Physical Chemistry of Foods

slower until the maximum height given by Eq. (10.13) is reached. In a
horizontal direction, there would be no limit to the liquid penetration.
For straight cylindrical pores and in the absence of counteracting
forces (like gravity), the penetration rate can readily be calculated.
According to Poiseuille, the mean linear flow ratevof a liquid of viscosity
Zin through a pore of radiusris given by

v¼

r^2

8 Z

Dp

L

ð 10 : 15 Þ

whereDpis the pressure difference acting over a distance (capillary length)
L. In the present caseDpis due to the Laplace pressure and is thus given by
(2/r)gcosy. Insertion into (10.15) then yields

v¼

rgcosy

4 ZL

ð 10 : 16 Þ

To give an example, if the displacing liquid is water (g¼72 mN?m
^1 ,
Z¼1mPa?s),y¼0, and pore radius and length are 1 mm and 10 cm,
respectively, we obtainv ¼18 cm?s
^1 , i.e., very fast. Forr¼ 10 mm,
g¼50 mN?m
^1 ,y¼ 458 ,Z¼10 mPa?s andL¼1 cm, we would have
v¼0.9 mm?s
^1 , still appreciable.
However, the pores in a porous solid are virtually never cylindrical.
They are tortuous, which increases their effective length; they vary in
diameter and shape, which causes the effective resistance to flow to be larger
than that given in Eq. (10.15) and the effective Laplace pressure to be
smaller than for the average pore radius. Most importantly, the effective
contact angle will be significantly larger than the true contact angle A–W–S.
This is similar to the situation depicted in Figure 10.27a. In a pore of
variable diameter and shape, it may well be that for a true value ofy¼ 458 ,
the meniscus of the liquid in the pore tends to become convex (as seen from
the air) at some sites rather than concave; this implies that the liquid will not
move at all. In many systems, the true contact angle has to be smaller than
about 30 8 for the effective angle to be acute, i.e., for capillary displacement
to occur.
Another complication is that equilibrium values ofgandyare often
not reached during displacement, for instance because adsorption of
surfactant is too slow.
An important example of capillary displacement concerns the
dispersion of powdersin water. Most powders have particles in the range
of 5 to 500mm. An example is flour, which has to be dispersed in an aqueous
liquid for the particles to swell and so obtain a dough. Another example is
milk powder, which has to be dispersed in water to achieve its dissolution,