Environmental Engineering Reference
In-Depth Information
Ry
@
þ
r
b
y
@
c
s
@c
@t
c ¼r y
ðÞrr
b
j
s
j
ð
Þ
with
R ¼
1
(6.16)
(Kinzelbach
1987
). For formulation (
6.16
) it is assumed that porosity and bulk
density are constant in time. One can interpret the role of the retardation factor on
the left side of the equation as changing the time scale (see below in this sub-chapter
for a more detailed discussion). As the factor is always greater than 1 (all terms
appearing in its defining equation are positive),
R
is responsible for
retardation
.
In the case of a linear isotherm
c
s
/c ¼ K
d
, there is no difference between the
factors
R
in (
6.15
) and (
6.16
):
þ
r
b
y
R ¼
1
K
d
(6.17)
there is a constant retardation, for which one often finds the
definition (
6.17
). Retardation factors range from values slightly above 1 up to 10
7
,
as measured for example by Luo et al. (
2000
) for Thorium 232.
In general,
R
depends on the concentrations and on porosity and thus may
change with time and space. Both definitions given above differ in the general
situation. The left hand side of the differential equations (
6.14
) and (
6.15
) is then
cause for nonlinearity. For the Freundlich-isotherm holds:
For constant
y
þ
r
b
y
a
F
1
a
F
2
ðcÞ
a
F
2
1
R ¼
1
(6.18)
and the Langmuir isotherm:
þ
r
b
y
a
L
1
a
L
2
a
L
2
þ c
R ¼
1
(6.19)
2
ð
Þ
The same procedure can be followed in modeling ion exchange. This will be
exemplified for two species, for which the cation exchange capacity CEC can be
noted as the sum of the solid phase concentrations of the two species:
CEC ¼ c
s
1
þ c
s
2
(6.20)
Using the Gapon isotherm (
6.8
) in addition one obtains:
c
s
1
¼ K CEC
c
1
1
=n
1
c
2
1
=n
2
c
s
2
¼
þ K
c
1
1
=n
1
c
2
1
=n
2
c
2
1
=n
2
c
1
1
=n
1
c
2
1
=n
2
c
1
1
=n
1
(6.21)
1
K
CEC
K
þ
1
and
1
in order to write the problem setting in two differential equations:
