Figure 5 Influence of the drop diameter on drop breakup frequency. (a) with different turbulent energy dissipation rates, σ = 35 mN/m; (b) with different interfacial tensions, ε = 3 m²s^-3

The shading regions in Figure 5 are where the drop breakup frequency can be accurately predicted by the proposed model in this study. It is noted that the breakup model constructed in the present work is based on the binary breakup assumption and the breakup time model adopted in this study is also valid for the breakup process with a small number of fragments generated (generally, no more than 4 fragments). According to our previous study²⁴, the predicted results of the breakup time are in good agreement with the experimental data whenWe_L is lower than 0.7 (corresponding to a breakup probability of approximately 0.5). Therefore, we can approximateWe_L = 0.7 as the criterion, whenWe_L < 0.7, the model can accurately predict the breakup frequency (corresponding to the shaded regions), and when We_L > 0.7, the predictive accuracy of the breakup frequency prediction will be limited, and the effect of multiple breakups needs to be quantified. Noteworthy, according to our previous experimental results^8–10,12,21, drop breakups are dominated by the low-fragment breakups (especially, the proportion of the binary breakup is more than 50%, and the breakups with the fragments number lower than 4 take the proportion of more than 90%) under steady-state operating conditions. Thus, the breakup model constructed in this study can be accurately applied to predict the drop breakup frequency in the practical apparatus.

The turbulent energy dissipation rate is one of the most important parameters affecting the drop breakup frequency. The larger the value of the turbulent energy dissipation rate, the more favorable the energy transfer from the eddies to the drop surface. Figure 6 showed that the drop breakup frequency gradually increases with the increase of the turbulent energy dissipation rate, which is consistent with the above analysis. Moreover, Figure 6a indicates that the breakup frequency versus the turbulent energy dissipation rate varies greatly for different drop diameters. The intersections of the gray straight line with each line in Figure 6a characterize values of drop breakup frequency at We_L = 0.7 for each case, and region below the intersection (or the shading region in Figure 6b) represents the applicable scope of the breakup model. Meanwhile, for the upper side of the intersection, the effect of the multiple breakups needs to be considered.