## 2 Modeling Approach
### Geometries, Grids, and Mesh Independency  
In this study, for the computational domain and grid generation, Fluent Meshing software is used. The dimensions of geometry are shown in Table 1 and Figure 3[42]. Additionally, 3D and 2D solution domain with uniform mesh is shown in Figure 3. Due to the mixing shock, the concentration of grid density is centered on the exit of nozzle and also the throat. To acknowledge the stability of numerical results (mesh independency), the eject with 3D design and d\({}_{\rm nozzle}=12\) mm is selected. The effect of three sets of grids (i.e., 107,793, 206,537, and 387,987 mesh elements) on the air entrainment rate, Q\({}_{\rm g}\), at motive liquid flow rate (Q\({}_{\rm L}=2.17\times 10^{-4}\) m\({}^{3}\)/s) is listed in Table 2. The error percentage is calculated by obtaining air entrainment rate for each grids and comparing it with experimental data [23]. It can be seen that the results achieved for grids with 206,537 and 387,987 cells confirm the stability of numerical results; however, the grid with 206,537 cells needs less computational time than the one with the 387,987 cells. Therefore, the grid with 206,537 cells was selected for this section.  
The predicted outcomes could be significantly affected by the bubbles' diameter. Based on the experimental work done by [PERSON] [23], which is the basis of this study, an asymmetry toward larger bubble sizes was observed in the bubble size distributions, which differed from a normal distribution. Thus, according to this paper, we chose d\({}_{\rm b}=1\) mm as the model parameter for the simulation. It is noted that d\({}_{\rm b}=1\) mm is also used for the independency of grid.  
\begin{table}
\begin{tabular}{l c} \hline \hline
**Description** & **Dimension (mm)** \\ \hline Height of the suction chamber & 50 \\ Diameter of suction chamber & 60 \\ Diameter of throat & 19 \\ Length of the throat & 183 \\ Length of the diffuser & 254 \\ Diameter of the contactor & 60 \\ Diameter of gas inlet & 10 \\ Length of the contactor & 25 \\ Diameter of the nozzle used & 6 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Dimensions of the liquid–gas ejector [42].  
To investigate the absorption of carbon dioxide by MEA inside the ejector, due to the existence of mass transfer and enhancing the convergence and analyzing time, a 2D-axisymmetric geometry has been considered with a quadrilateral and triangular-meshing scheme. For two-dimensional geometry, similar procedures to the three-dimensional geometry are presented.  
Figure 3: Dimensions, grid structures, and boundary conditions of 3D and 2D ejector.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{3}{c}{**Total Cell Amount**} & \multicolumn{1}{c}{**Parameter**} \\ \hline  & **107,793** & **206,537** & **387,987** & \multirow{3}{*}{
\begin{tabular}{c} CPU \\ (2.4 GHz and 8 cores) \\ \end{tabular} } \\ \cline{2-4}
3D geometry & 2.4 & 2.4 & 2.4 &
\begin{tabular}{c} CPU \\ (2.4 GHz and 8 cores) \\ \end{tabular} \\ \cline{2-4}
3D geometry & 6 & 9 & 14 & Time (day) \\ \cline{2-4}  & 20 & 6 & 6 & \
\begin{tabular}{c} Error: \\ \(\left(\frac{\left|Q_{c,Exponential}-Q_{c,c}\text{cp}\right|}{Q_{c,Exponential}}\times 100\right)\) \\ \end{tabular} \\ \hline  & 40,780 & 60,234 & 75,325 & \multirow{3}{*}{
\begin{tabular}{c} CPU \\ (2.4 GHz and 8 cores) \\ \end{tabular} } \\ \cline{2-4}
2D geometry & 2.4 & 2.4 & 2.4 &
\begin{tabular}{c} CPU \\ (2.4 GHz and 8 cores) \\ \end{tabular} \\ \cline{2-4}
2D geometry & 1 & 2.5 & 4 & Time (day) \\ \cline{2-4}  & 23 & 7 & 7 &
\begin{tabular}{c} Error: \\ \(\left(\frac{\left|Q_{c,Exponential}-Q_{c,c}\text{cp}\right|}{Q_{c,Exponential}}\times 100\right)\) \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table}
Table 2: Investigation of mesh independency of the designed 3D and 2D liquid–gas ejector.  
geometry were applied, and the grid with 60,234 cells was selected. The results are shown in Table 2. Additionally, a 2D solution domain with uniform mesh is shown in Figure 3.