## 2 Modeling  
### Establishment of Automatic Design Process  
To accelerate the design process, an automatic design and calculation process of the radial-inflow turbine is firstly established by calling batch modes in MATLAB. The flow chart is shown in Figure 1. The initial geometric model of the turbine is obtained based on design parameters, such as inlet pressure and temperature, outlet pressure, output power, rotation speed, etc. Then, the rotor solidity and working fluid used in the calculation are determined. The macro files of batch mode are called by MATLAB to execute the process in Workbench BladeGen, Turbogrid, CFX, and CFD-Post. The modeling, discretization, numerical simulation and post-processing are then completed. Finally, the program repeats this procedure to obtain the results of the required different cases.  
The thermodynamic design is based on the conservation of momentum, mass and energy equations. The fluid properties of SCO\({}_{2}\) and air are obtained from NIST REFPROP. The basic principal for the thermodynamic design is to ensure the highest efficiency at a given rotation speed [27]. Due to space limits, the following formulas are some of the calculations in the thermodynamic design program. The influence of friction loss at the wheel back and the flow loss in the volume are ignored. The loss producing by nozzle and impeller are considered. Hence, the isentropic efficiency can be estimated as Equation (1):  
\[\eta_{\text{is}}=1-(1-\alpha)\Big{(}1-\zeta^{2}\Big{)}-\bigg{(}\frac{\omega_{ \text{out}}}{C_{\text{is}}}\bigg{)}^{2}\bigg{(}\frac{1}{\psi^{2}}-1\bigg{)} \tag{1}\]  
where \(\alpha\) is the degree of reaction. \(\zeta\) and \(\psi\) are the velocity coefficient of nozzle and impeller respectively, which represents the loss in nozzle and impeller. We adopt 0.96 for \(\zeta\) and 0.84 for \(\psi\). \(\omega_{\text{out}}\) and \(C_{\text{is}}\) are, respectively, the relative velocity at turbine outlet and the isentropic ideal velocity.  
To gain the best isentropic efficiency in the thermodynamic design, we can correspondingly acquire the degree of reaction, the velocity ratio and etc. For example, the degree of reaction is chosen as 0.47 and the velocity ratio is 0.67. After the determination of design parameters, we can calculate the geometry parameters. For example, the impeller diameter is acquired by Equation (2):  
\[D=\frac{60v_{\text{in}}}{\pi r} \tag{2}\]  
where \(r\) is the rotating speed, which is 50,000 rpm in this case. \(v_{\text{in}}\) is the linear velocity at the inlet of the impeller.  
The blade height at the turbine inlet can be calculated as Equation (3):  
\[l_{\text{in}}=\frac{\dot{m}}{\pi D\rho_{\text{in}}\omega_{\text{in}}} \tag{3}\]  
Figure 1: Flow chart of design and simulation.  
where \(\dot{m}\) is the mass flow rate, \(\rho_{\text{in}}\) and \(\omega_{\text{in}}\) are, respectively, the density and the relative velocity at the rotor inlet.  
The area of turbine outlet is calculated in Equation (4):  
\[A_{\text{out}}=\frac{\dot{m}}{\rho_{\text{out}}\omega_{\text{out}}\sin\beta_{ \text{out}}} \tag{4}\]  
where \(\beta_{\text{out}}\) is the flow angle at turbine outlet, \(\rho_{\text{out}}\) and \(\omega_{\text{out}}\) are, respectively, the density and the relative velocity at rotor outlet. With the empirical coefficients regarding to impeller diameter, the average outlet diameter and axial length are determined. Then we can obtain the geometric parameters of meridian plane for the rotor.  
The geometric parameters are calculated by estimating the velocity at both inlet and outlet of the stator and the rotor. To compare the difference of air and SCO\({}_{2}\), we firstly conduct the thermodynamic design using SCO\({}_{2}\) and then, estimate the working condition with air to conduct the numerical simulation with the same radial-inflow turbine model.