FUNDAMENTALS OF FATE AND TRANSPORT - Water Quality Engineering in Natural Systems - page 74

is placed 4 m from the side of the channel (so
that it extends from 4 to 9 m from the side of
the channel and extends 1 m in the upstream/
downstream direction). An accident an the indus-
trial facility causes a highly toxic substance to be
released with the wastewater at a rate of 50 kg/s
for 10 seconds. Discretizing the 5-m 2 discharge
structure into 0.5 m × 0.5 m elements and dis-
cretizing the continuous discharge into 1-second
“puffs,” estimate the concentration of the toxic
substance at 30-second increments at a location
50 m downstream from the center of the discharge
structure. The maximum simulation time is 300
seconds. (Hint: It might be helpful to set up these
calculations on an electronic spreadsheet.)

use this result to calculate the steady-state con-
centration at a point 16 km downstream of the
beginning of the overflow pipes. Neglect longitu-
dinal diffusion.

3.18. A contaminated stream 10 m wide and 2 m deep
merges with an uncontaminated stream such that
the merged stream has an average velocity of
15 cm/s, and longitudinal and transverse diffusion
coefficients estimated as 1 and 0.05 m 2 /s, respec-
tively. The flux of a conservative contaminant from
the contaminated stream is 0.1 kg/s. Estimate the
concentration 15 m from the bank of the stream
at a section 20 m downstream of the confluence.
Neglect the influence of images in accounting for
the sides of the stream.

3.16. A bridge crosses a river 4.5 km upstream of a
water supply intake, and in recent years, there
have been several contaminant spills at the river
crossing, primarily on the side of the river oppo-
site the water supply intake. The river is 30 m wide,
3 m deep, has an average flow of 13.5 m 3 /s, the
longitudinal diffusion coefficient in the river is
estimated to be 1.27 m 3 /s, and the transverse dif-
fusion coefficient is 0.0127 m 3 is What mass of con-
taminant spilled at the bridge (on the opposite
side to the intake) would lead to a contaminant
concentration at the intake equal to 1 mg/L?

3.19. A 20-m-long diffuser discharges a contaminant
into the ocean such that after initial mixing over
the depth, the concentration is 5 mg/L. The ocean
current is 15 cm/s, and the initial diffusion coeffi-
cient is 0.1 m 2 is The first-order decay constant of
the contaminant is 0.05 min −1 . Estimate the
maximum concentration and the width of the
plume 200 m downstream of the outfall.

3.20. Five kilograms of a toxic contaminant is released
deep into the ocean and spreads in all three coor-
dinate directions.

3.17. Surface runoff is distributed by overflow pipes
along a 1.61-km length of stream, where the runoff
rate is 0.142 m 3 is and the BoD of the runoff is
300 mg/L. The average flow rate along the stream
can be taken as 1.56 m 3 is the BoD decay rate is
0.4 d −1 , the BoD of the stream water upstream of
the overflow pipes is zero, and the cross sectional
area of the stream is 18.6 m 2 . The longitudinal dif-
fusion coefficient is estimated to be 1 m 2 /s.

(a) If the N-S, E-W, and vertical diffusion coef-
ficients are 15, 20, and 0.5 m 2 /s, respectively,
find the concentration at a point 50 m north,
50 m east, and 5 m above the release point as
a function of time.
(b) What is the concentration at the release point
after 12 hours? Assume no reactions and no
advection.
3.21. A submarine releases 100 kg of waste at a location
25 m below the surface of the ocean. The ambient
current is 30 cm/s to the north and the compo-
nents of the diffusion coefficient are 12, 5, and
1 m 2 /s in the N-S, E-W, and vertical direction,
respectively. Determine the maximum concentra-
tion as a function of time and the concentration at
the release location after 1 hour. utilize the prin-
ciple of superposition to account for the presence
of the ocean surface. Explain how you would
account for the solubility of the waste in your
calculations.

(a) Write an analytic expression for the BoD
along the stream.
(b) Assess the validity of neglecting diffusion in
calculating the downstream concentrations.
(c) If diffusion is neglected, the 1-D advection-
diffusion equation for a stream with velocity
V , decay coefficient, k , and source flux S 0 is
given by

V dc
dx

+ kc S =

0

For the boundary condition that c = 0 when x = 0,
the solution to this equation is given by

3.22. Seventy-five kilograms of a tracer is released into
the ocean where the mean current is 10 cm/s, the
velocity shears in the horizontal-transverse and
vertical-transverse directions are 0.2 and 0.1 s −1 ,
respectively, and the diffusion coefficients in

S
k

0

c = ( − e kx V

− /

)