Environmental Engineering Reference
In-Depth Information
in the oil-contaminated areas due to the slow degradation of the oil in the marine
environment. Sufficient conditions for such a methodology are given in Sect.
2.3
.
2.2 Dispersion Model
3
and time interval
The concentration of nutrient
ˆ(
r
,
t
)
in a bounded domain
D
ↂ R
[
0
,
T
]
is estimated by the following dispersion model
N
∂ˆ
∂
t
+
U
·∇
ˆ
−∇·
μ
∇
ˆ
+
˃ˆ
+∇·
ˆ
s
=
Q
i
(
t
)ʴ(
r
−
r
i
)
(2.4)
i
=
1
ˆ
s
=−
v
s
ˆ
k
,
in
D
(2.5)
μ
∂ˆ
∂
n
=
ˆ
s
·
n
−
ʶˆ
k
·
n
on
S
T
(2.6)
μ
∂ˆ
∂
0on
S
+
n
=
(2.7)
μ
∂ˆ
∂
0on
S
−
n
−
U
n
ˆ
=
(2.8)
μ
∂ˆ
∂
n
=
0on
S
B
(2.9)
0
ˆ(
r
,
0
)
=
ˆ
(
r
)
in
D
(2.10)
∇·
U
=
0in
D
.
(2.11)
Here (
2.4
) is the advection-diffusion equation,
U
(
r
,
t
)
is the known current velocity
μ(
,
)
that satisfies the incompressibility condition (
2.11
),
r
t
is the turbulent diffusion
˃(
,
)
coefficient,
is the chemical transformation coefficient characterizing the decay
rate of nutrient in water. Note that the first-order (linear) kinetics
r
t
describing the
process of chemical transformation is a reasonable approximation for such nutrients
in water as the nitrogen and phosphorus. The term
˃ˆ
∇·
ˆ
s
in (
2.4
), describes the change
of concentration of nutrient per unit time because of sedimentation with constant
velocity
v
s
is the Dirac delta centred at the discharge point
r
i
.
Equation (
2.6
) is the boundary condition on the free surface
S
T
of domain
D
, where
ʶ(
>
0, and
ʴ(
r
−
r
i
)
is the coefficient characterizing the process of evaporation of nutrient, and (
2.9
)
represents the boundary condition on the bottom
S
B
of domain
D
. Equations (
2.7
)
and (
2.8
) are the corresponding conditions on the lateral boundary of
D
, besides,
S
+
r
,
t
)
0, and
S
−
is the rigid or outflow part of the boundary where
U
n
=
U
·
n
≥
is its inflow part where
U
n
<
0 (Fig.
2.1
). Finally, Eq. (
2.10
) represents the initial
distribution of the nutrient at
t
=
0. In all equations,
n
is the unit outward normal
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