## 5 Results and Discussion  
A parametric study is presented in this section to investigate the effect of the flow coefficient (\(\phi\)), degree of reaction \((\Lambda)\) and loading coefficient (\(\psi\)) on the turbine performance \(\eta_{tt}\) and design feasibility. Different turbine designs are generated assuming different values of the design parameters \(\phi\), \(\psi\), \(\Lambda\), based on the specified ranges in Table 1. Accordingly, a Smith chart is obtained as shown in Figure 4, which shows the normalised efficiency achieved for the sCO\({}_{2}\) turbine with a degree of reaction and rotational speed of 0.5 and 150 kRPM respectively. It is observed that the highest normalised efficiencies are obtained at low flow and loading coefficients, as in the original Smith chart [25]. According to Figure 4, a design point can be selected as a starting point for the parametric study. Though one of the benefits of using sCO\({}_{2}\) is having compact component designs, clearance losses will be proportionally larger compared to turbomachines of a comparable power rating owing to the high density of sCO\({}_{2}\) and small turbine dimensions. Additionally, windage losses could be expected to be significant on the turbomachinery wheel surfaces compared to large-scale gas turbines as reported from the tests conducted by the Naval Nuclear Laboratory and the Tokyo institute of technology's (TIT) [39; 40]. Therefore, whilst windage losses are not accounted for in the current study, it is important to include these in the future. Likewise, losses due to surface roughness are not considered in this analysis owing to the simplicity of the implemented loss model. Thus, the mean-line model will be extended to include both roughness and windage effects in future studies. The effect of changing the rotor rotational speed on \(\eta_{tt}\) at different flow and loading coefficients has also been examined. The efficiency is found to increase linearly with increasing rotational speeds \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Performance Parameters** & **Verified**[41] & **Results** & **Difference [\%]** & **Design Angles** & **Verified**[41] & **Results** & **Difference [\%]** \\ \hline \(\eta_{tt}\) & 0.903 & 0.9058 & 0.31 & \(\beta_{3}\) & 42.3 & 42.310 & 0.024 \\ \(\eta_{ts}\) & 0.835 & 0.8391 & 0.48 & \(\beta_{2}\) & 63.9 & 63.960 & 0.094 \\ \(\xi_{N}\) & 0.108 & 0.1077 & 0.28 & \(a_{2}\) & 70.7 & 70.703 & 0.004 \\ \(\xi_{N}\) & 0.070 & 0.0699 & 0.14 & \(a_{3}\) & 5.70 & 5.777 & 1.340 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: 100 kW supercritical carbon dioxide (sCO\({}_{2}\)) turbine design model validation. Figure 4: Contour plot for normalised sCO\({}_{2}\) turbine efficiency at \(\Lambda=0.5\) and 150 kRPM. over the range from 150 to 250 kRPM for the loading and flow coefficient domains specified in Table 1. The lowest flow coefficient, over the range from 0.2 to 1.0, results in the highest efficiency level, as confirmed in Figure 4. However, a low flow coefficient results in larger turbine annulus area and a larger deflection angle. A loading coefficient of 1.6 results in the maximum efficiency at a fixed flow coefficient and degree of reaction of 0.2 and 0.5 respectively. At a loading coefficient of 1.6, the swirl angle is kept close to the recommended value, which is recommended not to exceed 20\({}^{\circ}\)[26], whilst the optimum difference in the whirl velocity components at the inlet and exit of the rotor is achieved, which enables a high efficiency to be achieve. Thus, for this design, the flow coefficient and loading coefficient are set to 0.2 and 1.6 respectively to achieve the maximum efficiency at a degree of reaction of 0.5. The corresponding turbine power output is 116 kW, with a total-to-total isentropic efficiency of 78%. In the following set of results, Figure 5a-d, \(\phi\) and \(\Lambda\) have been set to 0.2 and 0.5 respectively, whilst \(\psi\) is varied from 0.8 to 3.0.