Environmental Engineering Reference
In-Depth Information
roughness is dominated by submerged vegetation,
values of
K
L
determined by the associated vertical
velocity profile can be significantly greater than pre-
dicted
b
y Equation (4.36) (Murphy et al., 2007). values
of
K du
The average velocity,
V
, can be estimated by
10
1
∑
V
=
v A
i
i
A
i
=
1
L
/
*
typically found in rivers are on the order of
20 (Fischer et al., 1979). It is important to note that the
formulas for longitudinal dispersion coefficients given
in Table 4.1 do not include parameters that measure the
sinuosity, sudden contractions, expansions, dead zones,
sand bars, pools, riffles, bridge piers, and other human
influences on natural streams; these characteristics tend
to increase the dispersion coefficient relative to straight
open channels. Since data from both natural streams
and straight open channels were used in deriving the
formulas in Table 4.1, these formulas yield a range of
values, with lower limits characteristic of straight open
channels and higher values characteristic of sinuous
natural streams.
It has been shown that the estimated dispersion coef-
ficient is most sensitive to the
mean velocity,
V
, with the
top width,
w
, average depth,
d
, and shear velocity,
u
*
, in
decreasing order of sensitivity (Deng et al., 2001). The
relative sensitivity of
V
i
s
about twice that of
w
, which
is roughly twice that of
d
. Typical values of
K
L
are on
the order of 0.05-0.3 m
2
/s for small streams to greater
than 1000 m
2
/s for large rivers.
where
v
i
and
A
i
are the velocity and area increments
measured across the channel. Therefore,
1
12.5
[(0.3)(0.2 1)
V
=
×
+
(0.6)(0.9 1)
×
+
(0.8)(1.2 1)
×
+
(1.4)(2.1 1)
×
+
(2.0)(3.0 1)
×
+
(1.6)(2.4 1)
×
+
(1.0)(1.5 1)
×
+
(0.5)(0.75 1)
×
+ .3)(0.45 1)]
(0
×
=
1.3
m/s
Since
f
= 0.03 and
V
= 1.3 m/s, the shear velocity,
u
*
, is
given by Equation (4.3) as
f
V
0.03
8
u
=
=
(1.3)
=
0.080
m/s
*
8
and the average depth,
d
, is given by
A
w
12.5
10
d
=
=
=
1.25 m
EXAMPLE 4.6
Substituting
d
= 1 2. m
,
u
*
= 0.080 m/s,
w
= 10 m, and
V
= 1.3 m/s into the formulas in Table 4.1 yields the fol-
lowing results:
Stream depths and vertically averaged velocities have
been measured at 1-m intervals across a 10-m-wide
stream; the results are as shown in Table 4.2.
If the friction factor of the flow is 0.03, estimate the
longitudinal dispersion coefficient using the expressions
in Table 4.1.
Method
K
L
(m
2
/s)
Fischer et al. (1979)
19
Liu (1977)
5
Koussis and Rodríguez-Mirasol (1998)
4
Iwasa and Aya (1991)
5
Solution
Seo and Cheong (1998)
115
Deng et al. (2001)
63
The flow area,
A
, can be estimated by summing the
trapezoidal areas between the measurement locations,
which yields
These results indicate that if the channel reach is rela-
tively straight and uniform, the longitudinal dispersion
coefficient is expected to be on the order of 10 m
2
/s,
while if the channel reach is sinuous with contractions,
expansions, and dead zones, the dispersion coefficient is
expected to be on the order of 100 m
2
/s.
A
=
(0 0.2 0.9 1.2 2.1 3.0 2.4 1.5
0.75 0.45 0)(1)
12.5
+
+
+
+
+
+
+
+
+
+
2
=
m
TABLE 4.2. Measurements of Depth and Vertically Averaged Velocity across a Stream
Distance from side,
y
(m)
0
1
2
3
4
5
6
7
8
9
10
Depth,
d
(m)
0.0
0.20
0.90
1.2
2.1
3.0
2.4
1.5
0.75
0.45
0.0
velocity,
v
(m/s)
0.0
0.30
0.60
0.80
1.4
2.0
1.6
1.0
0.50
0.30
0.0
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