Environmental Engineering Reference
In-Depth Information
solution
g
. Indeed, let us take
1
˄
|
ʩ
i
|
,
r
∈
ʩ
i
and
t
∈
(
T
−
˄,
T
)
p
(
r
,
t
)
=
0
,
otherwise
where
is the time required for the
nutrient to reach its critical concentration in the zone. Then the use of this formula
in (
2.18
) leads to
|
ʩ
i
|
denotes the volume of oil-polluted zone, and
˄
T
N
0
J
i
(ˆ)
=
g
i
(
r
j
,
t
)
Q
j
(
t
)
dt
+
g
i
(
r
,
0
)ˆ
(
r
)
dr
,
(2.25)
0
D
j
=
1
0
also known as the duality principle. Provided that
ˆ
(
r
)
=
0 for the first discharge
of nutrient, the last formula is reduced to
T
N
J
i
(ˆ)
=
g
i
(
r
j
,
t
)
Q
j
(
t
)
dt
.
(2.26)
0
j
=
1
, transforms the
variational problem (
2.1
)-(
2.3
) to a more convenient form for the analysis:
The use of (
2.26
)in(
2.2
) for each zone
ʩ
i
(
i
=
1
,...,
N
)
T
N
1
2
Q
j
(
minimize
m
(
Q
1
,...,
Q
N
)
=
t
)
dt
(2.27)
0
j
=
1
T
N
subject to:
c
i
−
ʱ
i
≤
g
i
(
r
j
,
t
)
Q
j
(
t
)
dt
≤
c
i
+
ʲ
i
,1
≤
i
≤
N
(2.28)
0
j
=
1
0
≤
Q
j
(
t
),
0
≤
t
≤
T
,
1
≤
j
≤
N
.
(2.29)
Note that problem (
2.27
)-(
2.29
)uses
N
adjoint functions
g
i
(
r
,
t
)
, which, when
N
, generate
N
2
restricted to the discharge points
r
j
,
j
=
1
,...,
temporal influ-
ence functions
g
i
(
compresses dynamical information
necessary to estimate how a signal emitted at point
r
j
impacts the zone
r
j
,
t
)
. Each function
g
i
(
r
j
,
t
)
ʩ
i
.Asa
consequence, the duality principle (
2.26
) quantifies the total effect on zone
ʩ
i
due
to the signals emitted at points
r
j
,
j
.
However, if a repeated discharge of nutrient is needed for degrading oil-residuals,
then the nonzero initial concentration of the nutrient must be taken into account (see
(
2.25
)). It should be noted that, due to microbial intake of nutrient in the oil-polluted
=
1
,...,
N
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