Environmental Engineering Reference
In-Depth Information
corresponding to an initial plume size of 35 m is 0.5 m
2
/s.
The first-order decay constant of the contaminant is
estimated as 0.04 min
−1
. Estimate the maximum concen-
tration and the width of the plume 100 m downstream
of the diffuser.
where
δ
(
x
,
y
,
z
) is the three-dimensional Dirac delta
function defined by
∞
x
=
0
,
y
=
0
,
z
=
0
(
)
=
δ
x y z
,
,
and
0
otherwise
(3.132)
+∞
+∞
+∞
∫
∫
−∞
∫
(
)
δ
x y z dxdydz
,
,
= 1
Solution
−∞
−∞
From the given data:
b
= 35 m,
c
0
= 100 mg/L,
V
=
10 cm/s = 0.10 m/s,
D
y
0
= 0.5 m
2
/s,
k
= 0.04 min
−1
=
0.000667 s
−1
, and
x
= 100 m. Since the source is an ocean
outfall diffuser, it is appropriate to used the 4/3 law
formulation in Table 3.1. Therefore,
The solution to the fundamental three-dimensional dif-
fusion problem is given by (Carslaw and Jaeger, 1959)
M
c x y z t
( ,
,
, )
=
3
2
(
4
π
t
)
D D D
x
D t
x
y
z
(3.133)
2
y
D t
2
z
D t
2
exp
−
−
−
4
4
4
kx
V
3 2
/
x
y
z
c x
( , )
0
=
c
exp
erf
0
2
8
D x
Vb
−
y
0
2
1
+
1
This concentration distribution is a three-dimensional
Gaussian distribution with mean,
μ
, and standard devia-
tions,
σ
x
,
σ
y
, and
σ
z
, given by
( .
0 000667 100
0 10
)(
)
c
(
100 0
, )
=
(
100
)exp
( .
)
µ= 0
(3.134)
(3.135)
σ
x
= 2
D t
3 2
/
erf
2
8 0 5
100
0 10 35
( . )
(
)
σ
y
= 2
D t
(3.136)
1
+
−
1
2
( .
)(
)
σ
z
= 2
D t
(3.137)
=
30 6
.
mg/L
Since 95% of the area under a Gaussian distribution is
within ±2
σ
of the mean, the extent of the contaminated
area in the
x
,
y
, and
z
directions,
L
x
,
L
y
, and
L
z
are com-
monly taken as
L
x
= 4
σ
x
,
L
y
= 4
σ
y
, and
L
y
= 4
σ
z
, respec-
tively. If the tracer undergoes first-order decay, then the
concentration distribution is given by
L
b
8
D x
Vb
8 0 5 100
0 10 35
( . )(
)
y
0
2
=
1
+
=
1
+
=
8 81
.
( .
)(
)
2
L
=
8 81
.
b
=
8 81 35
.
(
)
=
308
m
Me
−
kt
Based on these results, the maximum concentration
100 m downstream of the outfall is 30.6 mg/L, and the
width of the plume at this location is 308 m.
c x y z t
( ,
,
, )
=
3
2
(
4
π
t
)
D D D
x
D t
x
y
z
(3.138)
2
y
D t
2
z
D t
2
exp
−
−
−
4
4
4
3.3.3 Diffusion in Three Dimensions
x
y
z
The diffusion equation in three dimensions is given by
where
k
is the first-order decay constant.
∂
∂
c
t
∂
∂
2
c
∂
∂
2
c
∂
∂
c
z
2
=
D
+
D
+
D
(3.129)
x
y
z
x
2
y
2
2
EXAMPLE 3.13
one kilogram of a toxic contaminant is released deep
into the ocean and spreads in all three coordinate direc-
tions. The N-S, E-W, and vertical diffusion coefficients
are 10, 15, and 0.1 m
2
/s, respectively. (a) Find the
concentration at a point 100 m north, 100 m east, and
10 m above the release point as a function of time.
and the fundamental solution corresponds to the fol-
lowing initial and boundary conditions
c x y z
( ,
,
, )
0 =
M x y z
δ
( ,
, )
(3.130)
c
(
±∞ ± ∞ ± ∞
,
,
, )
t
= 0
(3.131)
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