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.