longitudinal dispersion in nonuniform isotropic porous media

186
where P = vL/4D L and Ak are the positive roots in order of increasing
magnitude of
tan(ZA) =
Note that an additional dimensionless parameter, P, has appeared due to
the additional length scale, L. If we only consider the exit
concentration (x = L), then we can write X = 4P and the solution may be
expressed in terms of X and T only.
c
C! o
co
1 - exp [i (2X-T)] L
k=l
16A k sin(2A k )
(16).. 2k +X2+4X)
exp
(
-4>..2 kT )
X2 (B.lS)
Table B.l summarizes the solutions according to the type of boundary
condition employed.
B.2 Comparison £i Solutions
Figures B.l-B.S present a comparison of the solutions for a range
of X from 0.8 to 80.0 as calculated from equations (B.6), (B.ll),
(B.12), and (B.lS). The values plotted for a finite medium are taken
from tabulated values given by Brenner (1962). As the value of X
increases, the different solutions tend to the same curve. For
X = 24., (Figure B.4), both solutions for a semi-infinite medium and
the solution for an infinite medium have less than 1% maximum error
among them. Maximum error is defined as the maximum difference in
concentration among the breakthrough curves being considered, expressed
as a per cent (i.e.,a maximum concentration difference of .03 is a 3%