## 3 Optimization of the Blade Profile
It consisted of five design variables: \(\alpha\), \(\gamma\), \(\beta_{1}\), \(D_{1}\), and \(D_{2}\). To select the major design variables, L12 orthogonal arrays were used for 11 factors, and screening analysis was performed by computational simulation. The main-effect analysis results of Layer 1 and Layer 4 are shown in Figure 9a, and the effects of \(\alpha\) at Layer 1 and \(D_{1}\), \(D_{2}\), and \(\gamma\) at Layer 4 were significant. The RSM was applied to the first optimization in Layers 1 and 4. Four factors that showed a significant effect by main effect analysis were selected as factors of the RSM, and each factor was divided into three levels. Table 2 shows each factor and level. In the design of the experiment, central composite design (CCD) [22] was applied. In Layers 2 and 3, we selected an arbitrary cross section of the blade, and these two layers consisted of six design variables such as \(\alpha\), \(\gamma\), \(\beta_{1}\), \(D_{1}\), \(D_{2}\), and height. To select the major design variables, L16 orthogonal arrays were used for 12 factors, and screening analysis was performed by computational simulation. The main-effect analysis results of Layer 2 and Layer 3 are shown in Figure 9b, and the effects of \(\gamma\) at Layer 2 and \(\gamma\), \(\beta_{1}\), and \(D_{2}\) at Layer 3 were significant. The second optimization in Layers 2 and 3 was performed by applying the RSM in the same way as the first optimization. After determining the initial shapes of Layers 2 and 3, four factors that showed significant effects in the main effect analysis were selected as factors of the RSM, and each factor was divided into three levels. The values of Layers 1 and 4 were used to optimize the results from the first optimization. Table 2 shows each factor and level. Figure 4:0 Flowchart of the iteration method for blade optimization. After the first and second optimizations, the shape of the initial model could be determined. Once the initial shape was determined, the optimal design values were obtained from Layer 1 using the RSM. The iteration was performed by applying this value to the optimization of Layer 2. The calculations were repeated and finally converged to the target static pressure efficiency of 78% to complete the optimization. The optimal design values of the blades designed by the iteration method are shown in Figure 4:1 The blade shapes of the original and optimized models are shown in Figure 4:2 \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Layer**} & \multirow{2}{*}{**Factor**} & \multicolumn{3}{c}{**Level**} & \multirow{2}{*}{**Layer**} & \multirow{2}{*}{**Factor**} & \multicolumn{3}{c}{**Level**} \\ \cline{3-3} \cline{5-8}  & & \(\mathbf{-1}\) & \(\mathbf{0}\) & \(\mathbf{1}\) & & \(\mathbf{-1}\) & \(\mathbf{0}\) & \(\mathbf{1}\) \\ \hline \multirow{4}{*}{Layer 1} & \(\alpha\) (Layer 1) & 20 & 35 & 50 & \multirow{4}{*}{Layer 2} & \(\gamma\) (Layer 2) & 20 & 35 & 50 \\ \cline{2-3} \cline{5-8}  & \(D_{1}\) (Layer 4) & 200 & 300 & 400 & & & & \(\beta_{1}\) (Layer 3) & 5 & 15 & 30 \\ \cline{2-3} \cline{5-8}  & \(D_{2}\) (Layer 4) & 450 & 500 & 550 & & & & \(D_{2}\) (Layer 3) & 450 & 500 & 550 \\ \cline{2-3} \cline{5-8}  & \(\gamma\) (Layer 4) & 20 & 35 & 50 & & & \(\gamma\) (Layer 3) & 20 & 35 & 50 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Factors and levels.