# Construction of a Numerical Model for Flow Flash Evaporation with Non-Condensable Gas
## 2 Construction of the Numerical Model
### Model Construction and Governing Equations  
In this study, a multiphase, multicomponent, and pressure-driven Lee phase-change-coupled model was proposed [24]. The governing equations and calculation flowchart are shown in Figure 2. The left side of the chart shows the main governing equations, including the transient momentum conservation equations, mass conservation equation, energy conservation equation, phase fraction conservation equation (VOF model), and the non-condensing gas fraction equation. The right side shows the calculated equations for the phase change and released non-condensing CO\({}_{2}\) source term, which is the original model of this work. The phase-change ratio in the conventional Lee model is calculated using the temperature difference (\(T\)-\(T_{\text{sat}}\)), whereas in this study, \(T_{\text{sat}}\) is not a fixed value and is calculated using the local pressure, as shown in Equations (18) and (19). Two types of dissolved components are released (CO\({}_{\text{2(aq)}}\)): supersaturation release and interfacial diffusion release, corresponding to Equations (20) and (23), respectively.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Materials** & \(\rho\) & \(c_{\text{P}}\) & \(\lambda\) & \(\mu\) \\ \hline water & 983.16 & 4185.1 & 0.65100 & 0.00046602 \\ vapor & — & 1964.8 & 0.02104 & 0.00001085 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Physical properties of saturated water and saturated steam at 19.9 kPa.  
Figure 2: Flow chart of the numerical model calculation.  
2.3.1. Mass, Momentum, and Energy Equations  
Equations (1)-(3) show the conservation of mass, momentum, and energy of the two phases, respectively. The calculated pressure, velocity, and temperature are shared between the two phases, and the relative motion of the two phases can be described by variations in the local phase fraction [25; 26].  
\[\frac{\partial\rho}{\partial t}+\
abla\cdot\rho\mathbf{v}=0 \tag{1}\]  
\[\frac{\partial}{\partial t}(\rho\mathbf{v})+\
abla\cdot(\rho\mathbf{v}\mathbf{v})=-\
abla p +\
abla\cdot\mu(\
abla\mathbf{v})+\rho\mathbf{g}+\mathbf{F}_{\text{CSF}} \tag{2}\]  
\[\frac{\partial}{\partial t}(\rho E)+\
abla\cdot[(\rho E+p)\mathbf{v}]\ =\
abla\cdot( \lambda\
abla T)+\dot{S}_{\text{W}} \tag{3}\]  
where \(t\) represents time, \(p\) is the local pressure, \(\mathbf{v}\) is the local velocity vector, \(\rho\) is the density of the mixture, \(\mu\) is the dynamic viscosity, \(\lambda\) is the thermal conductivity, \(T\) is the local temperature, \(g\) is the gravitational acceleration vector, \(E\) is the internal energy, and \(\dot{S}_{\text{W}}\) is the source term of energy.  
The density of the vapor is calculated using the ideal gas equation of state, which is updated before each calculation step, as shown in Equation (4), where M\({}_{\text{W}}\) is the molar mass of water, taken as 0.018 kg mol\({}^{-1}\), and R is the ideal gas constant of 8.314 J/(mol\(\cdot\)K).  
\[\rho_{\text{v}}=p\text{M}_{\text{w}}/\text{R}T \tag{4}\]  
The \(\mathbf{F}_{\text{CSF}}\) is the source term of the volume force used to characterize the surface tension, as shown in Equations (5) and (6). Its representation was derived from a continuum surface force model [27]. The model assumes that the surface tension is constant along the surface and models only the normal force. The magnitude of the pressure drop across the surface is obtained from the magnitude of the curvature of the interface \(\kappa\) and the surface tension coefficient \(\sigma\), and the calculation is added in the form of the source term of the momentum equation by deriving the divergence theorem.  
\[\mathbf{F}_{\text{CSF}}=\sigma\frac{\rho\kappa\
abla\alpha}{0.5(\rho_{\text{1}}+ \rho_{\text{v}})} \tag{5}\]  
\[\kappa=\
abla\cdot\frac{\
abla\alpha}{|\
abla\alpha|} \tag{6}\]