## 5 Preliminary Design  
### Preliminary Design Performance Estimation  
The design space was sorted based on efficiency and designs with geometric properties not conducive to 'good' blades were filtered out. These included high rotor inlet angles at the hub and unrealistically high stagger angles at the tip. The convergence graphs for the chosen preliminary design point can be seen in Figure 9a,b. The solution converged after 7 iterations and resulted in loss coefficients of \(\xi_{stator}=0.1130\) and \(\xi_{rotor}=0.1807\). This produces a turbine stage with predicted efficiencies of \(\eta_{ls}=74.94\) and \(\eta_{tt}=84.20\%\). The estimated stage reaction at mean span is R = 67.24%. The meanline losses are accurate for the rotor but show a slight overestimate for stator, resulting in slightly lower thermodynamic efficiencies than the Ainley-Mathieson predictions.  
Trailing edge blockage (\(TE_{b}\)) values were chosen as 0.06 for the stator and 0.08 for the rotor. [PERSON] [18] suggests that values in excess of 0.1 incur significant performance penalties. As this is the preliminary design, the optimum performance is not expected, nor is an acceptable FEA result, but it is important to use realistic values to base the optimisation ranges on. For this reason, a slightly higher value was used for the rotor, as it is under increased loading conditions from both aerodynamic loading and rotational forces ([PERSON] et al. [15]).  
Although the blade loss coefficients are directly comparable, they do not themselves identify the enthalpy loss caused by each blade row as the coefficient is multiplied by fluid properties at the relevant station point, as seen in Equation (A71). Figure 9a splits the system enthalpy losses into comparable portions, showing that the rotor is responsible for significantly more enthalpy loss than the stator compared to what the ratio between \(\xi_{stator}\) and \(\xi_{rotor}\) would suggest (Figure 10). As the total to static efficiency is the performance metric for a single stage turbine, the unrecovered pressure from the exhaust flow is lost. Figure 9a also shows that the magnitude of this enthalpy loss is similar to that of the rotor. It is this loss component that causes the significant efficiency drop between total to total and total to static.  
Figure 9b illustrates that stator row losses are dominated by the secondary flow. This explains earlier findings in Figure 7, that showed how influential the stator chord was to system efficiency. The exit loss share is calculated by the magnitude of the trailing edge thickness correction calculated from Figure 5. Had a lower trailing edge blockage value been chosen the effect of this correction could become negative, as the multiplier would drop below unity. Figure 9c depicts the overwhelming importance of minimising tip clearance on the efficiency of the rotor. This high share is due to the increased correlation for \(Y_{k}\) predicted by the Dunham & Came correction for low aspect ratio systems.  
Figure 9: Convergence plots for Ainley-Mathieson and meanline thermodynamic (**a**) losses (**b**) efficiencies.  
To assess the shaft power produced by the turbine the equations in Equations (4) and (5) [17] are used. To double check the MATLAB algorithm an additional method for calculating [6] shaft power was used for cross comparison.  
\[\tau=\dot{m}r_{mean}[V_{a}(\tan{(\alpha_{2,mean})}+\tan{(\beta_{3,mean})})- \omega r_{mean}] \tag{4}\]  
\[P_{s}=\tau\omega \tag{5}\]  
\[P_{s}=\dot{m}C_{p}\Delta T_{t} \tag{6}\]