## 1 Introduction  
The Brayton cycle with supercritical carbon dioxide (s-CO\({}_{2}\)) as a working fluid has raised universal attention in the last few decades [1]. The s-CO\({}_{2}\) Brayton cycle exhibits a few advantages compared with the Brayton cycle with the working fluid. A higher cycle thermal efficiency can be achieved with a relatively lower turbine inlet temperature, which could enhance the operation's dependability, reduce the system's complexity and save on costs. Furthermore, the lower turbine inlet temperature of the s-CO\({}_{2}\) Brayton cycle makes the applications possible in solar energy and geothermal power generation, waste heat recovery, and nuclear power generation [2; 3; 4; 5; 6]. Moreover, because it is an environmentally friendly working fluid, no pollution is generated in the power generation process of the s-CO\({}_{2}\) Brayton cycle. However, more challenges bring to the key power generation component, i.e., the s-CO\({}_{2}\) radial inflow turbine design, resulting from unique thermal-physical properties. The geometry scaling effect is a salient issue that needs to be resolved due to the smaller geometry of the s-CO\({}_{2}\) turbine [7]. Hence, the radial inflow design with high efficiency is a fundamental issue for applying the advanced s-CO\({}_{2}\) Brayton cycle. Sandia National Laboratory [8] performed the pioneering experimental work to test the s-CO\({}_{2}\) Brayton cycle and key component performance. In their work, the No. 2 radial inflow turbine was designed with a power of 213 kW and a total-to-static efficiency of 87%. Bechtel Marine Propulsion Corporation [9] designed the s-CO\({}_{2}\) radial inflow turbine with a total inlet temperature of 299 \({}^{\circ}\)C. However, due to several confines in their experimental test, the turbine inlet's total temperature was limited to 232 \({}^{\circ}\)C, and the rotating speed could not achieve the nominal value. In the experimental test carried out by the Tokyo Institute of Technology [10], the rotating speed of the s-CO\({}_{2}\) radial inflow was only 55,000 rpm, which was much lower than the design value of 100,000 rpm; this was attributed to the technical limitations of the shaft bearing and sealing. Hence, the output power and the efficiency measured were lower than the designed values. The experimental tests performed by the Korea Institute of Energy Research [11] found the same astriction. The designed rotating speed of their s-CO\({}_{2}\) radial inflow turbine was 70,000 rpm; however, the actual experimental value was limited to 30,000 rpm. [PERSON] et al. [12] designed the s-CO\({}_{2}\) radial inflow turbine with the power of 1.5 MW and performed the performance prediction based on the Computational Fluid Dynamics (CFD) method using the software package ANSYS CFX. They found that the predicted total-to-static efficiency is 83.49%, with a deviation less than 5% from the calculated value from their one-dimensional design code. [PERSON] et al. [13] presented an optimization work for the one-dimensional design of the s-CO\({}_{2}\) radial turbine based on the Sequential Quadratic Programming (SQP) optimization algorithm. They revealed the superiority of the optimization work with an improvement of 2.27% on the total-to-static efficiency. [PERSON] et al. [14] designed s-CO\({}_{2}\) turbines using deep neural networks. Results suggest that the employed multifaceted approach reduces computational time and resources significantly and is required to completely understand the effects of various turbine design parameters on its performance and sizing. [PERSON] et al. [15] developed a code to predict the aerodynamic performance of an s-CO\({}_{2}\) radial inflow turbine under the off-design working condition. The variations were less than 10% between the one-dimensional calculations and the three-dimensional CFD predictions. The energy loss empirical or semi-empirical correlations determine the predicted accuracy. However, limited correlations are available, especially for the s-CO\({}_{2}\) radial inflow turbine.