Environmental Engineering Reference
In-Depth Information
and
3
2
2
3
V
σ
−
−
σ
(0.63)
2
17 500 9580
700 400
,
−
−
t
2
t
1
K
=
=
=
3.30
m /s
2
L
2
x
x
2
1
27
∑
t c
i
=
612
mg min/L
⋅
2
i
Conservation of mass requires that the areas under the
concentration versus time curves at the upstream and
downstream measurement locations are equal. The
foregoing calculations show that the areas under the
concentration versus time curves at the upstream and
downstream locations (
∆
t
i
=
1
therefore
1
35.9
(612)
t
2
=
=
17.0
minutes
27
2
, respec-
tively) are both equal to 35.9 mg·min/L (where Δ
t
= 1
minute); hence, mass is conserved.
27
∑
=1
c
1
and
∆
t
∑
=1
c
i
i
i
i
the mean velocity in the river is therefore given by
x
−
−
x
700 400
17.0 9.1
−
−
It should be noted that if longitudinal mixing in a
river is Fickian, the tracer concentration resulting from
an instantaneous release at a channel section will have
a Gaussian distribution along the channel (i.e., relative
to
x
); however, the measured concentration at a fixed
location (
x
=
x
meas
) will not have a Gaussian distribution
in time.
Field observations commonly indicate that time con-
centration distribution curves resulting from spills are
highly skewed and heavily tailed. This typically results
from the tracer being trapped and released from within
boundary layers and zones of separated flow, and the
effect is commonly called the
dead-zone effect
or
storage-
zone effect
. The ADE does not generate the skewed
concentration distributions associated with the storage-
zone effect since it is based on the assumption that the
probability of tracer particles moving forward is the
same as the probability of tracer particles moving back-
ward. Alternative approaches used to solve this incon-
sistency include using the concept of transient storage
(Chong and Seo, 2003; Kazezyılmaz-Alhan, 2008), time-
variable dispersion coefficient (Hunt, 2006b), random
walk modeling (Boano et al., 2007), and using a general-
ized Fick's law that takes the flux to be proportional to
the
α
th derivative of the concentration, where
α
need
not be an integer (Kim and Kavvas, 2006). Specialized
field methods and associated analytical techniques must
be used to determine the dispersion-related parameters
in storage-zone models (e.g., Lambertz et al., 2006;
Wörman and Wachniew, 2007). Some field experiments
have shown these dispersion parameters to depend on
the flow rate (Schmid, 2008). Another process that is
neglected in the ADE, but is sometimes important, is
the process of sorption and desorption of a contaminant
on the bottom sediments of a river. In these cases, spe-
cialized numerical fate and transport models that
account for these bottom-sorption processes are avail-
able (e.g., Kumarasamy et al., 2011).
2
1
V
=
=
=
38.0
m/min
=
0.63
m/s
t
t
2
1
The variance of the concentration distribution at
x
= 400 m,
σ
t
2
, is given by
27
1
∑
2
2
σ
t
=
(
t
−
t
)
c
i
1
1
i
27
∑
c
i
=
1
1
i
i
=
1
where
27
∑
2
2
(
t
−
t
)
c
=
95.4
mg min /L
⋅
i
1
1
i
i
=
1
Hence,
1
35.9
(95.4)
σ
t
2
2
2
=
=
2.66
min
=
9580
s
The variance of the concentration distribution at
x
= 700 m,
σ
t
2
, is given by
27
1
∑
σ
t
2
=
(
t
−
t
)
2
c
2
i
2
2
i
∑
27
c
i
=
1
2
i
i
=
1
where
27
∑
(
t
−
t
)
2
c
=
175
mg
⋅
min
2
/L
i
2
2
i
i
=
1
Hence,
1
35.9
(175)
σ
t
2
=
=
4.87
min
2
=
17 500
,
s
2
Substituting into Equation (4.34) to determine the lon-
gitudinal dispersion coefficient,
K
L
, gives
Estimation of K
L
from Velocity Measurements.
Although
field measurements of
K
L
are the best way to determine
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