# Effects of an Inclined Blade on the Performance of a Sirocco Fan
## 3 Results and Discussion
### Qualitative and Quantitative Results
#### 3.2.2 Vorticity Distribution within the Fan
The modified models had no obvious improvement in the high vorticity region of the impeller passages, and the vorticity decreased significantly in the inner region of the volute. For model-B, the vorticity near the volute tongue decreased greatly, while the high vorticity region near the volute tongue of model-A and model-C increased. In Figure 20, we can see that the distribution of high vorticity in the impeller passages of the fan is approximately the same as that in Figure 19, but the region of high vorticity increases. The modified models played an important role in reducing the high vorticity region near the volute tongue, and model-B was the most effective. The vorticity distribution of the baseline fan and the optimization models was basically the same in the volute. Figure 18: Contour of vorticity magnitude for the \(z=75\) mm plane at a rated flow rate \(Q/Q_{n}=1.000\): (**A**) model-A; (**B**) model-B; (**C**) model-C.  
Figure 19: Contour of vorticity magnitude for the \(z=35\) mm plane at a high flow rate \(Q/Q_{n}=1.760\): (**A**) model-A; (**B**) model-B; (**C**) model-C.  
Figure 20: Contour of vorticity magnitude for the \(z=75\) mm plane at a high flow rate \(Q/Q_{n}=1.760\): (**A**) model-A; (**B**) model-B; (**C**) model-C. The three-dimensional distribution of vortices in the impeller and volute is further represented by the \(Q\)-criterion:  
\[Q=\frac{1}{2}\Big{(}\|S^{2}\|-\|\Omega^{2}\|\Big{)}=\frac{1}{2}\Bigg{(}\Bigg{(} \frac{\partial\mu}{\partial x}\Bigg{)}^{2}+\Bigg{(}\frac{\partial v}{\partial y }\Bigg{)}^{2}+\Bigg{(}\frac{\partial\omega}{\partial z}\Bigg{)}^{2}\Bigg{)}- \frac{\partial\mu}{\partial y}\frac{\partial v}{\partial x}-\frac{\partial\mu }{\partial z}\frac{\partial\omega}{\partial x}-\frac{\partial v}{\partial z} \frac{\partial\omega}{\partial y}. \tag{11}\]  
The tensor of strain rate and vorticity is:  
\[S_{ij}=\frac{1}{2}\Bigg{(}\frac{\partial\mu_{i}}{\partial x_{j}}-\frac{ \partial\mu_{j}}{\partial x_{i}}\Bigg{)}, \tag{12}\]  
\[\Omega_{ij}=\frac{1}{2}\Bigg{(}\frac{\partial\mu_{i}}{\partial x_{j}}+\frac{ \partial\mu_{j}}{\partial x_{i}}\Bigg{)}, \tag{13}\]  
in which \(\mu\), \(v\), and \(w\) are the velocities in the \(x\), \(y\), and \(z\) directions, respectively. As shown in Figure 21, there are vortices on the surface of the blade at low flow rate. The vortices of the optimized models decreased in size obviously in the impeller passages near the outlet of the volute (the dotted rectangular region), while there was no obvious change in other regions of the impeller. In Figure 22, it can be found that the volume of the vortices in the exit of the volute of model-A and model-B was diminished and it was enlarged for model-C, which prohibited the flow out of the volute (the dotted rectangular region). Figure 15:0 Three-dimensional distribution of vortices in impeller passages at \(Q/Q_{n}\) = 0.483: (**A**) model-A; (**B**) model-B; (**C**) model-C. Similarly, for flow conditions of maximum static pressure efficiency, the vortices of model-B and model-C were reduced in the impeller flow passages near the solute outlet (dotted rectangular region), as given in Figure 23, whereas the vortices of model-A were increased. At the exit of the solute, as given in Figure 24, the vortices of model-A in the vicinity of the solute tongue increased, and the volume of the vortices at the intermediate position of the model-B and model-C at the exit of the solute decreased (the dotted rectangular region). In the case of high flow rate (Q/Q\({}_{n}\) = 1.760), as shown in Figure 25, the vortices on both sides of the flow passages of the modified models were reduced. In Figure 26, the volume of the vortices at the exit of the solute of the model-B was reduced, while it increased for model-A; the volume of the vortices of the model-C was decreased, but the number of vortices was increased (the dotted rectangular region).