## 2 Method for Analysing Losses in Centrifugal Pumps with Different Prewhirl Angles
### Experimental Determination of the Head Characteristics with Varying Prewhirl Angles  
In order to validate the results of the numerical computer fluid dynamics (CFD) simulations, the head characteristic curves of two pumps were determined experimentally. The two pumps are referred to below as the test object (PO). Important parameters are summarised in Table 1. The two pumps exhibit different specific speeds and dimensions. In addition to the geometric parameters, the data for the BEP and the performance factor (\(\gamma\)) are also given. The performance factor is defined in accordance with Equation (1) [27]. In Equation (1), \(u_{2}\) represents the circumferential velocity at the impeller outlet, \(c_{\mathrm{u2}}\) denotes the circumferential component of the absolute velocity, and \(c_{\mathrm{u2}\infty}\) represents the circumferential component of the absolute velocity in the presence of blade-congruent flow. \[\gamma=1-\frac{c_{\mathrm{u2}\infty}-c_{\mathrm{u2}}}{u_{2}} \tag{1}\]  
Figure 2a depicts the velocity triangle within the pump impeller, encompassing all significant velocity components and the definition of the prewhirl angles, with positive and negative values of \(c_{\mathrm{u2}\infty}\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **PO001** & **PO002** \\ \hline specific speed \(\left(n_{\mathrm{q}}\right)\) & 36 min\({}^{-1}\) & 54 min\({}^{-1}\) \\ \hline outer diameter inlet \(\left(d_{\mathrm{1a}}\right)\) & 124 mm & 122 mm \\ \hline inner diameter inlet \(\left(d_{\mathrm{1i}}\right)\) & 46 mm & 47 mm \\ \hline diameter outlet \(\left(d_{\mathrm{2}}\right)\) & 210 mm & 180 mm \\ \hline width impeller outlet \(\left(b_{\mathrm{2}}\right)\) & 22 mm & 33 mm \\ \hline number of blades \(\left(z_{\mathrm{L}}\right)\) & 7 & 6 \\ \hline blade angle outlet \(\left(\beta_{\mathrm{2B}}\right)\) & 27\({}^{\circ}\) & 28\({}^{\circ}\) \\ \hline mechanical power at \(n_{\mathrm{N}}\left(P_{\mathrm{mech}}\right)\) & 330 W & 70 W \\ \hline flow rate at BEP \(\left(Q^{*}\right)\) & 98.6 \(\frac{\mathrm{m}^{3}}{\mathrm{h}}\) & 113.2 \(\frac{\mathrm{m}^{3}}{\mathrm{h}}\) \\ \hline head at BEP \(\left(H^{*}\right)\) & 13 m & 8.2 m \\ \hline pump power at BEP \(\left(P_{\mathrm{2}}{}^{*}\right)\) & 3143 W & 3375 W \\ \hline nominal speed \(\left(n_{\mathrm{N}}\right)\) & 1470 min\({}^{-1}\) & 1470 min\({}^{-1}\) \\ \hline performance factor \(\left(\gamma\right)\) & 0.81 & 0.78 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the two measured pumps (PO001 & PO002). The characteristic curve data in the BEP are indicated by an asterisk (*). Figure 1: Flowchart illustrating the research methodology, divided into the two main branches of experimental analysis and CFD analysis. negative signs. The absolute velocity (\(c\)) can be represented as the sum of its circumferential velocity (\(u\)) and relative velocity (\(w\)) or it can be written as the sum of a circumferential component (\(c_{\text{u}}\)) and a meridional component (\(c_{\text{m}}\)). The relative velocity describes the flow velocity relative to the rotating pump impeller. The angle \(\alpha\) is the angle between \(u\) and \(c\) and characterises the orientation of the absolute flow with respect to the pump impeller. The flow angle \(\beta\) can quantify at flow congruence the blade angle. Figure 2b shows the meridional view of a radial centrifugal pump impeller, highlighting its crucial geometric parameters. Figure 3 [26] depicts the experimental setup along with the measured variables utilized to assess the characteristic curves at varying prewhirl angles of the prewhirl bodies (\(\Delta\alpha_{\text{pb}}\)). The prewhirl angle of the flow is a function of the prewhirl bodies. This approach has previously been employed in [26] and is founded upon the proposal by [PERSON] [9]. The prewhirl bodies were manufactured using an additive process and have six blades distributed evenly around the circumference.