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Encyclopedia of Physical Science and Technology EN009H-407 July 18, 2001 23:34

172 Mass Transfer and Diffusion

provide the basic description of commercial separation volving diffusion in many dimensions are treated in detail
processes and hence supply an important topic of chemi- elsewhere (Crank, 1975 and Carslaw et al., 1986).
cal engineering.
Choosing between these three approaches is not always
B. Diffusion Across a Thin Film
easy. Diffusion problems normally give a concentration
profile as a function of position and time. Dispersion can We can explore the use of Fick’s law by considering three
do the same, but dispersion tends to be dependent solely key cases (Cussler, 1997). The easiest case for this vari-
on the physics, and not be affected by chemistry. Mass ation occurs across a thin film like that in Fig. 1. In this
transfer coefficients, on the other hand, tend to describe figure, we show one large well-stirred volume of a fluid
concentrations as a function of position or time, rather containing a solute at concentration, c 10 . It is separated by
than both variables at once. a thin film from another well-stirred volume of solution
In general, diffusion is most useful for fundamental at a different concentration, c 1l . We want to find how this
studies where we want to know the details about the sys- concentration varies between these two volumes.
tem. For example, if we were concerned with a plastisizer To find this variation, we make a mass balance on a thin
inside a polymer film, we might want to know where and layer z thick located at some arbitrary position z within
when the plasticizer is located. Diffusion will tell us. Dis- the thin film. The mass balance on this layer is
persion can be important when there is convection, as in
solute accumulation = diffusion in–out (3)
chromatography or atmospheric pollution. Mass transfer,
on the other hand, tends to be useful in less fundamental, Because the volumes adjacent to the film are large, the
more practical problems. For example, if we want to know process is in steady state and the accumulation is zero.
how to humidify and ventilate a house, we probably will The mass balance is thus
use mass transfer coefficients.
0 = j 1 | z − j 1 | z+ z (4)
We will emphasize diffusion and mass transfer in this
Dividing by z and taking the limit as z goes to zero,
article, for these are two of the more important processes in
we obtain
chemical engineering. We will mention dispersion simply
because insights into diffusion are often a valuable aid in 0 =− dj 1 (5)
understanding dispersion. We turn first to the subject of dz
diffusion itself. When we combine this with Fick’sLaw,weget
2
d c 1
0 = D (6)
I. DIFFUSION dz 2
This is subject to the boundary conditions
A. Basic Equations
z = 0 c 1 = c 10 (7)
The key equation describing diffusion, commonly called
z = l (8)
Fick’s law, asserts that the flux, that is, the amount of c 1 = c 1l
solute per area per time, is proportional to the concentra- The result is easily integrated to find the concentration
tion gradient, that is, the derivative of the concentration profile:
with respect to position (Graham, 1850 and Fick, 1855).
c 1 = c 10 − (c 10 − c 1l )z/l (9)
In quantitative terms, this relationship in one dimension
can be written as
dc 1
− j 1 = D (1)
dz
where j 1 is the flux in, for example, moles per area per
time; c 1 is the concentration in, for example, moles per
volume; z is the position, and D is a proportionality con-
stant called a diffusion coefficient. In three dimensions,
this can be written as
− j 1 = D∇c 1 (2)
which recognizes that the flux is a vector and the con-
centration can vary in all three dimensions. In this ar-
ticle we will almost always restrict our discussion to
FIGURE 1 Diffusion across a thin film. This is the simplest diffu-
one-dimensional diffusion because this is the most im- sion problem, basic to perhaps 80% of what follows. Note that the
portant case and the easiest to understand. Problems in- concentration profile is independent of the diffusion coefficient.