## 2 Method for Analysing Losses in Centrifugal Pumps with Different Prewhirl Angles
### 3D Model and Meshes for the Numerical Investigation of Hydraulic Losses  
The CFD simulation for the analysis of hydraulic losses is based on a three-dimensional mesh of the pump flow area. For PO001, three-dimensional geometries were generated for three prewhirl bodies with prewhirl angles \(\Delta\alpha_{\text{pb}}\in\{-30^{\circ};0^{\circ};+30^{\circ}\}\). Figure 4a depicts the side view, which includes all externally visible flow areas. The inlet pipe has a length that is six times the hydraulic diameter, which allows for the exclusion of influences of the boundary conditions. The flow area of the prewhirl body is behind the inlet pipe. The prewhirl body flow domain contains the six blades of the prewhirl bodies, which were employed in the experimental measurements. The inlet housing, which is connected to the gap as well as the impeller, follows the flow area of the prewhirl body. The front and back side gaps are formed between the discs of the impeller and the housing, each of which is represented by a flow area. Figure 4a illustrates the abrupt widening of the cross-section that occurs at the transition between the volute and the outlet pipe. This transition is in accordance with the measurement setup. Figure 4b depicts the three-dimensional geometries of the flow area in front view, omitting the inlet pipe, side gaps, gap and outlet pipe with volute, interface and impeller. The flow area of the interface is situated between the flow areas of the impeller and the volute. The interface connects the main flow from the impeller to the volute, with the front and back side gap. As illustrated, the flow area of PO001 was completely mapped. In some  
of the works discussed in the introduction, the mapping was limited to the flow areas of the inlet and outlet pipes, the volute and the impeller. This results in differences between the simulation and the experimental measurement data. Furthermore, the losses in the impeller side gaps cannot be analysed. Furthermore, the gap flow from the impeller outlet via the front side gap and the gap into the inlet housing, along with the associated losses, are not taken into account. The three-dimensional geometries illustrated in Figure 4 were individually meshed. The impeller was meshed using TurboGrid 2022 R1. The meshing was implemented for a passage situated between two blades and was subsequently replicated for all blades positioned around the pump's axis of rotation. The flow domains of the prewhirl bodies were also meshed using this method. The remaining flow domains were meshed using the SpaceClaim 2022 R1 meshing tool. The near wall layers were refined in a uniform manner for all geometries. The height of the final wall node layer was found to be \(\Delta y=0.05\) mm. [PERSON] et al. demonstrated that the flow patterns of centrifugal pumps can be accurately replicated with meshes comprising a lower number of nodes [28]. A comparison was conducted between three meshes comprising between 1.8 million and 16 million nodes. It was demonstrated that even at a node count of 1.8 million, it was possible to achieve small deviations between the simulation and the experiment [28]. [PERSON] et al. demonstrated that node numbers between 1.2 million and 4.1 million nodes exert only a minor influence on the results [19]. A mesh sensitivity study was conducted to demonstrate the independence of the results from the meshes. In order to achieve this objective, four meshes with varying node numbers were constructed. The flow area with \(\Delta\alpha_{\text{pb}}=0^{\circ}\) was meshed. The configuration of the simulations is detailed in the subsequent chapter. Table 2 presents the findings of the mesh sensitivity analysis. The meshes are designated as Mesh 1 to Mesh 4, respectively. Table 2 presents a summary of the number of nodes, the simulated head (\(H\)) and power consumption (\(P_{2}\)). In accordance with the findings of [PERSON]. [PERSON] et al. [28] and [PERSON] et al. [19], the outcomes pertaining to node numbers within the range of 1 to 5 Mio.