## 4 Designs of the Small-Scale Tower from Numerical Experiments  
### Comparisons between Four-Tray and Empty Tower (Case A and Case C)  
Case C has the same flow conditions (\(L/G\) = 4.9) as Case A, except all the sieve trays are removed. Figure 12a,b shows the liquid flow structures by \(\alpha_{l}\) and gas flow structure by streamlines for Cases A and C. As the implementation of sieve trays can provide deceleration of the liquid phase as it approaches the sieve tray, the liquid phase of Case A is apparently much denser at the tray locations than for Case C in Figure 12a. The gas flow within the liquid column of Case C remains downward because of the large liquid-to-density ratio. Instead of several pairs of counter vortexes, as shown in Figure 12b for Case A, the downward gas flow only forms one larger pair of counter vortexes with the uprising gas flow outside the liquid column. Therefore, the complex flow structure of Case A in Figure 12b can enhance the two-phase mixing and the ensuing chemical reactions in Equations (6)-(8) more efficiently. As a result, the SO\({}_{2}\) mass fraction within the liquid column of Case A is lower than that of Case C in Figure 12c. Figure 11: The schematic shows the instantaneous variation of mass fraction of SO\({}_{2}\) (Y\({}_{\text{SO2}}\)) at outlet for the full-scale tower. Figure 10: The figure displays (**a**) the liquid volume fraction and (**b**) the mass fraction of SO\({}_{2}\) (Y\({}_{\text{SO2}}\)) for the large-scale tower at different times, by the porous media model. However, from Table 6, the SO\({}_{2}\) removal efficiency at outlet of Case C (56.9%) is very close to that of Case A (57.1%), which seems to be inconsistent with previous results shown in Figure 12c. The contradiction can be explained in Figure 13. Figure 13 shows the top view of SO\({}_{2}\) mass fraction at different cross sections as \(y=0.615\) and 2.46 m from Figure 12c. The SO\({}_{2}\) mass fraction is indeed lower for Case A near the center where the liquid column passes through, but there exists a high SO\({}_{2}\) region to the right of the lateral at each cross section. This trend can also be seen in Figure 12c for Case A near the right wall, which is related to the size of liquid column and explained as follows. Figure 8 above displays the deceleration of liquid flow near the top surface of each sieve tray and acceleration as liquid flow penetrates it. Furthermore, the dilute liquid flow can accumulate into a denser liquid column by using sieve trays, as shown in Figure 7a. As a result of the continuity, the deceleration enlarges the region of the liquid column, while the acceleration reduces it. This trend can be seen every time the liquid column passes the tray regions from their top to bottom surfaces in Figure 9. Figure 8 further indicates that the velocity of liquid flow has not reached its terminal velocity between  
Figure 12: The figure displays (**a**) the liquid volume fraction, (**b**) streamlines of gas flow, and (**c**) the mass fraction of SO\({}_{2}\) within the right tower of a small-scale FGD facility for Case A and Case C (empty tower). Figure 13: This schematic shows the mass fraction of SO\({}_{2}\) (\(Y_{\text{SO2}}\)) at different cross sections from top view for Case A and Case C.  
each sieve tray, which implies that the acceleration remains even after the tray region. As a result, the size of the liquid column keeps decreasing in the region between two trays. These aforementioned mechanisms result in a continuous reduction of the radius of liquid column, as shown in Figure 9.  
Without a sieve tray, the liquid flow fails to accumulate to form a liquid column with a higher liquid volume fraction. The liquid column resembles several different discrete liquid flows. This scatter characteristic makes its variation in size less sensitive to the \(y\)-direction. The radius of liquid column is almost decided by the radial location of the outermost nozzle spray shown in Figure 3b.