# Surrogate-Based Optimization of a Centrifugal Pump with
## 6 Results and Discussion  
The DoE consists of 210 training points, but 38 designs have been rejected after their evaluation because of errors in the mesh generation or CFD calculation phase. The efficiency and the pressure head of these design are set to 0.8 and 3.2 bar, respectively, to force the search of the optimum far away. The analysis of the optimization convergence (see Figure 12) shows the absence of further unfeasible design. Figure 12 shows the optimization convergence in terms of efficiency and pressure head. The SBO reaches the convergence after 68 iterations. Both values predicted by CFD and estimated by the Kriging model are reported. Figure 13 deeply investigates the optimization process in terms of the global error (see Equation (2)) and the achieved maximum efficiency. The global error shows a decreasing behaviour, while the efficiency reaches a value above 0.9 starting from 16-th iteration. The optimum design is characterized by \(\eta=0.905\), i.e., an improvement of 3% with respect to the baseline is achieved, and \(\Delta p_{t}=3.35\) bar. Figure 12: Evaluation of the design from the SOGA with the surrogate and CFD simulation: efficiency (**top**) and total pressure head (**bottom**). Figure 13: Convergence of the optimization strategy: global error (**top**) and maximum efficiency (**bottom**). The profiles of the blade camberline in a conformal mapping plane are compared in Figure 14 for the baseline and optimal geometries. The abscissa of the coordinate system is  
\[A(x_{m})=\varphi(x_{m})r(x_{m}),\]  
where \(x_{m}=\sqrt{z^{2}+r^{2}}\) is the distance in the meridional direction (the blade leading edge is taken as origin), and represents also the ordinate, \(r\) is the distance from the rotation axis, and \(\varphi\) is the wrap angle, defined as  
\[\varphi=\int_{0}^{L_{m}}\frac{dx_{m}}{r\tan(\beta_{b})},\]  
where \(\beta_{b}\) is the blade angle, and \(L_{m}\) the meridional distance between leading and trailing edge. Profiles at 0%, 50%, and 100% of the span are depicted. The baseline profile shows a constant distribution of \(\beta_{b1}\) and \(\beta_{b2}\) along the span, while a variation is appreciable for the optimized geometry. The optimal geometry is compared with the baseline in Figure 15, where the blade profiles of the impeller at hub and shroud (right), the meridional channel (left), and the solute shape (bottom) at the outlet are shown. Figures 16-22 compare the flow field between the optimized and the baseline geometries to investigate the effect of the geometrical changes on the efficiency. Notice that maximize the efficiency corresponds to minimize the power at the impeller \(W\), as the \(\Delta p_{t}\) is set at less than a tolerance. Figure 14: Profiles of the blade camberline in the conformal mapping plane for the baseline and optimized geometry. Initially, the effect of the impeller blade geometry is investigated, comparing the pressure coefficient, \(c_{p}\), and the skin friction coefficient, \(c_{f}\), along the blade (see Figure 16) for the baseline and the optimal geometries. The coefficients are defined as follows:  
\[c_{p}=\frac{2(p-p_{o})}{\rho U_{o}^{2}}\qquad c_{f}=\frac{2\tau_{null}}{\rho U_ {0}^{2}}\]  
where \(p_{0}\) and \(U_{0}\) are the static pressure and the velocity magnitude at the inlet pipe and \(\tau_{null}\) is the wall shear stress along the blade. The \(c_{p}\) distribution on the suction side shows a reduction of the pressure drop near the leading edge, while it assumes lower values on the pressure side. The area defined by the pressure and suction side curves is lower for  
Figure 15: Comparison between baseline and optimized geometries. the optimal geometry, which corresponds to a decrease of the blade loading, and, as a consequence, of \(W\). The difference in the friction coefficient \(c_{f}\) at the leading edge between the suction and pressure side is reduced in the optimal configuration; this discrepancy is motivated by an erroneous blade angle at the leading edge for the baseline.