Page 120 - Separation process principles 2

3.3 One-Dimensional, Steady-State and Unsteady-State, Molecular Diffusion through Stationary Media 85

where Rearranging and simplifying,
ALM = log mean of the areas 2n-rL at rl and r2
L = length of the hollow cylinder
3. Spherical shell of inner radius rl and outer radius r2,
In the limit, as Az + 0,
with diffusion in the radial direction outward:

Equation (3-68) is Fick's second law for one-dimensional
diffusion. The more general form, for three-dimensional rec-
tangular coordinates, is

where AGM = geometric mean of the areas 4n-r2 at
rl and r2.
When rllrz < 2, the arithmetic mean area is no more For one-dimensional diffusion in the radial direction only,
than 4% greater than the log mean area. When rl/r2 < 1.33, for cylindrical and spherical coordinates, Fick's second law
the arithmetic mean area is no more than 4% greater than the becomes, respectively,
geometric mean area.

Unsteady State
and
Equation (3-56) is applied to unsteady-state molecular diffu-
sion by considering the accumulation or depletion of a
species with time in a unit volume through which the species
is diffusing. Consider the one-dimensional diffusion of Equations (3-68) to (3-71) are analogous to Fourier's sec-
species A in B through a differential control volume with dif- ond law of heat conduction where CA is replaced by temper-
fusion in the z-direction only, as shown in Figure 3.5. ature, T, and diffusivity, DAB, is replaced by thermal diffu-
Assume constant total concentration, c = CA + CB, constant sivity, ci = klp Cp. The analogous three equations for heat
diffusivity, and negligible bulk flow. The molar flow rate of conduction for constant, isotropic properties are, respec-
species A by diffusion at the plane z = z is given by (3-56): tively:

At the plane, z = z + Az, the diffusion rate is

The accumulation of species A in the control volume is

~CA
A- AZ (3-65) Analytical solutions to these partial differential equations in
at either Fick's law or Fourier's law form are available for a
Since rate in - rate out = accumulation, variety of boundary conditions. Many of these solutions are
derived and discussed by Carslaw and Jaeger [26] and Crank
-DABA($)z+DABA($) =A(%)AZ [27]. Only a few of the more useful solutions are presented
z+Az here.
(3-66)
FIOW in 1 Accumulation I FIOW out Semi-infinite Mediu111

Consider the semi-infinite medium shown in Figure 3.6,
which extends in the z-direction from z = 0 to z = oo. The x
and y coordinates extend from -oo to +oo, but are not of
interest because diffusion takes place only in the z-direction.
Thus, (3-68) applies to the region z >_ 0. At time t 5 0, the
concentration is CA~ for z > 0. At t = 0, the surface of the
z Z+AZ semi-infinite medium at = 0 is instantaneously brought to
Figure 3.5 Unsteady-state diffusion through a differential the concentration CA, > CA, and held there for t > 0. There-
volume A dz. fore, diffusion into the medium occurs. However, because