## 3 Numerical Modeling  
### The Dimensions of the Nozzle and Numerical Solution Settings  
In this study, the nozzle B from the works of [PERSON] et al. [11] was selected as the original structure (see Table 1) that was used in order to investigate the effects of the throat radius, expansion angle of the divergent section, and length of the divergent section on the steam's spontaneous condensation. The structures of the nozzles with different throat radii, expansion angles, and divergent section lengths are shown in Figure 2; all the geometric parameters are summarized in Table 1.  
In terms of the numerical solution settings, the commercial software Ansys Fluent 15.0, with an implicit density-based solver, was used to solve the flow-governing equations \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Coordinate Value (m)** & **Point A** & **Point B** & **Point C** & **Point D** \\ \hline Throat & x & \(-\)0.25 & \(-\)0.2 & 0 & 0.5 \\ \hline Radius & y & 0.0563 & 0.0563 & 0.025/0.03/0.035/ & 0.072 \\ \hline  & & & & 0.04/0.045/0.05 & \\ \hline Expansion & x & \(-\)0.25 & \(-\)0.2 & 0 & 0.5 \\ \hline Angle & y & 0.0563 & 0.0563 & 0.05 & 0.056/0.063/0.072 \\ \hline Divergent & x & \(-\)0.25 & \(-\)0.2 & 0 & 0.2\(-\)0.5 \\ \hline Section Length & y & 0.0563 & 0.0563 & 0.5 & 0.059/0.063/0.068 \\ \hline  & & & & 0.072/0.076/0.081 \\ \hline Nozzle B & x & \(-\)0.25 & \(-\)0.2 & 0 & 0.5 \\ \hline Structure & y & 0.0563 & 0.0563 & 0.05 & 0.072 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometric structure of the nozzle B.  
Figure 2: Geometric structure of the nozzle B with: (**a**) different throat radii, (**b**) different expansion angles of the divergent section, and (**c**) different lengths of the divergent section. and the wet steam model was selected. The conservation Equations (1)-(5) for the vapor phase were directly solved in the Fluent solver, while the governing Equations (6)-(22) for the liquid phase and the phase change were performed when the wet steam model was called in the solver. The Equations (23)-(30) were implanted into the Fluent solver by UDS and UDF functions and then the entropy generation rate and entropy production rate (the volume integrals for the entropy generation rate) could be solved in every cell of the computational domain and displayed in the post-processing stage. The k-\(\omega\) SST turbulence model was used. This model did not need the wall function to be set based on its special algorithm [32]. To save computing costs and time, a 2D axisymmetric geometry was adopted instead of a 3D geometry. It has been proven that there is no obvious difference between the two calculation results [40]. The nozzle inlet and outlet were set to the pressure inlet and outlet, respectively. The wall was set to adiabatic and no-slip. All the initial boundary conditions, boundary condition settings, and properties of the working fluid are listed in Tables 2-4, respectively. The second-order upwind scheme was selected for all of the convection terms and central difference discretization was selected for the diffusion terms. The mass flow rate difference between the inlet and outlet and all of the calculated residuals needed to be less than \(10-6\) in order to ensure iteration convergence. used, as illustrated in Figure 5. It was found that the CFD results are in close agreement with the present experiment's data. In particular, the wet steam model can accurately predict the onset position and the peak of the condensation shock. Moreover, the average relative error between the simulation result and the experiment value was also calculated and is shown in Figure 5. The average relative error can be obtained by Equation (31):  
\[\text{Average relative error}\left(\%\right)\ =\frac{1}{n}\sum_{n=1}^{n}\left(\left(Expvalue- CFDresult \right)/Expvalue\right) \tag{31}\]  
Among these three nozzles, the biggest average relative error (2.55) and maximum relative error (4.43) were both generated in nozzle C (see Figure 5c).