## 5 Results and Discussion
The gas bending and centrifugal tensile stresses have been  
Figure 7: Flow coefficient (\(\phi\)) versus (**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. evaluated for the selected material at a design condition of \(\phi\) = 0.2, \(\psi\) = 1.6, \(\Lambda\)= 0.5, 150 kRPM and aspect ratio of 1.At this point, the gas bending and centrifugal tensile stresses are predicted to be 122 and 35 MPa respectively; resulting into a total stress of 157 MPa, which falls within the material allowable limit (\(\sigma_{max}\) = 303 MPa). A creep failure criterion, which shows the continuous application of a steady stress over a period of time at various temperatures required to produce 0.2 percent strain, is used to assess blade life. Based on the results from [26], it is anticipated that a blade life of 10,000 hr can be achieved under these operating conditions. It is worth emphasising that creep failure is an important consideration, but a detailed analysis is outside the scope of this study. A noticeable difference between the magnitude of the gas bending (GB) stress in comparison to the centrifugal tensile (CT) stress, for tapered blades, is experienced owing to the density effect of sCO\({}_{2}\). Increasing the aspect ratio from 1 to 3 results in an increase in the gas bending stress by a factor of approximately 9. Furthermore, decreasing the degree of reaction from 0.5 to zero results in a decrease in the the gas bending stress by a factor of approximately 4. Hence, the aspect ratio and the degree of reaction should be minimised. However, decreasing the aspect ratio is associated with a drop in efficiency. Therefore, selecting the optimal aspect ratio is a trade-off between high efficiency and low stresses. Increasing the rotational speed from 150 to 250 kRPM results in centrifugal tensile stress increasing from 35 to 95 MPa as shown in Figure 8b. The last part of the parametric study aims to show how the turbine design parameters can be varied to obtain a feasible turbine geometry, defined by a minimum allowable rotor diameter and inlet blade height of 30.00 mm and 1.25 mm respectively, over the range of the rotational speeds from 150 to 250 kRPM. The feasibility criteria can be achieved by operating at a high loading coefficient, where the diameter and inlet blade height are largest. Increasing the rotational speed results in a decrease in the diameter and an increase in the blade height. Hence, this should be considered during the selection of the rotational speed and flow coefficient of the turbine as shown in Figure 9a,b. The results reported in Figure 9a refer to 0.5 degree of reaction, while those in Figure 9b refer to zero degree of reaction. Thus, it can be concluded that running at a zero reaction (impulse) results in a more feasible range of inlet blade heights and diameters. The maximum diameter that can be achieved is 56 mm with a maximum inlet blade height of 3.3 mm at a flow coefficient of 0.2. Figure 8: Aspect ratio of the rotor blades versus (**a**) normalised efficiency \(|\eta_{tt}|\) (**b**) centrifugal tensile [CT] and gas bending [GB] stresses at various rotational speeds. A comparison between two turbine designs, to depict the effect of the selected design variables on the final turbine geometry, is presented in Figure 10. Geometry (A) is obtained at \(\Lambda=0.5\), \(\psi=1.6\) and \(\phi=0.2\), while geometry (B) is obtained at \(\Lambda=0.0\), \(\psi=3.0\) and \(\phi=0.2\) at 150 kRPM. Reducing the degree of reaction and increasing the loading coefficient results in more feasible design dimensions (Design B).