**Hydrodynamic and Heat Transfer Simulation of Fluidized Bed Using CFD**

Osama Sayed Abd El Kawi Ali

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/52072

## **1. Introduction**

The nuclear energy is suffering from the lack of public acceptance everywhere mainly due to the issues relating to reactor safety, economy and nuclear waste. The Fluidized Bed Nuclear Reactor (FBNR) concept has addressed these issues and tried to resolve such problems. The FBNR is small, modular and simple in design contributing to the economy of the reactor. It has inherent safety and passive cooling characteristics. Its spent fuel being small spherical elements may not be considered nuclear waste, and can be directly used as a source of radiation for applications in industry and agriculture resulting in reduced environmental impact [1].

With the increase of computational power, numerical simulation becomes an additional tool to predict the fluid dynamics and the heat transfer mechanism in multiphase flow. A numerical hydrodynamic and heat transfer model has been developed to simulate the gas fluidized bed. All of CFD, in one form or another, is based on the fundamental governing equations of fluid dynamics (continuity, momentum and energy equations). These equa‐ tions speak physics. They are mathematical statements of three fundamental physical principals upon which all fluid dynamics is based: mass is conserved, Newton's second law and energy is conserved [2].

This chapter aims to study a mathematical modeling and numerical simulation of the hydro‐ dynamics and heat transfer processes in a two-dimensional gas fluidized bed with a vertical uniform gas velocity at the inlet. The velocity, volume fraction, temperature distribution for gas phase and particle phase are calculated. Also, gas pressure and a prediction of the average heat transfer coefficient are also studied.

Such a simulation technique allows performance evaluation for different bed input parame‐ ters, and can evolve into a tool for optimized design of fluidized beds for different industrial use.

© 2013 Abd El Kawi Ali; licensee InTech. This is an open access article 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. © 2013 Abd El Kawi Ali; licensee InTech. This is a paper 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.

The numerical setup consists of a two dimensional fluidized bed filled with particles.The cold gas enters to the bed to cool the hot particles. Based on conservation equations for both phases it is possible to predict particles and gas volume fractions, velocity distributions (for gas and particles), temperature distribution, heat transfer coefficient as well as gas pressure field.

## **2. General assumptions for the mathematical model**

Fluidized beds are categorized as multiphase flow problems. There are currently two ap‐ proaches to model multiphase flow problems as discussed in chapter two. The best overall balance between computational time and accuracy seems to be achieved by implementing an Eulerian-Eulerian approach. The following assumptions are introduced into the present analysis:

**Figure 1.** The schematic diagram of the present work model

( ) ( )0 *<sup>s</sup> ss ss u v*

( ) ( )0 *<sup>g</sup> gg gg u v*

++=

1.0 *g s*

( ) ( ) "x -direction " *<sup>s</sup>*

( ) ( ) "y -direction " *<sup>s</sup>*

¶¶ ¶ (4)

¶¶ ¶ (5)

++=

 e

> e

¶¶ ¶ (1)

Hydrodynamic and Heat Transfer Simulation of Fluidized Bed Using CFD

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157

¶¶ ¶ (2)

+ = (3)

*tx y*

*tx y*

e

¶ ¶ ¶

e e

*s s s s s s s s s s sx <sup>u</sup> uu vu F*

*s s s s s s s s s s sy <sup>v</sup> uv vv F*

 re

 re

++=

++=

*tx y*

*tx y*

¶ ¶ ¶

¶ ¶ ¶

e

¶ ¶ ¶

e

e

**3.1. Continuity equation**

**3.2. Volume fraction constraint**

**3.3. Momentum equation**

re

re

 re

 re

Particle phase:

Particle phase:

Gas phase :


## **3. Governing equations**

Due to the high particle concentration in fluidized beds the particles interactions cannot be neglected. In fact the solid phase has similar properties as continuous fluid. Therefore, the Eulerian approach is an efficient method for the numerical simulation of fluidized beds.

A hydrodynamic and thermal model for the fluidized bed is developed based on schematic diagram shown in Figure (1). The principles of conservation of mass, momentum, and energy are used in the hydrodynamic and thermal models of fluidization. The general mass conser‐ vation equations and the separate phase momentum equations and energy equations (for each phase) for fluid–solids, nonreactive transient and two-phase flow will be discussed in the following sections.

Hydrodynamic and Heat Transfer Simulation of Fluidized Bed Using CFD http://dx.doi.org/10.5772/52072 157

**Figure 1.** The schematic diagram of the present work model

#### **3.1. Continuity equation**

Particle phase:

The numerical setup consists of a two dimensional fluidized bed filled with particles.The cold gas enters to the bed to cool the hot particles. Based on conservation equations for both phases it is possible to predict particles and gas volume fractions, velocity distributions (for gas and particles), temperature distribution, heat transfer coefficient as well as gas pressure field.

Fluidized beds are categorized as multiphase flow problems. There are currently two ap‐ proaches to model multiphase flow problems as discussed in chapter two. The best overall balance between computational time and accuracy seems to be achieved by implementing an Eulerian-Eulerian approach. The following assumptions are introduced into the present

**8.** The expanded bed region is considered in the analysis in addition to the out flowing gas,

**9.** Viscous heat dissipation in the energy equation is negligible in comparison with conduc‐

Due to the high particle concentration in fluidized beds the particles interactions cannot be neglected. In fact the solid phase has similar properties as continuous fluid. Therefore, the Eulerian approach is an efficient method for the numerical simulation of fluidized beds.

A hydrodynamic and thermal model for the fluidized bed is developed based on schematic diagram shown in Figure (1). The principles of conservation of mass, momentum, and energy are used in the hydrodynamic and thermal models of fluidization. The general mass conser‐ vation equations and the separate phase momentum equations and energy equations (for each phase) for fluid–solids, nonreactive transient and two-phase flow will be discussed in the

**2. General assumptions for the mathematical model**

**6.** There is no mass transfer or chemical reaction between the two phases.

**7.** All particles are spherical in shape with the same diameter,dp.

analysis:

**1.** The bed is two-dimensional.

**4.** Uniform fluidization.

**5.** Constant input fluid flux.

tion and convection.

**3. Governing equations**

following sections.

**2.** Eulerian-Eulerian approach is applied.

156 Nuclear Reactor Thermal Hydraulics and Other Applications

**3.** The gas has constant physical properties.

i.e. suspension and free board regions.

$$\frac{\partial \mathcal{E}\_s}{\partial t} + \frac{\partial}{\partial \mathbf{x}} (\mathcal{E}\_s \boldsymbol{\mu}\_s) + \frac{\partial}{\partial y} (\mathcal{E}\_s \boldsymbol{\upsilon}\_s) = 0 \tag{1}$$

Gas phase :

$$\frac{\partial \mathcal{E}\_{\mathcal{S}}}{\partial t} + \frac{\partial}{\partial \mathbf{x}} (\mathcal{E}\_{\mathcal{S}} \boldsymbol{u}\_{\mathcal{S}}) + \frac{\partial}{\partial y} (\mathcal{E}\_{\mathcal{S}} \boldsymbol{v}\_{\mathcal{S}}) = 0 \tag{2}$$

#### **3.2. Volume fraction constraint**

$$
\varepsilon\_g + \varepsilon\_s = 1.0\tag{3}
$$

#### **3.3. Momentum equation**

Particle phase:

$$
\rho\_s \varepsilon\_s \frac{\partial u\_s}{\partial t} + \rho\_s \varepsilon\_s \frac{\partial}{\partial x} (u\_s u\_s) + \rho\_s \varepsilon\_s \frac{\partial}{\partial y} (v\_s u\_s) = F\_{sx} \text{ \textquotedbl{}x\textquotedbl{}direction\textquotedbl{}}\tag{4}
$$

$$
\rho\_s \varepsilon\_s \frac{\partial v\_s}{\partial t} + \rho\_s \varepsilon\_s \frac{\partial}{\partial x} (u\_s v\_s) + \rho\_s \varepsilon\_s \frac{\partial}{\partial y} (v\_s v\_s) = F\_{sy} \text{ \textquotedbl{}y\textquotedbl{}direction\textquotedbl{}}\tag{5}
$$

The total force acting on particle phase is the sum of the net primary force and the force resulting from particle phase elasticity. The x and y components of forces acting on particle phase are as following:

$$\begin{aligned} F\_{sy} &= C\_d \frac{3\varepsilon\_s \rho\_g (\upsilon\_g - \upsilon\_s) \Big| \upsilon\_g - \upsilon\_s \Big|}{4d\_p} (1 - \varepsilon\_s)^{-1.8} - \varepsilon\_s \rho\_s g \\ &- \varepsilon\_s \frac{\partial p}{\partial y} - \frac{\partial \varepsilon\_s}{\partial y} (3.2 \, g d\_p \varepsilon\_s (\rho\_s - \rho\_g)) \end{aligned} \tag{6}$$

tion led to the relation linking fluid and particle phase velocities at all location. By applying

( ) ( )0 *ss gg ss gg uu vv*

*V vv gin g g s s* = + e

*U uu gin g g s s* = + e

 ee

Equation (12) shows that the total flux (fluid plus particles) in x- direction and y-direction remains constant, equal to that of fluid entering the bed Ugin,Vgin. This result is a simple

> e

 e

Equations (13) and (14) enable the fluid velocity variables to be expressed in terms of the

In this section the combined momentum equation is produced by combining the fluid and particle momentum equations (4), (5), (8) and (9) by elimination of the fluid pressure gradient,

( ) ( ) ( ) ( )0 *s s*

*uu vu uu vu*

() () *<sup>g</sup> sy gy*

Equations (15) and (16) with the continuity equation for the two phase (1) and (2),now define

*v F F uv vv*

é¶ ¶ ¶¶ ¶ ¶ ù é <sup>ù</sup> <sup>ê</sup> + + ++ + = ú ê <sup>ú</sup> ¶¶ ¶ ¶¶ ¶ <sup>ë</sup> û ë <sup>û</sup> (15)

*s g*

(16)

e e

*s ss ss g gg gg*

 r

*tx y tx y*

¶ ¶ (12)

Hydrodynamic and Heat Transfer Simulation of Fluidized Bed Using CFD

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159

(13)

(14)

++ +=

the overall mass balance, which is obtained by summing equations (1) and (2):

*x y* ee

consequence of the particles and fluid being considered incompressible :

which appears in them. This yields the combined momentum equation:

[divide equation (4-4) by εs ] + [divide equation (4-8) by εg] which give us :

[divide equation (4-3) by εs ] – [divide equation (4-7) by εg] which give us :

*g gg gg*

é ù ¶ ¶ ¶ - + + =- ê ú ¶¶ ¶ ë û

*tx y*

the two phase system without account of fluid pressure variation.

*u u*

particle velocity at all points in the bed.

*3.3.2. Combined momentum equation*

x- direction:

y- direction:

*ρs* ∂*vs* <sup>∂</sup>*<sup>t</sup>* <sup>+</sup> r

∂ <sup>∂</sup> *<sup>x</sup>* (*usvs*) <sup>+</sup>

∂ <sup>∂</sup> *<sup>y</sup>* (*vsvs*)

r

¶ ¶

$$F\_{\rm sx} = C\_d \frac{3\varepsilon\_s \rho\_g (u\_g - u\_s) \left| u\_g - u\_s \right|}{4d\_p} (1 - \varepsilon\_s)^{-1.8} - \varepsilon\_s \frac{\partial p}{\partial \mathbf{x}} \tag{7}$$

Gas phase :

$$
\rho\_{\mathcal{S}} \varepsilon\_{\mathcal{S}} \frac{\partial u\_{\mathcal{S}}}{\partial t} + \rho\_{\mathcal{S}} \varepsilon\_{\mathcal{S}} \frac{\partial}{\partial \mathbf{x}} (u\_{\mathcal{S}} u\_{\mathcal{S}}) + \rho\_{\mathcal{S}} \varepsilon\_{\mathcal{S}} \frac{\partial}{\partial y} (v\_{\mathcal{S}} u\_{\mathcal{S}}) = \mathbf{F}\_{\mathbf{g} \mathbf{x}} \text{ "x-direction "y"} \tag{8}
$$

$$
\rho\_g \varepsilon\_g \frac{\partial v\_g}{\partial t} + \rho\_g \varepsilon\_g \frac{\partial}{\partial \mathbf{x}} (u\_g v\_g) + \rho\_g \varepsilon\_g \frac{\partial}{\partial y} (v\_g v\_g) = F\_{gy} \text{ \textquotedbl{}y-direction\textquotedbl{}}\tag{9}
$$

The fluid phase forces are readily obtained from the particle phase relations for fluid – particle interaction (drag and pressure gradient force), which act in the opposite direction on the fluid, together with gravity. The x and y components of forces acting on gas phase are as following:

$$F\_{gy} = -C\_d \frac{3\varepsilon\_s \rho\_g (v\_g - v\_s) \left| v\_g - v\_s \right|}{4d\_p} (1 - \varepsilon\_s)^{-1.8} - \varepsilon\_g \rho\_s g - \varepsilon\_g \frac{\partial p}{\partial y} \tag{10}$$

$$F\_{gx} = -F\_{sx} = -\mathbf{C}\_d \frac{3\boldsymbol{\sigma}\_s \rho\_g (\boldsymbol{u}\_g - \boldsymbol{u}\_s) \Big| \boldsymbol{u}\_g - \boldsymbol{u}\_s \Big|}{4d\_p} (\mathbf{1} - \boldsymbol{\sigma}\_s)^{-1.8} + \boldsymbol{\varepsilon}\_s \frac{\partial p}{\partial \mathbf{x}} \tag{11}$$

#### *3.3.1. Relation between fluid and particle velocities*

We assume that both particles and fluid are regarded as being incompressible. This was justified on the basis that only a gas phase is going to exhibit any significant compressibility, and the orders of magnitude differences in particle and fluid density for gas fluidization render quite insignificant the small change in gas density resulting from compression. This assump‐ tion led to the relation linking fluid and particle phase velocities at all location. By applying the overall mass balance, which is obtained by summing equations (1) and (2):

$$\frac{\partial}{\partial \mathbf{x}} (\varepsilon\_s \boldsymbol{u}\_s + \varepsilon\_g \boldsymbol{u}\_g) + \frac{\partial}{\partial y} (\varepsilon\_s \boldsymbol{v}\_s + \varepsilon\_g \boldsymbol{v}\_g) = 0 \tag{12}$$

Equation (12) shows that the total flux (fluid plus particles) in x- direction and y-direction remains constant, equal to that of fluid entering the bed Ugin,Vgin. This result is a simple consequence of the particles and fluid being considered incompressible :

$$V\_{gin} = \varepsilon\_g \upsilon\_g + \varepsilon\_s \upsilon\_s \tag{13}$$

$$
\Delta I\_{gin} = \varepsilon\_g \mu\_g + \varepsilon\_s \mu\_s \tag{14}
$$

Equations (13) and (14) enable the fluid velocity variables to be expressed in terms of the particle velocity at all points in the bed.

#### *3.3.2. Combined momentum equation*

In this section the combined momentum equation is produced by combining the fluid and particle momentum equations (4), (5), (8) and (9) by elimination of the fluid pressure gradient, which appears in them. This yields the combined momentum equation:

x- direction:

The total force acting on particle phase is the sum of the net primary force and the force resulting from particle phase elasticity. The x and y components of forces acting on particle

e

*d x*

e

¶¶ ¶ (8)

¶¶ ¶ (9)

 er


*d x*

e

 e

 e ¶ (11)

 er

 e- - - ¶ <sup>=</sup> - - ¶ (7)

(6)

1.8 3( ) (1 ) <sup>4</sup>

> r

( ) ( ) "x -direction " *<sup>g</sup>*

( ) ( ) "y -direction " *<sup>g</sup>*

The fluid phase forces are readily obtained from the particle phase relations for fluid – particle interaction (drag and pressure gradient force), which act in the opposite direction on the fluid, together with gravity. The x and y components of forces acting on gas phase are as following:

e

1.8 3( ) (1 ) <sup>4</sup>

We assume that both particles and fluid are regarded as being incompressible. This was justified on the basis that only a gas phase is going to exhibit any significant compressibility, and the orders of magnitude differences in particle and fluid density for gas fluidization render quite insignificant the small change in gas density resulting from compression. This assump‐

*u v vv F*

*u u vu F*

1.8 3( ) (1 ) <sup>4</sup>

*u uu u <sup>p</sup> F C*

(3.2 ( ))

*sg g s g s sx d s s p*

*g g g g g g g g g g gx*

*g g g g g g g g g g gy*

r e

r e

1.8 3( ) (1 ) <sup>4</sup>

*gy d s gs g*

*v vv v <sup>p</sup> F C <sup>g</sup> d y*

*sg g s g s gx sx d s s p u uu u <sup>p</sup> F FC*


*sg g s g s*

*p*

e r

++=

++=

*tx y*

*tx y*

¶ ¶ ¶

¶ ¶ ¶

er

*sg g s g s sy d s ss p*

*v vv v F C <sup>g</sup> <sup>d</sup>*


*s s ps s g*

e

e r

e

158 Nuclear Reactor Thermal Hydraulics and Other Applications

*u*

*v*

 re

 re

> e r

*3.3.1. Relation between fluid and particle velocities*

re

re

e r

*<sup>p</sup> gd y y*

¶ ¶ -- - ¶ ¶

phase are as following:

Gas phase :

[divide equation (4-4) by εs ] + [divide equation (4-8) by εg] which give us :

$$
\rho\_s \left[ \frac{\partial \boldsymbol{u}\_s}{\partial t} + \frac{\partial}{\partial \mathbf{x}} (\boldsymbol{u}\_s \boldsymbol{u}\_s) + \frac{\partial}{\partial y} (\boldsymbol{v}\_s \boldsymbol{u}\_s) \right] + \rho\_g \left[ \frac{\partial \boldsymbol{u}\_s}{\partial t} + \frac{\partial}{\partial \mathbf{x}} (\boldsymbol{u}\_g \boldsymbol{u}\_g) + \frac{\partial}{\partial y} (\boldsymbol{v}\_g \boldsymbol{u}\_g) \right] = 0 \tag{15}
$$

y- direction:

[divide equation (4-3) by εs ] – [divide equation (4-7) by εg] which give us :

$$
\rho\_s \left[ \frac{\partial v\_s}{\partial t} + \frac{\partial}{\partial x} (u\_s v\_s) + \frac{\partial}{\partial y} (v\_s v\_s) \right]
$$

$$-\rho\_{\mathcal{S}} \left[ \frac{\partial \upsilon\_{\mathcal{S}}}{\partial t} + \frac{\partial}{\partial \mathbf{x}} (\mu\_{\mathcal{S}} \upsilon\_{\mathcal{S}}) + \frac{\partial}{\partial y} (\upsilon\_{\mathcal{S}} \upsilon\_{\mathcal{S}}) \right] = \frac{F\_{\text{sy}}}{\mathcal{E}\_{s}} - \frac{F\_{\text{gy}}}{\mathcal{E}\_{\text{g}}} \tag{16}$$

Equations (15) and (16) with the continuity equation for the two phase (1) and (2),now define the two phase system without account of fluid pressure variation.

#### *3.3.3. Drag coefficient*

An important constitutive relation in any multiphase flow model is the formula for the fluidparticle drag coefficient, which is may be expressed by the empirical Dallavalle relation as reported in [3] :

$$C\_d = \left(0.63 + \frac{4.8}{\mathrm{Re}^{0.5}}\right)^2 \tag{17}$$

**Figure 2.** Total pressure drop in fluidized bed

( )( )

+ + -+

( )( )

*k hT T*

*g gg v s g*

*T*

+ +-

*3.4.1. Thermal conductivity values (kg and ks)*

*y y*

e

¶ ¶

¶ ¶

e

r

represented in general as:

*ss v g s s*

re

*<sup>T</sup> k hT T q*

*s*

*y y*

e

¶ ¶

¶ ¶

e

( ) ( )( )

( ) ( )( )

 e

Hydrodynamic and Heat Transfer Simulation of Fluidized Bed Using CFD

http://dx.doi.org/10.5772/52072

 e

(25)

(26)

(23)

161

(24)

*s s s*

*g g g*

*t x y xx*

The thermal conductivities of the fluid phase and the solid phase (kg and ks) in the two fluid model formulation should be interpreted as effective transport coefficients which means that the corresponding microscopic (or absolute) coefficients kg,o and ks,o cannot be used. It can be

> , , , , , particle geometry) ( *g g go so g k kk k* = e

> , , , , , particle geometry) ( *s s go so g k kk k* = e

However, such a general formulation is not yet available for fluidized beds and approximate constitutive equations have to be used. These approximate equations have been obtained on modeling of the effective thermal conductivity kb in packed beds. According to their model,

re

*T T*

*s ps s ps s s s s ps s s s s s*

rere

 e

*g pg g pg g g g g pg g g g g g*

¶ ¶ ¶ ¶¶ ++= ¶ ¶ ¶ ¶¶

*C C uT C vT k*

*T T C C uT C vT k t x y xx*

¶ ¶ ¶ ¶¶ ++= ¶ ¶ ¶ ¶¶

·

**3.4. Energy equation**

r

Particle phase:

GAS phase:

The particle Reynolds number, Re, based on particle diameter is given by :

$$\text{Re} = \frac{\varepsilon\_{\g} \rho\_{\g} d\_{p} \left| \overrightarrow{\mathcal{U}\_{r}} \right|}{\mu\_{\g}} \tag{18}$$

#### *3.3.4. Gas pressure drop*

Figure (2) shows the relation between the total pressure drop across the bed "ΔPB" and the input gas velocity [3].where:

$$
\Delta P\_B = \left(\rho\_g \varepsilon\_{g,mf} + \rho\_s \left(1 - \varepsilon\_{g,mf}\right)\right) \text{g} \, H\_{mf} \tag{19}
$$

The gas pressure at the entrance of the fluidized bed can be calculated from the following equation:

$$P\_{\rm g,in} = P\_{\rm atm} + \rho\_{\rm g} \text{g} \left( H - H\_{\rm mf} \right) + \left( \rho\_{\rm g} \varepsilon\_{\rm g,mf} + \rho\_{\rm s} \left( 1 - \varepsilon\_{\rm g,mf} \right) \right) \text{g} \, H\_{\rm mf} \tag{20}$$

and the pressure drop at any position and head "h" can be calculated from:

$$
\Delta P\_g = \rho\_{sus}gh\tag{21}
$$

Where *ρsus*is the suspension density which calculated from:

$$
\rho\_{\rm sus} = \rho\_s \varepsilon\_s + \rho\_g \varepsilon\_g \tag{22}
$$

**Figure 2.** Total pressure drop in fluidized bed

#### **3.4. Energy equation**

Particle phase:

*3.3.3. Drag coefficient*

160 Nuclear Reactor Thermal Hydraulics and Other Applications

reported in [3] :

*3.3.4. Gas pressure drop*

equation:

input gas velocity [3].where:

An important constitutive relation in any multiphase flow model is the formula for the fluidparticle drag coefficient, which is may be expressed by the empirical Dallavalle relation as

2

(17)

(18)

0.5

uur

4.8 0.63 Re

Re *g gp r*

D= + - *PB g g mf s g mf mf* (

 r

re

*P P gH H g in atm g* , =+ - + + -

and the pressure drop at any position and head "h" can be calculated from:

*g sus* D = *P gh* r

*sus s s g g*

 re r e

r

Where *ρsus*is the suspension density which calculated from:

r

e r *d U* m<sup>=</sup>

*g*

Figure (2) shows the relation between the total pressure drop across the bed "ΔPB" and the

 e

The gas pressure at the entrance of the fluidized bed can be calculated from the following

 ( *mf g g mf s g mf mf* ) ( re

 r  e

, , (1 )) *gH* (19)

, , (1 )) *gH* (20)

(21)

= + (22)

æ ö =ç + ÷ ç ÷ è ø

*Cd*

The particle Reynolds number, Re, based on particle diameter is given by :

$$\begin{aligned} \rho\_s \mathbf{C}\_{ps} \frac{\partial \varepsilon\_s T\_s}{\partial t} + \rho\_s \mathbf{C}\_{ps} \frac{\partial}{\partial \mathbf{x}} (\varepsilon\_s u\_s T\_s) + \rho\_s \mathbf{C}\_{ps} \frac{\partial}{\partial y} (\varepsilon\_s v\_s T\_s) &= \frac{\partial}{\partial \mathbf{x}} (\varepsilon\_s k\_s \frac{\partial T\_s}{\partial \mathbf{x}}) \\ + \frac{\partial}{\partial y} (\varepsilon\_s k\_s \frac{\partial T\_s}{\partial y}) + h\_v (T\_g - T\_s) + \varepsilon\_s q &\tag{23} \end{aligned} \tag{23}$$

GAS phase:

$$\begin{split} \rho\_{g}\mathbb{C}\_{\mathcal{P}\_{\mathcal{B}}}\frac{\partial\varepsilon\_{\mathcal{S}}T\_{\mathcal{S}}}{\partial t} &+ \rho\_{g}\mathbb{C}\_{\mathcal{P}\_{\mathcal{B}}}\frac{\partial}{\partial \mathbf{x}}(\varepsilon\_{g}\boldsymbol{u}\_{\mathcal{S}}T\_{\mathcal{S}}) + \rho\_{g}\mathbb{C}\_{\mathcal{P}\_{\mathcal{B}}}\frac{\partial}{\partial \mathcal{y}}(\varepsilon\_{g}\boldsymbol{v}\_{\mathcal{S}}T\_{\mathcal{S}}) = \frac{\partial}{\partial \mathbf{x}}(\varepsilon\_{g}\boldsymbol{k}\_{\mathcal{S}}\frac{\partial T\_{\mathcal{S}}}{\partial \mathbf{x}}) \\ &+ \frac{\partial}{\partial \mathbf{y}}(\varepsilon\_{g}\boldsymbol{k}\_{\mathcal{S}}\frac{\partial T\_{\mathcal{S}}}{\partial \mathbf{y}}) + \boldsymbol{h}\_{v}(T\_{\mathcal{S}} - T\_{\mathcal{S}}) \end{split} \tag{24}$$

#### *3.4.1. Thermal conductivity values (kg and ks)*

The thermal conductivities of the fluid phase and the solid phase (kg and ks) in the two fluid model formulation should be interpreted as effective transport coefficients which means that the corresponding microscopic (or absolute) coefficients kg,o and ks,o cannot be used. It can be represented in general as:

$$k\_{\boldsymbol{\chi}} = k\_{\boldsymbol{\chi}}(k\_{s,o'}, k\_{s,o'} \boldsymbol{\varepsilon}\_{\boldsymbol{\chi}'} \text{particle geometry}) \tag{25}$$

$$k\_s = k\_s(k\_{\chi, o'}, k\_{s, o'}, \varepsilon\_{\chi'} \text{ particle geometry}) \tag{26}$$

However, such a general formulation is not yet available for fluidized beds and approximate constitutive equations have to be used. These approximate equations have been obtained on modeling of the effective thermal conductivity kb in packed beds. According to their model, the radial bed conductivity kb consists of a contribution kb,g due to the fluid phase only and a contribution due to a combination of the fluid phase and the solid phase[4].

$$k\_b = \begin{array}{c} k\_{b,g} + \; k\_{b,s} \tag{27}$$

*3.4.2. Interphase volumetric heat transfer coefficient hv*

, and gives the Nusselt number:

)(1 <sup>+</sup> 0.7Re *<sup>p</sup>*

0.2Pr 1 3 )

,and the overall heat transfer coefficient is evaluated from;

**4. Boundary and initial conditions**

the system of equations.

**4.1. Boundary conditions**

conditions are imposed as follow:

*v*

*h*

to Re = 105

*Nu* =(7−10*ε<sup>g</sup>*

+ 5*ε<sup>g</sup>* 2

The heat transfer coefficient is modelled using a correlation by Gunn as reported [5]. This correlation is applicable for gas voidage in the range of 0.35 to 1 and for Reynolds numbers up

> , *gp p g o*

*<sup>k</sup>* <sup>=</sup> (36)

Hydrodynamic and Heat Transfer Simulation of Fluidized Bed Using CFD

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163

= (38)

= (39)

(37)

1

*h d*

*Nu*

( )

2 0.7 <sup>3</sup> 1.33 2.40 1.20 Re Pr *g gp* +- + e

where Reynolds number is defined by equation (18).The Prandlt number is defined by;

, , Pr *pg g g o*

6 1( *g gp* )

*d* e *h*

*p*

The system of conservation equations (1),(2), (3), (4), (5), (8), (9), (23), and (24), nine equations which are discussed in previous sections must be solved for the nine dependent variables: the gas-phase volume fraction εg, the particle-phase volume fraction εs, the gas pressure Pg, the gas velocity components *ug* and vg and the solids velocity components *us* and vs in *x*-direction and *y*-direction, respectively,the gas temperature Tg and particle temperature Ts.We need appropriate boundary and initial conditions for the dependent variables listed above to solve

In this section the boundary conditions for the above governing equations, which relate to two dimensional fluidized bed with width "L" and height "H" to allow bed expansion typically i.e. the height of the bed is enough to prevent the particles being ejected from the bed. Boundary

*C k* m  e

where :

$$k\_{\rm bg} = \left(1 - \sqrt{1 - \varepsilon\_{\rm g}}\right) k\_{\rm go} \tag{28}$$

$$k\_{bs} = \sqrt{1 - \varepsilon\_{\mathcal{g}}} \left(\alpha A + (1 - \alpha)\Gamma\right) k\_{\mathcal{g}^0} \tag{29}$$

$$\Gamma = \frac{2}{\left(1 - \frac{B}{A}\right)} \left[ \frac{A - 1}{\left(1 - \frac{B}{A}\right)^2} \frac{B}{A} \ln\left(\frac{A}{B}\right) - \frac{B - 1}{\left(1 - \frac{B}{A}\right)} - \frac{1}{2}(B + 1) \right] \tag{30}$$

$$B = 1.25 \left( \frac{1 - \varepsilon\_g}{\varepsilon\_g} \right)^{\frac{10}{9}} \tag{31}$$

$$A = \frac{k\_{s,o}}{k\_{\chi,o}}\tag{32}$$

$$
\alpha = 7.26X10^{-3} \tag{33}
$$

Thus the thermal conductivities of the fluid phase and the solid phase then are given by :

$$k\_{\mathcal{g}} = \frac{k\_{b,\mathcal{g}}}{\mathcal{e}\_{\mathcal{g}}} \tag{34}$$

$$k\_s = \frac{k\_{b,s}}{\mathcal{E}\_s} \tag{35}$$

## *3.4.2. Interphase volumetric heat transfer coefficient hv*

the radial bed conductivity kb consists of a contribution kb,g due to the fluid phase only and a

, , *b bg bs kk k* = + (27)

*g go* ) (28)

(29)

*<sup>k</sup>* <sup>=</sup> (32)

<sup>=</sup> (34)

<sup>=</sup> (35)

7.26 10 *X* - = (33)

(30)

(31)

contribution due to a combination of the fluid phase and the solid phase[4].

162 Nuclear Reactor Thermal Hydraulics and Other Applications

*k k bg* =- - (1 1

*bs* 1 1 *<sup>g</sup>* ( ( ) ) *go k Ak* = - +- G ew

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

*A BA B Ln <sup>B</sup>*

é ù

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

ê ú - - æ ö G = ç ÷ - -+ æ ö æ ö æ ö è ø ç ÷ - - - ç ÷ ç ÷ è ø ê ú è ø ë û è ø

*B B A B B A A A*

*B*

w

e

 w

( ) <sup>2</sup>

10 1 <sup>9</sup> 1.25 *<sup>g</sup> g*

e

e

, , *s o g o*

Thus the thermal conductivities of the fluid phase and the solid phase then are given by :

*b g*,

*k*

*g*

*b s*,

*k*

*s*

e

e

*g*

*s*

*k*

*k*

3

*k A*

æ ö - <sup>=</sup> ç ÷ ç ÷ è ø

where :

The heat transfer coefficient is modelled using a correlation by Gunn as reported [5]. This correlation is applicable for gas voidage in the range of 0.35 to 1 and for Reynolds numbers up to Re = 105 , and gives the Nusselt number:

$$\text{Nu} = \frac{h\_{\text{gp}} d\_p}{k\_{\text{g,o}}} \tag{36}$$

*Nu* =(7−10*ε<sup>g</sup>* + 5*ε<sup>g</sup>* 2 )(1 <sup>+</sup> 0.7Re *<sup>p</sup>* 0.2Pr 1 3 ) ( ) 1 2 0.7 <sup>3</sup> 1.33 2.40 1.20 Re Pr *g gp* +- + e e(37)

where Reynolds number is defined by equation (18).The Prandlt number is defined by;

$$\mathbf{Pr} = \frac{\mathbf{C}\_{p,\mathcal{g}}\mu\_{\mathcal{g}}}{k\_{\mathcal{g},o}} \tag{38}$$

,and the overall heat transfer coefficient is evaluated from;

$$h\_v = \frac{\Theta\left(1 - \varepsilon\_g\right)h\_{gp}}{d\_p} \tag{39}$$

#### **4. Boundary and initial conditions**

The system of conservation equations (1),(2), (3), (4), (5), (8), (9), (23), and (24), nine equations which are discussed in previous sections must be solved for the nine dependent variables: the gas-phase volume fraction εg, the particle-phase volume fraction εs, the gas pressure Pg, the gas velocity components *ug* and vg and the solids velocity components *us* and vs in *x*-direction and *y*-direction, respectively,the gas temperature Tg and particle temperature Ts.We need appropriate boundary and initial conditions for the dependent variables listed above to solve the system of equations.

#### **4.1. Boundary conditions**

In this section the boundary conditions for the above governing equations, which relate to two dimensional fluidized bed with width "L" and height "H" to allow bed expansion typically i.e. the height of the bed is enough to prevent the particles being ejected from the bed. Boundary conditions are imposed as follow:

$$\begin{aligned} \text{x = 0} \quad & \because v\_{\mathcal{g}} = v\_{\mathcal{g}} = u\_{\mathcal{g}} = u\_{s} = 0, \quad & \frac{\partial \varepsilon\_{\mathcal{g}}}{\partial \mathbf{x}} = \frac{\partial T\_{\mathcal{g}}}{\partial \mathbf{x}} = \frac{\partial T\_{s}}{\partial \mathbf{x}} = 0 \\ \text{x = \frac{L}{2} \quad & \because \frac{\partial \varepsilon\_{\mathcal{g}}}{\partial \mathbf{x}} = \frac{\partial T\_{\mathcal{g}}}{\partial \mathbf{x}} = \frac{\partial T\_{s}}{\partial \mathbf{x}} = \frac{\partial v\_{\mathcal{g}}}{\partial \mathbf{x}} = \frac{\partial u\_{\mathcal{g}}}{\partial \mathbf{x}} = \frac{\partial v\_{s}}{\partial \mathbf{x}} = \frac{\partial u\_{s}}{\partial \mathbf{x}} = 0 \quad \text{(symmetric)} \\ \text{y = 0} \quad & \because v\_{\mathcal{g}} = V\_{\mathcal{g}, i\nu'} = v\_{s} = 0, \quad & \varepsilon\_{\mathcal{g}} = 1, \quad P\_{\mathcal{g}} = P\_{\mathcal{g}, i\nu'} \quad T\_{\mathcal{g}} = T\_{\mathcal{g}, i\nu'} \quad \frac{\partial T\_{s}}{\partial \mathcal{g}} = 0 \\ \text{y = H} \quad & \frac{\partial v\_{\mathcal{g}}}{\partial \mathbf{y}} = \frac{\partial u\_{\mathcal{g}}}{\partial \mathbf{y}} = \frac{\partial T\_{\mathcal{g}}}{\partial \mathbf{y}} = 0, \quad & \varepsilon\_{\mathcal{g}} = \mathbf{1}, \quad v\_{\mathcal{g}} = u\_{s} = 0, \quad P\_{\mathcal{g}} = P\_{\mathcal{at}\mathcal{g}} \end{aligned}$$

#### **4.2. Initial conditions**

For setting the initial conditions, the model is divided into two regions: the bed and the freeboard. For each of the regions specified above, an initial condition is specified.

( ) ( ) <sup>1</sup> <sup>1</sup> , , <sup>1</sup> *<sup>n</sup> <sup>n</sup> g s i j i j*

The gas continuity equation residual, dg, is discretized at i; j in a fully implicit way:

( ) ( ) ( ) ( ) ( ) ( )

<sup>ì</sup> - ³ <sup>D</sup> ïï = -+ <sup>í</sup> <sup>D</sup> <sup>ï</sup> - < ïî

*gg g g ij ij i j nn n*

*if u <sup>t</sup> d u*

+ + -

e

e

*v*

**5.2. Discretization of combined momentum equations**

 e

1 1 , 1, , ,, , 11 1

*nn n gg g ij i j i j*

e

 e

e

( ) ( ) ( ) ( )

<sup>ì</sup> - ³ <sup>D</sup> ïï <sup>+</sup> <sup>í</sup> <sup>D</sup> <sup>ï</sup> - < ïî

*if v <sup>t</sup>*

+ -

*<sup>y</sup> if v*

+

( ) ( ) ( ) ( ) ( ) ( )

*u uu u u if u*

, ,, , 1, ,

*t x u u if u*

é ù <sup>ì</sup> - - ³ ê ú <sup>ï</sup> <sup>+</sup> <sup>í</sup> D D <sup>ï</sup> - < ë û <sup>î</sup>

+

*n nn nn n s ss ss s ij ij ij ij i j i j s nn n*

( ) ( ) ( ) ( )

*y u u if v*

( ) ( ) ( ) ( )

*<sup>y</sup> u u if v*

*v u u if v*

+

( ) ( ) ( )

ë û ïî

,1 , ,

*gg g ij ij i j*


, , ,1 ,

é ù <sup>ì</sup> - ³ ê ú ïï + = <sup>í</sup> <sup>D</sup> <sup>ï</sup> - <

*n nn n <sup>g</sup> gg g i j ij ij i j g nn n*

*v u u if v*

<sup>í</sup> <sup>+</sup> <sup>D</sup> <sup>ï</sup> - < <sup>î</sup>

+

( ) ( )


*u u*

+

1 ,

*n g g ij i*

*g*

r

r

y –direction:

+

r

r

( ) ( ) ( )

ë û

,1 , ,


, , ,1 ,

é ù <sup>ì</sup> - ³ ê ú <sup>ï</sup>

*n nn n s ss s i j ij ij i j nn n <sup>s</sup> ss s ij ij i j*

( ) ( ) ( )

The combined momentum equations may after a time discretization be expressed in the

*gg g ij ij i j*

1 , ,1 , , 11 1 ,1 , ,

*<sup>n</sup> gg g ij ij i j <sup>g</sup> i j nn n*

> e

 e

( ) ( ) ( )

11 1

++ +

( ) ( ) ( )


, 0.0

, 0.0

( ) ( ) ( ) ( )

*u u u if u*

, , , 1, ,

*t x u u if u*

é ù <sup>ì</sup> - ³ ê ú ïï <sup>+</sup> <sup>í</sup> D D <sup>ï</sup> - < ïî

ë û

+

*n n nn n <sup>g</sup> gg g j ij ij i j i j*

, 0.0

, 0.0

( ) ( ) ( )


1, , ,

*nn n gg g i j ij i j*

0

1, , ,

*ss s i j ij i j*

 e

> e

*nn n*

*<sup>x</sup> if u*

+ ++ +

11 1

++ +

*nn n*

1, , ,

++ +

*gg g i j ij i j*

, 0.0

, 0.0

, 0.0

, 0.0

, 0.0

, 0.0

<sup>+</sup> <sup>+</sup> = - (41)

Hydrodynamic and Heat Transfer Simulation of Fluidized Bed Using CFD

http://dx.doi.org/10.5772/52072

, 0.0

, 0.0

(42)

165

(43)

e

*5.1.2. Gas phase continuity equation*

e

following forms for x and y directions:

1

+

x –direction:

Bed region

$$\varepsilon\_{\mathcal{S}} = \varepsilon\_{\mathcal{g}, m\!\!\!\!\/, \prime} \qquad \upsilon\_{\mathcal{g}} = \frac{V\_{\mathcal{g}, m\!\!\!\/}}{\varepsilon\_{\mathcal{g}, m\!\!\!\/, \!\!\/}} , \qquad \mathsf{u}\_{\mathcal{g}} = \upsilon\_{\mathcal{s}} = \mathsf{u}\_{\mathcal{s}} = \mathsf{0} \,, \qquad T\_{\mathcal{g}} = T\_{\mathcal{g}, \mathsf{in}\prime} \qquad T\_{\mathcal{s}} = T\_{\mathcal{s}, \mathsf{in}\prime}$$

Freeboard region

$$\varepsilon\_{\mathcal{S}} = 1, \qquad \upsilon\_{\mathcal{S}} = V\_{\mathcal{S}, m^f \prime}, \qquad \mu\_{\mathcal{S}} = \upsilon\_s = \mu\_s = 0, \qquad T\_{\mathcal{S}} = T\_{\mathcal{S}, iu}$$

#### **5. Finite difference approximation scheme**

The conservation equations are transformed into difference equations by using a finite difference scheme.

#### **5.1. Discretization of continuity equations**

#### *5.1.1. Particle phase continuity equation*

The particle phase continuity equation,Eq.(1), is discretized at the node i; j in an explicit form as:

$$\begin{split} \left(\boldsymbol{\varepsilon}\_{s}\right)\_{i,j}^{n+1} &= \left(\boldsymbol{\varepsilon}\_{s}\right)\_{i,j}^{n} - \frac{\Delta t}{\Delta x} \left(\boldsymbol{u}\_{s}\right)\_{i,j}^{n} \begin{bmatrix} \left(\boldsymbol{\varepsilon}\_{s}\right)\_{i,j}^{n} - \left(\boldsymbol{\varepsilon}\_{s}\right)\_{i-1,j}^{n} & \operatorname{if}\left(\boldsymbol{u}\_{s}\right)\_{i,j}^{n} \geq 0.0\\ \left(\boldsymbol{\varepsilon}\_{s}\right)\_{i+1,j}^{n} - \left(\boldsymbol{\varepsilon}\_{s}\right)\_{i,j}^{n} & \operatorname{if}\left(\boldsymbol{u}\_{s}\right)\_{i,j}^{n} < 0.0 \end{bmatrix} \\ &- \frac{\Delta t}{\Delta y} \left(\boldsymbol{\upsilon}\_{s}\right)\_{i,j}^{n} \begin{cases} \left(\boldsymbol{\varepsilon}\_{s}\right)\_{i,j}^{n} - \left(\boldsymbol{\varepsilon}\_{s}\right)\_{i,j-1}^{n} & \operatorname{if}\left(\boldsymbol{\upsilon}\_{s}\right)\_{i,j}^{n} \geq 0.0\\ \left(\boldsymbol{\varepsilon}\_{s}\right)\_{i,j+1}^{n} - \left(\boldsymbol{\varepsilon}\_{s}\right)\_{i,j}^{n} & \operatorname{if}\left(\boldsymbol{\upsilon}\_{s}\right)\_{i,j}^{n} < 0.0 \end{cases} \end{split} \tag{40}$$

The gas phase volume fraction is then calculated explicitly as:

Hydrodynamic and Heat Transfer Simulation of Fluidized Bed Using CFD http://dx.doi.org/10.5772/52072 165

$$\left(\left(\boldsymbol{\varepsilon}\_{\mathcal{S}}\right)\_{i,j}^{n+1} = 1 - \left(\boldsymbol{\varepsilon}\_{s}\right)\_{i,j}^{n+1}\right.\tag{41}$$

#### *5.1.2. Gas phase continuity equation*

*<sup>x</sup>* =0 :*vg* <sup>=</sup>*vs* <sup>=</sup>*ug* <sup>=</sup>*us* =0, <sup>∂</sup>*ε<sup>g</sup>*

∂*ug* <sup>∂</sup> *<sup>y</sup>* <sup>=</sup>

∂*Tg* <sup>∂</sup> *<sup>x</sup>* <sup>=</sup> ∂*Ts* <sup>∂</sup> *<sup>x</sup>* <sup>=</sup>

164 Nuclear Reactor Thermal Hydraulics and Other Applications

∂*Tg*

*Vg*,*mf εg*,*mf*

*ε<sup>g</sup>* =1, *vg* =*Vg*,*mf* , *ug* =*vs* =*us* =0, *Tg* =*Tg*,*in*

**5.1. Discretization of continuity equations**

*5.1.1. Particle phase continuity equation*

e

**5. Finite difference approximation scheme**

,, ,

 e

∂*ε<sup>g</sup>* <sup>∂</sup> *<sup>x</sup>* <sup>=</sup>

∂*vg* <sup>∂</sup> *<sup>y</sup>* <sup>=</sup>

**4.2. Initial conditions**

*<sup>x</sup>* <sup>=</sup> *<sup>L</sup>* <sup>2</sup> :

*y* =*H* :

Bed region

*ε<sup>g</sup>* =*εg*,*mf* , *vg* =

Freeboard region

difference scheme.

<sup>∂</sup> *<sup>x</sup>* <sup>=</sup>

∂*vg* <sup>∂</sup> *<sup>x</sup>* <sup>=</sup>

∂*Tg* <sup>∂</sup> *<sup>x</sup>* <sup>=</sup>

> ∂*ug* <sup>∂</sup> *<sup>x</sup>* <sup>=</sup>

*<sup>y</sup>* =0 :*vg* <sup>=</sup>*Vg*,*in*, *ug* <sup>=</sup>*us* <sup>=</sup>*vs* =0, *<sup>ε</sup><sup>g</sup>* =1, *Pg* <sup>=</sup>*Pg*,*in*, *Tg* <sup>=</sup>*Tg*,*in*, <sup>∂</sup>*Ts*

<sup>∂</sup> *<sup>y</sup>* =0, *<sup>ε</sup><sup>g</sup>* =1, *vs* <sup>=</sup>*us* =0, *Pg* <sup>=</sup> *Patm*

freeboard. For each of the regions specified above, an initial condition is specified.

, *ug* =*vs* =*us* =0, *Tg* =*Tg*,*in*, *Ts* =*Ts*,*in*

∂*Ts* <sup>∂</sup> *<sup>x</sup>* =0

∂*vs* <sup>∂</sup> *<sup>x</sup>* <sup>=</sup> ∂*us*

For setting the initial conditions, the model is divided into two regions: the bed and the

The conservation equations are transformed into difference equations by using a finite

The particle phase continuity equation,Eq.(1), is discretized at the node i; j in an explicit form as:

*if u <sup>t</sup>*

 e

*x if u*

( ) ( ) ( ) ( )

<sup>ì</sup> - ³ <sup>D</sup> <sup>ï</sup> - <sup>í</sup> <sup>D</sup> <sup>ï</sup> - < <sup>î</sup>

+

<sup>ì</sup> - ³ <sup>D</sup> <sup>ï</sup> = - <sup>í</sup> <sup>D</sup> <sup>ï</sup> - < <sup>î</sup>

 e

*if v <sup>t</sup>*

*y if v*

+

( ) ( ) ( )

 e

( ) ( ) ( )

 e

, ,1 ,


,1 , ,

1, , ,

*ss s i j ij i j nn n n ss s ij ij i j s i j nn n ss s ij ij i j*

*nn n*

, 0.0

, 0.0

(40)

, 0.0

, 0.0

( ) ( ) ( ) ( ) ( ) ( )

e

e

,

*v*

The gas phase volume fraction is then calculated explicitly as:

*u*

+ -

1 , 1, ,

e

e

*nn n ss s ij i j i j ss s ij ij i j nn n*

<sup>∂</sup> *<sup>x</sup>* =0 (*symmetric*)

<sup>∂</sup> *<sup>y</sup>* =0

The gas continuity equation residual, dg, is discretized at i; j in a fully implicit way:

$$\begin{split} d\_{\mathcal{S}} &= \left(\boldsymbol{\mathcal{E}}\_{\mathcal{S}}\right)\_{i,j}^{n+1} - \left(\boldsymbol{\mathcal{E}}\_{\mathcal{S}}\right)\_{i,j}^{n} + \frac{\Delta t}{\Delta x} \Big(\boldsymbol{u}\_{\mathcal{S}}\Big)\_{i,j}^{n+1} - \left(\boldsymbol{\mathcal{E}}\_{\mathcal{S}}\right)\_{i-1,j}^{n+1} \quad , \boldsymbol{\mathcal{H}}\Big(\boldsymbol{u}\_{\mathcal{S}}\Big)\_{i,j}^{n+1} \geq 0.0\\ & \left(\boldsymbol{\mathcal{E}}\_{\mathcal{S}}\right)\_{i+1,j}^{n+1} - \left(\boldsymbol{\mathcal{E}}\_{\mathcal{S}}\right)\_{i,j}^{n+1} \quad , \boldsymbol{\mathcal{H}}\Big(\boldsymbol{u}\_{\mathcal{S}}\Big)\_{i,j}^{n+1} < 0.0\\ & + \frac{\Delta t}{\Delta y} \Big(\boldsymbol{\mathcal{v}}\_{\mathcal{S}}\Big)\_{i,j}^{n+1} \left[ \left(\boldsymbol{\mathcal{E}}\_{\mathcal{S}}\right)\_{i,j}^{n+1} - \left(\boldsymbol{\mathcal{E}}\_{\mathcal{S}}\right)\_{i,j-1}^{n+1} \quad , \boldsymbol{\mathcal{H}}\Big(\boldsymbol{\mathcal{v}}\_{\mathcal{S}}\Big)\_{i,j}^{n+1} \geq 0.0\\ & \left(\boldsymbol{\mathcal{E}}\_{\mathcal{S}}\right)\_{i,j+1}^{n+1} - \left(\boldsymbol{\mathcal{E}}\_{\mathcal{S}}\right)\_{i,j}^{n+1} \quad , \boldsymbol{\mathcal{H}}\Big(\boldsymbol{\mathcal{v}}\_{\mathcal{S}}\Big)\_{i,j}^{n+1} < 0.0 \end{split} \tag{42}$$

#### **5.2. Discretization of combined momentum equations**

The combined momentum equations may after a time discretization be expressed in the following forms for x and y directions:

x –direction:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 , ,, , 1, , 1, , , , , ,1 , ,1 , , 1 , , 0.0 , 0.0 , 0.0 , 0.0 *n nn nn n s ss ss s ij ij ij ij i j i j s nn n ss s i j ij i j n nn n s ss s i j ij ij i j nn n <sup>s</sup> ss s ij ij i j n g g ij i g u uu u u if u t x u u if u v u u if v y u u if v u u* r r r + - + - + + é ù <sup>ì</sup> - - ³ ê ú <sup>ï</sup> <sup>+</sup> <sup>í</sup> D D <sup>ï</sup> - < ë û <sup>î</sup> é ù <sup>ì</sup> - ³ ê ú <sup>ï</sup> <sup>í</sup> <sup>+</sup> <sup>D</sup> <sup>ï</sup> - < <sup>î</sup> ë û - + ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , , , 1, , 1, , , , , ,1 , ,1 , , , 0.0 , 0.0 , 0.0 0 , 0.0 *n n nn n <sup>g</sup> gg g j ij ij i j i j nn n gg g i j ij i j n nn n <sup>g</sup> gg g i j ij ij i j g nn n gg g ij ij i j u u u if u t x u u if u v u u if v <sup>y</sup> u u if v* r - + - + é ù <sup>ì</sup> - ³ ê ú ïï <sup>+</sup> <sup>í</sup> D D <sup>ï</sup> - < ïî ë û é ù <sup>ì</sup> - ³ ê ú ïï + = <sup>í</sup> <sup>D</sup> <sup>ï</sup> - < ë û ïî (43)

y –direction:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 , ,, , 1, , 1, , , , , ,1 , ,1 , , 1 , , 0.0 , 0.0 , 0.0 , 0.0 *n nn nn n s ss ss s ij ij ij ij i j i j s nn n ss s i j ij i j n nn n s ss s i j ij ij i j nn n <sup>s</sup> ss s ij ij i j n g g ij i g v vu v v if u t x v v if u v v v if v y v v if v v v* r r r + - + - + + é ù <sup>ì</sup> - - ³ ê ú <sup>ï</sup> <sup>+</sup> <sup>í</sup> D D <sup>ï</sup> - < ë û <sup>î</sup> é ù <sup>ì</sup> - ³ ê ú <sup>ï</sup> <sup>í</sup> <sup>+</sup> <sup>D</sup> <sup>ï</sup> - < <sup>î</sup> ë û - - ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , , , 1, , 1, , , , , ,1 , ,1 , , , 0.0 , 0.0 , 0.0 , 0.0 *n n nn n <sup>g</sup> gg g j ij ij i j i j nn n gg g i j ij i j n nn n <sup>g</sup> gg g i j ij ij i j g y nn n gg g ij ij i j u v v if u t x v v if u v v v if v F <sup>y</sup> v v if v* r - + - + é ù <sup>ì</sup> - ³ ê ú ïï <sup>+</sup> <sup>í</sup> D D <sup>ï</sup> - < ë û ïî é ù <sup>ì</sup> - ³ ê ú ïï - = <sup>í</sup> <sup>D</sup> <sup>ï</sup> - < ë û ïî (44)

( ) ( )

*T T*


*t*

e

e

e

e

2

( ) ( ) ( )

e

*T KT KT x KT KT KT*


( ) ( ) ( )

( )( )

*y*

*n n <sup>n</sup> <sup>n</sup> v g s s ij i j i j i j*

D

· + + <sup>+</sup>

( ) ( ) ( )

+ -+

( ) ( )

*T T*

D


*t*

e

e

e

e

2

2

, ,

*n n gg gg i j i j*

 e

1

+

1 1 1 , , , ,

*hT T q*

2

,

e

*s s*

*K*

e

e

,

e

*g pg*

*C*

r

r

e

e

=

+

r

=

+

Gas Phase:

*s ps*

*C*

r

r

r

1 , ,

+

*n n ss ss i j i j*

 e

( ) ( ) ( ) ( )

 e

, , , 1, ,

*x T T if u*

*C u T T if u*

<sup>ì</sup> - ³ <sup>ï</sup> <sup>+</sup> <sup>í</sup> <sup>D</sup> <sup>ï</sup> - < <sup>î</sup>

*C v T T if v*

+

( ) ( ) ( ) ( )

 e

1, , 1, 2

, 1 , , 1 2

+ -

( ) ( ) ( ) ( )

 e

, , , 1, ,

*g pg g gg gg <sup>g</sup> i j ij i j i j*

<sup>ì</sup> - ³ ïï <sup>+</sup> <sup>í</sup> <sup>D</sup> <sup>ï</sup> - < ïî

*<sup>x</sup> T T if u*

*C u T T if u*

( ) ( ) ( ) ( )

 e

, , , ,1 ,

*s pg g gg gg <sup>g</sup> i j ij ij i j*

<sup>ì</sup> - ³ ïï <sup>+</sup> <sup>í</sup> <sup>D</sup> <sup>ï</sup> - < ïî

*<sup>y</sup> T T if v*

*n nn g gg g gg g gg i j i j i j*

ee

+ -

*n nn g gg g gg g gg i j i j i j*

ee

+

( ) ( ) ( )

*KT KT KT*

D - +

*x KT KT KT*


*y*

D

( ) ( ) ( )

( ) ( ) ( )

*hT T*

+ -

1 1 1 , , ,

+ + +

*n n n vs g ij ij i j*

( )

*C v T T if v*

+

( ) ( ) ( )

*n nn n*

1, , ,

*n nn gg gg <sup>g</sup> i j ij i j*


, 0.0

, 0.0

, 0.0

, 0.0

,1 , ,

*n nn gg gg <sup>g</sup> i j i j i j*


( ) ( ) ( )

 e

1, , 1, 2

, 1 , , 1 2

+ -

*n nn n*

 e

, , , ,1 ,

*y T T if v*

+

*n nn s s ss s ss i j i j i j*

+ -

*n nn s ss s ss s ss i j i j i j*

ee

<sup>ì</sup> - ³ <sup>ï</sup> <sup>+</sup> <sup>í</sup> <sup>D</sup> <sup>ï</sup> - < <sup>î</sup>

( ) ( ) ( )

 e

*n nn n s ps s ss ss s i j ij i j i j*

1, , ,

*n nn ss ss s i j ij i j n nn n s ps s ss ss s i j ij ij i j*


, 0.0

Hydrodynamic and Heat Transfer Simulation of Fluidized Bed Using CFD

http://dx.doi.org/10.5772/52072

, 0.0

, 0.0

, 0.0

(46)

167

(47)

,1 , ,

*n nn ss ss s i j i j i j*


( ) ( ) ( )

 e

e

 e

where:

$$\begin{split} F\_{y} &= \left(\mathbb{C}\_{d}\right)\_{i,j}^{n} \frac{3\rho\_{\mathcal{S}}\left(\left(\boldsymbol{\upsilon}\_{\mathcal{S}}\right)\_{i,j}^{n+1} - \left(\boldsymbol{\upsilon}\_{\mathcal{S}}\right)\_{i,j}^{n+1}\right)\big|\left(\boldsymbol{\upsilon}\_{\mathcal{S}}\right)\_{i,j}^{n} - \left(\boldsymbol{\upsilon}\_{\mathcal{S}}\right)\_{i,j}^{n}\big|}{4d\_{p}} \Big|1 - \left(\boldsymbol{\varepsilon}\_{\mathcal{S}}\right)\_{i,j}^{n}\big|^{-18} \\ &- \frac{\left(\boldsymbol{\varepsilon}\_{\mathcal{S}}\right)\_{i,j}^{n} - \left(\boldsymbol{\varepsilon}\_{\mathcal{S}}\right)\_{i,j-1}^{n}}{\Delta y} \Big(3.2 \operatorname{gd}\_{p} \left(\boldsymbol{\rho}\_{\mathcal{S}} - \boldsymbol{\rho}\_{\mathcal{S}}\right)\big) \\ &+ \left(\boldsymbol{\mathsf{C}}\_{d}\right)\_{i,j}^{n} \frac{3\rho\_{\mathcal{S}}\left(\boldsymbol{\varepsilon}\_{\mathcal{S}}\right)\_{i,j}^{n} \Big|\left(\boldsymbol{\upsilon}\_{\mathcal{S}}\right)\_{i,j}^{n+1} - \left(\boldsymbol{\upsilon}\_{\mathcal{S}}\right)\_{i,j}^{n+1}\big|\left(\boldsymbol{\upsilon}\_{\mathcal{S}}\right)\_{i,j}^{n} - \left(\boldsymbol{\upsilon}\_{\mathcal{S}}\right)\_{i,j}^{n}\big|\_{i,j} \Big|1 - \left(\boldsymbol{\varepsilon}\_{\mathcal{S}}\right)\_{i,j}^{n}\big|^{-2.8} \\ \end{split} \tag{45}$$

#### **5.3. Discretization of energy equations**

Particle phase:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 , , , , , , 1, , 1, , , , , , ,1 , ,1 , , , 0.0 , 0.0 , 0.0 , 0.0 *n n ss ss i j i j s ps n nn n s ps s ss ss s i j ij i j i j n nn ss ss s i j ij i j n nn n s ps s ss ss s i j ij ij i j n nn ss ss s i j i j i j s s T T C t C u T T if u x T T if u C v T T if v y T T if v K* e e r r e e e e r e e e e e + - + - + - D <sup>ì</sup> - ³ <sup>ï</sup> <sup>+</sup> <sup>í</sup> <sup>D</sup> <sup>ï</sup> - < <sup>î</sup> <sup>ì</sup> - ³ <sup>ï</sup> <sup>+</sup> <sup>í</sup> <sup>D</sup> <sup>ï</sup> - < <sup>î</sup> = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1, , 1, 2 , 1 , , 1 2 1 1 1 , , , , 2 2 ( )( ) *n nn s s ss s ss i j i j i j n nn s ss s ss s ss i j i j i j n n <sup>n</sup> <sup>n</sup> v g s s ij i j i j i j T KT KT x KT KT KT y hT T q* e e e ee e + - + - · + + <sup>+</sup> - + D - + + D + -+ (46)

Gas Phase:

( ) ( ) ( ) ( ) ( ) ( )

*v vu v v if u*

, ,, , 1, ,

*t x v v if u*

é ù <sup>ì</sup> - - ³ ê ú <sup>ï</sup> <sup>+</sup> <sup>í</sup> D D <sup>ï</sup> - < ë û <sup>î</sup>

+

*n nn nn n s ss ss s ij ij ij ij i j i j s nn n*

( ) ( ) ( ) ( )

*y v v if v*

( ) ( ) ( ) ( )

*<sup>y</sup> v v if v*

*v v v if v*

+

( ) ( ) ( )

ë û ïî

( ) ( ) ( ) ( ) ( )

1 1

+ +

4

æ ö - - ç ÷ è ø <sup>=</sup> -

*d*

*v v vv*

*ps g*

( ( ))

r r

3.2

*gd*

( ) ( ) ( ) ( ) ( )

æ ö - - ç ÷ è ø + -

, ,

*nn n n n gs g s g s i j i j i j n n i j i j d s i j i j p*

*v v vv*

1 1

+ +

4

*d*

, ,

*n n n n gg s g s i j i j n n i j i j y d s i j i j p*

,1 , ,


, , ,1 ,

é ù <sup>ì</sup> - ³ ê ú ïï - = <sup>í</sup> <sup>D</sup> <sup>ï</sup> - <

*n nn n <sup>g</sup> gg g i j ij ij i j g y nn n gg g ij ij i j*

*v v v if v*

<sup>í</sup> <sup>+</sup> <sup>D</sup> <sup>ï</sup> - < <sup>î</sup>

+

( ) ( )


*v v*

+

1 ,

*n g g ij i*

*g*

r

r

( ) ( )


3

**5.3. Discretization of energy equations**

, ,1



*n n s s ij ij*

 e

*y*

r e

3

r

( )

*C*

Particle phase:

*F C*

e

where:


r

r

1

+

166 Nuclear Reactor Thermal Hydraulics and Other Applications

( ) ( ) ( )

ë û

,1 , ,


, , ,1 ,

é ù <sup>ì</sup> - ³ ê ú <sup>ï</sup>

*n nn n s ss s i j ij ij i j nn n <sup>s</sup> ss s ij ij i j*

( ) ( ) ( )


, 0.0

, 0.0

( ) ( ) ( ) ( )

*u v v if u*

, , , 1, ,

*t x v v if u*

ë û ïî

é ù <sup>ì</sup> - ³ ê ú ïï <sup>+</sup> <sup>í</sup> D D <sup>ï</sup> - <

+

*n n nn n <sup>g</sup> gg g j ij ij i j i j*

, 0.0

, 0.0

, , 1.8 , ,

,, , 2.8 , ,

( ) ( ) ( )


1, , ,

*F*

( ( ) )

e


1

( ( ) )

e


1

*nn n gg g i j ij i j*

1, , ,

*ss s i j ij i j*

, 0.0

, 0.0

, 0.0

(44)

(45)

, 0.0

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 , , , , , , 1, , 1, , , , , , ,1 , ,1 , , , 0.0 , 0.0 , 0.0 , 0.0 *n n gg gg i j i j g pg n nn n g pg g gg gg <sup>g</sup> i j ij i j i j n nn gg gg <sup>g</sup> i j ij i j n nn n s pg g gg gg <sup>g</sup> i j ij ij i j n nn gg gg <sup>g</sup> i j i j i j T T C t C u T T if u <sup>x</sup> T T if u C v T T if v <sup>y</sup> T T if v* e e r r e e e e r e e e e + - + - + - D <sup>ì</sup> - ³ ïï <sup>+</sup> <sup>í</sup> <sup>D</sup> <sup>ï</sup> - < ïî <sup>ì</sup> - ³ ïï <sup>+</sup> <sup>í</sup> <sup>D</sup> <sup>ï</sup> - < ïî = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1, , 1, 2 , 1 , , 1 2 1 1 1 , , , 2 2 ( ) *n nn g gg g gg g gg i j i j i j n nn g gg g gg g gg i j i j i j n n n vs g ij ij i j KT KT KT x KT KT KT y hT T* e ee e ee + - + - + + + - + D - + + D + - (47)

## **6. Dimensionless numbers**

In this section we define a group of four dimensionless numbers which mainly affect and control fluidization field.

#### **6.1. Archimedes number" Ar"**

Archimedes Number is used in characterization of the fluidized state and is defined as follow:

$$Ar = \frac{gd\_p^3 \rho\_g \left(\rho\_s - \rho\_g\right)}{\mu\_g^2} \tag{48}$$

**7. Methodology of solution**

**•** Total time of calculation

**•** Width of fluidized bed

**•** Initial height of the bed

**•** Entering gas velocity

**•** Acceleration of gravity

**•** Entering gas temperature

**•** Initial particles temperature

**•** Particle diameter

tion [7]:

procedures of solution are as following: **1.** Input the following parameters :

**•** Total nodes number in x and y direction

**•** Total height of the bed and free board

**•** Properties of gas phase (μg,ρg,kg,Cp,g )

**•** Properties of solid particles phase (ρs,ks,Cp,s )

*g mf*

The method of solution used in the present work is described in details in this section. The

Hydrodynamic and Heat Transfer Simulation of Fluidized Bed Using CFD

http://dx.doi.org/10.5772/52072

169

**2.** Calculate the minimum fluidized velocity from the following equation [6] :

*V X Ar*

, 2 1 200

**4.** Calculate the gas void fraction at entering gas velocity which given by:

, 2 1 200

*g mf*

*g in*

, 33.7 1 3.59 10 1 *<sup>g</sup>*

( ) ( ) 0.5 <sup>5</sup>

**3.** Calculate the gas void fraction at minimum fluidized velocity from the following equa‐

[0.4 ( ) ] 2.1 ( )

[0.4 ( ) ] 2.1 ( )

r r

*ps g*

r r

*ps g*

*p g*

r

*d* m

Î= + - (54)

Î= + - (55)

, 1/3

, 1/3

*g g in*

*V d g* m

*g g mf*

*V d g* m

é ù - =+ - ê ú ë û (53)

#### **6.2. Density number "De"**

Which define as the density ratio:

$$De = \frac{\rho\_g}{\rho\_s} \tag{49}$$

#### **6.3. Flow number "fl"**

Which is defined as:

$$\text{ff} = \frac{V\_{\text{g,in}}}{u\_t} \tag{50}$$

#### **6.4. Dimensionless gas velocity "***Ω* **1/3 "**

Which is defined as:

$$
\Omega^{1/3} = \left[\frac{\rho\_{\mathcal{g}}^2}{\mu\_{\mathcal{g}} g \left(\rho\_s - \rho\_{\mathcal{g}}\right)}\right]^{1/3} V\_{\mathcal{g}, in} \tag{51}
$$

#### **6.5. Dimensionless time "***τ***"**

Which is defined as:

$$
\tau = \frac{t u\_t}{d\_p} \tag{52}
$$

## **7. Methodology of solution**

**6. Dimensionless numbers**

168 Nuclear Reactor Thermal Hydraulics and Other Applications

control fluidization field.

**6.1. Archimedes number" Ar"**

**6.2. Density number "De"**

**6.3. Flow number "fl"**

Which is defined as:

Which is defined as:

Which is defined as:

**6.5. Dimensionless time "***τ***"**

Which define as the density ratio:

**6.4. Dimensionless gas velocity "***Ω* **1/3**

In this section we define a group of four dimensionless numbers which mainly affect and

Archimedes Number is used in characterization of the fluidized state and is defined as follow:

( ) <sup>3</sup> 2 *pg s g g*

 r


<sup>=</sup> (49)

*<sup>u</sup>* <sup>=</sup> (50)

= (52)

(51)

rr

m

*g s*

r

r

*g in*, *t*

( )

*t p tu d* t

*g*

r

é ù W = ê ú ê ú - ë û

*gsg*

*g*

m rr

1/3 <sup>2</sup>

,

*V*

*g in*

*V fl*

**"**

1/3

*De*

*gd Ar*

The method of solution used in the present work is described in details in this section. The procedures of solution are as following:


$$V\_{g,mf} = 33.7 \left[ \left( 1 + 3.59X10^{-5}Ar \right)^{0.5} - 1 \right] \frac{\mu\_g}{\left( d\_p \rho\_g \right)} \tag{53}$$

**3.** Calculate the gas void fraction at minimum fluidized velocity from the following equa‐ tion [7]:

$$\epsilon\_{g,mf} = \frac{1}{2.1} [0.4 + (\frac{200\,\mu\_g V\_{g,mf}}{d\_p^2 (\rho\_s - \rho\_g)g})^{1/3}] \tag{54}$$

**4.** Calculate the gas void fraction at entering gas velocity which given by:

$$\epsilon\_{g,in} = \frac{1}{2.1} \text{[} 0.4 + (\frac{200\,\mu\_g V\_{g,in}}{d\_p^2 (\rho\_s - \rho\_g)g})^{1/3} \text{]} \tag{55}$$

**5.** Calculate the particle terminal velocity from the following equation [2]:

$$
\mu\_t = \left[ -3.809 + \left( 3.809^2 + 1.832Ar^{0.5} \right)^{0.5} \right]^2 \frac{\mu\_g}{\rho\_g d\_p} \tag{56}
$$


$$H1 = \frac{\left(1 - \varepsilon\_{\text{g, }mf}\right)}{\left(1 - \varepsilon\_{\text{g, }int}\right)} H\_{mf} \tag{57}$$


**Note**: we use the excess solid volume correction [8] in a special subroutine:

The correction works out as a posteriori redistribution of the particle phase volume fraction in excess in each cell where:

$$
\mathcal{E}\_s \le \mathcal{E}\_{s,\text{max}} \tag{58}
$$

**Figure 3.** The excess solid volume correction

where:

new time step x-components velocities (*ug*

Equation (43) is reduced to the following form :

**11.** Call subroutine "dirx" to solve momentum equation in x-direction and evaluate at the

and (43) are solved to get values of x-component new step velocities (*ug*

1 1 11 , 12 , 13 () () *n n Au Au A s ij g ij*

> 11 *<sup>s</sup> A t* r

12 *<sup>g</sup> A t* r

*n*+1 *and us n*+1

). In this subroutine, equations (14)

+ + + = (62)

Hydrodynamic and Heat Transfer Simulation of Fluidized Bed Using CFD

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171

<sup>=</sup> <sup>D</sup> (63)

<sup>=</sup> <sup>D</sup> (64)

*n*+1 *and us n*+1 ).

$$
\varepsilon\_{s,\max} = 1 - \varepsilon\_{\text{g,mf}} \tag{59}
$$

and if *ε<sup>s</sup>* >*εs*,maxthen:

$$
\boldsymbol{\varepsilon}\_s^{e\chi} = \boldsymbol{\varepsilon}\_s - \boldsymbol{\varepsilon}\_{s,\max} \tag{60}
$$

The balance may be expressed in terms of particle volume fraction:

$$
\varepsilon\_{s,i,j}^{new} = \varepsilon\_{s,i,j}^{old} - \varepsilon\_{s,i,j}^{ex} + \frac{\varepsilon\_{s,i+1,j}^{ex}}{4} + \frac{\varepsilon\_{s,i-1,j}^{ex}}{4} + \frac{\varepsilon\_{s,i,j+1}^{ex}}{4} + \frac{\varepsilon\_{s,i,j-1}^{ex}}{4} \tag{61}
$$

Figure (3) shows the correction mechanism:

Hydrodynamic and Heat Transfer Simulation of Fluidized Bed Using CFD http://dx.doi.org/10.5772/52072 171

**Figure 3.** The excess solid volume correction

**11.** Call subroutine "dirx" to solve momentum equation in x-direction and evaluate at the new time step x-components velocities (*ug n*+1 *and us n*+1 ). In this subroutine, equations (14) and (43) are solved to get values of x-component new step velocities (*ug n*+1 *and us n*+1 ). Equation (43) is reduced to the following form :

$$A\_{11} (\mu\_s)\_{i,j}^{n+1} + A\_{12} (\mu\_g)\_{i,j}^{n+1} = A\_{13} \tag{62}$$

where:

**5.** Calculate the particle terminal velocity from the following equation [2]:

*u Ar*

*t*

170 Nuclear Reactor Thermal Hydraulics and Other Applications

**9.** Specify Initial and boundary conditions.

time step particles volume fractions ( *ε<sup>s</sup>*

**6.** Specify stability of fluidized bed.

**7.** Determine the Δt, Δx and Δy

(41 ).

in excess in each cell where:

and if *ε<sup>s</sup>* >*εs*,maxthen:

( ) <sup>2</sup> 0.5 <sup>2</sup> 0.5 3.809 3.809 1.832 *<sup>g</sup>*

**8.** Determine the fluidized bed height at the entering velocity using the equation [5]:

( ) ( ) , ,

e

*H H* e

*g mf*

*g in*

**10.** Call subroutine "cont" to solve particles phase continuity equations and evaluate the new

The correction works out as a posteriori redistribution of the particle phase volume fraction

*s s*,max

,max , 1 *<sup>s</sup> g mf*

,, ,, ,, 4444

*new old ex si j si j sij sij*

 e

,max

, 1, , 1, , , 1 , , 1

+- +- =-+ + + + (61)

*ex ex ex ex*

eeee

*n*+1

*mf*

1

1

1

**Note**: we use the excess solid volume correction [8] in a special subroutine:

e e

e

*ex s ss*

e ee

The balance may be expressed in terms of particle volume fraction:

*sij sij sij*

eee

Figure (3) shows the correction mechanism:

é ù =- + + ê ú

*g p*


£ (58)

= - (59)

= - (60)

*n*+1

) from equation

) consequently evaluate (*ε<sup>g</sup>*

*d* m

r

ë û (56)

$$A\_{11} = \frac{\rho\_s}{\Delta t} \tag{63}$$

$$A\_{12} = \frac{\rho\_g}{\Delta t} \tag{64}$$

$$\begin{aligned} A\_{13} &= \frac{\rho\_s}{\Delta t} \left( u\_s \right)\_{i,j}^n + \frac{\rho\_s}{\Delta t} \left( u\_s \right)\_{i,j}^n \\ &- \frac{\left( u\_s \right)\_{i,j}^n}{\Delta x} \left[ \left( u\_s \right)\_{i+1,j}^n - \left( u\_s \right)\_{i-1,j}^n \right. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \right \right. \right. \right. \right. \right. \right. \right. \right. \right. \right. \right. \\ & \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \right. \right. \right. \right. \right. \right. \right. \right. \right. \right. \right. \right. \right. \right. \right. \right. \right. \right. \right. \right. \right. \right. \\ & \left. \left. \left. \left( u\_s \right)\_{i+1,j}^n - \left( u\_s \right)\_{i,j-1}^n \right. \right. \right. \right. \right. \right. \right. \right. \right. \right. \\ & \left. \left. \left. \left( u\_s \right)\_{i,j}^n - \left( u\_s \right)\_{i,j}^n \right. \right. \right$$

where:

( *<sup>B</sup>*<sup>11</sup> (*εs*)*i*, *<sup>j</sup> n*+1

*B*<sup>12</sup> (*εg*)*i*, *<sup>j</sup>*

Which are solved to get (*vg*

*n*+1)((*vs*)*i*, *<sup>j</sup> n*+1

(*vg*)*i*, *<sup>j</sup>*

solid particles temperatures (*Tg*

( ) ( ) ( )

*d*

*t* r

( ) ( )

*<sup>n</sup> <sup>n</sup> <sup>g</sup> <sup>s</sup> s g i j i j n nn n s ss s i j ij i j i j*

r

<sup>13</sup> , ,

+

*Bv v t t*

= + D D

( ) ( ) ( ) ( )

*x v v if u*

<sup>ì</sup> - ³ <sup>ï</sup> - <sup>í</sup> <sup>D</sup> <sup>ï</sup> - < <sup>î</sup>

*u v v if u*

( ) ( ) ( ) ( )

*v v v if v*

+

+

( ) ( ) ( ) ( )

*u v v if u*


( ) ( ) ( ) ( )

*<sup>y</sup> v v if v*

<sup>ì</sup> - ³ ïï - <sup>í</sup> <sup>D</sup> <sup>ï</sup> - < ïî

*v v v if v*

*y v v if v*

<sup>ì</sup> - ³ <sup>ï</sup> - <sup>í</sup> <sup>D</sup> <sup>ï</sup> - < <sup>î</sup>

( ) ( ) ( )


1, , ,

*nn n ss s i j ij i j n nn n s ss s i j ij ij i j*

,1 , ,


1, , ,

*nn n gg g i j ij i j n nn n <sup>g</sup> gg g i j ij ij i j*

*nn n ss s ij ij i j n nn n <sup>g</sup> gg g i j ij i j i j*


, , 1, ,

( ) ( ) ( )

, , 1, ,

*v v if u*

( ) ( ) ( )

3.2

*gd dy*

So equation (66) and equation (13) result the following system of equations:

*n*+1 *and Ts n*+1



+

, , ,1 ,

( ( ))

**13.** Call subroutine "temp" to solve energy equation and evaluate the new time step gas and

r r

*ps g*

,1 , ,

*nn n gg g ij ij i j*



<sup>ï</sup> - < ïî

, , ,1 ,

*n n n*

*v v*

1

b

*s g i j*

=- - + ç ÷ <sup>D</sup> ç ÷ - è ø

3

e r

12

*B*

r

*x*

ì ïï í

( ) ( )


e

*<sup>n</sup>*+1) =( *<sup>B</sup>*<sup>13</sup> *Vgin* )

> *n*+1 *and vs n*+1 )

, ,1

*n n s s ij ij*

 e

*C*

b

( ) , , , 1.8

*s gg s i j i j i j <sup>n</sup> d s i j p*


(1 ) <sup>4</sup>

( ) ( ( ) ) ,

e

e

æ ö ç ÷

*n*

1

,

, 0.0

, 0.0

, 0.0

, 0.0

, 0

, 0.0

, 0.0

, 0.0

.0

).In this subroutine, equations (46) and (47) are

*n s i j* ,

Hydrodynamic and Heat Transfer Simulation of Fluidized Bed Using CFD

http://dx.doi.org/10.5772/52072

(69)

173

(70)

 e-

= - (68)

so equation (62) and equation (14) result the following system of equations:

$$
\begin{pmatrix} A\_{11} & A\_{12} \\ \binom{\varepsilon\_s}{\varepsilon\_s}\_{i\_{\ast',\tilde{j}}}^{n+1} & \binom{\varepsilon\_s}{\varepsilon\_{\tilde{i},\tilde{j}}}\_{i\_{\ast',\tilde{j}}}^{n+1} \end{pmatrix} \begin{pmatrix} \mu\_s \end{pmatrix}\_{i\_{\ast},\tilde{j}}^{n+1} = \begin{pmatrix} A\_{13} \\ \boldsymbol{\mu}\_{13} \end{pmatrix}\_{\mathcal{S}^{in}}
$$

Which are solved to get (*ug n*+1 *and us n*+1 )

**12.** Call subroutine "diry" to solve momentum equation in y-direction and evaluate at the new time step y-components velocities (*vg n*+1 *and vs n*+1 ). In this subroutine, equations (13) and (44) are used to get values of (*vg n*+1 *and vs n*+1 ). Equation (44) is reduced to the following form :

$$B\_{11}(\upsilon\_s)\_{i,j}^{n+1} + B\_{12}(\upsilon\_g)\_{i,j}^{n+1} = B\_{13} \tag{66}$$

where:

$$B\_{11} = \frac{\rho\_s}{\Delta t} + \beta \left( 1 + \frac{\left(\varepsilon\_s\right)\_{i,j}^n}{\left(1 - \left(\varepsilon\_s\right)\_{i,j}^n\right)}\right) \tag{67}$$

where:

( ) ( )

*<sup>n</sup> <sup>n</sup> <sup>g</sup> <sup>s</sup> s g i j i j n nn n s ss s i j ij i j i j*

r

<sup>13</sup> , ,

+

*Au u t t*

= + D D

r

172 Nuclear Reactor Thermal Hydraulics and Other Applications

*x*

( *<sup>A</sup>*<sup>11</sup> (*εs*)*i*, *<sup>j</sup> n*+1

*A*<sup>12</sup> (*εg*)*i*, *<sup>j</sup>*

Which are solved to get (*ug*

form :

where:

*n*+1)((*us*)*i*, *<sup>j</sup>*

(*ug*)*i*, *<sup>j</sup>*

*n*+1

*<sup>n</sup>*+1) =( *<sup>A</sup>*<sup>13</sup> *Ugin* )

> *n*+1 *and us n*+1 )

> > 11

*t*

r

*B*

new time step y-components velocities (*vg*

and (44) are used to get values of (*vg*

ì ïï í

( ) ( ) ( ) ( )

*x u u if u*

<sup>ì</sup> - ³ <sup>ï</sup> - <sup>í</sup> <sup>D</sup> <sup>ï</sup> - < <sup>î</sup>

*u u u if u*

( ) ( ) ( ) ( )

*v u u if v*

+

+

( ) ( ) ( ) ( )

*u u u if u*


( ) ( ) ( ) ( )

*<sup>y</sup> u u if v*

<sup>ì</sup> - ³ ïï - <sup>í</sup> <sup>D</sup> <sup>ï</sup> - < ïî

*v u u if v*

+

so equation (62) and equation (14) result the following system of equations:

*y u u if v*

<sup>ì</sup> - ³ <sup>ï</sup> - <sup>í</sup> <sup>D</sup> <sup>ï</sup> - < <sup>î</sup>

( ) ( ) ( )


1, , ,

*nn n ss s i j ij i j n nn n s ss s i j ij ij i j*

, 0.0

, 0.0

, 0.0

, 0.0

, 0

, 0.0

, 0.0

, 0.0

.0

). In this subroutine, equations (13)

). Equation (44) is reduced to the following

*s ij g ij Bv Bv B* + + + = (66)

(65)

(67)

,1 , ,


1, , ,

*nn n gg g i j ij i j n nn n <sup>g</sup> gg g i j ij ij i j*

,1 , ,

**12.** Call subroutine "diry" to solve momentum equation in y-direction and evaluate at the

*n*+1 *and vs n*+1

1 1 11 , 12 , 13 () () *n n*

1

b

*<sup>s</sup> i j <sup>s</sup>*

=+ + ç ÷ <sup>D</sup> ç ÷ - è ø

*n*+1 *and vs n*+1

( ) ( ( ) ) ,

e

e

æ ö ç ÷

*n*

1

,

*n s i j*

*nn n gg g ij ij i j*


*nn n ss s ij ij i j n nn n <sup>g</sup> gg g i j ij i j i j*


, , 1, ,

( ) ( ) ( )

, , 1, ,

*u u if u*

( ) ( ) ( )

, , ,1 ,


<sup>ï</sup> - < ïî

, , ,1 ,

$$\beta = \mathbb{C}\_d \frac{\mathbf{3} \left( \boldsymbol{\varepsilon}\_s \right)\_{i,j}^n \rho\_{\mathcal{S}} \left| \left( \boldsymbol{v}\_{\mathcal{S}} \right)\_{i,j}^n - \left( \boldsymbol{v}\_s \right)\_{i,j}^n \right|}{4d\_p} (\mathbf{1} - \left( \boldsymbol{\varepsilon}\_s \right)\_{i,j}^n)^{-1.8} \tag{68}$$

$$B\_{12} = -\frac{\rho\_{\mathcal{g}}}{\Delta t} - \beta \left( 1 + \frac{\left(\boldsymbol{\varepsilon}\_{s}\right)\_{i,j}^{n}}{\left(1 - \left(\boldsymbol{\varepsilon}\_{s}\right)\_{i,j}^{n}\right)}\right) \tag{69}$$

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) <sup>13</sup> , , , , 1, , 1, , , , , ,1 , ,1 , , , , 1, , , 0.0 , 0.0 , 0.0 , 0.0 , 0 *<sup>n</sup> <sup>n</sup> <sup>g</sup> <sup>s</sup> s g i j i j n nn n s ss s i j ij i j i j nn n ss s i j ij i j n nn n s ss s i j ij ij i j nn n ss s ij ij i j n nn n <sup>g</sup> gg g i j ij i j i j Bv v t t u v v if u x v v if u v v v if v y v v if v u v v if u x* r r - + - + - = + D D <sup>ì</sup> - ³ <sup>ï</sup> - <sup>í</sup> <sup>D</sup> <sup>ï</sup> - < <sup>î</sup> <sup>ì</sup> - ³ <sup>ï</sup> - <sup>í</sup> <sup>D</sup> <sup>ï</sup> - < <sup>î</sup> - ³ - <sup>D</sup> ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ( )) 1, , , , , ,1 , ,1 , , , ,1 .0 , 0.0 , 0.0 , 0.0 3.2 *nn n gg g i j ij i j n nn n <sup>g</sup> gg g i j ij ij i j nn n gg g ij ij i j n n s s ij ij ps g v v if u v v v if v <sup>y</sup> v v if v gd dy* e e r r + - + ì ïï í <sup>ï</sup> - < ïî <sup>ì</sup> - ³ ïï - <sup>í</sup> <sup>D</sup> <sup>ï</sup> - < ïî - - - (70)

So equation (66) and equation (13) result the following system of equations:

$$
\begin{pmatrix} B\_{11} & B\_{12} \\ \left(\varepsilon\_s\right)\_{i\_{\cdot,\cdot}j}^{n+1} & \left(\varepsilon\_{\mathcal{S}}\right)\_{i\_{\cdot,\cdot}j}^{n+1} \end{pmatrix} \begin{pmatrix} \left(\upsilon\_s\right)\_{i\_{\cdot,\cdot}j}^{n+1} \\ \left(\upsilon\_{\mathcal{S}}\right)\_{i\_{\cdot,\cdot}j}^{n+1} \end{pmatrix} = \begin{pmatrix} B\_{13} \\ V\_{\mathcal{S}^{in}} \end{pmatrix},$$

Which are solved to get (*vg n*+1 *and vs n*+1 )

**13.** Call subroutine "temp" to solve energy equation and evaluate the new time step gas and solid particles temperatures (*Tg n*+1 *and Ts n*+1 ).In this subroutine, equations (46) and (47) are used to get values of new time step temperatures for both phases (*Tg n*+1 *and Ts n*+1 ). Equation (46) is reduced to the following form :

$$\mathbf{C}\_{11}\mathbf{(}T\_s\mathbf{)}\_{i,j}^{n+1} + \mathbf{C}\_{12}\mathbf{(}T\_g\mathbf{)}\_{i,j}^{n+1} = \mathbf{C}\_{13} \tag{71}$$

( ) ( ) , <sup>1</sup> <sup>1</sup>

<sup>D</sup> (77)

Hydrodynamic and Heat Transfer Simulation of Fluidized Bed Using CFD

http://dx.doi.org/10.5772/52072

, 0.

, 0.0

, 0.0

(78)

175

(79)

, 0.0


12 , *g pg n n g v i j*

e<sup>+</sup> <sup>+</sup> = +

( ) ( ) ( ) ( )

 e

, , , 1, ,

*g pg g gg gg <sup>g</sup> i j ij i j i j*

<sup>ì</sup> - ³ ïï - <sup>í</sup> <sup>D</sup> <sup>ï</sup> - < ïî

*<sup>x</sup> T T if u*

*C u T T if u*

( ) ( ) ( ) ( )

 e

, , , ,1 ,

*s pg g gg gg <sup>g</sup> i j ij ij i j*

<sup>ì</sup> - ³ ïï - <sup>í</sup> <sup>D</sup> <sup>ï</sup> - < ïî

*<sup>y</sup> T T if v*

*n nn g gg g gg <sup>j</sup> i j i j*

*n nn g gg g gg g gg i j i j i j*

ee

+

*C v T T if v*

+

( ) ( ) ( )

*n nn n*

 e

*n nn n*

1, , ,

*n nn gg gg <sup>g</sup> i j ij i j*


,1 , ,

*n nn gg gg <sup>g</sup> i j i j i j*


( ) ( ) ( )

 e

( ) ( )

*KT KT*

 e

1, , 1, 2

, 1 , , 1 2

+ -

*C D h t* r

( )

e

*T*

*t*

e

e

e

e

2

e

2

( ) ( ) ( )

*x KT KT KT*

D - +


*y*

D

so equation (71) and equation (75) result the following system of equations:

,

*n g g i j*

( )

e

e

*<sup>n</sup>*+1) =(*C*<sup>13</sup> *D*<sup>13</sup> )

+

+

(*C*11 *D*<sup>11</sup> *C*<sup>12</sup> *D*<sup>12</sup>

**•** Else adjust *ε<sup>g</sup>*

)((*Ts*)*i*, *<sup>j</sup> n*+1

(*Tg*)*i*, *<sup>j</sup>*

Which are solved to get (*Tg*

*n*+1 by :

e

 e

*K T*

*g gg <sup>i</sup>*

+

*n*+1 *and Ts n*+1 )

**14.** Make gas residual check,which given from the equation (42):

**•** If |*dg*(*i*, *j*)| ≤*δ* go to step 15, where *δ* is a small positive value *δ*=5X10-3.

( ) ( ) ( ) ( ) ( ) ( )

e

e

*u*

+ + -

*v*

1 1 , 1, , ,, , 11 1

*nn n gg g ij i j i j gg g ij ij i j nn n*

e

e

( ) ( ) ( ) ( )

<sup>ì</sup> - ³ <sup>D</sup> ïï - <sup>í</sup> <sup>D</sup> <sup>ï</sup> - < ïî

+

<sup>ì</sup> - ³ ïï = - <sup>í</sup> <sup>D</sup> <sup>ï</sup> - < ïî

 e

*if v <sup>t</sup>*

+ -

*<sup>y</sup> if v*

+

( ) ( ) ( )

*gg g ij ij i j*

1 , ,1 , , 11 1 ,1 , ,

*<sup>n</sup> gg g ij ij i j <sup>g</sup> i j nn n*

> e

*if u dt*

*<sup>x</sup> if u*

( ) ( ) ( )

*nn n*

++ +

++ +

 e

 e

11 1

++ +

*nn n*

, 0.0

, 0.0

, 0.0

, 0.0

1, , , 11 1

++ +

*gg g i j ij i j*

13 ,

r

*D C*

r

r *g pg*

<sup>=</sup> <sup>D</sup>

where:

$$\mathbf{C}\_{11} = \frac{\rho\_s \mathbf{C}\_{p,s}}{\Delta t} \left( \boldsymbol{\varepsilon}\_s^{n+1} \right) + \left( \boldsymbol{h}\_v \right)\_{i,j}^{n+1} \tag{72}$$

$$\mathbf{C}\_{12} = \left(\boldsymbol{h}\_v\right)\_{i,j}^{n+1} \tag{73}$$

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , 13 , , , , 1, , 1, , , , , , ,1 , ,1 , , 1, , 0.0 , 0.0 , 0.0 , 0.0 *s ps n s s i j n nn n s ps s ss ss s i j ij i j i j n nn ss ss s i j ij i j n nn n s ps s ss ss s i j ij ij i j n nn ss ss s i j i j i j s ss i j C C T t C u T T if u x T T if u C v T T if v y T T if v K T* r e r e e e e r e e e e e - + - + + <sup>=</sup> <sup>D</sup> <sup>ì</sup> - ³ <sup>ï</sup> - <sup>í</sup> <sup>D</sup> <sup>ï</sup> - < <sup>î</sup> <sup>ì</sup> - ³ <sup>ï</sup> - <sup>í</sup> <sup>D</sup> <sup>ï</sup> - < <sup>î</sup> + ( ) ( ) ( ) ( ) ( ) , 1, 2 , 1 , , 1 2 , 2 2 ( ) *n nn s ss s ss i j i j n nn s ss s ss s ss i j i j i j <sup>n</sup> s ij KT KT x KT KT KT <sup>y</sup> q* e e e ee e - · + - - + D - + + + D (74)

#### Also equation (47) is reduced to the following form:

$$D\_{11}(T\_s)\_{i,j}^{n+1} + D\_{12}(T\_g)\_{i,j}^{n+1} = D\_{13} \tag{75}$$

where:

$$D\_{11} = \left(h\_v\right)\_{i,j}^{n+1} \tag{76}$$

Hydrodynamic and Heat Transfer Simulation of Fluidized Bed Using CFD http://dx.doi.org/10.5772/52072 175

$$D\_{12} = \frac{\rho\_{\mathcal{S}} \mathbb{C}\_{p, \mathcal{S}}}{\Delta t} \Big( \boldsymbol{\varepsilon}\_{\mathcal{S}}^{n+1} \Big) + \left( \boldsymbol{h}\_{v} \right)\_{i,j}^{n+1} \tag{77}$$

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , 13 , , , , 1, , 1, , , , , , ,1 , ,1 , , , 0. , 0.0 , 0.0 , 0.0 *n g g i j g pg n nn n g pg g gg gg <sup>g</sup> i j ij i j i j n nn gg gg <sup>g</sup> i j ij i j n nn n s pg g gg gg <sup>g</sup> i j ij ij i j n nn gg gg <sup>g</sup> i j i j i j g gg <sup>i</sup> T D C t C u T T if u <sup>x</sup> T T if u C v T T if v <sup>y</sup> T T if v K T* e r r e e e e r e e e e e - + - + + <sup>=</sup> <sup>D</sup> <sup>ì</sup> - ³ ïï - <sup>í</sup> <sup>D</sup> <sup>ï</sup> - < ïî <sup>ì</sup> - ³ ïï - <sup>í</sup> <sup>D</sup> <sup>ï</sup> - < ïî + ( ) ( ) ( ) ( ) ( ) 1, , 1, 2 , 1 , , 1 2 2 2 *n nn g gg g gg <sup>j</sup> i j i j n nn g gg g gg g gg i j i j i j KT KT x KT KT KT y* e e e ee - + - - + D - + + D (78)

so equation (71) and equation (75) result the following system of equations:

(*C*11 *D*<sup>11</sup> *C*<sup>12</sup> *D*<sup>12</sup> )((*Ts*)*i*, *<sup>j</sup> n*+1 (*Tg*)*i*, *<sup>j</sup> <sup>n</sup>*+1) =(*C*<sup>13</sup> *D*<sup>13</sup> )

used to get values of new time step temperatures for both phases (*Tg*

1 1 11 , 12 , 13 () () *n n CT CT C s ij g ij*

( ) ( ) , <sup>1</sup> <sup>1</sup>

( ) <sup>1</sup> 12 , *n*

( ) ( ) ( ) ( )

 e

, , , 1, ,

*x T T if u*

*C u T T if u*

<sup>ì</sup> - ³ <sup>ï</sup> - <sup>í</sup> <sup>D</sup> <sup>ï</sup> - < <sup>î</sup>

*C v T T if v*

+

( ) ( ) ( ) ( )

 e

, , , ,1 ,

*y T T if v*

+

*n nn*

*n nn s ss s ss s ss i j i j i j <sup>n</sup>*

ee

2

*x KT KT KT*


<sup>ì</sup> - ³ <sup>ï</sup> - <sup>í</sup> <sup>D</sup> <sup>ï</sup> - < <sup>î</sup>

( ) ( ) ( )

 e

*n nn n s ps s ss ss s i j ij i j i j*

1, , ,

*n nn ss ss s i j ij i j n nn n s ps s ss ss s i j ij ij i j*


,1 , ,

, 1,

2 ,


*<sup>y</sup> q*

*n nn ss ss s i j i j i j*


( ) ( ) ( )

*s ss s ss i j i j*

 e

 e

( ) ( )

*KT KT*

, 1 , , 1

+ -

1 1 11 , 12 , 13 () () *n n DT DT D s ij g ij*

> ( ) <sup>1</sup> 11 , *n*

11 , *s ps n n s v i j*

e<sup>+</sup> <sup>+</sup> = +

*C C h t* r

( )

e

e

e

e

2

e

2

( ) ( ) ( )

*s ps n s s i j*

e

, 13 ,

*C C T t*

r

<sup>=</sup> <sup>D</sup>

r

r

( )

e

e

+

*K T*

*s ss i j*

Also equation (47) is reduced to the following form:

1,

+

(46) is reduced to the following form :

174 Nuclear Reactor Thermal Hydraulics and Other Applications

where:

where:

*n*+1 *and Ts n*+1

+ + + = (71)

<sup>D</sup> (72)

*<sup>v</sup> i j C h* <sup>+</sup> <sup>=</sup> (73)

, 0.0

, 0.0

, 0.0

, 0.0

( )

e

*s ij*

+ + + = (75)

*<sup>v</sup> i j D h* <sup>+</sup> <sup>=</sup> (76)

·

). Equation

(74)

Which are solved to get (*Tg n*+1 *and Ts n*+1 )


$$\begin{split} \left(\boldsymbol{\varepsilon}\_{\mathcal{S}}\right)\_{i,j}^{n+1} &= \left(\boldsymbol{\varepsilon}\_{\mathcal{S}}\right)\_{i,j}^{n} - \frac{dt}{\Delta x} \big(\boldsymbol{u}\_{\mathcal{S}}\big)\_{i,j}^{n+1} - \left(\boldsymbol{\varepsilon}\_{\mathcal{S}}\right)\_{i-1,j}^{n+1} \quad \text{;} \boldsymbol{f}\big(\boldsymbol{u}\_{\mathcal{S}}\big)\_{i,j}^{n+1} \geq 0.0\\ & \left(\boldsymbol{\varepsilon}\_{\mathcal{S}}\right)\_{i+1,j}^{n+1} - \left(\boldsymbol{\varepsilon}\_{\mathcal{S}}\right)\_{i,j}^{n+1} \quad \text{;} \boldsymbol{f}\big(\boldsymbol{u}\_{\mathcal{S}}\big)\_{i,j}^{n+1} < 0.0\\ & -\frac{\Delta t}{\Delta y} \big(\boldsymbol{\upsilon}\_{\mathcal{S}}\big)\_{i,j}^{n+1} \left[ \left(\boldsymbol{\varepsilon}\_{\mathcal{S}}\right)\_{i,j}^{n+1} - \left(\boldsymbol{\varepsilon}\_{\mathcal{S}}\right)\_{i,j-1}^{n+1} \cdot \boldsymbol{f}\big(\boldsymbol{\upsilon}\_{\mathcal{S}}\right)\_{i,j}^{n+1} \geq 0.0\\ & \left(\boldsymbol{\varepsilon}\_{\mathcal{S}}\right)\_{i,j+1}^{n+1} - \left(\boldsymbol{\varepsilon}\_{\mathcal{S}}\right)\_{i,j}^{n+1} \quad \text{;} \boldsymbol{f}\big(\boldsymbol{\upsilon}\_{\mathcal{S}}\big)\_{i,j}^{n+1} < 0.0 \end{split} \tag{79}$$

and calculate *ε<sup>s</sup> n*+1 from equation (41).


## **8. Parametric study of the hydrodynamic and thermal results**

The present work results show the effect of variation of several bed parameters such as particle diameter, terminal velocity of the particle, minimum fluidized velocity of the particle input gas velocity, fluidized material type and heat generation by particles on the hydrodynamic and thermal behavior of fluidized bed. In this section the effect of different parameters in hydrodynamic and thermal performance of fluidized bed is analyzed in detail.

#### **8.1. Effect of particle diameter**

Particle diameter is the most influential parameter in the overall fluidized bed performance. In view of that fact, the bed material in a fluidized bed is characterized by a wide range of particle diameter, so that the effect of particle diameter is analyzed in details. In this section particle diameter is changed from 100 μm to 1000 μm for sand as fluidized material to study the effect of particle diameter on the fluidization performance.

One of the important parameter of the fluidized bed study is the total pressure drop across the bed. Although it is constant after beginning of fluidization and equal to the weight of the bed approximately. But its value changes with change of particle diameter.The effect of change of particle diameter on total pressure drop is very important in design and cost of fluidized bed. Figure (4) shows that effect for sand particles of different diameters (100, 200, 300, 400, 500, 600, 700, 800, 900 and 1000 μm). It is clear that the pressure drop increase with increase of particle diameter, which means that small particle is betters in design of fluidized bed cost.

parameters (εg, ρsus,vg,vs,ug,us,Tg,Ts) due to the bubbles formation. The disturbance decreases with increase of particles diameter. This may be due to come near D-type under uniform

**0.00 4.00 8.00 12.00 16.00 20.00 Tim e,(s)**

**0 0.0002 0.0004 0.0006 0.0008 0.001 dp,(m)**

> **dp = 100 micron dp = 200 micron dp = 300 micron dp = 400 micron dp = 500 micron dp = 600 micron dp = 700 micron dp = 800 micron dp = 900 micron dp = 1000 micron**

Hydrodynamic and Heat Transfer Simulation of Fluidized Bed Using CFD

http://dx.doi.org/10.5772/52072

177

**6800**

**Figure 4.** Effect of particle diameter on total pressure drop across fluidized bed

**2000**

**0**

**Figure 5.** Effect of particle diameter on average heat transfer coefficient in fluidized bed

**400**

**800**

**hav ,(W/m2 K**)

**1200**

**1600**

**7000**

**7200**

D**PB,(Pa)** **7400**

**7600**

Fluidized bed is used for wide range of particle diameter. It is better to fluidize particle Dtype in spouted bed to decrease the pressure drop. However, using uniform fluidization for D type give a good and stable thermal behavior of fluidized bed. So in some applications like nuclear

Figures from (14) to (21) show the effect of change particle diameter on hydrodynamic and thermal behavior for particle D-type. For this particles type, the behavior of fluidized bed is more uniform in performance than B-type [be consistent with usage and define what the different types are]. Although D-type gives butter fluidization in spouted bed but it gives good performance under uniform fluidization with high pressure drop as shown in figure (4). This

reactors the stability and safety is important than cost of pumping power.

means an increase in pumping power and costs to achieve uniform fluidization.

fluidization.

A key parameter in the Thermal analysis of fluidized bed is the average heat transfer coeffi‐ cient. Figure (5) shows the variation of average heat transfer coefficient with the time for different particles diameter from 100 to 1000 μm sand particles.It is observed from the figure that particles with small size show higher values of average heat transfer coefficient. The particles with diameter 100 μm show higher for average heat transfer coefficient reaches to about 3 times of particles with diameter 1000 μm.

Particles diameter determines the type of particles on Geldart diagram, consequently the behavior of the fluidized bed.

Figures from (6) to (13) illustrate the effect of particles diameter on hydrodynamic and thermal behavior for particle type-B. These figures explain the high disturbance in different bed Hydrodynamic and Heat Transfer Simulation of Fluidized Bed Using CFD http://dx.doi.org/10.5772/52072 177

**Figure 4.** Effect of particle diameter on total pressure drop across fluidized bed

and calculate *ε<sup>s</sup>*

**17.** End program.

*n*+1

**15.** Calculate the gas pressure values.

176 Nuclear Reactor Thermal Hydraulics and Other Applications

**16.** Calculate dimensionless numbers.

**8.1. Effect of particle diameter**

from equation (41).

**•** Go to step 11 and calculate new time step velocities

**8. Parametric study of the hydrodynamic and thermal results**

hydrodynamic and thermal performance of fluidized bed is analyzed in detail.

the effect of particle diameter on the fluidization performance.

about 3 times of particles with diameter 1000 μm.

behavior of the fluidized bed.

The present work results show the effect of variation of several bed parameters such as particle diameter, terminal velocity of the particle, minimum fluidized velocity of the particle input gas velocity, fluidized material type and heat generation by particles on the hydrodynamic and thermal behavior of fluidized bed. In this section the effect of different parameters in

Particle diameter is the most influential parameter in the overall fluidized bed performance. In view of that fact, the bed material in a fluidized bed is characterized by a wide range of particle diameter, so that the effect of particle diameter is analyzed in details. In this section particle diameter is changed from 100 μm to 1000 μm for sand as fluidized material to study

One of the important parameter of the fluidized bed study is the total pressure drop across the bed. Although it is constant after beginning of fluidization and equal to the weight of the bed approximately. But its value changes with change of particle diameter.The effect of change of particle diameter on total pressure drop is very important in design and cost of fluidized bed. Figure (4) shows that effect for sand particles of different diameters (100, 200, 300, 400, 500, 600, 700, 800, 900 and 1000 μm). It is clear that the pressure drop increase with increase of particle diameter, which means that small particle is betters in design of fluidized bed cost.

A key parameter in the Thermal analysis of fluidized bed is the average heat transfer coeffi‐ cient. Figure (5) shows the variation of average heat transfer coefficient with the time for different particles diameter from 100 to 1000 μm sand particles.It is observed from the figure that particles with small size show higher values of average heat transfer coefficient. The particles with diameter 100 μm show higher for average heat transfer coefficient reaches to

Particles diameter determines the type of particles on Geldart diagram, consequently the

Figures from (6) to (13) illustrate the effect of particles diameter on hydrodynamic and thermal behavior for particle type-B. These figures explain the high disturbance in different bed

**Figure 5.** Effect of particle diameter on average heat transfer coefficient in fluidized bed

parameters (εg, ρsus,vg,vs,ug,us,Tg,Ts) due to the bubbles formation. The disturbance decreases with increase of particles diameter. This may be due to come near D-type under uniform fluidization.

Fluidized bed is used for wide range of particle diameter. It is better to fluidize particle Dtype in spouted bed to decrease the pressure drop. However, using uniform fluidization for D type give a good and stable thermal behavior of fluidized bed. So in some applications like nuclear reactors the stability and safety is important than cost of pumping power.

Figures from (14) to (21) show the effect of change particle diameter on hydrodynamic and thermal behavior for particle D-type. For this particles type, the behavior of fluidized bed is more uniform in performance than B-type [be consistent with usage and define what the different types are]. Although D-type gives butter fluidization in spouted bed but it gives good performance under uniform fluidization with high pressure drop as shown in figure (4). This means an increase in pumping power and costs to achieve uniform fluidization.

**0 4 8 1 2 1 6 2 0 Time,(s)**

**0 4 8 1 2 1 6 2 0 Time,(s)**

**0 4 8 1 2 1 6 2 0 Time,(s)**

**dp=200 m dp=300 m dp=400 m dp=500 m**

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**dp=200 m dp=300 m dp=400 m dp=500 m**

**dp=200 m dp=300 m dp=400 m dp=500 m**

**-0.2**

**Figure 9.** Effect of change particle diameter on vertical particle velocity (B-type)

**-0.008**

**0**

**0.001**

**0.002**

**0.003**

**ug,(m/s)**

**Figure 11.** Effect of particle diameter on horizontal gas velocity (B-Type)

**0.004**

**0.005**

**-0.004**

**us,(m/s)**

**Figure 10.** Effect of particle diameter on horizontal particle velocity (B-type)

**0**

**0.004**

**0**

**0.2**

**vs,(m/s)**

**0.4**

**0.6**

**Figure 6.** Effect of particle diameter on gas volume fraction (B-type)

**Figure 7.** Effect of particle diameter on suspension density (B-type)

**Figure 8.** Effect of particle diameter on vertical gas velocity (B-type)

Hydrodynamic and Heat Transfer Simulation of Fluidized Bed Using CFD http://dx.doi.org/10.5772/52072 179

**Figure 9.** Effect of change particle diameter on vertical particle velocity (B-type)

**0 4 8 1 2 1 6 2 0 Time,(s)**

**0 4 8 1 2 1 6 2 0 Time,(s)**

**0 4 8 1 2 1 6 2 0 Time,(s)**

**dp=200 m dp=300 m dp=400 m dp=500 m**

**dp=200 m dp=300 m dp=400 m dp=500 m**

**dp=200 m dp=300 m dp=400 m dp=500 m**

**0.7**

**800**

**0**

**Figure 8.** Effect of particle diameter on vertical gas velocity (B-type)

**0.4**

**0.8**

**1.2**

**vg,(m/s)**

**1.6**

**2**

**900**

**1000**

r**sus,(kg/m3)**

**Figure 7.** Effect of particle diameter on suspension density (B-type)

**1100**

**1200**

**1300**

**Figure 6.** Effect of particle diameter on gas volume fraction (B-type)

**0.72**

**0.74**

e**g** **0.76**

**0.78**

**0.8**

178 Nuclear Reactor Thermal Hydraulics and Other Applications

**Figure 10.** Effect of particle diameter on horizontal particle velocity (B-type)

**Figure 11.** Effect of particle diameter on horizontal gas velocity (B-Type)

**0 4 8 1 2 1 6 2 0 Time,(s)**

**0 4 8 1 2 1 6 2 0 Time,(s)**

**0 4 8 1 2 1 6 2 0 Time,(s)**

**dp=700 m dp=800 m dp=900 m dp=1000 m**

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**dp=700 m dp=800 m dp=900 m dp=1000 m**

dp=700 m dp=800 m dp=900 m dp=1000 m

**800**

**0**

**-0.5**

**Figure 17.** Effect of change particle diameter on vertical particle velocity (D-Type)

**0**

**0.5**

**vs,(m/s)**

**1**

**1.5**

**2**

**Figure 16.** Effect of particle diameter on vertical gas velocity (D-type)

**1**

**2**

**vg,(m/s)**

**3**

**4**

**1000**

r**sus,(kg/m3)**

**Figure 15.** Effect of particle diameter on suspension density (D-type)

**1200**

**1400**

**Figure 12.** Effect of particle diameter on gas temperature (B-type)

**Figure 13.** Effect of particle diameter on particle temperature (B-type)

**Figure 14.** Effect of particle diameter on gas volume fraction (D-type)

Hydrodynamic and Heat Transfer Simulation of Fluidized Bed Using CFD http://dx.doi.org/10.5772/52072 181

**Figure 15.** Effect of particle diameter on suspension density (D-type)

**0 4 8 1 2 1 6 2 0 Time,(s)**

0 4 8 1 2 1 6 2 0

**0 4 8 1 2 1 6 2 0 Time,(s)**

**0 4 8 1 2 1 6 2 0 Time,(s)**

**dp=200 m dp=300 m dp=400 m dp=500 m**

**dp=200 m dp=300 m dp=400 m dp=500 m**

4 0

**dp=700 m dp=800 m dp=900 m dp=1000 m**

8 0

120

160

200

**0**

**4 0**

**0.68**

**Figure 14.** Effect of particle diameter on gas volume fraction (D-type)

**0.72**

**0.76**

e**g** **0.8**

**0.84**

**8 0**

**120**

**Ts,(**°**C)**

**Figure 13.** Effect of particle diameter on particle temperature (B-type)

**160**

**200**

**4 0**

**8 0**

**Tg(**°**C)**

**Figure 12.** Effect of particle diameter on gas temperature (B-type)

**120**

**160**

180 Nuclear Reactor Thermal Hydraulics and Other Applications

**Figure 16.** Effect of particle diameter on vertical gas velocity (D-type)

**Figure 17.** Effect of change particle diameter on vertical particle velocity (D-Type)

**0 4 8 1 2 1 6 2 0 Time,(s)**

In this section the input gas velocity is changed from one to nine times minimum fluidized velocityfor500μmsandparticlestostudythetheeffectofthisparameteronfluidizationbehavior.

Input gas velocity has an effective role in thermal performance of the fluidized bed. In order to illustrate its effect, the relation between Nusselt number and flow number is described in figure (22). It is clear from this figure that with increase in flow numbers, the Nusselt number increases until reached an optimum flow number where the Nusselt number reaches its maximum value. After this optimum value the increase in flow number is associated with a decrease in Nusselt number. This decrease in Nusselt number may be due to the increase of input gas velocity toward the terminal velocity, consequently the bed goes to be empty bed.

Figure (23) shows the variation of Nusselt number with velocity number. Also the relation between Nusselt number and velocity number has the same trend as the Nusselt number with flow number. This confirms the result from figure (22). This means that there is an optimum input gas velocity to yield the best heat transfer characteristics. This velocity is the target of the fluidized bed designer. The value of this velocity depends on the fluidized gas properties,

Type of fluidized material controls hydrodynamic and thermal performance of fluidized bed.It affects on the different parameters of fluidization such as gas volume fraction, suspension density, gas velocity distribution and particle velocity distribution, gas phase temperature and particle phase temperature. In this section different types of materials such as sand, marble, lead, copper, aluminum and steel of particle diameters 1mm are used to study the effect of

Figure (24) shows the change of gas volume fraction with time for different types of fluidized materials. The figure shows that the gas volume fraction of copper is the highest value

dp=700 m dp=800 m dp=900 m dp=1000 m

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183

**4 0**

**Figure 21.** Effect of particle diameter on particle temperature (D-type)

the fluidized material, particle diameter, and bed geometry.

**8.3. Effect of fluidized material type**

fluidized materials on the bed performance.

**8.2. Effect of input gas velocity**

**8 0**

**120**

**Ts(**°**C)** **160**

**200**

**Figure 18.** Effect of particle diameter on horizontal particle velocity (D-type)

**Figure 19.** Effect of particle diameter on horizontal gas particle velocity (D-type)

**Figure 20.** Effect of particle diameter on gas temperature (D-type)

**Figure 21.** Effect of particle diameter on particle temperature (D-type)

#### **8.2. Effect of input gas velocity**

**0 4 8 1 2 1 6 2 0 Time,(s)**

**0 4 8 1 2 1 6 2 0 Time,(s)**

**0 4 8 1 2 1 6 2 0 Time,(s)**

**dp=700 m dp=800 m dp=900 m dp=1000 m**

**dp=700 m dp=800 m dp=900 m dp=1000 m**

> **dp=700 m dp=800 m dp=900 m dp=1000 m**

**-0.0006**

**0**

**0.0002**

**Figure 19.** Effect of particle diameter on horizontal gas particle velocity (D-type)

**3 0**

**Figure 20.** Effect of particle diameter on gas temperature (D-type)

**4 0**

**5 0**

**Tg(**°**C)**

**6 0**

**7 0**

**0.0004**

**0.0006**

**ug,(m/s)**

**0.0008**

**0.001**

**-0.0004**

**-0.0002**

**us,(m/s)**

182 Nuclear Reactor Thermal Hydraulics and Other Applications

**Figure 18.** Effect of particle diameter on horizontal particle velocity (D-type)

**0**

**0.0002**

In this section the input gas velocity is changed from one to nine times minimum fluidized velocityfor500μmsandparticlestostudythetheeffectofthisparameteronfluidizationbehavior.

Input gas velocity has an effective role in thermal performance of the fluidized bed. In order to illustrate its effect, the relation between Nusselt number and flow number is described in figure (22). It is clear from this figure that with increase in flow numbers, the Nusselt number increases until reached an optimum flow number where the Nusselt number reaches its maximum value. After this optimum value the increase in flow number is associated with a decrease in Nusselt number. This decrease in Nusselt number may be due to the increase of input gas velocity toward the terminal velocity, consequently the bed goes to be empty bed.

Figure (23) shows the variation of Nusselt number with velocity number. Also the relation between Nusselt number and velocity number has the same trend as the Nusselt number with flow number. This confirms the result from figure (22). This means that there is an optimum input gas velocity to yield the best heat transfer characteristics. This velocity is the target of the fluidized bed designer. The value of this velocity depends on the fluidized gas properties, the fluidized material, particle diameter, and bed geometry.

#### **8.3. Effect of fluidized material type**

Type of fluidized material controls hydrodynamic and thermal performance of fluidized bed.It affects on the different parameters of fluidization such as gas volume fraction, suspension density, gas velocity distribution and particle velocity distribution, gas phase temperature and particle phase temperature. In this section different types of materials such as sand, marble, lead, copper, aluminum and steel of particle diameters 1mm are used to study the effect of fluidized materials on the bed performance.

Figure (24) shows the change of gas volume fraction with time for different types of fluidized materials. The figure shows that the gas volume fraction of copper is the highest value

**Figure 22.** Variation of Nusselt number with flow number

**Figure 23.** Variation of Nusselt number with velocity number

followed by lead and steel. Gas volume fraction of sand, marble and aluminum are at the same level.

The change of suspension density with time for several types of fluidized materials is shown in Figure (25). It is clear that the highest suspension density lies with the material of the highest density.

Figures (26) to (29) show the effect of change of fluidized material type on horizontal and vertical velocities of gas and particle.

**0 4 8 1 2 1 6 2 0 Time,(s)**

**0 4 8 1 2 1 6 2 0 Time,(s)**

**0 4 8 1 2 1 6 2 0 Time,(s)**

Sand Particles Aluminum Particles Marble Particles Steel Particles Lead Particles Copper Particles

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185

Sand Particles Aluminum Particles Marble Particles Steel Particles Lead Particles Copper Particles

Sand Particles Aluminum Particles Marble Particles Steel Particles Lead Particles Copper Particles

**-2**

**Figure 26.** Effect of fluidized material type on vertical particle velocity

**0**

**2**

**vs,(m/s)**

**4**

**6**

**Figure 25.** Effect of fluidized material type on suspension density

**0**

**2000**

**4000**

r**sus,(kg/m3)** **6000**

**8000**

**0.68**

**Figure 24.** Effect of fluidized material type on gas volume fraction

**0.72**

**0.76**

e**g** **0.8**

**0.84**

**0.88**

Figures (30) and (40) illustrate the effect of change of fluidized material type on particle and gas temperatures.

Hydrodynamic and Heat Transfer Simulation of Fluidized Bed Using CFD http://dx.doi.org/10.5772/52072 185

**Figure 24.** Effect of fluidized material type on gas volume fraction

**Figure 25.** Effect of fluidized material type on suspension density

followed by lead and steel. Gas volume fraction of sand, marble and aluminum are at the

**0 0.5 1 1.5 2 2.5 3** W1/3

**0 0.1 0.2 0.3 0.4 0.5 0.6 fl**

**Sand particles dp= 500** m**m**

> **Sand particles dp= 500** m**m**

**7.6**

**7.8**

**Figure 23.** Variation of Nusselt number with velocity number

**7.9**

**8**

**8.1**

**Nu**

**8.2**

**8.3**

**Figure 22.** Variation of Nusselt number with flow number

184 Nuclear Reactor Thermal Hydraulics and Other Applications

**7.8**

**8**

**Nu**

**8.2**

**8.4**

The change of suspension density with time for several types of fluidized materials is shown in Figure (25). It is clear that the highest suspension density lies with the material of the highest

Figures (26) to (29) show the effect of change of fluidized material type on horizontal and

Figures (30) and (40) illustrate the effect of change of fluidized material type on particle and

same level.

density.

gas temperatures.

vertical velocities of gas and particle.

**Figure 26.** Effect of fluidized material type on vertical particle velocity

**Figure 27.** Effect of fluidized material type on horizontal particle velocity

**Figure 28.** Effect of fluidized material type on vertical gas velocity

**8.4. Effect of heat generation by particles**

**Figure 31.** Effect of fluidized material type on gas temperature

approximately in range of 3 °

reference [9].

The aluminum particles of 1 mm diameter are fluidized with different value of heat generated in particles (0, 500,1000,1500,2000 and 3000 Watt), the effect of change of heat generated in particles is studied. With the increase of heat generated by particles the gas phase tempera‐ ture increases as shown in Figure (32). The value of the increase in gas temperature is

**0 4 8 1 2 1 6 2 0 Time,(s)**

**0 4 8 1 2 1 6 2 0 Time,(s)**

Sand Particles Aluminum Particles Marble Particles Steel Particles Lead Particles Copper Particles

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Sand Particles Aluminum Particles Marble Particles Steel Particles Lead Particles Copper Particles

**4 0**

**3 0**

**3 5**

**4 0**

**Tg,(**

**C)**

**o**

**4 5**

**5 0**

**5 5**

**8 0**

**120**

**Ts,(**

**C)**

o

**Figure 30.** Effect of fluidized material type on particle temperature

**160**

**200**

clear that the rate of increase in particle temperature is more than the rate of increase in gas temperature, consequently the temperature difference between the two phases increase. Figure (34) illustrates the relation between average heat transfer coefficient and heat generated by particles. The results of the present work shows that the average heat transfer coeffi‐ cient dos not depend on heat generated by particles and all heat generated by particles converts to temperature difference between the two phases. This result agrees with that of

with the increase of heat generated by particles. The range of increase is about 45°

C. Figure (33) shows that the particle phase temperature increases

C. It is

**Figure 29.** Effect of fluidized material type on horizontal gas velocity

Hydrodynamic and Heat Transfer Simulation of Fluidized Bed Using CFD http://dx.doi.org/10.5772/52072 187

**Figure 30.** Effect of fluidized material type on particle temperature

**0 4 8 1 2 1 6 2 0 Time,(s)**

0 4 8 1 2 1 6 2 0

Sand Particles Aluminum Particles Marble Particles Steel Particles Lead Particles Copper Particles

0

Sand Particles Aluminum Particles Marble Particles Steel Particles Lead Particles Copper Particles

2

4

6

8

1 0

**0 4 8 1 2 1 6 2 0 Time,(s)**

**0 4 8 1 2 1 6 2 0 Time,(s)**

Sand Particles Aluminum Particles Marble Particles Steel Particles Lead Particles Copper Particles

**-0.004**

**0**

**0**

**0.0004**

**0.0008**

**ug,(m/s)**

**Figure 29.** Effect of fluidized material type on horizontal gas velocity

**0.0012**

**0.0016**

**Figure 28.** Effect of fluidized material type on vertical gas velocity

**2**

**4**

**6**

**vg,(m/s)**

**8**

**1 0**

**-0.002**

**us,(m/s)**

186 Nuclear Reactor Thermal Hydraulics and Other Applications

**Figure 27.** Effect of fluidized material type on horizontal particle velocity

**0**

**0.002**

**Figure 31.** Effect of fluidized material type on gas temperature

#### **8.4. Effect of heat generation by particles**

The aluminum particles of 1 mm diameter are fluidized with different value of heat generated in particles (0, 500,1000,1500,2000 and 3000 Watt), the effect of change of heat generated in particles is studied. With the increase of heat generated by particles the gas phase tempera‐ ture increases as shown in Figure (32). The value of the increase in gas temperature is approximately in range of 3 ° C. Figure (33) shows that the particle phase temperature increases with the increase of heat generated by particles. The range of increase is about 45° C. It is clear that the rate of increase in particle temperature is more than the rate of increase in gas temperature, consequently the temperature difference between the two phases increase. Figure (34) illustrates the relation between average heat transfer coefficient and heat generated by particles. The results of the present work shows that the average heat transfer coeffi‐ cient dos not depend on heat generated by particles and all heat generated by particles converts to temperature difference between the two phases. This result agrees with that of reference [9].

#### **8.5. Terminal velocity effect**

Figure (35) shows the relation between terminal velocity and average heat transfer coefficient. It is clear from the figure that with the increase in terminal velocity the average heat transfer coefficient decreases.

#### **8.6. Minimum fluidized velocity effect**

Minimum fluidized velocity is the most important parameter in study of fluidization. This velocity distinguishes the fluidized bed from a packed bed and is an indicator that fluidization is occurred. Figure (36) shows the variation of average heat transfer coefficient with minimum fluidized velocity. The average heat transfer coefficient decreases with the increase of mini‐ mum fluidized velocity.

**0 1000 2000 3000 Heat Generated by Particles,(W/m3)**

**0 2 4 6 8 Particle Terminal Velocity,(m/s)**

**0 0.2 0.4 0.6 Minimum Fluidized Velocity,(m/s)**

**Refernce [6]** Present W ork

Hydrodynamic and Heat Transfer Simulation of Fluidized Bed Using CFD

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189

**100**

**Figure 34.** Effect of particle heat generation on average heat transfer coefficient

**200**

**200**

**400**

**600**

**hav,(w/m2K)**

**Figure 36.** Effect of Minimum Velocity on Average Heat Transfer Coefficient

**800**

**1000**

**400**

**600**

**hav,(w/m2K)**

**Figure 35.** Effect of terminal velocity on average heat transfer coefficient

**800**

**1000**

**200**

**300**

**hav,(W/m2K)**

**400**

**Figure 32.** Effect of particle heat generation on gas temperature

**Figure 33.** Effect of particle heat generation in particle temperature

Hydrodynamic and Heat Transfer Simulation of Fluidized Bed Using CFD http://dx.doi.org/10.5772/52072 189

**Figure 34.** Effect of particle heat generation on average heat transfer coefficient

**8.5. Terminal velocity effect**

**8.6. Minimum fluidized velocity effect**

188 Nuclear Reactor Thermal Hydraulics and Other Applications

coefficient decreases.

mum fluidized velocity.

Figure (35) shows the relation between terminal velocity and average heat transfer coefficient. It is clear from the figure that with the increase in terminal velocity the average heat transfer

Minimum fluidized velocity is the most important parameter in study of fluidization. This velocity distinguishes the fluidized bed from a packed bed and is an indicator that fluidization is occurred. Figure (36) shows the variation of average heat transfer coefficient with minimum fluidized velocity. The average heat transfer coefficient decreases with the increase of mini‐

> **0 4 8 1 2 1 6 2 0 Time,(s)**

**0 4 8 1 2 1 6 2 0 Time,(s)**

**q= 0 w/m q= 500 w/m q= 1000 w/m q= 1500 w/m q= 2000 w/m q= 3000 w/m**

q = 0 w/m q = 500 w/m q= 1000 w/m q= 1500 w/m q= 2000 w/m q= 3000 w/m

**3 0**

**4 0**

**Figure 33.** Effect of particle heat generation in particle temperature

**8 0**

**120**

**Ts,(**°**C)**

**160**

**200**

**Figure 32.** Effect of particle heat generation on gas temperature

**3 5**

**4 0**

**Tg,(**°**C)**

**4 5**

**5 0**

**5 5**

**Figure 35.** Effect of terminal velocity on average heat transfer coefficient

**Figure 36.** Effect of Minimum Velocity on Average Heat Transfer Coefficient

## **Nomenclature**


Re Reynolds number, Re= <sup>ε</sup>*g*ρ*gdp* <sup>|</sup>*Ur*

S Stability function

TFM Two fluid model

∈<sup>g</sup> Gas phase volume fraction

∈<sup>s</sup> Particle phase volume fraction

δ Small positive value = 5X10-3

Osama Sayed Abd El Kawi Ali1,2

1 Egyptian Nuclear Research Center, Egypt

2 Faculty of Engineering – Al Baha University, Saudi Arabia

**Author details**

∈g,mf Gas phase volume fraction at minimum fluidization

∈in Gas volume fraction at the input velocity

Greek Letters

→ | μ*g*

T Time s Tg Gas phase temperature C◦ Ts Particle phase temperature C◦

Hydrodynamic and Heat Transfer Simulation of Fluidized Bed Using CFD

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191

vg gas phase velocity in y direction m/s Vg,mf Gas minimum fluidized velocity m/s Vg,in Input gas velocity to the bed in y direction m/s vs Particle phase velocity in y direction m/s

ΔP<sup>B</sup> Total Pressure drop across the bed Pa Δt Time step s Δx Length of cell in the computational grid m Δy height of cell in the computational grid m

ρ<sup>g</sup> Density of gas phase Kg/m3 ρ<sup>s</sup> Density of particle phase Kg/m3 ρsus Suspension density Kg/m3 μ<sup>g</sup> Viscosity of gas Pa.s


#### Greek Letters

**Nomenclature**

Ar Archimedes number, *Ar* <sup>=</sup> *gdp*

190 Nuclear Reactor Thermal Hydraulics and Other Applications

Cd Drag coefficient, *Cd* =(0.63 <sup>+</sup> 4.8

fl Flow number , *fl* <sup>=</sup> *Vg*,*in*

Pr Prandtl number , Pr= μgCpg/kg

*q*

*ut*

hv Volumetric heat transfer coefficient , *hv* <sup>=</sup> 6(1−ε*g*)*<sup>h</sup> gp*

Nu Nusselt number based on particle diameter, (Nu= hgpdp/kg)

3 ρ*g*(ρ*<sup>s</sup>* −ρ*g*) μ*g* 2

Re0.5 ) 2

Cp,g Specific heat of fluidizing gas at constant pressure J/kg.K Cp,s Specific heat of solid particles J/kg.K dg Residue of the gas continuity equation Kg/m3 s dp Mean particle diameter M G Acceleration due to gravity m/s2 Fgx Total gas phase force in x direction per unit volume N /m3 Fgy Total gas phase force in y direction per unit volume N /m3 Fsx Total particle phase force in x direction per unit volume N /m3 Fsy Total particle phase force in y direction per unit volume N /m3

hgp Heat transfer coefficient between gas phase and particle phase W/m2.K

H Total height of the bed and freeboard *M* Hmf Minimum fluidized head of the bed *M* H1 Expansion head of bed at the input velocity *M* kg Thermal conductivity of gas phase W/m.K ks Thermal conductivity of particle phase W/m.K L Width of the bed *M*

Pg Gas pressure Pa

• Rate of heat generated within particle phase W/m3

ug Gas phase velocity in x direction m/s Ugin Input gas velocity to the bed in x direction m/s us Particle phase velocity in x direction m/s ut Particle terminal velocity m/s Ur Relative velocity between two phases m/s

*dp*

W/m3.K


## **Author details**

Osama Sayed Abd El Kawi Ali1,2

1 Egyptian Nuclear Research Center, Egypt

2 Faculty of Engineering – Al Baha University, Saudi Arabia

## **References**


**References**

(1995).

(2001).

London, (1998).

ras,Indian, (1986).

Mansoura, Egypt, (2001).

1

www.regg.ufrgs.br/fbnr\_ing.htm,.

192 Nuclear Reactor Thermal Hydraulics and Other Applications

[1] Farhang Sefidvash 'Fluidized Bed Nuclear Reactor" from internet. (2000).

[2] John, D. and Jr. Anderson, "Computational Fluid Dynamics",McGraw-Hill,Inc,

[3] Gibilaro, L. G. Fluidization Dynamics", BUTTER WORTH-HEINEMANN, Oxford,

[4] Kodikal, J. Nilesh Kodikal and H. Bhavnani Sushil," A Computer Simulation of Hy‐ drodynamics and Heat Transfer at Immersed Surfaces in a Fluidized Bed", Paper Proceedings of the 15th International Conference on Fluidized Bed Combustion, May

[5] Hans Enwald and Eric PeiranoGemini: A Cartesian Multiblock Finite Differ‐ ence",Code for Simulation of Gas-Particle Flows", Department of Thermo and Fluid Dynamics, Chalmers University of Technology, 412 96 Goteborg, Sweden,(1997). [6] Martin RhodesIntroduction to Particle Technology", Published by John Wiley &sons,

[7] Asit KumarDesign, construction,and Operation of 30.5 cm Square Fluidized Bed for Heat Transfer Study", master of technology,Indian institute of technology, Mad‐

[8] Paola LettieriLuca CammarataGiorgi, D. M. Micale and John Yates, " CFD Simula‐ tions of Gas Fluidized Beds Using Alternative Eulerian-Eulerian Modelling Ap‐ proaches ",International Journal Of Chemical Reactor Engineering, Article 5, (2003). ,

[9] Osama Sayed Abd ElkawiStudy of Heat Transfer in Fluidized Bed Heat Exchangers", M.Sc. thesis, Mechanical power department, Faculty of engineering, University of

16-19, Savannah, Georgia, Copyright by ASME,(1999). (FBC99-0077), 99-0077.

## *Edited by Donna Post Guillen*

This book includes contributions from researchers around the world on numerical developments and applications to predict fluid flow and heat transfer, with an emphasis on thermal hydraulics computational fluid dynamics. Our ability to simulate larger problems with greater fidelity has vastly expanded over the past decade. The collection of material presented in this book augments the ever-increasing body of knowledge concerning the important topic of thermal hydraulics. Featured topics include coolant channel analysis, thermal hydraulic transport and mixing, as well as hydrodynamics and heat transfer processes. The contents of this book will interest researchers, scientists, engineers and graduate students.

Nuclear Reactor Thermal Hydraulics and Other Applications

Nuclear Reactor Thermal

Hydraulics and Other

Applications

*Edited by Donna Post Guillen*