## 3 Preliminary Optimal Design Method  
### One-Dimensional Design Method  
The objective of the one-dimensional design method is to acquire the meanline geometric, aerodynamic, and thermodynamic parameters of each blade row and the initial flow path.  
The theoretical mechanical work \(W_{\text{th}}\) is expressed by Euler equation:  
\[W_{\text{th}}=C_{2,\theta}U_{2}-C_{1,\theta}U_{1}, \tag{1}\]  
where subscript 1 and 2 denote inlet and outlet of each row, respectively; subscript \(\theta\) denotes a tangential component; \(C\) is velocity; and \(U\) is local blade speed. \(W_{\text{th}}\) is converted into kinetic energy and pressure energy, and overcome the flow loss, which is written as:  
\[W_{\text{th}}=\int_{1}^{2}\frac{dp}{\rho}+\frac{C_{2}^{2}-C_{1}^{2}}{2}+h_{ \text{loss}}, \tag{2}\]  
where \(\rho\) is density, \(p\) is static pressure, and \(h_{\text{loss}}\) is flow loss. \(h_{\text{loss}}\) is calculated by using loss correlations, as will be introduced in Section 3.2. The one-dimensional mass conservation equation [21] is:  
\[\dot{m}=\rho C_{z}A(k_{\text{M}}-b), \tag{3}\]  
where the subscript \(z\) denotes an axial component; \(A\) is flow cross area; \(k_{\text{M}}\) is the flow coefficient; and \(b\) is an aerodynamic blockage factor. Deviation angle \(\delta\) and incidence angle \(i\) are defined respectively as the following:  
\[\delta=\beta_{2}-\kappa_{2}, \tag{4}\]  
\[i=\beta_{1}-\kappa_{1}, \tag{5}\]  
where \(\kappa\) is the blade angle and \(\beta\) is the flow angle.  
Based on Equations (1)-(3), together with the flow loss and aerodynamic blockage models, the meanline parameters and the initial flow path are calculated. The flow angles are converted into the blade angles according to incidence and deviation angle models.  
Once flow path is specified, blade height can be obtained. The corresponding blade number and chord can be obtained with the obtained solidity and aspect ratio. [PERSON] [24] pointed out that the flow loss \(h_{\text{loss}}\) is a function of wake momentum thickness, solidity, velocity distribution, and chord length. Additionally, solidity will influence the reflections of oblique shock in cascade passage [25]. Reaction of stage indicates the allocation of static pressure rise capacity between stator and rotor in a stage, which influences magnitudes of Mach number and stage flow quality. From Equation (1), rotation speed will directly determine the theoretical work, the rotor tip Mach number, and thus the shock loss of the transonic and supersonic stage. Outlet axial Mach number will greatly affect the Mach number distribution along the stagewise direction as well as the meridional flow path contraction ratio. Moreover, the outlet axial Mach number is an important factor affecting the downstream component, such as the combustor in the aero engine and the heat transfer device in the air separation system. Lastly, the inlet radius ratio determines the inlet area and thus the inlet axial Mach number, rotor tip Mach number, and the average blade height.  
Overall, aspect ratio, solidity, reaction, rotation speed, outlet axial Mach number, and inlet radius ratio are six key parameters and regarded as design parameters in the current preliminary optimal design phase. It should be noted here that beside the six design parameters mentioned above, [PERSON] et al. [5] adopted another three design parameters, i.e., the number of stages, the inlet flow/annulus area, and flow path shape to maximize the compressor design efficiency potential. Meeting the same overall design requirements, the axial-flow compressors with different number of stages are particularly investigated in Section 5. The constant outer radius is here adopted and set to be the same value for each design, and thus as long as the inlet radius ratio, design flow mass rate, and inletconditions are known, inlet flow/annulus area as well as the flow path shape are naturally determined. As a result, the inlet flow/annulus area and flow path shape have not been taken as the dependent design parameters in this work.