**Meet the editor**

Dr. *Mohsen Sheikholeslami* works in the Department of Mechanical Engineering, Babol Noshirvani University of Technology, Iran. His research interests are CFD, mesoscopic modeling of fluid, nonlinear science, nanofluid, magnetohydrodynamics, ferrohydrodynamics, and electrohydrodynamics. He has written several papers and books in various fields of mechanical engineering. Ac-

cording to the reports of Thomson Reuters, he has been selected as a Web of Science Highly Cited Researcher (Top 1%) in 2016. He is also the first author of the books *Applications of Nanofluid for Heat Transfer Enhancement*, *Hydrothermal Analysis in Engineering Using Control Volume Finite Element Method*, and *External Magnetic Field Effects on Hydrothermal Treatment of Nanofluid* which are published in Elsevier.

## Contents

**Preface XI**



**Filled with Saturated Porous Medium 239** Aaiza Gul, Ilyas Khan and Sharidan Shafie

## Preface

**Section 4 Application of Nanofluid for Combustion Engine 119**

**Applying Nanofluids 121**

**Section 5 Heat Transfer of Ferrofluid 139**

Chapter 7 **Heat Transfer of Ferrofluids 141**

Seval Genc

**VI** Contents

Chapter 6 **Enhancing Heat Transfer in Internal Combustion Engine by**

Wenzheng Cui, Zhaojie Shen, Jianguo Yang and Shaohua Wu

Chapter 8 **Nanofluid with Colloidal Magnetic Fe3O4 Nanoparticles and Its**

Lucian Pîslaru-Dănescu, Gabriela Telipan, Floriana D. Stoian, Sorin

**Applications in Electrical Engineering 163**

Chapter 9 **Magnetic Nanofluids: Mechanism of Heat Generation and Transport and Their Biomedical Application 199**

Chapter 10 **Energy Transfer in Mixed Convection MHD Flow of Nanofluid**

**Filled with Saturated Porous Medium 239** Aaiza Gul, Ilyas Khan and Sharidan Shafie

**Containing Different Shapes of Nanoparticles in a Channel**

Holotescu and Oana Maria Marinică

Prem P. Vaishnava and Ronald J. Tackett

In this book, nanofluid heat and mass transfer in engineering problems are investigated. The use of additives in the base fluid like water or ethylene glycol is one of the techniques ap‐ plied to augment heat transfer. Newly, innovative nanometer-sized particles have been dis‐ persed in the base fluid in heat transfer fluids. The fluids containing the solid nanometersized particle dispersion are called "nanofluids." Two main categories were discussed in detail: the single-phase modeling, in which the combination of nanoparticle and base fluid is considered as a single-phase mixture with steady properties, and the two-phase modeling, in which the nanoparticle properties and behaviors are considered separate from the base fluid properties and behaviors.

In Chapter 1, three-dimensional nanofluid heat and mass transfer over a sheet are presented. Chapter 2 deals with Cattaneo-Christov heat flux model that is used for simulation of nano‐ fluid flow over a sheet. Properties of nanofluid are provided in Chapters 3 and 4. Problems faced for simulating nanofluids are reported in Chapter 5. Heat transfer in magnetic fluids is presented in Chapter 6. Applications of ferrofluid in electrical and biomedical engineering are considered in Chapters 7 and 8, respectively. In Chapter 9, application of nanofluid in internal combustion engine is investigated. In chapter10, magnetic field effect on nanofluid mixed convection in a porous media has been investigated.

> **Mohsen Sheikholeslami Kandelousi** Department of Mechanical Engineering, Babol Noshirvani University of Technology, Iran

**Nanofluid Flow and Heat Transfer Over a Sheet**

#### **Numerical Analysis of Three‐Dimensional MHD Nanofluid Flow over a Stretching Sheet with Convective Boundary Conditions through a Porous Medium Numerical Analysis of Three-Dimensional MHD Nanofluid Flow over a Stretching Sheet with Convective Boundary Conditions through a Porous Medium**

Stanford Shateyi Stanford Shateyi

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/65803

#### **Abstract**

Numerical analysis has been carried out on the problem of three-dimensional magnetohydrodynamic boundary layer flow of a nanofluid over a stretching sheet with convictive boundary conditions through a porous medium. Suitable similarity transformations were used to transform the governing partial differential equations into a system of ordinary differential equations. We then solved the resultant ordinary differential equation by using the spectral relaxation method. Effects of the dimensionless parameters on velocity, temperature and concentration profiles together with the friction coefficients, Nusselt and Sherwood numbers were discussed with the assistance of graphs and tables. The velocity was found to decrease with increasing values of the magnetic, stretching and permeability parameters. The local temperature was observed to rise as the Brownian motion, thermophoresis and Biot numbers increased. The concentration profiles diminish with increasing values of the Lewis number and chemical reaction parameter.

**Keywords:** numerical analysis, MHD nanofluid, stretching sheet, convective boundary conditions, porous medium

## **1. Introduction**

Many researchers have over the past few years paid significant attention to the study of boundary layer flow heat and mass transfer over a stretching sheet due to its industrial and engineering applications. These applications include cooling of papers, glass-fibre production, plastic sheets and polymer extrusion, hot rolling wire drawing, metal spinning, stretching of

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons © 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

rubber sheets and crystal growing. The quality and final product formation in these processes are dependent on the rate of stretching and cooling.

Since the pioneering study by Crane [1] who presented an exact analytical solution for the steady two-dimensional flow due to a stretching surface in a quiescent fluid many studies on stretched surfaces have been done [1–5].

Thermal conductivity of nanoparticles has been shown in recent research on nanofluid to change the fluid characteristics. The thermal conductivity of the base liquid with the enhanced conductivity of nanofluid and the turbulence induced by their motion contribute to a remarkable improvement in the convective heat transfer coefficient. This feature of nanofluid makes them attractive to a wide variety of industries, ranging from transportation to energy production and supply to electronics. They can be used in welding equipment, high heat flux and to cool car engines, among other applications. Many researchers [6–10] have studied the boundary layer flow of a nanofluid caused by a stretching surface.

Shateyi and Prakash [11] carried out a numerical analysis on the problem of magneto hydrodynamic boundary layer flow of a nanofluid over a moving surface in the presence of thermal radiation. Kuznetsov and Nield [12] examined the influence of nanoparticles on natural convection boundary layer flow past a vertical plate, using a model in which Brownian motion and thermophoresis are accounted for. Aziz and Khan [13] investigated using a similarity analysis of the transport equations by their numerical computations to the natural convective flow of a nanofluid over a convectively heated vertical plate.

Makinde and Aziz [14] numerically studied the boundary layer flow induced in a nanofluid due to a linearly stretching sheet. Hayat et al. [15] addressed the MHD flow of second grade nanofluid over a nonlinear stretching sheet. Zhao et al. [16] studied the three-dimensional nanofluid bio-convection near a stagnation attachment. Sheikholeslami and Ganji [17] studied two-dimensional laminar-forced convection nanofluids over a stretching surface in a porous medium. The study used different models of nanofluid based on different formulas for thermal conductivity and dynamic viscosity. Nayak et al. [18] did a numerical study on the mixed convection of copper-water nanofluid inside a differentially heated skew enclosure. Recently, Mabood and Das [19] analysed MHD flow and melting heat transfer of a nanofluid over a stretching surface. Naramgari and Sulochana [20] analysed the momentum and heat transfer of MHD nanofluid embedded with conducting dust particles past a stretching surface in the presence of volume fraction of dust particles. Sandeep et al. [21] analysed the unsteady MHD radiative flow and heat transfer characteristics of a dusty nanofluid over an exponentially permeable stretching in the presence of volume fraction of dust and nanoparticles. Sheikholeslami et al. [22] computationally investigated nanofluid flow and heat transfer in a square heated rectangular body. Sheikholeslami and Ganji [23] provided a review of researches on nanofluid flow and heat transfer via semi-analytical and numerical methods. Lastly, Naramgari and Sulochana [20] analysed the three-dimensional MHD Newtonian and non-Newtonian fluid flow over a stretching surface in the presence of thermophoresis and Brownian motion.

The main objective of this chapter is to numerically analyse the influence of convective boundary conditions on the model of three-dimensional magnetohydrodynamic, nanofluid flow over a stretching sheet through a porous medium in the presence of thermophoresis and Brownian motion as well as thermal radiation. The governing partial differential equations use suitable similarity transformations. The transformed governing equations are solved numerically using the spectral relaxation method (SRM). The effects of dimensionless parameters on velocity components, temperature and concentration profiles together with the skin friction coefficients, local Nusselt and Sherwood numbers are discussed with the aid of tables and graphs.

## **2. Mathematical formulation**

rubber sheets and crystal growing. The quality and final product formation in these processes

Since the pioneering study by Crane [1] who presented an exact analytical solution for the steady two-dimensional flow due to a stretching surface in a quiescent fluid many studies on

Thermal conductivity of nanoparticles has been shown in recent research on nanofluid to change the fluid characteristics. The thermal conductivity of the base liquid with the enhanced conductivity of nanofluid and the turbulence induced by their motion contribute to a remarkable improvement in the convective heat transfer coefficient. This feature of nanofluid makes them attractive to a wide variety of industries, ranging from transportation to energy production and supply to electronics. They can be used in welding equipment, high heat flux and to cool car engines, among other applications. Many researchers [6–10] have studied the bound-

Shateyi and Prakash [11] carried out a numerical analysis on the problem of magneto hydrodynamic boundary layer flow of a nanofluid over a moving surface in the presence of thermal radiation. Kuznetsov and Nield [12] examined the influence of nanoparticles on natural convection boundary layer flow past a vertical plate, using a model in which Brownian motion and thermophoresis are accounted for. Aziz and Khan [13] investigated using a similarity analysis of the transport equations by their numerical computations to the natural convective

Makinde and Aziz [14] numerically studied the boundary layer flow induced in a nanofluid due to a linearly stretching sheet. Hayat et al. [15] addressed the MHD flow of second grade nanofluid over a nonlinear stretching sheet. Zhao et al. [16] studied the three-dimensional nanofluid bio-convection near a stagnation attachment. Sheikholeslami and Ganji [17] studied two-dimensional laminar-forced convection nanofluids over a stretching surface in a porous medium. The study used different models of nanofluid based on different formulas for thermal conductivity and dynamic viscosity. Nayak et al. [18] did a numerical study on the mixed convection of copper-water nanofluid inside a differentially heated skew enclosure. Recently, Mabood and Das [19] analysed MHD flow and melting heat transfer of a nanofluid over a stretching surface. Naramgari and Sulochana [20] analysed the momentum and heat transfer of MHD nanofluid embedded with conducting dust particles past a stretching surface in the presence of volume fraction of dust particles. Sandeep et al. [21] analysed the unsteady MHD radiative flow and heat transfer characteristics of a dusty nanofluid over an exponentially permeable stretching in the presence of volume fraction of dust and nanoparticles. Sheikholeslami et al. [22] computationally investigated nanofluid flow and heat transfer in a square heated rectangular body. Sheikholeslami and Ganji [23] provided a review of researches on nanofluid flow and heat transfer via semi-analytical and numerical methods. Lastly, Naramgari and Sulochana [20] analysed the three-dimensional MHD Newtonian and non-Newtonian fluid flow

over a stretching surface in the presence of thermophoresis and Brownian motion.

The main objective of this chapter is to numerically analyse the influence of convective boundary conditions on the model of three-dimensional magnetohydrodynamic, nanofluid flow over a stretching sheet through a porous medium in the presence of thermophoresis and Brownian

are dependent on the rate of stretching and cooling.

4 Nanofluid Heat and Mass Transfer in Engineering Problems

ary layer flow of a nanofluid caused by a stretching surface.

flow of a nanofluid over a convectively heated vertical plate.

stretched surfaces have been done [1–5].

We consider a three-dimensional steady incompressible MHD nanofluid flow, heat and mass transfer over a linearly stretching sheet through a porous medium. The sheet is assumed to be stretched along the *xy* -plane while the fluid is placed along the *z*-axis. A uniform magnetic field *B*<sup>0</sup> is applied normally to the stretched sheet and the induced magnetic field is neglected by assuming very small Reynolds number. We assume that the sheet is stretched with linear velocities *u* ¼ *ax* and *v* ¼ *by* along the *xy*-plane, respectively, with constants *a* and *b*. Under the above assumptions and the boundary approximation, the governing equations for the current study are given by:

$$
\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0,\tag{1}
$$

$$
\mu \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z} = v \frac{\partial^2 u}{\partial z^2} - \frac{\sigma B\_0^2}{\rho} u - \frac{v}{k\_1} u,\tag{2}
$$

$$
\mu \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} + w \frac{\partial v}{\partial z} = \nu \frac{\partial^2 v}{\partial z^2} - \frac{\sigma B\_0^2}{\rho} v - \frac{v}{k\_1} v,\tag{3}
$$

$$
\mu \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} + w \frac{\partial T}{\partial z} = \alpha \frac{\partial^2 T}{\partial z^2} - \frac{\partial q\_r}{\partial z} + \tau \left[ D\_\beta \frac{\partial C}{\partial z} \frac{\partial T}{\partial z} + \frac{D\_T}{T\_\infty} \left( \frac{\partial T}{\partial z} \right)^2 \right], \tag{4}$$

$$
\mu \frac{\partial \mathcal{C}}{\partial x} + v \frac{\partial \mathcal{C}}{\partial y} + w \frac{\partial \mathcal{C}}{\partial z} = D\_B \frac{\partial^2 \mathcal{C}}{\partial z^2} + \frac{D\_T}{T\_\approx} \frac{\partial^2 T}{\partial z^2}, \tag{5}
$$

where *u*, *v* and *w* are the velocity components in the *x*, *y* and *z*- directions, respectively, *T* is the fluid temperature, *C* is the fluid concentration, *k*<sup>1</sup> is the permeability, *v* is kinematic viscosity, *ρ* is the fluid density, *τ* is the ratio of the heat capacitances, *DB* and *DT* are the Brownian motion and thermopheric diffusion coefficients and *cp* is the specific heat capacity.

The corresponding boundary conditions for the flow model are:

$$\mu = a\mathbf{x}, \upsilon = by, \upsilon = 0,\\
\text{-}k\frac{\partial T}{\partial \mathbf{z}} = h\_f(T\_f - T), \ \text{-}D\_\mathbf{B}\frac{\partial \mathbf{C}}{\partial \mathbf{z}} = h\_s(\mathbf{C}\_s - \mathbf{C}) \text{ at } \mathbf{z} = \mathbf{0},\tag{6}$$

$$\mu \to 0, \upsilon \to 0, \upsilon \to 0, T \to T\_{\ast\ast}, \mathbb{C} \to \mathbb{C}\_{\ast\ast}, \text{as} \boldsymbol{z} \to \ast. \tag{7}$$

We have *hf* as the convective heat transfer coefficient, *hs* is the convective mass transfer coefficient and *Tf* and *Cf* are the convective fluid temperature and concentration below the moving sheet.

## **3. Similarity transformation**

In order to non-dimensionalise the governing equations, we introduce the following similarity equations [24]. These transformations also transform the partial differential equations into a system of ordinary differential equations which is then solved using the spectral relaxation method:

$$\eta = \sqrt{\frac{a}{v}} z, \mathfrak{u} = a \mathfrak{x} \mathfrak{f}'(\eta), \mathfrak{v} = b \mathfrak{z} \mathfrak{g}'(\eta), \mathfrak{w} = -\sqrt{a \mathfrak{v}} [f(\eta) + c \mathfrak{g}(\eta)], \mathfrak{G}(\eta) = \frac{T - T\_{\mathfrak{w}}}{T\_f - T\_{\mathfrak{w}}}, \phi(\eta) = \frac{\mathsf{C} - \mathsf{C}\_{\mathfrak{w}}}{\mathsf{C}\_{\mathfrak{s}} - \mathsf{C}\_{\mathfrak{w}}}.\tag{8}$$

Upon substituting the similarity variables into Eqs. (2)–(5), we obtain the following system of ordinary equations

$$(f'' + (f + c\mathbf{g})f' \neg f'^2 \mathbf{-} (M + K)f' = 0,\tag{9}$$

$$\text{g}'' + (f + \text{cg})\text{g}'' \text{-g}'^2 \text{-}(M + K)\text{g}' = 0,\tag{10}$$

$$\left(\frac{3+4R}{3P\_rR}\right)\boldsymbol{\theta}'' + (\boldsymbol{f} + \boldsymbol{c}\boldsymbol{g})\boldsymbol{\theta}' + Nb\boldsymbol{\theta}'\boldsymbol{\mathcal{Q}}' + Nt(\boldsymbol{\theta}')^2 = 0,\tag{11}$$

$$\left(\mathfrak{D}\right)'' + \mathrm{Le}(f + \mathrm{cg})\mathfrak{D}' + \frac{\mathrm{Nt}}{\mathrm{Nb}}\mathfrak{G}'' = 0.\tag{12}$$

The corresponding boundary conditions are

$$f = 0, \boldsymbol{f}' = 1, \boldsymbol{g} = 0, \boldsymbol{g}' = 1, \boldsymbol{\theta}' = -\mathrm{Bi}\_l(1 - \boldsymbol{\theta}), \boldsymbol{\mathfrak{Q}}' = -\mathrm{Bi}\_t(1 - \boldsymbol{\mathfrak{Q}}), \text{ at } \boldsymbol{\eta} = 0,\tag{13}$$

$$f'(\circ \circ) \to 0, \operatorname{g}'(\circ) \to 0, \theta(\circ) \to 0, \mathfrak{D}(\circ \circ) \to 0. \tag{14}$$

Primes denote differentiation with respect to *η* and parameters appearing in Eqs (9)–(14) are defined as: *Pr* ¼ *v=α* is the Prandtl number, *Le* ¼ *v=DB* is the Lewis number, *Nb* ¼ *τDB*ð*Cs*−*C∞*Þ*=v* is the Brownian motion parameter, *Nt* = *τDT*ð*Tf* −*T∞*Þ*=vT<sup>∞</sup>* is the thermophoresis parameter, *Bit* <sup>¼</sup> *hf k* ffiffiffiffiffiffiffi *<sup>v</sup>=<sup>a</sup>* <sup>p</sup> , *Bic* <sup>¼</sup> *hs DB* ffiffiffiffiffiffiffi *<sup>v</sup>=<sup>a</sup>* <sup>p</sup> are the Biot numbers and *<sup>c</sup>* <sup>¼</sup> *<sup>b</sup>=<sup>a</sup>* is the stretching parameter. The quantities of engineering interest are the skin-friction coefficient *Cf* along the *x*- and *y*-direction (*Cf x* and *Cf y*), the Nusselt number and Sherwood number. These quantities are defined as follows:

$$\mathbf{C}\_{fx} = \frac{\tau\_{wx}}{\rho u\_w^2}, \mathbf{C}\_{fy} = \frac{\tau wx}{\rho u\_w^2}, \mathbf{N}u = \mathbf{x} \frac{q\_w}{k(T\_f - T\_w)}, \mathbf{S}h = \mathbf{x} \frac{q\_w}{D\_\mathcal{B}(\mathbf{C}\_f - \mathbf{C}\_w)},\tag{15}$$

where *τwx*, *τwy* are the wall shear along *x*- and *y*-directions, respectively, and *qw* and *qm* are the heat flux and mass flux at the surface, respectively.Upon using the similarity variables into the above expressions, we obtain the following:

$$\text{Re}^2 \mathbb{C}\_{\text{fix}} = \text{g}''(0), \text{Re}^\natural \mathbb{C}\_{\text{f} \text{y}} = \text{g}''(0), \text{Re}^\natural \text{N} \text{u} = -\text{\textquotedbl{}}'(0), \text{Re}^{\natural \ddagger} \text{S} \text{h} = -\textsf{\textquotedbl{}} (0). \tag{16}$$

## **4. Method**

**3. Similarity transformation**

6 Nanofluid Heat and Mass Transfer in Engineering Problems

*η* ¼

ffiffiffi *a v* r

ordinary equations

*<sup>z</sup>*, *<sup>u</sup>* <sup>¼</sup> *axf* <sup>0</sup>

<sup>ð</sup>*η*Þ, *<sup>v</sup>* <sup>¼</sup> *byg*<sup>0</sup>

*f* 000

*g* 000

> *θ* 00

∅00

0 ¼ 1, *θ* 0

*k*

, *Cf y* <sup>¼</sup> *<sup>τ</sup>wx ρu*<sup>2</sup> *w*

> <sup>2</sup>*Cf y* ¼ *g* 00 <sup>ð</sup>0Þ, *Re*<sup>−</sup><sup>1</sup>

0

ffiffiffiffiffiffiffi *<sup>v</sup>=<sup>a</sup>* <sup>p</sup> , *Bic* <sup>¼</sup> *hs*

ð*∞*Þ ! 0, *g*

3 þ 4*R* 3*PrR* 

¼ 1, *g* ¼ 0, *g*

*f* 0

The corresponding boundary conditions are

*f* ¼ 0, *f* 0

thermophoresis parameter, *Bit* <sup>¼</sup> *hf*

These quantities are defined as follows:

*Cf x* <sup>¼</sup> *<sup>τ</sup>wx ρu*<sup>2</sup> *w*

above expressions, we obtain the following:

*Cf x* ¼ *f* 00 <sup>ð</sup>0Þ, *Re*<sup>1</sup>

*Re*<sup>2</sup>

In order to non-dimensionalise the governing equations, we introduce the following similarity equations [24]. These transformations also transform the partial differential equations into a system of ordinary differential equations which is then solved using the spectral relaxation method:

Upon substituting the similarity variables into Eqs. (2)–(5), we obtain the following system of

00 −*f* 0 2

00 −*g* 0 2

> 0 þ *Nbθ* 0 ∅0

*av* <sup>p</sup> <sup>½</sup>*f*ð*η*Þ þ *cg*ð*η*Þ, *<sup>θ</sup>*ð*η*Þ ¼ *<sup>T</sup>*−*T<sup>∞</sup>*

0

0

þ *Nt*ð*θ* 0 Þ

−ð*M* þ *K*Þ*f*

−ð*M* þ *K*Þ*g*

þ *Nt Nb <sup>θ</sup>* 00

<sup>¼</sup> <sup>−</sup>*Bit*ð1−*θ*Þ, <sup>∅</sup><sup>0</sup>

Primes denote differentiation with respect to *η* and parameters appearing in Eqs (9)–(14) are defined as: *Pr* ¼ *v=α* is the Prandtl number, *Le* ¼ *v=DB* is the Lewis number, *Nb* ¼ *τDB*ð*Cs*−*C∞*Þ*=v* is the Brownian motion parameter, *Nt* = *τDT*ð*Tf* −*T∞*Þ*=vT<sup>∞</sup>* is the

*DB*

the stretching parameter. The quantities of engineering interest are the skin-friction coefficient *Cf* along the *x*- and *y*-direction (*Cf x* and *Cf y*), the Nusselt number and Sherwood number.

, *Nu* <sup>¼</sup> *<sup>x</sup> qw*

where *τwx*, *τwy* are the wall shear along *x*- and *y*-directions, respectively, and *qw* and *qm* are the heat flux and mass flux at the surface, respectively.Upon using the similarity variables into the

*k*ð*Tf* −*Tw*Þ

<sup>2</sup> *Nu* ¼ −*θ*

0 <sup>ð</sup>0Þ, *Re*<sup>−</sup><sup>1</sup>

ffiffiffiffiffiffiffi

*Tf* −*T<sup>∞</sup>*

, *<sup>φ</sup>*ð*η*Þ ¼ *<sup>C</sup>*−*C<sup>∞</sup>*

¼ 0, (9)

¼ 0, (10)

¼ 0*:* (12)

¼ −*Bic*ð1−∅Þ, *at η* ¼ 0, (13)

*<sup>v</sup>=<sup>a</sup>* <sup>p</sup> are the Biot numbers and *<sup>c</sup>* <sup>¼</sup> *<sup>b</sup>=<sup>a</sup>* is

*DB*ð*Cf* −*Cw*Þ

, (15)

<sup>2</sup> *Sh* ¼ −∅ð0Þ*:* (16)

ð*∞*Þ ! 0, *θ*ð*∞*Þ ! 0, ∅ð*∞*Þ ! 0*:* (14)

, *Sh* <sup>¼</sup> *<sup>x</sup> qw*

<sup>2</sup> <sup>¼</sup> <sup>0</sup>, (11)

*Cs*−*C<sup>∞</sup>*

*:* (8)

<sup>ð</sup>*η*Þ, *<sup>w</sup>* <sup>¼</sup> <sup>−</sup> ffiffiffiffiffi

þ ð*f* þ *cg*Þ*f*

þ ð*f* þ *cg*Þ*g*

þ ð*f* þ *cg*Þ*θ*

<sup>þ</sup> *Le*ð*<sup>f</sup>* <sup>þ</sup> *cg*Þ∅<sup>0</sup>

To solve the set of ordinary differential Eqs. (9)–(12) together with the boundary conditions (13) and (14), we employ the Chebyshev pseudo-spectral method known as spectral relaxation method. This is a recently developed method, and the details of the method are found in Motsa et al. [25]. This method transforms sets of non-linear ordinary differential into sets of linear ordinary differential equations. The entire computational procedure is implemented using a program written in MATLAB computer language. The nanofluid velocity, temperature, the local skin-friction coefficient and the local Nusselt and Sherwood numbers are determined from these numerical computations.

To apply the SRM to the non-linear ordinary differential equations, we first set *f* 0 ð*η*Þ ¼ *p*ð*η*Þ and *g* 0 ð*η*Þ ¼ *q*ð*η*Þ. We then write the equations as follows:

$$f = p,\tag{17}$$

$$p'' + (f + c\mathbf{g})p' - p^2 - (M + K)p = 0,\tag{18}$$

$$\mathbf{g'} = \boldsymbol{\eta},\tag{19}$$

$$
\ddot{q}'' + (f + c\mathfrak{g})\dot{q}' - q^2 - (M + K)\mathfrak{q} = 0,\tag{20}
$$

$$\left(\frac{3+4\mathcal{R}}{3Pr\mathcal{R}}\right)\boldsymbol{\theta}\prime + (\boldsymbol{f}+\boldsymbol{c}\boldsymbol{g})\boldsymbol{\theta}\prime + \mathrm{Nb}\boldsymbol{\theta}\prime \boldsymbol{\mathcal{Q}}\prime + \mathrm{N}\mathrm{t}\boldsymbol{\theta}\prime^2 = \boldsymbol{0},\tag{21}$$

$$\text{L}\,\mathsf{\mathfrak{D}}'' + \text{L}\,\mathsf{e}(f + \mathsf{c}\,\mathsf{g})\mathsf{\mathfrak{D}}' + \frac{\mathsf{N}t}{\mathsf{N}b}\theta'' = 0.\tag{22}$$

The boundary conditions become

$$f(0) = 0, g(0) = 0, p(0) = 1, q(0) = 1,\tag{23}$$

$$\mathcal{O}'(0) = -Bit(1-\theta), \mathcal{Q}'(0) = -Bic(1-\phi),\tag{24}$$

$$p(^{\circ\circ}) = 0, \eta(^{\circ\circ}) = 0, \mathfrak{Q}(^{\circ\circ}) = 0, \theta(^{\circ\circ}) = 0,\tag{25}$$

In view of the SRM, we then obtain the following iterative scheme:

$$\boldsymbol{f}\_{r+1}^{\prime} = \boldsymbol{p}\_r \boldsymbol{f}\_{r+1}(0) = 0,\tag{26}$$

$$p\_{r+1}^{''} + (f\_{r+1} + cg\_{r+1})p\_{r+1} - (M+K)p\_{r+1} = p\_r^2,\\ p\_{r+1}(0) = 1,\\ p\_{r+1}(\*\*) = 0,\tag{27}$$

$$
\dot{\mathcal{g}}\_{r+1} = \boldsymbol{q}\_r, \mathcal{g}\_{r+1}(0) = 0,\tag{28}
$$

$$q\_{r+1}^{''} + (f\_{r+1} + cg\_{r+1})q\_{r+1}^{'} - (M+K)q\_{r+1} = q\_r^2, q\_{r+1}(0) = 1, q\_{r+1}(\circ) = 0,\tag{29}$$

$$\left(\frac{\mathfrak{J} + 4\mathcal{R}}{3\mathcal{P}r\mathcal{R}}\right)\boldsymbol{\theta}\_{r+1}^{\prime} + (\boldsymbol{f}\_{r+1} + \boldsymbol{c}\mathbf{g}\_{r+1})\boldsymbol{\theta}\_{r+1}^{\prime} = -\mathrm{Nb}\boldsymbol{\theta}\_{r}^{\prime}\boldsymbol{\mathcal{Q}}\_{r}^{\prime} - \mathrm{Nt}\boldsymbol{\theta}\_{r}^{\prime 2},\\\boldsymbol{\theta}\_{r+1}^{\prime}(0) = -\mathrm{Bit}(1 - \boldsymbol{\theta}\_{r+1}(0)),\\\boldsymbol{\mathcal{Q}}\_{r+1}(\boldsymbol{\simeq}) = 0,\tag{30}$$

$$\mathbf{e}\mathfrak{G}\_{r+1}^{''} + \mathrm{Le}(f\_{r+1} + \mathrm{cg}\_{r+1})\mathfrak{G}\_{r+1}^{'} = -\frac{\mathrm{Nt}}{\mathrm{Nb}}\mathfrak{G}\_{r+1}^{''}, \\ \phi\_{r+1}^{'}(0) = -\mathrm{Bic}(1 - \phi\_{r+1}(0)), \mathfrak{G}\_{r+1}(\circ \circ) = 0. \tag{31}$$

The above equations form a system of linear decoupled equations which can be solved iteratively for *<sup>r</sup>* <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, …. Starting from initial guesses *p*0ð*η*Þ, *q*0ð*η*Þ, *θ*ð*η*Þ, ∅ð*η*Þ . Applying the Chebyshev pseudo-spectral method to the above equations, we obtain

$$A\_1 f\_{r+1} = B\_1,\\ f\_{r+1}(\tau \mathbf{N}) = \mathbf{0},\tag{32}$$

$$A\_2 p\_{r+1} = B\_2, p\_{r+1}(\tau \mathcal{N}) = 1, p\_{r+1}(\tau\_0) = 0,\tag{33}$$

$$A\_3 \mathbf{g}\_{r+1} = B\_3, \mathbf{g}\_{r+1}(\pi \mathbf{N}) = 0,\tag{34}$$

$$A\_4 \eta\_{r+1} = B\_4, \eta\_{r+1}(\tau \mathcal{N}) = 1, \eta\_{r+1}(\tau\_0) = 0,\tag{35}$$

$$A\_5 \theta\_{r+1} = B\_4,\\ \theta\_{r+1}(\tau \mathbf{N}) = \frac{Bit}{1 + Bit},\\ \theta\_{r+1}(\tau\_0) = 0,\tag{36}$$

$$A\_6 \mathfrak{D}\_{r+1} = B\_6,\\ \mathfrak{D}\_{r+1}(\tau N) = \frac{Bit}{1 + Bit},\\ \mathfrak{D}\_{r+1}(\tau\_0) = 0. \tag{37}$$

where, *<sup>A</sup>*<sup>1</sup> <sup>¼</sup> *<sup>D</sup>*, *<sup>B</sup>*<sup>1</sup> <sup>¼</sup> *pr*, *<sup>A</sup>*<sup>2</sup> <sup>¼</sup> *<sup>D</sup>*<sup>2</sup> <sup>þ</sup> diagð*<sup>f</sup> <sup>r</sup>*þ<sup>1</sup> <sup>þ</sup> *cgr*þ<sup>1</sup>Þ*D*−ð*<sup>M</sup>* <sup>þ</sup> *<sup>K</sup>*Þ*I*, *<sup>B</sup>*<sup>2</sup> <sup>¼</sup> *<sup>q</sup>*<sup>2</sup> *<sup>r</sup>*þ<sup>1</sup>, *<sup>A</sup>*<sup>3</sup> <sup>¼</sup> *<sup>D</sup>*, *<sup>B</sup>*<sup>3</sup> <sup>¼</sup> *qr*, *<sup>A</sup>*<sup>4</sup> <sup>¼</sup> *<sup>D</sup>*<sup>2</sup> <sup>þ</sup> diagð*<sup>f</sup> <sup>r</sup>*þ<sup>1</sup> <sup>þ</sup> *cgr*þ<sup>1</sup>Þ*D*−ð*<sup>M</sup>* <sup>þ</sup> *<sup>K</sup>*Þ*I*, *<sup>A</sup>*<sup>5</sup> <sup>¼</sup> <sup>3</sup>þ4*<sup>R</sup>* <sup>3</sup>*PrR <sup>D</sup>*<sup>2</sup> <sup>þ</sup> diagð*<sup>f</sup> <sup>r</sup>*þ<sup>1</sup> <sup>þ</sup> *cgr*þ<sup>1</sup>Þ*D*, *<sup>B</sup>*<sup>5</sup> <sup>¼</sup> <sup>−</sup>*N*6*<sup>θ</sup>* 0 *r*∅0 *r* −*Ntθ* 0 *r* 2, *<sup>A</sup>*<sup>6</sup> <sup>¼</sup> *<sup>D</sup>*<sup>2</sup> <sup>þ</sup> diag½*Lef <sup>r</sup>*þ<sup>1</sup> <sup>þ</sup> *cLegr*þ<sup>1</sup>*D*, *<sup>B</sup>*<sup>6</sup> <sup>¼</sup> <sup>−</sup> *Nt Nb* ∅<sup>00</sup> *<sup>r</sup>*þ<sup>1</sup>,

where *I* is the identity matrix of size ð*N* þ 1Þð*N* þ 1Þ. The initial guesses are obtained as:

$$p\_0(\eta) = \mathbf{e}^{-n}, \eta\_0(\eta) = \mathbf{e}^{-n}, \theta\_0(\eta) = \frac{Bite^{-n}}{1 + Bit}, \mathfrak{D}\_0(\eta) = \frac{Bite^{-n}}{1 + Bic}.\tag{38}$$

#### **5. Results and discussion**

The system of ordinary differential Eqs. (9)–(12) subject to the boundary conditions (13) and (14) is numerically solved by applying the spectral relaxation method. The SRM results presented in this chapter were obtained using *N* ¼ 40 collocation points, and also the convergence was achieved after as few as six iterations. We also use these default values for the parameters *Pr* ¼ 0*:*71, *Nt* ¼ *Nb* ¼ 0*:*3, *Le* ¼ 2, *R* ¼ 1, *M* ¼ 1,*K* ¼ 0*:*5, *C* ¼ 0*:*1, *Bit* ¼ 0*:*2 ¼ *Bic*.

**Table 1** displays the validation of the present results with those obtained by the bvp4c results. As can be clearly observed from this table, there is an excellent agreement between the results obtained by bvp4c method giving confidence in the findings of this study. **Table 1** also shows the influence of the magnetic, permeability and stretching parameters on the skin friction coefficients. It is noticed that the skin friction coefficient increase with the increasing values of the parameters.

**Table 2** depicts the influence of Brownian motion thermophoresis, parameters and the Biot numbers on the Nusselt and Sherwood numbers. Both the rates of heat transfer and mass Numerical Analysis of Three‐Dimensional MHD Nanofluid Flow over a Stretching Sheet with Convective Boundary... http://dx.doi.org/10.5772/65803 9

∅00

−*Ntθ* 0 *r* *<sup>r</sup>*þ<sup>1</sup> <sup>þ</sup> *Le*ð*<sup>f</sup> <sup>r</sup>*þ<sup>1</sup> <sup>þ</sup> *cgr*þ<sup>1</sup>Þ∅<sup>0</sup>

8 Nanofluid Heat and Mass Transfer in Engineering Problems

tively for *r* ¼ 1, 2, …. Starting from initial guesses

*<sup>A</sup>*<sup>4</sup> <sup>¼</sup> *<sup>D</sup>*<sup>2</sup> <sup>þ</sup> diagð*<sup>f</sup> <sup>r</sup>*þ<sup>1</sup> <sup>þ</sup> *cgr*þ<sup>1</sup>Þ*D*−ð*<sup>M</sup>* <sup>þ</sup> *<sup>K</sup>*Þ*I*, *<sup>A</sup>*<sup>5</sup> <sup>¼</sup> <sup>3</sup>þ4*<sup>R</sup>*

**5. Results and discussion**

the parameters.

2, *<sup>A</sup>*<sup>6</sup> <sup>¼</sup> *<sup>D</sup>*<sup>2</sup> <sup>þ</sup> diag½*Lef <sup>r</sup>*þ<sup>1</sup> <sup>þ</sup> *cLegr*þ<sup>1</sup>*D*, *<sup>B</sup>*<sup>6</sup> <sup>¼</sup> <sup>−</sup> *Nt*

*<sup>r</sup>*þ<sup>1</sup> <sup>¼</sup> <sup>−</sup> *Nt*

Chebyshev pseudo-spectral method to the above equations, we obtain

*Nb* <sup>∅</sup><sup>00</sup> *<sup>r</sup>*þ<sup>1</sup>, *<sup>φ</sup>* 0

*<sup>A</sup>*5*θ<sup>r</sup>*þ<sup>1</sup> <sup>¼</sup> *<sup>B</sup>*4, *<sup>θ</sup><sup>r</sup>*þ<sup>1</sup>ð*τN*Þ ¼ *Bit*

*<sup>A</sup>*6∅*<sup>r</sup>*þ<sup>1</sup> <sup>¼</sup> *<sup>B</sup>*6, <sup>∅</sup>*<sup>r</sup>*þ<sup>1</sup>ð*τN*Þ ¼ *Bit*

where, *<sup>A</sup>*<sup>1</sup> <sup>¼</sup> *<sup>D</sup>*, *<sup>B</sup>*<sup>1</sup> <sup>¼</sup> *pr*, *<sup>A</sup>*<sup>2</sup> <sup>¼</sup> *<sup>D</sup>*<sup>2</sup> <sup>þ</sup> diagð*<sup>f</sup> <sup>r</sup>*þ<sup>1</sup> <sup>þ</sup> *cgr*þ<sup>1</sup>Þ*D*−ð*<sup>M</sup>* <sup>þ</sup> *<sup>K</sup>*Þ*I*, *<sup>B</sup>*<sup>2</sup> <sup>¼</sup> *<sup>q</sup>*<sup>2</sup>

*<sup>p</sup>*0ð*η*Þ ¼ *<sup>e</sup>*<sup>−</sup>*<sup>n</sup>*, *<sup>q</sup>*0ð*η*Þ ¼ *<sup>e</sup>*<sup>−</sup>*<sup>n</sup>*, *<sup>θ</sup>*0ð*η*Þ ¼ *Bite*<sup>−</sup>*<sup>n</sup>*

The above equations form a system of linear decoupled equations which can be solved itera-

3*PrR*

where *I* is the identity matrix of size ð*N* þ 1Þð*N* þ 1Þ. The initial guesses are obtained as:

The system of ordinary differential Eqs. (9)–(12) subject to the boundary conditions (13) and (14) is numerically solved by applying the spectral relaxation method. The SRM results presented in this chapter were obtained using *N* ¼ 40 collocation points, and also the convergence was achieved after as few as six iterations. We also use these default values for the parameters *Pr* ¼ 0*:*71, *Nt* ¼ *Nb* ¼ 0*:*3, *Le* ¼ 2, *R* ¼ 1, *M* ¼ 1,*K* ¼ 0*:*5, *C* ¼ 0*:*1, *Bit* ¼ 0*:*2 ¼ *Bic*. **Table 1** displays the validation of the present results with those obtained by the bvp4c results. As can be clearly observed from this table, there is an excellent agreement between the results obtained by bvp4c method giving confidence in the findings of this study. **Table 1** also shows the influence of the magnetic, permeability and stretching parameters on the skin friction coefficients. It is noticed that the skin friction coefficient increase with the increasing values of

**Table 2** depicts the influence of Brownian motion thermophoresis, parameters and the Biot numbers on the Nusselt and Sherwood numbers. Both the rates of heat transfer and mass

*Nb* ∅<sup>00</sup> *<sup>r</sup>*þ<sup>1</sup>,

*<sup>r</sup>*þ<sup>1</sup>ð0Þ ¼ <sup>−</sup>*Bic*ð1−*φ<sup>r</sup>*þ<sup>1</sup>ð0ÞÞ, <sup>∅</sup>*<sup>r</sup>*þ<sup>1</sup>ð*∞*Þ ¼ <sup>0</sup>*:* (31)

<sup>1</sup> <sup>þ</sup> *Bit* , *<sup>θ</sup><sup>r</sup>*þ<sup>1</sup>ð*τ*0Þ ¼ <sup>0</sup>*;* (36)

<sup>1</sup> <sup>þ</sup> *Bit* , <sup>∅</sup>*<sup>r</sup>*þ<sup>1</sup>ð*τ*0Þ ¼ <sup>0</sup>*:* (37)

*<sup>D</sup>*<sup>2</sup> <sup>þ</sup> diagð*<sup>f</sup> <sup>r</sup>*þ<sup>1</sup> <sup>þ</sup> *cgr*þ<sup>1</sup>Þ*D*, *<sup>B</sup>*<sup>5</sup> <sup>¼</sup> <sup>−</sup>*N*6*<sup>θ</sup>*

. Applying the

*<sup>r</sup>*þ<sup>1</sup>, *<sup>A</sup>*<sup>3</sup> <sup>¼</sup> *<sup>D</sup>*, *<sup>B</sup>*<sup>3</sup> <sup>¼</sup> *qr*,

<sup>1</sup> <sup>þ</sup> *Bic :* (38)

0 *r*∅0 *r*

*p*0ð*η*Þ, *q*0ð*η*Þ, *θ*ð*η*Þ, ∅ð*η*Þ

*<sup>A</sup>*1*<sup>f</sup> <sup>r</sup>*þ<sup>1</sup> <sup>¼</sup> *<sup>B</sup>*1, *<sup>f</sup> <sup>r</sup>*þ<sup>1</sup>ð*τN*Þ ¼ <sup>0</sup>*;* (32)

*<sup>A</sup>*3*gr*þ<sup>1</sup> <sup>¼</sup> *<sup>B</sup>*3, *gr*þ<sup>1</sup>ð*τN*Þ ¼ <sup>0</sup>*;* (34)

*<sup>A</sup>*2*pr*þ<sup>1</sup> <sup>¼</sup> *<sup>B</sup>*2, *pr*þ<sup>1</sup>ð*τN*Þ ¼ <sup>1</sup>, *pr*þ<sup>1</sup>ð*τ*0Þ ¼ <sup>0</sup>*;* (33)

*<sup>A</sup>*4*qr*þ<sup>1</sup> <sup>¼</sup> *<sup>B</sup>*4, *qr*þ<sup>1</sup>ð*τN*Þ ¼ <sup>1</sup>, *qr*þ<sup>1</sup>ð*τ*0Þ ¼ <sup>0</sup>*;* (35)

<sup>1</sup> <sup>þ</sup> *Bit* , <sup>∅</sup>0ð*η*Þ ¼ *Bice*<sup>−</sup>*<sup>n</sup>*


**Table 1.** Variation of the magnetic, permeability and stretching parameters on the skin friction coefficients.


**Table 2.** The influence of the Brownian motion and thermophoresis parameters as well as that of the Biot numbers on the Nusselt and Sherwood numbers.

transfer are increasing functions of the Brownian motion parameter. By definition, thermophoresis is the migration of a colloidal particle in a solution in response to a microscopic temperature gradient. The heat transfer is reduced while the mass transfer increases with increasing values of the thermophoresis parameter. Lastly, **Table 2** shows the influence of the Biot numbers on the heat transfer and mass transfer rates and they both increase with increasing values of the thermal Biot number. But we noticed opposite effects when the solutal Biot number increases.

**Figures 1** and **2** display the effect of permeability parameter on the velocity profiles. We observe that the tangential velocity profiles decrease as the values of the permeability parameter. Also, the transverse velocity (*f* 0 ð*η*Þ) is reduced by the increasing values of the permeability parameter as more nanofluid is taken away from the boundary layer. This explains the thinning of the velocity boundary layers as the values of *K* increases (**Figure 1**). **Figures 3** and **4** depict the effect of the magnetic field parameter on the velocity profiles. As expected, we observe that both velocity components are greatly reduced as the values of the magnetic parameter increase. This is because physically increasing the values of magnetic field strength produces a drag-like force known as the Lorentz force. This force acts against the flow when the magnetic field is applied in the normal direction, as in this chapter. **Figures 5** and **6** display the influence of the stretching parameter on the velocity fields. It is seen from **Figure 5** that the tangential velocity profiles *f* 0 ð*η*Þ are reduced by increasing values of the stretching parameter c. The transverse velocity is enhanced with the increasing values of the stretching parameter.

**Figure 7** displays the influence of the Biot number *Bi*, on the temperature profiles. It is clearly observed on this figure that the nanofluid temperature field rapidly increases near the boundary with increasing values of the Biot number, *Bit*, . It is also observed that as the Biot number increases the convective heating of the sheet also increases.

**Figures 8** and **9** reveal the effect of the stretching ratio parameter *c* on the temperature and concentration profile. It is observed that the temperature and concentration profiles are

**Figure 1.** Effect of permeability parameter on the tangential velocity profiles.

Numerical Analysis of Three‐Dimensional MHD Nanofluid Flow over a Stretching Sheet with Convective Boundary... http://dx.doi.org/10.5772/65803 11

**Figure 2.** Effect of permeability parameter on the transverse velocity profiles.

with increasing values of the thermophoresis parameter. Lastly, **Table 2** shows the influence of the Biot numbers on the heat transfer and mass transfer rates and they both increase with increasing values of the thermal Biot number. But we noticed opposite effects when the solutal

**Figures 1** and **2** display the effect of permeability parameter on the velocity profiles. We observe that the tangential velocity profiles decrease as the values of the permeability param-

parameter as more nanofluid is taken away from the boundary layer. This explains the thinning of the velocity boundary layers as the values of *K* increases (**Figure 1**). **Figures 3** and **4** depict the effect of the magnetic field parameter on the velocity profiles. As expected, we observe that both velocity components are greatly reduced as the values of the magnetic parameter increase. This is because physically increasing the values of magnetic field strength produces a drag-like force known as the Lorentz force. This force acts against the flow when the magnetic field is applied in the normal direction, as in this chapter. **Figures 5** and **6** display the influence of the stretching parameter on the velocity fields. It is seen from **Figure 5** that the

The transverse velocity is enhanced with the increasing values of the stretching parameter.

**Figure 7** displays the influence of the Biot number *Bi*, on the temperature profiles. It is clearly observed on this figure that the nanofluid temperature field rapidly increases near the boundary with increasing values of the Biot number, *Bit*, . It is also observed that as the Biot number

**Figures 8** and **9** reveal the effect of the stretching ratio parameter *c* on the temperature and concentration profile. It is observed that the temperature and concentration profiles are

ð*η*Þ) is reduced by the increasing values of the permeability

ð*η*Þ are reduced by increasing values of the stretching parameter c.

0

0

increases the convective heating of the sheet also increases.

**Figure 1.** Effect of permeability parameter on the tangential velocity profiles.

Biot number increases.

eter. Also, the transverse velocity (*f*

10 Nanofluid Heat and Mass Transfer in Engineering Problems

tangential velocity profiles *f*

**Figure 3.** Influence of the magnetic parameter on the tangential velocity.

reduced with increasing values of the stretching ration parameter. **Figures 10** and **11** display the effects of thermophoresis parameter on the dimensionless temperature and concentration profiles. It is observed that the temperature and concentration profiles increase as the values of

**Figure 4.** Influence of the magnetic parameter on the transverse velocity.

**Figure 5.** Influence of the stretching parameter on the tangential velocity.

the thermophoresis *Nt* increase. **Figure 12** depicts the influence of the Brownian motion parameter *Nb* on the temperature profiles. Increasing the values of the Brownian motion parameter *Nb* results in thicking of the thermal boundary layer, thus enhancing the Numerical Analysis of Three‐Dimensional MHD Nanofluid Flow over a Stretching Sheet with Convective Boundary... http://dx.doi.org/10.5772/65803 13

**Figure 6.** Variation of the stretching parameter on the velocity.

**Figure 7.** Effect of the Biot number on the temperature.

the thermophoresis *Nt* increase. **Figure 12** depicts the influence of the Brownian motion parameter *Nb* on the temperature profiles. Increasing the values of the Brownian motion parameter *Nb* results in thicking of the thermal boundary layer, thus enhancing the

**Figure 4.** Influence of the magnetic parameter on the transverse velocity.

12 Nanofluid Heat and Mass Transfer in Engineering Problems

**Figure 5.** Influence of the stretching parameter on the tangential velocity.

temperature of the nanofluid. **Figures 13** and **14** are plotted to depict the influence of the permeability *K* and magnetic *M*, parameters on the temperature profiles. The temperature of

**Figure 8.** Effect of varying the stretching parameter on the temperature.

**Figure 9.** Effect of varying the stretching parameter on the temperature.

Numerical Analysis of Three‐Dimensional MHD Nanofluid Flow over a Stretching Sheet with Convective Boundary... http://dx.doi.org/10.5772/65803 15

**Figure 10.** Effect of thermophoresis parameter on the temperature profiles.

**Figure 8.** Effect of varying the stretching parameter on the temperature.

14 Nanofluid Heat and Mass Transfer in Engineering Problems

**Figure 9.** Effect of varying the stretching parameter on the temperature.

**Figure 11.** Effect of thermophoresis parameter on the concentration profiles.

**Figure 12.** Effect of Brownian motion parameter on the temperature profiles.

**Figure 13.** Effect of magnetic parameter on the temperature profiles.

the nanofluid increases with increases values of the permeability parameter. From **Figure 14**, we observe that the temperature profiles increase with the increasing values of the magnetic field parameter. **Figure 15** displays the effect of thermal radiation parameter *R* on the Numerical Analysis of Three‐Dimensional MHD Nanofluid Flow over a Stretching Sheet with Convective Boundary... http://dx.doi.org/10.5772/65803 17

**Figure 14.** Influence of the permeability parameter on the nanofluid temperature.

**Figure 15.** Influence of thermal radiation on the temperature.

the nanofluid increases with increases values of the permeability parameter. From **Figure 14**, we observe that the temperature profiles increase with the increasing values of the magnetic field parameter. **Figure 15** displays the effect of thermal radiation parameter *R* on the

**Figure 12.** Effect of Brownian motion parameter on the temperature profiles.

16 Nanofluid Heat and Mass Transfer in Engineering Problems

**Figure 13.** Effect of magnetic parameter on the temperature profiles.

**Figure 16.** Influence of the Lewis number on the concentration.

temperature profiles. We observe in this figure that increasing the values of the thermal radiation produces a significant reduction in the thermal condition of the fluid flow.

Lastly, the effect of the Lewis number on the concentration profiles is depicted on **Figure 16**. Large values of the Lewis number implies increased values of the Schmidt number which results in the thinning of the solutal boundary layer.

## **6. Conclusion**

A three-dimensional magnetohydrodynamic nanofluid, heat and mass transfer over a stretching surface with convective boundary conditions through a porous medium. The transformed governing equations are solved numerically using the spectral relaxation method. The accuracy of the SRM was validated against the MATLAB in-built bvp4c routine for solving boundary value problems. The following conclusions are driven from this study:


## **Nomenclature**

temperature profiles. We observe in this figure that increasing the values of the thermal

Lastly, the effect of the Lewis number on the concentration profiles is depicted on **Figure 16**. Large values of the Lewis number implies increased values of the Schmidt number which

A three-dimensional magnetohydrodynamic nanofluid, heat and mass transfer over a stretching surface with convective boundary conditions through a porous medium. The transformed governing equations are solved numerically using the spectral relaxation method. The accuracy of the SRM was validated against the MATLAB in-built bvp4c routine for solving

• The effect of increasing the magnetic field parameter is to reduce the momentum boundary layer there and to increase the thermal and solutal boundary layer thickness. The same effect on the flow characteristics is also experienced by increasing values of the stretching

• We observed that the local temperature rises as the Brownian motion, thermophoresis, permeability parameter and Biot numbers intensify. But opposite influences are observed

when the values of the thermal radiation and stretching parameters increase.

boundary value problems. The following conclusions are driven from this study:

radiation produces a significant reduction in the thermal condition of the fluid flow.

results in the thinning of the solutal boundary layer.

**Figure 16.** Influence of the Lewis number on the concentration.

18 Nanofluid Heat and Mass Transfer in Engineering Problems

**6. Conclusion**

parameter (*c*).



## **Greek symbols**


## **Author details**

Stanford Shateyi

Address all correspondence to: stanford.shateyi@univen.ac.za

Department of Mathematics, University of Venda, Thohoyandou, South Africa

## **References**

*Pr* Prandtl number

*qr* radiative heat flux *R* thermal radiation *Re* Reynolds number *Sh* Sherwood number *T* fluid temperature

*Tw* temperature

**Greek symbols**

*ρ* fluid density

*τwx*, *τwy* wall shears

**Author details**

Stanford Shateyi

ν kinematic viscosity

*σ* electrical conductivity

τ ratio of heat capacities

*θ* dimensionless temperature *φ* dimensionless concentration

Address all correspondence to: stanford.shateyi@univen.ac.za

Department of Mathematics, University of Venda, Thohoyandou, South Africa

*u*, *v*, *w* velocity components *x*, *y*, *z* Cartesian coordinates

*qw*, *qm* heat and mass fluxes at the surface

20 Nanofluid Heat and Mass Transfer in Engineering Problems

*Tf* convective fluid temperature

*α* thermal expansion coefficient


## **Cattanneo-Christov Heat Flux Model Study for Water-Based CNT Suspended Nanofluid Past a Stretching Surface Cattanneo-Christov Heat Flux Model Study for Water-Based CNT Suspended Nanofluid Past a Stretching Surface**

Noreen Sher Akbar, C. M. Khalique and Zafar Hayat Khan Noreen Sher Akbar, C. M. Khalique and Zafar Hayat Khan

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/65628

#### **Abstract**

[14] Makinde, O. D., Aziz, A., Boundary layer flow of a nanofluid past a stretching sheet with a convective boundary condition. International Journal of Thermal Sciences. 2011; 50,

[15] Hayat, T., Aziz, A., Muhammad, T., Ahmad, B. On magnetohydrodynamic flow of second grade nanofluid over a nonlinear stretching sheet. Journal of Magnetism and

[16] Zhao, Q., Xu, H., Tau, L., Raees, A., Sun, Q. Three-dimensional free bio-convection of nanofluid near stagnation point on general curved isothermal surface. Applied Mathe-

[17] Sheikholeslami, M., Ganji, D. D. Heated permeable stretching surface in a porous medium using nanofluids. Journal of Applied Fluid Mechanics. 2014; 7(3), 535–542. [18] Nayak, R. K., Bhattacharyya, S., Pop, I. Numerical study on mixed convection and entropy generation of Cu–water nanofluid in a differentially heated skewed enclosure.

[19] Mabood, F., Das, K. Melting heat transfer on hydromagnetic flow of a nanofluid over a stretching sheet with radiation and second-order slip. European Physical Journal Plus.

[20] Naramgari, S., Sulochana, C. MHD flow of dusty nanofluid over a stretching surface with volume fraction of dust particles. Ain Shams Engineering Journal. 2016; 7, 709–716. [21] Sandeep, N., Sulochana, C., Kumar, B. R. Unsteady MHD radiative flow and heat transfer of a dusty nanofluid over an exponentially stretching surface. Engineering Science and

[22] Sheikholeslami, M., Ashorynejad, H. R., Rana, P. Lattice Boltzmann simulation of nanofluid heat transfer enhancement and entropy generation. Journal of Molecular Liq-

[23] Sheikholeslami, M., Ganji, D. D. Nanofluid convective heat transfer using semi analytical and numerical approaches: a review. Journal of the Taiwan Institute of Chemical Engi-

[24] Sulochana, C., Ashwinkumar, G. P., Sandeep, N. Similarity solution of 3D Casson nanofluid flow over a stretching sheet with convective boundary conditions. Journal of

[25] Motsa, S. S., Dlamini, P. G., Khumalo, M. Solving hyperchaotic systems using the spectral relaxation method. Abstract and Applied Analysis. 2012; doi:10.1155/2012/203461.

International Journal of Heat and Mass Transfer. 2015; 85, 620–634.

Technology, an International Journal. 2016; 19(1), 227–240.

the Nigerian Mathematical Society. 2016; 35, 128–141.

1326–1332.

2016; 131, 3.

uids. 2016; 214, 86–95.

neers. 2016; 65, 42–77.

Magnetic Materials. 2016; 408, 99–106.

22 Nanofluid Heat and Mass Transfer in Engineering Problems

matics and Mechanics. 2016; 37(4), 417–432.

This chapter discusses the magnetic field effects on the flow of Cattanneo-Christov heat flux model for water-based CNT suspended nanofluid over a stretching sheet. According to the authors, knowledge idea of Cattanneo-Christov heat flux model for water-based CNT suspended nanofluid is not explored so far for stretching sheet. The flow equations are modeled for the first time in the literature transformed into ordinary differential equations using similarity transformations. The numerical solutions are computed using shooting technique and compared with the literature for the special case of pure fluid flow and found to be in good agreement. Graphical results are presented to illustrate the effects of various fluid flow parameters on velocity, heat transfer, Nusselt number, Sherwood number, and skin friction coefficient for different types of nanoparticles.

**Keywords:** boundary layer flow, nanofluid, stretching sheet, Cattanneo-Christov heat flux model, numerical solution

## **1. Introduction**

From recent few decades, heat transfer enhancement of the nanofluid has turned out to be a topic of main interest for the researchers and scientists. The word "nanofluid" was derived by Choi [1]. He defines a liquid suspension comprising ultrafine particles whose diameter is less than 50 nm. Xuan and Roetzel [2] investigated the mechanism of heat transfer enhancement of the nanofluid. According to them, the nanofluid is a solid-liquid mixture in which metallic or

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

nonmetallic nanoparticles are suspended. The suspended ultrafine particles change transport properties and heat transfer performance of the nanofluid, which exhibits a great potential in enhancing heat transfer. They found that the reduced Nusselt number is a decreasing function of each nanofluid parameters. Khanafer et al. [3] discussed buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids. The natural convective boundary-layer flow of a nanofluid over a vertical plate is studied analytically by Kuznetsov and Nield [4]. Boundary layer laminar nanofluid flow over the stretching flat surface has been investigated numerically by Khan and Pop [5]. They show that the reduced Nusselt number is a decreasing function of each dimensionless number, while the reduced Sherwood number is an increasing function of higher Prandtl number. Ebaid and his co-authors [6–10] present boundary-layer flow of a nanofluid past a stretching sheet with different flow geometries and with different conditions. Wang [11] discussed free convection on a vertical stretching surface. Scaling group transformation for MHD(Magneto hydrodynamic) boundary-layer flow of a nanofluid past a vertical stretching surface in the presence of suction/injection was discussed by Kandasamy et al. [12].

Fourier [13] was the first who discussed the heat transfer phenomenon in 1822. The equation presented by him was parabolic in nature and has draw back that in initial disturbance is felt instantly throughout the whole medium. Cattaneo [14] modifies the "Fourier law of heat conduction in which he added the thermal relaxation term. The addition of thermal relaxation time causes heat transportation in the form of thermal waves with finite speed." Christov [15] in this contest discussed Oldroyd upper-convected derivative as an alternative of time plagiaristic to complete the material-in variant formulation. This model is known as Cattaneo-Christov heat flux model. Tibullo et al. [16] described the uniqueness of Cattaneo-Christov heat flux model for incompressible fluids. Mustafa [17] presented the Cattaneo-Christov heat flux model for Maxwell fluid over a stretching sheet. According to him, velocity is inversely proportional to the viscoelastic fluid parameter. Further, fluid temperature has inverse relationship with the relaxation time for heat flux and with the Prandtl number. Very recently, Salahuddin et al. [18] discussed MHD flow of Cattanneo-Christov heat flux model for Williamson fluid over a stretching sheet with variable thickness. They solved nonlinear problem numerically by using implicit finite difference scheme known as Keller box method. They observed that large values of wall thickness parameter and Weissenberg number are suitable for reduction in velocity profile. For further details, see Refs. [11, 12, 19–32].

The aim of this chapter is to discuss the magnetic field effects on the flow of Cattanneo-Christov heat flux model for water-based CNT suspended nanofluid over a stretching sheet. Because according to the authors, knowledge idea of Cattanneo-Christov heat flux model for waterbased CNT suspended nanofluid is not explored so far for stretching sheet. The flow equations are modeled for the first time in the literature transformed into ordinary differential equations using similarity transformations. The numerical solutions are computed using shooting technique and compared with the literature for the special case of pure fluid flow and found to be in good agreement. Graphical results are presented to illustrate the effects of various fluid flow parameters on velocity, heat transfer, Nusselt number, Sherwood number, and skin friction coefficient for different types of nanoparticles.

## **2. Formatting mathematical model**

nonmetallic nanoparticles are suspended. The suspended ultrafine particles change transport properties and heat transfer performance of the nanofluid, which exhibits a great potential in enhancing heat transfer. They found that the reduced Nusselt number is a decreasing function of each nanofluid parameters. Khanafer et al. [3] discussed buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids. The natural convective boundary-layer flow of a nanofluid over a vertical plate is studied analytically by Kuznetsov and Nield [4]. Boundary layer laminar nanofluid flow over the stretching flat surface has been investigated numerically by Khan and Pop [5]. They show that the reduced Nusselt number is a decreasing function of each dimensionless number, while the reduced Sherwood number is an increasing function of higher Prandtl number. Ebaid and his co-authors [6–10] present boundary-layer flow of a nanofluid past a stretching sheet with different flow geometries and with different conditions. Wang [11] discussed free convection on a vertical stretching surface. Scaling group transformation for MHD(Magneto hydrodynamic) boundary-layer flow of a nanofluid past a vertical stretching surface in the presence of suction/injection was discussed

Fourier [13] was the first who discussed the heat transfer phenomenon in 1822. The equation presented by him was parabolic in nature and has draw back that in initial disturbance is felt instantly throughout the whole medium. Cattaneo [14] modifies the "Fourier law of heat conduction in which he added the thermal relaxation term. The addition of thermal relaxation time causes heat transportation in the form of thermal waves with finite speed." Christov [15] in this contest discussed Oldroyd upper-convected derivative as an alternative of time plagiaristic to complete the material-in variant formulation. This model is known as Cattaneo-Christov heat flux model. Tibullo et al. [16] described the uniqueness of Cattaneo-Christov heat flux model for incompressible fluids. Mustafa [17] presented the Cattaneo-Christov heat flux model for Maxwell fluid over a stretching sheet. According to him, velocity is inversely proportional to the viscoelastic fluid parameter. Further, fluid temperature has inverse relationship with the relaxation time for heat flux and with the Prandtl number. Very recently, Salahuddin et al. [18] discussed MHD flow of Cattanneo-Christov heat flux model for Williamson fluid over a stretching sheet with variable thickness. They solved nonlinear problem numerically by using implicit finite difference scheme known as Keller box method. They observed that large values of wall thickness parameter and Weissenberg number are suitable

for reduction in velocity profile. For further details, see Refs. [11, 12, 19–32].

friction coefficient for different types of nanoparticles.

The aim of this chapter is to discuss the magnetic field effects on the flow of Cattanneo-Christov heat flux model for water-based CNT suspended nanofluid over a stretching sheet. Because according to the authors, knowledge idea of Cattanneo-Christov heat flux model for waterbased CNT suspended nanofluid is not explored so far for stretching sheet. The flow equations are modeled for the first time in the literature transformed into ordinary differential equations using similarity transformations. The numerical solutions are computed using shooting technique and compared with the literature for the special case of pure fluid flow and found to be in good agreement. Graphical results are presented to illustrate the effects of various fluid flow parameters on velocity, heat transfer, Nusselt number, Sherwood number, and skin

by Kandasamy et al. [12].

24 Nanofluid Heat and Mass Transfer in Engineering Problems

We discuss the two-dimensional nanofluid flow over a stretching sheet with water as based fluids surrounding single- and multi-wall CNTs. The flow is supposed to be laminar, steady, and incompressible. The base fluid and the CNTs are usual to be in updraft stability. Sheet is whispered to be stretched with the dissimilar velocity *Uw*, *Vw* along the *x*-axis and *y*-axis, correspondingly. We have taken the invariable ambient temperature *T*∞. Supplementary new heat model named as Cattanneo-Christov heat flux model is considered to analyze heat transfer phenomena. The *x*-axis is taken along the sheet, and *y*-axis is chosen normal to it. Magnetic field of strength *B*0 is applied normal to the sheet (as shown in **Figure 1**).

**Figure 1.** Physical model for the magnetohydrodynamic nanofluid stretching sheet problem.

With the above analysis, the boundary layer equations for the proposed model, i.e., continuity, momentum, and energy equations, can be written as follows:

$$
\frac{
\partial \mathbf{u}
}{
\partial \mathbf{x}
} + \frac{
\partial \mathbf{v}
}{
\partial \mathbf{y}
} = \mathbf{0},
\tag{1}
$$

$$\mathbf{v}\left(\mathbf{u}\frac{\partial\mathbf{u}}{\partial\mathbf{x}}+\mathbf{v}\frac{\partial\mathbf{u}}{\partial\mathbf{y}}\right)=\mathbf{v}\_{\text{u}}\left(\frac{\partial^{2}\mathbf{u}}{\partial\mathbf{y}^{2}}\right)-\frac{\sigma\_{\text{nd}}\mathbf{B}\_{\text{o}}^{2}}{\rho\_{\text{nd}}}\mathbf{u}\_{\text{v}}\tag{2}$$

$$\rho\_{\rm n\ell} \left( \mathbf{c}\_{\rm p} \right)\_{\rm n\ell} \overline{\mathbf{v}}.\nabla \mathbf{T} = -\nabla.\mathbf{q}\_{\prime} \tag{3}$$

where *u* and *v* are the velocity components along *x* and *y* directions, respectively, *T* is the temperature of the fluid, *B*<sup>0</sup> is the magnitude of magnetic field, and *q* is the heat flux. Equation (3) is the Cattaneo-Christov flux model and has the following form:

$$\mathbf{q} + \lambda\_2 \left( \frac{\partial \mathbf{q}}{\partial t} + \mathbf{V}.\nabla.\mathbf{q} - \mathbf{q}.\nabla \mathbf{V} + \left( \nabla.\mathbf{V} \right) \mathbf{q} \right) = -\mathbf{K}\_{\mathrm{nd}} \nabla \mathbf{T},\tag{4}$$

where *λ*2 is the thermal relaxation time. Eliminating *q* from Eqs. (3) and (4) gives as follows:

$$\begin{aligned} & \left( \mathbf{u} \frac{\partial \mathbf{T}}{\partial \mathbf{x}} + \mathbf{v} \frac{\partial \mathbf{T}}{\partial \mathbf{y}} \right) + \lambda\_2 \begin{pmatrix} \mathbf{u} \frac{\partial \mathbf{u}}{\partial \mathbf{x}} \frac{\partial \mathbf{T}}{\partial \mathbf{x}} + \mathbf{v} \frac{\partial \mathbf{v}}{\partial \mathbf{y}} \frac{\partial \mathbf{T}}{\partial \mathbf{y}} + \mathbf{u} \frac{\partial \mathbf{v}}{\partial \mathbf{x}} \frac{\partial \mathbf{T}}{\partial \mathbf{y}} + \mathbf{v} \frac{\partial \mathbf{u}}{\partial \mathbf{y}} \frac{\partial \mathbf{T}}{\partial \mathbf{x}} \\ + 2 \mathbf{u} \mathbf{v} \frac{\partial^2 \mathbf{T}}{\partial \mathbf{x} \partial \mathbf{y}} + \mathbf{u}^2 \frac{\partial^2 \mathbf{T}}{\partial \mathbf{x}^2} + \mathbf{v}^2 \frac{\partial^2 \mathbf{T}}{\partial \mathbf{y}^2} \\ = \frac{\mathbf{K}\_{\text{nf}}}{\rho\_{\text{nf}} \left( \mathbf{c}\_{\text{p}} \right)\_{\text{nf}}} \frac{\partial^2 \mathbf{T}}{\partial \mathbf{y}^2} \end{aligned} \tag{5}$$

Further, *ρnf* is the effective density, *μnf* is the effective dynamic viscosity, (*ρcp*)*nf* is the heat capacitance, *αnf* is the effective thermal diffusibility, and *knf* is the effective thermal conductivity of the nanofluid, which are defined as follows:

$$\begin{split} \boldsymbol{\rho}\_{\rm{nf}} &= \left(1 - \boldsymbol{\phi}\right) \boldsymbol{\rho}\_{\rm{t}} + \boldsymbol{\phi} \, \boldsymbol{\rho}\_{\rm{CNT}'} \mu\_{\rm{nf}} = \frac{\mu\_{\rm{t}}}{\left(1 - \boldsymbol{\phi}\right)^{2.5}}, \boldsymbol{\alpha}\_{\rm{nf}} = \frac{\mathbf{k}\_{\rm{nd}}}{\left(\boldsymbol{\rho}\_{\rm{cp}}\right)\_{\rm{nf}}}, \\ \left(\boldsymbol{\rho}\_{\rm{cp}}\right)\_{\rm{nf}} &= \left(1 - \boldsymbol{\phi}\right) \left(\boldsymbol{\rho}\_{\rm{cp}}\right)\_{\rm{t}} + \boldsymbol{\phi} \left(\boldsymbol{\rho}\_{\rm{cp}}\right)\_{\rm{CNT}'}, \\ \mathbf{k}\_{\rm{nd}} &= \mathbf{k}\_{\rm{t}} \bigg{(}\frac{\left(1 - \boldsymbol{\phi}\right) + \frac{2\boldsymbol{\phi} \mathbf{k}\_{\rm{CT}}}{\mathbf{k}\_{\rm{CT}} - \mathbf{k}\_{\rm{t}}} \log\left(\frac{\left(\mathbf{k}\_{\rm{CT}} \times \mathbf{k}\_{\rm{t}}}{2\mathbf{k}\_{\rm{t}}}\right)}{\left(1 - \boldsymbol{\phi}\right) + \frac{2\boldsymbol{\phi} \mathbf{k}\_{\rm{t}}}{\mathbf{k}\_{\rm{CT}} - \mathbf{k}\_{\rm{t}}} \log\left(\frac{\left(\mathbf{k}\_{\rm{CT}} \times \mathbf{k}\_{\rm{t}}}{2\mathbf{k}\_{\rm{t}}}\right)}\right)}\right) \end{split} \tag{6}$$

where *μf* is the viscosity of base fluid, *φ* is the nanoparticles fraction, (*ρCp*)*<sup>f</sup>* is the effective heat capacity of a fluid, (*ρCp*)*CNT* is the effective heat capacity of a carbon nanotubes, *kf* and *kCNT* are the thermal conductivities of the base fluid and carbon nanotubes, respectively, *ρf* and *ρCNT* are the thermal conductivities of the base fluid and carbon nanotubes, respectively.

Corresponding boundary conditions are as follows:

$$
\Delta u = a\mathbf{x} + N\nu\_f \frac{\partial u}{\partial \mathbf{y}}, \mathbf{v} = \mathbf{0}, T = T\_w, \text{ at } \mathbf{y} = \mathbf{0},\tag{7a}
$$

$$u \to 0, v \to 0, T \to T\_{\ll}, \text{ as } y \to \infty. \text{zd} \tag{7b}$$

where *T*, *Tw* and *N* are the ambient, wall fluid temperature, and slip parameter, respectively. Introducing the following similarity transformations, we have

Cattanneo-Christov Heat Flux Model Study for Water-Based CNT Suspended Nanofluid Past a Stretching Surface http://dx.doi.org/10.5772/65628 27

$$\mathbf{u} = \sqrt{\frac{\mathbf{a}}{\mathbf{v}\_t}} \mathbf{y}, \ \mathbf{u} = \operatorname{axf}'(\mathbf{u}), \ \mathbf{v} = -\sqrt{\operatorname{av}\_t} \mathbf{f}(\mathbf{u}), \theta = \frac{\mathbf{T} - \mathbf{T}\_\alpha}{\mathbf{T}\_t - \mathbf{T}\_\alpha} \tag{8}$$

Making use of Eqs. (6, 8) in Eqs. (1–5), we have

<sup>2</sup> ( ) nf <sup>q</sup> q V. .q q. V .V q K T, <sup>t</sup> æ ö ¶ +l + Ñ - Ñ + Ñ =- Ñ ç ÷

2 2 22

T T xx yy xy yx u v

æ ö ¶¶ ¶¶ ¶¶ ¶¶ ç ÷ +++ æ ö ¶ ¶ ¶¶ ¶¶ ¶¶ ¶¶ ç ÷ + +l

x y T TT 2uv u v

¶ ¶ ¶ ¶¶ è ø + ++

2 2 2 2

¶¶ ¶ ¶ è ø

( ) ( )

f nf

cp nf

(7a)

0, 0, , as . ® ® ® ®¥ ¥ *u v T T y zd* (7b)


f

uvuv

xy x y

Further, *ρnf* is the effective density, *μnf* is the effective dynamic viscosity, (*ρcp*)*nf* is the heat capacitance, *αnf* is the effective thermal diffusibility, and *knf* is the effective thermal conductivity

> <sup>k</sup> 1 , ,, 1

uT vT vT uT

( )

nf p nf

nf

¶ <sup>=</sup> r ¶

2

of the nanofluid, which are defined as follows:

( )

Corresponding boundary conditions are as follows:

Introducing the following similarity transformations, we have

f

 f

( ) ( )( ) ( )

f

cp cp cp nf <sup>f</sup> CNT

f

f

r =- r +r

( ) ( ) ( ) ( )

<sup>m</sup> r = - r+ r m = a =

nf f CNT nf 2.5 nf



capacity of a fluid, (*ρCp*)*CNT* is the effective heat capacity of a carbon nanotubes, *kf*

the thermal conductivities of the base fluid and carbon nanotubes, respectively, *ρf*

=+ = = = ¶*f w <sup>u</sup> u ax N v T T y y* n

the thermal conductivities of the base fluid and carbon nanotubes, respectively.

j + -

2 k k k k k 2k

CNT CNT f CNT f f f CNT f CNT f f

where *μf* is the viscosity of base fluid, *φ* is the nanoparticles fraction, (*ρCp*)*<sup>f</sup>* is the effective heat

, 0, , at 0, ¶

where *T*, *Tw* and *N* are the ambient, wall fluid temperature, and slip parameter, respectively.

j +

k k 2k

 f

1 ,

1 log k k , 1 log

æ ö - + ç ÷ <sup>=</sup>

nf f 2 k k k

<sup>K</sup> <sup>T</sup> , <sup>c</sup> <sup>y</sup>

26 Nanofluid Heat and Mass Transfer in Engineering Problems

2

where *λ*2 is the thermal relaxation time. Eliminating *q* from Eqs. (3) and (4) gives as follows:

è ø ¶ (4)

(5)

(6)

and *kCNT* are

and *ρCNT* are

$$\left[f''' + \left(1 - \phi\right)^{2.5}\right] \left[\left(1 - \phi + \phi \frac{\rho\_{CNT}}{\rho\_f}\right)\left\{f'' - f'^2\right\} - M^2 \left.f'\right|\right] = 0,\tag{9}$$

$$
\left(\frac{k\_{\rm af}}{k\_{\rm f}}\right)\theta'' + \Pr\left(\mathrm{l} - \phi + \phi \frac{\left(\rho c\_{\rho}\right)\_{\mathrm{CNT}}}{\left(\rho c\_{\rho}\right)\_{\mathrm{f}}}\right] \left[\left(f\theta'\right) - \gamma \left(f\theta'\theta' + f^{2}\theta''\right)\right] = 0,\tag{10}
$$

$$f\left(0\right) = 0, \ f'\left(0\right) = 1, \ f'\left(\infty\right) = 0, \theta(0) = 1, \ \theta\left(\infty\right) = 0,\tag{11}$$

where Pr = is the Prandtl number, *γ* = *aλ*2 is the non-dimensional thermal relaxation time, and <sup>=</sup> is the slip parameter.

The quantity of practical interest, in this study, is the skin friction coefficient *cf* and Nusselt number *Nux*, which is defined as follows:

$$\mathbf{c}\_{\mathbf{t}} = \frac{\mu\_{\mathrm{nf}}}{\rho\_{\mathrm{t}} \mathbf{U}\_{\mathrm{w}}^{-2}} \left( \frac{\partial \mathbf{u}}{\partial \mathbf{y}} \right)\_{\mathrm{y}=0}, \mathrm{Nu}\_{\mathrm{x}} = \frac{-\mathrm{x} \mathbf{K}\_{\mathrm{nf}}}{\mathbf{k}\_{\mathrm{t}} (\mathbf{T}\_{\mathrm{t}} - \mathbf{T}\_{\mathrm{w}})} \left( \frac{\partial \mathbf{T}}{\partial \mathbf{y}} \right)\_{\mathrm{y}=0} \tag{12}$$

where *qw* is the heat flux and *Knf* is the effective thermal conductivity. Using variables (8), we obtain:

$$\operatorname{Re}\_{x}^{1/2} \mathcal{C}\_{f} = \frac{f''(0)}{\left(1 - \phi\right)^{2.5}}, \quad \operatorname{Re}\_{x}^{-1/2} \operatorname{Nu}\_{x} = -\frac{k\_{sf}}{k\_{f}} \theta'(0). \tag{13}$$

#### **3. Numerical scheme**

The nonlinear ordinary differential equations (9)–(10) subject to the boundary conditions (11) have been solved numerically using an efficient Runge-Kutta fourth-order method along with shooting technique. The asymptotic boundary conditions given by Eq. (11) were replaced by using a value of 15 for the similarity variable *η*max. The choice of *η*max = 15 and the step size *Δη* = 0.001 ensured that all numerical solutions approached the asymptotic values correctly. For validating of the proposed scheme, a comparison for the Nusselt number with the literature [4, 8, 9] has been shown in **Table 2** for both active and passive control of *φ* in the special case when. Therefore, we are confident that the applied numerical scheme is very accurate.

## **4. Results and discussion**

In this section, the graphical explanation of the numerical results for velocity, temperature, skin friction coefficients, Nusselt number, and stream lines is expressed with respect to certain changes in the physical parameters through illustrations (**Figures 2**–**7**). A comparative study for pure water, SWCNT and MWCNT, is also depicted through **Tables 1**–**5**.

(**Figure 2a** and **b**) represents the changes in the fluid velocity profiles with respect to different values of solid nanoparticle volume fraction. **Figure 2(a)** shows the variation in solid volume fraction of nanoparticles with respect to Hartmann number M. As Hartmann number is the ratio of electromagnetic forces to the viscous forces. It is observed that when Hartmann number increases, electromagnetic forces will be dominant to the viscous forces that give declines in the velocity field (see **Figure 2(a)**). **Figure 2(b)** shows the variation in solid nanoparticle volume fraction with slip parameter β on velocity profile. It is analyzed that with an increase in slip parameter, velocity profile decreases. Further with an increase in solid nanoparticle volume fraction, velocity profile increases, and boundary layer thickness also increases with the increase in Hartmann number M, slip parameter, and solid nanoparticle volume fraction.

**Figure 2.** Velocity profile for different values of solid nanoparticle volume fraction. (a) Shows the variation with Hartmann number M. (b) Shows the variation with slip parameter β.

Temperature profile for different values of solid nanoparticle volume fraction with the variation in thermalrelaxation time γ, Hartmann number M, and slip parameter β is presented in **Figure 3**(**a**–**c**). Temperature profile decreases with the rise in thermal relaxation time, but thermal boundary layer increases with an increase in thermalrelaxation time (see **Figure 3(a)**). **Figure 3(b)** depicts that with an increase in electromagnetic forces as compared to the viscous forces, temperature profile and thermal boundary layer increase rapidly. Temperature profile and thermal boundary layer also increase rapidly with the rise in slip parameter (see **Fig‐ ure 3(c)**). Moreover, temperature profile and thermal boundary layerincrease with an increase in solid nanoparticle volume fraction.

*Δη* = 0.001 ensured that all numerical solutions approached the asymptotic values correctly. For validating of the proposed scheme, a comparison for the Nusselt number with the literature [4, 8, 9] has been shown in **Table 2** for both active and passive control of *φ* in the special case when. Therefore, we are confident that the applied numerical scheme is very

In this section, the graphical explanation of the numerical results for velocity, temperature, skin friction coefficients, Nusselt number, and stream lines is expressed with respect to certain changes in the physical parameters through illustrations (**Figures 2**–**7**). A comparative study

(**Figure 2a** and **b**) represents the changes in the fluid velocity profiles with respect to different values of solid nanoparticle volume fraction. **Figure 2(a)** shows the variation in solid volume fraction of nanoparticles with respect to Hartmann number M. As Hartmann number is the ratio of electromagnetic forces to the viscous forces. It is observed that when Hartmann number increases, electromagnetic forces will be dominant to the viscous forces that give declines in the velocity field (see **Figure 2(a)**). **Figure 2(b)** shows the variation in solid nanoparticle volume fraction with slip parameter β on velocity profile. It is analyzed that with an increase in slip parameter, velocity profile decreases. Further with an increase in solid nanoparticle volume fraction, velocity profile increases, and boundary layer thickness also increases with the increase in Hartmann number M, slip parameter, and solid nanoparticle volume fraction.

**Figure 2.** Velocity profile for different values of solid nanoparticle volume fraction. (a) Shows the variation with Hart-

Temperature profile for different values of solid nanoparticle volume fraction with the variation in thermalrelaxation time γ, Hartmann number M, and slip parameter β is presented in **Figure 3**(**a**–**c**). Temperature profile decreases with the rise in thermal relaxation time, but thermal boundary layer increases with an increase in thermalrelaxation time (see **Figure 3(a)**).

mann number M. (b) Shows the variation with slip parameter β.

for pure water, SWCNT and MWCNT, is also depicted through **Tables 1**–**5**.

accurate.

**4. Results and discussion**

28 Nanofluid Heat and Mass Transfer in Engineering Problems

**Figure 3.** Temperature profile for different values of solid nanoparticle volume fraction. (a). Shows the variation with thermal relaxation time γ. (b) Shows the variation with Hartmann number M. (c) Shows the variation with slip parameter β.

**Figure 4.** Skin friction coefficient for SWCNT and MWCNT. (a) Shows the variation with slip parameter β. (b) Shows the variation with Hartmann number M.

**Figure 5.** Nusselt number for SWCNT and MWCNT. (a) Shows the variation with Hartmann number M. (b) Shows the variation with thermal relaxation time γ. (c) Shows the variation with slip parameter β.

**Figure 6(a‐c).** Streamlines for different values of Hartmann number M other parameters are β = 0.4, γ = 0.3.

**Figure 7(a‐c).** Isotherms for different values of thermal relaxation time γ other parameters are M = 2, β = 0.2.

Cattanneo-Christov Heat Flux Model Study for Water-Based CNT Suspended Nanofluid Past a Stretching Surface http://dx.doi.org/10.5772/65628 31


**Table 1.** Thermophysical properties of different base fluid and CNTs.

**Figure 5.** Nusselt number for SWCNT and MWCNT. (a) Shows the variation with Hartmann number M. (b) Shows the

**Figure 6(a‐c).** Streamlines for different values of Hartmann number M other parameters are β = 0.4, γ = 0.3.

**Figure 7(a‐c).** Isotherms for different values of thermal relaxation time γ other parameters are M = 2, β = 0.2.

variation with thermal relaxation time γ. (c) Shows the variation with slip parameter β.

30 Nanofluid Heat and Mass Transfer in Engineering Problems


**Table 2.** Comparison of results for the skin friction for pure fluid (*φ* = 0).


**Table 3.** Comparison of results for the Nusselt number for pure fluid (*φ* = 0) with *M* = 0 and *γ* = 0.


**Table 4.** Skin friction coefficient for different values of Hartmann number M and slip parameter β.


**Table 5.** Nusselt number for different values of Hartmann number M and thermal relaxation time γ.

Variation in skin friction coefficient for SWCNT and MWCNT with slip parameter β and Hartmann number M is presented in **Figure 4**(**a** and **b**). It is seen that with the augment in M, electromagnetic strength is elevated in contrast to thick strength, skin friction coefficient rises for SWCNT as well as for MWCNT, but with the increase in slip parameter, skin friction coefficient decreases for both SWCNT and MWCNT. It is also seen that density and thermal conductivity of SWCNT are greater as compared to the MWCNT; therefore, the skin friction coefficient for SWCNT is greater as compared to the MWCNT.

Nusselt number for SWCNT and MWCNT shows the variation in Hartmann number M, thermal relaxation time γ, and slip parameter β. It is observed that the higher values of thermal relaxation time γ raise the Nusselt number for SWCNT as well as for MWCNT, and it is also analyzed that Nusselt number gives the larger values for SWCNT than MWCNT (see **Fig‐ ure 5**(**a**–**c**)). Increasing values of Hartmann number M and slip parameter β decrease the Nusselt number for both SWCNT and MWCNT. But due to high density and thermal conductivity of SWCNT, Nusselt number for SWCNT is higher than MWCNT.

Streamlines and Isotherms are presented in **Figures 6**(**a**–**c**) and **7**(**a**–**c**), respectively. It is analyzed from **Figures 6** and **7** that for increasing Hartmann number M and thermal relaxation time γ, streamlines and Isotherms are going close to origin.

## **5. Conclusions**

This chapter discussed the magnetic field effects on the flow of Cattanneo-Christov heat flux model for water-based CNT suspended nanofluid over a stretching sheet. Key points of the performed analysis are as follows:


## **Nomenclature**

*φ* **Nusselt number**

0.1 0.2

0.1

1.77095 2.66543 3.22397

32 Nanofluid Heat and Mass Transfer in Engineering Problems

1.77095 2.63339 1.33685 2.03231 2.49289

1.33685 2.02350

coefficient for SWCNT is greater as compared to the MWCNT.

tivity of SWCNT, Nusselt number for SWCNT is higher than MWCNT.

dominant to the viscous forces that give declines in the velocity.

parameter β, and solid nanoparticle volume fraction ϕ.

time γ, streamlines and Isotherms are going close to origin.

1.74513 2.60424 3.13585

1.74513 2.57329

**Table 5.** Nusselt number for different values of Hartmann number M and thermal relaxation time γ.

SWCNT 0.0

MWCNT 0.0

**5. Conclusions**

performed analysis are as follows:

*γ* **= 0** *γ* **= 0** *γ* **= 0.1** *γ* **= 0.1** *M* **= 0** *M* **= 0.5** *M* **= 1** *M* **= 2**

> 1.25641 1.89346 2.31928

> 1.25641 1.88386

Variation in skin friction coefficient for SWCNT and MWCNT with slip parameter β and Hartmann number M is presented in **Figure 4**(**a** and **b**). It is seen that with the augment in M, electromagnetic strength is elevated in contrast to thick strength, skin friction coefficient rises for SWCNT as well as for MWCNT, but with the increase in slip parameter, skin friction coefficient decreases for both SWCNT and MWCNT. It is also seen that density and thermal conductivity of SWCNT are greater as compared to the MWCNT; therefore, the skin friction

Nusselt number for SWCNT and MWCNT shows the variation in Hartmann number M, thermal relaxation time γ, and slip parameter β. It is observed that the higher values of thermal relaxation time γ raise the Nusselt number for SWCNT as well as for MWCNT, and it is also analyzed that Nusselt number gives the larger values for SWCNT than MWCNT (see **Fig‐ ure 5**(**a**–**c**)). Increasing values of Hartmann number M and slip parameter β decrease the Nusselt number for both SWCNT and MWCNT. But due to high density and thermal conduc-

Streamlines and Isotherms are presented in **Figures 6**(**a**–**c**) and **7**(**a**–**c**), respectively. It is analyzed from **Figures 6** and **7** that for increasing Hartmann number M and thermal relaxation

This chapter discussed the magnetic field effects on the flow of Cattanneo-Christov heat flux model for water-based CNT suspended nanofluid over a stretching sheet. Key points of the

**1.** It is observed that when Hartmann number increases, electromagnetic forces will be

**2.** It is analyzed that with an increase in slip parameter, velocity profile decreases. Further, with an increase in solid nanoparticle volume fraction, velocity profile increases. **3.** Boundary layer thickness also increases with the increase in Hartmann number M, slip

*β* **= 0** *β* **= 1** *β* **= 0** *β* **= 1** *β* **= 0** *β* **= 1** *β* **= 0** *β* **= 1**

1.71287 2.48958 2.95535

1.71287 2.46327 1.08479 1.58280 1.92489

1.08479 1.57551 1.52688 2.07111 2.37985

1.52688 2.05686 0.70076 0.92785 1.11466

0.70076 0.92736



## **Author details**

Noreen Sher Akbar1\*, C. M. Khalique2 and Zafar Hayat Khan3

\*Address all correspondence to: noreensher1@gmail.com

1 DBS&H CEME, National University of Sciences and Technology, Islamabad, Pakistan

2 International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, South Africa

3 Department of Mathematics, University of Malakand, Dir (Lower), Khyber Pakhtunkhwa, Pakistan

## **References**


[6] A. Ebaid, H.A. El-arabawy, Y. Nader, New exact solutions for boundary-layer flow of a nanofluid past a stretching sheet, Int. J. Differential Equ. 2013 (Article ID 865464) 8.

*η*: Similarity variable (transformed coordinate) *αnf*: Effective thermal diffusibility

*θ*: Dimensionless temperature *μnf*: Effective dynamic viscosity

(*ρc*)*p*: Effective heat capacity of the nanoparticle material (*ρc*)*<sup>f</sup>*

34 Nanofluid Heat and Mass Transfer in Engineering Problems

Noreen Sher Akbar1\*, C. M. Khalique2

of Mathematical Sciences, South Africa

\*Address all correspondence to: noreensher1@gmail.com

ASME, New York, vol. 66 (1995) 99–105.

Heat Mass Tranfer 53 (2010) 2477–2483.

Mass Transfer. 43 (2000) 3701–3707.

**Author details**

Pakistan

**References**

3639–3653.

*Nux*: Local Nusselt number *x, y*: Coordinate along and normal to the sheet

*γ*: Non-dimensional thermal relaxation time *knf*: Effective thermal conductivity of the nanofluid

1 DBS&H CEME, National University of Sciences and Technology, Islamabad, Pakistan

2 International Institute for Symmetry Analysis and Mathematical Modelling, Department

3 Department of Mathematics, University of Malakand, Dir (Lower), Khyber Pakhtunkhwa,

[1] S.U.S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, in: D.A. Siginer, H.P. Wang (Eds.), Developments and Applications of Non-Newtonian Flows,

[2] Y. Xuan, W. Roetzel, Conceptions for heat transfer correlation of nanofluids, Int. J. Heat

[3] K. Khanafer, K. Vafai, M. Lightstone, Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids, Int. J. Heat Mass Transfer. 46 (2003)

[4] A.V. Kuznetsov, D.A. Nield, Natural convective boundary-layer flow of a nanofluid

[5] W.A. Khan, I. Pop, Boundary-layer flow of a nanofluid past a stretching sheet, Int. J.

past a vertical plate, Int. J. Therm. Sci. 49 (2010) 243–247.

and Zafar Hayat Khan3

: Heat capacity of the fluid


**Thermal Physical Properties of Metal Oxides Nanofluid**

[21] N.S. Akbar, Z.H. Khan, S. Nadeem, W.A. Khan, Double-diffusive natural convective boundary-layer flow of a nanofluid over a stretching sheet with magnetic field, Int. J.

[22] M. Sheikholeslami, D.D. Ganji, M. Gorji-Bandpy, S. Soleimani, Magnetic field effect on nanofluid flow and heat transfer using KKL model, J. Taiwan Inst. Chem. Eng. 45 (2014)

[23] H. Togun, G. Ahmadi, T. Abdulrazzaq, A.J. Shkarah, S.N. Kazi, A. Badarudin, M.R. Safaei, Thermal performance of nanofluid in ducts with double forward-facing steps,

[24] M. Sheikholeslami, R. Ellahi, H.R. Ashorynejad, G. Domairry, Effects of heat transfer in flow of nanofluids over a permeable stretching wall in a porous medium, J. Comput.

[25] M. Sheikholeslami, D.D. Ganji, M.Y. Javed, R. Ellahi, Effect of thermal radiation on magnetohydrodynamics nanofluid flow and heat transfer by means of two phase

[26] M. Sheikholeslami, M. Gorji-Bandpy, Free convection of ferrofluid in a cavity heated from below in the presence of an external magnetic field, Powder Technol. 256 (2014)

[27] M. Sheikholeslami, M. Gorji-Bandpy, D.D. Ganji, Lattice Boltzmann method for MHD natural convection heat transfer using nanofluid, Powder Technol. 254

[28] M. Sheikholeslami, M. Gorji-Bandpy, D.D. Ganji, Numerical investigation of MHD effects on Al2O3-water nanofluid flow and heat transfer in a semi-annulus enclosure

[29] M. Sheikholeslami, R. Ellahi, M. Hassan, S. Soleimani, A study of natural convection heat transfer in a nanofluid filled enclosure with elliptic inner cylinder, Int. J. Numer.

[30] M. Sheikholeslami, M. Gorji, M. Bandpy, R. Ellahi, M. Hassan, S. Soleimani, Effects of MHD on Cu-water nanofluid flow and heat transfer by means of CVFEM, J. Magnet.

[31] M. Sheikholeslami, M. Bandpy, M.G.R. Ellahi, R. Zeeshan, A simulation of MHD CuOwater nanofluid flow and convective heat transfer considering Lorentz forces, J.

[32] R. Ellahi, M. Gulzar, M. Sheikholeslami, Effects of heat transfer on peristaltic motion of Oldroyd fluid in the presence of inclined magnetic field, J. Magnet. Magnet. Mater. 372

Numer. Methods Heat Fluid Flow 26(1) (2016) 108–121.

J. Taiwan Inst. Chem. Eng. 47 (2015) 28–42.

model, J. Magnet. Magnet. Mater. 374 (2015) 36–43.

Theor. Nanosci. 11(2) (2014) 486–496.

36 Nanofluid Heat and Mass Transfer in Engineering Problems

using LBM, Energy 60 (2013) 501–510.

Magnet. Mater. 349 (2014) 188–200.

Magnet. Magnet. Mater. 369 (2014) 69–80.

Methods Heat Fluid Flow 24 (2014) 1906–1927.

795–807.

490–498.

(2014) 82–93.

(2014) 97–106.

Provisional chapter

## **Thermophysical Properties of Metal Oxides Nanofluids** Thermophysical Properties of Metal Oxides Nanofluids

Zafar Said and Rahman Saidur Zafar Said and Rahman Saidur

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/65610

#### Abstract

Thermophysical properties of TiO2, Al2O3 and SiO2 nanofluids are experimentally investigated and compared with published data. Density has been measured over a range of 25–40°C for nanoparticle volumetric concentration of 0.05–4%. Viscosity experiments were carried out over a wide temperature range, from 25 to 80°C, to determine their applicability in such ranges. Nanofluids with particle volume fraction ranging from 0.02 to 0.03% and 1–4 kg/min were examined for the convective heat transfer and pumping power. The heat transfer coefficient of the nanofluid rises with rising mass flow rate, as well as rising volume concentration of metal oxide nanofluids; however, increasing the volume fraction results in increasing the density and viscosity of nanofluid, leading to a slight increase in friction factor which can be neglected. Addition of surfactants results in part of the increment in viscosity as well. An empirical formula for density is proposed, which also contributes to the novelty of this paper.

Keywords: nanofluids, surfactants, density, viscosity, pressure drop, heat transfer coefficient

## 1. Introduction

Thermophysical properties of nanofluids are significant to enhance the heat transfer behaviour. It is tremendously significant in controlling industrial and energy saving prospects. Great interest has been shown by the industry in nanofluids. Unlike conventional particle-fluid suspension (millimetre- and micrometre-sized particles), nanoparticles have great ability to enhance the thermal transport properties. In the last decade, due to the ability of improving thermal properties, nanofluids have gained prominent attention. Based on the broad research, it has been recognized that the suspension of metallic particles in a base fluid significantly increases the thermal conductivity of the mixture [1], therefore improving the heat transfer capability. Such observations have inspired the industrial as well as the science community to

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons © 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

discover the thermophysical properties of nanofluids, such as density, viscosity, thermal conductivity and heat capacity. Obtaining the viscosity of nanofluids is of importance for establishing sufficient pumping power. Besides, the convective heat transfer coefficients, Prandtl and Reynolds numbers, are also reliant on viscosity.

The available literature has suggested that the nanofluids tend to enhance the heat transfer performance, most importantly the heat transfer coefficient, thermal conductivity and viscosity, which in turn affects the pumping power. Das et al. [2] experimentally showed that thermal conductivity of a nanofluid can be augmented up to fourfold by increasing temperature. Xuan and Li [3] also found that heat transfer coefficient could be enhanced by the use of nanofluids, particularly when increasing the flow or nanoparticle concentration.

Different results on the density of nanofluids have been reported by the researchers, most of which are for a particular temperature or for a particular nanofluid [4–12]. There is no generalized method or model to obtain the density of different nanofluids theoretically. Density of the nanofluids strongly depends on nanoparticle material and increases with the increase in volume concentration. Base fluid also plays a significant role in the density of the nanofluids, whereas the other parameters such as nanoparticles shape, size, zeta potential and additives do not affect the density of the nanofluids. For engineering applications, larger density is more preferable [13]. The equation for the density of two phase mixtures for particles of micrometre size is available in the literature for slurry flows [14]. Densities of solids are greater than that of the liquids, with the increase in the concentration of nanoparticles in the fluid, and the density of the nanofluid is found to increase. Density of the nanofluids is proportional to the volume ratio of nanoparticles (solid) and base fluid (liquid) in a system. Pak and Cho [15] conducted an experiment at one temperature (25°C) for Al2O3 and TiO2 nanofluids of up to 4 vol.%.

Studies on viscosity have been reported by numerous researchers. Several researchers [16–19] observed a Newtonian behaviour in TiO2-ethylene glycol, Al2O3-water, single wall carbon nanohorns (SWCNH)-water and TiO2-water nanofluids, respectively. With the increase in particle concentration, increase in the viscosity is noted. Timofeeva et al. [20] observed that viscosity decreases with the increase in particle size; however, Pastoriza-Gallego et al. [21] observed a different behaviour. A non-Newtonian behaviour was found by other researchers. Some authors have also developed models to describe the rheological behaviour of nanofluids. Koo [22] introduced a model to predict the thermal conductivity and viscosity of nanofluid in terms of nanoparticle size, concentration and density. Masoumi et al. [23] also presented a model to calculate the effective viscosity by considering the Brownian motion and the relative viscosity between the fluid and particles.

Pumping power is deserted in more than a few studies although it directly effects the usefulness of the fluid in applications. This is due to the reason that heat transfer coefficients could be enhanced by merely increasing the flow velocity of the fluid, which results in additional pumping power due to the surged pressure losses. Any effort to improve heat transfer results in improved pressure losses. Also with the addition of nanoparticles to the base fluid, an antagonism between heat transfer growth and enhanced pressure losses is present. Hence, the real measure of efficiency of a heat transfer fluid is not only the convective heat transfer coefficient alone, but also the pressure losses need to be taken into account for calculations as well [24]. Heat transfer of a nanofluid flow which is squeezed between parallel plates is investigated analytically using homotopy perturbation method (HPM). It was reported from the findings that the Nusselt number has a direct relationship with the nanoparticle volume fraction, the squeeze number and Ecket number when two plates are separated, but an inverse effect is noted when the plates are squeezed [25]. In another study, heat and mass transfer characteristics of unsteady nanofluid flow between two parallel plates are investigated considering thermal radiation. Ordinary differential equations are solved numerically using the fourth-order Runge-Kutta method. Results indicated that the radiation parameter increased with the concentration boundary layer thickness. It was also reported that the Eckert number, Schmidt number, squeeze parameter and radiation parameter have direct relationship with Nusselt number [26].

For nanofluids to be applied for practical applications, it is essential to study the effect of these nanofluids on the flow features and their effect on the pumping power and pressure drop, in addition to heat transfer performance enhancement. Calculations are carried out for different volume fractions of nanoparticles and for changing mass flow rate. The effectiveness of nanofluids is investigated by comparing the required pumping power of oxides nanofluids and the base fluid. The reason these oxides are considered for this study is that they are easy to produce and cheaper compared to the CNTs and graphene.

On the basis of the inclusive literature review, the main objectives of this study are to examine the effects of nanofluids on the density, viscosity, the pressure drop and the convection heat transfer characteristics. The nanofluids contained Al2O3, SiO2, and TiO2 nanoparticles in water as a base fluid. The analyses were done for several nanofluids and then associated them with the water as the base fluid. The potential outcomes and their details were also described.

## 2. Methodology

## 2.1. Materials

discover the thermophysical properties of nanofluids, such as density, viscosity, thermal conductivity and heat capacity. Obtaining the viscosity of nanofluids is of importance for establishing sufficient pumping power. Besides, the convective heat transfer coefficients,

The available literature has suggested that the nanofluids tend to enhance the heat transfer performance, most importantly the heat transfer coefficient, thermal conductivity and viscosity, which in turn affects the pumping power. Das et al. [2] experimentally showed that thermal conductivity of a nanofluid can be augmented up to fourfold by increasing temperature. Xuan and Li [3] also found that heat transfer coefficient could be enhanced by the use of nanofluids,

Different results on the density of nanofluids have been reported by the researchers, most of which are for a particular temperature or for a particular nanofluid [4–12]. There is no generalized method or model to obtain the density of different nanofluids theoretically. Density of the nanofluids strongly depends on nanoparticle material and increases with the increase in volume concentration. Base fluid also plays a significant role in the density of the nanofluids, whereas the other parameters such as nanoparticles shape, size, zeta potential and additives do not affect the density of the nanofluids. For engineering applications, larger density is more preferable [13]. The equation for the density of two phase mixtures for particles of micrometre size is available in the literature for slurry flows [14]. Densities of solids are greater than that of the liquids, with the increase in the concentration of nanoparticles in the fluid, and the density of the nanofluid is found to increase. Density of the nanofluids is proportional to the volume ratio of nanoparticles (solid) and base fluid (liquid) in a system. Pak and Cho [15] conducted an experiment at one temperature (25°C) for Al2O3 and

Studies on viscosity have been reported by numerous researchers. Several researchers [16–19] observed a Newtonian behaviour in TiO2-ethylene glycol, Al2O3-water, single wall carbon nanohorns (SWCNH)-water and TiO2-water nanofluids, respectively. With the increase in particle concentration, increase in the viscosity is noted. Timofeeva et al. [20] observed that viscosity decreases with the increase in particle size; however, Pastoriza-Gallego et al. [21] observed a different behaviour. A non-Newtonian behaviour was found by other researchers. Some authors have also developed models to describe the rheological behaviour of nanofluids. Koo [22] introduced a model to predict the thermal conductivity and viscosity of nanofluid in terms of nanoparticle size, concentration and density. Masoumi et al. [23] also presented a model to calculate the effective viscosity by considering the Brownian motion and the relative

Pumping power is deserted in more than a few studies although it directly effects the usefulness of the fluid in applications. This is due to the reason that heat transfer coefficients could be enhanced by merely increasing the flow velocity of the fluid, which results in additional pumping power due to the surged pressure losses. Any effort to improve heat transfer results in improved pressure losses. Also with the addition of nanoparticles to the base fluid, an antagonism between heat transfer growth and enhanced pressure losses is present. Hence, the real measure of efficiency of a heat transfer fluid is not only the convective heat transfer

Prandtl and Reynolds numbers, are also reliant on viscosity.

40 Nanofluid Heat and Mass Transfer in Engineering Problems

TiO2 nanofluids of up to 4 vol.%.

viscosity between the fluid and particles.

particularly when increasing the flow or nanoparticle concentration.

Deionized water was used as the base fluid. The nanoparticles (TiO2, P25 ≥ 99.5% trace metals basis), (SiO2, 99.5% trace metals basis) and (Al2O3, 99.8% trace metals basis) were purchased from Sigma-Aldrich. Sodium dodecyl sulphate (SDS, 92.5–100.5%, Sigma-Aldrich), polyvinylpyrrolidone (PVP, Sigma-Aldrich), poly(ethylene glycol) 400 (PEG, Sigma-Aldrich) and hexadecytrimethyl-ammonium bromide (HTAB, ≥ 98%) were used as surfactants (Figures 1–4).

## 2.2. Nanofluids preparation

Ultra sound sonication was used to homogenize the suspensions. The particles can be mixed into many different liquids at preferable concentrations. Probe-type sonicators break particle agglomerates faster and more thoroughly than bath sonicators, and thus, it was chosen for our experiments. The nanoparticles were dispersed mechanically in distilled water at a concentration of 0.05, 0.5, 1, 2 and 4% by volume for density and 0.05–0.5% by volume for viscosity.

Figure 1. SEM images of (a) TiO2, (b) SiO2 and (c) Al2O3 nanoparticles.

Figure 2. TEM images of silica (SiO2) nanoparticles ~10 to 20 nm and with surfactant (0.14% PEG).

Aforementioned surfactants with 1:2, 1:3 and 1:5 nanoparticle to surfactant ratios (by volume) were used to prepare nanofluids for this research.

Figure 3. TEM images of titania (TiO2) nanoparticles ~21 nm and the surfactant (0.1 %vol. PEG).

Figure 4. TEM images of alumina (Al2O3) nanoparticles, average 13 nm and the surfactant (0.1%vol. HTAB).

#### 2.3. Nanofluid characterization

In order to characterize the prepared nanofluids, particle size, dynamic viscosity and density were measured as functions of temperature and particle volumetric fraction. Field emission scanning electron microscopy (FESEM) and transmission electron microscopy (TEM) were used to obtain the morphological characterization of the nanoparticles with SIGMA Zeiss instrument (Carl Zeiss SMT Ltd., UK). The Density Metre DA-130N from Kyoto Electronics Shinjuku-ku, Tokyo, Japan was used to measure the density of the nanofluids.

#### 2.4. Sedimentation

Aforementioned surfactants with 1:2, 1:3 and 1:5 nanoparticle to surfactant ratios (by volume)

Figure 2. TEM images of silica (SiO2) nanoparticles ~10 to 20 nm and with surfactant (0.14% PEG).

were used to prepare nanofluids for this research.

Figure 1. SEM images of (a) TiO2, (b) SiO2 and (c) Al2O3 nanoparticles.

42 Nanofluid Heat and Mass Transfer in Engineering Problems

Different nanofluids of 10 ml volume were used to investigate the sedimentation rate, and data were collected for 1 month from the date of preparation. Table 1 shows the detailed description of our investigation. It was also found that 0.1 %vol. HTAB for TiO2/DW and 0.1 %vol. PVP for Al2O3/DW and TiSiO4/DW work as the best surfactants for stability.

From Table 1, it is noticed that all the nanoparticles, dispersed in DW, showed stability for longer periods of time, except for SiO2.

#### 44 Nanofluid Heat and Mass Transfer in Engineering Problems


Table 1. Sedimentation rate of different nanofluids.

#### 2.5. Density of nanofluid

Experimental data on density measurements are not sufficient for various nanofluids at varying temperatures in the literature. Therefore, we carried out comprehensive measurements to obtain density and provide data as well as to verify the applicability of Eq. (1) (which is also called as mixing theory) [15] for various nanofluids. Figure 5 shows that density varies with temperature. Therefore, density equation should include temperature as a variable, whereas the mixing theory does not consider effect of temperature.

$$
\rho\_{\eta f} = \left(\frac{m}{V}\right)\_{\eta f} = \frac{m\_f + m\_p}{V\_f + V\_p} = \frac{\rho\_f V\_f + \rho\_p V\_p}{V\_f + V\_p} = (1 - \phi\_p)\rho\_{bf} + \phi\_p \rho\_p,\tag{1}
$$

where <sup>φ</sup><sup>p</sup> <sup>¼</sup> Vp Vf <sup>þ</sup>Vp is the volume fraction of the nanoparticles.

#### 2.6. Theoretical models

In this section, the existing theoretical models and correlations for the viscosity of nanofluid suspensions are presented. Each model is used for specific circumstances.

Figure 5. Densities of particle volumetric concentrations as a function of temperature [15, 27].

#### 2.6.1. Implemented models for viscosity and density

In this part, the correlations that we have implemented for comparison between the experimental data and the predicted data are indicated.

$$\begin{split} \frac{\mu\_{eff}}{\mu\_f} &= \frac{1}{1 - 34.87 (d\_{\mathbb{P}}/d\_f)^{-0.3} q^{1.03}}, \quad \text{where} \\ d\_f &= 0.1 \left( \frac{6M}{N \pi \rho\_{\sharp}} \right)^{1/3} \end{split} \tag{2}$$

It may be noted that once the base fluid is designated, the dimensionless effective viscosity of the nanofluid <sup>μ</sup>ef f μf increases with the decrease in particle diameter and increase in volume concentration.

#### 2.6.2. Pumping power

2.5. Density of nanofluid

where <sup>φ</sup><sup>p</sup> <sup>¼</sup> Vp

2.6. Theoretical models

the mixing theory does not consider effect of temperature.

Nanofluid Surfactant Sedimentation

44 Nanofluid Heat and Mass Transfer in Engineering Problems

TiO2/DW 0.1 %vol. PVP 95 0.317

Al2O3/DW 0.1 %vol. PVP 10 0.033

SiO2/DW 0.1 %vol. PVP 99 9.9

<sup>¼</sup> mf <sup>þ</sup> mp Vf þ Vp

Vf <sup>þ</sup>Vp is the volume fraction of the nanoparticles.

suspensions are presented. Each model is used for specific circumstances.

<sup>ρ</sup>nf <sup>¼</sup> <sup>m</sup> V nf

Table 1. Sedimentation rate of different nanofluids.

Experimental data on density measurements are not sufficient for various nanofluids at varying temperatures in the literature. Therefore, we carried out comprehensive measurements to obtain density and provide data as well as to verify the applicability of Eq. (1) (which is also called as mixing theory) [15] for various nanofluids. Figure 5 shows that density varies with temperature. Therefore, density equation should include temperature as a variable, whereas

after 30 days (%)

0.1 %vol. PEG 44 0.147 0.15 %vol. PEG 42 0.140 0.25 %vol. PEG 35 0.117 0.1 %vol. HTAB 31 0.103

0.1 %vol. PEG 20 0.066 0.15 %vol. PEG 27 0.090 0.25 %vol. PEG 21 0.070 0.1 %vol. HTAB 20 0.066

0.1 %vol. PEG 99 9.9 0.15 %vol. PEG 99 9.9 0.25 %vol. PEG 99 9.9 0.1 %vol. HTAB 99 9.9

Sedimentation rate (ml/day)

<sup>¼</sup> <sup>ρ</sup>fVf <sup>þ</sup> <sup>ρ</sup>pVp Vf þ Vp

In this section, the existing theoretical models and correlations for the viscosity of nanofluid

¼ ð1−φpÞρbf þ φpρp, (1)

The system counted is a forced flow nature. A pump is needed to mingle nanofluids throughout the system. The pump would require electrical energy. It is crucial to comprehend the entire energy needed by the pump to sustain a constant flow across the collector. The pumping power is analysed as follows [28]. The pressure drop throughout the collector is specified by Δp, which is determined from the subsequent equation [29]

$$
\Delta p = f \frac{\rho V^2}{2} \frac{\Delta l}{d} + \mathcal{K} \frac{\rho V^2}{2},\tag{3}
$$

where K is the loss coefficient because of entrance effects, exit effects, bends, elbows, valves, etc. V is the mean flow velocity of nanofluids in the system and is given by

$$V = \frac{\dot{m}}{\rho\_{\text{mf}} \pi D\_{\text{H}}^2 / 4},\tag{4}$$

where DH represents the hydraulic diameter. In present analysis consider DH = pipe diameter (d). ρnf was calculated from Eq. (1). The frictional factor, f, for laminar and turbulent flow, correspondingly, is as follows [30]

$$f = \frac{64}{Re} \quad \text{for laminar flow}$$

$$f = \frac{0.079}{Re^{1/4}} \quad \text{for turbulent flow}$$

The Reynolds number is composed as

$$\text{Re} = \frac{\rho V D\_H}{\mu}.\tag{5}$$

Now, the pumping power can be calculated using Eq. (20)

$$\text{Pumping power} = \left(\frac{\dot{m}}{\rho\_{nf}}\right) \Delta p. \tag{6}$$

#### 2.6.3. Convective heat transfer

Elementary forced convective heat transfer model utters the interactions in the middle of fluid flow and convective heat transfer in terms of correlations between the dimensionless Reynolds, Prandtl and Nusselt numbers (Nu, Re and Pr, respectively). For the regular instances of laminar flows inside a pipe of diameter d, the theoretically developed relations are given as follows [31]:

$$h\_{\eta f} = \frac{q}{T\_{\mathcal{W}} - T\_f} \tag{7}$$

$$Nu\_{\eta f} = \frac{h\_{\eta f} d}{k\_{\eta f}}.\tag{8}$$

The Nusselt number for the laminar flow throughout a circular pipe is the function of the Reynolds and Prandtl numbers and can be resolved by employing Eq. (8) [32]. Nusselt number,


Table 2. Environmental and analysis conditions for the flat plate solar collector.

$$Nu = 0.000972 Re^{1.17} Pr^{\circ\_3} \text{ for } \text{Re} < 2000,\tag{9}$$

where Pr and knf can be expressed as,

Δp ¼ f

etc. V is the mean flow velocity of nanofluids in the system and is given by

<sup>f</sup> <sup>¼</sup> <sup>64</sup>

<sup>f</sup> <sup>¼</sup> <sup>0</sup>:<sup>079</sup>

Now, the pumping power can be calculated using Eq. (20)

correspondingly, is as follows [30]

46 Nanofluid Heat and Mass Transfer in Engineering Problems

The Reynolds number is composed as

2.6.3. Convective heat transfer

follows [31]:

ρV<sup>2</sup> 2 Δl <sup>d</sup> <sup>þ</sup> <sup>K</sup> <sup>ρ</sup>V<sup>2</sup>

<sup>V</sup> <sup>¼</sup> <sup>m</sup>\_ <sup>ρ</sup>nf <sup>π</sup>D<sup>2</sup> <sup>H</sup>=4

where K is the loss coefficient because of entrance effects, exit effects, bends, elbows, valves,

where DH represents the hydraulic diameter. In present analysis consider DH = pipe diameter (d). ρnf was calculated from Eq. (1). The frictional factor, f, for laminar and turbulent flow,

Re for laminar flow

Re<sup>1</sup>=<sup>4</sup> for turbulent flow:

Re <sup>¼</sup> <sup>ρ</sup>VDH

Pumping power <sup>¼</sup> <sup>m</sup>\_

Elementary forced convective heat transfer model utters the interactions in the middle of fluid flow and convective heat transfer in terms of correlations between the dimensionless Reynolds, Prandtl and Nusselt numbers (Nu, Re and Pr, respectively). For the regular instances of laminar flows inside a pipe of diameter d, the theoretically developed relations are given as

> hnf <sup>¼</sup> <sup>q</sup> TW−Tf

Nunf <sup>¼</sup> hnf <sup>d</sup> knf

The Nusselt number for the laminar flow throughout a circular pipe is the function of the Reynolds and Prandtl numbers and can be resolved by employing Eq. (8) [32]. Nusselt number,

ρnf

! Δp:

<sup>2</sup> , (3)

, (4)

<sup>μ</sup> : (5)

: (8)

(6)

(7)

$$Pr = \frac{\mathbb{C}\_{p, \eta^f} \mu\_{\eta^f}}{k\_{\eta^f}} \tag{10}$$

$$\frac{k\_{n\not\!f}}{k\_{\not\!f}} = \frac{k\_p + (SH - 1)k\_{\not\!f} - (SH - 1)q(k\_{\not\!f} - k\_p)}{k\_p + (SH - 1)k\_{\not\!f} + q(k\_{\not\!f} - k\_p)}.\tag{11}$$

In Eq. (12), SH is the shape factor, which is given to be three for the spherical shape of nanoparticle [33] (Table 2).

#### 3. Results and discussions

#### 3.1. Size distribution of the nanoparticles

SEM nano-graphs of (a) TiO2, (b) SiO2 and (c) Al2O3 nanoparticles are presented in Figure 1. Figure 2 presents the TEM images of silica (SiO2) nanoparticles with surfactant (0.14% PEG). Figure 3 presents the TEM images of titania (TiO2) nanoparticles ~21 nm and the surfactant (0.1 %vol. PEG). Figure 4 presents the TEM images of alumina (Al2O3) nanoparticles, average 13 nm and the surfactant (0.1%vol. HTAB).

Figure 6. Size distribution of the nanoparticles in 0.05 %vol. water-based TiO2 nanofluids with surfactants. Inset is the full size distribution.

Figure 7. Size distribution of the nanoparticles in 0.05 %vol. water-based Al2O3 nanofluids with surfactants.

Sizes of the nanoparticles in all the prepared nanofluids were measured using Zetasizer3000HSa (Malvern), and results for the most stable nanofluids are shown in Figures 6 and 7.

The numbers placed at the apexes describe the average particle size as obtained from the machine. It should be noted that the Malvern Nanosizer measures hydrodynamic properties based on the Strokes-Einstein equation, which is expected to be slightly larger than the actual size. Incorporating the findings from those in Figures 6 and 7, it is found that the visual sedimentation rate of TiO2/water nanofluid decreases with increasing surfactant (PEG), whereas particle size increases with the increase in surfactant. For Al2O3/water nanofluid, the sedimentation rate for PEG in different concentrations is approximately similar and the size distribution of the particles is approximately also in the same region. The stability of the stable nanofluids as shown in Figures 6 and 7 was obtained for more than 1 week.

### 3.2. Density of nanofluids

Sizes of the nanoparticles in all the prepared nanofluids were measured using Zetasizer3000HSa

Figure 6. Size distribution of the nanoparticles in 0.05 %vol. water-based TiO2 nanofluids with surfactants. Inset is the full

size distribution.

48 Nanofluid Heat and Mass Transfer in Engineering Problems

(Malvern), and results for the most stable nanofluids are shown in Figures 6 and 7.

Figure 7. Size distribution of the nanoparticles in 0.05 %vol. water-based Al2O3 nanofluids with surfactants.

First, a benchmark test for the density of the base fluid is presented showing excellent agreement with the data presented in the handbook of the American Society of Heating, Refrigerating and Air-Conditioning Engineers (ASHRAE). Next, density measurements of the TiO2, Al2O3 and SiO2 nanofluids over a temperature range of 25–40°C for several particle volume concentrations are presented. These measured results were compared with a widely used theoretical equation, and good agreements between the theoretical equation and measurements were obtained for the TiO2, Al2O3 and SiO2 nanofluids.

We found the density decreasing with the increase in temperature. Eq. (1) is used to make the comparison with our experimental data. Figures 8 and 9 show the experimental values at different temperatures and the theoretical value obtained from Pak and Cho [15]. The trend lines generated from the experimental values clearly certify the linear relationship of density with concentration at a particular temperature. It also validates that the density increases with the increase in concentration. The pattern of the change in density is analysed, and the

Figure 8. Density vs. temperature graph of Al2O3–water nanofluid at different concentrations.

Figure 9. Density vs. temperature graph of TiO2–water nanofluid at different concentrations.

proposed empirical formula is presented below. The effective density of the nanofluids is measured using authors' proposed formula derived based on Einstein's theory. This model considers the changing temperature, volume fraction as well as the size of the nanoparticles. No such models have been presented previously, which also justifies the novelty of this proposed formula.

$$
\rho\_{\rm nf} = (1 - \phi\_p)\rho\_{\rm bf} + \phi\_p \rho\_p + \left(a - \frac{\ln(T)}{100}\right),
\text{where } a = 0.03358.\tag{12}
$$

Moreover, as we examine the average absolute percentage deviation, we notice a systematic increase in percentage deviation with the concentration, as shown in Figures 8–10. Further examination shows a gradual increase in percentage deviation with the temperature. Therefore, the proposed equation may have limitations for some nanofluids. The proposed formula is found valid for up to 2 %vol. concentration. The maximum relative difference between the experimental and theoretical values is 0.3% for Al2O3, 0.44% for TiO2 and 0.28% for SiO2. Figures 8–10 represent the comparative graph for experimental data, Pak and Cho model data and proposed model data (solid line).

#### 3.3. Viscosity

Viscosity of water-based nanofluids with and without surfactant was experimentally obtained. Viscosity of nanofluid was measured using Brookfield viscometer (DV-II + Pro Programmable Viscometer), which was connected with a temperature controlled bath. To verify the accuracy of our equipment and experimental procedure, viscosity of the ethylene glycol and water mixture (60:40 by mass) was measured and compared with the data from the American Society of Heating, Refrigerating and Air-Conditioning Engineers (ASHRAE) handbook [34].

Figure 10. Density vs. temperature graph of SiO2–water nanofluid at different concentrations.

proposed empirical formula is presented below. The effective density of the nanofluids is measured using authors' proposed formula derived based on Einstein's theory. This model considers the changing temperature, volume fraction as well as the size of the nanoparticles. No such models have been presented previously, which also justifies the novelty of this

Moreover, as we examine the average absolute percentage deviation, we notice a systematic increase in percentage deviation with the concentration, as shown in Figures 8–10. Further examination shows a gradual increase in percentage deviation with the temperature. Therefore, the proposed equation may have limitations for some nanofluids. The proposed formula is found valid for up to 2 %vol. concentration. The maximum relative difference between the experimental and theoretical values is 0.3% for Al2O3, 0.44% for TiO2 and 0.28% for SiO2. Figures 8–10 represent the comparative graph for experimental data, Pak and Cho model data

Viscosity of water-based nanofluids with and without surfactant was experimentally obtained. Viscosity of nanofluid was measured using Brookfield viscometer (DV-II + Pro Programmable Viscometer), which was connected with a temperature controlled bath. To verify the accuracy of our equipment and experimental procedure, viscosity of the ethylene glycol and water mixture (60:40 by mass) was measured and compared with the data from the American Society

of Heating, Refrigerating and Air-Conditioning Engineers (ASHRAE) handbook [34].

100 

, where a ¼ 0:03358: (12)

<sup>ρ</sup>nf ¼ ð1−φpÞρbf <sup>þ</sup> <sup>φ</sup>pρ<sup>p</sup> <sup>þ</sup> <sup>a</sup> <sup>−</sup> lnðT<sup>Þ</sup>

Figure 9. Density vs. temperature graph of TiO2–water nanofluid at different concentrations.

proposed formula.

3.3. Viscosity

and proposed model data (solid line).

50 Nanofluid Heat and Mass Transfer in Engineering Problems

Figure 11. Comparison of ASHRAE viscosity values of 60:40 ethylene glycol and water mixture (by mass) and experimental data. 1 cP (centipoise) = 1 mPa s.

Before conducting the density measurement, the equipment must be calibrated. Density measurements of a mixture of 60:40 EG/W by mass were first conducted to confirm the accuracy of our apparatus and the calibration procedure. The results of these measurements and the data from ASHRAE are presented in Figure 11 over a temperature range of 0–80°C. Excellent agreement is observed between the current measurements and the ASHRAE data. Figure 11 shows that the experimental values of viscosity for the ethylene glycol mixture and the ASHRAE data match fairly with a maximum alteration of ±2.0%. The deviations at low shear rates were due to machine error.

Figure 12. Viscosity of TiO2 at different temperatures and concentrations.

Figure 13. Viscosity of TiO2 at different shear rates and concentrations.

Viscosity of TiO2-H2O with different volume concentration and changing temperature is presented in Figure 12. It is observed that the viscosity reduces with the increase in temperature and rises with the increase in volume fraction. From Figure 13 at 0.5 vol.%, titanium

Figure 14. Viscosity of Al2O3 at different temperatures and concentrations.

Viscosity of TiO2-H2O with different volume concentration and changing temperature is presented in Figure 12. It is observed that the viscosity reduces with the increase in temperature and rises with the increase in volume fraction. From Figure 13 at 0.5 vol.%, titanium

Figure 12. Viscosity of TiO2 at different temperatures and concentrations.

52 Nanofluid Heat and Mass Transfer in Engineering Problems

Figure 13. Viscosity of TiO2 at different shear rates and concentrations.

behaves in a non-Newtonian way after 40°C. For the whole temperature range, this nanofluid demonstrates a Newtonian behaviour at 0.05 vol.%, while a non-Newtonian nature is found for 0.1 vol.%.

From Figure 14, the effect of surfactant is observed to increase viscosity. Addition of surfactant augments the viscosity of the nanofluid. Similarly, viscosity tends to increase in Figure 15, for alumina with the rise in the volume concentration as well as addition of surfactants. As opposed to thermal conductivity, viscosity of these nanofluids showed a decreasing trend with the increase in temperature. Furthermore, a nonlinear relation is observed between the viscosity of alumina nanofluid and particle concentration except for 0.05 vol.% and temperatures lower than 40°C.

As it is observed from Figure 15, in the case of nanoparticles with surfactants, viscosity tends to decrease due to augmenting the temperature. This can be explained by the change in the shape of micelles. Worm shapes change to spherical or vesicles which lead to the destruction of network structure and consequently a decrease in viscosity through temperature rise [35]. After the breakdown in network occurs, the attractive forces between the particles become dominant to produce aggregates.

Viscosity of SiO2 at different temperatures and concentrations is illustrated in Figure 16. It is witnessed that the rise in temperature results in reduced viscosity and addition of volume fraction also results in higher viscosity. Figure 17 presents the changes in the viscosity for SiO2 nanofluid. In this case, the viscosity keeps increasing up to the concentration of 0.1 vol.%, but remains unchanged afterwards. This is because SiO2 nanofluid is not stable, and due to the high rate of aggregation, sedimentation of nanoparticles takes place and consequently no

Figure 15. Viscosity of Al2O3 at different shear rate and concentrations.

Figure 16. Viscosity of SiO2 at different temperatures and concentrations.

Figure 17. Viscosity of SiO2 at different shear rate and concentrations.

increment is observed. Adding surfactant causes minor increments in the viscosity of the nanofluid, with no significant changes in the stability of the colloid even after a period of half an hour. Newtonian behaviour is found for this nanofluid at 0.05 vol.% for temperatures over 50°C, but non-Newtonian characteristics are shown beyond this point. A similar trend occurred for 0.5 vol.% and at 60°C. A non-Newtonian behaviour for the entire range of temperature is observed for 0.1 vol.%.

In the previous articles, it was claimed that the viscosity of nanofluids is mainly dependent on the concentration of nanoparticles, properties of base fluid and temperature. However, some also reported about the size of the nanoparticles. From the results of current study, it is observed that for the same concentration (0.5%vol.), viscosities of the nanofluids are different, and TiO2 results in higher viscosity followed by Al2O3 and SiO2 nanofluids for 0.5 vol.% concentration. Therefore, it can be said that viscosity depends on nanoparticles' properties such as the size and density. The available models are not appropriate to forecast or measure accurate viscosity of nanofluids as they are not related to temperature variation and nanoparticles' properties. Therefore, the results suggest the requirement of providing a more generalized viscosity model.

#### 3.4. Pumping power and convective heat transfer

Figure 16. Viscosity of SiO2 at different temperatures and concentrations.

Figure 15. Viscosity of Al2O3 at different shear rate and concentrations.

54 Nanofluid Heat and Mass Transfer in Engineering Problems

The resulting pressure losses are analysed in detail in this section. Figure 18 presents the pumping power and pressure drop with respect to volume fraction and volume flow rate for

Figure 18. Effect of volume fraction and mass flow rate on pumping power (solid line) and pressure drop (dotted line).

Figure 19. Effect of volume fraction and mass flow rate on the Reynolds number and Nusselt number.

the laminar flow, respectively. These two parameters are calculated using Eqs. (4) to (7) and Table 1.

The results show that the friction factor enhances with the increase in volume fraction and flow rate. It is observed from the figures that the friction factors of the nanofluids are almost the same as that of the base fluid under same nanoparticle volume concentrations, therefore resulting in little supplementary pumping power for the process [36–38].

Effect of volume fraction and mass flow rate is presented in Figure 19. These parameters are calculated using Eqs. (6) and (9). With the rising volume fraction of the nanoparticles suspended in water, it is observed that the both the Reynolds number and Nusselt number increase slightly in comparison with water. Reynolds number and Nusselt numbers of the nanofluid are greater compared to water, and the numbers are rising with the growing volume fraction as well as with the rising mass flow rate. This shows that increasing the molecular thermal diffusion due to increasing the nanoparticles volume fraction is the foremost purpose of the heat transfer enhancement for an exact Reynolds number. As a significance, the increase in the Nusselt number is found with a rise in Re. An alike trend was also witnessed by Maïga et al. [39].

The experimental results clearly show that the nanoparticles suspended in water enhance the convective heat transfer coefficient, although the volume fraction of nanoparticles is very low ranging from 0.01 to 0.3 vol.%. The convective heat transfer coefficient of waterbased Al2O3 nanofluids increases with volume fraction of Al2O3 nanoparticles as shown in Figure 20.

The overall heat transfer coefficient of water rises with the rising mass flow rate, with a maximum value of 804 W/m2 compared to that of water, which is 732 W/m2 K with the highest flow rate investigated. Heat transfer coefficient is improved with the addition of volume fraction as well as with the mass flow rate. Heat transfer coefficient is directly proportional to heat rate.

From the above results presented in the figures, it can be concluded that the heat transfer coefficient enhanced up to 15% with the suspended nanoparticles in the base fluid. The

Figure 20. Heat transfer coefficient with respect to changing volume fraction and mass flow rate.

the laminar flow, respectively. These two parameters are calculated using Eqs. (4) to (7) and

Figure 19. Effect of volume fraction and mass flow rate on the Reynolds number and Nusselt number.

Figure 18. Effect of volume fraction and mass flow rate on pumping power (solid line) and pressure drop (dotted line).

56 Nanofluid Heat and Mass Transfer in Engineering Problems

The results show that the friction factor enhances with the increase in volume fraction and flow rate. It is observed from the figures that the friction factors of the nanofluids are almost the

Table 1.

patterns shown by the oxide nanofluids are due to the fact that the addition of nanoparticles tends to enhance the thermal conductivity, density and viscosity of the base fluid. The enhancement in Reynolds number and Nusselt number was observed to be 8.4, 7.6 and 7.5% for TiO2, SiO2 and Al2O3, respectively. Nusselt number was improved by 6.8, 5.5 and 5.4%, for TiO2, SiO2 and Al2O3, respectively. This enhancement results in heat transfer performance. Enhancement in pumping power was observed to be 1.5%.

## 4. Future recommendations

The prospect of using nanofluids in different applications is related to their thermal and flow properties. Measurement of these properties will be performed in near future in order to access the total improvement in energy efficiency where these fluids are being used. Despite some undesirable changes such as rising viscosity or decreasing specific heat, we can consider nanofluids as good thermal fluids. However, researches to obtain a comprehensive formula to determine other physical properties of nanofluid must be continued.

## 5. Conclusion

The effect of volume fraction, temperature and mass flow rate was investigated on density, viscosity, pumping power and convective heat transfer of nanofluids in this article. Stability of nanofluids was obtained using different surfactants. Important conclusions have been attained and summarized as follows:


and 5.4%, for TiO2, SiO2 and Al2O3, respectively, which is greater than that of distilled water.

## Acknowledgements

patterns shown by the oxide nanofluids are due to the fact that the addition of nanoparticles tends to enhance the thermal conductivity, density and viscosity of the base fluid. The enhancement in Reynolds number and Nusselt number was observed to be 8.4, 7.6 and 7.5% for TiO2, SiO2 and Al2O3, respectively. Nusselt number was improved by 6.8, 5.5 and 5.4%, for TiO2, SiO2 and Al2O3, respectively. This enhancement results in heat transfer performance. Enhance-

The prospect of using nanofluids in different applications is related to their thermal and flow properties. Measurement of these properties will be performed in near future in order to access the total improvement in energy efficiency where these fluids are being used. Despite some undesirable changes such as rising viscosity or decreasing specific heat, we can consider nanofluids as good thermal fluids. However, researches to obtain a comprehensive formula to

The effect of volume fraction, temperature and mass flow rate was investigated on density, viscosity, pumping power and convective heat transfer of nanofluids in this article. Stability of nanofluids was obtained using different surfactants. Important conclusions have been attained

1. The density is found decreasing with the increase in temperature. An empirical model is proposed to describe the behaviour. The maximum deviation between the experimental values and the proposed model is 0.3% for Al2O3, 0.44% for TiO2 and 0.3% for SiO2 which

2. Viscosity of our nanofluids increases dramatically with the increase in particle concentra-

3. Friction factor rises with the rising volume fraction. This is due to the increasing density and viscosity with the addition of nanoparticles. Since this effect is slightly higher compared to base fluid, therefore little penalty in pressure drop and pumping power occurs.

4. At a particle volume concentration of 3%, the use of oxide nanofluid gives significantly higher heat transfer characteristics. For example, at the particle volume concentration of 3%, the overall heat transfer coefficient is 804 W/m<sup>2</sup> compared to that of water, which is 732 W/m<sup>2</sup> for a mass flow rate of 1 kg/min, so the overall heat transfer coefficient of the

5. The enhancement in Reynolds number Nusselt number was observed to be 8.4, 7.6 and 7.5% for TiO2, SiO2 and Al2O3, respectively. Nusselt number was improved by 6.8, 5.5

oxides nanofluid is 15% greater than that of distilled water as a base fluid.

ment in pumping power was observed to be 1.5%.

58 Nanofluid Heat and Mass Transfer in Engineering Problems

determine other physical properties of nanofluid must be continued.

is very small and well below the minimum acceptable limit (1%).

tion. Addition of surfactants results in part of the increment.

4. Future recommendations

5. Conclusion

and summarized as follows:

The authors would like to acknowledge the financial support from the Ministry of higher education (MOHE), project no: FP019-2011A & High Impact Research Grant (HIRG) project no: UM.C/HIR/MOHE/ENG/40.

## Nomenclature


## Greek symbols


## Subscripts


## Author details

Zafar Said1,2\* and Rahman Saidur2,3

\*Address all correspondence to: zaffar.ks@gmail.com; zsaid@masdar.ac.ae

1 Department of Sustainable and Renewable Energy Engineering (SREE), University of Sharjah, Sharjah, United Arab Emirates

2 Department of Mechanical Engineering, Faculty of Engineering, University of Malaya, Kuala Lumpur, Malaysia

3 Centre of Research Excellence in Renewable Energy (CoRE-RE), King Fahd University of Petroleum and Minerals (KFUPM), Dhahran, Saudi Arabia

## References


[9] Duangthongsuk, W. and S. Wongwises, Measurement of temperature-dependent thermal conductivity and viscosity of TiO2-water nanofluids. Experimental Thermal and Fluid Science. 33(4)(2009): p. 706–714.

Author details

Zafar Said1,2\* and Rahman Saidur2,3

60 Nanofluid Heat and Mass Transfer in Engineering Problems

Sharjah, Sharjah, United Arab Emirates

Kuala Lumpur, Malaysia

References

\*Address all correspondence to: zaffar.ks@gmail.com; zsaid@masdar.ac.ae

nanoparticles. Applied Physics Letters. 78(6)(2001): p. 718–720.

nanofluids. Journal of Heat Transfer. 125(2003): p. 151.

nal of Experimental Nanoscience. 5(5)(2010): p. 463–472.

Journal of Thermophysics. 30(4)(2009): p. 1213–1226.

nology Letters. 4(1)(2012): p. 105–109.

(2011): p. 024305.

Petroleum and Minerals (KFUPM), Dhahran, Saudi Arabia

1 Department of Sustainable and Renewable Energy Engineering (SREE), University of

2 Department of Mechanical Engineering, Faculty of Engineering, University of Malaya,

3 Centre of Research Excellence in Renewable Energy (CoRE-RE), King Fahd University of

[1] Eastman, J., S. Choi, S. Li, W. Yu, and L. Thompson, Anomalously increased effective thermal conductivities of ethylene glycol-based nanofluids containing copper

[2] Das, S.K., N. Putra, P. Thiesen, and W. Roetzel, Temperature dependence of thermal conductivity enhancement for nanofluids. Journal of Heat Transfer. 125(2003): p. 567.

[3] Xuan, Y. and Q. Li, Investigation on convective heat transfer and flow features of

[4] Reddy, M., V.V. Rao, B. Reddy, S.N. Sarada, and L. Ramesh, Thermal conductivity measurements of ethylene glycol water based TiO2 nanofluids. Nanoscience and Nanotech-

[5] Wamkam, C.T., M.K. Opoku, H. Hong, and P. Smith, Effects of ph on heat transfer nanofluids containing ZrO2 and TiO2 nanoparticles. Journal of Applied Physics. 109

[6] Xie, H., W. Yu, and W. Chen, Mgo nanofluids: higher thermal conductivity and lower viscosity among ethylene glycol-based nanofluids containing oxide nanoparticles. Jour-

[7] Turgut, A., I. Tavman, M. Chirtoc, H. Schuchmann, C. Sauter, and S. Tavman, Thermal conductivity and viscosity measurements of water-based TiO2 nanofluids. International

[8] Li, Y., J.e. Zhou, S. Tung, E. Schneider, and S. Xi, A review on development of nanofluid preparation and characterization. Powder Technology. 196(2)(2009): p. 89–101.


[37] Gherasim, I., G. Roy, C.T. Nguyen, and D. Vo-Ngoc, Heat transfer enhancement and pumping power in confined radial flows using nanoparticle suspensions (nanofluids). International Journal of Thermal Sciences. 50(3)(2011): p. 369–377.

[23] Masoumi, N., N. Sohrabi, and A. Behzadmehr, A new model for calculating the effective viscosity of nanofluids. Journal of Physics D: Applied Physics. 42(5)(2009): p. 055501. [24] Meriläinen, A., A. Seppälä, K. Saari, J. Seitsonen, J. Ruokolainen, S. Puisto, N. Rostedt, and T. Ala-Nissila, Influence of particle size and shape on turbulent heat transfer characteristics and pressure losses in water-based nanofluids. International Journal of Heat and

[25] Sheikholeslami, M. and D. Ganji, Heat transfer of Cu-water nanofluid flow between

[26] Sheikholeslami, M. and D.D. Ganji, Unsteady nanofluid flow and heat transfer in presence of magnetic field considering thermal radiation. Journal of the Brazilian Society of

[27] Vajjha, R.S. and D.K. Das, A review and analysis on influence of temperature and concentration of nanofluids on thermophysical properties, heat transfer and pumping power.

[28] Garg, H.P. and R.K. Agarwal, Some aspects of a pv/t collector/forced circulation flat plate solar water heater with solar cells. Energy Conversion and Management. 36(2)(1995): p.

[29] White, F.M., Fluid mechanics. 5th edition. Boston: McGraw-Hill Book Company. (2003). [30] Kahani, M., S. Zeinali Heris, and S. M. Mousavi. "Effects of curvature ratio and coil pitch spacing on heat transfer performance of Al2O3/water nanofluid laminar flow through helical coils." Journal of Dispersion Science and Technology 34.12 (2013): 1704-1712 [31] Li, Q., Y. Xuan, and J. Wang, Investigation on convective heat transfer and flow features

[32] Owhaib, W. and B. Palm, Experimental investigation of single-phase convective heat transfer in circular microchannels. Experimental Thermal and Fluid Science. 28(2)(2004):

[33] Li, Feng-Chen, et al. "Experimental study on the characteristics of thermal conductivity and shear viscosity of viscoelastic-fluid-based nanofluids containing multiwalled carbon

[34] Handbook, A., 1985 Fundamentals. American Society of Heating, Refrigerating, and Air

[35] Mingzheng, Z., X. Guodong, L. Jian, C. Lei, and Z. Lijun, Analysis of factors influencing thermal conductivity and viscosity in different kinds of surfactant solutions. Experimen-

[36] Mahian, O., A. Kianifar, S.A. Kalogirou, I. Pop, and S. Wongwises, A review of the applications of nanofluids in solar energy. International Journal of Heat and Mass Trans-

International Journal of Heat and Mass Transfer. 55(15)(2012): p. 4063–4078.

Mass Transfer. 61(2013): p. 439–448.

62 Nanofluid Heat and Mass Transfer in Engineering Problems

87–99.

p. 105–110.

parallel plates. Powder Technology. 235(2013): p. 873–879.

Mechanical Sciences and Engineering. 37(3)(2015): p. 895–902.

of nanofluids. Journal of Heat Transfer. 125(2003): p. 151–155.

nanotubes." Thermochimica Acta 556 (2013): 47-53.

Conditioning Engineers, Inc., Atlanta, Georgia. (1985).

tal Thermal and Fluid Science. 36(2012): p. 22–29.

fer. 57(2)(2013): p. 582–594.


#### **Measuring Nanofluid Thermal Diffusivity and Thermal Effusivity: The Reliability of the Photopyroelectric Technique Measuring Nanofluid Thermal Diffusivity and Thermal Effusivity: The Reliability of the Photopyroelectric Technique**

Monir Noroozi and Azmi Zakaria Monir Noroozi and Azmi Zakaria

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/65789

#### **Abstract**

It is important to study nanofluids to understand their extraordinary thermal properties and how the size, concentration and agglomeration of the nanoparticles affect those properties. Photopyroelectric (PPE) technique has been well established in the use of non-destructive measurement of thermal diffusivity and thermal effusivity, by using polyvinylidene fluoride (PVDF) films as sensitive pyroelectric sensors in thermally thick conditions instead of using very thick ceramic sensors. There have been two proposed practical configurations for the PPE technique, the back and the front PPE configurations, to obtain both the thermal diffusivity and effusivity, which are suitable thermal parameters of materials. This PPE technique involves the measurement of thermal waves in the sample due to absorption of optical radiation, by placing a pyroelectric sensor in thermal contact with the sample. This chapter provides a review of the back and the front PPE configurations to determine the thermal diffusivity and effusivity of nanofluids, sample preparation techniques using high-amplitude ultrasonic dispersion and data analysis for metal oxide-based nanofluid materials.

**Keywords:** nanofluids, thermal properties, photopyroelectric technique, thermal diffusivity, thermal effusivity

## **1. Introduction**

Water, ethylene glycol (EG) and oil are universally used for transfer heating, but unfortunately these fluids have extremely poor thermal conductivity; thus, smaller and lighter heat exchanges could be reduced leading to reduced power and the size of the required heat transfer. As can be seen in **Figure 1**, the thermal conductivity of copper is about 700 times more than the water and about 3000 times more than the engine oil.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons © 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

**Figure 1.** Thermal conductivity of the materials (solids and liquids) at room temperature [1].

The study of heat transport in solid-liquid dispersions (colloids) began as early as in the 1970s. The first thermal conductivity enhancement of nanoparticles (NPs) was reported by Masuda et al. [2]. Due to the size of the NPs, they are well suited for use in microsystems. It was observed that the thermal conductivity of an ultrafine suspension of metal oxides in water increased up to 30% for an NP volume fraction of 4.3%. The term 'nanofluid' was first coined by Choi and Eastman [3] when he reported a class of engineered fluids containing nanosized particles dispersed in ethylene glycol that had a thermal conductivity with almost a factor of 2 greater than the base fluid. Nanofluids have widespread usage in industry, including medicine applications, engineering applications, in cooling/heating systems and in micromechanical systems, due to their enhanced thermal management. Nanofluids are able to improve heat transfer and, thus, allow for smaller pumping, heaters and other elements. Nanofluids containing metals and metal oxides have been considered as the next generation of the heat transfer fluids, as they have shown an increase in their effective thermal conductivity compared to their base fluid [4]. Metal oxide nanofluids have potential applications in the main processing industries, such as the materials, chemistry, biomedicine, food and drink, oil and gas industries as they are able to enhance thermal transfer [4, 5].

The preparation of nanofluids is an important parameter in the investigation of the thermal properties of nanofluids. Nanofluids can be prepared by directly dispersed nanopowders in the base fluid, but this method could result in a large degree of NP agglomeration. Therefore, a clear understanding of the effect of concentration, dispersion/aggregation state and particle size on the thermal properties of prepared nanofluid is an essential assay step for particle validation. Data has shown that the thermal conductivity of nanofluids could significantly be increased with an increase in the volume fraction of NPs [6–8]. It is clear that four thermal parameters can be connected by two relationships, α = k/rC and (e = k/√α), where C, α, k and e are the volume specific heat, thermal diffusivity, conductivity and effusivity, respectively. Although the thermophysical properties of nanofluids are intensely researched at present, most of the studies have been focused on measuring the thermal conductivity of the nanofluids.

There are only a few reports on the thermal diffusivity measurements using thermal lens spectrometry and the transient double-hot-wire method [9]. However, these methods often require high temperatures to obtain reasonable signal-to-noise (SNR) ratios, which in turn, could increase the sample temperature and thus increase the measurement error. These techniques are also disadvantageous because they require high-volume samples, long measurement times without according the effect of aggregation time and are expensive. Nowadays, highly sensitive photopyroelectric (PPE) techniques have been designed to measure the thermal properties of samples and are actually one of the several available photothermal techniques [10–12]. The thermal parameters directly resulting from the PPE experiment are usually the 'fundamental' ones: the thermal diffusivity and also the thermal effusivity. The unique features of the back and front PPE techniques to measure the thermal diffusivity and the thermal effusivity of nanofluids in high resolution are that they are not possible with other existing techniques [13]. The advantages of this method include its relatively low cost, and only a small volume of the sample is required with a short measurement time, where the concentration of the nanofluid remains constant in the measurement process, thus making this technique suitable for nanofluids. In the principle by using both the back PPE and front PPE configurations, source of information (amplitude and phase of pyroelectric signal), cavity scanning or frequency scanning, it is possible to obtain both the thermal diffusivity and thermal effusivity. Recently, a new simplified front PPE configuration was designed using a metalized polyvinylidene fluoride (PVDF) sensor in a thermally thick condition instead of using the very thick ceramic sensors of typically 300 μm [14] or 500 μm in size [15] (usually LiTaO3) that have to be coated with gold with a very low chopping frequency facility.

The study of heat transport in solid-liquid dispersions (colloids) began as early as in the 1970s. The first thermal conductivity enhancement of nanoparticles (NPs) was reported by Masuda et al. [2]. Due to the size of the NPs, they are well suited for use in microsystems. It was observed that the thermal conductivity of an ultrafine suspension of metal oxides in water increased up to 30% for an NP volume fraction of 4.3%. The term 'nanofluid' was first coined by Choi and Eastman [3] when he reported a class of engineered fluids containing nanosized particles dispersed in ethylene glycol that had a thermal conductivity with almost a factor of 2 greater than the base fluid. Nanofluids have widespread usage in industry, including medicine applications, engineering applications, in cooling/heating systems and in micromechanical systems, due to their enhanced thermal management. Nanofluids are able to improve heat transfer and, thus, allow for smaller pumping, heaters and other elements. Nanofluids containing metals and metal oxides have been considered as the next generation of the heat transfer fluids, as they have shown an increase in their effective thermal conductivity compared to their base fluid [4]. Metal oxide nanofluids have potential applications in the main processing industries, such as the materials, chemistry, biomedicine, food and drink, oil and

**Figure 1.** Thermal conductivity of the materials (solids and liquids) at room temperature [1].

66 Nanofluid Heat and Mass Transfer in Engineering Problems

The preparation of nanofluids is an important parameter in the investigation of the thermal properties of nanofluids. Nanofluids can be prepared by directly dispersed nanopowders in

gas industries as they are able to enhance thermal transfer [4, 5].

The present chapter provides a review on the thermal properties of nanofluids measured by the PPE technique by using frequency scans of the signals employing PVDF as a pyroelectric (PE) sensor in thermally thick conditions, due to its low cost, light weight, flexibility and sensitivity. The back and front PPE configurations in 'thermally thick' conditions have been implemented to measure the thermal diffusivity and the thermal effusivity of nanofluids (containing Al2O3 and CuO NPs) as a function of the base fluid's particle size. To reduce agglomeration of the NPs, an ultrasonic dispersion technique was utilized for in low concentration of the NPs to produce stable nanofluids, and the effects of sonication type on the stability and thermal properties were investigated.

## **2. Thermal properties of the nanofluids**

## **2.1. Synthesis and stability of nanofluids**

The preparation of nanofluids is an important parameter in the investigation of the thermal properties of nanofluids. The preparation of nanofluid is not merely a simple mixing of liquid and nanopowder, and thus a good dispersion method of dispersing NPs in liquids or a direct production of stable nanofluids is crucial. A good dispersion of NP materials into liquids such as deionized water (DW), ethylene glycol (EG) or oil is needed for producing a stable nanofluid. There are primarily two methods for the synthesis of nanofluids, including the two-step process and the single-step process for the direct synthesis of nanofluids.

The two-step method is achieved by firstly synthesizing dry NPs with the preferred size and shape. In the second step, these particles are carefully mixed into the required base fluid in the desired volume fraction, typically with some additives to enhance the stability of the nanofluids. Thus, the small volume fraction of NPs and proper dispersion techniques are important for the preparation of stabile nanofluids in this technique. Many researchers have reported successful fabrication and testing of nanofluids using the two-step preparation method [16, 17]. Due to the high surface area of the NPs, they have the tendency to aggregate. A large degree of agglomeration in NPs may occur as a result of using this method. Thus, proper dispersion techniques, such as the ultrasonic dispersion technique [18] or the fragmentation process of NPs using laser irradiation, in low concentrations of NPs, are important for the production stability of nanofluids. Another technique to enhance the stability of NPs in fluids is the use of surfactants. To summarize, the optimization of thermal characteristics of nanofluids requires stabile nanofluids, which can be achieved by synthesis and dispersion processes.

## **2.2. Experimental investigation methods**

#### *2.2.1. Thermal diffusivity measurement techniques*

Highly sensitive photothermal methods using a laser as an optical source have been widely used in the thermal diffusivity measurements of nanofluids [19–28]. The photothermal effect in a material is a consequence of the deposition of heat in the sample following absorption of a laser beam and subsequent thermal de-excitations, which results in the indirect heating of the sample. Photoacoustics, photothermal deflection, thermal lens, photothermal radiometry and photopyroelectric methods are some of the techniques commonly used powerful for thermal and optical characterization of materials using lasers. The conventional techniques such as the 'hot-wire', 'laser flash', '3ω-wire method' and 'optical (forced Rayleigh light scattering)' techniques have also been utilized by some researchers [29–33], as seen in **Table 1**.

#### *2.2.2. Thermal effusivity measurement techniques*

Very few studies have been reported on the determination of the thermal effusivity of liquids. During the last two decades, the front PPE configuration and photoacoustic techniques have been used for determining the thermal effusivity [13–15]. For the front detection configuration, Measuring Nanofluid Thermal Diffusivity and Thermal Effusivity: The Reliability of the Photopyroelectric Technique http://dx.doi.org/10.5772/65789 69

**2. Thermal properties of the nanofluids**

The preparation of nanofluids is an important parameter in the investigation of the thermal properties of nanofluids. The preparation of nanofluid is not merely a simple mixing of liquid and nanopowder, and thus a good dispersion method of dispersing NPs in liquids or a direct production of stable nanofluids is crucial. A good dispersion of NP materials into liquids such as deionized water (DW), ethylene glycol (EG) or oil is needed for producing a stable nanofluid. There are primarily two methods for the synthesis of nanofluids, including the

The two-step method is achieved by firstly synthesizing dry NPs with the preferred size and shape. In the second step, these particles are carefully mixed into the required base fluid in the desired volume fraction, typically with some additives to enhance the stability of the nanofluids. Thus, the small volume fraction of NPs and proper dispersion techniques are important for the preparation of stabile nanofluids in this technique. Many researchers have reported successful fabrication and testing of nanofluids using the two-step preparation method [16, 17]. Due to the high surface area of the NPs, they have the tendency to aggregate. A large degree of agglomeration in NPs may occur as a result of using this method. Thus, proper dispersion techniques, such as the ultrasonic dispersion technique [18] or the fragmentation process of NPs using laser irradiation, in low concentrations of NPs, are important for the production stability of nanofluids. Another technique to enhance the stability of NPs in fluids is the use of surfactants. To summarize, the optimization of thermal characteristics of nanofluids requires stabile

Highly sensitive photothermal methods using a laser as an optical source have been widely used in the thermal diffusivity measurements of nanofluids [19–28]. The photothermal effect in a material is a consequence of the deposition of heat in the sample following absorption of a laser beam and subsequent thermal de-excitations, which results in the indirect heating of the sample. Photoacoustics, photothermal deflection, thermal lens, photothermal radiometry and photopyroelectric methods are some of the techniques commonly used powerful for thermal and optical characterization of materials using lasers. The conventional techniques such as the 'hot-wire', 'laser flash', '3ω-wire method' and 'optical (forced Rayleigh light scattering)' tech-

Very few studies have been reported on the determination of the thermal effusivity of liquids. During the last two decades, the front PPE configuration and photoacoustic techniques have been used for determining the thermal effusivity [13–15]. For the front detection configuration,

two-step process and the single-step process for the direct synthesis of nanofluids.

nanofluids, which can be achieved by synthesis and dispersion processes.

niques have also been utilized by some researchers [29–33], as seen in **Table 1**.

**2.1. Synthesis and stability of nanofluids**

68 Nanofluid Heat and Mass Transfer in Engineering Problems

**2.2. Experimental investigation methods**

*2.2.1. Thermal diffusivity measurement techniques*

*2.2.2. Thermal effusivity measurement techniques*


**Table 1.** Summary of experimental studies of thermal diffusivity enhancement.

two schemes were proposed, namely, the configuration with a thermally thin and the optically opaque PVDF sensor [34] and the configuration with a thermally thick and optically semitransparent sensor using LiTaO3 [15]. Balderas-López et al. [35] applied the front PE configuration to perform high precision measurements of thermal effusivity in transparent liquids in a very thermally thick regime.

Esquef et al. [36], in 2006, developed a method consisting essentially of a photoacoustic cell and a PE cell enclosed in a single compact gas analyzer for the measurement of thermal diffusivity and thermal effusivity. Concerning the front configuration, a simplified method to measure both the thermal diffusivity and thermal effusivity of sensor was proposed. For example, Streza et al., in 2009, [37]applied two PE detection configurations, 'back' and 'front', to the calorimetric studies of some liquids (liquid mixtures, magnetic material nanofluids, liquid foodstuffs, etc.). They demonstrated that if the back configuration used the phase of PE signal in the cavity scan method and the front configuration used the frequency scan, both thermal diffusivity and thermal effusivity could be measured. Dadarlat et al.[23], in 2008, measured the thermal diffusivity and thermal effusivity of Fe3O4 and CoFe2O4 nanofluids by using two PPE detection configurations (back and front). Their thermal diffusivity and effusivity measurements were obtained with high accuracy (within 0.5%), and the results were sensitive to changes in the relevant parameters of the nanofluid as the base fluid, concentration and type of NPs. Thus, the front PPE method [13–15, 35–39] was a suitable for accurate and simultaneous measurements of thermal diffusivity and effusivity of nanofluids.

#### **2.3. Theoretical background: photopyroelectric technique**

The photothermal method has been widely used for determining the thermal parameters of materials. This technique typically uses a modulation of laser beam for inducing a thermal-wave (TW) field in the sample. The obtained TW distribution is then detected by various photothermal methods, such as photoacoustics [40], photothermal spectroscopy [41] or PPE techniques [11, 42]. Recently, many useful applications of the photopyroelectric (PPE) effect have been reported with regard to the measurement of both thermal and optical absorption properties of a material [43]. The PPE effect has provided a calorimetric method in which a thin-film PE sensor produces a voltage proportional to its surface temperature change due to the propagation of TWs through a sample in intimate contact with the PVDF sensor. In this technique, the light modulation impinges on the front surface of a sample and the PE sensor, located in good thermal contact with the sample's backside so the PE signal can be measured by performing either a frequency or a cavity length scan. The back and front PPE configurations in 'thermally thick' conditions have been reported to measure the thermal diffusivity and thermal effusivity of a sample [39, 44]. A front PPE technique is the modification of the classical configuration of the PPE technique. In this technique, the TW is introduced to the rear of the PE detection [45]. In the back PPE technique, a very thin metal film is illuminated by a modulated laser beam, and the PE cell consisted of these two parallel walls, one the metallic foil as the TW generator and another the PE film as a PE signal sensor which was placed parallel to the TW generator surface at a fixed cavity length as a function of frequency in frequency scanning and at a given frequency as a function of cavity length in cavity scanning, respectively [46]. This experimental device has allowed the measurement of thermal properties of gas and liquid and liquid mixtures [47]. This expression is typically based on the general theory of PE detection. The experimental results can

then be obtained by using the PPE technique that is designed at different configurations in the measurement of the thermal properties of the nanofluids. The following results and discussion are divided into two parts: (i) the back and (ii) front PPE configurations to measure the thermal diffusivity and thermal effusivity of the nanofluid samples.

## *2.3.1. Back photopyroelectric theory*

two schemes were proposed, namely, the configuration with a thermally thin and the optically opaque PVDF sensor [34] and the configuration with a thermally thick and optically semitransparent sensor using LiTaO3 [15]. Balderas-López et al. [35] applied the front PE configuration to perform high precision measurements of thermal effusivity in transparent liquids in a

Esquef et al. [36], in 2006, developed a method consisting essentially of a photoacoustic cell and a PE cell enclosed in a single compact gas analyzer for the measurement of thermal diffusivity and thermal effusivity. Concerning the front configuration, a simplified method to measure both the thermal diffusivity and thermal effusivity of sensor was proposed. For example, Streza et al., in 2009, [37]applied two PE detection configurations, 'back' and 'front', to the calorimetric studies of some liquids (liquid mixtures, magnetic material nanofluids, liquid foodstuffs, etc.). They demonstrated that if the back configuration used the phase of PE signal in the cavity scan method and the front configuration used the frequency scan, both thermal diffusivity and thermal effusivity could be measured. Dadarlat et al.[23], in 2008, measured the thermal diffusivity and thermal effusivity of Fe3O4 and CoFe2O4 nanofluids by using two PPE detection configurations (back and front). Their thermal diffusivity and effusivity measurements were obtained with high accuracy (within 0.5%), and the results were sensitive to changes in the relevant parameters of the nanofluid as the base fluid, concentration and type of NPs. Thus, the front PPE method [13–15, 35–39] was a suitable for accurate and

simultaneous measurements of thermal diffusivity and effusivity of nanofluids.

The photothermal method has been widely used for determining the thermal parameters of materials. This technique typically uses a modulation of laser beam for inducing a thermal-wave (TW) field in the sample. The obtained TW distribution is then detected by various photothermal methods, such as photoacoustics [40], photothermal spectroscopy [41] or PPE techniques [11, 42]. Recently, many useful applications of the photopyroelectric (PPE) effect have been reported with regard to the measurement of both thermal and optical absorption properties of a material [43]. The PPE effect has provided a calorimetric method in which a thin-film PE sensor produces a voltage proportional to its surface temperature change due to the propagation of TWs through a sample in intimate contact with the PVDF sensor. In this technique, the light modulation impinges on the front surface of a sample and the PE sensor, located in good thermal contact with the sample's backside so the PE signal can be measured by performing either a frequency or a cavity length scan. The back and front PPE configurations in 'thermally thick' conditions have been reported to measure the thermal diffusivity and thermal effusivity of a sample [39, 44]. A front PPE technique is the modification of the classical configuration of the PPE technique. In this technique, the TW is introduced to the rear of the PE detection [45]. In the back PPE technique, a very thin metal film is illuminated by a modulated laser beam, and the PE cell consisted of these two parallel walls, one the metallic foil as the TW generator and another the PE film as a PE signal sensor which was placed parallel to the TW generator surface at a fixed cavity length as a function of frequency in frequency scanning and at a given frequency as a function of cavity length in cavity scanning, respectively [46]. This experimental device has allowed the measurement of thermal properties of gas and liquid and liquid mixtures [47]. This expression is typically based on the general theory of PE detection. The experimental results can

**2.3. Theoretical background: photopyroelectric technique**

very thermally thick regime.

70 Nanofluid Heat and Mass Transfer in Engineering Problems

In the back PPE technique, named the thermal-wave cavity (TWC) technique, a very thin metal film was illuminated by a modulated laser beam, and the PE cell consisted of these two parallel walls, one the metallic foil as a TW generator and another the PVDF film, as a PE signal sensor. The sample(s) converts the modulated laser beam into TWs.

The induced TWs then transmit through the intracavity medium (l) by TW transmission, and the reflection mechanism is detected by the PE sensor (p), as shown in **Figure 2**. TW's arrival at PVDF film gives rise to the surface temperature at the film (*x* = 0) [48]:

$$\Theta\_0 = \frac{\Theta\_{l\_s} T\_{sl} e^{-\upsilon l\_l}}{1 - R\_{ls} R\_{lp} e^{-2\upsilon\_l l\_l}} \tag{1}$$

The transmitted terms of TWs, of the solid-liquid interface at (*x* = − *l*), are given by

$$\Theta\_{\iota\_s} = \frac{Q\_0 T\_{\mathcal{S}^3} e^{-\iota\_s l\_s}}{1 - R\_{\mathcal{S}^2} R\_{sl} e^{-2\iota\_s l\_s}} \tag{2}$$

Hence, the surface temperature of the PVDF can be continued as

$$\Theta\_0 = \frac{Q\_0 T\_{\otimes^s} T\_{sl} e^{-(\sigma\_s l\_s + \sigma\_l l\_l)}}{\left(1 - R\_{s\S} R\_{sl} e^{-2\sigma\_s l\_s}\right)\left(1 - R\_{ls} R\_{lp} e^{-2\sigma\_l l\_l}\right)}\tag{3}$$

where *Qo* is the TW source intensity, σ*<sup>j</sup>* is the complex TW diffusion coefficient *σ<sup>j</sup>* = (1 + *i*)/*μ<sup>j</sup>* and *Tjk* and *Rjk* are TW transmission coefficient and TW reflection coefficient, respectively, at (*j*−*k*)

**Figure 2.** 1D configuration of TWC showed that the thermal waves are partially reflected and transmitted upon striking the boundaries (*g*, *s*, *l*, *p* and *b*) which stand for gas, solid, liquid sample, PVDF film and backing, respectively.

interface, defined as *Tjk* <sup>¼</sup> <sup>2</sup> <sup>1</sup>þ*bjk* ; *Rjk* <sup>¼</sup> <sup>1</sup>−*bjk* <sup>1</sup>þ*bjk* ; *bjk* <sup>¼</sup> *kk kj <sup>α</sup><sup>j</sup> αk* <sup>1</sup>*=*<sup>2</sup> . The following parameters were also defined: *αj*, the thermal diffusivity of *j* (= *g*, *s*, *l*, *p*, *b*); *μ<sup>j</sup>* (= (*αj*/*πf*) 1/2), the thermal diffusion length of *j* at modulation frequency *f* ; and *lj*, the thickness of *j*. The temperature distribution in PVDF film from two parts, the PVDF film-liquid interface and the PVDF filmbacking interface, can be written as

$$\Theta\_p(f, \mathbf{x}) = \Theta\_0 \frac{T\_{lp} \left( e^{-\sigma\_p \mathbf{x}} + R\_{pb} e^{\left(-2\sigma\_p l\_p + \sigma\_p \mathbf{x}\right)} \right)}{\left(1 - R\_{pb} R\_{pl} e^{-2\sigma\_p l\_p} \right)} \tag{4}$$

The average PE voltage is given by

$$V(f, l\_1) = \frac{p}{\varepsilon \varepsilon\_o} < \Theta\_p > = \frac{Q\_o T\_{sl} T\_{lp} p e^{-\varepsilon\_s l\_s} \left(1 - e^{-\varepsilon\_p l\_p}\right) \left(1 + R\_{pb} e^{-\varepsilon\_p l\_p}\right)}{\varepsilon \varepsilon\_o \sigma\_p \left(1 - R\_{sq} R\_{sl} e^{-2\varepsilon\_s l\_s}\right) \left(1 - R\_{pb} R\_{pl} e^{-2\varepsilon\_p l\_p}\right) \left(1 - R\_{ls} R\_{lp} e^{-2\varepsilon\_l l\_l}\right)} \tag{5}$$

If P is the PE coefficient, *lp* is the thickness of the PVDF sensor, *ε* is the dielectric constant of the pyroelectric sensor, *ε*<sup>0</sup> is the permittivity constant of vacuum, *ω* is the angular frequency of modulated light and *Rjk* is the interfacial thermal coefficients. Considering that for thermally thick condition *e*<sup>−</sup>2*σll <<* 1, Equation (5) can be written more simply as [49]

$$\mathcal{V}(f, L) = \text{Constant}\ (f)e^{\neg \vartheta L} \tag{6}$$

$$|\mathcal{V}(f, L)| = \text{Constant} \,(f)e^{-\mathcal{L}/\mu} \tag{6a}$$

$$\phi(f, L) = \text{Constant} \,(f) \text{-} L/\mu \tag{6b}$$

The thermal diffusivity of sample can be obtained by the slope liner fitting from the plot ln (amplitude) and phase versus both cavity length (from the cavity scan) and frequency square (from the frequency scan). In frequency scanning method, the cavity was at a fixed thickness *L*. By plotting the phase and ln(amplitude) as a function of frequency scan, the thermal diffusivity

$$\text{can be determined: } \alpha = \pi L^2 / \left(\frac{\phi}{\sqrt{f}}\right)^2, \alpha = \pi L^2 / \left(\frac{\ln(|V|)}{\sqrt{f}}\right)^2$$

#### *2.3.2. Front photopyroelectric theory*

Usually, the front PPE configuration can be obtained as shown in **Figure 3**, the sensor directly is irradiated and the sample in contact with its rear surface. Then, the cell structure, gas, PE sensor and sample, (g/p/s) becomes another variant of PPE technique.

Under this assumption, for a cell structure (*g*/*p*/*s*), the average PE voltage simplifies to [35]

$$V(f,l) = V\_s \frac{\left(1 - e^{-\sigma\_p l\_p}\right)\left(1 - R\_{ps}e^{-\sigma\_p l\_p}\right)}{\left(1 - R\_{ps}R\_{p\chi}e^{-2\sigma\_p l\_p}\right)}\tag{7}$$

where *Vs*(*f*) = *P*/*εεo*〈*θs*〉 with thermally thick sensor and sample, the signal defined by

Measuring Nanofluid Thermal Diffusivity and Thermal Effusivity: The Reliability of the Photopyroelectric Technique http://dx.doi.org/10.5772/65789 73

**Figure 3.** 1D geometry of the front PPE configuration, for a cell structure (g/p/s).

interface, defined as *Tjk* <sup>¼</sup> <sup>2</sup>

backing interface, can be written as

72 Nanofluid Heat and Mass Transfer in Engineering Problems

The average PE voltage is given by

*p εε<sup>o</sup>*

*V f* ð Þ¼ *; l*<sup>1</sup>

thick condition *e*<sup>−</sup>2*σll*

can be determined: *<sup>α</sup>* <sup>¼</sup> *<sup>π</sup>L*<sup>2</sup>

*2.3.2. Front photopyroelectric theory*

*<sup>=</sup> <sup>φ</sup>*ffiffi *f* p <sup>2</sup>

sensor and sample, (g/p/s) becomes another variant of PPE technique.

*V f*ð Þ¼ *; l Vs*

<sup>1</sup>þ*bjk* ; *Rjk* <sup>¼</sup> <sup>1</sup>−*bjk*

*θp*ð Þ¼ *f ; x θ*<sup>0</sup>

eters were also defined: *αj*, the thermal diffusivity of *j* (= *g*, *s*, *l*, *p*, *b*); *μ<sup>j</sup>* (= (*αj*/*πf*)

*<sup>&</sup>lt; <sup>θ</sup><sup>p</sup> <sup>&</sup>gt;*<sup>¼</sup> *QoTslTlppe*<sup>−</sup>*σsls* <sup>1</sup>−*e*<sup>−</sup>*σplp*

*εεoσ<sup>p</sup>* 1−*RsgRsle*<sup>−</sup>2*σsls*

If P is the PE coefficient, *lp* is the thickness of the PVDF sensor, *ε* is the dielectric constant of the pyroelectric sensor, *ε*<sup>0</sup> is the permittivity constant of vacuum, *ω* is the angular frequency of modulated light and *Rjk* is the interfacial thermal coefficients. Considering that for thermally

*<<* 1, Equation (5) can be written more simply as [49]

Vð Þ¼ *f ; L* Constant ð Þ*f e*

Vð Þj *f ; L* j ¼ Constant ð Þ*f e*

The thermal diffusivity of sample can be obtained by the slope liner fitting from the plot ln (amplitude) and phase versus both cavity length (from the cavity scan) and frequency square (from the frequency scan). In frequency scanning method, the cavity was at a fixed thickness *L*. By plotting the phase and ln(amplitude) as a function of frequency scan, the thermal diffusivity

> *=* ln j jð Þ *<sup>V</sup>* ffiffi *f* p <sup>2</sup>

Usually, the front PPE configuration can be obtained as shown in **Figure 3**, the sensor directly is irradiated and the sample in contact with its rear surface. Then, the cell structure, gas, PE

Under this assumption, for a cell structure (*g*/*p*/*s*), the average PE voltage simplifies to [35]

where *Vs*(*f*) = *P*/*εεo*〈*θs*〉 with thermally thick sensor and sample, the signal defined by

1−*e*<sup>−</sup>*σplp*

� 1−*Rpse*<sup>−</sup>*σplp*

1−*RpsRpge*<sup>−</sup>2*σplp*

�

� (7)

, *<sup>α</sup>* <sup>¼</sup> *<sup>π</sup>L*<sup>2</sup>

<sup>1</sup>þ*bjk* ; *bjk* <sup>¼</sup> *kk*

*Tlp <sup>e</sup>*<sup>−</sup>*σpx* <sup>þ</sup> *Rpbe*ð Þ <sup>−</sup>2*σplp*þ*σpx* 1−*RpbRple*<sup>−</sup>2*σplp*

� 1−*RpbRple*<sup>−</sup>2*σplp*

� <sup>1</sup> <sup>þ</sup> *Rpbe*<sup>−</sup>*σplp*

�

*φ*ð Þ¼ *f ; L* Constant ð Þ*f* −*L=μ* (6b)

�

diffusion length of *j* at modulation frequency *f* ; and *lj*, the thickness of *j*. The temperature distribution in PVDF film from two parts, the PVDF film-liquid interface and the PVDF film-

*kj <sup>α</sup><sup>j</sup> αk* <sup>1</sup>*=*<sup>2</sup>

� (4)

*e*<sup>−</sup>*σlll* 1−*RlsRlpe*<sup>−</sup>2*σlll*

<sup>−</sup>*σ<sup>L</sup>* (6)

<sup>−</sup>*L=<sup>μ</sup>* (6a)

� (5)

. The following param-

1/2), the thermal

$$V(f) = V\_s \left[ 1 - \left( 1 + R\_{sp} \right) e^{-\sigma\_p L\_p} \right] \tag{8}$$

$$R\_{sp} = \left(\mathbf{e}\_s \mathbf{-e}\_p\right) / \left(\mathbf{e}\_s + \mathbf{e}\_p\right) \tag{9}$$

where *es* and *ep* are the thermal effusivity of sample and PE sensor, respectively. The normalizing signal is determined (by using air), and the normalized signal becomes

$$V\_n(f) = 1 - \left(1 + R\_{sp}\right)e^{-\phi\_p L\_p} \tag{10}$$

The normalized phase and amplitude of the signal are defined by

$$\theta = \arctan\left[\frac{A e^{-L\_p/\mu\_p} \sin\left(L\_p/\mu\_p\right)}{1 - A e^{-L\_p/\mu\_p} \cos\left(L\_p/\mu\_p\right)}\right] \tag{10a}$$

$$|V\_n(f)| = \left\{ \left[ \mathbf{A} \sin \left( L\_p / \mu\_p \right) e^{-L\_p / \mu\_p} \right]^2 + \left[ \mathbf{1} \mathbf{-A} \cos \left( L\_p / \mu\_p \right) e^{-L\_p / \mu\_p} \right]^2 \right\}^{1/2} \tag{10b}$$

where *A* =1+ *Rsp*, A the constant can be obtained by optimizing the fit performed on the experimental data with the normalized signal phase by using Equation (10b). It can be shown that from the phase of the normalized signal, one can obtain the thermal effusivity of the liquid sample.

## **3. Experimental method**

#### **3.1. Preparation of metal oxide nanofluids**

Nanofluids were prepared by dispersing pre-synthesized NPs into fluids, and if necessary, in the presence of the stabilizer polyvinylpyrrolidone (PVP) to keep the NPs stable in the fluids. Nanopowders into base fluids were dispersed by stirring, and the suspensions were ultrasonicated by using probe-type or bath-type sonicator. Nanofluids were prepared using Al2O3 (Nanostructured & Amorphous Materials, Inc.), and copper oxide (Sigma-Aldrich) particles were dispersed in various base fluids, DW and EG. To make the desired volume concentration percentage of NPs in the nanofluids, the weights of the base fluid and NPs were measured using an electric balance (Ohaus Adventurer Balances). For example, 3.97g of Al2O3 NPs, which is 1 ml based on the density provided by the vendor, were added to 99g (99 ml) of DW to make 1 % volume concentration of the Al2O3/DW nanofluid. All nanofluids are processed by the same ultrasound power.

## *3.1.1. Ultrasonication dispersion process*

Physical dispersion of powders in a liquid can be achieved by ultrasonic irradiation, either in a bath or by direct irradiation using a probe sonication method. Probe sonication has been studied to determine its effect on the particle characteristics such as the average agglomerate size and the surface charges [50]. Probe sonication is expected to provide higher power to the suspension than the ultrasonic bath as the probe is directly immersed in the suspension. The bottles containing the nanofluid were placed in the ultrasonic bath which was filled with water. The influence of the main parameter of ultrasonication such as the irradiation type (probe and bath) to dispersion and reduced size was observed in the suspension of Al2O3 in low concentration in water, as shown in **Figure 4**. Al2O3 NPs (99%, 11 nm) 0.5 wt% were dissolved in DW and magnetically stirred vigorously until a clear solution was obtained in about 1 h. The suspension was sonicated for 30 min using an ultrasonic probe (VCX 500, 25 kHz, 500 W) and labelled as sample P or using an ultrasonic bath (Powersonic, UB-405, 40 KHz, 350 W), which was labelled as sample B, respectively. As energy transferred into the liquid, the liquid become heated, and a cooling system to control the temperatures between 35 and 40°C was required. This temperature range is favourable to produce a large cavity field that greatly accelerates the integration of NPs in fluids. Unlike the bath sonication that was performed at room temperature, the tip probe sonication had higher amplitudes and, thus, a more effective creation of cavitation and heating. In the case of the ultrasonic probe, the nanoparticle/DW mixture was placed in another larger container filled with ice cubes. This was to prevent the evaporation of fluids caused by elevated temperatures. It was found the

Measuring Nanofluid Thermal Diffusivity and Thermal Effusivity: The Reliability of the Photopyroelectric Technique http://dx.doi.org/10.5772/65789 75

**Figure 4.** Probe (VCX 500, 20 kHz, 500 W) (*left*) and bath (POWERSONIC, UB-405, 40 KHz, 350 W) (*right*) ultrasonic, respectively.

most appropriate power and conditions were obtained using the ultrasonic probe to achieve the highest dispersion and long-term stability.

#### *3.1.2. Sample characterization*

where *A* =1+ *Rsp*, A the constant can be obtained by optimizing the fit performed on the experimental data with the normalized signal phase by using Equation (10b). It can be shown that from the phase of the normalized signal, one can obtain the thermal effusivity of the liquid

Nanofluids were prepared by dispersing pre-synthesized NPs into fluids, and if necessary, in the presence of the stabilizer polyvinylpyrrolidone (PVP) to keep the NPs stable in the fluids. Nanopowders into base fluids were dispersed by stirring, and the suspensions were ultrasonicated by using probe-type or bath-type sonicator. Nanofluids were prepared using Al2O3 (Nanostructured & Amorphous Materials, Inc.), and copper oxide (Sigma-Aldrich) particles were dispersed in various base fluids, DW and EG. To make the desired volume concentration percentage of NPs in the nanofluids, the weights of the base fluid and NPs were measured using an electric balance (Ohaus Adventurer Balances). For example, 3.97g of Al2O3 NPs, which is 1 ml based on the density provided by the vendor, were added to 99g (99 ml) of DW to make 1 % volume concentration of the Al2O3/DW nanofluid. All nanofluids are

Physical dispersion of powders in a liquid can be achieved by ultrasonic irradiation, either in a bath or by direct irradiation using a probe sonication method. Probe sonication has been studied to determine its effect on the particle characteristics such as the average agglomerate size and the surface charges [50]. Probe sonication is expected to provide higher power to the suspension than the ultrasonic bath as the probe is directly immersed in the suspension. The bottles containing the nanofluid were placed in the ultrasonic bath which was filled with water. The influence of the main parameter of ultrasonication such as the irradiation type (probe and bath) to dispersion and reduced size was observed in the suspension of Al2O3 in low concentration in water, as shown in **Figure 4**. Al2O3 NPs (99%, 11 nm) 0.5 wt% were dissolved in DW and magnetically stirred vigorously until a clear solution was obtained in about 1 h. The suspension was sonicated for 30 min using an ultrasonic probe (VCX 500, 25 kHz, 500 W) and labelled as sample P or using an ultrasonic bath (Powersonic, UB-405, 40 KHz, 350 W), which was labelled as sample B, respectively. As energy transferred into the liquid, the liquid become heated, and a cooling system to control the temperatures between 35 and 40°C was required. This temperature range is favourable to produce a large cavity field that greatly accelerates the integration of NPs in fluids. Unlike the bath sonication that was performed at room temperature, the tip probe sonication had higher amplitudes and, thus, a more effective creation of cavitation and heating. In the case of the ultrasonic probe, the nanoparticle/DW mixture was placed in another larger container filled with ice cubes. This was to prevent the evaporation of fluids caused by elevated temperatures. It was found the

sample.

**3. Experimental method**

**3.1. Preparation of metal oxide nanofluids**

74 Nanofluid Heat and Mass Transfer in Engineering Problems

processed by the same ultrasound power.

*3.1.1. Ultrasonication dispersion process*

Various techniques have been applied to analyze the chemical and physical properties of the prepared nanofluids. The morphologies of the deposits were studied using an S-4700 field emission scanning electron microscope (FESEM) (Hitachi, Tokyo, Japan), operating at 5.0 kV. The size, distribution and morphology of the synthesized NPs were determined via TEM (H-7100, Hitachi, Tokyo, Japan), and the particle size distributions were determined using the UTHSCSA Image Tool software (version 3.00; UTHSCSA Dental Diagnostic Science, San Antonio, TX). In the characterization of the prepared nanofluids, the particle size and size distribution of spherical NPs in colloidal form were measured by the Nanophox particle size analyzer (Sympatec GmbH System-Partikel-Technik). This equipment is based on the principle of dynamic light scattering, which provides mean particle size as well as particle size distribution (PSD). The surface plasmon or absorption maximum in the colloidal solution spectrum provides information on the average size of the particles, and a UV-Vis spectrophotometer (Shimadzu-UV1650PC) was used to measure the absorption spectra at room temperature for wavelength range 200–800 nm.

#### **3.2. Experimental setup of the photopyroelectric methods**

The systematic experiments were to investigate the accuracy of thermal diffusivity and effusivity by the PE method using the back PPE and front PPE configurations as a special case of the different structures of the PE cell. The basic design of the analytical instrument consisted of only a laser, a TW generator and a PVDF, PE sensor. The thermal diffusivity and thermal effusivity of the nanofluids were obtained with both the back and front PPE configurations.

## *3.2.1. Back PPE configuration and experimental conditions*

The schematic diagram of the experimental setup is shown in **Figure 5**. Here, a 52 μm PVDF film PE sensor (MSI DT1-028K/L), which is an excellent choice for signal detection due to its low cost, low weight, flexibility and sensitivity, was used in signal detection [51]. A 30 mW He-Ne laser (05-HR-991) was modulated by an optical chopper (SR540) before illumination on copper foil of 50μm thickness and 0.8cm diameter. To maximize its optical to thermal conversion efficiency, a very thin layer of carbon soot was coated on the surface of the foil. When the laser was illuminated on the copper foil, TWs were generated in this foil.

In the cell, the initiated TWs propagated across the fluid and reached the PE sensor. Since the PVDF film is very flexible and any film wrap can cause a change of signal, it was fixed with silicon glue to a Perspex substrate. On its top side, a plastic ring of 1 cm diameter was glued to it to act as the sample container. A small volume of the liquid sample, *<*0.1 cm<sup>3</sup> , was simply filled in the inner side of the ring, with a sample depth or thickness of around 1 mm. The PE signal generated by PVDF sensor was analyzed by using a lock-in amplifier (SR-530) to produce the PE amplitude and phase. The electromagnetic noise was reduced by eliminating all the ground loops via proper grounding.

The typical PE signal was measured with respect to time to investigate the steady state of the signal. The sensitivity of the back PPE technique was tested by maintaining the cavity length at about 100 μm, and the PE signal was recorded over 300 s. The experiment was carried out with

**Figure 5.** Schematic diagram of back PPE configuration [51].

a single drop of DW. In **Figure 6**, it can be observed that the PE signal was quite stable around 1.49 ×10−<sup>3</sup> V with a standard deviation of 5.12 ×10−<sup>5</sup> V.

A frequency scan was carried out as it was important to choose the optimal value of frequency for thermophysical measurements of nanofluids. **Figure 7** displays the frequency behaviour of the signal amplitude obtained from the distilled water as a reference sample with known thermal properties. It can be observed that at frequencies above 7 Hz, the effect of thermally thick regime become obvious. The amplitude of the PE signal of the sample decreased exponentially to zero with increasing modulation of the frequency in the thermally thick regime. Therefore, the frequency range between 7 and 30 Hz was used for the frequency scan, which is shown in **Figure 5**. The noise level in the present setup was about 75 μV. The ln(amplitude) of the PE signal as a function of *f* 1/2 in this useful frequency range was linear. The thermal diffusivity was calculated from the slope of the linear part of the logarithmic amplitude of the signal curves by using Equation (6b).

## *3.2.2. Front photopyroelectric configuration*

case of the different structures of the PE cell. The basic design of the analytical instrument consisted of only a laser, a TW generator and a PVDF, PE sensor. The thermal diffusivity and thermal effusivity of the nanofluids were obtained with both the back and front PPE config-

The schematic diagram of the experimental setup is shown in **Figure 5**. Here, a 52 μm PVDF film PE sensor (MSI DT1-028K/L), which is an excellent choice for signal detection due to its low cost, low weight, flexibility and sensitivity, was used in signal detection [51]. A 30 mW He-Ne laser (05-HR-991) was modulated by an optical chopper (SR540) before illumination on copper foil of 50μm thickness and 0.8cm diameter. To maximize its optical to thermal conversion efficiency, a very thin layer of carbon soot was coated on the surface of the foil. When the

In the cell, the initiated TWs propagated across the fluid and reached the PE sensor. Since the PVDF film is very flexible and any film wrap can cause a change of signal, it was fixed with silicon glue to a Perspex substrate. On its top side, a plastic ring of 1 cm diameter was glued to

filled in the inner side of the ring, with a sample depth or thickness of around 1 mm. The PE signal generated by PVDF sensor was analyzed by using a lock-in amplifier (SR-530) to produce the PE amplitude and phase. The electromagnetic noise was reduced by eliminating

The typical PE signal was measured with respect to time to investigate the steady state of the signal. The sensitivity of the back PPE technique was tested by maintaining the cavity length at about 100 μm, and the PE signal was recorded over 300 s. The experiment was carried out with

, was simply

*3.2.1. Back PPE configuration and experimental conditions*

76 Nanofluid Heat and Mass Transfer in Engineering Problems

all the ground loops via proper grounding.

**Figure 5.** Schematic diagram of back PPE configuration [51].

laser was illuminated on the copper foil, TWs were generated in this foil.

it to act as the sample container. A small volume of the liquid sample, *<*0.1 cm<sup>3</sup>

urations.

In the new section design, a simplified front PPE configuration was setup using the similar PE sensor. The metalized PVDF sensor was used as an optically opaque sensor and in a thermally thick regime for both the sensor and sample, instead of a very thick sensor (usually LiTaO3) in

**Figure 6.** PVDF signals recorded versus time for distilled water; the baseline is a steady-state signal in various times.

**Figure 7.** Frequency behaviour of the amplitude of signal obtained from the distilled water [52].

the conventional front PPE configuration [53, 57]. The radiation from the similar He-Ne laser was modulated by the mechanical chopper, and the signal from the PVDF sensor was processed with the lock-in amplifier. The liquid sample was simply filled into a plastic ring and glued on the rear side of the sensor, and the overall thickness was about 1 mm. As the sample thickness decreased, the contribution from the reflected TW power increased. A schematic view of the experimental setup of the front PPE is presented in **Figure 8**. The scan was performed in thermally thick conditions in a frequency range of 7 to 30Hz with 1Hz steps. The S/N ratio of the experiment was more than 750. The LabVIEW software was used to capture the amplitude and phase data, and the data were analyzed using Microcal Origin 8. The following procedure describes the steps from the recorded experimental data up to obtaining thermal effusivity of the nanofluid by fitting the normalized phase of the PE signal versus frequency scan to obtain thermal effusivity (*ep* or *es*).

**Figure 9 (a–b)** displays the frequency behaviour of the normalized amplitude and phase of signal obtained from DW as a reference sample of known thermal effusivity, 1600Ws1/2m<sup>−</sup><sup>2</sup> K−<sup>1</sup> [53], to determine the thermal effusivity of the PVDF sensor. In **Figure 9 (a–b)**, the frequency range between 7 and 30 Hz was the best choice for fitting to find the parameters. However, here, the phase was used instead of the amplitude because it produced more accurate results as it did not change with source intensity fluctuations.

Measuring Nanofluid Thermal Diffusivity and Thermal Effusivity: The Reliability of the Photopyroelectric Technique http://dx.doi.org/10.5772/65789 79

**Figure 8.** Schematic view of experimental setup of front PPE configuration [54].

the conventional front PPE configuration [53, 57]. The radiation from the similar He-Ne laser was modulated by the mechanical chopper, and the signal from the PVDF sensor was processed with the lock-in amplifier. The liquid sample was simply filled into a plastic ring and glued on the rear side of the sensor, and the overall thickness was about 1 mm. As the sample thickness decreased, the contribution from the reflected TW power increased. A schematic view of the experimental setup of the front PPE is presented in **Figure 8**. The scan was performed in thermally thick conditions in a frequency range of 7 to 30Hz with 1Hz steps. The S/N ratio of the experiment was more than 750. The LabVIEW software was used to capture the amplitude and phase data, and the data were analyzed using Microcal Origin 8. The following procedure describes the steps from the recorded experimental data up to obtaining thermal effusivity of the nanofluid by fitting the normalized phase of the PE signal versus

**Figure 7.** Frequency behaviour of the amplitude of signal obtained from the distilled water [52].

**Figure 9 (a–b)** displays the frequency behaviour of the normalized amplitude and phase of signal obtained from DW as a reference sample of known thermal effusivity, 1600Ws1/2m<sup>−</sup><sup>2</sup>

[53], to determine the thermal effusivity of the PVDF sensor. In **Figure 9 (a–b)**, the frequency range between 7 and 30 Hz was the best choice for fitting to find the parameters. However, here, the phase was used instead of the amplitude because it produced more accurate results as

K−<sup>1</sup>

frequency scan to obtain thermal effusivity (*ep* or *es*).

78 Nanofluid Heat and Mass Transfer in Engineering Problems

it did not change with source intensity fluctuations.

**Figure 9.** Frequency behaviour of the normalized (a) amplitude and (b) phase measured for the PVDF sensor with water as substrate. Solid lines are the best fit of amplitude to Equation (10b) and phase to Equation (10a), respectively [54].

## **4. Results and discussion**

#### **4.1. Effect of ultrasonication on the thermal diffusivity of Al2O3 nanofluids**

#### *4.1.1. Sample preparation and characterization*

In the study, the influence of ultrasonication on the thermal diffusivity of low concentration of Al2O3 nanofluids in two sizes of NPs, size A (11 nm) and size B (30 nm) in water were investigated. Each nanofluid sample 0.125%, 0.25% and 0.5 wt% was dissolved in DW and magnetically stirred vigorously until a clear solution was observed after about 1 h. Two different ultrasonic systems were chosen to disperse the NPs in DW for 30 min using the bath sonicator, called sample B, or the probe sonicator, called sample P, respectively. The total

**Figure 10.** Particle size distributions determined using the Nanophox analyzer of Al2O3 particles in the nanofluids after three measurements at 15 min intervals, for NPs of size A (a,b) and B (c,d) prepared using the bath (a,c) and probe (b,d) sonicators. PDS just after sonication (□), after 15 min (○) and after 30 min (Δ) [51].

amount of energy delivered to the sample was constant for both sonicators. After each ultrasonication, the mean particle size was measured using the Nanophox particle size analyzer (Sympatec GmbH, D-38678), and an average was taken from at least three measurements. The morphology of the alumina clusters was characterized by TEM.

**4. Results and discussion**

*4.1.1. Sample preparation and characterization*

80 Nanofluid Heat and Mass Transfer in Engineering Problems

**4.1. Effect of ultrasonication on the thermal diffusivity of Al2O3 nanofluids**

In the study, the influence of ultrasonication on the thermal diffusivity of low concentration of Al2O3 nanofluids in two sizes of NPs, size A (11 nm) and size B (30 nm) in water were investigated. Each nanofluid sample 0.125%, 0.25% and 0.5 wt% was dissolved in DW and magnetically stirred vigorously until a clear solution was observed after about 1 h. Two different ultrasonic systems were chosen to disperse the NPs in DW for 30 min using the bath sonicator, called sample B, or the probe sonicator, called sample P, respectively. The total

**Figure 10.** Particle size distributions determined using the Nanophox analyzer of Al2O3 particles in the nanofluids after three measurements at 15 min intervals, for NPs of size A (a,b) and B (c,d) prepared using the bath (a,c) and probe (b,d)

sonicators. PDS just after sonication (□), after 15 min (○) and after 30 min (Δ) [51].

**Figure 10** shows the particle size distribution (PSD) and the hydrodynamic diameters of the Al2O3 NPs in the nanofluids. It can be seen that the NP agglomerates were only slightly broken up by the bath sonicator (**Figure 10 a, c**); however, the large agglomerates were completely broken down by the probe sonicator (**Figure 10 b, d**). The smallest mean PSD was recorded for samples with small particle size, A, prepared using probe sonication. There was no significant change in the mean particle size for the three measurements (**Figure 10 b**). However, in all cases, the NPs agglomerated in water were not completely broken up using sonication, whether by using the bath or probe sonicators.

The UV-Vis absorption spectra of the Al2O3 NPs prepared in DW, using bath- and probe-type ultrasonicator for the dispersion of the particles, are shown in **Figure 11**. The increase of absorption behaviour of the sample prepared using the ultrasonic probe could be attributed

**Figure 11.** UV-Vis absorption spectra of the Al2O3 nanofluids, in the treatment by bath (sample B) and probe (sample P) sonication, respectively.

to the increase in quantity of Al2O3 NPs assembled within the fluid, as proven by the Nanophox results. This indicated that for the sample prepared using the ultrasonic probe, the absorption of nanofluids was at a maximum; therefore, the stability of the nanofluid was high, and the agglomeration between particles was reduced [50].

The effect of ultrasonic irradiation on the synthesized Al2O3 nanofluids was analyzed by TEM. **Figure 12** shows the TEM images of Al2O3 NPs of two sizes A and B prepared in DW without sonication (a,b) and prepared using the bath (c,d) and the probe (e,f) sonicators, for NPs of size A (a,c,e) and B (b,d,f), respectively. It can be seen that most of the NPs were spherical and were connected to each other to form a porous structure. The size of the NPs was well distributed in both ultrasonic sonicators, as shown in **Figure 12** (c,d). However, the probe sonicator was more effective in reducing particle sizes to below 100 nm, as shown in **Figure 12** (e,f). As previously mentioned, in all nanofluids, the measured particle sizes were larger than the nominal particle sizes claimed by the vendor. This indicated that the oxide NPs agglomerated in water and the hard aggregates could not be broken down into individual NPs under these operating conditions or even with very high-energy input [18].

## *4.1.2. Enhancement of thermal diffusivity*

Before measuring thermal diffusivity of the nanofluids, the PPE setup was tested with DW as the base fluid. The recorded *α* value was (1.431 ± 0.030)×10−<sup>3</sup> cm<sup>2</sup> /s, which differed by less than 2% from the values reported in literature [49]. The thermal diffusivity of the Al2O3 nanofluids prepared using different sonication techniques at different concentrations of NPs of sizes A and B was obtained. **Figure 13** shows the typical behaviour of the (a) amplitude of the PE signal versus the frequency and (b) the plot of ln(amplitude) of PE signal versus square root of frequency. The thermal diffusivity can be calculated from the fitting slope of the linear part of the signal curves. The thermal diffusivity data are summarized in **Tables 2** and **3**. The data indicated that the thermal diffusivity of the Al2O3 nanofluids was higher than that of water.

The data also proved that the thermal diffusivity enhancement was greater for the smallersized NPs. This was because smaller particles have larger surface area (the heat transfer area), thus increasing the thermal diffusivity [55]. Hence, smaller particles helped form a stable nanofluid, and the probe sonicator had a substantial effect on the thermal diffusivity. At a given particle concentration, the thermal diffusivity enhancement was greater for the probe than the bath sonicator. This was because the NPs were more widely dispersed in water through probe sonication, generating a larger NP surface area and thus increasing the thermal diffusivity. The beneficial effect of using the probe sonicator on the thermal diffusivity of Al2O3 nanofluids was more pronounced at high particle concentrations and small particle sizes. For example, the greatest enhancement of thermal diffusivity of 6% was achieved for the probe sonicator with NPs of size A at a concentration of 0.5 wt%. The smallest enhancement was about ≈1% for NPs of size B at 0.125 wt% with the bath sonicator. These findings are possibly attributable to the rapid particle clustering at a high concentration, which necessitates using a more powerful sonication tool to break up large agglomerates into smaller-sized particles.

Measuring Nanofluid Thermal Diffusivity and Thermal Effusivity: The Reliability of the Photopyroelectric Technique http://dx.doi.org/10.5772/65789 83

to the increase in quantity of Al2O3 NPs assembled within the fluid, as proven by the Nanophox results. This indicated that for the sample prepared using the ultrasonic probe, the absorption of nanofluids was at a maximum; therefore, the stability of the nanofluid was high,

The effect of ultrasonic irradiation on the synthesized Al2O3 nanofluids was analyzed by TEM. **Figure 12** shows the TEM images of Al2O3 NPs of two sizes A and B prepared in DW without sonication (a,b) and prepared using the bath (c,d) and the probe (e,f) sonicators, for NPs of size A (a,c,e) and B (b,d,f), respectively. It can be seen that most of the NPs were spherical and were connected to each other to form a porous structure. The size of the NPs was well distributed in both ultrasonic sonicators, as shown in **Figure 12** (c,d). However, the probe sonicator was more effective in reducing particle sizes to below 100 nm, as shown in **Figure 12** (e,f). As previously mentioned, in all nanofluids, the measured particle sizes were larger than the nominal particle sizes claimed by the vendor. This indicated that the oxide NPs agglomerated in water and the hard aggregates could not be broken down into individual NPs under these operating condi-

Before measuring thermal diffusivity of the nanofluids, the PPE setup was tested with DW as

2% from the values reported in literature [49]. The thermal diffusivity of the Al2O3 nanofluids prepared using different sonication techniques at different concentrations of NPs of sizes A and B was obtained. **Figure 13** shows the typical behaviour of the (a) amplitude of the PE signal versus the frequency and (b) the plot of ln(amplitude) of PE signal versus square root of frequency. The thermal diffusivity can be calculated from the fitting slope of the linear part of the signal curves. The thermal diffusivity data are summarized in **Tables 2** and **3**. The data indicated that the thermal diffusivity of the Al2O3 nanofluids was higher than that of water. The data also proved that the thermal diffusivity enhancement was greater for the smallersized NPs. This was because smaller particles have larger surface area (the heat transfer area), thus increasing the thermal diffusivity [55]. Hence, smaller particles helped form a stable nanofluid, and the probe sonicator had a substantial effect on the thermal diffusivity. At a given particle concentration, the thermal diffusivity enhancement was greater for the probe than the bath sonicator. This was because the NPs were more widely dispersed in water through probe sonication, generating a larger NP surface area and thus increasing the thermal diffusivity. The beneficial effect of using the probe sonicator on the thermal diffusivity of Al2O3 nanofluids was more pronounced at high particle concentrations and small particle sizes. For example, the greatest enhancement of thermal diffusivity of 6% was achieved for the probe sonicator with NPs of size A at a concentration of 0.5 wt%. The smallest enhancement was about ≈1% for NPs of size B at 0.125 wt% with the bath sonicator. These findings are possibly attributable to the rapid particle clustering at a high concentration, which necessitates using a more powerful sonication tool to break up large agglomerates into smaller-sized

/s, which differed by less than

and the agglomeration between particles was reduced [50].

82 Nanofluid Heat and Mass Transfer in Engineering Problems

tions or even with very high-energy input [18].

the base fluid. The recorded *α* value was (1.431 ± 0.030)×10−<sup>3</sup> cm<sup>2</sup>

*4.1.2. Enhancement of thermal diffusivity*

particles.

**Figure 12.** TEM images of Al2O3 NPs prepared in DW without (a,b) and with (c,d) the bath sonicator and (e,f) probe sonicators, for NPs of size 11 nm (a,c,e) and 30 nm (b,d,f) [51].

**Figure 13.** (a) Amplitude of the PE signal as a function of the chopping frequency *f* and (b) natural log of the amplitude of the PE signal as a function of the square root of the chopping frequency and its fitting by using Equation (6b), for one of the samples [51].


**Table 2.** Thermal diffusivity of Al2O3 nanofluids, NP type A (11 nm), prepared by using different sonication techniques at different NP concentrations [51].


**Table 3.** Thermal diffusivity of Al2O3 nanofluids, NP type B (30 nm), prepared by using different sonication techniques at different NP concentrations [51].

## **4.2. Effect of base fluids on thermal effusivity of nanofluids**

## *4.2.1. Sample preparation and characterization*

**Figure 13.** (a) Amplitude of the PE signal as a function of the chopping frequency *f* and (b) natural log of the amplitude of the PE signal as a function of the square root of the chopping frequency and its fitting by using Equation (6b), for one of

**Table 2.** Thermal diffusivity of Al2O3 nanofluids, NP type A (11 nm), prepared by using different sonication techniques at

**Table 3.** Thermal diffusivity of Al2O3 nanofluids, NP type B (30 nm), prepared by using different sonication techniques at

**Thermal diffusivity enhancement %**

**Thermal diffusivity enhancement %**

**0.125** 1.446 ± 0.003 0.9 1.448 ± 0.001 1.1 **0.25** 1.461 ± 0.002 2.1 1.473 ± 0.002 2.9 **0.5** 1.478 ± 0.004 3.2 1.498 ± 0.003 4.6

**0.125** 1.476 ± 0.002 3.1 1.482 ± 0.004 3.5 **0.25** 1.483 ± 0.003 3.5 1.494 ± 0.002 4.3 **0.5** 1.492 ± 0.004 4.2 1.515 ± 0.003 5.8

**Bath Probe**

**Bath Probe**

**Thermal diffusivity** cm<sup>2</sup> ð Þ *<sup>=</sup>*<sup>s</sup> <sup>10</sup>�<sup>3</sup>

**Thermal diffusivity** cm<sup>2</sup> ð Þ *<sup>=</sup>*<sup>s</sup> <sup>10</sup>�<sup>3</sup>

**Thermal diffusivity enhancement %**

**Thermal diffusivity enhancement %**

the samples [51].

**Concentration wt%**

**Concentration wt%**

different NP concentrations [51].

different NP concentrations [51].

**Thermal diffusivity** cm<sup>2</sup> ð Þ *<sup>=</sup>*<sup>s</sup> <sup>10</sup>�<sup>3</sup>

84 Nanofluid Heat and Mass Transfer in Engineering Problems

**Thermal diffusivity** cm<sup>2</sup> ð Þ *<sup>=</sup>*<sup>s</sup> <sup>10</sup>�<sup>3</sup>

The thermal effusivity of Al2O3 (11 nm) and CuO (50 nm) NPs dispersed in three different base fluids, DW, EG and olive oil, in the presence of the stabilizer polyvinylpyrrolidone (PVP) was investigated. In each nanofluid, sample 0.125 wt% was dissolved in each base fluid and magnetically stirred vigorously until a clear solution were observed after about 1 h. The solution was then sonicated by probe sonicator for 30 min to ensure a uniform dispersion of NPs in the fluids. TEM was employed to obtain the morphology of the CuO and Al2O3 particles and to determine the average particle size.

**Figure 14** shows the TEM images and their corresponding size distributions of (a) CuO and (b) Al2O3 nanofluids prepared in water. It can be seen that most of the NPs were well dispersed and some agglomerates were present. The CuO and Al2O3 NP sizes were about 52.3 ± 4.2 nm and 7.5 ± 2.5 nm, respectively. These commercial NPs determined from TEM images were

**Figure 14.** TEM images and their size distributions of (a) CuO particles and (b) Al2O3 nanofluids prepared in water [56].

**Figure 15.** The experimental data and the best fit of the PE normalized phase versus modulation frequency in (a) Al2O3/ olive oil and (b) CuO/olive oil, obtained by using Equation (10a) [56].

slightly different from those reported by the vendors. This indicated that some of particles in each sample were aggregated with some uniform size distribution as reported by them.

#### *4.2.2. Thermal effusivity measurements*

**Figure 15.** The experimental data and the best fit of the PE normalized phase versus modulation frequency in (a) Al2O3/

olive oil and (b) CuO/olive oil, obtained by using Equation (10a) [56].

86 Nanofluid Heat and Mass Transfer in Engineering Problems

**Figure 15** shows the PE-normalized phase versus modulation frequency in (a) Al2O3/olive oil and (b) CuO/olive oil. It was observed that from this fit the values of A from Equation (9) were obtained at (1.112 ± 0.005) and (1.175 ± 0.006), corresponding to the values of thermal effusivity of Al2O3/olive oil (0.614 ± 0.003) × 10<sup>3</sup> Ws1/2m<sup>−</sup><sup>2</sup> K−<sup>1</sup> and CuO/olive oil (0.697 ± 0.003) × 10<sup>3</sup> Ws1/ 2 m−<sup>2</sup> K−<sup>1</sup> , respectively, obtained by using Equation (10a). The values of thermal effusivity measured for all nanofluids and their comparison with pure solvents are summarized in **Table 3** and **Figure 16**. The comparisons indicated that the thermal effusivity of the various base fluids mixed with NPs in the presence of PVP were reduced as compared to pure fluids, possibly due to the effect of the surfactant that inhibited the thermal effusivity of the nanofluids [56]. The results also showed that the base fluids had more influence on effusivity than the NPs. The relative standard deviation for measuring the thermal effusivity of

**Figure 16.** Thermal effusivity of Al2O3 and CuO nanofluids with (DW, EG and olive oil) and their pure solvents [56].


nanofluids was below 2%, as shown in **Table 4**. Therefore, the front PPE technique is a promising high-accuracy alternative for this measurement.

**Table 4.** Experimental thermal effusivity of Al2O3 and CuO nanofluids and their pure solvents and their literature values [56].

## **5. Conclusions**

The PPE technique is a sensitive method to measure the thermal properties of nanofluids in small volumes. Following this, the back PPE configuration was used to obtain the influence of ultrasonic irradiation modes (either bath or probe sonication) such as the cluster size of Al2O3 nanofluids in low concentrations on the thermal diffusivity. The ultrasonic bath proved to be almost ineffective in size reduction, as most of the Al2O3 particles were spherical and were connected to each other to form a porous structure ranging in size from 1 μm to larger, and the probe sonication effectively reduced the particle size to below 100 nm. This showed that the oxide NPs in water were agglomerated and some hard aggregates could not be broken into individual NPs under these operating conditions or even at very high-energy inputs. The proposed front PPE technique, with a metalized PVDF sensor in a thermally thick regime, was applied to measure thermal effusivity by utilizing the phase signal of nanofluids that contained Al2O3 and CuO NPs dispersed in different solvents, water, ethylene glycol and olive oil. As expected, the relative standard deviation of this measurement, 2%, confirmed that this method was also suitable for measuring the thermal effusivity of nanofluid with a high degree of accuracy.

## **Abbreviations and Nomenclature**


nanofluids was below 2%, as shown in **Table 4**. Therefore, the front PPE technique is a

**Relative error%**

**K−<sup>1</sup>**

**) literature**

**Thermal effusivity × 10<sup>3</sup> (Ws1/2 m−<sup>2</sup>**

**) measurement**

The PPE technique is a sensitive method to measure the thermal properties of nanofluids in small volumes. Following this, the back PPE configuration was used to obtain the influence of ultrasonic irradiation modes (either bath or probe sonication) such as the cluster size of Al2O3 nanofluids in low concentrations on the thermal diffusivity. The ultrasonic bath proved to be almost ineffective in size reduction, as most of the Al2O3 particles were spherical and were connected to each other to form a porous structure ranging in size from 1 μm to larger, and the probe sonication effectively reduced the particle size to below 100 nm. This showed that the oxide NPs in water were agglomerated and some hard aggregates could not be broken into individual NPs under these operating conditions or even at very high-energy inputs. The proposed front PPE technique, with a metalized PVDF sensor in a thermally thick regime, was applied to measure thermal effusivity by utilizing the phase signal of nanofluids that contained Al2O3 and CuO NPs dispersed in different solvents, water, ethylene glycol and olive oil. As expected, the relative standard deviation of this measurement, 2%, confirmed that this method was also suitable for measuring the thermal effusivity of nanofluid with a high degree

**Table 4.** Experimental thermal effusivity of Al2O3 and CuO nanofluids and their pure solvents and their literature values

promising high-accuracy alternative for this measurement.

**Thermal effusivity × 10<sup>3</sup> (Ws1/2 m−<sup>2</sup> K−<sup>1</sup>**

Al2O3 Water 1.523 ± 0.014 1.566 ± 0.015 0.95 – Al2O3 EG 1.223 ± 0.009 0.773 ± 0.006 0.77 – Al2O3 Olive 1.112 ± 0.005 0.614 ± 0.003 0.48 – CuO Water 1.519 ± 0.028 1.547 ± 0.029 1.87 – CuO EG 1.202 ± 0.021 0.738 ± 0.012 1.75 – CuO Olive 1.081 ± 0.018 0.577 ± 0.009 1.56 – – Water 1.528 ± 0.011 1.586 ± 0.011 0.69 1.579 [10] – EG 1.263 ± 0.008 0.839 ± 0.005 0.59 0.810 [11] – Olive 1.175 ± 0.006 0.697 ± 0.003 0.43 0.621 [11]

**5. Conclusions**

**NPs**

[56].

**Base fluid** **Fitting parameter(***A***)**

88 Nanofluid Heat and Mass Transfer in Engineering Problems

of accuracy.

**Abbreviations and Nomenclature**

*ω* Angular frequency of modulated light

*Qo* TW source intensity

*f* Modulation frequency


## **Author details**

Monir Noroozi\* and Azmi Zakaria

\*Address all correspondence to: monir.noroozi@gmail.com

Physics Department, Faculty of Science, Universiti Putra Malaysia, Malaysia

## **References**


[19] Dadarlat D, Longuemart S, Turcu R, Streza M, Vekas L, Hadj Sahraoui A. Photopyroelectric calorimetry of Fe3O4 magnetic nanofluids: effect of type of surfactant and magnetic field. International Journal of Thermophysics, 2014;**35**: 2032–2043.

[5] Sheikholeslami M, Hayat T, Alsaedi A. MHD free convection of Al2O3–water nanofluid considering thermal radiation: a numerical study. International Journal of Heat and Mass

[6] Eastman JA, Choi SUS, Li S, Yu W, & Thompson LJ. Anomalously increased effective thermal conductivities of ethylene glycol-based nanofluids containing copper

[7] Evans W, Prasher R, Fish J, Meakin P, Phelan P, Keblinski P. Effect of aggregation and interfacial thermal resistance on thermal conductivity of nanocomposites and colloidal

[9] Murshed SMS, Leong KC, Yang C. Determination of the effective thermal diffusivity of nanofluids by the double hot-wire technique. Journal of Physics D: Applied Physics 2006;

[10] Shen J, and Mandelis A. Thermal-wave resonator cavity. Review of Scientific Instruments

[11] Chirtoc M, Mihilescu G. Theory of the photopyroelectric method for investigation of optical and thermal materials properties. Physical Review B 1989; **40**: 9606–9617.

[12] Kwan CH, Matvienko A, Mandelis A. Optimally accurate thermal-wave cavity photopyroelectric measurements of pressure-dependent thermophysical properties of air: theory and experiments. Review of Scientific Instruments 2007; **78**: 104902–104910.

[13] Chirtoc M, Bentefour EH, Glorieux C., Thoen J. Development of the front-detection photopyroelectric (FPPE) configuration for thermophysical study of glass-forming liq-

[14] Dădârlat D, Neamtu C. Detection of molecular associations in liquids by photopyroelectric measurements of thermal effusivity. Measurement Science and Technology 2006;

[15] Longuemart S, Quiroz AG, Dadarlat D, Sahraoui AH, Kolinsky C, Buisine JM. An application of the front photopyroelectric technique for measuring the thermal effusivity of

[16] Wei Y, Xie H. A review on nanofluids: preparation, stability mechanisms, and applica-

[17] Taylor R, Phelan P, Otanicar T, Adrian R, Prasher R. Nanofluid optical property characterization: towards efficient direct absorption solar collectors. Nanoscale Research Let-

[18] Nguyen VS, Rouxel D, Hadji R, Vincent B, Fort Y. Effect of ultrasonication and dispersion stability on the cluster size of alumina nanoscale particles in aqueous solutions. Ultrason-

some foods. Instrumentation Science & Technology 2002; **30**: 157–165.

nanofluids. International Journal of Heat and Mass Transfer. 2008; **51**: 1431–1438. [8] Sheikholeslami M, Ellahi R. Three dimensional mesoscopic simulation of magnetic field effect on natural convection of nanofluid. International Journal of Heat and Mass Trans-

nanoparticles. Applied Physics Letters. 2001; **78**: 718–720.

uids. Thermochimica Acta. 2001; **377**: 105–112.

tions. Journal of Nanomaterials, 2012; **2012**: 17 pages

Transfer, 2016; **96**: 513–524.

90 Nanofluid Heat and Mass Transfer in Engineering Problems

fer, 2015;**89**: 799–808.

1995; **66**: 4999–5005

**39**: 5316.

**17**: 3250.

ters, 2011; **6**: 225.

ics Sonochemistry, 2011; **18**: 382–388.


temperature dependence using the photopyroelectric method. Review of Scientific Instruments. 2002; **73**: 2773–2780.

[45] Longuemart S, Sahraoui AH, Dadarlat D, Daoudi A, Laux V, & Buisine JM. Investigations of the thermal parameters of ferroelectric liquid crystals using the pyroelectric effect in the S C \* phase. Europhysics Letters (EPL). 2003; **63**: 453.

[32] Murshed SS, de Castro CN, Lourenço MJV, Lopes MM, Santos FJV. Experimental investigation of thermal conductivity and thermal diffusivity of ethylene glycol-based nanofluids. Proceedings of the International Conference on Mechanical Engineering

[33] Rondino F, D'Amato R, Terranova G, Borsella E, Falconieri M. Thermal diffusivity enhancement in nanofluids based on pyrolytic titania nanopowders: importance of

[34] Dadarlat D, Frandas A. Inverse photopyroelectric detection of phase transitions. Applied

[35] Balderas-López, JA, & Mandelis A. New photopyroelectric technique for precise measurements of the thermal effusivity of transparent liquids. International Journal of

[36] Esquef IA, Siqueira APL, da Silva MG, Vargas H, Miranda LCM. Photothermal gas analyzer for simultaneous measurements of thermal diffusivity and thermal effusivity.

[37] Streza M, Pop MN, Kovacs K, Simon V, Longuemart S, Dadarlat D. Thermal effusivity investigations of solid materials by using the thermal-wave-resonator-cavity (TWRC) configuration. Theory and mathematical simulations. Laser Physics, 2009; **19**: 1340–1344.

[38] Balderas-López JA, Jaime-Fonseca MR, Díaz-Reyes J, Gómez-Gómez YM, Bautista-Ramírez ME, Muñoz-Diosdado A, et al. Photopyroelectric technique, in the thermally thin regime, for thermal effusivity measurements of liquids. Brazilian Journal of Physics,

[39] Gutiérrez-Juárez G, Ivanov R, Pichardo-Molina JP, Vargas-Luna M, Alvarado-Gil JJ, & Camacho A. Metrological aspects of auto-normalized front photopyroelectric method to measure thermal effusivity in liquids. International Journal of Thermophysics 2008; **29**:

[40] George NA, Vallabhan C P G, Nampoori VPN, George AK, Radhakrishnan P. Use of an open photoacoustic cell for the thermal characterisation of liquid crystals. Applied Phys-

[41] De Albuquerque JE, Balogh DT, Faria RM. Quantitative depth profile study of polyaniline

[42] Coufal H and Mandelis A. Pyroelectric sensors for the photothermal analysis of con-

[43] Caerels J, Glorieux C, Thoen J. Absolute values of specific heat capacity and thermal conductivity of liquids from different modes of operation of a simple photopyroelectric

[44] Delenclos S, Chirtoc M, Sahraoui AH, Kolinsky C, Buisine JM. Assessment of calibration procedures for accurate determination of thermal parameters of liquids and their

films by photothermal spectroscopies. Applied Physics A 2007; **86**: 395–401

densed phases. Ferroelectrics 1991; **118**: 379–409

setup. Review of Scientific Instruments 1998; **69**: 2452–2458.

aggregate morphology. Journal of Raman Spectroscopy2014; **45**: 528–532

2011 (ICME2011), 18–20 December 2011, Dhaka, Bangladesh

Physics A, 1993; **57**: 235–238.

92 Nanofluid Heat and Mass Transfer in Engineering Problems

2016; **46**: 105–110.

ics B 2001; **73**: 145–149.

2102–2115.

Thermophysics. 2003; **24**: 463–471

Analytical Chemistry, 2006; **78**: 5218–5221


**Problems of Simulating Nanofluid**

**Provisional chapter**

## **Problems Faced While Simulating Nanofluids Problems Faced While Simulating Nanofluids**

## Adil Loya Adil Loya

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/66495

#### **Abstract**

Problems are faced when something is already been adopted for a considerable amount of time–here the problem that is discussed is related with nanofluids. The nanofluids have been considered for different engineering applications since last three decades; however, the work on its simulation has been started since last two decades. With the time, nanofluid simulations are increasing as compared to experimental testing. Researchers conducting nanofluid simulations do find difficulties and problems while trying to simulate this system. In addition to this, most of the time researchers are unaware of some basic problems and they find themselves stuck in relentless difficulties. Most of the time, these problems are very basic and can waste a lot of useful time of a research. Therefore, this chapter introduces some fundamental problems which a researcher can find while simulating nanofluids and with a simple way of dealing with it. Moreover, the chapter withholds lots of information regarding the way to design and to model a nanofluid system. Not only this, it also tends to elaborate the nanofluid simulation methodology in a precise manner. Moreover, the literature shows that nanofluid simulation has gained high consideration since last two decades, as experimental techniques are out of reach for everyone. In addition to experimental techniques, they are expensive, time-consuming and require high skills. However, it seems the simulation is picking pace with the due time and is considerably being adopted by the expertise dealing with nanofluids. This opens a high prospect of simulating nanofluids in future. Nevertheless, it seems there will be user-friendly software to conduct nanofluid simulations. Finally, issues and their resolution have also been conveyed which is the main aspect of this topic.

**Keywords:** nanoparticles, nanofluids, molecular dynamics, simulations, problems

## **1. Introduction**

Couple of decades back, nanofluid research was mostly conducted using experimental techniques. With time, as the computational power acquired drastic developments, new algorithms were designed, and therefore, today, we have got sophisticated software and mathematical models to solve and simulate the nanofluid environment.

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## **1.1. Background knowledge**

Nanofluids comprises of two constitutes, i.e. Nano comes from nanoparticles and fluid comes from base fluid. The need of combining nanoparticles with fluid was necessary for enhancing the properties of the base fluid. Addition of nanoparticles to the base fluid helps in altering and optimizing properties such as physiochemical [1], thermo-physical [2], rheological [2–4], etc.; to give a new composite performance. The initial mixing of nanofluids can be dated back to the time of the US choi in 1995, he was the first one to form nanofluid at Argonne laboratories USA [5, 6]. He used the nanofluid for optimization of thermal conductivity. Since then there have been several experimental studies over the thermal conductivity analysis of different nanoparticles in various base fluids [5, 7–9].

By looking at thermal conductivity improvement, other researchers came up with different ideas and formulations for utilization of this technique in various fields of science. Today, nanofluids are being used in biological, pharmaceuticals and medicine [10], engineering [7], lubrication industries [11, 12]. The major work on experimental side in all these industries has been carried out; however, these experiments of nanofluid require high skilled labour and expensive equipment. Furthermore, material purchase and characterization are costly. Due to this, researchers and industrialists working with nanofluids are trying to develop a model that can replicate mechanisms dealing with nanoparticle and fluid interactions. However, this subject is wide and requires huge expertise to deal with.

Currently, as the computational power has enhanced to a level where people are finding it easy to simulate and replicate systems within their personal computers, it is now becoming quite manageable task to simulate nanofluids. But the task is not as simple as it seems, it requires a lot of understanding of physiochemical interactions with thermo-physical boundary conditions. There are many algorithms and mathematical models to be considered. As the number of these models and algorithms increases, higher the computational power is required for solving. Nevertheless, the endless applications and usage makes it convincible for an end-user to adopt this creativity, as it enables one to understand the process and makes it visually quantifiable.

Before moving forward, it is necessary to understand some basic theory that is behind the dispersion of nanoparticles within a certain fluid.

## *1.1.1. Theory behind dispersion of nanoparticle*

Dispersion of nanoparticles is a process in which they are dispersed in a medium like fluid. These fluids are of different grades such as biological, aerospace, automotive and buffering solutions. According to the kinetic theory of molecules, as the molecule interacts with other molecule, it starts to generate some heat due to kinetic molecular movement of the particle. This movement is accountable for the dispersion of nanoparticles in different fluids; thereby, this model causes anomalous increase in the heat transfer of the nanofluids. Furthermore, using this model, four major effects produced by nanoparticles dispersion can be explained i.e. (a) Brownian motion of nanoparticle, (b) liquid layering at liquid particle interface, (c) nature of heat transport between nanoparticles and (d) the clustering effect of nanoparticles in fluid. These factors are responsible for inducing random motion within particle and liquid layers, and this phenomenon is Brownian motion. During the interaction between nanoparticle and fluid, heat is evolved, causing nanoparticles to cluster and agglomerate. These mechanisms have already been replicated by various researchers for analysing properties such as; (a) rheological, (b) thermo-physical and (c) physiochemical as mentioned in Section 1.2.

## **1.2. Applications**

**1.1. Background knowledge**

98 Nanofluid Heat and Mass Transfer in Engineering Problems

it visually quantifiable.

ent nanoparticles in various base fluids [5, 7–9].

subject is wide and requires huge expertise to deal with.

dispersion of nanoparticles within a certain fluid.

*1.1.1. Theory behind dispersion of nanoparticle*

Nanofluids comprises of two constitutes, i.e. Nano comes from nanoparticles and fluid comes from base fluid. The need of combining nanoparticles with fluid was necessary for enhancing the properties of the base fluid. Addition of nanoparticles to the base fluid helps in altering and optimizing properties such as physiochemical [1], thermo-physical [2], rheological [2–4], etc.; to give a new composite performance. The initial mixing of nanofluids can be dated back to the time of the US choi in 1995, he was the first one to form nanofluid at Argonne laboratories USA [5, 6]. He used the nanofluid for optimization of thermal conductivity. Since then there have been several experimental studies over the thermal conductivity analysis of differ-

By looking at thermal conductivity improvement, other researchers came up with different ideas and formulations for utilization of this technique in various fields of science. Today, nanofluids are being used in biological, pharmaceuticals and medicine [10], engineering [7], lubrication industries [11, 12]. The major work on experimental side in all these industries has been carried out; however, these experiments of nanofluid require high skilled labour and expensive equipment. Furthermore, material purchase and characterization are costly. Due to this, researchers and industrialists working with nanofluids are trying to develop a model that can replicate mechanisms dealing with nanoparticle and fluid interactions. However, this

Currently, as the computational power has enhanced to a level where people are finding it easy to simulate and replicate systems within their personal computers, it is now becoming quite manageable task to simulate nanofluids. But the task is not as simple as it seems, it requires a lot of understanding of physiochemical interactions with thermo-physical boundary conditions. There are many algorithms and mathematical models to be considered. As the number of these models and algorithms increases, higher the computational power is required for solving. Nevertheless, the endless applications and usage makes it convincible for an end-user to adopt this creativity, as it enables one to understand the process and makes

Before moving forward, it is necessary to understand some basic theory that is behind the

Dispersion of nanoparticles is a process in which they are dispersed in a medium like fluid. These fluids are of different grades such as biological, aerospace, automotive and buffering solutions. According to the kinetic theory of molecules, as the molecule interacts with other molecule, it starts to generate some heat due to kinetic molecular movement of the particle. This movement is accountable for the dispersion of nanoparticles in different fluids; thereby, this model causes anomalous increase in the heat transfer of the nanofluids. Furthermore, using this model, four major effects produced by nanoparticles dispersion can be explained i.e. (a) Brownian motion of nanoparticle, (b) liquid layering at liquid particle interface, (c) nature of heat transport between nanoparticles and (d) the clustering effect of There are various applications in the area of nanofluid simulation. Currently, nanofluid simulation is being applied for analysing the rheological properties of nanofluid environment, which is useful for biological, oil and gas, lubrication and chemical industries. Now, by the help of simulation, it is possible to test those undesirable conditions that could not be tested before, such as testing viscosity at low and very high temperatures. Properties of ideal nanofluid can be tested and their results can also be validated using autocorrelation functions for satisfaction.

The use of molecular dynamics has enabled us to test and quantify thermo-physical quantities of nanofluid at obnoxious level. The chemical interactions that were complicated to understand from the real interface, now it has become straightforward to know how the atoms of fluid and nanoparticle interacts together, nevertheless, Brownian dynamics is more appreciably demonstrated and visualized. Having this all, analysing different properties of fluid and nanoparticle interaction, now it is easy to know other parameters such as specific heat [13], total energy, bond formation at molecular level, chemical interactions, etc. [14]. Furthermore, various effects that could not be judged by experimental testing can now easily be known such as the effect of liquid layering on thermal conductivity as investigated by Li et al. [15]. Particle effect on thermal conductivity analysis can now be determined as carried out by Lu and Fan [16]. Nevertheless, effect of surfactant addition in nanofluid system can also be tested using molecular dynamics, which can better tell about the chemical interaction and aggregation dynamics within this system as conveyed by Mingxiang and Lenore [17]. Rudyak also succeeded in showing that by changing nanoparticle size and shape effects the viscosity [18]. Therefore, by looking at the vast applications of nanofluid simulation, it is necessary to know some overview about how these simulations can easily be conducted.

## **2. The literature review**

## **2.1. Need of simulations over experiments**

Simulations are being preferred over experimental practices in the twenty-first century. As experiments require a lot of man power and material, which is costly and time-consuming, therefore, researchers are favouring simulations, as it saves material, money and time. With the advancement in computational technology, simulations are being approached to replicate the nanofluids. Simulations are not an old technique, and it has got a firm ground. Currently, the area of simulation to replicate the real phenomena of dispersion is through the int ermediate stages. Before moving to simulations, it is important to understand dispersion and interaction mechanism of nanoparticles with fluids. For this, the major phenomena that is used for dispersion is Brownian motion, which is an important aspect that controls the r andom factor of nanoparticle dispersion.

## **2.2. Simulations of nanofluids**

Nowadays, the necessity of using simulation techniques is increasing due to its cost-effectiveness and time-saving capabilities. Simulations for nanofluids are mostly referred to as molecular dynamics simulation (MDS). However, before MDS, researchers adopted theoretical and numerical calculation method for computing thermo-physical quantities. Earlier theoretical formation, related to MDS research, has not established a strong hold position for replicating the mechanism of heat transfer, rheology and thermo-physics involved for nanofluid dispersion. This is because several researchers had modelled system using various assumptions rather using a definite formulation. This creates ambiguity in collecting results; however, they were well utilized for initial prediction of thermal transfer properties of nanofluid at the cost of wide inaccuracies. Experimental results that are representing actual system sometime are way off from the ideal method, in addition to this, researchers apply various differential equations for equating the system to realistic results as possible.

These methods are single-phase and two-phase methods [19] of nanofluid heat convection. They are still being used for predicting several properties related to heat transfer, convection and conduction within nanofluid systems [19–21]. Now these two methods are being embedded in computation fluid dynamic and molecular dynamics for heat transfer analysis [21]. The single-phase method of heat convection in nanofluid is an old method and is good for initial prediction of the thermal properties of nanofluid; however, the second-phase method is costlier as it requires higher computing power. In addition to the second-phase method, it is quite versatile as its prediction is in higher accuracy to the experimental results. Numerical approach simulates the nanofluid system using classical thermodynamics principles, which is more close to the single-phase model. Different correlations are applied to estimate the imbalance between the heat propagation values from actual to the ideal system. Physical interaction kinetics involved in real nanofluid system are not mimicked. This is why the real prediction is hard to achieve by this approach; moreover, two-phase fluid heat transfer involves higher mathematical complexity, which requires high computational power for general analysis of nanofluid heat transfer, rheology and thermo-physical quantities.

It was investigated by Sergis Antonis that due to not standardizing the procedure of nanofluid preparation diversifies accuracy of the experimental results obtained [2]. In this respect, MDS comes in to play, as it helps in simulating both nanoparticle and fluid particle system in one single domain, enabling us to mimic reaction kinetics of both materials in one single domain. However, these simulations require high computational power for simulating the system as it involves kinetic molecular movement of different atoms. Initially, MDS involved heat transfer within a nanofluid system in which it did not involve analysis with respect to the geometrical features or spherical with no surface texture. It used to be simple analysis in a uniform and homogeneous system. Earlier, properties of SiO<sup>2</sup> nanoparticles were calculated using Stillinger-Weber [22] and later fluid particles were represented by L-J potential.

int ermediate stages. Before moving to simulations, it is important to understand dispersion and interaction mechanism of nanoparticles with fluids. For this, the major phenomena that is used for dispersion is Brownian motion, which is an important aspect that controls the

Nowadays, the necessity of using simulation techniques is increasing due to its cost-effectiveness and time-saving capabilities. Simulations for nanofluids are mostly referred to as molecular dynamics simulation (MDS). However, before MDS, researchers adopted theoretical and numerical calculation method for computing thermo-physical quantities. Earlier theoretical formation, related to MDS research, has not established a strong hold position for replicating the mechanism of heat transfer, rheology and thermo-physics involved for nanofluid dispersion. This is because several researchers had modelled system using various assumptions rather using a definite formulation. This creates ambiguity in collecting results; however, they were well utilized for initial prediction of thermal transfer properties of nanofluid at the cost of wide inaccuracies. Experimental results that are representing actual system sometime are way off from the ideal method, in addition to this, researchers apply various differential equa-

These methods are single-phase and two-phase methods [19] of nanofluid heat convection. They are still being used for predicting several properties related to heat transfer, convection and conduction within nanofluid systems [19–21]. Now these two methods are being embedded in computation fluid dynamic and molecular dynamics for heat transfer analysis [21]. The single-phase method of heat convection in nanofluid is an old method and is good for initial prediction of the thermal properties of nanofluid; however, the second-phase method is costlier as it requires higher computing power. In addition to the second-phase method, it is quite versatile as its prediction is in higher accuracy to the experimental results. Numerical approach simulates the nanofluid system using classical thermodynamics principles, which is more close to the single-phase model. Different correlations are applied to estimate the imbalance between the heat propagation values from actual to the ideal system. Physical interaction kinetics involved in real nanofluid system are not mimicked. This is why the real prediction is hard to achieve by this approach; moreover, two-phase fluid heat transfer involves higher mathematical complexity, which requires high computational power for general analysis of

It was investigated by Sergis Antonis that due to not standardizing the procedure of nanofluid preparation diversifies accuracy of the experimental results obtained [2]. In this respect, MDS comes in to play, as it helps in simulating both nanoparticle and fluid particle system in one single domain, enabling us to mimic reaction kinetics of both materials in one single domain. However, these simulations require high computational power for simulating the system as it involves kinetic molecular movement of different atoms. Initially, MDS involved heat transfer within a nanofluid system in which it did not involve analysis with respect to the geometrical features or spherical with no surface texture. It used to be simple analysis in

r andom factor of nanoparticle dispersion.

100 Nanofluid Heat and Mass Transfer in Engineering Problems

tions for equating the system to realistic results as possible.

nanofluid heat transfer, rheology and thermo-physical quantities.

**2.2. Simulations of nanofluids**

There are two different dispersion prospects of MDS i.e. (1) non-equilibrium MDS (NEMD) and (2) equilibrium MDS (EMD). The macroscopic MDS mimics the molecular interactions between different molecules of various elements; in compound or ionic form. These different thermo-physical types of interactions of molecular dynamic quantities can be tailored and analysed by true boundary conditions. These boundary conditions are related to the physical settings, chemical interactions, charges, viscosity of the system and motion exhibition of particles. The interaction between the molecules is exhibited by Brownian motion as this mimics the random forces in the system. The system relies on different algorithms behind the scene to design a virtual nanoparticles dispersion in fluid. Furthermore, this is because the interaction kinetics of nanofluid system adhere with nanoparticle surface interacting with the surrounding fluid; this involves exchange of energy, surface tension between two, orientation of nanoparticle, surface energy, bonding configuration, nanoparticle dynamics and kinematics (including nanoparticle spin), liquid layering between nanoparticle and fluid molecule, and diffusion rate.

To explain the trajectories and velocities of a fluidic system, it is necessary to adopt a hydrodynamic framework. Computer simulations for mimicking trajectory of hydrodynamic dispersion of a dispersed particle in a fluid system was used by Ermak [23]. Nevertheless, Ermak and McCammon [24] work was more focused on the hyrdrodynamically concentrated system. The hydrodynamical system exhibited that the inter-particle distance is much greater than the range of hydrodynamic interactions. However, by implementation of Brownian dynamics by Ermak gave highly concurrent results with the experimental values achieved. The hydrodynamics of the system display combinations of Coulomb interactions; i.e. long range interactions as well as the Vander Waal interactions; short range interactions. Furthermore, the dynamics of the system is more convincing after applying the Derjaguin, Landau, Verwey and Overbeek (DLVO) [25] theory/factor in the system to mimic the charges and to enhance the realistic intermolecular attractions and repulsions.

Currently, there are different nanoparticles being considered for various applications. Therefore, for simulating nanofluids, modelling the nanoparticle is important, for that nanoparticle structure, shape and its properties should be known.

Subsequently, the mimicking of interaction potentials; i.e. using force fields such as embedded atom method (EAM), COMPASS, universal, etc; and the other forces between the atoms and molecules, the velocity verlet theorem is implemented. The velocity verlet theorem is a time-dependent movement of the atoms from one position to another using an algorithm for defining the movement, which is based on Brownian dynamics (BD). In addition to this, velocities or movements of atoms are controlled using thermal ensembles i.e. canonical (NVT), grand canonical (ΔPT), isobaric and isothermal (NPT) and micro canonical (NVE). These ensembles support in conducting thermal and physical perturbation to change the dynamical position of the atoms and molecules within a desired system. This causes the system to move to an un-equilibrium state. After starting and moving from an un-equilibrium state, the system is then equilibrated for convergence to equilibrium state. Finally, by this convergence, the system acquires stability of temperature and physical quantity fluctuations. However, this convergence is an iterative process for which time steps are varied to achieve the real convergence results [26, 27].

Currently, there are various simulation of nanofluids, for example; CuO, TiO<sup>2</sup> and CeO<sup>2</sup> nanoparticle dispersion in water [3, 4]; furthermore, there are also studies of dispersing nanoparticles in hydrocarbons [28]. By having two different simulation strategies, a perspectives and robust methodology can be formulated. As these simulations are performed on two different types of fluids i.e. polar and non-polar, so a concurrent methodology for both fluids can be deduced. Furthermore, up to the date, investigators have carried out various researches on nanofluid MDS, in addition to this, last two decades of work has been cumulated in **Figure 1**. Following are the details of their work in the field of nanofluid simulations.

In 1998, Malevanets and Kapral [29] formulated a method for computing complex fluidic systems using H theorem, which helped in solving hydrodynamics equations and transport coefficients. Colloidal model and random stochastic movement algorithm was established using Brownian dynamics which was formulated by Lodge and Heyes [30].

Francis W. Starr investigated effect of glass transition temperature on the bead spring polymer melts with a nanoscopic particle. He found that the surface interaction dominates due to nanoparticle diffusion within the melted polymeric system [31].

Simulation of chemical interactions was also carried out, and the bond length and structural orientation was noted for Silica nanoparticles in poly ethylene oxide (PEO) oligomer system. By this study, Barbier et al. concluded that the silica nanoparticles influence structural properties of PEO up to two to three layers [32].

Mingxiang and Lenore worked on hydrocarbon surfactant in an aqueous environment with a nanoparticle diffused within this system. It was observed from interactions that the agglomeration created between water molecules and surfactant was independent of nanoparticle i.e. it does not matter whether it is present or not [17].

**Figure 1.** Timeline showing work carried out by different researchers since last two decades.

Sarkar and Selvam designed a nanofluid system of Cu nanoparticle and Argon as basefluid, for this, he used EAM potential and Green Kubo technique to find the thermal conductivity of this system. He examined that the periodic oscillation existed due to the heat fluxes imposed by Leonard Jones (L-J) potential [9].

the system acquires stability of temperature and physical quantity fluctuations. However, this convergence is an iterative process for which time steps are varied to achieve the real

nanoparticle dispersion in water [3, 4]; furthermore, there are also studies of dispersing nanoparticles in hydrocarbons [28]. By having two different simulation strategies, a perspectives and robust methodology can be formulated. As these simulations are performed on two different types of fluids i.e. polar and non-polar, so a concurrent methodology for both fluids can be deduced. Furthermore, up to the date, investigators have carried out various researches on nanofluid MDS, in addition to this, last two decades of work has been cumulated in **Figure 1**. Following are the details of their work in the field of nanofluid simulations. In 1998, Malevanets and Kapral [29] formulated a method for computing complex fluidic systems using H theorem, which helped in solving hydrodynamics equations and transport coefficients. Colloidal model and random stochastic movement algorithm was established

Francis W. Starr investigated effect of glass transition temperature on the bead spring polymer melts with a nanoscopic particle. He found that the surface interaction dominates due to

Simulation of chemical interactions was also carried out, and the bond length and structural orientation was noted for Silica nanoparticles in poly ethylene oxide (PEO) oligomer system. By this study, Barbier et al. concluded that the silica nanoparticles influence structural proper-

Mingxiang and Lenore worked on hydrocarbon surfactant in an aqueous environment with a nanoparticle diffused within this system. It was observed from interactions that the agglomeration created between water molecules and surfactant was independent of nanoparticle i.e.

and CeO<sup>2</sup>

Currently, there are various simulation of nanofluids, for example; CuO, TiO<sup>2</sup>

using Brownian dynamics which was formulated by Lodge and Heyes [30].

**Figure 1.** Timeline showing work carried out by different researchers since last two decades.

nanoparticle diffusion within the melted polymeric system [31].

ties of PEO up to two to three layers [32].

it does not matter whether it is present or not [17].

convergence results [26, 27].

102 Nanofluid Heat and Mass Transfer in Engineering Problems

Li et al. later worked on similar system of Cu nanoparticle with Ar base fluid; however, they investigated Brownian dynamics induces a thin layer around a particle, giving a hydrodynamic effect to the particle dispersion [33].

Lu and Fan investigated thermo-physical quantities of Alumina nanoparticles dispersed in water and concluded that the particle volume fraction and size effects the viscosity and thermal conductivity [16].

Sankar et al. examined and formulated an algorithm for calculating metallic nanoparticle thermal conductivity in fluid. They articulated that the volume fraction of nanoparticles and temperature of the system effects the overall thermal conductivity [8].

Moreover, Cheung carried out research on L-J nanoparticles within solvent and quantified that the detachment energy decreases as the nanoparticle solvent attraction rises [1].

Sun et al. devised a technique using EMD using Green Kubo method to find the effective thermal conductivity of the Cu nanoparticles in Ar liquid. It was found that there was a linear increase in the effective thermal conductivity of shearing nanofluid due to micro-convection [34].

Rudyak and Krasnolutskii later on worked on Aluminium and Lithium nanoparticles with liquid Ar and suggested that the size and material of nanoparticle considerably effects the viscosity [18].

Lin Yun Sheng et al. also detected increment in thermal conductivity by Cu nanoparticle dispersion in Ethylene glycol fluid. In this study, he used Green kubo formulation for finding thermal conductivity using NEMD [35]. Furthermore, Mohebbi investigated a method to calculate thermal conductivity of nanoparticles in fluid using a non-periodic boundary conditions with EMD and NEMD [14].

Kang H et al. carried out work on coupling factor between nanoparticle of Copper and Ar as base fluid, his investigations suggest that coupling factor is proportional to the volume concentration of particles, nevertheless, he also suggested the that there is no effect of temperature change from 90 to 200 K on coupling factor [36].

Rajabpour et al. investigated the specific heat capacity of Cu nanoparticles within water and he found that the specific heat capacity of this system decreases by increasing the volume fraction of particles in base fluid [13].

Loya et al. initiated work on CuO nanoparticles dispersion in water focusing on the change of viscosity due to temperature increase, he figured that temperature increment decreases the viscosity of nanofluid as also initially predicted using experimental testing [37].

In addition to above, further rheological analysis of CuO nanoparticles in straight chain alkanes [28] and water [4] and CeO<sup>2</sup> in water [3] was carried out by Loya et al. For conducting these simulations, molecular dynamics was used and studies provided highly accurate results of viscosity to experimental findings.

 Finally, after knowing the perspective of nanofluid simulation, a simple and general way is deduced for researcher, industrialist and their co-worker in Section 2.3.

## **2.3. Mimicking different properties of nanofluids using simulation**

Several studies about simulation work were reported on the diffusion of polymeric, ionic and mineral nanoparticles [38–40]. An example of this is calcite nanoparticles. These have been simulated in water for salt molecular dynamics for thermal energy storage nanofluidic simulations [38]. Simulations such as these are mostly conceiving diffusions of the polymeric nanoparticles or di-block polymers represented by spheres. The major diffusion phenomena that have been implemented on the nanoparticle or the polymer dispersion is with the help of BD, targeting the random motion of the particles in a solvent or any solution system. Some further surveys show that one of the best simulations for the dispersion of the metal oxide nanoparticle in the water system was carried out using the DPD potential [41–43]. This potential has the power to disperse nanoparticles as well as replicating the phenomena of the BD [44]. DPD was first carried out on nano-water systems by Hooggerbrugge and Koelman [44, 45]. Moreover, the work was carried out by Español and Warren for implementing the DPD technique using statistical mechanics. DPD technique imparts stochastic phenomena on particle dynamics [46]. This is how BD was integrated into DPD technique. However, the random forces will only be in pairwise interaction since DPD at the same time imparts the hydrodynamic effect on the system. Many studies of DPD for complex fluidic systems [41–43] show that the dispersion of nanoparticles in water exhibits complex properties and to simulate this, initial selection of boundary conditions are important to replicate the real scenario. Thereby, the best way to simulate is to acquire the boundary conditions of the existing experimental system and then use a molecular dynamic simulator to further implement it [47]. The considerations of boundary conditions are particle sizes, force field for particleto-particle interactions, solvent in which the particles will be diffused, and physiochemical nature of the system [48, 49]. Within the simulation system, force field plays an important role since it provides charges on atoms for interaction. The force field is a mathematical parameter that governs the energies and potentials between interactive atoms. The physiochemical settings of the system refer to the thermal, chemical and physical properties of the system such as initial temperature settings, charges and dynamics. Finally, the temperature is controlled using different ensembles.

## **3. Methodology**

## **3.1. Simulation strategy**

The nanofluid interactions are carried out at molecular level. Therefore, by keeping this in mind to conduct nanofluid simulations, it is necessary to have a simulation technique which allows us to do simulation at molecular level. Hence, the technique use for this is molecular dynamics and package that is focused through this chapter is Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). Furthermore, how to approach this is mentioned in the next section of this chapter i.e. Section 3.1.1.

## *3.1.1. Approach*

these simulations, molecular dynamics was used and studies provided highly accurate results

Finally, after knowing the perspective of nanofluid simulation, a simple and general way is

Several studies about simulation work were reported on the diffusion of polymeric, ionic and mineral nanoparticles [38–40]. An example of this is calcite nanoparticles. These have been simulated in water for salt molecular dynamics for thermal energy storage nanofluidic simulations [38]. Simulations such as these are mostly conceiving diffusions of the polymeric nanoparticles or di-block polymers represented by spheres. The major diffusion phenomena that have been implemented on the nanoparticle or the polymer dispersion is with the help of BD, targeting the random motion of the particles in a solvent or any solution system. Some further surveys show that one of the best simulations for the dispersion of the metal oxide nanoparticle in the water system was carried out using the DPD potential [41–43]. This potential has the power to disperse nanoparticles as well as replicating the phenomena of the BD [44]. DPD was first carried out on nano-water systems by Hooggerbrugge and Koelman [44, 45]. Moreover, the work was carried out by Español and Warren for implementing the DPD technique using statistical mechanics. DPD technique imparts stochastic phenomena on particle dynamics [46]. This is how BD was integrated into DPD technique. However, the random forces will only be in pairwise interaction since DPD at the same time imparts the hydrodynamic effect on the system. Many studies of DPD for complex fluidic systems [41–43] show that the dispersion of nanoparticles in water exhibits complex properties and to simulate this, initial selection of boundary conditions are important to replicate the real scenario. Thereby, the best way to simulate is to acquire the boundary conditions of the existing experimental system and then use a molecular dynamic simulator to further implement it [47]. The considerations of boundary conditions are particle sizes, force field for particleto-particle interactions, solvent in which the particles will be diffused, and physiochemical nature of the system [48, 49]. Within the simulation system, force field plays an important role since it provides charges on atoms for interaction. The force field is a mathematical parameter that governs the energies and potentials between interactive atoms. The physiochemical settings of the system refer to the thermal, chemical and physical properties of the system such as initial temperature settings, charges and dynamics. Finally, the temperature is controlled

The nanofluid interactions are carried out at molecular level. Therefore, by keeping this in mind to conduct nanofluid simulations, it is necessary to have a simulation technique which allows us to do simulation at molecular level. Hence, the technique use for this is molecular

deduced for researcher, industrialist and their co-worker in Section 2.3.

**2.3. Mimicking different properties of nanofluids using simulation**

of viscosity to experimental findings.

104 Nanofluid Heat and Mass Transfer in Engineering Problems

using different ensembles.

**3. Methodology**

**3.1. Simulation strategy**

The simulation of nanoparticle dispersion is related to the MDS. For this, the software or the package that needs to be selected was based on the criteria of the conditions that were needed to be simulated, and the flexibility was a major concern for the applicability of different systems. The LAMMPS can be a best molecular dynamics package for simulating the nanofluidic system. This is the code generated by the Sandia Laboratories by Plimpton [50]. This molecular dynamics software has high viability over other available software like Montecarlo and Gromacs.

After selection of the MD package, to simulate a desired system with realistic features, it is highly vital to know and understand initial boundary conditions. These initial conditions for a dispersion of nanoparticles are related to charges within the system for interaction, molecular bonding, forces of attraction i.e. Vander Waal or electrostatic coulombs interactions, forcefields, pair potentials (i.e. molecular mechanics constants) and molecular weight. To perform MD simulation, initial boundary conditions are major and fundamental parameters to devise actual dynamics that exist in a real system. After setting the initial parameters, velocity of the system is equilibrated and ensembles are applied to mimic the real thermo-physical conditions.

In addition to above, after setting all the boundary conditions related to chemical and thermophysical parameters, the system is then equilibrated for certain time steps. Simulations are processed until converging results are obtained as that of the actual system. Over here, "time step" is the major dependent factor. This accounts for equilibrating the kinetics of the system that takes place; i.e. movement of system from an un-equilibrated state to equilibrium conditions. The above explained method has been compressed and illustrated using a flowchart for better understanding as shown in **Figure 2**.

After suggesting how to approach and initiate your work for simulation of nanofluids, it is also important to know the briefed-out details about the steps like force field, pair potentials, ensembles, etc.

After setting up the atoms in a coordinate system using a molecular modelling software, then force field is applied on the system (i.e. Universal, COMPASS, OPLS, etc.) by this atomic charges and bond configurations are setup. These force fields are interlinked with pair potentials (such as DPD, BD, Smoothed Particle Hydrodynamic, LJ, etc.), they are parameters which are used to describe vibrational and oscillation settings between two different atoms. Finally, ensembles are applied on the molecular dynamic system for equilibrating the actual thermal settings for example NVT, NPT, NPH, etc.

## *3.1.2. Techniques and tools*

As of now, it is known from the previous sections that to simulate and perform MDS it is necessary to know techniques and tools that can be beneficial for use and executing the work.

**Figure 2.** Flow chart of molecular dynamics simulation [26].

Today, there are several tools and ways to perform this; however, still researchers are unsure about "what are the clear steps for conducting nanofluid simulations using molecular dynamics?" Therefore, through this section, a brief and concise way is illustrated and conveyed for better and easier understanding for people working under the horizon of nanofluid simulations. These steps are as follow:

**a.** Firstly, for creating nanofluid simulation system, it is required to setup a nanoparticle and fluid, then combine them together, for which material studio is the best software for designing a nanoparticle. Now, the nanoparticles can be inserted and replicated in a box containing fluid particles, however, this may be tedious for bigger systems. Therefore, it is suggested to use Packmol after creating the Protein Data Bank (PDB) file from material studio and then create an input script for Packmol to replicate the system with as many particles and fluid molecules as per required. This software automatically packs up the overall molecular arrangement with in a confined imaginary box.


Finally, the data quantification, visualizing the effects and properties that can be analysed have been jotted below in different sections.

## *3.1.3. Data quantification*

 Now, the data obtained by using different compute commands can be quantified on MATLAB or Excel. MATLAB initially requires more time for developing its script for computing the mathematical problem or graphs. However, on a long run, it does save time. Whereas, excel is easy going but requires more time for plotting graph each time you feed the new data.

MATLAB scripting helps in formulating the work in a precise manner, and digitalise the work with high quality publishing of the data for journal publications. However, MATLAB requires good command over the MATLAB scripting and functions. By using MATLAB, it is easy to apply discrete as well as continuous algorithms and equations for refining and optimization of results. Furthermore, it helps in applying the regression on the noisy data for refinement.

In Excel, similar stuff is possible as in MATLAB, but in excel, it is quite complicated as you need to apply macros. These days the computation of MATLAB can be computed in parallel mode; again for excel, it is quite difficult. However, for graphical representation of data, excel is quite versatile.

Vice versa both tools have their own benefits over each other; it depends totally on a userfriendliness with certain software. In addition to excel, to compute or establish complex calculations, it will be required to interlink its macros with visual basic scripting, which is under a developer's tool library, mostly hidden from newbies.

## *3.1.4. Visualizing the effects*

Today, there are several tools and ways to perform this; however, still researchers are unsure about "what are the clear steps for conducting nanofluid simulations using molecular dynamics?" Therefore, through this section, a brief and concise way is illustrated and conveyed for better and easier understanding for people working under the horizon of nanofluid simula-

**a.** Firstly, for creating nanofluid simulation system, it is required to setup a nanoparticle and fluid, then combine them together, for which material studio is the best software for designing a nanoparticle. Now, the nanoparticles can be inserted and replicated in a box containing fluid particles, however, this may be tedious for bigger systems. Therefore, it is suggested to use Packmol after creating the Protein Data Bank (PDB) file from material studio and then create an input script for Packmol to replicate the system with as many particles and fluid molecules as per required. This software automatically packs up the

overall molecular arrangement with in a confined imaginary box.

tions. These steps are as follow:

**Figure 2.** Flow chart of molecular dynamics simulation [26].

106 Nanofluid Heat and Mass Transfer in Engineering Problems

After the successful execution of simulation, you will get dump files from LAMMPS, here a software that can read LAMMPS trajectories can be used for reading the file and visualizing it. For which Visual Molecular Dynamic (VMD) can be used. However, OVITO is also a good software for visualizing your trajectories.

The results generated by OVITO are represented as small spheres merged together to form a particular system representation, as shown in **Figure 3**, i.e. of a CuO-water nanofluid system.

In the similar way for showing how the VMD gives visual output is shown in **Figure 4**. It is similar to that of OVITO, however, VMD has capability of representing the trajectories in the form of molecular structure. This gives an extra possibility for researchers working in the area of Biochemistry, pharmacy, drug delivery and biomedical to represent and observe the chemical kinetics in real-time, i.e. how one atom reacts and interacts with another atom within a confined system.

## *3.1.5. Properties that can be analysed*

Some properties and parameters can directly be analysed using VMD using trajectories dump files. VMD has option for analysing the radial distribution function (RDF) and mean square displacement (MSD), they indicate about the agglomeration and dispersion rate, respectively.

**Figure 3.** Representation of OVITO output of molecular dynamics of CuO nanoparticles in water system [4].

**Figure 4.** Visual output showing two CuO nanoparticles in a water-based nanofluid.

In the similar way for showing how the VMD gives visual output is shown in **Figure 4**. It is similar to that of OVITO, however, VMD has capability of representing the trajectories in the form of molecular structure. This gives an extra possibility for researchers working in the area of Biochemistry, pharmacy, drug delivery and biomedical to represent and observe the chemical kinetics in real-time, i.e. how one atom reacts and interacts with another atom within

Some properties and parameters can directly be analysed using VMD using trajectories dump files. VMD has option for analysing the radial distribution function (RDF) and mean square displacement (MSD), they indicate about the agglomeration and dispersion rate, respectively.

**Figure 3.** Representation of OVITO output of molecular dynamics of CuO nanoparticles in water system [4].

a confined system.

*3.1.5. Properties that can be analysed*

108 Nanofluid Heat and Mass Transfer in Engineering Problems

When nanofluids are concerned the major parameters or properties researcher are interested to investigate are viscosity, thermal conductivity, specific heat capacity, thermal diffusivity, diffusion coefficient, total energy, heat loss, etc. To find these properties LAMMPS provide versatile options to compute what you require, using different algorithms or previously established techniques. Currently, main concerned variables out of above mentioned ones are viscosity, diffusion coefficient and thermal conductivity. Therefore, in the next section, we will discuss about how to validate and quantify your results obtained from the simulation.

## **4. Validation and quantification of results**

To validate the three major properties mentioned in Section 3.1.5, it is required to know initial experimental results, however, sometime it is hard to obtain those results as some simulation condition cannot be tested, either due to lack of experimental device or it is not possible to meet the boundary conditions as setup over the simulation platform.

Now, in this case, the best way is to analyse using autocorrelation function; which is a time series modelling of a function of a variable dependent on time fluctuation. Let us take the case of viscosity, as it is related with shearing stress, there are shear forces acting between the layers of molecular interaction causing pressure function to be induced. This pressure function is dependent on stress due to shearing force. If this stress is analysed using the function of time, this becomes stress tensor. This stress tensor is used for analysing stresses exiting between the molecular layers. Therefore, this is known as stress autocorrelation function (SACF). The SACF accounts for the stresses imparted on the system due to the diffusion of molecules and intermolecular kinetics; i.e. molecular stresses caused by attraction and repulsion of molecules. During the intermolecular kinetics drag is created between the molecular layers, this drag is due to the effect of shearing forces. Ultimately as the system is equilibrated, it shows unstable response of the SACF, however, as it approaches stability the SACF starts to converge to a monotonic level, which satisfies that the viscosity analysed is acceptable.

 In the similar manner, thermal conductivity is quantified, but here instead of stress and shear forces, heat is considered. Therefore, this is known as heat autocorrelation function (HACF), which quantifies or validates the thermal conductivity obtained is satisfactory.

In addition to HACF and SACF for thermal conductivity and viscosity, respectively, for diffusion coefficient, velocity autocorrelation function is used for its quantification. As diffusion coefficient is measured by taking the slope of the MSD. So to quantify and validate it, displacement with respect to time i.e. velocity can be used.

**Figure 5.** Autocorrelation output gained by running a molecular dynamics simulations [26].

The accuracy of results equilibrated for measuring the viscosity and thermal conductivity of a system can be justified in a better way with the estimation of heat autocorrelation function and stress autocorrelation function as show in **Figure 5**. The graphical result in **Figure 5** explains the process of the integration of non-equilibrated system to equilibration.

At step (a), the system starts with a thermodynamic equilibrium, but the system is not at equilibrium state. At step (b), the thermodynamic conditions are changed due to implementation of thermal ensemble so the system tends to go towards equilibrium. At step (c), the non-equilibrium system moves to equilibrated level of convergence at this level the system satisfies the convergences. This process is followed during the equilibration of the thermophysical quantities, the convergence time steps depend on the volume and quantity of the atoms in that system. For the larger system, large amount of computational power and time step will be required for convergence.

## **5. Discussion**

molecular layers, this drag is due to the effect of shearing forces. Ultimately as the system is equilibrated, it shows unstable response of the SACF, however, as it approaches stability the SACF starts to converge to a monotonic level, which satisfies that the viscosity analysed

 In the similar manner, thermal conductivity is quantified, but here instead of stress and shear forces, heat is considered. Therefore, this is known as heat autocorrelation function (HACF),

In addition to HACF and SACF for thermal conductivity and viscosity, respectively, for diffusion coefficient, velocity autocorrelation function is used for its quantification. As diffusion coefficient is measured by taking the slope of the MSD. So to quantify and validate it, dis-

The accuracy of results equilibrated for measuring the viscosity and thermal conductivity of a system can be justified in a better way with the estimation of heat autocorrelation function and stress autocorrelation function as show in **Figure 5**. The graphical result in **Figure 5**

At step (a), the system starts with a thermodynamic equilibrium, but the system is not at equilibrium state. At step (b), the thermodynamic conditions are changed due to implementation of thermal ensemble so the system tends to go towards equilibrium. At step (c), the

explains the process of the integration of non-equilibrated system to equilibration.

**Figure 5.** Autocorrelation output gained by running a molecular dynamics simulations [26].

which quantifies or validates the thermal conductivity obtained is satisfactory.

placement with respect to time i.e. velocity can be used.

110 Nanofluid Heat and Mass Transfer in Engineering Problems

is acceptable.

## **5.1. Problems faced for simulating nanofluids**

So far the topic has been conveying the techniques, approach and method for carrying out nanofluid simulations. Moreover, there has been no data available for the expertise to know what are the problems faced when these simulations are conducted, number of questions can arise, for example, (1) Till what level, computational power can support our simulations? (2) Is there any other way out rather than this? (3) How larger systems can be simulated? etc.

Therefore, to answer these questions, it is necessary to understand the material and knowledge given before, however, as the number of atoms are increased within a nanofluid system the molecular dynamics demonstrates sluggish performance due to less computational capabilities i.e. either central processing unit (CPU) power or graphic processing unit (GPU). Furthermore, it is not just simulation that need to be carried out but for the data quantification, the data that are gathered requires huge memory for storage. Thereby, requiring the random access memory (RAM) and hard disk drive (HDD) to be large enough to store the required data easily [51].

After hardware issues, the second set of problems faced by nanofluid simulation is the use of multiple software for designing, modelling, processing and visualization, which needs a lot of understanding of computer for a new geek. Furthermore, if this all is combined in one package, this can marvellously save time and money for purchasing different software for data acquisition. It is slightly known at the moment that there are few software in market for helping in simulating nanofluid; however, academia is not yet aware of it due to less versatility such as Medea and Scienomics MAPS.

One of the major problem is that, people of twenty-first century like working using graphical user interface (GUI), as it is easy and you can do everything by just clicks rather than using complicated commands, however, most of the molecular dynamics package are used on Linux operating system, moreover, commands are used for computing and feeding the data for computation.

In addition to high computing power, it should be known that before attempting to simulate large scale molecular dynamics (i.e. with more than 0.1 million atoms), it is required to have parallel processing enabled on the PC. For that high end, CPU or GPU is required with multi cores for processing the data in parallel mode. However, this processing has some drawbacks that are loop holes for simulations, one such kind is that sometimes the algorithm is not designed in a way to parallel the process efficiently, which in turn gives ambiguous simulation output and convergence. For avoiding this, it is necessary for the user to know the correct working of the algorithm. Moreover, the field programmable gate array (FPGA) is good outbreak technology that is being implemented for paralleling the process [52, 53], nevertheless, again this technology requires new stuff and bits coding to be learned before operating or using this module for rapidly solving the simulation.

## **6. Conclusion**

The chapter has brought about marvellous information and the literature for new geeks for conducting a nanofluid simulation. However, this chapter acts as a guide for a newbie for initialising the nanofluid simulation.

## **Nomenclature**



## **Author details**

## Adil Loyai

simulation output and convergence. For avoiding this, it is necessary for the user to know the correct working of the algorithm. Moreover, the field programmable gate array (FPGA) is good outbreak technology that is being implemented for paralleling the process [52, 53], nevertheless, again this technology requires new stuff and bits coding to be learned before

The chapter has brought about marvellous information and the literature for new geeks for conducting a nanofluid simulation. However, this chapter acts as a guide for a newbie for

Words **Abbreviation**

File format output from material studio CAR and.COR

Molecular dynamics simulation MDS Non-equilibrium molecular dynamics NEMD Equilibrium molecular dynamics EMD Embedded atom method EAM Condensed-phase optimized molecular potentials for atomistic simulation studies COMPASS Brownian dynamics BD Canonical **NVT** Grand canonical ΔPT Isobaric and isothermal NPT Micro canonical NVE Normal pressure hydrocephalus NPH Copper oxide CuO Titanium oxide TiO<sup>2</sup> Cerium oxide CeO<sup>2</sup> Poly ethylene oxide PEO Leonard Jones L-J Copper **Cu** Argon **Ar** Kelvin **K** Discrete particle dynamics DPD Large-scale atomic/molecular massively parallel simulator LAMMPS Optimized potential for liquid simulation sOPLS Protein Data Bank PDB

operating or using this module for rapidly solving the simulation.

**6. Conclusion**

**Nomenclature**

initialising the nanofluid simulation.

112 Nanofluid Heat and Mass Transfer in Engineering Problems

Address all correspondence to: loya\_adil@yahoo.com

PAF-Karachi Institute of Economics and Technology, Karachi, Pakistan

## **References**


[7] Karthik R, Harish Nagarajan R, Raja B, Damodharan P. Thermal conductivity of CuO– DI water nanofluids using 3-ω measurement technique in a suspended micro-wire.

[8] Sankar N, Mathew N, Sobhan C. Molecular dynamics modeling of thermal conductivity enhancement in metal nanoparticle suspensions. International Communications in Heat

[9] Sarkar S, Selvam RP. Molecular dynamics simulation of effective thermal conductivity and study of enhanced thermal transport mechanism in nanofluids. Journal of Applied

[10] Esmaeilzadeh P, Fakhroueian Z, Beigi M, Akbar A. Synthesis of biopolymeric α-lactalbumin protein nanoparticles and nanospheres as green nanofluids using in drug delivery and

[11] Rapoport L, Leshchinsky V, Lvovsky M, Lapsker I, Volovik Y, Feldman Y, et al. Superior tribological properties of powder materials with solid lubricant nanoparticles. Wear.

[12] Peng DX, Kang YA, Chen SK, Shu FC, Chang YP. Dispersion and tribological properties of liquid paraffin with added aluminum nanoparticles. Industrial Lubrication and

[13] Rajabpour A, Akizi FY, Heyhat MM, Gordiz K. Molecular dynamics simulation of the specific heat capacity of water-Cu nanofluids. International Nano Letters. 2013;3:1–6.

[14] Mohebbi A. Prediction of specific heat and thermal conductivity of nanofluids by a combined equilibrium and non-equilibrium molecular dynamics simulation. Journal of

[15] Li L, Zhang YW, Ma HB, Yang M. Molecular dynamics simulation of effect of liquid layering around the nanoparticle on the enhanced thermal conductivity of nanofluids.

[16] Lu W-Q, Fan Q-M. Study for the particle's scale effect on some thermophysical properties of nanofluids by a simplified molecular dynamics method. Engineering Analysis

[17] Mingxiang L, Lenore LD. Molecular dynamics simulations of surfactant and nanoparticle self-assembly at liquid–liquid interfaces. Journal of Physics: Condensed Matter.

[18] Rudyak VY, Krasnolutskii SL. Dependence of the viscosity of nanofluids on nanoparticle

food technology. Journal of Nano Research: Trans Tech Publ. 2011;16:89–96.

Experimental Thermal and Fluid Science. 2012;40:1–9.

and Mass Transfer. 2008;35:867–72.

114 Nanofluid Heat and Mass Transfer in Engineering Problems

Physics. 2007;102:074302.

2003;255:799–800.

2007;19:375109.

Tribology. 2010;62:341–48.

Molecular Liquids. 2012;175:51–8.

Journal of Nanoparticle Research. 2010;12:811–21.

size and material. Physics Letters A. 2014;378:1845–9.

with Boundary Elements. 2008;32:282–9.


[48] Bunte SW, Sun H. Molecular modeling of energetic materials: the parameterization and validation of nitrate esters in the COMPASS force field. Journal of Physical Chemistry B. 2000;104:2477–89.

[33] Li L, Zhang Y, Ma H, Yang M. An investigation of molecular layering at the liquidsolid interface in nanofluids by molecular dynamics simulation. Physics Letters A.

[34] Sun C, Lu W-Q, Liu J, Bai B. Molecular dynamics simulation of nanofluid's effective thermal conductivity in high-shear-rate Couette flow. International Journal of Heat and

[35] Lin YS, Hsiao PY, Chieng CC. Roles of nanolayer and particle size on thermophysical characteristics of ethylene glycol-based copper nanofluids. Applied Physics Letters.

[36] Kang H, Zhang Y, Yang M, Li L. Nonequilibrium molecular dynamics simulation of coupling between nanoparticles and base-fluid in a nanofluid. Physics Letters A. 2012;376:521–4.

[37] Loya A, Stair JL, Ren G. The study of simulating metaloxide nanoparticles in aqueous fluid. International Journal of Engineering Research & Technology. 2014;3:1954–60.

[38] Cooke DJ, Elliott JA. Atomistic simulations of calcite nanoparticles and their interaction

[40] Kowsari MH, Alavi S, Ashrafizaadeh M, Najafi B. Molecular dynamics simulation of imidazolium-based ionic liquids. I. Dynamics and diffusion coefficient. Journal of

[41] Spaeth JR, Kevrekidis IG, Panagiotopoulos AZ. A comparison of implicit- and explicitsolvent simulations of self-assembly in block copolymer and solute systems. Journal of

[42] Symeonidis V, Em Karniadakis G, Caswell B. Dissipative particle dynamics simulations of polymer chains: scaling laws and shearing response compared to DNA experiments.

[43] Gao L, Shillcock J, Lipowsky R. Improved dissipative particle dynamics simulations of

[44] Hoogerbrugge PJ, Koelman JMVA. Simulating microscopic hydrodynamic phenomena

[45] Koelman JMVA, Hoogerbrugge PJ. Dynamic simulations of hard-sphere suspensions

[46] Español P, Warren P. Statistical mechanics of dissipative particle dynamics. EPL

[47] Cheng S, Grest GS. Structure and diffusion of nanoparticle monolayers floating at liquid/ vapor interfaces: a molecular dynamics study. Journal of Chemical Physics. 2012;136:214702.

with dissipative particle dynamics. Europhysics Letters. 1992;19:155–60.

nanoparticles. Nano. 2007;02:301–3.

with water. Journal of Chemical Physics. 2007;127:104706–9.

lipid bilayers. Journal of Chemical Physics. 2007;126:015101–8.

under steady shear. Europhysics Letters. 1993;21(3):363–368.

2008;372:4541–4.

2011;98(15):153105.

Mass Transfer. 2011;54:2560–7.

116 Nanofluid Heat and Mass Transfer in Engineering Problems

[39] Hoang VV. Diffusion in simulated SiO<sup>2</sup>

Chemical Physics. 2008;129:224508–13.

Chemical Physics. 2011;134:164902–13.

Physical Review Letters. 2005;95:076001.

(Europhysics Letters). 1995;30:191.


**Application of Nanofluid for Combustion Engine**

## **Enhancing Heat Transfer in Internal Combustion Engine by Applying Nanofluids Enhancing Heat Transfer in Internal Combustion Engine by Applying Nanofluids**

Wenzheng Cui, Zhaojie Shen, Jianguo Yang and Shaohua Wu Wenzheng Cui, Zhaojie Shen, Jianguo Yang and Shaohua Wu

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/65554

#### **Abstract**

Nanofluids exhibit novel properties including significant heat transfer properties that make them potentially useful in internal combustion engine cooling. However, although there is a substantial number of mechanisms proposed, modeling works related to their enhanced thermal conductivity, systematic mechanisms, or models that are suitable for nanofluids are still lacked. With molecular dynamics simulations, thermal conductivities of nanofluids with various nanoparticles have been calculated. Influence rule of various factors for thermal conductivity of nanofluids has been studied. Through defining the ratio of thermal conductivity enhancement by nanoparticle volume fraction, *Κ*, the impacts of nanoparticle properties for thermal conductivity are further evaluated. Furthermore, the ratio of energetic atoms in nanoparticles, *E*, is proposed to be an effective criterion for judging the impact of nanoparticles for the thermal conductivity of nanofluids. Mechanisms of heat conduction enhancement are investigated by MD simulations. Altered microstructure and movements of nanoparticles in the base fluid are proposed to be the main reasons for thermal conductivity enhancement in nanofluids. Both the static and dynamic mechanisms for heat conduction enhancement in nanofluids have been considered to establish a prediction model for thermal conductivity. The prediction results of the present model are in good agreement with experimental results.

**Keywords:** nanofluids, internal combustion engine, heat transfer, mechanism

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## **1. Introduction**

In recent years, nanofluids (NFs) have received much attention due to their strengthening heat transfer properties, which possess important application in heat transfer. The concept of nanofluidswasfirstproposedbyChoiin1995,whichindicates thefluids containingnanometer‐ sizedparticles, callednanoparticles (NPs)[1].Thesefluids are engineeredcolloidal suspensions of nanoparticles in a base fluid. Numerous experimental studies discovered that nanofluids exhibit thermal properties superior to those of base fluid or conventional solid‐liquid suspen‐ sions.Mostofthethermalpropertiesofnanofluidsmeasuredgreatlyexceedthevaluespredicted by classical macroscopic theories and models. Nanofluids possess significantly increased thermal conductivity and improved convective heat transfer coefficient. Therefore, they are potentiallyusefulinmanyenhancedheattransfer application,includingengine cooling,vehicle thermal management, and power battery. Researchers are working to explain the significant high thermal properties of nanofluids [2–4]. However, although there is a substantial number of mechanisms proposed, and modeling works related to their enhanced thermal conductivity, systematic mechanisms, or models that are suitable for nanofluids are still lacked.

Regarding the excellent thermal properties of nanofluids, researchers are interested in the application of nanofluids in internal combustion engine, and began the study of applying nanofluids in internal combustion engine. In 1999, Wambsganss in Argonne national labora‐ tory proposed the idea of applying nanofluids in car engine to improve the vehicle thermal management performance [5]. Choi indicates in a report that in Argonne national laboratory a research program of enhanced heat transfer by nanofluids is launched aiming at the cooling and heat transfer problems in the heavy‐duty engine [6]. The results show that due to the excellent heat transfer performance of nanofluids, the size and weight of the engine can be reduced by 10%. Choi pointed out that the application of nanofluids in engine is one of the best methods of improving heat transfer performance of the cooling system. Saripella et al. studied the heat transfer performance of nanocoolant (nanofluids) in Volvo truck engine, and the results indicate that with nanofluids the temperatures of combustion chamber components and coolant are lowered [7]. Lockwood et al. in Valvoline Company reported the application of nanofluids in the cooling for internal combustion engine [8]. The experiments found that adding 1% vol. carbon nanotube in engine oil could increase the thermal conductivity by 150%. Wallner et al. in Delphi Company found that applying nanofluids can efficiently improve the efficiency of internal combustion engine and decrease the size and weight of the cooling system [9]. Huminic et al. studied the performance of nanofluids in a car radiator with a numerical method and found that the convective heat transfer performance is distinctly better than that of single‐phase fluids [10]. Furthermore, the heat transfer properties of nanofluids are influenced by many factors, including the volume concentrations, temperatures, and fluid velocities. Vajjha et al. reported their research on the flow and heat transfer properties of Al2O3 and CuO nanofluids when applying them in the car radiator [11]. Their results reveal that nanofluids possess improved convective heat transfer properties and the increase rate is increased by increased volume concentrations. Leong et al. found that the heat transfer coefficient and heat transfer rate in the cooling system of internal combustion engine are improved by using nanofluids [12]. Peyghambarzadeh et al. experimentally verified that the application of nanofluids improves heat transfer efficiency of the car radiator by 45% when using Al2O3‐H2O nanofluids [13].

The authors have focused on the application of nanofluids in internal combustion engine (ICE) for heat transfer enhancement. In order to apply nanofluids in ICE, the mechanisms of heat transfer enhancement and the rules of enhanced heat transfer by nanofluids should be clarified first. The original cause of heat transfer enhancement is due to the adding of nanoscale particles. Therefore, we have attempted to use molecular dynamics (MD) simula‐ tions to study these microscopic mechanisms [14]. By using MD simulation, we have calcu‐ lated thermal conductivity of nanofluids via the Green‐Kubo equation and proposed an effective criterion for predicting the enhancement of apparent thermal conductivity. Fur‐ thermore, possible mechanisms of heat conduction increase in nanofluids are studied by MD simulation, including: (1) the micromotions of nanoparticles, (2) changed microstruc‐ ture of base fluid by adding nanoparticles, and (3) the influence of absorption layer of base fluid at the surface of nanoparticles. On the basis of the microscopic mechanisms found by MD simulations, we have also proposed a revised thermal conductivity model, which con‐ sidered both the static and dynamic mechanisms. The revised model is verified by experi‐ mental data, which has been proved to be quite accurate for predicting thermal conductivity of common types of nanofluids.

## **2. Influence rule and criterion for nanofluids' thermal conductivity**

## **2.1. Simulation results of thermal conductivity**

**1. Introduction**

122 Nanofluid Heat and Mass Transfer in Engineering Problems

In recent years, nanofluids (NFs) have received much attention due to their strengthening heat transfer properties, which possess important application in heat transfer. The concept of nanofluidswasfirstproposedbyChoiin1995,whichindicates thefluids containingnanometer‐ sizedparticles, callednanoparticles (NPs)[1].Thesefluids are engineeredcolloidal suspensions of nanoparticles in a base fluid. Numerous experimental studies discovered that nanofluids exhibit thermal properties superior to those of base fluid or conventional solid‐liquid suspen‐ sions.Mostofthethermalpropertiesofnanofluidsmeasuredgreatlyexceedthevaluespredicted by classical macroscopic theories and models. Nanofluids possess significantly increased thermal conductivity and improved convective heat transfer coefficient. Therefore, they are potentiallyusefulinmanyenhancedheattransfer application,includingengine cooling,vehicle thermal management, and power battery. Researchers are working to explain the significant high thermal properties of nanofluids [2–4]. However, although there is a substantial number of mechanisms proposed, and modeling works related to their enhanced thermal conductivity,

systematic mechanisms, or models that are suitable for nanofluids are still lacked.

Regarding the excellent thermal properties of nanofluids, researchers are interested in the application of nanofluids in internal combustion engine, and began the study of applying nanofluids in internal combustion engine. In 1999, Wambsganss in Argonne national labora‐ tory proposed the idea of applying nanofluids in car engine to improve the vehicle thermal management performance [5]. Choi indicates in a report that in Argonne national laboratory a research program of enhanced heat transfer by nanofluids is launched aiming at the cooling and heat transfer problems in the heavy‐duty engine [6]. The results show that due to the excellent heat transfer performance of nanofluids, the size and weight of the engine can be reduced by 10%. Choi pointed out that the application of nanofluids in engine is one of the best methods of improving heat transfer performance of the cooling system. Saripella et al. studied the heat transfer performance of nanocoolant (nanofluids) in Volvo truck engine, and the results indicate that with nanofluids the temperatures of combustion chamber components and coolant are lowered [7]. Lockwood et al. in Valvoline Company reported the application of nanofluids in the cooling for internal combustion engine [8]. The experiments found that adding 1% vol. carbon nanotube in engine oil could increase the thermal conductivity by 150%. Wallner et al. in Delphi Company found that applying nanofluids can efficiently improve the efficiency of internal combustion engine and decrease the size and weight of the cooling system [9]. Huminic et al. studied the performance of nanofluids in a car radiator with a numerical method and found that the convective heat transfer performance is distinctly better than that of single‐phase fluids [10]. Furthermore, the heat transfer properties of nanofluids are influenced by many factors, including the volume concentrations, temperatures, and fluid velocities. Vajjha et al. reported their research on the flow and heat transfer properties of Al2O3 and CuO nanofluids when applying them in the car radiator [11]. Their results reveal that nanofluids possess improved convective heat transfer properties and the increase rate is increased by increased volume concentrations. Leong et al. found that the heat transfer coefficient and heat transfer rate in the cooling system of internal combustion engine are improved by using nanofluids [12]. Peyghambarzadeh et al. experimentally verified that the

MD simulation is used to calculate thermal conductivity of nanofluids via the Green‐Kubo equation [14]. A series of influencing factors for the thermal conductivity of nanofluids have been considered, including: nanoparticles' volume concentrations, sizes, materials, and shapes [15]. In this work, inert liquid Ar is chosen as the base fluid because of the mature and credible potential function. The NP materials include Cu, Ag, Fe, and Au. We have chosen these types of nanoparticles because they are commonly reported in the litera‐ tures on nanofluids. The MD simulation results reveal that the thermal conductivity of nanofluids can be obviously increased by adding nanoparticles, as shown in **Table 1**. How‐ ever, the contributions of several influencing factors for thermal conductivity of nanofluids are different.

**Figure 1** shows the MD simulation results of thermal conductivities for nanofluids contain‐ ing spherical nanoparticles. In this case, the nanoparticle volume fractions, nanoparticle di‐ ameter, and thermal conductivity of nanoparticles are considered. For the influencing factors that have been considered in this work, the influencing rules are regular. The thermal conductivity of nanofluids is increased with increased volume fraction of NPs, decreased NP sizes, and higher thermal conductivity of NPs. For instance, the ratios of thermal con‐ ductivity enhancement for Ag, Cu, Fe, and Au nanofluids are 1.41, 1.15, 1.11, and 1.08 se‐ quentially when the other conditions are the same.


**Table 1.** Ratios of nanofluids' thermal conductivity enhancement with different influence factors.

**Figure 1.** MD simulation results for thermal conductivity of nanofluids.

To examine the influencing rules for thermal conductivity of nanofluids in depth, the ratio of thermal conductivity enhancement by nanoparticle volume fraction, *Κ*, is defined:

$$K = \frac{k \,/\, k\_f - 1}{V\_{\rm np} \,/\, V} \, \text{} \tag{1}$$

where *V*np and *V* are the volume of nanoparticles and nanofluids, respectively; *k* and *kf*represent the thermal conductivities of nanofluids and base fluid, respectively.

**Diameters (nm)**

**Volume fractions**

124 Nanofluid Heat and Mass Transfer in Engineering Problems

**Cu Ag Au Fe** 

2 0.5 1.112 1.326 1.062 1.036

4 0.5 1.095 1.287 1.055 1.041

6 0.5 1.065 1.244 1.049 1.045

To examine the influencing rules for thermal conductivity of nanofluids in depth, the ratio of

*V V* (1)

thermal conductivity enhancement by nanoparticle volume fraction, *Κ*, is defined:

*K*

np / 1 , / - <sup>=</sup> *<sup>f</sup> k k*

**Table 1.** Ratios of nanofluids' thermal conductivity enhancement with different influence factors.

**Figure 1.** MD simulation results for thermal conductivity of nanofluids.

1 1.149 1.405 1.112 1.059 2 1.359 1.498 1.205 1.079 3 1.414 1.570 1.322 1.096

1 1.134 1.325 1.096 1.065 2 1.303 1.396 1.175 1.085 3 1.376 1.462 1.202 1.112

1 1.097 1.265 1.086 1.079 2 1.266 1.326 1.152 1.098 3 1.336 1.369 1.256 1.132

**(%)**

The physical significance of *K* is to evaluate the ratio of thermal conductivity increase and nanoparticle volume fraction. In other words, *K* could be used to evaluate the impact of nanoparticle properties for thermal conductivity. With *K*, the contributions from volume fraction and thermal conductivity of nanoparticles could be compared, as shown in **Figures 2** and **3**. From the figures i,t could be easily found that the contributions of nanoparticle materials are in the order of Ag, Cu, Au, and Fe. Furthermore, through comparing *K* values, it is also found that the influence of nanoparticle materials is weakened when the volume fraction or nanoparticle size is increased. With the help of *K* value the other influencing factors could be further evaluated, please refer to reference [15].

**Figure 2.** Comparison of *K* values against nanoparticle volume fraction for various nanofluids.

**Figure 3.** Comparison of *K* values against nanoparticle size for various nanofluids.

#### **2.2. Criterion for the increased thermal conductivity**

It is found that the nanoparticles containing higher atomic potential energy (energetic atoms) are better for thermal conductivity enhancement of nanofluids [15]. The ratio of energetic atoms in a nanoparticle *E* is proposed as a criterion for enhanced thermal conductivity of nanofluids, which is written as:

$$E = \frac{N\_E}{N} \,\prime \tag{2}$$

where *N* and *NE* are the quantity of atoms and energetic atoms in a nanoparticle, respectively.

If we set a standard for delimiting the energetic atoms in a nanoparticle, *E* can be calculated according to Eq. (2). **Figure 4** illustrates *E* of different types of nanoparticles. The ratio of energetic atoms in an Ag nanoparticle is the largest, and that of Fe nanoparticle is the lowest. The larger *E* value in a nanoparticle is better for thermal conductivity enhancement in nanofluids. **Figure 5** shows the influence of nanoparticle shapes (surface area to volume ratio S/V) for the *E* value. It is found that nanoparticle with larger S/V value possesses larger *E* value. **Figure 6** illustrates that the *E* value of spherical Ag nanoparticle is larger than that of Cu nanoparticle with larger S/V value. Therefore, the thermal conductivity of nanofluids with spherical Ag nanoparticles is higher than that of nanofluids with nonspherical Cu nanoparti‐ cles.

**Figure 4.** Atomic potential energy distributions of various nanoparticles.

Enhancing Heat Transfer in Internal Combustion Engine by Applying Nanofluids http://dx.doi.org/10.5772/65554 127

**2.2. Criterion for the increased thermal conductivity**

126 Nanofluid Heat and Mass Transfer in Engineering Problems

**Figure 4.** Atomic potential energy distributions of various nanoparticles.

nanofluids, which is written as:

cles.

It is found that the nanoparticles containing higher atomic potential energy (energetic atoms) are better for thermal conductivity enhancement of nanofluids [15]. The ratio of energetic atoms in a nanoparticle *E* is proposed as a criterion for enhanced thermal conductivity of

where *N* and *NE* are the quantity of atoms and energetic atoms in a nanoparticle, respectively. If we set a standard for delimiting the energetic atoms in a nanoparticle, *E* can be calculated according to Eq. (2). **Figure 4** illustrates *E* of different types of nanoparticles. The ratio of energetic atoms in an Ag nanoparticle is the largest, and that of Fe nanoparticle is the lowest. The larger *E* value in a nanoparticle is better for thermal conductivity enhancement in nanofluids. **Figure 5** shows the influence of nanoparticle shapes (surface area to volume ratio S/V) for the *E* value. It is found that nanoparticle with larger S/V value possesses larger *E* value. **Figure 6** illustrates that the *E* value of spherical Ag nanoparticle is larger than that of Cu nanoparticle with larger S/V value. Therefore, the thermal conductivity of nanofluids with spherical Ag nanoparticles is higher than that of nanofluids with nonspherical Cu nanoparti‐

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

<sup>=</sup> , *NE <sup>E</sup>*

**Figure 5.** Comparison of atomic potential energy distributions for nanoparticles with different shapes.

**Figure 6.** Comparison of atomic potential energy distributions for nanoparticles with different materials.

## **3. Proposed mechanisms of heat conduction in nanofluids**

## **3.1. Altered microstructure of nanofluids**

In order to analyze the microscopic structure characteristics of nanofluids, number density distribution, radial distribution function (RDF), coordination number, and potentials of mean force (PMF) should be considered [16].

Number density distribution represents the distribution of liquid atoms around a centered nanoparticle. **Figure 7** illustrates the number density distributions of base fluid atoms in different types of nanofluids. It is found that at the positions of 0.25 and 0.5 nm all the curves show the first and second peak values. But for different types of nanoparticles, the first peak values of curves are different. The order of first peak values is of the same order of thermal conductivity of bulk materials of nanoparticles.

**Figure 7.** Statistical result of absorption layers around nanoparticles of different materials.

RDF represents the probability of finding an atom of a specified type near the central atom. Through RDF, the microscopic structure of fluid could be examined. **Figure 8** illustrates the RDF of Cu nanofluids with a 2 nm‐diameter nanoparticle. In the figure, the RDF curve of "Ar‐ Ar" represents the chance of finding an Ar atom near the central Ar atom. It could be found that the Ar‐Ar RDF shows typical characteristics of the liquid: "short‐range order and long‐ range disorder." In the figure, the "total" RDF represents the chance of finding an atom of any type near the central atom, which represents the microscopic structure of nanofluid. It is found that the first curvilinear peak in the RDF of nanofluids is larger than that of base fluid, which means the probability of finding an atom is higher than that in a single‐phase base fluid. It could also be found that there are several diminutive curvilinear peaks in the RDF of nano‐ fluids, which is due to the adding of nanoparticles in base fluid. In general, the microscopic structure of nanofluid exerts a mixed up structure characteristics of liquid and solid. Both the liquid characteristic of "short‐range order and long‐range disorder" and the solid character‐ istic of "long‐range order" have been found in the microscopic structure of nanofluids. Therefore, the microscopic structure of nanofluids is always ordered.

**3. Proposed mechanisms of heat conduction in nanofluids**

**Figure 7.** Statistical result of absorption layers around nanoparticles of different materials.

RDF represents the probability of finding an atom of a specified type near the central atom. Through RDF, the microscopic structure of fluid could be examined. **Figure 8** illustrates the RDF of Cu nanofluids with a 2 nm‐diameter nanoparticle. In the figure, the RDF curve of "Ar‐ Ar" represents the chance of finding an Ar atom near the central Ar atom. It could be found that the Ar‐Ar RDF shows typical characteristics of the liquid: "short‐range order and long‐ range disorder." In the figure, the "total" RDF represents the chance of finding an atom of any type near the central atom, which represents the microscopic structure of nanofluid. It is found that the first curvilinear peak in the RDF of nanofluids is larger than that of base fluid, which

In order to analyze the microscopic structure characteristics of nanofluids, number density distribution, radial distribution function (RDF), coordination number, and potentials of mean

Number density distribution represents the distribution of liquid atoms around a centered nanoparticle. **Figure 7** illustrates the number density distributions of base fluid atoms in different types of nanofluids. It is found that at the positions of 0.25 and 0.5 nm all the curves show the first and second peak values. But for different types of nanoparticles, the first peak values of curves are different. The order of first peak values is of the same order of thermal

**3.1. Altered microstructure of nanofluids**

128 Nanofluid Heat and Mass Transfer in Engineering Problems

force (PMF) should be considered [16].

conductivity of bulk materials of nanoparticles.

**Figure 8.** Radial distribution functions of nanofluids with spherical copper nanoparticle.

Coordinate number indicates the average adjacent atomic number for a certain atom within an interval of *r*. By comparing the coordinate number curve of Cu nanofluids with a 2 nm‐ diameter spherical nanoparticle, it is found that the coordinate number curves of Cu‐Ar and Ar‐Ar cross at 0.35 nm, which confirms that the local density of Ar atoms near a Cu atom is larger (**Figure 9**). Therefore, the existence of absorption layer is verified.

PMF reflects the combining capacity between particles in pairs. The value of PMF could be used to investigate the combining capacity between different particle pairs. **Figure 10** shows the PMF curve for Cu‐Ar nanofluids. The contact minimum (CM), separated minimum (SM), and the layer barrier (LB) could easily be found in the PMF curve of nanofluids. But the positions and values of CM, SM, and LB are different for disparate atom pairs. The cis‐trans direction of energy barrier between molecular layers of liquid Ar is different. When an atom is approaching the central particle, then it needs to conquer the energy barrier between the first and second molecular layers. But the atom is harder to leave the central particle. An atom of base fluid needs to conquer greater energy barrier to reenter the base fluid. The PMF of nanofluids is different from that of base fluid. At 0.3 nm, there is a huge energy barrier in the PMF of nanofluids, which indicates the surrounding atom needs to conquer two energy barriers to get close to the central atom. The cis‐trans direction of the first energy barrier is nearly the same, but the cis‐trans direction of the second energy barrier is obviously different. Once a base fluid atom enters the adjacent area of the central atom, then it is very hard to reenter the base fluid because of the large energy barrier.

**Figure 9.** The coordination number of nanofluids with spherical copper nanoparticle.

**Figure 10.** PMF of nanofluids with spherical copper nanoparticle.

### **3.2. Movements of nanoparticles in the base fluid**

barriers to get close to the central atom. The cis‐trans direction of the first energy barrier is nearly the same, but the cis‐trans direction of the second energy barrier is obviously different. Once a base fluid atom enters the adjacent area of the central atom, then it is very hard to

reenter the base fluid because of the large energy barrier.

130 Nanofluid Heat and Mass Transfer in Engineering Problems

**Figure 9.** The coordination number of nanofluids with spherical copper nanoparticle.

**Figure 10.** PMF of nanofluids with spherical copper nanoparticle.

Through MD simulation, the nanoparticles are observed to move chaotically at high speed in the base fluid. Through MD simulation, the instantaneous velocity and position coordinates of each atom could be obtained [17]. The translational and rotational velocity of nanoparticles could be acquired by defining a group for the Cu atoms within the nanoparticle. With commands provided by LAMMPS the time‐averaged translational and rotational velocity of the atom group could be calculated and output derived. For the case of imposed shearing velocity *v* = 50 m/s on the fluid, the translational velocity components of nanoparticles are statistically analyzed, as shown in **Figure 11**. Along *x*‐directions, the average translational velocity components are ‐2 m/s∼2 m/s, and the instantaneous peak value can reach 5 m/s. The translational velocity components of nanoparticles oscillate sharply, which demonstrate the chaotic movements of nanoparticles are mainly caused by their Brownian motion.

**Figure 11.** Translational velocity component of nanoparticle along *x*‐axis in shearing flow.

Rotation of nanoparticles is also statistically analyzed. For the case of imposed shearing velocity 50 m/s, the angular velocity component along the *x*‐axis of nanoparticle is shown in **Figure 12**. The peak angular velocity of nanoparticles can reach 6 × 109 rad/s, meanwhile the rotational directions of nanoparticles change randomly because of their nanoscale size. It is found that imposing shearing velocity affects little for rotation of nanoparticles. Imposed shearing velocity or not, the angular velocity components of nanoparticles are of the order of magnitude of 109 rad/s.

**Figure 12.** Angular velocity component of nanoparticle around *x*‐axis.

## **4. Modeling thermal conductivity of nanofluids**

Jeffrey applied Green's function method and relaxed the requirement of uniform configuration for particles. The formula, which is suitable for predicting suspensions with nonuniformly distributing nanoparticles and relatively large volume concentration, is written as [18],

$$\frac{k}{k\_f} = 1 + 3\beta \theta \phi\_p + 3\beta \left(\beta + \Sigma\right) \phi\_p^{-2},\tag{3}$$

where ∑ is a convergent series, which depends on the specific value of thermal conductivity of nanoparticle and base fluid /. *β* is a coefficient, which is determined by /:

$$
\beta = \frac{\alpha - 1}{\alpha + 2},
\tag{4}
$$

where *α* is the specific value of thermal conductivity of nanoparticle and base fluid /:

Enhancing Heat Transfer in Internal Combustion Engine by Applying Nanofluids http://dx.doi.org/10.5772/65554 133

$$\alpha = \frac{k\_m}{k\_f}.\tag{5}$$

Through considering the above‐mentioned mechanisms of thermal conductivity enhancement in nanofluids, a revised model for predicting thermal conductivity of nanofluids that takes into account both the static and dynamic mechanisms is proposed, which is written as[19]:

$$\frac{k\_{\eta'}}{k\_f} = \frac{k\_S}{k\_f} + \frac{k\_D}{k\_f} = 1 + 3\beta \left( 1 + \frac{2t\_{ab}}{d\_\rho} \right)^3 \phi\_\rho + c \left( 1 + \frac{2t\_{ab}}{d\_\rho} \right)^6 \phi\_\rho^{-2} + \frac{\phi\_\rho \left( \rho c\_\rho \right)\_\rho}{2k\_f} \sqrt{\frac{k\_g T}{3\pi r\_{cl}\mu\_f}} \tag{6}$$

In the equation, *k*nf is thermal conductivity of nanofluids; *kf* is thermal conductivity of base fluid; *kS* is the static part of thermal conductivity model for nanofluids, which takes into account the static mechanisms of heat conduction enhancement; *kD* is the static part of thermal conductivity model for nanofluids; = 1 / + 2 and = cl , where *kcl* is thermal conductivity of nanoparticle cluster; *t*ab is the thickness of absorption layer formed by absorbed liquid molecules which can be estimated by Langmuir monolayer equation; *dp* is the average diameter of nanoparticles; *Φp* is the initial volume concentration of nano‐ particles in nanofluids; = 32 + 3 is a coefficient in Jeffrey Model; *ρ* is the density of nanoparticle bulk material; *cp* is the specific heat of nanoparticle bulk material; *kB* is the Boltzmann constant and = 1.381 × 1023/; *T* is thermodynamic temperature; *rcl* is the radius of the nanoparticle cluster; and *μf* is the viscosity of base fluid.

**Figure 12.** Angular velocity component of nanoparticle around *x*‐axis.

132 Nanofluid Heat and Mass Transfer in Engineering Problems

**4. Modeling thermal conductivity of nanofluids**

*f k k*

Jeffrey applied Green's function method and relaxed the requirement of uniform configuration for particles. The formula, which is suitable for predicting suspensions with nonuniformly distributing nanoparticles and relatively large volume concentration, is written as [18],

( ) <sup>2</sup> = + + +å 13 3 , *p p*

of nanoparticle and base fluid /. *β* is a coefficient, which is determined by /:

a

 b b

where ∑ is a convergent series, which depends on the specific value of thermal conductivity

1 , 2 - = + a b

where *α* is the specific value of thermal conductivity of nanoparticle and base fluid /:

 f

(3)

(4)

bf

Compared with existing prediction models for thermal conductivity of nanofluids, the present model takes into account the static and dynamic mechanisms of strengthened heat conduction in nanofluids simultaneously and possesses more definite physics meaning. In addition, parameters used in the current model are more precise that ensures the veracity of prediction result. For instance, the thermal conductivity of nanoparticles *kp* is distinguished with that of bulk material of nanoparticles *km* and used in the prediction model as an independent param‐ eter. And the thermal conductivity of nanoparticles‐cluster *k*cl is introduced as an independent parameter to include the heat conduction of absorption layer, which further improves the prediction accuracy of the present model.

Through comparing the prediction results of the present model and existing experimental data, the present prediction model is proved to be quite effective for predicting thermal conductivity of common nanofluids, as shown in **Table 2**. For various types of nanofluids (with different materials including: metal, metallic oxide, and nonmetallic oxide, different volume fractions, or different nanoparticle diameters), the present model gives good pre‐ dictions.


**Table 2.** Comparison between the prediction results and experimental data.

## **5. Concluding remarks**

Thermal conductivities of nanofluids with various nanoparticles have been calculated through MD simulations. Influence rule of various factors for thermal conductivity of nanofluids has been studied. Through defining the ratio of thermal conductivity enhancement by nanoparticle volume fraction, *Κ*, the impacts of nanoparticle properties for thermal conductivity are further evaluated. Furthermore, the ratio of energetic atoms in nanoparticles, *E*, is proposed to be an effective criterion for judging the impact of nanoparticles for the thermal conductivity of nanofluids.

Mechanisms of heat conduction enhancement are investigated by MD simulations. Altered microstructure and movements of nanoparticles in the base fluid are proposed to be the main reasons for thermal conductivity enhancement in nanofluids. Number density distribution, radial distribution function (RDF), coordination number, and potentials of mean force (PMF) are used to analyze the microscopic structure characteristics of nanofluids. Through MD simulation, the average translational and rotational velocities of nanoparticles are obtained.

Both the static and dynamic mechanisms for heat conduction enhancement in nanofluids have been considered to establish a prediction model for thermal conductivity. The parameters in the model have definite physical meaning and are more precise. The prediction results of the present model are in good agreement with experimental results.

## **Acknowledgements**

**References NF NP size**

134 Nanofluid Heat and Mass Transfer in Engineering Problems

(60:40% EG/W)

**Table 2.** Comparison between the prediction results and experimental data.

Vajjha [24] Al2O3/

**5. Concluding remarks**

nanofluids.

**(nm)**

**Volume fraction (%)**

Masuda [20] Al2O3/Water 13 1.3–4.3 1.107–1.318 1.097–1.322 0.9% Teng et al. [21] Al2O3/Water 20 0.001–0.005 1.018–1.065 1.010–1.043 2.1% Chon et al. [22] Al2O3/Water 13 1 1.081 1.072 0.9% Xie et al. [23] Al2O3/EG 25 1.7–5 1.097–1.294 1.103–1.303 0.8%

Wang et al. [25] Al2O3/EG 28 5–8 1.246–1.404 1.276–1.445 2.9% Eastman et al. [26] CuO/EG 35 1–4 1.050–1.227 1.061–1.245 3.7% Lee et al. [27] CuO/EG 24 1–4 1.060–1.242 1.047–1.212 3.2% Xuan and Li [28] Cu/Water 100 1–5 1.078–1.434 1.090–1.459 3.1% Li et al. [29] Cu/Water 20 1–3 1.120–1.289 1.096–1.293 2.1% Eastman et al. [26] Cu/EG 10 0.33–0.55 1.041–1.101 1.061–1.122 2% Hwang et al. [30] CuO/Water 33 1 1.05 1.08 2.9% Zhang et al. [31] SiO2/Water 7 0.5–3 1.016–1.084 1.014–1.088 0.4% Hwang et al. [30] SiO2/Water 12 1 1.03 1.03 0

Thermal conductivities of nanofluids with various nanoparticles have been calculated through MD simulations. Influence rule of various factors for thermal conductivity of nanofluids has been studied. Through defining the ratio of thermal conductivity enhancement by nanoparticle volume fraction, *Κ*, the impacts of nanoparticle properties for thermal conductivity are further evaluated. Furthermore, the ratio of energetic atoms in nanoparticles, *E*, is proposed to be an effective criterion for judging the impact of nanoparticles for the thermal conductivity of

Mechanisms of heat conduction enhancement are investigated by MD simulations. Altered microstructure and movements of nanoparticles in the base fluid are proposed to be the main reasons for thermal conductivity enhancement in nanofluids. Number density distribution, radial distribution function (RDF), coordination number, and potentials of mean force (PMF) are used to analyze the microscopic structure characteristics of nanofluids. Through MD simulation, the average translational and rotational velocities of nanoparticles are obtained.

Both the static and dynamic mechanisms for heat conduction enhancement in nanofluids have been considered to establish a prediction model for thermal conductivity. The parameters in

**Thermal conductivity ratio (***k***nf/***kf* **)**

53 1–4 1.069–1.159 1.048–1.195 3.1%

**Present model Maximum error**

The authors have received funds from the National Natural Science Foundation of China (Grant Nos. 51506038, 51606052), Shandong Provincial Natural Science Foundation, China (Grant No. ZR2015EQ003), and China Postdoctoral Science Foundation (Grant Nos. 2016T90284, 2015M571411). We acknowledge the reviewers' comments and suggestions very much. Many thanks to INTECH Publishing and the editor and staffs who helped us publish this chapter.

## **Nomenclature**


## **Author details**

Wenzheng Cui1\*, Zhaojie Shen1 , Jianguo Yang1 and Shaohua Wu2

\*Address all correspondence to: cuiwenzheng@hit.edu.cn


## **References**


[10] Huminic G, Huminic A. The cooling performances evaluation of nanofluids in a compact heat exchanger. In: SAE Technical Paper; 2012. Paper number: 2012‐01‐1045. DOI: 10.4271/2012‐01‐1045

**Author details**

**References**

5978

2004.07.012

10.4271/2007‐01‐2141

01‐2141. DOI: 10.4271/2007‐01‐2141

Wenzheng Cui1\*, Zhaojie Shen1

136 Nanofluid Heat and Mass Transfer in Engineering Problems

, Jianguo Yang1

1 School of Automotive Engineering, Harbin Institute of Technology, Weihai, China

2 School of Energy Science and Engineering, Harbin Institute of Technology, Harbin, China

[1] Lee S, Choi SUS, Li S, Eastman JA. Measuring thermal conductivity of fluids containing oxide nanoparticles. Journal of Heat Transfer. 1999;121(2):280–289. DOI: 10.1115/1.282

[2] Wen DS, Ding YL. Experimental investigation into convective heat transfer of nano‐ fluids at the entrance region under laminar flow conditions. International Journal of Heat and Mass Transfer. 2004;47(24):5181–‐5188. DOI: 10.1016/j.ijheatmasstransfer.

[3] Hussein AM, Sharma KV, Bakar RA, Kadirgama K. A review of forced convection heat transfer enhancement and hydrodynamic characteristics of a nanofluid. Renewable and

[4] Sundar LS, Singh MK. Convective heat transfer and friction factor correlations of nanofluid in a tube and with inserts: A review. Renewable and Sustainable Energy

[5] Wambsganss MW. Thermal management concepts for higher‐efficiency heavy vehicles. In: SAE Technical Paper ; 1999. Paper number: 1999‐01‐2240. DOI: 10.4271/1999‐01‐2240

[6] Choi SUS. Nanofluids for improved efficiency in cooling systems. In: Heavy Vehicle

[7] Saripella SK, Yu W, Routbort JL, France DM. Effects of nanofluid coolant in a class 8 truck engine. In: SAE Technical Paper; 2007. Paper number: 2007‐01‐2141. DOI:

[8] Lockwood FE, Zhang ZG, Forbus TR, Choi SUS, Yang Y, Grulke EA. The current development of nanofluid research. In: SAE Technical Paper; 2005. Paper number: 2007‐

[9] Wallner E, Sarma DHR, Myers B, Shah S, Ihms D, Chengalva S, Parker R, Eesley G, Dykstra C. Nanotechnology application in future automobiles. In: SAE Technical Paper;

2010. Paper number: 2010‐01‐1149. DOI: 10.4271/2010‐01‐1149

Sustainable Energy Reviews. 2014;29:734–743. DOI: 10.1016/j.rser.2013.08.014

Reviews. 2013;20:23–35. DOI: 10.1016/j.rser.2012.11.041

Systems Review; April 18–20,2006; Chicago. 2006.

\*Address all correspondence to: cuiwenzheng@hit.edu.cn

and Shaohua Wu2

