## 4 Conclusions  
Currently approximation functions are used in algorithms for calculating the flow rate of centrifugal pumps. Polynomials are often used; however, their parameters are not physically interpretable. When calculating the set of performance curves, the affinity laws are primarily employed by the approximation functions. This approach leads to deviations, particularly when the pump speed is significantly reduced. Procedures to improve the calculation of the set of performance curves often require additional measurements. In addition, the approximation functions cannot calculate changes in the set of performance curves due to prewhirl. Through the utilization of the developed physical model to depict the set of performance curves, enhancements can be achieved in the calculation of these curves across a broad speed spectrum. Moreover, the model's physical interpretability enables one to adjust the set of performance curves to account for the influence of prewhirl. The findings of this study can be summarized as follows:  
* The physical model relies on established empirical equations. The model can be adapted to specific pump performance curves via parameter tuning. A modified Levenberg-Marquardt method is used to fit the tuning vectors.
* The accuracy of mapping the performance curve by the physical model is in the range of fourth-degree polynomial accuracy. However, in comparison to the polynomial approximation, the model can be parameterized with significantly fewer measurement points.
* Due to the separate consideration of the mechanical losses of the pumps, the physical model offers superior accuracy in flow rate determination compared to the affinity  
Figure 7: (**a**) Spectrogram of the error \(|Q-Q_{calc}|\) with variation in the flow rate and the prewhirl angle for PO001; (**b**) Spectrogram of the error \(|Q-Q_{calc}|\) with variation in the flow rate and the prewhirl angle for PO002.  
laws. Deviations of less than 2% can be achieved. It can be concluded that the need for additional measurements can be avoided using a physical model.
* It is possible to calculate the changes in the set of performance curves resulting from prewhirl. This also resulted in deviations of less than 2% when determining the flow rate. The improved accuracy of the physical model in determining the flow rate is due to its ability to reproduce the set of performance curves with minimal deviation despite the large number of influencing variables.  
Methodology N.R.; Software N.R.; Validation N.R.; Formal analysis N.R.; Investigation N.R.; Writing--original draft N.R.; Writing--review and editing G.M. and P.J.K.; Visualization 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 Herbormer 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 a potential conflict of interest.