## 2 Establishment of Prediction Model  
### Design Variables and Optimization Objectives  
The structural parameters of the impeller and volute casing exert a decisive influence on the aerodynamic performance of the multi-wing centrifugal fan. Table 1 summarizes the key structural and operational parameters of the multi-wing centrifugal fan designed for use in the automotive air conditioning system in this study. Before constructing the predictive model, the design variables and optimization objectives should be clearly defined. Under the design conditions, as the outlet angle of the blades increases, the total pressure and efficiency of the fan increase, which will lead to an increase in noise, because the airflow can be sufficiently accelerated to form a \"jet-wake\" when the outlet angle is large. Therefore, a reasonable outlet angle can effectively improve the performance of the fan. The match between the blade inlet angle and the airflow inlet angle determines the quality of flow in the blade channel. Flow separation in the blade channel typically produces shedding vortices that interfere with the main flow, degrade fan performance, and generate noise. Therefore, it is necessary to determine the optimal blade inlet angle to minimize flow losses in the blade channel, while also reducing the noise generated by inflowing airflow over the blades. The single-arc blade in this paper is presented in Figure 1. The relationship between the arc radius \(R_{c}\) and central angle \(\alpha\) of the blade and the inner diameter \(D_{1}\), outer diameter \(D_{2}\), inlet angle \(\beta_{i}\), and outlet angle \(\beta_{o}\) of the impeller can be obtained:  
\[R_{c}=\frac{R_{2}^{2}-R_{1}^{2}}{2[R_{2}\text{cos}(\pi-\beta_{o})-R_{1}\text{ cos}(\pi-\beta_{i})]} \tag{1}\]  
\[R_{oo^{\prime}}=\sqrt{{R_{c}}^{2}+{R_{2}}^{2}-2R_{c}R_{2}\text{cos}(\pi-\beta_ {o})} \tag{2}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Category** & **Parameter** & **Value** \\ \hline \multirow{4}{*}{Structural Parameters} & Blade inlet angle & 90\({}^{\circ}\) \\ \cline{2-3}  & Blade outlet angle & 139\({}^{\circ}\) \\ \cline{2-3}  & Volume tongue radius & 13 mm \\ \cline{2-3}  & Number of blades & 41 \\ \hline \multirow{2}{*}{Operational Parameters} & Impeller speed & 4202 r/min \\ \cline{2-3}  & Rated flow rate & 500 m\({}^{3}\)/h \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Initial parameters of the centrifugal fan. \[\alpha=\text{acos}\left(\frac{\left|{R_{c}}^{2}+{R_{oo}}^{2}-{R_{2}}^{2}\right|}{2R _{c}R_{oo^{\prime}}}\right)-\text{acos}\left(\frac{\left|{R_{c}}^{2}+{R_{oo^{ \prime}}}^{2}-{R_{1}}^{2}\right|}{2R_{c}R_{oo^{\prime}}}\right) \tag{3}\]  
where \(R_{c}\) is the arc radius of the blade, \(R_{1}\) and \(R_{2}\) are the inner and outer diameters of the impeller, respectively, \(\beta_{i}\) and \(\beta_{o}\) are the inlet and outlet angles of the blade, respectively, \(R_{oo^{\prime}}\) is the distance from the center of the blade arc to the center of the impeller, and \(\alpha\) is the blade central angle. The solute tongue prevents air recirculation within the volute during fan operation. However, residual airflow bypassing the volute tongue disrupts the impeller mainstream, inducing flow separation within the blade passage and cross-impeller flow, thereby causing performance degradation and noise generation. Furthermore, the tongue's close proximity to the impeller subjects it to periodic high-speed flow interactions, generating time-varying pressure pulsations that render the tongue structure the primary noise source in the fan system. As shown in Figure 2, the inclined tongue structure features a radius varying along the axial direction, leading to a phase difference in the airflow effects from the front to the rear disc sides of the impeller on the volute tongue. The superposition of noise from unsteady forces with different phases yields smaller amplitudes compared to forces with the same phase in constant-radius tongue structures.