Figure 8 Comparison of the predicted results with the initial breakup experimental data, c_vis = 3.0

The construction of the accurate drop breakup model is crucial for the application of the population balance model and the study of liquid-liquid dispersion characteristics. Based on the drop oscillation breakup mechanism, in the present work, we proposed a theoretical model of drop breakup probability by adopting the three-dimensional Maxwell velocity distribution. The model considers the breakup constraints of both the maximum increase of the interfacial energy and the increase of drop viscous energy during the breakup process. The results show that for low-viscous drops, drop breakup probability is only related to the Weber number (We_L ), which characterizes the relative magnitude of the external turbulent stress and the interfacial stress. When the drop viscosity exceeds a certain limit, the effect of drop viscosity needs to be considered and the Ohnesorge number (Oh ), which characterizes the relative magnitude of the viscous stress of the drop and the interfacial stress, is introduced.

By combining the theoretical model of drop breakup time constructed in our previous work, the breakup frequency model is obtained based on the statistical description framework. The accuracy and generality of the model were then validated using the direct experimental data. Moreover, we systematically analyzed the effects of the drop diameter, turbulent energy dissipation rate, and interfacial tension on the predicted drop breakup frequency. Results showed that the breakup frequency increases monotonically with increasing turbulent energy dissipation rate. Differently, with the increase of the drop diameter or the interfacial tension, the breakup frequency firstly increases and then decreases. The reason for the above results is related to the coupling effect of the parameters on the breakup time and breakup probability. The results of this study are applicable to predict drop breakup frequency for drop breakups with low fragments generated. Combined with the results in our previous studies, the proportion of the breakups with the number of fragments lower than 4 accounts for more than 90% under steady-state operating conditions. Therefore, the model constructed in the present work can directly serve for the quantitative characterization of drop breakup behaviors under steady-state operating conditions.

We gratefully acknowledge the support of the National Natural Science Foundation of China (21776151, 21576147).

c_f coefficient of surface area,

d drop diameter, m

D the diameter of the impeller, m

E (k ) the energy spectrum of turbulence, m³/s²

the energy constraint, J

k the wavenumber of the turbulence, 1/m

L the maximum scale of the local turbulent structure, m

N the rotating speed of the impeller, 1/s

N (d ) the number of mother drops

Oh the Ohnesorge number

p_os the average oscillation kinetic energy per unit drop volume, J/m³

p_t the turbulent stress, Pa

p_vis the viscous energy per unit drop volume, J/m³

P_b (d ) drop breakup probability

t_b drop breakup time, s

the critical oscillation velocity, m/s

the root-mean-square velocity of the surface oscillation, m/s

We_L the Weber number