## 2 Derivation of the Physical Model of the Set of Performance Curves
### Experimental Setup to Measure the Set of Performance Curves by Varying the Prewhirl Angles  
To validate the physical model, measurements were conducted on two radial centrifugal pumps. The pumps are denoted as test objects (PO) in the following context. They exhibited varying specific speeds. [PERSON] et al. [12] validated their approach based on data from [PERSON] et al. [26]. These data can be obtained by measuring the set of characteristic curves of a centrifugal pump subjected to a significant speed reduction. Incorporating inequality constraints into the parameter identification process for tuning parameters enables the development of a physical model, even when assuming pump parameters. This model's validation will be carried out using the data from [26]. Table 1 shows the parameters of the three PO. PO003 indicates the pump measured at [26]. As explained in the introductory section, centrifugal pump characteristics can be affected by prewhirl. The PO001 and PO002 characteristics with prewhirl were measured by using prewhirl bodies. This approach was proposed by [PERSON] [21]. The experimental setup for the measurements is shown in Figure 2. A torque-measuring shaft is installed between the pumps and their motor. Through this, the pump power can be measured. The pump head can be determined by measuring the differential pressure across the pump (\(\Delta p_{P}\)) in accordance with Equation (24). The differential pressure is determined using a differential pressure gauge. The velocities of the flow on both the pressure and suction sides are computed using the measured flow rate (\(Q_{exp}\)) and the cross-sectional areas of the pipes. A magnetic inductive flow meter (MID) is used to measure \(Q_{exp}\). The flow rate can be adjusted by regulating a control valve. Moreover, the difference in geodetic height (\(\Delta h\)) between the differential pressure gauge and the pump's centreline is accounted for. As depicted in Figure 2, the pressure loss across the prewhirl bodies (\(\Delta H_{pb}\)) is factored into the head calculation. Both the pressure losses and the effective prewhirl angles following the existence of the prewhirl bodies were determined through a CFD simulation. \[H_{exp}:=\frac{\Delta p_{p}}{sp}+\frac{v_{p}^{2}-v_{s}^{2}}{2g}+\Delta h+ \Delta H_{pb} \tag{24}\]  
The prewhirl angle was altered within a range of \(\Delta a_{1}\in[-34^{\circ};34^{\circ}]\). Additionally, the pump's speed can be modified. It varies within the interval \(n\in\left[1180\,\text{min}^{-1};1475\,\text{min}^{-1}\right]\) by a variable frequency converter (VFD). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **PO001** & **PO002** & **PO003** \\ \hline specific speed (\(n_{q}\)) & 43 min\({}^{-1}\) & 54 min\({}^{-1}\) & 24 min\({}^{-1}\) \\ outer diameter inlet (\(d_{1a}\)) & 124 mm & 122 mm & 65 mm \\ inner diameter inlet (\(d_{1i}\)) & 46 mm & 47 mm & 23 mm \\ diameter outlet (\(d_{2}\)) & 210 mm & 180 mm & 158 mm \\ width impeller outlet (\(b_{2}\)) & 22 mm & 33 mm & 10 mm \\ number of blades (\(z_{L}\)) & 7 & 6 & 6 \\ blade angle outlet (\(\beta_{2B}\)) & 32\({}^{\circ}\) & 35\({}^{\circ}\) & 30\({}^{\circ}\) \\ mechanical power at \(n_{N}\) (\(P_{m}\)) & 330 W & 70 W & 240 W \\ flow rate at BEP (\(Q^{*}\)) & 98.6 \(\frac{\text{m}^{3}}{\text{h}}\) & 113.2 \(\frac{\text{m}^{3}}{\text{h}}\) & 39\(\frac{\text{m}^{3}}{\text{h}}\) \\ head at BEP (\(H^{*}\)) & 10 m & 8.2 m & 30 m \\ pump power at BEP (\(P_{K^{*}}\)) & 3143 W & 3375 W & 4586 W \\ nominal speed (\(n_{N}\)) & 1475 min\({}^{-1}\) & 1475 min\({}^{-1}\) & 2920 min\({}^{-1}\) \\ power factor (\(\gamma\)) & 0.798 & 0.767 & 0.782 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the two measured pumps (PO001 and PO002) and the assumed parameters of the pump in the work of [26] (PO003) for the initialization of the physical model.