# Mechanisms of Sweep on the Performance of Transonic Centrifugal Compressor Impellers
## 5 Aerodynamic Effects of Sweep Feature
### Effect on the Front Loading
\[\lambda_{A}=\lambda_{B}+\int_{r_{B}}^{r_{A}}\frac{\partial\lambda}{\partial r}dr \tag{3}\]  
\[\lambda_{A^{\prime}}=\lambda_{B^{\prime}}+\int_{r_{B^{\prime}}}^{r_{A^{\prime}} }\frac{\partial\lambda}{\partial r}dr. \tag{4}\]  
Because the radial gradient of blade loading is decided by the radial gradient of pressure, it can be deduced from the radial equivalent equation:  
\[F_{r}+\frac{V_{t}^{2}}{r}=V_{x}\frac{\partial V_{r}}{\partial x}+V_{r}\frac{ \partial V_{r}}{\partial r}+\frac{1}{\rho}\frac{\partial p}{\partial r}, \tag{5}\]  
where terms on the left hand side represent the radial blade force and the centrifugal force respectively, while terms on the right hand side represent the radial inertial force and the pressure gradient respectively. For simplicity, the radial blade force is ignored because it is usually introduced by blade lean. The radial velocity is usually much smaller than the other two components, and its axial gradient is also considered small. Equation (5) is then simplified into the equation of simple radial equilibrium:  
\[\frac{\partial p}{\partial r}=\rho\frac{V_{t}^{2}}{r}. \tag{6}\]  
Therefore, the radial gradient of blade loading is obtained:  
\[\frac{\partial\lambda}{\partial r}=-\rho\frac{V_{t,s}^{2}-V_{t,p}^{2}}{r}. \tag{7}\]  
Equation (7) reveals that the loading gradient mainly depends on the tangential velocity of the incoming flow, which does not change with the sweep angle. Comparing Equations (3) and Equations (4):  
Figure 15.0  
Because the loading at point \(B\) is positive and the loading at point \(B^{\prime}\) is zero (no blade), the front loading near the tip of an aft-swept blade will be higher than that of a forward swept blade:  
Figure 15.1  
Similarly, it can be deduced that the front loading near the hub of an aft-swept blade will be lower than that of a forward swept blade. For axial-to-radial parts of centrifugal compressors, illustration of loading is shown in Figure 17b. The spanwise equivalent equation and the spanwise gradient of blade loading are deduced as follows:  
Figure 15.2  
Figure 15.3  
where \(m\) and \(n\) are streamwise and spanwise coordinates, and \(\delta\) is the streamline slope angle. The loading gradient near the hub remains constant with changing sweep angle. Thus, similar conclusions can be drawn for the hub front loading. The loading gradient near the tip increases when sweeping forward the blade, because the incoming flow has a lower tangential velocity at upstream positions. Comparing Equations (3) and (4):  
Figure 15.4It shows that the unloading effect of forward sweep near the tip has been hindered by the altered incoming flow condition. Limited change in tip loading is therefore expected:  
Figure 15.5  
The above equivalent explanations support most of the loading changes in Figure 15 except for the hub loading of splitter I. Since all blades sweep together in the current investigation, the upstream swept blades will influence downstream blades. When increasing the sweep angle, the increased hub front loading of the main blade leads to a thickened boundary layer or even flow separations near the pressure surface, which reduces the channel diffusion ratio and decreases hub incidence of splitter I. Its hub loading is therefore decreased.