Physical Chemistry of Foods

concentrationNwill be

dN

dt

¼

1

2

JorthoN¼

2

3

d^3 N^2 C¼

4

p

jNC ð 13 : 8 Þ

where C is the velocity gradient, which here equals the shear rate.
Integration of (13.8) and puttingN¼N 0 /2 yields the halving time

t 0 : 5 ¼

pln 2

4 jC

&

0 : 54

jC

ð 13 : 9 Þ

Comparing (13.8) with perikinetic aggregation [Eq. (13.2)] shows that
the particle size dependence is now retained, and that it is to the third power;
this means that the equation can be written involvingj, as shown. Sincej
remains in principle constant during aggregation, the equation is first-order
with respect to time, rather than second-order as Eq. (13.2). Moreover,
viscosity is not in the equation; however, in an agitated vessel a higher
viscosity often leads to a smaller velocity gradient. The ratio of the initial
rates of orthokinetic over perikinetic aggregation is given by

J 0 ;ortho

J 0 ;peri

¼

d^3 ZC

2 kBT

ð 13 : 10 Þ

For particles in water at room temperature, this ratio reduces to 0.12d^3 Cif
dis expressed inmm. Taking, for instance,C¼10 s
^1 , the ratio is slightly
greater than unity for d¼ 1 mm. This would mean that encounters by
Brownian motion and by the velocity gradient are about equally frequent.
However, because Eq. (13.10) hasd^3 in the numerator, the preponderance of
orthokinetic over perikinetic aggregation will rapidly increase during
ongoing aggregation. It should further be realized that slight temperature
fluctuations, as usually occur in a laboratory, readily lead to convection
currents withC¼ 0 :1s
^1 (or higher), which then would lead to significant
orthokinetic aggregation for particles over 4mm. In other words,
orthokinetic aggregation often is more important than perikinetic aggrega-
tion in food dispersions.

Capture Efficiency. The value of the capture efficiencyais nearly
always smaller than unity, for a variety of reasons. In many cases the
magnitude of a can be reasonably predicted, although the theory is
complicated.Hydrodynamiceffects are in principle like those in perikinetic
aggregation: they impede close encounter of the particles. Moreover, the
velocity gradient exerts a shear stress, order of magnitudeZC, onto the