Environmental Engineering Reference
In-Depth Information
temperature is 10°C and the wastewater temperature is
20°C, estimate the temperature of the mixed river water.
tion and dispersion of a substance that undergoes
first-order decay:
Solution
c
t
c
x
c
x
+
V
=
x K
kc
(4.29)
L
(a) From the data given, Q r = 50 m 3 /s, c r = 10 mg/L,
Q w = 2 m 3 /s, c w = 1 mg/L, and the concentration of
Do in the mixed river water is given by Equation
(4.22) as
This equation makes the usual assumption that the
velocity, V , in the stream remains constant in space and
time. If this cannot be reasonably assumed, then more
complex dispersion models are available (e.g., Liang
and Kavvas, 2008).
Q c Q c
Q Q
+
+
(50)(10) (2)(1)
50 2
+
+
r
r
w w
c
=
=
=
9.7
mg/L
0
r
w
4.2.2.1  Field Measurement of K L .  Dye studies are the
most accurate method for determining the longitudinal
dispersion coefficients in rivers. These studies typically
involve releasing a conservative tracer, such as rhoda-
mine WT dye or lithium, into a river and measuring the
concentration distribution as a function of time at
several downstream locations along the river. According
to the moment property of the ADE presented in
Section 3.2.3, the dispersion coefficient is related to the
variance of the concentration distribution by Equations
(3.50-3.52). In the context of the one-dimensional ADE
that is applicable to rivers, the moment property is given
by Equation (3.50), which can be written as
(b) Since T r = 10°C and T w = 20°C, the temperature, T 0 ,
of the mixture is given by Equation (4.26) as
QT Q T
Q Q
+
+
(50)(10) (2)(20)
50 2
+
+
r
r
w w
T
=
=
=
10.4
°
C
0
r
w
Therefore, in this case, the wastewater discharge has
a relatively small effect on the Do and temperature
of the river.
4.2.2 Longitudinal Dispersion
Longitudinal mixing in streams is caused primarily by
shear dispersion, which results from the “stretching
effect” of both vertical and transverse variations in the
longitudinal component of the stream velocity. The lon-
gitudinal dispersion coefficient is used to parameterize
the longitudinal mixing of a tracer that is well mixed
across a stream, in which case the advection and disper-
sion of the tracer is described by the one-dimensional
advection-diffusion equation (ADE)
σ
L = 1
2
d
dt
x
(4.30)
K
where σ 2 is the variance of the concentration distribu-
tion (L 2 ) in the streamwise, x , direction. Equation (4.30)
is useful in determining K L from field tests in which a
slug of tracer is introduced instantaneously into a
stream, and the concentration as a function of time is
measured at two downstream locations, x 1 and x 2 . If V
is the mean flow velocity in the channel, then at any
location, x , in the channel, the travel time, t , of the tracer
can be estimated by the relation
c
t
c
x
c
x
+
+
V
=
x K
S
(4.27)
L
m
where c is the cross-sectionally averaged tracer concen-
tration (ML −3 ), V is the mean velocity in the stream
(LT −1 ), K L is the longitudinal dispersion coefficient
(L 2 T −1 ), x is the coordinate measured along the stream
(L), and S m is the net influx of tracer mass per unit
volume of water per unit time (ML−3T−1). −3 T −1 ). If the tracer
is conservative, S m = 0. If the tracer undergoes first-
order decay,
x
V
t
=
(4.31)
where Equation (4.31) assumes that transport due to
dispersion is negligible compared with advective trans-
port, which occurs in flows that have high Péclet
numbers. Equation (4.31) requires that the variance of
the temporal distribution of concentration, σ t 2 (T 2 ), be
related to the variance of the spatial distribution of
concentration, σ 2 (L 2 ), by
S
m = −
kc
(4.28)
where k is a first-order rate constant or decay factor (T −1 ).
Combining Equations (4.27) and (4.28) gives the follow-
ing equation for one-dimensional (longitudinal) advec-
1
(4.32)
σ
2
=
σ
2
t
x
V
2
 
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