**4.1.1 Continuity equations**

The gas and solid continuity equations are represented by:

$$\frac{\partial}{\partial t} \left( \alpha\_{\mathcal{g}} \rho\_{\mathcal{g}} \right) + \nabla \cdot \left( \alpha\_{\mathcal{g}} \rho\_{\mathcal{g}} \vec{v}\_{\mathcal{g}} \right) = 0 \tag{1}$$

Fluid Dynamics of Gas – Solid Fluidized Beds 45

*<sup>r</sup> g g g g gg g gg g s g g g r*

*s s s s ss s ss s H vH T TT <sup>g</sup> <sup>s</sup> <sup>t</sup>*

 

quantities and their gradients in some physically plausible manner.

The turbulence models are summarized in Table 4

Zero equation models

One equation models Two equation models

Smagorinsky-Lilly

RNG – LES model WALLE model

Dynamic subgrid-scale

κ – ε κ – ω

Reynolds Stress Model (RSM)

model

 

Turbulence is that state of fluid motion which is characterized by random and chaotic threedimensional vorticity. When turbulence is present, it usually dominates all other flow phenomena and results in increased energy dissipation, mixing, heat transfer, and drag. The physical turbulence models provide the solution the closure problem in solving Navier – Stokes equations. While there are ten unknown variables (mean pressure, three velocity components, and six Reynolds stress components), there are only four equations (mass balance equation and three velocity component momentum balance equations). This disparity in number between unknowns and equations make a direct solution of any turbulent flow problem impossible in this formulation. The fundamental problem of turbulence modeling is to relate the six Reynolds stress components to the mean flow

*<sup>C</sup> <sup>H</sup> v H T TT <sup>H</sup> t t*

(11)

 

Models Description and advantages

model Provides excellent results for all flow systems.

scale motions.

flows.

 

*r*

 

(12)

The most widely used models. Its main advantages are short computation time, stable calculations and reasonable results for many

Provides good predictions for all types of flows, including swirl, and separation. Longer calculation times than the RANS models.

LES solves the Navier-Stokes equations for large scale motions of the flow models only the small

 

**4.1.3 Energy equation** 

*H* = Specific enthalpy *T* = Temperature

λ = Thermal conductivity

**4.2 Turbulence models** 

Family group

Reynolds – Averaged Navier – Stokes (RANS)

Large Eddy Simulation (LES)

Where

The gas and solid energy equations can be written as:

 

γ = Interface heat transfer coefficient: / *Nu dp*

$$\frac{\partial}{\partial t}(a\_s \rho\_s) + \nabla \cdot \left(a\_s \rho\_s \vec{v}\_s\right) = 0 \tag{2}$$

Where α, �� and *v* are volume fraction, density and the vector velocity, respectively. No mass transfer is allowed between phases.

#### **4.1.2 Momentum equations**

The gas phase momentum equation may be expressed as:

$$\frac{\partial}{\partial t} \left( a\_g \rho\_g \vec{v}\_g \right) + \nabla \cdot \left( a\_g \rho\_g \vec{v}\_g \vec{v}\_g \right) = -a\_g \nabla p + \nabla \cdot \left[ \pi\_g \right] + a\_g \rho\_g \vec{g} + \mathcal{J} \left( \vec{v}\_s - \vec{v}\_s \right) \tag{3}$$

 *p* and *g* are fluid pressure and gravity acceleration. *β* is the drag coefficient between the phases *g* and *s*. The stress tensor is given by:

$$
\sigma\_{\mathcal{g}} = \alpha\_{\mathcal{g}} \mu\_{\mathcal{g}} \left[ \nabla \vec{v}\_{\mathcal{g}} + \left( \nabla \vec{v}\_{\mathcal{g}} \right)^{T} \right] - \frac{2}{3} \alpha\_{\mathcal{g}} \mu\_{\mathcal{g}} \nabla \vec{v}\_{\mathcal{g}} \tag{4}
$$

The solid phase momentum equation may be written as:

$$\frac{\partial}{\partial t} \left( a\_s \rho\_s \vec{v}\_s \right) + \nabla \cdot \left( a\_s \rho\_s \vec{v}\_s \vec{v}\_s \right) = -a\_s G \nabla a\_s + \nabla \cdot \left[ \tau\_s \right] + a\_s \rho\_s \vec{g} + \mathcal{J} \left( \vec{v}\_g - \vec{v}\_s \right) \tag{5}$$

$$
\sigma\_s = \alpha\_s \mu\_s \left[ \nabla \vec{v}\_s + \left( \nabla \vec{v}\_s \right)^T \right] - \frac{2}{3} \alpha\_s \mu\_s \nabla \vec{v}\_s \tag{6}
$$

*G* is the modulus of elasticity given by:

$$G = \exp\left[C\_G \left(\alpha\_s - \alpha\_{s,\text{max}}\right)\right] \tag{7}$$

Where αs,max is the maximum solid volume fraction and β is the interface momentum transfer proposed by Gidaspow, (1994):

$$\begin{cases} \beta = 150 \frac{\alpha\_s \left(1 - \alpha\_{\mathcal{S}}\right) \mu\_{\mathcal{S}}}{\alpha\_{\mathcal{S}} d\_p^2} + 1.75 \frac{\alpha\_s \rho\_{\mathcal{S}} \left| \vec{v}\_s - \vec{v}\_{\mathcal{S}} \right|}{d\_p} & \left| \alpha\_{\mathcal{S}} \le 0.8 \\\\ \beta = \frac{3}{4} C\_D \frac{\alpha\_s \alpha\_{\mathcal{S}} \rho\_{\mathcal{S}} \left| \vec{v}\_s - \vec{v}\_{\mathcal{S}} \right|}{d\_p} \alpha\_{\mathcal{S}}^{-2.65} & \left| \alpha\_{\mathcal{S}} > 0.8 \end{cases} \tag{8}$$

Where *dp* and *CD* are the particle diameter and the drag coefficient, based in the relative Reynolds number (*Res*)

$$\text{C}\_{D} = \left| \frac{24 \left( 1 + 0.15 \,\text{Re}\_{s}^{0.687} \right)}{\text{Re}\_{s}} \right| \, |\, \text{Re}\_{s} \le 1000 \,\tag{9}$$

$$0.44 \,\qquad \left| \, \text{Re}\_{s} > 1000 \,\right.$$

$$\text{Re}\_s = \frac{\rho\_{\mathcal{g}} \left| \vec{v}\_s - \vec{v}\_{\mathcal{g}} \right|}{\mu\_{\mathcal{g}}} \tag{10}$$

### **4.1.3 Energy equation**

The gas and solid energy equations can be written as:

$$\frac{\partial}{\partial t} \left( a\_g \rho\_g H\_g \right) + \nabla \cdot \left( a\_g \rho\_g \vec{v}\_g H\_g \right) = \nabla \cdot \left( a\_g \lambda\_g \nabla T\_g \right) + \gamma \left( T\_s - T\_g \right) + a\_g \rho\_g \sum\_r \Delta H\_r \frac{\partial \mathcal{C}\_r}{\partial t} \tag{11}$$

$$\frac{\partial}{\partial t}(a\_s \rho\_s H\_s) + \nabla \cdot \left(a\_s \rho\_s \vec{v}\_s H\_s\right) = \nabla \cdot \left(a\_s \lambda\_s \nabla T\_s\right) + \gamma \left(T\_g - T\_s\right) \tag{12}$$

Where

44 Advanced Fluid Dynamics

0 *ss sss v*

*g gg g ggg g v v <sup>v</sup> p g g gg v v <sup>s</sup> <sup>g</sup> <sup>t</sup>*

 *p* and *g* are fluid pressure and gravity acceleration. *β* is the drag coefficient between the

 

*s ss s sss s s s s s v vv G <sup>g</sup> v v <sup>g</sup> <sup>s</sup> <sup>t</sup>*

 

*G C* exp *Gs s* ,max

is the maximum solid volume fraction and

 2

 

 

1

*C*

*C*

<sup>2</sup>

<sup>2</sup>

*vv v*

150 1.75 | 0.8

 

*s g g sg s g*

*g p p*

*d d v v*

*sgg s g*

Where *dp* and *CD* are the particle diameter and the drag coefficient, based in the relative

*s*

Re *<sup>g</sup> <sup>s</sup> <sup>g</sup>*

 

 *v v* 

*g*

*p*

0.687 24 1 0.15Re

*<sup>s</sup> <sup>D</sup> <sup>s</sup>*

*s*

*d*

*T s ss s s ss s*

*vv v*

*T g gg g g gg g*

 

 

are volume fraction, density and the vector velocity, respectively. No

 

3

 

3

2.65

*s*

*v v*

<sup>3</sup> | 0.8 <sup>4</sup>


*D gg*

 

(5)

 

(3)

   

(4)

 

(7)

is the interface momentum

(9)

(10)

(6)

β

*g*

(8)

 

(2)

*t* 

The gas phase momentum equation may be expressed as:

 

The solid phase momentum equation may be written as:

 

 

 

Where α,

Where

αs,max

Reynolds number (*Res*)

��and *v*

**4.1.2 Momentum equations** 

mass transfer is allowed between phases.

*G* is the modulus of elasticity given by:

transfer proposed by Gidaspow, (1994):

 

phases *g* and *s*. The stress tensor is given by:

*H* = Specific enthalpy

*T* = Temperature

γ = Interface heat transfer coefficient: / *Nu dp* 

λ = Thermal conductivity

#### **4.2 Turbulence models**

Turbulence is that state of fluid motion which is characterized by random and chaotic threedimensional vorticity. When turbulence is present, it usually dominates all other flow phenomena and results in increased energy dissipation, mixing, heat transfer, and drag. The physical turbulence models provide the solution the closure problem in solving Navier – Stokes equations. While there are ten unknown variables (mean pressure, three velocity components, and six Reynolds stress components), there are only four equations (mass balance equation and three velocity component momentum balance equations). This disparity in number between unknowns and equations make a direct solution of any turbulent flow problem impossible in this formulation. The fundamental problem of turbulence modeling is to relate the six Reynolds stress components to the mean flow quantities and their gradients in some physically plausible manner.

The turbulence models are summarized in Table 4


Fluid Dynamics of Gas – Solid Fluidized Beds 47

Fig. 2. Gas flow over a flat solid surface (left to right) experimental picture, refined mesh

*P P PP <sup>C</sup> SV S S* 

*<sup>P</sup>* is the value of source term in the center of the cell *P* and *VP* is the volume of

(16)

*<sup>P</sup>* was suggested by

computational cell centered on node *P.* The method to represent *S*

near the wall and contrast between experiment and discretization.

Fig. 3. Finite volume representation and notation.

**4.4 Source term linearization** 

Where *S*

Patankar, 1980

A generic source term may be written as


Table 4. Summary of turbulence models.
