## 2 Boundary Layer Models of Steam-CO\({}_{2}\) Mixture Condensation
### Numerical Method  
Through the introduction of the above model, we transformed the partial differential equation, which is not easy to solve into the ordinary differential equation and easier to solve through the intermediate variable. Different from the integral method adopted by [PERSON] [14; 15], this paper uses a discrete case to solve the equation. This method expands the computational scope of the model and can find the convergence solution quickly. By solving the NG or CG model, the gas-liquid interface data \(T_{ix}\) and \(\alpha_{ix}\) can be obtained.  
The solution process is shown in Figure 2, and the fourth-order Runge-Kutta (RK4) and Euler iterative methods are respectively used to calculate and solve the ordinary differential equations on the horizontal and vertical plate surfaces. The process is to assume the physical property of the gas-liquid interface at any position of the plate first, solve the assumed actual value in relation to the existing solution conditions described in the NG model or CG model, complete the cycle, and repeat the iterative calculation until the assumed value and the actual value meet the specified deviation, after which it is considered to solve the actual solution of the gas-liquid interface. For the solution of the physical properties of the mixture, a program in MATLAB R2021a was written that calls  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Pressure** & **Temperature** & **Solution Enthalpy** & **Solution Enthalpy** & **\(h_{\text{so}}\)/\(h_{\text{fg}}\)** & \(h_{\text{so}}\)/\(h_{\text{fg}}\) \\ (MPa) & **(K)** & **(k/mol)** & **(k/kg)** & **(k\({}_{\text{so}}\))** & **(k\({}_{\text{so}}\))** \\ \hline
0.101325 & 298.15 & 19.75 [40] & 448.864 & 0.1838 & 0.0002698 \\
2.06 & 323.1 & 14.8 [41] & 336.364 & 0.1412 & 0.002223 \\
5.10 & 323.1 & 13.1 [41] & 297.727 & 0.1250 & 0.004401 \\
10.53 & 323.1 & 7.5 [41] & 170.455 & 0.07155 & 0.003569 \\
5.05 & 373.1 & 6.3 [41] & 143.182 & 0.06345 & 0.001235 \\
10.08 & 373.1 & 4.6 [41] & 104.545 & 0.04633 & 0.001561 \\ \hline \end{tabular}
\end{table}
Table 3: The solution enthalpy of CO\({}_{2}\).  
the NIST 9.1 [42] database to check the physical parameters to be calculated at the same time in the process of iterative calculation.  
After \(T_{ix}\) and \(\alpha_{ix}\) are acquired at the gas-liquid interface, condensation heat transfer features can be calculated, as shown below [15].  
The average heat transfer coefficient (HTC) on the horizontal plate:  
\[\overline{h}=\frac{1}{L}\int_{0}^{L}h_{x}\mathrm{d}x= \frac{1}{L}\int_{0}^{L}\frac{q}{(T_{\infty}-T_{w})}\mathrm{d}x= \frac{2k_{l}}{\eta_{i,l}}\sqrt{\frac{U_{\infty}}{v_{l}L}}\frac{T_{i}-T_{w}}{T_ {\infty}-T_{w}} \tag{21}\]  
The average HTC on the vertical plate:  
\[\overline{h}=\frac{1}{L}\int_{0}^{L}h_{x}\mathrm{d}x= \frac{h_{fS}\mu_{l}}{T_{\infty}-T_{w}}\frac{4}{3}[\frac{C_{p}(T_{i}-T _{w})}{h_{fS}Pr}]^{3/4}(\frac{gL^{3}}{4v_{l}^{2}})^{1/4} \tag{22}\]  
where \(L\) is the length of the flat plate and \(h_{x}\) is the local heat transfer coefficient. \(T_{\infty}-T_{w}\) is the temperature difference between the bulk and the wall. For saturate inlet conditions, this indicates the subcooling of condensation. \(T_{i}\) is the average temperature along the plate.  
For condensation heat transfer reduction, \(q/q_{0}\) (subscript 0 is the data for pure streams) on the horizontal plate is:  
\[\frac{q}{q_{0}}=\frac{k_{l}}{k_{l,0}}\cdot\sqrt{\frac{v_{l,0}}{v_{l}}}\frac{ \eta_{i,0}}{\eta_{i}}\frac{T_{i}-T_{w}}{T_{\infty}-T_{w}} \tag{23}\]  
On the vertical plate, this is:  
\[\frac{q}{q_{0}}=[\frac{C_{p}(T_{i}-T_{w})}{C_{p,0}(T_{\infty}-T_{w})}]^{3/4}( \frac{h_{fS}\mu_{l}}{h_{fS,0}\mu_{l,0}})^{1/4}\sqrt{(\frac{v_{l,0}}{v_{l}})} \tag{24}\]  
Figure 2: Iterative for \(T_{ix}\) and \(\alpha_{ix}\).