Environmental Engineering Reference
In-Depth Information
[
t
n
−
ʔ
,
t
n
+
ʔ
]
in each subinterval
t
t
. The sum of such relations over all subintervals
[
,
]
in
(i.e., over all
n
) and the use of conditions (
2.10
) and (
2.24
) leads to a discrete
version of adjoint estimate (
2.26
).
0
T
2.6 Theoretical Results: Existence, Uniqueness
and Formulation of Discharge Parameters
2.6.1 First Stage: Discharge Points and Basic Form
of Discharge Rates of Nutrient
In order to find the optimal discharge points
r
i
in
D
, and the basic shape of discharge
rates
Q
i
(
t
)
at these points, we consider here the variational problem (
2.27
)-(
2.29
)
for
N
0. That is we consider in the first stage of the strategy
just a local problem of remediation in which the critical concentration
c
1
is reached
exactly. Thus, taking into account the corresponding adjoint function
g
1
(
=
1 and
ʱ
1
=
ʲ
1
=
r
,
t
)
for the
oil-polluted zone
ʩ
1
, the variational problem becomes
T
1
2
Q
2
minimize
m
(
Q
)
=
(
t
)
dt
(2.50)
0
T
subject to:
g
1
(
r
1
,
t
)
Q
(
t
)
dt
=
c
1
(2.51)
0
Q
(
t
)
≥
0
,
0
≤
t
≤
T
(2.52)
where, for simplicity, we have omitted the subindex in the release rate, that is
Q
(
t
)
=
Q
1
(
t
)
. At first, the site
r
1
∈
D
is considered as any point such that
T
P
(
r
1
)
=
g
1
(
r
1
,
t
)
dt
>
0
.
(2.53)
0
The set of points where condition (
2.53
) holds is called support of function
P
[
11
].
Note that condition (
2.53
) is necessary to satisfy constraint (
2.51
) and that such
condition is fulfilled for any point
r
1
in the polluted zone
ʩ
1
. Moreover, condition
(
2.53
) is also satisfied for points that are outside
ʩ
1
but fairly close to this area;
such points are adjacent to
ʩ
1
and are located on the streamlines coming into the
zone. The size of such set of points depends on how large is the parameter
T
and the
velocity of the flow
U
2
in a neighbourhood of the zone
ʩ
1
.
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