## 5 Results and Discussion
At low reaction values, the stator outlet velocity is high as a result of the large acceleration and thus the Mach number is expected to be high. A higher degree of reaction results in a thin boundary layer and less tendency to secondary flow as result of having a good acceleration at the stator outlet [44]. A noticeable decrease in the efficiency has been experienced while increasing the degree of reaction from 0.0 to 0.5, and this is found for all three of the rotational speeds considered. Particularly, the normalised efficiency decreased from 0.97 to 0.91 at a rotational speed of 150 kRPM. At high degree of reaction values, the swirl angle is small which results in less rotor losses. However, high degree of reaction results in low rotor inlet blade angles (\(\alpha_{2}\)), and hence an overall reduction in the efficiency as observed in Figure 5(c). In view of the fact that high reaction leads to higher pressure and high density at the rotor inlet, the blade heights decrease as the degree of reaction is increased, as shown in Figure 5(d). Figure 5: Loading coefficient (\(\psi\)) versus (a) flow angles [\(\beta_{2}\) and \(\alpha_{2}\)] (b) Mach number at the rotor inlet [\(\text{Ma}_{2}\)] and exit [\(\text{Ma}_{3}\)] (c) normalised efficiency [\(\eta_{tt}\)] and swirl angle [\(\alpha_{3}\)] (d) blade heights [\(\text{b}_{2}\) and \(\text{b}_{3}\)] at various rotational speeds. A maximum value of 2.20 mm was found at zero reaction and 250 kRPM and a minimum of 0.74 mm at 150 kRPM and reaction of 0.5. To investigate the effect of the flow coefficient on the turbine performance, the analysis has been repeated at \(\Lambda=0.5\) and\(\psi\)= 1.60 respectively. Reducing \(\phi\) over the range from 0.2 to 1 results in an increase in rotor outlet flow angle \(\alpha_{2}\) from 43 to 77\({}^{\circ}\) as shown in Figure 6(a). However, increasing the flow coefficient also resulted in an increase in both Mach numbers, as shown in Figure 6(b), where Ma\({}_{2}\) increased from 0.88 to 1.30 and Ma\({}_{3}\) increased from 0.93 to 1.34 at 250 kRPM; thus, supersonic conditions occur in both the rotor and stator. To ensure subsonic flow at the rotor inlet, the flow coefficient should be kept below 0.45 for a loading coefficient of 1.6 and degree of reaction of 0.5. Figure 6: Degree of reaction versus (\(\Lambda\)) (**a**) flow angles [\(\beta_{2}\) and \(\alpha_{2}\)] (**b**) Mach number at the rotor inlet [Ma\({}_{2}\)] and exit [Ma\({}_{3}\)] (**c**) normalised efficiency [\(\eta_{tt}\)] and swirl angle [\(\alpha_{3}\)] (**d**) blade heights [b\({}_{2}\) and b\({}_{3}\)] at various rotational speeds. Furthermore, increasing the flow coefficient results in an efficiency decrease at all rotational speeds, as observed in Figure 7c. Additionally, it results in a decrease in swirl angle from 26 to 5\({}^{\circ}\). However, at high flow coefficients, low swirl angles and low rotor inlet blade angles (\(\alpha_{2}\)) are achieved. Increasing the flow coefficient results in higher Mach number at the rotor exit and hence higher losses incurred by the formation of shock waves in the rotor blade passages. Additionally, profile losses will be higher owing to the boundary layer growth, whilst friction losses at the exit are expected to be high. Consequently, a drop-in efficiency is observed at higher flow coefficients. The effect of changing the flow coefficient on the blade heights is shown in Figure 7d. Increasing the flow coefficient resulted in a decrease in both blade heights at all speeds. A minimum inlet blade height of 0.24 mm is found at a rotational speed of 150 kRPM and flow coefficient of 1.0. The effect of changing the aspect ratio on the turbine efficiency is shown in Figure 8a. Increasing the aspect ratio from 1 to 3 resulted in the normalised efficiency increasing from 0.987 to 1.000 at 250 kRPM. Higher aspect ratios result in lower rotor and stator losses and higher efficiency. For high aspect ratios, secondary effects are confined to the end-wall region. However,it affects the whole passage for small aspect ratios [44].