# Hydrodynamics of an Airlift Column for Aeration in Molten Sulfur
## 4 Results and Discussion
### Bubble Diameter  
The equilibrium value of the average bubble diameter is determined by the local hydrodynamic conditions. The well-known correlation for the bubble diameter is as follows [29; 30]:  
\[d_{B}=4.15\epsilon_{r}^{0.5}\frac{\sigma^{0.6}}{\varepsilon^{0.4}\rho^{0.2}}+ 0.0009 \tag{13}\]  
The bubble diameter distribution is an important parameter of hydrodynamics, which depends on the coalescence and breakup of bubbles [31]. Figure 14 shows the average bubble diameter distribution with superficial velocity at a total column height of 15 m. The bubble diameter gradually increases from the nozzle to  
Figure 12: Comparison of liquid circulation velocity calculated from the correlations and CFD simulation with the \(U_{Gr}\). Figure 13: Comparison of liquid circulation velocity calculated from the correlations and CFD simulation at different total column heights. the top of the column, and the large bubbles in the downcomer are mainly concentrated at the top. The bubble diameter at the middle and bottom of the downcomer is smaller than the riser. Figure 15 shows the average bubble diameter distribution in the riser for different total column heights with \(U_{Gr}\) of 0.0389 m/s. As the total column height increases, the average bubble diameter gradually increases up to a height of 20 m and decreases when the height is 25 m. It shows that the column height has a significant effect on the average bubble diameter. The bubble diameter grows gradually from the bottom to the top. As the bubble reaches the maximum critical stable size, the bubble begins to breakup. Herein, the breakup of the bubbles will increase the turbulence in the surrounding flow field, which causes additional bubbles to break. Figure 14: Average bubble diameter distribution with \(U_{Gr}\) at a total column height of 15 m.  
Figure 15: Average bubble diameter distribution at different total column heights with \(U_{Gr}\) of 0.0389 m/s. The average bubble diameter at different cross sections was analyzed to investigate the change in the average bubble diameter along the axial direction of the column. Figure 16 shows the average bubble diameter at different cross sections with \(U_{Gr}\) at a total column height of 15 m. The average bubble diameter gradually increases from 10 to 22 mm from the first to the seventh cross section. However, the average bubble diameter decreases in the upper part of the column when the \(U_{Gr}\) is more than 0.0333 m/s. As the \(U_{Gr}\) increases, the turbulence degree of the column increases, which results in the decrease of the maximum stable bubble diameter and the average bubble diameter [32; 33]. A few researchers argued that the average bubble diameter slightly increases with the increasing \(U_{Gr}\)[34]. A maximum stable bubble diameter exists at a critical value of the \(U_{Gr}\). When the \(U_{Gr}\) is less than the critical value, the average bubble diameter slightly increases with the \(U_{Gr}\). However, when the \(U_{Gr}\) is larger than the critical value, a severe bubble breakup occurs and the average bubble diameter decreases. Figure 17 shows the average bubble diameter at different cross sections for various total column heights with the \(U_{Gr}\) of 0.0389 m/s. The average bubble diameter increases with the local height up to a total column height of 20 m. In addition, the growth rate of the bubble diameter from the first to the third section increases with the total column height. At the total column heights of 20 and 25 m, the bubble diameter decreases from the fifth and the third section onwards, respectively. This is investigated on the basis of the turbulent dissipation in each section. Figure 16: Average bubble diameter at different cross sections for various \(U_{Gr}\) at a total column height of 15 m.  
Figure 17: Average bubble diameter at different cross sections for various total column heights with \(U_{Gr}\) of 0.0389 m/s. Figure 18 shows the turbulent eddy dissipation at different cross sections for various total column heights with the \(U_{Gr}\) of 0.0389 m/s.