Environmental Engineering Reference
In-Depth Information
In Fig.
6.5
the concentration profiles for linear and Freundlich isotherms are
compared. There is the same marginal retardation factor for both cases, i.e. for
concentration
c ¼
1 the retardation for the Freundlich isotherm is identical to
R ¼
2.3 of the linear isotherm. The figure illustrates the higher retardation for the
Freundlich isotherm for low concentrations, with the effect that for the same time
instant the concentration values for the Freundlich case are always below those of
the linear sorption case.
6.5 Slow Sorption
In the derivation of differential equations, as presented above, it was assumed that
the interphase exchange processes are fast compared to the other relevant processes.
The equilibrium between solid phase and fluid phase concentrations is reached
at all times. Such an assumption is valid in many field situations where transport
time scales are long, for example, in aquifers or aquatic sediments It is surely not
valid in other cases.
Concerning slow sorption one may keep the original set of two differential
equations (
6.13
). In the following we will show how to treat such a system in
case of no transport processes for the solid phase (
j
s
¼
0):
@
@t
yðÞ¼ry
ðÞylc e
fs
j
(6.31)
@
@t
r
b
c
s
ð
Þ¼ r
b
l
s
c
s
þ e
fs
The (
6.31
) describe transport, sorption and degradation. The latter is allowed to
be different in the dissolved and in the solid phases. Replacing the detailed
formulation for the transport fluxes, and using the assumption of constant
y
and
r
b
,
yields the formulation:
@
@t
c ¼r
ð
D
rc
Þ
v
rc lc e
fs
=y
@
@t
c
s
¼l
s
c
s
þ e
fs
=r
b
(6.32)
The exchange term is assumed to have the following form:
e
fs
¼ k
f
c k
s
c
s
(6.33)
The following describes an extension of the already developed M-file to account
for slow sorption:
