i} Q(\vec{r}, t) dV, \quad q\_{\vec{i}}^{\varepsilon} = \int\_{\Gamma} \boldsymbol{\psi}\_{i} \left[ q\_{laser} - h \left( T - T\_{\infty} \right) - \varepsilon \sigma\_{\rm s} \left( T^{4} - T\_{\infty}^{4} \right) \right] dS, \end{aligned} \tag{8}$$

where the dependencies of *T* on*r* and *t* have been omitted for simplicity. *Q*(*r*, *t*) represents the heat flux across the common surfaces of neighbouring elements and the equations of the

Substitution of Equation 13 in Equation 14 yields the solution to the problem:

and Phase Transformations in the Rapid Manufacturing of Titanium Components

**<sup>F</sup>** [**T**(*ts*+1)] <sup>−</sup> **<sup>K</sup>**<sup>ˆ</sup> [**T**(*ts*+1)] **<sup>T</sup>**(*ts*+1)

<sup>321</sup> Modelling of Heat Transfer

In general Equation 15 leads to an implicit scheme that requires iterative solutions to be found within each time step. The forward difference method is the only one of the above which is an explicit method and is the easiest to implement. It results in a simple iterative solution where **T**(*ts*+1) is readily obtained from the solution at the previous step **T**(*ts*), and is given by:

Starting from **T**(0) = **T**0, the solution at subsequent steps can be calculated from Equation 16. Equation 16 is a general expression that relates the temperatures at various points of a geometry by requiring the balance of heat fluxes across the boundaries between neighbouring elements and the continuity of the temperature field, governed by the weak form of the heat conduction equation. The temperature evolution during additive manufacture for a component of arbitrary geometry can be found by implementing Equation 16 as a computer

A finite element model should ideally describe the geometry of the substrate and the tracks as closely as possible. Frequently the substrate is a parallelepiped which can be easily represented in the form of a finite element mesh. However, it is more difficult to develop a mesh which allows a step wise description of the deposition of tracks with complex 3-D features, such as curved cross sections or curved fronts. To describe the full detail of track overlap during manufacture, the finite element mesh becomes complex and requires many elements for the proper representation of the 3-D features of the tracks, as shown in Figure

(a) (b)

Fig. 2. (a) Finite element mesh of substrate and tracks. (b) Step-wise approach to simulate the addition of material. New elements are activated at liquidus temperature. Adapted from

One commonly applied strategy to reduce the number of elements is to use a fine mesh only in regions which have complex geometries or where thermal gradients are expected to be high (in the vicinity of interaction zone between the energy source and the material), while using a coarser mesh away from these zones (Figure 2.a). The level of refinement shown in Figure

**<sup>F</sup>** [**T**(*ts*)] <sup>−</sup> **<sup>K</sup>**<sup>ˆ</sup> [**T**(*ts*)] **<sup>T</sup>**(*ts*)

**<sup>F</sup>** [**T**(*ts*)] <sup>−</sup> **<sup>K</sup>**<sup>ˆ</sup> [**T**(*ts*)] **<sup>T</sup>**(*ts*)

(*ts*−<sup>1</sup> <sup>−</sup> *ts*). (15)

(*ts*−<sup>1</sup> − *ts*)

(*ts*−<sup>1</sup> − *ts*). (16)

**<sup>T</sup>**(*ts*+1) = **<sup>T</sup>**(*ts*)+(<sup>1</sup> <sup>−</sup> *<sup>α</sup>*)**M**−<sup>1</sup> [**T**(*ts*)]

**T**(*ts*+1) = **T**(*ts*) + **M**−<sup>1</sup> [**T**(*ts*)]

**2.3 Representation of the physical domain**

Crespo and Vilar (Crespo & Vilar, 2010)

*α***M**−<sup>1</sup> [**T**(*ts*+1)]

code.

2.a.

1 2 3

different elements are assembled together by requiring the balance of this flux from each element to its neighbours and the continuity of the temperature field *T*(*r*, *t*). This system of equations is commonly written in matrix form as:

$$\mathbf{M}^{\varepsilon}\dot{\mathbf{T}}^{\varepsilon} + \mathbf{\hat{K}}^{\varepsilon}\mathbf{T} = \mathbf{F}^{\varepsilon},\tag{9}$$

where

$$\begin{aligned} \mathbf{M}^{\varepsilon} &= M\_{ij}^{\varepsilon} \\ \hat{\mathbf{K}}^{\varepsilon} &= K\_{ij}^{\varepsilon} - I\_{ij} \int\_{\Gamma\_2} \boldsymbol{\psi}\_i \left( -\hbar - \epsilon \sigma\_s T^3 \right) dS \\ \mathbf{F}^{\varepsilon} &= Q\_i^{\varepsilon} + \int\_{\Gamma\_2} \boldsymbol{\psi}\_i \left( q\_{laser} + \hbar T\_{\infty} + \epsilon \sigma\_s T\_{\infty}^4 \right) dS, \end{aligned} \tag{10}$$

with *Iij* the identity matrix.

The equations of the single elements are assembled by summing the element equations corresponding to the same nodes:

$$\mathbf{M} = \sum\_{\varepsilon} \mathbf{M}^{\varepsilon}, \quad \mathbf{\hat{K}} = \sum\_{\varepsilon} \mathbf{\hat{K}}^{\varepsilon}, \quad \mathbf{F} = \sum\_{\varepsilon} \mathbf{F}^{\varepsilon}, \tag{11}$$

resulting in the global equation:

$$\mathbf{M}\dot{\mathbf{T}} + \dot{\mathbf{K}}\mathbf{T} = \mathbf{F}.\tag{12}$$

The system of ordinary differential equations expressed by the matrix equation 12 must be completed by providing an initial condition **T**(0) = **T**0. Therefore we seek to solve the initial value problem defined by:

$$\mathbf{M}\dot{\mathbf{T}} + \mathbf{\hat{K}T} = \mathbf{F}\_{\prime}$$

$$\mathbf{T}(0) = \mathbf{T}\_{0}.\tag{13}$$

This can be converted to a system of algebraic equations by dividing the time domain into steps and using finite differences to approximate the time derivatives. Equation 13 can be solved by considering a weighted average of the time derivatives at two consecutive time steps (*ts* and *ts*+1) and developing an iterative procedure to find the solution at each step (Reddy & Gartling, 1994):

$$\begin{aligned} \mathbf{T}(t\_{s+1}) &= \mathbf{T}(t\_s) + \dot{\mathbf{T}}(t\_{s+a})(t\_{s+1} - t\_s), \\ \mathbf{T}(t\_{s+a}) &= (1 - a)\dot{\mathbf{T}}(t\_s) + a\dot{\mathbf{T}}(t\_{s+1}). \end{aligned} \tag{14}$$

Different choices of *α* lead to well known approximation schemes that are commonly found in the literature:


Substitution of Equation 13 in Equation 14 yields the solution to the problem:

$$\begin{aligned} \mathbf{T}(t\_{s+1}) &= \mathbf{T}(t\_s) + (1 - a)\mathbf{M}^{-1}\left[\mathbf{T}(t\_s)\right] \left\{ \mathbf{F}\left[\mathbf{T}(t\_s)\right] - \hat{\mathbf{K}}\left[\mathbf{T}(t\_s)\right] \mathbf{T}(t\_s) \right\} (t\_{s-1} - t\_s) \\ a\mathbf{M}^{-1}\left[\mathbf{T}(t\_{s+1})\right] \left\{ \mathbf{F}\left[\mathbf{T}(t\_{s+1})\right] - \hat{\mathbf{K}}\left[\mathbf{T}(t\_{s+1})\right] \mathbf{T}(t\_{s+1}) \right\} (t\_{s-1} - t\_s) . \end{aligned} \tag{15}$$

In general Equation 15 leads to an implicit scheme that requires iterative solutions to be found within each time step. The forward difference method is the only one of the above which is an explicit method and is the easiest to implement. It results in a simple iterative solution where **T**(*ts*+1) is readily obtained from the solution at the previous step **T**(*ts*), and is given by:

$$\mathbf{T}(t\_{s+1}) = \mathbf{T}(t\_s) + \mathbf{M}^{-1} \left[ \mathbf{T}(t\_s) \right] \left\{ \mathbf{F} \left[ \mathbf{T}(t\_s) \right] - \hat{\mathbf{K}} \left[ \mathbf{T}(t\_s) \right] \mathbf{T}(t\_s) \right\} (t\_{s-1} - t\_s). \tag{16}$$

Starting from **T**(0) = **T**0, the solution at subsequent steps can be calculated from Equation 16. Equation 16 is a general expression that relates the temperatures at various points of a geometry by requiring the balance of heat fluxes across the boundaries between neighbouring elements and the continuity of the temperature field, governed by the weak form of the heat conduction equation. The temperature evolution during additive manufacture for a component of arbitrary geometry can be found by implementing Equation 16 as a computer code.

#### **2.3 Representation of the physical domain**

6 Will-be-set-by-IN-TECH

different elements are assembled together by requiring the balance of this flux from each element to its neighbours and the continuity of the temperature field *T*(*r*, *t*). This system

**T** = **F***<sup>e</sup>*

<sup>−</sup>*<sup>h</sup>* <sup>−</sup> *�σsT*<sup>3</sup>

*qlaser* + *hT*<sup>∞</sup> + *�σsT*<sup>4</sup>

**K**ˆ *e*

The equations of the single elements are assembled by summing the element equations

The system of ordinary differential equations expressed by the matrix equation 12 must be completed by providing an initial condition **T**(0) = **T**0. Therefore we seek to solve the initial

**MT**˙ + **KT**ˆ = **F**,

This can be converted to a system of algebraic equations by dividing the time domain into steps and using finite differences to approximate the time derivatives. Equation 13 can be solved by considering a weighted average of the time derivatives at two consecutive time steps (*ts* and *ts*+1) and developing an iterative procedure to find the solution at each step

**<sup>T</sup>**(*ts*+1) = **<sup>T</sup>**(*ts*) + **<sup>T</sup>**˙ (*ts*+*α*)(*ts*+<sup>1</sup> <sup>−</sup> *ts*),

Different choices of *α* lead to well known approximation schemes that are commonly found

*α* = 2/3, Galerkin scheme

, **<sup>K</sup>**<sup>ˆ</sup> <sup>=</sup> <sup>∑</sup>*<sup>e</sup>*

 *dS*

, **<sup>F</sup>** <sup>=</sup> <sup>∑</sup>*<sup>e</sup>*

∞ *dS*,

**F***e*

**MT**˙ + **KT**ˆ = **F**. (12)

**<sup>T</sup>**(0) = **<sup>T</sup>**0. (13)

**<sup>T</sup>**˙ (*ts*+*α*)=(<sup>1</sup> <sup>−</sup> *<sup>α</sup>*)**T**˙ (*ts*) + *<sup>α</sup>***T**˙ (*ts*+1). (14)

*α* = 0, forward difference, or Euler, scheme

*α* = 1/2, Crank-Nicholson scheme

*α* = 1, backward difference scheme.

, (9)

, (11)

(10)

**<sup>T</sup>**˙*<sup>e</sup>* + **<sup>K</sup>**<sup>ˆ</sup> *<sup>e</sup>*

**M***<sup>e</sup>*

of equations is commonly written in matrix form as:

**M***<sup>e</sup>* = *M<sup>e</sup> ij*

**<sup>K</sup>**<sup>ˆ</sup> *<sup>e</sup>* = *<sup>K</sup><sup>e</sup>*

**F***<sup>e</sup>* = *Q<sup>e</sup>*

*ij* − *Iij* Γ2 *ψi* 

*<sup>i</sup>* + Γ2 *ψi* 

**<sup>M</sup>** <sup>=</sup> <sup>∑</sup>*<sup>e</sup>*

**M***<sup>e</sup>*

where

with *Iij* the identity matrix.

corresponding to the same nodes:

resulting in the global equation:

value problem defined by:

(Reddy & Gartling, 1994):

in the literature:

A finite element model should ideally describe the geometry of the substrate and the tracks as closely as possible. Frequently the substrate is a parallelepiped which can be easily represented in the form of a finite element mesh. However, it is more difficult to develop a mesh which allows a step wise description of the deposition of tracks with complex 3-D features, such as curved cross sections or curved fronts. To describe the full detail of track overlap during manufacture, the finite element mesh becomes complex and requires many elements for the proper representation of the 3-D features of the tracks, as shown in Figure 2.a.

Fig. 2. (a) Finite element mesh of substrate and tracks. (b) Step-wise approach to simulate the addition of material. New elements are activated at liquidus temperature. Adapted from Crespo and Vilar (Crespo & Vilar, 2010)

One commonly applied strategy to reduce the number of elements is to use a fine mesh only in regions which have complex geometries or where thermal gradients are expected to be high (in the vicinity of interaction zone between the energy source and the material), while using a coarser mesh away from these zones (Figure 2.a). The level of refinement shown in Figure

Fig. 3. Phase transformations during rapid manufacturing of Ti-6Al-4V.

*eq <sup>α</sup>* (*T*),

of parts by rapid manufacturing are represented in the diagram of Figure 3.

**3.1 Phase transformations during cooling from the liquid phase**

different phase transformations depending on the cooling rate.

the *β*-transus, and is given by (R. Castro, 1966):

*<sup>β</sup>* (*T*) = 1 − *f*

*f eq <sup>α</sup>* (*T*) =

*f eq*

a mixture of *α* and *β* phases for temperatures between room temperature and 980 ◦C, which is called the *β*-transus temperature (Polmear, 1989). The proportion of *β* phase in equilibrium depends on the temperature, varying from approximately 0.08 at room temperature to 1.00 at

<sup>323</sup> Modelling of Heat Transfer

and Phase Transformations in the Rapid Manufacturing of Titanium Components

with T in ◦C. In titanium alloys the *β*-transus temperature represents the minimum temperature above which *β* is the only equilibrium phase. The phase transformations that can occur due to the consecutive thermal cycles generated by layer overlap during build-up

Prior to incorporation into the part the feedstock Ti-6Al-4V is melted, and after solidification its structure consists of *β* phase. During cooling to room temperature *β* may undergo two

0.925 <sup>−</sup> 0.925.*e*[0.0085(980−*T*)], *<sup>T</sup>* <sup>≤</sup> <sup>980</sup>◦*C*/*<sup>s</sup>*

0, *T* > 980◦*C*/*s*

(17)

2.a is necessary if certain aspects of the fabrication process such as the formation of hot-spots or the solidification rate must be predicted, which require the precise shape of the melt pool and of the incorporated material to be taken into account (Bontha et al., 2006; Crespo et al., 2006). When the purpose of the simulation does not demand such a rigorous description of the track shape, simpler meshes may be used by assuming that the shapes of the melt pool and of the tracks can be approximated by simpler geometries. This has the advantage of reducing considerably the number of elements in the mesh, and as a consequence the number of calculations and the computational time necessary to resolve the problem. Several authors have developed finite element models which use simple cubic elements to simulate the addition of material and have demonstrated the validity of this approach (Costa et al., 2005; Deus & Mazumder, 2006), which is also used in the present work. If the deposition if assumed to take place in the mid-plane of the substrate, there is a symmetry plane in respect of which heat flow is symmetrical and one needs only consider half the geometry of the problem, as illustrated in Figure 2.b, further reducing the computational time needed to achieve the solution for the heat transfer problem.

In the model proposed in this chapter, Equation 16 is solved iteratively for each element in the step by step approach described in the previous section. Addition of material is taken into account by activating at each new time step elements with a volume corresponding to the volume of material incorporated into the part during the duration of that step (Figure 2.b), based on a methodology first presented by Costa et al. (Costa et al., 2005). Taking into consideration the results of Neto and Vilar (Neto & Vilar, 2002), who showed that in blown powder laser cladding the powder flying through the laser beam often reaches the liquidus temperature before impinging into the part, the newly active elements are assumed to be at the liquidus temperature.

#### **3. Phase transformations during the rapid manufacturing of titanium components**

Titanium alloys are being increasingly used in a wide range of applications due to properties such as high strength to weight ratio, excellent corrosion resistance, high temperature strength and biocompatibility. These properties have made titanium alloys a widespread material in industries such as the aerospace, automotive, biomedical, energy production, chemical, off-shore and marine industries, among others (Boyer et al., 1994; Donachie, 2004).

In the last decade, the Ti-6Al-4V alloy has accounted for more than half the production of titanium alloys worldwide, a market estimated at more than \$2,000 million (Leyens & Peters, 2003). This predominance is mainly due to Ti-6Al-4V having the best all-around mechanical characteristics for numerous applications. This alloy is extensively used in the aerospace industry for the production of turbine engines and airframe components, which account for approximately 80% of its total usage. Additionally, Ti-6Al-4V presents excellent biocompatibility and osseointegration properties which have made it a natural choice as a biomaterial for the fabrication of implants and other biomedical devices (Brunette, 2001; Yoshiki, 2007). When compared to other materials usually used for the same purpose, such as stainless steel or CoCr alloys, Ti-6Al-4V allows the production of much stronger, lighter and less stiff implants and with improved biomechanical behaviour.

Ti-6Al-4V is an *α*/*β* titanium alloy that contains 6% of the *α*-phase stabilising element Al, and 4% of the *β*-phase stabilising element V in its composition. As a result of the combined effect of these two alloying elements, the equilibrium microstructure of Ti-6Al-4V consists of 8 Will-be-set-by-IN-TECH

2.a is necessary if certain aspects of the fabrication process such as the formation of hot-spots or the solidification rate must be predicted, which require the precise shape of the melt pool and of the incorporated material to be taken into account (Bontha et al., 2006; Crespo et al., 2006). When the purpose of the simulation does not demand such a rigorous description of the track shape, simpler meshes may be used by assuming that the shapes of the melt pool and of the tracks can be approximated by simpler geometries. This has the advantage of reducing considerably the number of elements in the mesh, and as a consequence the number of calculations and the computational time necessary to resolve the problem. Several authors have developed finite element models which use simple cubic elements to simulate the addition of material and have demonstrated the validity of this approach (Costa et al., 2005; Deus & Mazumder, 2006), which is also used in the present work. If the deposition if assumed to take place in the mid-plane of the substrate, there is a symmetry plane in respect of which heat flow is symmetrical and one needs only consider half the geometry of the problem, as illustrated in Figure 2.b, further reducing the computational time needed to achieve the

In the model proposed in this chapter, Equation 16 is solved iteratively for each element in the step by step approach described in the previous section. Addition of material is taken into account by activating at each new time step elements with a volume corresponding to the volume of material incorporated into the part during the duration of that step (Figure 2.b), based on a methodology first presented by Costa et al. (Costa et al., 2005). Taking into consideration the results of Neto and Vilar (Neto & Vilar, 2002), who showed that in blown powder laser cladding the powder flying through the laser beam often reaches the liquidus temperature before impinging into the part, the newly active elements are assumed to be at

**3. Phase transformations during the rapid manufacturing of titanium components** Titanium alloys are being increasingly used in a wide range of applications due to properties such as high strength to weight ratio, excellent corrosion resistance, high temperature strength and biocompatibility. These properties have made titanium alloys a widespread material in industries such as the aerospace, automotive, biomedical, energy production, chemical,

In the last decade, the Ti-6Al-4V alloy has accounted for more than half the production of titanium alloys worldwide, a market estimated at more than \$2,000 million (Leyens & Peters, 2003). This predominance is mainly due to Ti-6Al-4V having the best all-around mechanical characteristics for numerous applications. This alloy is extensively used in the aerospace industry for the production of turbine engines and airframe components, which account for approximately 80% of its total usage. Additionally, Ti-6Al-4V presents excellent biocompatibility and osseointegration properties which have made it a natural choice as a biomaterial for the fabrication of implants and other biomedical devices (Brunette, 2001; Yoshiki, 2007). When compared to other materials usually used for the same purpose, such as stainless steel or CoCr alloys, Ti-6Al-4V allows the production of much stronger, lighter and

Ti-6Al-4V is an *α*/*β* titanium alloy that contains 6% of the *α*-phase stabilising element Al, and 4% of the *β*-phase stabilising element V in its composition. As a result of the combined effect of these two alloying elements, the equilibrium microstructure of Ti-6Al-4V consists of

off-shore and marine industries, among others (Boyer et al., 1994; Donachie, 2004).

less stiff implants and with improved biomechanical behaviour.

solution for the heat transfer problem.

the liquidus temperature.

Fig. 3. Phase transformations during rapid manufacturing of Ti-6Al-4V.

a mixture of *α* and *β* phases for temperatures between room temperature and 980 ◦C, which is called the *β*-transus temperature (Polmear, 1989). The proportion of *β* phase in equilibrium depends on the temperature, varying from approximately 0.08 at room temperature to 1.00 at the *β*-transus, and is given by (R. Castro, 1966):

$$f\_{\alpha}^{eq}(T) = \begin{cases} 0.925 - 0.925e^{[0.0085(980 - T)]}, & T \le 980^{\circ} \text{C/s} \\ 0, & T > 980^{\circ} \text{C/s} \end{cases} \tag{17}$$
  $f\_{\beta}^{eq}(T) = 1 - f\_{\alpha}^{eq}(T),$ 

with T in ◦C. In titanium alloys the *β*-transus temperature represents the minimum temperature above which *β* is the only equilibrium phase. The phase transformations that can occur due to the consecutive thermal cycles generated by layer overlap during build-up of parts by rapid manufacturing are represented in the diagram of Figure 3.

#### **3.1 Phase transformations during cooling from the liquid phase**

Prior to incorporation into the part the feedstock Ti-6Al-4V is melted, and after solidification its structure consists of *β* phase. During cooling to room temperature *β* may undergo two different phase transformations depending on the cooling rate.

where *t f*

Figure 4.

transformations.

into martensite (*α*

**3.1.2 Martensitic transformations**

�

*<sup>s</sup>* is given by:

*t f <sup>s</sup>* = *ns* 

and Phase Transformations in the Rapid Manufacturing of Titanium Components

<sup>−</sup> *ln*[<sup>1</sup> <sup>−</sup> *<sup>f</sup>α*(*ts*)/ *<sup>f</sup>*

<sup>325</sup> Modelling of Heat Transfer

The application of the additivity rule to the Johnson-Mehl-Avrami equation is illustrated in

Fig. 4. Generalization of the Johnson-Mehl-Avrami equation for anisothermal

temperature (*Ms*) and is given by (Koistinen & Marburger, 1959):

*f*

For cooling rates higher than 410 ◦C/s the *β* → *α* diffusional transformation is suppressed and *β* decomposes by a martensitic transformation. The proportion of *β* transformed

) depends essentially on the undercooling below the martensite start

*<sup>α</sup>*�(*T*) = 1 − *exp* [−*γ*(*Ms* − *T*)] . (24)

*ks*

*eq <sup>α</sup>* (*Ts*)]

. (23)

#### **3.1.1 Diffusional transformations**

For cooling rates lower than 410 ◦C/s, a *β* → *α* transformation takes place controlled by a diffusional mechanism, starting at the *β*-transus temperature (980 ◦C). At room temperature, the final microstructure consists of *α* and *β* because the transformation does not reach completion. In isothermal condition the kinetics of this transformation is described by the Johnson-Mehl-Avrami (JMA) equation:

$$f(t) = 1 - \exp\left(-jt^n\right),\tag{18}$$

where *fα*(*t*), *k* and *n* are the fraction of *α* formed after time *t*, the reaction rate constant and the Avrami exponent, respectively. The values for *k* and *n* were determined as a function of the temperature by Malinov et al. (Malinov, Markovsky, Sha & Guo, 2001). The Johnson-Mehl-Avrami equation cannot be used to describe the kinetics of anisothermal transformations because the reaction rate constant *k* depends on the temperature. As a consequence, the direct integration of the Johnson-Mehl-Avrami equation to calculate the transformed proportion during cooling is not possible. Nevertheless, good results have been achieved by generalising the Johnson-Mehl-Avrami equation to anisothermal conditions using the additivity rule (Malinov, Guo, Sha & Wilson, 2001; S. Denis, 1992). In this method, continuous cooling is replaced by a series of small consecutive isothermal steps where the Johnson-Mehl-Avrami equation can be applied. During the first isothermal time step, [*t*0, *t*1[, at temperature *T*0, the fraction of *α* phase formed can be calculated from Equation 18 and is given by:

$$f\_{\mathfrak{a}}(t\_1) = \left\{1 - \exp\left[-k\_0(t\_1 - t\_0)^{\mathfrak{a}\_0}\right]\right\} f\_{\mathfrak{a}}^{\mathfrak{a}q}(T\_0) \tag{19}$$

where *k*<sup>0</sup> and *n*<sup>0</sup> are the reaction rate constant and Avrami exponent at the temperature *T*0, respectively. In the next interval, [*t*1, *t*2[, the transformation is assumed to take place at the temperature *T*1, but one must take into consideration the fact that a fraction *fα*(*t*1) of *α* phase has already formed in the previous step. Substituting the fraction *fα*(*t*1) in Equation 18, one can calculate the time it would take to form the proportion *fα*(*t*1) of *α* phase if the whole transformation had taken place at the temperature *T*1:

$$t\_1^f = \sqrt[n]{-\frac{\ln[1 - f\_a(t\_1)/f\_a^{eq}(T\_1)]}{k\_1}},\tag{20}$$

where *k*<sup>1</sup> and *n*<sup>1</sup> are the reaction rate constant and Avrami exponent at the temperature *T*1. The additivity principle requires that *t f* <sup>1</sup> be the initial time for the new transformation step. Therefore, for the time interval [*t*1, *t*2[, one gets:

$$f\_{\mathfrak{a}}(t\_2) = \left\{ 1 - \exp\left[ -k\_1 (t\_1^f + t\_2 - t\_1)^{n\_1} \right] \right\} f\_{\mathfrak{a}}^{\mathfrak{a}q}(T\_1). \tag{21}$$

Equation 21 can be generalised for an arbitrary time step [*ts*, *ts*+1[ at temperature *Ts*, leading to a fraction of *α* formed during that step given by:

$$f\_{\mathfrak{A}}(t\_{\mathfrak{s}+1}) = \left\{1 - \exp\left[-k\_{\mathfrak{s}}(t\_{\mathfrak{s}}^{f} + t\_{\mathfrak{s}+1} - t\_{\mathfrak{s}})^{n\_{\mathfrak{s}}}\right]\right\} f\_{\mathfrak{A}}^{\mathfrak{e}q}(T\_{\mathfrak{s}}) \,. \tag{22}$$

where *t f <sup>s</sup>* is given by:

10 Will-be-set-by-IN-TECH

For cooling rates lower than 410 ◦C/s, a *β* → *α* transformation takes place controlled by a diffusional mechanism, starting at the *β*-transus temperature (980 ◦C). At room temperature, the final microstructure consists of *α* and *β* because the transformation does not reach completion. In isothermal condition the kinetics of this transformation is described by the

where *fα*(*t*), *k* and *n* are the fraction of *α* formed after time *t*, the reaction rate constant and the Avrami exponent, respectively. The values for *k* and *n* were determined as a function of the temperature by Malinov et al. (Malinov, Markovsky, Sha & Guo, 2001). The Johnson-Mehl-Avrami equation cannot be used to describe the kinetics of anisothermal transformations because the reaction rate constant *k* depends on the temperature. As a consequence, the direct integration of the Johnson-Mehl-Avrami equation to calculate the transformed proportion during cooling is not possible. Nevertheless, good results have been achieved by generalising the Johnson-Mehl-Avrami equation to anisothermal conditions using the additivity rule (Malinov, Guo, Sha & Wilson, 2001; S. Denis, 1992). In this method, continuous cooling is replaced by a series of small consecutive isothermal steps where the Johnson-Mehl-Avrami equation can be applied. During the first isothermal time step, [*t*0, *t*1[, at temperature *T*0, the fraction of *α* phase formed can be calculated from Equation 18 and is

<sup>1</sup> <sup>−</sup> *exp* [−*k*0(*t*<sup>1</sup> <sup>−</sup> *<sup>t</sup>*0)*n*<sup>0</sup> ]

where *k*<sup>0</sup> and *n*<sup>0</sup> are the reaction rate constant and Avrami exponent at the temperature *T*0, respectively. In the next interval, [*t*1, *t*2[, the transformation is assumed to take place at the temperature *T*1, but one must take into consideration the fact that a fraction *fα*(*t*1) of *α* phase has already formed in the previous step. Substituting the fraction *fα*(*t*1) in Equation 18, one can calculate the time it would take to form the proportion *fα*(*t*1) of *α* phase if the whole

<sup>−</sup> *ln*[<sup>1</sup> <sup>−</sup> *<sup>f</sup>α*(*t*1)/ *<sup>f</sup>*

where *k*<sup>1</sup> and *n*<sup>1</sup> are the reaction rate constant and Avrami exponent at the temperature *T*1.

Equation 21 can be generalised for an arbitrary time step [*ts*, *ts*+1[ at temperature *Ts*, leading

*f*

 −*k*1(*t f*

 −*ks*(*t f* *k*1

<sup>1</sup> <sup>+</sup> *<sup>t</sup>*<sup>2</sup> <sup>−</sup> *<sup>t</sup>*1)*n*<sup>1</sup>

*<sup>s</sup>* <sup>+</sup> *ts*+<sup>1</sup> <sup>−</sup> *ts*)*ns*

*<sup>f</sup>*(*t*) = <sup>1</sup> <sup>−</sup> *exp* (−*jtn*), (18)

 . *f eq*

*eq <sup>α</sup>* (*T*1)]

<sup>1</sup> be the initial time for the new transformation step.

 . *f eq*

> . *f eq*

*<sup>α</sup>* (*T*0) (19)

, (20)

*<sup>α</sup>* (*T*1). (21)

*<sup>α</sup>* (*Ts*), (22)

**3.1.1 Diffusional transformations**

Johnson-Mehl-Avrami (JMA) equation:

*fα*(*t*1) =

transformation had taken place at the temperature *T*1:

The additivity principle requires that *t*

Therefore, for the time interval [*t*1, *t*2[, one gets:

*fα*(*t*2) =

to a fraction of *α* formed during that step given by:

*fα*(*ts*+1) =

*t f* <sup>1</sup> <sup>=</sup> *<sup>n</sup>*<sup>1</sup> 

1 − *exp*

1 − *exp*

given by:

$$t\_s^f = \sqrt[n\_s]{-\frac{\ln[1 - f\_a(t\_s)/f\_a^{eq}(T\_s)]}{k\_s}}.\tag{23}$$

The application of the additivity rule to the Johnson-Mehl-Avrami equation is illustrated in Figure 4.

Fig. 4. Generalization of the Johnson-Mehl-Avrami equation for anisothermal transformations.

#### **3.1.2 Martensitic transformations**

For cooling rates higher than 410 ◦C/s the *β* → *α* diffusional transformation is suppressed and *β* decomposes by a martensitic transformation. The proportion of *β* transformed into martensite (*α* � ) depends essentially on the undercooling below the martensite start temperature (*Ms*) and is given by (Koistinen & Marburger, 1959):

$$f\_{\mathfrak{a}'}(T) = 1 - \exp\left[-\gamma(M\_s - T)\right].\tag{24}$$

**4. Results**

laser.

**4.1 Experimental confirmation**

**4.1.1 Simulation results**

locations (Laeng et al., 2000; R.Vilar, 1999; 2001).

The model was first validated by comparing the calculation results with the experimental distributions of microstructure and properties found in Ti-6Al-4V walls produced by laser powder deposition (LPD), a rapid manufacturing technique that uses a focused laser beam to melt a stream of metallic powder and deposit the molten material continuously at precise

<sup>327</sup> Modelling of Heat Transfer

and Phase Transformations in the Rapid Manufacturing of Titanium Components

The model was applied to simulate the phase transformations occurring during the deposition of a 75 layer Ti-6Al-4V wall with 0.32 mm width, 10.00 mm length and 3.50 mm height, represented in Figure 5. The scanning speed was 4 mm/s, the laser beam diameter 0.3 mm, the idle time between the deposition of consecutive layers 6 s and the initial substrate temperature 20 ◦C. The laser beam power was varied according to the plot of Figure 6.a, reflecting the power adjustments performed by a closed loop online control system utilised during the manufacture of the experimental sample, which acts to keep the size of the melt pool generated by the laser beam at the surface of the workpiece constant. An initial beam power of 130 W was used and progressively decreased with each new deposited layer up to the 20th layer, where a beam power of 50 W was reached and kept constant for the rest of the process. An average absorptivity of 15 % was considered in the calculations, according to the results of Hu *et al.* (Hu & Baker, 1999) regarding the laser deposition of Ti-6Al-4V using a CO2

Fig. 5. View of the substrate and the wall with a detail of the wall mesh.

The calculated phase distribution is shown in Figures 6.b and 7. The highest volume fractions of *α* and *β* phases (0.03 and 0.07 respectively) occur close to the substrate, and decrease as the distance from the substrate increases, reaching zero in the uppermost layers of the part. Conversely, the volume fraction of martensite is lowest near the substrate (approximately 0.9) and has a maximum at the top of the wall, where the structure is fully martensitic. The cooling rates experienced by the material during the deposition process are always higher than 410 ◦C/s (Figure 8.a), and, as a consequence, after solidification the material undergoes a martensitic transformation during cooling to room temperature. Figure 8.a

The values of *γ*, *Ms* and *Mf* used in the present work (0.015 ◦C−1, 650 ◦C and 400 ◦C respectively) were calculated on the basis of the results of Elmer et al. (Elmer et al., 2004). If the material cools below *Mf* its microstructure is fully martensitic.

#### **3.2 Phase transformations during re-heating**

When new layers are added to the part, the previously deposited material undergoes heating/cooling cycles that may induce microstructural and properties changes. If the microstructure formed in first thermal cycle is composed of *α* + *β*, reheating will lead to diffusion controlled *α* → *β* transformation with a kinetics described by the JMA equation generalised to anisothermal processes (Equation 22). If, on the other hand, the microstructure is martensitic, heating up the material into the tempering range (> 400 ◦C) will cause the decomposition of *α* � into a mixture of *α* and *β*. This transformation is also diffusion controlled and its kinetics are also described by the JMA equation (Equation 18). The values of *k* and *n* in Equation 18 for this reaction were determined by Mur et al. (Mur et al., 1996). If the decomposition is incomplete, tempering results in a three-phase microstructure consisting of *α* � + *α* + *β*.

#### **3.3 Phase transformations during second cooling**

During cooling down to room temperature at cooling rates lower than 410 ◦C/s *β* phase decomposes into *α* by a diffusion controlled mechanism. For cooling rates in excess of 410 ◦C/s *β* may undergo a martensitic transformation or be retained at room temperature, depending on the volume fraction of this phase in the alloy. Several authors have observed that *β* is completely retained upon quenching if its proportion in the alloy is lower than 0.25, because the *β* phase is enriched in vanadium, a *β* stabiliser (Fan, 1993; Lee et al., 1991; R. Castro, 1966). If the volume fraction is higher than 0.25 a proportion of *β* given by (Fan, 1993):

$$f\_r = 0.25 - 0.25.f\_{\beta}(T\_0)\_{\prime} \tag{25}$$

is retained at room temperature, where *fb*(*T*0) is the volume fraction of *β* prior to quenching. The remaining *β* ( *fb*(*T*0) − *fr* ) undergoes a martensitic transformation. As a result, cooling an alloy consisting only of *β* phase at rates higher than 410 ◦C/s originates a fully martensitic structure, while materials with smaller volume fractions of this phase retain a variable proportion of *β* (Figure 3). Thus, the martensite volume fraction is given by:

$$f\_{\mathfrak{n}'}(T) = f\_{\mathfrak{n}'}(T\_0) + (f\_{\mathfrak{f}}(T\_0) - f\_r) \left[1 - \exp\left(-\gamma (M\_\mathfrak{s} - T)\right)\right],\tag{26}$$

with *f <sup>α</sup>*�(*T*0) the volume fraction of *α* � phase present in the alloy prior to quenching. Similar phase transformations will occur during subsequent thermal cycles and the final microstructure will result from all the consecutive transformations occurring at each point.

#### **3.4 Calculation of mechanical properties**

The Young's modulus and hardness were calculated from the phase constitution of the alloy using the rule of mixtures (Costa et al., 2005; Fan, 1993; Lee et al., 1991). The Young's moduli of *α*, *β* and *α* � are 117, 82 and 114 GPa respectively and the Vickers hardnesses are 320, 140 and 350 HV.

#### **4. Results**

12 Will-be-set-by-IN-TECH

The values of *γ*, *Ms* and *Mf* used in the present work (0.015 ◦C−1, 650 ◦C and 400 ◦C respectively) were calculated on the basis of the results of Elmer et al. (Elmer et al., 2004).

When new layers are added to the part, the previously deposited material undergoes heating/cooling cycles that may induce microstructural and properties changes. If the microstructure formed in first thermal cycle is composed of *α* + *β*, reheating will lead to diffusion controlled *α* → *β* transformation with a kinetics described by the JMA equation generalised to anisothermal processes (Equation 22). If, on the other hand, the microstructure is martensitic, heating up the material into the tempering range (> 400 ◦C) will cause the

and its kinetics are also described by the JMA equation (Equation 18). The values of *k* and *n* in Equation 18 for this reaction were determined by Mur et al. (Mur et al., 1996). If the decomposition is incomplete, tempering results in a three-phase microstructure consisting of

During cooling down to room temperature at cooling rates lower than 410 ◦C/s *β* phase decomposes into *α* by a diffusion controlled mechanism. For cooling rates in excess of 410 ◦C/s *β* may undergo a martensitic transformation or be retained at room temperature, depending on the volume fraction of this phase in the alloy. Several authors have observed that *β* is completely retained upon quenching if its proportion in the alloy is lower than 0.25, because the *β* phase is enriched in vanadium, a *β* stabiliser (Fan, 1993; Lee et al., 1991; R. Castro, 1966). If the volume fraction is higher than 0.25 a proportion of *β* given by (Fan,

is retained at room temperature, where *fb*(*T*0) is the volume fraction of *β* prior to quenching. The remaining *β* ( *fb*(*T*0) − *fr* ) undergoes a martensitic transformation. As a result, cooling an alloy consisting only of *β* phase at rates higher than 410 ◦C/s originates a fully martensitic structure, while materials with smaller volume fractions of this phase retain a variable

Similar phase transformations will occur during subsequent thermal cycles and the final microstructure will result from all the consecutive transformations occurring at each point.

The Young's modulus and hardness were calculated from the phase constitution of the alloy using the rule of mixtures (Costa et al., 2005; Fan, 1993; Lee et al., 1991). The Young's moduli

are 117, 82 and 114 GPa respectively and the Vickers hardnesses are 320, 140 and

proportion of *β* (Figure 3). Thus, the martensite volume fraction is given by:

�

into a mixture of *α* and *β*. This transformation is also diffusion controlled

*fr* = 0.25 − 0.25. *fβ*(*T*0), (25)

*<sup>α</sup>*�(*T*0)+(*fβ*(*T*0) − *fr*)[1 − *exp* (−*γ*(*Ms* − *T*))] , (26)

phase present in the alloy prior to quenching.

If the material cools below *Mf* its microstructure is fully martensitic.

**3.2 Phase transformations during re-heating**

�

*f*

**3.4 Calculation of mechanical properties**

*<sup>α</sup>*�(*T*) = *f*

*<sup>α</sup>*�(*T*0) the volume fraction of *α*

**3.3 Phase transformations during second cooling**

decomposition of *α*

+ *α* + *β*.

*α* �

1993):

with *f*

of *α*, *β* and *α*

350 HV.

�

#### **4.1 Experimental confirmation**

The model was first validated by comparing the calculation results with the experimental distributions of microstructure and properties found in Ti-6Al-4V walls produced by laser powder deposition (LPD), a rapid manufacturing technique that uses a focused laser beam to melt a stream of metallic powder and deposit the molten material continuously at precise locations (Laeng et al., 2000; R.Vilar, 1999; 2001).

#### **4.1.1 Simulation results**

The model was applied to simulate the phase transformations occurring during the deposition of a 75 layer Ti-6Al-4V wall with 0.32 mm width, 10.00 mm length and 3.50 mm height, represented in Figure 5. The scanning speed was 4 mm/s, the laser beam diameter 0.3 mm, the idle time between the deposition of consecutive layers 6 s and the initial substrate temperature 20 ◦C. The laser beam power was varied according to the plot of Figure 6.a, reflecting the power adjustments performed by a closed loop online control system utilised during the manufacture of the experimental sample, which acts to keep the size of the melt pool generated by the laser beam at the surface of the workpiece constant. An initial beam power of 130 W was used and progressively decreased with each new deposited layer up to the 20th layer, where a beam power of 50 W was reached and kept constant for the rest of the process. An average absorptivity of 15 % was considered in the calculations, according to the results of Hu *et al.* (Hu & Baker, 1999) regarding the laser deposition of Ti-6Al-4V using a CO2 laser.

Fig. 5. View of the substrate and the wall with a detail of the wall mesh.

The calculated phase distribution is shown in Figures 6.b and 7. The highest volume fractions of *α* and *β* phases (0.03 and 0.07 respectively) occur close to the substrate, and decrease as the distance from the substrate increases, reaching zero in the uppermost layers of the part. Conversely, the volume fraction of martensite is lowest near the substrate (approximately 0.9) and has a maximum at the top of the wall, where the structure is fully martensitic. The cooling rates experienced by the material during the deposition process are always higher than 410 ◦C/s (Figure 8.a), and, as a consequence, after solidification the material undergoes a martensitic transformation during cooling to room temperature. Figure 8.a

(a) (b)

<sup>329</sup> Modelling of Heat Transfer

and Phase Transformations in the Rapid Manufacturing of Titanium Components

(a) (b)

Fig. 9. (a) Temperature evolution of the interface between the wall and the substrate after the

its deposition, which amounts to approximately 250 s in the tempering range, leading to a decomposition of approximately 10% of the previously formed martensite. The evolution of the phase constitution of the material is presented in Figure 9.a and the Young's modulus and

The distributions of microstructure, Young's modulus and hardness predicted by the model were compared to the values measured on a sample manufactured with similar processing parameters to validate the model. The system used in the experimental tests was developed by C. Meacock and R. Vilar (Meacock, 2009) to manufacture small to medium size parts for biomedical applications and the experimental work described in this section was carried out in collaboration with these authors. The system uses a CO2 laser with a maximum beam power of 130 W which can be focused to a spot of 0.3 mm in diameter by means of a ZnSe lens with a focal length of 63.5 mm. The system employs a closed loop online control system whereby

deposition of each layer. (b) Time above 400 ◦C during the deposition of each layer,

measured at the interface between the wall and the substrate.

**4.1.2 Experimental results**

hardness variations along the wall height are presented in Figure 10.b

Fig. 8. (a) Cooling rates experienced by the material deposited in the different layers. (b) Temperature evolution of the material deposited in the first layer for the first 120 s of the fabrication process. Tempering of the martensite takes place at temperatures higher than 400

◦C.

Fig. 6. (a) Laser beam power used to deposit each layer. (b) Phase constitution as a function of the distance from the fusion line.

Fig. 7. *β* phase distribution.

shows that the cooling rate progressively decreases as the number of layers increases and asymptotically approaches a value below the martensite critical cooling rate (410 ◦C/s). Therefore, the deposition of additional layers would likely lead to the suppression of the martensitic transformation in the top layers of the part.

The thermal cycles originated by layer overlap heat up the previously deposited material to temperatures in the tempering range (*T* > 400◦C), causing the progressive decomposition of the martensite into *α* and *β* (Figure 8.b).

The idle time between the deposition of consecutive layers used (6 s) is too short to allow the part to cool down to room temperature before the deposition of a new layer. As a result the temperature of the workpiece increases progressively as the deposition advances, eventually stabilising at approximately 270 ◦C after the deposition of the 15th layer, as depicted in the plot of Figure 9.a.

This facilitates tempering because, as heat accumulates in the part, the material residence time in the tempering temperatures range increases from less than 1 s in the first cycles to approximately 4 s from the 15th cycle onwards (Figure 9.b).

The cumulative effect of the consecutive thermal cycles is sufficient for significant tempering to take place, particularly in the layers deposited at the beginning of the buildup process. For example, the material in the first layer is subjected to 74 thermal cycles subsequent to

Fig. 8. (a) Cooling rates experienced by the material deposited in the different layers. (b) Temperature evolution of the material deposited in the first layer for the first 120 s of the fabrication process. Tempering of the martensite takes place at temperatures higher than 400 ◦C.

Fig. 9. (a) Temperature evolution of the interface between the wall and the substrate after the deposition of each layer. (b) Time above 400 ◦C during the deposition of each layer, measured at the interface between the wall and the substrate.

its deposition, which amounts to approximately 250 s in the tempering range, leading to a decomposition of approximately 10% of the previously formed martensite. The evolution of the phase constitution of the material is presented in Figure 9.a and the Young's modulus and hardness variations along the wall height are presented in Figure 10.b

#### **4.1.2 Experimental results**

14 Will-be-set-by-IN-TECH

(a) (b)

Fig. 6. (a) Laser beam power used to deposit each layer. (b) Phase constitution as a function

shows that the cooling rate progressively decreases as the number of layers increases and asymptotically approaches a value below the martensite critical cooling rate (410 ◦C/s). Therefore, the deposition of additional layers would likely lead to the suppression of the

The thermal cycles originated by layer overlap heat up the previously deposited material to temperatures in the tempering range (*T* > 400◦C), causing the progressive decomposition of

The idle time between the deposition of consecutive layers used (6 s) is too short to allow the part to cool down to room temperature before the deposition of a new layer. As a result the temperature of the workpiece increases progressively as the deposition advances, eventually stabilising at approximately 270 ◦C after the deposition of the 15th layer, as depicted in the

This facilitates tempering because, as heat accumulates in the part, the material residence time in the tempering temperatures range increases from less than 1 s in the first cycles to

The cumulative effect of the consecutive thermal cycles is sufficient for significant tempering to take place, particularly in the layers deposited at the beginning of the buildup process. For example, the material in the first layer is subjected to 74 thermal cycles subsequent to

of the distance from the fusion line.

Fig. 7. *β* phase distribution.

plot of Figure 9.a.

martensitic transformation in the top layers of the part.

approximately 4 s from the 15th cycle onwards (Figure 9.b).

the martensite into *α* and *β* (Figure 8.b).

The distributions of microstructure, Young's modulus and hardness predicted by the model were compared to the values measured on a sample manufactured with similar processing parameters to validate the model. The system used in the experimental tests was developed by C. Meacock and R. Vilar (Meacock, 2009) to manufacture small to medium size parts for biomedical applications and the experimental work described in this section was carried out in collaboration with these authors. The system uses a CO2 laser with a maximum beam power of 130 W which can be focused to a spot of 0.3 mm in diameter by means of a ZnSe lens with a focal length of 63.5 mm. The system employs a closed loop online control system whereby

from the tempering of martensite, which is a slow process when compared to the typical time scales involved in laser processes. However, the deposition of the 75 layers takes approximately 450 s, which is long enough for tempering to occur and a noticeable volume

<sup>331</sup> Modelling of Heat Transfer

The Young's modulus and hardness of the material were measured by depth sensing indentation testing carried out on the longitudinal section of the wall at 1 mm intervals starting at a distance of 0.5 mm from the fusion line, and the results are presented in Figures 12.b and 12.c, respectively. The Young's modulus is seen to increase slightly with increasing distance from the fusion line, from 110 GPa at 0.5 mm to 114 GPa at 2.5 mm. Likewise, the hardness increases with increasing distance from the fusion line, from 330 HV at a distance of 0.5 mm to 365 HV at 2.5 mm. The values of *β* volume fraction, Young's modulus and hardness calculated by the model are compared to the experimental values and plotted as a function of

(a) (b) (c)

The variation of the volume fraction of *β* along the height of the wall is small but compares well with the values predicted by the model. The calculated Young's modulus and hardness show an overall good correlation with the experimental values and are within the experimental error limits, although the Young's modulus varies only slightly in the material.

To optimise the deposition process in order to obtain parts fulfilling specific requirements it is necessary to assess how the choice of processing parameters affects the properties of the deposited material. To this end, the model was used to obtain processing maps relating the scanning speed (*v*), idle time between the deposition of consecutive layers (Δ*t*) and substrate temperature (*Tsub*) to the microstructure, hardness and Young's modulus distributions in parts produced by laser powder deposition. A summary of these results has been published elsewhere (Crespo & Vilar, 2010) but a more detailed analysis of the heat transfer and metallurgical phenomena is presented in this section. A thin wall geometry was considered with a width of 1 mm, a length of 14 mm and a height of 5 mm, produced by overlapping 10 layers of Ti-6Al-4V on a substrate of the same material with the dimensions 100\*25\*140 mm (Figure 13). The deposition was assumed to take place along the longitudinal

Fig. 12. Comparison between the values obtained using the model and the experimental measurements for: (a) volume fraction of *β* phase, (b) Young's modulus and (c) Vicker's

fraction of *β* phase is observed in the deposited material.

and Phase Transformations in the Rapid Manufacturing of Titanium Components

the distance from the fusion line in Figure 12.

**5. Development of processing maps**

hardness.

Fig. 10. (a) Evolution of the phase constitution of the material in the first layer. (b) Distribution of properties along the wall height.

the intensity of the infra-red radiation emitted in the range 1.0-1.7 *μ*m is monitored by a GaAs In doped photodiode. The acquired information is processed by a control function which acts to adjust the laser power in order to maintain constant melt pool dimensions during buildup, allowing for a high stability and dimensional accuracy in the manufacture of the parts. The deposition was conducted using a Ti-6Al-4V powder with a particle size in the range 25-75 *μ*m fed through a capillary at a mass flow rate of 0.14 g/min.

An optical micrograph of the cross section of the manufactured sample reveals an acicular morphology in the upper region of the wall (Figure 11.a). This is observed in the last 15 layers and is consistent with the hexagonal *α* -martensite microstructure of Ti-6Al-4V, which typically presents a morphology consisting of long orthogonally oriented plates. Close to the bottom of the wall, the material presents a different microstructure (Figure 11.b), consisting of martensite needles interspersed with regions of *α* + *β*. To quantify the volume fraction of the different phases, X-ray diffraction was conducted on the deposited material. The volume fraction of *β* phase was calculated from the X-ray diffractograms by the direct comparison method, with the error being the standard deviation of the averaged intensities method (Meacock, 2009). The volume fraction of *β* phase decreases with increasing distance to the substrate from 0.06 at 0.5 mm to 0.04 at 2.5 mm (Figure 12.a). The *β* phase results primarily 16 Will-be-set-by-IN-TECH

(a) (b)

the intensity of the infra-red radiation emitted in the range 1.0-1.7 *μ*m is monitored by a GaAs In doped photodiode. The acquired information is processed by a control function which acts to adjust the laser power in order to maintain constant melt pool dimensions during buildup, allowing for a high stability and dimensional accuracy in the manufacture of the parts. The deposition was conducted using a Ti-6Al-4V powder with a particle size in the range 25-75

(a) (b)

Fig. 11. (a) Optical micrograph taken approximately 250 *μ*m from the wall apex. (b) Optical micrograph taken approximately 500 *μ*m from the fusion line. Adapted from C. Meacock

An optical micrograph of the cross section of the manufactured sample reveals an acicular morphology in the upper region of the wall (Figure 11.a). This is observed in the last 15

typically presents a morphology consisting of long orthogonally oriented plates. Close to the bottom of the wall, the material presents a different microstructure (Figure 11.b), consisting of martensite needles interspersed with regions of *α* + *β*. To quantify the volume fraction of the different phases, X-ray diffraction was conducted on the deposited material. The volume fraction of *β* phase was calculated from the X-ray diffractograms by the direct comparison method, with the error being the standard deviation of the averaged intensities method (Meacock, 2009). The volume fraction of *β* phase decreases with increasing distance to the substrate from 0.06 at 0.5 mm to 0.04 at 2.5 mm (Figure 12.a). The *β* phase results primarily


Fig. 10. (a) Evolution of the phase constitution of the material in the first layer. (b)

Distribution of properties along the wall height.

layers and is consistent with the hexagonal *α*

(Meacock, 2009)

*μ*m fed through a capillary at a mass flow rate of 0.14 g/min.

from the tempering of martensite, which is a slow process when compared to the typical time scales involved in laser processes. However, the deposition of the 75 layers takes approximately 450 s, which is long enough for tempering to occur and a noticeable volume fraction of *β* phase is observed in the deposited material.

The Young's modulus and hardness of the material were measured by depth sensing indentation testing carried out on the longitudinal section of the wall at 1 mm intervals starting at a distance of 0.5 mm from the fusion line, and the results are presented in Figures 12.b and 12.c, respectively. The Young's modulus is seen to increase slightly with increasing distance from the fusion line, from 110 GPa at 0.5 mm to 114 GPa at 2.5 mm. Likewise, the hardness increases with increasing distance from the fusion line, from 330 HV at a distance of 0.5 mm to 365 HV at 2.5 mm. The values of *β* volume fraction, Young's modulus and hardness calculated by the model are compared to the experimental values and plotted as a function of the distance from the fusion line in Figure 12.

Fig. 12. Comparison between the values obtained using the model and the experimental measurements for: (a) volume fraction of *β* phase, (b) Young's modulus and (c) Vicker's hardness.

The variation of the volume fraction of *β* along the height of the wall is small but compares well with the values predicted by the model. The calculated Young's modulus and hardness show an overall good correlation with the experimental values and are within the experimental error limits, although the Young's modulus varies only slightly in the material.

#### **5. Development of processing maps**

To optimise the deposition process in order to obtain parts fulfilling specific requirements it is necessary to assess how the choice of processing parameters affects the properties of the deposited material. To this end, the model was used to obtain processing maps relating the scanning speed (*v*), idle time between the deposition of consecutive layers (Δ*t*) and substrate temperature (*Tsub*) to the microstructure, hardness and Young's modulus distributions in parts produced by laser powder deposition. A summary of these results has been published elsewhere (Crespo & Vilar, 2010) but a more detailed analysis of the heat transfer and metallurgical phenomena is presented in this section. A thin wall geometry was considered with a width of 1 mm, a length of 14 mm and a height of 5 mm, produced by overlapping 10 layers of Ti-6Al-4V on a substrate of the same material with the dimensions 100\*25\*140 mm (Figure 13). The deposition was assumed to take place along the longitudinal

(a) (b)

<sup>333</sup> Modelling of Heat Transfer

and Phase Transformations in the Rapid Manufacturing of Titanium Components

Fig. 14. (a) Young's modulus (GPA) and (b) Vickers hardness (HV) distributions in a part

Fig. 15. Temperature variation during build-up for the 2nd and 6th layers of deposited material. The total time above 400 ◦C, where tempering takes place, is approximately 1s for each of the 5 layers deposited subsequently, which is not sufficient for significant tempering

> *∂T <sup>∂</sup><sup>t</sup>* <sup>=</sup> *<sup>k</sup> cpρ*

to be conducted away from the interaction zone and reducing the temperature gradient in the wall, as shown in Figure 17. As a consequence of heat conduction to the substrate being the main mechanism of heat extraction from the interaction zone, a lower temperature gradient in the build direction slows down the heat flow, causing a reduction of the cooling rate which

where *xx*� is the build-up (vertical) direction. Figure 18.a shows the variation of the cooling rate experienced during the deposition of the 10*th* layer of material, with the scanning speed for two different values of the idle time. For low scanning speeds, the material in the last layers cools from above the *β*-transus at rates lower than 410 ◦C/s and the martensitic transformation is replaced by the diffusional *β* → *α* transformation, leading to a microstructure composed of 0.92 *α* + 0.08 *β* in this region (Figure 18.b). As a result, the final part presents a non-uniform distribution of hardness, 350 HV in the bottom layers and 305 HV

*∂*2*T*

*<sup>∂</sup>x*<sup>2</sup> , (27)

produced using a scanning speed of 20 mm/s.

to occur.

is approximately given by:

direction of the substrate and on its mid plane so that a symmetry plane exists and only half of the geometry needs to be considered for calculation purposes. A laser beam with a power P = 1000 W focused to a spot *dbeam* = 1.5 mm in diameter (at *e*−<sup>2</sup> of the maximum intensity) was used so that a melt pool of approximately 1 mm in diameter is created in the laser / material interaction zone, matching the track width. An average absorptivity of 15 % was used in the calculations, assuming the utilisation of a CO2 laser (Hu & Baker, 1999).

Fig. 13. Finite element mesh.

#### **5.1 Influence of scanning speed and idle time**

Figure 14 shows the computed Young's modulus and hardness distributions along the wall height using a scanning speed of 20 mm/s and an idle time of 10 s. The final part presents a fully martensitic microstructure and uniform distributions of Young's modulus and hardness, 114 GPa and 350 HV, respectively (Figure 14). During the fabrication process the material undergoes cooling rates in excess of 10<sup>3</sup> ◦C/s, which favour the transformation of the *β* phase formed upon solidification by a martensitic mechanism. Some tempering occurs due to re-heating caused by the overlapping of the following layers, but its extent is small because it takes several minutes for significant martensite decomposition to occur, whereas the residence time of the material within the tempering temperature range (above 400 ◦C) during the complete build-up process is less than 10 s (around 1 s for each subsequently deposited layer, Figure 15). A 10 s idle time is sufficient for the part to cool down to approximately 20 ◦C before the deposition of each new layer, therefore the average substrate temperature increases only slightly during build-up of the part (Figure 15). Using lower idle times leads to a progressive increase of the workpiece temperature during the deposition process (Figure 16.a), but the deposited material still presents a martensitic microstructure because the cooling rates are not significantly affected by the temperature increase in the part at the scanning speed used (20 mm/s). For this scanning speed the cooling rates are much higher than the critical cooling rate for the martensitic transformation (410 ◦C/s) and asymptotically approach a limit value between 1500 ◦C/s (for Δ*t* = 2 s) and 1900 ◦C/s (for Δ*t* > 30 s) as the number of deposited layers increases (Figure 16.b).

Another critical parameter to play a role in the formation of the microstructure of the material is the scanning speed used to perform the manufacture. Low values of this parameter can cause the suppression of the martensitic transformation because they lead to longer interaction times between the heat source (laser radiation) and the material, allowing more time for heat 18 Will-be-set-by-IN-TECH

direction of the substrate and on its mid plane so that a symmetry plane exists and only half of the geometry needs to be considered for calculation purposes. A laser beam with a power P = 1000 W focused to a spot *dbeam* = 1.5 mm in diameter (at *e*−<sup>2</sup> of the maximum intensity) was used so that a melt pool of approximately 1 mm in diameter is created in the laser / material interaction zone, matching the track width. An average absorptivity of 15 % was used in the

Figure 14 shows the computed Young's modulus and hardness distributions along the wall height using a scanning speed of 20 mm/s and an idle time of 10 s. The final part presents a fully martensitic microstructure and uniform distributions of Young's modulus and hardness, 114 GPa and 350 HV, respectively (Figure 14). During the fabrication process the material undergoes cooling rates in excess of 10<sup>3</sup> ◦C/s, which favour the transformation of the *β* phase formed upon solidification by a martensitic mechanism. Some tempering occurs due to re-heating caused by the overlapping of the following layers, but its extent is small because it takes several minutes for significant martensite decomposition to occur, whereas the residence time of the material within the tempering temperature range (above 400 ◦C) during the complete build-up process is less than 10 s (around 1 s for each subsequently deposited layer, Figure 15). A 10 s idle time is sufficient for the part to cool down to approximately 20 ◦C before the deposition of each new layer, therefore the average substrate temperature increases only slightly during build-up of the part (Figure 15). Using lower idle times leads to a progressive increase of the workpiece temperature during the deposition process (Figure 16.a), but the deposited material still presents a martensitic microstructure because the cooling rates are not significantly affected by the temperature increase in the part at the scanning speed used (20 mm/s). For this scanning speed the cooling rates are much higher than the critical cooling rate for the martensitic transformation (410 ◦C/s) and asymptotically approach a limit value between 1500 ◦C/s (for Δ*t* = 2 s) and 1900 ◦C/s (for Δ*t* > 30 s) as the number of deposited

Another critical parameter to play a role in the formation of the microstructure of the material is the scanning speed used to perform the manufacture. Low values of this parameter can cause the suppression of the martensitic transformation because they lead to longer interaction times between the heat source (laser radiation) and the material, allowing more time for heat

calculations, assuming the utilisation of a CO2 laser (Hu & Baker, 1999).

Fig. 13. Finite element mesh.

layers increases (Figure 16.b).

**5.1 Influence of scanning speed and idle time**

Fig. 14. (a) Young's modulus (GPA) and (b) Vickers hardness (HV) distributions in a part produced using a scanning speed of 20 mm/s.

Fig. 15. Temperature variation during build-up for the 2nd and 6th layers of deposited material. The total time above 400 ◦C, where tempering takes place, is approximately 1s for each of the 5 layers deposited subsequently, which is not sufficient for significant tempering to occur.

to be conducted away from the interaction zone and reducing the temperature gradient in the wall, as shown in Figure 17. As a consequence of heat conduction to the substrate being the main mechanism of heat extraction from the interaction zone, a lower temperature gradient in the build direction slows down the heat flow, causing a reduction of the cooling rate which is approximately given by:

$$\frac{\partial T}{\partial t} = \frac{k}{c\_p \rho} \frac{\partial^2 T}{\partial \mathbf{x}^2} \tag{27}$$

where *xx*� is the build-up (vertical) direction. Figure 18.a shows the variation of the cooling rate experienced during the deposition of the 10*th* layer of material, with the scanning speed for two different values of the idle time. For low scanning speeds, the material in the last layers cools from above the *β*-transus at rates lower than 410 ◦C/s and the martensitic transformation is replaced by the diffusional *β* → *α* transformation, leading to a microstructure composed of 0.92 *α* + 0.08 *β* in this region (Figure 18.b). As a result, the final part presents a non-uniform distribution of hardness, 350 HV in the bottom layers and 305 HV

*a b*

<sup>335</sup> Modelling of Heat Transfer

and Phase Transformations in the Rapid Manufacturing of Titanium Components

Fig. 18. (a) Cooling rate experienced by the material deposited in the last layer as a function of the scanning speed for Δt = 2 s and Δt = 10 s; (b) volume fraction of martensite in the

*a b*

transformation before the deposition of the next layer. For example, if the substrate is pre-heated to 500 ◦C, the *β* → *α*� transformation is stopped at about 75% of its extent, and at this temperature vanadium diffuses into the *β* phase (less than 20 s are necessary to achieve a concentration of 10 wt.%), stabilising and retaining this phase upon cooling to room temperature and leading to a material with roughly 0.75 of *α* and 0.25 of *β* (Fan,

2. Secondly, increasing the temperature of the substrate acts to reduce the temperature gradient throughout the material. This leads to lower cooling rates which facilitate the

The joint influence of the scanning speed, idle time and substrate temperature on the final microstructure and properties of the parts is best analysed in terms of processing maps that relate this information. One of the most relevant applications of the model is the construction

Fig. 19. (a) Young's modulus (GPA) and (b) Vickers hardness (HV) distributions in a part

1993; Katzarov et al., 2002; Malinov, Markovsky, Sha & Guo, 2001);

decomposition of *β* into *α* by a diffusional process.

microstructure of a part fabricated using a scanning speed of 5 mm/s.

fabricated using a scanning speed of 5 mm/s.

Fig. 16. (a) Average temperature in the part immediately before the deposition of each layer for different values of idle time, *vscan* = 20 mm/s and an initial substrate temperature of 20 ◦, and (b) cooling rates experienced by the material in each of the layers deposited (plotted in logarithmic scale).

in the upper region (Figure 19.b). The Young's modulus is practically uniformly distributed (Figure 19.a), 113 GPa in the lower region (practically only *α*� ) and 114 GPa in the upper layers (0.92 *α* + 0.08 *β*), because *α* and *α*� have similar Young's modulus and the proportion of *β* in the alloy is very low. From the plot of Figure 18.a it is apparent that scanning speeds equal to 12 mm/s or higher result in cooling rates above 410 ◦C/s and lead to *β* transforming by a martensitic mechanism, originating an *α*� structure. Scanning speeds lower than 12 mm/s lead to lower cooling rates and to parts with two microstructurally distinct regions, a bottom region composed of *α*� , and a top region composed of 0.92 *α* and 0.08 *β*, resulting from the diffusion controlled *β* → *α* transformation.

Fig. 17. Temperature (◦C) distribution at the end of the last deposition step using scanning speeds of (a) 5mm/s and (b) 20 mm/s. 20 mm/s. Adapted from Crespo and Vilar (Crespo & Vilar, 2010)

#### **5.2 Influence of substrate temperature**

In addition to the scanning speed and idle time, the substrate temperature has an important influence on the microstructure and properties of the material. Increasing the temperature of the substrate has two principal effects:

1. Firstly, pre-heating the substrate to temperatures close to or above *Mf* (400 ◦C) gives rise to different microstructures because the material cannot complete the martensitic 20 Will-be-set-by-IN-TECH

(a) (b)

Fig. 16. (a) Average temperature in the part immediately before the deposition of each layer for different values of idle time, *vscan* = 20 mm/s and an initial substrate temperature of 20 ◦, and (b) cooling rates experienced by the material in each of the layers deposited (plotted in

in the upper region (Figure 19.b). The Young's modulus is practically uniformly distributed

(0.92 *α* + 0.08 *β*), because *α* and *α*� have similar Young's modulus and the proportion of *β* in the alloy is very low. From the plot of Figure 18.a it is apparent that scanning speeds equal to 12 mm/s or higher result in cooling rates above 410 ◦C/s and lead to *β* transforming by a martensitic mechanism, originating an *α*� structure. Scanning speeds lower than 12 mm/s lead to lower cooling rates and to parts with two microstructurally distinct regions, a bottom

*a b*

Fig. 17. Temperature (◦C) distribution at the end of the last deposition step using scanning

In addition to the scanning speed and idle time, the substrate temperature has an important influence on the microstructure and properties of the material. Increasing the temperature of

1. Firstly, pre-heating the substrate to temperatures close to or above *Mf* (400 ◦C) gives rise to different microstructures because the material cannot complete the martensitic

speeds of (a) 5mm/s and (b) 20 mm/s. 20 mm/s. Adapted from Crespo and Vilar

, and a top region composed of 0.92 *α* and 0.08 *β*, resulting from the

) and 114 GPa in the upper layers

(Figure 19.a), 113 GPa in the lower region (practically only *α*�

logarithmic scale).

region composed of *α*�

(Crespo & Vilar, 2010)

**5.2 Influence of substrate temperature**

the substrate has two principal effects:

diffusion controlled *β* → *α* transformation.

Fig. 18. (a) Cooling rate experienced by the material deposited in the last layer as a function of the scanning speed for Δt = 2 s and Δt = 10 s; (b) volume fraction of martensite in the microstructure of a part fabricated using a scanning speed of 5 mm/s.

Fig. 19. (a) Young's modulus (GPA) and (b) Vickers hardness (HV) distributions in a part fabricated using a scanning speed of 5 mm/s.

transformation before the deposition of the next layer. For example, if the substrate is pre-heated to 500 ◦C, the *β* → *α*� transformation is stopped at about 75% of its extent, and at this temperature vanadium diffuses into the *β* phase (less than 20 s are necessary to achieve a concentration of 10 wt.%), stabilising and retaining this phase upon cooling to room temperature and leading to a material with roughly 0.75 of *α* and 0.25 of *β* (Fan, 1993; Katzarov et al., 2002; Malinov, Markovsky, Sha & Guo, 2001);

2. Secondly, increasing the temperature of the substrate acts to reduce the temperature gradient throughout the material. This leads to lower cooling rates which facilitate the decomposition of *β* into *α* by a diffusional process.

The joint influence of the scanning speed, idle time and substrate temperature on the final microstructure and properties of the parts is best analysed in terms of processing maps that relate this information. One of the most relevant applications of the model is the construction

*a b*

<sup>337</sup> Modelling of Heat Transfer

and Phase Transformations in the Rapid Manufacturing of Titanium Components

A thermo-kinetic finite element model coupling heat transfer calculations and phase transformation kinetics theory which predicts the microstructure and properties distributions in titanium parts produced by rapid manufacturing was developed. The results of the model were compared to experimental measurements of the phase constitution and properties performed on samples produced by a laser based technique using identical parameters to the ones utilised in the model and showed a good agreement. The model was also applied to the development of processing maps relating the deposition parameters to the microstructure and

1. Computational methods are an efficient way to study heat transfer and the associated

2. This type of approach provides a cheap and rapid way to construct processing maps for

3. Pre-heating the substrate allows controlling the final distribution of microstructure and

4. By changing the processing parameters during the build-up process it possible to control

The model can be easily modified to describe the microstructural transformations of other Ti alloys. In the case of *α*/*β* alloys these modifications simply amount to changing the kinetic parameters used to describe the various transformations simulated in the model and future work will include the application of the model to simulate the microstructural evolution of different titanium alloys. The phase transformations kinetics subroutine is a generic algorithm that describes the phase transformations of Ti-6Al-4V as a function of the heat treatment and is not specific to rapid manufacturing processes. This subroutine can be used to describe other processing techniques and, more generally, any heat treatment of the Ti-6Al-4V alloy. Additionally, more experimental work will be carried out to achieve a thorough validation of

properties of the parts. From the results achieved it was concluded that:

properties in Ti-6Al-4V parts produced by rapid manufacturing;

the properties in Ti-6Al-4V produced by rapid manufacturing processes

the optimisation of fabrication processes;

the model for different processing conditions.

metallurgical phenomena which occur during manufacture of components;

Fig. 21. (a) Young's modulus (GPA) and (b) Vickers hardness (HV) distributions in a part fabricated using a scanning speed of 15 mm/s on a substrate at 500 ◦C. Adapted from Crespo

and Vilar (Crespo & Vilar, 2010).

**6. Conclusion**

**6.1 Future work**

of such processing maps to allow the prediction of the microstructure of the material and its properties given any set of processing parameters. The maps in Figure 20.a and 20.b (Crespo & Vilar, 2010) show the dependence of the cooling rate on the scanning speed and on the idle time for substrate temperatures of 20 and 500 ◦C, respectively, and allow finding the processing windows leading to specific microstructures.

Fig. 20. (a) Contour plot of the cooling rate as a function of *v* and Δ*t* for *Tsub* = 20◦*C*. The cooling rate is mostly dependent on the scanning speed, as evidenced by the constant cooling rate lines being almost vertical. (b) Contour plot showing the dependence of the cooling rate on the scanning speed and the idle time using a substrate pre-heated to 500 ◦*C*. Adapted from Crespo and Vilar (Crespo & Vilar, 2010).

For example, when the deposition takes place on a substrate pre-heated at 500 ◦C, a scanning speed of 20 mm/s results in a martensitic transformation if an idle time of 5 s is used, while for 16 mm/s, idle times longer than 15 s are necessary to achieve the martensite critical cooling rate. More generally, any set of parameters to the right of the bold line (*∂T*/*∂t* = 410◦*C*/*s*) shown in the plot of Figure 20.b leads to a martensitic transformation in the material. For these sets of parameters, after the deposition of the last layer, the material is at 500 ◦C and its microstructure consists approximately of 0.75 *α*� + 0.25 *β*, with a vestigial proportion of *α* formed during tempering. During cooling to room temperature, the *β* phase remains stable and is retained due to the enrichment in vanadium and the final part has a uniform microstructure containing 0.75 *α*� + 0.25 *β* (Fan, 1993). This material presents a uniform hardness of 300 HV and a Young's modulus of 106 GPa, which is the minimum that can be obtained for the Ti-6Al-4V alloy. The distributions of Young's modulus and Vickers hardness are shown in Figure 21.a and b, respectively.

These results are in agreement with results published by other authors. In particular the diffusional transformation of *β* phase in Ti-6Al-4V components produced by laser assisted deposition at low scanning speeds was verified by Kelly et al. (*v*=0.625 - 2.5 mm/s) (Kelly & Kampe, 2004a) and Qian et al. (*v*=6.7 - 10.0 mm/s) (Qian et al., 2005), whereas the occurrence of a martensitic transformation at higher speeds was confirmed by Groh (d. Groh, 2006).

Fig. 21. (a) Young's modulus (GPA) and (b) Vickers hardness (HV) distributions in a part fabricated using a scanning speed of 15 mm/s on a substrate at 500 ◦C. Adapted from Crespo and Vilar (Crespo & Vilar, 2010).

#### **6. Conclusion**

22 Will-be-set-by-IN-TECH

of such processing maps to allow the prediction of the microstructure of the material and its properties given any set of processing parameters. The maps in Figure 20.a and 20.b (Crespo & Vilar, 2010) show the dependence of the cooling rate on the scanning speed and on the idle time for substrate temperatures of 20 and 500 ◦C, respectively, and allow finding

(a) (b)

For example, when the deposition takes place on a substrate pre-heated at 500 ◦C, a scanning speed of 20 mm/s results in a martensitic transformation if an idle time of 5 s is used, while for 16 mm/s, idle times longer than 15 s are necessary to achieve the martensite critical cooling rate. More generally, any set of parameters to the right of the bold line (*∂T*/*∂t* = 410◦*C*/*s*) shown in the plot of Figure 20.b leads to a martensitic transformation in the material. For these sets of parameters, after the deposition of the last layer, the material is at 500 ◦C and its microstructure consists approximately of 0.75 *α*� + 0.25 *β*, with a vestigial proportion of *α* formed during tempering. During cooling to room temperature, the *β* phase remains stable and is retained due to the enrichment in vanadium and the final part has a uniform microstructure containing 0.75 *α*� + 0.25 *β* (Fan, 1993). This material presents a uniform hardness of 300 HV and a Young's modulus of 106 GPa, which is the minimum that can be obtained for the Ti-6Al-4V alloy. The distributions of Young's modulus and Vickers hardness

These results are in agreement with results published by other authors. In particular the diffusional transformation of *β* phase in Ti-6Al-4V components produced by laser assisted deposition at low scanning speeds was verified by Kelly et al. (*v*=0.625 - 2.5 mm/s) (Kelly & Kampe, 2004a) and Qian et al. (*v*=6.7 - 10.0 mm/s) (Qian et al., 2005), whereas the occurrence of a martensitic transformation at higher speeds was confirmed by Groh (d. Groh,

Fig. 20. (a) Contour plot of the cooling rate as a function of *v* and Δ*t* for *Tsub* = 20◦*C*. The cooling rate is mostly dependent on the scanning speed, as evidenced by the constant cooling rate lines being almost vertical. (b) Contour plot showing the dependence of the cooling rate on the scanning speed and the idle time using a substrate pre-heated to 500 ◦*C*. Adapted

the processing windows leading to specific microstructures.

from Crespo and Vilar (Crespo & Vilar, 2010).

are shown in Figure 21.a and b, respectively.

2006).

A thermo-kinetic finite element model coupling heat transfer calculations and phase transformation kinetics theory which predicts the microstructure and properties distributions in titanium parts produced by rapid manufacturing was developed. The results of the model were compared to experimental measurements of the phase constitution and properties performed on samples produced by a laser based technique using identical parameters to the ones utilised in the model and showed a good agreement. The model was also applied to the development of processing maps relating the deposition parameters to the microstructure and properties of the parts. From the results achieved it was concluded that:


#### **6.1 Future work**

The model can be easily modified to describe the microstructural transformations of other Ti alloys. In the case of *α*/*β* alloys these modifications simply amount to changing the kinetic parameters used to describe the various transformations simulated in the model and future work will include the application of the model to simulate the microstructural evolution of different titanium alloys. The phase transformations kinetics subroutine is a generic algorithm that describes the phase transformations of Ti-6Al-4V as a function of the heat treatment and is not specific to rapid manufacturing processes. This subroutine can be used to describe other processing techniques and, more generally, any heat treatment of the Ti-6Al-4V alloy. Additionally, more experimental work will be carried out to achieve a thorough validation of the model for different processing conditions.

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<sup>339</sup> Modelling of Heat Transfer

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#### **7. Acknowledgements**

The author thankfully acknowledges the valuable help and contributions from Dr. C. Meacock and Prof. R. Vilar.

#### **8. References**


24 Will-be-set-by-IN-TECH

The author thankfully acknowledges the valuable help and contributions from Dr. C. Meacock

Bontha, S., Klingbeil, N. W., Kobryn, P. A. & Fraser, H. L. (2006). Thermal process maps for

Boyer, R., Welsch, G. & Coolings, E. W. (1994). *Materials Properties Handbook : titanium alloys*,

Costa, L., Vilar, R., Reti, T. & Deus, A. M. (2005). Rapid tooling by laser powder deposition:

Crespo, A. (2010). *Mathematical Simulation of Laser Powder Deposition Applied to the Production of Prosthesis and Implants of Ti and Ti-6Al-4V*, PhD thesis, Instituto Superior Técnico. Crespo, A., Deus, A. M. & Vilar, R. (2006). Finite element analysis of laser powder deposition

Crespo, A., Deus, A. M. & Vilar, R. (2008). Modeling of phase transformations and internal

Crespo, A. & Vilar, R. (2010). Finite element analysis of the rapid manufacturing of ti-6al-4v

d. Groh, H. C. (2006). Development of laser fabricated Ti-6Al-4V, *Technical report*, National Aeronautics and Space Administration John H. Glenn Research Center. Desai, C. S. & Abel, J. F. (1972). *Introduction to the Finite Element Method*, Van Nostrand Reinhold

Deus, A. M. & Mazumder, J. (2006). Three-dimensional finite element models for the

Elmer, J. W., Palmer, T. A., Babu, S. S., Zhang, W. & DebRoy, T. (2004). Phase transformations dynamics during welding of Ti-6Al-4V, *Journal of Applied Physics* 95: 8327–8339. Fan, Z. Y. (1993). On the young moduli of Ti-6Al-4V alloys, *Scripta Metallurgica Et Materialia*

Ghosh, S. & Choi, J. (2005). Three-dimensional transient finite element analysis for residual

Hu, C. & Baker, T. N. (1999). A semi-empirical model to predict the melt depth developed in

Kahlen, F. J. & Kar, A. (2001). Residual stresses in laser-deposited metal parts, *Journal of Laser*

stresses in the laser aided direct metal/material deposition process, *Journal of Laser*

overlapping laser tracks on a Ti-6Al-4V alloy, *Journal of Materials Processing Technology*

calculation of temperature and residual stress fields in laser cladding, *Proceedings of ICALEO 2006-25th International Congress on Applications of Lasers and Electro-Optics*,

Process using finite element analysis, *Acta Materialia* 14: 3987–3999.

*Lasers and Electro-Optics*, Scottsdale, Arizona, pp. 1016–1021.

parts by laser powder deposition, *Scripta Materialia* 63: 140–143.

Donachie, M. J. (2004). *Titanium, A Technical Guide - 2nd Edition*, ASM International.

*Journal of Materials Processing Technology* 178: 135–142.

predicting solidification microstructure in laser fabrication of thin-wall structures,

of titanium, *Proceedings of ICALEO 2006-25th International Congress on Applications of*

stresses in laser powder deposition, *Proceedings of GCL-HPL 2008*, Lisbon, Portugal,

**7. Acknowledgements**

ASM International.

pp. 713120–713120–10.

Scottsdale, Arizona, pp. 496–505.

Company.

29: 1427–1432.

94: 116–122.

*Applications* 17: 144–158.

*Applications* 13: 60–69.

Brunette, D. M. (2001). *Titanium in Medicine*, Springer.

and Prof. R. Vilar.

**8. References**


**16** 

Wei Liu

*Japan* 

*Japan Atomic Energy Agency* 

**Measurement of Boundary Conditions -** 

**Surface Heat Flux and Surface Temperature** 

In order to understand a heat transfer phenomenon, it is sometimes very important to know the change in heat flux and temperature with time or the distribution of the heat flux and temperature on the surface of a heating block. Furthermore, with the development of computational thermal-fluid dynamics, experimental data that are of the CFD level, both spatially and temporally, are needed to validate the simulation results. However, it is difficult to measure the surface heat flux and surface temperature if no sensors are allowed setting on the surface, which may be a disturbance to the phenomenon itself occurring on the surface. This chapter introduces a technique for the measurement of surface temperature and surface

**2. Technique for the measurement of surface temperature and surface heat** 

The technique for the measurement of surface temperature and surface heat flux involves two steps: (1) measurement of the inner block temperatures using special micro temperature sensors, which are placed at a depth of several micrometers beneath the surface; and (2) solving an inverse heat conduction problem (IHCP) to obtain the surface heat flux and surface temperature, by using the measured inner block temperature data. In other words, the introduction of the IHCP enables us to derive the boundary conditions, which include both surface heat flux and surface temperature, without necessitating the use of a sensor on

When compared to a normal heat conduction problem, that enables us to calculate temperature distributions inside a block by solving the heat conduction equation, with the initial and boundary conditions as input, the IHCP is an analysis method that enables us to calculate the boundary conditions by solving the heat conduction equation with Laplace transform technique, with the initial conditions and inner temperature as input. A considerable number of research studies (Burggraf, 1964; Sparrow et al., 1964; Imber, 1974; Shoji, 1978; Monde, 2003a, 2003b; Woodfield, 2006a, 2006b) have adopted analytical approaches for solving the IHCPs. Although limited to fairly simple geometry, explicit analytical solutions to IHCP have been derived and are very fast computational. If a one- (two-) dimensional heating block comprises two unknown boundaries, as shown in Fig. 1,

**1. Introduction** 

heat flux.

the surface of interest.

**2.1 Inverse heat conduction problem (IHCP)** 

**flux** 


### **Measurement of Boundary Conditions - Surface Heat Flux and Surface Temperature**

Wei Liu *Japan Atomic Energy Agency Japan* 

#### **1. Introduction**

26 Will-be-set-by-IN-TECH

340 Convection and Conduction Heat Transfer

Toyserkani, E., Khajepour, A. & Corbin, S. (2004). 3-d finite element modeling of laser cladding

Vasinonta, A., Beuth, J. L. & Grifith, M. (2001). A process map for consistent build conditions

Wang, L., Felicelli, S., Gooroochurn, Y., Wang, P. & Horstemeyer, M. (2008). Optimization

Zienkiewicz, O. C. & Taylor, R. L. (2000). *The Finite Element Method*, Butterwoth Heinemann.

*Science and Engineering - Transactions of the AIME* 123(4): 615–622.

Yoshiki, O. (2007). *Bioscience and Bioengineering of Titanium Materials*, Elsevier.

*in Engineering* 41: 849–867.

474: 148–156.

by powder injection: effects of laser pulse shaping on the process, *Optics and Lasers*

in the solid freeform fabrication of thin-walled structures, *Journal of Manufacturing*

of the lens process for steady molten pool size, *Materials Science and Engineering A*

In order to understand a heat transfer phenomenon, it is sometimes very important to know the change in heat flux and temperature with time or the distribution of the heat flux and temperature on the surface of a heating block. Furthermore, with the development of computational thermal-fluid dynamics, experimental data that are of the CFD level, both spatially and temporally, are needed to validate the simulation results. However, it is difficult to measure the surface heat flux and surface temperature if no sensors are allowed setting on the surface, which may be a disturbance to the phenomenon itself occurring on the surface. This chapter introduces a technique for the measurement of surface temperature and surface heat flux.

#### **2. Technique for the measurement of surface temperature and surface heat flux**

The technique for the measurement of surface temperature and surface heat flux involves two steps: (1) measurement of the inner block temperatures using special micro temperature sensors, which are placed at a depth of several micrometers beneath the surface; and (2) solving an inverse heat conduction problem (IHCP) to obtain the surface heat flux and surface temperature, by using the measured inner block temperature data. In other words, the introduction of the IHCP enables us to derive the boundary conditions, which include both surface heat flux and surface temperature, without necessitating the use of a sensor on the surface of interest.

#### **2.1 Inverse heat conduction problem (IHCP)**

When compared to a normal heat conduction problem, that enables us to calculate temperature distributions inside a block by solving the heat conduction equation, with the initial and boundary conditions as input, the IHCP is an analysis method that enables us to calculate the boundary conditions by solving the heat conduction equation with Laplace transform technique, with the initial conditions and inner temperature as input. A considerable number of research studies (Burggraf, 1964; Sparrow et al., 1964; Imber, 1974; Shoji, 1978; Monde, 2003a, 2003b; Woodfield, 2006a, 2006b) have adopted analytical approaches for solving the IHCPs. Although limited to fairly simple geometry, explicit analytical solutions to IHCP have been derived and are very fast computational. If a one- (two-) dimensional heating block comprises two unknown boundaries, as shown in Fig. 1,

Measurement of Boundary Conditions - Surface Heat Flux and Surface Temperature 343

the boundary conditions. As an example, a comparison of the surface heat fluxes between the predictions obtained from the Woodfield's analytical approach and the exact values at different frequencies is shown in Fig. 3 (Woodfield, 2006a). In the calculation, copper is used for the heating block and the exact solution calculated from Eq. (1) with h1 = 2 mm is used as the input data. h1 = 2 mm represents the depth at which the temperature sensor is placed. As shown in the figure, the agreement between the sine waves of the actual and predicted heat fluxes is remarkably good at low frequencies of 1 and 3 Hz; however, the accuracy decreases as the frequency increases. This means that when the sensor is placed at a depth of 2 mm beneath the surface, it cannot sense the effect of the surface heat flux with a frequency higher than 5 Hz. Therefore, a new technique through which the temperature sensors can be

<sup>1</sup> <sup>2</sup> 0 0 2 1

2 4 *<sup>h</sup> <sup>q</sup> <sup>a</sup> <sup>a</sup> q a uh au t TT e t h e du*

<sup>2</sup> cos( ) sin

*a a u*

πλ

ω

ω

<sup>+</sup> ⎝ ⎠ <sup>∫</sup> (1)

0 1 <sup>0</sup> 2 24

<sup>−</sup> <sup>∞</sup> <sup>−</sup> ⎛ ⎞ − = ⎜ ⎟ − −+

ω

h1

ω π

Boundary 1: qw = q0sin(ω t)

Temperature sensor

Boundary 2: adiabatic

As already mentioned, in order to accurately calculate the boundary conditions by using the IHCP, the temperature sensors in the heating block should be positioned as close as possible to the surface of interest. Furthermore, in order to determine the distribution of heat flux and temperature on the boundary in detail, we need to solve the two-dimensional IHCP,

We therefore introduced special micron-sized thermocouples that share a single positive pole (Enomoto & Furuhama, 1984; Buchholz et al., 2004; Liu & Takase, 2009). A schematic illustration of the thermocouples is shown in Fig. 4. In a heating block made of copper, the temperature junctions are located at multiple points from No. 1 to No. N. Each junction has

which requires the temperature sensors to have a high spatial resolution.

Fig. 2. A One-dimensional block with the sinusoidal heat flux boundary imposed on one

placed very close to the surface of interest is necessary.

λ ω

**2.2 Special micro temperature sensors** 

surface

ω

then it needs two (rows of) inner-temperature readings to close the heat conduction equation and to obtain the heat fluxes and temperatures on the two boundaries. However, if the heating block has only a single unknown boundary, theoretically, then it needs only a single (line of) inner-temperature reading to close the heat conduction equation. Noting that the recent introduction of new concepts such as moving window (for two rows of measuring data to close two unknown boundary conditions) (Woodfield, 2006a) and superposition of successive corrections in approximating the temperature readings (for one row of measuring data to close one unknown boundary condition) (Woodfield, 2006b) has made the use of the IHCP practical.

Fig. 1. A Heating block with two unknown boundaries

To solve the IHCP accurately, there is a demand for the measured inner temperature data. That is, the data should have sensed the effect of the boundary under study. This necessitates the use of highly sensitive temperature sensors located in close proximity to the surface. This is very important in tracing a fast-changing transient phenomenon. In a highfrequency process such as boiling, the high-frequency effects are strongly damped within the heating block because of the thermal inertia of the heating block. Thus, the temperature sensors must be located as close as possible to the surface of interest. If a sinusoidal heat flux boundary q0(ωt) is imposed on one surface of a one-dimensional solid with initial temperature T0 and the other surface is kept adiabatic, as shown in Fig. 2, the exact solution for the temperature within the solid is given by Eq. (1) (Carslaw, 2003), where λ and α are the thermal conductivity and thermal diffusivity, respectively. Equation (1) clearly shows that the effect of the oscillation diminishes quickly with an increase in the depth into the heating block, which is more pronounced for high frequencies. This sets an upper limit to the frequency of the fluctuations that can be detected by a sensor placed within the heating block. In other words, if a temperature sensor is placed at a distance from the surface, it may be impossible to detect the effects from a high-frequency mode. Further, using the readings taken at that point as the input data of the IHCP will result in the inaccurate prediction of

then it needs two (rows of) inner-temperature readings to close the heat conduction equation and to obtain the heat fluxes and temperatures on the two boundaries. However, if the heating block has only a single unknown boundary, theoretically, then it needs only a single (line of) inner-temperature reading to close the heat conduction equation. Noting that the recent introduction of new concepts such as moving window (for two rows of measuring data to close two unknown boundary conditions) (Woodfield, 2006a) and superposition of successive corrections in approximating the temperature readings (for one row of measuring data to close one unknown boundary condition) (Woodfield, 2006b) has made the use of the

Z

Adiabatic

To solve the IHCP accurately, there is a demand for the measured inner temperature data. That is, the data should have sensed the effect of the boundary under study. This necessitates the use of highly sensitive temperature sensors located in close proximity to the surface. This is very important in tracing a fast-changing transient phenomenon. In a highfrequency process such as boiling, the high-frequency effects are strongly damped within the heating block because of the thermal inertia of the heating block. Thus, the temperature sensors must be located as close as possible to the surface of interest. If a sinusoidal heat flux boundary q0(ωt) is imposed on one surface of a one-dimensional solid with initial temperature T0 and the other surface is kept adiabatic, as shown in Fig. 2, the exact solution for the temperature within the solid is given by Eq. (1) (Carslaw, 2003), where λ and α are the thermal conductivity and thermal diffusivity, respectively. Equation (1) clearly shows that the effect of the oscillation diminishes quickly with an increase in the depth into the heating block, which is more pronounced for high frequencies. This sets an upper limit to the frequency of the fluctuations that can be detected by a sensor placed within the heating block. In other words, if a temperature sensor is placed at a distance from the surface, it may be impossible to detect the effects from a high-frequency mode. Further, using the readings taken at that point as the input data of the IHCP will result in the inaccurate prediction of

T1-1, T1-2, … T1-n

Unknown

h

2

h

Adiabatic

1

T2-1, T2-2, … T2-n

0 X

Unknown

IHCP practical.

h1

Z

T2

T1

Unknown

Unknown

h2

(a) One-dimensional (b) Two-dimensional

Fig. 1. A Heating block with two unknown boundaries

the boundary conditions. As an example, a comparison of the surface heat fluxes between the predictions obtained from the Woodfield's analytical approach and the exact values at different frequencies is shown in Fig. 3 (Woodfield, 2006a). In the calculation, copper is used for the heating block and the exact solution calculated from Eq. (1) with h1 = 2 mm is used as the input data. h1 = 2 mm represents the depth at which the temperature sensor is placed. As shown in the figure, the agreement between the sine waves of the actual and predicted heat fluxes is remarkably good at low frequencies of 1 and 3 Hz; however, the accuracy decreases as the frequency increases. This means that when the sensor is placed at a depth of 2 mm beneath the surface, it cannot sense the effect of the surface heat flux with a frequency higher than 5 Hz. Therefore, a new technique through which the temperature sensors can be placed very close to the surface of interest is necessary.

$$T - T\_0 = \frac{q\_0}{\lambda} \sqrt{\frac{a}{a\nu}} e^{-h\_1} \sqrt{\frac{\alpha}{2a}} \sin \left(\alpha t - h\_1 \sqrt{\frac{a\nu}{2a}} - \frac{\pi}{4}\right) + \frac{2q\_0 a\alpha}{\pi \lambda} \int\_0^\infty \frac{\cos(uh\_1)}{a^2 + a^2 u^4} e^{-m^2 t} du \tag{1}$$

Boundary 2: adiabatic

#### **2.2 Special micro temperature sensors**

As already mentioned, in order to accurately calculate the boundary conditions by using the IHCP, the temperature sensors in the heating block should be positioned as close as possible to the surface of interest. Furthermore, in order to determine the distribution of heat flux and temperature on the boundary in detail, we need to solve the two-dimensional IHCP, which requires the temperature sensors to have a high spatial resolution.

We therefore introduced special micron-sized thermocouples that share a single positive pole (Enomoto & Furuhama, 1984; Buchholz et al., 2004; Liu & Takase, 2009). A schematic illustration of the thermocouples is shown in Fig. 4. In a heating block made of copper, the temperature junctions are located at multiple points from No. 1 to No. N. Each junction has

Measurement of Boundary Conditions - Surface Heat Flux and Surface Temperature 345

By using teflon-coated (76μm) and polyimide-coated (50μm) constantan wires, we succeeded in developing temperature sensors with a pitch of 0.5 mm and at a depth of 3.1 μm, and temperature sensors with a pitch of 0.16 mm and at a depth of 1.3 μm, respectively, beneath the surface of interests. Figure 5 shows a comparison between the imposed sinusoidal heat flux and the sinusoidal heat flux predicted from the IHCP on the basis of the temperature calculated using Eq. (1) with h1 = 3.1 μm. As shown in the figure, there is a very good agreement between the imposed and predicted sinusoidal heat fluxes at the frequencies of 20 and 50 Hz. These cases correspond to the bubble frequencies under subcooled and saturated atmospherical pressure pool boiling conditions, respectively. The figure theoretically clarifies that the high-frequency changed boundary conditions can be traced by using the temperature sensor located at a depth of 3.1 �m beneath the surface. The new sensors are of the wire type and are therefore highly sensitive. With sharing a common positive pole, the number of wiring lines is decreased and it is easier to obtain a high spatial resolution. The response time of the sensors is assumed to be zero. These characteristics, together with the fact that temperature junctions can be easily placed several microns beneath the surface, contribute to the effectiveness of the IHCP technique in tracing

0.0 0.1 0.2 0.3

Fig. 5. Effect of inner-temperature measuring position on the accuracy of the IHCP predictions (temperature measured at a depth of 3.1 μm beneath the surface is used as the input of IHCP) Note that as shown in Fig. 1, the standard IHCP requires the sensors to be placed at two different depths. However, in the case where only one boundary is unknown (with the adiabatic or semi-infinite assumption valid on the other boundary), only one row of sensors is required in order to obtain the boundary conditions about the surface of interest. This is highly desirable since the overall cost incurred when allocating the sensors is reduced. As

the heating block, then for practical use, the block can be treated as being semi-infinite. Further, note that the thin copper film, which acts as the common pole of the temperature sensors, is to be coated with a protection film to prevent it from oxidation. Selection of the type of protection film is very important for the realization of durable temperature sensors.

50s**-1**

< , where H is the thickness of

Time, t [s]

20s**-1** qw = sin(40πt)

qw = sin(100πt)

*H*

Imposed sinusoidal heat flux boundary

the boundary conditions.


1.5◯ Calculation from the IHCP


recommended by Woodfield, if the Fourier number 2 0.1 *at*

qw [MW/m2

]

qw [MW/m2

]

a constantan wire as its negative pole, but shares a common positive pole made of copper. The temperature sensors are thus of the T-type. The common positive pole can be formed by using a film-formation technique - such as sputtering - after the constantan wires are set into the copper block. The thickness of the sputtering film can be controlled accurately within several micrometers. This means that the temperature junctions can be formed at a depth of several microns beneath the surface of interest.

Fig. 3. Effect of inner-temperature measuring position on the accuracy of the IHCP predictions (temperature measured at a depth of 2 mm beneath the surface is used as the input of IHCP, (Woodfield, 2006a))

Fig. 4. Schematic illustration of the special micron-sized T-type thermocouples that share a single positive pole

a constantan wire as its negative pole, but shares a common positive pole made of copper. The temperature sensors are thus of the T-type. The common positive pole can be formed by using a film-formation technique - such as sputtering - after the constantan wires are set into the copper block. The thickness of the sputtering film can be controlled accurately within several micrometers. This means that the temperature junctions can be formed at a depth of

Fig. 3. Effect of inner-temperature measuring position on the accuracy of the IHCP predictions (temperature measured at a depth of 2 mm beneath the surface is used as the

1 2 …… N-1 N

Common pole , Cu

block (Cu)

Constantan wire

Heating

Insulation





Fig. 4. Schematic illustration of the special micron-sized T-type thermocouples that share a


Copper wire

several microns beneath the surface of interest.

input of IHCP, (Woodfield, 2006a))

single positive pole

By using teflon-coated (76μm) and polyimide-coated (50μm) constantan wires, we succeeded in developing temperature sensors with a pitch of 0.5 mm and at a depth of 3.1 μm, and temperature sensors with a pitch of 0.16 mm and at a depth of 1.3 μm, respectively, beneath the surface of interests. Figure 5 shows a comparison between the imposed sinusoidal heat flux and the sinusoidal heat flux predicted from the IHCP on the basis of the temperature calculated using Eq. (1) with h1 = 3.1 μm. As shown in the figure, there is a very good agreement between the imposed and predicted sinusoidal heat fluxes at the frequencies of 20 and 50 Hz. These cases correspond to the bubble frequencies under subcooled and saturated atmospherical pressure pool boiling conditions, respectively. The figure theoretically clarifies that the high-frequency changed boundary conditions can be traced by using the temperature sensor located at a depth of 3.1 �m beneath the surface. The new sensors are of the wire type and are therefore highly sensitive. With sharing a common positive pole, the number of wiring lines is decreased and it is easier to obtain a

high spatial resolution. The response time of the sensors is assumed to be zero. These characteristics, together with the fact that temperature junctions can be easily placed several microns beneath the surface, contribute to the effectiveness of the IHCP technique in tracing the boundary conditions.

Fig. 5. Effect of inner-temperature measuring position on the accuracy of the IHCP predictions (temperature measured at a depth of 3.1 μm beneath the surface is used as the input of IHCP)

Note that as shown in Fig. 1, the standard IHCP requires the sensors to be placed at two different depths. However, in the case where only one boundary is unknown (with the adiabatic or semi-infinite assumption valid on the other boundary), only one row of sensors is required in order to obtain the boundary conditions about the surface of interest. This is highly desirable since the overall cost incurred when allocating the sensors is reduced. As

recommended by Woodfield, if the Fourier number 2 0.1 *at H* < , where H is the thickness of

the heating block, then for practical use, the block can be treated as being semi-infinite. Further, note that the thin copper film, which acts as the common pole of the temperature sensors, is to be coated with a protection film to prevent it from oxidation. Selection of the type of protection film is very important for the realization of durable temperature sensors.

Measurement of Boundary Conditions - Surface Heat Flux and Surface Temperature 347

The data collected by two micron-sized thermocouples that share a common positive pole are shown in Fig. 8. In the figure, T4 and T10 are the temperatures measured at the same depth but at different radial positions. The figure shows that the thermocouples are effective in measuring the temperatures. The data taken at T4, which is just below a bubble, were used as the input data for solving a semi-infinite one-dimensional IHCP in order to obtain the surface heat flux and surface temperature, the results of which are shown in Fig. 9. The

012345

012345

Fig. 9. Measured surface heat flux and surface temperature during the boiling process

Time, t[s]

Time, [s]

T4 T10

peak of the heat flux corresponds to the formation of a large, deformed bubble.

Fig. 8. Temperatures measured during the boiling processes by the micron-sized

*P* = 0.1 MPa

102

thermocouples sharing a common positive pole

102

0.6

0.0

0.2

0.4

103

104

Surface temperature,

Surface heat flux,

qw [MW/m2

]

Tw [oC]

105

106

103

104

Measured

inner temperatures, T [oC]

105

#### **3. Measurement example**

The above mentioned measurement technique was successfully used in the measurement of the surface heat flux and surface temperature during a boiling process. The experimental apparatus used is shown in Fig. 6. It includes a boiling vessel and a heating block. The heating block is made of copper and is peripherally insulated. The heating block is composed of two parts, upper and lower. The image of the upper part is shown in Fig. 7, in which 10 special micro temperature sensors are placed radially at a pitch of 0.5 mm and at a depth of 3.1 μm from the boiling surface. The upper and lower parts of the heating block are joined together by using a high-temperature adhesive. A cartridge heater is inserted into the lower heating block.

Fig. 6. Experimental apparatus

Fig. 7. The upper heating block

The above mentioned measurement technique was successfully used in the measurement of the surface heat flux and surface temperature during a boiling process. The experimental apparatus used is shown in Fig. 6. It includes a boiling vessel and a heating block. The heating block is made of copper and is peripherally insulated. The heating block is composed of two parts, upper and lower. The image of the upper part is shown in Fig. 7, in which 10 special micro temperature sensors are placed radially at a pitch of 0.5 mm and at a depth of 3.1 μm from the boiling surface. The upper and lower parts of the heating block are joined together by using a high-temperature adhesive. A cartridge heater is inserted into the

Boiling vessel

Drain

Boiling surface

Heating block (Cu)

camera Boiling surface

+ -

Tw qw

T1 , T2, … T10

Semi-infinitive

R = 5 mm

Insulation

3.1 μm

Lower heating block (Cu)

Cartridge heater

(Cu)

Immersion heater

Upper heating block

**3. Measurement example** 

lower heating block.

High- speed

Fig. 6. Experimental apparatus

Fig. 7. The upper heating block

The data collected by two micron-sized thermocouples that share a common positive pole are shown in Fig. 8. In the figure, T4 and T10 are the temperatures measured at the same depth but at different radial positions. The figure shows that the thermocouples are effective in measuring the temperatures. The data taken at T4, which is just below a bubble, were used as the input data for solving a semi-infinite one-dimensional IHCP in order to obtain the surface heat flux and surface temperature, the results of which are shown in Fig. 9. The peak of the heat flux corresponds to the formation of a large, deformed bubble.

Fig. 8. Temperatures measured during the boiling processes by the micron-sized thermocouples sharing a common positive pole

Fig. 9. Measured surface heat flux and surface temperature during the boiling process

**17** 

*Czech Republic* 

**Properties and Numerical** 

Pavel Fiala, Ivo Behunek and Petr Drexler

**Modeling-Simulation of Phase Changes Material** 

There exists a domain of models that is principially classified into the linear and non-linear fields of modelling. In the field of non-linear modelling, significant progress has been made since the 1960s thanks to the widespread and regularly available computer technology. This dynamic development influenced a large number of problems including the description of physical behaviour of non-trivial tasks. Non-linear models are solved in the material, spatial, and time domains. However, certain non-linear model domains are not sufficiently developed or regularly used for analysis of the more simple tasks. This group includes task models using non-linear discontinuous characteristics of materials, which can be exemplified by the change of state of a material during heating or cooling. In this section, we would like to use several descriptive examples to expose the problem of thermal tasks solution utilizing applied materials with a phase change (PCM) (Gille, T.; at al. 2007, Volle, F. at al. 2010, Shi, L.P.; at al. 2006)*.* These are mostly coupled tasks (Fiala,P. December 1998). Within the specification of different aspects of the solution process, emphasis will be placed on the final accuracy of the results of numerical analyses, and therefore (rather than focusing on a complete description of the model) the text will accentuate problematic spots within the

The PCM characteristics are demonstrated on the task of designing a low-temperature accumulator, an efficient cooler of electronic components, and a separator of impurities in

Within the last decade, scientific interest in the fields of basic and applied research has been focused more intensively on the problem of increasing the share of renewable sources of energy in total energy consumption per capita (Solar energy 2010). In this context, we have seen major development in the field of energy harvesting (Murat Kenisarin, & Khamid Mahkamov May 2006, John Greenman at al. May 2005, Junrui Liang & Wei-Hsin Liao 2010, Vijay Raghunathan at al. April 2005, Jirku T at al. May 2010), or the acquirement of energy from hitherto unused forms. The reason for such processes in technology naturally consists in the fact that the reserves of classical primary sources of energy and fossil fuels (Behunek, I. 2002, WORLD ENERGY STATISTICS from the IEA 2002) available to current industrial

**1. Introduction** 

solution of such tasks.

an industrial oil emulsion.

**1.1 Energy, transformation, accumulation** 

*Brno univerzity of technology, Faculty of electrical engineering and communication, Department of theoretical and experimental electrical engineering, Kolejni 4, Brno* 

### **4. Conclusions**

In this chapter, a technique for the measurement of surface temperature and surface heat flux was introduced. This technique involves two steps: (1) measurement of the inner block temperatures near the surface using special micro temperature sensors; and (2) solving an IHCP to obtain the surface heat flux and surface temperature by using the measured inner block temperature data as input. For the inner block temperature measurement, special Ttype temperature sensors with a common positive pole were introduced. By using the film formation technique of sputtering, the temperature junctions can be placed at a depth of several micrometers beneath the surface of interest.

The developed system was used to determine the change in the surface heat flux and surface temperature during a boiling process. The results from the experiments showed that the special T-type micro temperature sensors can effectively trace the temperature change during a boiling process. By using the measured inner temperatures, one-dimensional IHCP was solved to obtain the surface heat flux and surface temperature. The increase in the surface heat flux with the formation of big bubbles was calculated successfully.

#### **5. References**


## **Properties and Numerical Modeling-Simulation of Phase Changes Material**

Pavel Fiala, Ivo Behunek and Petr Drexler

*Brno univerzity of technology, Faculty of electrical engineering and communication, Department of theoretical and experimental electrical engineering, Kolejni 4, Brno Czech Republic* 

#### **1. Introduction**

348 Convection and Conduction Heat Transfer

In this chapter, a technique for the measurement of surface temperature and surface heat flux was introduced. This technique involves two steps: (1) measurement of the inner block temperatures near the surface using special micro temperature sensors; and (2) solving an IHCP to obtain the surface heat flux and surface temperature by using the measured inner block temperature data as input. For the inner block temperature measurement, special Ttype temperature sensors with a common positive pole were introduced. By using the film formation technique of sputtering, the temperature junctions can be placed at a depth of

The developed system was used to determine the change in the surface heat flux and surface temperature during a boiling process. The results from the experiments showed that the special T-type micro temperature sensors can effectively trace the temperature change during a boiling process. By using the measured inner temperatures, one-dimensional IHCP was solved to obtain the surface heat flux and surface temperature. The increase in the

Buchholz M., Luttich T., Auracher H., Marquardt W., (2004). Experimental investigation of local process in pool boiling along the entire boiling curve, *Int. J. Heat Fluid*, Vol. 25, 243 -261 Burggraf, O.R., (1964). An exact solution of the inverse problem in heat conduction theory

Liu W. & Takase K. (2009). Measurement of surface heat flux and surface temperature in

Monde M., Arima H. & Mitsutake Y., (2003a) Estimation of surface temperature and heat

Monde M., Arima H., Liu W., Mitsutake Y. & Hammad J.A., (2003b). An analytical solution

Shoji M. (1978). Study of inverse problem of heat conduction, *Transactions of Japan Society of Mechanical Engineering, series B,* Vol. 44, No.381, 1633-1643 (in Japanese). Sparrow E.M., Haji-Sheikh A. & Lundgren T.S., (1964). The inverse problem in transient heat

Woodfield P.L., Monde M. & Mitsutake Y., (2006a). Implementation of an analytical inverse

Woodfield P.L., Monde M., Mitsutake Y., (2006b). Improved analytical solution for inverse

*International Conference on Nuclear Engineering,* ICONE17-75880

temperature measurement in internal combustion engine, *Transactions of Japan Society of Mechanical Engineering, series B*, Vol.50, No.453, 1353-1362, (In Japanese). Imber M., (1974). Temperature extrapolation mechanism for two-dimensional heat flow,

Nucleate Pool Boiling Using Micro-Thermocouples, *Proceedings of the 17th* 

flux using inverse solution for one-dimensional heat conduction, *ASME J. Heat* 

for two-dimensional inverse heat conduction problems using Laplace transform,

heat conduction technique to practical problems, *Int. J. Heat Mass Transfer,* Vol. 49,

heat conduction problems on thermally thick and semi-infinite solids, Int. *J. Heat* 

Carslaw H.S., Jaeger J.C., Conduction of Heat in Solids, Oxford University Press, 2003. Enomoto Y. & Furuhama S., (1984). Research on thin film thermocouple for transient surface

surface heat flux with the formation of big bubbles was calculated successfully.

and applications, *ASME J. Heat Transfer*, Vol. 86, 373-382

**4. Conclusions** 

**5. References** 

several micrometers beneath the surface of interest.

*AIAA J*. Vol.12, No.8, 1089-1093

*Transfer,* Vol *.125,* 213-223.

*Mass Transfer,* Vol. 49, 2864-2876.

187-197.

*Int. J. Heat Mass Transfer,* Vol. 46, 2135-2148

conduction, *ASME J. Appl. Mech.,* Vol. 31, 369-375

There exists a domain of models that is principially classified into the linear and non-linear fields of modelling. In the field of non-linear modelling, significant progress has been made since the 1960s thanks to the widespread and regularly available computer technology. This dynamic development influenced a large number of problems including the description of physical behaviour of non-trivial tasks. Non-linear models are solved in the material, spatial, and time domains. However, certain non-linear model domains are not sufficiently developed or regularly used for analysis of the more simple tasks. This group includes task models using non-linear discontinuous characteristics of materials, which can be exemplified by the change of state of a material during heating or cooling. In this section, we would like to use several descriptive examples to expose the problem of thermal tasks solution utilizing applied materials with a phase change (PCM) (Gille, T.; at al. 2007, Volle, F. at al. 2010, Shi, L.P.; at al. 2006)*.* These are mostly coupled tasks (Fiala,P. December 1998). Within the specification of different aspects of the solution process, emphasis will be placed on the final accuracy of the results of numerical analyses, and therefore (rather than focusing on a complete description of the model) the text will accentuate problematic spots within the solution of such tasks.

The PCM characteristics are demonstrated on the task of designing a low-temperature accumulator, an efficient cooler of electronic components, and a separator of impurities in an industrial oil emulsion.

#### **1.1 Energy, transformation, accumulation**

Within the last decade, scientific interest in the fields of basic and applied research has been focused more intensively on the problem of increasing the share of renewable sources of energy in total energy consumption per capita (Solar energy 2010). In this context, we have seen major development in the field of energy harvesting (Murat Kenisarin, & Khamid Mahkamov May 2006, John Greenman at al. May 2005, Junrui Liang & Wei-Hsin Liao 2010, Vijay Raghunathan at al. April 2005, Jirku T at al. May 2010), or the acquirement of energy from hitherto unused forms. The reason for such processes in technology naturally consists in the fact that the reserves of classical primary sources of energy and fossil fuels (Behunek, I. 2002, WORLD ENERGY STATISTICS from the IEA 2002) available to current industrial

Properties and Numerical Modeling-Simulation of Phase Changes Material 351

being consistently developed (Juodkazis, Saulius; at al. November 2004, Zhen Ren at al.

Some of the proposed approaches are based on classical solutions. These include the accumulation of energy utilizing the potential energy of mass in a gravitational field (water), the kinetic energy of mass (flywheels), the non-linearities in the state of a mass phase – the compression of gases. Another group of accumulators is based on the solution utilizing the energy of an electromagnetic field. In this case, the most common is the application of electric accumulators or microbiological systems [mikrobiolog akumulator]. Yet another one of the fields to be quoted comprises energy accumulation using the properties of chemical bonds of non-trivial chemical systems (the production of synthetic fuels), electrochemical bonds, and the utilization of photochemical energy (the accumulation of low-potential heat

The process of designing chemical accumulator forms utilizes the physical effects of nonlinear behaviour of materials at phase changes. (Behunek,I. April 2004, Behunek I & Fiala P.

Principles are known (Baylin, F. 1979) for the utilization of characteristics of chemicalphysical effects, and in this context there exist four basic methods of thermal energy accumulation. The first method consists in the utilization of specific thermal capacity of substances (sensible heat), the second one is built on the application of change in the state of substances (latent heat) , the third one lies in the thermochemical reaction, and the fourth

Generally, the thermochemical reactions method provides a higher density of accumulated energy than the sensible heat or phase change options (Mar, R.W. & Bramletta, T.T. 1980). An endothermic reaction product contains energy in the form of a chemical bond that is released retroactively during an exothermic reaction. The energy release occurs through the action of a catalyst, which is a suitable characteristic for long-term accumulation. Other advantages of thermochemical accumulation include the possibility of transporting the products over long distances, the possibility of product storage at both low (with a low rate of loss) and very high temperatures (Goldstein, M. 1961), the low cost, and the fact that products of the reaction can be used as the medium in thermodynamic cycles (The Australian National University 2004). Currently, research (Mar, R.W. 1978, Mar, R.W. 1980) is conducted in this field. In order to accumulate energy, we can utilize heat balance at sorption/desorption of moisture in the working substance. The difference with respect to other types of heat accumulation consists in the fact that sorption does not directly depend on the temperature,. but rather on relative humidity of the surrounding air. Therefore, the described method of accumulation may be realized at a constant temperature, which is an aspect utilizable in discharging the accumulator. In the progress of charging, relative humidity of air is decreased to the required level through heating the air to achieve a higher

The classical accumulation of heat utilizes the so-called sensible heat of substances (Kleczek, J., 1981) being the simplest one of all the methods, this approach was historically used in the first place. Traditional materials applied for the accumulation of heat are water and gravel.

one applies the sorption and desorption of gas / water vapour.

temperature (Close, D.J. & Dunkle, R.V. 1977, Verdonschet, J.K.M. 1981).

**2.1 Classical heat accumulation methods** 

Januar 2010, Liu, Y.-T.; at al. 2008 ).

in solar-powered systems).

**2. Heat accumulation** 

Jun 2007).

society are limited. Moreover, such classification applies also to the possibilities of utilizing the energy of water and wind.

A large number of countries have committed themselves to the reduction of greenhouse gas emissions and the related increase of renewable sources of energy (Ministery of industry and trade of Czech Republic, Stat energetic conception 2004, Ministery of industry and trade of Czech Republic 2000) share in total energy consumption. However, the effort to comply with these commitments may be realized in absurd ways such as an uncontrolled surge in the number of constructed solar photovoltaic systems, which is further aggravated by the related problem of their integration into the energy production system of a country, Fig. 1.

Fig. 1. A photovoltaic power plant design, the Czech Republic

One of the applicable alternative source solutions consist in utilizing solar radiation ( Solar energy, 2010) within its entire spectrum. Among the advantages of this energy harvesting method there are mainly the low cost of the impinging energy, the unimpeded availability of the source in many regions of the Earth, and the polution-free operation. Conversely, the related disadvantages can be identified in the low density of the impinging radiation power flow in the visible and the near-visible spectrum (λ∈<440,780> nm), the comparatively low efficiency of transformation into other forms of energy (considering the currently used photovoltaic elements), and the fact that the cost of a produced energy unit is often rather high when compared to other clean sources of electrical energy (such as nuclear power plants) (Kleczek, J. 1981). In consequence of the uneven power flow density of the solar source within the daily or yearly cycle and owing to weather changes, the solar energy application method is affected by the problems of effective utilization, regulation in power systems, and necessity of accumulating the energy acquired from solar radiation.

A feasible technique of energy accumulation seems to consist in direct exploitation of physical effect of material properties as related to metals, liquids and gases (Gille, T. 2007, Shi, L.P. 2006). Accumulators will facilitate power take-off during any time period depending on the needs of the consumer or the power system operator, which provides for the balance in the cost/power take-off relation within the required time interval. Thus, the power distribution network stability will be improved, with substantial reduction of the probability of black-out (Black-out 2003). There occurs the compensation of time disproportion between the potential of the sources and the eletrical energy on the one hand and the consumption in time and place on the other. Solar energy accumulation can be technologically realized through a wide variety of methods; also, research in the field is being consistently developed (Juodkazis, Saulius; at al. November 2004, Zhen Ren at al. Januar 2010, Liu, Y.-T.; at al. 2008 ).

Some of the proposed approaches are based on classical solutions. These include the accumulation of energy utilizing the potential energy of mass in a gravitational field (water), the kinetic energy of mass (flywheels), the non-linearities in the state of a mass phase – the compression of gases. Another group of accumulators is based on the solution utilizing the energy of an electromagnetic field. In this case, the most common is the application of electric accumulators or microbiological systems [mikrobiolog akumulator]. Yet another one of the fields to be quoted comprises energy accumulation using the properties of chemical bonds of non-trivial chemical systems (the production of synthetic fuels), electrochemical bonds, and the utilization of photochemical energy (the accumulation of low-potential heat in solar-powered systems).

The process of designing chemical accumulator forms utilizes the physical effects of nonlinear behaviour of materials at phase changes. (Behunek,I. April 2004, Behunek I & Fiala P. Jun 2007).

#### **2. Heat accumulation**

350 Convection and Conduction Heat Transfer

society are limited. Moreover, such classification applies also to the possibilities of utilizing

A large number of countries have committed themselves to the reduction of greenhouse gas emissions and the related increase of renewable sources of energy (Ministery of industry and trade of Czech Republic, Stat energetic conception 2004, Ministery of industry and trade of Czech Republic 2000) share in total energy consumption. However, the effort to comply with these commitments may be realized in absurd ways such as an uncontrolled surge in the number of constructed solar photovoltaic systems, which is further aggravated by the related problem of their integration into the energy production system of a country, Fig. 1.

One of the applicable alternative source solutions consist in utilizing solar radiation ( Solar energy, 2010) within its entire spectrum. Among the advantages of this energy harvesting method there are mainly the low cost of the impinging energy, the unimpeded availability of the source in many regions of the Earth, and the polution-free operation. Conversely, the related disadvantages can be identified in the low density of the impinging radiation power flow in the visible and the near-visible spectrum (λ∈<440,780> nm), the comparatively low efficiency of transformation into other forms of energy (considering the currently used photovoltaic elements), and the fact that the cost of a produced energy unit is often rather high when compared to other clean sources of electrical energy (such as nuclear power plants) (Kleczek, J. 1981). In consequence of the uneven power flow density of the solar source within the daily or yearly cycle and owing to weather changes, the solar energy application method is affected by the problems of effective utilization, regulation in power

systems, and necessity of accumulating the energy acquired from solar radiation.

A feasible technique of energy accumulation seems to consist in direct exploitation of physical effect of material properties as related to metals, liquids and gases (Gille, T. 2007, Shi, L.P. 2006). Accumulators will facilitate power take-off during any time period depending on the needs of the consumer or the power system operator, which provides for the balance in the cost/power take-off relation within the required time interval. Thus, the power distribution network stability will be improved, with substantial reduction of the probability of black-out (Black-out 2003). There occurs the compensation of time disproportion between the potential of the sources and the eletrical energy on the one hand and the consumption in time and place on the other. Solar energy accumulation can be technologically realized through a wide variety of methods; also, research in the field is

Fig. 1. A photovoltaic power plant design, the Czech Republic

the energy of water and wind.

Principles are known (Baylin, F. 1979) for the utilization of characteristics of chemicalphysical effects, and in this context there exist four basic methods of thermal energy accumulation. The first method consists in the utilization of specific thermal capacity of substances (sensible heat), the second one is built on the application of change in the state of substances (latent heat) , the third one lies in the thermochemical reaction, and the fourth one applies the sorption and desorption of gas / water vapour.

Generally, the thermochemical reactions method provides a higher density of accumulated energy than the sensible heat or phase change options (Mar, R.W. & Bramletta, T.T. 1980). An endothermic reaction product contains energy in the form of a chemical bond that is released retroactively during an exothermic reaction. The energy release occurs through the action of a catalyst, which is a suitable characteristic for long-term accumulation. Other advantages of thermochemical accumulation include the possibility of transporting the products over long distances, the possibility of product storage at both low (with a low rate of loss) and very high temperatures (Goldstein, M. 1961), the low cost, and the fact that products of the reaction can be used as the medium in thermodynamic cycles (The Australian National University 2004). Currently, research (Mar, R.W. 1978, Mar, R.W. 1980) is conducted in this field. In order to accumulate energy, we can utilize heat balance at sorption/desorption of moisture in the working substance. The difference with respect to other types of heat accumulation consists in the fact that sorption does not directly depend on the temperature,. but rather on relative humidity of the surrounding air. Therefore, the described method of accumulation may be realized at a constant temperature, which is an aspect utilizable in discharging the accumulator. In the progress of charging, relative humidity of air is decreased to the required level through heating the air to achieve a higher temperature (Close, D.J. & Dunkle, R.V. 1977, Verdonschet, J.K.M. 1981).

#### **2.1 Classical heat accumulation methods**

The classical accumulation of heat utilizes the so-called sensible heat of substances (Kleczek, J., 1981) being the simplest one of all the methods, this approach was historically used in the first place. Traditional materials applied for the accumulation of heat are water and gravel.

Properties and Numerical Modeling-Simulation of Phase Changes Material 353

A prospective method of heat accumulation consists in utilizing the change of state of a substance used in a heat reservoir. Such reservoir charges (fillings) are described by the abbreviation PCM (Phase Change Material). According to Ehrenfest (Mechlova, E., Kostal, K. 1999), the changes of state are among the first type of phase changes, where the change of internal energy and substance volume occurs through a jump. If specific melting heat or solidification of the given substance is utilized, the calorimetry equation assumes the

m e

0 m <sup>m</sup> d d *T T*

is the density, *V* the volume, *c* the specific heat, Δ*h*m the enthalpy, *Q* the heat, and

ρρ

∫ ∫ (2)

Δ

*Q* (J)

**phase- 1** 

**phase- 2** 

*Q* necessary to change the

*T T Q Vh VcT VcT* = Δ+ +

*T*m , *T*e the temperature according to Figure 3. If heat is supplied to the material, there occurs the transformation from the liquid into the solid state. Phase transition appears when crystal lattice is disrupted, namely when the amplitude of the crystal lattice particles oscillation is comparable with relative distance between the particles. At this moment, the oscillation energy rises above the value of the crystal binding energy, the bond is broken and the crystal transforms into the liquid phase. However, if heat is removed from the substance, there occurs the solidification (crystallization) of material. During crystallization, the orderly motion of molecules gradually assumes the character of thermal oscillations around certain middle positions, namely crystal lattice is formed. In pure crystalline substances, melting and solidification proceed at a constant temperature *Tm*, which does not vary during the phase transition. In amorphous substances, the phase transition temperature is not constant and the state change occurs within a certain range of temperatures, Figure 4. In simplified terms for a macroscopic description of the numerical model, the phase change of a material is understood as a state in which the material changes its physical characteristics on the basis of variations (external) of its thermodanymic system. This state is often accompanied by a nonlinear effect, Figure 3. The effect involves energy *Q* supplied to the thermodynamic

externalmacroscopic state of the material, initial state temperature *T*0, phase change

Δ*Q*

**2.4 Utilizing the change of state in substances for the accumulation of heat** 

ρ

system of the material, temperature *T*, latent energy

*T* (K)

*T*e

Δ*T*

*T*<sup>m</sup>

*T*0

Fig. 3. The PCM macroscopic characteristics

temperature *T*m , and temperature *Te* limiting the low-temperature mode.

form

where ρ

The weight and specific heat capacity of these materials indicate the accumulable quantity of heat. This quantity is given by the calorimetric equation

$$Q = \int\_{T\_1}^{T\_2} V \rho \text{cd}\, T \tag{1}$$

where *T*1 is the temperature at the beginning and *T*2 the temperature at the end of charging.

#### **2.2 Water heat reservoirs**

If water is utilized in the process of accumulation, it is usually held in a suitable container during heating (Garg, H.P., at al. 1985). Even though the applicaton of water has proved to be advantageous in many respects, there are also many drawbacks, especially for the preservation of low-potential heat. Water has the highest specific heat capacity of all known substances (Vohlídal J. et al. 1999). It can be applied as an accumulation and working medium (exchangers are not necessary); charging and discharging can be simulated in an exact manner. If water is applied, heat storage or offtake causes temperature fluctuation and the thermal potential is lost (namely the accumulator is charged at a sufficiently high input temperature *T*, which is averaged in the reservoir to a mean temperature *Tstř* and, during the subsequent heat offtake, the original temperature *T* can not be reached) (Fisher L.S. 1976, Lavan, Z. & Thomson, J. 1977). With water accumulators, liquid photothermal collectors need to be applied; this means that expensive and rather complicated technologie and installation methods are used as opposed to the hot.-air option) .

#### **2.3 Gravel heat accumulators**

For multi-day or seasonal accumulation (Behunek, I. April 2004), heat reservoris utilizing gravel are preferred; here, the air used as the heating medium is heated in hot-air collectors. This system eliminates some of the disadvantages of the previously described method. In regular realizations, Figure 2, heat transfer by conduction is also minimal (the individual gravel pieces touch one another only at the edges); here, however, the characteristics include low heat capacity of the crushed stone (excessive dimensions of the reservoir) as well as a very difficult (even impossible as per (Garg, H.P., et al. 1985 ) simulation of charging and discharging.

Fig. 2. Schematic description of a heat accumulator using gravel

The weight and specific heat capacity of these materials indicate the accumulable quantity of

d

∫ (1)

hot gravel

cool gravel

2

*T*

1

where *T*1 is the temperature at the beginning and *T*2 the temperature at the end of charging.

If water is utilized in the process of accumulation, it is usually held in a suitable container during heating (Garg, H.P., at al. 1985). Even though the applicaton of water has proved to be advantageous in many respects, there are also many drawbacks, especially for the preservation of low-potential heat. Water has the highest specific heat capacity of all known substances (Vohlídal J. et al. 1999). It can be applied as an accumulation and working medium (exchangers are not necessary); charging and discharging can be simulated in an exact manner. If water is applied, heat storage or offtake causes temperature fluctuation and the thermal potential is lost (namely the accumulator is charged at a sufficiently high input temperature *T*, which is averaged in the reservoir to a mean temperature *Tstř* and, during the subsequent heat offtake, the original temperature *T* can not be reached) (Fisher L.S. 1976, Lavan, Z. & Thomson, J. 1977). With water accumulators, liquid photothermal collectors need to be applied; this means that expensive and rather complicated technologie and

For multi-day or seasonal accumulation (Behunek, I. April 2004), heat reservoris utilizing gravel are preferred; here, the air used as the heating medium is heated in hot-air collectors. This system eliminates some of the disadvantages of the previously described method. In regular realizations, Figure 2, heat transfer by conduction is also minimal (the individual gravel pieces touch one another only at the edges); here, however, the characteristics include low heat capacity of the crushed stone (excessive dimensions of the reservoir) as well as a very difficult (even impossible as per (Garg, H.P., et al. 1985 ) simulation of charging and

> hot gravel

cool gravel

thermal accumulation heating

*T Q VcT* = ρ

heat. This quantity is given by the calorimetric equation

installation methods are used as opposed to the hot.-air option) .

gravel

Fig. 2. Schematic description of a heat accumulator using gravel

**2.2 Water heat reservoirs** 

**2.3 Gravel heat accumulators** 

discharging.

#### **2.4 Utilizing the change of state in substances for the accumulation of heat**

A prospective method of heat accumulation consists in utilizing the change of state of a substance used in a heat reservoir. Such reservoir charges (fillings) are described by the abbreviation PCM (Phase Change Material). According to Ehrenfest (Mechlova, E., Kostal, K. 1999), the changes of state are among the first type of phase changes, where the change of internal energy and substance volume occurs through a jump. If specific melting heat or solidification of the given substance is utilized, the calorimetry equation assumes the form

$$Q = \rho V \Delta l\_{\rm m} + \int\_{T\_0}^{T\_{\rm m}} V \rho \text{cd}T + \int\_{T\_{\rm m}}^{T\_{\rm e}} V \rho \text{cd}T \tag{2}$$

where ρ is the density, *V* the volume, *c* the specific heat, Δ*h*m the enthalpy, *Q* the heat, and *T*m , *T*e the temperature according to Figure 3. If heat is supplied to the material, there occurs the transformation from the liquid into the solid state. Phase transition appears when crystal lattice is disrupted, namely when the amplitude of the crystal lattice particles oscillation is comparable with relative distance between the particles. At this moment, the oscillation energy rises above the value of the crystal binding energy, the bond is broken and the crystal transforms into the liquid phase. However, if heat is removed from the substance, there occurs the solidification (crystallization) of material. During crystallization, the orderly motion of molecules gradually assumes the character of thermal oscillations around certain middle positions, namely crystal lattice is formed. In pure crystalline substances, melting and solidification proceed at a constant temperature *Tm*, which does not vary during the phase transition. In amorphous substances, the phase transition temperature is not constant and the state change occurs within a certain range of temperatures, Figure 4. In simplified terms for a macroscopic description of the numerical model, the phase change of a material is understood as a state in which the material changes its physical characteristics on the basis of variations (external) of its thermodanymic system. This state is often accompanied by a nonlinear effect, Figure 3. The effect involves energy *Q* supplied to the thermodynamic system of the material, temperature *T*, latent energy Δ*Q* necessary to change the externalmacroscopic state of the material, initial state temperature *T*0, phase change temperature *T*m , and temperature *Te* limiting the low-temperature mode.

Fig. 3. The PCM macroscopic characteristics

Properties and Numerical Modeling-Simulation of Phase Changes Material 355

PCM materials applicable for the accumulation of heat utilizing state change ought to meet the following criteria (Behunek, I. 2002): Physical (a suitable phase diagram in the transition area, a suitable phase transition temperature, small changes of volume during the change of state, high density of the substance, supercooling tolerance, high specific melting heat, good thermal conductivity), chemical (nonflammability, nontoxicity, chemical stability, anticorrosive properties), economical (low asking price, availability, low cost of a suitable accumulator). The structure of PCM reservoirs must conform to standard requirements placed on thermal containers. In general, with respect to the provision of a suitable speed of heat transfer, it is necessary to encase the actual PCM material and insert the resulting containers that hold the PCM into an external envelope; this insertion should be realized in such a way that, through its circulation, the heating medium ensures an optimum transfer of

Consequently, there exists substantial similarity to caloric reservoirs containing crushed stone (aggregate) and therefore the rules governing the construction of these reservoirs can

If we are to futher consider the properties of PCMs, it is necessary to describe their

a. The advantages of *anorganic substances* (GARG, H.P. et al. 1985, VENER, C. 1997) mainly consist in the high value of specific melting heat, good thermal conductivity, nonflammability, and low cost. The negative characteristics include corrosivity of the substances to most metals, decomposition, loss of hygroscopic water, and the possibility

CaCl2.6H2O, Na2SO4.10H2O, Na2CO3.10H2O, MgCl2.6H2O, CaBr2.6H2O, Mg(NO3)2.6H2O,

b. *Organic substances* (GARG, H.P. et al. 1985, VENER, C. 1997) offer advantages such as a high value of specific melting heat, chemical stability, elimination of supercooling, and no corrosivity. The disadvantages consist in the inferior thermal conductivity, relatively significant variations of volume during the change of state, flammability). Examples of organic PCMs include paraffin, wax, polyethylene glycol, high-density polyethylene,

c. *Other substances* include compounds, combinations of amorphous and crystalline substances, kombinace amorfních a krystalických látek, clathrates, and other items. The advantage of low-potential heat accumulation in PCM application consists in the variability. A comparison of PCM and classical materials together with a listing of several PCMs (] LANE, G.A. 1983, FAVIER, A. 1999) is provided in Table 1. The elementary reference quantity is the density of accumulated energy. We assume the initial charging temperature as *T*0 20 °C and the final termperature as *T*e 50 °C. The course of accumulator loading with various types of filling (charge) is shown in Figure 4; a realization example of a

( ) () ( ) teplo X Y mH O X Y pH O m p H O n n 2 2 <sup>2</sup> ⋅ ←⎯⎯→ ⋅ +− (3)

**2.5 Requirements placed on the PCM, reservoirs and casings** 

heat energy in both directions (during charging and discharging), Figure 5.

minimum classicification and properties in phase changes, relation (3).

stearic acid (C17H35COOH), and palmitic acid (C15H31COOH).

be applied.

of supercooling.

**2.6 Basic classification of PCM materials** 

Examples of anorganic PCMs are as follows:

PCM-based accumulator is provided in Figure 5.

LiNO3.3H2O, KF.4H2O, Na2SO4.10H2O, Na2HPO4.12H2O.

Fig. 4. The course of accumulator charging with PCMs and classical materials

Fig. 5. Experimental model of a heat accumulator

#### **2.5 Requirements placed on the PCM, reservoirs and casings**

PCM materials applicable for the accumulation of heat utilizing state change ought to meet the following criteria (Behunek, I. 2002): Physical (a suitable phase diagram in the transition area, a suitable phase transition temperature, small changes of volume during the change of state, high density of the substance, supercooling tolerance, high specific melting heat, good thermal conductivity), chemical (nonflammability, nontoxicity, chemical stability, anticorrosive properties), economical (low asking price, availability, low cost of a suitable accumulator). The structure of PCM reservoirs must conform to standard requirements placed on thermal containers. In general, with respect to the provision of a suitable speed of heat transfer, it is necessary to encase the actual PCM material and insert the resulting containers that hold the PCM into an external envelope; this insertion should be realized in such a way that, through its circulation, the heating medium ensures an optimum transfer of heat energy in both directions (during charging and discharging), Figure 5.

Consequently, there exists substantial similarity to caloric reservoirs containing crushed stone (aggregate) and therefore the rules governing the construction of these reservoirs can be applied.

#### **2.6 Basic classification of PCM materials**

354 Convection and Conduction Heat Transfer

20 40 60 80 100 120 140

Fig. 4. The course of accumulator charging with PCMs and classical materials

1. layer 2. layer 3. layer 4. layer 5. layer 6. layer 7. layer 8. layer

*T* (K)

20

outlet of air (ventilators)

Fig. 5. Experimental model of a heat accumulator

30

40

50

*q* (kWh m-3)

CaCl2.6H2O

gravel

H2O parafin

inlet of air

Na2CO3.10H2O Na2HPO4.12H2O

enclosures with PCM

heat insulation

ducts with flaps

If we are to futher consider the properties of PCMs, it is necessary to describe their minimum classicification and properties in phase changes, relation (3).

a. The advantages of *anorganic substances* (GARG, H.P. et al. 1985, VENER, C. 1997) mainly consist in the high value of specific melting heat, good thermal conductivity, nonflammability, and low cost. The negative characteristics include corrosivity of the substances to most metals, decomposition, loss of hygroscopic water, and the possibility of supercooling.

Examples of anorganic PCMs are as follows:

CaCl2.6H2O, Na2SO4.10H2O, Na2CO3.10H2O, MgCl2.6H2O, CaBr2.6H2O, Mg(NO3)2.6H2O, LiNO3.3H2O, KF.4H2O, Na2SO4.10H2O, Na2HPO4.12H2O.

$$\text{X(Y)}\_{\text{n}} \cdot \text{mH}\_{2}\text{O} \xleftarrow{\text{teplo}} \text{X(Y)}\_{\text{n}} \cdot \text{pH}\_{2}\text{O} + (\text{m} - \text{p})\text{H}\_{2}\text{O} \tag{3}$$


The advantage of low-potential heat accumulation in PCM application consists in the variability. A comparison of PCM and classical materials together with a listing of several PCMs (] LANE, G.A. 1983, FAVIER, A. 1999) is provided in Table 1. The elementary reference quantity is the density of accumulated energy. We assume the initial charging temperature as *T*0 20 °C and the final termperature as *T*e 50 °C. The course of accumulator loading with various types of filling (charge) is shown in Figure 4; a realization example of a PCM-based accumulator is provided in Figure 5.

Properties and Numerical Modeling-Simulation of Phase Changes Material 357

owing to weak nucleation. Crystallization was initiated thanks to a solid particle of the PCM added to the measured sample. Otherwise, the crystallization would not have occurred. The group of materials for the encasing of hexahydrate may include plastics, mild steel or

In some cases, temperature fluctuation above *Tm* may occur during solidification (Figure 7).

Figure 4 indicates the binary phase diagram of calcium chloride and water. The hexahydrate contains 50,66 wt% CaCl2, and the tetrahydrate 60,63 wt%. The melting point of the hexahydrate is 29,6 °C, with that of the tetrahydrate being 45,3 °C. The hexahydrate-α tetrahydrate peritectic point is at 49,62 wt% CaCl2-50,38 wt% H2O, and 29,45 °C. In addition to the stable form, there are two monotropic polymorphs of the tetrahydrate salt, β and γ. The latter two are rarely encountered when dealing with the hexahydrate composition; however, the α tetrahydrate is stable from its liquidus temperature, 32,78 °C, down to the peritectic point, 29,45 °C, thus showing a span of 3,33 °C. When liquid CaCl6.6H2O is cooled at the equilibrium, CaCl2.4H2O can begin to crystallize at 32,78 °C. When the peritectic is reached at 29,45 °C, the tetrahydrate hydrates further to form hexahydrate, and the material freezes. The maximum amount of tetrahydrate which can be formed is 9,45 wt%, calculated by the lever rule. This process is reversed when solid CaCl6.6H2O is heated at the equilibrium. At 29,45 °C the peritectic reaction occurs, forming 9,45% of CaCl2.4H2O and the liquid of the peritectic composition. With increasing temperature, the tetrahydrate melts, disappearing completely at 32,78 °C. Under actual freezing and melting conditions, the equilibrium processes described above may occur only partially or not at all. Supercooling of the tetrahydrate may lead to initial crystallization of the hexahydrate at 29,6 °C (or lower if this phase also supercools). It is possible to conduct modification by additives. From a number of potential candidates, Ba(OH)2, BaCO3 and Sr(OH)2 were chosen as they seemed to be feasible. When we used Ba(OH)2 and Sr(OH)2 at 1% part by weight, there was no supercooling. We were able to increase the stability of the equilibrium condition by adding KCl (2 wt%) and NaCl, Figure 8. NaCl is a weak soluble in CaCl2.6H2O, therefore the part by weight is only about 0,5%. The related disadvantage is that the melting point decreases by

The effectivity of transferring the heat to active elements in the accumulator consists in the optimum setting of dimensions and shapes in the process of circulation of the medium that transfers energy in the accumulator. Therefore, a necessary precondition of the design consisted in solving the air circulation model under the condition of change in its temperature and thermodynamical variations in the PCM material. The actual model of active elements and its temperature characteristic is not fundamental to this task; the

A geometric model of one layer of the accumulator is shown in Figure 10 (Behunek, I. 2004) It consists of 26 PVC pipes in a square configuration (Lienhard, J.H. IV & Lienhard, J.H. V. 2004). Inside of the pipes there are 9,36 litres of modified CaCl2.6H2O. The air flows through the layer and transfers heat into the pipes. The related numerical solution was realized in two parts. First, we solved the turbulence model and obtained the heat transfer film coefficients. These results constituted the input for the solution of the second part namely the calculation of the thermal model. The time dependence of temperature distribution in

characteristic is known and realizable through commonly appplied methods.

copper; aluminium or stainless steel are not suitable.

The explanation was found in the binary diagram.

ca. 3 °C at 26-27 °C.

the layer is the final result.

**2.8 Numerical model of heat accumulators** 


Table 1. A comparison of accumulated energy density for different substances

Fig. 6. Phase changes of CaCl2.6H2O

Fig. 7. Phase change of CaCl2.6H2O (CaCl2.4H2O crystallization)

#### **2.7 Calcium chloride hexahydrate and its modification**

In Figure 6, the phase change of CaCl2.6H2O during heating and cooling is shown. The dashed lines show the theoretical behaviour under the condition when the melting and freezing were realized at constant temperature *Tm* – the case of pure crystallic substances. Impurity and the methodology of measuring are the main cause of variations (the probe has to be placed only in small amounts of hexahydrate. During solidification, ing occurred

In Figure 6, the phase change of CaCl2.6H2O during heating and cooling is shown. The dashed lines show the theoretical behaviour under the condition when the melting and freezing were realized at constant temperature *Tm* – the case of pure crystallic substances. Impurity and the methodology of measuring are the main cause of variations (the probe has to be placed only in small amounts of hexahydrate. During solidification, ing occurred

°C

°C

Water 34,5 Aggregate of stones 23,0 Paraffin 62,4 CaCl2 . 6H2O 117,4 Na2CO3 . 10H2O 131,7 Na2HPO4 . 12H2O 134,7 Table 1. A comparison of accumulated energy density for different substances

**Accumulated energy density** *q* **[kWh.m-3]** 

0:00 0:36 1:12 1:48 2:24 3:00 3:36 4:12 čas [hodiny]

0:00 0:07 0:14 0:21 0:28 0:36 0:43 0:50 čas [minuty]

**Material used for accumulation in heat reservoir**

0:00 0:21 0:43 1:04 1:26 1:48 2:09 čas [hodiny]

0:00 0:28 0:57 1:26 1:55 2:24 2:52 čas [hodiny]

Fig. 7. Phase change of CaCl2.6H2O (CaCl2.4H2O crystallization)

**2.7 Calcium chloride hexahydrate and its modification** 

Fig. 6. Phase changes of CaCl2.6H2O

°C

°C

owing to weak nucleation. Crystallization was initiated thanks to a solid particle of the PCM added to the measured sample. Otherwise, the crystallization would not have occurred. The group of materials for the encasing of hexahydrate may include plastics, mild steel or copper; aluminium or stainless steel are not suitable.

In some cases, temperature fluctuation above *Tm* may occur during solidification (Figure 7). The explanation was found in the binary diagram.

Figure 4 indicates the binary phase diagram of calcium chloride and water. The hexahydrate contains 50,66 wt% CaCl2, and the tetrahydrate 60,63 wt%. The melting point of the hexahydrate is 29,6 °C, with that of the tetrahydrate being 45,3 °C. The hexahydrate-α tetrahydrate peritectic point is at 49,62 wt% CaCl2-50,38 wt% H2O, and 29,45 °C. In addition to the stable form, there are two monotropic polymorphs of the tetrahydrate salt, β and γ. The latter two are rarely encountered when dealing with the hexahydrate composition; however, the α tetrahydrate is stable from its liquidus temperature, 32,78 °C, down to the peritectic point, 29,45 °C, thus showing a span of 3,33 °C. When liquid CaCl6.6H2O is cooled at the equilibrium, CaCl2.4H2O can begin to crystallize at 32,78 °C. When the peritectic is reached at 29,45 °C, the tetrahydrate hydrates further to form hexahydrate, and the material freezes. The maximum amount of tetrahydrate which can be formed is 9,45 wt%, calculated by the lever rule. This process is reversed when solid CaCl6.6H2O is heated at the equilibrium. At 29,45 °C the peritectic reaction occurs, forming 9,45% of CaCl2.4H2O and the liquid of the peritectic composition. With increasing temperature, the tetrahydrate melts, disappearing completely at 32,78 °C. Under actual freezing and melting conditions, the equilibrium processes described above may occur only partially or not at all. Supercooling of the tetrahydrate may lead to initial crystallization of the hexahydrate at 29,6 °C (or lower if this phase also supercools). It is possible to conduct modification by additives. From a number of potential candidates, Ba(OH)2, BaCO3 and Sr(OH)2 were chosen as they seemed to be feasible. When we used Ba(OH)2 and Sr(OH)2 at 1% part by weight, there was no supercooling. We were able to increase the stability of the equilibrium condition by adding KCl (2 wt%) and NaCl, Figure 8. NaCl is a weak soluble in CaCl2.6H2O, therefore the part by weight is only about 0,5%. The related disadvantage is that the melting point decreases by ca. 3 °C at 26-27 °C.

#### **2.8 Numerical model of heat accumulators**

The effectivity of transferring the heat to active elements in the accumulator consists in the optimum setting of dimensions and shapes in the process of circulation of the medium that transfers energy in the accumulator. Therefore, a necessary precondition of the design consisted in solving the air circulation model under the condition of change in its temperature and thermodynamical variations in the PCM material. The actual model of active elements and its temperature characteristic is not fundamental to this task; the characteristic is known and realizable through commonly appplied methods.

A geometric model of one layer of the accumulator is shown in Figure 10 (Behunek, I. 2004) It consists of 26 PVC pipes in a square configuration (Lienhard, J.H. IV & Lienhard, J.H. V. 2004). Inside of the pipes there are 9,36 litres of modified CaCl2.6H2O. The air flows through the layer and transfers heat into the pipes. The related numerical solution was realized in two parts. First, we solved the turbulence model and obtained the heat transfer film coefficients. These results constituted the input for the solution of the second part namely the calculation of the thermal model. The time dependence of temperature distribution in the layer is the final result.

Properties and Numerical Modeling-Simulation of Phase Changes Material 359

direction of air flow <sup>Ω</sup>

Fig. 10. Geometric model of a layer with the mesh of elements

For a steady state of flow there holds the continuity equation

equilibrium of forces the Navier-Stokes equation for the fluid element

υ

ρμ

dimension of tensor. In equation (7) we substitute pressure losses

z z z zzz h

*<sup>f</sup> Kv v Cv D*

⎛ ⎞ −++ ⎜ ⎟ ⎝ ⎠

v vu

ρμ

derived for the incompressible fluid with the condition

0 *div*

2 *curl w*

*v*

∂

where *A* is the external acceleration,

ρ

ρ

diameter of ribs, *C* the air permeability of system,

**2.8.1 Mathematical and numerical model** 

We assume a turbulent flow

750

The mathematical model of air velocity distribution uses fluid equations which were

ν

ν

theorem for the moving particle and the continuity equation, we can formulate from the

−=⋅+ Δ⋅+

<sup>∂</sup> *grad pgrad <sup>t</sup> T*

*f f gradp K v v C v K v v C v D D*

where *K* are the suppressed pressure losses, *f* the resistance coefficient, *D*h the hydraulic

vector of the Cartesian coordinate system. The resistance coefficient is obtained from the

is the angular velocity of fluid. If we use the Stokes theorem, the Helmholtz

*Avv υ v*

x x x xxx y y y yyy h h

μ

⎛ ⎞ ⎛ ⎞ =− + + − + + ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠

v vu v vu

 ρ

<sup>1</sup> )( (7)

the vector of kinematic viscosity, and (*grad v*) has the

*<sup>b</sup> aRef* <sup>−</sup> = (9)

ρμ

the dynamic viscosity, and *u*x,y,z the unit

(8)

ρ

*div* ρν 600

100

ΓPCM ΓAir

= (4)

= 0 (5)

= (6)

CaCl2.6H2O

where

ω

Boussinesq theorem

PVC pipe

air

Γm

Fig. 8. Binary diagram of CaCl2.6H2O [9]

čas [hodiny]

Fig. 9. Modification of CaCl2.6H2O at a different rate of heat removal

čas [hodiny]

Fig. 10. Geometric model of a layer with the mesh of elements

#### **2.8.1 Mathematical and numerical model**

The mathematical model of air velocity distribution uses fluid equations which were derived for the incompressible fluid with the condition

$$d\dot{\nu}\text{ }\nu=0\tag{4}$$

For a steady state of flow there holds the continuity equation

$$\text{div } \rho \mathbf{v} = \mathbf{0} \tag{5}$$

We assume a turbulent flow

358 Convection and Conduction Heat Transfer

Fig. 8. Binary diagram of CaCl2.6H2O [9]

0:00 0:15 0:31 0:47 1:03 1:19 1:35 1:50 2:06 2:22 čas [hodiny]

Fig. 9. Modification of CaCl2.6H2O at a different rate of heat removal

°C

0:00 0:11 0:23 0:34 0:46 0:57 1:09 1:20 1:32 1:43 čas [hodiny]

°C

$$\text{curl } \mathbf{v} = \mathbf{2}w \tag{6}$$

where ω is the angular velocity of fluid. If we use the Stokes theorem, the Helmholtz theorem for the moving particle and the continuity equation, we can formulate from the equilibrium of forces the Navier-Stokes equation for the fluid element

$$\frac{\partial \mathbf{v}}{\partial t} + (\operatorname{grad} \boldsymbol{\nu}) \cdot \mathbf{v}^{\top} = \boldsymbol{A} - \frac{1}{\rho} \operatorname{grad} \, p + \mathbf{v} \cdot \boldsymbol{\Delta \mathbf{v}} \tag{7}$$

where *A* is the external acceleration, υ the vector of kinematic viscosity, and (*grad v*) has the dimension of tensor. In equation (7) we substitute pressure losses

$$\begin{aligned} \frac{1}{2}\rho \boldsymbol{\sigma} \boldsymbol{d} \boldsymbol{p} &= -\Big(\mathbf{K}\_{\mathbf{x}} \rho \boldsymbol{\upsilon}\_{\mathbf{x}} \left| \mathbf{v} \right| + \frac{f}{D\_{\mathbf{h}}} \rho \boldsymbol{\upsilon}\_{\mathbf{x}} \left| \mathbf{v} \right| + \mathbf{C}\_{\mathbf{x}} \mu \boldsymbol{\upsilon}\_{\mathbf{x}} \Bigg) \mathbf{u}\_{\mathbf{x}} - \Big(\mathbf{K}\_{\mathbf{y}} \rho \boldsymbol{\upsilon}\_{\mathbf{y}} \left| \mathbf{v} \right| + \frac{f}{D\_{\mathbf{h}}} \rho \boldsymbol{\upsilon}\_{\mathbf{y}} \left| \mathbf{v} \right| + \mathbf{C}\_{\mathbf{y}} \mu \boldsymbol{\upsilon}\_{\mathbf{y}} \left| \mathbf{u} \right| + \mathbf{C}\_{\mathbf{y}} \mu \boldsymbol{\upsilon}\_{\mathbf{y}} \left| \mathbf{u}\_{\mathbf{y}} \right| \\ &- \Big(\mathbf{K}\_{\mathbf{z}} \rho \boldsymbol{\upsilon}\_{\mathbf{z}} \left| \mathbf{v} \right| + \frac{f}{D\_{\mathbf{h}}} \rho \boldsymbol{\upsilon}\_{\mathbf{z}} \left| \mathbf{v} \right| + \mathbf{C}\_{\mathbf{z}} \mu \boldsymbol{\upsilon}\_{\mathbf{z}} \Bigg) \mathbf{u}\_{\mathbf{z}} \end{aligned} \tag{8}$$

where *K* are the suppressed pressure losses, *f* the resistance coefficient, *D*h the hydraulic diameter of ribs, *C* the air permeability of system, μ the dynamic viscosity, and *u*x,y,z the unit vector of the Cartesian coordinate system. The resistance coefficient is obtained from the Boussinesq theorem

$$f\_- = aRe^{-b} \tag{9}$$

Properties and Numerical Modeling-Simulation of Phase Changes Material 361

Fig. 11. Velocity distribution of the air

Fig. 12. Velocity distribution of the air (vectors, detail)

processor

Fig. 14. Example of a processor cooler with a phase change material)

PCM

ribs

Ω

Fig. 13. Distribution of temperature

where *Re* is Reynolds number and *a*, *b* are coefficients from [40]. The model of short deformation field is formulated from the condition of steady-state stability, which is expressed

$$\int\_{\Omega} f \, \mathrm{d}\mathcal{Q} + \int\_{\Gamma} t \mathrm{d}\Gamma = 0 \tag{10}$$

where *f* are the specific forces in domain Ω, and *t* the pressures, tensions and shear stresses on the interface area Γ. By means of the transformation into local coordinates, we obtain the differential form for the static equilibrium

$$\mathbf{f} + d\mathbf{\dot{v}}^{\mathsf{T}} \mathbf{T}\_{\mathsf{v}} = \mathbf{0} \tag{11}$$

where *div*2 stands for the div operator of tensor quantity and *T*v is the tensor of internal tension

$$\mathbf{T\_{v}} = \begin{bmatrix} X\_{\mathbf{x}} & X\_{\mathbf{y}} & X\_{\mathbf{z}} \\ Y\_{\mathbf{x}} & Y\_{\mathbf{y}} & Y\_{\mathbf{z}} \\ Z\_{\mathbf{x}} & Z\_{\mathbf{y}} & Z\_{\mathbf{z}} \end{bmatrix} \tag{12}$$

where *X*, *Y*, *Z* are the stress components which act on elements of the area. It is possible to add a form of specific force from (4)-(7) to the condition of static equilibrium. The form of specific force is obtained by means of an external acceleration *A*, on the condition that pressure losses and shear stresses τare given as

$$
\rho \left( \frac{\partial \mathbf{v}}{\partial t} + \cdot (\text{grad} \boldsymbol{\nu}) \cdot \mathbf{v}^{\boldsymbol{T}} \right) - \rho \boldsymbol{A} - \sum\_{l=1}^{N\_s} \mathbf{F}\_l + d \boldsymbol{\dot{\nu}}^2 \mathbf{T}\_{\mathbf{v}} = \mathbf{0} \tag{13}
$$

where *F*1 are the discrete forces and *div*2 is the divergence operator of tensor. The model which covers the forces, viscosity, and pressure losses is

$$
\rho \left( \frac{\partial \mathbf{v}}{\partial t} + \left( \text{grad} \, \mathbf{v} \right) \cdot \mathbf{v}^T \right) - \rho \mathbf{A} - \sum\_{l=1}^{N\_s} \mathbf{F}\_l + \text{grad} \, p - \mu \cdot \mathbf{A} \, \mathbf{v} = \mathbf{0} \tag{14}
$$

We can prepare the discretization of equation (7) by means of the approximation of velocity *v* and acceleration *a* (Behunek I, Fiala P. Jun 2007). On the interface there are defined boundary and initial conditions. Initial and boundary conditions can be written; the initial temperature of the air is 50 °C, the initial velocity of the air is 0,4 m.s-1, the outlet pressure is 101,3 kPa + 10 Pa, and the initial temperature of the air inside the accumulator, PVC and CaCl2.6H2O is 20 °C. There are the distribution of velocity values indicated in figures 11, 12, and other results for the distribution of turbulent kinetic energy, dissipation, temperature and pressure follow on Figures 13. Calculation of the thermal model (finite element methode (FEM), finite volume methode (FVM), Ansys User's Manual) was realized under the same conditions as the previous turbulence model.

Figure 13 shows the time dependence of temperature in CaCl2.6H2O in the pipe marked with a black cross (Figure 11). We can compare the result of the numerical simulation with the measurement. Differences between the simulation and the measurement are caused by the inaccuracy of the model with respect to reality.

Fig. 11. Velocity distribution of the air

where *Re* is Reynolds number and *a*, *b* are coefficients from [40]. The model of short deformation field is formulated from the condition of steady-state stability, which is

where *f* are the specific forces in domain Ω, and *t* the pressures, tensions and shear stresses on the interface area Γ. By means of the transformation into local coordinates, we obtain the

where *div*2 stands for the div operator of tensor quantity and *T*v is the tensor of internal

⎢ ⎢ ⎢

v

τ

<sup>⎞</sup> <sup>⎜</sup>

<sup>⎛</sup> ⋅⋅+

⎛ ⎞ ∂

⎡ =

⎣

are given as

<sup>l</sup> )(

*<sup>T</sup> grad div*

ρ

where *F*1 are the discrete forces and *div*2 is the divergence operator of tensor. The model

+ ⋅ ⋅ − − + − ⋅Δ = ⎜ ⎟

We can prepare the discretization of equation (7) by means of the approximation of velocity *v* and acceleration *a* (Behunek I, Fiala P. Jun 2007). On the interface there are defined boundary and initial conditions. Initial and boundary conditions can be written; the initial temperature of the air is 50 °C, the initial velocity of the air is 0,4 m.s-1, the outlet pressure is 101,3 kPa + 10 Pa, and the initial temperature of the air inside the accumulator, PVC and CaCl2.6H2O is 20 °C. There are the distribution of velocity values indicated in figures 11, 12, and other results for the distribution of turbulent kinetic energy, dissipation, temperature and pressure follow on Figures 13. Calculation of the thermal model (finite element methode (FEM), finite volume methode (FVM), Ansys User's Manual) was realized under the same

Figure 13 shows the time dependence of temperature in CaCl2.6H2O in the pipe marked with a black cross (Figure 11). We can compare the result of the numerical simulation with the measurement. Differences between the simulation and the measurement are caused by

*0TFAvv <sup>v</sup>*

∂ <sup>s</sup>

( ) <sup>s</sup>

*grad gradp <sup>t</sup>*

*T*

0 <sup>v</sup>

zyx zyx zyx

*ZZZ YYY XXX*

where *X*, *Y*, *Z* are the stress components which act on elements of the area. It is possible to add a form of specific force from (4)-(7) to the condition of static equilibrium. The form of specific force is obtained by means of an external acceleration *A*, on the condition that

⎥ ⎥ ⎥

⎤

⎦

∑= <sup>⎟</sup> =+−−⎠

*l*

l 1 <sup>v</sup> vv A F v 0 *N*

ρμ

⎝ ⎠ <sup>∂</sup> ∑ (14)

*l*

=

*N*

1

=+ ∫ *tf ΓΩ* 0dd (10)

<sup>2</sup> *div Tf* =+ (11)

*T* (12)

v 2

(13)

∫ Ω

expressed

tension

differential form for the static equilibrium

pressure losses and shear stresses

⎝

ρ

ρ

conditions as the previous turbulence model.

the inaccuracy of the model with respect to reality.

∂

*t*

which covers the forces, viscosity, and pressure losses is

Fig. 12. Velocity distribution of the air (vectors, detail)

Fig. 13. Distribution of temperature

Fig. 14. Example of a processor cooler with a phase change material)

Properties and Numerical Modeling-Simulation of Phase Changes Material 363

λs, ρ<sup>s</sup> λ<sup>l</sup> , cs , ρl, cl liquid PCM solid PCM

Tif

T0

s (x,t)

**s (x,t)**

Fig. 16. Solidification of a semi-infinite plate of PCM

the interface between the solid and the liquid (*x* = *s*)

0

x0

Fig. 15. Heat transfer on the interface between the solid and the liquid parts

λs, ρs, cs λl, ρl, cl solid PCM liquid PCM

**ds**

We consider a semi-infinite mass of liquid PCM at initial temperature *T*0, which was cooled by a sudden drop of surface temperature *T*p = 0 °C. This temperature is constant during the whole process of solidification. The simplifying assumptions are as follows: The body is a semi-infinite plane, the heat flux is one-dimensional in the *x*-axis, the interface between the solid and the liquid is planar, there is an ideal contact on the interface, the temperature of the surface is constant (*T*p = 0 °C), the crystallization of PCM is at a constant temperature *T*m, the thermophysical properties of the solid and the liquid are different but independent of the temperature, there is no natural convection in the liquid. The initial and boundary conditions involve initial temperature *T*0 for *x* ≥ 0 at time 0; the temperature equals *T*m on

Tb

x

∞

Ts (x,t)

Tw

We used tabular values of pure CaCl2.6H2O; howerer, the pipes contain modified hexahydrate with 1,2% of BaCO3.

#### **3. Cooling system**

The PCM may be used for active or passive electronic cooling applications with high power at the package level (see Figure 14).

#### **3.1 Analytical description and solution of heat transfer and phase change**

We analyze the problem of heat transfer in a 1D body during the melting and freezing process with an external heat flux or heat convection, which is given by boundary conditions. The solution of this problem is known for the solidification of metals. We tried to apply this theory to the melting of crystalline salts. The 1D body could be a semifinite plane, cylinder or sphere. As the solid and the liquid part of PCM have different temperatures, there occurs heat transfer on the interface. According to Fig. 16, the origin of *x* is the axis of pipe, centre of sphere, or the origin of plate. Liquid starts to solidify if the surface is cooled by the flowing fluid (*T*w < *T*m). The equation describing the solid state is

$$\frac{\partial T\_s}{\partial t} = \frac{a\_s}{\mathbf{x}''} \frac{\partial}{\partial \mathbf{x}} \left( \mathbf{x}'' \frac{\partial T\_s}{\partial \mathbf{x}} \right) \tag{15}$$

where for the plate *n* = 0, cylinder *n* = 1 and sphere *n* = 2; *a*s is the thermal diffusion coefficient in the solid state. For *x* = *x*0 we can assume the following boundary conditions: constant temperature

$$T = T\_{\text{w}} \tag{16}$$

or constant heat flux

$$
\mathcal{A}\_s \frac{\partial T\_s}{\partial \mathbf{x}} = -q\_w \tag{17}
$$

or for convective cooling

$$
\mathcal{A}\_{\text{s}} = \frac{\widehat{\boldsymbol{\sigma}} \boldsymbol{T}\_{\text{s}}}{\widehat{\boldsymbol{\sigma}} \boldsymbol{x}} = -k(\boldsymbol{T} - \boldsymbol{T}\_{\text{b}}) \tag{18}
$$

where *q*w is the specific heat flux and λs is the thermal conductivity coefficient. Initial condition (*t* = 0) for (24) is

$$T\_s(0) = T\_0. \tag{19}$$

For the interface between the solid and the liquid we obtain

$$
\rho\_s \Delta h\_\text{m} \frac{\text{ds}}{\text{dt}} = \mathcal{A}\_s \frac{\partial T\_s}{\partial \text{x}} + \alpha (T\_\text{m} - T\_0). \tag{20}
$$

The analytical solution is exact but we consider several simplifying assumptions. The most important of these is that we can solve the solidification of PCM only in a one-dimensional body.

We used tabular values of pure CaCl2.6H2O; howerer, the pipes contain modified hexahydrate

The PCM may be used for active or passive electronic cooling applications with high power

We analyze the problem of heat transfer in a 1D body during the melting and freezing process with an external heat flux or heat convection, which is given by boundary conditions. The solution of this problem is known for the solidification of metals. We tried to apply this theory to the melting of crystalline salts. The 1D body could be a semifinite plane, cylinder or sphere. As the solid and the liquid part of PCM have different temperatures, there occurs heat transfer on the interface. According to Fig. 16, the origin of *x* is the axis of pipe, centre of sphere, or the origin of plate. Liquid starts to solidify if the surface is cooled

> ∂ <sup>∂</sup> <sup>=</sup> <sup>∂</sup>

where for the plate *n* = 0, cylinder *n* = 1 and sphere *n* = 2; *a*s is the thermal diffusion coefficient in the solid state. For *x* = *x*0 we can assume the following boundary conditions:

> s <sup>s</sup> *<sup>q</sup> <sup>x</sup> <sup>T</sup>* −= <sup>∂</sup> ∂ λ

s <sup>s</sup> *TTk x*

λ

∂

λ

d

=Δ

*t*

For the interface between the solid and the liquid we obtain

ρ

*<sup>T</sup>* −−= <sup>∂</sup>

).( <sup>d</sup>

s ms <sup>s</sup> *TT x T*

αλ

*<sup>s</sup> <sup>h</sup>* −+ ∂ ∂

The analytical solution is exact but we consider several simplifying assumptions. The most important of these is that we can solve the solidification of PCM only in a one-dimensional

0m

*a*

*T <sup>n</sup> n*

*<sup>T</sup> <sup>x</sup> <sup>x</sup> <sup>x</sup>*

⎟ ⎠ <sup>⎞</sup> <sup>⎜</sup> ⎝ ⎛

ss <sup>s</sup> (15)

= *TT* <sup>w</sup> (16)

(17)

= (18)

s is the thermal conductivity coefficient. Initial

.)0(s = *TT* <sup>0</sup> (19)

(20)

∂ ∂

w

)( <sup>b</sup>

*x*

**3.1 Analytical description and solution of heat transfer and phase change** 

by the flowing fluid (*T*w < *T*m). The equation describing the solid state is

∂

*t*

with 1,2% of BaCO3.

**3. Cooling system** 

constant temperature

or constant heat flux

or for convective cooling

condition (*t* = 0) for (24) is

body.

where *q*w is the specific heat flux and

at the package level (see Figure 14).

Fig. 15. Heat transfer on the interface between the solid and the liquid parts

Fig. 16. Solidification of a semi-infinite plate of PCM

We consider a semi-infinite mass of liquid PCM at initial temperature *T*0, which was cooled by a sudden drop of surface temperature *T*p = 0 °C. This temperature is constant during the whole process of solidification. The simplifying assumptions are as follows: The body is a semi-infinite plane, the heat flux is one-dimensional in the *x*-axis, the interface between the solid and the liquid is planar, there is an ideal contact on the interface, the temperature of the surface is constant (*T*p = 0 °C), the crystallization of PCM is at a constant temperature *T*m, the thermophysical properties of the solid and the liquid are different but independent of the temperature, there is no natural convection in the liquid. The initial and boundary conditions involve initial temperature *T*0 for *x* ≥ 0 at time 0; the temperature equals *T*m on the interface between the solid and the liquid (*x* = *s*)

Properties and Numerical Modeling-Simulation of Phase Changes Material 365

If we compare the results of the analytical solution with the experimental measurement of the materials (Behunek, I. 2002), we can see a good agreement. Outside domain Ω, is the air velocity and pressure are zero. We can write the form for an element of mesh related to the (Behunek I & Fiala P. Jun 2007) Cu-cooler with PCM. For the description of different turbulent models see (Piszachich, W.S. 1985, Wilcox, D.C. 1994). The numerical solution consists of two parts. Firstly, we solve the turbulence model and obtain heat transfer coefficients on the surface of the ribs. These results constitute the input for the second part, in which the thermal model is calculated. We obtain the time dependence of temperature distribution in the PCM. A geometric model of a copper cooler in shown in Figure 19. The CaCl2.6H2O is closed inside of the bottom plate (see Figure 14). The size of the plate is 30x30x5 mm, and the ribs are 20 mm in height. The PCM volume is approximately 3,8.10-6 m3. The plate takes the heat from the processor up and the crystalline salt starts to melt at *T*m. The air flows through the ribs and extracts heat from the cooler. In Figure 20, the distribution of air velocity module is indicated. We can see the effective rise of air flow velocity at the bottom of the ribs (detail A in Figures 19, 20). Temperature distribution in the ribs is shown in Figure 20, Figure 21 compares the results of numerical simulation with the measuring in the middle of PCM enclosure (casing). We measured the temperature by

The differences between the simulation and the measurement are due to the inaccuracy of the model with respect to reality. We used tabular values of pure CaCl2.6H2O but we modified the hexahydrate with 1,2% of BaCO3. to avoid supercooling and deformation of cooling curves after more cycles of melting and freezing. In order to obtain exact results, we would nevertheless need to obtain exact knowledge of the temperature dependence of

processor

PCM

thermal conductivity, specific heat and density during phase change (see Figure 22).

n

W

y x

z

copper cooler

fan

air

Г

Cu

Fig. 18. Geometric model of a Cu-cooler with PCM elements

Г

PCM

Г

**3.2 Nymerical analysis of heat transfer and phase change** 

means of a probe.

$$\mathbf{x} = \mathbf{s} \land t > \mathbf{0} \quad \Longrightarrow \quad T\_s = T\_l = T\_m = \text{const} \tag{21}$$

The evolved latent heat during the interface motion (the thickness of volume element *ds*, area 1 m2, time 1 s) is

$$dQ\_{\Delta h\_m} = \Delta h\_m \rho\_l \mathbf{1} \frac{ds}{dt} \tag{22}$$

Position of the interface is a function of time

$$\mathbf{s} = \mathbf{s}(t) = 2\varepsilon \sqrt{a\_s t},\tag{23}$$

This dependence is called the parabolic law of solidification, where ε is the root of equation describing the freezing. The boundary and initial condition for the phase change is

$$
\mathcal{A}\_s \left( \frac{\partial T\_s}{\partial \mathbf{x}} \right)\_{\mathbf{x}=\mathbf{s}} = \mathcal{A}\_l \left( \frac{\partial T\_l}{\partial \mathbf{x}} \right)\_{\mathbf{x}=\mathbf{s}} + \Delta h\_m \rho\_l \frac{d\mathbf{s}}{dt} \tag{24}
$$

$$1\ x \to \infty \land t > 0 \quad \Longrightarrow \quad T\_l = T\_0 = \text{const} \tag{25}$$

$$\text{Let } \mathbf{x} = \mathbf{0} \land t \ge \mathbf{0} \quad \Longrightarrow \quad T\_p = T\_s(\mathbf{x} = \mathbf{0}) = \mathbf{0} \text{ } \mathbf{0}^\circ \text{C} \tag{26}$$

If we solve the Fourier relations of heat conduction under the above-given conditions for the solid and the liquid, we get the equations below which allow for the calculation of temperatures in the solid, liquid PCMs as well as the location of interface. The results are shown in Figure 17, (Behunek I & Fiala P. Jun 2007).

Fig. 17. Position between the solid and the liquid PCM

#### **3.2 Nymerical analysis of heat transfer and phase change**

364 Convection and Conduction Heat Transfer

The evolved latent heat during the interface motion (the thickness of volume element *ds*,

*tatss* ,2)( *<sup>s</sup>* == ε

> *x T*

If we solve the Fourier relations of heat conduction under the above-given conditions for the solid and the liquid, we get the equations below which allow for the calculation of temperatures in the solid, liquid PCMs as well as the location of interface. The results are

⎟ Δ+

⎠ <sup>⎞</sup> <sup>⎜</sup> ⎝ ⎛ ∂

*sx l*

0 500 1000 1500 2000 2500 3000 3500 4000 4500 time (s)

<sup>Δ</sup>*hm* Δ= *hdQ*

describing the freezing. The boundary and initial condition for the phase change is

<sup>∂</sup> <sup>⎟</sup> <sup>=</sup> <sup>⎠</sup>

*l*

= =

λ

This dependence is called the parabolic law of solidification, where

*x T*

<sup>⎞</sup> <sup>⎜</sup> ⎝ ⎛ ∂ ∂

λ*s*

shown in Figure 17, (Behunek I & Fiala P. Jun 2007).

Fig. 17. Position between the solid and the liquid PCM

0

0.005

0.01

0.015

s (m)

0.02

0.025

0.03

*sx s*

area 1 m2, time 1 s) is

Position of the interface is a function of time

= ∧ *tsx* 0 ⇒> = = *mls* = *constTTT* (21)

1 (22)

(23)

is the root of equation

(24)

ε

*dt ds <sup>h</sup>*

*lm*

ρ

*tx* 0 *<sup>l</sup>* <sup>0</sup> ==⇒>∧∞→ *constTT* (25)

= ∧ *tx* 00 ⇒≥ = *xTT* = = °C0)0(*sp* (26)

CaCl2∙6H2

Na2CO3∙10H2O

Na2HPO4∙12H2

*dt ds*

ρ*lm* If we compare the results of the analytical solution with the experimental measurement of the materials (Behunek, I. 2002), we can see a good agreement. Outside domain Ω, is the air velocity and pressure are zero. We can write the form for an element of mesh related to the (Behunek I & Fiala P. Jun 2007) Cu-cooler with PCM. For the description of different turbulent models see (Piszachich, W.S. 1985, Wilcox, D.C. 1994). The numerical solution consists of two parts. Firstly, we solve the turbulence model and obtain heat transfer coefficients on the surface of the ribs. These results constitute the input for the second part, in which the thermal model is calculated. We obtain the time dependence of temperature distribution in the PCM. A geometric model of a copper cooler in shown in Figure 19. The CaCl2.6H2O is closed inside of the bottom plate (see Figure 14). The size of the plate is 30x30x5 mm, and the ribs are 20 mm in height. The PCM volume is approximately 3,8.10-6 m3. The plate takes the heat from the processor up and the crystalline salt starts to melt at *T*m. The air flows through the ribs and extracts heat from the cooler. In Figure 20, the distribution of air velocity module is indicated. We can see the effective rise of air flow velocity at the bottom of the ribs (detail A in Figures 19, 20). Temperature distribution in the ribs is shown in Figure 20, Figure 21 compares the results of numerical simulation with the measuring in the middle of PCM enclosure (casing). We measured the temperature by means of a probe.

The differences between the simulation and the measurement are due to the inaccuracy of the model with respect to reality. We used tabular values of pure CaCl2.6H2O but we modified the hexahydrate with 1,2% of BaCO3. to avoid supercooling and deformation of cooling curves after more cycles of melting and freezing. In order to obtain exact results, we would nevertheless need to obtain exact knowledge of the temperature dependence of thermal conductivity, specific heat and density during phase change (see Figure 22).

Fig. 18. Geometric model of a Cu-cooler with PCM elements

Properties and Numerical Modeling-Simulation of Phase Changes Material 367

0:00 0:19 0:38 0:57 1:17 1:36 1:55 2:15 2:34 2:53 3:13 3:32 3:51 time [hours]

Fig. 22. Comparison between the measurement and the simulation

measuring simulation

Fig. 21. Temperature distribution in the cooler (the cross section)

temperature [°C]

Fig. 19. Geometric model of a Cu-cooler with the mesh of elements

Fig. 20. The distribution of air velocity module

T0,air =

> 20 °C, v0,air =

airflow

0.1

A

**A**

Fig. 19. Geometric model of a Cu-cooler with the mesh of elements

Fig. 20. The distribution of air velocity module

m.s-1

PCM enclosure T0,PCM = 20 °C

> copper cooler T0,Cu = 50 °C

0 .469E-02 .859E-02 .125E-01 .172E-01 .211E-01 .250E-01 .297E-01 .336E-01 .375E-01 .414E-01 .461E-01 .500E-01 .539E-01 .586E-01 .625E-01 .664E-01 .703E-01 .750E-01 .789E-01 .828E-01 .875E-01 .914E-01 .953E-01 .100

air fan

Fig. 21. Temperature distribution in the cooler (the cross section)

Fig. 22. Comparison between the measurement and the simulation

Properties and Numerical Modeling-Simulation of Phase Changes Material 369

described separators may utilize classical properties of H2O and oil (the mechanical – fluid separation); alternatively, the access of heating or, for example, microwave heating may be applied. In order to use this variant, however, we need to know the process of the material phase change- H2O to vapour (steam), and the related diversion of the vapour from the separator. Here, the numerical model proved to be superior to all experiments as it enabled us to examine the details of behaviour and states within individual operating modes of the separator. By means of this method, it is possible to model various states of the emulsion as well as fault conditions in the apparatus; thus, we may identify critical sections of the separator design and perform sensitivity analysis of the system. The reactor exploiting active porous substances was designed to enable oil preparation. The reactor is fed with an industrially produced mixture of oil and water; the desired reaction proceeds in the ceramic porous material. To achieve the desired reaction condition, it is necessary to heat the material and, simultaneously, remove the products of the reaction. After the reaction of water, further heating is undesirable with respect to side reactions. Considering the above mentioned requirements, microwave heating was chosen. The microwave heating effect is selective for the reaction of water. The designed reactor operates at the frequency of *f* = 2.4 GHz, with the magnetron output power of *P*=800W. This allows selective heating in the active porous material of the chamber. The basic scheme of the reactor is shown in

It is possible to carry out an analysis of an MG model as a numerical solution by means of the FEM. The electromagnetic part of the model is based on the solution of full Maxwell's

where *E* and *H* are the electrical field intensity vector and the magnetic field intensity vector, *D* a *B* are the electrical field density vector and the magnetic flux density vector, *J* is the

conductivity of the material, and Ω is the definition area of the model. The model is given in manual (Ansys User's Manual). The set of equations (27) is independent of time and gives *E*.

> Re{ } *<sup>j</sup> <sup>t</sup> E Ee* ω

The results were obtained by the solution of the non-linear thermal model with phase change of the medium. The phase change occurs via the phase conversion of water to steam. Figure 28 shows the phase-change time characteristic of water. The thermal model is based

( ) *<sup>T</sup> q c divT div k gradT c <sup>t</sup>*

+ ⋅− = ⎜ ⎟ ⎝ ⎠ <sup>∂</sup>

temperature, *t* is the time, *k* is the thermal conductivity coefficient, *v* is the medium flow

 ρ

is the electric charge density,

 ρ

is the specific weight, *c* is the specific heat capacity, *T* is the

⎛ ⎞ ∂

*v* , (29)

= . (28)

σ

is the electric

<sup>∂</sup> <sup>∂</sup> in region Ω. (27)

B D E - , H E J, D , B 0 *<sup>s</sup> t t* σ

ρ

<sup>∂</sup> <sup>∂</sup> ∇× = ∇× = + + ∇⋅ = ∇⋅ =

Figure 23.

equations

**4.1 Mathematical model** 

current density vector of the sources,

For the transient vector *E* we can write

on the first thermodynamic law

where *q* is the specific heat,

ρ

ρ

velocity. If we consider the Snell's principle, the model can be simplified as

#### **3.3 Results of the analysis related to accumulators and coolers**

The part of the analysis of one layer of a heat accumulator which is derived from the gravel accumulator. We used pure CaCl2.6H2O with an additional 1,2% of BaCO3 to increase heat capacity and avoid supercooling. The numerical model was solved by the help of the FEM/FVM in ANSYS software (Ansys User's Manual). If we compare the results of the simulation and the experimental measuring, we will obtain comparatively good congruence. Exact knowledge of the material properties has a crucial effect on the accuracy of the numerical model.

Fig. 23. The basic scheme of the reactor

Fig. 24. The phase-change characteristics of water

#### **4. Microwave heating for the separation of the water/oil emulsion**

Industrial applications of distribution transformers used in high and mid-performance power distribution systems are supported by maintenance-free systems constructed for the purposes of separation of water bound to the transformer oil. In this case, oil is the insulator whose quality determines the life of the equipment as well as its fault or interference states. The oil is cleaned and filled with adjectives to be reused in functional parts of the transformer, and the related steps are realized in the course of the machine operation. The described separators may utilize classical properties of H2O and oil (the mechanical – fluid separation); alternatively, the access of heating or, for example, microwave heating may be applied. In order to use this variant, however, we need to know the process of the material phase change- H2O to vapour (steam), and the related diversion of the vapour from the separator. Here, the numerical model proved to be superior to all experiments as it enabled us to examine the details of behaviour and states within individual operating modes of the separator. By means of this method, it is possible to model various states of the emulsion as well as fault conditions in the apparatus; thus, we may identify critical sections of the separator design and perform sensitivity analysis of the system. The reactor exploiting active porous substances was designed to enable oil preparation. The reactor is fed with an industrially produced mixture of oil and water; the desired reaction proceeds in the ceramic porous material. To achieve the desired reaction condition, it is necessary to heat the material and, simultaneously, remove the products of the reaction. After the reaction of water, further heating is undesirable with respect to side reactions. Considering the above mentioned requirements, microwave heating was chosen. The microwave heating effect is selective for the reaction of water. The designed reactor operates at the frequency of *f* = 2.4 GHz, with the magnetron output power of *P*=800W. This allows selective heating in the active porous material of the chamber. The basic scheme of the reactor is shown in Figure 23.

#### **4.1 Mathematical model**

368 Convection and Conduction Heat Transfer

The part of the analysis of one layer of a heat accumulator which is derived from the gravel accumulator. We used pure CaCl2.6H2O with an additional 1,2% of BaCO3 to increase heat capacity and avoid supercooling. The numerical model was solved by the help of the FEM/FVM in ANSYS software (Ansys User's Manual). If we compare the results of the simulation and the experimental measuring, we will obtain comparatively good congruence. Exact knowledge of the material properties has a crucial effect on the accuracy of the

> Chamber filled with emulsion and active material

> > *t* (s)

**Vapour phase** 

**3.3 Results of the analysis related to accumulators and coolers** 

Δ

**4. Microwave heating for the separation of the water/oil emulsion** 

*q=*30MJ/m3

*<sup>T</sup>*=60K **Liquid phase** 

Industrial applications of distribution transformers used in high and mid-performance power distribution systems are supported by maintenance-free systems constructed for the purposes of separation of water bound to the transformer oil. In this case, oil is the insulator whose quality determines the life of the equipment as well as its fault or interference states. The oil is cleaned and filled with adjectives to be reused in functional parts of the transformer, and the related steps are realized in the course of the machine operation. The

Controlled microwave power source

numerical model.

Ω

Fig. 23. The basic scheme of the reactor

*T* (K)

Δ

Fig. 24. The phase-change characteristics of water

It is possible to carry out an analysis of an MG model as a numerical solution by means of the FEM. The electromagnetic part of the model is based on the solution of full Maxwell's equations

$$
\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}\mathbf{\prime}\prime\prime\prime \qquad \nabla \times \mathbf{H} = \sigma \cdot \mathbf{E} + \frac{\partial \mathbf{D}}{\partial t} + \mathbf{J}\_s \cdot \nabla \cdot \mathbf{D} = \rho \, \nabla \cdot \mathbf{B} = 0 \quad \text{in region } \Omega. \tag{27}
$$

where *E* and *H* are the electrical field intensity vector and the magnetic field intensity vector, *D* a *B* are the electrical field density vector and the magnetic flux density vector, *J* is the current density vector of the sources, ρ is the electric charge density, σ is the electric conductivity of the material, and Ω is the definition area of the model. The model is given in manual (Ansys User's Manual). The set of equations (27) is independent of time and gives *E*. For the transient vector *E* we can write

$$E = \text{Re}\left\{\underline{E}e^{j\alpha t}\right\}.\tag{28}$$

The results were obtained by the solution of the non-linear thermal model with phase change of the medium. The phase change occurs via the phase conversion of water to steam. Figure 28 shows the phase-change time characteristic of water. The thermal model is based on the first thermodynamic law

$$
\rho \, d + \rho \quad \text{c} \quad \text{v} \cdot d \\
\text{iv} \\
T - \text{div} \begin{pmatrix} k & \text{grad} T \\ \end{pmatrix} = \rho \text{c} \left( \frac{\partial T}{\partial t} \right) \, \tag{29}
$$

where *q* is the specific heat, ρ is the specific weight, *c* is the specific heat capacity, *T* is the temperature, *t* is the time, *k* is the thermal conductivity coefficient, *v* is the medium flow velocity. If we consider the Snell's principle, the model can be simplified as

Properties and Numerical Modeling-Simulation of Phase Changes Material 371

heat. The non-linear thermal model including the phase change solves the temperature distribution. The analysis was performed for the time interval *t*∈<0,300> s and the results were experimentally verified. The simulated results were found to correspond to measured values. A middle electrode was used in the model. The purpose of the middle electrode was to ensure the homogenous distribution of electromagnetic power and, subsequently, to

[°C]. In Figure

Θ

Magnetron Teflon

Emulsion Air

Electric conductive housing

Presure spring

increase the reaction efficiency.

Fig. 26. The geometric model of the HF chamber

Fig. 27. The distribution of the electric field intensity vector module *E*

generated through the Joule loss in material *W*jh,. Figure 29 shows heating

Thesic geometrical model of a separator is shown in Figure. 26; in Figures 26 and 27 we can see the distribution of the electric field vector modules with intensities *E* as well as the heat

26, for the given instant of time, the distribution of elements is shown in which phase

$$q - di\upsilon \begin{pmatrix} k & \mathsf{grad}T \end{pmatrix} = \rho c \begin{pmatrix} \hat{\mathscr{O}}T \\ \hat{\mathscr{O}}t \end{pmatrix} \tag{30}$$

The solution was obtained by the help of the ANSYS solver. The iteration algorithm (FEM/FVM) was realized using the APDL language as the main program. The simplified description of the algorithm is shown in Figure 25.

Fig. 25. The simplified iteration algorithm of the model evaluation

#### **4.2 FEM/FVM model**

A geometric model using HF119, HF120 and SOLID70 (Ansys User's Manual) in ANSYS software was built – Figure 26. A solution of the coupled field model was performed using the APDL program. The microwave model solution is evaluated according to the specific

⎛ ⎞ <sup>∂</sup> − = ⎜ ⎟

The solution was obtained by the help of the ANSYS solver. The iteration algorithm (FEM/FVM) was realized using the APDL language as the main program. The simplified

**HF 120, solution** 

**K**

**The density of heat energy** 

**SOLID 70, solution T** 

**Output: T,Q,…,PRe, Pdie**<sup>l</sup>

*T(*K*)*

Δ*T=60*K

description of the algorithm is shown in Figure 25.

**Z** 

*t*=0; 300

i=1

if i <sup>e</sup>

Fig. 25. The simplified iteration algorithm of the model evaluation

**+** 

*<sup>Q</sup> <sup>q</sup> <sup>V</sup>* <sup>=</sup> ( ) e 3 Δ ≥ *q T MJ / m* <sup>30</sup> **+** 

**-** 

Te;

**4.2 FEM/FVM model** 

i=i+1

e e

File

A geometric model using HF119, HF120 and SOLID70 (Ansys User's Manual) in ANSYS software was built – Figure 26. A solution of the coupled field model was performed using the APDL program. The microwave model solution is evaluated according to the specific

i <sup>e</sup> *q* = 0

i e e e *<sup>Q</sup> q ; <sup>V</sup>* <sup>=</sup>

> if <sup>e</sup> Δ > *T* 60<sup>D</sup>

> > **-**

( ) *<sup>T</sup> q div k gradT c <sup>t</sup>*

ρ

⎝ ⎠ <sup>∂</sup> (30)

Phase change implementation

Δ*q=*30MJ/m3

*t(*s*)*

heat. The non-linear thermal model including the phase change solves the temperature distribution. The analysis was performed for the time interval *t*∈<0,300> s and the results were experimentally verified. The simulated results were found to correspond to measured values. A middle electrode was used in the model. The purpose of the middle electrode was to ensure the homogenous distribution of electromagnetic power and, subsequently, to increase the reaction efficiency.

Fig. 26. The geometric model of the HF chamber

Fig. 27. The distribution of the electric field intensity vector module *E*

Thesic geometrical model of a separator is shown in Figure. 26; in Figures 26 and 27 we can see the distribution of the electric field vector modules with intensities *E* as well as the heat generated through the Joule loss in material *W*jh,. Figure 29 shows heating Θ [°C]. In Figure 26, for the given instant of time, the distribution of elements is shown in which phase

Properties and Numerical Modeling-Simulation of Phase Changes Material 373

Fig. 31. The emulsion exsiccation process depending on time; emulsion heating generated by

Dielectric losses

the dielectric and Joule losses; the distribution of temperature rise in the PCM model

Real (Joule) losses

Fig. 30. The emulsion exsiccation process depending on time

Emulsion

time [s]

Exsiccated elements

Heating in progress according to the scale Max 118°C

change occurred, namely exsiccation and separation of the water from the oil. *t*=10.8 s, with the magnetron output of *P*=800W and frequency *f*=2.4GHz. Figure 31 shows the distribution of temperature rise in the process of exsiccation; here, the indicated aspects include the heat generated through the Joule loss in the material and through dielectric heating for the instant of time *t*=3.6 s. The distribution of temperature was, for the individual instants, compared with the laboratory measurement.

Fig. 28. The distribution module of the Joule heat module *W*jh

Fig. 29. The distribution of temperature rise Θ

change occurred, namely exsiccation and separation of the water from the oil. *t*=10.8 s, with the magnetron output of *P*=800W and frequency *f*=2.4GHz. Figure 31 shows the distribution of temperature rise in the process of exsiccation; here, the indicated aspects include the heat generated through the Joule loss in the material and through dielectric heating for the instant of time *t*=3.6 s. The distribution of temperature was, for the individual instants,

> Temperature exceeding all limits max (100000°C)

time [s]

Heating in progress according to the scale Max 350°C

compared with the laboratory measurement.

Fig. 28. The distribution module of the Joule heat module *W*jh

Compare the emulsion volume (heating not significant)

Fig. 29. The distribution of temperature rise Θ

Fig. 30. The emulsion exsiccation process depending on time

Fig. 31. The emulsion exsiccation process depending on time; emulsion heating generated by the dielectric and Joule losses; the distribution of temperature rise in the PCM model

Properties and Numerical Modeling-Simulation of Phase Changes Material 375

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#### **5. Results of the thermal analysis**

We may conclude that, in relation to the desiccated emulsion system, the numerical model enabled us to identify weak points of the design. These drawbacks consist in the fact that, under certain conditions, local spots might occur where uncontrolled temperature rise could result in explosion of the equipment. This status was experimentally verified.

Different variants of the reactor design were analyzed with the aim to evaluate the process and time of desiccation.

Even though modelling realized under the application of PCM produces certain theoretical complications, the actual model is reliable for the assumed boundary and initial conditions if the PCM macroscopic and microscopic properties are respected. This fact was verified experimentally and in laboratory conditions.

#### **6. Acknowledgement**

The research described in the paper was financially supported by the FRVS grant, research plan No MSM 0021630516, No MSM 0021630513.

#### **7. References**

Ansys User's Manual. Huston (USA): Svanson Analysys System, Inc., 1994-2011.


We may conclude that, in relation to the desiccated emulsion system, the numerical model enabled us to identify weak points of the design. These drawbacks consist in the fact that, under certain conditions, local spots might occur where uncontrolled temperature rise could

Different variants of the reactor design were analyzed with the aim to evaluate the process

Even though modelling realized under the application of PCM produces certain theoretical complications, the actual model is reliable for the assumed boundary and initial conditions if the PCM macroscopic and microscopic properties are respected. This fact was verified

The research described in the paper was financially supported by the FRVS grant, research

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**18** 

*Brazil*

**Finite Element Methods to Optimize by** 

Throughout the manufacturing industry, casting process simulation has been widely accepted as an important tool in product design and process development to improve yield and casting quality. Casting simulation requires high-quality information concerning thermo-physical and physical properties during solidification. Some properties have been measured for specific alloys, but the number of alloys for which information is available is limited. Furthermore, the information may be incomplete in the sense that not all properties have been measured and sometimes, disparate information from a variety of sources is used to build up the database for one specific alloy. To overcome the lack of data and achieve a better understanding of how changes in composition within a specification range of an alloy may affect solidification properties, it is highly desirable to develop experimental techniques or computer models for calculation of the thermo-physical and physical properties of multi-

The computer simulation of cooling patterns in castings has done much to broaden our understanding of casting and mold system design. The structural integrity of shaped castings is closely related to the time–temperature history during solidification, and the use of casting simulation could do much to increase this knowledge in the foundry industry.

The ability of heat to flow across the casting and through the interface from the casting to the mold directly affects the evolution of solidification and plays a notable role in determining the freezing conditions within the casting, mainly in foundry systems of high thermal diffusivity such as chill castings. Gravity or pressure die castings, continuous casting and squeeze castings are some of the processes where the product's soundness is more directly affected by heat transfer at the metal/mold interface (Ferreira et al., 2005). According to the authors Atwood and Lee (Atwood & Lee, 2003), the mechanical properties of metal products depend upon the phenomena occurring during production. Defects formed during each stage of production can persist or modify the behavior of the metal during subsequent processing steps. Therefore, ensuring that an appropriate microstructure is formed at each stage with minimal defects has always been a focus in the study of metal production. In aluminum alloy shape castings, the final microstructure is directly dependent upon the as-cast microstructure since the only post-casting processing is normally a heat treatment. One microstructural feature that can affect the final properties of aluminum alloy

component alloys for the process of reliable solidification (Guo et al., 2005).

**1. Introduction**

(Ferreira et al., 2005).

**Factorial Design the Solidification of** 

**Cu-5wt%Zn Alloy in a Sand Mold** 

Moisés Meza Pariona and Viviane Teleginski

*State University of Ponta Grossa – UEPG* 

*Phase Change Random Access Memory,* Non-Volatile Memory Technology Symposium, 2006. NVMTS 2006. 7th Annual, Digital Object, Page(s): 76 - 80


### **Finite Element Methods to Optimize by Factorial Design the Solidification of Cu-5wt%Zn Alloy in a Sand Mold**

Moisés Meza Pariona and Viviane Teleginski *State University of Ponta Grossa – UEPG Brazil*

#### **1. Introduction**

376 Convection and Conduction Heat Transfer

Symposium, 2006. NVMTS 2006. 7th Annual, Digital Object, Page(s): 76 - 80

Vener, C. (1997). *Phase change thermal energy storage*. The Department of The Built Environment

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Vijay Raghunathan; Kansal, A.; Hsu, J.; Friedman, J.; Mani Srivastava. (April 2005) *Design* 

Vohlídal, Julák, Štulík. (1999). *Chemické a analytické tabulky*. In Czech, Prague: GRADA

Volle, F.; Garimella, S.V.; Juds, M.A. (2010). *Thermal Management of a Soft Starter: Transient* 

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Zhen Ren; Wen-Yan Yin; Yan-Bing Shi; Qing Huo Liu. (Januar 2010). *Thermal Accumulation* 

*considerations for solar energy harvesting wireless embedded systems*, Information Processing in Sensor Networks, 2005. IPSN 2005. Fourth International Symposium

*Thermal Impedance Model and Performance Enhancements Using Phase Change Materials*, Power Electronics, IEEE Transactions on, Volume: 25, Issue: 6, Page(s): 1395 - 1405.

*Effects on the Transient Temperature Responses in LDMOSFETs Under the Impact of a Periodic Electromagnetic Pulse*, Electron Devices, IEEE Transactions on, vol.57, no.1,

The Australian National University (2004), Department of Engineering, FEIT,

*solar heating system.* The Hague (Holland): C. Den Ouden.

Brighton, University of Brighton, Brighton.

Solar energy, (2010), www.ausra.com

on, vol., no., pp. 457- 462.

Publishing.

Industries, Inc.

pp.345-352.

http://solar.anu.edu.au/.

*Phase Change Random Access Memory,* Non-Volatile Memory Technology

Throughout the manufacturing industry, casting process simulation has been widely accepted as an important tool in product design and process development to improve yield and casting quality. Casting simulation requires high-quality information concerning thermo-physical and physical properties during solidification. Some properties have been measured for specific alloys, but the number of alloys for which information is available is limited. Furthermore, the information may be incomplete in the sense that not all properties have been measured and sometimes, disparate information from a variety of sources is used to build up the database for one specific alloy. To overcome the lack of data and achieve a better understanding of how changes in composition within a specification range of an alloy may affect solidification properties, it is highly desirable to develop experimental techniques or computer models for calculation of the thermo-physical and physical properties of multicomponent alloys for the process of reliable solidification (Guo et al., 2005).

The computer simulation of cooling patterns in castings has done much to broaden our understanding of casting and mold system design. The structural integrity of shaped castings is closely related to the time–temperature history during solidification, and the use of casting simulation could do much to increase this knowledge in the foundry industry. (Ferreira et al., 2005).

The ability of heat to flow across the casting and through the interface from the casting to the mold directly affects the evolution of solidification and plays a notable role in determining the freezing conditions within the casting, mainly in foundry systems of high thermal diffusivity such as chill castings. Gravity or pressure die castings, continuous casting and squeeze castings are some of the processes where the product's soundness is more directly affected by heat transfer at the metal/mold interface (Ferreira et al., 2005).

According to the authors Atwood and Lee (Atwood & Lee, 2003), the mechanical properties of metal products depend upon the phenomena occurring during production. Defects formed during each stage of production can persist or modify the behavior of the metal during subsequent processing steps. Therefore, ensuring that an appropriate microstructure is formed at each stage with minimal defects has always been a focus in the study of metal production. In aluminum alloy shape castings, the final microstructure is directly dependent upon the as-cast microstructure since the only post-casting processing is normally a heat treatment. One microstructural feature that can affect the final properties of aluminum alloy

Finite Element Methods to Optimize by Factorial

**1.1 Mathematical solidification heat transfer model** 

�� (���) <sup>=</sup> �

�� = ��

relationship (Shi & Guo, 2004; Pericleous et al., 2006):

liquid but no diffusion in the solid (Shi & Guo, 2004):

where ko the equilibrium partition coefficient of the alloy.

(�� � �) <sup>+</sup> �

���

��� �� = ��

�

Santos et al, 2005; Shi & Guo, 2004).

[m] and ��

Design the Solidification of Cu-5wt%Zn Alloy in a Sand Mold 379

The mathematical formulation of heat transfer to predict the temperature distribution during solidification is based on the general equation of heat conduction in the unsteady state, which is given in two-dimensional heat flux form for the analysis of the present study

where ρ is density [kgm-3]; c is specific heat [J kg-1 K-1]; k is thermal conductivity [Wm-1K-1]; ∂T/∂t is cooling rate [K s-1], T is temperature [K], t is time [s], x and y are space coordinates

In this equation, it was assumed that the thermal conductivity, density, and specific heat vary with temperature. In the current system, no external heat source was applied and the only heat generation was due to the latent heat of solidification, L (J/kg) or ΔH (J/kg). ��

proportional to the changing rate of the solidified fraction, fs, as follow (Ferreira et al, 2005;

��� �� = ��

�� = � <sup>0</sup> � ���� (�� � �)�(�� � ��)� �� ����� � �� � ��

�� =����� � �

Therefore, Eq. (2) is actually dependent on two factors: temperature and solid fraction. The solid fraction can be a function of a number of solidification variables. But in many systems, especially when undercooling is small, the solid fraction may be assumed as being dependent on temperature only. Different forms have been proposed to the relationship between the solid fraction and the temperature. One of the simple forms is a linear

where �� and �� are, respectively, the liquid and solid temperature (K). Another relation is the widely used Scheil relationship, which assumes uniform solute concentration in the

> �� � �� � ��

Eq. (1) defines the heat flux (Radovic & Lalovic, 2005), which is released during liquid cooling, solidification and solid cooling in classical models. The heat evolved after solidification was assumed to be equal zero, i.e. for ����� �� = 0. However, experimental investigations have showed that lattice defects and vacancy are condensed in the already solidified part of the crystal and the enthalpy of the solid increases and thus the latent heat will decrease (Radovic & Lalovic, 2005). Due to this fact, another way to represent the change of the solid fraction during solidification can be written as (Radovic & Lalovic, 2005):

� (�� � ��)(� � ��� ��(����)

(�� � ��)(� � ���)

�(�����) �

�� ���

represents the term associated to the latent heat release due to the phase change.

��� �� ��

��

���+�� (1)

�� (2)

���� (4)

is

(3)

(5)

(Ferreira et al., 2005; Santos et al., 2005; Shi & Guo, 2004; Dassau et al., 2006).

�� ��� + �

�� ���

shape castings is microporosity, formed due to the combined effects of volumetric shrinkage upon solidification. Developing a model to predict the formation of microporosity in solidifying metal castings is one way of helping. Porosity in castings is a defect that results from the interaction of a number of processes: volume change, nucleation and growth of the solid phase, diffusion of dissolved species, and the interaction of interphase surfacial. Although some of these individual processes may be treated analytically, combining the processes into a predictive tool requires the numerical calculation power that has only become available over the last few decades due to the development of digital computers

The use of multivariate experimental design techniques is becoming increasingly widespread in analytical chemistry. Multivariate experimental design techniques, which permit the simultaneous optimization of several control variables, are faster to implement and more cost-effective than traditional univariate (one at a time) approaches (Khajeh, 2009). One of the most popular multivariate design techniques is two level full/fractional factorial, in which every factor is experimentally studied at only two levels. Due to their simplicity and relatively low cost, full factorial design techniques are very useful for preliminary studies or in the initial steps of an optimization, while fractional factorial designs are almost mandatory when the problem involves a large number of factors (Khajeh, 2009). On the other hand, since only two levels are used, the models that may be fit to these designs are somewhat restricted. If a more sophisticated model is needed, as for the location of an optimum set of experimental conditions, then one must resort to augment response surface designs, which employ more than two factor levels. Among these, Box–Behnken is a secondorder multivariate design technique based on three-level incomplete factorial designs that received widespread application for evaluation of critical experimental conditions, that is, maximum or minimum of response functions (Khajeh, 2009).

Experimental design is a systematic, rigorous approach to engineering problem solving that applies principles and techniques at the data collection stage so as to ensure the generation of valid, precise, and accurate engineering conclusions (Xiao & Vien, 2004). It is a very economic way of extracting the maximum amount of complex information and saving a significant experimental time and the material used for analyses and personal costs as well (Kincl et al., 2005). Different experimental designs are used for different objectives. For example, randomized block designs can be used to compare data sets, full or fractional factorial design can be used for screening relevant factors (Xiao & Vien, 2004).

The design of mixture experiments configures a special case in response to surface methodologies using mathematical and statistical techniques, with important applications not only in new products design and development, but also in the improvement of the design of existing products. In short, the methodology consists firstly to select the appropriate mixtures from which the response surface might be calculated; having the response surface, a prediction of the property value can be obtained for any design, from the changes in the proportions of its components (Aslan, 2007). The other important issue is for engineering experimenters who wish to find the concentration conditions under which a certain process attains the optimal results. That is, they want to determine the levels of the operational factors at which the response reaches its optimum. The optimum could be either a maximum or a minimum of a function of the design parameters (Aslan, 2007). Factorial design is a useful tool in order to characterize multivariable processes. It gives the possibility to analyze the important influent factors of the process, and to identify any possible interactions among them.

shape castings is microporosity, formed due to the combined effects of volumetric shrinkage upon solidification. Developing a model to predict the formation of microporosity in solidifying metal castings is one way of helping. Porosity in castings is a defect that results from the interaction of a number of processes: volume change, nucleation and growth of the solid phase, diffusion of dissolved species, and the interaction of interphase surfacial. Although some of these individual processes may be treated analytically, combining the processes into a predictive tool requires the numerical calculation power that has only become available over the last few decades due to the development of digital computers The use of multivariate experimental design techniques is becoming increasingly widespread in analytical chemistry. Multivariate experimental design techniques, which permit the simultaneous optimization of several control variables, are faster to implement and more cost-effective than traditional univariate (one at a time) approaches (Khajeh, 2009). One of the most popular multivariate design techniques is two level full/fractional factorial, in which every factor is experimentally studied at only two levels. Due to their simplicity and relatively low cost, full factorial design techniques are very useful for preliminary studies or in the initial steps of an optimization, while fractional factorial designs are almost mandatory when the problem involves a large number of factors (Khajeh, 2009). On the other hand, since only two levels are used, the models that may be fit to these designs are somewhat restricted. If a more sophisticated model is needed, as for the location of an optimum set of experimental conditions, then one must resort to augment response surface designs, which employ more than two factor levels. Among these, Box–Behnken is a secondorder multivariate design technique based on three-level incomplete factorial designs that received widespread application for evaluation of critical experimental conditions, that is,

Experimental design is a systematic, rigorous approach to engineering problem solving that applies principles and techniques at the data collection stage so as to ensure the generation of valid, precise, and accurate engineering conclusions (Xiao & Vien, 2004). It is a very economic way of extracting the maximum amount of complex information and saving a significant experimental time and the material used for analyses and personal costs as well (Kincl et al., 2005). Different experimental designs are used for different objectives. For example, randomized block designs can be used to compare data sets, full or fractional

The design of mixture experiments configures a special case in response to surface methodologies using mathematical and statistical techniques, with important applications not only in new products design and development, but also in the improvement of the design of existing products. In short, the methodology consists firstly to select the appropriate mixtures from which the response surface might be calculated; having the response surface, a prediction of the property value can be obtained for any design, from the changes in the proportions of its components (Aslan, 2007). The other important issue is for engineering experimenters who wish to find the concentration conditions under which a certain process attains the optimal results. That is, they want to determine the levels of the operational factors at which the response reaches its optimum. The optimum could be either a maximum or a minimum of a function of the design parameters (Aslan, 2007). Factorial design is a useful tool in order to characterize multivariable processes. It gives the possibility to analyze the important influent factors of the process, and to identify any

factorial design can be used for screening relevant factors (Xiao & Vien, 2004).

maximum or minimum of response functions (Khajeh, 2009).

possible interactions among them.

#### **1.1 Mathematical solidification heat transfer model**

The mathematical formulation of heat transfer to predict the temperature distribution during solidification is based on the general equation of heat conduction in the unsteady state, which is given in two-dimensional heat flux form for the analysis of the present study (Ferreira et al., 2005; Santos et al., 2005; Shi & Guo, 2004; Dassau et al., 2006).

$$\frac{\partial}{\partial t} \{ \rho cT \} = \frac{\partial}{\partial \mathbf{x}} \left( k\_{\mathbf{x}} \frac{\partial T}{\partial \mathbf{x}} \right) + \frac{\partial}{\partial \mathbf{y}} \left( k\_{\mathbf{y}} \frac{\partial T}{\partial \mathbf{y}} \right) + \dot{Q} \tag{1}$$

where ρ is density [kgm-3]; c is specific heat [J kg-1 K-1]; k is thermal conductivity [Wm-1K-1]; ∂T/∂t is cooling rate [K s-1], T is temperature [K], t is time [s], x and y are space coordinates [m] and �� represents the term associated to the latent heat release due to the phase change. In this equation, it was assumed that the thermal conductivity, density, and specific heat vary with temperature. In the current system, no external heat source was applied and the only heat generation was due to the latent heat of solidification, L (J/kg) or ΔH (J/kg). �� is proportional to the changing rate of the solidified fraction, fs, as follow (Ferreira et al, 2005; Santos et al, 2005; Shi & Guo, 2004).

$$\partial Q = \Delta H \frac{\partial f\_s}{\partial t} = \rho L \frac{\partial f\_s}{\partial t} = \rho L \frac{\partial f\_s}{\partial T} \frac{\partial T}{\partial t} \tag{2}$$

Therefore, Eq. (2) is actually dependent on two factors: temperature and solid fraction. The solid fraction can be a function of a number of solidification variables. But in many systems, especially when undercooling is small, the solid fraction may be assumed as being dependent on temperature only. Different forms have been proposed to the relationship between the solid fraction and the temperature. One of the simple forms is a linear relationship (Shi & Guo, 2004; Pericleous et al., 2006):

$$f\_s = \begin{cases} 0 & T > T\_l \\ (T\_l - T)/(T\_l - T\_s) & T\_s \le T \le T\_l \\ 1 & T < T\_s \end{cases} \tag{3}$$

where �� and �� are, respectively, the liquid and solid temperature (K). Another relation is the widely used Scheil relationship, which assumes uniform solute concentration in the liquid but no diffusion in the solid (Shi & Guo, 2004):

$$f\_s = 1 - \left(\frac{T\_s - T}{T\_s - T\_l}\right)^{\frac{-1}{1 - k\_0}}\tag{4}$$

where ko the equilibrium partition coefficient of the alloy.

Eq. (1) defines the heat flux (Radovic & Lalovic, 2005), which is released during liquid cooling, solidification and solid cooling in classical models. The heat evolved after solidification was assumed to be equal zero, i.e. for ����� �� = 0. However, experimental investigations have showed that lattice defects and vacancy are condensed in the already solidified part of the crystal and the enthalpy of the solid increases and thus the latent heat will decrease (Radovic & Lalovic, 2005). Due to this fact, another way to represent the change of the solid fraction during solidification can be written as (Radovic & Lalovic, 2005):

$$f\_{s=} \frac{(T\_l - T) + \frac{2}{\pi}(T\_s - T\_l)(1 - \cos\left[\frac{\pi(T - T\_l)}{2(T\_s - T\_l)}\right]}{(T\_l - T\_s)(1 - 2/\pi)}\tag{5}$$

Finite Element Methods to Optimize by Factorial

�=�� + ���� + ���� + ���� + �����

*β33* are the quadratic coefficients (Kwak, 2005).

**2. Methodology of the numerical simulation** 

Midea and Shah (Midea and Shah, 2002).

In general Eq. (9) can be written in matrix form (Aslan, 2007).

Y = bX +

b = (X'X)-1X'Y

where *X'* is the transpose of the matrix *X* and *(X'X)-1* is the inverse of the matrix *X'X*.

finally an analysis by the factorial design of the considered parameters was made.

The finite elements method was used in this study (Su, 2001; Shi & Guo, 2004; Janik & Dyja, 2004; Grozdanic, 2002). Software program Ansys version 11 (Handbook Ansys, 2010) was used to simulate the solidification of alloy Cu-5 wt %Zn in green-sand mold. Effects due to fluid motion and contraction are not considered in the present work. The geometry of the cast metal and the greensand mold is illustrated in Figure 1(a), which is represented in three-dimensions. However, in this work the analysis was accomplish in 2-D, which is illustrated in Figure 1(b). Some material properties of Cu-5 wt %Zn alloy were taken from the reference Miettinen (Miettinen, 2001), the other properties were taken from Thermo-calc software (Thermo-calc software, 2010), and in Figure 2 the enthalpy and the phase diagram of alloy Cu-Zn are presented (Thermo-calc software, 2010). Three pseudo specific heat (c´) obtained from the equations (3), (4) and (5) were used and these equations were denoted respectively by models A, B and C, and the sand thermo-physical properties was given by

In this study, the Box–Behnken factorial design in three levels (Aslan, 2007; Paterakis et al., 2002; Montgomery, 1999) was chosen to find out the relationship between the response

Design the Solidification of Cu-5wt%Zn Alloy in a Sand Mold 381

The model based on Eq. (9), if m=3 (three variables) this equation is of the following form:

� + �����

where *y* is the predicted response, *βo* model constant; *x1, x2* and *x3* independent variables; *β1*, *β2* and *β3* are linear coefficients; *β12*, *β13* and *β23* are cross product coefficients and *β11*, *β22* and

where *Y* is defined to be a matrix of measured values, *X* to be a matrix of independent variables. The matrixes *b* and *ε* consist of coefficients and errors, respectively. The solution of Eq. (11) can be obtained by the matrix approach (Kwak, 2005; Gunaraj & Murugan, 1999).

The objective of this work was to study the solidification process of the alloy Cu-5 wt %Zn during 1.5 h of cooling. It was optimized through the factorial design in three levels, where the considered parameters were: temperature of the mold, the convection in the external mold and the generation of heat during the phase change. The temperature of the mold was initially fixed in 298, 343 and 423 K, as well as the loss of heat by convection on the external mold was fixed in 5, 70 and 150 W/m2.K. For the generation of heat, three models of the solid fraction were considered: the linear relationship, Scheil´s equation and the equation proposed by Radovic and Lalovic (Radovic & Lalovic, 2005). As result, the transfer of heat, thermal gradient, flow of heat in the system and the cooling curves in different points of the system were simulated. Also, a mathematical model of optimization was proposed and

+������� + � (10)

� + �����

� + ������� + �������

*ε* (11)

(12)

Considering c´, as pseudo specific heat, as �� =��� ��� �� and combining Eqs. (1) and (2), one obtains (Shi & Guo, 2004 ; Radovic & Lalovic, 2005):

$$\frac{\partial(\rho c'T)}{\partial t} = \nabla(k\nabla T) \tag{6}$$

The boundary condition applied on the outside of the mold is:

$$-k\frac{\partial T}{\partial n} = h(T - T\_o) \tag{7}$$

Here *h* is the heat transfer coefficient for air convection and *T*o is the external temperature.

#### **1.2 The factorial design technique**

The factorial design technique is a collection of statistical and mathematical methods that are useful for modeling and analyzing engineering problems. In this technique, the main objective is to optimize the response surface that is influenced by various process parameters. Response surface methodology also quantifies the relationship between the controllable input parameters and the obtained response surfaces (Kwak, 2005). The design procedure of response surface methodology is as follows (Gunaraj & Murugan, 1999):


$$\mathbf{y} = \mathbf{f}(\mathbf{x}, \mathbf{x}; \mathbf{x}, \mathbf{x}; \dots \mathbf{x}) \tag{8}$$

where *y* is the answer of the system, and *xi* the variables of action called variables (or factors).

The goal is to optimize the response variable *y*. It is assumed that the independent variables are continuous and controllable by experiments with negligible errors. It is required to find a suitable approximation for the true functional relationship between independent variables (or factors) and the response surface. Usually a second-order model is utilized in response surface methodology:

$$\mathbf{y} = \beta\_0 + \sum\_{l=1}^{m} \beta\_l \mathbf{x}\_l + \sum\_{l=l}^{m} \beta\_{ll} \mathbf{x}\_l^2 + \sum\_{l=l}^{m-1} \sum\_{j=2}^{m} \beta\_{lj} \mathbf{x}\_l \mathbf{x}\_j + \varepsilon \tag{9}$$

where *x1, x2,…,xk* are the input factors which influence the response *y; βo, βii (i=1, 2,…,m), βij (i=1, 2,…,m; j=1,2,…,m)* are unknown parameters and *ε* is a random error. The *β* coefficients, which should be determined in the second-order model, are obtained by the least square method.

Here *h* is the heat transfer coefficient for air convection and *T*o is the external temperature.

The factorial design technique is a collection of statistical and mathematical methods that are useful for modeling and analyzing engineering problems. In this technique, the main objective is to optimize the response surface that is influenced by various process parameters. Response surface methodology also quantifies the relationship between the controllable input parameters and the obtained response surfaces (Kwak, 2005). The design procedure of response surface methodology is as follows (Gunaraj & Murugan, 1999): i. Designing of a series of experiments for adequate and reliable measurement of the

ii. Developing a mathematical model of the second-order response surface with the best

iii. Finding the optimal set of experimental parameters that produce a maximum or

iv. Representing the direct and interactive effects of process parameters through two and three-dimensional plots. If all variables are assumed to be measurable, the response surface can be expressed as follows (Aslan, 2007; Yetilmezsoy et al., 2009; Pierlot et al.,

where *y* is the answer of the system, and *xi* the variables of action called variables (or

The goal is to optimize the response variable *y*. It is assumed that the independent variables are continuous and controllable by experiments with negligible errors. It is required to find a suitable approximation for the true functional relationship between independent variables (or factors) and the response surface. Usually a second-order model is utilized in response

� +

where *x1, x2,…,xk* are the input factors which influence the response *y; βo, βii (i=1, 2,…,m), βij (i=1, 2,…,m; j=1,2,…,m)* are unknown parameters and *ε* is a random error. The *β* coefficients, which should be determined in the second-order model, are obtained by the least square

� �������� + �

�

���

���

���

y= f(x1, x2, x3, …xk)

�=�� +����� +������

�

���

�

���

�� and combining Eqs. (1) and (2), one

(8)

(9)

�� = �(���) (6)

�� = �(� � ��) (7)

Considering c´, as pseudo specific heat, as �� =��� ���

�(��� �)

�� ��

The boundary condition applied on the outside of the mold is:

obtains (Shi & Guo, 2004 ; Radovic & Lalovic, 2005):

**1.2 The factorial design technique** 

response of interest.

minimum value of response.

2008; Dyshlovenko et al. 2006):

fittings.

factors).

method.

surface methodology:

The model based on Eq. (9), if m=3 (three variables) this equation is of the following form:

$$\begin{array}{c} \mathbf{y} = \boldsymbol{\beta}\_{0} + \boldsymbol{\beta}\_{1}\mathbf{x}\_{1} + \boldsymbol{\beta}\_{2}\mathbf{x}\_{2} + \boldsymbol{\beta}\_{3}\mathbf{x}\_{3} + \boldsymbol{\beta}\_{11}\mathbf{x}\_{1}^{2} + \boldsymbol{\beta}\_{22}\mathbf{x}\_{2}^{2} + \boldsymbol{\beta}\_{33}\mathbf{x}\_{3}^{2} + \boldsymbol{\beta}\_{12}\mathbf{x}\_{1}\mathbf{x}\_{2} + \boldsymbol{\beta}\_{13}\mathbf{x}\_{1}\mathbf{x}\_{3} \\ + \boldsymbol{\beta}\_{23}\mathbf{x}\_{2}\mathbf{x}\_{3} + \varepsilon \end{array} \tag{10}$$

where *y* is the predicted response, *βo* model constant; *x1, x2* and *x3* independent variables; *β1*, *β2* and *β3* are linear coefficients; *β12*, *β13* and *β23* are cross product coefficients and *β11*, *β22* and *β33* are the quadratic coefficients (Kwak, 2005).

In general Eq. (9) can be written in matrix form (Aslan, 2007).

$$Y = \mathbf{b}X + \varepsilon \tag{11}$$

where *Y* is defined to be a matrix of measured values, *X* to be a matrix of independent variables. The matrixes *b* and *ε* consist of coefficients and errors, respectively. The solution of Eq. (11) can be obtained by the matrix approach (Kwak, 2005; Gunaraj & Murugan, 1999).

$$b = \langle X'X \rangle^{\perp} X'Y \tag{12}$$

where *X'* is the transpose of the matrix *X* and *(X'X)-1* is the inverse of the matrix *X'X*.

The objective of this work was to study the solidification process of the alloy Cu-5 wt %Zn during 1.5 h of cooling. It was optimized through the factorial design in three levels, where the considered parameters were: temperature of the mold, the convection in the external mold and the generation of heat during the phase change. The temperature of the mold was initially fixed in 298, 343 and 423 K, as well as the loss of heat by convection on the external mold was fixed in 5, 70 and 150 W/m2.K. For the generation of heat, three models of the solid fraction were considered: the linear relationship, Scheil´s equation and the equation proposed by Radovic and Lalovic (Radovic & Lalovic, 2005). As result, the transfer of heat, thermal gradient, flow of heat in the system and the cooling curves in different points of the system were simulated. Also, a mathematical model of optimization was proposed and finally an analysis by the factorial design of the considered parameters was made.

#### **2. Methodology of the numerical simulation**

The finite elements method was used in this study (Su, 2001; Shi & Guo, 2004; Janik & Dyja, 2004; Grozdanic, 2002). Software program Ansys version 11 (Handbook Ansys, 2010) was used to simulate the solidification of alloy Cu-5 wt %Zn in green-sand mold. Effects due to fluid motion and contraction are not considered in the present work. The geometry of the cast metal and the greensand mold is illustrated in Figure 1(a), which is represented in three-dimensions. However, in this work the analysis was accomplish in 2-D, which is illustrated in Figure 1(b). Some material properties of Cu-5 wt %Zn alloy were taken from the reference Miettinen (Miettinen, 2001), the other properties were taken from Thermo-calc software (Thermo-calc software, 2010), and in Figure 2 the enthalpy and the phase diagram of alloy Cu-Zn are presented (Thermo-calc software, 2010). Three pseudo specific heat (c´) obtained from the equations (3), (4) and (5) were used and these equations were denoted respectively by models A, B and C, and the sand thermo-physical properties was given by Midea and Shah (Midea and Shah, 2002).

In this study, the Box–Behnken factorial design in three levels (Aslan, 2007; Paterakis et al., 2002; Montgomery, 1999) was chosen to find out the relationship between the response

Finite Element Methods to Optimize by Factorial

x1 Mold Temperature

Design the Solidification of Cu-5wt%Zn Alloy in a Sand Mold 383

1 -1 -1 -1 806.799 2 -1 -1 0 775.945 3 -1 -1 +1 862.902 4 -1 0 -1 800.301 5 -1 0 0 769.408 6 -1 0 +1 855.752 **7 -1 +1 -1** 798.197 **8 -1 +1 0** 767.562 **9 -1 +1 +1** 854.967 10 0 -1 -1 840.174 11 0 -1 0 809.199 12 0 -1 +1 897.176 13 0 0 -1 833.699 14 0 0 0 802.835 15 0 0 +1 890.279 16 0 +1 -1 832.200 17 0 +1 0 801.430 18 0 +1 +1 890.110 19 +1 -1 -1 899.860 20 +1 -1 0 868.171 21 +1 -1 +1 958.587 22 +1 0 -1 893.996 23 +1 0 0 862.277 24 +1 0 +1 953.026 25 +1 +1 -1 893.136 26 +1 +1 0 861.015 27 +1 +1 +1 952.674

x3 Mathematic model

Z - Temperature after 1.5 h of solidification (K)

x2 Convection phenomenon (hf)


Table 1. Factorial design of the solidification process parameters

Fig g. 1. The cast part t and mold in (a) three dimensiona al and (b) bi dime ensional

Fig 5w g. 2. (a) Enthalpy wt%Zn alloy (Ther and phase diagra rmo-calc softwar am of Cu-5 wt %Z e, 2010) Zn alloy and (b) p phase diagram of f Cu-

fun the of sol wa int nctions. Indepen e mold temperatu the latent heat lidification. The f as adopted, wher termediate state b dents variables ( ure (x1), the conve release, (Z) rep factorial design is re for the inferio by (0) and for the (factors) and thei ection phenomen presents the resu s shown in Table or state of the v superior state by ir coded/actual l non (x2) and the m ult of the temp e 1. For this desig variable it was de y (+1). levels considered mathematical mod erature after 1.5 gn type a nomenc enoted by (-1), f d were del (x3) 5 h of clature for the 38

2

Fig

g. 1. The cast part

(a)

three dimensiona

Convection and

d Conduction Heat T

Transfer

(b)

al and (b) bi dime

ensional

phase diagram of

f Cu-

d were del (x3) 5 h of clature for the

levels considered mathematical mod erature after 1.5 gn type a nomenc enoted by (-1), f

Zn alloy and (b) p

ir coded/actual l non (x2) and the m ult of the temp e 1. For this desig variable it was de

y (+1).

am of Cu-5 wt %Z

(factors) and thei ection phenomen presents the resu s shown in Table or state of the v superior state by

e, 2010)

t and mold in (a)

and phase diagra rmo-calc softwar dents variables ( ure (x1), the conve release, (Z) rep factorial design is re for the inferio by (0) and for the

Fig 5w fun the of sol wa int

g. 2. (a) Enthalpy wt%Zn alloy (Ther nctions. Indepen e mold temperatu the latent heat lidification. The f as adopted, wher termediate state b


Table 1. Factorial design of the solidification process parameters

Finite Element Methods to Optimize by Factorial

(line 9 of Table 1)

profiles were compared.

Fig. 5. Reference points for the mold/metal system

modes of latent heat release, when the alloy mushy zone is narrow.

**2 1**

Design the Solidification of Cu-5wt%Zn Alloy in a Sand Mold 385

(a) (b) Fig. 4. (a) Thermal gradient (K/m) in vector form and (b) Heat flux (W/m2) in vector form

In order to simulate the cooling curves, two points were considered, as shown in Figure 5: one located in the core (point 2) and the other in the metal (point 1). The three forms of latent heat release were applied into the mathematical model and the resulting thermal

The cooling curves were studied for condition of line 7, 8 and 9 from Table 1 as shown in Figure 6. Figure 6 (a) shows a comparison of temperature evolution at point (2) for the three formulations of latent heat release: linear (model A), Scheil (model (B) and Radovic and Lalovic (model C). It can be observed that the highest temperature profile corresponds to model A, followed by model C and last by model B, mainly after the solidification range. Although not presented, a similar behavior has occurred at other positions in the casting. Chen and Tsai (Chen and Tsai, 1990) analyzed theoretically four different modes of latent heat release for two of alloys solidified in sand molds: Al-4,5wt%Cu (wide mushy region, 136K ) and a 1wt% Cr steel alloy (narrow mushy region, 33.3K). In their work, they conclude that no significant differences can be observed in the casting temperature for different

The alloy used in the present work, Cu-5wt%Zn, as shown in Figure 2(b), has a narrow mushy zone (less than 10K). Figure 6(a) shows that there is a significant temperature profile difference due to the three different latent heat release modes. In addition, it is important to remark that the latent heat release form has strongly influenced the local solidification time.

The initial and boundary conditions were applied to geometry of Figure 1 according to Table 1. The boundary condition was the convection phenomenon and this phenomenon was applied to the outside walls of the sand mold, as shown in Table 1. The convection transfer coefficient at the mold wall was considered constant in this work, due to lack of experimental data. The effects of the refractory paint and of the gassaging process were not taken into consideration either. The final step consisted in solving the problem of heat transfer of the mold/cast metal system using equation (6), in applied boundary condition and in controlling the convergence condition. Heat transfer is analyzed in 2-D form, as well as the heat flux, the thermal gradient, and in addition, the thermal history for some points in the cast metal and in the mold is discussed.

#### **3. Result and discussion**

The result for solidification was discussed for some particular cases, at condition given in lines 7, 8 and 9 from Table 1, which correspond respectively to the lowest temperatures for each mathematical model of latent heat release. Each one of the lines corresponds to the temperature of the mold for the lower state (-) and for convection phenomenon for the higher state (+).

Fig. 3. Temperature distribution in (a) sand mold system, (b) cast metal (line 9 of Table 1)

The condition mentioned on line 9 of Table 1 was chosen to present heat transfer results, where the temperature field is shown in Figure 3(a) in all the system mold and in the cast metal (Figure 3(b)). This last case can be visualized in more detail in part (b), where an almost uniform temperature is observed. In the geometric structure of the mold there is a core constituted of sand that is represented by a white circle in Figure 3(b), which can be verified also in Figure 1(a). In Figure 4 the results of the thermal gradient and the thermal flux are shown, where the thermal gradient goes from the cold zone to the hot zone. On the other hand, the thermal flux goes from the hot zone to the cold zone. Also the convergence of the solution was studied; this point is discussed in more detail by Houzeaux and Codina (Houzeaux & Codina, 2004).

The initial and boundary conditions were applied to geometry of Figure 1 according to Table 1. The boundary condition was the convection phenomenon and this phenomenon was applied to the outside walls of the sand mold, as shown in Table 1. The convection transfer coefficient at the mold wall was considered constant in this work, due to lack of experimental data. The effects of the refractory paint and of the gassaging process were not taken into consideration either. The final step consisted in solving the problem of heat transfer of the mold/cast metal system using equation (6), in applied boundary condition and in controlling the convergence condition. Heat transfer is analyzed in 2-D form, as well as the heat flux, the thermal gradient, and in addition, the thermal history for some points in

The result for solidification was discussed for some particular cases, at condition given in lines 7, 8 and 9 from Table 1, which correspond respectively to the lowest temperatures for each mathematical model of latent heat release. Each one of the lines corresponds to the temperature of the mold for the lower state (-) and for convection phenomenon for the

(a) (b)

The condition mentioned on line 9 of Table 1 was chosen to present heat transfer results, where the temperature field is shown in Figure 3(a) in all the system mold and in the cast metal (Figure 3(b)). This last case can be visualized in more detail in part (b), where an almost uniform temperature is observed. In the geometric structure of the mold there is a core constituted of sand that is represented by a white circle in Figure 3(b), which can be verified also in Figure 1(a). In Figure 4 the results of the thermal gradient and the thermal flux are shown, where the thermal gradient goes from the cold zone to the hot zone. On the other hand, the thermal flux goes from the hot zone to the cold zone. Also the convergence of the solution was studied; this point is discussed in more detail by Houzeaux and Codina

Fig. 3. Temperature distribution in (a) sand mold system, (b) cast metal (line 9 of Table 1)

the cast metal and in the mold is discussed.

**3. Result and discussion** 

(Houzeaux & Codina, 2004).

higher state (+).

Fig. 4. (a) Thermal gradient (K/m) in vector form and (b) Heat flux (W/m2) in vector form (line 9 of Table 1)

In order to simulate the cooling curves, two points were considered, as shown in Figure 5: one located in the core (point 2) and the other in the metal (point 1). The three forms of latent heat release were applied into the mathematical model and the resulting thermal profiles were compared.

Fig. 5. Reference points for the mold/metal system

The cooling curves were studied for condition of line 7, 8 and 9 from Table 1 as shown in Figure 6. Figure 6 (a) shows a comparison of temperature evolution at point (2) for the three formulations of latent heat release: linear (model A), Scheil (model (B) and Radovic and Lalovic (model C). It can be observed that the highest temperature profile corresponds to model A, followed by model C and last by model B, mainly after the solidification range. Although not presented, a similar behavior has occurred at other positions in the casting. Chen and Tsai (Chen and Tsai, 1990) analyzed theoretically four different modes of latent heat release for two of alloys solidified in sand molds: Al-4,5wt%Cu (wide mushy region, 136K ) and a 1wt% Cr steel alloy (narrow mushy region, 33.3K). In their work, they conclude that no significant differences can be observed in the casting temperature for different modes of latent heat release, when the alloy mushy zone is narrow.

The alloy used in the present work, Cu-5wt%Zn, as shown in Figure 2(b), has a narrow mushy zone (less than 10K). Figure 6(a) shows that there is a significant temperature profile difference due to the three different latent heat release modes. In addition, it is important to remark that the latent heat release form has strongly influenced the local solidification time.

Fin De

nite Element Method esign the Solidificati

ds to Optimize by F on of Cu-5wt%Zn A

Factorial

Alloy in a Sand Mold

d

effects for the full built based on th the main effects e dispersed poin ed region points st positive influen havior, followed rm). Parameter x design and the o 8. For this analys

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:

quadratic equati itical points of th hen, the derivatio uations, being eq

In coe coe in fol

A cri Th equ

Su sec sol cor arm et uch solidification condary dendriti lidification (tSL) rrelating ultimate m spacings have al., 2000). n parameter affe c arm spacings. are well known e tensile strength shown that (ߪ ( cts the microstr Correlations betw n in the literatu (ߪ (and seconda increases with de ructure character ween dendritic s ure (Rosa et al. ary (SDAS) or pr ecreasing (SDAS) rized by primar spacings and loca , 2008). Investig imary (PDAS) de ) or (PDAS) (Qua ry and al time gations endrite aresma

Fig for pro is r gure 6 (b) show rmulations of lat ofile corresponds repeated for the o ws a comparison ent heat release. s to model A, follo other points in th n of temperature It can be observ owed by model C e mold. e evolution at p ved again, that th C and last by mod point (1) for the he highest tempe del B, and this be e three erature havior

Th lin wh fac he significant var near regression a hich is a respon ctorial design. iables indicated nd analysis of v nse surface meth by the Pareto ch variance) were op hodology, based hart (which was o ptimized using a on a highly fra obtained after m a Box-Behnken d actionalized thre ultiple design, e-level

Fig 1. ( g. 6. Thermal prof (a) Inside of the c files for the mold cast – point 1, (b)I d/metal system co Inside of the mold oncerning conditi d – point 2 ion 7, 8 and 9 of T Table

A Ye an Fig lev ad the pa rel three level Boxetilmezsoy et al., d x3, based in Ta gure 7. In this fig vel (p) of 95%, sho opted for this an e main effect of rameter 2 (conve lease form). -Behnken desig ( 2009) was used able 1. The resul gure the estimated owing the variab nalysis was, "L" m the first factor ection phenomen (Aslan, 2007; Pat to determine the lt of this analysis d valor of the res bles with and with means linear, "Q" and "2L by 3Q non) with the qua terakis et al., 200 e responses of th s is shown in th sult Z is presente hout significant in " means quadrati Q" means the lin adratic effect of p 02; Montgomery, he three variables he Pareto's diagra ed with the signif nfluences. The no ic. For example, " near interaction parameter 3 (laten , 1999; s x1, x2 am of ficance otation "(1)" is of the nt heat

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6

uch solidification condary dendriti lidification (tSL) rrelating ultimate m spacings have al., 2000). gure 6 (b) show rmulations of lat ofile corresponds repeated for the o he significant var near regression a hich is a respon ctorial design.

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Convection and

d Conduction Heat T

Transfer

ry and al time gations endrite aresma

e three erature havior

ultiple design, e-level

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Table

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d/metal system co Inside of the mold (Aslan, 2007; Pat to determine the lt of this analysis d valor of the res bles with and with means linear, "Q" and "2L by 3Q non) with the qua

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gnificant influenc ical model) with bles and interact e of standard gra

Su sec sol cor arm et Fig for pro is r Th lin wh fac

Fig 1. ( A Ye an Fig lev ad the pa rel In an ind an

g. 6. Thermal prof (a) Inside of the c three level Boxetilmezsoy et al., d x3, based in Ta gure 7. In this fig vel (p) of 95%, sho opted for this an e main effect of rameter 2 (conve lease form). Figure 7, two sig d x3 (mathemati dependent variab alysis, other type

(a)

Fig g. 7. Pareto chart of standardized e effects for the full l factorial design

fig (M inf po ob ini pa ne aro giv gure of the factor Montgomery, 1999 fluence can be o oints that belong t served in Figure itial temperature rameter 3 (latent gative influence ound zero, as pre ven Z the followin rial design was b 9). In this figure bserved by those to the concentrate 8 that the bigges ) with linear be t heat release for on the factorial esented in Figure ng equation: built based on th the main effects e dispersed poin ed region points st positive influen havior, followed rm). Parameter x design and the o 8. For this analys he Student's prob s and their intera nts (around of th are of the negligi nce is due to the d by the linear a 2 with linear beh other effects had sis a mathematica bability distribut actions with sign he straight line). ible influence. It main effect of x1 and quadratic eff havior presents a d a negligible beh al model was pro tion (t) nificant Those can be (mold fect of a small havior, posed,

$$\begin{array}{l} Z = 802.7889 + 46.4582\mathbf{x}\_1 - 3.9348\mathbf{x}\_2 - 28.3638\mathbf{x}\_3 - 13.0766\mathbf{x}\_1^2 - 2.5486\mathbf{x}\_2^2 \\ \quad + 59.2232\mathbf{x}\_3^2 + 0.4273\mathbf{x}\_1\mathbf{x}\_2 + 0.7476\mathbf{x}\_1\mathbf{x}\_3 + 0.1988\mathbf{x}\_2\mathbf{x}\_3 \\ \quad + 1.27212\mathbf{x}\_1\mathbf{x}\_3^2 + 0.2196\mathbf{x}\_1^2\mathbf{x}\_3 + 0.6696\mathbf{x}\_1^2\mathbf{x}\_3^2 + 0.3273\mathbf{x}\_2^2\mathbf{x}\_3 \end{array} \tag{13}$$

In coe coe in fol this equation th efficients are neg efficients belong Figures 7 and 8 llowing equation: he linear and qu gligible (they are to the variables w 8. Then accordin :uadratic coefficie e considered as r which strongly in ng to this consid ents are most im residue). Precisel nfluence the resul deration, equatio mportant and the ly the most sign lt, as it can be obs on (13) reduces e other nificant served to the

$$\begin{array}{l} Z = 802.7889 + 46.4582\mathbf{x}\_1 - 3.9348\mathbf{x}\_2 - 28.3638\mathbf{x}\_3 - 13.0766\mathbf{x}\_1^2 - 2.5486\mathbf{x}\_2^2 \\ \quad + 59.2232\mathbf{x}\_3^2 + 0.4273\mathbf{x}\_1\mathbf{x}\_2 + 0.7476\mathbf{x}\_1\mathbf{x}\_3 + 0.1988\mathbf{x}\_2\mathbf{x}\_3 \end{array} \tag{14}$$

A cri Th equ quadratic equati itical points of th hen, the derivatio uations, being eq on that correlate his equation can on of this equatio quations 15, 16 an es the variables a n be estimated th on in relation to nd 17: and the response hrough this math (x1), (*x2*) and (*x* was obtained, an hematical relatio *x3*) results in thre nd the onship. ee new

Finite Element Methods to Optimize by Factorial

(a)

(b)

(c)

Design the Solidification of Cu-5wt%Zn Alloy in a Sand Mold 389

Fig. 9. Response surface plots showing the effect of the (a) x2 and x3 factors, x1 was held at zero level, (b) x1 e x3 factors, x2 was held at zero level and (c) x1 and x2 factors, x3 was held at

zero level

$$\frac{d\mathbf{u}}{\mathbf{x}\_1} = 46.4582 \, + \, 26.1532 \mathbf{x}\_1 + \, 0.4273 \mathbf{x}\_2 + \, 0.7476 \mathbf{x}\_3 \tag{15}$$

$$\frac{dZ}{dx\_2} = -3.9348 + 0.4273x\_1 + 5.0972x\_2 + 0.1988x\_3 \tag{16}$$

$$\frac{dZ}{\mathcal{X}\_3} = 28.3638 \, + \, 0.7476 \mathbf{x}\_1 + 0.1988 \mathbf{x}\_2 \, + \, 118.4464 \mathbf{x}\_3 \tag{17}$$

Fig g. 8. Curve of stan ndardized effects s of the factorial d design

Th the rec va mo he critical point in e condition of � � commendations o lues for the critic odel. n the surface res Z � = 0, �Z �� = 0 and of the authors Ma cal point are: x1= sponse are found d �Z �� = 0. This c artendal et al. (M =-1.7850 K, x2= 0. d by solving thes riterion of solut Martendal el al., 2 9306 W/m2.K, x3 e equation system tion was based o 007) and the calc 3=-0.2298 mathem ms for on the culated matical

An acc ap are equ for he nalyzing this resu cording to the cr proach to these v e a bit different uation (13). Then rm, where x1 =-1 at release, 150 W/ ult, we know that riteria adopted in values adopted, b of these values n according to th 1.7850 ≈-1 (mold /m2.K) and x3 =-0 t the variables x1, n factorial design but it can be obser adopted, this is hese consideration temperature, in 0.2298 ≈ 0 (mathe x2 and x3 must ta n. Because the ca rved that some so possible by sim ns, the approxim environment), x2 ematic model, Sch ake values -1 or 0 alculated values s olutions of x1, x2 mplifying conside mations must be in 2 = 0.9306 ≈ + 1 heil relationship). 0 or +1, should and x3 ered in n such (latent

Th tem mi for mo fin he values of x1 =-1 mperature after 1 inimum temperat r the optimization odeling represent nite elements, see 1, x2 = + 1 and x3 1.5h of solidificati ture of factorial d n process of the c ts a proof trivial. Figure 6. 3 = 0 in Table 1 c ion that is 767.562 design. This solut casting by factori . Also this result orresponds to th 2 K, justly this va tion proved the v ial design, despit is to agree with e line 8, that mea alue corresponds validity of the mo te that this prove the result obtain .ans Z s to the odeling of the ned by

38

8

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> � �

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6.1532x� + 0.427

Convection and

d Conduction Heat T

Transfer

(15)

(16)

(17)

73x� + 0.7476x�

2�� + 0.1988��

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0.4273�� + 5.0972

7476�� + 0.1988�

s of the factorial d sponse are found

t the variables x1, n factorial design but it can be obser adopted, this is hese consideration temperature, in 0.2298 ≈ 0 (mathe 3 = 0 in Table 1 c ion that is 767.562 design. This solut casting by factori . Also this result

= 0. This c artendal et al. (M =-1.7850 K, x2= 0.

design

d by solving thes riterion of solut Martendal el al., 2 9306 W/m2.K, x3

e equation system tion was based o 007) and the calc 3=-0.2298 mathem

ms for on the culated matical

0 or +1, should and x3 ered in n such (latent .ans Z s to the odeling of the ned by

ake values -1 or 0 alculated values s olutions of x1, x2 mplifying conside mations must be in 2 = 0.9306 ≈ + 1 heil relationship). e line 8, that mea alue corresponds validity of the mo te that this prove the result obtain

x2 and x3 must ta n. Because the ca rved that some so possible by sim ns, the approxim environment), x2 ematic model, Sch orresponds to th 2 K, justly this va tion proved the v ial design, despit is to agree with

= �3.9348 + 0

= 28.3638 + 0.7

�� ��

Fig Th the rec va mo An acc ap are equ for he Th tem mi for mo fin

odel.

g. 8. Curve of stan he critical point in e condition of �

commendations o lues for the critic

nalyzing this resu cording to the cr proach to these v e a bit different uation (13). Then rm, where x1 =-1 at release, 150 W/ he values of x1 =-1 mperature after 1 inimum temperat r the optimization odeling represent nite elements, see

�

Z �

ndardized effects n the surface res

of the authors Ma cal point are: x1=

ult, we know that riteria adopted in values adopted, b of these values n according to th 1.7850 ≈-1 (mold /m2.K) and x3 =-0 1, x2 = + 1 and x3 1.5h of solidificati ture of factorial d n process of the c ts a proof trivial. Figure 6.

= 0 and

d �Z ��

= 0, �Z ��

Fig. 9. Response surface plots showing the effect of the (a) x2 and x3 factors, x1 was held at zero level, (b) x1 e x3 factors, x2 was held at zero level and (c) x1 and x2 factors, x3 was held at zero level

Finite Element Methods to Optimize by Factorial

few decades due to the development of digital computers.

**4. Conclusion** 

product.

**5. Acknowledgment**

5910.

8375.

**6. References** 

Council) and Araucária Foundation (AF).

Design the Solidification of Cu-5wt%Zn Alloy in a Sand Mold 391

This work has done the numerical simulation by finite element method to the copper alloy solidification in sand mold and also this simulation was accompanied with the process of optimization of solidification parameters. This type of study accompanied with the cellular automata or cellular automaton can help minimize the defects generated during solidification. One microstructural feature that can affect the final properties of copper alloy shape castings is microporosity, formed due to the combined effects of volumetric shrinkage upon solidification. Porosity in castings is a defect that results from the interaction of a number of processes: volume change, nucleation and growth of the solid phase, diffusion of dissolved species, and the interaction of interphase surfaces. Although some of these individual processes may be treated analytically, combining the processes into a predictive tool requires the numerical calculation power that has only become available over the last

In this study, a three-level Box-Behnken factorial design in qualitative form and combining with a response surface methodology was employed for modeling and optimizing three operations parameters of the casting process. According to this study, it was observed when the parameters of the solidification process are in the following state, such as, mold temperature in the environment, convection phenomenon in its fullest expression and the latent heat release to the model Sheil shows a minimum temperature after 1.5 h of solidification. Also this result was verified by finite element method. The factorial design method is a useful tool to determine what factors are crucial in the solidification process and thus, a special care needs to be taken during the project elaboration of the casting. Also this optimization tool can be used in other research areas, optimizing and predicting industrial processes, saving manpower, material and time in order to improve cost and quality of the

The authors acknowledge financial support provided by CNPq (The Brazilian Research

Aslan, N. (2007). Modeling and optimization of multi-gravity separator to produce celestite

Atwood, R.C. & Lee, P.D. (2003). Simulation of the three-dimensional morphology of

Chen, J.H. & Tsai, H.L. (1990). Comparison on different modes of latent heat release for

(October 2003), pp. 5447–5466, ISSN 1359-6454.

concentrate. *Powder Technology*, Vol. 174, No. 3, (may 2007), pp. 127–133, ISSN 0032-

solidification porosity in an aluminum–silicon alloy. *Acta Materialia,* Vol. 51, No. 18,

modeling casting solidification. *AFS Transactions*, (1990), pp. 539-546. ISSN 0065-

While a quantitative analysis of equation (14) was made, will soon be made a qualitative analysis of equation (13) by means of graphic representation of this equation and this discussion will be confronted with factorial design of the solidification process parameters (Table 1) and Figure 8. Figure 9 presents the response surface plots (Aslan, 2007; Paterakis et al., 2002), obtained from equation (13), that describe the influence of the factors on the overall desirability, Figure 9(a) shows the 3D response surface relationship between convection phenomenon (x2) and latent heat release form (x3) at zero level of mold temperature (x1). Note that, for a given value of x2, as the x3 increases and the Z decreases until a minimum value for the interval of x3 between -0.8 and 0.2. After this minimum point, Z changes its behavior and start to increase as the x3 increases reaching the highest value of 897.176, as can be seen in Table 1. For a given value of x3, it can be observed that Z profile is almost constant in relation to x2 increase. Another way to visualize Z variation is to project Z on the x2 and x3 plane, in terms of color band. The region limited by the white points on this curve represents the Z confidence interval. For other levels of x1, the surface graph behavior has the same characteristic as previously mentioned.

Figure 9(b) shows the effect of mold temperature (x1) and latent heat release form (x3) at zero level of convection phenomenon (x2). In this case, x1 e x3 generated a complex surface of paraboloid type. According to the surface projection Z on the x1 and x3 plane, for a given x3 value, it can be observed that the variation of Z is linear in relation to x1 and this fact can be confirmed by the point x1(L) mentioned at the graph of Figure 8. On the other hand, the same cannot be affirmed in relation to x3, if the same analysis is done.

For a given value of x1, the variation of Z is parabolic in relation to x3 and this fact can be observed by the points x3(L) and x3(Q) mentioned at the graph of Figure 8. Note that, for a given value of x1, as the x3 increases and the Z increases reaching the highest value of 953.026, as can be seen in Table 1.

Figure 9(c) shows the effect of mold temperature (x1) and convection phenomenon (x2) at zero level of latent heat release (x3). Note that, as the x1 factor increases, the Z increases. But, for a given value of x1, it is observed that for every x2 value, Z is almost constant. As a result, the surface geometry is not a complex one, if we compare to the surface geometry of Figure 9(b).

In this type of analysis, one can realize that the parameters x1 and x3 had variations more accentuated than the parameter x2. This behavior is also verified in Figures 7 and 8. Note that, for a given value of x1, as the x2 increases and the Z increases reaching the highest value of 868.171, as can be seen in Table 1.

Three level Box-Behnken design referred to Table 1 showed as results Pareto's diagram of Figure 7 and curve of standardized effect of the factorial design of Figure 8. According to these results was estimated a mathematic model for Z (represents the result of the temperature after 1.5 h of solidification) based in equation (9) and through the process of maximizing and minimizing of this model was found the optimal values of x1, x2 and x3. It was also shown the graphical interpretation of equation (13) so qualitatively and this interpretation this related to factorial design of the solidification process parameters (Table 1). Consequently it was shown that three level Box-Behnken is a very powerful tool to optimize and predict results in quantitative and qualitative way. This tool can be used in research, optimization and prediction of industrial processes, saving manpower, material and time in order to improve cost and quality of the product.

This work has done the numerical simulation by finite element method to the copper alloy solidification in sand mold and also this simulation was accompanied with the process of optimization of solidification parameters. This type of study accompanied with the cellular automata or cellular automaton can help minimize the defects generated during solidification. One microstructural feature that can affect the final properties of copper alloy shape castings is microporosity, formed due to the combined effects of volumetric shrinkage upon solidification. Porosity in castings is a defect that results from the interaction of a number of processes: volume change, nucleation and growth of the solid phase, diffusion of dissolved species, and the interaction of interphase surfaces. Although some of these individual processes may be treated analytically, combining the processes into a predictive tool requires the numerical calculation power that has only become available over the last few decades due to the development of digital computers.

### **4. Conclusion**

390 Convection and Conduction Heat Transfer

While a quantitative analysis of equation (14) was made, will soon be made a qualitative analysis of equation (13) by means of graphic representation of this equation and this discussion will be confronted with factorial design of the solidification process parameters (Table 1) and Figure 8. Figure 9 presents the response surface plots (Aslan, 2007; Paterakis et al., 2002), obtained from equation (13), that describe the influence of the factors on the overall desirability, Figure 9(a) shows the 3D response surface relationship between convection phenomenon (x2) and latent heat release form (x3) at zero level of mold temperature (x1). Note that, for a given value of x2, as the x3 increases and the Z decreases until a minimum value for the interval of x3 between -0.8 and 0.2. After this minimum point, Z changes its behavior and start to increase as the x3 increases reaching the highest value of 897.176, as can be seen in Table 1. For a given value of x3, it can be observed that Z profile is almost constant in relation to x2 increase. Another way to visualize Z variation is to project Z on the x2 and x3 plane, in terms of color band. The region limited by the white points on this curve represents the Z confidence interval. For other levels of x1, the surface graph behavior

Figure 9(b) shows the effect of mold temperature (x1) and latent heat release form (x3) at zero level of convection phenomenon (x2). In this case, x1 e x3 generated a complex surface of paraboloid type. According to the surface projection Z on the x1 and x3 plane, for a given x3 value, it can be observed that the variation of Z is linear in relation to x1 and this fact can be confirmed by the point x1(L) mentioned at the graph of Figure 8. On the other hand, the

For a given value of x1, the variation of Z is parabolic in relation to x3 and this fact can be observed by the points x3(L) and x3(Q) mentioned at the graph of Figure 8. Note that, for a given value of x1, as the x3 increases and the Z increases reaching the highest value of

Figure 9(c) shows the effect of mold temperature (x1) and convection phenomenon (x2) at zero level of latent heat release (x3). Note that, as the x1 factor increases, the Z increases. But, for a given value of x1, it is observed that for every x2 value, Z is almost constant. As a result, the surface geometry is not a complex one, if we compare to the surface geometry of Figure

In this type of analysis, one can realize that the parameters x1 and x3 had variations more accentuated than the parameter x2. This behavior is also verified in Figures 7 and 8. Note that, for a given value of x1, as the x2 increases and the Z increases reaching the highest

Three level Box-Behnken design referred to Table 1 showed as results Pareto's diagram of Figure 7 and curve of standardized effect of the factorial design of Figure 8. According to these results was estimated a mathematic model for Z (represents the result of the temperature after 1.5 h of solidification) based in equation (9) and through the process of maximizing and minimizing of this model was found the optimal values of x1, x2 and x3. It was also shown the graphical interpretation of equation (13) so qualitatively and this interpretation this related to factorial design of the solidification process parameters (Table 1). Consequently it was shown that three level Box-Behnken is a very powerful tool to optimize and predict results in quantitative and qualitative way. This tool can be used in research, optimization and prediction of industrial processes, saving manpower, material

has the same characteristic as previously mentioned.

953.026, as can be seen in Table 1.

value of 868.171, as can be seen in Table 1.

and time in order to improve cost and quality of the product.

9(b).

same cannot be affirmed in relation to x3, if the same analysis is done.

In this study, a three-level Box-Behnken factorial design in qualitative form and combining with a response surface methodology was employed for modeling and optimizing three operations parameters of the casting process. According to this study, it was observed when the parameters of the solidification process are in the following state, such as, mold temperature in the environment, convection phenomenon in its fullest expression and the latent heat release to the model Sheil shows a minimum temperature after 1.5 h of solidification. Also this result was verified by finite element method. The factorial design method is a useful tool to determine what factors are crucial in the solidification process and thus, a special care needs to be taken during the project elaboration of the casting. Also this optimization tool can be used in other research areas, optimizing and predicting industrial processes, saving manpower, material and time in order to improve cost and quality of the product.

### **5. Acknowledgment**

The authors acknowledge financial support provided by CNPq (The Brazilian Research Council) and Araucária Foundation (AF).

### **6. References**


Finite Element Methods to Optimize by Factorial

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### *Edited by Amimul Ahsan*

The convection and conduction heat transfer, thermal conductivity, and phase transformations are significant issues in a design of wide range of industrial processes and devices. This book includes 18 advanced and revised contributions, and it covers mainly (1) heat convection, (2) heat conduction, and (3) heat transfer analysis. The first section introduces mixed convection studies on inclined channels, double diffusive coupling, and on lid driven trapezoidal cavity, forced natural convection through a roof, convection on non-isothermal jet oscillations, unsteady pulsed flow, and hydromagnetic flow with thermal radiation. The second section covers heat conduction in capillary porous bodies and in structures made of functionally graded materials, integral transforms for heat conduction problems, non-linear radiativeconductive heat transfer, thermal conductivity of gas diffusion layers and multicomponent natural systems, thermal behavior of the ink, primer and paint, heating in biothermal systems, and RBF finite difference approach in heat conduction. The third section includes heat transfer analysis of reinforced concrete beam, modeling of heat transfer and phase transformations, boundary conditions-surface heat flux and temperature, simulation of phase change materials, and finite element methods of factorial design. The advanced idea and information described here will be fruitful for the readers to find a sustainable solution in an industrialized society.

> ISBN 978-953-307-582-2 ISBN 978-953-51-6066-3

Convection and Conduction Heat Transfer

Convection and Conduction

Heat Transfer

*Edited by Amimul Ahsan*

Photo by sakkmesterke / iStock