Page 121 - Separation process principles 2

86 Chapter 3 Mass Transfer and Diffusion

Thus, using the Leibnitz rule for differentiating the integral
of (3-76), with x = z/247a,

Figure 3.6 One-dimensional diffusion into a semi-infinite
medium. Thus.

the medium is infinite in the z-direction, diffusion cannot
extend to z = oo and, therefore, as z -+ oo, CA = CA~ for all We can also determine the total number of moles of
t >_ 0. Because the partial differential equation (3-68) and its
solute, NA, transferred into the semi-infinite medium by in-
one boundary (initial) condition in time and two boundary
tegrating (3-78) with respect to time:
conditions in distance are linear in the dependent variable,
CA, an exact solution can be obtained. Either the method of
combination of variables [28] or the Laplace transform
method [29] is applicable. The result, in terms of the frac-
tional accomplished concentration change, is

Determine how long it will take for the dimensionless concentra-
where the complementary error function, erfc, is related to tion change, 0 = (cA - cpb)/(cA, - cA,), to reach 0.01 at a depth
the error function, erf, by z = 1 m in a semi-infinite medium, which is initially at a solute
concentration CA,, after the surface concentration at z = 0 increases
to CA, , for diffusivities representative of a solute diffusing through
a stagnant gas, a stagnant liquid, and a solid.

The error function is included in most spreadsheet programs SOLUTION
and handbooks, such as Handbook of Mathematical Func-
For agas, assume DAB = 0.1 cm2/s. We know that z = 1 m = 100 cm.
tions [30]. The variation of erf(x) and erfc(x) is as follows:
From (3-75) and (3-76),

0 = 0.01 = 1 - erf -
(2&)
Therefore,

From tables of the error function,
Equation (3-75) is used to compute the concentration in
the semi-infinite medium, as a function of time and distance
fromthe surface, assuming no bulk flow. Thus, it applies most Solving,
rigorously to diffusion in solids, and also to stagnant liquids
and gases when the medium is dilute in the diffusing solute.
In (3-73, when (z/247a) = 2, the complementary
error function is only 0.0047, which represents less than a In a similar manner, the times for typical gas, liquid, and solid
1% change in the ratio of the concentration change at z = z media are:
to the change at z = 0. Thus, it is common to refer to
Semi-infinite
z = 4 m as the penetration depth and to apply (3-75) to Medium DAB, cm2/s Time for 0 = 0.01 at 1 m
media of finite thickness as long as the thickness is greater
Gas 0.10 2.09 h
than the penetration depth.
Liquid 1 x 2.39 year
The instantaneous rate of mass transfer across the surface Solid 239 centuries
of the medium at z = 0 can be obtained by taking the deriv- 1 x
ative of (3-75) with respect to distance and substituting it These results show that molecular diffdsion is very slow, espe-
into Fick's first law applied at the surface of the medium. cially in liquids and solids. In liquids and gases, the rate of mass