## 5 Conclusions  
A physical model was developed to determine the characteristic curve of the head for radial centrifugal pumps. To assess the impact of varying prewhirl angles on hydraulic losses, experimental and numerical investigations were conducted. The findings from this study enabled the development of modelling approaches for the various sources of loss. As a result, it has been shown that the model can achieve higher accuracies in mapping the characteristics with prewhirl than with a polynomial-based approach. This study addressed the identified research gap by providing a mathematical description of the dependence of hydraulic losses on the pre-twist angle. The following conclusions can be drawn:  
* The hydraulic losses in radial centrifugal pumps are influenced by the prewhirl. The observed changes in losses can be attributed primarily to the pump impeller.
* The extent of the losses depends on the sign of the prewhirl angle. Negative prewhirl angles increase hydraulic losses, while positive prewhirl angles reduce them.
* The majority of the hydraulic losses in the impeller occur at the leading edge of the blades and in the area of the impeller outlet.
* The degree of prewhirl angle has a significant impact on the whirl of the impeller, as well as the relative velocities. The Coriolis force increases in direct proportion to the relative velocity, which can result in the generation of secondary flows. These factors contribute to an increase in the likelihood of loss, particularly at the point of impeller inlet.
* In addition to the secondary flows, the shock losses are also influenced by the prewhirl angles.
* Furthermore, the relative velocities exert an influence on the friction losses in the impeller, whereby these increase with negative prewhirl angles.
* The modelling of impeller losses based on relative velocity allows for the consideration of the differing influences of positive and negative prewhirl angles.
* The model enhances the mapping of the characteristic curves of the head. It can therefore be utilised in the construction of soft sensors for the determination of flow rates in instances of disrupted inflows. This facilitates more accurate flow rate determination.  
Figure 15: (**a**) Norm of the objective functional (\(\phi_{i}\)) of the individual models plotted against the investigated swirl body angles (\(\Delta\alpha_{\text{pb}}\)) of PO001 and PO002; (**b**) norm of the objective functional of the gradient (\(\phi_{\text{grad},i}\)) of the individual models plotted over the investigated prewhirl body angles (\(\Delta\alpha_{\text{pb}}\)) of PO001 and PO002.  
ization N.R., G.M. and P.J.K.; Supervision G.M. and P.J.K.; Project administration G.M. and P.J.K. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
The data presented in this study are available on request from the corresponding author. The data are not publicly available due to private ownership.  
Author [PERSON] was employed by the company Herborner Pumpentechnik GmbH & Co KG. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as potential conflicts of interest.