## 3 Computational Model
### Two-Phase Model  
The mixture two-phase model in Ansys Fluent is used for the present computations [31]. This model is the best representation of the homogenous two-phase flow for water, air and water vapor flow in the U-tube. In the mixture model, all phases share the same cell volume and can interchange energy and momentum. The cell volume fraction for the phase is \(\alpha\), and all phase volume fraction summation is equal to 1. In the following equations, \(\alpha_{m}\) is the volume ratio of the side mixture in a combination of the mixture (water vapor and air) and water flows. \(\rm X_{w}\) and \(\rm X_{v}\) are the water and water vapor mass flow rate ratios concerning the side mixture mass flow rate, respectively. \(\rm Y_{a}\) is the ratio of air mass flow rate to the total fluid mass flow rate, while \(\rm V_{a,v}\) is the volume for the side mixture [29; 32; 33]. \[\sum_{j=1}^{n}\alpha_{j}=1 \tag{10}\]  
\[\alpha_{m}=\frac{V_{a,v}}{\left(V_{w}+V_{a,v}\right)},\ \dot{m}_{a,v}=\dot{m}_{a}+\dot{m}_{v} \tag{11}\]\[X_{\rm w}=\frac{\dot{\rm m}_{\rm w}}{\dot{\rm m}_{\rm a,v}},\,X_{\rm v}=\frac{\dot{ \rm m}_{\rm v}}{\dot{\rm m}_{\rm a,v}},\,Y_{\rm a}=\,\frac{\dot{\rm m}_{\rm a}}{ \dot{\rm m}_{\rm w}+\dot{\rm m}_{\rm a}} \tag{12}\]  
For the filled cells with water in two-phase flows, the volume fraction of water, \(\alpha_{\rm w}\), is 1 while \(\alpha_{\rm v}\) and \(\alpha_{\rm a}\) are taken to be 0 and the same way for the filled water vapor cells. If the cell combines the mixing of air, water, and water vapor, then these cell volume fractions are between 0 and 1. For calculating the secondary and primary phase volume fractions, FLUENT(r) solves separate continuity equations for secondary phases, while Equation (10) is utilized for computing the primary phase volume fraction. In the condensation process, the water is generated from water vapor due to phase change. Therefore, the water is taken to be the primary phase while water vapor and air are the secondary phases. The continuity equation for calculating the secondary phase volume fraction is given as follows [29]. \[\
abla.\alpha_{\rm sec}\rho_{\rm sec}\stackrel{{\rightarrow}}{{ \
abla}}_{\rm m}=\,\sum\limits_{j=1}^{n}\dot{\rm m}_{pri-\rm sec}-\dot{\rm m}_ {\rm sec-pri} \tag{13}\]  
\(\dot{\rm m}_{pri-\rm sec}\), \(\dot{\rm m}_{\rm sec-pri}\) is a mass source term that represents the mass transfer rate from primary phase to secondary phase in the case of evaporation or mass transfer rate from secondary phase to primary phase in the case of condensation, respectively. The average velocity of the mixture is, \(\stackrel{{\rightarrow}}{{\
abla}}_{\rm m}=\frac{\sum\limits_{j =1}^{n}\dot{\rm m}_{\rm j}\dot{\rm m}_{\rm j}\dot{\rm v}_{\rm j}}{\rho_{\rm m}}\) and the mixture density is computed for all phase densities, \(\rho_{\rm m}=\sum\limits_{j=1}^{n}\alpha_{\rm j}\rho_{\rm j}\). All the mixture properties are correlated, likewise, with these mixture density calculations. The source term added in the energy equation and the internal energy is found as  
\[S_{\rm I}=\Big{(}\dot{\rm m}_{pri-\rm sec}-\dot{\rm m}_{\rm sec-pri}\Big{)}h_{ \rm fg} \tag{14}\]  
\[E_{\rm j}=h_{\rm j}-\frac{P}{\rho_{\rm j}}+\frac{\Big{(}u_{\rm j}^{2}+v_{\rm j }^{2}+w_{\rm j}^{2}\Big{)}}{2} \tag{15}\]  
In Equation (15), \(h_{\rm j}\) can be correlated as \(h_{\rm j}=h_{\rm ssej}+\int_{T_{\rm ref}}^{\rm T}c_{\rm pj}{\rm d}T\), where \(h_{\rm ssej}\) is defined as the phase standard state enthalpy and \(h_{\rm fg}\) is the vaporization latent heat, which is the difference between water vapor and water-sensible heat. The Evaporation-Condensation model is utilized in this work to describe the mass transfer rate between phases. This model relies on the relation of the phase temperature inside the cell to the saturation temperature. Based on this temperature comparison, a mass transfer occurs with releasing or gaining the latent heat in the case of condensation or evaporation, respectively [34].