## 2 Design of the S-CO\({}_{2}\) Compressor
### Optimization Method  
At present, the empirical coefficients and estimated parameters matching the thermal design of the S-CO\({}_{2}\) centrifugal compressor have not been disclosed. The size of the flow passage obtained by the 1-D flow method does not necessarily conform to the actual flow requirements. At the same time, the design and optimization period of the traditional design method is long. Therefore, based on the 1-D flow theory, a fast thermal design method of S-CO\({}_{2}\) compressor was established in this study. Combined with the high-precision three-dimensional aerodynamic analysis method, a design-optimization method based on Gauss process regression was proposed, as shown in Figure 3.  
According to the thermal design results and aerodynamic design results, and based on the Gauss process regression assumption, the true isentropic efficiency of an unknown design condition y can be estimated:  
\[f_{r}(\mathbf{y})=f_{a}(\mathbf{y})+\Delta f_{i}(\mathbf{y}) \tag{8}\]  
where \(\Delta f_{i}(\mathbf{y})\) is the deviation between the thermal design \(f_{a}(\mathbf{y})\) and CFD accurate calculation results. The calculation deviation satisfies Gauss distribution at each point in the whole design space:  
\[\Delta f(\mathbf{y})\sim N(\mu(\mathbf{y}),\sigma^{2}(\mathbf{y})) \tag{9}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{**Key Parameters**} & **Value** & **Unit** \\ \hline \multirow{4}{*}{Thermodynamic parameters} & Speed & 40,000 & rpm \\  & Inlet pressure & 7.8 & MPa \\  & Inlet temperature & 309.15 & K \\  & Mass flow rate & 6.66 & kg/s \\ \hline \multirow{4}{*}{Geometric parameters} & Wheel diameter & 76 & mm \\  & Inlet diameter & 15 & mm \\  & Blade height (inlet) & 11.05 & mm \\  & Blade height (outlet) & 2.4 & mm \\  & Tip clearance & 0.24 & mm \\  & Inlet geometric angle & 50 & \({}^{\circ}\) \\  & Outlet geometric angle & 30 & \({}^{\circ}\) \\  & Leading edge sweep angle & 0 & \({}^{\circ}\) \\  & Axial length & 15 & mm \\  & Blade thickness & 1.0 & mm \\  & Main blades & 12 & pc. \\  & Splitter blades & 12 & pc. \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Key parameters of the initial design of the compressor. Figure 2: The initial design of the S-CO\({}_{2}\) centrifugal compressor: (**a**) rotor’s meridian face profile and (**b**) 3-D model. The mean \(\mu(\mathbf{y})\) and variance \(\sigma(\mathbf{y})\) of Gaussian distribution are:  
\[\left\{ \begin{array}{l}\mu(\mathbf{y})=\mu+\mathbf{R}_{\mathcal{D}}\mathbf{R}_{ DD}^{-1}(\Delta f_{D}-1\mu)\\ \sigma(\mathbf{y})=\sigma^{2}(1-\mathbf{R}_{\mathcal{D}}\mathbf{R}^{-1}{}_{DD} \mathbf{R}_{\mathcal{D}\mathcal{Y}})\end{array} \right. \tag{10}\]  
where \(D\) is the training set, \(\mathbf{R}_{\mathcal{D}\mathcal{D}}\) and \(\mathbf{R}_{\mathcal{D}\mathcal{Y}}\) are the covariance vector of the calculated condition and the new sampling point, \(\mathbf{R}_{DD}\) is the covariance matrix and \(\Delta f_{D}\) is the deviation between the thermal design results and CFD accurate calculation results of all samples in the training set. Therefore, the real isentropic efficiency (aerodynamic analysis result) of an unknown design condition y is estimated as follows:  
\[\left\{ \begin{array}{l}\widetilde{f}_{r}(\mathbf{y})=f_{a}(\mathbf{y})+\mu( \mathbf{y})+\xi\sigma(\mathbf{y})\\ \widetilde{f}_{r}(\mathbf{y})=f_{a}(\mathbf{y})+\mu(\mathbf{y})-\xi\sigma( \mathbf{y})\end{array} \right. \tag{11}\]  
where \(\widetilde{f}_{r}(\mathbf{y})\) is the lower limit estimation, \(\widehat{f}_{r}(\mathbf{y})\) is the upper limit estimation and \(\xi\) is the confidence constant. This method has the characteristics of having fast speed and being self-adaptive. The detailed optimization theory and method can refer to Reference [17]. Key geometric parameters were used as optimization variables, including: the relative width of the impeller outlet, inlet and outlet geometric angle, leading edge sweep angle, etc.