United States
Environmental Protection
Agency
Environmental Monitoring and Suppo
Laboratory
Cincinnati OH 45268
Research and Development
EPA-600/S4-83-021 Aug. 1983
SERA Project Summary
Mathematical Models |
Associated with Point and Line^
Source Discharges in Rivers
Philip C. L Lin
A literature search was conducted in
the area of mass dispersion from in-
stantaneous and continuous sources
in bounded and unbounded flows. This
revealed that most of the studies are
concentrated on flows in channels with
simple geometries such as constant
depth and width, and with constant
fluid velocity and dispersion coefficients.
Very often the depth-averaged value of
concentration, rather than point value,
is assumed to approximate the com-
plicated nature of mixing, especially in
bounded flows in natural rivers. Analyt-
ical solutions often require tedious and
repetitious calculations. This study
presents a mathematical model con-
sisting of only two nondimensionalized
variables capable of generating uni-
versal concentration profiles of the
pollutants in a rectangular channel
flow for specified initial and boundary
conditions. Determination of diffusion
coefficients, mixing distance, sampling
location, and number of sampling points
in a cross-section is possible.
In other aspects, determination of
the minimum sampling frequency re-
quired to collect a representative com-
posite sample based on the previous
effort is reexamined. It is concluded
that a minimum sampling frequency of
eight (8) for most of the flow and
concentration patterns in the complete
mixing zone is sufficient to provide
enough accuracy for a representative
sample collected by three out of four
known compositing techniques.
This Project Summary was developed
by EPA's Environmental Monitoring and
Support Laboratory, Cincinnati, OH, to
announce key findings of the research
project that is fully documented in a
separate report of the same title (see
Project Report ordering information at
back).
Introduction
When pollutants are discharged into a
river, it is often desirable to be able to
predict their dispersion downstream from
the release point This is because of the
importance of obtaining representative
samples for any monitoring or enforcement
program. Hence, the more accurately one
can predict the behavior of the pollutant
dispersion, the more representative a
sample one can expect to collect
For the prediction of the pollutant disper-
sion, one must know the hydraulics of the
stream, the physical, chemical, or bio-
logical change, and the rate of the pollutant
mixing. Because of the complexities of
variations of depth and width in most
natural rivers, fluid velocity distribution,
and the rate of mixing may vary from one
location to another. It is, therefore, dif-
ficult to fully describe the mixing phe-
nomenon in natural rivers. The approach
generally used to predict pollutant mixing
in a stream is based on assumptions that
an average velocity, an average depth and
even a constant width for the river are
adopted to characterize the flow in an
irregularly shaped water channel. Many
unknown aspects of the mixing in the flow
are then implicitly lumped into coefficients
of dispersion. A review of the literature
reveals that most of the analytical solu-
tions of pollutant mixing are either for
instantaneous point and line sources in an
unbounded fluid or for continuous line
sources in both unbounded and bounded
fluid. Among the two categories of sources,
the instantaneous sources are only of
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theoretical interest in a practical sense.
The continuous sources, however, provide
a deeper insight into the behavior of the
mass being transported, as actual pollutant
spills are never instantaneous, but are
spread over a finite time interval. Although
analytical solutions for predicting pollutant
mixing are closed forms, they often re-
quire tedious and time-consuming calcula-
tions. Furthermore, repetitious calcula-
tions are needed for any change of values
in variables such as velocity, depth, or
dispersion coefficients. Most of the solu-
tions also have the aforementioned restric-
tions such as a constant channel width
and constant velocity.
It is the intention in this study to develop
a mathematical model which combines
variables such as distance, dispersion co-
efficients, depth, and kinematic viscosity
into groups of nondimensionalized vari-
ables and, therefore, to produce universal
concentration profiles for the river pollu-
tants under specified conditions in the
area of mass dispersion. This model also
has its limitations and advantages. It is
restricted to a rectangular channel flow,
but, however, is allowed to have a variable
velocity profile across a section of the
channel.
Also included in this study is a further
examination of the minimum sampling
frequency for collecting representative
samples under completely mixed condi-
tions for variable flow and concentration
patterns.
Discussion
Most predictions of concentration dis-
tributions from point or line sources in
bounded or unbounded flows are expres-
sed as closed-form solutions under condi-
tions of uniform or average flow velocity,
constant channel width or dispersion co-
efficients. Only those solutions from con-
tinuous sources in bounded flows seem
applicable to simulate problems in rivers.
Conclusions from the selected literature
review in this area may be summarized as
follows:
For flow in a rectangular channel
with constant velocity and constant
dispersion coefficient in the trans-
verse direction of the flow
Eqs. (1) and (2) may be used to
evaluate the concentration distribu-
tions at any location of the river. Eq.
(3) may be used to determine the
mixing distance where the material
being transported in the channel is
completely mixed.
qx,z) =
h(7iezux)1/2 n = -
u(z-2nH)2
2 exp
a(q's2+2n-q')
4ezx
(1)
«(q'sif2n-q')
v/2
a(q's2-l-2n+q')
1 On 20mn=acosn-^ |
|C (x,z) = + 2 exp i
i hHu hHu n=1
ux
2ez
0.445 H2u
(2)
(3)
{erf
-erf
+erf
a(qs,+2n+q) gg
-erf } t?
v/2 Si
n=°° a(q'S2-2n-q')
+ 2 { erf
n=0
where
C ^concentration of material being
transported in the channel
X =coordinate in the flow direction
in the channel
z =coordinate in the transverse di-
rection in the channel
CVn=a constant release of material
emanating at the origin of the
coordinates
h =depth of the channel
?§ ez =dispersion coefficient in the trans- g
$;! verse direction, z, in the channel $
u =constant flow velocity
H =constant channel width
2ezmr 2
°"-(-TH '
Xm= mixing distance
For flow in a river with variable
depth and width
Eq- (4)| can be used to calculate the
concentration distributions if the dis-
tributions of flow velocity, river depth
and dispersion coefficients across
the river at any section are known.
-erf
erf
- erf
o(q's1-2n-q')
^2
«(q's2-2n+q')
v/2
o(q'sf 2n+q')^
-^-
where
erf q =
V/V
e 'P2 dp
Q2
a =
q =
2x ezuh2
C =-
C'(a,q') =
n=c
[2
1 Q
-/ Cdq
QO
q =/z uh dz
o
Q =/H uh dq
o
q'si,q's2 = the two end points c
the line source e>
pressed as fraction
of the total discharge'
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This study presents simple universal
concentration profiles for a rectangular
channel flow. The profiles are valid for
steady state condition from continuous
sources in the steam. The applications for
this model can be summarized as follows:
Determination of Schmidt number,
Sz, and dispersion coefficient ez
The concentration distributions a-
cross a water channel at any location
x are measured and also are obtained
from the model. Comparison of the
results will indicate that an appropri-
ate value of nondimensionalized dis-
tance X* should be selected. There-
fore, Schmidt number, Sz, and dis-
persion coefficient, ez can be deter-
mined as follows:
Sz =
X
X*ReH
(5)
where
1.5UH
Re = ' Reynolds number
V
v = kinematic viscosity of water
U = average velocity in the channel
H = channel width
Although this effort is restricted to a
rectangular channel flow, it also provides
an approximate solution for channel flows
; with constant width and moderate variation
| of depth. This study is a continuous effort
: in this area and will be helpful to field :
personnel in selecting the sampling loca-
tion and number of sampling points.
In the study of sampling frequency, it is
found that a minimum sampling frequency
of m = 8 is sufficient to provide enough
accuracy for samples being collected for
most of the flow and concentration patterns
studied.
Conclusions
The capability to analytically solve the
mixing phenomenon in rivers is very limited,
as can be seen from the literature being
reviewed. The present mathematical
model which includes velocity variations
also has its limitations such as the restric-
tions of constant channel width and depth.
There is, therefore, a need to develop a
model which takes variations of depth,
width, and velocity in a river into considera-
tion. The possible solution may only be
numerically obtained by a computer.
The EPA author Philip C. L. Lin is with the Environmental Monitoring and Support
Laboratory, Cincinnati. OH 45268.
The complete report, entitled "Mathematical Models Associated with Point and
Line Source Discharges in Rivers," (Order No. PB 83-207 373; Cost: $10.00,
subject to change) will be available only from:
National Technical Information Service
5285 Port Royal Road
Springfield, VA 22161
Telephone: 703-487-4650
The EPA author can be contacted at:
Environmental Monitoring and Support Laboratory
U.S. Environmental Protection Agency
Cincinnati, OH 45268
ftUS GOVERNMENT PRINTING OFFICE 1983-659-017/7224
Determination of sampling location
X
Once the Schmidt number is known,
one can choose a known concentra-
tion distribution of X* and locate the
sampling location X as follows:
X = X* Re Sz H (6)
Determination of number of samp-
ling points at location X
Since the concentration distributions
are known at a certai n value of X*, the
value of X can be calculated. The
number of sampling points are then
determined from the known concen-
tration distributions by Simpson's
rule or any other approximate tech-
nique.
Determination of the mixing distance
The mixing distance for a continuous
discharge in a rectangular channel is
Xm = K ReSzH
(7)
where
K= 0.30 for a continuous discharge
at one of the banks
K = 0.14 for a continuous discharge
in the center fo the channel.
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United States
Environmental Protection
Agency
Center for Environmental Research
Information
Cincinnati OH 45268
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