## 3 Results and Discussion
### Pareto Front Points
#### 3.1.4 Cyclone Performance  
In general, the performance of the cyclones is determined based on the pressure drop values (\(\Delta P\)), collection efficiencies (\(\eta\)), and cut-off diameters (\(d_{50}\)). The pressure drop refers to the (total) pressure difference between the inlet and outlet, whereas the cut-off diameter is defined as the particle size corresponding to 50% collection efficiency on the grade efficiency curve (GEC). Separation efficiency denotes the ratio of particles retained by the cyclone separator to the injected particles at the inlet of the cyclone separator. In cyclones, the pressure losses take place mainly inside the outlet tube due to the contraction experienced by the swirling fluid as well as wall friction. The pressure drop for all the models of the cyclones is presented in Figure 10. The maximum pressure drop was found in model C and lowest in model A, with the intermediate values for the model Std and B. This is due to the reduction in the diameter of the vortex finder, which significantly increases the magnitude of the tangential velocity. With respect to the Std  
Figure 8: Radial profiles: From left to right at axial locations \(Z/D\) = 1.0, 2.0, and 2.5; from top to bottom, rmse static pressure, rmse axial velocity, and rmse tangential velocity, respectively, in cyclone models Std, A, B, and C.  
model, the pressure drop in models A, B, and C increases by \(-43.22\%\), \(-4.67\%\), and \(36.62\%\), respectively. The GECs for all the models operating are presented in Figure 11. Compared to the Std model, a significant shifting of the curves toward the left takes place in models B and C, whereas model A gradually shifts toward the right. Figure 12a summarizes the \(d_{50}\) values for the different cyclone models. It is observed that the largest value of \(d_{50}\) exists for model A, while the smallest value is found for model C, with the intermediate value for model B. Compared to the Std model, \(d_{50}\) increases by \(12.3\%\), \(-15.71\%\), and \(-41.94\%\) in models A, B, and C, respectively. Figure 12b denotes the overall collection efficiency for all the models. It becomes apparent that model C has the highest value of \(\eta\), whereas model A has the lowest value. Compared to the Std model, \(\eta\) increases by \(-2.16\%\), \(19.05\%\), and \(24.69\%\) in models A, B, and C, respectively. The enhancement in \(\eta\) or reduction in \(d_{50}\) is due to the largest magnitude of the tangential velocity, which creates a strong centrifugal field that enhances the particle separation process. Figure 10: The pressure drop values in all the cyclone variants. Figure 9: The vortex core representation based on \(\lambda_{2}\) criteria set at a level of 0.25 in all the cyclone models. The vortex core is colored with the velocity magnitude. Due to the change in the cyclone height, the cyclone volume changes significantly, which in turn influences the residence time (\(T_{res}\)) of particulate matter--here, \(T_{res}\) is the time for which the particles stay inside the cyclone. Figure 13 illustrates the \(T_{res}\) for all cyclone models (the calculations consider solids that exit the cyclone via the outlet plane). For all particle sizes, the Std model possesses the smallest \(T_{res}\), whereas for all other models, variations in \(T_{res}\) are dramatic. Maximum value of \(T_{res}\) for particle sizes of 0.5, 1.0, 1.5, 2.0, 2.5, 4.0, 6.0, 8.0, and 10.0 \(\upmu\)m is observed in cyclone models C, B, A, B, C, A, A, and A, respectively. A complete detail is provided in Figure 13. Interestingly, the solids having a size greater than 2.5 \(\upmu\)m are collected with 100% efficiency. Hence, no particle greater than this size reaches the outlet plane, due to which their \(T_{res}\) values are zero. Figure 11: The grade efficiency curves of all cyclone variants. Figure 12: (**a**) Cut-off particle size, and (**b**) total collection efficiency in the different cyclone models.