Physical Chemistry of Foods

may eventually become visible to the eye (1); the volume fraction of particles
jdoes not alter. Nearly the same situation arises when the particles upon
aggregation immediately rearrange into compact aggregates of dimension-
ality close to three, althoughjwill then somewhat increase. Ift 24 t 1 ,
fractal aggregation occurs,jmarkedly increases, and eventually a gel tends
to form (2). If large particles or aggregates formed do sediment before (1) or
(2) can occur, layer separation (3) is the result. In case (2) a fractal
dimensionality applies. Of course, intermediate situations can occur.
Prediction of the aggregation time, i.e., the critical time for occurrence
of a perceptible changetcr, is often desirable. The halving timet0.5[e.g., Eq.
(13.3)] is often called the coagulation time, assuming thattcrwill generally be
a small multiple, say by a factor of 10, oft0.5. However, that may turn out
to be very misleading. The way to calculatetcris to use the equation for

dN/dt, rearrange it into an equation for dR/dt, invert the latter, and
calculate

tcr¼

ZRcr

a

dt

dR

dR ð 13 : 19 Þ

Applying this toperikinetic aggregation[Eqs. (13.2) and (13.3)] leads to

tcr¼t 0 : 5 ðqD
 1 Þ&t 0 : 5 qD ð 13 : 20 Þ

whereq¼Rcr/a. The stability factorWcan be included in the expression of
t0.5. For case (1),D¼3andRcr¼Rvis& 0 :2 mm. For case (2),Dis mostly
between 1.8 and 2.4, andRcr¼Rg,as given by Eq. (13.15). For case (3), see
Eqs. (13.11) and (13.17).
A sample calculation may be enlightening. Assume aggregation of
spherical particles in water at room temperature;W¼1 (rapid perikinetic
aggregation),a¼0.2mm,j¼0.01, andDr¼0. The halving time would then
be 0.6 s. Aggregation according to case (1), whereD¼3, would lead to

FIGURE13.9 Aggregation of particles followed by coalescence.