Hindered Sedimentation. Strictly speaking, Eq. (13.23) is only
valid for one particle in an infinite amount of liquid, and not for finite values
ofj. Moreover, it gives the speed relative to the liquid, whereas that relative
to the vessel is desired. To correct for this effect, i.e.,displacement of liquid
by the sedimenting particles, vS has to be multiplied by (1j). The
sedimentation rate decreases much more steeply withj, however. In Section
5.1.2 we discussed how theeffective viscosityincreases with increasingj
value. An even much larger correction stems from the frictional effect
caused by the counterflow of displaced liquid. Furthermore, there is a
positive contribution to the sedimentation rate bygroup sedimentation(see
below). Altogether, theory leads to an equation of the type
vðjÞ
vS¼ð 1 jÞn ð 13 : 26 Þwhere the exponentnwould equal 6.55. This would imply that forj¼0.1,
the velocity is reduced by a factor of 2. The relation has been confirmed for
strictly monodisperse suspensions. In practice, highernvalues are observed,
up to 9. The main cause appears to be polydispersity, which further impedes
sedimentation rate.
Group Sedimentation. Sedimenting particles also show Brownian
motion. This implies that two particles may for a little while be close
together, i.e., form a temporary doublet. If, moreover, the doublet is
oriented in the direction ofg, it will sediment faster than a single particle: the
net gravitational force is doubled, but the frictional force will be less than
doubled. If the doublet remains for a period over which significant
sedimentation occurs, sedimentation will be enhanced. It is useful to
estimate aPe ́clet numbergiving the ratio of the time scales involved. The
numerator is the time it takes for two particles to diffuse away over the
distance of a particle diameter, given byd^2 /4D. The denominator is the time
needed for sedimentation over the same distance, to be derived from Eq.
(13.23). Using also Eq. (5.16) for the diffusion coefficient, we arrive at
Pe¼pd^4 gjDrj
6 kBTð 13 : 27 ÞIf Pe>1, the sedimentation rate of the doublet is enhanced, but it will only
be quantitatively important if (a) it lasts relatively long, i.e., for Pe 4 1, and
(b) if a sufficient proportion of the droplets is at any time present in
doublets. The latter proportion is, in first approximation, proportional toj.
Figure 13.12 gives results on the creaming of O–W emulsions of
various oil contents, both by gravity and by centrifuging. The duration of