The non-dimensional form of Equation (13) was produced using dimensional
analysis on the given parameter space, which results in a single
non-dimensional term equal to proportionality constant (k ). If
this is repeated with the addition of viscosity
(μL ) into the parameter space, then Equation (14)
relates the bubble size (d32 ) with the input
power (Pm = gUSG ) and
liquid properties (surface tension, liquid viscosity, and liquid
density). Equation (14) suggests that the unknown functional formf () needs to be found experimentally from bubble size
(d32 ) data. Detailed inspections show that at
lower specific input powers the bubble column is still operating in the
homogenous regime; consequently, in the absence of shear breakage bubble
size cannot be predicted from Equation (13). Figure 9 also shows that
the d32 from conditions tested in water increase
with increasing gas superficial velocity (specific input power), this is
due to homogenous operation regime. The non-dimensional terms in
Equation (14) are well established dimensionless terms; the scaled
bubble size (left hand side) is the Ohnesorge number (Oh ), which
is the ratio of the product of the inertia and surface tension forces to
viscous forces. The scaled specific input power, which is related to the
shear breakage, is the product of the Morton number\(\left(Mo=\frac{g\mu_{L}^{4}}{\rho_{L}\sigma^{3}}\right)\) and the
Capillary number\(\left(Ca=\frac{\mu_{L}U_{\text{SG}}}{\sigma}\right)\); here the
scaled Pm term is a combination of viscous,
inertia, surface tension, and gravitational forces.