## 3 Results
### Computational Fluid Dynamics Analyses
#### 3.3.2 Design Point Turbine Analysis  
Performance data obtained from the numerical 3D turbulent simulations is listed in Table 9 which can be compared to the 2D analysis results found in Table 7. Global variables of mass flow, power, and non-dimensional parameters all show good agreement; the small difference in thermodynamic properties between the analyses outcomes (enthalpies and temperatures) lead to a greater difference in power output. The total pressure at turbine outlet is 62.9 bar with a total temperature of 975 K which are both slightly higher than the values initially anticipated. The most pronounced discrepancy can be observed in the values of efficiencies that can reach 20%. This change is not fully uncommon with comparisons of 15% efficiency difference between meanline design and CFD results noted by [PERSON] et al. (2016) in their design of a R134A radial turbine, and almost 24% efficiency difference reported by [PERSON] (2016) between 1D design and 3D CFD modelling for a supercritical CO\({}_{2}\) radial turbine. The source of this variance can be attributed to a number of factors, the turbine 3D blade geometries not being fully optimised, and the effects of three-dimensional viscous losses and tip clearance not accounted for in earlier analysis stages. The current design corresponds to an intermediate development stage and requires further adjustment based on the following observations made hereafter. \begin{table}
\begin{tabular}{l l l l l} \hline \hline \multicolumn{4}{c}{**Boundary Condition:**\(\dot{\mathbf{u}}_{in}\) **\& P\({}_{out}\)} \\ \hline
**Number of Elements [-]** & P\({}_{0_{w}}\) **[bar]** & \(\dot{W}\) **[MW]** & \(\mathbf{\eta}_{tt}\) **[\%]** & \(\mathbf{\eta}_{ts}\) **[\%]** \\ \hline
51,110 & 224 & 83.97 & 72.66 & 67.50 \\
209,308 & 281 & 93.16 & 73.48 & 67.98 \\
971,613 & 282 & 102.2 & 69.77 & 63.17 \\ \hline \multicolumn{4}{c}{**Boundary Condition: P\({}_{0_{w}}\) **\& P\({}_{out}\)**} \\ \hline
**Number of Elements [-]** & _in_ **[kg/s]** & \(\dot{W}\) **[MW]** & \(\mathbf{\eta}_{tt}\) **[\%]** & \(\mathbf{\eta}_{ts}\) **[\%]** \\ \hline
51,110 & 377.3 & 81.48 & 76.28 & 70.42 \\
209,308 & 375.2 & 101.3 & 73.01 & 67.21 \\
971,613 & 372.5 & 108.5 & 69.87 & 63.05 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Performance results for 3D CFD mesh sensitivity study using two different boundary condition definitions. Figure 5 shows relative Mach number (M\({}_{rel}\)) distribution (absolute for stator) at three locations across the span. At the throat of the stator, the flow reaches its supersonic conditions with the onset of the trailing edge shock; high losses are associated with shockwaves which are mitigated by changing airfoil shape to lower Mach number values. Low Mach number values near the rotor pressure side (PS) leading edge is caused by incorrect incidence and the propagating suction side acceleration interacting with the presence of the rotor tip clearance; the area of incidence loss near the pressure side lessens from 10% to 90% span with the mean relative flow angle at rotor inlet \(\beta_{4}=28.22^{\circ}\) which is lower than the calculated flow angle from throughflow analysis (\(\beta_{4}=45.50^{\circ}\)). The low M\({}_{rel}\) values on the suction side indicate a region of flow separation that begins at SS leading edge and grows along the flow as end-span is approached (also reported in [PERSON] et al. 's work [24]). \begin{table}
\begin{tabular}{l l l} \hline \hline
**Variable** & **Unit** & **Value** \\ \hline \(\dot{m}\) & kg/s & 372.5 \\ \(\dot{W}\) & MW & 108.5 \\ \(\eta_{tt}\) & \% & 69.87 \\ \(\eta_{ts}\) & \% & 63.05 \\ \(\psi\) & - & 1.34 \\ \(\phi\) & - & 0.511 \\ R & - & 0.355 \\ \(\Gamma_{ts}\) & - & 5.51 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: 3D CFD performance results. Figure 5: Relative Mach number distribution at three spanwise turbine passage locations: (**a**) 10% span, (**b**) 50% span, (**c**) 90% span. Figures 6 and 7 can be viewed in conjunction to further investigate loss zones in the radial turbine rotor passage.