## 2 Derivation of the Physical Model of the Set of Performance Curves  
### Derivation of the Physical Model Equations  
The functional relationships of the pump performance curve (\(P_{K}\)) are represented by a physical model expressed as \(P_{K}:\mathbb{R}^{2}\rightarrow\mathbb{R}\). The physical model is adapted to a specific pump with the help of a tuning parameter vector (\(t\)). The derivation of the physical model starts with the calculation of the specific impeller work (\(\tilde{Y}\)) using Equation (2), known as the Euler Equation for turbomachinery. The left term represents the whirl at the impeller blade outlet, consisting of the circumferential speed (\(u_{2}\)) and the circumferential component of the absolute speed (\(c_{\mu 2}\)). Similarly, the right term describes the whirl at the inlet to the impeller blade with the circumferential speed (\(u_{1}\)) and the circumferential component of the absolute speed (\(c_{\mu 1}\)). \[\tilde{Y}:=c_{\mu 2}u_{2}-c_{\mu 1}u_{1} \tag{2}\]  
The circumferential speed at position \(i\) of the pump impeller (\(u_{i}\)) is generally determined from Equation (3) if the pump speed \(n\) and the diameter \(d_{i}\) are known. \[u_{i}:=\pi\;n\;d_{i} \tag{3}\]  
In Figure 0(a), a meridian view of an impeller for a radial centrifugal pump is presented. Index 1 at the parameters denotes the inlet to the impeller blade. Index 2 indicates the exit of the impeller blade. Figure 0(b) illustrates the velocity triangle, showing the velocity components and angles at each location within the pump impeller. The absolute velocity (\(c\)) splits into its circumferential velocity (\(u\)) and relative velocity (\(w\)) components. The absolute velocity can be further decomposed into a circumferential component (\(c_{u}\)) and a meridional component (\(c_{m}\)). The meridian portion refers to the portion of the absolute velocity that follows the flow along the meridian section. Similarly, the circumferential component refers to the portion of the absolute velocity magnitude that is parallel to the circumferential velocity direction. The relative velocity describes the flow velocity relative to the rotating pump impeller. The angle \(\alpha\) is the angle between \(u\) and \(c\) and characterizes the orientation of the absolute flow to the pump impeller. Angle \(\beta\) quantifies the blade angle. After several transformations, Equation (2) can be expressed as follows, assuming an incident flow without prewhirl (\(c_{u1}=0\)):  
\[\tilde{Y}=\left(\gamma-\frac{\tilde{Q}}{\pi(b\;d\;u)_{2}}\text{cot}(\beta_{2B} )\right){u_{2}}^{2}. \tag{4}\]  
The inclusion of the power factor (\(\gamma\)) considers the variation between the actual flow and the flow congruent with the blades. When considering a flow congruent with the blades, it is assumed that the pump impeller has an infinite number of blades (denoted by the index \(\infty\)) [10]. The flow then precisely follows the shape of the blade contour. The power factor is defined by Equation (5) [10]. \[\gamma:=1-\frac{c_{u2\infty}-c_{u2}}{u_{2}} \tag{5}\]  
Considering the power factor, the blade angle at the outlet (\(\beta_{2B}\)) can be used in Equation (4). To estimate the power factor \(\gamma\in[0;1]\), empirical equations such as in [10; 11] can be used. In Equation (4), \(\tilde{Q}\) is the flow rate delivered by the impeller. This exceeds the pump flow (\(Q\)) due to the presence of the gap flow (\(Q_{Sp}\)):  
\[\tilde{Q}:=Q+Q_{Sp}. \tag{6}\]  
This is due to the backflow from the blade outlet to the blade inlet via the side spaces between the pump impeller and pump casing.