# Surrogate-Based Optimization of a Centrifugal Pump with
## 7 Conclusions  
A procedure for the automated design of a centrifugal pump for the cooling of car engines is presented. It involves a shape optimization of the impeller, vaneless diffuser and volute, with the objective of maximize the efficiency of the pump, keeping constant the operating conditions. The baseline geometry is obtained with a lumped parameter code. The baseline geometry is enhanced with a 3D surrogate based optimization. The proposed approach demonstrates its robustness, as it provides for the prescribed operating condition a 3D design from scratch, which was optimized to enhance the pump hydraulic efficiency (\(\eta\) is increased by 3% at the end of the optimization). The optimal design presents a decrease of the blade loading, and a smoother acceleration along the pressure side. Moreover, the low pressure area at the leading edge is reduced, and, consequently, the related cavitation problems. The velocity field across the meridional channel is modified by changing the hub and shroud profile to obtain a more uniform distribution. In particular, the reduction of the section at the end of the inlet pipe avoids the stall at the hub. Moreover, at the impeller outlet the velocity peak is reduced and the velocity distribution is more uniform along the blade height. In the volute the pressure distribution is quite similar for both designs. However, the optimal design shows a reduction of the low pressure zone at the volute tongue, and smaller velocity values for the increased cross-section area. Moreover, the small recirculation zone after the volute tongue is removed in the optimal geometry. The components that have the greater impact to improving efficiency are the impeller and the vaneless diffuser. Only a slight improvement is obtained in the volute, probably for the reduced number of DVs used for its geometrical parametrization. Each simulation has been run in parallel on 32 cores, and 1 h is needed to complete the workflow (mesh generation and CFD simulation). DoE required 210 simulations, while 68 iterations were needed by the optimization procedure to converge to an optimum design. The computational cost of the whole optimization process was approximately 280 h (\(\approx\)12 days) on 32 cores. Ongoing work is devoted to enhance the parametrization of the volute, and to reduce the global number of DVs with an active subspace technique. Conceptualization, A.G. and R.D.D. ; methodology, A.F. ; software, R.D.D. ; validation, A.F. and A.M.; formal analysis, G.N. and A.M.; investigation, A.F. ; resources, R.D.D. and A.M.; data curation, G.N. ; writing--original draft preparation, A.F. and R.D.D. ; writing--review and editing, A.G., G.N. and A.M.; visualization, G.N. ; supervision, A.G. and R.D.D. ; project administration, A.G. All authors have read and agreed to the published version of the manuscript. This research received no external funding. \begin{table}
\begin{tabular}{l l l l l} \hline \hline  & & & **Baseline** & **Optimal** \\ \hline \multirow{3}{*}{**Impeller**} & \(\Delta p_{imp}\) & [bar] & 2.901 & 2.781 \\  & \(\Delta p_{t,imp}\) & [bar] & 3.887 & 3.555 \\  & \(C\) & [Nm] & 3.900 & 3.458 \\  & \(\eta_{imp}\) & [-] & 0.931 & 0.960 \\ \hline \multirow{3}{*}{**Diffuser**} & \(\Delta p_{diff}\) & [bar] & 0.127 & 0.133 \\  & \(\Delta p_{t,diff}\) & [bar] & \(-\)0.063 & \(-\)0.030 \\ \hline \multirow{3}{*}{**Volute**} & \(\Delta p_{vol}\) & [bar] & 0.426 & 0.391 \\  & \(\Delta p_{t,sol}\) & [bar] & \(-\)0.173 & \(-\)0.171 \\ \hline \multirow{3}{*}{**Global**} & \(\Delta p\) & [bar] & 3.453 & 3.304 \\  & \(\Delta p_{t}\) & [bar] & 3.651 & 3.354 \\  & \(\eta\) & [-] & 0.875 & 0.905 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparison of the baseline and the optimal design for the impeller, the vaneless diffuser and the volute. \(\eta_{imp}\) is the impeller efficiency, \(C\) the torque absorbed by the pump shaft, \(\Delta p\) the static pressure variation, \(\Delta p_{tot}\) the total pressure variation, \(\eta\) the pump hydraulic efficiency.