United States
Environmental Protection
Agency
Environmental Research
Laboratory
Athens GA 30613
Research and Development
EPA-600/S3-82-049 Sept. 1982
Project Summary
Analysis of Mathematical
Models for Pollutant
Transport and Dissipation
W. F. Ames
Four realistic nonlinear models of
pollutant transport with turbulent
diffusion and reaction in rivers were
studied. Exact solutions for all the
kinetics models (no transport, no
diffusion — the so-called stirred tank
reactor) are described. An algorithm
for calculating the rate constantsf rom
the exact solutions is given.
Exact solutions for all systems are
also provided when transport terms
are included with the kinetics (plug
flow model). The inclusion of turbulent
diffusion prevents exact solution, but
the methods of perturbation and the
maximum (minimum) principle provide
approximate solutions and bounds on
the traveling wave solution. The
steady state is also analyzed by the
bounding technique. These bounds,
which may be used independently,
demonstrate how the various param-
eters affect the solutions.
This Project Summary was developed
by EPA's Environmental Research
Laboratory. Athens. GA, 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
Mathematical models of pollutant
reaction, diffusion and transport in
rivers and streams are nonlinear partial
differential equations, primarily because
of the nonlinear kinetics of the bio-
chemical reactions. In previous studies,
exact solutions have been given for, at
most, simplified and/or linearized
problems. These solutions, usually, are
not good approximations for the true
situation. For the nonlinear problems,
numerical solutions are usually obtained.
Although often useful, numerical solu-
tions are not easily employable in
analyzing the adequacy of a model and
evaluating parameters. Numerical
solutions, without error analysis, must
be viewed with caution and even
suspicion — doubly so for nonlinear
problems that may develop singularities,
bifurcations, etc.
In this study, four realistic models of
pollutant transport, turbulent diffusion
and reaction are employed. In the
absence of turbulent diffusion and
transport, the kinetic equations are
solved exactly in all four cases. When
transport is added, the equations are
also solved in all four cases. For the full
system, exact solutions do not seem
possible. For travelling wave problems,
however, models I and II can be exactly
solved for the active carbon and the
bacteria and approximately for the
pollutant.
For all travelling wave cases and all
steady state cases, upper and lower
bounds involving all parameters of the
problem are constructed using the
maximum (minimum) principle. These
bounds are simple negative exponentials
and may be used independently — that
is, they are not coupled together as are
the equations. This is the most useful
and interesting result to come out of
this work.
-------
Procedure
With C,, i =1,2,3 as the concentra-
tions of pesticide, bacteria and organic
carbon; D,, i = 1,2,3 as the respective
diffusion coefficients; k,, i = 1,2,3 as the
appropriate rate constants; v as the
mean stream velocity; T asthe time; and
x as the distance along the river, the
model equations are:
dr dx.
9x2
9C2 + y 9C2 -
dr dx
3x2
9r ax-
i+k3f3(Ci, c2,c3)
dx2
Model I: fi = CiC2,
f2 = fa =
Model II: fi ^ CiC2C3,
fa = fa = C2C3
Model III: fi = C,C2,
K + C3
Model IV: fi = CiC2 C3,
The analysis had four stages.
1. Stirred tank reactor - Kinetics only.
With D, = 0, and V = 0, the kinetic
equations are solved exactly using
the methods of ordinary differential
equations. An algorithm is described
for the evaluation of rate constants,
which are employed in subsequent
analyses. Solutions are displayed for
several models.
2. Plug flow reactor - Kinetics and
transport. With D, = 0, the transport
equations with kinetics are solved
exactly using the method of charac-
teristics of hyperbolic first order
equations. The resulting solutions
have a number of arbitrary functions
that depend upon initial and/or
boundary conditions.
3. The Full System. In this case all
physical processes, kinetics, trans-
port and turbulent diffusion are
included. These parabolic equations
admit the important ca.se of travelling
wave solutions. In cases I and II,
exact solutions are obtainable for
the bacteria and active carbon but
only a perturbation solution is
possible for the pollutant. For all
cases uncoupled analytic upper and
lower bounds are obtained by means
of maximum (minimum) principle.
These bounds contain the parameters
of the problem and may be used
independently.
4. The Steady State. Solutions of the
steady state are not obtainable
analytically. The maximum (mini-
mum) principle is used to derive
upper and lower bounds. These
bounds contain the parameters of
the problems and are uncoupled.
In this problem the exact values of
parameters, such as rate constants,
turbulent diffusion constants, etc. are
not known exactly but do lie on an
interval- i.e. ,/}
-------
W. F. Ames is with Georgia Institute of Technology, Atlanta. GA 30332.
James W. Falco is the EPA Project Officer (see below).
The complete report, entitled "Analysis of Mathematical Models for Pollutant
Transport and Dissipation," (Order No. PB 82-256 900; Cost: $10.50, subject
to change) will be available only from:
National Technical Information Service
5285 Port Royal Road
Springfield, V'A 22161
Telephone: 703-487-4650
The EPA Project Officer can be contacted at:
Environmental Research Laboratory
U.S. Environmental Protection Agency
College Station Road
Athens, GA 30613
I U£.GOVERNMENTPRINTimOFFICE: 1«M-559-017/0826
-------
United States
Environmental Protection
Agency
Center for Environmental Research
Information
Cincinnati OH 45268
Postage and
Fees Paid
Environmental
Protection
Agency
EPA 335
Official Business
Penalty for Private Use $300
------- |