## 4 Development of a Physical Model for the Head Characteristic Curve  
### Derivation of the Model  
In a previous publication, we have already derived a model for determining the specific impeller work \(\widetilde{(\gamma)}\)[26]. It is calculated according to Equation (18) [26]. The angle of the impeller blade at the impeller outlet is designated as \(\beta_{\text{2B}}\). The two tuning factors, \(t_{0}\) and \(t_{2}\), are employed to align the specific impeller work with the respective pump. \[\widetilde{Y}=\left(\gamma+t_{0}-\frac{Q+t_{2}\sqrt{H}}{\pi\left(b\ d\ u \right)_{2}}\left(\cot(\beta_{\text{2B}})+\frac{4b_{2}d_{\text{1,m}}}{d_{\text {1a}}^{2}-d_{\text{1i}}^{2}}\tan(\Delta\alpha_{1})\right)\right)u_{2}^{2} \tag{18}\]  
In addition to the specific hydraulic work \((Y)\), the specific impeller work also encompasses the hydraulic losses that occur in the pump. As demonstrated by Equation (19), the specific hydraulic work is calculated by subtracting the total hydraulic losses from the specific impeller work. In addition to the mentioned losses in the impeller (\(Z_{\text{LN,St}},\ Z_{\text{LN,Ka}}\)), losses also occur in the volute. These are divided into two categories: friction losses (\(Z_{\text{GL,R}}\)) and shock losses (\(Z_{\text{GL,St}}\)). During partial load operation, a recirculation area develops in front of the pump impeller [27]. This can result in an additional prewhirl, which typically results in a reduction in the head [27]. For this reason, partial load recirculation is considered a loss and incorporated into Equation (19) as recirculation loss (\(Z_{\text{Rez}}\)). \[Y=\widetilde{Y}-Z_{\text{LN,St}}-Z_{\text{LN,Ka}}-Z_{\text{GL,St}}-Z_{\text{ GL,R}}-Z_{\text{Rez}} \tag{19}\]  
In order to map the head characteristic curve using the approach outlined in Equation (19), it is necessary to model the individual loss components. The loss models are considered in accordance with physical approaches and interpretable tuning factors (\(t_{i}\)). This extends the model presented earlier [26], which considers the tuning factors \(t_{i}\in L\;\;i\in\{0;4\},\;i\in\mathbb{N}\) in the tuning vector (\(\underline{t}\)). We identified tuning factors by the Levenberg-Marquardt method with barrier terms [26], the same approach is used to identify the new tuning factors. The preceding chapters have demonstrated that the relative velocity plays an important role. This not only affects the friction losses but also the Coriolis acceleration and the associated secondary flows. This implies that the relative velocity must be incorporated into the loss models for the impeller. A methodology for modelling friction and turbulence losses was presented by [PERSON] [27]. The relationship \(Z_{\text{LN,Ka}}\propto w_{\text{m}}{}^{2}\) can be derived from this model. Upon consideration of the tuning factor \(t_{5}\), the resulting equation is Equation (20). The mean relative velocity in the impeller (\(w_{\text{m}}\)) is calculated using Equation (21) [27]. \[Z_{\text{LN,Ka}}=t_{5}w_{\text{m}}{}^{2} \tag{20}\]  
\[w_{\text{m}}=\frac{w_{\text{1,m}}+w_{2}}{2} \tag{21}\]  
In addition to the influence of the relative velocity on the secondary flows, an influence on the shock losses may also be determined. [PERSON] developed a model for calculating the shock losses (\(Z_{\text{LN,St}}\)) [34], as follows:  
\[Z_{\text{LN,St}}=\varphi_{\text{St}}\frac{w_{\text{s}}{}^{2}}{2}. \tag{22}\]  
The shock factor, denoted by \(\varphi_{\text{St}}\), is assumed to lie within the interval \(\varphi_{\text{St}}\in\{0.5;0.7\}\)[34]. The shock velocity (\(w_{\text{s}}\)) included in Equation (22) forms an impulse when the vector direction of the relative velocity deviates from the blade entry angle. Equation (22) and the named relationship \(Z_{\text{LN,St}}\propto w_{\text{1}}{}^{2}\)[27] demonstrate the parabolic trajectory of shock losses. Nevertheless, the superimposed offset of the shock losses, as illustrated in Figure 12a, is not reflected in the models.