## 2 Boundary Layer Models of Steam-CO\({}_{2}\) Mixture Condensation
### Steam-CO\({}_{2}\) Condensation Model Based on CO\({}_{2}\) as an NCG
\[P_{\rm H_{2}O}=\frac{1-\alpha_{ix}}{1-\alpha_{ix}(1-M_{\rm H_{2}O}/\,M_{\rm CO_{2 }})}P \tag{8}\]  
Furthermore, in the NG model, in the liquid-phase boundary layer, the condensation mass flux (\(m\)) can be represented as:  
\[m=\rho_{l}(u_{l}\frac{\mathrm{d}\delta}{\mathrm{d}x}-v_{l})=\rho_{v}(u_{v}-v_{ v}) \tag{9}\]  
combined with Equations (4a) and (4c) and Equations (5a) and (5c), we can obtain the following:  
\[m=\frac{1}{2}\sqrt{\frac{v_{l}U_{\infty}}{x}}f(\eta_{i})=\frac{1}{2}\sqrt{ \frac{v_{p}U_{\infty}}{x}}F(0) \tag{10}\] \begin{table}
\begin{tabular}{c c c} \hline
**Plate** & **Phase** & **Derivation and Solution** \\ \hline \multirow{10}{*}{**Horizontal**} & \multirow{6}{*}{**Liquid**} & \multirow{6}{*}{**Liquid**} & Introduce intermediate variables, \\  & & \(\eta=y\sqrt{\frac{U_{\infty}}{v_{l}}};\psi_{l}=\sqrt{U_{\infty}v_{l}}\bar{x}f( \eta);\theta=\frac{T_{l}-T_{\
u}}{T_{l_{\infty}}-T_{l\
u}}\) \\  & & \(u_{l}=\frac{\partial\psi}{\partial y}=U_{\infty}f^{\prime},v_{l}=-\frac{ \partial\psi}{\partial x}=\frac{1}{2}\sqrt{\frac{U_{\infty}v_{l}}{x}}(\eta f^{ \prime}-f)\) \\  & & Ordinary differential equation equations, \\  & & \(f^{\prime\prime}(\eta)=0;\theta^{\prime\prime}(\eta)=0\) \\  & & Solution, \\  & & \(f=\frac{1}{2}f^{\prime\prime}(0)\eta^{2}\); \(\theta=\frac{\eta}{\eta_{i}};\eta_{i}=\delta\sqrt{\frac{U_{\infty}}{v_{l}}}\) \\  & & Introduce intermediate variables, \\  & & \(\xi=(y-\delta)\sqrt{\frac{U_{\infty}}{v_{x}}},\quad\varphi_{v}(\xi)=\frac{\alpha _{v}-\alpha_{v}}{\delta_{xv}-\alpha_{v}},\quad\varphi_{v}=\sqrt{U_{\infty}v_{ v}x}F(\xi)\) \\  & & \(u_{v}=\frac{\partial\varphi}{\partial y}=U_{\infty}F^{\prime},v_{v}=-\frac{ \partial\varphi}{\partial x}=\frac{1}{2}\sqrt{\frac{U_{\infty}v_{v}}{x}}(\xi F^{ \prime}-F)+U_{\infty}\frac{\mathrm{d}\xi}{\mathrm{d}x}F^{\prime}\) \\
**Gas** & Ordinary differential equation equations, \\  & & \(F^{\prime\prime\prime}+\frac{1}{2}FF^{\prime\prime}=0;\phi^{\prime\prime}+\frac {1}{2}ScF\phi^{\prime}=0\) \\  & & Solution, \\  & & \(F^{\prime}(0)=0;\sqrt{\frac{p_{\rm H}i}{p_{\rm H}p_{\rm H}}}\frac{C_{\rm r}(T_{l _{\infty}}-T_{l\
u})}{T_{l_{\rm H}p}F^{\prime}}=\sqrt{\frac{[F(0)]^{3}}{2F^{ \prime}(0)}};\eta_{i}=\sqrt{\frac{2F(0)}{F^{\prime}(0)}}\) \\  & & \(F(0)=\)2\(\frac{u_{l}}{U_{\infty}}\sqrt{\frac{x-U_{\infty}}{v_{l}}};\alpha_{ix}=\frac{ \alpha_{v}}{1+1/2 Sc}\frac{\alpha_{v}(0)}{\varphi_{v}^{2}(0)}\) \\  & & Solve directly, \\  & & \(u_{l}x=[\frac{C_{\rm r}(T_{l_{\infty}}-T_{l_{\infty}})}{h_{l_{\rm H}p_{\rm H}}} ]^{1/2}(g\cdot x)^{1/2};m=\frac{q}{h_{l_{\rm H}}}\) \\  & & Introduce intermediate variables, \\  & & \(\xi=\frac{c(y-\delta)}{x^{3/4}};c=[\frac{\delta(M_{\rm CO_{2}}-M_{\rm H_{2}O})/ 4\omega_{v}^{2}}{M_{\rm CO_{2}}-(M_{\rm CO_{2}}-M_{\rm H_{2}O})\alpha_{m}}]^{ 1/4}\) \\
**Vertical** & & \(\psi_{v}=4v_{\varphi}cx^{3/4}F(\xi);\varphi_{v}=\alpha_{v}-\alpha_{\infty}\) \\  & & Ordinary differential equation equations, \\  & & \(F^{\prime\prime}+3F^{\prime\prime}-2(F^{\prime})^{2}+\phi=0\) \\  & & \(\phi^{\prime\prime}+3 Sc\cdot F\phi^{\prime}=0\) \\  & & Solution, \\  & & \(F^{\prime}(0)=\frac{\sqrt{3}}{4v_{\varphi}c^{2}}\frac{(C_{\rm r}(T_{l_{\infty}}-T_ {l_{\infty}})}{h_{l_{\rm H}p_{\rm H}}}]^{1/2};F(0)=\frac{\sqrt{3}\sqrt{2v_{l}v_{ l}}}{3\sqrt{2v_{l}v_{l}}}\sqrt{\frac{p_{\rm H}i_{l}}{p_{\rm H}p_{\rm H}}}[ \frac{C_{\rm r}(T_{l_{\infty}}-T_{l_{\infty}})}{h_{l_{\rm H}g_{l}}F^{\prime }}]^{3/4}\) \\  & & \(\varphi_{v}{}^{\prime}(0)=-3 Sc\cdot F(0)[\varphi_{v}(0)+\alpha_{\infty}]\) \\ \hline \end{tabular} Note: \(f\) (for liquid) and \(F\) (for gas) are the functions of \(\eta\) and \(\xi\), and \(\eta\) and \(\xi\) are the dimensionless coordinates. \(\theta\) is the dimensionless temperature, \(\varphi\) is the concentration function, \(\psi\) is the stream function, \(C_{P}\) is the specific heat, \(h_{l_{\rm H}}\) is the latent heat, \(P_{P}\) is the Prandtl number, \(Sc\) is the Schmidt number, \(q\) is the heat flux, \(m\) is the condensation mass flux, \(M\) is the molar mass, and \(p\) is the dynamic viscosity.  
\end{table}