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

where
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X = y/
Unfortunately, this integral, which is known as the Gauss Error integral or
probability function or error function, cannot be solved analytically, thus tabulated
values must be used. The error function has the property that it is zero when X is
zero, and it approaches unity when X is 3 or larger. Its value can be found in tables
of mathematical functions, or it can be evaluated using built-in approximations in
spreadsheet software. A convenient approximation is
erf(X) = 1 – exp(–0.746X – 1.101 X 2 )
which is quite accurate when X exceeds 0.75. When X is less than 0.5, erf(X) is
approximately 1.1X. The penetration solution shown in Figure 7.4 illustrates the
very rapid initial transfer close to the interface, followed by slower penetration that
occurs later as the concentration gradient becomes smaller. Now the transfer rate at
the boundary (y = 0) can be shown to be
B(dC/dy) y=0 = C sA
Over a time t, the total flux (mol) becomes
C SA
The average flux is then obtained by dividing by t
CsA 4B/pt mol/h
But, since the average flux is CSAkM, the average mass transfer coefficient kM, which
applies over this time, must be 4B/pt .
The mass transfer coefficient, kM, under these transient conditions, thus depends
on the time of exposure (short exposures giving a large kM) and on the square root
of diffusivity. This contrasts with the steady-state solution, in which kM is independent
of time and proportional to diffusivity. The reason for this behavior is that kM is apparently very large initially, because the concentration gradient is large. It falls
in inverse proportion to t , thus the average also falls in this proportion. The lower
dependence on diffusivity (to the power of 0.5 instead of 1.0) arises, because not
all the transferring mass has to diffuse the total distance; much of it goes into
“storage” during the transient concentration buildup.
A problem now arises in environmental calculations: which definition of kM applies, B/Dy or 4B/pt ? Contact time is the key determinant. If the contact time
between phases is long, and the amount transferred exceeds the capacity of the
phases, it is likely that a steady-state condition applies, and we should use B/Dy.
4Bt
4Bt/p
B/p t