## 3 Numerical Method
In turn, the gas velocity at the interface of the control volume can be written as:  
\[B_{x}=\frac{\varepsilon_{P}\Delta x\Delta y}{\Delta t}(u_{x}^{0})_{P} -\frac{\varepsilon_{P}\Delta y}{\rho_{g}}(p_{e}-p_{w})-\frac{(S_{P _{P}})_{p}\Delta x\Delta y}{\rho_{g}} \tag{49a}\]8 \[B_{x}=\frac{\varepsilon_{P}\Delta x\Delta y}{\Delta t}(u_{x}^{0})_{P} -\frac{\varepsilon_{P}\Delta y}{\rho_{g}}(p_{e}-p_{w})-\frac{(S_{P _{P}})_{p}\Delta x\Delta y}{\rho_{g}} \tag{49a}\]9 (53b) \[+\frac{\mu_{g}}{\beta_{g}}\frac{\Delta y}{\Delta x}\left[ \varepsilon_{e}((u_{x})_{P}-(u_{x})_{P})-\varepsilon_{w}((u_{x})_{P}-(u_{x})_{ W})\right]\]0 (53c) \[+\frac{\mu_{g}}{\beta_{g}}\frac{\Delta y}{\Delta x}\left[ \varepsilon_{e}((u_{x})_{P}-(u_{x})_{P})-\varepsilon_{w}((u_{x})_{P}-(u_{x})_{ W})\right]\]1  
By inserting Equation (53) into Equation (50), we get:  
\[ \begin{array}{ll}\Big{[}\frac{\varepsilon_{e}}{a_{e}}+\frac{\varepsilon_{w }}{a_{w}}\Big{]}\Delta y&+\Big{(}\frac{\varepsilon_{w}}{a_{e}}+\frac{ \varepsilon_{s}}{a_{e}}\Big{)}\Delta x\Big{]}p_{P}\\ &=\Big{[}\frac{\varepsilon_{e}}{a_{e}}p_{E}+\frac{\varepsilon_{w}}{a_{w}}p_{ W}\Big{]}\Delta y+\Big{(}\frac{\varepsilon_{w}}{a_{P}}p_{N}+\frac{ \varepsilon_{s}}{a_{P}}p_{S}\Big{)}\Delta x\Big{]}\\ &-\Big{[}\big{(}(\varepsilon\overline{u}_{x})_{e}-(\varepsilon\overline{u}_{x })_{w}\big{)}\Delta y+\Big{(}(\varepsilon\overline{u}_{y})_{n}-(\varepsilon \overline{u}_{y})_{s}\Big{)}\Delta x\Big{]}\end{array}  \tag{54}\]  
Now if a guessed velocity \(\mathbf{u}\) is given, the pseudo velocity \(\overline{\mathbf{u}}\) can be calculated from Equation (51) and hence a guessed pressure \(p^{*}\) can be solved from Equation (54). By employing \(p^{*}\), a velocity can be solved from Equation (48). Because the pressure \(p^{*}\) is not correct in general, the resulting velocity will not satisfy the continuity equation. Such an imperfect velocity field based on the guessed pressure \(p^{*}\) is denoted by \(\mathbf{u}^{*}\). Suppose the correct pressure is \(p=p^{*}+p^{*}\), where \(p^{*}\) is called the pressure correction. Next, the corresponding velocity correction \(\mathbf{u}^{{}^{\prime}}\) can be introduced in a similar manner \(\mathbf{u}=\mathbf{u}^{*}+\mathbf{u}^{{}^{\prime}}\). At this point, the velocity at the interface of the control volume, taking \((u_{x})_{e}\) for example, can be expressed as:  
\[(u_{x})_{e}=(\overline{u}_{x})_{e}-\frac{(p_{E}-p_{P})}{a_{e}}=\left[(\overline {u}_{x})_{e}-\frac{(p^{*}_{E}-p^{*}_{P})}{a_{e}}\right]-\frac{(p^{\prime}{}_{E} -p^{\prime}{}_{P})}{a_{e}}=(u^{*}_{x})_{e}-\frac{(p^{\prime}{}_{E}-p^{\prime}{ }_{P})}{a_{e}} \tag{55}\]By inserting the interfacial velocities in form of Equation (55) into the discrete continuity Equation (50), we obtain an equation for \(p^{\prime}\):  
\[ \begin{array}{ll}\left[\left(\frac{\varepsilon_{e}}{a_{e}}+\frac{\varepsilon_{ w}}{a_{w}}\right)\Delta y&+\left(\frac{\varepsilon_{a}}{a_{w}}+\frac{\varepsilon_{e}}{a_{ s}}\right)\Delta x\right]p^{\prime}p\\ &=\left[\left(\frac{\varepsilon_{e}}{a_{e}}p^{\prime}+\frac{\varepsilon_{w}}{a _{w}}p^{\prime}w\right)\Delta y+\left(\frac{\varepsilon_{w}}{a_{w}}p^{\prime} +\frac{\varepsilon_{e}}{a_{w}}p^{\prime}s\right)\Delta x\right]\\ &-\left[\left(\left(\varepsilon u^{\prime}_{e}\right)_{e}-\left(\varepsilon u ^{\prime}_{w}\right)_{w}\right)\Delta y+\left(\left(\varepsilon u^{\prime}_{ w}\right)_{n}-\left(\varepsilon u^{\prime}_{y}\right)_{s}\right)\Delta x\right] \end{array}  \tag{56}\]  
Now the numerical method for solving the CFD model for gas phase can be summarized as follows:  
1. Calculate the pseudo velocity \(\overline{\mathbf{u}}\) from Equation (51) by substituting the initial velocity \(\mathbf{u}^{0}\). 2. Solve Equation (54) to obtain the pressure \(p^{*}\). 3. Employing the pressure \(p^{*}\), solve Equation (48) to obtain the velocity \(\mathbf{u}^{*}\). 4. Solve Equation (56) to obtain \(p^{\prime}\). 5. Correct the velocity to get \(\mathbf{u}\) by use of equations in form of Equation (55). 6. Return to step 1 and repeat until convergence. 7.