# Mechanisms of Sweep on the Performance of Transonic Centrifugal Compressor Impellers
## 4 Performance Comparison
### Efficiency  
The CFD results of efficiency with different sweep angle are presented in Figure 10a. When the sweep angle is gradually enlarged from \(-25^{\circ}\) to \(+11^{\circ}\), the efficiency tends to be increased by 1.2%. The efficiency decreases moderately by 0.2% when the sweep angle further increases from \(+11^{\circ}\) to \(+25^{\circ}\). The optimal sweep angle at \(+11^{\circ}\) and the maximum sweep angle at \(+25^{\circ}\) are referred as OSA (optimal sweep angle) and MSA (maximum sweep angle) respectively. The resulting design speed efficiency comparison between datum, OSA and MSA cases is presented in Figure 10b. It is observed that a 0.8% increase in efficiency occurs at the whole speed line of OSA. The high mass flow conditions of MSA are similar to OSA, but a deficit of 0.2% in efficiency occurs at low mass flow conditions.  
For the purpose of understanding the loss reduction in detail, the parameter efficiency deviation (\(\Delta\eta\)) is applied to investigate the loss reduction process, which is defined based on the concept of loss of efficiency [38]:  
\[\Delta\eta(M)=(\frac{T_{02}\Delta s(M)}{\Delta t_{02}})|_{swep}^{ datum} \tag{1}\]  
Figure 10: Efficiency: (**a**) trend with sweep angle and (**b**) design speed performance comparison. OSA: optimal sweep angle; MSA: maximum sweep angle.  
Figure 9: Choke mass flow rate of cases with different sweep angle.  
\[\Delta\eta(N)=(\frac{T_{02}\Delta s(N)}{\Delta h_{02}})\big{|}_{\text{sweep}}^{ \text{}{datum}}, \tag{2}\]  
where \(M\) and \(N\) are the corrected streamwise and spanwise coordinates from 0 to 1 (inlet to exit and hub to shroud), respectively, \(\Delta s(M)\) is the mass-averaged entropy rise of a cross-section cut at certain streamwise position, and \(\Delta s(N)\) is the pitch-averaged entropy rise at a certain spanwise position of a cross-section cut. The increase of \(\Delta\eta\) along the streamwise direction represents loss reduction at a corresponding location, and a positive value of \(\Delta\eta\) along the spanwise direction represents a higher efficiency at a corresponding location.  
The streamwise \(\Delta\eta\) distribution of OSA and MSA cases is shown in Figure 11. From inlet to Section I near the leading edge of splitter I, \(\Delta\eta\) drops suddenly at first and finally increases quickly. The sudden changes of \(\Delta\eta\) are caused by the sweep of main blade leading edge, where the incoming flow encounters the leading edge and generates entropy at a more upstream location. The negative value of \(\Delta\eta\) at Section I is due to additional friction loss by the increased blade surface area. Comparing OSA with MSA, it is illustrated that the too large sweep angle induces extra loss near Section I, which is responsible for the efficiency drop from +11\({}^{\circ}\) to +25\({}^{\circ}\) of sweep angle. From Section I to Section II near the leading edge of splitter II, \(\Delta\eta\) increases gradually. From Section II to the exit, \(\Delta\eta\) increases at a much higher rate. Generally, the efficiency improvement focuses near the radial part of the impeller, although the blade modification by sweep is made near the axial part.  
The pitch-averaged spanwise \(\Delta\eta\) distribution at Section I and the impeller exit is illustrated in Figure 12. At Section I in Figure 12a, it is observed that the induced extra loss from OSA to MSA occurs near the hub, where the \(\Delta\eta\) of MSA is obviously lower than that of OSA. At impeller exit in Figure 12b, it can be seen that the efficiency significantly improves near the casing in both case. When increasing sweep angle from OSA to MSA, the loss near the casing is further reduced, but the loss near the hub is enhanced. The trade-off effect determines the best sweep angle in terms of efficiency. To pinpoint the trade-off effect, \(\Delta\eta\) contour of both cases at impeller exit is presented, where the change of \(\Delta\eta\) focus on passages around the main blade.  
Figure 11: Streamwise efficiency deviation distribution.