## 4 Development of a Performance-Based Design Method Using Airfoil Cascades  
### Design Parameters for the Optimum Design of Airfoils  
To enhance the operational efficiency of the current fan configuration, a systematic approach is employed to optimize the geometry of the airfoil blades. This optimization is conducted through a rigorous two-dimensional airfoil cascade analysis. Specifically, the focal region for the optimization lies in the mid-span segment of the blade, as depicted in Figure 6. The selection is made based on the predicted streamline analysis, resulting in the identification of three crucial strips located at the radial positions of r/R: 0.45, 0.65, and 0.8. The inlet and outlet camber angles establish the primary design parameters that govern the airfoil shape. These angles play a critical role in regulating the inflow conditions, mitigating adverse pressure gradients along the suction surface of airfoil and managing the formation of trailing edge vortices. The optimization procedure utilizes the central composite design response surface methodology to determine the optimal combination of design parameters that results in enhanced performance. The response surface methodology (RSM) has been developed based on the research conducted by [PERSON] and [PERSON] [16]. A wide range of research publications and academic textbooks explore the foundational principles of the RSM [17; 18; 19; 20; 21]. Operating as a collection of mathematical and statistical techniques, the RSM operates with the purpose of analyzing lies in optimizing the relationships between the response variables and a set of quantitative experimental variables or factors. Under the assumption of \(f\) being differentiable \(d\) times, the Taylor polynomial of degree \(d\) is employed to approximate the function within a local vicinity of a point. Practical considerations limit the use of polynomials exceeding the second degree. This second-order model effectively approximates the response surface in a localized domain centered around a point of interest. The mathematical relationship between the expected response value and the experimental variables is of great importance when determining the optimal operating conditions to achieve a maximum yield. The toolkit of the RSM serves multiple purposes, encompassing the identification of factor settings that yield the desired responses, alignment of factors with management specifications, and the construction of models illustrating the nexus between the quantitative aspects and responses. Central composite designs (CCDs) emerge as the most prevalent response surface designs [22; 23; 24; 25]. In two dimensions, CCDs incorporate factorial points (\(2k\), where \(k\) denotes the number of factors, each featuring two levels), axial points (positioned at a specific alpha from the design centre along each axis, designed to capture curvature, \(\alpha=(2k)^{1/4}\)), and the center point (typically 5 in a number of \(k=2\)). The aggregate of the factorial, axial, and center points determines the total number of experimental runs, amounting to \(13\) (\(4+4+5\)) for designs involving \(k=2\). In the context of response surface designs, distinct low and high levels (\(-\alpha\) and \(+\alpha\)) are defined for each factor. The experimental setup is outlined in Table 1, where the independent variables are specified alongside their respective levels--these levels, spanning approximately 7 degrees from the prevailing design value, factor in the geometrical constraints. The objective function adopted to gauge  
Figure 6: Three radial strips of the blade and the design parameters of the sectional airfoil for optimization. the performance for optimization is the lift-to-drag (L/D) ratio. The relationship between the lift force produced by the airfoil and the drag force opposing the forward motion of the airfoil through the flow is denoted by the L/D. Prior to reaching the stall condition, a higher L/D typically denotes better efficiency in producing lift with less resistance or drag.