## 3 Parameterization the Centrifugal Fan Blade Based on CST Function  
[PERSON] and [PERSON] proposed using a Class Function and a Shape Function (CST function) to represent the geometric shape of an aircraft. This parametric representation method uses the Class Function and the Shape Function to represent the airfoil. It is highly accurate and requires few parameters, making it intuitive and convenient when managing important design parameters such as a leading-edge airfoil radius. The CST method is mainly used to parameterize airfoils in the aviation and military fields and has shown good curve-fitting capabilities [24; 25]. Multi-blade centrifugal fan airfoil designs are mostly based on traditional concepts. An improved CST function was applied here to parameterize the multi-blade centrifugal fan blade: \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Size** \\ \hline Impeller inlet radius, \(R_{1}\) (mm) & 115 \\ Impeller outlet radius, \(R_{2}\) (mm) & 140 \\ Impeller width, \(B_{1}\) (mm) & 100 \\ Volute width, \(B_{2}\) (mm) & 135 \\ Single-arc blade radius, r (mm) & 15 \\ Blade inlet angle, \(\beta_{1A}\) (deg) & 67 \\ Blade outlet angle, \(\beta_{2A}\) (deg) & 163 \\ Number of blades, \(z\) & 60 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The profile data of impeller. Figure 3: Optimization flow chart. \[\frac{z}{c}\Big{(}\frac{x}{c}\Big{)}=C\Big{(}\frac{x}{c}\Big{)}\cdot S \Big{(}\frac{x}{c}\Big{)}+\frac{x}{c}\frac{z_{TE}}{c} \tag{1}\] \[C(x/c)=\Big{(}\frac{x}{c}\Big{)}^{N_{1}}\Big{[}1-\frac{x}{c}\Big{]} ^{N_{2}}\] (2) \[S(x/c)=\sum_{i=0}^{N}\Big{[}v_{i}\cdot\Big{(}\frac{x}{c}\Big{)}^ {i}\cdot\Big{(}1-\frac{x}{c}\Big{)}^{N-i}\Big{]} \tag{3}\]  
where \(C(x/c)\) is the class function of the leaf shape; \(N_{1}\) and \(N_{2}\) indicate the type of leaf shape. When \(N_{1}\) = 0.5, \(N_{2}\) = 0.5, the category function is expressed as an elliptical airfoil with a round nose and round tail. When \(N_{1}\) = 0.75, \(N_{2}\) = 0.75, the category function is represented as a Sears-Haack body, as shown in Figure 4 below. \(S(x/c)\) is the shape function of the airfoil, \(\sum\limits_{i=0}^{N}\Big{[}v_{i}\cdot\Big{(}\frac{x}{c}\Big{)}^{i}\Big{]}\) controls the curve shape of the airfoil from the leading edge to the trailing edge, and \(\frac{z_{TE}}{c}\) is the thickness of the trailing edge of the airfoil, which can be determined according to the design requirements. The airfoil of the multi-blade centrifugal fan can be considered to have no trailing edge, so the thickness is 0; \(c\) is the chord length of the airfoil. The Q35 fan blade airfoil was decomposed by the CST parameter using a third-order Bernstein polynomial in this study. \(N_{1}\) and \(N_{2}\) were fixed at 0.75 and 0.75 to obtain a curve that is close to the original impeller. \[a_{i}(x)=C(x)\cdot S(x) \tag{4}\]  
\[a_{i}(x)=x^{0.75}\cdot(1-x)^{0.75}\cdot\sum_{i=0}^{n}v_{i}x^{i}(1-x)^{n-i} \tag{5}\]  
During the entire optimization process, the parameterization of the blade profile centerline is the most important. The parameterized blade profile centerline has fixed endpoints M1 and M2 so that the blade chord length \(L\) remains unchanged. Figure 5 shows the Schematic diagram of CST function superimposed blade forming. The value of \(n\) in Formula (5) not only determines the number of functions in the curvature distribution but also ensures smooth adjustment of key points of the leaf shape. An oversize \(n\) value, however, creates an increase in the number of design variables that complicates the optimization process. If the value of \(n\) is too small, the leaf shape cannot be adjusted effectively. After many comparisons, \(n\) = 3 was found to allow the CST function to be expressed as the sum of four divided subfunctions (Figure 5). Figure 6 shows the blade structure of a multi-blade centrifugal fan. The CST parameterization method was used to fit the airfoil of the multi-blade centrifugal fan to validate the proposed method.