**5. Case studies**

48 Advanced Fluid Dynamics

*d* 

This type of linearization is recommended since the source term decreases with increasing

 

\* \* *P C P P P dS S S V d*

 

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

> 

*v dA v A C w wW wW*

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

31 1

 *w W WW W W WW* (22)

*A MAX C MAX C D w w e w* (24)

 

3 1 2 2

*e PW* 

Temporal discretization involves the integration of every term in the differential equations

*<sup>F</sup> <sup>t</sup>* 

 

*t.* A generic expression for the time evolution of a variable *Φ* is given by

<sup>1</sup> ,0 ,0 <sup>2</sup>

*S S*

value *(Φw)* are equal to the cell-center value of *Φ* in the upstream cell.

 

The east face coefficient and matrix coefficient are shown below

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

cell centroid. Thus, the face value *Φ<sup>w</sup>* is computed using the following expression:

 

22 2

 

*Φ.* The source term coefficients are represented by:

**4.4.1 Spatial discretization** 

**4.4.2 Temporal discretization** 

∆

over a time step

\* \* \* *P P P P P dS*

> 

(20)

*A MAX C D w ww* ,0 (21)

(17)

(18)

(19)

(23)

(25)

\*

\* *P P P dS S V d*

 

> 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 package.

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