## 1 Introduction  
In addition to the excellent properties of supercritical fluid, supercritical carbon dioxide (S-CO\({}_{2}\)) is characterized by a lower critical temperature of 31.1 \({}^{\circ}\)C and a critical pressure of 7.38 MPa [1], which is an accessible critical condition. Meanwhile, supercritical carbon dioxide is colorless, odorless, non-toxic, and chemically inert. It is an environmentally friendly natural working fluid with characters of non-flammable, non-explosive, safe, cheap, low values of ozone depletion potential and climate warming potential [2, 3]. The Brayton cycle power generation system using this working fluid has the advantages of compact structure, high cycle efficiency and low cost [4, 5], which makes supercritical carbon dioxide receive extensive attention in the field of solar power generation in recent years. For the S-CO\({}_{2}\) Brayton cycle, [PERSON] [6] proposed the use of supercritical cycles to substitute the Rankine cycle and the regenerative Brayton cycle as early as 1968. He pointed out that the supercritical cycle working fluid can be any fluid, especially indicating that S-CO\({}_{2}\) has many advantages as a working fluid. Some scholars [7; 8] compared the S-CO\({}_{2}\) cycle under different boundary conditions with the He Brayton cycle and the water vapor Rankine cycle system, and found that the S-CO\({}_{2}\) cycle system has certain advantages in terms of economy and efficiency. In the use of solar energy, [PERSON], [PERSON] et al. [9] studied the concentrated solar S-CO\({}_{2}\) cycle under different season conditions. They optimized the parameters of the system and found the optimization efficiency was 21%, and the maximum output power was 1.6 MW. [PERSON] et al. [10] proposed a \"combination\" cycle including recompression, reheating, and intercooling in order to adapt the concentrated solar S-CO\({}_{2}\) cycle to fluctuations in available solar intensity. [PERSON] et al. [11] studied the thermodynamic performance of five S-CO\({}_{2}\) Brayton cycles integrated with solar towers using an evolutionary algorithm. The study concluded that the highest thermal efficiency was achieved by recompression Brayton cycle at noon in June. The maximum integrated system thermal efficiency is 40%. In the theoretical design and experimental research of the S-CO\({}_{2}\) turbine, the US Sandia National Laboratory (SNL) [12; 13] developed the S-CO\({}_{2}\) turbine with a power of 92 kW in 2010, which can test various cycles including a regenerative cycle, intercooling and recompression cycle. [PERSON] et al. [14] designed a 6-stage S-CO\({}_{2}\) turbine with power up to 100 MW and inlet temperature up to 1350 K, and optimized the parameters of each velocity triangle.. Based on the Sunshot project, the Southwest Research Institute of the United States [15] established a supercritical carbon dioxide closed recompression Brayton cycle system using solar energy, and designed an efficient multi-stage axial turbine for the S-CO\({}_{2}\) Brayton cycle. [PERSON] et al. [16] designed a single-stage MW-stage S-CO\({}_{2}\) axial and radial turbine. The axial turbine power is 15 MW, the rotational speed is 10,000 rpm, the total static efficiency is 83.96%. The radial turbine power is 1.5 MW, the rotational speed is 45,000 rpm, and the total static efficiency is 80.02%. [PERSON] et al. [17] achieved the non-design effect of S-CO\({}_{2}\) radial turbine by changing the diameter of the leading edge of the nozzle, and carried out comprehensive numerical analysis to obtain the non-design performance of the radial turbine. [PERSON] et al. [18] proposed an improved method for mechanical design of S-CO\({}_{2}\) turbines using programming methods, and compared the thermal performance of axial turbines and radial turbines under the same conditions. [PERSON] et al. [19] designed an S-CO\({}_{2}\) gas turbine with a net output power of 10 kW and a working fluid recirculation flow rate of 1.2 kg/s, and conducted a principle and demonstration test.