## 2 Mathematical Model
### Wet Steam Modeling  
The wet steam flow was modeled by the Eulerian-Eulerian approach. Compressible Navier-Stokes equations were adopted for modeling of the condensation process of the two-phase flow. In addition, two more transport equations accounting for the liquid-phase mass-fraction (\(\beta\)) and the number of liquid-droplets per unit volume (\(\eta\)) were used to capture and describe the distribution of water droplets within the field [38]. The two-equation model is a viable approach which governs the proportion and particle numbers by resolving two conservative transporting properties, and it has been proven to be effective in applications such as turbulence and smoke particles modeling [36; 39; 40]. Based on the classical non-isothermal nucleation theory, the formation of liquid-droplets in a homogeneous nonequilibrium condensation process is described in the phase change model. As for the non-equilibrium state in supersonic flows, the nucleation and condensation of condensable gases are overwhelmingly complicated [41; 42]. Based on the single-fluid Eulerian model [41], several assumptions were implemented to ensure the simulation was numerically stable  
Figure 3: (**a**) Enthalpy-entropy diagram of the working process in the nozzle of the steam ejector. (**b**) Main view of the ejector refrigeration system (**c**) Steam ejector. (**d**) Three-dimensional isometric side view of the system. and efficient. Since the mass fraction of the condensed phase is quite small (<0.2), the effect of droplets size to the main flow stream is negligible and the interactions between droplets are not considered. Moreover, since the droplets sizes are sufficiently small, the volume of the condensed phase and thermal capacity are negligible [27]. A non-slip condition was also assumed, which means that the velocity difference between the droplets and gaseous-phase is negligible. Within the phase change model, the droplet was assumed to be spherical and surrounded by infinite vapor space due to the small liquid volume fraction. The droplet growth was based on average representative mean radii. With the aforementioned assumptions, the Euler Equations [43] can be written as:  
\[\frac{\partial U}{\partial t}+\frac{\partial F}{\partial x}+\frac{\partial G}{ \partial y}=0 \tag{2}\]  
where \(x\) and \(y\) represent the special coordinates while \(t\) depicts the physical time. The unsteady term \(U\), flux matrices \(F\) and \(G\) respectively in \(x\)-, \(y\)-directions can be expressed as:  
\[U=\left[ \begin{array}{c}\rho\\ \rho u\\ \rho v\\ \rho E\end{array} \right]_{\Gamma}F=\left[ \begin{array}{c}\rho u\\ \rho u^{2}+P\\ \rho uv\\ (\rho E+P)u\end{array} \right],G=\left[ \begin{array}{c}\rho v\\ \rho uv\\ pv^{2}+P\\ (\rho E+P)v\end{array} \right] \tag{3}\]  
The mixture density (\(\rho\)) is given by:  
\[\rho=\frac{\rho_{v}}{(1-\beta)} \tag{4}\]  
where (\(\rho_{v}\)) is the vapor density and (\(\beta\)) is the liquid-phase mass fraction. And in Equation (3), the internal energy (\(E\)) and the specific enthalpy (\(h\)) are given by:  
\[E=h_{0}-\frac{P}{\rho} \tag{5}\]  
\[h_{0}=h+\frac{\left(u^{2}+v^{2}\right)}{2} \tag{6}\]  
\[h=(1-\beta)h_{v}+h_{l} \tag{7}\]  
In order to describe the distribution of nucleating droplets within the computational field, two transport properties were introduced and solved. The first being the mass fraction of fluid condensed liquid phase (\(\beta\)), where the transport equation can be expressed as:  
\[\frac{\partial\rho\beta}{\partial t}+\
abla\cdot\left(\rho\overrightarrow{v }\beta\right)=\Gamma \tag{8}\]  
The second additional transport equation governs the number density of the droplets per unit volume, which is given by:  
\[\frac{\partial\rho\eta}{\partial t}+\
abla\cdot\left(\rho\overrightarrow{v }\eta\right)=\rho I \tag{9}\]  
In Equation (8), \(\Gamma\) is the mass generation rate, related to the nucleation rate (\(I\)).