## 2 Design of the S-CO\({}_{2}\) Compressor  
### Design Method  
Based on our previous design experience and methods [14; 15], the thermodynamic design software of S-CO\({}_{2}\) centrifugal compressor (S-CO\({}_{2}\)CPTD) was developed. The software was adopted in the compressor thermodynamic design of this research. The software has five main functions: data input, core calculation, data output, drawing and secondary exploration. The software is designed by Visual Basic 6.0 and Intel Fortran 2017. Based on 1-D flow theory, the following main equations were used in the core calculation. (1) Euler Equation  
According to the energy transformation and conservation law, the mechanical energy transferred to rotor blades is converted into fluid energy. Thus, the energy gained by one kilogram of fluid is:  
\[h_{th}=\frac{1}{g}(c_{2u}u_{2}-c_{1u}u_{1}) \tag{1}\]  
where \(h_{th}\) (J/kg) is the energy head gained by one kilogram of fluid in the impeller blade channel, \(g\) (m/s\({}^{2}\)) is the acceleration of gravity, \(c\) (m/s) is the absolute speed and \(u\) (m/s) is the circumferential speed. The subscript 1 represents the impeller inlet, the subscript 2 represents the impeller outlet and the subscript \(u\) represents the circumferential component. (2) Energy Equation  
The total power of per kilogram of fluid consumed in centrifugal compressor stages is considered to consist of three parts: the work of the impeller on the fluid, the loss of internal leakage and the loss of impeller resistance. Based on the energy equation, the work of the impeller on fluid is converted to the fluid energy. \[h_{tot}=h_{th}+h_{df}+h_{t} \tag{2}\]  
where \(h_{tot}\) (J/kg) is the total energy head, \(h_{df}\) (J/kg) is the loss of impeller resistance and \(h_{t}\) (J/kg) is the loss of internal leakage. In Formula (3), the theoretical energy head is converted to the working substance in the form of mechanical energy. The loss of internal leakage and the loss of impeller resistance are converted to fluid in the form of heat. Therefore, the total energy equation can be written as:  
\[h_{tot}=\frac{c_{p}}{A}(T_{2}-T_{1})+\frac{c_{2}^{2}+c_{1}^{2}}{2g} \tag{3}\]  
where \(c_{p}\) (J/(kg\(\cdot\)K)) is the specific heat capacity at constant pressure, \(A\) is the thermal equivalent of work (J/cal) and \(T\) (K) is the temperature. (3) Bernoulli Equation  
For the whole compressor stage, the work of the impeller on the fluid is converted to the following three parts: (1) Improving the static pressure energy of fluid. (2) Improving the kinetic energy of fluid, but in general, the kinetic energy increases little, and is often negligible. (3) Overcoming the flow loss of working fluids in stages. By adopting the Bernoulli equation in the impeller, the following equation can be obtained:  
\[h_{th}=\int_{1}^{2}\frac{dp}{y}+\frac{c_{2}^{2}-c_{1}^{2}}{2g}+h_{f1} \tag{4}\]where \(p\) (Pa) is the pressure, \(\gamma\) (N/m\({}^{3}\)) is the specific weight and \(h_{f1}\) (J/kg) is the flow loss of fluid flowing in the impeller. When adopting the Bernoulli equation in the diffuser, \(h_{th}\) is zero:  
\[\frac{c_{2}^{2}-c_{3}^{2}}{2g}=\int_{2}^{3}\frac{dp}{\gamma}+h_{f2} \tag{5}\]  
where \(h_{f2}\) (J/kg) represents the flow loss of fluid flowing in the diffuser. The subscript 2 represents the diffuser inlet and 3 represents the diffuser outlet. (4) Key geometric design Equations  
The thermal design is mainly to calculate the flow in the centrifugal compressor impeller to determine the geometric parameters. According to the design conditions and estimated parameters, the impeller outer diameter can be obtained:  
\[D=\sqrt{\frac{\dot{m}}{\rho_{2}u_{2}\pi\tau_{1}\tau_{2}}} \tag{6}\]  
where \(\dot{m}\) (kg/s) is the mass flow rate, \(\rho\) (kg/m\({}^{3}\)) is density, \(\tau_{1}\) is the blocking factor of the impeller outlet, \(\tau_{2}=l_{2}/D\) is the relative width of the impeller outlet and \(l_{2}\) (mm) is the outlet blade height of impeller. The subscript \(r\) represents the radial component.