## 3 Methodology
### Validation of Fluid Property  
For the working fluid air, the fixed composition mixture is set up in CFX 18.2 with the state equation of Peng Robinson. This option is commonly used in CFX to model the real gas properties for gas density, enthalpy, etc. In the simulation of a SCO\({}_{2}\) turbine, it is of great importance to precisely obtain the fluid property of SCO\({}_{2}\). For the simulated condition, the state of working fluid lies in the supercritical region. To simulate the fluid property precisely, we use the real gas property (RGP) format table to implement the sharply variable density and specific heat capacity near the critical point. The tabulated pressure and temperature region are, respectively, 7 MPa to 16 MPa and 300 K to 600 K in order to simulate the fluctuated state parameter during the converging process. The parameters for the RGP table are obtained from NIST REFPROP, which is widely referred to as a fluid property database [33]. MATLAB is used to generate the RGP format file with a predefined  
Figure 6: Validation of different turbulence models. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Mesh**} & **Total Element** & **Mass Flow** & **GCI/\%** & **Output** & **GCI/\%** \\  & **Number/10\({}^{4}\)** & **Rate/kg.s\({}^{-1}\)** & **Power/kW** & **GCI/\%** \\ \hline Coarse(\(f_{3}\)) & 113.2 & 6.581 & \multirow{2}{*}{\begin{tabular}{c} GCI\({}_{23}\) = 1.02 \\ GCI\({}_{12}\) = 0.64 \\ \end{tabular} } & 343.03 & \multirow{2}{*}{\begin{tabular}{c} GCI\({}_{23}\) = 0.16 \\ GCI\({}_{12}\) = 0.05 \\ \end{tabular} } \\ Medium(\(f_{2}\)) & 193.6 & 6.549 & \multirow{2}{*}{\
\begin{tabular}{c} GCI\({}_{23}\) = 1.02 \\ GCI\({}_{12}\) = 0.64 \\ \end{tabular} } & 342.16 & \multirow{2}{*}{
\begin{tabular}{c} GCI\({}_{23}\) = 0.16 \\ GCI\({}_{12}\) = 0.05 \\ \end{tabular} } \\ Fine(\(f_{1}\)) & 335.4 & 6.529 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Result of grid convergence index. temperature and pressure range. Figure 7 gives the density variation with the temperature and pressure range for the SCO\({}_{2}\) turbine. As shown in Figure 7, the density of the working fluid changes sharply around the critical point. The CFX solver calculates the SCO\({}_{2}\) property by bilinear interpolation between parameter points. Hence, we choose to densely generate the RGP table near the critical point for the balance of accuracy and computation time. The region is shown in the dashed box in Figure 7. For the temperature range, we adopt a gradient of 2 K for 300-350 K while for 350-600 K, we use a gradient of 5 K. For the pressure range, we utilize 0.1 MPa as a gradient between 7-8 MPa while 0.2 MPa as a gradient is employed between 8-16 MPa. Compared with 2 K for all temperature range and 0.1 MPa for all pressure range, the computation time is effectively reduced by 20.4%. The accuracy of RGP format table using in ANSYS CFX has been validated by [PERSON] et al. [34] and [PERSON] et al. [35]. We conduct a validation in the selected range of this research. Three points are chosen to validate the precision of RGP file, which are, respectively, the inlet, outlet and the interface between stator and rotor. We consider four parameters to test the relative error between CFX calculation and data from NIST, which are density, enthalpy, \(C_{\text{P}}\) and dynamic viscosity. As shown in Table 4, the maximum relative error is 0.009%. Hence, the RGP table is sufficiently accurate for the simulation of SCO\({}_{2}\) working fluid.