Figure 4 Sensitivity analysis of model parameters. (a) prediction results of the breakup probability versus We_L ; (b) prediction results of the breakup probability versus

In the constructed breakup model, L characterizes the maximum scale of the local turbulence structure and is directly related to the parameters of the apparatus. For practical liquid-liquid dispersion equipment, L values as the minimum scale of the local runner. For example, in a pulsed disc and doughnut column extraction column, the value is taken as the distance between plates. For mixing equipment, since drop breakup occurs mainly in the region of the blade runner,L is taken as the blade height. For static mixers, L is counted as the runner width of the minimum mixing unit, and so forth. Based on the above criteria, analysis in Section 4.1 shows that the predicted results of the model are in good agreement with the experimental data, which indicates the accuracy and applicability of the criteria valuing L . However, due to the diversity and complexity of the practical apparatus, the minimum scale of the local runner may deviate from the maximum scale of the local turbulence structure, thereby influencing the predictive accuracy of the proposed breakup model. In such a scenario, the sensitivity of the value of L to the predicted breakup probability is analyzed in this section, as shown in Figure 4. Herein, denotes the value of the maximum scale of the local turbulence structure and L denotes the actual value used for the model calculation. Figure 4a shows the prediction results of the breakup probability versus We_L , it indicates that the larger the value of , the larger the predicted result of the breakup probability, which is because the larger the value of means the higher the turbulent stress and thus the drop is more prone to break up. Besides, Figure 4b shows that the deviation of the predicted results of the breakup probability model is approximately within ±30% when the value of fluctuates between 0.5 and 2.0.

Figure 5 shows the calculated drop breakup frequency versus drop diameter. It indicates that the drop breakup frequency firstly increases with the increase of the drop diameter and then reaches it’s maximum, further increase the drop diameter will decrease the predicted drop breakup frequency. The definition of the breakup frequency (see Eq. in Section 2) indicates that it is the product of the reciprocal of the breakup time and the breakup probability. As both the drop breakup probability and breakup time increase with the increase of drop diameter when the drop diameter is relatively small, with the increase of the drop diameter, the breakup probability increases more quickly than the breakup time, and the overall performance is that drop breakup frequency increases with the increase of drop diameter. However, when the drop diameter exceeds a certain value, the effect of drop diameter on the breakup time is more significant than that of the breakup probability, and then the performance is that the breakup frequency gradually decreases with the increase of drop diameter. Moreover, Figure 5 shows that as the turbulent energy dissipation rate increases or the interfacial tension decreases, the peak value of the breakup frequency gradually increases and moves toward smaller drop diameters, indicating that higher turbulent kinetic energy dissipation rate and lower interfacial tension are more favorable for the drop to break up, which is consistent with the experimental conclusions in our previous studies^8,9.