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
 
Search WWH ::




Custom Search