## 2 Derivation of the Physical Model of the Set of Performance Curves
### Derivation of the Physical Model Equations
If we assume that \(c_{u1}\
eq 0\), Equation (4) can be expanded to give the following:  
\[\tilde{Y}=\left(\gamma-\frac{\tilde{Q}}{\pi(b\;d\;u)_{2}}\;\left(\cot(\beta_{2 B})+\frac{4\;b_{2}d_{1m}}{d_{1a}^{2}-d_{1i}^{2}}\;\tan(\Delta\alpha_{1})\right) \right){u_{2}}^{2}. \tag{7}\]  
Figure 1: (**a**) Meridian view of the impeller for a radial centrifugal pump and geometrical parameters, where index 1 denotes the blade inlet, index 2 denotes the blade outlet and the shaded area represents the impeller blade; (**b**) velocity triangle of the impeller showing the different velocity components at positive (black lines) and negative (grey lines) whirl angles (\(\Delta\alpha\)). Equation (7) outlines the inflow conditions when the prewhirl is engaged, with the prewhirl angle (\(\Delta\alpha_{1}\)) serving as the defining parameter, as shown in Figure 1b. The mean diameter at the blade inlet (\(d_{1m}\)) is calculated from Equation (8). \[d_{1m}:=\sqrt{\frac{1}{2}\big{(}d_{1a}^{2}+d_{1i}^{2}\big{)}} \tag{8}\]  
\(\tilde{Y}\) does not include any losses. The flow losses due to hydraulic efficiency (\(\eta_{h}\)) can be considered using Equation (9). The parameter \(Y\) signifies the pump delivery work, which can be transformed into the pump head in accordance with Equation (10), where \(g\) denotes the acceleration due to gravity (\(g=9.81\frac{m}{g^{2}}\)). \[Y:=\tilde{Y}\,\eta_{h} \tag{9}\]  
\[H:=\frac{Y}{g} \tag{10}\]  
Apart from hydraulic losses, the pump also incurs mechanical losses. The mechanical power (\(P_{m}\)) includes the mechanical losses that arise from components like bearings or the mechanical seal. Moreover, recirculation could potentially manifest within the pump. The recirculation efficiency (\(\eta_{Rez}\)) factors in any rise in pump power that could arise from the presence of recirculation zones within the pump impeller. The efficiency \(\eta_{Rez}\) is calculated using Equation (11). In addition, the previously discussed gap loss can be accounted for by the volumetric efficiency (\(\eta_{V}\)), as shown in Equation (12). \[u_{i}:=\pi\;n\;d_{i} \tag{3}\]0  
\[u_{i}:=\pi\;n\;d_{i} \tag{3}\]1  
In addition to the losses mentioned earlier, the pump also requires power for the impeller friction (\(P_{RR}\)). This is caused by friction between the impeller discs and the pump housing. The pump efficiencies, power factor, impeller friction power, mechanical power, and geometric parameters are indispensable for the \(P_{K}(Q,n)\) calculation. The power factor and geometric parameters can both be extracted from design data. Due to the difficulty of determining efficiencies, tuning parameters (\(t\)) are used to adjust efficiencies for a specific pump. To achieve this, knowledge of the pump's discrete measurement points (\(Q_{exp,i}\), \(H_{exp,i}\), \(P_{K,exp,i}\), \(n_{exp,i}\)) is essential. These points can often be extracted from the pump manufacturer's data booklets. Recognizing that the power factor calculation is an approximation, it is necessary to introduce the tuning parameter \(t_{0}\). This estimates the actual reduced power factor by summing it with the calculated reduced power factor. For the gap current (\(Q_{Sp}\)), an empirical equation for calculation is given in [10]. In this equation, the relation \(Q_{Sp}\propto\sqrt{H}\) is specified. When calculating the actual gap current for the pump under certain measurements, it is assumed that  
\[u_{i}:=\pi\;n\;d_{i} \tag{3}\]2  
Since the tuning factors can be identified using the manufacturer's measurements, the prewhirl term with the specification of \(\Delta\alpha_{1}\) in Equation (7) is omitted. Therefore, the specific impeller work can be parameterized with the tuning parameters \(t_{0}\) and \(t_{2}\) to the specific pump using Equation (14). \[u_{i}:=\pi\;n\;d_{i} \tag{3}\]3The wheel friction power is replaced by the parameter \(t_{1}\) (\(t_{1}\triangleq P_{RR}\)). The power of the recirculation (\(P_{Rez}\)) diminishes linearly from \(Q=0\) to \(Q_{Rez,min}\) and becomes negligible at a flow rate (\(Q_{Rez,ic}\)) [10].