## 5 Analysis of the Numerical Results
### Velocity Field  
Once the fluid particles are separated by centrifugal force, the axial velocity has an important role since it is primarily responsible for transporting the particulate matter towards the bottom of the cyclone [14; 42]. In addition to this, this velocity component is responsible for the direction of both flow currents inside the cyclone [11]. Figure 5 shows the axial velocity contour plots for models A, B, C, and D at the middle plane. Starting from these contour plots, the effect of the double vortex can be seen in the flow of the cyclone, formed by an initial spiral in a descending direction that forms close to the walls of the cyclone, which transports the particles toward its bottom, along with a second spiral in an upward direction in the central region whose purpose is to extract the gas flow through the vortex finder. It is noted that the axial velocity contours for the four models are qualitatively very similar; however, there are quantitative differences between the proposed models and the reference one (Model A); there is the higher axial velocity in models B and C, while in the model D, the axial velocity contours are very similar to the reference model. On the other hand, Figure 6 illustrates the axial velocity profiles for the four cyclone models. It shows that the axial velocity profiles of the models are similar in the magnitude of their values as well as their shape (an inverse W). However, it is possible to see an increase in axial velocity in the central region of the profiles of models B, C, and D, as well as a slight decrease in its fall as a result of the increase in kinetic energy that is the result of the passing of flow in the cavities of the cylindrical section of the cyclone. In addition to this, through the axial velocity profiles, the effects of a double vortex in the flow of the cyclone mentioned previously can be seen, because near the wall of the cyclone the values of the axial velocity are negative, thus demonstrating that the direction of the first spiral is descending. Additionally, in the direction of the central region of the cyclone, there is a point where the velocity is zero, indicating the change of direction between both spirals. Finally, in the central region of the cyclone are positive values which confirm that the direction of the second spiral is ascending towards the vortex finder. Apart from that, Figure 6 shows the asymmetry present in each of the profiles. This asymmetric factor is due  
Figure 4: Axial stations and middle planes of cyclones; (**a**) Stairmand cyclone separator, (**b**) Model B cyclone separator, (**c**) Model C cyclone separator, (**d**) Model D cyclone separator. in part to the spiral flow in the interior of the cyclone and its geometry since the tangential shape inlet section does not allow for its design to be symmetric. In addition, this could be due to instability in the ascending vortex. However, it is worth mentioning that in studies such as [10; 41; 43] this asymmetric condition is seen in the axial velocity profiles. Figure 5: Contour plots of axial velocity of cyclone models; (**a**) Stairmand cyclone separator, (**b**) Model B cyclone separator, (**c**) Model C cyclone separator, (**d**) Model D cyclone separator. Figure 6: Axial velocity profiles of cyclone models A, B, C y D.  
Tangential velocity is the most important parameter regarding flow velocity because it has a great impact on the separation efficiency since it determines the magnitude of the centrifugal force produced in the cyclone [11; 22]. Figure 7 illustrates the tangential velocity profiles for all the cyclone models in a 217.5 mm axial station. According to the profiles, it is seen that this model, a Rankine vortex, which is made of an internal forced vortex with a tendency to rotate like a solid, where tangential velocity increases radially from the central axis of the cyclone, as well from a free external vortex, where tangential velocity values decrease as the radial distance increases [10; 22].