## 4 Results and Discussion  
### Sensitivity Analysis  
Figure 7 shows the generated response surface for sensitivity analysis of the design variables. The x-axis and y-axis represent the elliptic aspect ratios of the leading and trailing edge of the blade, respectively, while the z-axis indicates the power, head, and efficiency from left to right. Between the elliptic aspect ratios of the leading and trailing edge of the blade, the more sensitive design variable in terms of performance is the latter, and there is a tendency that as the blade edges sharpen, the power and head decrease. In addition, the response surface in the region of 2-4 for \(\mathrm{A}_{\mathrm{LE}}\) is rather steep, especially for the efficiency. This can be considered for several reasons. Here, the figures used in the explanation are represented at 50% span which is the middle of the hub and shroud (see Figure 8). First, for the blunt leading edge (particularly \(\mathrm{A}_{\mathrm{LE}}\) = 2 or 3), the flow separation occurs due to the strong adverse pressure (see Figure 9). This is because of the drastic change of curvature radius. Furthermore, the sharper the trailing edge, the more the flow separation is delayed (see Figure 10), and the lower the pressure on the pressure side (see Figures 11 and 12). In addition, in the case of the leading edge, drastic variation of the pressure is inevitably formed due to the stagnation point, regardless of the change of the leading edge shape of the blade (see Figure 13). The sharper the leading edge, the greater the tendency of the drastic pressure fluctuation on the suction side to be resolved, but the more insignificant the overall pressure difference between the pressure and suction sides becomes (see Figure 14). Moreover, the results can be considered using the velocity triangle at the inlet and outlet of the blade. The water flows in a straight direction until just before the runner entrance, due to the absence of a guide vane. Figure 15 shows that the absolute velocity and blade linear velocity are perpendicular at the leading edge. Equation (7) indicates [PERSON]'s turbine equation, which enables calculation of the theoretical turbine output based on [PERSON]'s second law of motion [46]:  
\[P=\rho Q(\mathbf{V_{1}}\cdot\mathbf{U_{1}}-\mathbf{V_{2}}\cdot\mathbf{U_{2}}) \tag{7}\]  
Figure 6: Process of genetic algorithm.  
Figure 8: Location of 50% span.  
Figure 9: Velocity vector distributions at the leading edge at 50% span of the different elliptic aspect ratios of the leading edge (**a–c**).  
Figure 11: Pressure distribution at 50% span of the different elliptic aspect ratios of the trailing edge (**a**,**b**).  
Figure 12: Blade loading at 50% span of the different elliptic aspect ratios of the trailing edge.  
Figure 10: Streamlines at 50% span of the different elliptic aspect ratios of the trailing edge (**a**,**b**).  
In the equation, \(\mathbf{V_{1}}\cdot\mathbf{U_{1}}\), corresponding to the velocity component at the leading edge, has the value of 0, due to the orthogonality. As a result, the variation of the curvature of the leading edge configuration does not significantly affect the performance.  
For these reasons, there are many studies focused on blade trailing edge rather than leading edge to improve blade performance. For a case in point, there is a study that modified the blunt trailing edge thickness of the airfoil to improve the aerodynamic performance of a wind turbine [47]. In addition, the blade trailing edge profile also has a significant effect on pump performance, and there is a case where pressure pulsations in  
Figure 14: Blade loading at 50% span of the different elliptic aspect ratios of the leading edge.  
Figure 13: Stagnation point at 50% span of the different elliptic aspect ratios of the leading edge (**a,b**).  
Figure 15: Velocity triangle at the leading and trailing edge of the blade.  
a centrifugal pump were reduced through modification of the blade trailing edge profile [48].