## 2 Theory and Method
### CFD Analysis Method  
The general forms of the three control equations mass conservation equation, momentum conservation equation, and energy conservation equation can be expressed as follows [23; 24]:  
\[\frac{\partial(\rho\phi)}{\partial t}+\text{div}(\rho U\phi)=\text{div}\Big{(} \Gamma_{\phi}\text{grad}\phi\Big{)}+S_{\phi} \tag{1}\]  
where \(\rho\) is density, \(t\) is time, \(U\) is velocity, \(\Gamma_{\phi}\) is the generalized diffusion coefficient, \(S_{\phi}\) is the generalized source term, and \(\phi\) is the general variable. In this study, the zonal shear stress transport (SST) \(k-\omega\) turbulence model was adopted. It was raised by [PERSON] [25] on the basis of standard \(k-\omega\) turbulence model. This turbulence model is considered by more and more scholars as the preferred choice in the field of fluid machinery. Additionally, it has a good agreement with measurement data [26; 27]. The transport equation is:  
\[\frac{\partial(\rho k)}{\partial t}+\frac{\partial(\rho k\overline{u_{i}})}{ \partial x_{i}}=\frac{\partial}{\partial x_{j}}\bigg{[}\Gamma_{k}\frac{ \partial k}{\partial x_{j}}\bigg{]}+G_{k}-Y_{k} \tag{2}\]  
\[\frac{\partial(\rho\omega)}{\partial t}+\frac{\partial(\rho\omega\overline{u_ {j}})}{\partial x_{j}}=\frac{\partial}{\partial x_{j}}\bigg{[}\Gamma_{\omega} \frac{\partial\omega}{\partial x_{j}}\bigg{]}+G_{\omega}-Y_{\omega}+D_{\omega} \tag{3}\]  
where \(\overline{u_{i}}\) and \(\overline{u_{j}}\) are the average turbulent velocity, \(x_{i}\) and \(x_{j}\) are the coordinate component, \(G_{k}\) is the generation term of turbulent kinetic energy \(k\) based on the average velocity gradient, \(G_{\omega}\) is the generation term of dissipation rate \(\omega\), \(Y_{k}\) is the dissipation term of \(k\), \(Y_{\omega}\) is the dissipation term of \(\omega\), \(\Gamma_{k}\) and \(\Gamma_{\omega}\) are the effective diffusion coefficients of \(k\) and \(\omega\), respectively, \(D_{\omega}\) is cross-diffusion term, which coordinates the interface between the standard \(k-\varepsilon\) turbulence model and the standard \(k-\omega\) turbulence model. The formula of turbulent dynamic viscosity coefficient \(\mu_{t}\) of the modified turbulence model is as follows:  
\[\mu_{t}=\frac{\rho k}{\omega}\frac{1}{\text{max}\Big{[}\frac{1}{\alpha^{ \prime}},\frac{c\Gamma_{2}}{a_{0}\omega}\Big{]}} \tag{4}\]  
Figure 1: The overall architecture. where \(\alpha^{*}\) is the low Reynolds number correction coefficient for reducing turbulent eddy viscosity, \(c\) is the constant term of shear stress tensor, \(\alpha_{0}\) is the empirical constant, and \(F_{2}\) is the mixed function. Based on a large number of physical characteristics data of carbon dioxide, the explicit equation of Helmholtz energy equation, the improved Benedict Webb Rubin (MBWR) state equation and the extended corresponding state (ECS) model were adopted. The MBWR state equation is as follows:  
\[P=\sum_{n=1}^{9}a_{n}\rho^{n}+\exp\left(-\left(\frac{\rho}{\rho_{c}}\right)^{ 2}\right)\sum_{n=10}^{15}a_{n}\rho^{2n-17} \tag{5}\]  
where \(P\) is pressure, \(\rho_{c}\) is critical density, and \(a_{n}\) is characteristic parameters related to temperature. In this study, \(\mathbf{x}=[T_{in},P_{in},\alpha_{1},\dot{m},\omega_{R}]\) is taken as the design variable of the turbine, including: inlet temperature \(T_{in}\), inlet pressure \(P_{in}\), inlet air flow angle \(\alpha_{1}\), mass flow rate \(\dot{m}\), and rotating speed \(\omega_{R}\). The real result field \(\mathbf{f}\) obtained by 3-D CFD analysis is as follows:  
\[\mathbf{f}=\mathbb{F}_{\text{ctd}}(\mathbf{x})=[\mathbf{P},\mathbf{T}] \tag{6}\]  
where \(\mathbf{P}\) is pressure fields and \(\mathbf{T}\) is the temperature fields. According to 3-D numerical results, the pressure and temperature distribution on the blade surface can be obtained.