FUNDAMENTALS OF FATE AND TRANSPORT - Water Quality Engineering in Natural Systems - page 73

3.6. Fifty kilograms of a tracer is dumped over a 3-m
length of stream, where the tracer is initially well
mixed across the stream, and the tracer has a first-
order decay constant of 0.5 d −1 . The stream is 5 m
wide and 3 m deep, with an average velocity of
1 cm/s and a longitudinal diffusion coefficient of
1 m 2 /s. Estimate the concentration in the stream
900 m downstream of the center of the spill after
1 day.

the start of the contaminant release? (d) What is
the steady-state concentration at the source?

3.11. Two kilograms of a contaminant is spilled at a
point in a 3-m-deep reservoir. The contaminant is
spilled at x = 0, y = 0, and is instantaneously mixed
over the entire depth (in the z direction).
(a) If the diffusion coefficients in the x (E-W)
and y (N-S) directions are 10 and 20 m 2 /s,
respectively, calculate the concentration as a
function of time at a point 50 m north and
50 m east of the spill.
(b) What is the concentration at the spill location
after 1 minute?

3.7. Two kilograms of a contaminant is spilled into an
open channel at a location 20 m from the end of
the channel. The channel has a rectangular cross
section 8 m wide and 2 m deep, and the diffusion
coefficient along the channel is estimated as 5 m 2 /s.

3.12. A vertical diffuser discharges wastewater at a rate
of 3 m 3 /s uniformly over a 4-m-deep reservoir. If
the wastewater contains 100 mg/L of a conserva-
tive tracer for 20 hours, calculate the concentra-
tion of the contaminant as a function of time
during this 20-hour period at a distance of 150 m
from the diffuser. Assume that the diffusion coef-
ficient in the reservoir is equal to 15 m 2 /s.

(a) Assuming that the contaminant is initially
well mixed across the channel, express the
concentration as a function of time at the end
of the channel.
(b) How long will it take for the contaminant con-
centrations 10 m upstream (in the direction of
the channel end) to be 20% higher than the
concentration 10 m downstream of the spill
(in the direction away from the channel end)?

3.13. A vertical line source discharges a toxic effluent
at a rate of 3 m 3 /s uniformly over a 3-m-deep res-
ervoir in which the mean velocity is 20 cm/s, and
the longitudinal and transverse diffusion coeffi-
cients are 10 m 2 /s and 1 m 2 /s, respectively. If the
wastewater contains 100 mg/L of a conservative
toxic contaminant, determine the steady-state
concentration 50 m downstream of the source.
Compare the exact steady-state concentration
with the approximate steady-state concentration
obtained by neglecting longitudinal diffusion.

3.8. Fifty kilograms of a tracer is instantaneously
released across over the entire width of a lake that
is 25 m wide, 60 m long, and 3 m deep. If the tracer
is released across the midpoint of the lake and the
diffusion coefficient is estimated as 30 m 2 /d, esti-
mate the concentration on the lake shorelines
across the width of the lake after 60 days.

3.9. At the location of a wastewater discharge, a con-
taminant is well mixed across a river that is 5 m
wide and 1 m deep. The flow in the river is 2 m 3 /s,
the contaminant concentration is 200 mg/L, the
longitudinal diffusion coefficient is 3 m 2 /s, and the
decay constant is 50 d −1 . (a) What is the maximum
contaminant concentration that is expected at a
location 2 km downstream of the wastewater dis-
charge? (b) How would this maximum concentra-
tion be affected if the waste discharge is limited to
operating for only 30 minutes per day.

3.14. Consider the case in which 500 kg of a conserva-
tive contaminant is spilled 5 m from the side of a
20-m-wide river. The longitudinal and transverse
diffusion coefficients are estimated to be 10 and
0.1 m 2 is respectively, the depth of the river is 2 m,
and the velocity of the river is 20 cm/s. Determine
the maximum concentration that can be expected
on either side of the river 300 m downstream from
the spill.

3.15. Wastewater from an industrial facility is dis-
charged into a river channel that is approximately
rectangular and is 50 m wide with a depth of flow
of 3 m. The velocity of flow in the river is 30 cm/s,
and the longitudinal and transverse diffusion coef-
ficients are 5 and 0.5 m 2 /s, respectively. The 5-m-
long by 1-m-wide discharge structure is placed on
the bottom of the channel such that the discharged
wastewater is immediately well mixed over the
depth above the discharge structure. The structure

3.10. A contaminant is discharged uniformly across a
10 m × 3 m stream at a rate of 1 kg/s. The stream
has an average flow velocity of 10 cm/s and a lon-
gitudinal diffusion coefficient of 5 m 2 /s. The decay
rate of the contaminant is 0.2 min −1 . (a) What is
the concentration 60 m downstream of the source
5 minutes after the start of the contaminant
release. (b) What is the steady-state concentration
60 m downstream of the source? (c) What is the
concentration at the source 5 minutes after