Environmental Engineering Reference
In-Depth Information
2.7 Numerical Examples of Remediation in a Channel
In order to illustrate themethod developedwe now consider a two-dimensional exam-
ple of remediation in a channel of one hundred and twenty metres long
[
0
,
120
]
and
ten metres wide
[
0
,
10
]
. The channel contains three oil-polluted zones
ʩ
i
(
N
=
3).
]
−
3
The critical nutrient concentrations
c
i
in the zones vary from one ex-
periment to another according to Table
2.1
. The zones under consideration are:
ʩ
1
(
grm
)
.
The parameters of the adjoint model (
2.18
)-(
2.24
) have been taken as follows: the
velocity vector
U
is directed along the channel and is equal to 30 mh
−
1
, the dif-
fusion coefficient
=[
20
,
30
]×[
9
,
10
]
,
ʩ
2
=[
40
,
60
]×[
9
,
10
]
and
ʩ
3
=[
96
,
100
]×[
0
,
2
]
is 6m
2
h
−
1
, the coefficient of chemical decay
is 1 h
−
1
, and
μ
˃
ʶ
=
0. The discharge of nutrient is performed from the optimal points during
four hours,
v
s
=
(
0
,
T
)
≡
(
0
,
4
),
and the mean concentration is controlled within the last
one-hour interval
1
h
.
For each oil polluted zone the adjoint model (
2.18
)-(
2.24
) was solved by means
of the bidimensional version of the splitting-up method (
2.45
)-(
2.46
) which is de-
scribed in Sect.
2.5
. The parameters of discretization are the same in all the numerical
experiments. The mesh size is the same in both directions, namely,
(
3
,
4
)
, i.e.,
˄
=
ʔ
x
=
ʔ
y
=
0
.
4,
and the corresponding mesh size in the time direction is
005. The function
I
,
given by Eq. (
2.63
), was built for each polluted zone through the respective adjoint
solution. In each case, by themaximization of function
I
we determined the following
optimal discharge points:
r
1
ʔ
t
=
0
.
,
r
2
and
r
3
.
For this grid (as well as for finer grids) we obtained that the optimal discharge site
tends to be the point at the left-superior corner of the zones
=
(
20
.
2
,
9
.
8
)
=
(
40
.
2
,
9
.
8
)
=
(
96
.
2
,
0
.
2
)
ʩ
1
and
ʩ
2
, and the
left-inferior corner of zone
ʩ
3
, as it must be in order to have the maximum impact
of nutrient in each polluted zone.
The adjoint solutions
g
ij
=
r
j
,
,forthe
i
th polluted zone and the
j
th optimal
discharge point, are plotted in Figs.
2.3
,
2.4
and
2.5
. According to Eq. (
2.64
), the
basic discharge rate for each polluted zone
g
i
(
t
)
ʩ
i
is a multiple of the adjoint function
r
i
,
g
ii
. From the shape of these functions, given in Figs.
2.3
,
2.4
and
2.5
,
one concludes that the basic discharge rates are equal to zero in the time interval
[
=
g
i
(
t
)
. According to Eq. (
2.25
), this means that a basic discharge rate influences
the nutrient concentration of a polluted zone only if the adjoint function of the zone
is non-zero in the time interval
0
,
2
.
25
]
. Figure
2.3
shows that
g
12
and
g
13
do not
satisfy this condition, and therefore the discharge of nutrients at points
r
2
and
r
3
has
no influence on its concentration in zone
[
2
.
25
,
4
.
0
]
ʩ
1
, as it was to be expected due to the flow
direction and the location of zones in the channel.
Table 2.1
Concentrations
c
i
(grm
−
3
) in the three polluted
zones
Concentration
\
Experiment
1
2
3
4
5
c
1
0
.
8
1
.
0
0
.
5
1
.
2
0
.
6
.
.
.
.
.
c
2
0
8
0
8
1
0
0
5
1
2
c
3
0
.
8
0
.
5
1
.
5
1
.
2
0
.
6
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