Environmental Engineering Reference
In-Depth Information
∑
∆
=
∑
/(
∑
′
y
u
′
h
Δ
y
I
1
=
u h y
I
1
/(
ε
y
h
)Δ
y
I
I
ε ∆
h y
I
2
u
′
h
Δ
y
I
=
I u h y
∆
2
1
y
3
2
(m)
(m
3
/s)
(m
3
/s)
(m)
(m)
(m
4
/s)
(m
4
/s)
0
0.000
0.000
0.00
0.00
0.00
0.00
1
−0.219
−0.219
−9.13
−9.13
2.00
2.00
2
−0.163
−0.382
−7.34
−16.47
2.68
4.68
3
−0.044
−0.426
−6.08
−22.25
0.99
5.67
4
0.180
−0.245
−3.23
−25.78
−4.65
1.02
5
0.270
0.024
0.30
−25.47
−6.87
−5.85
6
0.243
0.267
3.71
−21.76
−5.28
−11.13
7
0.131
0.398
5.69
−16.07
−2.11
−13.24
8
−0.088
0.310
5.75
−10.33
0.91
−12.33
9
−0.310
0.000
0.00
−10.33
3.20
−9.13
10
0.000
0.000
0.00
−10.33
0.00
−9.13
released into the top 3 m of the ocean at coordi-
nates
x
= 0 m,
y
= 0 m, and the distribution of
tracer concentration (in mg/L) sampled by
fishermen (contracted to help in simultaneous
sampling) 1 hour after release is shown in
Table 3.5. use the sample concentrations to esti-
mate the components of the diffusion coefficient
and the mean ocean current. of what practical use
are the diffusion coefficients you have
Based on the tabulated results, the value of
I
3
at
y
= 10 m
is −9.13 m
4
is and hence according to Equation (3.244),
K
x
is given by
I
A
= −
−
9 13
26 6
.
.
3
K
x
= −
=
0 34
. m /s
2
Hence, the estimated longitudinal dispersion coefficient
in the stream is 0.34 m
2
/s.
just
determined?
3.4.
Fifty kilograms of toxic material is spilled uni-
formly across a canal. The canal is trapezoidal with
a bottom width of 3 m, side slopes of 2.5:1 (H :v),
and a depth of flow of 1.7 m. The discharge in the
canal is 16 m
3
is and the longitudinal diffusion
coefficient is 7 m
2
/s.
PROBLEMS
3.1.
Show that ∇·
V
= 0 for incompressible fluids,
where
V
is the velocity field in the fluid.
3.2.
The fundamental solution to the diffusion
equation for the instantaneous release of a non-
conservative tracer of mass
M
over an area
A
is
given by
(a) How soon after the spill will the peak concen-
tration be observed 12 km downstream of the
spill, and what is the maximum concentration
expected at that location?
(b) If a safe level of this contaminant in recre-
ational waters is 15
μ
g/L, for approximately
how long will the water at the downstream
location be unsafe?
(c) What length of canal is contaminated 2 hours
after the spill?
−
kt
2
Me
x
D t
c x t
( , )
=
exp
−
A
4
π
D t
4
x
x
where
k
is the first-order decay constant, and
D
x
is the diffusion coefficient. If the nonconservative
tracer is released uniformly across a river with a
flow velocity
V
and a reference length
L
, express
the resulting concentration distribution in nondi-
mensional form. Identify the key dimensionless
parameters, and discuss their significance.
3.5.
Water stored in a reservoir behind a gate contains
a toxic contaminant at a concentration of 1 mg/L.
If the gate is opened and water flows downstream
at a velocity of 30 cm/s and with a longitudinal
diffusion coefficient of 10 m
2
/s, how long after
the gate is opened will the concentration of
the contaminant 1 km downstream be equal to
10
μ
g/L?
3.3.
You have been put in charge of a tracer test to
determine the oceanic diffusion coefficients 8 km
offshore of Miami Beach. Congratulations! Previ-
ous to your assignment, 10 kg of a tracer were
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