## 4 Results and Discussion
### Aerodynamic Performance  
Output power, isentropic efficiency and mass flow rate are vital parameters in the assessment of the turbine aerodynamic performance. The isentropic efficiency discussed in this research is defined as Equation (8):  
\[\eta_{\rm{ls}}=\frac{2T_{Z}\times 2\pi\times r}{60\times\left(m_{in}+m_{out} \right)\Delta h_{\rm{ls}}} \tag{8}\]  
where \(T\) stands for torque, \(r\) represents the rotation speed, \(h\) is enthalpy and \(\dot{m}\) is mass flow rate. As for the subscripts, is stands for isentropic, in and out represent the inlet and outlet of the turbine respectively, \(z\) corresponds to axial direction. The mass flow rate of inlet and outlet are slightly different due to the random error in the numerical simulation. To be precise, we adopt the average to calculate the efficiency. High solidity usually results in large weight and axial thrust of the wheel. The working fluid \(\rm{SCO}_{2}\) has more special application scenarios (such as aerospace and warships) than air due to its small size. In these cases, the weight of the wheel is limited. The problems of overlarge axial thrust and trans-critical phenomenon need to be solved. Hence, we concentrate on the improvement of \(\rm{SCO}_{2}\) turbine. Three new splitters structures with and without offsetting are introduced. All solidity structures are calculated and compared in this section to test the effectiveness in the \(\rm{SCO}_{2}\) radial-inflow turbine. Figure 17 gives the variation of output power and isentropic efficiency with different solidity. For both \(\rm{SCO}_{2}\) and air, the output power of the turbine first increases and then decreases with the augment of solidity. For \(\rm{SCO}_{2}\), the maximum output power is 341.87 kW and it is realized when solidity is 11.94. While in Figure 17b, the maximum output power is 10.41 kW and it happens when solidity is 13.93 for the air turbine. Comparing (a) and (b), the radial-inflow turbine working with air has apparently lower power capacity, which results from a lower power density. With the augment of solidity, the output power descends more rapidly after the maximum in \(\rm{SCO}_{2}\) turbine than that in air turbine. The decline of output power is owing to the blocking effect of turbine outlet when solidity reaches a certain extent. Obviously, it has a greater influence in a \(\rm{SCO}_{2}\) turbine due to the fluid property similar to liquid. As for the isentropic efficiency, the tendency is similar for \(\rm{SCO}_{2}\) and air. In general, the isentropic efficiency first increases and then decreases with the augment of solidity and the efficiency of \(\rm{SCO}_{2}\) turbine is higher in this power scale. In a \(\rm{SCO}_{2}\) turbine, when solidity is 11.94, the isentropic efficiency reaches the maximum 90.54%. When solidity is 13.93, the efficiency is 90.52%. Although the highest solidity corresponds to the lowest output power, its efficiency is acceptable due to the lowest mass flow rate. The solidity 7.96 has the minimum efficiency owing to the unsatisfactory flow condition at the inlet of impeller, which is caused by the large pitch at leading edge. In the air turbine, the highest efficiency is 85.02% when solidity is 13.93. Likewise, the isentropic efficiency is the lowest 82.46% when  
Figure 17: Output power and isentropic efficiency of different solidity and splitters: (a) \(\rm{SCO}_{2}\); (b) air. solidity is 7.96. The efficiency declines slower than that in SCO\({}_{2}\) turbine after the solidity reaches the maximum. The blocking effect is smaller and the large pitch at the leading edge affects the efficiency in a large scale. The ss offset splitter can increase the output power and efficiency to the greatest extent. The power is increased to 347.22 kW, and the efficiency is increased to 90.18%. The normal splitter can improve the performance to a small extent while the ps offset splitter is ineffective. Overall, in the preliminary design of a SCO\({}_{2}\) radial-inflow turbine, the best solidity is around 12 and for an air turbine, it is around 14.