## 2 Mathematical Modelling and CFD Modelling  
### Mathematical Modelling  
In this paper, the governing equations of the vapor-liquid two-phase flow were described using the Eulerian-Eulerian model, and the following assumptions were considered when establishing the model.  
1. The flow of fluid was considered to be compressible, adiabatic and continuous in the flow field.
2. The nucleation of the steam was spontaneous nucleation without the influence of external condensation nuclei.
3. The generated droplets were very small and incompressible spheres. Regardless of the slip flow between the vapor-liquid phases, the droplets were uniformly dispersed in the vapor phase.
4. There was no collision or polymerization between droplets or between droplets and the walls.  
Based on the above assumptions, the vapor-phase flow governing the equations of the wet steam model were as follows:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial(\rho u_{j})}{\partial x_{j}}=- \dot{m}, \tag{1}\]  
\[\frac{\partial}{\partial t}(\rho u_{i})+\frac{\partial}{\partial x_{j}}(\rho u _{i}u_{j})=-\frac{\partial p}{\partial x_{i}}+\frac{\partial\tau_{ij}}{\partial x _{j}}-u\dot{m}, \tag{2}\]  
\[\frac{\partial}{\partial t}(\rho E)+\frac{\partial}{\partial x_{j}}(u_{j}(\rho E +p))=-\frac{\partial}{\partial x_{j}}(\lambda_{eff}\frac{\partial T}{\partial x _{j}})+\frac{\partial}{\partial x_{j}}(u_{i}\tau_{ij})+h_{lt}\dot{m}, \tag{3}\]  
where \(\rho\) is the density, \(u\) is the velocity, \(p\) is the pressure, \(h_{lt}\) is the latent heat of vaporization and \(E\) is the total enthalpy.  
The liquid-phase condensation control equation included the liquid-phase mass fraction equation and the liquid drop number equation [25], and were as follows:  
\[\frac{\partial(\rho Y)}{\partial t}+\frac{\partial(\rho Yu_{j})}{\partial x_ {j}}=\dot{m}, \tag{4}\]  
\[\frac{\partial(\rho N)}{\partial t}+\frac{\partial(\rho Nu_{j})}{\partial x _{j}}=\rho J, \tag{5}\]  
where \(Y\) is the mass fraction of the liquid phase, \(N\) is the number of droplets, \(\dot{m}\) is the mass generation rate of the liquid phase and \(J\) is the nucleation rate.  
The nucleation rate adopted the classical nucleation theory modified by [PERSON] [26]:  
\[J=\frac{q_{c}}{1+\phi}(\frac{\rho_{v}{}^{2}}{\rho_{l}})\sqrt{\frac{2\sigma}{ M_{mole}\pi}}\exp^{-(\frac{4\pi\pi^{2}\sigma}{4\kappa_{B}T_{v}})}, \tag{6}\]  
where \(k_{B}\) is the Boltzmann constant, \(\sigma\) is the droplet surface tension and \(\phi\) is the correction coefficient. Additionally, the correction coefficient [27] was as follows:  
\[\phi=\left[1+2\frac{\gamma-1}{\gamma+1}\frac{h_{lv}}{RT_{V}}(\frac{h_{lv}}{RT _{V}}-\frac{1}{2})\right]^{-1}, \tag{7}\]  
In Equation (6), the droplet surface tension had an important effect on the nucleation rate. The droplet surface tension was calculated by adopting [PERSON]'s modified surface tension model [28].  
\[\sigma=\sigma_{P}(1-\frac{\sqrt[3]{\rho_{l}/m}}{4.836\pi}), \tag{8}\]\[\sigma_{p}=235.8(1-\frac{T_{d}}{647.3})^{1.256}\bigg{[}1-0.625(1-\frac{T_{d}}{647.3}) \bigg{]}, \tag{9}\]  
The mass generation rate of the liquid phase was as follows:  
\[\dot{m}=\frac{4\pi r^{*3}}{3}\rho_{l}I+4\pi\rho_{l}nr^{2}\frac{dr}{dt}, \tag{10}\]  
where \(r^{*}\) is the critical radius of the droplet, \(\rho_{l}\) is the density of the liquid phase, \(r\) is the droplet radius and \(\frac{dr}{dt}\) is the droplet growth rate.  
Based on the energy conservation equation proposed by [PERSON] [29], the droplet growth rate equation was obtained as follows:  
\[\frac{dr}{dt}=\frac{p}{h_{l\tau}\rho_{l}\sqrt{2\pi RT_{\tau}}}\frac{\gamma+1}{ 2\gamma}Cp(T_{d}-T_{\tau}) \tag{11}\]  
The physical properties of the steam, such as the enthalpy, entropy, viscosity and specific heat capacity at constant pressure, were calculated using empirical functions based on the second and third virial coefficients proposed by [PERSON] [30].