Environmental Engineering Reference
In-Depth Information
v
@
c
1
@x
R
1
@
c
1
@t
¼
@
D
@
c
1
@x
@x
v
@
c
2
@x
(6.22)
R
2
@
c
2
@t
¼
@
D
@
c
2
@x
@x
with
Kc
1
1
þ
1
=n
1
c
2
1
=n
2
Kc
1
1
=n
1
c
2
1
=n
2
þ
r
b
y
CEC
n
1
R
1
¼
1
1
þ Kc
1
1
=n
1
þ Kc
1
1
=n
1
(6.23)
c
2
1
þ
1
=n
2
c
2
1
=n
2
c
2
1
=n
2
þ
r
b
y
CEC
n
2
R
2
¼
1
1
þ Kc
1
1
=n
1
c
2
1
=n
2
þ Kc
1
1
=n
1
Using post-processing the solute phase concentrations can then be obtained by
and
c
s
1
¼ K CECc
1
1
=n
1
c
2
1
=n
2
þ Kc
1
1
=n
1
=
(6.24)
c
s
2
¼ CECc
2
1
=n
2
c
2
1
=n
2
þ Kc
1
1
=n
1
=
In systems with one static phase as it is in groundwater, formulations (
6.14
) and
(
6.15
) have profound advantage in comparison with the (
6.13
). If the solid phase is
fixed in space (if it is static), the genuine processes of advection and diffusion
are not present, or in mathematical formulation:
j
s
¼
0. Then (
6.16
) is a differential
equation for the unknown variable
c
:
Ry
@
@t
c ¼r yDrc
ð
Þ vgrc
(6.25)
On the right side appear terms for diffusion, dispersion and advection in the fluid
phase, but there is no contribution from the solid phase. When the differential
equation is solved, the other unknown variable
c
s
and its change in space and time
can be computed easily with the help of the isotherm. Division by
y
yields:
R
@
@t
c ¼r
ð
D
rc
Þ
v
rc
(6.26)
For a further interpretation, (
6.25
) is compared to (
6.26
) (the latter without
sources, sinks and exchange, as it is valid for tracers). In (
6.25
) retardation is
nothing more than the validity of a prolonged time scale in comparison to the
tracer. In mathematical analysis one could formally express that by the notation
R @=@t ¼ @=@ðt=RÞ
. Using the new timescale
t ¼ t R
one can say that the spatial
concentration distribution of the retarded component at time
t
is equal to the
distribution of the tracer at time
t
.
Following the concept of pure retardation sorption has no effect on stationary
concentration distributions. This can easily be seen in (
6.26
)and(
6.25
): the left side
