## 4 Optimization of the Mixed Flow Pump
### Optimization Setting  
The accuracy of RSM prediction is greatly affected by the number of design parameters \(N\) and sampling points \(S\). Generally, the minimum number of sampling points required for RSM is \(S_{\text{min}}=\frac{(N+s)!}{N!s!}\), where \(s\) is the order of the interpolation. To improve the accuracy of RSM, the number of sampling points used in this study was \(2S_{\text{min}}\). As mentioned earlier, in the first optimization case, the influence of SDIEC was ignored, while in the second case, the influence of SDIEC was considered. Table 2 shows the design parameters, constraints, and optimization objectives for both cases.  
In the first case, only the stacking condition and blade loading were adopted as design variables. To widen the mixed flow pump high-efficiency region, the pump efficiency at \(1.2Q_{\text{des}}\) and \(0.8Q_{\text{des}}\) were optimized. To ensure that the hydraulic performance of the optimized pump at the design point was truly better than that in the baseline model, the pump head and efficiency at the design point were set as constraints. The pump head and efficiency are calculated by Equations (8) and (9):  
\[H{=}\ \frac{P_{\text{out}}-P_{\text{in}}}{\rho g} \tag{8}\]  
\[\eta{=}\ \frac{(P_{\text{out}}-P_{\text{in}})Q}{M\omega} \tag{9}\]  
where \(g\) represents the acceleration of gravity; \(\rho\) represents the fluid density; \(P_{\text{in}}\) and \(P_{\text{out}}\) represent the inlet and outlet total pressures; \(\omega\) represents the impeller angular velocity; and \(M\) represents the torque acting on the impeller by the motor shaft.  
In the second case, the influence of SDIEC on the optimization results was considered, and \(rV_{\text{h}}\) and \(rV_{\text{s}}\) were added to the design parameters as shown in Table 2. To reduce the number of calculation times, the two design parameters \(N\text{C}_{\text{h}}\) and \(N\text{C}_{\text{s}}\) were set as 0.18 and 0.10, respectively, because they have the least influence on the optimization objectives. Same as in the first case, the optimization targets were the pump efficiency at \(1.2Q_{\text{des}}\) and \(0.8Q_{\text{des}}\) and the constraints were the pump efficiency and head at the design point.  
Therefore, to achieve the purpose of making the second-order RSM more accurate in both cases, 110 sample points were generated in each case. As shown in Table 3, parameter setting of NSGA-II was the same in both cases. A total of 24,000 impellers with different configurations were generated in each case.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**TypeParameters** & \multicolumn{2}{c}{**First Case**} & \multicolumn{3}{c}{**Second Case**} \\ \cline{2-6}  & **Name** & **Variable** & **Range** & **Name** & **Variable** & **Range** \\ \hline \multirow{7}{*}{Design Parameters} & \multirow{7}{*}{Blade Loading} & \(N\text{C}_{\text{h}}\) & 0.1–0.5 & \multirow{7}{*}{Circulation} & \(RV_{\text{h}}\) & 0.29–0.34 \\  & & \(N\text{C}_{\text{s}}\) & 0.1–0.5 & & \(RV_{\text{s}}\) & 0.29–0.34 \\  & & \(DRVT_{\text{h}}\) & \(-\)0.25–0.25 & & \(DRVT_{\text{h}}\) & \(-\)0.25–0.25 \\ \cline{3-6}  & & \(ND_{\text{h}}\) & 0.5–0.9 & & \(ND_{\text{h}}\) & 0.5–0.90 \\  & & \(K_{\text{h}}\) & \(-\)2.0–2.0 & & \(K_{\text{h}}\) & \(-\)2.0–2.0 \\  & & \(DRVT_{\text{s}}\) & \(-\)0.25–0.25 & & \(DRVT_{\text{s}}\) & \(-\)0.25–0.25 \\  & & \(ND_{\text{s}}\) & 0.5–0.9 & & \(ND_{\text{s}}\) & 0.5–0.90 \\  & & \(K_{\text{s}}\) & \(-\)2.0–2.0 & & \(K_{\text{s}}\) & \(-\)2.0–2.0 \\ \cline{2-6}  & & Stacking & \(\beta\) & \(-\)20.0–20.0 & Stacking & \(\beta\) & \(-\)20.0–20.0 \\ \hline \multirow{2}{*}{Constraints} & \multirow{7}{*}{Plump efficiency at \(1.0Q_{\text{des}}\)} & \multirow{7}{*}{Plump head at \(1.0Q_{\text{des}}\)} & \multirow{7}{*}{Plump efficiency at \(1.2Q_{\text{des}}\)} \\ \cline{1-1} \cline{3-6}  & & & & \(P_{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text