Environmental Engineering Reference
In-Depth Information
distributed in the
z
direction, and
δ
(
x
,
y
) is the two-
dimensional Dirac delta function defined by
From the data given,
M
= 1 kg,
L
= 4 m,
D
x
= 5 m
2
/s
(N-S),
D
y
= 10 m
2
/s (E-W). At 100 m north of the
spill,
x
= 0 m,
y
= 100 m, and the concentration as a
function of time is given by
∞
x
=
0
,
y
=
0
+∞
+∞
∫
∫
(
)
=
(
)
δ
x y
,
and
δ
x y dxdy
,
=
1
0
otherwise
−∞
−∞
1
100
4 5
2
(3.112)
c
( ,
0 100
, )
t
=
exp
−
( )
t
4
π
t
( ) ( )(
4
5 10
)
The source in this case can be characterized as an instan-
taneous line source of length
L
, and the the solution to
this fundamental two-dimensional diffusion problem is
given by (Carslaw and Jaeger, 1959)
0 00281
.
5
00
kg/m
=
exp
−
3
t
t
At 100 m east of the spill,
x
= 100 m,
y
= 0 m,
and the concentration as a function of time is
given by
2
2
M
tL D D
x
D t
y
D t
c x y t
( ,
, )
=
exp
−
−
(3.113)
4
4
4
π
x
y
x
y
1
100
4 10
2
c
(
100 0
,
, )
t
=
exp
−
(
)
t
4
π
t
( ) ( )(
4
5 10
)
This concentration distribution is a two-dimensional
Gaussian distribution with mean,
μ
, and standard devia-
tions,
σ
x
and
σ
y
, given by
0 00281
.
250
kg/m
=
exp
−
3
t
t
µ= 0
(3.114)
(b) At the spill location,
x
= 0 m and
y
= 0 m, and the
concentration as a function of time is given by
(3.115)
σ
x
= 2
D t
(3.116)
σ
y
= 2
D t
1
0 00281
.
c
( ,
0 0
, )
t
=
=
t
4
π
t
( ) ( )(
4
5 10
)
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
and
y
directions,
L
x
and
L
y
, are commonly
taken as
L
x
= 4
σ
x
and
L
y
= 4
σ
y
, respectively. If the tracer
undergoes first-order decay, then the concentration dis-
tribution is given by
At
t
= 5 minutes = 300 seconds,
0 00281
300
.
−
6
3
c
( ,
0 0 300
,
)
=
=
9 37 10
.
×
kg/m
=
9 37
.
µ
g/L
−
kt
2
2
Me
tL D D
x
D t
y
D t
Therefore, the concentration at the spill location
after 5 minutes is 9.37
μ
g/L.
c x y t
( ,
, )
=
exp
−
−
(3.117)
4
4
4
π
x
y
x
y
where
k
is the first-order decay constant.
3.3.2.1 Spatially and Temporally Distributed
Sources.
The principle of superposition can be applied
to the fundamental solution of the two-dimensional dif-
fusion equation to yield the concentration distribution,
c
(
x
,
y
,
t
), resulting from an initial mass distribution,
g
(
x
,
y
), as
EXAMPLE 3.9
one kilogram of a contaminant is spilled at a point in a
4-m-deep reservoir and is instantaneously mixed over
the entire depth. (a) If the diffusion coefficients in the
N-S and E-W directions are 5 and 10 m
2
/s, respectively,
calculate the concentration as a function of time at loca-
tions 100 m north and 100 m east of the spill. (b) What
is the concentration at the spill location after 5 minutes?
2
2
(
)
(
)
(
)
g
ξη ξ η
π
d d
tL D D
,
x
−
ξ
y
−
η
x
y
2
2
∫
∫
c x y t
( ,
, )
=
exp
−
−
4
4
D t
4
D t
y
x
y
1
1
x
x
y
(3.118)
where the contaminant source is located in the region
x
∈ [
x
1
,
x
2
],
y
∈ [
y
1
,
y
2
]. An example of diffusion from a
two-dimensional rectangular source is shown in Figure
3.18, where taking
D
x
=
D
y
leads to a symmetrical diffu-
sion pattern. Superposition in time can also be applied
to yield the concentration distribution,
c
(
x
,
y
,
t
), result-
ing from a continuous mass input
m
( )
as
Solution
(a) The concentration distribution is given by
M
tL D D
x
D t
2
y
D t
2
c x y t
( ,
, )
=
exp
−
−
4
4
4
π
x
y
x
y
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