## 2 Boundary Layer Models of Steam-CO\({}_{2}\) Mixture Condensation
### Model Modification Considering CO\({}_{2}\) Solubility in the Condensate  
Figure 1c shows the difference in mixture condensation under the NCG and solubility assumptions of CO\({}_{2}\). The solubility of CO\({}_{2}\) in water (\(S\)) has complete data [35; 36; 37]. Equation (12) gives a method that is used to calculate \(S\) in this paper, which is a correlation of CO\({}_{2}\) solubility under different temperatures and pressures [35]. \[\left\{ \begin{array}{l}Par=c_{1}+c_{2}T+c_{3}/T+c_{4}T^{2}+c_{5}/T^{2}+c_{6} P+c_{7}PT+c_{8}P/T+c_{9}PT^{2}+c_{10}P^{2}T+c_{11}P^{3}\\ \frac{S}{M_{\text{CO}_{2}}}=\frac{P-P_{\text{H}_{\text{LO}}}}{\exp\left(Par- R_{\text{CO}_{2}}\right)}\end{array} \right. \tag{12}\]  
where \(Par\) depends on \(T\) and \(P\), and Equation (12) is an empirical formula showing piecewise functions, where \(T\) and \(P\) are different values, and the coefficients will be different. Moreover, coefficients c\({}_{1}\)\(\sim\)c\({}_{11}\) can be acquired from the study conducted by [PERSON] et al. [35], \(R_{\text{CO}2}\) is the fugacity coefficient of CO\({}_{2}\), and bar is the unit of pressure in Equation (12). Based on the solubility data, CO\({}_{2}\) is a soluble gas in the pressure range of 101.325 kPa-10 MPa, and the temperature is calculated in the range of 345.15 K-583.2 K; thus, \(S\) is in the range of 0.3545-63.12 g/kg\({}_{\text{(water)}}\), which increases with pressure and decreases with temperature. Given the solubility, the low solubility of CO\({}_{2}\) in the condensate heat transfer of steam-CO\({}_{2}\) cannot be simply ignored. When considering that CO\({}_{2}\) is slightly soluble, the expression of mass transfer at the gas-liquid interface changes, which may affect the solution results of the heat transfer. It is necessary to modify the boundary layer model in Section 2.1. In this paper, the steam-CO\({}_{2}\) condensation model based on low CO\({}_{2}\) solubility is referred to as the low-solubility condensable gas (CG) model. In the CG model, the compositions of the condensate are different. The amount of CO\({}_{2}\) dissolved into the condensate through the gas-liquid interface, \(f(\alpha_{l})\), is related to the distribution of CO\({}_{2}\) in the condensate, \(\alpha_{l}\). Equation (11) in the NG model becomes:  
\[\left[\rho_{\text{CO}_{2}}(u_{v}\frac{d\delta}{dx}-v_{v})-j_{\text{CO}_{2}} \right]\big{|}_{y=\delta}=f(\alpha_{l}) \tag{13}\]  
The ionization of CO\({}_{2}\) in water [32; 38] and the condensation and upstream CO\({}_{2}\) dissolution can all influence the \(\alpha_{l}\) distribution in the condensate. However, to simplify the analysis, it is assumed that the change in the CO\({}_{2}\) percentage in the near-thin film region is small, which is:  
\[\frac{\partial\alpha_{l}}{\partial x}=0,\frac{\partial\alpha_{l}}{\partial y}=0 \tag{14}\]  
Thus, \(\alpha_{l}\) is constrained by:  
\[\frac{\partial^{2}\alpha_{l}}{\partial y^{2}}=0,y=0:\frac{\partial\alpha_{l}} {\partial y}=0,y=\delta:\alpha_{l}=\frac{S}{1+S} \tag{15}\]  
In Equation (15), the form of the conservation equation is the same as Equations (2a) and (3a), but the boundary conditions are different from those of momentum and energy equations. The solution of Equation (15) is shown below:  
\[\alpha_{l}=\frac{S}{1+S}(0\leq y\leq\delta) \tag{16}\]This uniform distribution of CO\({}_{2}\) at the \(y\) direction is obtained under simplifications. However, this is acceptable because the liquid film is thin and the condensate flow rate is small. To form the CO\({}_{2}\) distribution in Equation (16), the following can be obtained:  
\[f(\alpha_{l})=m\cdot S \tag{17}\]  
This expression is similar to the condensation of the steam-NH\({}_{3}\) mixture, in which the condensation proportion of ammonia and steam is also determined by the phase equilibrium [39].