**7. Simple algorithm**

As discussed in the preceding section, the governing equation for the flow may be solved in terms of derived variables, or in term of primitive variables consisting of the velocity components and the pressure.

**Figure 3.** Staggered location for the velocity components in a two dimensional flow (Patankar, 1980)

However, in the advent of Simple (Semi Implicit Method for Pressure Linked Equations) algorithm, along with its revised version Simpler and the enhancement such as Simplec, the solution of the equations using primitive variable approach has become very attractive. In fact, Simple and Simple like algorithms are extremely popular for the solution of problems involving convective flow and transport. The basic approach involves the control volume formulation, with the staggered grid, as outlined in the proceeding section. This avoids the appearance of physically unrealistic wavy velocity fields in the solution to equations. The pressure at a chosen point is taken at arbitrary value and the pressures at other points are calculated as differences from the chosen pressure value.

Following (Patankar, 1980), if a guessed pressure field *p\** is taken, the corresponding velocity field can be calculated from the discretised equations for the control volume shown in Fig. 4 These equations are of the form:

$$a\_e u\_e = \Sigma a\_{nb} \ u\_{nb}^\* \ + \ \ \ \{p\_p^\* - p\_E^\*\} A\_e,\tag{43}$$

where the asterisk on the velocity indicates the erroneous velocity field based on guessed pressure field. Here, *anb* is a coefficient that accounts for the combined convection-diffusion at the faces of the control volume, with *nb* referring to the neighbors e to the control volume, b includes the source terms except the pressure gradient, and *Ae* is the area on which pressure acts, being *Δy\*Δz* for 3D. The numbers of neighbor terms are 6 for three dimensional ones. Similar equations can be written for *vn\** and *wt \** , where t lies on the zdirection grid line between grid points P and T. if p is the correct pressure and *p* is the correct pressure and *p'* the pressure correction, we may write:

$$\mathbf{p} = \mathbf{p}^\* + \mathbf{p}', \; \mathbf{u} = \mathbf{u}^\* + \mathbf{u}', \; \mathbf{v} = \mathbf{v}^\* + \mathbf{v}', \; \mathbf{w} = \mathbf{w}^\* + \mathbf{w}' \tag{44}$$

where the prime indicate corrections needed to reach the correct values that satisfy the continuity equation. Omitting the correction terms due to the neighbors, an iterative solution may be developed to solve for the pressure and the velocity field. Then, the velocity correction formula becomes:

$$u\_{\varepsilon} = u\_{\varepsilon}^{\*} + \frac{A\_{\varepsilon}}{a\_{\varepsilon}} (p\_p' - p\_E').$$

$$v\_n = v\_{\varepsilon}^{\*} + \frac{A\_n}{a\_{\varepsilon}} (p\_p' - p\_N'), \tag{45}$$

And similarly for *wt*. From the time dependent continuity equation, the pressure correction equation in then developed as:

$$\mathbf{a}\_{\mathsf{P}}\mathbf{p}\_{\mathsf{P}}^{\prime} = \mathbf{a}\varepsilon\mathbf{p}\_{\mathsf{e}}^{\prime} + \mathbf{a}\_{\mathsf{W}}\mathbf{p}\_{\mathsf{W}}^{\prime} + \mathbf{a}\mathbf{v}\mathbf{p}\_{\mathsf{N}}^{\prime} + \mathbf{a}\_{\mathsf{S}}\mathbf{p}\_{\mathsf{s}}^{\prime} + \mathbf{a}\mathbf{r}\mathbf{p}\_{\mathsf{i}}^{\prime} + \mathbf{a}\mathbf{n}\mathbf{p}^{\prime}\mathbf{v} + \mathbf{b},\tag{46}$$

where *b* is a mass source which must be eliminated through pressure correction so that continuity is satisfied. Here, T and B are neighboring grid points on the z direction grid line.

**Figure 4.** Control volume for driving the pressure correction equation (Patankar, 1980)

The simple algorithm has the following main steps:

1. Guess the pressure field *p\*.* 

542 Numerical Simulation – From Theory to Industry

ap = aw+ aE + as + aN + aB+ aT + aº

calculated as differences from the chosen pressure value. Following (Patankar, 1980), if a guessed pressure field *p\**

���� = ∑ ��� ��� 

These equations are of the form:

For more information interested reader can see (Mohammadi, 2010).

As discussed in the preceding section, the governing equation for the flow may be solved in terms of derived variables, or in term of primitive variables consisting of the velocity

**Figure 3.** Staggered location for the velocity components in a two dimensional flow (Patankar, 1980)

However, in the advent of Simple (Semi Implicit Method for Pressure Linked Equations) algorithm, along with its revised version Simpler and the enhancement such as Simplec, the solution of the equations using primitive variable approach has become very attractive. In fact, Simple and Simple like algorithms are extremely popular for the solution of problems involving convective flow and transport. The basic approach involves the control volume formulation, with the staggered grid, as outlined in the proceeding section. This avoids the appearance of physically unrealistic wavy velocity fields in the solution to equations. The pressure at a chosen point is taken at arbitrary value and the pressures at other points are

field can be calculated from the discretised equations for the control volume shown in Fig. 4

where the asterisk on the velocity indicates the erroneous velocity field based on guessed pressure field. Here, *anb* is a coefficient that accounts for the combined convection-diffusion at the faces of the control volume, with *nb* referring to the neighbors e to the control volume, b includes the source terms except the pressure gradient, and *Ae* is the area on which pressure acts, being *Δy\*Δz* for 3D. The numbers of neighbor terms are 6 for three

<sup>∗</sup> � � � ���

<sup>∗</sup> � ��

and ��� = �� � ����

p +�F- Sp (42)

is taken, the corresponding velocity

<sup>∗</sup> � ��, (43)

where:

with ��

° <sup>=</sup> �� �� º �

**7. Simple algorithm** 

components and the pressure.


The revised version Simpler is quite similar to preceding algorithm and was developed mainly to improve the rate of convergence. In this case, the mail steps are:

Numerical Simulation of Combustion in Porous Media 545

Prior to CFD simulation, computational mesh of cylinder was generated with Kiva-Prep (preprocessor for mesh generation for KIVA-3V main code). The geometry of a mesh is composed of one block. Fig. 5 shows the grid configuration of porous tube, About 300000

The test section was a vertical quartz glass-tube with 1.3 m in length and 0.076 m in diameter, that was isolated from the environment. The test section was filled with a packed bed of 0.0056 m solid alumina spheres. For simulation a cylinder with 0.076 m in diameter and 0.60 m in length that filled with PM, was considered (Fig. 5). The boundary condition applied to the momentum and energy equation with the assumption of zero gradients for temperature of both phase of PM and for species transport through the downstream boundary. At the upstream boundary, the gas temperature is 300K, composition is premixed methane-air with equivalence ratio 0.15, and velocity is 0.43 m/s of the premixed reactants and zero gradient for solid phase, were specified. For initial temperature for both phase of PM experimental measured data was used. Fuel is methane, porosity of PM is 0.4. The laminar flow considered for simulation. For validation of numerical simulation, modified KIVA code was used for simulation of unsteady combustion is a cylindrical tube with the experiments of Zhdanok et al. Fig. 6 plots a comparison of computation results to the experimental results of Zhdanok at the time of 147 s, which shows that the computed speed of combustion wave agrees well with the same condition of the experimental results. It is seen that methane is completely

**9. Examples** 

**9.1. Mesh preparation** 

grids were generated for computational studies.

**Figure 5.** Computational mesh for the CFD calculation

consumed in flame front that has maximum temperature.

**9.2. Initial and boundary conditions** 


The pressure at any arbitrary point in the computational domain is specified and pressure differentials from this value are computed. The boundary condition may be a given pressure, which makes *p' = 0*, or a given normal velocity which makes the velocity a known quantity at the boundary and not a quality to be corrected so that *p'* at the boundary is not needed. For further details, (Patankar, 1980) may be consulted.
