Environmental Engineering Reference
In-Depth Information
for initial concentration c0. The degradation rate
can directly be obtained from the
optimized parameter set of the previous example. The retardation factor
R
is
obtained from a mass balance at time
t ¼
l
0:
y
c
in
¼
y
c
0
þ
r
b
c
s
(10.16)
The initial concentration
c
in
after the fast 'splitting' is divided in a fluid and
a solid phase part. We re-write the right hand side and utilize formula (6.15) under
the assumption of a linear isotherm:
þ
r
b
c
s
y
c
y
c
in
¼
y
c
0
1
¼
y
Rc
0
(10.17)
from which we get an initial guess for the retardation, even without the observation
of any temporal changes:
c
in
c
0
R ¼
(10.18)
According to formula (
10.18
), microcystin has a retardation of
R ¼
1.2 in the
above described batch experiment.
Example 2
For nuclide chains the decay rates can be subsequently determined by using the
described parameter estimation strategy. We exemplify this here for a system of
two species, one the mother and one the daughter nuclide for example. The system
of differential equations is given by:
¼
c
1
c
2
@
@t
c
1
c
2
l
1
0
(10.19)
gl
1
l
2
denotes a partition parameter, which is lower than 1, if only a part of specie 1
decays into specie 2. The analytical solution for given initial conditions
c
10
and
c
20
is than given by:
g
c
1
ðtÞ¼c
10
exp
ð
l
1
tÞ
gl
1
l
2
l
1
c
2
ðtÞ¼c
20
exp
ð
l
2
tÞþc
10
ð
exp
ð
l
1
tÞ
exp
ð
l
2
tÞ
Þ
(10.20)
In the first step we determine the decay constant for specie 1, using the condition
(
10.8
) with the derivative (
10.9
). After that we proceed with specie 2. We can also
