## 2 Research Objects and Numerical Methods
### Thermal Design  
The turbine's thermal calculation is done by self-developed design software (S-CO2 TBTD) designed by Visual Basic 6.0 and Intel Fortran 2013. The software consists of five modules: (1) Data Input: input parameters, and exchange data with computing program after processed by data management module. (2) Core Calculation: calculate the state parameters and input power of the compressor by one-dimensional thermal design method, then output the calculation results after convergence. (3) Data Output: after calculating each module, the main contents are exported to the software interface, also supporting the derivation of the result files. (4) Drawing: showing the relation curves between the investigation parameters when designing the turbine and the operating parameters, also supporting the derivation of the result files. (5) Secondary Exploration: for user extension functions, such as adding  
Figure 1: Research flow chart. more performance impact parameters and more available working materials, designing more detailed computing processes, and even adding optimization processes for specific parameters. According to the one-dimensional flow theory, the main ideas and methods of Core Calculation are as follows:  
1. The series \(z\) and Reaction coefficient \(\Omega\) are determined to obtain the enthalpy drop distribution of each row of cascades. \[\Omega=\frac{h_{2s}}{h_{s}^{*}}\] (1) where \(h_{2s}\) is enthalpy drop in rotor cascade, \(h_{s}^{*}\) is total stagnation enthalpy drop. 2. The ideal speed \(c_{s}\) of the stage is determined by the enthalpy drop of the stage, and the speed ratio \(x_{a}\) is prefabricated in the optimum range to obtain the wheel circumference velocity \(U\). \[x_{a}=\frac{\varphi\cos\alpha_{1}\sqrt{1-\Omega}}{2(1-\Omega)}\] (2)
3. The outlet velocities of nozzles (\(c_{1}\)) and rotor blades (\(c_{2}\)) are obtained by prefabricating the speed coefficient and prefabricating the inlet and outlet flow angle in the optimum range. 4. The average diameter \(d_{m}\) of the stage is preliminarily determined by the wheel circumference velocity \(U\). 5. Calculating the circumference loss and the actual loss of the whole machine. The efficiency of the wheel periphery \(\eta_{u}\): \[\eta_{u}=1-\xi_{n}-\xi_{b}-\xi_{c_{2}}\] (3) where relative loss of nozzle \(\xi_{n}=\left(1-\varphi^{2}\right)\), relative loss of rotor \(\xi_{b}=\left(1-\varphi^{2}\right)\), exit losses \(\xi_{c_{2}}=\frac{\xi_{c}^{2}}{2}/h_{s}^{*}\). 6. Determining relative internal efficiency of the turbine. \[\eta_{i}=(1-\xi_{g})\eta_{u}-\xi_{f}\] (4) where \(\xi_{g}\) is structural loss coefficient, \(\xi_{f}\) is friction loss coefficient. 7. Checking the speed coefficient and flow coefficient \(\mu\), and revising the speed ratio \(x_{a}\) and the inlet and outlet flow angle in the optimum range to obtain the optimum turbine efficiency. 8. According to the aerodynamic parameters of each stage, the geometric parameters of the nozzle and impeller are obtained. Firstly, entering the five determined initial design parameters in the thermal design software: power, rotation speed, total inlet pressure, total inlet temperature, and static outlet pressure, as shown in Table 1.  
Secondly, according to the existing published literature, empirical formula and experimental curve, the five pre-fetch parameters including the speed coefficient, reaction coefficient, flow coefficient, installation angle and relative pitch of each stage cascade are pre-fetched and input into the thermal design software. The specific value is shown in Table 2. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Power/MW & 10 \\ Rotation speed/rpm & 10,000 \\ Total inlet pressure/MPa & 15 \\ Total inlet temperature/K & 773.15 \\ Static outlet pressure/MPa & 9.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Initial design parameters.