## 2 Condensation Heat Transfer Model
For the film heat transfer coefficient \(h_{\rm film}\), generally, the Nusselt theory and the modified equation based on the Nusselt theory are widely used to calculate the heat transfer coefficient of liquid film. However, according to the Nusselt theory, the liquid film is assumed to be laminar flow; when the vertical surface is too long, the liquid film fully develops, and the flow in the liquid film gradually changes from laminar flow to wavy laminar flow and then to turbulence flow. Using the solution based on the Nusselt theory will simply produce a certain deviation, so the following formula is adopted for laminar flow when \(Re<30\)[29]:  
\[Re=3.78\left[\frac{k_{\rm L}L(T_{\rm i}-T_{\rm w})}{\mu_{\rm i}h_{\rm Ig}^{*} (v_{\rm 1}^{2}/g)^{1/3}}\right]^{3/4}Re<30 \tag{3}\]  
[PERSON]'s relation is adopted for wavy laminar flow when \(30\leq Re<1800\):  
\[Re=\left[\frac{3.70\rm k_{\rm I}L(T_{\rm i}-T_{\rm w})}{\mu_{\rm i}h_{\rm Ig}^ {*}(v_{\rm 1}^{2}/g)^{1/3}}+4.81\right]^{0.820}30\leq Re<1800 \tag{4}\]  
[PERSON]'s relation is adopted for turbulent condensate flow when \(Re\geq 1800\):  
\[Re=\left[\frac{0.069k_{\rm I}L(T_{\rm i}-T_{\rm w})}{\mu_{\rm i}h_{\rm Ig}^{* }(v_{\rm 1}^{2}/g)^{1/3}}Pr^{0.5}-151 Pr^{0.5}+253\right]^{4/3}Re\geq 1800 \tag{5}\]  
where: \(k_{\rm I}\)--thermal conductivity of the liquid;  
\(L\)--height of the vertical plate;  
\(\mu_{\rm I}\)--dynamic viscosity of the liquid;  
\(v_{\rm 1}\)--kinematic viscosity of the liquid;  
\(g\)--gravitational acceleration;  
\(Pr\)--Prandtl number. Actually, the condensation process is cooled further to some average temperature between \(T_{\rm i}\) and \(T_{\rm w}\), releasing more heat in the process. Therefore, the actual heat transfer will be larger. [PERSON] suggested that the cooling of the liquid below the saturation temperature can be accounted for the modified latent heat of vaporization \(h_{\rm Ig}^{*}\), defined as \(h_{\rm Ig}^{*}=h_{\rm fg}+0.68c_{p,\rm I}(T_{\rm i}-T_{\rm w})\), where \(c_{p,\rm I}\) is the specific heat of the liquid at the average film temperature. After calculating \(Re\), the liquid film heat transfer coefficient \(h_{\rm film}\) is finally calculated as follows:  
\[h_{\rm film}=\frac{Re\mu_{\rm i}h_{\rm fg}^{*}}{4L(T_{\rm i}-T_{\rm w})} \tag{6}\]  
For the convective heat transfer coefficient \(h_{\rm conv}\), according to the numerical correlation of the convective heat transfer similarity criterion without considering the convective heat transfer coefficient under the influence of normal mass transfer, it can be calculated as follows:  
\[h_{\rm conv}=Nu\cdot k/l \tag{7}\]  
The \(Nu\) number in the equation above refers to the relationship between turbulent natural convection and forced convection over a flat plate according to different convection forms:  
\[Nu^{\rm FC}=0.037 Re^{0.8}Pr^{1/3} \tag{8}\]\[Nu^{NC}=0.13(Gr\cdot Pr)^{1/3} \tag{9}\]  
For mixed convection, the following equation is adopted [29]:  
\[h_{\rm tot}\big{(}T_{\rm g}-T_{\rm w}\big{)}=h_{\rm film}(T_{\rm i}-T_{\rm w} )=(h_{\rm conv}+h_{\rm cond})\big{(}T_{\rm g}-T_{\rm i}\big{)} \tag{2}\]0  
where \(Nu_{\text{mixed}}\) is the Nusselt number for the mixed flow, and \(Nu_{\text{FC}}\) and \(Nu_{\text{NC}}\) are the Nusselt number calculated via forced convection correlation and natural convection correlation under given conditions, respectively. A positive sign was taken when the two flow directions were the same, a negative sign was taken when they were opposite, and for an uncertain flow direction, the minimum value of \(Nu^{n}_{\text{mixed}}\) was adopted via conservative estimation. The exponent, n, is usually 3.