# Optimization of a Centrifugal Compressor Using the Design of Experiment Technique
## 6 Results and Discussions
### Response Surface for Finding the Optimal Geometry  
The above mentioned analysis for different cases show that the controlling point at the hub plays an important role in the objective functions which are the pressure ratio and isentropic efficiency. This result is may be due to following reasons:  
* The wider ranges are considered for the parameters in the hub section
* The hub section variation has more effect on whole impeller geometry in comparison to the changes in the shroud section
* The shroud characteristics are dependent on hub geometry hence the most effective parameters in hub also have effects on the shroud  
To find the optimal geometry which considers all the aforementioned analyses, just the parameters on the hub are considered for the next analysis. These are \(M_{1}\) and \(M_{2}\) in stage I, \(B_{1}\) and \(B_{2}\) in stage II and \(I_{1}\) and \(I_{3}\) in stage III. The other factors are set in the values which show the best performance in the previous case analysis. This leads to construction a design space for \(13\) (\(=5+4+4\)) parameters. A full factorial design space with each parameter at two levels leads to \(2^{13}\) runs whereas to better capture the design space, three levels for each parameter are more helpful. However, using the full factorial technique for six parameters each at three levels consequences \(3^{6}\) runs which also imposes high computational cost. In stage IV, as mentioned before, the Box-Benken technique is used which reduces the design space to 54 points. For each factor the levels are shown by \(-1\), \(0\) and \(1\) in which the center value (\(0\)) is set the value was obtained by the previous results. For obtaining the desired values from the parametric study, both isentropic efficiency and total pressure ratio are considered as the objective functions by following relation [26],  
\[f=C_{1}(\frac{1-\eta_{tt}}{1-\eta_{tt}^{i}})+C_{2}(\frac{Pr^{i}}{Pr}) \tag{11}\]  
in which the superscript \(i\) stands for the initial value and the coefficients \(C_{1}\) and \(C_{2}\) are set to 0.5 for considering the same share for efficiency and pressure ratio. Minimizing \(f\) results in the best  
Figure 15: Streamlines of the two cases at different spans. performance results in each case. Using the aforementioned objective function, in case I, \(M_{1}M_{3}M_{4}\) run, in Case II, \(B_{1}B_{3}\) run and \(I_{3}\) effect show the best performance results. Pressure ratio, isentropic efficiency and also the value of \(f\) for all design points are shown in Figure 16. The design points with maximum efficiency and with maximum pressure ratio are also shown in Figure 17. The results show that efficiency variation is about 1.2% and and pressure ratio variation in about 13.2% among all points. The point with maximum efficiency shows a 0.4% improvement in comparison to initial run and the point with maximum pressure ratio has 2.5% increase in this ratio. Figure 16: All design point performances and the values of \(f\) function. Second-order polynomial regression is used for pressure ratio, \(Pr\), efficiency, \(\eta\), and for both simultaneously, \(f\), using Equation (6), and the surrogate functions are noted by \(\hat{y}_{Pr}\), \(\hat{y}_{\eta}\) and \(\hat{y}_{f}\) respectively. Response surface method analysis is used to find the variables \(\{A,B,C,D,E,F\}\) to maximize \(\hat{y}_{Pr}\) and \(\hat{y}_{\eta}\) and to minimize the \(\hat{y}_{f}\). Table 3 shows the results. Three extra performance analyses are performed to investigate the design parameter values \(\{A,B,C,D,E,F\}\) in Table 3 on the efficiency and the pressure ratio. The CFD results show an additional 0.2% improvement for pressure ratio while optimum conditions are not changed regarding the efficiency and \(f\) function. Hence related to initial condition, \(f_{0}\), the isentropic efficiency improves by about 0.4% and the pressure ratio could be increased by 2.7%. Figures 18 and 19 show the performance curve for the optimized impeller and the original impeller.