## 3 Results: Optimization Workflow
### Numerical Results
The resulting operating range of the pump in the WLTC is given in Table 1. For efficient operation, it is crucial to appropriately select the best efficiency point of the pump along with the number of impeller blades and impeller vane exit angle. The installation space of the pump impeller (\(f_{1}\) in Figure 15) is calculated from its diameter and width, which are provided by the pump dimensioning. The pump dimensioning provides a total of nine quantities for calculating the mechanical shaft power (Figure 15). The mechanical shaft power of the pump \(P_{shaft}\) is composed of four components (Equation (5)). \(P_{h}\) is the hydrodynamic power and takes into account power for conveying the net flow rate and the occurring gap loss flows. The disk friction power \(P_{R}\) includes all losses caused by fluid friction on rotating components wetted by the pumped coolant. The mechanical power \(P_{m}\) includes all losses that occur at the bearings and dynamic seals. Exchange losses \(P_{a}\) occur mainly when the pump is operated at a significantly lower flow rate than the best efficiency point. The pump is then in partial load operation, characterized by turbulence behind the pump impeller. Backflows occur, which act back on the impeller and are captured in the additional power \(P_{a}\). \[P_{shaft}=P_{h}+P_{R}+P_{m}+P_{a} \tag{5}\]  
The losses of the electric motor \(P_{el,\ loss}\) are described by the analytical approximate Equation (6) [46]. The torque \(T\) and the angular velocity \(\omega\) correspond to the operating point of the electric motor and are determined by the calculation step of the mechanical \begin{table}
\begin{tabular}{c c c} \hline \hline
**Variable** & \(\mathbf{Q}\) & \(\mathbf{H}\) \\ \hline Lower limit & \(30\ \frac{l}{\mathrm{min}}\) & \(4.9\ \mathrm{kPa}\) \\ Upper limit & \(363.4\ \frac{l}{\mathrm{min}}\) & \(30\ \mathrm{kPa}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Maximum and minimum flow rate and head values in the WLTC. Figure 16: Pump characteristic and system head curves as well as pump efficiency curves. power demand (Figure 15). The parameters \(k_{c}\), \(k_{i}\), \(k_{\omega}\), and \(C\) were determined by numerical fitting of the efficiency map of the known and unvaried electric motor. \[P_{el,\ loss}=k_{c}\cdot T^{2}+k_{i}\cdot\omega+k_{\omega}\cdot\omega^{3}+C \tag{6}\]  
The consumption of electrical energy (\(f_{2}\) in Figure 15) is determined by time integration of the mechanical power and electrical power loss (Equation (7)). \[E_{el}=\int P_{shaft}+P_{el,loss}dt \tag{7}\]  
The design variables are varied within the limits specified in Table 2. The limits of \(n_{Opt}\), \(Q_{Opt}\) and \(H_{opt}\) are selected based on trial executions. The limits of the number of impeller blades and impeller vane exit angle \(\beta_{2}\) are selected based on empirical values according to the literature [39]. In this paper, we present the results of the multiobjective optimization using pareto fronts. A pareto front contains all variants which do not dominate each other with respect to their objectives. If two variants are compared on the pareto front, one variant will be better in one objective and worse in the other [40]. The pareto front of the optimization problem regarding the constraints is shown in Figure 17. It can be seen that spatially compact pumps have a higher energy consumption. Efficient pumps, on the other hand, take up a larger installation space. The influence of the design parameter \(n_{Opt}\) of the best efficiency point is shown as an example. Here, higher rotational speeds in the best efficiency point lead to more compact but less efficient pumps. In this paper, the best tradeoff (cf. select in Figure 14) is selected by a weighted rating of the two objectives (Equation (8)). For this purpose, the objective values \(E_{El}\) and \(V_{Pump}\) of each technical variant on the pareto front are first scored. An objective with the lowest value is assigned the score \(S\) of 10, an objective with the highest value is assigned the score \(S\) of 1.