**4.3 System discretization**

The most important numerical methods used to approximate the partial differential equations by a system of algebraic equations in terms of the variables at some discrete locations in space and time (called "discretization method") are the Finite Volume (FV), the Finite Difference (FD) and the Finite Element (FE) methods. In this book, the finite volume method and the commercial software CFX® 12.0 were chosen; the solution domain is discretized in a computational mesh that can be structured or unstructured.

Finite volume (FV) method

The FV discretization method is obtained by integrating the transport equation around a finite volume. The general form of transport equations is given by:

$$\underbrace{\frac{\partial\left(\rho\phi\right)}{\partial t}}\_{\widetilde{V}} + \underbrace{\nabla\cdot\left(\rho\vec{v}\phi\right)}\_{\widetilde{W}} = \underbrace{\nabla\cdot\left(\Gamma\_{\phi}\nabla\phi\right)}\_{\widetilde{W}} + \underbrace{S\_{\phi}}\_{\widetilde{W}}\tag{13}$$


The transport equations are integrated in each computational cell using the divergence theorem over a given time interval ∆*t:* 

$$\int\_{t}^{t+\Lambda t} \left\{ \int\_{V} \frac{\partial \left(\rho \phi\right)}{\partial t} dV + \oint \rho \phi \vec{v} \cdot d\vec{A} = \oint \Gamma\_{\phi} \nabla \phi \cdot d\vec{A} + \int\_{v} S\_{\phi} dV \right\} dt \tag{14}$$

Linearization and interpolation techniques can be clarified considering the finite volume *P*  shown in Figure 3.

In agreement with Figure 3 notation, diffusive term can be represented as

$$\oint \Gamma\_{\phi} \nabla \phi \cdot d\vec{A} = \frac{\Gamma\_{\phi} A\_w}{h\_w} \left(\phi\_{\mathcal{P}} - \phi\_{\mathcal{W}}\right) = D\_w \left(\phi\_{\mathcal{P}} - \phi\_{\mathcal{W}}\right) \tag{15}$$

46 Advanced Fluid Dynamics

Models Description and advantages

solution strategy.

processes.

The most important numerical methods used to approximate the partial differential equations by a system of algebraic equations in terms of the variables at some discrete locations in space and time (called "discretization method") are the Finite Volume (FV), the Finite Difference (FD) and the Finite Element (FE) methods. In this book, the finite volume method and the commercial software CFX® 12.0 were chosen; the solution domain is

The FV discretization method is obtained by integrating the transport equation around a

The transport equations are integrated in each computational cell using the divergence

Linearization and interpolation techniques can be clarified considering the finite volume *P* 

*t V v*

*w A dA D*

 

*h* 

 

*II III IV*

*dV v dA dA S dV dt*

 *<sup>w</sup> P W wP W*

 

(15)

*v S*

  

 

(14)

(13)

discretized in a computational mesh that can be structured or unstructured.

finite volume. The general form of transport equations is given by:

*I*

∆*t:* 

In agreement with Figure 3 notation, diffusive term can be represented as

*t*

The difficulties associated with the use of the standard LES models, has lead to the development of hybrid models (like that DES) that attempt to combine the best aspects of RANS and LES methodologies in a single

The most exact approach to turbulence simulation without requiring any additional modeling beyond accepting the Navier–Stokes equations to describe the turbulent flow

Family group

Detached Eddy Simulation (DES)

Direct Numerical Simulation (DNS)

Table 4. Summary of turbulence models.

**4.3 System discretization** 

Finite volume (FV) method

*i.* Transient term *ii.* Convective term *iii.* Diffusive term *iv.* Source term

shown in Figure 3.

theorem over a given time interval

*t t*

*t*

 

Fig. 2. Gas flow over a flat solid surface (left to right) experimental picture, refined mesh near the wall and contrast between experiment and discretization.

Fig. 3. Finite volume representation and notation.

#### **4.4 Source term linearization**

A generic source term may be written as

$$S\_{\phi P} V\_P = S\_C^{\phi} + S\_P^{\phi} \phi\_P \tag{16}$$

Where *S<sup>P</sup>* is the value of source term in the center of the cell *P* and *VP* is the volume of computational cell centered on node *P.* The method to represent *S<sup>P</sup>* was suggested by Patankar, 1980

Fluid Dynamics of Gas – Solid Fluidized Beds 49

Where the function *F* incorporates any spatial discretization. The first-order accurate

 *n n* <sup>1</sup> *F*

*F*

(26)

(27)

*t*

1 1 3 4 2 *n nn*

 

*t*

In order to give a better introduction with regards to the simulation of fluidized beds, in this chapter there are presented three case studies that were carried out by using a CFD software

The case studies were carried out using simulations in dynamic state. These simulations were set up taking into account the average value of the Courant number, which is recommended to be near 1. Besides this, it was used a constant step time, in this way was

Lab scale riser reactor (Samuelsberg and B. H. Hjertager 1996; V Mathiesen 2000). Riser height, 1 m; riser diameter, 0.032 m. Experimental data and LES - Smagorinsky simulations

In addition, tests were made with a 500.000 control volume mesh with same block distribution (the description of volume distribution in the meshes, are presented in Table 7). Obtaining similar results with the 100.000 control volume mesh. Both meshes are shown in

In Gas velocity = 0.36; 1.42 m/s

Out *Opening* = atmospheric pressure

Gas = no slip

Particle mass flow equal to the output

Particles = *free slip* and *No slip*

possible to have numerical stability during the execution of each of the simulations.

were compared for three velocities with initial particle bed, 5cm.

The boundary conditions for both cases are shown in Table 5 and Table 6.

Initial height Bed height = 0,05 m Particles 60 μm; 1600 kg/m3

**5.1.1 Mesh parameters and boundary conditions** 

Matrix determinant > 0.5 and minimum angle > 50°

Control volumes number: 100.000

Wall

Table 5. Boundary conditions for the Case 1.

 

temporal discretization is given by

**5. Case studies** 

**5.1 Cases 1 and 2** 

∆x = 2 mm

Figure 4.

package.

And the second-order discretization is given by

$$\boldsymbol{S}\_{\phi P} = \boldsymbol{S}\_{\phi P}^\* + \left(\frac{d\boldsymbol{S}\_{\phi P}}{d\phi}\right)^\* \left(\phi\_P - \phi\_P^\*\right) \tag{17}$$

This type of linearization is recommended since the source term decreases with increasing *Φ.* The source term coefficients are represented by:

$$S\_{\dot{C}}^{\phi} = \left[ S\_{\phi P}^{\*} - \left( \frac{d S\_{\phi P}}{d \phi} \right)^{\*} \phi\_p^{\*} \right] V\_P \tag{18}$$

$$S\_P^\phi = \left(\frac{dS\_{\phi P}}{d\phi}\right)^\* V\_P \tag{19}$$

#### **4.4.1 Spatial discretization**

The most widely used in CFD is first and second order Upwind methods. In the first order one, quantities at cell faces are determined by assuming that the cell-center values of any field variable represent a cell-average value and hold throughout the entire cell. The face value *(Φw)* are equal to the cell-center value of *Φ* in the upstream cell.

$$
\oint \rho \phi \vec{v} \cdot d\vec{A} = \rho v\_w A\_w \phi\_W = C\_w \phi\_W \tag{20}
$$

Where, *Cw* is the west face convective coefficient. *Aw* can be represented by:

$$A\_w = MAX(\mathbb{C}\_{w'}0) + D\_w \tag{21}$$

In the second order one, quantities at cell faces are computed using a multidimensional linear reconstruction approach (Jespersen and Barth 1989). In this approach, higher-order accuracy is achieved at cell faces through a Taylor series expansion of the cell-centered solution about the cell centroid. Thus, the face value *Φ<sup>w</sup>* is computed using the following expression:

$$
\phi\_w = \frac{3}{2}\phi\_W - \frac{1}{2}\phi\_{\rm MW} = \phi\_W + \frac{1}{2}(\phi\_W - \phi\_{\rm MW}) \tag{22}
$$

The east face coefficient and matrix coefficient are shown below

$$
\phi\_e = \frac{3}{2}\phi\_P - \frac{1}{2}\phi\_W \tag{23}
$$

$$A\_w = MAX\left(\mathbb{C}\_{w'}0\right) + \frac{1}{2}MAX\left(\mathbb{C}\_{\varepsilon'}0\right) + D\_w\tag{24}$$

#### **4.4.2 Temporal discretization**

Temporal discretization involves the integration of every term in the differential equations over a time step ∆*t.* A generic expression for the time evolution of a variable *Φ* is given by

$$\frac{\partial \phi}{\partial t} = F\left(\phi\right) \tag{25}$$

Where the function *F* incorporates any spatial discretization. The first-order accurate temporal discretization is given by

$$\frac{\phi^{n+1} - \phi^n}{\Delta t} = F\left(\phi\right) \tag{26}$$

And the second-order discretization is given by

$$\frac{3\phi^{n+1} - 4\phi^n + \phi^{n-1}}{2\Delta t} = F\left(\phi\right) \tag{27}$$
