WAT]ER
POLLUTION CONTROL RESEARCH SERIES • ORD 6
Use of
Mathematical Models
in
Water Quality Control
Studies
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WATER POLLUTION CONTROL RESEARCH SERIES
The Water Pollution Control Research Reports describe the
results and progress in the control and abatement of
pollution in our Nation's Waters. They provide a central
source of information on the research development and
demonstration activities in the Federal Water Pollution
Control Administration, in the u.S. Dept. of the Interior
(both inhouse and through grants and contracts with Federal,
State, and local agencies, research institutions, and in-
dustrialorganizations). The exchange of such data should
contribute toward the long range development of economical,
large-scale management of our Nation's water resources.
Water Pollution Control Research Series will be distributed
to requesters as supplies permit. Requests should be sent
to the Publications Office, Dept. of the Interior, Federal
Water Pollution Control Administration, Washington, D.C.
20242.
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USE OF MATHEMATICAL MODELS
IN
WATER QUALITY CONTROL STUDIES
FEDERAL WATER POLLUTION CONTROL ADMINISTRATION
DEPARTMENT OF THE INTERIOR
by
Alvin S. Goodman, Principal Investigator
and
Richard J. Tucker, Research Assistant
Department of Civil Engineering
Northeastern University
Boston, Massachusetts
Program No. 16090
Grant No. WP-01090
July. 1969
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FWPCA Review Notice
This report has been reviewed by the Federal
Water Pollution Control Administration and
approved for publication. Approval does not
signify that the contents necessarily reflect
the views and policies of the Federal Water
Pollution Control Administration.
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ABSTRACT
Mathematical models were utilized to study water
pollution control programs in a river basin. Sensitivity
analyses, with a steady state model, showed substantial
variation of cost for sewage treatment, depending upon stream
purification parameter selections. \fuen actual parameters
are less favorable than design values, quality standards may
not be met~ these effects are more serious with lower levels
of treatment.
An unsteady state model was developed to trace a time
profile at any specified station in terms of flow and quality
as BOD, dissolved oxygen, coliforms, and chlorides while up-
stream discharge, water temperature, and solar radiation vary.
The techniques assume that, for short reaches and/or times,
steady state conditions apply without undue loss of accuracy.
A new empirical procedure was developed to route unsteady
stream flow.
The time varying model was used to investigate the
effectiveness of an assumed configuration of treatment plants
when the stream's assimilative capacity varies with distance
and time. Studies showed that DO values are worse at times
than the steady state value, and that susceptibility to poorer
conditions increases with higher BOD releases. Lower treatment
levels also result in a greater range of river conditions than
high levels. Sensitivity analyses of stream parameters were
also made with the time varying model.
This report was submitted in fulfillment of Grant No.
WP-Ol090 between the Federal Water Pollution Control Admini-
stration and Northeastern University, Department of Civil
Engineering.
iii
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CONTENTS
ABSTRACT. . . . . . . . . . . . . . . . . . . . . . . .
StJl.n.iA.RY . . . . . . . . . . . . . . . . . . . . . . . .
Chapter I INTRODUCTION. . . . . . . . . . . . . . . .
A. Research Objectives. . . . . . . . . . . . . . .
B. Authority. . . . . . . . . . . . . . . . . . . .
C. Outline of Research Procedures. . . . . . . . .
D. Needs for Better Mathematical Tools. . . . . . .
E. Significant Results. . . . . . . . . . . . . . .
Chapter II SETTING FOR RESEARCH. . . . . . . . . . . .
A. Description of Goodman-Dobbins Mathematical Model
B. Optimization Routine. . . . . . . . . . . . . .
Chapter III STUDIES WITH STEADY STATE MODEL. . . . . .
A. Introduction. . . . . . . . . . . . . . . . . .
B. Formulas for Evaluation of BOD and DO . . . . . .
C. Values of Parameters Selected in Surveys. . . .
D. Techniques for Evaluation of Stream Parameters
E. Sensitivity Analyses by Computer Simulation. . .
F. Discussion. . . . . . . . . . . . . . . . . . .
Chapter IV TIME VARYING MATHEMATICAL MODEL. . . . . .
A. Introduction. . . . . . . . . . . . . . . . . .
B. General Methodology for Time Varying Model. . .
C. Factors Affecting Oxygen Variations. . . . . . .
D. Streamflow Routing Procedure. . . . . . . . . .
E. Deoxygenation. . . . . . . . . . . . . . . . . .
F. Reaeration . . . . . . . . . . . . . . . . . . .
G. Photosynthetic Oxygen Production. . . . . . . .
H. Description of Computer Program. . . . . . . . .
Chapter V SIMULATION STUDIES WITH TIME VARYING MODEL
A. Introduction. . . . . . . . . . . . . . . . . .
B. Results Obtained with Time Varying Model. . . .
C. Comparison with Steady State Model. . . . . . .
D. Sensitivity Analysis. . . . . . . . . . . . . .
E. Effect of Statistical Variations. . . . . . . .
Chapter VI CURRENT SAMPLING PRACTICES BY NEW ENGLAND
REGULATORY AGENCIES. . . . . . . . . . . . . . . . .
A. Introduction. . . . . . . . . . . . . . . . . .
B. Current Sampling Practices. . . . . . . . . . .
ACKN~DGEMENTS . . . . . . . . . . . . .
. . . . . . .
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BIBLIOGRAPHY
APPENDIX I
APPENDIX II
APPENDIX III
APPENDIX IV
. . . . . . .
.........
. . . . . .
INPUT-OUTPUT DATA DESCRIPTIONS. . . . . .
SUMMARY OF EQUATIONS AND PROCEDURES FOR
FUNCTIONS. . . . . . . . . . . . . . . .
COMPUTER PROGRAM STATEMENTS CORRESPONDING
TO FUNCTIONS. . . . . . . . . . . . . . .
STREAMFLOW ROUTING FOR WATER POLLUTION
STUDIES. . . . . . . . . . . . . . . . .
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107
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FIGURES
Figure
1.
Sketch of Study Stream Showing Locations of Sewage
Discharges and Water Supply Intakes. . . . . . . .
2.
Effect of Varying Z7 - (~) Parameter on Oxygen
Deficit of Stream with Primary Treatment. . . . .
3.
Effect of Varying Z7-(a) Parameter on oxygen
Defici t of Stream with Seamdaxy Treatment. . . . .
4.
Example of Configuration of Fixed and Moving
S tat ion s . . . . . . . . . . . . . . . . . .
. . .
5.
Paths for Mathematical Analysis - Time varying
Ri ve r Mode 1 . . . . . . . . . . . . . . . . . . . .
6.
Gross Oxygen Production vs. Sunlight Intensity
7 .
Gross oxygen Production vs. Depth for Indicated
Values of Solar Radiation in Langleys . . . . . . .
8.
General Flow Diagram - Time Varying River Model. .
9.
Simulation of River Conditions at Station 41,
July 1964 .................
. . .
10. Simulation of River Conditions at Station 41,
August 1964 . . . . . . . . . . . . . . . . . . . .
11. Simulation of River Conditions at Station 41,
August 1965 . . . . . . . . . . . . . . . . . . . .
12. Simulation of River Conditions for Station 41 -
Modified Input Data - Chart A . . . . . . . . . . .
13.
Simulation of River Conditions for Station 41 -
Modified Input Data - Chart B . . . . . . . . . . .
14. Sensitivity Analysis for kl . . . . . . . . . . . .
15. Sensitivity Analysis for k2 . . . . . . . . . . . .
16. Sensitivity Analysis for a . . . . . . . . . . . .
17. Sensitivity Analysis for Water Temperature . . . .
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Table
10.
TABLES
1.
General Information for Communities for River
Mode 1 s . . . . . . . . . . .
. . . . . . . .
. . .
2.
River parameters for Existing Conditions (CDM 1963)
3.
River parametern for Existing Conditions (FWPCA
1964- 5) .....................
4.
River Parameters for Year 2010 (CDM)
. . . . . . .
5.
River Paramete1'S for Year 1985 (FWPCA)
. . . . . .
6.
River Parameters Adopted for CARM-l . . . .
. . . .
7.
Sensitivity Analysis Using Optimizing Routine
8.
DO Deficits with Primary Treatment and with Optimum
Treatment. . . . . . . . . . . . . . . . . . . .
9.
Sensitivity Analysis Using Optimizing Routine
DO Deficits with Primary Treatment Plants and with
Optimum Treatment. . . . . . . . . . . . . . . .
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SUMMARY
Introduction
Mathematical models, including both steady state and
time varying versions, were developed and utilized to study
water pollution control programs. Although hypothetical, these
models are based to a large extent upon population and other
river and community data for the Merrimack River Basin in
northeastern Massachusetts. This basin is 302 square miles in
area, with a design population of 579.000 persons. The stream
is 50.9 miles long and has a discharge of 650 mgd during the
critical low flow period. It may be stated, without substantial
qualification. that the models were developed with realistic
data such as those obtained with an actual river basin. The
studies were limited to effects for non-tidal streams.
In the initial proposal for this project, it was stated
that an analysis of data collection and processing programs of
the water pollution control agencies of the New England States
would permit a comprehensive study to establish a scientific
basis for judging the economic efficiency of such programs.
Because adequate information was not available for a systematic
statistical analysis, the research focused on methodologies for
utilizing data rather than on a quantitative evaluation of the
costs of data collection and processing. The emphasis was on
the relationships between stream parameter selection and the
cost and effectiveness of systems of waste water treatment
plants.
Although the models were studied within a rather limited
framework of assumptions and results, the capabilities of such
models, in terms of producing better design decisions, have been
demonstrated. It is most desirable to coordinate individual
field and laboratory studies, automatic monitoring programs, and
mathematical model investigations. All of these methods are
needed in the efficient operation of water pollution control
programs.
Significant Results
The fOllowing are the most significant conclusions of
this research project:
1. While a steady state river model is a practicable
tool for selecting levels of waste water treatment in a system,
the levels of treatment and the consequent costs can be quite
sensitive to the river parameters chosen for the analysis.
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2. An unsteady state river model incorporating vari-
ations, with distance and time, of flow, temperature, solar
radiation, and stream parameters can be developed, and can be
used to produce a time profile of river quality at any specified
location.
3. Considering the present inadequate state of the art
in estimating stream parameters, there can be no assurance that
treatment plant systems will maintain specified river quality.
If parameters are less favorable than assumed, susceptibility
to stream conditions which do not meet quality standards in-
creases with lower levels of waste water treatment.
Studies with Steady State Mathematical Model
Input data for the steady state model include hydraulic
constants and stream purification rate constants for each reach
of the river, population and economic information for each com-
munity, and assumptions with respect to administrative decisions.
If the levels of sewage treatment are specified for each com-
munity, a computer program evaluates water quality throughout
the length of the stream, and the costs and benefits of the water
pollution control program. If a single water quality criterion
is specified for each river section (e.g. in terms of dissolved
oxygen), an optimizing routine determines desirable levels of
sewage treatment.
Two surveys made prior to this research served as the
basis of field data. Both surveys used the dissolved oxygen con-
centration as the principal criterion of stream quality. Equations
determining changes in BOD and DO are by Streeter and Phelps, as
modified by Camp. These formulas require values of rate parameters
for deoxygenation (kl)' reaeration (k2)' settling out of BOD to
bottom deposits (k3)' re ension of BOD from bottom d
(p),_and oxygenation by photosynthesis 0 Each para-
meter chosen as a basis for design affects the planning of a
system of treatment plants, each with an appropriate performance
level in terms of BOD removal fraction. More than one set of
parameters, even if individual parameters are incorrect, can give
a satisfactory agreement of estimated and measured BOD and DO
values for existing stream conditions. Considering the inadequate
state of the art in adjusting parameters for future conditions,
however, good projections are not necessarily assured.
A sensitivity analysis by computer simulation was carried
out, in which each of the stream parameters entering in the BOD
and DO equations was investigated within a possible range, in
order to include many of the possible combinations which might
occur. The resulting river water qualities, degrees of treat-
ment, and economic effects, in combination, form a picture of
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how different assumptions might affect the ultimate design.
The studies demonstrated that the k3 and p values were the
insensitive, while the a value was the most sensitive.
most
Of particular interest was the effect of parameter se-
lection on sewage treatment plant cost. The technique used
was to hold all parameters constant at the basic design values
except one, and to test that parameter at the upper and lower
bounds of its range. In each case, an optimum set of treatment
plants was established to achieve a 3 ppm DO deficit at all
points in the river. For all testing, the investment cost
varied between $30.5 million and $43.0 million, a range of 92
to 130 percent of an investment cost of $33.1 million based on
the basic design values. The highest cost corresponded to an
assumption that photosynthesis is not active at all, showing
that for the modeled stream the algae can have a large favorable
effect if the growth can be controlled to avoid a nuisance.
Similar results were obtained with an objective of 4 ppm
DO deficit at all points in the river. It was found, however,
that the costs of treatment appear to be less sensitive to
parameter selection for lower standards of river water quality.
While cost is important in water pollution control
planning, a more essential consideration is the effectiveness
of treatment. If treatment plant levels are selected on the
basis of assumed values of parameters, and the actual values of
parameters turn out later to be unfavorable, the water quality
standards would not be met. It was found that the effects on
stream quality for such an unfavorable set of parameters were
more serious with lower levels of treatment than with higher
levels. This would indicate that if the treatment process for
a plant were limited to primary treatment on the basis of com-
putation with a mathematical model, the plant ought to be con-
structed to permit expansion to secondary or higher levels of
treatment. Furthermore, communities which construct primary
treatment plants should be advised that such plants may turn
out to be inadequate.
On the basis of these studies, it would appear to be
prudent to base every system design on the premise that the
estimated values of parameters may be inaccurate, and possible
ranges of parameters should be investigated.
The methods used to determine stream parameters should
be improved. Emphasis should be placed on the development of
good techniques to evaluate parameters individually and as
directly as possible. Only then, can engineers make dependable
adjustments to parameters in order to predict future conditions.
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While steady state models for estuarine streams are
often mwatisfactory, it is judged that steady state models can
continue to be useful to determine approximate plans for ~ater
pollution control for many n~dal rivers. Such models can be
developed uSing the data from brief field surveys, and can
provide comparisons of alternative plans for pollution control
at reasonable cost for the engineering studies. It is clear,
however, that adequate margins of safety over the results indi-
cated by the model studies should be employed.
Time Varying Mathematical Model
For important streams ~ith existing or potential trouble
spots ~ith respect to ~ater quality, an approximate model of
river conditions based on the steady state does not provide a
means for a sufficiently adequate simulation of conditions ~hen
streamflow and environmental factors vary substantially over a
period of operation. An unsteady state model can be used to
investigate the effectiveness of an assumed configuration of
treatment plants ~hile a stream's assimilative capacity changes
~ith distance and time. Also ~hen a simulation showing the
variation of quality ~ith time is available, the suitability of
the stream for fish propagation or other specific purpose may
be studied in detail.
The techniques ~hich have been developed can traoe a
time profile at any specified station for flow and quality in
terms of BOD, DO, coliforms and chlorides ~hile the upstream
discharge, ~ater temperature, and solar radiation are changing.
With a change in programming instructions, values could be
shown for all stations at any specified time.
Most previous investigators of time varying models have
developed sophisticated differential equations and numerical
methods for computer solutions of these equations. When "plug"
flow can be assumed, as is the case for almost all non-tidal
conditions, different methods can be used based upon relatively
uncomplicated elaborations of a classical approach such as the
oxygen sag expression employed by Streeter and_Phelps.
The principal objectives in the development of the time
varying mathematical model described in this report have been
to make maximum use of the types of data ~hich can typically be
obtained from records and field surveys, and to produce the
kinds of results that are needed to examine the effectiveness
of ~ater pollution control programs. It ~as found possible to
accomplish these objectives by ~orking ~ith short reaches and/or
times, over ~hich steady state conditions could be assumed to
apply ~ithout undue loss of accuracy. It ~as also found possible
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to accept input data in any available form (such as a table,
when an equation cannot be created without compromising the
data) .
The adopted system utilizes moving stations in addition
to the fixed stations of the steady state model. The moving
stations are located so that the time of travel between two
successive stations for any flow is an arbitrary time interval.
It was found that the analysis of six intervals per day was
sufficient to show the diurnal variation of dissolved oxygen
due to photosynthesis.
The computations proceed in a downstream direction. A
set of "distance varying parameters" are assumed to remain at
these values until the next fixed river station is reached.
At the beginning of each time interval which corresponds to a
moving station location, one or all of the "time varying para-
meters" may change, and these are assumed to remain constant
until the beginning of the next time interval. Many of the
functions used for evaluating river conditions utilize both time
varying and distance varying parameters.
Six values of flow are routed for each day- All of the
computed values for the time, coliforms, BOD, DO, chlorides, and
discharge remain in computer storage until a flow increment has
passed all the way downstream, after which these values may be
printed for as many of the fixed stations as desired. After the
printing operation is complete, the procedure cycles back to the
upstream end of the stream, and a new incremental flow is pro-
cessed.
Stream factors influencing the oxygen balance which have
been considered for the time varying model include: deoxygenation,
reaeration, algae activity,benthal demand, settling and resuspension
of BOD, temperature, sunlight, and streamflow.
A new empirical procedure was developed and tested for
the representation of unsteady streamflow. The procedure is es-
pecially suited for studying normal and low flow periods. The only
input data required are an upstream hydrograph and velocity-
discharge relationships for the reaches of the stream. unlike
many flood routing procedures, this method of low flow routing is
concerned with the time displacement of pollutants for different
discharges rather than representing the movement of translatory
waves. Much of the computer time required for processing with
the time varying model is attributable to the routing procedure.
The report discusses appropriate relationships to handle
variations with time of the other input data for the stream
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parameters. The programming methods are qUite flexible, and can
facilitate revisions in computational procedures to accommodate
different relationships for the parameters which may be pre-
ferred by other analysts.
Features which could have been added to the model but
which have not been included for these studies are: the varying
of treatment plant operation with time, the inclusion of over-
land flow and major tributaries along the stream, changing
population configurations, the direct addition of molecular
oxygen to the stream, the employment of low flow augmentation,
and thermal discharge effects. In general, these were not ana-
lyzed because they were not of particular importance in previous
studies of the Merrimack River, on which the model was patterned.
They can be added, however, without much difficulty if the
variations with time and/or distance are known.
The time varying model was based on a deterministic
approach which primarily considers the changes due to flow,
temperature, and sunlight. As with practically any simulation
vehicle, however, the model can be modified to include stoch-
astic fluctuations in the rate parameters and other input data,
or by adding random components to the output values.
Simulation Studies with Time VarYing Model
The examples described in the report are limited to a
few basic demonstrations. Due to the comprehensive nature of
the time varying model and its flexibility in accepting modifi-
cations, however, many other input-output studies are possible.
Although output was normally available from a computer run for
every preselected ("fixed") river station within the study
stream, the examples present results for only one river station,
which was shown by the steady state model studies to be critical
with respect to DO for the specified design period.
Although the time varying mathematical model and the
input data are patterned on information for the Merrimack River
in Massachusetts, it is not claimed that the results reproduce
values for this stream. It is also emphasized that the con-
clusions are based upon only limited testing work. Neverthe-
less, the research has effectively studied different techniques
for analyses, and the implications of uSing available data and
design assumptions on the planning and operation of water pOllu-
tion control programs. Furthermore, the conclusions conform to
what may be expected from experience and judgment.
Initial time varying studies, in which typical input vs.
output relationships were studied, demonstrated that by merely
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inspecting the input data, it may not be possible to determine
which monthls output will contain the lowest single DO value,
the lowest average DO value, or the longest period with the
DO values below some specified value.
It was not possible to obtain a natural record with a
coincidence of low flow, high temperature, and little sunlight,
that would occur for a long enough period of time, to cause the
kinds of stream conditions indicated by the steady state
analysis. In order to provide results for the time varying
model which could be used for comparison with the steady state
model, it was necessary to modify the input. Two months of
synthetic sequential input data were constructed, one in which
the averages for the worst seven day period conformed to the
input values for the steady state model, and the other in which
the averages for the month conformed to steady state values.
For each of these modified months, biological conditions
were predicted for three different plant configurations--all
primary treatment with 38% BOD removal, an optimum set of plants
to satisfy an objective DO deficit of 3 ppm for the whole stream
which was obtained from the steady state model, and all secondary
treatment with 90% BOD removal.
At the critical station, and for the first modified
month, the lowest DO for the unsteady state model for the "all
secondary treatment" assumption was virtually the same as that
for the steady state model, but the lowest DO's for the "optimum
treatment" and for the "all primary treatment" assumptions were
about 0.7 ppm less and 1.4 ppm less, respectively, than the
steady state results. For the second modified month, the lowest
DOls were 0.3 ppm less for "all secodary treatment," 1.0 ppm
less for "optimum treatment," and 1.9 ppm less for "primary treat-
ment." These results demonstrate that worse values than the
steady state value can occur at times, and that the susceptibility
and degree of such differences increases with higher BOD loads,
corresponding to lower treatment plant performance levels.
The ranges of DO results are also significant. For the
second modified month, values were between 1.82 ppm and 4.42 ppm
for primary treatment, 4.48 ppm and 5.79 ppm for optimum treat-
ment, and 6.28 ppm and 7.30 ppm for secondary treatment. There-
fore, DO ranges for the three forms of treatment (primary,
optimum, and secondary) were 2.60 ppm, 1.31 ppm, and 1.02 ppm,
respectively. It would appear from these studies that a
greater range of river conditions over a period of time may
result when higher BOD loads are applied to the stream.
The methods used to test the sensitivity of the various
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model parameters in the unsteady state model were similar to
that used with the steady state model. Such analyses were
performed for the deoxygenation (kl> and reaeration (k2>
constants, the amount of algae activity (~). and the water
temperature. In each case, the values of the parameter being
studied were altered, while the values of the other parameters
remained the same.
The most apparent indication of the sensitivity analyses
for kl and k2 was that with lower levels of treatment (allowing
greater BOD loads to enter the stream), the DO is prone to much
greater variation. Thus, design values chosen for these stream
parameters are very critical. Another way of looking at this
is that if the planner hopes to use the stream's assimilative
capacity to adjust for lower forms of treatment, it is likely
that even if the average values for a 7 day or 30 day period
are acceptable, worse conditions will probably occur for a
portion of the time.
Although the concept has
gated, the testing work suggests
in considering BOD concentration
waters.
not been thoroughly investi-
that there may be some merit
as a criterion for receiving
As for the steady state model, the effect of the ~
assumption is marked and is emphasized in the time variable
studies which take account of the diurnal changes in photo-
synthesis. It was assumed that the ~ values are not affected
by the treatment levels chosen.
For the ranges studied, changes in temperature produced
about the same average change in DO for the different levels of
treatment. A 30e change in temperature for the stream produced
a fairly consistent change in DO of nearly 1 ppm at the critical
station.
The work of other investigators was reviewed in order
to gain some insight in the matter of statistical variations.
Since the model is deterministic, it does not explicitly con-
sider either the random variations in the input or the random
variations in local environmental conditions. It would appear
reasonable to expect that the results for DO obtained for the
time varying model represent the most probable values, while the
actual values of DO would range above and below these values.
A tentative judgment is that the differences between probable
and actual values would be less than 1 ppm for at least 90 per-
cent of the time. Additional studies would be needed to es-
tablish a firmer relationship.
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CHAPl'ER I
INTRODUCTION
A.
Research Objectives
The principal objective of this research project has
been to study relationships between data collection programs
and water pollution control plans. The ultimate aim of re-
search such as this is to improve the efficiency of data
collection and processing programs, to make them as responsive
as practicable (within budget constraints) to the planning and
operation requirements of water pollution control agencies.
Water pollution control agencies have programs for
cOllecting water quality data and other physical and economic
data, and for processing them in accordance with procedures
that have evolved over the years. Certain data are collected
more or less routinely (e.g. data on performance of sewage
treatment plants) while other data are obtained to meet specific
needs (e.g. sanitary surveys of river water quality and pollution
discharges for purposes of river basin planning).
The agencies do not have unlimited budgets. It is,
therefore, desirable for the agencies to know the relative im-
portance of various field and laboratory data used in making
studies leading to: 1) selection of required performance levels
for waste treatment plants: 2) selection of objective qualities
for watercourses: and 3) administrative regulations to control
the operation of water pollution control programs.
In the initial proposal for this project, it was stated
that information on data collection and processing programs would
be obtained from the water pollution control agencies of the New
England states, whose work is partially coordinated by the New
England Interstate Water Pollution Control Commission. It had
been hoped that the analysis of such information would indicate
the feasibility of a comprehensive study to establish a sci-
entific basis for judging the economic efficiency of data col-
lection and processing programs. Discussions with these agencies
were rather disappointing, however, in that it appeared that
adequate information was not available for systematic statistical
and economic analyses.
Seme attempt was made to obtain useful data from other
states but this was not successful. The scope of this project
did not include a more extensive survey of possible sources.
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In view of the above, this research has focused on
methodologies for utilizing data .rather than on a quantitative
evaluation of the costs of data collection and processing.
The vehicles for the studies have been mathematical models.
Steady state and time varying simulation models have been used,
both patterned on community and stream data for the Merrimack
River Basin in northeastern Massachusetts.
The preceding discussion has referred to the needs of
governmental agencies. To a large extent, the results of this
research are also applicable to the activities of planning and
design groups under private control.
B.
Authority
A proposal entitled "Data Collection Economics for Water
Pollution Studies" was submitted to the Federal Water Pollution
Control Administration on 25 May 1966. On March 31, 1967, a
notice of grant awarded for Grant Number WP-Ol090-0l was issued
covering the two year period March 1, 1967 through February 28,
1969. Because graduate research assistance was not available
until July 1967, the termination of the project was extended by
the FWPCA to June 30, 1969.
C.
Outline of Research Procedures
Water pollution control criteria for wastes treatment
and for quality of receiving waters were reviewed. Methods for
predicting river water quality were reviewed. Data used for
predictive formulas and for program examination were classified.
Two studies, by the FWPCA (l)a and Camp Dresser and
McRee (2), Consulting Engineers, have recently been made for
the Merrimack River in Massachusetts. These studies were ex-
amined in order to establish possible ranges of river parameters
that could be selected for planning water pOllution control pro-
grams. Using a slightly modified version of a steady state
mathematical model previously developed by the Principal In-
vestigator, electronic computer studies were made which varied
data within possible ranges of values and determined resulting
river water qualities and economic effects.
The steady state studies indicated the need for a model
to study the unsteady state effects. The development and test-
ing of the time varying model has been the major effort of this
aNote: Numbers in parentheses refer to corresponding
items in bibliography.
-------
11
research project. Operating studies were made to compare
results with those of the steady state model and to investigate
the sensitivity of model parameters in terms of river water
quality predictions.
This report includes a brief discussion of the data
collection and processing programs of the New England regu-
latory agencies. As mentioned above, however, this portion of
the project could not be pursued at a desirable level.
Needs for Better Mathematical Tools
D.
In the following chapters, steady state and unsteady
state models are discussed within a rather limited framework of
assumptions and results. However, the capabilities of such
models, in terms of producing better design decisions, are
demonstrated.
Improved mathematical tools should serve as aids in
activities such as the following:
1. Selection of treatment plant capacities to meet
water quality criteria.
2. Selection of operating modes for plants to meet
water quality criteria.
3. Prediction of ecological effects (e.g., fish kills)
with varying plant treatment levels, flows, and
climatological elements.
4. Establishing field and laboratory programs to obtain
required data for initial selections of plant
capacities and operating modes.
5. Establishing monitoring programs to ensure compliance
with water quality requirements, and to measure
precision of mathematical techniques.
It is most desirable to coordinate the individual field
and laboratory studies, automatic monitoring programs, and
mathematical model investigations. All of these methods are
needed in the efficient operation of water pollution control
programs.
E.
Significant Results
It is believed that the following are the most signifi-
cant conclusions of this research project:
1. Nhile a steady state river model is a practicable
tool for selecting levels of waste water treatment in a system,
the levels of treatment and the consequent costs can be quite
sensitive to the river parameters chosen for the analysis.
-------
12
2. An unsteady state river model incorporating vari-
ations with distance and time of .flow, temperature, solar
radiation, and stream parameters can be developed, and can be
used to produce a record of river quality at any specified
location.
3. Considering the present state of the art in est-
imating river purification parameters, there can be no assur-
ance that treatment plant systems will operate as planned to
maintain river quality- If parameters are less favorable than
assumed, susceptlbility to stream conditions which do not meet
quality standards increases with lower levels of waste water
treatment.
-------
13
CHAPl'ER II
SETTING FOR RESEARCH
A.
Description of Goodman-Dobbins Mathematical Model
In a doctoral thesis (3) completed at New York
university in 1965, a methodology was proposed by the Principal
Investigator for studying the physical, economic, and admini-
strative interrelationships of water pollution control programs.
The work was later outlined in a paper co-authored with Dobbins,
published by the American Society of Civil Engineers (4).
The methodology featured a steady state mathematical
model for a stretch of river where bordering populations and
industries use the flowing water for municipal water supply,
disposal of treated sewage, and recreation. A computer program
was used with simulation techniques to obtain numerical values
for the characteristics of importance in the planning, design
and operation of water pollution control programs. The computer
program was comprised of three essential components: I} state-
ments to control data processing~ 2} equations for a "community
and river model," and 3} statements for an "optimizing routine."
Subprograms were included for processing "functions" for
repetitive calculations, such as for the oxygen sag curve and
for the cost estimates.
Input data for the model included hydraulic constants
and stream purification rate constants for each reach of the
river, population and other economic information for each com-
munity, and assumptions with respect to administrative decisions.
If the levels of sewage treatment were specified for each com-
munity, the program evaluated the water quality throughout the
length of the stream, and the costs and benefits of the water
pollution control program. If a single water quality criterion
for the stream was specified for each river section (e.g. in
terms of dissolved oxygen), an optimizing routine was used to
determine desirable levels of sewage treatment. The work de-
scribed in detail in the aforementioned publications was based ~.
upon operating a model designated "CARM-l" (for "community and
river model number I"). -
The steady state studies which are discussed in the
next chapter were made with essentially the same model CARM-1.
An improved time varying river model, whose development, testing,
and operation are discussed in Chapters IV and V, is patterned
upon the same stream in Massachusetts as was used for the steady
state model.
-------
14
Although hypothetical, these models are based to a
large extent upon the geographical and population con-
figurations and other river and community data for the
Merrimack River Basin in northeastern Massachusetts. The
Merrimack River has been studied in recent years by Massachusetts
governmental agencies (5): the U. S. Public Health Service (6):
Camp Dresser, & McKee (2,7), Consulting Engineers, Boston: and
the Federal Water Pollution Control Administration (1). It may
be stated without substantial qualification, that the models
have been developed with realistic data such as those obtained
with an actual river basin.
The area serving as the basis for these models is shown
on Figure 1. The distance covered by the stream is 50.9 miles.
There are nine communities with a total area of 302 square miles,
and a total population of 579,000 persons. Each community is
shown with locations of water supply intake facilities (desig-
nated I) and sewage discharge facilities (designated D).
Table 1 contains information for each community with
respect to area, population, industry, type of sewage system,
treatment and disposal, and water supply. The steady state
model CARM-l also included locations for recreational facilities.
The studies with the time varying model have not included all
aspects of water supply and recreation potential. Community
"THREE" does not front on the river but instead discharges its
treated sewage to the river via a tributary. This tributary is
assumed to have an insignificant flow in comparison to the
larger stream. Therefore, the mouth of this tributary can be
considered to be the sewage outfall for this community.
The time setting for this study is assumed to be the
year 2000 or later when urban pressures are expected to in-
tensify the need to exploit many rivers for water supply,
recreation, and wastewater disposal. The use of this projected
time with its estimated population values also permits more
interesting model studies, because the current populations do
not exert as much stress on what is a fairly large stream.
The time varying model discussed in this report is
functional for the warm months of the year (May-october). To
extend its use requires only the addition of more detailed data
for colder and higher flow periods. Generally, the warm months
contain the periods of low-flow and high temperature, with
associated high water supply demands. Such conditions provide
the setting for waste water treatment design and for critical
examination of river water quality.
-------
t
3tI--r
.
--
o 1 2 3 ~ 5
~
Scale in Miles
f Iqlllf' I.
~ k elL h 0 f ~ t lid Y ... t r ... . .
.) e.,m ,lOW I nQ l OCi1' lOll,> () f
~f'W;tqf' Oi,;chi1rc1P<; .1nd Wi11
cr
~lJprly Int.lk('<,
-
V1
-------
TABLE 1
General Information for Communities for River Models
% Homes % BOD Removal by
Communi ty Area Population Employees in Water Sewage Sewage Served by Treatment Plant
Name J (sq. mi.) Community Supply Disposal System Public Set 1 Set 2 Set 3
Sewers
ONE 1 30.8 14,000 0 On-river On-river Sanitary 100 38 38 90
TWO 2 43.4 67,000 5,650 On-river On-river Sanitary 100 38 80 90
THREE 3 25.5 48,000 2,750 On-river On-river Sanitary 100 38 80 90
FOUR 4 13.4 94,000 29,500 On-river On-river Sanitary 100 38 38 90
FIVE 5 39.9 142,000 55,000 On-river On-river Sanitary 100 38 65 90
SIX 6 22.4 45,000 41,650 On-river On-river Sanitary 100 38 38 90
SEVEN 7 51.7 70,000 33,000 On-river On-river Sanitary 100 38 38 90
EIGHT 8 36.4 49,000 4,300 On-river On-river Sanitary 100 38 38 90
NINE 9 38.5 50,000 9,550 On-river On-river Sanitary 100 38 38 90
302.0 579,000 144,400
....
CJ\
-------
17
B.
Optimization Routine
In the next chapters of the report, references are
made to an "optimum" set of plants, required in order to meet
specified ~ater quality criteria. In each case, the optimum
set is derived for the steady state model, using an optimizing
routine based on the concept of "path of steepest ascent."
This is an operations research procedure ~hich makes changes
leading to a specified set of objectives by means of a series
of steps, each step being the most efficient that can be taken.
Some studies using this method have been made for hydraulic
and sanitary engineering problems (8,9).
The optimizing routine makes successive improvements
to the quality of ~ater (in terms of DO deficit) at one river
station at a time. The objective quality is reached for the
most upstream station for ~hich a quality is specified, before
proceeding to adjust the quality at the next station do~nstream
~ith a specification.
The ~ater quality at a river station can be adjusted
by raising the level of treatment at one or more upstream
se~age treatment plants. Only substantial increments of treat-
ment are practical from the standpoint of engineering economy.
up to nine levels of treatment may be considered ~ith the
routine, ranging from a minimum of comminution and chlorination
(~ith 10% BOD removal) to a maximum of tertiary treatment (~ith
99% BOD removal).
Starting ~ith an initial state of performance levels
for the treatment plants, one increment of level is examined
for each plant and the most efficient of these increments is
selected. Proceeding from the ne~ state of performance levels,
one increment is again examined for each plant and the most
efficient selected. The process is repeated until the ob-
jective quality specified for the river station is reached.
The routine then goes on to make adjustments for the next station.
The "efficiency" of an increment (increase in level of
treatment) requires a definition by the planner. The term
"effectiveness" is preferred as being more descriptive of the
intent of an increment. Three alternative measures of effective-
ness are accepted by the routine and may be specified by the
planner as part of the input data. For all of the ~ork done
for this report, the effectiveness ~as defined in terms of
least cost of se~age treatment. Thus, the optimization process
examined a ratio of the reduction of DO deficit at a station
to increase in annual cost of se~age treatment.
-------
18
In the discussion of the. Goodman-Dobbins model presented
in the ASCE paper, several reviewers criticized the method of
optimization on the grounds that it does not guarantee a global
minimum. As pointed out in the closure discussion (10), the
difference is small between the results of the method based on
path of steepest ascent and a dynamic programming method de-
veloped by Liebman and Lynn (11). A linear programming method
has been proposed for this problem by Revelle, Louks and Lynn
(12). And a mixed integer programming method is now being de-
veloped by Camp, Dresser and McKee for a study of the Merrimack
River, on which the principal author is a consultant.
The path of steepest ascent has been retained for the
optimizing procedures used in this report. The authors are
confident that changes in the interpretation of results would
be negligible if another slightly more precise method were used
for optimizing.
-------
19
CHAPl'ER :III
STUDIES WITH STEADY STATE MODEL
A.
Introduction
A brief description of the mathematical model by
Goodman, patterned after the Merrimack River in Massachusetts,
was given in Chapter II. The sensitivity analyses discussed
in this chapter are based on the assumption of the steady
state. Stream purification parameters have been estimated
for this stream and reported on by Camp, Dresser and McKee in
December 1963 (2) and by the staff of the Merrimack River
Project, Northeast Region of the Federal Water Pollution Ad-
ministration in August 1966 (1). For convenience, these study
groups will be referred to as "CDM" and "FWPCA." The CDM
work is also the subject of a paper by Camp (13), published in
October 1965.
The principal reasons for using the Goodman model and
the CDM and FWPCA surveys as the basis of the steady state
studies are as follows: 1) The Merrimack River, while larger
than most, is typical of the New England rivers being studied
for pollution abatement; 2) The river has been the object of
several extensive surveys which have developed considerable
data; 3) Only minor modifications to the Goodman model are
needed for the studies.
Both surveys, like most stream investigations, have
used the dissolved oxygen concentration as the principal cri-
terion of stream quality. Equations determining changes in
BOD and DO are by Streeter and Phelps (14), ~s modified by
,Camp (15). ~~ormulas require values of rate p~r_~~~~~s I
for deoxygena~, ~ation, ~~t~ling_Qu~Q~OD ~~~~~m
aepos1ts, ~esuspension or-Bab from bottom deposits, and
oxygenation by pho'tosyn'the'sii-ot-a:lgae.---" --.--.
- ---. -'--~- '- - ----.. -'- --' ~ ._--+- - --. --~.-
Various approaches may be taken for a sensitivity
analysis of the rate parameters. For the 50 mile stream with
9 sewage treatment plants, the focus was on the effects of
parameter selection on the cost of sewage treatment necessary
to maintain a specified DO throughout the stream length. Thus,
each stream parameter chosen as a basis for design affects the
planning of a system of treatment plants, each with an appro-
priate performance level in terms of BOD removal fraction.
The values of parameters proposed by CDM and by the FWPCA
were examined. In addition, a range of values of each of the
-------
20
parameters was considered. Al~ of these values were proposed
as estimates by responsible engineers. Nevertheless, all in-
vestigators in this field work under the restrictions of an
inadequate state of knowledge concerning field and laboratory
- --~ .~- ._~~--~o~~~1on of data
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-------
21
5 day measurements. Current studies on the Merrimack by CDM,
with which the Principal Investigator is familiar, also
consider the nitrogenous demand. If the nitrogenous demands
were included in the model studies, individual values of
results would be expected to change somewhat, but it is judged
that no concepts or significant conclusions would be modified.
C.
Values of parameters Selected in Surveys
For purposes of the computer work, it was convenient
to designate each parameter as a "z" value. Thus Z(3) is k1'
Z(4) is k2' Z(5) is k3, Z(6) is p, and Z(7) is a. The 50 mile
length of study river was divided into 62 river stations at
unequal distances. Values of parameters for existing con-
ditions are shown in Table 2, based on the CDM report, and in
Table 3, based on the FWPCA report. Values are projected for
the year 2010 by CDM as shown in Table 4, and for the year
1985 by FWPCA in Table 5. Goodman adopted the values shown in
Table. 6 for use in previous mathematical modeling studies.
Comparing Tables 2 and 3, it will be noted that there
are large differences between values of the parameters determined
by CDM and FWPCA. The values in Tables 4 and 5 for future con-
ditions are also different. Despite this, design recommendations
made by CDM and Ft~PCA based on these parameters were similar.
More than one set of parameters, even if individual
parameters are incorrect, can give the same overall results.
This is true when reproducing actual oxygen sag cruves, and
also if approximately parallel adjustments are applied for fore-
casting future conditions. Considering the inadequate state of
the art in adjusti.ng parameters for future conditions, however,
good projections are not necessarily assured.
D.
Techniques for Evaluation of Stream Parameters
For completeness, the procedures used by each study
group for estimating values of stream parameters for conditions
during the field surveys will be outlined. The CDM studies,
based on surveys in the summers of 1962 and 1963, will be de-
scribed first.
The k1 value was obtained from measurements of oxygen
uptake in BOD bottle tests in the laboratory at constant temper-
ature. Camp suggests that a better procedure would have been to
submerge the bottles in the river water at the locations of the
samples so that the sample temperatures would vary with the
stream temperature. The p value was determined by measuring
the oxygen demand of samples of bottom deposits: the demand in
-------
TABLE 2
River Parameters for Existing Conditions
(Camp, Dresser and McKee Report 1963)
River
Station Z(3)-(k1) Z(4)-(k2) Z(5)-(k3) Z(6)-(p) Z (7 ) -@)
1
0.06 0.04 0.11 1.0 0.5
27
28
0.06 0.07 0.13 1.0 1.0
34
35
0.07 0.15 0.05 0.5 0.5
62
N
N
-------
TABLE 3
River Parameters for Existing Conditions
(Federal Water Pollution Control Administration August 1964 & August 1965)
Ri ver
Station Z(3)-k1) Z(4)-k2) Z(5)-(k3) Z(6)-(p) Z(7)-(~)
1
1964 (1125 cfs) 0.130 0.210 0.140 0.50 1.70
1965 (770 cfs) 0.095 0.230 0.040 1.0 2.0
6
19
0.161 0.160 0.010 0.50 0.80
30
31
0.175 0.220 0.010 0.20 1.0
40
41
0.175 0.140 0.00 0.90 1.70
58
~
w
-------
TABLE 4
River Parameters for Year 2010
(Camp, Dresser and McKee Report)
River
Station Z(3)-(k1) Z(4)-(k2) Z(5)-(k3) Z(6)-(p) Z(7)-(a)
1
0.05 0.04 0.07 0.5 0.3
27
28
0.05 0.07 0.08 0.5 0.5
34
35
0.05 0.15 0.05 0.3 0.3
62
N
~
-------
TABLE 5
River Parameters for 1985
(Federal Water Pollution Control Administration Report)
River
Station Z(3)-(kl) Z(4)-(k2) Z(5)-(k3) Z(6)-(p) Z(7)-(a)
1
0.080 0.170 0.010 0.30 0.20
19
20
0.080 0.170 0.010 0.30 0.20
30
31
0.100 0.230 0.010 0.10 0.40
41
42
0.100 0.150 0.010 0.50 0.10
58
~
U1
-------
TABLE 6
River Parameters
(Adopted for CARM 1)
River
Station
Z(3)-(k1)
Z(4)-(k2)
Z(5)-(k3)
Z(6)-(p)
Z (7 ) - (~)
1
.05
.03
.07
.50
.30
8
9
.05
.04
.07
.50
.30
27
28
.05
.06
.08
.50
.40
34
35
.05
.08
.08
.50
.50
62*
*
Note that in this section (Haverhill to the Mouth) the Merrimack is an
estuary and because the original model was not designed to cope with
this, these are ficticious values.
~
en
-------
27
gms per square meter was divided by the hydraulic depth in
meters to obtain the parameter in the proper units, where the
hydraulic depth is defined as the total volume in a reach
divided by the water surface area. La and ~ were determined
from BOD bottle tests for samples collected at stations a and
b. With the value of kl, p, La and ~ thus obtained and the
travel time estimated, the value of k3 was determined from the
BOD equation given above. The method of determining the travel
time was not discussed in the CDM report.
The value of ~ was evaluated by the light- and dark-
bottle technique, with the bottles suspended at the points
where the samples were collected: values of ~ for various lo-
cations were averaged for the reach in accordance with the
hydraulic depth calculations. The k2 value can then be esti-
mated from the oxygen sag equation, if the oxygen deficits at
the ends of the reach are measured and all of the other para-
meters are as found above. Instead, and the CDM report does
not indicate why this was done, it was assumed that at a point
in the receiving waters, both D and L are constant ("preferably
averaged over a period of several days") and the rate of supply
of oxygen is equal to the rate of deoxygenation, as:
2.3 k2D + ~
=
2,3 kl L
and the value of k2 may be determined if the other parameters
are known.
As indicated in the FWPCA report, it was decided to use
the basic approach of CDM but "with more emphasis on certain of
the variables considered to be weak. In addition, gaps in water
quality information, such as the biological condition of the
river, were to be filled." The field work was done in the
summers of 1964 and 1965.
The ~ value was determined by the light- and dark-bottle
technique. Data were plotted as oxygen production per day versus
depth to obtain a parabolic curve extending from a maximum effect
in the first foot of depth to zero effect at 8 feet depth. The
value of a was obtained by dividing the area of the curve by the
hydraulic depth. Studies were also made to relate measured sun-
light intensity to the oxygen production.
Values of p were determined in several ways. The first
method was based upon running bottle tests and Warburg analyses
on samples taken from the benthic deposits. The second method
was based upon pilot plant studies of bottom sediments. The
FWPCA report states that a representative value of p was selected
-------
28
for each reach based upon the t~o methods and that the "se-
lection was influenced by field observations of the area, and
the relationship of p with the observed oxygen sag equations."
Comparisons of values of p are not available in the report.
Time of travel was determined by the use of the
Rhodamine B dye and fluorometer technique. Values of Kl were
available from laboratory bottle tests. With Known values of
La' ~. Kl' t, and p, the K3 was obtained from the BOD equation.
The K2 was found from the oxygen sag equation with the Known
values of Da. Db, Kl, k3' p, a and t.
The kJ obtained as described above was generally nega-
tive or of very low positive value. The report states that
"considering the low dissolved oxygen levels and physical
characteristics of the MerrimacK River, such K2 results were not
considered representative." Of the various key parameters, it
was felt that the ~ and p were acceptable but that the values
of Kl and kJ should be adjusted to give reasonable values of
K2. Bottle kl values were used to compute L values. By
plotting L vs time of flow, a combined (Kl + kJ) term was
calculated. Values of kl' kJ and K2 were then determined that
would result in duplicating observed field conditions for the
oxygen sag curve, on the assumption that the "bottle Kl" was
not representative of the "river kl'" It was made clear that
various values of kl' K2 and kJ would result in satisfactory
matching of the sag curve.
There is no basis at this time for judging which
study's values best represent the present and anticipated
stream conditions. The studies were made in different years
and projected conditions for different periods. It may be
questioned, however, why two surveys using similar sampling
techniques would obtain values of parameters over ten times
different.
It may be reasoned that perhaps the differences are
attributable to the data collection systems. These studies
relied on standard bottle tests. Those parameters which could
not be evaluated directly were determined by substituting into
the BOD and DO equations those values which could be determined
directly- Bottle tests do not simulate the current of the
stream and the benthal material, in producing a "tricKling
filter" action. Furthermore, these values of parameters are
so dependent on the experimental values that a reasonable
degree of accuracy may not be obtained. For example, KJ
(constant for the settling out of BOD to bottom deposits)
depends on the correct evaluation of Kl (deoxygenation constant),
p (addition of BOD to the stream from bottom deposits) and
-------
29
La and ~ (the BOD at upstream and downstream locations).
The separation of the parameters affecting dissolved
oxygen into a number of components is conceptually satisfying,
but implies a degree of precision in the estimate of the
parameters which does not currently exist. Due to the way in
which these parameters are evaluated, no way exists of checking
their accuracy- Although it has been impossible to make an
accurate diagnosis of the degree of error, the magnitudes of
the differences between the results of the two studies would
indicate that it might have been large.
The search for improved methods of laboratory testing
is documented by many reports in the literature. Much less
has been written on the improvement of field measuring tech-
niques. In a report dated April 1967, the FWPCA described ex-
periments with more sophisticated sampling devices (16). Using
the Blue River in Oklahoma, they attempted to better separate
the stream parameters by evaluating them individually- It was
shown earlier in this chapter that light and dark bottle tests
employed in the two river surveys of the Merrimack River can
only include the effects of suspended plant material. In the
Oklahoma study, not only was an attempt made to include the
benthos, but a system of paddles was devised to simulate the
current as well. Methods of this type should provide better
estimates of Kl.
In view of the difficulties of obtaining good field
measurements of stream parameters, continued research to
improve field sampling is needed.
E.
Sensitivity Analyses by Computer Simulation
With the availability of a new advanced electronic
computer facility recently installed at Northeastern University
and methodologies based on those previously developed by the
Principal Investigator, a large number of runs were made for a
sensitivity analysis. The procedure was to multiply each of
the river parameters (Table 6) selected for design, by co-
efficients to individually vary them within possible ranges.
These ranges were taken from the extremes of the values measured
in each survey studied. The resulting river water qualities,
degrees of treatment, and economic effects were graphed and
charted to form a picture of how different assumptions might
affect the ultimate design.
These studies have shown that for the Merrimack River
the constants for the settling out and resuspension of BOD are
the most insensitive, while the rate for the production of
-------
30
oxygen by photosynthesis of algae is the most sensitive. With
primary ~aste treatment for all se~age discharging communities,
DO deficit ranges for a particular station ~ere observed to
vary from 1 ppm to over 6 ppm ~hile the parameter for the
oxygen production by algae varied from approximately 1.5 ppm/day
to 0.0 ppm/day~ These values ~ould correspond to an assumption
of sunlight for the former and darkness for the latter. These
conclusions are in agreement ~ith those determined by the
FWPCA, ~ho applied a type of sensitivity analysis to a single
oxygen sag curve. The CDM and FWPCA studies both decided to
employ values for design corresponding to a cloudy day.
The results ~ere even moreimpreSBlve~hen the data vere
processed by passing values, assumptions, and criteria through
a computer routine ~hich derives an optimum configuration of
treatment plants. As discussed in Chapter II, this selection
is based on a required DO minimum in the river being met by
one or more up-stream systems of treatment plants each having
incremental performance levels giving the most benefits down-
stream for the invested capital. These results are best ex-
plained by Tables 7 through 10. Again, the parameter associ-
ated ~ith photosynthesis of algae produces the most significant
changes in the results. For a condition of darkness, secondary
treatment in all plants ~as not sufficient to keep the DO
deficit below 3 ppm, ~hile ~ith a condition associated ~ith much
sunlight, primary treatment in all plants was more than suf-
ficient, as the DO deficit remained substantially belo~ 2 ppm
throughout the study area. The following paragraphs describe
the results in greater detail.
Table 7 presents the results of a sensitivity analysis
in terms of the effect of parameter on se~age treatment plant
cost. The technique used ~as to hold all parameters constant
at the design values of Table 6 except one, and to test the
remaining parameter at the upper and lower bounds of its range.
In each case, a set of treatment plants is established by the
optimizing routine, ~ith allowable levels of treatment bet~een
38% and 90% BOD removal, to achieve a 3 ppm DO deficit at all
points in the river. The investment cost for se~age treatment
plants based upon the design values of Table 6 was estimated
at $33.1 million. For all testing, the investment cost varied
bet~een $30.5 million and $43.0 million, a range of 92 to 130
percent of the design investment cost of $33.1 million. The
highest cost corresponds to an assumption that photosynthesis
is not active at all: this sho~s that for the Merrimack River,
the activity of algae has a large favorable economic effect
provided that the growth can be controlled to avoid a nuisance.
With respect to individual parameters, the following
-------
TABLE 7
Sensitivity Analysis
(using optimizing routine to determine treatment plant removal fraction and location)
Each river parameter (Z value) is varied through given range, w}1ile holding constant the other river
parameter values originally adopted for CA~l
Minimum allowable treatment = 38% BOD removal
Maximum allowable treatment - 90% BOD removal
Objective DO deficit = 3 ppm
Investment Cost of
Community - Percent Treatment Sewage Treatment
Assumed
Parameter Values 1 2 3 4 5 6 7 8 9
0.01 38 38 38 38 38 38 38 38 38 30,512,828
Z3 0.05 38 80 80 38 65 38 38 38 38 33, 074, 04Q
0.10 80 90 85 80 80 65 65 38 38 39,536,362
0.010-0.027 80 80 80 80 80 65 38 38 38 36,473,606
Z4 0.030-0.080 38 80 80 38 65 38 38 38 38 33,074,040
0.10-0.267 38 38 38 38 38 38 38 38 38 30,512,828
0.014-0.016 65 80 80 65 80 65 38 38 38 35,624,589
Z5 0.070-0.080 38 80 80 38 65 38 38 38 38 33,074,040
0.i40-0.160 38 65 65 38 65 38 38 38 38 32,365,717
0.00 38 80 80 38 65 38 38 38 38 33,074,040
Z6 0.50 38 80 80 38 65 38 38 38 38 33,074,040
1.50 65 80 80 80 80 38 38 38 38 36,004,233
0.00 80 90 90 90 80 65 38 38 38 43,012,646
Z7 0.30-0.50 38 80 80 38 65 38 38 38 38 33,074,040
1.0 -1.667 38 38 38 38 38 38 38 38 38 30,512,828
w
....
-------
32
table shows the results obtained:
Range of Investment Costs
Parameter Million $ % of $33.1 Million % Difference
Kl 30.5-39.5 92-120 8-20
K2 30.5-36.5 92-110 8-10
K3 32.4-35.6 98-108 2-8
P 33.1-36.0 100-109 0-9
a 30.5-43.0 92-130 8-30
Table 8 shows the maximum DO deficit upstream and down-
stream of each of the sewage treatment plants, for two con-
ditions - all primary treatment and with the optimum set of
plants from Table 7 (selected in each case based upon the
assumed parameters). For the optimum sets, this table merely
confirms that each set will result in meeting the criterion of
3 ppm maximum DO deficit at all points on the stream. For the
set of primary plants, the maximum DO deficit with the design
parameters is 5.0 ppm while the DO deficit can go to as much as
7.9 ppm if the parameter turns out to be less favorable than
assumed. Figures 2 and 3 show the effects of varying the
photosynthesis parameter, for all primary treatment, and all
secondary treatment.
The tables and exhibits show that lower levels of treat-
ment, which correspond to discharge of higher BOD loads to the
stream, may result in stream conditions which are quite sensitive
to the stream parameters that actually obtain.
Table 9 shows results similar to those on Table 7, except
that the maximum DO deficit specified is 4 ppm rather than 3 ppm.
For 4 ppm, optimum configurations of plants range in investment
cost from $30.5 million to $35.2 million, compared with $32.4
for design values of parameters. For 3 ppm, the costs were
$30.5, $43.0, and $33.1 million respectively. This might indi-
cate that costs are less sensitive to parameter selection, when
standards of quality are lower. Table 10 iSs1m1lar to Table 8,
except for uSing a criterion of 4 ppm DO defi~it.
F.
Discussion
The sensitivity analyses have been made in terms of the
-------
TABLE 8
DO Deficits with Primary Treatment and with Optimum Treatment*
Minimum allowable treatment ~ 38% BOD removal
Maximum allowable treatment - 90% BOD removal
Objective D~ aef1c1t - 3 ppm
Maximum DO Deficit (ppm) for stations
Parameter
Assumed
Values
1-6
7-16
17-23
24-26
27-32
33-43
44-47
48-57
58-59
60-62
0.01 3.0 2.5 2.4 2.3 2.4 2.1 1.5 1.6 0.6 0.4
0.05 3.0 2.8 3.0 3.2 3.8 4.8 4.9 .5.0 4.1 3.9
Z3 3.0 2.8 2.7 2.6 2.9 2.8 2.8 3.0 2.5 2.4
0.10 3.1 3.2 3.8 4.3 5.8 7.9 7.9 7.9 6.7 6.3
3.1 3.1 3.1 3.1 3.1 2.9 2.8 2.8 2.2 2.4
0.010-0.027 3.0 3.0 3.3 3.6 4.8 6.6 6.8 7.3 7.1 7.1
3.0 2.9 2.9 2.9 2.9 2.8 2.7 3.0 2.8 2.9
Z4 0.030-0.080 3.0 2.8 3.0 3.2 3.8 4.8 4.9 5.0 4.1 3~9
3.0 2.8 2.7 2.6 2.9 2.8 2.8 3.0 2.5 2.4
0.10-0.267 3.0 2.4 2.2 2.2 2.4 2.5 2.2 2.2 1.1 1.0
0.014-0.016 3.0 2.8 3.1 3.4 4.4 6.1 6.2 6.7 6.6 6.6
3.0 2.8 2.7 2.7 2.7 2.7 2.6 2.9 2.8 3.0
Z5 0.070-0.080 3.0 2.8 3.0 3.2 3.8 4.8 4.9 5.0 4.1 3.9
3.0 2.8 2.7 2.6 2.9 2.8 2.8 3.0 2.5 2.4
0.140-0.160 3.0 2.8 2.9 3.1 3.4 3.9 3.8 3.9 2.5 2.2
3.0 2.8 2.7 2.7 2.9 2.6 2.4 2.5 1.5 1.6
0.00 3.0 2.8 2.9 3.1 3.5 4.4 4.4 4.5 3.5 3.2
3.0 2.8 2.6 2.5 2.7 2.5 2.4 2.6 1.8 1.8
Z6 0.50 3.0 2.8 3.0 3.2 3.8 4.8 4.9 5.0 4.1 3.9
3.0 2.8 2.7 2.6 2.9 2.8 2.8 3.0 2.5 2.4
1.50 3.0 2.9 3.2 3.5 4.4 5.7 5.8 5.9 5.4 5.2
3.0 2.9 2.9 2.9 2.8 2.8 2.8 3.0 2.8 2.9
(concluded on next page)
w
w
-------
Table 8--concluded
Parameter
Assumed
V~lues
1-6
7-16
17-23
24-26
27-32
33-43
44-47
48-57
58-59
60-62
0.0 3.1 3.2 3.6 3.9 5.0 6.3 6.4 6.5 6.5 5.9
3.1 3.1 3.1 3.1 3.0 2.9 2.9 2.9 3.0 3.0
Z7 0.30-0.50 3.0 2.8 3.0 3.2 3.8 4.8 4.9 5.0 4.1 3.9
3.0 2.8 2.7 2.6 2.9 2.8 2.8 3.0 2.5 2.4
1.00-1.67 3.0 2.0 1.8 1.7 1.0 1.8 1.4 1.4 0.1 0.2
*
When two rows are shown, first row is for primary treatment and second row is for
opt1lliu18
treatment.
w
.
-------
1
Effect of Varying Z7 (a) parameter
on the oxygen deficIt of a stream
with primary treatment.
Z7 = 9 = Constant for rate of DO
6 .' roduction by algae through
photosynthesis in Ppm/day
5
4
-
i
-
~3
-
u
-
u..
w
o
- - - - - - - - - - - - - --- - - -R-ecomme-naecr Maxfmum" DO bejicit - - - ... - - - -- - -
g2
\AI
\11
50
4
3 RIVER MILE
20
10
o
Figure 2.
-------
17 = q = Conltant for rat. of DO
produdion by a lea. th rough
photosynthesis in Ppm/day
Effect of Varying Z (a) parameter on
oxygen deficit of slream with secondary
treatment.
5
Secondary sewage treatment for a11
STP = 90% BOD remova1.
-
.e
!
....
-
U
-
&L.
au
o
~7_-.P___-
MaXimum- -Dcf beficit"
82
Z7 ==1.0 -1.667
\AI
G'
~
3 RIVER MILE
1
F'1__r-- 3-
~~~~~~ ~-
-------
TABLE 9
Sensitivity Analysis
(using optimizing routine to determine treatment plant removal fraction and location)
Each river parameter (Z value) is varied through given range, while holding constant the other river
parameter values originally adopted for CA~l
Minimum allowable treatment = 38% BOD removal
Maximum allowable treatment - 90% BOD removal
Objective DO deficit - 4 ppm
Community - Percent Treatment Investment Cost of
Sewage Treatment
Assumed
Parameter Values 1 2 3 4 5 6 7 8 9
Z3 0.01 38 38 38 38 38 38 38 38 38 30,512,828
0.05 38 65 65 38 65 38 38 38 38 32,365,717
0.10 38 80 80 65 80 38 38 38 38 35,173,474
0.010-0.027 38 80 80 65 80 38 38 38 38 35,173,474
Z4 0.030-0.080 38 65 65 38 65 38 38 38 38 32,365,717
0.01-0.267 38 38 38 38 38 38 38 38 38 30,512,828
0.014-0.016 38 80 80 65 65 38 38 38 38 33,692,019
Z5 0.070-0.080 38 65 65 38 65 38 38 38 38 32,365,717
0.14-0.16 38 38 38 38 38 38 38 38 38 30,512,828
0.000 38 65 65 38 38 38 38 38 38 31,112,178
Z6 0.50 38 65 65 38 65 38 38 38 38 32,365,717
1.50 38 80 80 38 65 38 38 38 38 33,074,040
0.00 38 80 80 65 80 38 38 38 38 35,173,474
Z7 0.30-0.50 38 65 65 38 65 38 38 38 38 32,365,717
1.0 -1.667 38 38 38 38 38 38 38 38 38 30,512,828
w
....a
-------
TABLE 10
DO Deficits with Primary Treatment and with Optimum Treatment*
Minimum allowable treatment - 3~ BOD removal
Maximum allowable treatment. 90% BOD removal
Objective DO deficit. 4 ppm
Maximum DO Deficit (ppm) for Stations
Assumed
Parameter Values 1-6 7-16 17-23 24-26 27-32 33-43 44-47 48-57 58-59 60-62
0.01 3.0 2.5 2.4 2.3 2.4 2.1 1.5 1.6 0.6 0.4
0.05 3.0 2.8 3.0 3.2 3.8 4.8 4.9 5.0 4.1 3.9
Z3 3.0 2.8 2.7 2.8 3.1 3.1 3.2 3.3 2.7 2.8
0.10 3.1 3.2 3.8 4.3 5.8 7.9 7.9 7.9 6.7 6.3
3.1 3.2 3.3 3.4 3.6 3.6 3.5 3.7 3.2 3.3
~). 010-0.027 3.0 3.0 3.3 3.6 4.8 6.6 6.8 7.3 7.1 7.1
3.0 3.0 3.0 3.0 3.1 3.1 3.1 3.4 3.3 3.4
Z4 0.030-0.080 3.0 2.8 3.0 3.2 3.8 4.8 4.9 5.0 4.1 3.9
3.0 2.8 2.7 2.8 3.1 3.1 3.2 3.3 2.7 2.8
0.10-0.267 3.0 2.4 2.2 2.2 2.4 2.5 2.2 2.2 1.1 1.0
0.014-0.016 3.0 2.8 3.1 3.4 4.4 6.1 6.2 6.7 6.6 6.6
3.0 2.8 2.8 2.8 2.8 3.2 3.3 3.8 3.8 3.9
Z5 0.070-0.080 3.0 2.8 3.0 3.2 3.8 4.8 4.9 5.0 4.1 3.9
3.0 2.8 2.7 2.8 3.1 3.1 3.2 3.3 2.7 2.8
D.140-0160 3.0 2.8 2.9 3.1 3.4 3.9 3.8 3.9 2.5 2.2
0.00 3.0 2.8 2.9 3.1 3.5 4.4 4.4 4.5 3.5 3.2
3.0 2.8 2.6 2.1 2.9 3.8 3.8 4.0 3.1 3.2
Z6 0.50 3.0 2.8 3.0 3.2 3.8 4.8 4.9 5.0 4.1 3.9
3.0 2.8 2.7 2.8 3.1 3.1 3.2 3.3 2.7 2.8
1.50 3.0 2.9 3.2 3.5 4.4 5.7 5.8 5.9 5.4 5.2
3.0 2.9 2.9 2.9 3.4 3.6 3.7 3.9 3.7 3.8
(concluded on next page)
w
CD
-------
Table 10--concluded
Assumed
Parameter Values 1-6 7-16 17-23 24-26 27-32 33-43 44-47 48-57 58-59 60-62
0.0 3.1 3.2 3.6 3.9 5.0 6.3 6.4 6.5 6.1 5.9
3.1 3.2 3.2 3.3 3.5 3.5 3.6 3.8 3.7 3.8
Z7 0.30-0.50 3.0 2.8 3.0 3.2 3.8 4.8 4.9 5.0 4.1 3.9
3.0 2.8 2.7 2.8 3.1 3.1 3.2 3.3 2.7 2.8
0.0-1.667 3.0 2.0 1.8 1.7 1.9 1.8 1.4 1.4 0.1 0.2
* When two rows are shown, first row is for primary treatment and second row is for optimum treatment.
W
\0
-------
40
effect of parameter on design decision. In each case, the
river quality criteria were spec~fied, and it was assumed
that the designer would have to provide the appropriate levels
of treatment in order to meet this standard.
The sensitivity analyses showed that the selection of
parameters can greatly influence the cost of treatment, if the
plants are permitted to vary from primary treatment to
secondary treatment. For a model based on the Merrimack River
in Massachusetts, the investment cost ranged from $30.5 million
to $43.0 million for nine plants, with $33.1 million based on
the "design" values of parameters. The estimates were particu-
larly sensitive to the ~ assumption, indicating that as much
as 30 percent more would be spent on treatment if photosynthesis
were assumed inoperative. The costs of treatment appear to be
more sensitive to parameter selection for higher standards of
river water quality.
While cost is an important factor in water pollution
control planning, the following is undoubtedly a more essential
consideration. If treatment plant levels are selected on the
basis of assumed values of parameters, and the actual values of
parameters turn out later to be unfavorable, the water quality
standards would not be met. The effects on stream quality for
such an unfavorable set of parameters are more serious with
lower levels of treatment than with higher levels. This would
indicate that if the treatment process for a plant is limited
to primary treatment on the basis of computation with a mathe-
matical model, the plant ought to be constructed for possible
expansion to secondary or higher levels of treatment. Further-
more, communities which construct primary treatment plants should
be advised that they may turn out to be inadequate.
While the Camp-Dobbins equations for BOD and DO sag are
theoretically an improvement over the Streeter-Phelps equation,
they are only as good for predictive purposes as the stream
parameters employed for the solutions. Because existing tech-
nology does not include satisfactory procedures for measuring
all parameters separately, the practice is to estimate some of
the parameters by trial-and-error, so that computed oxygen sag
curves match field measured curves. More than one set of para-
8eters are possible from this approach. If judgments are used
to adjust the parameters for assumed future conditions, such
procedures are risky and can lead to highly unreliable results.
Both CDM and FWPCA attempted to employ the latest tech-
niques for estimating parameters. While some analysts would
prefer other techniques, there is really no way to check results.
Such estimates by experts do not assure that future conditions
-------
41
will agree closely with predictions. On the basis of the
studies for this report, it would appear to be prudent to
base every system design on the premise that the assumed
values of parameters may be inaccurate, and possible ranges
of parameters should be investigated.
The methods used to determine stream parameters should
be improved. Emphasis should be placed on the development of
good techniques to evaluate parameters individually and as
directly as possible. Only then, can engineers make confident
adjustments to parameters in order to predict future conditions.
-------
42
CHAPTER IV
TIME VARYING MATHEMATICAL MODEL
A.
Introduction
The preceding two chapters have discussed a steady
state mathematical model for studying water pollution control
programs for a non-tidal river. The authors started with a
steady state model because, despite its obvious limitations,
most regulatory agencies assume a steady state corresponding
to a low flow summer period with a specified return interval as
the basis for estimating the efficacy of water pollution
control programs.
While steady state models for estuarine streams are
rarely satisfactory. it is judged that steady state models can
continue to be useful to determine approximate plans for water
pollution control for many non-tidal rivers. Such models can
be developed using the data from brief field surveys, and can
provide comparisons of alternative plans for pollution control
at reasonable cost for the engineering studies. It is clear,
however, that adequate margins of safety over the results indi-
cated by the model studies should be employed.
For important streams with existing or potential trouble
spots with respect to water quality, an approximate model of
river conditions based on the steady state will not provide the
means for a sufficiently adequate simulation of conditions when
streamflow and environmental factors vary substantially over a
period of operation. Thus, whereas tentative plans for con-
struction and operation of waste water treatment plants can be
formulated for steady state assumptions, changes in construction
and/or operation may be indicated to be necessary or desirable
after studies are made with a time varying mathematical model.
This chapter discusses studies to develop a mathematical
model for unsteady conditions, which represents system per-
formance more realistically than the steady state mathematical
model. The unsteady state model can be used to investigate the
effectiveness of an assumed configuration of treatment plants
while a stream's assimilative capacity changes with distance
and time. Effectiveness can be determined in terms of com-
pliance with specified stream quality criteria. Also, when a
simulation showing the variation of "quality with time is
available, the suitability of the environment for fish propa-
gation or other specific purpose may be studied in detail.
-------
43
For any river station over the length of stream upon
which the model is patterned, the techniques can trace a time
profile of flow and quality in terms of BOD, dissolved oxygen,
coliforms and chlorides while the upstream discharge, water
temperature and solar radiation are changing. In the next
Chapter, the results for model operation are discussed for a
single station which was shown to be critical by the steady
state studies. It would be possible, however, to develop a
record of flows and any or all of the water quality parameters
at any other specified station.
With a change in the printing instructions, values
could be shown for all stations at any specified time. This
is the form in which some of the other investigators have
shown their results.
Many investigators have recognized the need for a satis-
factory method of determining the spatial and temporal variations
of dissolved oxygen in a flowing stream. It is only in recent
years, however, that methods of "operations research" and the
electronic computer have been available for practical in-
vestigations of unsteady conditions. Significant advances in
this regard have been made by Thomann and collaborators (17,
18, 19, 20, 21, 22) who, since 1963, have presented results for
mathematical models designed for the estuarine conditions of
the Delaware River. Such models determine the dissolved oxygen
at various locations in a body of water in response to the
variations of BOD inputs (and other changing environmental
effects such as temperature and photosynthesis). A feature of
Thomann's approach has been the partitioning of a body of water
into segments which are affected, not only by local factors,
but also by interrelationships with all other adjacent segments
and through them with more distant segments. In order to employ
his approach, it is necessary to have equations to describe the
variation of any parameter with respect to distance, or time,
or both. The ~niques lead to a number of linear differential
equations with time included as a variable, which are solved by
means of numerical integration using a high speed computer.
The computer program developed for the Delaware River (23) was
adapted for the simulation of the Connecticut River by Arnoldi
and Hoover (24): this application was limited by an assumption
of constant values of river flow (except for tidal effects) and
other parameters.
Another serious investigator has been O'Connor (25, 26,
27, 28). Like Thomann, he has aimed at developing differential
equations and appropriate methods of computer solution. Again,
the approach requires sophisticated mathematical forms in-
corporating the parameters which vary with distance and/or time.
-------
44
Dobbins, with Dresnack and Bella (28, 29) have focused
on the methods of numerical analyses to solve the differential
equations underlying the BOD and DO profiles for streams.
All of the investigators, whose work is outlined above,
have developed models applicable to both estuarine and non-tidal
streams. under estuarine conditions, the diffusion of BOD and
DO, and any other substance may be considerable at certain times.
With the non-tidal stream, however, even with typical low
velocities in the downstream direction during critical low flow
periods, the effect of diffusion may be ignored. This may be
considered to be a special case, "plug flow," which would
partially reduce the complexities of the methods employed by
these investigators.
It has been found, however, that when "plug" flow can
be assumed, as is the case for almost all non-tidal conditions,
different methods can be used, based upon relatively uncompli-
cated elaborations of a classical approach such as the oxygen
sag expression employed by streeter and Phelps. This can be
accomplished with virtually no loss of precision, compared with
the more sophisticated methods. Thus Frankel and Hansen (31,
32) have used an expanded version of the Streeter and Phelps
equation, particularly as modified to include diurnal variations
of photosynthesis. The DO deficit may be plotted versus time,
or versus distance, when the velocity of flow along the reach is
known. The reaches are processed successively in the downstream
direction. By making a number of simulations with different
conditions at different times at the starting point of each
tributary, enough results are produced to plot variations with
respect to distance and time.
The Frankel and Hansen model assumes that the stream
flows for respective calculations are available as input. Le
Feuvre and pogge (33), with Mac Roberts, have worked on simu-
lation procedures in which flow variations are routed from
reach to reach throughout a river system: the results are being
used to study quality variations on the Lower Kansas River
System.
The examples cited above are believed to be representa-
tive of the most advanced work, and some features have in-
fluenced the direction of development of the time varying
mathematical model described herein. The principal objectives
have been to make maximum use of the types of data which can
typically be obtained from records and field surveys, and to
produce the kinds of results that are needed to examine the
effectiveness of water pol~ution control programs. It has been
found possible to accomplish these objectives without sophisti-
-------
45
cated manipulations of differential equations -- by working
with short reaches and/or times, . over which steady state con-
ditions may be assumed to apply without undue loss of accuracy,
and by accepting input data in any available form (such as a
table when an equation cannot be created without compromising
the data).
While the procedures have been developed with the
Merrimack River data and previous investigations as background,
it is believed that the time varying mathematical model should
have wide application. The remainder of this chapter is devoted
to the details of the procedures, while the results of the com-
puter runs are described in the next chapter.
It is noted, at the outset of this exposition, that the
procedures are deterministic. Random variations of input data
may, however, easily be accommodated in a simulation model.
Alternatively, the output values may be assumed to be the means
in a band of values encompassing the variations of an output
having stochastic properties.
B.
General Methodology for Time Varying River Model
It was decided to develop the methodology to show the
effects of time varying river parameters, for use with currently
accepted steady state formulas for determining the quality
characteristics. For this purpose, it was necessary to work
with sufficiently small time intervals during which these para-
.eters can be assumed to be constant. The previous steady state
8tudy divided the stream into 62 reaches with distances between
stations varying between 0.01 and 5.90 miles. The travel times
for the constant design discharge of 650 mgd varied from 0.02
to 1.04 days. Preliminary studies for the time varying river
model indicated the desirability of providing additional river
8tations, in order to properly process the input data.
The adopted system utilizes moving stations in addition
to the 62 fixed stations of the steady state model. The moving
8tations are located so that the time of travel between two
successive stations for any flow is an arbitrary time interval.
For this study, it was found that the analysis of six intervals
per day was sufficient to show the diurnal variation of dis-
80lved oxygen due to photosynthesis. Because the locations of
the 80ving stations are dependent only on the travel times,
which are in turn dependent on the discharge, it follows that
the moving stations are in different locations for every dis-
Charge. A procedure was devised which performs an oxygen sag
analysis for the short stretches between fixed and moving
8tations.
-------
46
Figure 4 shows a possible configuration of fixed and
moving stations. These stations' are identified by "I" and
"Nil designations, respectively. The figure also shows the sub-
programs used to compute the changes in oxygen concentration
between the stations: the appropriate subprogram depends on
the relative positions of fixed and moving stations.
The locations of the fixed river stations remain un-
changed for the particular river studied. The moving river
stations, however, may be closer or farther apart depending on
the magnitude of the flow. If, as in the case of Figure 4,
there are no changes in the discharge or hydraulic character-
istics of the stream channel, the moving stations will be
evenly spaced. This is because the velocity would remain
constant and therefore the distance traveled in each constant
time interval would remain constant.
For a larger discharge, corresponding to a greater
velocity. the distance traveled in a specified time interval
would be greater: thus, the moving river station N+3 may be
reached after the flow has passed fixed river station I+3.
a smaller discharge, N+3 may be reached before station I+2.
For
Six values of flow are routed for each day. As each
incremental volume of flow, which is assumed to originate at the
upstream end of the stream, moves downstream it must pass all
of the fixed river stations, which are indexed by I, Itl.....,
etc. At these stations, sewage discharges carrying pollutants
may be added: the river flow may decrease because of water
supply intakes: the discharge rate may change in accordance
with the routing procedure for unsteady flow: and any or all of
the parameters which vary with distance and time may change.
A description of all of the elements of the model and
a summary of the respective functions used for repetitive evalu-
ations are contained in Appendix I and Appendix II. Functions
1, 2, 5, and 6 were especially developed or were obtained by
modifying similar functions for the steady state model, in order
to accommodate the needs of the time varying model. The remaining
functions are the same for the steady and unsteady state models.
It is possible to bypass those functions which are not needed for
desired results, and/or to print only those results which are of
interest. This is a matter of detailed programming.
An outline of the step-by-step computation procedure
which takes account of changing conditions at both fixed and
moving stations is shown in Figure 5.
The computations proceed in a downstream direction.
A
-------
FIXED RIVER STATIONS
1+1 1+2 1+3
. I I I
I . I .
I ' I .
.
I I I I
, " .
, I I
I I I .
I . I
, I . I
I ' I I
I I I
, ' I I
, I . I
I , I .
, I I I
, , I I
, " ,
I I I I
, I I U.
I U. U. u. .
U. u.I U. ' . w
u. , , w w w
w ' W wI W , N M - ' N
I N _I ~ U. . lot.
- I u.. u..
u. ' u.. u.. I I
. I ' I
, " ,
.
, . , I
I " I
I ' ,
I I
I I ' I
. , I
N N+1 N+2 N+3 N+4
MOVING RIVER STATIONS
Figure 1+
Examp1e of Configuration of .Axed and Moving Stations
~
......
-------
NOTES:
(1) PATH S SHOWN FOR ONLY ONE FLOW
INCREMENT AS IT PASSES BETWEEN
2 FIXED RIVER STATIONS. RE-
PEATED FOR 6 INCREMENTS PER
DAY, AND FOR All REACHES.
(2) FLOW ROUTING COEFFICIENTS ESTAB-
LISHED PRIOR TO BEGINNING
OP£RATIONS SHOWN.
I-corresponds to river station number
N-corresponds to time interva1 number
I=Number of next river station
Ca 11 F 1T (I)
Compute Q (I)
Compute T (I)
Ca 11 F2EF (I, N)
1sT ( I) less
YES
Call F4EF (I,N)
Call F5AI (I,N)
Compute ZZ3(N); ZZ4(N);
Compute F(I-I) for new temp.
Compute E(I); F(I); 00(1)
ZZ7(N)
NO
Call F5AI (I,N)
Compute ZZ3(N); ZZ4(N); ZZ7(N)
Compute F(I-I) for new Temp.
N=N+I
Compute TZ(N); EZ(N); FZ(N); DOZ(N)
Ca 11 F3EF (I ,N)
Is TZ(N) + 1/6 less
NO/
than~
Call FIEF (I,N)
- Call F5AI (I,N)
Compute ZZ3(N); ZZ4(N); ZZ7(N)
Compute FZ(N) for new temp.
Compute E(I); F(I); DO(I)
~
YES
Call F5AI (I.N)
Compute ZZ3(N); ZZ4(N); ZZ7(N)
Compute FZ(N) for new temp.
N= N+I
Compute TZ(N); FZ(N); EZ(N); DOZ(N)
Call statements for routines computing coliforms (MPN), chlorides (PPM), water supply for
community (MGD), quantity and quality of sewage discharge if any, and resulting condition
of the stream Q(I), F(I), DO(I), E(I), etc. at outfal1
.I::-
00
Figure 5 - Paths for Mathematica1 Ana1ysis - Time Varying River Mode1
-------
49
set of "distance varying parameters" are assumed to remain at
these values until the next fixed river station is reached.
At the beginning of each time interval which corresponds to a
moving station location, one or all of the "time varying
parameters" may change. These are assumed to remain constant
until the beginning of the next time interval. Many of the
functions utilize both time varying and distance varying para-
meters; thus, a reach of the river where all parameters are
constant is always equal to or less than the distance between
moving stations.
With the configuration of fixed and moving river
stations described above, it was possible to use the oxygen sag
equation developed by Streeter and Phelps (14) as modified by
Camp (15), and other steady state formulas previously applied
by the Principal Investigator. Four subprograms are available
for implementing Function 2 to compute the BOD and dissolved
oxygen (F1EF, F2EF, F3EF, or F4EF) and the appropriate one in
each case depends on the relative locations of the fixed and
moving stations (see Figures 4 and 5).
Because the location of a moving station is derived from
the stream flow and, therefore, may be in a single place only
once there is little reason to print out the computed values
for such a station. These values are merely used as the initial
values for the next river reach which will end at either a
fixed or another moving station, whichever comes first.
All of the computed values for the time, co1iforms, BOD,
DO, chlorides and discharge remain in computer storage until a
flow increment has passed all the way downstream, after which
these values may be printed for as many of the fixed stations
as desired. After the printing operation is complete, the pro-
cedure cycles back to the upstream end of the stream, and a
new incremental flow is processed. This procedure may be re-
peated for as long a period as data are available.
Because of the limitations on computer storage and the
computer time required, the authors have generally processed
only one month of data at a time. To obtain approximately one
month of output for three systems of treatment plants, about
ten minutes of computer time were required. Input data for
several days before the start of the period for which output
data is desired are needed in order to process the first flows
of the period.
Previous analyses using a steady state model have demon-
strated that particular stations in this river are critical with
respect to river quality criteria. For the operational studies
-------
50
of the time varying mathematical model, the most downstream
of these stations was retained for the production of results.
C.
Factors Affecting Oxygen Variations
Most of the stream factors which influence the oxygen
balance have been studied extensively by many investigators.
Those parameters which are most often considered are as follows:
1. Deoxygenation
2. Reaeration
3. Algae Activity
4. Benthal Demand
5. Settling and Resuspension of BOD
6. Temperature
7. Sunlight
8. Streamflow
All of these factors were included in the steady state
model, but they varied only with distance and were assumed
constant with time. The rate of benthal demand was included in
the parameter defining the rate of resuspension of BOD: the
temperature was assumed to be 250C: the stream flow was 650 mgd:
and the sunlight condition was assumed to be cloudy.
For the development of a realistic unsteady state model,
acceptable relationships had to be developed to demonstrate the
time variation of the parameters. Satisfactory existing re-
lationships between water temperature and oxygen saturation,
reaeration, and deoxygenation could easily be adopted for the
model. Water temperature is easily measured and was in terms of
a daily input value. Water temperature data for the operation
of the model were taken from the records of treatment plants
along the river (34). A relationship between algae activity
and sunlight based on experimental data for the study stream was
also available. Daily values for sunlight intensity at Boston,
Mass. were used for input (35).
For benthal demand, and the settling and resuspension of
BOD, little useful information was available. A relationship for
these parameters would have to consider discharge as it effects
scour, and account would also have to be kept of available
benthal material subject to resuspension as the discharge
changes. If better data were available, it might be possible
to develop an empirical relationship. The studies with the
steady state mathematical model (see Chapter III) indicated
that the dissolved oxygen values for the stream were not very
sensitive to changes in the rate at which BOD is settling or is
resuspended. Accordingly, it was decided to make these factors
constant with time for each computer run. Since each run covers
-------
51
a limited portion of the year, any inaccuracies due to this
assumption are small.
As noted in the next section, the representation of
unsteady streamflow, using a new empirical procedure, was de-
veloped and tested. An upstream hydrograph (36) was determined
by combining the recorded flows of the main stream and tribu-
taries upstream of the study reach.
Succeeding sections provide details on the relationships
which were adopted for deoxygenation, reaeration, and photo-
synthetic oxygen production.
It is emphasized that the programming methods for the
time varying mathematical model are quite flexible, and can
facilitate revisions in computational procedures to accommodate
different relationships for the parameters which may be pre-
ferred by other analysts.
variable inputs from waste treatment plants have not
been included in the time varying model. These were not con-
sidered to be of importance for a model patterned on treatment
arrangements for the Merrimack River, but could easily be ac-
commodated by relatively minor programming modifications.
Similarly, it would not be difficult to include the effects of
contributions of pollutants from overland flow and tributaries,
if estimates were available.
D.
Streamflow Routing Procedure
For the biological analysis of a stream where the time
varying nature of streamflow is taken into account~ two concepts
should be considered. One of these is the changing of the
dilution of pollutants. The other is the differing travel times
for pollutants. To properly consider these effects, an empiri-
cal flow routing procedure has been developed, which is es-
pecially suited for studying normal and low flow periods.
The only input data required are an upstream hydrograph
and velocity-discharge relationships for the reaches. unlike many
flood routing procedures, this method of low flow routing is
concerned with the time displacement of pOllutants for different
discharges rather than representing the movement of translatory
waves. This procedure proved to be quite complex and a major
portion of the time used for the study was spent on its develop-
ment. Also, much of the computer time required for processing
with the time varying model is attributable to the routing pro-
cedure. A detailed explanation of this procedure is contained
in Appendix IV. The establishment of coefficients for flow
routing is completed prior to the operations in Figure 5.
-------
52
It is noted that the effects of overland flow and
ground water flow on stream discharge are included only im-
plicitly in the streamflow routing which aims at correlation
of available stream records. If these types of data are
available, they can be included explicitly with little
difficulty.
E.
Deoxygenation
For a flowing stream, the measurement of the oxidation
coefficient or deoxygenation constant, usually abbreviated kl in
the literature, is very difficult. The rate of removal of BOD
in a stream may be determined from the BOD values at an upstream
and a downstream station and the travel time between them. How-
ever, because BOD may be removed in a stream by processes other
than oxidation, the oxidation coefficient may not equal the
computed stream rate. Current methods of analysis cannot
separate the oxidation of the organic matter as determined in
a BOD test, from the other effects in a flowing stream. Usually
the kl factor for a stream is higher than the laboratory kl but
less than the total BOD removal rate of a stream. This problem
in evaluating kl is evidenced by the fact that Camp Dresser and
McKee (2) and the FWPCA (1) each ran extensive surveys of the
study stream and determined very different values.
O'Connor (37) lists the factors which may influence the
oxidation rate as turbulence, biological growths on the stream
bed, algae, immediate or chemical demands, nutrients and lag or
adaptive periods. Other factors which may affect the rate of
BOD removal, but not necessarily the oxidation rate, are sedi-
mentation and flocculation, scour, and volatilization. It is
obviously impractical to quantify the effects of all of these
factors. Some distinction was recognized by Streeter and Phelps
(14) and others who included effects other than oxidation by
incorporating the coefficient k3. The k3 constant is usually
interpreted as the BOD which settles out to bottom deposits
rather than exerting its demand on the flowing stream.
For the basic data in the time varying model, the authors
have adopted the values of kl used for the steady state model
(3, 4) which were taken from the Camp Dresser and McKee report (2).
It is generally accepted that as the temperature of a
body of water increases, factors affecting the kl value also
increase their activity. Recent work by Zanoni (38, 39) demon-
strates that a modified Arrhenius expression using two breaks,
one at lSoC and the other at 32oC, gives a good approximation
of the effect of temperature on the value of kl.
-------
53
These equations are as follows:
Equation
Range
I
-------
54
k2 =
5.026 VO.969
H1.673
T-20
(1.024)
k2 = reaeration coefficient as a constant per day
T
= temperature in degrees centigrade
v
= velocity in ft. per sec.
H
= depth in feet
k2 (To) = 3.739
Gaudy (43) in one of the most recent studies
following formula:
V TO-20
H 3/2 (1.0241)
Issacs and
have developed the
k2 (To) = reaeration coefficient as a constant per day
at a temperature TO
V
. average stream velocity in ft. per sec.
H
= average stream depth in feet
All of these formulas are basically similar in that they
show that the reaeration coefficient depends on the stream
turbulence, which by fluid mechanics theory would be expected
to increase with increasing velocity and with decreasing depth.
Temperature also enters into each formula since it affects the
diffusivity of oxygen. Some of the studies used data for actual
streams while other investigations were based on laboratory
simulations. The nature of the equations implies that each
class of stream could have a unique formula for relating k2' V,
H, and T.
Another approach by Camp (2, 13) estimated the value of
k2' along with other parameters entering the standard oxygen sag
formulas, from 1ight- and dark-bottle oxygen analysis. Camp
demonstrated that the Merrimack River, on which the mathematical
model has been based, exhibits k2 values which are much less
than values indicated by the above formulas for streams of this
type. The FWPCA (1), using similar techniques, estimated higher
values for the reaeration constant while also determining
proportionately higher values for the deoxygenation constant
(k1). Since this method for evaluating k2 depends upon first
estimating k1' the differences between the CDM and FWPCA esti-
mates are attributable to the determinations of k1.
Theoretical formulas are particularly useful for mathe-
-------
55
matical model studies. To use them effectively, however, more
stream factors would have to be developed which vary with
distance and time. Channel roughness and velocity vs. stage
data appear to be most important although wind, precipitation,
ice cover, and barometric pressure also have an effect. None
of the above formulas gives results for the Merrimack River
consistent with those determined in the two stream studies.
All the effects except temperature have been grouped
into one constant which is unique to each river stretch. This
is essentially what the two steady state studies have done.
This factor implicitly includes the channel roughness and
average values of other parameters affecting aeration.
The following formula is based on the concept that an
interfacial liquid film is in a continuous state of random re-
newal (44, 45):
k Q(t-20)
kt = ~20
kt = reaeration constant at any temperature "t"
k20 = measured reaeration constant at 200 centigrade
Q
- temperature coefficient
In regard to an appropriate Q value, Dobbins and Metzger
(46) have observed in experiments with helium and nitrogen that
the absorption coefficients for gases of different diffusivity
are affected differently by temperature, and that for a given
gas the effects of temperature on the absorption coefficient
depends on the level of turbulence. Metzger (47) extended
these results to include oxygen absorption and interpreted the
variation in reported Q values. He has proposed a series of
graphs and formulas for predicting this value. It seems likely.
in light of this recent work, that a different value of Q may
be appropriate for each river stretch.
Considering the available information, the following
values for Q were adopted:
Range centigrade Temp. Q Value
10-20 1.025
20-30 1.035
With this formula, a table of coefficients was developed
-------
56
to modify the values of k2 which. were originally estimated for
the steady state model at 25 degrees centigrade. These original
values, as mentioned earlier, vary from one river stretch to
another and probably reflect the physical characteristics of
the river during the low flow period. The table of coefficients
applies to temperatures from 100 to 300 centigrade.
By using a table rather than an inflexible formula, it
is simple to introduce modifications as more accurate information
becomes available. In the future, it should be possible to de-
velop additional tables of coefficients which would allow the k2
parameter to vary with flow and with other stream and atmospheric
parameters which vary with distance and time.
G.
Photosynthetic Oxyqen Production
There appears to be no uniformity in design practice as
to whether to evaluate the effects of algae activity on stream
water quality- This is probably due largely to difficulties
in obtaining appropriate basic data and incorporating them in a
predictive formula that can be used with confidence. The
sensitivity of algae to changes in light intensity, as in-
fluenced by cloud cover in the atmosphere and turbidity in the
water, have been cited as reasons why photosynthetic oxygen pro-
duction is too variable to be a reliable addition to stream
oxygen resources.
Literature is in general agreement that algae growths
which are concentrated at a river location are not desirable
from an oxygen balance standpoint. O'Connell and Thomas (48)
concluded that the oxygen production by benthic algae and other
attached plants have little beneficial effect on the oxygen
balance of streams. On the contrary, they state that oxygen
demand associated with algal respiration and decay negate any
temporary gains made as a result of pbotb~nthesis. Since these
algae must undergo their entire life cycle in the same place,
this location must eventually be subject to oxygen demands of
the same magnitude as the amount of oxygen originally produced
due to photosynthesis.
The
nificant in
will not be
oxygen. In
ocean.
effects of phytoplankton may, however, be more sig-
the stream environment. When these plants die they
in the same place they were when they were producing
fact, they could decompose quite harmlessly in the
Hull (4) has the following interesting comment regarding
these concepts:
-------
57
"I have been frequently reminded that for every gram
of oxygen released by photosynthesis, a gram of bio-
chemical oxygen demand is produced by the synthesis of
organic matter, and therefore there can be no net gain
of oxygen. This is a gross oversimplification---it must
be qualified by consideration of the relative rates,
periods, and locations of the oxygen liberation and
subsequent oxygen demand. To support this, one need only
point to the vast standing crop of fixed carbon on this
earth in the form of petroleum, coal, plants and animals,
including several billions of human beings. All of this
carbon was fixed by photosynthesis, but some of it has
waited millions of years to be oxidized, before it will
eventually take back the oxygen liberated at its birth."
Although the authors do not mean to imply its general
applicability for all streams, it appears that in the Merrimack
River, which was used as a basis for this model, some benefit
can be gained from the action of phytoplankton. Camp (13)
attributes two thirds of the available oxygen in the Merrimack
to photosynthesis and states that reliance must be placed on
the photosynthetic production of dissolved oxygen. Both the
FWPCA and the CDM reports have included an analysis of the
effects of algae. They have assumed positive alpha (~) values
of approximately 0.3 - 0.5 ppm per day corresponding to a
cloudy day period.
The oxygen model which has been formulated represents
as closely as possible the effect of algae on the stream. Using
records from the u.s. Weather Bureau (34) and correlative
studies by the FWPCA, relationships were developed between total
daily solar radiation in Langleys (gm-cal/cm2/day) and the
photosynthetic oxygen production in ppm per day (see Figure 6).
A series of parabolic curves were drawn (see Figure 7) of gross
oxygen production vs depth, similar to those shown in the FWPCA
report and developed in other studies (50). The area above one
of these curves represents the total oxygen production for the
indicated solar radiation value. The alpha (~) value can be
determined by dividing the area over the curve by the average
depth of the reach in question. With this method, an estimate
of the oxygen contribution from algae can be determined from
daily solar radiation (time varying) and depth (distance varying).
Also incorporated in this procedure is a technique which
takes account of the diurnal variation in the rate of photo-
synthesis. Each daily solar radiation value has six rate
constants which correspond to the six 4-hour periods occurring
each day. For example, the first value is for 12 midnight to
4 A.M. and the second is from 4 A.M. to 8 A.M. In the absence
-------
58
9
8
-8 7
.......
E
Q. 6
Q.
.
Z
0
_.
U s
::)
Q
2
A. 4
Z
",0
~.
~ 3
en
S 2
1
o
o
~
~
CD
SUNLIGHT INTENSITY - Langleys = gm -cal/ana,
ICiay
Figure 6
Gross Oxygen Production vs. Sun1ight Intensity
-------
~
-
tJ
J!
I
DIi: 5
w
~
J
"-
0 6
:J:
~
Q.
~
7
11 gure 7
59
o
1
2
3
o
2
~
5
1
3
GROSS OXYGEN - ppm Iday
Gross Oxygen Production vs. Depth for Indicated Va1ues
of So1ar Radiation in Lang1eys
-------
60
of light energy it is known that plants respire so night-time
values must be negative. Also cloudy day values must be low.
Most studies of the Merrimack River have observed stream
dissolved oxygen variations of about 1.0 ppm for sunny days
and something less than that for cloudy periods.
In the literature it has been reported that respiration
accounts for 10 percent of the gross photosynthesis (51): how-
ever, this percentage should vary considerably for different
sunlight intensities. A matrix of values was developed which
approximates this variation while maintaining the measured
average daily rates.
The use of matrices permits flexibility in the compu-
tations. For example, while maintaining the same total average
daily oxygen contribution, it is possible to increase the ac-
tivity and to simulate the venting to the atmosphere of mole-
cular oxygen when the stream becomes supersaturated. The corre-
sponding night-time demand will be proportionately greater and,
therefore, lower early morning values of dissolved oxygen will
be observed. Although studies have indicated that significant
supersaturation does occur because of algae (52), this effect
has not been included because it can be quite localized and
not readily estimated. For the initial studies, a constant
night-time alpha value of -0.5 ppm per day was selected.
There were some factors which were not taken into account
in this study because of insufficient information. In the
natural case the peaks and valleys in the dissolved oxygen curve
lag about one day behind the extreme values of the sunshine curve.
This has been explained by the fact that with a large amount of
sunshine for two or more consecutive days, there may be an in-
creasing production potential caused by the development of a
young and vigorously growing phytoplankton population. Con-
secutive cloudy days result in a loss of phytoplankton and are
followed by increased BOD and decreased DO (51). Recent studies
have tried to clarify the factors which control this kind of
activity but further research is needed before these phenomena
are fully understood (53). The possibility was also investigated
of including the effects of varying temperature and nutrient
concentrations on the rate of oxygen production by algae (54)
but, like the lag effect described above, the proposed methods
have not been sufficiently proven to warrant their inclusion at
this time.
The time varying model is based on an actual stream.
studies made of this stream previously indicated that benthic
algae would not be a significant factor: however, it would be
quite simple to include these effects.
-------
61
The matrices were developed for summer conditions.
It would be a simple matter to ~nsert other input tables
which would be applicable to fall or winter temperatures.
In summary, the model is based on the following assump-
tions: 1) the rate of algae respiration is constant during all
dark periods: 2) any instantaneous rate is independent of the
algae's previous history: 3) algal synthessis is independent of
temperature: and 4) there is no significant relationship between
nutrient concentrations, due to differing sewage treatment
levels, and algal synthesis.
H. . Description of Computer Program
The computer program developed for the time varying
river model consists of a main program and seventeen subroutines.
These source programs were written using Fortran II language and
programming rules, and were processed by the CDC 3300 computer
system installed at Northeastern University in 1967. The
computer as of January 1, 1969 had a core storage of 64,000
words and a processing speed of more than 300,000 basic in-
structions per second.
The main computer program has three major components
in addition to the standard statements required by the specific
computer system used. They are as follows:
1. Statements to control data processing--to place data
into storage: to cause a "path" for data processing:
and to print results
2. Procedure for the routing of the unsteady inflow
3. Equations for the river model which perform simple
calculations or call upon the subroutines where the
more complex analyses are handled.
The subroutines are coordinated with the main program
with respect to the use of common symbols and common areas of
computer storage.
Appendix I contains precise descriptions of the various
input data needed for this river model. The input data are of
several types and are designated by the following symbols:
1. "Z" Data - hydraulic and stream purification con-
stants associated with river location
2. "CZ" Data - temperature coefficients to vary "z"
data with temperature
3. "ALA" Data - data varying with solar radiation
4. liNT" Data - input data varying daily (sunlight,
temperature, streamflow)
5. "G" Data - physical and population data for com-
munities
-------
62
9.
"Y" Data - economic oata for communities
"A" Data - administrative assumptions for communities
"KD" Input Data - Table defining coefficients in
equations evaluating treatment plant efficiencies
Miscellaneous -- miscellaneous input data (for com-
puter processing).
6.
7.
8.
All of these data are used in the solution of the functions
contained in the main program and subroutines. A summary of these
functions appears in Appendix II. A general flow diagram for the
entire process is shown in Figure 8.
The river modeling statements are of three types: 1) id-
entities; 2) simple additions and subtractions; and 3) "call"
statements, each of which brings a corresponding subroutine into
play. A given subroutine may "call" another subroutine which
may "call" still another subroutine etc. Appendix III lists
the computer call" statements for the seventeen subroutines and
gives a precise description of the arguments of these statements.
A total of 269 statements were required to model the
portion of the river used for the studies of time-varying re-
lationships, which covered slightly over thirty miles and had
41 stations. The model was developed for a full fifty miles and
62 stations but it was found that the critical sections occurred
above station 41. These modeling statements are the only part
of the program which is unique for a specific stream. Once the
principles of the model and the pattern for these modeling state-
ments are fully understood, the programming of the model for
another stream should not be difficult.
Each river station has a group of equations which evalu-
ate the quantity of flow and various quality properties. Each
community has two groups of equations -- for water supply and
sewage treatment. The water supply equations are placed just
after the equations for the river station where the raw water
quality is established. The sewage treatment equations are
placed just before the equations for the discharge of treated
sewage.
Appendix I also contains descriptions of computed output.
Generally the only computed results which are printed are the
flow and water quality characteristics at one of more specified
river stations. Depending on the routing of unsteady flow an
average of six values of each parameter are printed daily- The
number may depart from the six values in accordance with the time
intervals selected for discharge values, as discussed in the
description of the river routing procedure (Appendix IV).
-------
ARead and print fixed Input data
(KD; ALA; CZ) l
BRead and print variable input data
(NSTA; NCOM; NDAY; NOPT)
(Z; G; V; Z; and NT)
,
Read river routing input data TS, ZN
River routing procedure
Compute CQ va1ue for every input f10w
t
Read new initial values at upstream end ~Read and print all or a portion of new variable
(Q; T; E; F; DO; H; and V) input (B) according to option number (NOPT)
t
Run river model for one input flow
~
Print rEsults for specified downstream stations
(Q; T; E; F; DO; H; V; WI; WR)
t
Have all study periods been evaluated?
/"
NO
YES
Are the re more
/
NO
t
END
options
"
YES
to be run?
Figure 8 - General Flow Diagram of Time Varying River Mode1
~
\AI
-------
64
The print-out instructions provide for printing all
input data as well as output results. This enables the
verification of card punching and the debugging of errors
which may be caused by incorrect data or data presented in a
form which is unacceptable to the computer.
This program required the use of nearly all available
core storage, and six minutes were needed to complete output
from one month of daily input values. About two thirds of
this time is required for the river routing procedure; there-
fore, once the inflows and river routing data are available,
to run the program for different sewage treatment combinations,
administrative assumptions, or river parameters takes only
about two more minutes per run. At current computer prices of
from $200 to $300 per hour, the initial input set for one month
can be processed for about $25 and each additional option can
be processed for about $10 more.
The computer output obtained was found to be very
voluminous and, therefore, it was quite difficult to examine
the relationships between input and output data. Because
graphing of the results proved to be quite tedious, computer
graphing techniques were investigated. The C.D.C. 3300 com-
puterhasa Cal-Comb plotter, but the techniques for its use were
not yet sufficiently developed for use in this study. It is
judged, however, that automatic graphing with the computer can
be shown to be feasible and can greatly facilitate the analysis
of the computer output.
-------
6S
CHAPTER V
SIMULATION STUDIES WITH TIME VARYING MODEL
A.
Introduction
The preceding chapter has discussed the need for a
time varying model and has described the general methodology
and details for its employment. This chapter shows various
types of results which can be obtained with this model.
Results are compared with those obtained with a steady state
model patterned on the same stream. Sensitivity analyses are
also made to show the effect of stream parameter selection on
the prediction of stream quality conditions.
A steady state model based on a seven day low flow
period corresponding to a specified return interval cannot
give a complete simulation of stream conditions. A time varying
model can be used not only to study conditions for the variations
within the worst seven day low flow period, but can also cover
longer periods at other times of the year.
The time varying model is based on a deterministic
approach which primarily considers the changes due to flow,
temperature, and sunlight. Although not included in the studies
for this report, the model could be modified to include stoch-
astic fluctuations in the rate parameters and other input data.
Time varying results should enable planners to investi-
gate the effectiveness of treatment plant configurations in
terms of a realistic appraisal of stream conditions as they may
be expected to occur in a stream. The model described herein
enables the analyst to use the types of data which are typically
available, without the need for sophisticated construction and
manipulation of differential equations. Another advantageous
feature of the model is its flexibility. which permits the re-
placement of any of the individual functions used for repetitive
computations as better relationships become available.
Although the time varying mathematical model and the
input data used to produce the results are patterned on in-
formation for the Merrimack River in Massachusetts, it is not
claimed that the results reproduce values for this stream.
Rather, the intent has been to study different techniques for
analyses, and the implications of using available data and
design assumptions in the planning and operation of water
pollution control programs.
-------
66
The examples described in the following sections are
limited to a few basic demonstrations. It is important,
however, to emphasize that, due to the comprehensive nature
of the time varying model and its flexible procedures, many
other input-output studies are possible. Although output is
normally available from a computer run for every preselected
("fixed") river station within the study stream, the examples
included herein present results for only one river station.
This station was chosen because it was shown by the steady
state model studies to be critical with respect to DO for the
specified design period. It is judged that, unless unfore-
seen combinations of unusual river conditions and/or treat-
ment plant configurations were to exist, this station would
remain critical for the time varying model.
Features which could have been included for the studies
are: the varying of treatment plant operations with time, the
inclusion of overland flow and major tributaries along the
stream, changing population configurations, the direct addition
of molecular oxygen to the stream, the employment of low flow
augmentation, and thermal discharge effects. In general, these
have not been analyzed because they have not been of particular
importance in previous studies of the Merrimack River, on which
the model is patterned. They could have been included, however,
without much. difficulty.
The representative results for the critical station
which are described in this chapter are of three types. The
first demonstrates the form of input vs. output which can be
obtained with the model. The second set compares the results
obtainable with the time varying model with those previously
determined with the steady state model. Finally, the third set
comprises the results of a sensitivity analysis performed on
the model parameters.
The final section of this chapter discusses various
results which have been obtained by other investigators who
have incorporated a statistical approach to evaluate the vari-
ation of results. This is used as a basis for discussing the
possibility of expanding the time varying model to consider
random variations in the input, and the variations of output
produced by effects other than those included thus far by the
mathematical relationships.
It is emphasized that the conclusions discussed in this
chapter are based upon only limited testing work with the time
varying model. They conform, however, to what may be expected
from experience and intuition.
-------
67
B.
Results Obtained with Time Varying Model
Figures 9, 10, and 11 are charts which represent
typical input vs. output conditions which may occur during
periods when the heaviest demands are being placed on a
stream. These charts show simulations of conditions similar
to those for the Merrimack River for July and August 1964, and
for August 1965. These months were chosen because they are
drought periods and the inputs for the model contain the worst
combinations which could be found in the records for recent
years. By merely inspecting the input data, it may not be
possible to determine which month's output will contain the
lowest single DO value, the lowest average DO value, or the
longest period with the DO values below some specified value.
As shawn, the output with "optimum" treatment for August 1964,
with an average DO of 7.41 ppm, was generally better than
July 1964, which had an average DO of 7.10 ppm. The con-
trolling factor in this case seemed to be the temperature
which averaged 24.80 C for July and 21.80 F for August, even
though the discharges averaged 819 mgd and 700 mgd, respectively.
The month of August 1965 contained the worst input which
could be found from the natural records. This month was used as
a basis for the remaining analyses described in this chapter.
Since each chart has 32 vertical spaces available, the
first space on each chart shows the input and output for the
last day of the preceding month. The upper graphs of the
chart show the daily averages for input data (flow, temperature,
and solar radiation) as they occur at the upstream station. The
bottom four graphs of the chart show output for the critical
downstream station, comprising continuous graphs of discharge
(mgd) , coliform (mpn per 100 ml), BOD (ppm), and DO (ppm). To
facilitate comparison with upstream discharge values, output
discharge values are shown as daily averages, instead of the
computed output which was more variable. Also computed but
not shown here are chlorides (ppm), DO deficit (ppm). and two
index values which could be used by expanded versions of the
program to determine the utility of the stream for water supply
and/or recreation.
Outputs pertaining to biological conditions are shown
for three different treatment plant configurations - all
primary treatment with 38% BOD removal, all secondary treatment
with 90% BOD removal, and an "optimum" (least cost) configura-
tion. The optimum configuration was previously determined by
the steady state model studies, and was based on satisfying
an objective DO deficit of 3 ppm for the whole stream including
the critical station. For the steady state model, the DO
-------
18 19 20 21 22 23 24 25 26 27 28
+--~~=-J~=_==i-nn -*~~ -j~~ =t_t~==- ~r - -::}~ --
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Simulation of ~iver Conditions at Station 41
Treatment Plant Configuration
All Primary ----
Optimum
All Secondary- ------
1500 1 2 3 4
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81
8D
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J(
MONTH OF
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1964
Figure 9,
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19_~
MONTH OF
AUGUST
"
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_.--..............,-- ----
Treatment plant Configuration
70
---
-
12 13 14 15 16 17 18 19 20 21 22- 23 24 25 26 27 28 29 30 31
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1965
'I
26 u
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29-~-- -30---- -- 3\
5
Simulation of River Conditions at Station 41
MONTH OF
August
F'i gure 11
-------
71
deficit for the critical station for the three forms of treat-
ment (primary, optimum, and secondary) equalled 4.63 ppm,
2.94 ppm, and 1.83 ppm, respectively- At 250 C, the oxygen
saturation value is 8.38 ppm, which would give corresponding
DO values of 3.75 ppm, 5.44 ppm, and 6.55 ppm, respectively.
C.
Comparison With Steady State Model
As shown by Figures 9, 10, and 11, the resulting con-
ditions in the stream for the three months were generally
better than those projected with the steady state model. This
is interesting because, individually, each input value meets
the steady state input assumption at certain times during
these periods. However, the coincidence of low flow, high
temperatures, and little sunlight did not occur for a long
enough period of time to cause the kinds of stream conditions
indicated by the steady state analysis. For example, the
average DO values for July and August 1964 with only primary
treatment were at a generally acceptable 5.85 ppm and 6.28 ppm
respectively.
In order to make a comparison between the steady and
unsteady state models, synthetic sequential input data whose
averages would conform to the steady state design values were
developed. August 1965 was used as a start for developing the
synthetic data. The averages conforming to the steady state
design values were 650 mgd for discharge, 250 C for temperature,
and 200 Langleys per day for sunlight. The latter value was
chosen to represent the average cloudy day- The steady state
study was based on these values which were estimated by previous
investigators to occur for seven days once every ten years.
For the month of August 1965, the worst seven days occurring
naturally appeared to be the 21st to the 27th, inclusive (see
Figure 12). In constructing the first set of synthetic data,
the input data for the entire month were multiplied by co-
efficients so that the averages for this seven day period con-
formed to the input values for the steady state model. It is
interesting to note, however, that the lowest values for DO
occurred during a high temperature period near the beginning
of the month, and not during the low flow period. The lowest
values of DO for the three forms of treatment (primary, optimum,
and secondary) for the month were 2.34 ppm, 4.79 ppm, and 6.48
ppm respectively: for the "all secondary treatment" assumption,
the lowest DO for the unsteady state model was virtually the
same as that for the steady state model, but the lowest DO's for
the "all primary treatment" and the "optimum treatment" assump-
tions were about 1.4 ppm less and 0.7 ppm less, respectively,
than the steady state results.
-------
. !--! .-I~ 'c-I ;1=1-1 ~==-;! ...f lc:'.: -II :~'- -I~ _I ~
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500 o~' !' 1----. : ... I' j , 90
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s:: r i' . I ..-- ... -" _n' T L... i " "" 70
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~
Special imput with 7 day 10 year return period
Treatment plant Configuration
All Primary - - --
Optimum
All Secondary --------
'P
1500 1 2
7
8
9 10 11 12 13
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conform1ng tc ~teady state des1gn-- Chart A
72
14
15
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25 I
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18
19
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21
22
23
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31
28
29
24
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MONTH OF
Figure 12
19
'\
-------
73
It is important to note that while with the steady state
model it was sufficient to consider merely the DO deficit, the
unsteady state model operations make it necessary also to con-
sider the relationships between dissolved oxygen, the DO
saturation value, and the DO deficit as they vary with water
temperature. At 250 C, the steady state design temperature, a
DO deficit of 3 ppm is equal to a DO of 5.38 ppm. At 280 C,
however, the same deficit equals a DO of 4.92 ppm. Therefore,
a true comparison of the results requires the inclusion of
water temperature.
Another approach utilized for comparing steady and un-
steady state models was to adjust the average input values for
the entire month to the steady state design values. As shown
in Figure 13, the average values of DO for the various con-
figurations of treatment were approximately equal to the steady
state values. On this figure, the coliform output has not been
included and DO results have been shown to larger scale to give
better representation of the variation in these values. For
primary treatment, the DO range was 1.82 ppm to 4.42 ppm. With
optimum treatment, this range was 4.48 ppm to 5.79 ppm, and,
finally, for all secondary treatment, the range was 6.28 ppm
to 7.30 ppm. This gives ranges between extremes for the three
forms of treatment (primary, optimum, and secondary) of 2.60
ppm, 1.31 ppm, and 1.02 ppm, respectively. It would appear
from these studies that a greater range of river conditions
over a period of time may result when higher BOD loads are
applied to the stream.
The lower bounds of the ranges were 1.93 ppm less than
the steady state value for primary treatment, 0.96 ppm less for
optimum treatment, and 0.27 less for secondary treatment. This
again appears to demonstrate that worse values than the steady
state value can result at times, and that the susceptibility
and degree of such differences increases with higher BOD loads.
D.
Sensitivity Analysis
The method used to test the sensitivity of the various
model parameters in the unsteady state model was similar to
that used with the steady state model (see Chapter III). Each
of the river parameters recommended for design was multiplied
by coefficients to individually vary them within their possible
ranges of values. For this study, the analysis of temperature
required that a given number of degrees were added or subtracted.
The ranges are similar to those for the steady state analysis
but, due to the nature of the way input data is presented to
the program, it was not possible to make them identical in all
cases.
-------
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Treatment Plant Configuration
All Primary -... --
Optimum
All Secondary- - - - - ---
74
l50n 1
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Special i.put with averages conforming to steady
state design Chart B
5
I ..... -_u
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MONTH OF
Figure 13
19
,~
-------
75
This sensitivity analysis was performed on the para-
meters which were assumed to vary with time. These included
the deoxygenation and reaeration constants, the amount of
algae activity and the water temperature. In each case, the
values of the parameter being studied were altered while the
values of the other parameters remained the same.
Figures 14 and 15 show the results of the sensitivity
analysis of the deoxygenation (kl) and reaeration (k2) constants.
The most apparent indication of this analysis is that with
lower levels of treatment (allowing greater BOD loads to enter
the stream). the DO is prone to much greater variation. Thus,
design values chosen for these stream parameters are very
critical. Another way of looking at this is that if the
planner hopes to use the stream's assimilative capacity to
adjust for lower forms of treatment, it is likely that even if
the average values of DO are acceptable, worse conditions will
probably occur for a portion of the time.
Although the concept has not been thoroughly investi-
gated, it is suggested that the analysis may indicate some merit
in considering BOD concentration as a criterion for receiving
waters. In the above case, the BOD values for the critical
river station during the study period for three levels of treat-
ment (primary, optimum, and secondary) averaged approximately
16 ppm, 10 ppm and 4 ppm respectively.
In the sensitivity analysis, a range of values of the
parameter ~ implies that the amount of algae in the stream would
either increase or decrease, while the ratio of photosynthesis
to respiration would remain the same. In other words, this
method assumes an increase in the numbers of algae while their
individual synthesis rates remain unchanged. Due to the nature
of the oxygen sag equation and because for this study algae
concentrations were assumed to be independent of treatment
plant levels, different algae concentrations would produce
about the same effect on the stream regardless of treatment
levels.
Figure 16 demonstrates the effect of different algae
concentrations on the study stream. It is reiterated that
the methodology assumes supersaturation cannot occur, so that
excess oxygen vents to the atmosphere. This is the reason
Why, for this stream, which obtains a net gain in dissolved oxy-
gen from algae, the DO values for different forms of treatment
at high algae concentrations is not greatly different. It is
pointed out that the variations caused by different algae
concentrations may be quite different for another stream. Using
-------
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Sensitivity Analysis
MONTH OF AUGUST
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0.0 = ---
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SENSITIVITY ANALYSIS
MONTH OF AUGUST
19 65
Figure 16
-------
79
relationships for this stream previously developed between
sunlight intensity and the action of algae (see Chapter IV)
and the photosynthesis-respiration ratios reported in the
literature, it was found that many times the normal algae
concentration would have to occur in this stream before night-
time respiration would become a problem in reducing DO levels.
Figure 17 shows the effect of varying the temperature
of the stream on the resultant DO at the critical station.
The procedure was to add or subtract a given number of degrees
to each daily temperature for the entire study period. For
the ranges studied, the changes in temperature produced about
the same average change in DO for the different levels of
treatment. A 30 C change in temperature for the stream pro-
duced a fairly consistent change in DO of nearly 1 ppm at the
critical station. This fact stresses the need for accurate
data on temperature profiles for a stream before an effective
treatment plant system can be designed.
Effect of Statistical Variations
E.
Although the time varying model described herein pro-
duces continuous estimates of river conditions which demon-
strate the true nature of their variations, it is not claimed
that they are the exact values of DO or other quality character-
istics which would occur. It is recognized that this model is
deterministic and, therefore, does not consider either the
random variations in the input or the random variations in
local environmental conditions.
Thomann and Sobel (18) state that a good forecasting
scheme must contain an adequate representation of output which
may be caused by combinations of periodic, transient, and
random phenomena. O'Connor and Thomann (21) recognize in
their work with estuaries, that the various processes are
composed of largely deterministic components on which are
superimposed random elements. Thomann (2) made estimates of
the random variations of dissolved oxygen in the Delaware
estuary and determined a standard deviation of 0.85 ppm.
Thayer and Krutchkoff (55) determined a variation of about 0.3
ppm above and below predicted DO's, which were found in
laboratory analyses. For tests with values on the Sacramento
River, they determined a range of about 0.5 ppm between the
mean and the 10% lower confidence limit. Kothandaraman and
Ewing (56) made sensitivity analyses using a steady state model.
They determined the probability distributions for the model
rate parameters kl and k2' With these results, a band of
values for DO was created having limits up to 1 ppm above and
below the most probable values.
-------
80
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81
81
e
0-
0-
I
o
r:::I
s
e
0-
0-
I
o
r:::I
WATER TEMPERATURE
+3 degrees =-.-
Normal temperatures =-
-3 degrees = -.-.-
Sensitivity Analysis
MONTH OFAUGUST
19 65
Figure 17
-------
81
In light of the above, it would appear reasonable to
expect that the results for DO obtained for the time varying
model represent the most probable values, while the actual
values of DO would range above and below these values. A
tentative judgment is that the differences between probable
and actual values would be less than 1 ppm for at least 90
percent of the time. Additional studies comparing predicted
and actual values would be needed to establish a firmer
relationship.
An extension of the time varying model could super-
impose additional statistical fluctuations on the computed
values. These fluctuations could be estimated by generating
stochastic input, or by adding random components to the
output. The use of a Monte Carlo procedure could be one
approach to generating these values randomly.
-------
82
CHAPl'ER VI
CURRENT SAMPLING PRACTICES BY NEW ENGIAND
REGULATORY AGENCIES
A.
Introduction
When this project was originally conceived it was
planned to collect all available information pertaining to
the data collection and processing programs of the water
pollution control agencies of the New England states. It was
thought that this information could be used to make a compre-
hensive study to establish a scientific basis for judging the
economic efficiency of data collection programs. It was
planned to determine the importance of various field and
laboratory data in making studies leading to: 1) selection of
required performance levels for waste treatment plants; 2) se-
lection of objective qualities for watercourses; and 3) ad-
ministrative regulations for reports of participants in water
resources operational programs. An ultimate objective for this
work was to be able to indicate 1) for portions of such programs
where field and laboratory techniques are well developed, the
extent of sampling needed for useful results and 2) for portions
of such programs where field and laboratory techniques are not
well developed, the relative importance that ought to be placed
on researching and developing such techniques.
B.
Current Sampling Practices
The discussions with the pollution control agencies
were quite disappointing in that they did not produce the
amount and/or kinds of information necessary to make the
planned analyses. Generally, all of the New England states have
data collection programs, but most of them have not been in
existence long enough to adequately study their performance.
These programs have been set up more or less intuitively with
sampling stations at the "trouble spots." These locations
have usually been specified by complaints, which would indicate
that they would tend to be in the most populated areas. The
frequency of sampling is quite variable and is often limited to
only a few times a year because of insufficient personnel.
Depending on their available laboratory facilities, these
samples may be either analyzed by the agency or contracted to
a private laboratory. The use of automatic samplers by the
states is quite new and in most cases are not being used or
are just used experimentally.
-------
83
It should be pointed out that several government and
private agencies have conducted special surveys which have
been quite extensive. Some rivers where such studies have
been made include: the Merrimack in New Hampshire and
Massachusetts, the Penobscot in Maine, and the Hudson in New
York. In these cases a great deal of data has been obtained
and, in the case of the Merrimack, it has been possible to
make a comparison of two different studies in this report.
The only NEIWPCC member state which currently appears
to have a program directed toward an optimum sampling program
is New York (57). New York currently has established a net-
work of more than 100 stations which were selected as being
representative of the general quality of the water in a given
basin or as being indicative of the effect of upstream dis-
charges on water quality. A total of 33 different parameters
are evaluated to demonstrate the condition of a receiving body
of water. The cost for a single sample is from $150 to $200--
which includes the transportation from the sampling site to
the lab. Several automatic samplers have been installed and
seem to provide a worthwhile supplement for the periodic
sampling data. For the manual program, a sampling frequency
of once per month has been adopted. This frequency may be
changed if conditions demonstrate that more or less frequent
sampling is necessary. As with the other states, New York's
data collection program has not been operated long enough for
an evaluation of its performance by the authors.
For the New York program, much use is made of data pro-
cessing systems. All water quality data is placed on magnetic
tape and is therefore readily retrievable for processing by
computer to produce stream analyses, raw data listings, summary
listings, statistical evaluations, and reports. New York
has also been attempting to analyze this data from a statistical
standpoint (58). They have not as yet found any theoretical
statistical distribution which is applicable. They have, how-
ever, been studying the applicability of control charts to
administer water quality control programs. Although these
studies have some promise, the results are still inconclusive
and more data will be needed before a complete evaluation can
be made.
In view of the above, the work for this report has con-
centrated on methodologies for utilizing data rather than a
quantitative evaluation of the cost and efficiency of data
collection and processing. The models discussed in the previous
chapters have been used to analyze the effect of assuming various
values of stream purification parameters and other factors on
the cost and effectiveness of water pollution control.
-------
84
ACKN~LEDGEMENTS
The following individuals have furnished data and advice
concerning the research and their assistance is gratefully
acknowledged:
Mr. William Albert, Vermont Water Resources Board.
Mr. David L. Clough, Chief, Water Quality Section, Vermont
Dept. of Water Resources.
Mr. Kenneth I. Darmer, Chief, Hydraulic Studies Section,
United States Department of the Interior, Geological
Survey. Albany, N.Y.
Mr. Terrence P. Frost, Chief Aquatic Biologist, New
Hampshire Water Supply and Pollution Control Commission.
Mr. James F. Gleason, Senior Biostatistician, Office of
Biostatistics, New York State Department of Health.
Mr. Irving Grossman, Associate Sanitary Engineer, Division
of Pure Waters, New York State Department of Health.
Mr. Merwin E. Hupfer, Principal Sanitary Engineer,
Connecticut Water Resources Commission.
Mr. Pearce M. Klazer, Principal Sanitary Engineer, Division
of Water Pollution Control, Rhode Island Department of
Heal th .
Mr. Henry Mann, Maine Water Improvement Commission.
Mr. Ronald E. Maylath, Associate Sanitary Engineer,
Division of Pure Waters, New York State Department of
Heal th.
Mr. Thomas C. McMahon, Director, Division of Water Pollu-
tion Control, Massachusetts Water Resources Commission.
Mr. Herbert R. pahren, Director, Merrimack River Project,
Federal Water Pollution Control Administration, Boston,
Massachusetts.
Mr. Alfred E. Peloquin, Executive Secretary. New England
Interstate Water Pollution Control Commission.
-------
10.
85
BIBLIOGRAPHY
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34.
35.
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43.
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88
Lowell Water Treatment Plant., Lowell, Massachusetts,
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u. S. Department of Commerce, Weather Bureau, "Clima-
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1963-1968.
U. S. Department of the Interior, Geological Survey.
"\'later Resources Data for Mass.. New Hampshire, Rhode
Island, and Vermont," Boston, Mass., 1964-1968.
O'Connor, Donald J., "The Effect of Stream Flow on Waste
Assimilation Capacity," Proceedinqs Seventeenth Purdue
Industrial Waste Conference, May 1962.
Zanoni, A. E.. "Effluent Deoxygenation at Different
Temperatures," Marquette univ., Milwaukee, Wisconsin,
Dept. of Civil Engineering, August 31, 1967.
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Temperatures." Water Research, Pergamon Press, Vol. 1,
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the Sanitary Engineering Division, ASCE, Vol. 88, No. SA4,
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-------
57.
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90
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70.
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united States Department of the Interior, Geological
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which references No. 36 are compiled.
-------
92
APPENDIX I
a
INPUT - OUTPUT DATA DESCRIPTIONS
Flow Characteristics and Water Quality at River Station I
Q(I)
T(I)
E(I)
River flow in MGD
Time in days
BOD in ppm
F(I) Oxygen deficit in ppm
DO(D)
H(I)
V(I)
WI(I)
WR(I)
Dissolved oxygen in ppm
Coliform count in MPN/lOO ml
Chlorides in ppm
Water quality index for water supply
Water quality index for recreation
Flow Characteristics and Water Quality at End of Time Interval (Not Printed)
TZ(N) = Time in days
EZ(N) = BOD in ppm
FZ(N) = Oxygen deficit ppm
DOZ(N) = Dissolved oxygen in ppm
"z" Data (Hydraulic and Stream Purification Constants Associated with
River Location)
Zl(I), Z2(I) Constants for velocity expression, in which velocity equals
-------
Z2
Zl Q ;
Z3 (I)
Z4(1)
ZS (I)
Z6(1)
Z7 (I)
Z8 (I)
Z9(I)
93
Appendix I
-- Continued
velocity is miles per day and Q is MeD
Deoxygenation constant in ppm/day
Reaeration constant in ppm/day
Rate constant for settling out of BOD to bottom deposits in ppm/day
Rate constant for addition of BOD to overlying water from bottom
deposits in ppm/day
Constant to alter oxygen production by algae in ppm/day
Constant defining the die-off rate of coliform bacteria in
fraction of coliforms per day
River station in miles
"cz" Data
.(e:}!mperatuY"e Coefficients to Vary "z" Data with Temperature)
CZ3(MT)
CZ4(MT)
OSAT(MT)
Deoxygenation coefficient
Reaeration coefficient
Oxygen saturation value
"ALA D~.tc~\O (Data Varying with Solar Radiation)
Constant for rate or production of dissolved oxygen by algae
ALA(NSR.NTN)
through photosynthesis in ppm/day NSR = Daily solar radiation
(Langleys) value /10
NTN = time period during day.
"NT" Data
(Input Data Varying Daily)
SR(NT)
TH(NT)
RQ(NT)
Total solar radiation in Langleys
Average river water temperature degrees Centigrade
Average daily flow at upstream station 1 . 1
-------
94
Appendix I--Continued
"zz" Data (Significant Internally Computed Data - Not Printed)
ZZ3(N) Value of Z3 as modified by CZ3
ZZ4(N) Value of Z4 as modified by CZ4
ZZ7(N) Value of ALA(NSR, NTN) as modified by Z7
CQ(NTC)
Constant for modification of Q as flow moves downstream to
compensate for varying Q
"G" Data (physical and population data for communities)
Maximum feasible off-river water supply in MGD
Gl(J)
G2(J)
Community area in square miles (or drainage area in sq. mi.)
G3(J)
G4(J)
Ratio of average infiltration to 1000 gal. per acre per day
Ratio of infiltration to average infiltration for period of study
G5(J)
G6(J)
DO for saturation of raw sewage discharge
Chlorides in off-river water supply in ppm
G14(J) Residential population
G15(J) Employees of commercial enterprises
G16(J) Employees of fabrication industries
G17(J) Employees of wet-process industries
"y" Data (economic data for communities)
Yl(J) Ratio of
per day
Y2(J) Ratio of
per day
residential water supply demand to 90 gal. per capita
commercial water supply demand to 250 gal. per employee
-------
95
A'pendix 1 -- Continued
'ty.. Data -- Continued
Y3(J) Ratio of fabrication industry water demand to 250 gal. per
employee per day
Y4(J) Ratio of wet-process industry water demand to 900 gal. per
employee per day
Y5 (J) Ratio of public water supply demand to 25 gal. per capita per
day
Y6(J). Y7(J)
Maximum weekly municipal water supply demand equals
Y QY7 where Q is average water supply demand
6
Ratio of residential average sewage discharge to 75 gal. per
Y8(J)
capita per day
Y9(J)
Ratio of commercial average sewage discharge to 225 gal. per
employee per day
Y10(J)
Ratio of fabrication industry average sewage discharge to 235 gal.
per employee per day
Yll (J)
Ratio of wet-process industry average sewage discharge to 800 gal.
per day.
Y12 (J)
Ratio of public average sewage discharge to 20 gal. per capita per
day
Y13(J)
Y 14 (J)
Maximum daily municipal sewage discharge (without infiltra-
tion equals Y13QY14 where Q is average discharge without
infi ltration
Y15(J)
Rltio of expected BOD to 0.2 pounds per resident per day for all
sewage except wet-process industrial waste.
-------
96
Appendix I--Continued
Y16(J)
Ratio of expected BOD to 0.0025 pounds per gallon of wet-
process industrial waste
Y17(J)
Ratio of expected coliforms in raw sewage to 80 billion MPN
per capita per day
Y18(J) Ratio of expected coliforms in raw sewage to 0.07 pounds per
capita per day
Y19(J) Factor to adjust efficiency of BOD removal for operation of
sewage treatment plant
Y20(J) Factor to adjust efficiency of DO additions for operation of
sewage treatment plant
Y2l(J) Factor to adjust efficiency of coliform removals for operation
of sewage treament plant
Y42(J), Y43(J). Y44(J), Y45(J) Constants used to define annual cost
y43
on-river water supply (Qh in MGD), where CH = Y42~
(in $) of
Y 0 Y45
44 "H
+
The value of Cu may be the estimated actual value
or a relative value suitable for making a comparison with an
off-river water supply.
Y46(J), Y47(J), Y48(J), Y49(J)
Constants used to define annual cost
of off-river water
Y49
Y48QB
Y47
supplies (Q in MGD). Where: C = Y Q +
B B 46 B
of C may be computed to reflect credits or
B
penalities for intangible values entering into judgment of off-
The value
river supplies.
"A" Data (administrative assumptions for cOlll11unities)
Ai (J)
Defines the extent to which the average municipal water supply
demand (QO) will be met by on-river supply (QH) and off-river
-------
97
Aopendix 1-- Continued'
"A" Data (administrative assumptions for communities)
Supplies (Q ). as follows:
B
If
o
A
1
1.0.
Q = A Q
B 1 0
If A = 2.0, split between Q and Q is determined by
1 B H
analysis using standard subroutine.
A2(J)
Define whether there is to be a municipal sewage treatment
plant discharging to the river, as follows:
If A
2
= 0, Q
S
= 0 and Q
D
= 0
If A = 1.0, Q and Q are defined by standard sub-
2 S D
routines
A3(J)
Defines the degree of sewage treatment; A equals the fraction
3
of BOD to be removed, and this in turn fixes the quality of
the effluent in terms of BOD, DO deficit and coliforms.
"KD" Input Data
Table defining coefficients in equations evaluating treatment plant ef-
fic hnc ies
Miscellaneous Input Data (for computer processing)
NSTA Total number of river stations
NCOM Total number of communities
NOPT
Two digit number defining path for computer processing
NDAY
Total n8mber of consecutive study days
ZN
Exponent controlling pattern of flow addit'ons and subtractions
in river routing procedure
-------
Appendix I--Concluded
98
TS - Time last flow reaches downstream station, before start of study
period.
a
Using notation for oomputer statements in Fortran, in which I
is river station number and J is community number;
N ~ number of time interval, NT = number of the study day, MT =
day temperature, NTC = number of time interval when flow passes
upstream station
study
most
-------
99
APPENDIX II
SUMMARY OF EQUATIONS AND PROCEDURES FOR FUNCTIONS
FUNCTION -1-- TIME AT RIVER STATION (1) -- DAYS, Q AT RIVER STATION (1) --
HeD
Qi - Qk + (Mk - Hi) CQn (Hi I LN) ZN
Ti - Tk + (~ - Hi) I 81 ~Qk + Qi) I ~ 82
K is fir8t station upstream of 1
Input data: 81' 82' ZN, LN, Mk (same a8 89 at Sta. K), Hi (888e as
89 at Sta. 1)
Determined previously: Tko Qk' CQn
Output: Ti' Qi
FUNCTIONS - 2A; 2B; 2C; 2D - BOD AND 00 AT RIVER STATION (1) OR (N) - PPM
Ei . ~ - (86 I 2.3) I (883 + 85~X 10-(883 + 85) (Ti - Tk) +
(86 I 2.3) I (883 + 85)
Fk - OSAT Mt - DOk
F i . [Z83 I (884 - 883 - 85~ [Ek - (86 I 2.3) I (883 + 85~
X~0-(883 + 85) (Ti - Tk) -10 -884 (Ti - T~
+ (883 I 84) ~86 I 2.3) I (883 + 85) - 887 I 2.3 ~
X ~ - 10 -884 (Ti - TJ
- Fk X 10 -884 (Ti - Tk)
OOi = OSAT - F
Mt i
-------
Appendix II--Conrinued
K i8 first station upstream of I
Input data: OSAT Mt' zs' z6
Determined previously: Ek' DOk' Tk' Ti' zZ3'
Output: E , F , DO
i i i
100
ZZ , zz
4 7
FUNCTION -S- DEOXYGENATION REAERATION AND PHOTOSYNTHETIC CONS~S
AT RIVER STATION I OR N
t-TM
Nt
ZZ3 .. z3 X CZ3t
ZZ ..
4
Z X CZ4
4 t
Sr = Sr
Nt
/ 10
Td .. T
k
- Nt
ZZ
7
.. AlA X Z
Sr,Td 7
K is first station upstream of I
Nt is day number
Fixed Input data: Z , Z , Z
3 4 7
Daily input data: SR, TM
Input data from matrices: CZ3' CZ4' ALA
Determined previously:
T , Nt
k
Output: zz , zz , zz
347
-------
101
A~pendix II--Continued
FUNCTION 6 -- COLIFORMS AT RIVER STATION (I) -- MPN/100 ML
H Q = N
c c c
for inflow at Blost upstre881 "control" section
HIQl .. Nl
~ Qk = Nk
for first sewage discharge Ql below control section
for last sewage discharge Qk before reaching Station I
n = N
c c
x 10 - Zs (Ti - Tc)
nl = Nl x 10 (Ti - Tl), if (Ti - Tl) ~ 0.5
nl = Nl x 5 x lO-zS (Ti - Tl - 0.5), if (Ti - Tl) > 0.5
£!.
n " .
2
. ~ are each computed as for nl
Then:
Hi = (nc + nl + . . . . . nk) / Qi
Input data: ZS' H , Q , T
c c c
Determined previously: Hl. . . .~; Ql. . . .Qi; Tl. . .
.Ti
Output: H
i
FUNCTION 11 -- QUALITY OF MIXTURE OF TWO FLOWS AT RIVER STATION (1) --
BOD (PPM); DO DEFICIT (PPM), COLIFORHS (MPN/IOO ML).
CHLORIDES (PPM)
Ei .. (EkQk + EjQj) / (Qk + Qj)
Fi = (FkQk + FjQj) / (Qk + Qj)
Hi - (~Qk + HjQj) / (Qk + Qj)
Vi .. (VkQk + VjQj) / (~ + Qj)
Qk is river flow just upstream of point of sewage discharge Qj
Qi is river flow just downstream of Qj (and Qi .. Qk + Qj)
Input data:
None
Determined previously:
Qk' Ek' Fk' Hk' Vk;Qj' Ej' Fj' Hj" Vj
Output: Ei' Fi' Hi' Vi
-------
102
Appendix II--Continued
FUNCTION 12 -- WATER QUALITY RATING AT RIVER STATION (1) FOR MUNICIPAL
WATER SUPPLY -- INDEX NUMBER
Wli is given a base value of 0.00
0.75 is credited for meeting ~ of the following criteria:
Ei ~ 4
F i ~ 4
Hi ~ 5000
V .(. 250
i -
Maximum value for WIi is 3.00
Input data:
None
Determined previously: Ei' Fi' Hi' Vi
Output: Wli
FUNCTION 13 -- WATER QUALITY RATING AT RIVER STATION (I) FOR RECREATION --
INDEX NUMBER
WR is given a base value of 0.00
i
1.00 is credited for meeting each of following!!!! of criteria:
E i ~ 50, F i ~ 7, and Hi ~ 20,000
E i ~ 15, F i ~ 4, and Hi <. 2,000
Ei ~ 10, Fi ~ 3, and Hi ~ 1,000
Maximum value for WRi is 3.00
Input data: None
Deterained previously: Ei' Fi' Hi
Output:
WRi
-------
103
Arypendix Il--Continued
FUNCTION 15-- AVERAGE MUNICIPAL WATER SUPPLY DEMAND OF COMMUNITY (J) --
MGD
QO e (90 n Y + 250 n Y + 250 n Y + 900 n Y + 25 n Y) I
j 11 22 33 44 15
1,000,000
n (same as g for J). n
1 14 2
Y 'Y2' Y , Y , Y5
1 3 4
Determined previously: None
Input data:
(815)' n3(816)' n4 (817)
Output: QO.
J
FUNCTION 16 -- AVERAGE OFF-RIVER MUNICIPAL WATER SUPPLY OF COMMUNITY (J) --
HGD
If a .. Q, QB. - 0
1 J
If a1 :> 0 and \ ~ 1.0, QBj .. \ QOj
Ifa
1
.. 2, procedure follows:
QB .. 0.05 pQO
j j
QH = QO - QB
j j j
QH y43 Y45
CH = Y j + Y44 QHj
j 42
CB .. Y QB Y47 + Y QB Y49
j 46 j 44 j
CO .. CH + CB
j j j
QB is determined
j
0, I,. . . . 20 and for QB ~
j -
for minimum CO
j
8
1
by comparing values for p ..
Input data: a , Y , Y , Y , Y Y Y Y, Y J g
1 42 43 44 45' 46' 47' 48 49 1
-------
104
Appendix II--Continued
FUNCTION 19 -- Continued
Input data: g , g , g ; Y . Y
2 3 4 13 14
Determined previously QS
j
Output: QD
j
FUNCTION 26 -- QUALITY OF RAW SEWAGE OF COMMUNITY (J) -- BOD (PPM1,
DO DEFICIT (PPM). COLIFORMS (MPN/I00 ML), CHLORIDES (PPM)
If QS. = 0, dummy values are as follows:
J
ES = O. FS = O. HS = O. VS = 0
j j j j
If QS. > 0, equations follow:
J
ES = (0.2 nl Y + 2.0 n4 Y Y ) I 8.34 QS.
j 15 11 16 J
FS = g
j 5
HS = 2100 nl Y I QS
j 17 j
VS = (0.5 V X QH + g QB) I (QH + QB ) + 0.0085
j ij j 6 j j j
nl Y 18 I QS j
Input data: n (same as g 4 for J), n (g ) g g. Y Y
1 1 4 17; 5' 6' 11' 15'
Y , Y , Y
16 17 18
Determined previously: QS., V ,
J ij
Output: ES, FS , HS , VS
j j j j
QH , QB.
j J
-------
105
Appendix II--Continued
FUNCTION 16 -- Continued
Determined previously: QO
j
Output: QB
j
FUNCTION 17 -- MAXIMUM WEEny ON-RIVER MUNICIPAL WATER SUPPLY OF COMMUNITY
(J) -- MGD
QI = (QH. I QO.)
j J J
(Y QO Y7)
6 j
Input data: Y6' Y7
Determined previously: QO , QH
j j
Output: QI
j
FUNCTION IS -- AVERAGE SEWAGE DISCHARGE OF COMMUNITY (J) -- MGD
If a = 0, QS = 0
2 j
If a2 = I, equation follows:
QSj - (75 n Y + 225 n Y + 235 n Y + SOO n4 Y + 20 nl Y )
1 S 2 9 3 10 11 12
+ 0.64 8283
Input data: nl (same as 814 for J), n2 (815)' n3 (816) n4 (817);
I 1,000,000
YS' Y9' Y10' Yll' Y12; 82' 83'; az
Determined previously: None
Output: QS
j
FUNCTION 19 -- MAXIMUM DAILY SEWAGE DISCHARGE OF COMMUNITY (J) -- MGD
If a = 0, QD = 0
2 j
If a2 - 1, equation follows:
QD = Y (QS - 0.64 8 83)Y14 + 0.64 8
j 13 j 2 2 &3 84
-------
106
Appendix II--Concluded
FUNCTION 27 --QUALITY OF TREATED SEWAGE OF COMMUNITY (J) -- BOD (PPM) ,
DO DEFICIT (PPM). COLIFORMS (MPN/100 ML). CHLORIDES (PPM)
If QD = 0, dummy values are as follows:
j
ED = Ot FD = 0t HD = 0t VD = 0
j j j j
If QD ") 0t equations follow:
j
ED = (1.0 - d ) Y ESj
j 1 19
FDj = d2 &5 Y20
HDj = 5,000 d3 Y2l
VD = (V X QH + g QB) I (QH + QB ) + 0.0085 n
j ij j 6 j j j 1
Y / QS
18 j
In the above equations, dlt d , and d are obtained from a fixed
2 3
input table and depend upon the value of a (given in
3
the input data or generated by special routine)
Input data: nl (same a8 g14 for J); &5' g6; Y18' Y19' Y20' Y2l; a3
Determined previously:
QS V QH, QB , ES
j' ij' j j j
Output:
ED,ro,HD,VD
j j j j
-------
APPENDIX III COMPUTER PROGRAM STATEMENTS CORRESPONDING TO FUNCTIONS
Function Description
Arguments of Call Statement
Title of Subprogram with
Instructions for EvaluAtIon
Call Statement in Main
Program or Subprogram
Function I - Time at river sta.
(I)-days; at river sta. (I) CFS
Function 2A - BOD and DO in ppm
at river sta. when no river
stations have passed since com-
pletion of last time interval
Function 2B - BOD and DO in ppm
at end of time interval when no
time intervals have elapsed since
last river station
Function 2C - BOD and DO in ppm
at end of time interval when two
or more time intervals will elapse
between river stations
Function 2D - BOD and DO in ppm at
river sta. when two or more river
stations will be passed betwwen
time intervals
CALL FIT (I)
CALL FIEF (I,N)
CALL F2EF
CALL F3EF (I,N)
CALL F4EF (I,N)
I
I = river station number
SUBROUTINE FIT (I)
I = river station number
N = time interval number
SUBROUTINE FIEF (I,N)
I = river station number
N = time interval number
SUBROUTINE F2EF (I,N)
I = river station number
N = time interval number
SUBROUTINE F3EF (I,N)
I - river station number
N = time interval number
SUBROUTINE F4EF (I,N)
~
o
~
-------
Appendix III - Continued
Function Description
Title of Subprogram with
Instructions for Evaluation
Call Statement in Main
Program or Subprogram
Arguments of Call Statement
Function 5 - Time changing of
river parameters; reaeration
constant, deoxygenation constant
and constant for oxygen pro-
duction by algae
Function 6 - Coliforms at River
Station (I) - MPN/100 ml.
Function 11 - Quality of mixture of
two flows at River station (I) -
BOD (ppm), Do deficit (ppm),
Coliforms (MPN/100 ml.), Chloirdes
(ppm)
Function 12 - Water Quality Rating
at river station (I) for municipal
water supply (index ranging from 0
to 3)
Function 13 - Water quality rating
at river station (I) for recreation
(index ranging from 0 to 3)
CALL F 5Al (I, N)
CALL F6H (I,J,JST 6)
CALL FUP (I,J)
CALL F12WI (I)
CALL F13WR (I)
I = river station number
N = time interval number
I = river station number
J = community number for
first sewage discharge
upstream of Station I.
JST6 - river station
number for first sewage
discharge upstream of
station I
I = river station number
J = community number for
sewage discharge mixing with
river flow at station I
I = river station number
I - river station number
SUBROUTINE F5Al (I,N)
SUBROUTINE F6H (I,J, JST6)
SUBROUTINE FllP (I,J)
SUBROUTINE F12WI (I)
SUBROUTINB F13WR (I)
....
o
m
-------
Appendix III - Continued
Function Description
Title of Subprogram with
Instructions for Evaluation
Call Statement in Main
Program or Subprogram
Arguments of Call Statement
Function 15 - Average munici-
pal water supply demand of
community (J) - mgd
Function 16 - Average off-
river municipal water supply
of community (J) - mgd
Function 17 - Maximum weekly
on-river municipal water supply
of community (J) - mgd
Function 18 - Average sewage
discharge of community (J)~
mgd
Function 19 - Maximum daily
sewage discharge of community
(J) - mgd
Function 26 - Quality of raw
sewage of community (J) - BOD
(ppm). DO deficit (ppm). coli-
forms (MPN/IOO mI.), chlorides
(ppm)
CALL Fl5QO (J)
J = community number
CAU. F16QB (J)
J = community number
CALL F17QI (J)
J = community number
CAu. F18QS (J)
J = community number
CAU. F19QD (J)
J = community number
CAu. F26PS (J,I)
J = community number
I = river station number
of municipal water supply
intake of community J
SUBROUTINE F15QO (J)
SUBROUTINE F16QB (J)
SUBROUTINE F17QI (J)
SUBROUTINE Fl8QS (J)
SUBROUTINE F19QD (J)
SUBROUTINE F26PS (J.I)
....
o
\D
-------
Appendix III-Concluded
Function Description
Call Statement in Main
Program or Subprogram
Arguments of Call Statement
Title of Subprogram with
Instructions for Evaluation
SUBROUTINE F27PD (J,I)
Function 27 - Quality of treat-
ed sewage of community (J) -
BOD (ppm), DO deficit (ppm),
coli forms (MPN/100 ml.),
chlorides (ppm)
CALL F27PD (J,I)
J = community number
I = river station number
of municipal water supply
intake of community J
....
....
o
-------
111
APPENDIX IV
STREAMFLOW ROUTING FOR WATER POLLUTION STUDIES
Introduction
For the analysis of the flow regimen in a channel re-
ceiving discharges of pollutants, two concepts are important.
First, account must be taken of the change in concentration
of pollutants at each location where a pollutant enters the
stream. Second, the relationship between discharge and
velocity for any reach must be maintained. For example, the
change in BOD between two points on a stream, in which the
pollutional load is delivered at the upstream end, depends on
the travel time for the reach. For water pollution studies,
this is estimated by determining the modified discharge and
concentration at the point receiving the pollutant, estimating
the velocity for the reach, and computing the time from
velocity and distance traveled. In the natural case, the BOD
and other stream quality characteristics are continuously
changing with both time and distance.
As part of a larger study which utilized a time varying
model patterned on the Merrimack River in northeastern
Massachusetts, a special flow routing procedure has been de-
veloped to study these effects. Although the time varying
model also included other time varying parameters, the unsteady
flow condition present in natural streams appears to have been
the major problem in the development of practical models of
this type.
The engineering literature describes many different
procedures for solving flow-routing problems. The theoretical
analysis of the movement of flood waves is quite complex, and
methods based on a strict mathematical treatment are not
practical for routing through natural river channels. Methods
applicable to a river basin such as the ones studied for this
work rely on some simplification of the concepts in the theo-
retical approach. Most of these methods utilize a relationship
between stage and storage or between discharge and storage.
Procedures which assume an invariable discharge-storage
relationship are useful only for reservoir routing (59). Of
the methods using a variable discharge-storage relationship,
both the "Muskingum method" (60) and the "working value method"
(59) have been satisfactory for routing in a natural stream.
The more empirical ",lag method" (61) has also been found to
yield adequate approximations. These methods are discussed ex-
tensively in publications by the u.S. Army Corps of Engineers
-------
112
(62) and by Chow (59), and compu~er programs have been written
to facilitate the great number of computations which must be
made (63, 64).
Recently. several newer methods have been proposed
which recognize more of the parameters involved in the theo-
retical approach. Baltzer and Lai (69) use a set of non
linear, partial differential equations describing one
dimensional translatory wave motion. They have included such
effects as fluid friction, variable channel geometry, wind,
lateral inflow and outflow, Corio1is acceleration, and over
bank storage. Another method by Himme1b1au and Yates (66)
represents the excess flow for pulse-like stream releases.
Amein and Fang (67) have reviewed three methods in detail and
evaluated their utility for computing the unsteady flow in
artificial and natural channels. They note that the advent
of the digital computer has made it feasible to obtain
complete solutions of the equations by numerical methods.
They have also found that different methods, although pro-
ducing almost identical solutions, vary considerably with
respect to speed, reliability, simplicity, and convenience.
After completing the investigation of the existing
flow routing procedures, it was concluded that none of them
would give the discharge-ve10city-time results necessary to
study the effects of unsteady flow on the pollution of a
stream. It was, therefore, necessary to develop a new pro-
cedure which would be especially suited for pollution studies.
The continued use of the Merrimack River data created
several problems for the development of a routing procedure.
The stream is subject to reservoir regulation at many sections,
making short term correlations between gaging stations im-
practical. Also, although much information is available on
the routing of unsteady flows occurring during high flow periods,
a thorough search of literature revealed no satisfactory pro-
cedure for the routing of normal and low flows in a natural
stream.
It was decided to use the river data for the Merrimack
which was available and in the form obtained for many other
streams where pollution abatement is being studied. For the
Merrimack, the available low flow information comprised a
hydrograph upstream of the region studied and measurements of
the travel times of various steady discharges through the
study reach. The upstream hydrograph was available from the
USGS (36) and the time of travel studies were made in field
surveys by the FWPCA (1), in 1965, and Camp, Dresser and
MCKee (2), in 1963. This information was used as the basis
-------
113
for deriving a downstream flow configuration. After the
routing procedure for the Merrimack River was well developed,
similar stream flow information was obtained for the
Susquehanna River in New York and Pennsylvania (68, 69).
For the Susquehanna, a downstream hydrograPh unaffected by
reservoir regulation was also available; this additional in-
formation was used to confirm that the method is a valid
routing technique.
Description of River Routing Procedure
This routing procedure accepts existing hydrograph
data at a station, and estimates hydrograph data at any down-
stream location. In the next portion of t~e discussion, dis-
charges at the upstream and downstream ends of a reach will
be referred to as "inflows" and "outflows" respectively.
Because recorded daily inflows at the station where
hydrograph data are available often change radically between
two successive values, the routing procedure must take account
of their differing travel times. A key premise of the procedure
is that the travel time required for any inflow to reach a
downstream station may be related to a weighted average of the
travel times for succeeding inflows. This is a reasonable
assumption because the merging of successive flows is obvious.
The principle employed is similar to the "lag method" (61)
which makes approximations by time displacement of average
inflows.
The flow-velocity equation which was used for this pro-
cedure is as follows:
Z
V = Z 0 2
1
(1)
Where V is velocity in miles per day
o is discharge in mgd
Zl and Z2 are distance varying river constants
This equation, used by Goodman (3, 4) in his steady state
mathematical model of the Merrimack was demonstrated by Leopold
and Maddock (7) to have general applicability for many streams.
Between two successive stations on the river, the discharge and
velocity are assumed to be constant. This approximation is
reasonable for the Merrimack River model because the distance
between stations averages less than 1 mile in length, and a
new station is always located where any significant change in
flow characteristics can occur. The time of travel can be
computed from the following simple relationship:
-------
114
Z2
T . D/V = D/Zl ° .
(2)
where T is time in days
D is distance in miles
In order to estimate a change in discharge between
successive stations, an increase or decrease of discharge
i's computed using the following empirical formula:
Zn
Q2 = 01 t (Dl - D2) x COy x (D2/Ln)
(3)
where 01 - inflow at section 1, in mgd
02 . flow at section 2, the next station
downstream, in mgd
D2 = distance of river station 2, above
the farthest downstream station in
miles
Dl ~ distance of river station 1 above
the farthest downstream station in
miles
COy = dimensionless coefficient determined
by the flow routing procedure, which
may be either positive or negative
Ln = total length of river channel studied
in miles
Zn . "Fitting coefficient" which controls
the pattern of flow additions or sub-
tractions
The essentials of the routing procedure are as follows.
First, an "objective" travel time is found for an inflow. This
travel time is evaluated by equation (2) for some weighted
average of the succeeding inflows. In order for this inflow
or a portion thereof to reach a downstream station in the ob-
jective travel time, while maintaining the integrity of
equation (1), the discharge as it moves downstream changes by
amounts given by equation (3).
The incremental flow computed according to equation (3)
is added or subtracted to obtain an adjusted discharge. This
adjusted flow applies for the stretch to the next station, where
another adjustment in discharge is made. As noted previously,
equations (1) and (2) assume a steady discharge between stations.
It is recognized that more than one pattern of flow
additions or subtractions will give the same travel time. The
exponent "Zn," which has been designated as a "fitting co-
efficient," determines the distribution of these flow alter-
ations. As shown on Figure 1, by increasing the "fitting co-
-------
115
efficient" (Zn) a greater amount of the total change in dis-
charge takes place near the beginning'of the reach. With
lower values, the additions or subtractions are distributed
more evenly- Also, Figure 1 shows that less total change of
flow over a reach is required to meet the objective time
with higher values of Zn. The corresponding effects on dis-
charge over a period of time at a downstream station are
shown on Figure 2.
To ensure that the general shape of the inflow hydro-
graph is retained for downstream stations, maximum values for
compression or spreading of time intervals are specified. For
example, the normal time interval between successive inflows
is 0.167 days (4 hours), whereas the allowable intervals at
the farthest downstream station could range from 0.02 to 0.314
days. An inflow, when followed by a sequence of larger in-
flows, is "compressed" so that the corresponding outflow rate,
after routing, increases and is maintained for a shorter
interval. When an inflow is followed by a sequence of smaller
flows, a "spreading" of time intervals is effected.
Hydrograph data used for the initial studies were ob-
tained from USGS records (36), and were in terms of daily
values. Six flow routings are made per day (at 4 hour inter-
vals) in which each inflow is set equal to the daily value.
Also available were 2 hour discharge values (71): these could
be used to refine the routing procedure, but are not discussed
herein.
The method has a great deal of flexibility since, de-
pending on the particular flow conditions, different values
may be used for the parameters in the routine. For less vari-
ation of inflows and/or shorter reach of river, a weighted
average of the succeeding discharges may be computed from
perhaps the next two days of inflow values. other conditions
may require 5 or more days of inflow values in order to ade-
quately adjust for the downstream effects. The selection of
the "fitting coefficient" in equation (3) determines the
pattern by which the incremental flow changes are to be made.
And the specification of allowable time intervals can also be
modified.
After working with this procedure, one develops an
understanding of the combined effects of changes in the
parameters. These parameters can be employed advantageously
to obtain good agreement between a derived outflow hydrograph
and a recorded hydrograph when available. Because appropriate
outflow hydrograph data were not available for the initial
studies, parameters were selected to obtain an outflow hydro-
-------
147.
&
'45
..
m
..
.142
"
'"
.-
25
20
River Mile
15
Figure 1
116
--
6
5'"
>-
4D
a
31
.
2&
-
~
1
10
5
Effect of Fitting Coefficient on Routing
800
-a
m
&60
I
.
m
..
D500
.I:
X
.-
a
400
Zn:3
Zn=2
;;-,
I
I
I
I
I
I
I
r-
I
I
I
I
I
I
I
I
I r-
L, I
-. I
I I
I I
L.... I
I
Average
River Mile- 0
Daily Discharge for 18 Consecutive
Record lagged 1/2 day' for Zn =3
Days
300
Figure 2
Effect of Fitting Coefficient on Computed
Discharges at Downstream Station
-------
117
graph which seemed most reasonable. Figure 3 shows daily
inflows together with computed outflows for a 30 mile long
reach of the Merrimack. This figure also includes the out-
flows in terms of daily values, in order to compare inflows
and outflows on the same basis.
In the absence of measured outflows, an approximate
checking procedure was developed to continuously compare
cumulative inflow and outflow (see Figure 4). By adjusting the
weighted average of succeeding discharges, the mass balance can
be maintained for as long a study period as desired.
This work employed a C.D.C. 3300 computer at Northeastern
university. The computer time required ranges widely depending
upon the precision desired for the outflows. The computer
routine determines the values of COy in equation (3) by
successive increments. If the adjustments of COy are small,
the results will have improved accuracy but the computer time
will be correspondingly greater. Increments requiring about
6 minutes of computer time per month gave the best results for
developing good fit with an actual, downstream hydrograph.
Reasonable approximations, however, were obtained with larger
increments which required only 2 minutes of computer time.
The latter computations were found to be adequate to demonstrate
the effect of time varying discharges on the oxygen balance of
a stream. Based on the present charges of 200 to 300 dollars
per hour for use of a medium sized computer, it would cost about
5 to 15 dollars to process one month of data for use in a time
varying mathematical model.
Checking of Procedure
The data used for the validation of this procedure was
for the Susquehanna River in New York and Pennsylvania. In-
formation included a daily discharge hydrograph at Vestal,
New York (69) which was just upstream of a reach where a
recent study by the USGS (68) has related discharge and time
of travel. The magnitudes of the discharges and the corre-
sponding velocities were not greatly different from those ob-
tained for the Merrimack. The checking was made possible by
the availability of another hydrograph located at waverly,
Pennsylvania (69) which was just downstream of the reach for
the USGS study. The total length of this reach was just over
30 miles.
Comparisons of the hydrographs at Vestal and Waverly
indicated that in this reach the Susquehanna was very much
affected by ground water and local surface water inflows
-------
1200
1000
600
800
200
o
Inflow - m d
1200
80
60
40
200
Outflow - mgd
20
o
2
4
6
8
Com uted Outflow
10 12 1 16 18
August 1965
Figure 3
Comparison of Inflow and Outflow Hydrographs
-------
60
-
Ct
Ct
...
Ct 50
~
"
C
"
c
a
1ft
~
0 40
.I:
...
I
Ct
m
~
a 30
.I:
"
1ft
a
..
>
.-
-
a 2
-
~
E
~
u
10
119
70
Inflow
----- Outflow
8
12
12
16
16 20
20 24
Time - Days
24
28
28
32
F i gu re 4
Mass. Diagram of Inf10w and Outf:1_..
-------
120
during the low flow period. This caused some complications
in trying to obtain a good correlation between the computed
and observed hydrographs.
Figure 5 shows the upstream measured hydrograph at
Vestal and the corresponding measured downstream hydrograph
at Waverly. Superimposed on the Waverly hydrograph are the
instantaneous values determined by the routing procedure.
The instantaneous values computed for Waverly were derived
from the daily mean discharges measured at Vestal. Because
of the nonrecorded fluctuations about the daily averages at
the measured hydrographs, and because of the effects of ground
water and local surface inflow, one could not expect to
obtain a perfect fit, and the results are judged to be satis-
factory.
Conclusions
For water pollution studies the travel times of the
pollutants are quite important. Unlike available flood routing
methods, this low flow routing procedure emphasizes this fact.
Reasonable approximations of unsteady flow effects are obtained
by utilization of an inflow hydrograph and information on the
travel times of pollutants for varying discharges. If an out-
flow hydrograph is available, the method can be adapted to
provide reasonable correlation of computed and measured outflows.
Although the method is somewhat empirical, and is de-
rived from intuitive concepts rather than a rigorous analysis
of unsteady flow theory, it is suitable for incorporation in
a mathematical model used in the biological analysis of a
stream environment.
-------
Average Daily Inflow at Vestal, New York - cfs
20
Waverly, Pennsylvania - cfs
Figure 5
Comparison of Measured and Computed Hydrographs for Susquehanna River
,'\
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60-
-------
As the Nation's principal conservation agency, the Department of the
I nterior has basic responsi bi I ities for water, fish, wi Id life, m i nera I, land,
park, and recreational resources. Indian and Territorial affairs are other
major concerns of America's "Department of Natural Resources."
The Department works to assure the wisest choice in managing all our
resources so each will make its full contribution to a better United
States-now and in the future.
ORD.6
------- |