## 5 Conclusions  
In this research, an automatic design and simulation process of the radial-inflow turbine is established in MATLAB. The design reference of rotor solidity for both SCO\({}_{2}\) and air are provided. A new splitter structure to improve the aerodynamic performance of low solidity case is proposed. Concretely:  
1. For both SCO\({}_{2}\) and air, with the increase of rotor solidity, the working capacity for each blade declines. The cross section of hub has the highest power ability. Comparing five kinds of solidity, the pressure along spanwise tends to be more uniform as the solidity decreases. Small rotor solidity can result in the chaotic flow at the inlet of impeller, which further leads to the nonuniform pressure distribution. Large solidity disturbs the flow at the outlet and the blocking flow increases the trans-critical area in SCO\({}_{2}\) turbine. Comparing two working fluids, the air turbine has no trans-critical area and the low-pressure region at the suction side of the rotor blade trailing edge vanishes.
2. The aerodynamic loss of the SCO\({}_{2}\) and air turbine mainly concentrates on the hub and shroud area due to the boundary layer effect of the impeller. For SCO\({}_{2}\), when solidity is 7.96, the highest static entropy locates at the shroud due to the vacant flow passage. When solidity is 15.92, the highest static entropy locates at the hub on account of the limited flow area. For air, the increase of solidity less affects the aerodynamic loss and the entropy production mainly concentrates on the shroud.
3. For both SCO\({}_{2}\) and air, the solidity mainly affects the pressure distribution on the suction side and the pressure on the pressure side is of minor variation. In the SCO\({}_{2}\) turbine, large solidity corresponds to lower power capacity and greater trans-critical area of each blade. The high density of SCO\({}_{2}\) working fluid causes much higher axial thrust than air.
4. For both SCO\({}_{2}\) and air, the output power and isentropic efficiency of the turbine first increases and then decreases with the augment of solidity. In the SCO\({}_{2}\) turbine, the isentropic efficiency reaches the maximum 90.54% and the maximum output power is 341.87 kW when solidity is 11.94. While in the air turbine, the maximum isentropic efficiency is 85.02% and the maximum output power is 10.41 kW when solidity is 13.93. 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. Arranging the ss offset splitter can improve the flow condition and the aerodynamic performance of the low solidity case.  
Conceptualization: [PERSON] and D.Z.; Methodology: D.Z.; Software: Y.X.; Validation: [PERSON] and J.L.; Formal analysis: Y.W.; Investigation: Y.W. and J.L.; Resources: Y.X.; Data curation: D.Z.; Writing--original draft preparation: Y.W.; Writing--review and editing: J.L.; Visualization: J.L.; Supervision: [PERSON]; Project administration: [PERSON]; funding acquisition: Y.X. All authors have read and agreed to the published version of the manuscript.  
This research was funded by 111 project, grant number B16038.  
The authors declare no conflict of interest.