Physical Chemistry of Foods

depends on the boundary conditions, i.e., the geometrical constraints. For
the fairly simple case of diffusion through an infinite plane surface, on one
side of which a constant concentrationc 1 is maintained, whereas at the other
side initiallyc¼0, the amount of mass transported is given by

m¼ 2 Ac 1

ffiffiffiffiffiffi

Dt

p

r

ð 5 : 19 Þ

The concentration as a function of the distancexfrom the surface then is
given by

cðxÞ¼c 1 ð 1 
erfyÞ¼c 1 1 

2

ffiffiffi

p

p

Z y

0

expð
z^2 Þdz



y¼

x

2

ffiffiffiffiffiffi

Dt

p ð 5 : 20 Þ

wherezis an integration variable; erfy, the error function, is tabulated in
the Appendix J. Some results are illustrated in Figure 5.13, which also gives
the distancex^0 over which the original concentration difference is precisely
halved. From Eq. (5.20) it follows that the relation with the time needed for
this to occur, thehalving time t 0 : 5 , is given by

ðx^0 Þ^2 &Dt 0 : 5 ð 5 : 21 Þ*

This is a very useful equation. Although it is often only approximately
correct, because the boundary conditions are not fully met, it gives the order
of magnitude of the time scales or distance scales one has to reckon with.
For many molecules in water,Dis of order 10
^10 m^2 ?s
^1. This yields for
x^0 ¼ 1 mm a halving time of 10 ms; for 1 mm it is 10^4 s or about 17 min, and
for 1 m it is 10^10 s or about 300 years. When a cube of sugar is put into a cup
of tea, and if convective transport of dissolving sugar would not occur, it
would take on the order of several timest 0 : 5 , i.e., several months, before the
sugar is more or less evenly distributed.
In practice, however, convection will of course occur, due to currents
arising from temperature differences, if not due to stirring, and convection
greatly enhances mixing rate. In Section 5.1.1 flow is considered. It follows
from Figure 5.1b that mass transport (and transport of heat, for that
matter) in a direction perpendicular to the flow rate will not be enhanced by
simple shear flow. Elongational flow is more effective to achieve mixing. A
look at Figure 5.4 will make it clear that turbulent flow is superior in
speeding up mixing; the thickness of the laminar boundary layer will often
be a limiting factor for the transport rate perpendicular to the overall flow
direction. Pure diffusional transport can be observed by putting the sugar