## 3 Results
### Preliminary Design
Filter limits are specified to automatically and rapidly remove unsatisfactory points; the imposed criteria is to discard any turbines that have \(\dot{W}<100\) MW and efficiencies \(<80\%\). 78 out of the 130 solutions do not meet the imposed criteria and are removed from the selection process. The remaining solutions are individually selected for further assessment of the general geometric features and velocity triangles in order to ensure that the restrictions on tip speeds and flow angles are applied. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Variable** & **Unit** & **Value** \\ \hline \multicolumn{3}{c}{**Boundary Conditions**} \\ \hline Inlet total pressure & bar & 297 \\ Inlet total temperature & K & 1173 \\ Outlet total pressure & bar & 60 \\ Mass flow rate & kg/s & 354.2 \\ Inlet rotor metal angle & \({}^{\circ}\) & 90 \\ Shaft rotational speed & rpm & 17,000–24,000 \\ \hline \multicolumn{3}{c}{**Geometric Parameters and Constraints**} \\ \hline Stator outlet mean diameter & m & 0.4–0.6 \\ Rotor blade height ratio & - & 2–3 \\ Rotor diameter ratio & - & 1.5–3 \\ Rotor inlet height & m & 0.01–0.04 \\ Hub reaction & - & 0.3–0.5 \\ Stator inlet diameter & m & 0.4–0.8 \\ Inlet flow angle & \({}^{\circ}\) & 1–179 \\ Rotor outlet hub diameter & m & 0.05–0.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Input data for preliminary turbine design. From the few selected designs, some are discarded because they either exhibit high circumferential rotor blade tip speed, or did not meet performance metrics, or the exit absolute flow angle was far from zero, or they had narrow passages which would incur high blockage. One particular design that meets the defined criteria, and which is characterised by a smooth rotor turning from the radial to the axial direction is presented in Figure 3, with the meridional dimensions given in Figure 4. Additional geometric parameters of the radial turbine are found in Table 6; the blade metal angle (\(\theta\)) is defined as the angle between the tangential reference plane and the blade camberline extension at leading edge for inlet, and trailing edge for outlet. In the case of no volute, the flow at vane inlet is assumed almost purely radial with straight radial blades at rotor inlet as well (\(\theta_{in}\approx 90^{\circ}\)). The shaft rotational speed is 21,409 rpm and the rotor tip clearance is 0.782 mm. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline \multicolumn{2}{c}{**Stator**} & \multicolumn{4}{c}{**Rotor**} \\ \hline
**Variable** & **Unit** & **Value** & **Data** & **Unit** & **Value** \\ \hline Blade count (N\({}_{S}\)) & - & 19 & N\({}_{R}\) & - & 16 \\ Chord (\(c_{S}\)) & mm & 99.90 & \(c_{R}\) & mm & 176.0 \\ Pitch (q\({}_{S}\)) & mm & 70.24 & q\({}_{R}\) & mm & 47.81 \\ Intel thickness (t\({}_{2}\)) & mm & 9.150 & t\({}_{4}\) & mm & 3.520 \\ Outlet thickness (t\({}_{3}\)) & mm & 1.830 & t\({}_{5}\) & mm & 3.520 \\ Throat (\(o_{S}\)) & mm & 16.82 & \(o_{R}\) & mm & 28.35 \\ Inlet angle (\(\theta_{2}\)) & \({}^{\circ}\) & 112.42 & \(\theta_{4}\) & \({}^{\circ}\) & 90 \\ Outlet angle (\(\theta_{3}\)) & \({}^{\circ}\) & 13.86 & \(\theta_{5_{u}}\) & \({}^{\circ}\) & 36.37 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Geometric parameters of turbine stator (\(s\)) and rotor (\(g\)) blades. Figure 3: 3D model of selected turbine: (**a**) Front view, (**b**) Perspective view. Figure 2: Preliminary design space: (**a**) Flow coefficient vs velocity ratio, (**b**) Power vs velocity ratio. The performance parameters obtained upon completion of 2D throughflow analysis for the turbine case are listed in Table 7. These results are obtained assuming inviscid flow.