## 2 Computational Models
### Eulerian-Eulerian Two-Phase Flow Model  
The Favre-Averaged Navier-Stokes model [5; 18; 22] has been used to account for the density variation between liquid and gas phases, and the Eulerian-Eulerian two-phase model [5; 10] is utilized to handle the slip velocity. Two sets of Navier-Stokes equations are solved in this study. For each phase, the continuity in Equation (1) and Navier-Stokes equations in Equation (2) based on the Eulerian-Eulerian model are presented as follows:  
\[ \begin{array}{l}\frac{\partial\left(\alpha_{g}\rho_{g}\right)}{\partial t}+ \frac{\partial\left(\alpha_{g}\rho_{g}u_{g,i}\right)}{\partial x_{j}}=0\\ \frac{\partial\left(\alpha_{g}\rho_{l}\right)}{\partial t}+\frac{\partial \left(\alpha_{l}\rho_{l}u_{l,j}\right)}{\partial x_{j}}=0\end{array}  \tag{1}\]  
\[ \begin{array}{l}\frac{\partial\left(\alpha_{g}\rho_{g}u_{g,i}\right)}{ \partial t}+\frac{\partial\left(\alpha_{g}\rho_{g}u_{g,i}u_{g,i}\right)}{ \partial x_{j}}=-\alpha_{g}\frac{\partial P}{\partial x_{i}}+\frac{\partial \alpha_{g}\left(\tau_{g,i}\right)+\tau_{g,i}^{R}}{\partial x_{j}}+\alpha_{g} \rho_{g}G_{i}+K_{i}(u_{g,i}-u_{i,i})+\mathrm{M}_{g,i}\\ \frac{\partial\left(\alpha_{i}\rho_{l}u_{l,j}\right)}{\partial t}+\frac{ \partial\left(\alpha_{l}\rho_{l}u_{l,j}\right)}{\partial x_{j}}=-\alpha_{l} \frac{\partial P}{\partial x_{i}}+\frac{\partial\alpha_{l}\left(\tau_{l,ij}+ \tau_{l,ij}^{R}\right)}{\partial x_{j}}+\alpha_{l}\rho_{l}G_{i}+K_{i}(u_{l,i}- u_{g,i})+\mathrm{M}_{l,i}\end{array}  \tag{2}\]  
In Equations (1) and (2), the gas and liquid phases are represented by the subscripts \(g\) and \(l\), respectively, and the subscripts \(i\) and \(j\) are the Einstein notations. The symbol \(u\) represents velocity, \(\rho\) is density, \(P\) is pressure, \(\alpha\) is volume fraction, \(\tau\) is laminar stress, \(\tau^{R}\) is Reynolds stress, \(G\) is gravity (9.8 m/s\({}^{2}\) in -y direction), \(K_{i}\) is the exchange coefficient, and \(M\) is a momentum sink term for the porous media model. For simulation by real perforated structure, the porous media model is not activated. As shown in Figure 1, air and SO\({}_{2}\) gas are the components of gas phase, and the sulfurous slurry and magnesium hydroxide (Mg(OH)\({}_{2}\)) are the components of liquid phase. The Mg(OH)\({}_{2}\) solution removes the SO\({}_{2}\) gas phase by absorption, and new chemical by-product, namely sulfurous slurry, is generated in the liquid phase. The SO\({}_{2}\) gas is less than 130 ppm, and hence it is reasonable to assume that the corresponding phase change in Figure 1 does not really affect the continuity and momentum equations. Therefore, the source term related to chemical reactions is absent in Equations (1) and (2) [1; 5; 10; 29]The symbols \(\rho_{g}\) and \(\rho_{l}\) in Equation (3) are the mixture densities for the gas and liquid phases, respectively:  
\[ \begin{array}{l}\frac{\rho_{g}Y_{Air}}{\rho_{Air}}+\frac{\rho_{g}Y_{\text{SO}_{2 }}}{\rho_{SO_{2}}}=1\\ \frac{\rho_{l}Y_{Slamy}}{\rho_{Slamy}}+\frac{\rho_{l}Y_{M}}{\rho_{M}}=1\end{array}  \tag{3}\]  
The value of \(\rho_{g}\) can be calculated from the mass fractions of SO\({}_{2}\) and air within the gas phase (\(Y_{\text{SO}2}\) and \(Y_{air}\)) and their densities (\(\rho_{\text{SO}2}\) = 2.384 kg/m\({}^{3}\) and \(\rho_{air}\) = 1.066 kg/m\({}^{3}\)). Similarly, \(\rho_{l}\) is obtained from the mass fractions of slurry and Mg(OH)\({}_{2}\) solution within the liquid phase (\(Y_{slurry}\) and \(Y_{M}\)) and the corresponding densities (\(\rho_{Slamy}\) = 998.2 kg/m\({}^{3}\) and \(\rho_{M}\) = 983 kg/m\({}^{3}\)). The laminar stress \(\tau\) in Equation (2) is based on the Newtonian assumption, which is the product of the velocity gradient and the mixture viscosity \(\mu\). The turbulent stress \(\tau^{R}\) in Equation (2) is computed by standard \(k\)-\(\epsilon\) turbulence model [5; 22; 30; 31]. The turbulent viscosity stems from the empirical relations between turbulence kinetic energy \(k\) and turbulent dissipation rate \(\varepsilon\).