Figure 8 . A correlation for scaling the void fraction in the homogenous regime (water).

3.3 Heterogeneous regime

The heterogeneous operation regime features frequent breakup and coalescence events. Coalescence produces larger bubbles, which are more susceptible to deformation and breakage. Generally, coalescence increases the number of large bubbles and breakage increases the number of small bubbles; therefore, in a statistically stationary bubble size population the coalescence skews the PDF negatively (towards the right tail) and breakage skews the PDF positively (towards the left tail). This explains the shift in PDF shape from a bell (hump) shape to a positively skewed spike shape when the operation regime changes from homogenous to heterogeneous regime. To approach the physical scaling of the bubble size characteristic length scale, it was hypothesized that in heterogeneous regime the interfacial momentum transfer sets the stable bubble size. Therefore, the energy supplied to the liquid phase from the injection of the gas phase is expected to power the interfacial momentum transfer. In the current work, statistically stationary samples of bubble size were used to test this hypothesis. Sauter mean diameter was measured according to the test matrix in Table 1 to test the relationship between bubble size and specific input power per unit mass (Pm = gUSG ). Hinze38 studied the breakage of drops and recommended using the maximum stable drop size (d95 ) under shear breakage for scaling and argues that d95 is the characteristic length that constrains 95% of the dispersed phase volume. Alves et al.51 argue that the Sauter mean diameter is proportional to the maximum stable bubble size; therefore, in the present work d32 was used as the bubble size characteristic length scale for bubble size scaling. Figure 9 shows the measured d32 at variousPm levels, which shows that for the glycerin conditions (G1-G3) the Sauter mean diameter decreases with increasing specific input power. Hinze38 proposed a correlation (Equation 13) to predict the maximum stable bubble size as a function of specific power input, surface tension, and density of the continuous phase. In Equation (13), the proportionality coefficient (k ) is a function of the critical Weber number (Wecr ). It has been demonstrated that the proportionality constant corresponds to different mechanisms, including k = 0.725 for isotropic turbulent38 and k ~ 1.7 for shear bubble breakup.52,53 Figure 9 compares the predicted bubble size from Equation (13) (k = 0.45) with the measured bubble size (Sauter mean diameter) from all cases tested in the present study.