## 3 Analysis and Discussion of the Results
### Validation of the CFD Simulation and Analysis of the Loss Behaviour
As the fluctuating components are described in different ways depending on the turbulence model employed in the URANS, there are also different models for the definition of turbulent entropy production. In this study, the SST model was employed, so Equation (12) [22; 32] was used to determine the turbulent entropy production. The quantity \(k\) represents the turbulent kinetic energy, while \(\omega\) denotes the turbulent eddy frequency. \[S_{\rm pro}^{\prime}=0.09\rho\omega k \tag{12}\]The entropy production can be analysed locally or for control volumes. In order to analyse the flow losses in control volumes, the entropy production is integrated over the control volume in accordance with Equation (13) [25]. \[P_{\text{loss}}=\iiint S_{\text{pro}}\;dV \tag{13}\]  
Firstly, the flow losses identified through entropy production are analysed for the volume and the impeller. Figure 10 illustrates the power losses calculated in accordance with Equation (13) for the volume (Figure 10a) and the impeller (Figure 10b). The power losses were quantified for the three prewhirl body angles \(\Delta\alpha_{\text{pb}}\in\{-30^{\circ};0^{\circ};+30^{\circ}\}\). The power  
Figure 9: Comparison of the head (\(H\)) (left-hand side) and the power consumption (\(P_{2}\)) (right-hand side) between the experimental results (Exp) and the simulation (CFD) for the prewhirl body angles \(\Delta\alpha_{\text{pb}}\in\{-30^{\circ};0^{\circ};+30^{\circ}\}\). loss in the volume exhibits a minimum for all three \(\Delta\alpha_{\text{pb}}\) in the range of \(Q=100\frac{\text{m}^{3}}{\text{h}}\). The curvature of the curves for \(Q>100\frac{\text{m}^{3}}{\text{h}}\) is larger than for the smaller flow rates. Moreover, the deviations observed in the analysed \(\Delta\alpha_{\text{pb}}\) for \(Q<120\frac{\text{m}^{3}}{\text{h}}\) are minimal. It can be assumed that the power loss in the volume is independent of the prewhirl angle within the limits of this range. Only at \(Q=140\frac{\text{m}^{3}}{\text{h}}\) do larger deviations occur between the prewhirl body angles. In contrast to the volume, the impeller's curves do not exhibit a minimum. Furthermore, there is a clear correlation between the power loss and \(\Delta\alpha_{\text{pb}}\). While the power losses for \(\Delta\alpha_{\text{pb}}\in\{0^{\circ};+30^{\circ}\}\) exhibit only minor deviations from each other, the losses for \(\Delta\alpha_{\text{pb}}=-30^{\circ}\) exhibit a significant increase. The varying losses in the pump, which depend on the prewhirl angle, can therefore be attributed primarily to the impeller. [PERSON] et al. examined the entropy production of the individual components over the flow rate [23]. At the BEP, they demonstrated that the impeller and housing exhibited a comparable entropy production. This finding is in accordance with the observations presented in Figure 10.  
To gain a more detailed understanding of the losses occurring within the impeller, mass flow averaged values were calculated using the meridional streamlines of the impeller. Figure 11a depicts the mass flow averaged whirl (\(\overline{u}\overline{c_{\text{u}}}\)) within the impeller over the dimensionless streamline length (\(s\)). The curves were subjected to analysis regarding the flow rate at \(Q=100\frac{\text{m}^{3}}{\text{h}}\). As shown in Table 1, for \(\Delta\alpha_{\text{pb}}=0^{\circ}\), this flow rate is the BEP of the pump. At \(\Delta\alpha_{\text{pb}}=0^{\circ}\), whirl is built up from the blade leading edge at \(s=0.15\). For \(\Delta\alpha_{\text{pb}}=+30^{\circ}\), it can be observed that the whirl build-up occurs at a later point, approximately at \(s=0.25\). However, a comparison of the two curves, with \(\Delta\alpha_{\text{pb}}=-30^{\circ}\), reveals that a slight whirl build-up is already evident at \(s<0.15\). Figure 11b illustrates the course of the mass flow averaged relative velocity (\(\overline{w}\)) over \(s\) for all three prewhirl body angles. While the curves for \(s>0.45\) exhibit a high degree of correlation, there are significant discrepancies for \(s<0.45\).