## 3 Multi-Objective Aerodynamic Design Optimisation of Centrifugal Impeller
### Optimisation Objectives and Constraints  
Centrifugal impeller optimised design conditions are usually selected at rated speed for geometric parameter improvement. Further, non-design conditions are often selected at other mass flow at the same speed, which can meet the requirements for industrial compressors working in a single environment. However, the working speed range of the vehicle-mounted centrifugal impeller is wide, and the performance of the rated speed can be improved by optimising the rated speed only. However, the performance of the common operating conditions cannot be guaranteed. The isentropic efficiency of the aerodynamic performance of the centrifugal impeller of FC at the rated operating point and common operating point is optimised by multi-objective optimisation in this paper to improve them at the same time. Take 100,000 rpm, mass flow of 118.32 g/s and pressure ratio of 2.7 as the rated working condition. The selection of the common condition point is based on the U.S. Department of Energy (DOE) technical requirements for the same class of automotive FC power systems. A total of 70% of the rated speed of the centrifugal impeller, 77.36 g/s mass flow and 1.7 total pressure ratio are set as the common operating point. The original performance curve under two working conditions is shown in Figure 6.  
Figure 5: CAD of the original impeller.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Fluid medium & Air \\ Rated output power of FCS (kW) & 100 \\ Rated speed (rpm) & 100,000 \\ Design point Mass flow (g/s 100,000 rpm) & 118.32 \\ Design point Total pressure ratio (100,000 rpm) & 2.7 \\ Design point Isentropic efficiency (100,000 rpm) & 83.5\% \\ Normal condition speed (rpm) & 70,000 \\ Design point Mass flow (g/s 70,000 rpm) & 77.36 \\ Design point Total pressure ratio (70,000 rpm) & 1.7 \\ Design point Isentropic efficiency (70,000 rpm) & 85.5\% \\ Non-dimensional speed coefficient & 0.577 \\ Blade numbers & 16 (8 + 8) \\ Blade thickness (mm) & 1 \\ Tip clearance (mm) & 2.5 \\ Radial exit angle (\({}^{\circ}\)) & 85 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Aerodynamic and geometric parameters of the centrifugal compressor.  
According to the above description, maximising the isentropic efficiency at two operating points is the optimisation objective, with the optimised mass flow not decreasing and the optimised total pressure ratio in-between 1 and 1.01 times the original total pressure ratio as the constraint. The mathematical expression is as follows:  
Objective function:  
\[\max Eff_{d\_70k} \tag{3}\]  
\[\max Eff_{d\_100k} \tag{4}\]  
Constraint function:  
\[Mass_{o\_70k}\geq\ Mass_{d\_ori\_70k} \tag{5}\]  
\[Mass_{o\_100k}\geq\ Mass_{d\_ori\_100k} \tag{6}\]  
\[0\leq\frac{Trp_{o\_70k}-Trp_{d\_ori\_70k}}{Trp_{d\_ori\_70k}}\leq 0.01 \tag{7}\]  
\[0\leq\frac{Trp_{o\_100k}-Trp_{d\_ori\_100k}}{Trp_{d\_ori\_100k}}\leq 0.01 \tag{8}\]  
\[x_{i}^{L}\leq x_{i}\leq x_{i}^{II} \tag{9}\]  
\(Eff_{d\_70k}\) is the isentropic efficiency at the design point of 70,000 rpm, \(Eff_{d\_100k}\) is the isentropic efficiency at the design point of 100,000 rpm, and \(Mass_{d\_ori\_70k}\) and \(Mass_{o\_70k}\) are the mass flow at the original impeller design point and the optimised impeller design point at 70,000 rpm, respectively. \(Mass_{d\_ori\_100k}\) and \(Mass_{o\_100k}\) are the mass flow at the original impeller design point and the optimised impeller design point at 100,000 rpm, respectively. \(Trp_{d\_ori\_70k}\) and \(Trp_{o\_70k}\) are the total pressure ratios at the original impeller design point and the optimised impeller design point at 70,000 rpm, respectively. \(Trp_{d\_ori\_100k}\) and \(Trp_{o\_100k}\) are the total pressure ratios at the original impeller design point and the optimised impeller design point at 100,000 rpm, respectively. \(x_{i}\) is the optimisation variable, b \(x_{i}^{II}\) is the upper bound of the optimisation variable, and \(x_{i}^{L}\) is the lower bound of the optimisation variable.