## 2 Methodology
### Governing Equations  
The volume of fluid (VOF) method is employed in this work, which has been proven to be suitable for simulating the local thermodynamic and fluid dynamic characteristics of boiling flows [21; 22; 23]. The phase fraction, \(\alpha\), of the liquid phase is tracked in the computational domain to determine the position of the phase interface to capture the motion of the two phases. This study uses the VOF model for numerical calculations, aiming to track the \(\alpha\) of the liquid phase in the computational domain to determine the position of the phase interface to capture the motion of the two phases. In the VOF model, the sum of the \(\alpha\) of the two phases in any control unit is 1. The physical properties of the mixed phase are calculated by weighing the physical properties of the two phases as follows:  
\[y=\alpha_{i}y_{i}+(1-\alpha_{i})y_{v}\;y\in\left[\rho,\mu,K,C_{p}\right] \tag{1}\]  
where \(\alpha\) represents the phase fraction. \(\rho\), \(\mu\), \(K\), and \(C_{p}\) represent density (kg/m\({}^{3}\)), viscosity (kg/(m\(\cdot\)s)), thermal conductivity (W/(m\(\cdot\)K)), and specific heat capacity (J/(kg\(\cdot\)K)), respectively.  
The momentum equation and energy equation are as follows:  
\[\frac{\partial}{\partial t}(\rho\mathbf{U})+\
abla\cdot(\rho\mathbf{U}\mathbf{ U})=-\
abla P+\
abla\cdot\left[\mu_{eff}(\
abla\mathbf{U}+\
abla\mathbf{U}^{T}) \right]+\rho g+F_{vol} \tag{2}\]  
where \(\mathbf{U}\) and \(g\) are velocity (m/s) and gravitational acceleration (m/s\({}^{2}\)), respectively; \(P\), \(F_{vol}\), and \(\mu_{eff}\) are pressure (Pa), surface tension per unit volume (N/m\({}^{3}\)), and effective viscosity (kg/(m\(\cdot\)s)), respectively.  
\[\frac{\partial}{\partial t}(\rho E)+\
abla\cdot\left[\mathbf{U}(\rho E+P) \right]=\
abla\cdot\left(K_{eff}\
abla T\right)+S_{E} \tag{3}\]  
where \(E\), \(S_{E}\), and \(K_{eff}\) are the energy density, the energy transfer rate per unit volume between the two phases, and the effective thermal conductivity, respectively.  
Since evaporation and condensation cause the conversion between the gas phase and the liquid phase, considering the continuity equation of each phase, the phase fraction equations of the liquid phase and the gas phase are expressed as follows:  
\[\frac{\partial}{\partial t}(\rho_{l}\alpha_{l})+\
abla\cdot(\rho_{l}\alpha_{l }\mathbf{U}_{l})=S \tag{4}\]  
\[\frac{\partial}{\partial t}(\rho_{v}\alpha_{v})+\
abla\cdot(\rho_{v}\alpha_{v }\mathbf{U}_{v})=S \tag{5}\]  
where \(S\) represents the mass transfer source term (kg/(m\({}^{3}\cdot\)s)) due to phase change. The above two-phase equations can be further simplified into one equation as follows:  
\[\frac{\partial}{\partial t}(\rho_{l}\alpha_{i})+\
abla\cdot(\rho_{l}\alpha_{i }\mathbf{U}_{i})=\pm S \tag{6}\]  
Furthermore, the continuity equation can be obtained as presented below:  
\[\
abla\mathbf{U}=S(1/p_{l}-1/p_{v}) \tag{7}\]  
The interphase mass transfer model is the key to accurately predicting mass transfer and heat transfer in boiling heat transfer. The mass transfer model proposed by [PERSON] assumes that both the saturated phase and the phase interface are kept at a saturated temperature, which can simulate the heat and mass transfer process between the two phases well. This study uses the [PERSON] model [24] for simulation. It is worth noting that in the [PERSON] model, the evaporation condensation coefficient (_coeff_) needs to be determined in combination with experimental results to better conform to the actual situation. The _coeff_ can be obtained as follows:  
\[coeff=\frac{6}{d_{b}}\beta\sqrt{\frac{M}{2\pi RT_{sat}}}L\left(\frac{\alpha_{v} \rho_{v}}{\rho_{l}-\rho_{v}}\right) \tag{8}\]This work uses the finite volume method (FVM) in commercial software ANSYS Fluent 2021R1 to conduct the simulation. The meshes used are finite volume meshes, and they are unstructured tetra meshes generated by commercial software ICEM CFD 2021R1.