## 3 Analysis and Discussion of the Results  
### Analysis and Discussion of the Experimental Results  
Figure 7 illustrates the characteristic curves of the head plotted against the flow rate for PO001 (figure above) and PO002 (figure below). The circles (\(\Delta\alpha_{\rm pb}=0^{\circ}\) and \(\Delta\alpha_{\rm pb}>0^{\circ}\)) and crosses (\(\Delta\alpha_{pb}<0^{\circ}\)) represent the measured data points, which were connected linearly. The characteristic curves for the prewhirl body with \(\Delta\alpha_{\rm pb}=0^{\circ}\) are depicted in red, while all characteristic curves measured with \(\Delta\alpha_{\rm pb}>0^{\circ}\) and \(\Delta\alpha_{\rm pb}<0^{\circ}\) are shown in black. The characteristic curves are arranged in accordance with the motor frequencies (\(f_{\rm M}\)) indicated on the VFD. As the measured speed (\(n_{\rm exp}\)) is dependent on the power consumption, the value of \(f_{\rm M}\) was specified. The characteristic curves of the heads are demonstrably influenced by the prewhirl body angle. As illustrated in Figure 2b, the circumferential components of the absolute velocity at the impeller inlet (\(c_{\rm u1}\)) are subject to influence from the prewhirl angle. It can be demonstrated that positive prewhirl angles (\(\Delta\alpha_{\rm 1}\)) result in a reduction of the Euler equation of turbomachines, as shown in Equation (5) [27]. Conversely, when negative prewhirl angles are present, an increase in the specific impeller work (\(\overset{\sim}{Y}\)) is observed. The velocity components \(u_{\rm 1,m}\) and \(c_{\rm u1,m}\) are defined as the velocity components on the mean meridian streamline of the impeller. \[\overset{\sim}{Y}=u_{\rm 2}c_{\rm u2}-u_{\rm 1,m}c_{\rm u1,m} \tag{5}\]  
\[H=\overset{\sim}{Y}\eta_{\rm h} \tag{6}\]  
Figure 7: Characteristic curves of the head (\(H_{\rm exp}\)) plotted against the flow rate (\(Q_{\rm exp}\)) for PO001 (top figure) and PO002 (bottom figure) with varying motor frequencies (\(f_{\rm M}\)) and varying inflow without (\(\Delta\alpha_{\rm pb}=0^{\circ}\)), with positive (\(\Delta\alpha_{\rm pb}>0^{\circ}\)) and with negative (\(\Delta\alpha_{\rm pb}<0^{\circ}\)) prewhirl angles. The head of the pump is adjusted according to Equation (6) [27]. In Equation (6), \(\eta_{\rm h}\) represents the hydraulic efficiency of the pump which includes all hydraulic losses in the pump. It can be observed that while the head decreases continuously with positive prewhirl angles (\(\Delta\alpha_{\rm pb}>0^{\circ}\)), the increase in head appears to reach a plateau with increasing negative prewhirl angles (\(\Delta\alpha_{\rm pb}<0^{\circ}\)). This phenomenon is more pronounced in the case of PO002 than in the case of PO001. As indicated by Equation (6), this suggests a variable hydraulic efficiency. This phenomenon can be attributed to the occurrence of varying flow losses within the pump. A further investigation of the hydraulic losses in the pump can be conducted by analysing the pressure in the impeller back side gap (\(p_{\rm PSR,exp}\)). Figure 8 illustrates the ratio of the static pressure difference between the static pressure at the pressure side (\(p_{\rm ps,exp}\)) and the static pressure in the back side gap (\(p_{\rm PSR,exp}\)) for PO001 at varying motor frequencies (\(f_{\rm M}\)). The representation of the characteristic curves is consistent with the depiction in Figure 7. At all motor frequencies, the pressure ratio between the individual characteristic curves exhibits only slight deviations when varying \(\Delta\alpha_{\rm pb}\). The back side gap is directly coupled to the impeller outlet. The pressure in the back side gap can be calculated using Equation (7) [27]. \(c_{\rm p,RSR}\) denotes the pressure reduction coefficient [27], which quantifies the pressure loss between the pressure at the impeller outlet (\(p_{2}\)) and \(p_{\rm RSR}\). The value of \(p_{\rm RSR}\) therefore depends on \(p_{2}\), \(u_{2}\) and \(c_{\rm p,RSR}\). It can be demonstrated that variations in \(p_{\rm RSR}\) are attributable to \(p_{2}\) at a constant speed and geometry.