Physical Chemistry of Foods

13.3 SEDIMENTATION

The Stokes Equation. By Archimedes’ principle, the forceFBdue
to buoyancy and gravity acting on a submerged sphere of diameterdis given
by

FB¼

1

6

pd^3 gðrd
rcÞð 13 : 21 Þ

wheregis the acceleration due to gravityð 9 :81 m?s
^1 Þ,ris mass density,
and the subscripts d and c refer to dispersed particles and continuous liquid,
respectively. Hence the sphere will move downward (or upward ifrd<rc)
through the liquid and sense a drag force FSthat equals, according to
Stokes,

FS¼fv¼ 3 pdZcv ð 13 : 22 Þ

wherefis the friction factor,vis the linear velocity of the particle with
respect to the continuous phase, and Z is viscosity. The particle will
accelerate until FB¼FS, and from this equality the Stokes velocity is
obtained:

vS¼

FB

f

¼

gDrd^2

18 Zc

ð 13 : 23 Þ

For instance, an oil droplet in water of 2mm andDr¼
70 kg?m
^3 at room
temperature would attain a sedimentation rate v¼
0 : 15 mm?s
^1 , i.e.,
cream by 13 mm per day.
Forcentrifugalsedimentation, the accelerationgshould be replaced by
Ro^2 , whereRis centrifuge radius andorevolution rateðrad?s
^1 Þ. The
centrifugal acceleration is often expressed in units ofg. It should be realized
that the magnitude ofR, hence the effective acceleration, generally varies
throughout the liquid.

Conditions. For Eq. (13.23) to hold, a number of conditions must
be met, including the following:

The particles should be perfect homogeneous spheres. If they are

inhomogeneous—e.g., small oil droplets with a thick protein coat—

a correction can often be calculated. For nonspherical particles of

equal volume, the numerical factor in the equation will be smaller

than 1/18, the more so for greater anisometry.

The particlesurface should be immobile. Even for fluid particles this is

nearly always the case; see Section 10.7.