tvis¼ 63? 107 s&20 years. For case (2), assuming D¼2, we obtain
tg¼6300 s& 1 :8 h, i.e., smaller than for case (1) by a factor of 10^5. (Make
these calculations yourself to check whether you have understood the
derivations.)
However, there are several complicating factors, and both results given
will be too high:
It is very difficult to avoidconvection currentswith a velocity gradient
of the order of 0.1 s^1 to occur. As soon as the particles have grown
to a size for which Eq. (13.10) equals unity,orthokinetic aggregation
will take over, which soon proceeds very much faster than
perikinetic aggregation.
If fractal aggregation occurs, itsrate constant will increaseduring the
process, because the value ofjkeeps increasing, decreasing the
diffusional distances involved. Moreover, the value ofWtends to
decrease during aggregation.
Sedimentationcan readily occur, especially if the particles andjDrjare
large. In such a case,tsedshould be estimated and compared totvis
ortg.Altogether, it is far from simple to predict aggregation times. This is
because (a) there are so many variables involved (D,a,j,Dr,C,Z,W); (b) it
depends on combinations of their magnitudes what equation is valid; and (c)
the theory has some uncertainties. Nonetheless, some understanding is
useful. Knowledge about trends as to what will happen if certain variables
are changed can be important, and that can often be deduced. One trend is
that the formation of a gel tends to occur much faster than the formation of
visible particles, because it takes far fewer aggregation steps to obtain a large
fractal aggregate than a large compact particle, unless the particle size is
large and the volume fraction of the particles very small. Another trend is
that for largejDrj, layer separation due to sedimentation is quite likely,
unlessais very small andjis large.
If the capture efficiency is close to unity, aqueous dispersions tend to
have short aggregation times, often a few minutes and nearly always below
an hour, provided thatjis larger than about 0.02 and the viscosity is not
much greater than that of water. It should further be realized that food
systems can be subject to slow chemical changes that affect the magnitude of
W; for instance, some bacteria can make lactic acid, whereby the pH is
decreased, which can decrease the negative charge of particles, which then
will significantly decrease electrostatic repulsion.