Environmental Engineering Reference
In-Depth Information
conditions. In cases where the source mass flux (
M
) is
known, the boundary condition
c
(
r
0
) =
c
0
can be replaced
by a condition that the source mass flux at
r
= 0 is equal
to
2
2
∂
∂
c
+
∂
∂
c
−
D
kc
=
0
(7.82)
x
2
y
2
M
, which yields
where
D
is the diffusion coefficient (L2T−1),
2
T
−1
),
c
is the
contaminant concentration (ML
−3
), and
k
is the first-
order decay constant (T
−1
). Equation (7.82) can be
written in polar (
r
,
θ
) coordinates as
2
M
HD
K
kr
D
c
=
(7.89)
0
π
where
H
is the depth of water (L). In Equation (7.89),
the source is idealized as a vertical line source of strength
M
(MT
−1
), and care should be taken in applying this
equation near the source, since
c
→ ∞ as
r
→ 0.
∂
∂
2
c
1
∂
∂
c
r
1
∂
∂
2
c
k
D
c
(7.83)
+
+
−
=
0
2
2
2
r
r
r
θ
For a radially symmetric concentration distribution,
∂
c
/∂
θ
and ∂
2
c
/∂
θ
2
are both equal to zero, and Equation
(7.83) reduces to
EXAMPLE 7.19
An industrial plant discharges wastewater through a
single-port outfall into the shoreline region of a lake.
Field measurements indicate that the dispersion coeffi-
cient in the lake is 1 m
2
is the decay rate of the contami-
nant is 0.1 d
−1
, and the effluent dilution 30 m from the
outfall is 12. Estimate the distance from the outfall
required to achieve a dilution of 100.
d c
dr
2
1
dc
dr
k
D
c
=
(7.84)
+
−
0
2
r
which has the general solution
+
2
2
kr
D
kr
D
c r
( ) =
AI
BK
(7.85)
0
0
Solution
where
A
and
B
are constants (dimensionless), and
I
0
and
K
0
are modified Bessel functions of the first and second
kind, respectively, both of order zero (see Appendix
D.2). Taking the boundary conditions as
Dilution is defined as the initial concentration
divided by the final concentration; hence, if
c
30
is the
contaminant concentration 30 m from the outfall and
c
i
is the concentration at the discharge location,
c
c
c r
( )
0
=
c
(7.86)
i
=
12
0
30
c
( ∞ = 0
(7.87)
and Equation (7.88) gives the concentration at a dis-
tance
r
from the outfall as
requires that the concentration is equal
c
0
on the bound-
ary of a mixing zone at a distance
r
0
from the discharge
location and that the pollutant concentration decays to
zero at a large distance from the discharge location.
Imposing the boundary conditions given by Equations
(7.86) and (7.87) on Equation (7.85) gives the
concentration distribution in the lake as (O'Connor,
1962)
2
2
kr
D
kr
D
K
K
0
0
c
i
12
c
r
=
c
=
30
2
2
k
30
k
30
K
K
0
0
D
D
Hence the dilution,
S
r
, at a distance
r
from the outfall is
given by
kr
D
2
K
0
2
k
30
c
=
c
0
(7.88)
K
0
kr
D
2
D
c
c
0
K
i
S
=
=
12
0
r
2
kr
D
r
K
0
Application of Equation (7.88) is somewhat limited in
that it is based on the concentration at the edge of
a mixing zone (
c
0
) rather than the actual source
Since
S
r
= 100,
k
= 0.1 d
−1
, and
D
= 1 m
2
/s = 8.64 ×
10
4
m
2
/d, we are looking for
r
such that
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