## 4 Parametric Study  
The impeller is a key element of a centrifugal pump, responsible for converting the mechanical rotational energy into the kinetic and potential energy of the fluid flow. For a detailed analysis of its operation, hydrodynamic modeling of a single blade sector, without taking into account the volute, was conducted at a flow rate of \(Q=300\) m\({}^{3}\)/h and a rotation speed of \(n=1500\) rpm. The modeling results showed that the pressure created by the impeller was \(H=36.9\) m, and the efficiency coefficient (EC) reached \(\eta=95.7\%\). These data confirm the high efficiency of the impeller under the specified conditions, which indicates the correct design and optimal hydrodynamic characteristics. Then, a parametric analysis of the impeller was carried out, during which the geometric parameters of the blades were varied (see Table 4), including the blade LE angles (\(\beta_{1b}\), \(\beta_{1 bsh}\), \(\beta_{1 bh}\)), blade TE angle (\(\beta_{2b}\)), blade wrap angles (\(\varphi\), \(\varphi_{sh}\), \(\varphi_{h}\)), and the TE shape (Figure 8). The analysis is aimed at assessing the influence of these parameters on the hydraulic characteristics, such as the pressure and efficiency. The initial parameters of the impeller were set as follows: blade LE angle \(\beta_{1b}=20^{\circ}\), \(\beta_{1 bsh}=14^{\circ}\), \(\beta_{1 bh}=32^{\circ}\); blade TE angle \(\beta_{2b}=34^{\circ}\); blade wrap angles \(\varphi_{sh}=112.2^{\circ}\), \(\varphi_{h}=127^{\circ}\), \(\varphi=119.2^{\circ}\); the TE shape was defined as \"Simple\". Changing these parameters allows for a deeper understanding of  
Figure 6: Pressure contour distribution in the radial cross section of a pump (**a**) and impeller (**b**). Figure 7: Velocity contour distribution in the radial cross section of a pump (**a**) and impeller (**b**). Figure 10 analyzes the influence of the blade TE shape on the impeller hydraulic characteristics. Three alternative shapes were considered: ellipse, Bezier, and linear. All the new shapes demonstrated a lower pressure compared to the original shape (simple). The maximum pressure drop of 1.57 m was recorded for the ellipse shape, while the minimum drop of 0.53 m was observed for the linear shape. Despite this, the efficiency of all three shapes was higher than the original, with the linear shape demonstrating the greatest increase in efficiency by 0.76%. Figure 11 shows the blade loadings at 20%, 50%, and 80% span. In each case, despite the blade profile changes, there is a tendency for the pressure to increase from an initial value of 0.3 \(\times\) 10\({}^{5}\) Pa (0.3 atm) at the inlet to more than 3 \(\times\) 10\({}^{5}\) Pa at the blade tip. Figure 11: Blade loadings. (**a**) Original impeller; (**b**) \(\beta_{1b}=14^{\circ}\), \(\beta_{1 bsh}=8^{\circ}\), \(\beta_{1 bth}=16^{\circ}\); (**c**) linear TE form. Figure 10: Comparison of the hydraulic performances of impeller. (**a**) Head (\(H\))—TE form; (**b**) efficiency (\(\eta\))—TE form. A significant pressure drop is recorded at the leading edge of the blade. For the original impeller, these drops are \(-\)148,065 Pa, \(-\)204,238 Pa, and \(-\)261,739 Pa at 20%, 50%, and 80% of the hub, respectively (Figure 10(a)). This phenomenon is due to the increase in the fluid velocity caused by acceleration near the blade inlet. The minimum pressure drop values of \(-\)168,964 Pa, \(-\)182,457 Pa, and \(-\)214,801 Pa were recorded at the leading edge at the inlet angles \(\beta_{1b}\) = 14\({}^{\circ}\), \(\beta_{11 sh}\) = 8\({}^{\circ}\), and \(\beta_{1 bh}\) = 16\({}^{\circ}\) (Figure 10(b)). A pressure drop of up to 2 \(\times\) 10\({}^{5}\) Pa is also observed at the trailing edge, which is associated with the formation of a flow separation zone and local liquid recirculation. Changing the shape of the blade trailing edge resulted in a reduction in the pressure drop at the blade exit, which contributed to flow stabilization at the outlet (Figure 10(c)).