## 3 Optimization of the Blade Profile  
In this study, to design the optimum airfoil of the plenum fan, the blade was divided into four layers, and the optimum airfoil was designed for each layer. Figure 6 shows a three-dimensional curved shape and an image of the blade with four layers. The airfoil was designed using NACA (National Advisory Committee for Aeronautics) four-digit. The basic design parameters required to design the airfoil are shown in Figure 7. \(D_{1}\) and \(D_{2}\) are the values that determine the radial positions of the leading and trailing edges, and can be defined as the  
Figure 4: (**a**) Schematic diagram of multi-nozzle chamber; (**b**) Test scene. Figure 5: Comparison between the numerical and experimental results (at 1100 rpm). Figure 6: Layer definition of the plenum fan blade. inner and outer diameters. \(\gamma\), \(\alpha\), and \(\beta_{1}\) are all angles, where \(\gamma\) is the relative position of the airfoil's leading edge, \(\alpha\) is the code angle corresponding to the airfoil length, and \(\beta_{1}\) is the incidence angle. Note that when defining \(\beta_{1}\), the minimum value of \(\beta_{1}\) (\(\beta_{1,min}\)) is determined using \(D_{1},D_{2}\), and \(\alpha\). This value can be calculated using the following equations:  
\[c=\sqrt{\left(\frac{D_{1}}{2}\right)^{2}+\left(\frac{D_{2}}{2}\right)^{2}-2 \frac{D_{1}}{2}\frac{D_{2}}{2}\cos(\alpha)} \tag{6}\]  
\[2\frac{D_{1}}{2}c\cos(\beta_{1,min}+90)=\left(\frac{D_{1}}{2}\right)^{2}+l_{c }^{2}-\left(\frac{D_{2}}{2}\right)^{2} \tag{7}\]  
where \(c\) is the code length. \(\theta_{1},\ \theta_{2},\ \text{and}\ \varphi\) are defined using the geometric relationship as follows:  
\[\varphi=\cos^{-1}\left[\frac{C^{2}+\left(\frac{D_{2}}{2}\right)^{2}-\left( \frac{D_{1}}{2}\right)^{2}}{2c\left(\frac{D_{2}}{2}\right)}\right]\theta_{1}= \alpha+\varphi-\left(\frac{\pi}{2}-\beta_{1}\right)\theta_{2}=\frac{\pi}{2}- \beta_{2}-\varphi \tag{8}\]  
The thickness distribution for the NACA four-digit section was selected to correspond closely to that for the wing sections and is given by:  
\[y_{t}=\frac{t}{0.2}c\left[0.2969\sqrt{\frac{x}{c}}-0.1260\left(\frac{x}{c} \right)-0.3516\left(\frac{x}{c}\right)^{2}+0.2843\left(\frac{x}{c}\right)^{3} -0.1015\left(\frac{x}{c}\right)^{4}\right] \tag{9}\]  
where c is the code length, \(x\) is a position along the code from 0, \(y\) is half the thickness at the \(x\) position, and \(t\) is the maximum thickness. In this study, the iteration method was introduced to optimally design the plenum fan with a three-dimensional blade. The flowchart of the iteration method used in the calculation is shown in Figure 8. The Taguchi method [20,21] was used for initial factor selection as a powerful statistical tool for designing an experiment utilizing orthogonal array tables. Orthogonal arrays that depend on the number of factors and levels are used to study parameter space. Numerous input variables can be assessed to determine their contribution to the output response variable using a fewer number of experimental runs. In this study, an experimental plan was established using the Taguchi method to select the initial factor values for each layer. Experimental plans were established using L12 (\(2^{11}\)) orthogonal arrays in Layers 1 and 4, and L16 (\(2^{12}\)) orthogonal arrays in Layers 2 and 3. Figure 7: Airfoil design parameters. After selecting the initial values using the Taguchi method, the response surface method (RSM) was used to optimize the variables selected as the main factors through main-effect analysis. In using the RSM, the dependent variable to be optimized was selected efficiently. The parameters of Layers 1, 4, and 2, 3 were simultaneously considered to attain the initial values for each layer. Layer 1 was selected as the plane where the blade met the shroud, and consisted of six design variables: \(\alpha\), \(\gamma\), \(\beta_{1}\), \(D_{1}\), \(D_{2}\), and height. Layer 4 was selected as the plane where the blade met the back plate.