## 3 Validations of the Two-Phase and Porous Media Model
### Boundary Conditions, Grid Layouts, and Validations for the Small-Scale Tower
Table 6 presents the case name, flow condition, parameters in porous media model, and SO\({}_{2}\) removal efficiency at outlet for all the different cases. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \begin{tabular}{c} **Case** \\ **Number** \\ \end{tabular} & \begin{tabular}{c} **Number of** \\ **Trays** \\ \end{tabular} & \
\begin{tabular}{c} **Q** \\ **(m\({}^{3}\)/min)** \\ \end{tabular} & \(Re_{h,g}\) & \(Re_{h,l}\) & \(C_{g,y}\) & \(C_{l,y}\) &
\begin{tabular}{c} **Efficiency** \\ **(\%)** \\ \end{tabular} \\ \hline A & 4 & 24 & 5130 & 860 & 15 & 10.3 & 57.1 \\ \hline B & 4 & 24 & 5130 & 570 & 15 & 10.5 & 51.5 \\ \hline C & 0 & 24 & 5130 & 860 & Empty tower & 56.9 \\ \hline D & 1 & 24 & 5130 & 860 & 15 & 10.3 & 61.6 \\ \hline E & 0 & 36 & 7700 & 860 & Empty tower & 48 \\ \hline F & 1 & 36 & 7700 & 860 & 15.7 & 10.3 & 58.8 \\ \hline G & 4 & 36 & 7700 & 860 & 15.7 & 10.3 & 63 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: This table highlights the different cases of small-scale tower that computed by porous media model with SO\({}_{2}\) removal efficiencies at outlet. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **GT1** & **GT2** & **GT3** \\ \hline Case 1 & 62.7\% & 72.6\% & 60.2\% \\ \hline Case A & 66.4\% & 77.7\% & 67.2\% \\ \hline Exp. & 62.2\% & 71.4\% & 69.1\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: This table displays the SO\({}_{2}\) removal efficiencies of small-scale tower by numerical and experimental results [36] at different sampling points as \(L/G\) = 4.9. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **GT1** & **GT2** & **GT3** \\ \hline Case 2 & 58.7\% & 67.9\% & 54.9\% \\ \hline Case B & 60.4\% & 70.9\% & 61.9\% \\ \hline Exp. & 48.6\% & 68.4\% & 57.7\% \\ \hline \hline \end{tabular}
\end{table} 
Table 5: This table displays the SO\({}_{2}\) removal efficiencies of small-scale tower by numerical and experimental results [36] at different sampling points as \(L/G\) = 3.2. For the small-scale tower of \(L/G=4.9\), the hole Reynolds number for gas phase \(Re_{h_{t}}\) is 5130, porosity \(f_{h}\) is 34.3%, and \(t_{h}/D_{h}\) is 0.75 for Case 1, as shown in Table 1. These values are used to tabulate in ref. [28], and obtain values of \(\xi_{1}\)= 0.123, \(\xi_{3}=0.658\), \(\xi_{4}=0.525\), and \(\lambda=0.037\) in Equation (16). The inertial loss coefficient in Equation (2) for gas flow in \(y\)-direction \(C_{g,y}\) is 11.33 after inserting these values into Equation (16). As for liquid phase in Table 1, the hole Reynolds number for liquid phase \(Re_{h,t}\) is 860 for Case 1, corresponding to \(\xi_{1}=0.25\), \(\xi_{3}=0.513\), \(\xi_{4}=0.525\), and \(\lambda=0.074\). From Equation (16), the inertial loss coefficient for liquid phase \(C_{Ly}\) is calculated as 10.31. As the main direction of flow within the perforated hole aligns with the \(y\)-direction, the inertial loss coefficients of these two phases in the \(x\)- and \(z\)-directions are amplified by 100 times from their corresponding values in the \(y\)-direction [4; 22]. From our preliminary results, based on the aforementioned setups (\(C_{g,y}=11.33\) and \(C_{l,y}=10.31\)) for porous media model, the pressure drop within the right tower of the small-scale tower as \(L/G=4.9\) is 86.7 Pa, while that computed by real perforated structures (Case 1) is 104.6 Pa. This deficiency by using porous media model is caused by Equation (16) being calibrated based on experiments with single-phase flow [4; 25; 28] and the two-phase flows within the tower have opposite direction. As a result, it is reasonable to increase the inertial loss coefficients for the two-phase counterflow. Considering that gas flow occupies most of the domain within the tower, only the value of gas flow \(C_{g,y}\) increases by 33% from 11.3 to 15 after a series of tests. The numerical results of Case A by \(C_{g,y}=15\) and \(C_{l,y}=10.31\) yields a pressure drop 103.5 Pa, which is exactly the same as that by Case 1 with perforated structures.