McKay, Donald. "Front matter" Multimedia Environmental Models ...

sionally, the term flux rate is used in the literature. This is definitely wrong, because
flux contains the concept of rate just as does speed. Flux rate is as sensible as speed
rate.
It is worthwhile digressing to examine how the mixing process leads to diffusion
and eventually to Fick’s first law. This elucidates the fundamental nature of diffusivity
and the reason for its rather strange units of m2/h.
Much of the pioneering
work in this area was done by Einstein in the early part of this century and arose
from an interest in Brownian movement—the erratic, slow, but observable motion
of microscopic solid particles in liquids, which is believed to be due to multiple
collisions with liquid molecules.
7.3.2 Fick’s Law and Diffusion at Steady State
We consider a square tunnel of cross-sectional area A m2
containing a nonuniform
solution, as shown in the middle of Figure 7.2, having volumes V1,
V2,
etc., separated
by planes 1–2, 2–3, 3–4, etc., each y metres apart.
We assume that the solution consists of identical dissolved particles that move
erratically, but on the average travel a horizontal distance of y metres in t hours. In
time t, half the particles in volume V3
will cross the plane 2–3, and half the plane
3–4. They will be replaced by (different) particles that enter volume V3
by crossing
these planes in the opposite direction from volumes V2
and V4.
Let the concentration
of particles in V3
and V4
be C3
and C4
mol/m3
such that C3
exceeds C4.
The net
transfer across plane 3–4 will be the sum of the two processes: C3
yA/2 moles from
left to right, and C4
yA/2 moles from right to left. The net amount transferred in
time t is then
©2001 CRC Press LLC
C3yA/2
– C4
yA/2 = (C3
– C4)
yA/2 mol
Note that CyA is the product of concentration and volume and is thus an amount
(moles).
The concentration gradient that is causing this net diffusion from left to right is
(C3
– C4)/y
or, in differential form, dC/dy. The negative sign below is necessary,
because C decreases in the direction in which y increases. It follows that
The flux or diffusion rate is then N or
(C3
– C4)
= –ydC/dy
N = (C3
– C4)
yA/2t = –(y2A/2t)
dC/dy = –BAdC/dy mol/h
which is referred to as Fick’s first law. The diffusivity B is thus (y2/2t),
where y is
the molecular displacement that occurs in time t.
In a typical gas at atmospheric pressure, the molecules are moving at a velocity
of some 500 m/s, but they collide after traveling only some 10–7
m, i.e., after 10–7/500
or 2 ¥ 10–10
s. It can be argued that y is 10–7
m, and t is 2 ¥ 10–10;
therefore, we