The eddies have lifetimes given bytðleÞ&
le
u^0 ðleÞ¼l^2 e=^3 e^1 =^3 r^1 =^3 ð 11 : 9 ÞHence the smaller the eddies, the shorter their lifetime. Moreover, smaller
eddies have a higher local power density. Inside a small eddy, Re is small,
hence the flow is laminar, and in laminar floweequalsZ?C^2 ; localCwill
equalu^0 ðleÞ=le&l=tðleÞ. Eddies below a certain size cannot be formed, since
the local value ofewould be so high that the kinetic energy would be fully
dissipated as heat.
Regime TI. The very simple Eq. (11.8) provides the basis for the
relations for drop or bubble disruption in regime TI. Because of their
abundance and short lifetime, the energy-bearing eddies cause rapid local
velocity fluctuations, which according to the Bernoulli equation (5.4) cause
pressure fluctuations given by
DpðleÞ&r½u^0 ðleÞ^2 &e^2 =^3 l^2 e=^3 r^1 =^3 ð 11 : 10 ÞIf nowDpnear a drop is larger than its Laplace pressure, the drop will be
broken up. A more detailed analysis of the eddy size spectrum shows that
drop disruption is most effective ifd¼le. Hence putting the Laplace
pressure equal to Eq. (11.10), and insertingdforle, results fordin the
maximum size that a drop can have in the turbulent field, because larger
ones will be disrupted. The relation is
dmax&e^2 =^5 g^3 =^5 r^1 =^5 ð 11 : 11 ÞDisruption probably occurs via the sudden formation of a local protrusion
on a drop, which then breaks off. This would mean that fairly wide droplet
size distributions result, as is indeed observed: see Figure 11.9a.
In most cases, Eq. (11.11) would also be valid for the average diameter
obtained (generallyd 32 is used), albeit with a somewhat smaller proportion-
ality constant. If power input is varied, power density may vary
proportionally, which then implies that the average drop size obtained
would be proportional to energy input to the power0.4. In a stirrer,eis
proportional to revolution rate cubed. In a high-pressure homogenizer, the
homogenizing pressure pH can be varied, and the power density is
proportional top^1 H:^5 ; it follows that average drop size is proportional to
pH^0 :^6. These relations are well obeyed for dilute emulsions with an excess of
surfactant. Figure 11.9 gives some practical results. In part b, lower curve,
the slope is indeed exactly0.6.