Environmental Engineering Reference
In-Depth Information
Therefore, the calculated centerline concentration
using the exact solution of the diffusion equation
(15.8 mg/L) is only slightly different from that cal-
culated using the approximate solution (15.9 mg/L).
The corresponding concentrations at the offset
point,
c
25
and
{
+
=
(0.288)
2 625
π
(
)
c
0
exp
2
(
625 2 62 10
)( .
×
−
5
)
[
(
)
−
(
)
]
erf
1.076 0.119
erf 1.021 0.125
+
(
)
−
5
+
exp
−
2
(
625 2 62 10
)( .
×
)
c
25
, are determined using these same
calculations with the only difference being that
y
= 25 m. The results are as follows,
′
]
}
[
(
)
−
(
)
erf
1.076 0.119
−
erf
1.021 0.125
−
c
25
=
13 7
. mg/L
=
7 41 10
.
×
−
4
kg/m
3
c
25
′ =
15 . mg/L
=
0 741
.
mg/L
It is apparent from these results that there is a sig-
nificant difference between the concentration esti-
mated using the exact solution (13.7 mg/L) and the
concentration estimated using the approximate
“slab” solution (15.4 mg/L). This 12% difference
can be mostly attributed to the fact that the longi-
tudinal diffusive flux is not negligible compared
with the advective flux, since Pe =
Vx
/
D
x
=
(0.01)(50)/(1) = 0.5. However, is spite of this rela-
tively low Péclet number, the centerline concentra-
tions are in close agreement.
Therefore, the expected concentration 50 m down-
stream from the release point after 10 minutes is
0.741 mg/L. The expected concentration,
c
25
, at a
point 25 m off the plume centerline is obtained by
repeating the above calculations with the only dif-
ference being that
y
= 25 m. These calculations yield
c
25
= .
0 563
mg/L
,
and hence the expected concentration at the offset
point is 0.563 mg/L.
(b) For a continuous release of tracer at 10 kg/s, the
derived parameters for the centerline point (50 m,
0 m, 0 m) are the same as those calculated in (a).
Substituting these derived parameters into Equa-
tion (3.163) gives the steady-state concentration,
c
0
,
on the centerline point (50 m, 0 m, 0 m) as
3.3.3.4 Continuous Point Source with Variable Dif-
fusion Coefficient.
In some aqueous environments, par-
ticularly in the ocean and in groundwater, the diffusion
coefficient increases with distance from the source. This
happens in these environments because the range of
ambient velocities experienced by the tracer cloud
increases with distance traveled, and since the diffusion
coefficient depends directly on the range of velocities
experienced by the cloud, the diffusion coefficient must
necessarily increase with distance traveled. However,
after some distance, the range of velocities experienced
by the tracer cloud no longer increases and the diffusion
coefficient asymptotes to a constant value. A diffusion
process in which the diffusion coefficient is not constant
is called a
non-Fickian process
.
Consider a continuous point source in which the
transverse diffusion coefficients are functions of the dis-
tance from the source and longitudinal diffusion is negli-
gible compared with the advective mass flux (i.e., Pe >> 1),
the governing diffusion equation is then given by
d
π
(
)
c
=
exp
−
2
ab
0
a
( .
0 288
625
)
π
(
)
=
exp
−
2
(
625 2 62 10
)( .
×
−
5
)
(
)
3
=
0.015
kg/m
mg/L
8
=
15 8
.
In contrast, if the approximate “slab” equation is
used, the estimated centerline concentration,
c
0
, is
′
given by Equation (3.152) as
2
2
m
V y y
D x
(
−
)
V z z
D x
(
−
)
1
1
c
′ =
exp
−
−
0
4
(
−
x
)
4
(
−
x
)
4
π(
x
−
x D D
)
y
1
z
1
1
y
z
10
=
4
π(
50 0
−
) ( . )( . )
1 0 1 0
2
2
∂
∂
c
x
∂
∂
c
y
∂
∂
c
z
V
=
D x
( )
+
D x
( )
(3.164)
y
z
2
2
0 01 0 0
4 1 0 50 0
( .
)(
−
)
2
( .
01 0 0
4 1 0 50 0
0
)(
−
)
2
exp
−
−
( . )(
−
)
( . )(
−
)
where
D
y
(
x
) and
D
z
(
x
) are the horizontal-transverse
and vertical-transverse diffusion coefficients as a func-
tion of the longitudinal coordinate
x
, respectively. rep-
resenting
D
y
(
x
) and
D
z
(
x
) by the relations
=
0 0159
.
kg/m
3
=
15.9
mg/L
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