## 2 Methodology
### Indexes Determination  
Gas-liquid contact with the co-current flow can be found in Figure 3a, the simulated flue gas, CO\({}_{2}\)(A) + N\({}_{2}\)(B), go through the distributor forming small bubbles and mixing with absorbent in the bubble-column scrubber. The CO\({}_{2}\) gas from the gas film diffuses into the liquid film and is absorbed by the liquid as shown in Figure 3b. At the interface, the CO\({}_{2}\) gas follows Henry's law and the absorption rate accompanying with two-film model becomes:  
\[-r_{A}=(K_{G}a)_{loc}(C_{A}-HC_{LA}). \tag{15}\]  
In order to obtain the absorption rate and overall mass-transfer coefficient, the isothermal mass balance at a steady-state condition is used:  
Figure 3: Absorption of CO\({}_{2}\) in the scrubber accompanied with two-film model. (Mass transfer of CO\({}_{2}\) at inlet) \(-\) (Mass transfer of CO\({}_{2}\) at outlet) = Absorption rate of CO\({}_{2}\). (16)  
The equations can be expressed below:  
\[(N_{A}|_{\rm z}-N_{A}|_{\rm z+\Delta z})A=(-r_{A})(A\Delta z)\epsilon_{L} \tag{17}\]  
where \(N_{A}\) is the molar flux of component A. Alternatively, Equation (17) becomes:  
\[(F_{A}|_{\rm z}-F_{A}|_{\rm z+\Delta z})=(-r_{A})(A\Delta z)\epsilon_{L} \tag{18}\]  
where \(N_{A}A\) = \(F_{A}\), the molar flow rate. Take limit, Equation (18) becomes  
\[-\frac{dF_{A}}{dz}=(-r_{A})(A\epsilon_{L}). \tag{19}\]  
Integrate the Equation (19), it is  
\[-\int_{F_{A1}}^{F_{A2}}dF_{A}=\int_{0}^{L}A\epsilon_{L}dz=\int_{0}^{V_{L}}{(-r _{A})}dV_{L}, \tag{20}\]  
\[F_{A1}-F_{A2}=R_{A}V_{L}, \tag{21}\]  
or  
\[R_{A}=\frac{F_{A1}-F_{A2}}{V_{L}}. \tag{22}\]  
Equation (22) shows that the overall absorption rate can be calculated from the measured input and output CO\({}_{2}\) gas molar rates and liquid volume. Because the molar flow rate of inert gas (N\({}_{2}\)) is equal to \(F_{A1}(1-y_{A1})/y_{A1}\), the molar flow rate of carbon dioxide at the outlet (\(F_{A2}\)) is equal to \(F_{A1}[(1-y_{A1})/y_{A1}][y_{A2}/(1-y_{A2})]\). Therefore, Equation (22) can be rewritten as:  
\[R_{A}=\frac{F_{A1}}{V_{L}}[1-(\frac{1-y_{A1}}{y_{A1}})(\frac{y_{A2}}{1-y_{A2}} )]. \tag{23}\]  
Equation (23) shows that absorption rate can be evaluated when \(F_{A1}\), \(V_{L}\), \(y_{A1}\), and \(y_{A2}\) are given. In addition, Equation (17) can be written as  
\[(u_{z}C_{A}|_{\rm z}-u_{z}C_{A}|_{\rm z+\Delta z})A=(-r_{A})(A\Delta z) \epsilon_{L} \tag{24}\]  
where \(u_{z}\) is the linear velocity of gas through the column. Substitute Equation (15) into Equation (24) and it is divided by \(\Delta z\) and take the limit, the equation becomes  
Figure 3:0  
For most systems, \(C_{A}\) > \(HC_{LA}\)[49]. Therefore, Equation (25) can be rearranged and integration, which is  
Figure 3:1  
Finally, the overall mass-transfer coefficient becomes  
Figure 3:2  
where \(K_{G}a\) is defined as the following:  
Figure 3:3Equation (27) shows that overall mass-transfer coefficient can be estimated when \(C_{A1}\), \(C_{A2}\), \(V_{L}\), and \(Q_{g}\) are measured. In addition, the absorption efficiency and scrubbing factor are defined below:  
Figure 3:4  
Figure 3:5  
Heat duty of solvent regeneration includes three parts, i.e., heat of adsorption (\(q_{ads}\)), sensitive heat (\(q_{sen}\)), and heat of evaporation (\(q_{sol}\)):  
\[ \begin{array}{ll}q(G&Jt^{-1})=q_{ads}(G/t)+q_{sen}(G/t)+q_{sol}(G/t)\\ &=\Delta H^{ad}(G/t)+\frac{m_{sol}(kg)C_{p}(kJkg^{-1}-K^{-1})\Delta T(K)}{ \Delta mC_{\text{CO}_{2}}(kg)}+\frac{\Delta m_{1}(kg)}{\Delta mC_{\text{CO}_{2 }}(kg)}\Delta H^{rap}(G/t)\end{array} . \tag{31}\]  
They can be determined when thermal data are available [32; 50; 51; 52; 53; 54]. Thermal data used here include those of heat capacity [50], the heat of absorption [32; 52; 53; 54], and latent heat [51].