Environmental Engineering Reference
In-Depth Information
vanishes, and the concentration
c
is determined as solution of the remaining terms on the
right side of the differential equation, in which the retardation factor does not appear.
The concept of retardation can also be maintained if degradation or decay have
to be taken into account. The equations above have to be extended by decay terms.
Instead of (
6.14
) one obtains:
@
@t
yc þ r
b
c
s
ð
Þ ¼ r y
ðÞylc r r
b
j
s
j
ð
Þr
b
l
s
c
s
(6.27)
and instead of (
6.15
):
@
@t
Ryc
þ
r
b
y
l
s
l
c
s
c
ÞRylc
with
R ¼
ð
Þ¼ r y
ðÞrr
b
j
s
j
ð
1
(6.28)
If there is the same decay constant in both phases (which is surely valid for
the radioactive decay of radio-nuclides), both
R
-factors are identical:
R ¼ R
. For
a fixed porous matrix, instead of (
6.26
) the following differential equation results:
R
@
@t
c ¼r
ð
D
rc
Þ
v
rc Rylc
(6.29)
6.3 Analytical Solution
For a homogeneous 1D constant flow field and constant parameters, the differential
equation for a retarded species (
6.29
) has an analytical solution. For the inflow of a
front with concentration
c
in
into a region with concentration
c
0
holds:
erfc
Rx
vt
2
vx
D
Rx þ vt
2
1
2
erfc
1
2
exp
cðx; tÞ¼c
0
exp
ð
lt
Þ
1
p
p
:::
DRt
DRt
erfc
erfc
c
in
2
v
u
2
D
x
Rx
ut
2
v
þ
u
2
D
x
Rx þ ut
2
þ
exp
p
þ
exp
p
DRt
DRt
(6.30)
lR
p
(Kinzelbach
1987
). This is an extension of the formula of
Ogata and Banks (
1961
), which was presented in Chaps. 4 and 5. In contrast to the
original formula there are two terms, one describing the decline of the original
concentration
c
0
and the second concerning the change of the inflow concentration
c
in
. If one of these two concentrations is zero, the formula becomes less lengthy as
one of the two terms can be omitted.
First the already introduced MATLAB
v
2
with
u ¼
þ
4
M-file
'analtrans.m'
is extended to
account for fast sorption. The retardation factor
R
is a new input parameter:
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