## 4 Development of a Physical Model for the Head Characteristic Curve
### Comparison between the Models with Different Loss Treatments and the Experimental Results  
The physical model derived in the previous chapter for mapping the characteristic curves of the head can be utilised to calculate the characteristic curves of radial centrifugal pumps for varying prewhirl angles. For this purpose, the experimentally determined characteristic curves of the two analysed pumps are calculated using the physical model. In contrast, the characteristic curves are calculated using an approach function based on a third-degree polynomial, without modelling the losses. The objective of this analysis is to examine the impact of the loss treatment of the derived model based on the relative velocity in greater detail. Equation (31) shows the formula used to calculate the head on a polynomial basis (\(H_{\text{poly}}\)). The letters \(a,b,c\) and \(d\) represent the individual coefficients of the polynomial. The right-hand term in Equation (31) is employed in order to calculate the influence of the prewhirl on the head. In Equation (31), \(\eta_{\text{Sp}}\) represents the volumetric efficiency used to account for gap flow losses. For this, \(\eta_{\text{Sp}}=0.96=\text{const. }\) is assumed. \[H_{\text{poly}}=aQ^{3}+bQ^{2}+cQ+d-\frac{4 Qd_{1,\text{m}}u_{2}}{g\eta_{\text{ Sp}}\pi d_{2}\big{(}d_{1\text{a}}^{2}-d_{1\text{i}}^{2}\big{)}}\tan(\Delta \alpha_{1}) \tag{31}\]  
In order to assess the deviations between the various models and the experiments, an objective functional of the respective model \(i\) (\(\underline{z}_{i}\)) is defined based on the identification of the tuning vectors by the Levenberg-Marquardt method, as outlined in Equation (32) [26]. The objective functional is a vector with length \(n\). In this context, \(n\) represents the number of measurement points derived from the experimental determination of the characteristic curves. \[\underline{z}_{i}:=\big{(}H_{i,1}-H_{\text{exp},1},H_{i,2}-H_{\text{exp},2}, \cdots,H_{i,n}-H_{\text{exp},n}\big{)}^{T} \tag{32}\]  
To facilitate a comparative analysis of the models for different prewhirl body angles, the objective functional norm for model \(i\) (\(\phi_{i}\)) is formed using the equation proposed by [26]:  
\[\phi_{i}=\|\underline{z}_{i}\|. \tag{33}\]  
The prewhirl angle at the impeller inlet (\(\Delta\alpha_{1}\)) is required for the calculation of the characteristic curves. \(\Delta\alpha_{1}\) was identified by CFD simulations and is presented in Table 4 for the investigated prewhirl body angles. Figure 14 illustrates the characteristic curves of the head for PO001 and PO002 at the prewhirl body angles \(\Delta\alpha_{\text{pb}}\in\{-30^{\circ},0^{\circ};+30^{\circ}\}\). The characteristic curve of the experimentally determined head (\(H_{\text{exp}}\)) is represented by the circles. The blue lines represent the characteristic curves of the head, which have been calculated by the polynomial model derived from Equation (31) (\(H_{\text{poly}}\)). The black characteristic curves of the head have been calculated by the physical model derived in the previous chapter (\(H_{\text{phys}}\)). For both pumps, \(H_{\text{poly}}\) overestimates the head, particularly for negative prewhirl angles. This is because increased hydraulic losses were not taken into account. For PO002, a deviation can also be recognized for a positive prewhirl body angle. It is assumed that the hydraulic losses are too high in this instance. The qualitative analysis indicates that the determination of the head using the derived physical model (\(H_{\text{phys}}\)) more accurately reflects the measurements.