## 4 Optimization and Analysis of Multi-Blade Centrifugal Fan
### Optimization Results and Analysis  
In this study, a collaborative optimization design of a prototype multi-wing centrifugal fan was carried out using the wPSO-BP prediction model and the NSGA-III-LBWO multi-objective optimization algorithm. The algorithm parameters were set as follows: the population size was set to 200, the number of iterations was 100, and the computational program was developed on the MATLAB 2022a platform. The core process includes the following:  
1. Parameter initialization: 200 sets of initial samples are generated using the Latin hypercubic sampling technique and input into the wPSO-BP model for the prediction of full pressure (\(P_{total}\)), efficiency (\(\eta\)), and noise (\(L_{p}\));
2. Pareto front search: the reference point stratification mechanism of NSGA-III is used to screen the non-dominated solutions, and the Logistic chaos mapping of LBWO is combined to enhance the global search capability. The Pareto frontiers obtained through optimization (see Figure 13) show that the optimal solution set exhibits an obvious trade-off between full pressure enhancement, efficiency improvement, and noise suppression, thus verifying the effectiveness of the proposed multi-objective optimization framework. As shown in Figure 13, the sound pressure level of the fan increases with both efficiency and total pressure, indicating that the total pressure, efficiency, and noise performance of the multi-blade centrifugal fan cannot be simultaneously optimized, necessitating trade-off decisions. Compared to the improvements in aerodynamic performance, noise reduction during fan operation holds higher practical significance. The optimization configuration and corresponding results are summarized in Table 6. The optimized fan achieves a total pressure increase of 34.79 Pa, an efficiency improvement of 0.67%, and a reduction in sound pressure level of 1.73 dB compared to the baseline prototype. These results demonstrate that the optimization approach based on the wPSO-BP prediction model and NSGA-III-LBWO \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Model** & **RMSE** & **MAE** & **R\({}^{2}\)** \\ \hline \multirow{2}{*}{Total pressure} & BP & 35.82 & 29.18 & 0.74 \\ \cline{2-5}  & wPSO-BP & 17.74 & 13.74 & 0.93 \\ \hline \multirow{2}{*}{Efficiency} & BP & 2.30 & 1.71 & \textless{}0 \\ \cline{2-5}  & wPSO-BP & 0.58 & 0.50 & 0.92 \\ \hline \multirow{2}{*}{
\begin{tabular}{c} Sound \\ pressure level \\ \end{tabular} } & BP & 1.81 & 1.56 & 0.43 \\ \cline{2-5}  & wPSO-BP & 0.70 & 0.61 & 0.91 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison of predicted results. algorithm is effective, and the optimization goal of improving aerodynamic performance while reducing noise is achieved. For the convenience of analyzing the fans before and after optimization, three sections are selected, namely the front disc side (z = 75 mm), middle position (z = 50 mm), and rear disc side (z = 25 mm) (Figure 14). The static pressure nephograms before and after optimization are shown in Figures 15 and 16. After optimization, the range of low static pressure near the center of the impeller in the z = 75 mm section is significantly reduced, the pressure gradient \begin{table}
\begin{tabular}{c c c c c c c c} \hline  & \(\beta_{t}J^{o}\) & \(\beta_{o}J^{o}\) & \(R_{\text{v,g}}\)/mm & \(\theta\)/\({}^{\text{c}}\) & \(P_{total}\)/Pa & \(\eta\)/\(\%\) & \(L_{p}\)/dB \\ \hline Prototype fan & 90 & 139 & 12.87 & 0 & 1275.89 & 60.07 & 57.75 \\ \hline Optimized fan & 98.81 & 147.06 & 17.47 & 39.85 & 1310.68 & 60.74 & 56.02 \\ \hline \end{tabular}
\end{table} 
Table 6: Comparison of results before and after optimization. Figure 14: Sections at different heights perpendicular to the axis of the impeller. Figure 13: Pareto front. decreases, the interference of the airflow entering the blade channel is reduced, and the flow loss inside the blade channel is reduced, resulting in an improvement in aerodynamic performance.