## 2 Method for Analysing Losses in Centrifugal Pumps with Different Prewhirl Angles
### Experimental Determination of the Head Characteristics with Varying Prewhirl Angles
Figure 3: The experimental setup with the variables that are measured in order to obtain the characteristic curves at varying prewhirl body angles (\(\Delta\alpha_{\text{pb}}\)), which are imposed on the inflow by prewhirl bodies as described in [26]. Figure 2: (**a**) Velocity triangle of the impeller showing the different velocity components at positive (black lines) and negative (grey lines) whirl angles (\(\Delta\alpha\)); (**b**) meridian view of the impeller for a radial centrifugal pump and geometrical parameters, where the index \(1\) denotes the blade inlet and \(2\) the blade outlet. In order to determine the head (\(H_{\text{exp}}\)) experimentally, Equation (2) [26] was employed. The differential pressure (\(\Delta p_{\text{P}}\)) between the static pressure on the pressure side (\(p_{\text{ps,exp}}\)) and the suction side (\(p_{\text{ss,exp}}\)) is measured by pressure transmitters. The dynamic pressure component is calculated from the flow velocities in the pressure (\(v_{\text{ps}}\)) and suction (\(v_{\text{ss}}\)) pipe sections at the pressure measurement points. Furthermore, the geodetic height between the suction pipe and the pressure transmitters (\(\Delta h\)) is also considered. The head loss of the prewhirl bodies (\(\Delta H_{\text{pb}}\)) was quantified for each prewhirl body and subsequently incorporated into the head. \(g\) denotes the acceleration due to gravity (\(g=9.81\frac{\text{m}}{g^{2}}\)). \[H_{\text{exp}}=\frac{\Delta p_{\text{P}}}{g\rho}+\frac{v_{\text{ps}}^{2}-v_{ \text{ss}}^{2}}{2g}+\Delta h+\Delta H_{\text{pb}} \tag{2}\]  
In addition to the pressures \(p_{\text{ps,exp}}\) and \(p_{\text{ss,exp}}\), a measured pressure value was also recorded in the impeller side gap between the pump impeller back disc and the pump housing (\(p_{\text{PSR,exp}}\)). To determine the power consumption of the pump, it is necessary to measure the torque (\(M_{\text{exp}}\)) and the speed (\(n_{\text{exp}}\)) via a torque-measuring shaft. The speed of the pump can be adjusted by means of a frequency converter (VFD). The pump power consumption (\(P_{2,\text{exp}}\)) is calculated according to Equation (3). The power consumption of the pump includes the mechanical power loss of the bearing and the mechanical seal, which is designated as \(P_{\text{mech}}\). This was specified in Table 1 and determined experimentally. \[P_{2,\text{exp}}=2\pi M_{\text{exp}}n_{\text{exp}} \tag{3}\]  
The flow rate (\(Q_{\text{exp}}\)) can be adjusted by a control valve in the pressure pipe through the process of throttling and is then measured by a magnetic inductive flow meter (MID). The pumped liquid is taken from a tank with a volume of \(V=16\) m\({}^{3}\) and returned by the pump. Given the considerable volume of the tank and the brief measurement period, it may be reasonably assumed that the pumped medium will not experience a significant degree of heating. Consequently, the density of the pumped medium was assumed to be constant at \(\rho=998\frac{\text{kg}}{\text{m}^{3}}\). For each measured operating point, all values were sampled at a sampling rate of \(f_{\text{S}}=500\) Hz over a measurement period of \(f_{\text{S}}=10\) s. Subsequently, the mean values were calculated for the operating points. The maximum relative errors of the individual measured variables and the calculated variables can be found in Appendix A. The maximum relative errors of the calculated variables were determined using [PERSON]'s law of error propagation.