## 3 Results
### Computational Fluid Dynamics Analyses
#### 3.3.3 Off-Design Turbine Analysis  
The turbine performance maps are generated using 2D CFD simulations rather than the full 3D ones, on account of the computational resources available as well as the long time required for the 3D simulations. The conditions extend over the range of total-to-static pressure ratio between 2 and 7 at three rotational speeds of 19,268 rpm, 21,409 rpm and 23,550 rpm. The variation of mass flow rate and stator outlet Mach number with the off-design conditions is shown in Figure 10. In general, the mass flow is expected to increase with higher pressure ratios but have lower values at higher rotational speeds, until choking occurs whereby the mass flow rate reaches its maximum asymptote. The radial turbine becomes choked at around \(\Pi_{ts}=3\), thus for the operating range of the design, the turbine will almost always experience supersonic flows. Both total-to-total and total-to-static efficiency maps are plotted in Figure 11. The difference between the efficiency values is solely attributed to the loss in the available kinetic energy at rotor outlet, i.e., related to the exit swirl (\(\alpha_{5}\)), because the axial velocity is not a function of rotational speed (but is related to pressure ratio). The trend for both efficiencies are broadly similar except for the value difference due to the mentioned loss. The highest efficiency value occurs with a pressure ratio of about  
Figure 10: Variation of parameters with total-to-static pressure ratio at three rotational speeds: (**a**) Mass flow, (**b**) Stator outlet Mach number. Figure 9: Relative total pressure contours at rotor exit stream section. 3 at 23,550 rpm and the lowest with \(\Pi_{ts}\) = 7 at 29,268 rpm. The maximum Mach number at stator outlet is lower for higher \(\omega\) (as deduced from Figure 10) meaning that the entropy rise in the stator exit channel is also lower, hence the higher efficiencies. Table 10 shows the change in relative flow angle at rotor inlet and absolute flow angle at rotor outlet at the off-design conditions. The flow angle at rotor inlet (\(\beta_{4}\)) plays a role in determining incidence losses. The recommended values of rotor incidence angles for a radial turbine are in the range of \(-40^{\circ}\)--\(20^{\circ}\)[36]; incidence angles around the suggested values are only obtained in two cases (\(-37.01^{\circ}\) & \(-44.19^{\circ}\)), all which correspond to pressure ratios below choking conditions and those cases correspond to relatively high efficiencies. Ideally, the closer the absolute flow angle at rotor outlet is to zero, the lower the kinetic energy loss will be; therefore the lower the difference between \(\eta_{tt}\) and \(\eta_{ts}\) is. Values of \(\alpha_{5}\approx 0^{\circ}\) are attained at higher rpms because of the lower Mach number at stator outlet/rotor inlet which in turn leads to a lower relative Mach at rotor outlet and thus can have lower absolute flow angles for similar pressure ratios at different speeds. The total-to-static pressure ratio and rotational speed of the most and least efficient points are used in 3D CFD to visualise the flow behaviour for each case. Figure 12 shows the relative Mach number distribution for both cases at mid-span. One can note the absence of the shockwave and supersonic flows at the stator outlet channel for the conditions with high rpm which leads to much improved efficiency values (\(\eta_{tt}\) = 82.40% \(\eta_{ts}\) = 72.39%). The right-hand-side figure (**b**) also shows that the inefficient turbine (\(\eta_{tt}\) = 65.19% \(\eta_{ts}\) = 57.22%) exhibits loss features that include the presence of a vortex, incidence losses (\(\beta_{4}\) = 35.44\({}^{\circ}\)) and suction side separation or flow reversal as explained in Section 3.3.2. Whereas, the left-hand-side figure (**a**) shows smooth flow acceleration and deceleration as would be ideal in a radial turbine case with proper rotor incidence of \(\beta_{4}\) = \(-12.93^{\circ}\). The aim would be to design a turbine that coincides with such flow behaviour, however, operating conditions