## 2 Modeling Approach
### Species Transport Equation  
In this study, to observe the changes in the concentration of carbon dioxide in the gas phase, it is necessary to consider the species transport equation in the gas phase [25]:  
\[\frac{\partial\big{(}\alpha_{G}\rho_{G}Y_{G}{}^{CO_{2}}\big{)}}{\partial t}+ \
abla.\Big{(}\alpha_{G}\rho_{G}u_{G}Y_{G}{}^{CO_{2}}-\alpha_{G}\Gamma_{G}{}^{ CO_{2}}\
abla\ Y_{G}{}^{CO_{2}}\Big{)}=-\dot{m}_{G\to l} \tag{19}\]  
In this equation, \(\Gamma_{G}\) shows the diffusion coefficient, \(\alpha_{G}\) is the volume fraction of gas that is defined based on C_VOF (cell, gas) function, \(\rho_{G}\) is the gas density that is defined based on C_R (cell, gas) function, \(u_{G}\) is the gas velocity, \(Y_{G}{}^{CO_{2}}\) is the CO\({}_{2}\) mass fractions in the gas phase, and \(-\dot{m}_{G\to l}\) is the mass transfer of CO\({}_{2}\) from the gas to liquid phase that is determined by Equation (20) [25; 35]. As shown schematically in Figure 2, chemical reactions in a gas-liquid system include extremely complicated interconnections between the main processes. The pace of chemical reaction is regulated by the local concentration of the species and the mixing caused by the interphase mass transfer mechanism and scattered bubbles. The mass transfer coefficient, meanwhile, is a function of local hydrodynamics, which is itself impacted by bubble shrinkage as a result of physical or chemical absorption, as well as differences in physical characteristics as a result of inhomogeneous distributions of chemical species [36]. \[\dot{m}_{G\to l}=K_{L}aE\rho_{L}\Big{(}Y_{l}{}^{CO_{2}*}-Y_{L}{}^{CO_{2}}\Big{)} \tag{20}\]  
where \(K_{L}\) shows the overall mass transfer coefficient of liquid phase, \(a=\frac{6\alpha_{G}}{d_{B}}\) is interfacial area per unit volume, \(d_{B}\) is bubble diameter that is equal to 1 mm, and \(Y_{L}{}^{CO_{2}}\) and \(Y_{G}{}^{CO_{2}}\)(=0.191) are CO\({}_{2}\) mass fractions in liquid and gas phases, respectively. Due to the point that there is a mass transfer resistance in the liquid phase, with respect to [PERSON]'s law, the mass fraction of liquid phase that is in equilibrium with the mass fraction of the gas phase is calculated by [25; 35]:  
\[Y_{l}{}^{CO_{2}*}=H^{CO_{2}}Y_{G}{}^{CO_{2}}\frac{\rho_{G}}{\rho_{L}} \tag{21}\]  
Figure 2: Schematic representation of inter-dependency of fluid flow, mass transfer, and chemical reaction. In this study, the value of \(H^{\text{CO}2}\)(=0.64) has been considered based on the data of an investigation carried out by [PERSON] et al. [37] and \(\rho_{G}\) and \(\rho_{L}\) are gas and liquid densities that are defined based on C_R (cell, gas) and C_R (cell, liq) functions, respectively. Additionally, in Equation (6), the enhancement factor \(E\) is a constant related to chemical reactions and should be considered in calculations. In this study, this term is determined for modeled system via [25,35]:  
\[E_{\text{CO}_{2}}=\Bigg{(}\frac{D_{\text{CO}2,liq}}{D_{MEAJ,liq}}\Bigg{)}^{1/2}+ \Bigg{(}\frac{D_{MEA,liq}}{D_{CO2,liq}}\Bigg{)}^{1/2}\frac{C_{MEA}}{v}\frac{C_{ CA}}{C_{G}^{\text{CO}^{\ast}}} \tag{22}\]  
where \(v=1\) denotes the stoichiometry coefficient for CO\({}_{2}\) based on the reaction between CO\({}_{2}\) and MEA as follows:  
\[CO_{2}(l)+2 HOC_{2}H_{4}NH_{2}\to HOC_{2}H_{4}NH_{3}{}^{+}+HOC_{2}H_{4}NHCOO^{-}. \tag{23}\]  
\(C_{MEA}=4.86\) kmol/m\({}^{3}\) is the concentration of MEA; besides, the diffusivity constant for CO\({}_{2}\) in the MEA (\(D_{CO2,liq}\)) is calculated by N\({}_{2}\)O analogy, that is \(6.7\times 10^{-10}\) m\({}^{2}\)/s [38], while the diffusivity constant for MEA in water (\(D_{MEA,liq}\)) is considered based on data of [PERSON] et al., which is equal to \(1.12\times 10^{-10}\) m\({}^{2}\)/s [39].