United States
Environmental Protection
Agency
Environmental Research
Laboratory
Athens GA 30605
EPA-600 3-78-105
December 1978
Research and Development
Rates, Constants, and
Kinetics
Formulations in
Surface Water
Quality Modeling
-------
RESEARCH REPORTING SERIES
Research reports of the Office of Research and Development, U.S Environmental
Protection Agency, have been grouped into nine series. These nine broad cate-
gories were established to facilitate further development and application of en-
vironmental technology Elimination of traditional grouping was consciously
planned to foster technology transfer and a maximum interface in related fields.
The nine series are:
1 Environmental Health Effects Research
2 Environmental Protection Technology
3. Ecological Research
4 Environmental Monitoring
5 Socioeconomic Environmental Studies
6 Scientific and Technical Assessment Reports (STAR)
7 Interagency Energy-Environment Research and Development
8. "Special" Reports
9. Miscellaneous Reports
This report has been assigned to the ECOLOGICAL RESEARCH series. This series
describes research on the effects of pollution on humans, plant and animal spe-
cies, and materials. Problems are assessed for their long- and short-term influ-
ences. Investigations include formation, transport, and pathway studies to deter-
mine the fate of pollutants and their effects. This work provides the technical basis
for setting standards to minimize undesirable changes in living organisms in the
aquatic, terrestrial, and atmospheric environments.
This document is available to the public through the National Technical Informa-
tion Service, Springfield, Virginia 22161.
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EPA-600/3-78-105
December 1978
RATES, CONSTANTS, AND KINETICS FORMULATIONS
IN SURFACE WATER QUALITY MODELING
Stanley W. Zison
William B. Mills
Dennis Deimer
Carl W. Chen
Tetra Tech, Incorporated
Lafayette, California 94549
Grant No. R804450-01-2
Project Officers
James W. Falco/Robert Ambrose
Technology Development and Applications Branch
Environmental Research Laboratory
Athens, Georgia 30605
ENVIRONMENTAL RESEARCH LABORATORY
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
ATHENS, GEORGIA 30605
-------
DISCLAIMER
This report has been reviewed by the Environmental Research Laboratory,
U.S. Environmental Protection Agency, Athens, Georgia, and approved for
publication. Approval does not signify that the contents necessarily reflect
the views and policies of the U.S. Environmental Protection Agency, nor does
mention of trade names or commercial products constitute endorsement or
recommendation for use.
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FOREWORD
As environmental controls become more costly to implement and the
penalties of judgment errors become more severe, environmental quality
management requires more efficient analytical tools based on greater know-
ledge of the environmental phenomena to be managed. As part of this Labora-
tory's research on the occurrence, movement,transformation, impact, and con-
trol of environmental contaminants, the Technology Development and Applica-
tions Branch develops management or engineering tools to help pollution con-
trol officials achieve water quality goals through watershed management.
Basin planning requires a set of analysis procedures that can provide
an assessment of the current state of the environment and a means of predic-
ting the effectiveness of alternative pollution control strategies. This
report contains a compilation and discussion of rates, constants, and kine-
tics formulations that have been used to accomplish these tasks. It is
directed toward all water quality planners who must interpret technical in-
formation from many sources and recommend the most prudent course of action
that will, minimize the cost of implementation and maximize the environmental
benefits to the community.
David W. Duttweiler
Director
Environmental Research Laboratory
Athens, Georgia
-------
PREFACE
Over recent years, the number of water quality modeling studies has
grown substantially. Conceptually, commonly used models (e.g., SWMM, QUAL)
have undergone major revision and improvement, and through application, data
on rate constants and coefficients have also proliferated. Yet, data and
information on simulation approaches tend to be scattered among technical
reports, journal articles, and government publications, and it is the need to
consolidate such information which led to the preparation of this volume.
The document represents many months of work spent in reviewing models
and application studies, compiling data, examining equations for validity,
reviewing conceptualizations and, of course, seemingly endless editing. Yet,
the work invested was nominal compared to the efforts of the thousands of
researchers the results of whose labors are so summarily presented. It is
those researchers to whom we are most indebted in the preparation of this
manual.
A comment is in order here regarding works cited in this document. We
have made every effort to provide appropriate citations acknowledging the
reports, articles, and materials used. Yet, we observed several cases,
especially with respect to figures, where materials were used in several
reports without proper source citations. This made it infeasible to search
to locate the original reference, and we regret that we were obliged to
cite the specific report in which the figure was found, although it might
not have been the primary reference.
Throughout the preparation of a document such as this, one thought
has recurred to us time and again namely, how desirable it would be to
have some feedback from users. We would like to know what proves useful,
what areas of special interest have not been covered in sufficient detail,
and what studies would be useful in expanding this kind of document in the
future. If there is sufficient interest, it may be possible at a later date
to expand coverage into additional areas of importance.
Finally, we would like to express our sincere hope that this document
is both convenient and useful to water quality modelers. Every effort has
been made to provide a balanced coverage, logical presentation and appropriate
organization. Yet, the final test comes in actual application. This is why
we hope users will provide us with both general and technically specific
feedback. Such inputs will be most sincerely appreciated.
IV
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ABSTRACT
Recent studies are reviewed to provide a comprehensive volume on
state-of-the-art formulations used in surface water quality modeling along
with accepted values for rate constants and coefficients. Topics covered
include system geometric representation (spatial and temporal), physical
processes (mass transport, heat budgets, ice formation, light extinction),
biological systems (fish, benthic organisms), and chemical processes
(nutrient cycles, carbonate system). A detailed discussion is also presented
on issues which are ordinarily of primary interest in modeling studies.
These include reaeration, dissolved oxygen saturation, photosynthesis,
deoxygenation, benthic oxygen demand, coliform bacteria, algae, and zoo-
plankton. These discussions incorporate factors affecting the specific
phenomena and methods of measurement in addition to data on rate constants.
This report was submitted in fulfillment of Grant No. R804450-01-2
by Tetra Tech, Incorporated, under the sponsorship of the U.S. Environmental
Protection Agency. The report covers the period May 10, 1976, to November 9,
1978, and work was completed as of November 9, 1978.
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CONTENTS
Foreword -j T i
Preface iv
Abstract v
Figures ix
Tables xii
Conversion Factors xvii
Acknowledgments xviii
1. Introduction 1
1 .1 Background 1
1.2 Scope and arrangement of manual 2
1.3 Some general observations 2
1.4 Use of this manual 4
2. A Process-Oriented Literature Review of Rate Parameters
Used in Surface Mater Quality Models 5
2.1 Introduction 5
2.2 Model conceptualization 5
2.3 Physical processes 19
2.4 Biological processes 61
2.5 Chemical processes 76
3. A Detailed Review of Selected Model Formulations and
Parameters 121
3.1 Introduction 121
3.2 Reaeration 122
3.3 Dissolved oxygen saturation 122
3.4 Photosynthesis 161
3.5 Carbonaceous deoxygenation 169
VI
-------
3.6 Nitrogenous deoxygenation 188
3.7 Benthic oxygen demand 199
3.8 Coliform bacteria 210
3.9 Algae 236
3.10 Zooplankton 287
vm
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FIGURES
Number page
2-1 One-dimensional geometric representation for river
systems (Chen and Wells, 1975) 7
2-2 One-dimensional geometric representation for lake
systems (U.S. Army Corps of Engineers, 1974) 9
2-3 Two-dimensional geometric representation for lake
systems (Baca and Arnett, 1976) 11
2-4 Pseudo-two-dimensional geometric representation
for estuary systems (redrawn from Orlob, 1974) 12
2-5 One-dimensional, advective-dispersive vertical
transport 21
2-6 Two-dimensional, advective-diffusive transport in
the longitudinal and vertical directions 22
2-7 Oscillation of velocity component about a mean
value (redrawn after Bird, ejt a]_., 1960) 28
2-8 Diffusion coefficients characteristic of various
environments (redrawn after Lerman, 1971) 30
2-9 Log of effective diffusion versus log of density
gradient (Water Resources Engineers, 1969) 35
2-10 Assumed vertical dispersion coefficients for three-
dimensional model (after Thomann, et_ al_., 1975) 36
2-11 Horizontal diffusivities used in Lake Ontario
modeling studies, illustrating the "thermal bar
effect" (after Thomann, et_ al_., 1975) 37
2-12 Factors contributing to tidally averaged dispersion
coefficients in the estuarine environment
(modified after Zison, et al_., 1977) 41
2-13 Pathways used in modeling the nitrogen cycle
(modified after Canale, et al_., 1976) 78
-------
FIGURES (continued)
Number Page
2-14 Pathways used in modeling the phosphorus cycle in
Lake Michigan (modified after Canale, ejt aj_. , 1976). . . 93
2-15 Pathways of phosphorus in the model by Lorenzen,
et al_. (1974) ................... ... 95
2-16 Pathways used in modeling the silicate cycle
(modified after Canale, et al . , 1976) .......... 104
3-1 Effects of the surface transfer coefficient (K[_)
on the value of 6 in Equation (3-19) (Metzger, 1968) . . 133
3-2 Field data considered by three different
investigators (Covar, 1976) ............... 145
3-3 k£ vs. depth and velocity using the suggested
method of Covar (1976) ................. 146
3-4 Diurnal variation of (P-R) in Truckee River near
Station 2B (O'Connell and Thomas, 1965) ......... 162
3-5 Deoxygenation parameter (kdH/v) versus the
Reynolds Number and Froude Number (pUH/y)/(uVgh~)
(from Bansal, 1975) ................... 172
3-6 Deoxygenation coefficient as a function of depth
(after Hydroscience, 1971) ............... 179
3-7 Example of computation of kr from stream data
(from Hydroscience, 1971) ................ 184
3-8 Nitrification parameter knH^/v versus the ratio
of the Reynolds number pVH/y and the Froude
number (Bansal, 1976) .................. 195
3-9 Typical mortality curves for col i forms as a function
of time (redrawn from Velz, 1970). .,....,„„.. 222
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Number
FIGURES (continued)
3-10 Effect of light levels on growth rates of
Gymnodinium (Thomas, 1966)
3-11 Effects of light levels on growth rate of
Nannochloris and Chaetoceros (Thomas, 1966)
3-12 The mean daily division rate of Detonula confervacea
grown for six days at various combinations 6T
light, temperature, and salinity, following pre-
conditioning at these experimental conditions
(after Smayda, 1969a) 245
3-13 Growth kinetics of three hypothetical algal species
as a function of substrate concentration 263
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TABLES
Number Page
2-1 Values for Empirical Coefficients a-| and 33 33
2-2 Tidally Averaged Dispersion Coefficients for
Selected Estuaries (from Hydroscience, 1971) 43
2-3 Tidally Averaged Dispersion Coefficients
(from Officer, 1976) 44
2-4 Values for Shortwave Radiation Coefficients A and B
(Lombardo, 1972) 50
2-5 Values for Empirical Coefficients a, and 3, 53
2-6 Range of Values for Various Component Heat Fluxes
(Baca and Arnett, 1976) 54
2-7 Range of Values for Various Component Heat Fluxes
(Lombardo, 1972) 55
2-8 Values for Coefficients Used in Fish Mass
Balance Formulations 64
2-9 Temperature Tolerances for Various Fish Groups
(after Leidy and Jenkins, 1977) 66
2-10 Chemical Composition of Fish (after Leidy and
Jenkins, 1977) 67
2-11 Estimated Half-Saturation Constants for Fish
Growth (after Leidy and Jenkins, 1977) 68
2-12 Fish Food Expressed as a Percentage of the Diet
by Volume (after Leidy and Jenkins, 1977) 69
2-13 Values for Coefficients Used in Detritus Formulations. . . 71
2-14 Values for Coefficients Used in Benthic Organism
Formulations 73
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TABLES (continued)
Number page
2-15 Release Rates from Sediment Deposits
(Lombardo, 1972) 74
2-16 Values for Coefficients in Nitrogen Formulations 90
2-17 Values for Coefficients in Phosphorus Formulations. ... 103
2-18 Values for Stoichiometric Equivalence for
Dissolved Oxygen Formulations 108
3-1 Reaeration Coefficients for Streams 125
3-2 Reported Values of Temperature Coefficient
(from Metzger, 1968) 132
3-3. Surface Transfer Ratio a for Various Substances
(from Poon and Campbell, 1967) 134
3-4 Reaeration Coefficients and Other Characteristics
of Streams and Rivers (after Hydroscience, 1971). ... 136
3-5 Hydraulic Characteristics and Reaeration Rates
Observed by Tsivoglou-Wallace for Five Rivers
(from Elmi, 1975) 140
3-6 Reaeration Coefficients and Other Characteristics
of Tidal Rivers and Estuaries (after
Hydroscience, 1971) 141
3-7 Solubility of Oxygen in Water (APHA, 1971) 155
3-8 Dissolved Oxygen Saturation Values in Distilled
Water (Elmore and Hayes, 1960) 156
3-9 Solubilities of Oxygen in Water for Different
Temperatures (CRC, 1967) 158
3-10 Some Average Values of Gross Photosynthetic
Production of Dissolved Oxygen and Average
Respiration (after Thomann, 1972, and Thomas
and O1 Cornell, 1966) 166
3-11 Chlorophyll-.a and Assimilation Number of Various
Communities (after Odum, e_t a]_., 1958) 167
xm
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TABLES (continued)
Number Page
3-12 Values of Temperature Correction Factor, 6
and Deoxygenation Coefficient, kj 171
3-13 Deoxygenation Rate Constants (from Bansal, 1975) 173
3-14 Deoxygenation Rates for Some Selected Rivers
(Eckenfelder and O'Connor, 1961) 180
3-15 Changes in Coefficient of Bed Activities by
Stream Slope 181
3-16 Temperature Correction Factor 6 for Ammonia
Oxidation (McCarty, Unpublished Notes) 190
3-17 Nitrification Rate Constants (from Bansal, 1976) 192
3-18 The Effect of Temperature on Benthic Oxygen Uptake .... 202
3-19 Source and General Characteristics of Benthal
Deposits (Hunter, et_ al_., 1973) 203
3-20 Relationship Between Benthal Oxygen Demand and
Tubifex Worm Population (Hunter, e_t al_., 1973) 204
3-21 Oxygen Consumption of Sediments in the Laboratory
and In Situ (from Edeberg and Hofsten, 1973) 205
3-22 Rates of Oxygen Consumption and Chemical Properties
of Mud Deposits Sampled from Various Sites
(after Rolley and Owens, 1967) 206
3-23 Average Values of Oxygen Uptake Rates of River
Bottoms (after Thomann, 1972) 207
3-24 Factors Affecting Coliform Die-Off Rate
(Abstracted from Mitchell and Chamber!in, in press). . . 215
3-25 Coliform Bacteria Freshwater Die-Off Rates Measured
In Situ (after Mitchell and Chamberlin, in press). . . . 218
3-26 Values for Coliform Specific Die-Off Rates Used
in Several Modeling Studies 219
3-27 Nutrient K? Values for E. Coll (after Mitchell and
Chamber! in, in press) 220
xnv
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TABLES (continued)
Number page
3-28 Values of C0, CQ, k, and k" from the Ohio River
(Phelps, 1944). .... ...... 221
3-29 Rate of Disappearance of Coliform and Fecal
Streptococcus Bacteria (from Lombardo, 1972). ..... 224
3-30 Comparison of k^ estimates (after Chamberlin
and Mitchell, in press) 228
3-31 Experimental Hourly T-90 Values (after Wall is,
et,al_.s 1977) 232
3-32 Optimal Growth Illumination (All Values in Ft-C)
for various measurements (Brown and
Richardson, 1968) 243
3-33 Qualitative Differences Among Phytoplankton
Types Modeled by Bierman (1976) 250
3-34 Values for the Half-Saturation Constant in
Michael is-Menton Growth Formulations 266
3-35 Michaelis-Menton Half-Saturation Constants for
Nitrogen and Phosphorus (from Di Toro,
et aj_., 1971) 268
3-36 Michaelis-Menton Half-Saturation Constants (Ks) for
Uptake of Nitrate and Ammonium by Cultured Marine
Phytoplankton at 18°C (after Eppley,
et al_., 1975) 269
3-37 Half-Saturation Constants for N, P, and Si Uptake
(yM) Reported for Marine and Freshwater Plankton
Algae (after Lehman, et a]_., 1975) 270
3-38 Minimum Cell Nutrient Quotas (yMoles Cell-1)
of P, Si, and N for Some Marine and Freshwater
Phytoplankton (after Lehman, e_t al_., 1975). ...... 271
3-39 Maximum (Saturated) Growth Rates as a Function
of Temperature (from Di Toro, e_t aj_., 1971) 272
3-40 Values for Coefficients in Phytoplankton Specific
Death Rate Expressions 278
xv
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TABLES (continued)
Number Page
3-41 Values for Endogenous Respiration Rates of
Phytoplankton (Di Toro, et al., 1971) 279
3-42 Values for Grazing Rates of Zooplankton
(Di Toro, et al., 1971) 279
3-43 References for Methods of Measurement of
Various Algal Modeling Parameters 280
3-44 Values for Coefficients in Michaelis-Menton
Zooplankton Growth Formulations 298
3-45 Values for Zooplankton Filtering Rates
(after Lombardo, 1972) 299
3-46 Comparison of Filtering Rates of Various
Cladoceran Zooplankters (after Wetzel, 1975) 300
3-47 Filtering Rates and Contribution to Total Grazing
of Species. Dominant Zooplankton of Acidic
Drowned Bog Lake, Ontario, in Early September
(after Wetzel, 1975) 301
3-48 Comparison of Filtering Rates of Various
Copepods (after Wetzel, 1975) 302
3-49 Examples of the Production of Herbivorous and
Predatory Zooplankton Communities (after
Wetzel, 1975) 303
3-50 Examples of Productivity of Herbivorous and
Predatory Forms of Zooplankton (after
Wetzel, 1975) 304
3-51 Exemplary Estimates of Efficiencies of Food
Utilization by Various Animals (after
Wetzel, 1975) 306
3-52 Values for Coefficients in Zooplankton Specific
Death Rate Expressions 310
3-53 Endogenous Respiration Rates of Zooplankton
(after Di Toro, et al_., 1971) 311
3-54 Values for Endogenous Respiration Rates of
Zooplankton (after Lombardo, 1972) 312
xvi
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CONVERSION FACTORS
English Unit
acre
acre
acre-ft
Btu
Btu
Btu/hr/sq ft
Btu/lb
cfm
cfs
cfs/sq miles
cu ft
cu ft
cu 1n.
cu yd
°F
°C
ft
ft-lb
gal
gal
gpd/acre
Multiplier
X4. 046. 724:
* 2.471X10" X
X0.405-
*• 2.471X
Xl.233.5-
<- 8. 11X10" "X
XI .055-
- 0.9478X
X0.252-
- 3.968X
X3.158-
••- 0.316X
X0.555-
*• 1.80X
X0.028-
-- 35.71X
XI. 7-
«- 0.588X
X0.657-
«- 1.522X
X0.028-
* 35.314X
X28.32-
*• 0.0353X
X16.39-
+ 0.061X
X0.75-
-i- 1.3709X
O.E55(°F-32)-
- 1.8(°C)+32
plus 273-
- minus 273
XO. 3048-
- 3.28X
XI. 356 +
^0.737X
X3.785-
*• 0.264X
XO. 003785-
•H264.2X
X0.9365-
i- 1.068X
SI Unit
m2
ha*
m3
kJ
kg-cal*
J/s-m2
kg-cal/kg*
mVmin
m Vmin
mVinin km2
m3
1*
cm3
m3
CC
K
m
J
1*
m1
m3/day km2
English Unit
gpd/ft
gpd/sq ft
gpm
gpm
gpm/sq ft
hp
hp-hr
in.
Ib/day/acre-ft
lb/1 ,000 cu ft
Ib/day/cu ft
Ib/mil gal
mil gal
mgd
ngd
mi le
ppb
ppm
sq ft
sq in.
sq miles
Multipl ier
X0.0124+
^80.65X
X0.0408-
-24. SIX
X0.0631-
- 15.85X
X0.0631-
•>- 15.85X
X40.7-
*• 0.0245X
X0.7454-*
* 1.341 X
X2.684-
* 0.372X
X2.54 +
•^0.3937X
X3.68-
- 0.2717X
XI 6.0+
-H 0.0625X
X16+
«- 0.0625X
X0.92+
-H 8.333X
X3.785-
* 2.64X10-"X
X3.785-
->- 2.6iX10""X
X0.0438-
^ 22.82X
XI. 61-
*• 0.621X
X10"3-
* 1 ,OOOX
approximately
equal to
X0.0929-
- 10.76X
X645.2-
-^0.00155
X2.590-
-0.3861X
SI Unit
m3/day m
m Vday m2
dmVs
TVs
l*/min m^
kW
MJ
cm
g/day m3
g/m3
kg/day m3
g/m3
m3
m3/day
m3/s
km
mg/1*
mg/1*
m2
mm2
km2
Other coffinQnly used conversions:
1 MOO • 1.55cfs
vc • 62.4 BTU/ft!/°F
p
1 MW -3.414 X 10' BTU/hr
1 BTU • 778 ft-lb
1 BTU « 252 cal
1 Ungley/day = 3.7 BTU/ft.
'Not an SI unit, but a term cotmonly used and preferred as a wastewater unit of expression.
Vday
XV11
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ACKNOWLEDGMENTS
The authors would like to first thank Dr. Marc Lorenzen for providing
technical guidance early in the study when goals and direction were being
defined and such guidance was so vitally important. Dr. Lorenzen1s input
was especially useful in determining appropriate coverage and in collecting
and organizing materials for review.
Thanks are due the participants in the Symposium on Rate Constants,
Coefficients, and Kinetics formulations which was held in the San Francisco
Bay Area from February 23-25, 1977, and whose inputs provided a significant
part of the materials used in preparing this document. The authors would
also like to acknowledge the efforts of Mrs. Anna Zison who provided all of
the graphics work in redrawing figures from reports reviewed and providing
several new ones, to Mr. Steve Gherini, and Mr. David Dykstra for their
editing efforts, and to Mrs. Bernice Bujacich for preparing the drafts.
Finally, thanks are perhaps most due the U.S. Environmental Protection
Agency-Environmental Research Laboratory-Athens for its financial support
(Grant number R804450-01-2) and in particular, to Dr. James Falco and
Mr. Robert Ambrose for their input and technical advice.
xvi 11
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CHAPTER 1
INTRODUCTION
1.1 BACKGROUND
The use of mathematical models to simulate ecological and water
quality interactions in surface waters has grown dramatically over the
past decade. To some extent, the recent increase in the use of models
as well as in model development is attributable to Section 208 of
PL92-500, the Water Pollution Control Act Amendments of 1972. To meet
the requirements of Section 208, which calls for areawide wastewater
planning in both designated and nondesignated areas across the United
States, 208 planners have often found that simulation techniques offer
the best course for evaluating wasteload abatement alternatives. Pre-
dictions of system behavior based upon mathematical simulation techniques
may be misleading, however, particularly if the physical mechanisms in-
volved are not accurately reflected in the model. Furthermore, even where
the model does faithfully describe mechanisms in the prototype, poor results
may be obtained where insufficient data are available to estimate rate
constants and coefficients.
Much of the recent work done in the water quality modeling field has
been oriented toward improvement of models--toward incorporating better
numerical solution techniques, toward an expanded complement of water
quality constituents simulated, and toward realistic representations of
modeled physical, chemical, and biological phenomena.
In contrast, although a substantial body of data has been developed
on rates and coefficients for use in the various important models, the
data are scattered throughout journal articles, government documents and
technical reports. This, of course, makes it difficult for the modeler to
obtain necessary guidance in assigning values to the various constants
and coefficients required by his model. The solution has commonly been
to gather data with which to estimate the coefficients or to undertake a
time consuming literature search—expensive in either case.
Recognizing these problems, the United States Environmental Protection
Agency and Tetra Tech, Inc. sponsored a symposium in the San Francisco Bay
Area, entitled "Rate Constants, Coefficients, and Kinetics Formulations in
Surface Water Modeling." The symposium was held on February 23 through
February 25, 1977. It was a major step in the preparation of a single,
comprehensive compilation of data on surface water quality modeling for-
mulations, and values for rate constants and coefficients. In addition
to the symposium, and also supported by U.S. Environmental Protection
-------
Agency funds, an extensive literature search was undertaken to provide
background material for the manual to be prepared. This document repre-
sents the culmination of these efforts, and it is hoped a substantial
beginning in the solution of difficulties inevitably encountered by mod-
elers when they attempt to select appropriate values for model parameters.
1.2 SCOPE AND ARRANGEMENT OF MANUAL
In preparing this manual, an attempt has been made to present as com-
prehensive a set of formulations and associated constants as possible.
Guidance in the selection of topics has beei? provided primarily by the
extent of use by workers in the field instead of through a critical assess-
ment of technical validity. Rather than exclude some methods, the philo-
sophy has been to present arguments for and against any controversial
approach.
The manual is divided into
chapter, Chapter 2 is largely a
of processes and formulations.
Model conceptualization
Physical processes
Biological processes
Chemical processes
References.
three chapters. Following this introductory
literature review covering a broad spectrum
The arrangement is:
Chapter 3 contains a very detailed discussion of selected parameters and
formulations not included in Chapter 2. Topics relegated to Chapter 3 in-
clude those which, it was felt, would be of greatest interest to the reader,
and about which questions most often arise. Chapter 3 includes discussions
on the following subjects:
Reaeration
Phytoplankton
Zooplankton
Biochemical oxygen demand
Coliform bacteria
Benthic oxygen demand
Dissolved oxygen saturation
Photosynthesis (without modeling phytoplankton).
References for the various discussion topics are included at the end of each
section.
1.3 SOME GENERAL OBSERVATIONS
In the course of compiling this manual, several issues arose which
warrant discussion here, First, each rate value or expression used in a
model should not be chosen as an "afterthought," but should be considered
as an integral part of the modeling process. A substantial portion of any
-------
modeling effort should go into selecting specific approaches and formula-
tions based upon the objectives of modeling, the kinds and amounts of data
available, and the strengths and weaknesses of the approach or formulation.
Once formulations have been selected, a significant effort should be made
to determine satisfactory values for model parameters. Even where the
parameter is to be "calibrated," it is clearly important to establish
whether the calibrated value is within a reasonable range or not. Further,
it should always be remembered that the value of any constant or coeffi-
cient is dependent upon the way it is used in the formulation.
Second, there is rarely consensus on how best to select a particular
rate, coefficient, or expression for a given prototype. Generally, there
are a great many environmental factors influencing a given rate parameter.
The factors can be complex, and as a result, may be poorly understood, and
their influence on rate constants inadequately quantified. In some cases,
such as in modeling stormwater runoff quality, there may be so many physical
and chemical factors involved in the prototype that developing a satisfactory
mechanistic model may be impractical or beyond the "state of the art."
Furthermore, it is not uncommon for researchers to disagree on both the
direction and extent to which a particular physicochemical or biological
factor influences a given rate. Accordingly, further study on rates of
various surface water phenomena is certainly to be encouraged.
Third, based on the commonly large differences in observed rates from
system to system, some researchers believe that some surface water quality
parameters are highly system-specific. Accordingly, it may be difficult
to establish or use general rate constant selection guidelines. It is
apparent, however, that so long as the phenomena modeled are basically
mechanistic rather than stochastic in nature, the key to intersystem gen-
erality lies in understanding and quantifying the actual determinative
factors and causal relationships influencing the process modeled. It may
be that apparent system specificity of rate constants is only a manifesta-
tion of the researcher relying on too few and/or incorrectly selected
predictive variables.
Fourth, modelers should be keenly aware of the range of applicability
of a particular expression or value, conditions under which the field
measurement technique is valid, and the limitations of both the expression
and the measurement technique. However, a simplified representation of
the prototype is often adequate where proper constraints are placed on
temporal or spatial conditions of the simulation. For example, a linear
estimate may be a sufficient representation of a nonlinear phenomenon if
the range over which the phenomenon is simulated is carefully selected.
Fifth, there is currently some disagreement among researchers as to
whether there is a need for increased model sophistication, particularly
where simulation of complex ecosystems are concerned. On the one hand,
by making ecosystem representations more sophisticated, the number of model
parameters almost certainly will increase. Since it is likely that many
of the new parameters may be understood only qualitatively, the new, more
-------
sophisticated model may be less reliable and accurate than the older version,
On the other hand, however, some systems and phenomena represented in ex-
isting models (e.g., fish, pe'sticide transport, runoff water quality) are so
complex in nature that unless these models are made more detailed and rea-
listic, there is often little reason to believe that good simulations are
other than fortuitous.
Finally, a comment is in order here regarding assumed singularity of
constants, coefficients, and results of simulations. In general, in model-
ing studies, a narrow range of values or even a single value for each model
coefficient is selected and used, and a single estimated value for each
parameter is determined as a function of time and location within the proto-
type. Similarly, in field studies and in interpretation of physical,
chemical, and biological data, individual values are taken as precisely
representing actual conditions when, in fact, all of the data and coeffi-
cients represent samples drawn from large populations. Although costs
usually preclude advanced statistical analysis and delineation of population
parameters for all input and, consequently, output of modeling studies, it
is important that the fact that model output also represents a sample from
a large population be kept in mind. The ramifications of this are very
important. What may appear to be an excellent calibration or an excellent
simulation may, in fact, be well outside the envelope of expected values
based on field conditions (if these conditions were known). Conversely,
what appears to be a poor simulation, and what might lead to extensive re-
working and rethinking of the adopted model, may be more a manifestation of
a large dispersion in field data than mathematical misrepresentations of
underlying physical mechanisms. Accordingly, the importance of visualizing
input data and the resulting output as a sample from a larger population
must be stressed.
1.4 USE OF THIS MANUAL
This document is intended for use as a handbook—a convenient reference
on modeling formulations, constants, and rates commonly used in surface
water quality simulations. It was impossible to encompass all formulations
or to examine and discuss even the majority of recent reports containing
rates data. It is hoped, therefore, that the user will recognize the
desirability of seeking additional sources where questions remain about
formulations or values. Data used from within this volume should be
recognized as representing only a small sampling from a much larger set of
data. It should also be noted that there are very real limitations involved
in using literature values for rates rather than observed system values.
It is hoped, finally, that this document will find its main use as a guide
in the search for "the correct value" rather than as the sole source of
that value.
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CHAPTER 2
A PROCESS-ORIENTED LITERATURE REVIEW OF RATE
PARAMETERS USED IN SURFACE WATER QUALITY MODELS
2.1 INTRODUCTION
The objective of this chapter is to present the results of a literature
survey which reviewed ecological modeling studies. The surveyed literature
emphasized recent works in which lakes, rivers, and estuaries were simulated.
Mathematical models used in studies of this type generally consist of a set
of equations that describe water and waste movement within the water column,
together with appropriate boundary conditions and empirical parameters. The
models cited were examined in terms of adopted simplifications and the assump-
tions under which each is appropriate. Emphasis was placed on gathering
information concerning rate constants, coefficients, and kinetics formula-
tions used in these models of water systems.
The kinetics formulations and associated rate parameters reviewed in
this chapter are grouped according to the kind of phenomena represented.
The categories are physical, chemical, biochemical, and biological processes.
Although the subdivision is admittedly artificial, it does provide a con-
venient basis for organization and allows for a logical flow from topic to
topic. In some cases, it has been difficult to select the proper niche for
a particular topic, but it is hoped the arrangement will nevertheless prove
convenient to the user.
Emphasis in this chapter is on model conceptualization, followed by a
fairly detailed coverage of formulations and parameters which, though impor-
tant, are not necessarily in the very general mainstream of ecological
modeling. Formulations and issues warranting special emphasis have been
relegated to Chapter 3.
2.2 MODEL CONCEPTUALIZATION
2.2.1 Geometric Representation
Model design is, by its nature, a pragmatic process — the simulation
objectives determine the basic form of the model, and then the mathematical
system description and solution techniques are formulated. In developing
the system description, several major issues must be addressed. These
include:
• What kind of spatial representation is needed to accurately
describe the prototype?
-------
• What temporal scale should be used?
• Should the model be dynamic or is a steady-state
approximation adequate?
• What kind of kinetic formulations are most appropriate
to describe the time-based changes in pollutant
concentrations?
In addressing these issues, it is reasonable to begin with the spatial rep-
resentation. Normally, the first step is to divide the water body into a
series of volume elements or segments, such that spatial homogeneity is
attained without violating mathematical requirements or posing unnecessary
computational complexity. Ideally, the proper segmentation of the system
allows for a realistic portrayal of both advective and dispersive trans-
port. It also permits inclusion of chemical and/or biological kinetics
where water quality is to be modeled. Invariably, the resulting spatial
representation is a compromise considering these criteria, the capacity of
computational facilities, and other resources available.
In water quality modeling, the ideal description of the pollutant con-
centration field would be a continuous accounting of concentrations at all
points throughout the system. That is, the instantaneous concentration of
each pollutant, Cp would be defined as a function of spatial location (x, y,
z axes) and time (t):
Cp = f(x,y,z,t) (2-1)
It is obviously infeasible to compute the infinite set of Cp. Fortunately,
however, feasible representations, often using fewer than three spatial ordi-
nates and discrete time steps, have proven satisfactory in most applications,
2.2.1.1 jero-Dimensional Models
Zero-dimensional models are used to estimate spatially averaged pollu-
tant concentrations at minimum cost. These models predict a concentration
field of the form C = g(t), where t represents time. They cannot predict
the fluid dynamics of a system, and the representation is usually such that
an analytical solution is possible.
As an example, the simplest representation of a lake is to consider it
as a continuously stirred tank reactor (CSTR). For such a coarse represen-
tation, the model generally cannot adequately simulate seasonal algal
growth dynamics. Such models are often used for nutrient budget computa-
tions using an annual time step.
2.2.1.2 One-Dimensional Mode1_s
Most river models use a one-dimensional representation, where the sys-
tem geometry is formulated conceptually as a linear network of segments or
volume sections (see Figure 2-1). Within each segment, the water is assumed
to be completely mixed, so that there is no water quality variation laterally
-------
Control Volume, X;
X. = volume element
QO = water withdrawals from element X.
xi 1
QI = water discharged to element X-
x. i
E = evaporation
P = precipitation
Q-j+1 = advective flow to downstream element X
Q. -I = advective flow from upstream element X
AX = longitudinal dimension of element
-. QO
'. QI
X;
Figure 2-1. One-dimensional geometric representation for river systems (Chen and Wells, 1975)
-------
or with depth. Variation occurs longitudinally (in the x-direction) as the
water is transported out of one segment and into the next. These river
models thus calculate the horizontal water quality profiles along a river
system, considering each segment as a CSTR.
As one-dimensional stream models have become relatively sophisticated,
the number of phenomena simulated, both in terms of hydrology and water
quality, has grown substantially. Early models, such as DOSAG, considered
only DO and BOD. QUAL-II, a model developed over the years by Water
Resources Engineers and the Texas Water Development Board, simulates roughly
20 constituents and will accommodate 90 reaches of up to 20 CSTR's each. A
more recent one-dimensional river model employed for a study of the Kings
River, California by Tetra Tech (1976) is similar in capability to QUAL-
II. It utilizes a river system representation which includes tributary
branches. At any section of the river, this model accepts reservoir
releases, natural runoff, and diverted flows. The river can also receive
discharges from a power plant and water can be diverted for power
generation.
For lakes exhibiting stratification and long residence times, a one-
dimensional approach in which the water body is represented by a stack
of horizontal slices or layers (Figure 2-2) has been used extensively
(Lombardo, 1972, 1973; U.S. Army Corps of Engineers, 1974; Chen and Orlob,
1975; Thomann, ejt al_., 1975; Tetra Tech, 1976b). Two-layer, three-layer,
and multiple-layer representations have all been used in one-dimensional
lake system applications (Chen, 1975). Within each slice or layer, the
properties of water are assumed to be completely uniform. This type of
representation assumes that the temperature gradients exist only along the
vertical axis.
One-dimensional estuary representations, similar to the one-dimensional
river systems, imply that the estuary is well-mixed vertically and laterally.
According to Callaway (1971), "borderline" cases are not at all uncommon and
assumptions of one-dimensionality for estuarial systems should be made
cautiously.
Setting the number of horizontal segments* to be used in a one-dimen-
sional estuarine simulation is a complex issue. Important considerations
include the homogeneity of the water mass, the variations in physical,
chemical, and biological parameters, the distribution of inputs along the
length of the estuary, and the desired simulation accuracy. In addition,
the tidal influences are a significant consideration in estuary systems,
whether simulations are performed on a real-time or tidally-averaged basis.
Real-time means that the hydrodynamics of each tidal cycle are actually
simulated. Tidally averaged models consider o.nly net flow, generally over
a standardized tidal cycle, and incorporate a tidal dispersion coefficient
to simulate flushing.
*"Horizontal segments" will be used in this document to mean segmentation
in the sense shown in Figure 2-1.
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tributary
inflow
evaporation
tributary ±s~ -ram
inflow
vertidal
advection
outflow
Figure 2-2. One-dimensional geometric representation for lake systems
(U.S. Army Corps of Engineers, 1974).
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2.2.1.3 Two-Dimensional Models
A wide variety of two-dimensional conceptualizations of lake systems
have been employed. Baca and Arnett (1976) have considered a quasi-two-
dimensional approach, first longitudinally segmenting the lake and then
considering each segment to be vertically layered (see Figure 2-3). The
Fisheries Research Board of Canada (Lam and Jacquet, 1976, and Lam and
Simons, 1976) has applied a one-layer, two-dimensional computer model
which describes the complete longitudinal and lateral pollutant concen-
tration regime while assuming homogeneity in the vertical direction.
The physical definition of an estuarine system restricts, to a large
degree, the type of geometric representations which may be used. Most
commonly, estuarine geometry is represented in mathematical models by a
two-dimensional or quasi-two-dimensional approximation.
One particular quasi-two-dimensional approach that has been used
extensively is sometimes referred to as a "link-node" system (see Figure
2-4) (Feigner and Harris, 1970; Crim and Lovelace, 1973; Chen and Orlob,
1975; Tetra Tech, 1976a). A "link-node" network may be constructed to
represent a two-dimensional embayment, a system of discrete estuarine
channels, or combinations of channels and shallow bays. The estuarine
water body is subdivided into discrete volume units or "nodes." Nodes are
characterized by surface area, depth, volume, and in some instances, side
slopes. Nodes are interconnected by channels or "links." Links are
defined by length, width, cross-sectional area, hydraulic radius (depth),
and friction factor. Water is constrained to flow from one node to another
through the defined channel. While the network is comprised of these one-
dimensional channels, the representation is quasi-two-dimensional in that
the description of quality provided is essentially two-dimensional
(Figure 2-4).
Several other two-dimensional modeling approaches are coming out of
the experimental stages and into operational use (Nihoul, 1975). Callaway
(1971) reports an estuarine model developed by Leendertse that is capable
of simulating two-dimensional, vertically integrated cases. Hinwood and
Wall is (1975b), in their review of tidal models, report the use of two-
dimensional side elevation models laterally averaged in density strati-
fied estuaries, where the transverse variation in waste concentration is
much smaller than the longitudinal and vertical variations.
2.2.1.4 Three-Dimensional Models
Three-dimensional spatial representations have been used to model over-
all lake circulation patterns. Until recently, much of the work on predict-
ing circulation in lakes dealt with the overall hydraulic flow regime and
not specifically with the nearshore area. A recent development which has
contributed significantly to three-dimensional model conceptualization has
been the use of a variable grid for the physical representation of a large
lake. Several investigators (Chen, 1975; Tetra Tech, 1975; Lick, et al.,
10
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tributary
inflow
tributary
inflow
horizontal
segmentation
outflow
Figure 2-3. Two-dimensional geometric representation for lake systems
(Baca and Arnett, 1976).
11
-------
Link
Length
Width
Cross sectional area
Friction factor
Hydraulic radius
Volume
Surface area
Depth
Surface elev.
Node
Figure 2-4. Pseudo-two-dimensional geometric representation for
estuary systems (redrawn from Orlob, 1974).
12
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1976)^have proposed such systems where detailed horizontal spatial descrip-
tion is provided for nearshore zones, and in the central core of the lake,
where variation in the vertical direction is more important, larger hori-
zontal segments with more numerous vertical layers are used.
The variable size grid employed in the hydrodynamic model can also be
used with a water quality model which utilizes either the same spatial
representation or a less detailed one having selected elements in common
with the^hydrodynamic model. Nihoul (1975) reports that three-dimensional
computations for estuaries are now being approached by using finite differ-
ence approximation, and also by analytical solutions of the vertical
velocity field.
2.2.2 Temporal Variation
Water quality models may be distinguished according to the temporal
scale used. The current state of modeling places an emphasis on short-
term responses of the ecosystem as opposed to the longer term responses.
2.2.2.1 Long-Term vs. Short-Term Models
The "long-term" models generally rely on a simplified geometric repre-
sentation and simplified expressions for complex biological processes. This
type of model has been useful in predicting long-term changes that result
from increased or decreased nutrient loading rates (Tetra Tech, 1975;
Lorenzen, et_ al_., 1976). Long-term models may also be useful in predict-
ing chronic problems which are slow to develop within an ecosystem.
"Short-term" models, which this review emphasizes, have been used to
identify and solve acute pollution problems. Models of this type simu-
late a shorter time span than the long-term models, quite commonly empha-
sizing from diurnal to seasonal variations in system constituents. Such
models provide an understanding of the behavior of the quality indices and
of the aquatic ecosystem.
2.2.2.2 Dynamic vs. Steady-State Model_s_
Ecological models are further distinguished on a temporal basis as
being either "dynamic" or "steady-state." A strict steady-state assumption
implies that the variables in the system equations do not change with time.
Forcing functions, or exogenous variables, that describe environmental con-
ditions which are unaffected by internal conditions of the system, have con-
stant values. Inflows and outflows are discharged to and drawn from the
system at a constant rate and any other hydrologic phenomena are also steady
Insolation, light intensity, photoperiods, extinction coefficients, and
settling rates are a few examples of additional forcing functions which are
held constant in a steady-state model. Constant forcing functions repre-
sent mean conditions observed in a system, and therefore the model cannot
simulate cyclic phenomena.
13
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A wide variety of planning problems can be analyzed by use of steady-
state mathematical models which provide the necessary spatial detail for
important water quality variables. Certain phenomena can achieve steady-
state conditions within a short time interval and therefore can be modeled
rather easily. Steady-state representations are particularly useful
because of their simplicity. Examples of phenomena which have been
modeled on a steady-state basis are: 1) bacterial die-off, 2) dissolved
oxygen concentrations (under certain conditions), and 3) nutrient distri-
bution and recycle.
The assumption of steady-state conditions may also be imposed in order
to observe system response to extreme conditions (e.g., low summer flows).
In these simulations, once steady-state has been reached, it persists indef-
initely because the controlling variables all remain constant. In such
cases, when arbitrary initial conditions are imposed, the output of the
model can be used to determine when the system has reached steady-state.
Linear programming concepts are also often used in conjunction with steady-
state simulations to ensure certain water quality standards are met a
given percent of the time at the least expense to the dischargers in-
volved. Kelly (1975) reports of such an application to the Delaware Estu-
ary. Long-term models are pseudo steady-state formulations, with a time
step of computation on the order of months to a year (Tetra Tech, 1975).
In estuarine steady-state models, the effect of tidal height varia-
tions on cross-sectional areas (and hence on water volume changes) and
tidal current fluctuations are not simulated. A steady-state estuarine
model will predict waste concentrations corresponding to steady inflow and
unvarying tidal conditions. Hinwood and Wall is (1975b) note that this type
of model is unsatisfactory in short estuaries. Steady-state estuarine
models are also unsatisfactory if the waste loads, river inflows, or tidal
range vary appreciably with a period close to the flushing time of the area
of interest.
The majority of dynamic models for rivers and lakes are concerned with
eutrophication and its control. Thus the seasonal distribution of certain
biological species and related abiotic substances is of major importance.
The time step used in these models is on the order of one day or less for
water quality computations (Hydroscience, 1971; Lombardo, 1973; U.S. Army
Corps of Engineers, 1974; Tetra Tech, 1975; Chen and Orlob, 1975; Baca
and Arnett, 1976; Tetra Tech, 1976b; Bierman, 1976; Simons, 1976).
The influence of tides in an estuary system poses a special problem
in dynamic simulations. The hydrodynamic simulation must often be carried
out with smaller time steps than the water quality computations to ensure
computational stability (Hydroscience, 1971; Simons, 1976). If an intra-
tidal model is used, the time step is largely governed by the grid spacing
and the numerical formulation on which the model is based and is usually
from 5 to 20 minutes. Time steps longer than about 30 minutes require
more complicated numerical formulations to avoid averaging over the tidal
period in an incorrect manner (Hinwood and Wallis, 1975a).
14
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Stability criteria for estuary models are generally expressions which
include channel length (indicative of grid spacing) and the celerity of
shallow waves (Battelle, 1974; Tetra Tech, 1976a). Battelle (1974) used
the following stability criteria in its estuarine simulation:
At
gn2L
2.2
(2-2)
where At
L
V
R
n
g = gravitational constant, ft/sec
= maximum allowable time step, seconds
= channel length, feet
= channel velocity, feet/sec
= hydraulic radius, feet
= Manning's n, feet '
In certain instances, "quasi-dynamic" estuarine models have been used
successfully (Crim and Lovelace, 1973; Tetra Tech, 1976a). In these models
the quality equations are effectively integrated over time, assuming net,
steady-state flows. The tidally averaged models use a tidal cycle as the
basic time step and yield average daily results.
Another approach used in estuarine simulations is to solve for the
quality interactions based on a "dynamic steady-state" hydraulic condition
predicted in the hydraulic program (Feigner and Harris, 1970). In this
approach, flows and heads are determined in a separate hydraulic subroutine
and averaged for the larger time step used in the water quality model.
These condensed hydraulic parameters for the full tidal cycle are stored on
tape or disk for input into the quality program and thus are the basis for
any number of quality runs. The quality solution proceeds over a full tidal
cycle at which point the hydraulic input tape is rewound and used again as
the basis for the next cycle.
The dynamic steady-state solution can be useful in estimating the
flushing time for either continuous or instantaneous releases of pollutant.
For the continuous case, after the pollutant has been distributed by allow-
ing the program to run long enough (i.e., once dynamic steady state has been
achieved), the given distribution at a given time, t, is simply used as the
initial distribution (t=0) on a second run. The waste input is set to zero
and the concentration decrease is recorded. To simulate the flushing of an
instantaneous release, the initial distribution in a given section or grid
of the model is set for a short time at a certain level and the program
runs with no further input required.
15
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2.2.3 Hydrodynamic Considerations^
Virtually all water quality simulation models are based on the conser-
vation of mass and the interaction of constituents. In general terms, the
mass rate of change of a constituent, c, in any system segment may be written
as (O'Connor, et al_., 1975):
V 4| = J ± ER ± ZW (2-3)
where R = chemical and biological transformation rates
within the segment
W - rates of input to and withdrawal from segment
J = transport rate through segment
The latter two terms are sometimes combined within a single source-sink term.
In order to solve Equation (2-3) the transport term, J, must be known. It is
the purpose of the hydrodynamic model to determine J.
One of the most important questions which must be answered when formu-
lating a modeling approach for a specific problem application, is the rigor
with which the hydrodynamic processes must be represented in the model. This
determination is made in conjunction with the geometric representation and
temporal scheme chosen and is governed primarily by the same considerations.
Generally two options are available to represent prototype hydrodynamics.
The most sophisticated approach is to compute the velocity field in a
separate hydrodynamic model. Hydrodynamic models are primarily used in con-
junction with water quality models containing detailed spatial representa-
tions. Hydrodynamic models can also be used in geometric representations of
less than three spatial dimensions, however. For example, if a vertically
averaged, two-dimensional representation is used, empirical resistance
coefficients are used to represent vertical shear stresses and the turbu-
lence effects are averaged over depth. When a separate hydrodynamic model
is used, the flow field information which is computed must be retained for
input into the water quality mass balance expressions.
This process of data transfer from the hydrodynamic to the water qual-
ity model is most easily accomplished if the same geometric grid is used
for both model simulations. Frequently, however, the use of the identical
grid is not practical because the hydrodynamic model grid may provide more
detail than required for the water quality simulation. In these cases, it
is the usual practice to provide a separate, less-detailed water quality
grid, which may easily incorporate the hydrodynamic model output. This
procedure requires the inclusion of a more sophisticated data exchange
mechanism, but saves computational time in the water quality model.
16
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The actual type and amount of data which are passed from the hydro-
dynamic model to the water quality model will vary with the particular
application and investigation. A set of values is required which will des-
cribe the velocity field throughout the system. Generally, this means
enough information to define the advection process.
The second approach to hydrodynamic simulation, which may be partic-
ularly advantageous in certain applications, is to input velocity values
describing the circulation patterns directly into the water quality model.
This Approach may be used in water quality models at all levels of spatial
detail. Often, circulation information is available from previous studies
on the same prototype system. Depending on the needs of the application,
previous results may suffice entirely or may be supplemented with additional
data. Supplemental data can usually be obtained by direct measurements in
the prototype system. The cost of this additional in situ sampling com-
pared with the cost of alternative means of hydrodynamic simulation might
determine the approach taken.
Due to the physical nature of typical estuaries, even the simplest of
geometries often requires relatively sophisticated hydrodynamic information
to accurately describe mass transport. The hydraulic behavior of estuaries
and other coastal waters are usually influenced significantly by ocean tides
and by the freshwater inflow to the system, in addition to the shape of the
estuary. Coriolis and wind forces may also be significant in certain
estuarine systems.
In modeling the hydrodynamic behavior of an estuary, the problem is
essentially one of solving the equations describing the propagation of a
tidal wave through a shallow water system. In an estuary, the principle
features to be reproduced by the model are flow velocities and tidal stages
induced by the river flow, the tidal currents, and the dispersion effects.
2.2.4 Solution Techniques
Mass balance expressions are formulated as differential equations, to
be solved for each hydraulic element and for each of the quality constit-
uents considered. In general a mass balance differential equation includes
advection, diffusion, input, output, and sources and sinks. Where micro-
organisms are involved, for example, the metabolic processes of growth,
respiration, and mortality can become the sources and sinks. A typical
water quality model is composed of a set of simultaneous nonlinear, partial
differential equations in both time and space. In order to solve these
equations, it is usually necessary to resort to numerical solution
techniques.
A specific method of solution, which was used in the majority of the
water quality models reviewed in this study, is known as the finite-
difference technique. In a finite-difference approach, the equations are
integrated by approximating the actual solution with small forward time
steps of finite dimension. For example, to integrate a basic differential
17
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equation over time the following approximation to the constituent concen-
tration can be used for each element (U.S. Army Corps of Engineers, 1974):
where C ,C. . = values of the dependent variable
(typically concentration) at the
beginning and end of the integration,
respectively
• •
Ct'Ct+At = values of the rate °f change of the
dependent variable at the beginning
and end of an integration interval,
respectively
At = length of integration interval
Because of the complexity of the processes affecting the concentration
c, the set of simultaneous differential equations are usually interdependent
Therefore, it is often required that the solution of one equation be avail-
able for computation in the next. Generally, a recursive scheme is estab-
lished to integrate the equations with respect to time. Judicious selection
of a solution order for the differential equations minimizes the ordering
impact on the solution. Quite commonly, all constituent concentrations are
computed sequentially for the hydraulic elements for each time step (Chen
and Orlob, 1975). Then after all elements have been considered, another
step forward in time is taken, and the computation process is repeated,
The finite difference approach can use either an implicit or an ex-
plicit solution technique. In an implicit solution the variables and their
derivatives are both considered unknowns and are determined simultaneously.
A commonly employed implicit solution technique is based on the Thomas
algorithm, an adaptation of Gaussian elimination (Tetra Tech, 1975; Baca
and Arnett, 1976).
The implicit technique is contrasted with explicit methods in which
the derivatives (or the differential terms) are evaluated before the
projections of concentrations are made. Call away (1971) reports that
trouble often arises in explicit methods of solution because of the short
time periods required to ensure stability as well as a problem of conser-
vation of mass. With regard to conservation of mass, momentum conservation
can be violated resulting in abnormal velocity and water transport volumes.
Often, modified explicit solution techniques, such as a "leap frog explicit
solution" are used, which add a degree of accuracy and stability. Implicit
methods of solution, on the other hand, although they tend to ensure
stability, may introduce inaccuracies.
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_A slight preference for the implicit method of solution over the ex-
plicit method has been shown by investigators in water quality models. Baca
and_Arnett (1976), Chen and Orlob (1975), and the U.S. Army Corps of
Engineers (1974) all use implicit solution techniques for their finite-
difference schemes. Lombardo (1973) and Harper, as reported by Hydrocomp
(Lombardo, 1972), employ explicit finite-difference solutions.
_Another type of numerical solution technique, the finite element method,
originally developed for structural analysis applications, has recently been
applied quite successfully in several water quality modeling studies.
Callaway, et_ al_. (1971, 1976), constructed a finite element formulation for
the geometry of the Columbia River system. Significant advantages to be
gained from use of the finite element method are in its flexibility to more
accurately portray boundary conditions and improved numerical stability.
2.3 PHYSICAL PROCESSES
2.3.1 Advective Transport
The concentration of a substance at a particular site within a system
is continually modified by the physical processes of advection and diffu-
sion which transport fluid constituents from location to location. However,
the total amount of a substance in a closed system remains constant unless
it is modified by physical, chemical, or biological processes. Employing
a Fickian type expression for turbulent mass flux, the three-dimensional
advection-diffusion (mass balance) equation can be written as:
3s u3s + v3s + w3s _ _3__ , K 3jL\
(2-5)
3 /i/ 3s% _ 9 /1^ "S\ - Y.S
where s = concentration of constituent
u = velocity in x-direction
v = velocity in y-direction
w = velocity in z-direction
K ,K ,K = eddy diffusion coefficients
x y z
ZS = sum of source/sink rates
t = time
19
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Difficulties exist in trying to correctly quantify the terms in this equa-
tion. The unsteady velocity field (u,v,w) must be evaluated separately
from equation (2-5) so that the pollutant concentration, s, can be pre-
scribed. Even then, the solution to the three-dimensional system repre-
sentation is difficult to obtain. Hence simplified versions of equation
(2-5) are used instead.
2.3.1.1 Vertical Advective Transport
The basic vertical transport mechanisms include the following: 1)
water flow in the direction of the velocity field (advection), 2) disper-
sion by diffusional processes, 3) buoyant forces, and 4) settling of
particulates.
A coarse representation of the water system as a CSTR is often suffi-
cient for problem applications to some lakes where detailed hydrodynamics
are not required. Since in this zero-dimensional type of representation
there is only a single element, no transport direction can be specified.
The quantity of flow entering and leaving the system alone determines
water volume changes within the element.
For lake systems with long residence times and stratification in the
vertical direction, one-dimensional representations are common (see Section
2.2.1). Horizontal layers are imposed and advective transport is assumed
to occur only in the vertical direction. The basic vertical advection/
diffusion transport equation typically used in conjunction with this system
representation is:
It- IT !Y + Dz T? + < c)z±s (2-6)
where C = concentration of specific constituent
C.j = concentration of constituent in upstream flow
DZ = effective diffusion coefficient
A = horizontal surface area of hydraulic element
Qv = vertical flow through element (layer)
Qh n-Qh n = horizontal inflow and outflow to element, respectively
n 5 1 n 5 o
Az _ = element thickness
S = source/sink rates
20
-------
The first term on the right-hand side of the equation represents advective
vertical transport, the second term represents diffusional transport based
on a concentration gradient, the third term represents the horizontal inflow-
outflow, and the fourth term (the source/sink term) represents the rate of
internal production or consumption of the constituent C.
Generally the tributary inflows and outflows are assumed to enter and
leave the lake at water levels of equal density. Since water is essentially
incompressible the inflow is assumed to generate advective flow between all
elements above the level of entry. The elements below this level, contain-
ing higher density water, are assumed to be unaffected. A simplified dia-
gram of this type of lake representation, which illustrates the physical
transport processes, is shown in Figure 2-5.
Qh,
Qh.c
A Dispersion
A Advection
Settli ng
Figure 2-5. One-dimensional, advective-dispersive
vertical transport.
The source/sink term of a mass balance expression generally includes
settling, often of significance in vertical mass transport in lakes. Mix-
ing and turbulence associated with fresh water flow in rivers and tidal
action in estuaries are frequently of significant magnitude to minimize
the settling effect in riverine and estuarine systems (O'Connor, et al.,
1975). Specific settling formulations and values are discussed elsewhere
in this report under specific constituent source/sink expressions.
21
-------
A two-dimensional estuarine model having vertical and longitudinal
spatial differentiation is another type of representation which requires
a vertical transport formulation. A simplified representation illustrat-
ing the transport mechanisms is shown in Figure 2-6.
Dispersion
Advection
Settling
Figure 2-6. Two-dimensional, advective-diffusive transport
in the longitudinal and vertical directions.
In this particular representation, the advection and diffusion pro-
cesses are variable in space in the longitudinal direction as well as in
the vertical. Generally, if vertical differentiation is necessary in a
given model application, settling effects will also be significant.
Three-dimensional system representations, incorporating vertical,
lateral, and longitudinal transport, are rarely employed. The use of
three spatial dimensions introduces substantial additional complexity
over one- and two-dimensional representations. The larger computing
facilities needed for a complete spatial description often severely limits
the amount of biological and chemical detail which can be built into a
model. Frequently, therefore, prototype systems are often represented
with the velocity and concentration field averaged over at least one
spatial dimension.
In Tetra Tech's (1975) three-dimensional model of Lake Ontario, the
wind stress generates advective flow from one element to the next in the
horizontal direction. Horizontal transport is expressed in terms of a
stream function. Once the horizontal velocities through an element are
established, vertical transport is addressed on a basis of continuity.
22
-------
2-3.1.2 Horizontal Advective Transport
All stream models reviewed represent stream systems conceptually as
horizontal, linear networks of segments or volume elements. The process of
advection is assumed to transport a constituent horizontally by movement of
the parcel of water containing the constituent.
The simplest situation encountered is that of a one-dimensional flow-
ing stream or river where the dispersion of mass is neglected because of its
small magnitude relative to the flow. In this case, advection is the major
transport mechanism. This simplification is important in terms of compu-
tational complexity and the amount of information required. The fundamental
equation that governs the transport of material in this type of non-dis-
persive system is:
8C _ 1 9(QC) v r (2_7]
31 ~ ' A T7~~ kcL ^ /;
where C = concentration of substance
A = cross-sectional area
k = first-order decay coefficient
For a river with a constant flow rate and constant dispersion, the
basic form of the one-dimensional equation for horizontal transport
becomes:
3£= _ MiC + E92C + + s (2_8)
31 3x 3 xz
where E = dispersion coefficient (assumed constant over x)
S = source/sink rate for substance C
The term u8C/9x represents the mass flux due to advection.
In horizontally layered, one-dimensional lake models, inputs and out-
puts of water at various depths in the geometric representation are speci-
fied. Models of this type commonly include empirical relationships for
densimetric flow in the vertical direction.
Baca and Arnett (1976) have used a pseudo-two-dimensional, laterally
mixed, analytical-empirical approach to characterize the complex "inter-
flows" through a reservoir and at its spillway(s). The "interflow" is
the horizontal flow between segments, and interflow depth is based on
density compatibility. In this approach, the outflow history of one seg-
ment becomes the inflow regime for the next. The interflow half thickness,
23
-------
labeled D, is computed from Debler's criterion (U.S. Corps of Engineers,
1974):
1/2
D =f 42.0 £ —^-) (2-9)
| 42.
where D = half the thickness of the inflow or withdrawal flow
field, m
Q = inflow or outflow rate, m3/sec
W = the width of the reservoir at inflow or withdrawal
depth, m
(3 = the density gradient at the point of inflow or
withdrawal, kg/m4 (6 negative)
The U.S. Army Corps of Engineers (1974) has two different methods for
distributing inflows and determining the location from which outflows arise,
depending on whether or not the reservoir is stratified. For the stratified
reservoir situation, Debler's criterion is used to determine the flow field
thickness, similar to the approach reported by Baca and Arnett. Outflows
arise from the elements within the region D meters above and below the
centerline of the outlet.
For a partially stratified situation, the water is added to, or with-
drawn from, the zone of complete mixing. The maximum amount of flow which
will remain contained in the completely-mixed zone without encroaching into
the stratified zone is determined using Craya's theory. This maximum flow,
labeled "Craya's critical flow" is defined by:
Q = CwD3/2AP1/2 (2-10)
where Q = Craya's critical flow or the maximum amount of flow which
will remain contained in the zone of complete mixing, m3/sec
C = an empirical constant, C = 0.0742 for surface outlets and
C = 0.1505 for submerged outlets
w = width of reservoir at the inflow or withdrawal depth, m
D = depth of the zone of convective mixing, m
Ap = maximum density difference between the zone of complete mixing
and the stratified zone, kg/m3.
If Craya's critical flow is greater than the rate of inflow or the
rate of withdrawal, the inflow or withdrawal is distributed throughout the
zone of complete mixing, assuming a uniform velocity distribution. Where
the rate of inflow or withdrawal is greater than the maximum that can be
24
-------
accommodated by the zone of complete mixing, the excess is added to or
withdrawn from the stratified zone using Debler's criterion. When the
reservoir is unstratified and complete mixing occurs throughout the
reservoir, inflows and withdrawals are distributed throughout the re-
servoir, assuming a uniform velocity distribution.
In an approach used by Tetra Tech (1976) for distribution of inflow and
outflow, the greatest quantity of water is deposited at the center of the
entry region and the remainder is distributed to the adjacent layers using
a Gaussian distribution. The thickness of the withdrawal layer, d, is
calculated as follows:
d = 2.0
1/2
(2-11)
where q = volumetric discharge per unit width of reservoir
p = water density at centerline of outlet
3 = 9p/9z, the density gradient with depth at the
centerline of outlet (g positive)
g = gravitational constant
This model was used in simulating the effects of pumped storage and
the introduction of water from an external source into an impoundment. It
simulates the effect on the far field and does not seek to represent the
near field phenomena of a thermal plume. Ambient water is taken from the
reservoir and mixed with the released water at the rising zone. To satisfy
continuity, downward advection is generated at the zone of entrainment, and
upward advection is assumed to occur at and above the zone of deposition.
Simons (1976) employs two separate models, one for vertically homog-
enous and one for vertically stratified lake systems. Each of these two
models provides a complete horizontal description (both lateral and longi-
tudinal differentiation). The vertically integrated model assumes a hydro-
static, homogeneous, and incompressible body of water in the vertical
direction and is generally applicable in simulations of storm surges. The
stratified lake model prescribes depths and temperatures for each layer as
a function of horizontal coordinates based on actual observations from a
specified time period. To facilitate the transition from homogeneous to
stratified dynamics, the equations for the stratified model are formulated
in terms of vertically integrated flow and the velocity difference across
the thermocline.
The governing equations for the vertically-mixed model are given as
follows:
25
-------
3t
9x
_
9t 9y
- fU + - + AV2V (2-14)
where H = local undisturbed depth of lake
C = elevation of the free surface above a
specified equilibrium level
U,V = components of vertically integrated volume
transport along x-axis and y-axis of coordinate
system superimposed on the lake surface
f = Coriolis parameter (taken to be 10" /sec)
T ,T = wind stress at surface in x and y
^ directions, respectively
T, ,T, = bottom frictional stress in x and y
y directions, respectively
p = density
A = coefficient of sub-grid-scale diffusion of
momentum (estimated to be lO^cm^/sec)
2
V = horizontal Laplace operator
The nonlinear inertial terms were removed from Equations (2-13) and (2-14)
based on earlier studies (Simons and Jordan, 1971).
Equation (2-12) states that the water surface must rise where currents
converge and fall where they diverge. Equations (2-13) and (2-14) repre-
sent current oscillation along the two coordinates resulting from the
following forces and effects:
1) the pressure gradient experienced by the water mass below a
sloping surface
2) the tendency of currents to deviate to the right of flow (north-
ern hemisphere) under the influence of earth's rotation (Coriolis
effects)
26
-------
3) the force exerted by the wind acting on the water surface
4) bottom stress
5) mixing of water masses with different momenta (turbulent
diffusion)
As discussed in Section 2.2.1.3, a particularly effective approach
for modeling certain types of estuaries is the construction of a pseudo-two-
dimensional "link-node" network. The modified tidal hydrodynamics program
of Chen and Orlob (1975) solves a set of motion equations written for the
links of the system and a set of continuity equations written for the nodes,
both sets of equations being one-dimensional in the horizontal direction.
The basic motion equation is given as:
3u_
3t
3x
9 ^7
(2-15)
where u = mean velocity in the x-direction
x = distance
g = acceleration due to gravity
h = height of water surface above a reference plane
S = friction gradient along x-axis
/\
The equation of continuity is expressed by:
(2-16)
3t 9x
where q = tidal flow per unit width along the x-axis
X
2.3.2 Dispersive Transport
2.3.2.1 Introduction
The major emphasis of this introduction will be to show how disper-
sive effects are incorporated into the equations of motion and continuity
for turbulent flow situations, where these equations are simplified
through either temporal or spatial averaging. If it were feasible to
solve for instantaneous velocities or concentrations without simplifying
the governing equations of motion, there would be no need for dispersive
terms in the equations. However, since the exact equations of motion are
intractable, some modifications to these equations becomes necessary, and
dispersive transport terms arise.
27
-------
A consequence of temporal averaging of either instantaneous velocity or
concentration is to produce a smoothed velocity or concentration response
curve over time. Figure 2-7 illustrates both instantaneous velocity and
time-smoothed curves. The time smoothed velocity curve is not to be con-
fused with a steady-state velocity, but instead is given by:
v = V + r (2-17)
where V = instantaneous velocity
V = time-smoothed velocity
V = velocity deviation from the time-smoothed velocity
The velocity component \T is a random component of velocity which vanishes
when averaged over time (i.e., V" = 0). Note that the time period of
averaging can be quite short, and is small when compared to the smallest
time intervals over which velocity predictions are generally made in estu-
aries, lakes, or streams. The latter time intervals typically range from
minutes to days or even greater between successive predictions.
>v
o
"o>
V = time smoothed velocity
V = instantaneous velocity
V
V = V- V
0
Time, t
0.05 sec
Figure 2-7. Oscillation of velocity component about a mean
value (redrawn after Bird, et al_., 1960).
28
-------
By using a temporal averaging process (or performing a similar process
called ensemble averaging) the stochastic component of motion is smoothed
out of the momentum and continuity equations, while the deterministic com-
ponent of velocity and concentration are retained. However, an additional
difficulty arises in both the momentum and continuity equations. Cross-
product terms result, such as V^Vx and V^Vy in the case of the momentum
equation, and VXC" and VyC' in the case orthe continuity equation (where C'
is the instantaneous concentration fluctuation). These terms are not iden-
tically zero, since one term in the product is not independent of the other.
In the case of the momentum equation these terms are called turbulent momen-
tum flux terms, while in the case of the continuity equation they are called
turbulent mass flux terms. It is through these terms that eddy viscosity
and eddy diffusivity enter into the three-dimensional momentum and continu-
ity equations.
In order to solve the time-smoothed equation, the time averaged cross
product terms must be expressed as functions of the time averaged variables.
Numerous empirical expressions have been developed to do this. The expres-
sions most often applied are analogous to Newton's law of viscosity for
turbulent momentum transport and Fick's law of diffusion for turbulent mass
transfer, although no strong theoretical arguments exist for choosing these
expressions. Expressed quantitatively these relationships are of the form:
t 3V"
= " - (2-18)
r— D (2-19)
where u = eddy viscosity
D - eddy diffusivity
Both the eddy viscosity and diffusivity are functions of position, and hence
vary in the x,y, and z directions. One might expect there to be a relation-
ship between ut and Dt. Indeed, Bird, et_ aj_. (I960) has suggested the ratio
of eddy viscosity to eddy diffusivity is generally in the range 0.5 to 1.0,
although the extent of turbulent motion may affect this value.
In natural water bodies the values given by Equations (2-18) and (2-19)
swamp their counterparts on the molecular level. Hence the Newtonian
expression for momentum transport in laminar flow, and the Fickian expres-
sion for molecular diffusion may be disregarded for natural water bodies.
This relationship of eddy diffusivities and molecular diffusion coefficients
is depicted graphically in Figure 2-8.
The process of spatial averaging often is done over one or two dimen-
sions. For example, vertically averaged two-dimensional momentum and mass
29
-------
I04+
io2+
o
CD
CM
h-
z
UJ
^^J O
_ ~£
LL
LJ
O
O
5i54-
LJ_
5 io^t
Id8+
-10
\Q-L-
EDDY DIFFUSION'
' Horizontal, Surface Waters
EDDY DIFFUSION'
' Vertical, Thermocline and Deeper
Regions in Lakes and Oceans
-Heat in H20
MOLECULAR DIFFUSION-'
' Salts and Gases in HoO
I
-Proteins in
THERMAL DIFFUSION
' Salts in H20
;onic Solutes in
Porous Media
[Sediments, Soils)
Figure 2-8. Diffusion coefficients characteristic of
various environments (redrawn after
Lerman, 1971).
30
-------
conservation equations are often useful. The vertically averaged mass
balance equation only will be discussed here for illustrative purposes.
For the vertically averaged case, the concentration is expressed by a
vertically averaged term and a deviation term:
C = Ch + C^ (2-20)
where C = concentration of constituent at a single point
in space (as appears in the three-dimensional mass
conservation equation)
C^ = vertically averaged concentration
C^j = deviation from C, at any point in the water column
The vertically averaged concentration term is given as the integral of
C over depth:
, /«h
r = —
h h
h
/ Cdz (2-21)
where h = water depth
z = vertical coordinate
When this integration process is performed on the three-dimensional mass
conservation equation, cross product terms appear in the resulting equation,
just as they do when temporal averaging is done in the three-dimensional
case. These terms are produced because vertical gradients generally exist
in both the concentration and velocity fields. As was done in the three-
dimensional case, the cross product terms can be replaced by Fickian-type
diffusion terms. The resulting "effective" diffusion coefficients, as they
are sometimes called, can be expected to differ considerably from those
appearing in the three-dimensional equations.
By continuing the process of spatial averaging, one-dimensional mass
continuity equations can be obtained. These equations express changes only
along the main flow axis, termed the longitudinal direction. The diffusion
term appearing in the one-dimensional mass transport equation is the sum of
two components. The first term is an integral involving the cross product
of the longitudinal velocity and concentration deviations and represents
the mass transport associated with the nonuniform velocity distribution.
The second term is a derivative involving the spatial mean value of the
three-dimensional eddy diffusivity. The integral component is represented
by a Fickian expression, and the sum of the two diffusive effects is termed
longitudinal dispersion.
31
-------
It is important to understand that although a specified water body at
a given instant has a particular degree and pattern of turbulence, the
method by which that water body is modeled will dictate the type and magni-
tude of the diffusion term appearing in the simulation equations. Thus, a
one-dimensional model uses different diffusion terms than does a three-
dimensional model of the same system.
2.3.2.2 Vertical Dispersive Transport
In a lake environment, dispersion is generally caused by a different
type of diffusional phenomenon than those predominant in a river or estuary
system. Whereas turbulent diffusion resulting from shear stresses and
irregular channel configuration and dispersion due to tidal mixing are often
the principal mechanisms affecting dispersion in the latter system types,
eddy diffusivity caused by wind action on the surface waters is often the
primary mechanism controlling dispersion in lakes. The wind action creates
eddies on the water surface and the dispersive effects are then trans-
mitted to the deeper layers by the action of shear stresses.
Tetra Tech (1975) has used the following empirical expression for
computation of the vertical eddy thermal diffusivity, Kv, in their three-
dimensional hydrodynamic simulation model of Lake Ontario:
(2-22)
-*-
where T = wind stress
w
PQ = density of fresh water at 4°C
R. = Richardson number
In this model eddy conductivity was ignored in the horizontal plane, being
assumed small in comparison to the advective effects.
Lake systems that are represented geometrically as a series of com-
pletely mixed horizontal slices consider advective and dispersive transport
processes to occur in the vertical direction alone. Baca and Arnett (1976),
in their one-dimensional hydrothermal lake model, proposed the following
expression for determining the one-dimensional vertical dispersion coef-
ficient:
(2-23)
32
-------
2
where DV = vertical dispersion coefficient, m /sec
z = depth, m
Vw = wind speed, m/sec
d = depth of thermocline, m
2
a-|,a2 = empirical constants, m /sec and m respectively
The following table of values (Table 2-1) for a-j and a2, as given by
Baca and Arnett (1976), were obtained from previous model applications.
TABLE 2-1. VALUES FOR EMPIRICAL COEFFICIENTS a] AND a2
Lake
American Falls
Lake Washington
Lake Mendota
Lake Wingra
Long Lake
Description
well -mixed
stratified
stratified
we 11 -mixed
1 inearly
stratified
Max.
Depth (m)
18
65
24
5
54
2
a-^m /sec)
c
1 x 10 b
1 x 10"6
_7
5 x 10 '
_R
5 x 10 b
5 x 10"6
a2 (m)
-d
1 x 10 4
1 x 10"5
5
5 x 10 °
-4
2 x 10 ^
5 x 10"5
For reservoir systems, the U.S. Army Corps of Engineers (1974) computes
the effective vertical diffusion rate between elements as a function of the
density gradient. This is done in a manner similar to that in the Lake
Ontario model (Tetra Tech, 1975). When the density gradient is less than
the critical, as required for stability between layers, the effective
diffusion coefficient is assumed constant. Mathematically, this is stated
as follows:
2
where D = effective diffusion coefficient, m /sec
2 4-4
A-, = maximum diffusion rate, m /sec: 0.1xlO~
-------
E = density gradient, m~
E . - density gradient at which Dc attains its maximum
cnt value (critical stability), 1/m
These relationships are shown graphically in Figure 2-9 (Water Resources
Engineers, 1969).
Lick, et aj_. (1976) applied a three-dimensional water quality model to
Lake Michigan in order to simulate the dispersion of heat in the discharge
from the Point Beach Power Plant. They assumed that the vertical eddy
viscosity and eddy thermal conductivity were identical with vertical eddy
diffusivity. The specific relationship they used was:
Dv = 50 - 200 |j (2-26)
2
where D = vertical eddy diffusivity, cm /sec
T = water temperature, °C, at depth z, m
z = depth, m
Thomann, e_t aj_. (1975) have used time- and space-varying vertical
diffusivities in modeling studies for Lake Ontario. They varied the verti-
cal diffusivity on a yearly basis, and also accounted for spatial differ-
ences between the main lake and near-shore regions. Figure 2-10 shows these
relationships. Murphy, during the International Field Year for the Great
Lakes, as reported by Thomann, et_ al_. (1975), found that values of the ver-
tical diffusivity coefficient generally ranged from 0.1 cm2/sec to 22 cm2/sec,
depending on effects such as wind conditions and density stratification.
2.3.2.3 Horizontal Eddy Diffusive Transport
Generally, horizontal eddy diffusivity is several orders of magnitude
greater than the vertical eddy diffusivity. The Journal of the Fisheries
Research Board of Canada (Lam and Jacquet, 1976) reported a range of values
for the horizontal diffusivity in lakes from 10^ to 10° cm2/sec. Thomann,
e_t ail_. (1975) used a horizontal diffusivity of 9,000 cm^/sec for most parts
of Lake Ontario. However, the horizontal coefficients were set to zero for
the stratification period to simulate the thermal bar effect which produces
a restriction of the mixing of near-shore waters with main lake waters (see
Figure 2-11).
Lam and Jacquet (1976) obtained Equation (2-27) for the horizontal
eddy diffusivity for lakes, Dn (Crn2/sec), based on experimental results:
1.3
.UUODL
where L = length scale of grid, cm
Dh = .0056L1'3 (2-27)
34
-------
CO
en
10.0
8.0
6.0
- 4.0
X
O
UJ
I 2.0
^> i.O
— .8
z .6
UJ
O 4
u.
Ui
O
(O
.2
± .08
O
uj -06
5 -04
UJ
u.
u.
111 .02
.01
—?JL
•^Mi ^BMMH
S
y^
\
POINT OF CRITICAL STABILITY^ ^
•^ •••
^N
(
•^•••^ a^BB
(?5W//
HHH^ HM^H
;
^^^M •
«•••»
^IBBB
,1
<-
X
N
^
X,
y
L
s
— N
vj
s
%
X
h
X
UPPER
ENVELOPE
L /
N'
v!
/
v.
N
«
X
C
X
y
>
.7
\
-7
X
LOWER
ENVELOPE
\
s.
V
\
^
X
X
\
^ y
X,
^
X
X
V
^ V
^<
X
.01 .02 .04.06.08.1 .2 .4 .6.81.0 2.0 4.06.08.010 20
DENSITY GRADIENT (E), I/meters x I06
40 6080100 200 400
Figure 2-9. Log of effective diffusion versus log of density gradient
(Water Resources Engineers, 1969),
-------
CO
en
O
O>
CO
\
(M
O
20-
10-
0
20
CO
:D
U.
U.
Q
_J
y io-
QL
UJ
0-
VERTICAL EXCHANGE-MAIN LAKE
I I I I I I I I I I I I I
VERTICAL EXCHANGE - NEAR SHORE
1 F I M I A I
M
0 30
M'J'J'A'S'O'N'D1
60 90 120 150 180 210 240 270 300 330 360
Figure 2-10. Assumed vertical dispersion coefficients for three-
dimensional model (after Thomann, et al., 1975).
-------
HORIZONTAL EXCHANGE
o
CD
CO
X.
CM
£
o
H
CO
9000
6000
3000
NEAR SHORE WITH NEAR SHORE
MAIN LAKE WITH MAIN LAKE
-
-
I i i i i i i i i i i
Q
_l 9000
2 6000
O
g 3000
O
X
_L
NEAR SHORE WITH MAIN LAKE
_L
_L
J_
_L
_L
_L
J
M
AMJ JASOND
I I I I I I I I I I I I I
0 30 60 90 120 150 180 210 240 270 300 330 360
Figure 2-11. Horizontal diffusivities used in Lake Ontario
modeling studies, illustrating the "thermal bar
effect" (after Thomann, et_ al_., 1975).
-------
For a grid size larger than 20 km, the diffusivity is expected to be
essentially constant (106 cm2/sec).
2.3.2.4 Longitudinal Dispersjcir[
As previously discussed in Section 2.3.2.1, longitudinal dispersion
is the "effective diffusion" that occurs in one-dimensional mass transfer
equations that have been integrated over the cross sectional area perpen-
dicular to flow. This one-dimensional approach to modeling has often been
applied to tidal and nontidal rivers, and to some estuaries. Dispersion
in estuaries and tidal rivers will be discussed first, in this section,
followed by a brief discussion of dispersion in streams.
The magnitude of the one-dimensional' dispersion coefficient in estu-
aries and tidal rivers is determined in part by the time scale for which
the simulation is performed. The time scale specifies the interval over
which quantities that generally change instantaneously, such as tidal
current, are averaged. For shorter time scales the simulated hydrodynamics
and water quality relationships are more accurately resolved and hence, in
such models, smaller dispersion coefficients are needed than in those which,
for example, have hydrodynamics averaged over a tidal cycle.
The magnitude of the dispersion coefficient can also be expected to
change as a function of location within an estuary. Since the one-dimen-
sional dispersion coefficient is the result of spatial averaging over a
cross section perpendicular to flow, the greater the deviation between
actual velocity and the area-averaged velocity, and between actual constit-
uent concentrations and area-averaged concentrations, the larger will be
the dispersion coefficient. These deviations are usually largest near the
mouths of estuaries due to density gradients set up by the interface be-
tween fresh and saline water. Strong tidal currents may also result in
large dispersion coefficients.
Because of the time scale and location dependency of the dispersion
coefficient, it is convenient to divide the discussion of dispersion into
real time and tidally averaged time expressions, and then to subdivide these
according to estuarine location, i.e., the salinity intrusion region and
the freshwater tidal region. The salinity intrusion region is that portion
of the estuary where a longitudinal salinity gradient exists. The loca-
tion of the line of demarcation between the salinity intrusion region and
the freshwater tidal region varies throughout the tidal cycle, and also
depends on the volume of freshwater discharge. It should also be noted
that the freshwater tidal region can contain saline water, if the water
is of uniform density throughout the region (TRACOR, 1971). There is at
present no analytical method for predicting dispersion in the salinity
intrusion region of estuaries. However, because of the presence of a con-
servative constituent (salinity), empirical measurements are easily per-
formed. In the freshwater tidal region, analytical expressions have been
developed, while empirical measurements become more difficult due to the
lack of a naturally occurring conservative tracer.
38
-------
For real time simulations in the constant density region of estuaries
and tidal rivers, the following expression has been proposed (TRACOR, 1971):
EL = 10° n Umax R
where E^ = longitudinal dispersion coefficient
in the constant density region, ft^/sec
n = Manning's roughness coefficient, ft '
Umax = maximum tidal velocity, ft/sec
RH = hydraulic radius, ft
Feigner and Harris (1970) used the following empirical expression to
represent real time longitudinal dispersion for the constant density region
in their estuary water quality model:
= C1 E1/3Le4/3 (2-29)
where E^ = longitudinal dispersion coefficient,
lengths/time
E - rate of energy dissipation per unit mass
L = mean size of eddies participating in the
mixing process
C-, = function of relative channel roughness
For computational purposes, Feigner used the following simplification:
EL = 0.042 u R (2-30)
where R - hydraulic radius, ft
|u| = absolute value of velocity, ft/sec
The determination of real time dispersion coefficients in the salinity
intrusion region requires field data on salinity distribution. Once the
field data have been collected, the magnitudes of the dispersion coeffi-
cients are found by fitting the solution of the salinity mass transfer
equation to the observed data. As reported in TRACOR (1971), this tech-
nique has been applied to the Rotterdam Waterway, an estuary of almost
uniform depth and width. The longitudinal dispersion coefficient was found
to be a function of x, the distance measured from the mouth (ft), as follows:
39
-------
EL = 13000
where E, = real time longitudinal dispersion
coefficient in salinity intrusion
region, ft^/sec
L = length of entire tidal region of
the estuary
(2-31)
At the estuary mouth, E|_ was found to be 13:
using the technique described above. Under
000
2 2
ft /sec or 40 mi /day by
same conditions in a con-
_ oirt i i_/v_\_i M ksw v \_ • jwm\_ \^wii<»iiuiwii.j 111 u \_/w 1 1 •
stant density region, Equation (2-28) would predict an E|_ of 175 ft2/sec
or 0.5 mi'2/day. This illustrates the large difference that can be expected
between the real time dispersion coefficient in the salinity intrusion
region of an estuary and the constant density region.
For tidal ly averaged or net nontidal flow simulations, since the system
hydrodynamics are averaged over a tidal period or longer, the dispersion
coefficient must include the additional effects of tidal mixing, which do
not need to be included for real time simulations. No general analytical
expressions exist for this coefficient. Hence, values must be determined
based on observed data, or based on empirical equations having parameters
that are determined from observed data. This limitation exists for both the
constant density and salinity intrusion regions of the estuary.
In their one-dimensional tidally averaged estuary model, Johanson,
e_t aJL (1977) used an empirical expression, comprised of three principal
components (tidal mixing, salinity gradient, and net freshwater advective
flow) for the dispersion coefficient. The relative location in an estuary
where each of these factors is significant, and their relative magnitudes,
are shown in Figure 2-12.
The expression used by Tetra Tech is:
u + a.
y +
+ c,
/AS)
U*/
-y/ "4
t
where E, = tidally averaged dispersion coefficient, ft'"/sec
C-j = tidally-induced mixing coefficient
(dimensionless)
y = tidally averaged depth, ft
\u\ = tidally averaged absolute value of
velocity, ft/sec
(2-32)
40
-------
f-
LU
O
u.
LJ
O
O
(f)
cr
UJ
Q.
CO
Salinity Gradient Mixing
Freshwater Mixing
7-.-.-.-.-.-.•.•.•.-.•.V'.•.'.-.
•I-:-:-:-:-:-:-:-:-:-:/:-:-:-:-
MOUTH
HEAD
Figure 2-12.
Factors contributing to tidally averaged dispersion coefficients
in the estuarine environment (modified after Zison, et al., 1977).
-------
a = standard deviation of velocity, ft/sec
a = standard deviation of depth, ft
C, = density-induced mixing coefficient,
ft3/sec/mg/l-salinity
-7—= salinity gradient, mg/l/ft
AX
The first term on the right side of Equation (2-32) represents mixing
brought about by the oscillatory flows associated with the ebbing and flood'
ing of the tide. The second term represents additional mixing when longi-
tudinal salinity gradients are present.
One common method of experimentally determining the tidally averaged
dispersion coefficient is by the "fraction of freshwater method," as
explained by Officer (1976), The expression is:
v - Rs = _R(t£L) (7
\ Alds/dxJ Afd~f/dx7 [
2
where K. = tidally averaged dispersion coefficient, ft /sec
s = mean salinity at a particular location averaged
over depth, mg/1
2
A = cross-sectional area normal to flow, ft
R = total river runoff flow rate, cfs
f = freshwater fraction = —- , unitless
a = normal ocean salinity of the coastal water into
which the estuary empties, mg/1
x = distance along estuary axis, ft
K|_ can be calculated at any location within the estuary if the river flow,
cross-sectional area, and salinity or freshwater fraction distributions
are known.
Hydroscience (1971) has collected values of tidally averaged dis-
persion coefficients for numerous estuaries, and these values are shown
in Table 2-2. Some additional information for these and other estuaries
will be given in Table 3-6.
In his book, Officer (1976) reviews studies performed in a number of
estuaries throughout the world. He discusses the dispersion coefficients
which have been determined, and a summary of values for these estuaries
is contained in Table 2-3. Many values were developed using the fraction
of freshwater method just discussed.
42
-------
TABLE 2-2. TIDALLY AVERAGED DISPERSION COEFFICIENTS FOR SELECTED ESTUARIES
(FROM HYDROSCIENCE, 1971)
CO
Estuary
Delaware River
Hudson River (N.Y.)
East River (N.Y.)
Cooper River (S.C. )
Savannah R. (Ga., S.C.)
Lower Raritan R. (N.J.)
South River (N.J.)
Houston Ship Channel (Texas)
Cape Fear River (N.C. )
Potomac River (Va. )
Compton Creek (N.J.)
Wappinger and
Fishkill Creek (N.Y.)
Freshwater
Inflow
(cfs)
2,500
5,000
0
10,000
7,000
150
23
900
1,000
550
10
2
Low Flow
Net Nontidal
Velocity (fps)
Head - Mouth
0.12-0.009
0.037
0.0
0.25
0.7-0.17
0.047-0.029
0.01
0.05
0.48-0.03
0.006-0.0003
0.01-0.013
0.004-0.001
Dispersion
Coefficient
9 *\
(mi^/day )
5
20
10
30
10-20
5
5
27
2-10
1-10
1
0.5-1
*1 mi2/day = 322.67 ft2/sec
-------
TABLE 2-3. TIDALLY AVERAGED DISPERSION COEFFICIENTS
(FROM OFFICER, 1976)
Estuary
Dispersion
Coefficient
Range
(ft2/sec)
Comments
San Francisco Bay
Southern Arm
Northern Arm
200-2,000
500-20,000
Hudson River
4,800-16,000
Narrows of Mercey
1,430-4,000
Potomac River
65-650
Severn Estuary
75-750
(by Stommel)
580-1,870
(Bowden)
Measurements were made at slack
water over a period of one to a
few days. The fraction of
freshwater method was used.
Measurements were taken over
three tidal cycles at 25
locations.
The dispersion coefficient was
derived by assuming K|_ to be
constant for the reach studied,
and that it varied only with
flow. A good relationship
resulted between K|_ and flow,
substantiating the assumption.
The fraction of freshwater
method was used by taking mean
values of salinity over a tidal
cycle at different cross
sections.
The dispersion coefficient was
found to be a function of dis-
tance below the Chain Bridge.
Both salinity distribution
studies (using the fraction of
freshwater method) and dye
release studies were used to
determine K|_.
Bowden recalculated K|_ values
originally determined by
Stommel, who had used the
fraction of freshwater method.
Bowden included the fresh-
water inflows from tributaries,
which produced the larger
estimates of K|_ -
(continued)
44
-------
TABLE 2-3 (continued)
Dispersion
Coefficient
Range
Estuary (ft2/sec) Comments
Tay Estuary 530-1,600 The fraction of freshwater
(up estuary) method was used. At a given
1,600-7,500 location, KL was found to vary
(down estuary) with freshwater inflow rate.
Thames Estuary 600-1,000 Calculations were performed
(low flow) using the fraction of fresh-
3,600 water method, between 10 and
(high flow) 30 miles below London Bridge.
Yaquina Estuary 650-9,200 The dispersion coefficients for
(high flow) high flow conditions were sub-
140-1,060 stantially higher than for low
(low flow) flow conditions, at the same
locations. The fraction of
freshwater method was used.
45
-------
In streams, one of the more popular equations for estimating dis-
persion is the Elder equation, which is given as:
EL = 22.6 n u D°'833 (2-34)
where n = Manning's coefficient, ft
u" = mean velocity, ft/sec
D - depth of flow, ft
2
E. = longitudinal dispersion coefficient, ft /sec
Bansal (1976), however, suggests that the longitudinal dispersion rates pre-
dicted using the above formulation tend to be low, resulting in higher con-
centrations of shorter duration than when better estimates of the dispersion
coefficients are used. He offers the following alternative expression for
computing the longitudinal dispersion coefficient, based on the Reynold's
number for flow and the channel configuration of the stream:
log K ^= 6.45 - 0.762 log t^-\ (2-35)
o
where EL = longitudinal turbulent diffusion coefficient, ft /sec
K = regional dispersion factor (equal to 1 for
Big Blue River), unitless
u" = average velocity (Q/A), ft/sec
v = x/tp where x = reach length and tp = time
to peak arrival of constituent concentration
at sampling station, ft/sec
A
H = „- where A = cross-sectional area and B = top
width of flow, ft
p = density, .lb/ft3
y = coefficient of viscosity, Ib-sec/ft
Using this alternative expression, Bansal reported considerably higher
values for the longitudinal dispersion coefficient than obtained with the
same conditions on the Big Blue River using the Elder expression. Bansal's
values ranged from 28.6 to 380.7 ft^/sec, where previously they had been
0.91 to 2.29 ft2/sec using the Elder expression.
Gloyna, ejt al_. (1971) also calculated longitudinal dispersion coeffi-
cients for stream velocities ranging from 0.33 to 3.30 ft/min using time-
46
-------
concentration curves from Rhodamine-B dye studies. The dispersion coeffi-
cients were found to be representable by an empirical relationship of the
form:
DL = 3.26 u°'607 (2-36)
where u = average velocity in ft/min
D| = longitudinal dispersion coefficient,
ft2/min
Battelle (1974), in their Gray's Harbor/Chehalis River application,
assumed diffusion to be negligible because "of the nebulous nature of
hydrodynamic dispersion coefficients." These investigators further contend
that finite-difference solutions to advection-diffusion equations introduce
"pseudo-dispersion effects" which can completely mask the actual solution.
Hydroscience (1971) has also suggested disregarding longitudinal dispersion
in streams upstream of tidal influence, for simplified applications.
2.3.2.5 Tidal Exchange
The hydrodynamic behavior of an estuary is controlled primarily by
tides at the mouth, although at times of high runoff, tributary flows may
dominate. During a tidal day (about 25 hours) normally two high and two
low tides occur. However, because of the complexity of tidal patterns,
diurnal tides (occurring once daily) and mixed tides (varying between once
and twice daily) can occur.
During flood tide, ocean water moves into the system and estuarial
water is pushed inland and mixed with fresh water entering through the
landward boundary. The ebbing tide is characterized by a reversal of flow
and a discharge of the freshwater-ocean water mixture through the seaward
boundary to the ocean. Some fraction of estuarial water is lost to the
ocean. The remainder mixes with new ocean water and returns to the estuary
on the succeeding flood.
The transition between flood and ebb conditions is called "slack water",
or "slack tide"; it occurs at different times throughout the system depending
on the velocity of propagation of the tidal wave from the mouth of the
estuary. Since tidal motions are dominated by the moon, a complete tidal
cycle has a period of approximately 25 hours, a lunar day. Variations in
amplitude and phase of the tide are associated with the combined effects
of the sun and the moon from month to month over the year.
Generally, in the modeling of estuarine systems, tidal information is
necessary to provide an indication of the amount of energy available for
mixing (being proportional to the square of the tidal range), and to
determine the amount of water available for dilution. In addition, the
excursion of pollutants depends on the range and duration of ebb and flood
47
-------
tides. The tidal heights or currents are often required, therefore, as
natural boundary conditions.
The primary purpose of a mathematical tidal hydrodynamics model,
accordingly, is to provide quantitative temporal descriptions of tidal
flows, current velocities, water levels, and tidal volumes. Prediction of
tidal heights by harmonic analysis is well-established, with discrepancies
from predicted heights occurring due to storm surges and freshets from
local precipitation (Callaway, 1971), In the absence of desired tidal
data, predictions in the Coast and Geodetic Survey Tide Tables (now pub-
lished by the National Oceanic and Atmospheric Administration) for the
tidal elevations at the four extreme stages of the tide can be used along
with the time of occurrences (Feigner and Harris, 1970). After the tidal
elevations have been referenced to the model datum, a harmonic regression
(or Fourier analysis) can then be performed.
Tidal current prediction is reportedly less well behaved from the
viewpoint of predictability than are heights (Callaway, 1971). Tidal
currents are composed primarily of a transitory component associated with
the mean hydrologic flux and an oscillatory component induced by an imposed
tide at the estuarine boundary. In addition, there may be components of
the current attributed to wind stress, density differences, local runoff
and the effect of the earth's rotation. Of the secondary effects, wind
stress is usually of interest in shallow, well-mixed estuaries like San
Francisco Bay and should be considered in special cases.
Chen and Orlob (1975) have adopted a simple and stable method for
representing tidal fluctuations. The method is based on the following
assumptions:
1. The first increment of ebbing water at the estuary
mouth is considered lost to sea.
2. The last increment of ebbing water before a tide
reversal at the estuary mouth returns to the
estuarine system with no change in quality; i.e.,
the first increment entering on the flooding tide
is equivalent to the last leaving on the preceding
tide (boundary condition no. 1).
3. The last increment of water to enter a system on a
flooding tide is equivalent in quality to raw sea
water (boundary condition no. 2).
4. Water entering the system between the beginning
and end of the flood tide may be described by
interpolating linearly between the two bounds
according to time of entry.
Based on this procedure, one need only specify the quality of sea water
to compute tidal exchange.
48
-------
2.3.3 Heat Budgets
The transfer of energy which occurs at the air-water interface is
generally handled in one of two ways in river, lake, and estuary models. A
simplified approach is to input temperature values directly and avoid a
more complete formulation of the energy transfer phenomena. This approach
is most often applied to those aquatic systems where the temperature regime
is_easily described. Alternatively, the various energy transfer phenomena
which occur at the air-water interface can be considered in a heat budget
formulation.
In a complete atmospheric heat budget formulation, three mechanisms
are generally considered: radiation (solar and atmospheric), evaporation,
and convection. Typically, the combined influence of these mechanisms is
represented by a net external heat exchange (flux) term, H. This term may,
in some instances, be input to the model as a specified value (Lick, et al.,
1976; Tetra Tech, 1975), but the importance of time and spatial variations
should be considered carefully before so doing.
Lombardo (1973) defines the heat exchange across the air-water inter-
face, H, with th\e following empirical expression:
H = S. - (60 + 6.2 T - 5,4 T )- 1634E - 0.89u (T - T, ) (2-37)
1 \ W a /
where S- = incident solar radiation
T = air temperature, °F
T = water temperature, °F
u = wind velocity, mph
E = evaporation, in/hr = 0.000073u (p,, - pj
W d
p = the vapor pressure for water in mb
w
p = the vapor pressure for air in mb
a
The net external heat flux, H, is also often formulated as an algebraic
sum of several component energy fluxes in some models (e.g., Baca and
Arnett, 1976; U.S. Army Corps of Engineers, 1974; Thomann, e_t aj_., 1975).
A typical expression is given as (all units are Kcal/m2-hr):
H = Qs ' Qsr + Qa - Qar - %r - Qe * Qc (2-38)
49
-------
where H = net surface heat flux
Q = shortwave radiation incident to water surface
0 = reflected shortwave radiation
xsr
Q = incoming long wave radiation from the
a atmosphere
Q = reflected long wave radiation
ar
Q = back radiation emitted by the body of water
br j
Q = energy utilized by evaporation
Q = energy convected to or from tne body of water
c at the surface
These flux components can be calculated within the models from semi-
theoretical relations, empirical equations, and basic meteorological data.
Depending on the algebraic formulation used for the net heat flux term and
the particular empirical expressions chosen for each component, all or some
of the following meteorological data may be required: atmospheric pressure,
cloud cover, wind speed and direction, wet and dry bulb air temperatures,
dew point temperature, shortwave solar radiation, relative humidity, water
temperature, latitude, and longitude.
Lombardo (1972) represents the net shortwave solar radiation, Qsw
(langleys/day), with the following expression:
(2-39)
where Q = shortwave radiation at che surface
(langleys/day)
R = reflectivity of water = 0.03, or alternately:
R = AaB (A,B given below in Table 2-4)
a = sun's altitude in degrees
TABLE 2-4, VALUES FOR SHORTWAVE RADIATION COEFFICIENTS A AND B
(LOMBARDO, 1972)
Cloudiness
A
B
Clear
1.18
-0.77
Scattered
2.20
-0.97
Broken
0.95
-0.75
Overcast
0.35
-0.45
50
-------
Various methods are available by which solar radiation can be deter-
mined. Hydrocomp has two computerized methods available for this compu-
tation (Lombardo, 1972).
The U.S. Army Corps of Engineers (1974) considers the net shortwave
solar radiation rate (Qs - Qsr) as a function of sun angle, cloudiness,
and the level of particulates in the atmosphere. Chen and Orlob, as reported
by Lombardo (1973), determine the net shortwave solar radiation by consider-
ing absorption and scattering in the atmosphere.
The U.S. Army Corps of Engineers (1974), Lombardo (1972), and Chen and
Orlob (1975), consider the net long wave radiation rate (Qa - Qar) as a
function of air temperature and cloudiness according to the following
expression:
Qat - 1.23 x 10"16 (l + 0.17 - CLOUD2 jrra + 273J6 (2-40)
where Qat = net long wave atmospheric radiation,
kcal/m^/sec
CLOUD = cloud cover, fraction
Ta = dry bulb air temperature, °C
This expression assumes the reflectivity of water for atmospheric radiation
is 0.03.
The Stefan-Boltzman expression is most often used to determine back
radiation by the water body:
Qbr= 0.97 a Tw4 (2-41)
where Qur = long wave back radiation, cal/m^-sec
T = water temperature, °K
a - Stefan-Boltzman constant =
1.357 x TO'8, eal/m2-see/°K4
The U.S. Army Corps of Engineers (1974) uses the following lineari-
zation of Equation (2-41) to express the back radiation emitted by the
water body:
Q. = 73,6 + 1.17 T (2-42)
vbr
where T = water temperature, °C
In the range of 0° to 30°C, this linear function has a maximum error of less
than 2.1% relative to Equation (2-41).
51
-------
Heat loss due to water evaporation, Qe, is represented by the U.S.
Army Corps of Engineers (1974), Chen and Orlob (1975), and Lombardo (1972),
with the following expression:
Qe = P Lw E (2-43)
2
where Qe = heat loss due to evaporation, kcal/m -sec
3
p = fluid density, kg/m
L = latent heat of vaporization, kcal/kg
E = evaporation rate, m/sec
The evaporation rate, E, is approximated by the semi -empirical function:
E = (a + bvWes - ej (2-44)
where a,b = empirical coefficients
V = wind speed over water surface, m/sec
e = saturation vapor pressure at the surface
water temperature, mb
ea = vapor pressure of the overlying
atmosphere, mb
The empirical coefficient, a, has often been taken to be zero, while b
ranges from 1 x 10~9 to 5 x 10~9.
The value of es is a nonlinear function of the surface temperature.
However es can be estimated in a piecewise linear fashion as follows:
(2-45)
where a.^8. = empirical coefficients with values
as given in Table 2-5 below.
T = water temperature, °C
The following empirical expression (Lombardo, 1972) can be used for
determining the latent heat of vaporization, Lw, in kcal/kg:
- °'57Tw
where T = water temperature, °C
52
-------
TABLE 2-5. VALUES FOR EMPIRICAL COEFFICIENTS a-, AND 3-|
^___ —
Temperature Range, °C
0-5
5-10
10-15
15-20
20-25
25-30
30-35
35-40
a1
6.05
5.10
2.65
-2.04
-9.94
-22.29
-40.63
-66.90
*1
0.522
0.710
0.954
1.265
1.659
2.151
2.761
3.511
The U.S. Army Corps of Engineers (1974) defines the convective heat
transfer, Q , by the following relationship:
\*
Qc = R Qe (2-47)
where R - Bowen ratio = (6.19 x 10"4) p w a
_ £
S 3
p = atmospheric pressure, mb
T = dry bulb air temperature, °C
a
Chen and Orlob, as reported by Lombardo (1973), consider similar
variables in their formulation for convective transfer, given as:
U Ta-Tw
where p - local barometric pressure, mb
p = water density
C = specific heat of air, kcal/kg-°C
N = empirical constant
u = wind speed , m/sec
53
-------
Similarly, convective heat exchange is given by Lombardo (1972) in
kcal/m2-hr, as:
CONVK p x 10~4/pQ ufla - Twj (2-49)
where p = barometric pressure at the site, mb
p = sea level pressure, mb
CONVK = coefficient of convection, ranging
from 1-20
u = wind speed, m/hr
T, = air temperature, °C
a
T = water temperature, °C
Baca and Arnett (1976) present the following table (Table 2-6), which
compares the amount of heat loss and gain by the various components:
TABLE 2-6. RANGE OF VALUES FOR VARIOUS COMPONENT HEAT FLUXES
(BACA AND ARNETT, 1976)
Component
Shortwave solar radiation
Reflected shortwave
Net shortwave
Long wave atmospheric radiation
Reflected long wave radiation
Net long wave radiation
Net surface radiation
Evaporative heat loss
Convective heat loss/heat gain
Range of Loss or Gain
Symbol (kcal/m2-hr)
Qs
Qsr
Qsn = Qs - Qsr
Qa
Qar
Qan = Qa - Qar
V
Qe
QC
30 •>
5 -»•
25 +
225 +
5 -»-
220 +
225 +
25 +
-35 ->-
300
25
375
360
15
345
400
900
50
54
-------
A similar range of values for the various component heat fluxes
(originally developed by Parker and Krenkel (1970)) is given by Lombardo
(1972) (Table 2-7):
TABLE 2-7. RANGE OF VALUES FOR VARIOUS COMPONENT HEAT FLUXES
(LOMBARDO, 1972)
Range of Loss or Gain
Component Symbol (kcal/m2-day)
Solar radiation
Reflected solar radiation
Net long wave radiation
Net surface radiation
Evaporative heat loss
Convective heat loss (heat gain)
QS
Qr
Qan
Q-br
Qe
Qc
1100 +
110 -»•
6350 ->
6500 -»•
5500 -»-
-870 ->
7600
550
8150
9800
20000
1100
For lake simulations, the net external heat flux, H, converted to a
single volumetric heat source, is typically assigned to the top layer of
fluid with the exception of the net shortwave radiation which is distributed
vertically according to Beer's law (see Section 2.3.5). The flow and dis-
persion regimes then transport this external heat input throughout the
entire lake.
The temperature values can either be computed along with the lake
circulation in the hydrodynamic model and then be transferred to the water
quality model (Tetra Tech, 1975), or a heat budget equation for temperature,
analogous to a mass transfer equation formulation, may be used for tempera-
ture regime prediction (U.S. Army Corps of Engineers, 1974).
2.3.4 Ice Formation
The formation of a surface ice layer is simulated only in a limited
number of lake models (Baca and Arnett, 1976; Tetra Tech, 1976b). The
procedure used in modeling ice formation was similar in the two models
reviewed.
Baca and Arnett's technique is to monitor the water surface temperature
until a temperature of 4°C or less is observed. At this point a continuous
accounting is kept of net heat loss (or gain) across the interface. Then
when the total heat loss to the atmosphere is equivalent to the heat of
fusion for a 5 cm depth of ice, an ice cover is assumed to be formed on the
lake's surface. Thereafter, cover thickness is computed by means of the
following expression:
A = - QT/Hp
(2-50)
55
-------
where A = ice thickness, m
QT = total heat loss to atmosphere from surface
layer, kcal/m2
HF = heat of fusion for water (7.97 x 104 kcal/m )
Once ice is formed, assumed atmospheric heat exchange is limited to
conductance through ice. The heat flux across the air-ice interface is then
given by:
where Q = heat flux transferred across ice by
n conduction, kcal/m^-sec
K = conductivity of ice (5 x 10 kcal/sec-m-°C)
Tc = temperature of the ice water interface,
typically 0°C, bt
dissolved sol ids
s typically 0°C, but a function of
Ta = air temperature, °C
Over a period of days the total heat flux transferred is equal to the sum
of the daily Qn values where Ta, Ts, and A can change on a daily basis.
The approach used by Tetra Tech is very similar. Heat loss is moni-
tored as the water temperature is lowered to 0°C and the overall heat flux
is negative. As in Baca and Arnett's formulation, the heat conductance
term, Hc, considers heat conductance through the ice layer as the limiting
factor:
/T T, \
"-K--^ (2-52)
where KT = thermal conductivity of ice (usually
1 5 x 10~4 kcal/sec-m-°C)
T = temperature of ice at ice-water
interface, °C (generally 0°C)
Td = dry bulb air temperature, °C
D = thickness of ice, m
56
-------
Once ice is formed, the back radiation from the ice, H^j, can be
approximated by:
Hbl = 0.074 + 0.0012 Td (2-53)
where Hbl = back radiation, kcal/m2-sec
T^ = dry bulb air temperature, °C
Equation (2-53) is a linear approximation of the Stefan-Boltzman equation
(Equation (2-41)).
The problem of estimating ice cover, being related to air temperature,
sometimes makes it necessary to forecast long-range air temperatures. Rogers
(1976) has analyzed four techniques for making monthly air temperature
predictions using wintertime data around Lakes Superior, Huron, and Michigan.
Rogers suggested that once an accurate air temperature forecasting technique
is devised, it could be applied to ice forecasting since the number of
freezing degree-days needed to cause ice formation over many parts of the
Great Lakes has been established (Snider, 1974). Accurate ice forecasting
in this region can prove beneficial by predicting the number of shipping
days available on the Great Lakes and the St. Lawrence Seaway. The four
forecasting techniques Rogers examined were:
1. the use of data on climatological cycles
and oscillations,
2. the extrapolation and kinematic process used by
the National Meteorological Center which provides
forecasts in the Average Monthly Weather Outlook,
3. the use of conditional probabilities, and
4. the use of a Markov chain equation (where the
outcome of a given trial depends only on the
outcome of the immediately preceding trial).
Rogers found that only techniques 1 and 2 improved estimates over that
due to chance alone. By combining these two methods, Rogers stated that
an improved air temperature forecasting method might be established for
application to ice simulation.
2.3.5 Light Attenuation
As discussed previously in Section 2.3.3, only shortwave solar radia-
tion can penetrate the water column. Therefore, the shortwave solar radia-
tion, Qs, is the only term in the heat budget equation which reaches
hydraulic elements below the surface. The amount of light (radiation)
reaching a given point within the modeled system is measured in terms of
light intensity, which is known to vary as a function of both the time of
day and the depth.
57
-------
The "extinction coefficient," together with the incident shortwave
solar radiation, are the variables controlling the light intensity. Empir-
ical fits are usually made to the distribution of these variables in order
to describe their effect. Although a single approach is commonly used in
river, lake, and estuary modeling studies, coefficients in some instances
are system specific.
Baca and Arnett (1976), Lombardo (1972), Di Toro, et al_. (1971),
Bierman (1976), and Thomann, e_t al_. (1975), all used a modified form of
the Beer-Lambert law to describe light attenuation with depth. This law
can be described by an equation of the following form:
where I
I,
Z
K,
I =
(2-54)
light intensity at depth,
langleys/min
Z;
intensity of light energy passing the
air-water interface after allowances for
absorption, scattering, or cloudiness,
and reflection at the water surface,
langleys/min
= depth, m
= extinction coefficient, m
-1
Orlob (1974) suggests that values of I0 can be determined by using a
pyroheliometer (on a clear day) at the water surface elevation and correct-
ing for cloud interception (when it exists) by multiplying by the factor
(1 - 0.65 C^) in which C is the cloudiness as a decimal part of the sky
covered by clouds. Reflection may also reduce the incident light by an
additional 3 to 5%.
Baca and Arnett (1976) modified the extinction coefficient, Ke, to
account for algal light absorption as follows:
Ke =
where a = extinction coefficient of water, m"
self-shading factor, per m per mg-C/1
(2-55)
3
IP
average phytoplankton concentration
above Z, mg-C/1
58
-------
The light extinction depth (euphotic depth), d, is defined as that
depth at which the light intensity is 1% of the incident value. Baca and
Arnett (1976) suggest that when the algal attenuation is insignificant,
the extinction coefficient, a0, may be assumed to equal 4.6/d. The extinc-
tion coefficient can also be estimated by means of Secchi disk readings
where a0 is defined as 1.89/ds and ds is the Secchi depth, in meters.
Values reported for the extinction coefficient range from 0.05 to 0.9 m~^
(Baca and Arnett 1976). Baca and Arnett also report values for the self-
shading factor, g, ranging from 0.2 to 0.6.
Thomann, et aj_. (1975), in a modeling study of Lake Ontario, and
Lombardo (197277 suggested the following expression for the extinction
coefficient:
Ke = aQ + 0.0088P + 0.054 P°'66 (2-56)
where aQ = light extinction coefficient at zero
chlorophyll concentration
P = phytoplankton chlorophyll, yg/1
Bierman (1976), also distributing light with depth according to the
Beer-Lambert law, proposed the following formulation for the extinction
coefficient:
Ke = ]-9/ds + °'17 * TCROP (2-57)
where d = Secchi depth, m
TCROP = total phytoplankton biomass,
mg dry weight/1
Lombardo (1972) also reports a range of values for the self-shading
factor, 3, given by Tailing as: 0.00304 to 0.00608 (ft-yg Chl/1)'1-
Tetra Tech (1976b) uses a term similar to that suggested by Baca and
Arnett (1976) for determining the extinction coefficient, Ke:
Ke = ao + sr. S. (2-58)
where a - light extinction coefficient of water
0 with no suspended matter, rrH
r. = self-shading coefficient of constituent i
S. = concentration of suspended matter, i,
1 mg/1 (detritus, sediment, algae)
59
-------
In describing the variability of light intensity with time, Di Toro,
et_ al_. (1971) express the incident solar radiation, I0, as a function of the
time of day according to the following criteria:
I0(day) = Ia (2-59)
I0(night) = 0 (2-60)
where I = average incident solar radiation intensity
a during photoperiod
Baca and Arnett (1976) used the following equation of "standard light
day" proposed by Vollenweider to account for the diurnal pattern of incident
light:
I0(t)=l/2Imaxl + cos , - A/2 < t < A/2
I (t) =0, for remaining 24-A hrs of day
(2-61)
where t = time, hrs
A = day length factor, hrs
I = maximum instantaneous light intensity
max
I max is calculated from the total daily solar radiation such that the
integral of I0 over one day equals the total daily radiation. The day
length factor, A, is given as a function of time and latitude as follows*:
- 24 - (2_62)
cos"1 /-tan(y0) tan (6)\
where YO = latitude, radians
6 = declination, radians = 0.409 sin (0.0172 Nd - 1.3762)
and N,j equals the Julian day number
In their formulation for describing the variation of light intensity
with time, Tetra Tech (1975) uses the following expression in order to
obtain a "constant average daily light intensity":
*Equation (2-62) as given by Baca and Arnett (1976) was in error. Version
is correct as shown here.
60
-------
LI - Tyw <2-63'
where LI = constant average daily light intensity
TSR = total shortwave radiation
DH = duration of daylight hours
2.4 BIOLOGICAL PROCESSES
2.4.1 Fish
The behavior of fish, as a function of food supply, concentration of
oxygen and toxic substances, and temperature, is known only in general
terms. Where fish are included in models it is primarily to indicate
population trends rather than to predict precise biomass levels. Fish are
generally modeled in a manner similar to that used for phytoplankton and
zooplankton, and are assumed to graze primarily on zooplankton and/or
benthic animals.
Since they are actively motile, fish are generally not assumed to be
advected or diffused, but to migrate to locations favorable for feeding or
reproduction and away from locations that are unfavorable for their life
processes. This is usually simulated in the models by distributing fish to
elements in proportion to the densities of the food within the element.
Fish are usually grouped in one of two ways in modeling studies. They
can be distinguished according to their feeding habits and temperature
preference, as well as by various life stages. In its Lake Ontario model,
Tetra Tech (1975) ^establ ished fish groups according to both criteria.
The following four groups of fish were chosen: 1) cold water, 2) warm
water, 3) forage, and 4) scavenger. For each group, three life stages
were assumed: 1) eggs and larvae, 2) young, and 3) adult. The eggs and
larvae group is assumed to behave like zooplankton--grazing on algae and
detritus. Members of the group are consumed by all young fishes and adult
alewife. Young fish graze on zooplankton and fish larvae, and are in
turn preyed upon by adult fishes.
In another modeling study, Tetra Tech (1976b) considered only three
groups of fish: 1) cold water, 2) warm water, and 3) scavenger. Similarly,
the U.S. Army Corps of Engineers (1974) and Chen and Orlob (1975) suggest
dividing the fish population into three separate groups.
Generally, a mass balance formulation describing the rate of change
of each fish group's biomass is included in the water quality model. Kelly
(1975) uses the following typical representation to describe changes in
fish biomass:
d(Fish) = FEEDING _ RESPIRATION - DEATH - EXCRETION (2-64)
dt
61
-------
where each term on the right-hand side may be further described by an
empirical expression.
Tetra Tech (1975) suggests the following similar type of formulation:
S - FISH • (FG-FM-FR) + GRADIN - GRADOT - FGRAZ - HARVST (2-65)
where FISH - fish biomass at a given life stage,
mg/1 or g/m^
FG = growth rate, day"
FM = mortality rate, day"
FR = respiration rate, day"
GRADIN = spawning or recruitment rate, mg/1-day
GRADOT = maturation rate to next life stage, mg/1-day
FGRAZ = grazing rate, mg/1-day
HARVST = harvest rate, mg/1-day
S = change in fish biomass day"
The fish growth rate, FG (mg/1-day), may be defined as follows (U.S.
Army Corps of Engineers, 1974):
(2-66)
where FMAX = maximum specific growth rate of fish
at 20°C, day1
ZB = quantity of zooplankton or benthic
animals available for grazing, mg/1
IB 2 ~ half-saturation constant for fish grazing
on zooplankton or benthic animals, mg/1
The fish growth rate in some approaches is also assumed to be a
function of temperature and food density. For each fish group, a satis-
factory temperature range is provided. Fish growth computations are not
performed when the simulated temperature is outside their growth range,
or when DO is too low to sustain a healthy fish population.
Tetra Tech (1975), in their Lake Ontario model, suggested that prefer-
ence factors be specified in accordance with the known food preference
of each group of fish. The effective food density would then be calculated
as follows:
62
-------
FOOD - E (PREY - PREF] (2-67)
where FOOD = effective food density, mg/1
PREF = preference factor, unitless
PREY = density of prey, mg/1
Table 2-8 presents a listing of coefficient values commonly used in
mass balance formulations for various fish groups.
Recently, Leidy and Jenkins (1977) have prepared an extensive document
on fish in ecosystem modeling. Tables 2-9 through 2-12 have been reproduced
from that report.
2.4.2 Detritus
Detritus is the term used to describe organic material suspended in
the water column. Sources of detritus may be phytoplankton and zooplankton
mortality, fish excretions, and tributary contributions. Losses or sinks
of detritus are generally through sedimentation and decay. The bacterial
and physical breakdown of detritus involves a large number of complex reac-
tions, most of which have been inadequately investigated (Canale, et al.,
1976).
The amount of detritus present at any time is usually simulated by
means of a mass balance expression (Tetra Tech, 1975; Tetra Tech, 1976b;
U.S. Army Corps of Engineers, 1974; Chen and Orlob, 1975; Chen and Wells,
1975). Detritus decay is generally formulated by means of a linear first
order differential equation. A typical mass balance expression, similar
to those used by Tetra Tech (1975, 1976b) and the U.S. Army Corps of
Engineers (1974), is given as:
dDET/dt - -DET (DQTEN + DSETL) + FISH • FM + ZFEX
(2-68)
• FDET + ZOO • ZM + ZEX • ZDET
where DET = detritus concentration, mg/1
DQTEN = temperature adjusted decay rate, day"
DSETL = detritus settling velocity', day~
FISH = fish larvae , mg/1
FM = larvae mortality rate, day"
ZFEX = high excretion rate (young and adult)
mg/1-day
63
-------
TABLE 2-8. VALUES FOR COEFFICIENTS USED IN FISH MASS BALANCE FORMULATIONS
System
rish Group '•'uJeled
Cold Water N. Pork,
v. i n a s
Rivc-r, CA
Warm Water
Scavenger
Cold Water Trinity
R. , TL'*as
Warm Water
Scavenger
Cold Water Lake
Wj s'\ i ng ton
Warn Water
Scavenger
Plankton S.F, Bay
ceeder Estuary
Scavenger
Co!d Water Boise P
Idaho
Warm Water
Scavenger
Max imum
Speci f i c
Growth Kate ,
? 20°C, days
0.025
0.03
0.025
0.02-0.03
0.02-0.03
0.02-0.03
0.02
0.03
0.02
0.03
0.02
0.03
0.03
0.025
Hal f-Saturation
Constants for Grazing
Fish on Fish on Fish on
7oo. Benthos Insects
mg/ 1 iny/iiN mu/ni2
0.20
0.20
500
0.05-0.01
0.05-0.01
50-2000
0.05
0.01
500
0.05
500
0.05
0.05
500 500
Resoi ration
i.. -L' > atiy
Active Inactive
0.002 0.000?
0.002 0.0002
0.002 0.0002
0.001- (l.Of)Ol-
0.01 U.OU5
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
R
Tpi.m
TetiM
1 C 1. I'd
u s
I'.S.
U.S.
Chen
Chen
Clien
Chen
Chen
Ciien
Chen
Chen
inference
Tech (1976L)
'U'cli (ll)/fi!>)
Tech Mr)76b)
Anny Corps ot Engineers (1970;
Army Corps of Engineers (19701
Anny Corps of Engineers (!9V-iv
din! Orlob (1975)
and Orlob (1975)
ami Orlob (1 975)
(1970)
(1970)
dnd Wells (1975)
and Wells (1975)
Jiui Wells (1975)
(continued)
-------
TABLE 2-8 (continued)
cn
Fish Group
Cold Water
Warm Water
Scavenger
Cold Water
Warm Water
Scavenger
Cold Water
Warm Water
Scavenger
Plankton
Feeder
Scavenger
Cold Water
Warm Water
Scavenger
Mortality Chemical
System Rate Compos! ti on
Modeled day-1 C N f
N. Fork, 0.002 0.4 0.08 0.015
Kings
River, CA
0.002 0.4 0.08 0.015
0.002 0.4 0.08 0.015
Trinity 0.001- 0.4- 0.07- 0.01-
R., Texas 0.005 0.5 0.09 0.012
Lake
Washington
--
--
S.F. Bay
Estuary
--
Boise R. -- 0.5 0.09 0.015
Idaho
0.5 0.09 0.015
0.5 0.09 0.015
Temperature
Tolerance
Range, °C Reference
5-20 Tetra Tech (1976b)
10-30 Tetra Tech (1976b)
5-30 Tetra Tech (1976b)
U.S. Army Corps of Engineers
U.S. Army Corps of Engineers
U.S. Army Corps of Engineers
Chen and Orlob (1975)
Chen and Orlob (1975)
Chen and Orlob (1975)
Chen (1970)
Chen (1970)
Chen and Wells (1975)
Chen and Wells (1975)
Chen and Wells (1975)
(1974)
(1974)
(1974)
-------
TABLE 2-9. TEMPERATURE TOLERANCES FOR VARIOUS FISH GROUPS*
(AFTER LEIDY AND JENKINS, 1977)
Species
Group
Pickerel
Minnows
Catfish
Sunfish
Black bass
Crappies
Yellow perch
Average values
Lower
Lethal
0
0
0
£2.5
<1.6
0
0
Optimum
for Growth
25.4
27
30
27.5
27
=23
24.2
26.3
Upper
Lethal
34.4
33.4
37.1
35.7
36.5
32.5
30.9
34.8
*
All values expressed in degrees C.
66
-------
TABLE 2-10. CHEMICAL COMPOSITION OF FISH (AFTER LEIDY AND JENKINS, 1977)
Carbon (C)
Ocean sunfish (Mola mola)
Phosphorus (?)
Salmon
Trout
Cod
Eel
Haddock
Halibut
Herring
Mackerel
Turbot
Average of above species
Bluegill
Bluegill
Channel catfish
Car?
Northern squawfish
Largescale sucker
Rainbow trout
General average (for fish flesh)
% Composition
Element and Species
Nitrogen (N)
Ocean sunfish (Mola mola)
Bluegill
Bluegill
Carp
Northern squawfish
Largescale sucker
Rainbow trout
Channel catfish
General average
Dry Weight
16.6-18.2
16.7
16.3
Wet Weight
2.72
2.6
2.5 ± 0.1
2.4
2.9
2.35
Reference
Green (1899)
Calculated from data by Geng (1925),
(1962), and Maynard (1951)
Gerking (1962)
Bull and MacKay (1976)
Worsham (1975)
Bailey (1937), Nottingham (1952)
Gerking
48.2
4.75 ± 0.70
4.73 ± 0.85
4.2
0.59
0.81
0.60
0.68
0.97
0.44
0.56
0.56
0.48
0.63
0.86
0.5 +
0.4
0.3
0.4
0.22 (range:
0.01
Green (1899)
Atwater (1892) as P.,05
0.1-0.4)
Kitchell, et al_. (1975)
± 1 S.E.
Hall, et al. (1970)
Worsham (1975)
Bull and MacKay (1976)
Clauseret (1962)
-------
TABLE 2-11. ESTIMATED HALF-SATURATION CONSTANTS FOR FISH GROWTH (AFTER LEIDY AND JENKINS, 1977)
CD
Species
Largemouth bass
Smallmouth bass
Muskel lunge
Reticulate
sculpin
Channel catfish
Sockeye salmon
Sockeye salmon
Water
Length and/ Temperature
or Weight °C
24.5 cm 21
8.3-20.2 cm 21.3
4-112 g
(mean wt.=40 g)
17.0 cm 19.5
17.0 g
1.2 g 11.6
4 g 30
6.9 g 10
7.1 g 15
Calculated Maxi-
mum Growth Rate
Expressed as %
of Body Weight
Gained Per Day
3.9
4.7
3.9
1.7
3.4
1.8
4.2
Calculated Half-
Saturation Con-
stant (Ks) Ex-
pressjd as % of
Body Weight Con-
sumed Per Day
4.6
7.2
5.6
4.4
3.1
3.9
7.9
Type
of Food Reference
minnows Thompson (1941)
minnows Williams (1959)
minnows Gammon (1963)
midge Davis and Warren
larvae (1965)
mixed diet Andrews and
Stickney (1972)
mixed diet Brett, et al .
(1969T
mixed diet Brett, et al.
(1969"T~~
-------
TABLE 2-12. FISH FOOD EXPRESSED AS A PERCENTAGE OF THE DIET BY VOLUME*
(AFTER LEIDY AND JENKINS, 1977)
en
Species or
Species Group
Gars
Bowfin
Gizzard shad
Threadfin shad (young)
Threadfin shad (adult)
Rainbow trout
Brook trout
Pickerels
Carp
Minnows
Carpsuckers
Suckers
Hog suckers
Buffalofishes
Redhorses
Bullheads
Catfishes
Madtoms
Si Iversides
Temperate basses
Sunfishes
Black basses
Crappies
Perches
Freshwater drum
All other species
Food categories are described
Plant
10
30
30
5
30
20
15
15
5
10
10
10
5
in the text of
Detritus
80
50
5
40
65
65
80
40
25
27
5
5
8
the original reference.
F 0
Benthos
5
10
15
60
90
20
20
5
5
5
5
100
50
10
55
20
20
65
8
20
20
58
100
0 D
Zooplankton
5
10
55
15
5
10
60
15
15
15
50
80
10
15
20
Fish Terrestrial
100
100
10 10
5
100
15
80
18
70
5 15
86 6
55
60
34
-------
FDET = detritus fraction of fish excreta, unitless
ZOO = zooplankton concentration, mg/1
ZM = zooplankton mortality, day
ZEX = zooplankton excretion rate, mg/1 -day
ZDET = detritus fraction of zooplankton
excreta, unitless
The detritus decay rate term, DQTEN, is temperature adjusted with a
typical correction expression as follows:
DQTEN = DQTEN, o e~ (2-69)
where 0 = temperature coefficient
T = temperature, °C
Table 2-13 is a list of coefficient values commonly used in mass
balance formulations for detritus. The decay of detritus is typically
associated with a resulting oxygen demand. The usual procedure in account-
ing for this demand is to include a term for detritus in the dissolved oxy-
gen mass balance expression. With this term, the amount of detritus decayed
is converted to oxygen utilized by means of a stochiometric coefficient.
Various values used for this coefficient are also listed in Table 2-13.
2.4.3 Benthic Organisms
Tetra Tech (1975, 1976b). Chen and Orlob (1975), Chen and Wells (1975),
and the U.S. Army Corps of Engineers (1974) are some of the relatively few
investigators who include mass balance formulations for benthic animals in
their water quality models. The benthic animal expressions are formulated
in the same manner as for zooplankton except that settled detritus (organic
sediment) is the main food source rather than algae. Benthic animals are
assumed to be preyed upon by scavenger fish species. A typical mass balance
expression is as follows:
3B/3t = B • KB • (BG-BM-BR) - BGZ (2-70)
where B = equivalent concentration of benthic
animals = BEN • AV, g/m3
BEN = benthic animal biomass per unit area, g/m
AV = ratio of average cross-sectional area to
volume of fluid element, m~^
KB = biota activity rate coefficient at
local temperature, day~^
70
-------
TABLE 2-13. VALUES FOR COEFFICIENTS USED IN DETRITUS FORMULATIONS
Detritus Temperature
Decay Rate Coefficient,
@ 20°C (day"1} 6
0.005 1.03
0.001-0.02
0.001
0.001 1.02
Settling
Velocity
m/day
0.25
0.-2.
0.2
0.5 ft/day
Stoichiometric Chemical
Equivalence Composition
(02/Detritus) C N P
1.2 0.4 0.08 0.015
-
2.0 ...
2.0 ...
Location
of
Study
N. Fork Kings
River, California
-
Lake Washington
San Francisco
Bay Estuary
Reference
Tetra Tech (1976b)
U.S. Army Corps of
Engineers (1974)
Chen ar.d Orlob (1975)
Chen (1970)
0.01
0.5 ft/day
2.0
0.2 0.05 0.007
Boise River, Idaho
Chen and Wells (1975)
-------
BG = benthic animal growth rate @ 20°C,
day'1 = BMAX • (SED/SED2 + SED)
BMAX = maximum specific growth rate
@ 20°C, day'1
SED = quantity of organic sediment per
unit area, mg/m^
SED2 = half-saturation constant for benthic
animals grazing on organic sediment,
mg/m^
BM = benthic animal mortality rate
@ 20°C, day-1
BR = benthic animal respiration rate
@ 20°C, day'1
BGZ = quantity of benthic animals grazed
by fish - F2B - FBEN/FEFF
FpB = benthic animal portion of the bottom
feeding fish diet, unitless fraction
FBEN = fish growth attributed to grazing in
benthic animals, g/m3-day
FEFF = digestive efficiency of fish, unitless fraction
Table 2-14 presents a listing of the values used for the various
coefficients in the above formulation for benthic organisms.
2.4.4 Sediment Deposits
Sediment deposits are composed of organic and inorganic material which
has settled to a river, lake, or estuary bottom. The source of this organic
material may be sewage effluents, settling of organic matter from run-off
and inflow streams, and/or photosynthetic production. Settling of algae
and detritus, and mortality of fish and benthic animals can also contribute
to organic sediment concentrations.
Nutrients in the sediment, both the organic and inorganic fractions,
can significantly influence the nutrient content of the water column. The
direction of nutrient flux often oscillates with time, and nutrients in
the system may be either deposited in, or released from, the bottom environ-
ment depending on the controlling conditions.
Fillos and Molof, as reported by Lombardo (1972), obtained data on
releases from sediment deposits, as given in Table 2-15 below.
72
-------
TABLE 2-14. VALUES FOR COEFFICIENTS USED IN BENTHIC ORGANISM FORMULATIONS
Benthic Animal
Maximum
Growth Rate
G>20°C (days'1)
0.03
0.02-0.03
0.03
0.05
Benthic Animal
Respiration Rate
@ 20°C (days'1)
0.002 0.001
(active) (inactive)
0.001-0.01 0.0001-0.001
(active) (inactive)
0.001
0.001
Michael is Benthic
Constants Animal
Benthos on Mortality Chemical
Sediment, Rate @20°C Composition
mg/rn^ (days'! ) C N P
1,000 0.003 0.4 0.08 0.015
50-2,000 0.001 -
100 - ...
100 - ...
Location
of
Study
N. Fork Kings
River, California
-
Lake Washington
San Francisco
Bay Estuary
Reference
Tetra Tech (1976b)
U.S. Army Corps of
Engineers (1974)
Chen and Orlob (1975)
Chen (1970)
0.05
0.01
50
0.5 0.09 0.015
Boise River, Idaho
Chen and Wells (1975)
-------
TABLE 2-15. RELEASE RATES FROM SEDIMENT DEPOSITS
(LOMBARDO, 1972)
Constituent
BOD
COD
P04-P
NH3-N
Aerobic Rate
0.061 g/m2-hr
0.154 g/m2-hr
0.125 mg/m-hr
?
0.1083 mg/m-hr
Anaerobic Rate
0.203 g/m2-hr
0.495 g/m-hr
0.250 mg/m2-hr
0.3125 mg/m2-hr
Tetra Tech (1975, 19765), Chen and Wells (1975), and the U.S. Army
Corps of Engineers (1974) formulate a separate mass balance expression for
organic sediment, assuming organic sediment decays by first order kinetics
and releases nitrogen, phosphorus, and C02 according to its stoichiometric
composition. A typical formulation is as follows:
d (cSED)/dt = -SQTEN • CSED • (l-SSINK) -
SGRAZ
PEFF
Decay Benthic
Grazers
+ DET • DSETL • BOTTOM + CBEN • BM (2-T(}
Detritus Benthic
Death
+ BEX • BDET + FISH • FM • BOTTOM + ALG • PSETL • BOTTOM
v /
Benthic Fish Death Algal Death
Excretion
where SQTEN = temperature adjusted decay rate, day"
2
CSED = organic sediment concentration, mg/m
SSINK = refractory fraction of organic sediment, unit!ess
2
SGRAZ = grazing rate by benthic animals and fish, mg/m -day
PEFF = digestive efficiency of grazer, unitless
DET = detritus concentration, mg/1
DSETL = detritus settling rate, day
74
-------
BOTTOM
CBEN
BM
BEX
conversion factor from mg/1 to mg/rn^
2
= benthic animal density, mg/m
= benthic animal mortality rate, day
= benthic animal excretion rate, day
-1
-1
BDET = organic participate fraction of
excreta, unitless
FM
= larval mortality, day
-1
FISH = fish larvae, mg/1
ALG
PSETL = settling, day
= algal density, mg/1
-1
The value used by Tetra Tech (1976b) for the sediment decay rate coeffi-
cient in their modeling study of the North Fork Kings River, California was
0.001 day' at 20°C. The U.S. Army Corps of Engineers (1974) reported a
range of values for the sediment decay rate from 0.001 to 0.02 day"1. Chen
and Wells (1975) report that the lower rate (0.001 day'1) is applicable to
organic matter associated with algal cell decomposition. In their Boise
River, Idaho modeling study, the organic deposits resulted primarily from
sewage inflows and were readily oxidizable. Therefore, a decay rate value
of 0.01 day"' was chosen.
Several other model formulations for sediment deposits (Baca and
Arnett, 1976; and Lerman, 1971) are primarily concerned with the nutrient
changes which occur within the sediments. Mass balance expressions specif-
ically for organic sediment are generally not used by these investigators.
Rather, the usual procedure is to include terms for sediment release and
deposition within the complete mass balance expression for the specified
nutrients (constituents) of concern. The formulations for the sediment term
or terms vary from model to model, and from study to study.
Baca and Arnett's (1976) model considers the loss of both organic and
inorganic phosphorus to the sediments, and anaerobic releases of certain
nutrients from the sediments. The sediment nutrient terms are included in
a mass balance expression for sediment phosphorus as follows:
-dF=Il Dl
D2 -
D
(2-72)
where D^ = sediment phosphorus, mg/1
D? = water column organic phosphorus, mg/1
D, = inorganic phosphorus, mg/1
75
-------
I-, = rate constant for loss to sediment,
day-1; typically 0.1-0.7
TO = rate constant for sediment release,
day-1; typically 0.1-0.7
I, = rate for organic phosphorus loss to
sediments, day1; typically 0.01-0.1
A similar type of formulation was used by Baca and Arnett (1976) for sediment
nitrogen.
Lam and Jacquet (1976) have developed a model which includes a detailed
simulation of the physical transport and regeneration of phosphorus in Lake
Erie. In their model, sediment resuspension is considered to occur as a
result of movement of the water at the bottom caused by surface waves only.
Resuspension from nonperiodic currents was not considered for the following
reasons:
1. The general current pattern and velocities are
computed from water transport and velocities are
depth averaged. This assumption makes computation
of actual bottom current velocities difficult.
2. For practically all time periods considered in
applications to Lake Erie, the bottom
orbital velocities were about one order of
magnitude higher than computed current velocities.
The following steps were followed by Lam and Jacquet in order to repre-
sent the physical regeneration process: 1) starting with wind speed and
direction, relevant wave parameters were obtained from a wave hindcasting
method and used to compute bottom orbital velocities; 2) sediment resuspen-
sion was then expressed as a function of this velocity; and finally 3)
phosphorus regeneration was estimated by taking into account its distribu-
tion in sediments.
Lerman (1971) reports that his model, designed for simulation of lakes
or impoundments, may be applied to situations where chemical species are
formed at the sediment-water interface. The model is capable of simulating
the transport of these released materials upwards by eddy diffusion.
Lombardo's (1972) model simulates scour in rivers by doubling the release
rates when the river velocity is greater than 10 fps.
2.5 CHEMICAL PROCESSES
2.5.1 Nitrogen Cycle
The major constituents comprising the nitrogen cycle are detrital or-
ganic nitrogen (org-N), ammonia nitrogen (Nh^-N), and nitrate nitrogen
(N03~N), with gaseous nitrogen being important when organisms are present
76
-------
having nitrogen fixation capabilities. In natural aerobic waters there is a
stepwise transformation from organic nitrogen to ammonia, to nitrite, and to
nitrate, a process which yields nutrients for phytoplankton growth. The
chain of reactions comprising the nitrogen cycle are illustrated in Figure
2-13, as given by Canale, ejb aj_. (1976). Also depicted in this figure are
some of the mechanisms of nutrient recycling.
Different authors, depending on their particular concern, will emphasize
various aspects of the nitrogen cycle, or emphasize a specific nitrogen
cycle applicable to the system they are studying. As examples, Kormondy
(1969) discusses a more general nitrogen cycle, from a biological and ecolog-
ical viewpoint, than that shown in Figure 2-13, Brezonik (1973) emphasizes
the nitrogen cycle reactions occurring in an idealized lake environment,
while Harleman, et a]_. (1977) illustrate a nitrogen cycle applicable to
aerobic estuaries, and apply it in water quality model development.
Kinetics of the transformations of nitrogen forms are generally assumed
to be first order reactions with temperature-dependent rate coefficients.
Source/sink formulations are developed to describe the rates of change of
the major nitrogen forms. These formulations are incorporated into complete
mass balance expressions which also consider the physical transport of the
various nitrogen constituents throughout the system by means of advection
and diffusion.
Sources of organic nitrogen, which contribute to the nitrogen concen-
tration in aquatic systems, are generally considered to be due to the
following processes: 1) respiration of algae and zooplankton, 2) natural
death of zooplankton, and 3) external sources of organic nitrogen such as
wastewater discharges. In a few extensive nutrient submodels, a source of
organic nitrogen due to the grazed but unassimilated portion of phyto-
plankton which is excreted by zooplankton is considered (see, for example,
Thomann, et al_. (1975)).
A typical formulation for the organic nitrogen source/sink term is
given as (Lorenzen, et al_., 1974):
dC4/dt= - J4 C4+ Dp PAnp+DzZAnz (2-73)
where D = specific algal death rate, day"
D = specific zooplankton death rate, day"
P = phytoplankton concentration, mg-C/1
Z = zooplankton concentration, mg-C/
A = N to C ratio for zooplankton, mg-N/mg-C
77
-------
EXTERNAL ORGANIC NITROGEN
SYSTEM BOUNDARY
HYPOLIMNION
EPILIMNION
Growth
CO
EXTERNAL LOADS OP--
ORGANIC |
NITROGEN 1
AMMONIA 1
NITRATE 1
TO HYPOLIMNION
Sinking
Growth
-• PHYTOPLANKTON *
Respiration
TOTAL
NONACCESSIBLE-*-
NITROGEN
SEDIMENTS
Sinking
LOAD TO EPILIMNION
EXTERNAL AMMONIA
I LOAD TO EPILIMNION
NITROGEN
'DETRITUS
* "
ORGANIC
"NITROGEN
I
EXTERNAL NITRATE
LOAD TO EPILIMNION
I
• AMMONlAi
NITRATES
1
PI f
[1
J
c
o
I/)
en
LLJ
(
\NKTOfs.
L I
O
ZJ
o
2
f T
I
o
a
Respiration
Respiration
L ;
L
J
Figure 2-13.
Pathways used in modeling the nitrogen cycle (modified after
Canal e, et al ., 1976).
-------
A = N to C ratio for phytoplankton, mg-N/mg-C
J4 = rate constant for organic nitrogen decay, day"
C4 = organic nitrogen concentration, mg/1
In this simplified expression, the endogenous respiration of phyto-
plankton is represented in the specific death rate term for phytoplankton.
Similarly, the endogenous respiration of zooplankton is incorporated into
the zooplankton specific death term. Settling effects and the unassimilated
portion of grazed algae are not included in the formulation.
Various investigators have used similar expressions to suit the needs
of their particular model application. Baca and Arnett (1976) use the
following formulation, which neglects organic nitrogen sources due to zoo-
plankton excretion but additionally considers zooplankton grazing and
settling of organic nitrogen.
dC4/dt = -J4 C4 + Dp - Cg Z P Anp + Dz Z Anz J6 C4 (2-74)
where C = zooplankton grazing rate, day"
A = nitrogen to carbon ratio for zooplankton,
mg-N/mg-C; typically 0.05-0.17
Jg = sediment uptake rate, day" ; typically 0.0-0.01
Both Anderson, et al_. (1976) and Lombardo (1972) have included source/
sink formulations for organic nitrogen in their nitrogen submodels, which
consider processes similar to those simulated by Baca and Arnett (1976).
Both submodels consider organic nitrogen concentration changes to be due to:
1) zooplankton excrement, death, respiration, and decay; and 2) phytoplankton
respiration and decay.
Canale, e_t aJL (1976), in their nitrogen cycle submodel, consider
nitrogen detritus and organic nitrogen as separate constituents (Figure
2-13). The nitrogen detritus is considered to be removed from the system
by both settling and decay to dissolved organic nitrogen. Canale noted
that the reaction rate for the breakdown of detrital forms into dissolved
organic forms was not readily available ifi the literature. For the Lake
Michigan application, this rate was set equal to the rate used for the
conversion of organic nitrogen to ammonia. Canale's inclusion of two
different formulations for the organic nitrogen concentration effectively
separates the entire set of reactions into two distinct groups, and allows
application of different rate coefficients to each group of reactions.
79
-------
Canale, e_t aj_. (1976) used the following equation to describe changes
in detrital nitrogen:
dDN/dt = NCR
+ NCR
("assimilation
- [efficiency
1 featingl I
J.[ rate J - cj
L '
["natural
(2-75)
death
- A18 • T •
- A23 - SINK (t)
where NCR = nitrogen to carbon ratio by weight, unitless
z = zooplankton species subscript
C = zooplankton concentration for species z, mg/1
A18 = detrital nitrogen to dissolved organic
nitrogen decay rate, day"'
A23 = detrital sinking rate, day'
DN - detrital nitrogen concentration, mg/1
T - water temperature, °C
SINK = seasonal variation adjustment factor, unitless
The first term in the expression represents the contribution due to
zooplankton excretion and the second term refers to the natural death of the
various zooplankton species. The third and fourth terms represent a first
order decay to organic nitrogen and settling losses, respectively.
The expression for dissolved organic nitrogen is given by Canale et al.
(1976) as:
dORGN/dt = NCR
+ A18
I
respiration
loss
p
T • DN - A20 • "
+ NCR - A21
ORGN + LOAD
Z
z
15
respiration
loss
(2-76)
where ORGN = dissolved organic nitrogen concentration, mg/1
p = phytoplankton species subscript
A21 = fraction of zooplankton respired nitrogen
that is organic, unitless; typically 0.7
80
-------
A20 = organic nitrogen to ammonia nitrogen
decay rate, (day - °C)~1
LOAD-|g = loading rate of dissolved organic
nitrogen, mg/epilimnion 1-day
_Dissolved organic nitrogen forms are generated during phytoplankton
respiration. Approximately 70% of the nitrogen respired by zooplankton is
assumed to enter the water in an organic form, with the remaining as ammonia
nitrogen.
Thomann, e_t a]_. (1975) considered two sources of detrital organic
nitrogen in their submodel: 1) organic nitrogen produced by the endogenous
respiration of phytoplankton and zooplankton (assuming only organic forms
of nitrogen result from this process), and 2) the organic nitrogen equiva-
lent of grazed, but nonmetabolized phytoplankton excreted by zooplankton.
They use a single expression of the following form to describe the total
organic nitrogen balance:
dORGN/dt = a, D P + — D Z + — P
IP ac z ac
1 -
azpKmp
C Z
- Ko C4 - Ks C4
where a-, = nitrogen to chlorophyll ratio, unitless
D = specific phytoplankton death rate, day"
P = phytoplankton concentration as chlorophyll, mg/1
a - zooplankton carbon to phytoplankton
Z'D chlorophyll efficiency, unitless
a = carbon to chlorophyll ratio, unitless
L»
D = specific zooplankton death rate, day
Z - zooplankton concentration as carbon, mg/1
K = Michael is constant for zooplankton grazing
mp on phytoplankton, mg/1
C = zooplankton grazing rate, day'1 (mg/1)"1
K = decay rate of organic nitrogen to ammonia, I/day
81
(2-77)
-------
K - removal rate due to settling or additional
decay, day~^
C, = organic nitrogen concentration, mg/1
The first and second terms represent organic nitrogen released through
endogenous respiration by the phytoplankton and zooplankton respectively.
The third term represents the organic nitrogen of the grazed but unassim-
ilated phytoplankton. The last two terms represent decomposition,
settling, and other effects that contribute to the overall removal of organic
nitrogen.
In their study of the Gray's Harbor/Chehalis River, Washington,
Battelle (1974) included two nitrogen submodels with different degrees of
complexity within their model. One submodel, called the "algal nitrogen"
model, incorporates the effects of algae on nitrogen concentrations and is
similar to the nitrogen formulations used by Baca and Arnett (1976). The
other submodel is a simple first order ammonia-to-nitrate model which can
be used when algal effects are not significant.
Tetra Tech (1975, 1976b), Chen and Orlob (1975), and the U.S. Army Corps
of Engineers (1974), do not consider a mass balance for organic nitrogen
directly in their nitrogen submodels. However, the nitrogenous portion of
detritus, which is obtained from the detritus mass balance expression, is
considered as a source term for ammonia nitrogen.
Ammonia, another important form of nitrogen, is formed in the aquatic
environment from other nitrogen species through several processes. One is
nitrogen fixation, a process by which certain bacteria (such as Azotobacter)
and blue-green algae convert gaseous nitrogen to inorganic nitrogen, one
form of which is ammonia nitrogen. Second, through the process of ammoni-
fication or mineralization, wastes containing organic nitrogen are converted
by certain organisms to ammonia. Third, denitrification can occur whereby
nitrate nitrogen is converted to a more reduced nitrogen form, such as
ammonia.
The process of nitrification, which constitutes one of the major sinks
of ammonia, refers to the sequential oxidation of ammonia, NH3, to nitrite,
N02 , and finally to nitrate, N03~. As far as is known, only certain auto-
trophic bacteria are responsible for nitrification. These are Nitrosomonas
species for ammonia oxidation and Nitrobacter species for nitrite oxidation:
Nitrosomonas
Nitrobacter
82
-------
The rate of nitrification is primarily dependent on temperature and pH.
Wezerak and Gannon (1968) give the optimum temperature as 30°C and the
optimum pH as 8.5.
A major difficulty in determining the extent of nitrification in a
water body is in verifying the presence of nitrifying bacteria (Lombardo,
1972). Stratton and McCarty (1967), as discussed by Lombardo (1972), have
presented a method to describe the nitrification process based on the
principles of biological kinetics. They represent the rate of change of
nitrifying bacterial mass, M, with the following expression:
3M/3t - -a 3C/3t - bM (2-78)
where 3M/3t = rate of change of bacterial
mass, mg/l-hr
C - substrate (NHs or
concentration, mg/1
b = organic decay parameters,
hr-1 ; typically 0.002-0.004
M = bacterial mass, mg/1
a = yield constant, mg bacterial
mass/mg substrate: (0.29 mg/mg for NH3 oxidation
0.084 mg/mg for N02~ oxidation)
The rate of change of substrate is given by:
3C/3t = - p-rp (2-79)
Km L
where k = substrate utilization constant,
hr-1; NH3 + 0.0613 @ 20°C
N02" + 0.204 @ 20°C
k = half-saturation constants,
m mg/1: NH3 -> 1.848
N02~ -> 0.69
The use of these Michael is-Menton type formulations for ammonia and
nitrite oxidation includes the effect of bacterial mass on the rate of the
nitrification reaction. Hence, knowledge of the bacterial population is
necessary if this approach is used. A major difficulty in applying
formulations of this type is in obtaining a value for the initial bacterial
popul ation .
83
-------
The nitrification process can also be modeled using several other
kinetics formulations. In addition to the Michaelis-Menton type formula-
tion, Huang and Hopson (1974) discuss the zero-order and first-order
kinetics assumptions, and autocatalytic growth equation as methods for
simulating nitrification. Wezernak and Gannon (1968) use an integrated
form of the Robertson growth equation to model instream nitrification.
Stratton, ejt al_. (1969) comment on the work of Wezernak and Gannon, and
discuss the relative merits of the Robertson, Monod, and zeroth order
models. In Section 3.6, a more detailed discussion is given of the single
step first-order reaction coefficient.
The effects of the nitrification process are generally considered, in
nutrient models, to be part of the source/sink terms for the major inorganic
nitrogen forms. Generally, simulation of the inorganic nitrogen forms
includes the following processes in addition to the nitrification consider-
ations: 1) BOD degradation (detrital decomposition), 2) algal uptake and
respiration, 3) inorganic portion of zooplankton excretion, and 4) bacterial
regeneration.
Phytoplankton are known to utilize both ammonia and nitrate for primary
production. Models by Thomann, et_ a]_. (1974), and Canale, et al. (1976)
distinguish between algal utilization of these two nutrients.
Auer (Canale, e_t aj_. (1976)) found that the uptake of inorganic
nitrogen forms appears to be controlled by complex environmental and intra-
cellular conditions including trace element availability, light and carbon
dioxide levels, internal and external pH, and cell age. Auer found some
reports of ammonia preference over nitrate, although his review found no
consensus on the subject.
Typical formulations for the inorganic nitrogen forms--ammonia, nitrite,
and nitrate--as given by Lorenzen, el: al_. (1974) are:
t J4 C4 (2-80)
dC2/dt = J1 C1 - J2 C2 (2-81)
03 C3 (2-82)
where C-, = ammonia nitrogen concentration, mg/1
C2 = nitrite nitrogen concentration, mg/1
C3 = nitrate nitrogen concentration, mg/1
84
-------
1 = rate constant for ammonia oxidation,
dav-1
day
OP = rate constant for nitrite oxidation,
day-1
J3 - denitrification rate constant, day"
A = ratio of nitrogen to carbon in algal
cells, mg-N/mg-C
J^ = rate constant for organic nitrogen
oxidation, day"'
C^ = organic nitrogen concentration, mg/1
G = net algal growth rate, day"
P = phytoplankton concentration, mg-C/1
Battelle's Gray's Harbor/Chehalis River model (Battelle, 1974) has
expressions for representing the inorganic nitrogen forms in the algal
nitrogen model which are identical to the general forms presented above.
They also have an alternate ammonia-nitrate model which has the following
formulation for the inorganic nitrogen forms:
dfydt = -J2 C1 (2-83)
dC3/dt = J2 C1 (2-84)
where J? = rate constant for ammonia oxidation, day~
Baca and Arnett (1976) use an expression for ammonia that is modified
slightly, as compared with Equation (2-80), to include a preference factor
for ammonia versus nitrate and an ammonia sediment term, as follows:
- P Gp Anp yC +(1.Y)C + J4 C4 + J5 C5 (2'85)
where C,- = sediment nitrogen concentration, mg/1
b
Jc = rate of ammonia formation in sediments,
b day'1; typically 0.01-0.1
Y = preference factor, decimal, unitless
85
-------
Baca and Arnett's nitrite and nitrate formulations (Baca and Arnett, 1976)
are identical to the general formulations given previously except the nitrate
formulation contains an uptake preference factor, y> as used in Equation
(2-85) above.
The model of Canale, et a]_. (1976) considers a one-step nitrification
process where ammonia decays directly to nitrate. Their formulation for
ammonia is given as:
("respiration 1
= NCR • (1-A21) • I [ loss J + A20 • T • ORGN
- A22 • T • C
(2-86)
(ANH3)- C
- NCR
. C] +(1.-ANH3). C3
+ LOAD1C
I b
where A22 = decay rate of ammonia nitrogen to nitrate,
(day-'C)-"1
o = preference factor, unitless
= waste loading rate of ammonia nitrogen, mg/epil imnion 1-day
The first term in the expression refers to the ammonia form of inorganic
nitrogen obtained from zooplankton respiration. The second term refers to
ammonia received from organic nitrogen decay and the third term is the sink
due to ammonia decay to nitrate. The fourth term refers to ammonia loss
due to algal uptake. The final term accounts for external sources of
ammonia.
The formulation used for the nitrate source/sink expression is given
by Canale, e_t al_. (1976) as:
- A22 • T • C1
MTP 1
NLK I ANH^ • C,
\
+ LOAD17
ANH3). C3
+ (1-ANH3) • C3
Z [growth] (2-87)
P L JP
86
-------
where LOAD17 = waste loading rate of nitrate nitrogen, mg/epil imnion 1 -day
The first term in the above equation is the source of nitrate due to
ammonia oxidation. The second term considers algal uptake of nitrate and
the final term accounts for external inputs.
Canale, et^ al_. (1976) also include a balance expression for "total
inaccessible nitrogen," which considers detrital and phytoplankton nitrogen
losses to the bottom layers due to settling:
= A23 • SINK(t)
* [sinking 1
DN - VOLEP + NCR • I [ loss J
VOLEP
|~LOAD1
LOAD
VOLHY
(2-88)
5 16 1
where VOLEP = epil imnion volume, 1
VOLHY = hypo! imnion volume, 1
LOAD = same as defined earlier, except mg/hypol imnion 1-day
Thomann, e_t al_. (1974), in a manner similar to Canale, represent the
inorganic nitrogen forms as a single step oxidation, considering source/sink
formulations for ammonia and nitrate only. For ammonia, C] , and nitrate, C%,
the source/sink terms are:
= KQ C4 - K12 C1 - anpX Gp P
dC3/dt = K12 C1 - anp (1 - X) Gp P
where K = rate of production of ammonia
from organic nitrogen, day"'
~
(2-89)
(2-90)
K-|2 = rate of oxidation of ammonia to nitrate, day
X - preference coefficient, unitless
a = nitrogen to phytoplankton chlorophyll ratio
If the ammonia is preferentially assimilated by phytoplankton the
preference coefficient, X, is introduced, specifying that ammonia is used
until its concentration reaches the range of the inorganic nitrogen half-
saturation constant, at which point the nitrogen source shifts to nitrate.
_
The factor VOLEP was not included here in the equation in the original
reference. However, based upon a personal communication with R.P., Canale
(July 14, 1978) TIN is in mass units. Accordingly, volume must multiply
every concentration flux term to keep units consistent.
87
-------
Tetra Tech (1975), Chen and Orlob (1975), and the U.S. Army Corps of
Engineers (1974) use a two step oxidation representation for the inorganic
nitrogen forms, and include separate source/sink formulations for ammonia,
nitrite, and nitrate. The representations proposed for ammonia are similar
to the general formulations previously described in most respects. They
include a source of ammonia nitrogen from detrital decay, and losses of
ammonia due to oxidation to nitrite and preferential algal uptake. In
addition to these commonly considered sources and sinks, the nitrogen models
of Chen and Orlob (1975) also consider sources of ammonia due to fish and
benthic animal respiration. The formulations used by Chen and Orlob for
nitrite and nitrate source/sinks are conceptually identical to those given
by Baca and Arnett (1976).
Anderson, et al . (1976) represent the inorganic forms of nitrogen with
a two step oxidaTioTT process, and consider source/sink terms for ammonia,
nitrite, and nitrate. Anderson's ammonia formulation neglected algal uptake
as a sink of ammonia, considering ammonia oxidation to nitrite as the sole
ammonia removal mechanism. The single source of ammonia was considered to
be from decaying organic nitrogen. In Anderson's nitrate formulation,
nitrate was considered to be utilized exclusively for synthesis reactions
and the single source term was considered to be nitrite oxidation to nitrate,
Lombardo (1973) considers the following processes to influence ammonia
concentrations: 1) BOD decay, 2) nitrification (either first-order or
Michaelis-Menton type reaction as proposed by Stratton and McCarty (1967),
both with temperature dependence), 3) algal growth with ammonia used as the
sole source of algal nitrogen, and 4) denitrification.
Denitrification is the process by which highly oxidized forms of
nitrogen, such as nitrate, are converted to more reduced forms, such as
ammonia. Denitrification occurs mostly under anaerobic conditions. It
occurs in muds where anaerobic conditions may exist, either below or at
the surface layer. Vollenweider (Lombardo, 1972) estimated denitrification
rates in some Swiss lakes, with the highest value being 56.5 mg-N/m^-day.
Denitrification has been observed to proceed with traces of oxygen and when
dissolved oxygen was as high as 5 ppm (Lembard®, 1972).
Two commonly proposed mechanisms for denitrification are:
and
2N®
3
The important difference between these two possible reaction mechanisms is
that in the first expression, nitrate is reduced to soluble and readily
oxidizable ammonia, while in the second it is reduced to gaseous nitrogen
-------
and eventually escapes the aquatic system. Lombardo (1972), in discussing
the nitrogen balance in lakes, stated that it is safe to assume that the
gaseous nitrogen end product predominates from the quantitative standpoint,
In his Delaware Estuary model, Kelly (1975) initially considered only
total kjeldahl nitrogen (organic plus ammonia nitrogen) in his nitrogen
formulation. However, early runs of the model indicated that the conver-
sion of organic nitrogen and ammonia nitrogen to other forms in the estuary
had to be included in the simulation. When the conversion of ammonia to
nitrate, as a function of temperature and dissolved oxygen and toxic sub-
stances concentrations, was included in the model, it caused the model to
behave more realistically with respect to nitrogen concentrations measured
in the estuary.
Table 2-16 presents a listing of coefficient values commonly used in
nitrogen formulations. Decay rates, reported at 20°C unless otherwise
noted, are temperature dependent. They are usually adjusted with the
following temperature correction expression:
kT = k28 6(T"28) (2-91)
where ky = rate at T°C
k2f = rate at 2|°C
6 = temperature coefficient
The utilization of oxygen by nitrification is an area of concern in
many water quality models since a sufficient oxygen content in a water sys-
tem is crucial to most biota. The usual procedure in water quality models
is to use the N:0 stoichiometric equivalence which can be combined with a
sink term in the dissolved oxygen mass balance formulation (e.g., see Chen
and Orlob, 1975). This stoichiometric equivalence gives an estimate of
the amount of oxygen utilized in oxidizing a specific amount of nitrogen.
Values of stoichiometric equivalence will vary depending on the form
of nitrogen being converted. The oxidation of ammonia to nitrite requires
3.43 grams of oxygen for one gram of nitrogen oxidized to nitrite. The
reaction which completes the oxidation requires 1.14 grams of oxygen per
gram of nitrite oxidized to nitrate. Therefore the total dissolved oxygen
utilization in the entire nitrification process is 4.47 grams of oxygen, per
gram of ammonia oxidized to nitrate.
Lombardo (1972) reports, however, that due to fixation of carbon
dioxide by bacteria, oxygen utilization is generally less than these
theoretical values. Wezernak and Gannon (1968) suggest values of 3.22
gram of oxygen per gram of ammonia oxidized to nitrite and 1.11 gram oxygen
per gram nitrite oxidized to nitrate.
-------
TABLE 2-16. VALUES FOR COEFFICIENTS IN NITROGEN FORMULATIONS
Organic
(Detrital)
Nitrogen
Decay Rate
(day-1)
0.02-0.4
0.005
0.001-0.02
-
0.024
-
-
0.14
0.04
0.035
-
-
0.1-0.4
Ammonia Ammonia
Nitrogen Nitrogen
Decay Rate Decay Rate
to Nth to NO?
(day-lj (day-1)
.0.1-0.5
0.10
0.05-0.02
0.03
0.16
-
-
0.2
0.052
0.04
0.1
0.003
0.1-0.5
Nitrite
Nitrogen
Decay Rate
to N03
(day-1)
3.0-10.0
0.5
0.2-0.5
0.09
-
-
-
-
-
-
0.3
0.009
5.0-10.0
Organic
(Detrital)
Nitrogen
Settling
Rate
(day-1)
-
-
-
-
0.05
-
-
0.10
0.028
0.001
-
-
-
Location of Study
-
N. Fork Kings River,
Ca 1 i f o rn i a
-
Lake Washington
Lake Michigan
San Joaquin River,
California
San Joaquin River
Estuary, California
Potomac Estuary
Lake Erie
Lake Ontario
Boise River,
Idaho
San Francisco Bay
Estuary
Gray's Harbor/
Chehalis River,
Washington
Reference
Baca and Arnett (1976)
Tetra Tech (1976b)
U.S. Array Corps of
Engineers (1974)
Chen and Orlob (1975)
Canale, et *\_. (1976)
O'Connor, e_t al_. (1975)
O'Connor, et al . (1975)
O'Connor, et al_. (1975)
O'Connor, ejt al_. (1975)
O'Connor, e_t aK (1975)
Chen and Wells (1975)
Chen (1970)
Battelle (1974)
(continued)
-------
TABLE 2-16. (continued)
Denitri- N to C N to C
fication Ratio for Ratio for
Rate Zooplankton Phytoplankton
(day1) (mg-N/mg-C) (mg-N/mg-C)
0.0-1.0 0.05-0.17 0.05-0.17
-
-
.
«.2 0.2
8.14 0.14
8.14 0.14
®.2 0.2
7/5@ 7/50
@.2 0.2
.
-
0.17
N to
Chlor
Ratio for
Phyto-
plankton
-
-
-
-
-
7
7
10
7
10
-
-
-
Location of Study
-
N. Fork Kings River,
California
-
Lake Washington
Lake Michigan
San Joaquin River,
California
San Joaquin River
Estuary, California
Potomac Estuary
Lake Erie
Lake Ontario
Boise River,
Idaho
San Francisco Bay
Estuary
Gray's Harbor/
Chehalis River,
Washington
Reference
Baca and Arnett (1976)
Tetra Tech (1976b)
U.S. Army Corps of
Engineers (1974)
Chen and Orlob (1975)
Canale, et al_. (1976)
O'Connor, et al_. (1975)
O'Connor, et al. (1975)
O'Connor, et al_. (1975)
O'Connor, et al . (1975)
O'Connor, e_t al_. (1975)
Chen and Wells (1975)
Chen (1970)
Battelle (1974)
-------
2,5.2 Phosphorus Cycle
The phosphorus cycle operates like the nitrogen cycle in many respects.
Organic forms of phosphorus are generated by the death of phytoplankton and
zooplankton. Therefore, organic phosphorus, like organic nitrogen, is often
present as detritus. Generally phosphorus in this form is assumed to be
converted to the inorganic state in which form it is available to algae for
primary production. Figure 2-14 shows phosphorus cycle pathways. With the
exception of sediment exchange, the pathways shown are those of major
importance.
Most model representations of the phosphorus systems consider both
organic and inorganic fractions. Occasionally a separate formulation for
sediment phosphorus is also included. The usual procedure in water quality
models, as with the nitrogen system, is to formulate source/sink terms for
these phosphorus groups. The source/sink terms are then included in complete
mass balance expressions, which also include expressions for the transport
of the various suspended or dissolved phosphorus forms by advection and
diffusion.
The internal processes which are generally assumed to affect the con-
centrations of the various phosphorus forms are: 1) algal uptake and
release, 2) zooplankton release, 3) detrital decay, 4) loss to sediments
(through adsorption, precipitation, and/or settling), 5) bacterial regen-
eration, and 6) benthal deposits release. A set of formulations which
represents the phosphorus system is given as (Lorenzen, e_t a_K» 1974):
dD^dt = -6 P App + I2 D2 - l} D1 + I3 D3 (2-92)
dD3/dt = Dp P App + Dz Z Apz - !1 D3 - h D3 (2'93)
and if sediment phosphorus is considered:
dD2/dt = I-, D1 - I2 D2 (+ ^ D3)* (2-94)
where D, = soluble inorganic phosphorus, mg/1
Dp = total sediment phosphorus, mg/1
D- = organic phosphorus in water column, mg/1
P = phytoplankton carbon concentration, mg/1
Z = zooplankton carbon concentration, mg/1
Note that in Lorenzen, et_ al. (1974), due to an oversight (Lorenzen, M.W.,
pers. comm., July 10, 1978)7 the term "+ I-, DV' was not included.
92
-------
EXTERNAL INORGANIC PHOSPHORUS
LOAD TO EPILIMNION
EXTERNAL ORGANIC PHOSPHORUS
SYSTEM BOUNDARY
HYPOLIMNION
LOAD TO EPILIMNION
EPILIMNION
EXTERNAL
ORGANIC
PHOSPHORUS
CJ
AND INORGANIC
PHOSPHORUS
LOADS TO
HYPOLIMNION
Sinking
Growth
PHYTOPLANKTON •-
Respiration
TOTAL
-NONACCESSIBLE
^PHOSPHORUS
SEDIMENTS
Sinking
PHOSPHORUS
DETRITUS '
A
ORGANIC
"PHOSPHORUS
INORGANIC
"PHOSPHORUS
ZOOPLANKTON
Respi ration
L
J
Figure 2-14.
Pathways used in modeling the phosphorus cycle in Lake Michigan
(modified after Canale, et al., 1976).
-------
A = phosphorus to carbon ratio in phytoplankton,
PP mg/mg
A = phosphorus to carbon ratio in zooplankton,
P mg/mg
I, = rate constant for loss to sediments by
sorption, precipitation, day"1
I2 = sediment release (desorption, solvation)
rate constant, day1
Io = rate constant for release of soluble
inorganic phosphorus from organic
phosphorus, day'1
D = specific death rate of phytoplankton, day-1
D = specific death rate of zooplankton, day1
G = specific growth rate of algae, day'1
Lorenzen, ejt aj_. (1974) assumes that the rate of loss of water column
organic phosphorus to the sediment (I]) is the same as the rate of loss of
soluble inorganic phosphorus to the sediment. Figure 2-15 shows schemati-
cally the pathways in this model,
Baca and Arnett (1976) use nearly identical formulations in their model
for the phosphorus cycle as those given above. Expressions for organic
phosphorus and inorganic phosphorus are essentially the same, but their
expression for sediment phosphorus is slightly different. In Baca and
Arnett's formulation a distinct rate constant (14) is used for loss of
organic phosphorus to the sediment.
dD2/dt = ^ D1 + I4 D3 - I2 D2 (2-95)
where I, = rate of organic phosphorus loss to
sediments, day-1; typically 0.01-0.03
\2 = rate constant for sediment release of
inorganic phosphorus, day'1; typically 0.1-0.7
I-j = rate constant for loss of soluble phosphorus
to sediments, day1; typically 0.1-0.7
O'Connor, Di Toro, and Thomann (1975) use the following formulations
for simulating the phosphorus system. The formulations are analogous to
their expressions representing nitrogen kinetics:
94
-------
Death
ZOOPLANKTON
^UUKLAIMMUIM
to
en
Dealh » ORGANIC
* PHOSPHORUS
SOLUBLE INORGANIC Growth
PHOSPHORUS
PHYTOPLANKTON
\
"1*
r i
'
'1
'
*
l£
SEDIMENT PHOSPHORUS
This pathway was not included in Equation (2-94)
as shown in the original reference.
Figure 2-15. Pathways of phosphorus in the model by Lorenzen, ejb a]_. (1974)
-------
dt "pp
D
1 -
Kmf
K
mp
(2-96)
- Ko D3 " Ks D3
where D^ = organic phosphorus concentration, mg-P/1
a = phosphorus to chlorophyll ratio, mg P/mg Chi
a = carbon to chlorophyll ratio, mg C/mg Chi in
zooplankton excreta
a = carbon to chlorophyll ratio, mg C/mg Chi in
p zooplankton biomass
D = specific death rate for phytoplankton, day"
D = specific death rate for zooplankton, day"
P = phytoplankton chlorophyll concentration in
the water column, mg/1
Z - zooplankton chlorophyll concentration in
the water column, mg/1
K = half-saturation constant for zooplankton
^ grazing, mg/1
C = zooplankton grazing rate, day" (mg/l)~
K = decay rate of organic phosphorus to
inorganic phosphorus, day'
K = settling removal or additional decay rate
to inorganic forms, day-1
The first and second terms in the above expression represent contri-
butions to the organic phosphorus concentration due to phytoplankton and
zooplankton composite death and respiration, respectively. The third
term represents a source of organic phosphorus due to zooplankton excretions,
In the model, phytoplankton are assumed to provide all zooplankton excre-
tion organic phosphorus. The fourth term is a loss of organic phosphorus
due to first order decay and the last term is any additional decay or
settling removal of organic phosphorus.
For inorganic phosphorus, which is assumed to be available to the
phytoplankton for assimilation, the equation is:
96
-------
dD1/dt = KQ D3 - app Gp P - K] D1 (2-97)
where Gp = phytoplankton specific growth rate, day
K-| = rate constant for loss of inorganic
phosphorus other than to phytoplankton
growth, day~'
The first term represents a source of inorganic phosphorus due to
organic phosphorus decay. The second and third terms are losses due to
algal uptake and settling (or additional decay), respectively.
The phosphorus representation used by Thomann, et_ ajk (1975) differs
from that of Lorenzen, et a_]_, (1974) and that used by Baca and Arnett (1976)
in two respects. First, sediment phosphorus is not considered in a separate
expression, although settling terms are included in the above two equations.
Second, an additional term for contributions to the organic phosphorus con-
centration due to zooplankton excretions is included in the Thomann
formulation.
The representation of the phosphorus system used by Canale, et al.
(1976) in their water quality model is analogous to their nitrogen system
representation. The phosphorus cycle submodel considers detrital phosphorus
and organic phosphorus as separate constituents (see Figure 2-14). As in
the nitrogen representation, the phosphorus detritus is considered to be
removed from the system by both settling and decay to dissolved organic
phosphorus. Canale reports that the reaction rates governing detrital-to-
organic phosphorus transfer and organic-to-inorganic phosphorus transfer
far exceed the fastest biological rates observed in the system.
The detrital phosphorus equation, as given by Canale, ejt a]_., is as
follows:
dDetP/dt = PCR • ElVl -
assimi
effi
imilation 1 j eating1
ficiency J / • rate J •
\ z
J I-A17 • T •
z/
zooz
z
'•I
/[" natural
+ PCR • £l[ death | J-A17 • T • DetP (2-98)
- A23 - SINK(t) - DetP
where PCR = phosphorus to carbon ratio,in detritus,
mg P/mg C; typically 0.2
A17 - detrital phosphorus to dissolved
organic phosphorus decay rate,
(day - °cH; typically 0.01
97
-------
A23 = maximal detrital sinking rate, day"
ZOO = zooplankton concentration, mg/1
T = temperature, °C
DetP = detrital phosphorus concentration, mg/1
Sink(t) = seasonal correction for sinking rate, unitless
For the dissolved organic phosphorus concentration, Canale, et al.
(1976) propose the following expression:
("respiration"]
dDisP/dt = PCR • I [ loss J + A17 • T • DetP
P 'P (2-99)
- A19 • T - DisP + LOADig
where A19 = organic phosphorus to inorganic ,
phosphorus decay rate, (day - °C)~
DisP = dissolved organic phosphorus concentration, mg/1
LOAD^g = external dissolved organic phosphorus
loading, mg/epilimnion 1-day
Canale's representation of the total organic phosphorus with two
separate formulations, namely detrital phosphorus and dissolved organic
phosphorus, enables contributions from the various algal and zooplankton
sources to be entered into the system in one of two possible forms, with
each form displaying a different decay characteristic. By separating the
organic forms, Canale's more complex formulation attempts to simulate the
decay of the various phosphorus forms more exactly.
Canale's formulation for the dissolved inorganic phosphorus concentra-
tion is given as:
dlnorgP/dt = PCR • E
respiration!
loss + A19 • T • DisP
z ~ -'z
(2-100)
- PCR • E growth + LOAD
20
p , , <:u
r
where InorgP = dissolved inorganic phosphorus
concentration, mg/1
98
-------
= external dissolved inorganic
phosphorus loading, mg/epilimnion 1-day
In addition to the above expressions, Canale, e_t aj_. (1976) also in-
cludes a source/sink formulation for total nonaccessible phosphorus. This
formulation is somewhat analogous to the sediment phosphorus expressions
as presented previously in Equations (2-94) and (2-95) with the exception
that phosphorus release from the sediments is not considered. Canale1s
total nonaccessible phosphorus formulation is given as:
* [sinking]
dNonP/dt = A23 - SINK(t) • DetP • VOLEP + PCR - Z |_ loss J
P P
(2-101)
• VOLEP + [~LOAD19 + LOAD20j • VOLHY
where NonP = nonaccessible phosphorus, mg
LOAD = same as above but mg/hypolimnion 1-day
Lam and Jacquet (1976) have also recognized a need to distinguish
between processes involving particulate phosphorus and soluble phosphorus.
They consider the following processes to be of fundamental importance in
modeling the phosphorus system: 1) the advective and diffusive processes
which are primarily responsible for horizontal movement of both soluble and
particulate phosphorus; 2) the phenomenon of downward settling of particu-
late phosphorus; and 3) the regeneration of soluble and particulate phos-
phorus from sediments due to biological, chemical, and physical causes.
Superimposed on these individual mechanisms are the conversions of
phosphorus from one form to another and the interrelations of each form
with other water quality parameters. Lam and Jacquet's representation of
the phosphorus cycle concentrates on accurately describing the physical
phenomena, and sacrifices some detailed representation of biological proc-
esses and constituent interactions.
Their formulations for the phosphorus system include mass balance
expressions for total phosphorus and particulate phosphorus. The total
phosphorus formulation considers settling, regeneration flux of phosphorus
from the sediment, and external inputs, in addition to advective and dif-
fusive transports. The particulate phosphorus mass balance formulation, in
addition to including the terms considered for the total phosphorus forma-
tion, also includes separate terms for soluble phosphorus conversion to
particulate phosphorus due to photosynthesis and particulate phosphorus
conversion to soluble phosphorus due to respiration.
*The factor VOLEP was not included here in the original reference. However,
based upon a personal communication with R.P. Canale (July 14, 1978) NonP
is in units of mass. Accordingly, volume must multiply every concentration
flux term to keep units consistent.
99
-------
Tetra Tech (1975), Chen and Orlob (1975), and the U.S. Army Corps of
Engineers (1974), consider a single mass balance formulation for their
phosphorus systems. Opinions in the literature differ as to which forms of
phosphorus are available to phytoplankton for uptake. Chen and Orlob assume
that all dissolved phosphorus forms are available for uptake. Their single
mass balance expression for total dissolved phosphorus is given as follows:
dTDP/dt = KDET • SD • DP + KB • |ZP • ZR • Z + FP • FR • F
L -i (2-102)
+ BP • BR • B + PP • (PR - PG) • P I
where TOP = total dissolved phosphorus concentration, mg/1
KDET = detritus decay rate @ local temperature,
day-"I
SD = effective detritus concentration - detritus
concentration + equivalent concentration, mg/1
DP = phosphorus portion of detritus, unitless
KB = biota activity rate coefficient, unitless
ZP = phosphorus portion of zooplankton, unitless
Z = zooplankton concentration, mg/1
ZR = zooplankton respiration rate at 20°C,
day1
FP = phosphorus portion of fish, unitless
FR = fish respiration rate @20°C, day'1
F = fish concentration, mg/1
BP = phosphorus portion of benthic animal
population, unitless
BR = benthic animal respiration rate, day"
B = benthic animal concentration, mg/1
PP = phosphorus portion of phytoplankton, unitless
PR = phytoplankton respiration rate, day
PG = phytoplankton specific growth rate,
day-1
P = phytoplankton concentration,
100
-------
In the above expression, participate phosphorus in the form of
detrital phosphorus, is assumed to decay to a dissolved form, thus con-
tributing to the dissolved phosphorus concentration.
Sediment phosphorus is not represented in a separate expression by
Chen and Orlob, in contrast to the representation suggested by Baca and
Arnett (1976). In Tetra Tech's phosphorus system modeling approach (Tetra
Tech, 1975) the sediment contributions are not specifically formulated in
the mass balance expression. Sediment contributions are included indirect-
ly, however, by considering the phosphorus portion of respired benthic
algae to be a source of dissolved phosphorus. Tetra Tech considers detri-
tal settling in the mass balance expression for detritus which in effect
accounts for particulate phosphorus settling. A separate settling term is
not included in the phosphorus mass balance.
Lombardo (1973) considers two groups of phosphorus in his submodel on
orthophosphate and potential phosphorus. Orthophosphate concentrations
are determined as a function of the following processes: 1) BOD decay,
2) algal growth and respiration, 3) zooplankton excretion, and 4) benthic
deposits release. The other phosphorus form, "potential phosphorus"
defined as that portion of phosphorus which is degradable to orthophos-
phate, is assumed to be controlled by the following: 1) BOD changes, 2)
phytoplankton and zooplankton population changes, and 3) zooplankton
excretion.
As in their nitrogen submodel, Battelle (1974) offers a choice of sub-
models for the phosphorus system. For the phosphorus system, three differ-
ent models are available: 1) an algal phosphorus model, 2) a first order
phosphorus model, and 3) a second order phosphorus model. The first order
model assumes that only reactions between soluble and sediment phosphorus
occur. This model uses the following equations:
-IT D] + I2 D2 (2-103)
dD2/dt - -I2 D2 + I-, D-, (2-104)
where I-, - decay rate for sediment phosphorus to
soluble phosphorus, day~'
I? = decay rate for soluble phosphorus to
sediment phosphorus, day'1
D-, = sediment phosphorus concentration, mg/1
Do - soluble phosphorus concentration, mg/1
The second order model uses the following single expression:
2
cHydt = -I-, D-, (2-105)
101
-------
Sediment exchange is not considered in the second order model.
The algal phosphorus model is similar to that of Lorenzen, et al_.
(1974) for the phosphorus system and to that of Baca and Arnett TT976) as
discussed previously. Battelle's algal model assumes that the reactions
between the various forms of phosphorus can be described by a first order
relationship.
O'Melia (1974) has also developed a lake phosphorus model that has a
stratification period lasting six months. The lake is then assumed to be
unstratified the other six months of the year. A two box model is used
for the summer period (with the epilimnion and hypolimnion comprising each
box). Both boxes are well-mixed. Two phosphorus compartments, soluble
orthophosphate and particulate phosphorus, are used for each of the boxes.
The two compartment assumption was found useful for modeling phosphorus
over a time period of months and seasons.
For the summer model, four interdependent linear differential equations
are needed for mass balances of orthophosphate and particulate phosphorus in
the two boxes, while in the winter only two differential equations are
needed for the single box. Phosphorus inputs from the sediment are not
considered.
Table 2-17 presents a listing of commonly used coefficient values for
phosphorus formulations. The decay rates, reported at 20°C, are generally
temperature dependent, and can be adjusted with the same temperature correc-
tion expression as used for the nitrogen rates.
2.5.3 Silicate Cycle
Since green algae, blue-green algae, and zooplankton require only trace
amounts of silicon, the silicon cycle is usually considered to involve only
the diatoms, particulate (detrital) silicon forms, and dissolved silicate.
Diatoms, unlike other commonly considered phytoplankton groups, require the
element silicon for continued production. A typical representation of the
silicon cycle, as given by Canale, ejt al_. (1976), is shown in Figure 2-16.
Because the presence of silicon is very rarely a dominating factor and is
limited to such a restricted area of concern, it is generally simulated only
in models of considerable complexity.
Canale, ejt al_. (1976) consider three forms of silicon in their repre-
sentation: detrital silicon, dissolved silicon, and total nonaccessible
silicon. The mathematical formulations for these three forms are given as:
dDETSIL
dt
- SCR
respiration
loss
diatoms
predati
loss
(2-106)
- A16 • T • DETSIL - A23 • SINK(t) • DETSIL
102
-------
TABLE 2-17. VALUES FOR COEFFICIENTS IN PHOSPHORUS FORMULATIONS
o
CO
Organic
Phosphorus
Decay Rate
(day1)
0.1-0.7
0.14
0.40
0.14
0.20
Detrital Phos-
Phosphorus Detrital phorus
Decay (Particulate) Phosphorus Phosphorus to
Rate to Settling to Carbon to Carbon Chloro-
Dissolved Rate Ratio for Ratio for phyll
Form (day"1) (day"1) Zooplankton Phytopl ankton Ratio Location of Study
0.01-0.1 0.024-0.24 0.024-0.24
0.005 - - - - N. Fork Kings River,
California
0.10 - - 1 Potomac Estuary
0.028 - - 1 Lake Erie
0.001 - - 1 Lake Ontario
0.001-0.02 - - -
0.20 0.05 0.2 0.2 - Lake Michigan
Reference
Baca and Arnett
(1976)
Tetra Tech
(1976b)
O'Connor, et al .
(1975)
0' Connor, et al .
(1975)
0 ' Connor, et al .
(1975)
U.S. Army Corps of
Engineers (1974)
Canale, et al .
(1976) ~~
0.1-0.7
BatteHe (1974)
-------
EXTERNAL DISSOLVED SILICON
LOAD TO EPI LIMNION
SYSTEM BOUNDARY
HYPOLIMNION
EPILIMNION
Growth
EXTERNAL
DISSOLVED SILICON
DIATOMS.
Respiration
LOAD TO HYPOLIMNION
TOTAL
NONACCESSIBLE-
*-SILICON
Sinking
SI LICON
DETRITUS
ZOOPLANKTON
SEDIMENTS
Egest ion
L
DISSOLVED
SILICON
Figure 2-16. Pathways used in modeling the silicate cycle
(modified after Canale, et a/K, 1976).
-------
dDISSIL
= A16 • T • DETSIL - SCR • I [growth] + LOAD22 (2-107)
dt diatoms
= A23 • SINK(t) • DETSIL • VOLEP* + SCR • I
sinking]
. 10SSJ
VOLEP + LOAD22 • VOLHY
diatoms p
(2-108)
where SCR = silicon to carbon ratio, unitless; typically 0.2
A16 = decay rate for detrital silicon to
dissolved silicon, (day - "C)-"1; typically 0.0015
A23 = maximum detrital sinking rate, day~ ; typically 0.05
T = temperature, °C
DETSIL = detrital silicon concentration, mg/1
DISSIL = dissolved silicon concentration, mg/1
NONSIL = total nonaccessible silicon, mg
VOLEP = epilimnion volume, 1
VOLHY = hypolimnion volume, 1
LOAD99 = loading rate of dissolved silicon, mg/epilimnion 1-day
* (Eq. (2-107)) and mg/hypolimnion 1-day (Eq. (2-108))
It should be noted that the silicon model of Canale, et al. (1976) is
coupled with their phytoplankton model, and that in order to simulate
silicon, diatoms must be among the modeled phytoplankton species.
Bierman (1976) considered the following mass balance formulation for
the dissolved silicon concentration in his model:
= Q/V(SCMBD - SCM) - £ A(L) • SPGR(L) • SSA(L)
V ' diat°mS L J (2-109)
+ RDCMP • T • TOS + WSCM/V
Note that VOLEP was not included here in the original reference. Through
a personal communication with Canale (July 14, 1978) it was determined that
NONSIL is in mass units. In order to make units correct, equation (2-108)
has been changed accordingly.
105
-------
where SCM = silicon concentration in solution,
moles/1
= diatom concentration, mg dry wt/1
SCMBD = boundary value of SCM, moles/1
A(L)
SPGR(L) = diatom specific growth rate, day
SSA(L)
-1
silicon stoichiometry for diatoms
(moles/mg dry wt)
RDCMP = decomposition rate from unavailable to
available nutrient pool (day - °C)~^
TOS = concentration of unavailable silicon,
moles/1
T = temperature, °C
WSCM = external point loading rates of
available silicon, moles/day
V = system volume
Q = water circulation rate, volume/day
Although silicon representation has generally not been considered by
Thomann, O'Connor, and Di Toro in their previous modeling efforts, work is
under way to include simulation of the nutrient silicon in their Lake
Ontario model (O'Connor, et_ al_. (1975)). Under peak growing conditions in
Lake Ontario, surface silica values are about 0.1-0.2 mg Si/1. At the
lower concentration levels, the silicon nutrient could be of importance
(Thomann, e_t aj_., 1975).
2.5.4 Dissolved Oxygen
The oxygen balance in an aquatic ecosystem depends on the capacity of
the system to reaerate itself, which is a function of the advection and
diffusion processes occurring within the system, and the internal sources
and sinks of oxygen. The major sources of oxygen, in addition to atmos-
pheric reaeration, are photosynthetic oxygen production and the oxygen
contained in incoming flow. The sinks of dissolved oxygen include bio-
chemical oxidation of carbonaceous and nitrogenous organic matter, benthal
oxygen demand, and the oxygen demand of respiring organisms. Available
oxygen, therefore, is required for many of the chemical and biological
reactions occurring in the ecosystem.
The complete dissolved oxygen mass balance equation involves two basic
components: 1) the mass transport portion involving advective and diffusive
processes, and 2) a source/sink expression for the individual reactive con-
stituents formulated in a linear fashion. The assertion that dissolved
106
-------
oxygen concentration is a function of reaeration and BOD decay alone has,
in many instances, been shown to be an inadequate approach for modeling D.O.
fluctuations (see Lombardo (1973) for a discussion of several cases).
A typical source/sink formulation for dissolved oxygen used in current
water quality models, as given by Baca and Arnett (1976), is:
L -a, ^ C] -cu, J3C2
Carbon- Ammonia Nitrite Benthic Reaeration
aceous Oxida- Oxida- Uptake
BOD tion tion
(2-110)
Algal
Productivity
where DO = dissolved oxygen concentration, mg/1
DO = dissolved oxygen saturation
concentration, mg/1
C-j = ammonia concentration, mg/1
G£ = nitrite concentration, mg/1
L = carbonaceous BOD, mg/1
2
L. = benthic ^2 uptake rate, g/m -day
P = phytoplankton concentration, mg-C/1
G ,D = phytoplankton specific growth and
p p death rates, day'
P = phytoplankton concentration, mg/1
aijOUja., = stoichiometric constants, unitless
K, ,J-|,J3 = deoxygenation rates, day"
K = reaeration coefficient, day~
AZ = bottom layer thickness, m
107
-------
The saturated oxygen concentration, DOS, is mainly a function of temperature,
and to a lesser degree, a function of salinity and barometric pressure. It
is usually sufficient to obtain the saturated oxygen values by means of an
empirical formulation. Expressions for obtaining dissolved oxygen satura-
tion are discussed in Section 3.3.
Table 2-18 presents a listing of the various stoichiometric equivalence
values used in dissolved oxygen formulations. The various substrate decay
rates, usually converted to an oxygen demand by means of these stoichio-
metric equivalence values, are more fully discussed in the various sections
dealing with the particular substrate processes. Values for these decay
rates are reported in the appropriate sections and are not duplicated here.
TABLE 2-18. VALUES FOR STOICHIOMETRIC EQUIVALENCE
FOR DISSOLVED OXYGEN FORMULATIONS
Stoichiometric
Equivalence
Value
Reference
02/NH3
02/NH3
02/NH3
3.0-3.43
3.5
3.5
3.5
3.5
Baca and Arnett (1976)
Tetra Tech (1976b)
U.S. Army Corps of Engineers (1974)
Chen and Wells (1975)
Chen (1970)
02/N02
02/N02
02/N02
02/N02
1.5
1.2
1.0-1.14
1.2
1.2
Chen (1970)
Chen and Wells (1975)
Baca and Arnett (1976)
Tetra Tech (1976b)
U.S. Army Corps of Engineers (1974)
02/Algae
02/Algae
02/Algae
02/Algae
1.6-2.66
1.6
1.6
1.6
Baca and Arnett (1976)
U.S. Army Corps of Engineers (1974)
Chen and Wells (1975)
Chen (1970)
02/Detritus
02/Detritus
02/Detritus
C02/BOD
1.2
2.0
2.0
0.2
Tetra Tech (1976b)
Chen (1970)
Chen and Wells (1975)
Chen and Orlob (1975)
108
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Several investigators have proposed modifications to Baca and Arnett's
(1976) basic dissolved oxygen formulation. Formulations for dissolved
oxygen balance due to Tetra Tech (1975), the U.S. Army Corps of Engineers
(1974), and Chen and Wells (1975) differ from Baca and Arnett's general
expression in that they include additional sink terms for the respiration
of fish and zooplankton and a term for detritus decay. Lombardo (1973)
includes a source term for contributions from the denitrification process.
Anderson, e_t al_. (1976) consider the production of dissolved oxygen due to
photosynthesis to be contributed by two separate sources: 1) the phyto-
plankton population, and 2) the attached algae concentration. They propose
the following formulation for determining the 02 production (SOURCE,
mg/l-day) by algae:
] + A2 j
SOURCE = Ka ( A] + A., ) (2-111 )
where K = average rate coefficient of oxygen
production by algae, day"'
A-, = phytoplankton concentration as
chlorophyll-a_, mg/1
l\2 ~ attached algae concentration as
chlorophyll-a_, mg/1
Similarly, Battelle (1974), in its Gray's Harbor/Chehalis River study, in-
cluded separate terms in its D.O. formulation for suspended and attached
algae.
The formulations used by various investigators for benthic oxygen
demand differ sufficiently to warrant special mention. Lombardo (1973)
uses a constant rate expression similar to the approach of Baca and Arnett
(1976). Bansal (1975) considers the overall BOD decay rate, KI, to be
composed of a decay rate fraction for suspended matter, k], and a decay
rate fraction for benthal demands, k3 (K] - k] +
Harper, as reported by Lombardo (1972), uses an expression which sets
the benthal oxygen demand, Dg, constant until the dissolved oxygen concen-
tration reaches a level where it limits exertion of the demand. At this
point, the benthal oxygen demand is determined as follows:
bi
where K = demand coefficient
C = D.O. concentration, mg/1
109
-------
H = water depth
0(T-20) _ temperature correction factor
Tetra Tech (1975) and the U.S. Army Corps of Engineers (1974) consider
the total benthic oxygen demand to consist of two separate components - an
organic sediment oxygen demand and a benthic organism oxygen demand. In
their dissolved oxygen formulation, terms for both of these oxygen sinks
are included. Section 3.7 considers further the question of benthic oxygen
demand.
Dissolved oxygen is very important in water quality models in the
sense that it affects or controls the rate of change, and therefore the
concentration, of a significant number of other system constituents.
Oftentimes, in formulations for dissolved oxygen, the various source/sink
components are computed elsewhere in the model and then made available for
use in the oxygen balance expression. For example, the process of nitrifi-
cation is generally simulated as part of the constituent mass balance for
nitrogen and the necessary part of this computed information is then uti-
lized in the oxygen computations. It follows, therefore, that when comput-
ing the dissolved oxygen sources and sinks that are incorporated in the
dissolved oxygen mass balance expression, the sequence of reactions must
be carefully considered.
2.5.5 Toxic Materials
The behavior of toxic compounds in an aquatic environment is quite
variable and depends upon the chemical properties of the compound or sub-
stance being modeled. For simplicity, toxic compound source/sink formula-
tions used in water quality models generally assume the following form (Baca
and Arnett, 1976; Lorenzen, et al., 1974):
dT
where T = concentration of toxic substance, mg/1
K. = decay rate, day"
The above expression represents a first-order decay source/sink term
which is incorporated into a total mass balance expression. The total mass
balance expression for toxic compounds would include, in addition to the
source/sink term, advectiye and diffusive transport of toxic compounds and
the contributions from tributary inputs.
Battelle (1974), in their model application to the Gray's Harbor/
Chehalis River, included a simple nth order decay source/sink formulation
to describe toxic compounds as follows:
110
-------
dT
where TC = concentration of toxic compound, tng/1
a = decay rate constant, day"
n = order of decay
Chen and Orlob (1975) and the U.S. Army Corps of Engineers (1974) use
a mass balance expression to account for the "toxicity level" of an aquatic
system. This toxicity level or content of the water body is then multiplied
by a toxicity coefficient, 3, in order to describe the mortality due to
toxic effects of the various biological constituents.
The majority of the model studies reviewed did not consider specific
applications where toxic compounds were simulated. In general, formulations
for handling toxic compounds were included should a need arise for them in
future applications. Consequently, no toxic substance coefficients or decay
rate values were available.
2.5.6 Carbonate System
The carbonate system is of great importance in lakes, rivers, and
estuaries. The carbon dioxide (C02) - bicarbonate (HC03~) - carbonate
(C03=) equilibrium is the major buffer system in aquatic environments.
This equilibrium directly affects the pH, which in turn can affect the bio-
logical and chemical constituents. Since algae use carbon dioxide as a
carbon source during photosynthesis, this is a nutrient which can reduce
the growth rate when its concentration is low.
To date, the inclusion of the carbonate system in models has been
limited to formulations employing simplified kinetic expressions. Tetra
Tech (1975), the U.S. Army Corps of Engineers (1974), and Chen and Orlob
(1975) include a carbonate system representation which calculates the total
inorganic carbon (TIC) as the sum of bicarbonate, carbonate, and carbon
dioxide. Carbon dioxide is assumed to be produced by respiration and con-
sumed by algal growth. The major source is atmospheric exchange. Harper's
model, as reported by Lombardo (1973), includes a total carbon-carbon
dioxide-pH system and simulates the sources and sinks accordingly.
Thomann, e_t aj_. (1975) plan to include a complete chemical model,
including a representation of the kinetics of the dissolved carbon dioxide
system, into their water quality models of the Great Lakes. A complete
chemical representation poses a new set of problems due to the rapid rates
and highly non-linear forms of the governing equations. Additional program-
ming problems can be introduced when representing the chemical species with
traditional mass balance formulations.
Ill
-------
Thomann, £t aj_. currently treat the carbonate equilibrium in a conven-
tional manner. The major species considered are dissolved carbon dioxide,
bicarbonate, and carbonate ion, together with the hydrogen and hydroxyl
ions. Mass balance equations for each of these species are complicated by
the fact that they undergo reversible ionization reactions. The individual
species are extremely reactive so that a direct mass balance formulation
results in equations which are nonlinear and numerically quite badly
behaved.
The crux of the formulation problem, therefore, is in producing a more
tractable formulation in terms of quantities which are conservative relative
to these ionization reactions. The computational feasibility of including
these chemical reactions in the water quality model has been explored by
Thomann, e_t a^L (1975) in a series of computations using the Rand Chemical
Composition Program. It is expected that this computer program will form
the basis for their chemical submodel calculation in the Great Lakes water
quality models.
2.5.7 Conservative Sub_s_tances
Conservative constituents may be defined as those constituents that
do not undergo significant chemical decay. Substances which are commonly
considered to be conservative in water quality simulations include total
dissolved solids (IDS), chloride ion, alkalinity, and salinity.
Conservative substances are generally assumed to move through a water
body by advection and diffusion processes alone in simulation models. Trib-
utary inflows may also contain conservative substances which are assumed to
mix with the main water body. Models typically include a mass balance
expression for conservative substances with the decay rate assumed equal to
zero (e.g., Tetra Tech, 1975; U.S. Army Corps of Engineers (1974), and Lam
and Simons (1976)). The governing equations for conservative substances
which are considered in a water quality model, therefore, are simply time-
dependent advection-diffusion equations as discussed previously.
Tetra Tech (1975) uses the results from the calculation of the concen-
tration of alkalinity, which is treated as a conservative substance, to
perform a calculation for pH and carbon dioxide. Battelle (1974) treats
salinity as a conservative constituent, which is advected and transported
in a mass balance equation. Salinity is included in their oxygen saturation
concentration calculations because it influences the rate of dissolved
oxygen reaeration.
Tetra Tech (1976b), the U.S. Army Corps of Engineers (1974), and
Lombardo (1973), all consider total dissolved solids as a conservative
substance in their models. Total dissolved solids is generally assumed
to behave as a conservative substance in models because of the difficulty
of simulating all of the processes affecting total dissolved solids con-
centration, such as adsorption and desorption from sediments. Biological
uptake of certain ions may also influence total dissolved solid concen-
trations, and could be a source of error when considering total dissolved
112
-------
solids as a conservative substance (Lombardo, 1973). Total dissolved
solids concentration is of great importance because it influences the
possible uses of water. The Public Health Service recommends that the
total dissolved solids concentration of drinking water be less than 500 ppm
and water with a concentration of total dissolved solids greater than
1,000 ppm is generally unfit for industrial purposes.
Some biological mechanisms - algal and bacterial growth for example -
will decrease the concentration of certain ions; while others, such as zoo-
plankton excretion and bacterial nutrient regeneration, will increase ion
concentrations. Rainfall may also influence the total dissolved solids
content of a water body when its total dissolved solids concentration is
different from that of the water body. Lombardo (1973) stresses the need
for verification of the conservative constituent assumption for each sub-
stance modeled as such.
In collecting field data, specific electrical conductance is used for
measuring total dissolved solids concentration. The conversion relation-
ship between conductivity and dissolved solids, however, is not perfectly
linear. For most natural waters, the conversion factor ranges from 0.54
to 0.96 mg/1 per micromho/cm and is usually taken as 0.65 ± 0.1 (Lombardo,
1972). Tanji and Beggar, as reported by Lombardo (1972), found the con-
version factor to be 0.727 for a number of western U.S. rivers. Conductance
is usually reported at 25°C. Frequently, when the dissolved solids concen-
tration becomes high, as in seawater, the relationship between conductance
and concentration is not well defined.
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the Phytoplankton Population in the Sacramento-San Joaquin Delta," Advances
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Feigner, K.D. and H.S. Harris, 1970. Documentation Report FWQA Dynamic
Estuary Model, Federal Water Quality Administration, Washington, D.C.,
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to Production Dynamics and Structure of Freshwater Animal Communities,"
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and R.A. Ferrara, 1977. User's Manual for the M.I.T. Transient Watejr
Quality Network Model. For Corvallis Environmental Research Laboratory,
EPA-600/3-77-010.
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Processes," Journal of the Environmental Engineering Division, ASCE, EE2,
pp 409-421.
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Compartments and Population Rate Coefficients for Use in Reservoir Ecosystem
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Engineers, U.S. Army, Washington, D.C.
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120
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CHAPTER 3
A DETAILED REVIEW OF SELECTED
MODEL FORMULATIONS AND PARAMETERS
3.1 INTRODUCTION
Chapter 3 covers several aspects of water quality models not presented
in Chapter 2. The approach is also different. In Chapter 2, material was
organized along process lines. For example, coverage included geometric
representations, mass transport, and chemical processes. Other subjects
presented in Chapter 2 included such issues as numerical solution techniques
and brief discussions of formulations and parameters which are either not
universally simulated (e.g., "fish" and "toxic substances") or for which
there are little rate data available. Chapter 3 presents several topics
which are ordinarily of major importance in surface water quality modeling
and which merit extended coverage. Also, discussions are arranged by water
quality parameter rather than by process, as they generally were in Chapter 2.
Chapter 3 addresses the problem of selecting appropriate values for
model rates and parameters. This is done by discussing the formulations and
incorporated constants, factors affecting those constants, and ways of meas-
uring the phenomena involved. Definite guidelines for the selection of rate
constant values are not always available, however. Further, an effort has
generally been made to avoid including material recommending a specific value
under a given set of circumstances. It was felt that ordinarily, the
selection of parameter values is too case-dependent, far too many factors
must be considered for each situation, and presenting values to be used
under set conditions would more often mislead than guide the user to a
satisfactory value.
For each rate parameter discussed, mathematical expressions are
presented to show how the parameter relates to the mass balance expression
of which it is a part. This illustrates precisely what parameter is being
discussed, and minimizes the ambiguity and possible misinterpretation of
the parameter's significance and normal use. Also incorporated are values
for the parameters being discussed as reported in the literature. The data
presented include, in addition, various related factors which may be of use
to the modeler. For example, in selecting specific growth rate values, it
is important to compensate for the effect of temperature. This, in turn
requires data on temperature response. In general, such temperature response
data are presented in conjunction with the rate data themselves.
121
-------
Finally, at the end of each section, references are presented. This
was done because of the large number of sources cited and in order to
facilitate the user's locating of reference material.
3.2 REAERATION
3.2.1 Introduction
The dissolution of gaseous oxygen in water can be viewed as a mass
transfer process occurring in sequential steps. According to the "Two Film
Theory," beginning in the vapor phase, oxygen travels through a gas film on
the vapor side of the gas-liquid interface. It then passes through a liquid
film on the liquid side of the interface, and is finally dispersed through-
out the bulk solution.
Each of these steps requires a finite period of time. However, one
particular step may take significantly longer than the other and is thus
termed the rate limiting step. Under perfectly quiescent conditions the
diffusion of the oxygen through the bulk liquid is the slowest step, and
here, molecular diffusion expressions may be used to predict the rate of
transfer of dissolved oxygen. However due to turbulance in natural systems,
the bulk diffusion rate is not ordinarily rate limiting. For oxygen, or any
other sparingly soluble gas, the rate limiting step becomes the passage
through the liquid film. The rate limiting step for highly soluble carbon
dioxide, in contrast, is passage through the gas film.
Lewis and Whitman (1924) assumed that transport through the inter-
facial film was affected solely by molecular action and that the process
obeyed Pick's first law (Bird, et^ a]_., 1960). Assuming the concentration
gradient to be linear through the film, Fick's law can be written as:
q =
cs-c
Jm 6
(3-1)
where q = rate of transport of oxygen through a
surface, mg min"1
D = molecular diffusivity of oxygen in water,
m cm^ min-1
6 = film thickness, cm
C = dissolved oxygen concentration below
film, mg ml"'
C = saturation dissolved oxygen concentration,
mg ml'1
2
A = surface area, cm
122
-------
Letting
K. = L (3-2)
L o
Equation (3-1) can be rewritten as
q = AKL (CS-C) (3-3)
where K^ is commonly known as the oxygen transfer coefficient. Researchers
have expressed KL in terms of more readily measurable parameters, as will be
shown shortly. Holley (1975) has discussed potential ramifications of the
assumptions used by Lewis and Whitman in the above derivation.
Equation (3-3) can be used to determine the flux of dissolved oxygen
entering (or leaving) a receiving water body through the mechanism of
reaeration, and incorporates no assumptions about whether or not the water
body is well -mixed. If, however, the water body is assumed to be well mixed
vertically, so that a single concentration, C, exists over depth, the time
rate of accumulation of dissolved oxygen due to reaeration can be expressed
as:
V!T=AKL(CS-C)
or
or
^r / \
(3-6)
where V = volume of water under the surface area, A, cm
The ratio V/A equals average depth, H. The quantity K[_/H is usually
expressed by a single coefficient (as shown in Equation 3-6), most commonly
designated as ka or k^ in the literature, and commonly having units of day~^
or hour'l. The notation k£ will be used here.
3.2.2 Expressions for Reaeration Coefficient
Much research has been done to develop both empirical and mechanistically
justifiable expressions for k2 and K[_. This section will discuss a number of
these expressions, and will be divided essentially into discussions of coeffi-
cients applicable to streams, to lakes, and to estuaries. Some overlap of
expressions is unavoidable, however. For more exhaustive discussions on
123
-------
the expressions presented here (along with some others that are not) the
user is referred to Covar (1976), Elmi (1975), Holley (1975), Kramer (1974),
Bennett and Rathbun (1972)., and Lau (1972).
Weber (1972) showed, by using dimensional analysis, that the reaeration
coefficient for streams is a function of stream depth and velocity, each
variable being raised to some power. Many researchers have developed
expressons for ^
have based their
these expressions
to the conditions
that contain these variables, although other researchers
ormulas on other variables. Table 3-1 shows a number of
In addition, the table contains information pertaining
under which the formulas were developed, the units in which
the variables should be expressed, the original reference in which each
formulation was reported, and more recent references discussing the expres-
sions in detail. In Section 3.2.6 a method devised by Covar (1976) will be
discussed that is useful in selecting a particular reaeration coefficient
for streams.
Many rivers and streams have small to moderate sized dams crossing them
in one or more places. Reaeration occurs as the water flows over the dam.
Based on experimental data (Gameson, et_ a]_. , 1958), and later verified with
field data (Barrett, et_ al_. , 1960), the following relationship for reaera-
tion over dams has been developed:
Da - Db =
1 + 0.11 ab(l + 0.046T)H
(3-7)
where D = dissolved oxygen deficit above dam, mg/1
a
D, = dissolved oxygen deficit below dam, mg/1
T = temperature, °C
H = height through which the water falls, ft
a =1.25 in clear to slightly polluted water:
1.00 in polluted water
b =1.00 for weir with free fall: 1.3 for step
weirs or cascades
An alternate equation developed from data on the Mohawk River and Barge
Canal in New York State (Mastropietro, 1968) is as follows:
- 0.037HD
(3-8)
Equation (3-8) is valid for H up to 15 feet and for temperatures in the
range of 20° to 25°C. Under these conditions, it is recommended that this
equation be used in preference to Equation (3-7) because it was specifically
developed for such conditions.
124
-------
TABLE 3-1. REAERATION COEFFICIENTS FOR STREAMS
IN)
c_n
Original Reference
O'Connor and Dobbins
(1958)
Churchill , et al.
(1962)
Owens , et al .
( 19647"
Langbein and Durum
(1967)
Isaacs and Gaudy
(1968)
Negulescu and Rojanski
(1969)
Tsivoglou (1967, 1972)
Tsivoglou and Neal
(1976)
Foree (1976)
Gloyna, et a]_. (1971)
Formulation for kp (base e) Units of Variables
/ \ Any compatible set of units
(v ur5
H1.5
U-fps
n £,,0.969 H-feet
11 '6U k 1/ri-w
lfi7T r\ o - I / u ay
H
U-fps
,, 7|,0.67 H-feet
k 1 ' H -i • -
,,1.85 K2-l/day
n
7.6U U-fps
,1.33 H-feet
k2-l/day
7.03U U-fps
H1.5 H-feet
k2-l/day
0 85 U"fps
A 74 I u I ' H-feet
H'"\H/ k2-l/day
(, \ Ah-feet
^ 1 @25°C t-hours
/ k2-l/hour
0.30+0.19S1'2 S-feet/raile
K£- I /oay
@ 25°C
U-fps
6.86U0'703 »-™
,,1.05 k2-l/day
Development Conditions
For streams displaying isotropic
turbulence. The observed data
had the following characteristics:
1 '
-------
TABLE 3-1 (continued)
ro
en
Original Reference Formulation for k? (base e) Units of Variables
Development Conditions
Thackston and Krenkel
(1969)
0'Connor and Dobbins
(1956)
Krenkel and Orlob
(1962)
0.000299 1
•
-------
Foree (1976) has more recently investigated the reaeration effects of
small dams in Kentucky under low flow conditions. The expression he
developed is:
Db = F Da 0-9)
where
r = e°-16H (G> 25°C) (3-10)
The value r was corrected for temperature by:
In r25oc = (in ry ) (l .022)(25'T) (3-11)
Foree developed this relationship within the following ranges of values:
2 •< Q _< 108 cfs
2.6 <_ H <_ 14.0 feet
17.0 ^ T ^ 24.0°C
A limited number of expressions have been developed for K|_ to be used in
stratified systems, such as stratified lakes or estuaries. Generally these
expressions contain terms for wind velocity in addition to, or in place of,
hydraulic parameters, since wind can be the major driving force inducing
turbulence into the flow field. This is especially true in lake systems
where the net advective velocity may approach zero. Of course Ki can be
obtained from ^2 for completely mixed systems from Equation (3-6). However,
for stratified systems the relationship between K[_ and k2 as expressed from
Equations (3-5) and (3-6) is only valid for a depth over which the dissolved
oxygen concentration can be considered constant. This depth is almost always
less than the depth of the system.
Baca and Arnett (1976) have employed the surface transfer coefficient
to simulate reaeration in the mixed surface layer of lakes. The expression
used is:
(3-12)
where V = wind speed, m/day
a-! - 0.005-0.01 day"1
a2 = 10"6 - 10"5 nf1
K, = surface transfer coefficient, day
127
-------
Kanwisher (1963) has developed an expression for the transfer coeffi-
cient that has been applied to lakes (Chen, e_t aj_., 1976). It is given by:
KL = V
(
200-60
(3-13)
where D = molecular diffusivity of oxygen, m^/sec
V,, = wind speed, m/sec.
w
Note from Equation (3-2) that the denominator of Equation (3-13) is
equivalent to 6, the liquid film thickness.
Smith (pers. comm.) has used open-sea data previously collected by
Kanwisher (1963) to develop an expression (Equation (3-14)) for the transfer
coefficient applicable to estuaries. This expression is given as:
KL = 1.57 + 0.32 Vw2 (3-14)
where KL and Vw are in units of feet/day and feet/second, respectively. The
use of this expression is discussed by Johanson, et_ al_. (1977).
For shallow lakes Banks (1975) developed several expressions for KL,
each applicable over a limited range of wind speeds. The relationships are:
KL = 0.362 V^ for 0 5 Vw < 5.5 m/sec (3-15)
and
K - 0.0277 Vw2 for 5.5 m/sec < V (3-16)
where K, is in meters/day.
Although Banks actually developed three relationships, the third one
is applicable over such a narrow range of wind speeds that the above two
formulations are sufficient. By "shallow lakes" Banks implies that the
vertical turbulent diffusion coefficient is constant throughout the depth
of a lake.
Eloubaidy and Plate (1972) have performed experiments in the wind wave
facility of Colorado State University for the purpose of determining the
effect of wind on the reaeration rate in flowing water. Their ultimate
objective was to develop a practical method of predicting reaeration in
streams with wind blowing across the surface. They arrived at the following
expression for the transfer coefficient, K, , in feet per day:
128
-------
CU* h U*
KL = s c (3-17)
where C - a constant of proportionality
2
v = kinematic viscosity of water, m /sec
U* = surface shear velocity, m/sec = 0.0185 V '
s w
V = wind, m/sec
U* = shear velocity defined as »/ gh S , m/sec
c \ c
h = normal depth (i.e., depth with uniform flow), m
1 dP
S = pressure-adjusted channel slope, unitless, S + — -r-
^ up y u A
3
p = mass density of water, kg/m
2
g = gravitational constant, m/sec
S = slope of energy gradient (channel slope for
uniform flow), unitless
dP
-j- = air pressure gradient in the longitudinal
direction, kg/m2-sec2
From their experiments Eloubaidy and Plate found that C = .0027.
The variables comprising Equation (3-17) are readily obtainable, with
the exception of the pressure gradient. For short river reaches this can be
set to zero. The authors determined that an error on the order of 2% was
obtained in k2 (- KL/h) by neglecting the pressure gradient.
A summary of the conditions under which Equation (3-17) was developed
is as follows:
channel slope: .00043, .001
air velocity: 22, 30, 38 fps for each slope
discharge: 0.79, 0.83, 0.91 cfs at 0.001 slope
0.58, 10.63, 0.75 cfs at 0.0043 slope
water depth: 0.385 feet
129
-------
Downing and Truesdale (1955) performed a laboratory study using a
small tank and fan to generate wind currents for the purpose of comparing
their results with estuary data. Their findings were not in good agreement
with observed reaeration rates, and they attributed differences partly to
the turbulent nature of the wind generated by the fan. Their data also
indicated that below 7 mph, wind had little effect on the reaeration rate.
Eloubaidy and Plate (1972), on the other hand, had found that wind began to
play a significant role in reaeration at speeds above 1.6 mph. In accu-
rately quantifying the effect of wind on reaeration Kramer (1974) hypoth-
esized that it is not sufficient to consider wind speed alone but direction
as well when dealing with inland water bodies protected by complex sur-
rounding topographic features.
As well as using predictive relationships based on wind velocities to
express reaeration in estuaries, numerous investigators have based reaeration
on velocity and depth (see, for example, Johanson, e^t a/K (1977) and
Hyer, ert al . (1971)). Each of these workers uses an expression developed by
O'Connor TT960). O'Connor arrived at his formula from relationships
developed specifically for non-tidal streams and rivers. The expression is:
(D u )
=Vj
°'5
where D = molecular diffusivity of oxygen
U - mean tidal velocity over a complete tidal cycle
H = average depth at a section over the tidal cycle
and any consistent units are used. Equation (3-18) is valid for nonstrati-
fied estuaries and was verified on the Delaware and James Rivers.
In modeling estuarine reaeration, other researchers have used expres-
sions such as those for k2 found in Table 3-1, which were developed for
streams (Kramer, 1974). Since the reaeration expressions were not developed
under estuarine flow conditions, their application to such conditions may
not be justifiable.
3.2.3 Factors Affecting Reaeration
In the predictive equations presented in Section 3.2.2 it was tacitly
assumed that the reaeration rate, in each case, depends only on the vari-
ables appearing in the particular expression. In most instances the
variables were depth, flow velocity, and wind velocity. For streams, the
flow velocity is the instantaneous velocity. For estuaries the instan-
taneous velocity can be used in real-time models, while for tidally averaged
models, the tidally averaged velocity must be used.
130
-------
Wind speed has most often appeared in expressions for the transfer
coefficient for lakes and estuaries, although Eloubaidy and Plate (1972)
developed an expression containing wind speed applicable to streams, as
discussed earlier. A smaller number of researchers have used other variables
for stream reaeration, such as channel slope (Foree, 1976) and change in
water surface elevation with respect to travel time (Tsivoglou, 1967).
Other factors are also known to influence reaeration rates. These
effects have been satisfactorily quantified, in some instances, while in
others, they have not. The factors (from Hoi ley, 1975) are listed below,
followed by a discussion of each.
1. The Schmidt number, S , where S^ = v/D , the kinematic viscosity
of the liquid divided by the molecular diffusivity of the dis-
solved gas. Since Sc is a function of temperature, the dependence
of the reaeration coefficient on Sc can be, and often has been,
replaced by an Arrhenius temperature relationship:
'™
where T = temperature at which the reaeration coefficient is to be
evaluated, °C
0 = temperature adjustment factor
2. Surfactants.
3. Suspended particles.
4. Dissolved substances.
5. Artificial mixing.
In Equation (3-19), the base temperature used to evaluate reaeration
is 20°C. Some researchers have preferred to use 25°C as the base tempera
ture. The value of k£ at 25°C can be found from Equation (3-19) to be:
(3-20)
25 V "/20
Although k2 has been used in both of the above two equations, KL could as
well have been used.
Table 3-2 shows values of the temperature coefficient 6 for different
types of systems and the reference from which each value of 8 was obtained.
It is important to note that the data are for a base temperature of 25°C,
131
-------
TABLE 3-2. REPORTED VALUES OF TEMPERATURE COEFFICIENT
(FROM METZGER, 1968)
Temperature
Coefficient, 9
1.047
1.0241
1.0226
1.020
1.024
1.016
1.016
1.018
1.015
1.008
Surface Transfer
Coefficient,
K.L, in
Centimeters
Per Minute
0.01
0.02
-
0.02
0.03
0.06
0.15
0.028
0.042
0.37
Aeration
System
channel
stirred
stirred
stirred
sti rred
sti rred
stream
channel
channel
channel
Reference
Streeter, et al_. (1936)
Elmore and West (1961)
Elmore and West (1961)
Downing and Truesdale (1955)
Downing and Truesdale (1955)
Downing and Truesdale (1955)
Streeter (1926)
Truesdale and Van Dyke (1958)
Truesdale and Van Dyke (1958)
Truesdale and Van Dyke (1958)
-------
The range of e's presented in Table 3-2 (1.008 to 1.047) produce reaeration
rate ratios of 146 percent and 68 percent at 30° and 10°C, respectively,
when a base temperature of 20°C is used. The ramifications of this are
that by using an incorrect 6 for a particular system, the dissolved oxygen
resource estimates for the system can be quite inaccurate.
Metzger (1968) undertook a study to determine the reasons behind the
variation in 9 values reported from the literature. He found that for
higher mixing (more turbulence) the surface transfer coefficient increased.
This, in turn, decreased 0. Figure 3-1 shows the results of Metzger1s work,
along with the 6 values shown earlier in Table 3-2. Metzger's formula for
K|_, using film penetration theory, will not be presented here. However,
Metzger found that for low mixing intensity (surface water smooth), K|_ ranged
from 0.10 to 0.12 cm/min at 20°C, and for moderate mixing intensity (water
surface mildly rippled) K|_ was approximately 0.40 cm/mi n.
CD
<4—
O
o>
D
1.050
1.040
1.030
1.020
1.010
1.000
0,005 0.01 0.02 0.04 0.06 O.I 0.2 0.4 0.6 1.0
Surface Transfer Coefficient (cm/min)
Figure 3-1. Effects of the surface transfer coefficient
(KL) on the value of 6 in Equation (3-19) (Metzger, 1968)
133
-------
r
Surface active agents (SAA) tend to decrease atmospheric reaeration by
reducing the surface transfer coefficient, KL (Mancy and Barlage, undated).
The extent of reduction depends upon the hydrodynamic characteristics of
the liquid phase and the physicochemical characteristics of the SAA.
Resistance to oxygen transfer has been shown to result from two kinds of
effects:
1. physical interference of SAA molecules, and
2. increase in viscosity of the interfacial liquid.
Conflicting results have been obtained in the few studies which
examine the effects of suspended solids on reaeration rates (Holley, 1975),
These conflicting results prompted Alonso, et^ al_. (1975) to perform a study
of the effect of suspended sediment loads on reaeration rates in sediment-
laden flows in straight open channels. Alonso, et. a\_. compared the results
with the surface reaeration rates observed in clear water open channel flows
having the same bulk-flow characteristics. From their experiments they
concluded that the reaeration rate decreased about 35% over the sediment
concentration range used (up to 3,562 ppm). This decrease in k£ was attri-
buted to the decrease in the intensity of turbulent mixing at higher sus-
pended sediment loads, which, in turn, decreases reaeration.
Poon and Campbell (1967) investigated the effects of dissolved chemi-
cals on the rate of oxygen transfer using water column aeration by sub-
merged diffusers. They found that by increasing the concentrations and
molecular weights of dissolved chemicals, the rate of oxygen transfer due
to the rising bubbles was reduced. The results of Poon and Campbell are
summarized in Table 3-3, where a is defined as the ratio of the diffusion
coefficients with to without the dissolved substances.
TABLE 3-3. SURFACE TRANSFER RATIO a
FOR VARIOUS SUBSTANCES (FROM POON AND CAMPBELL, 1967)
Dissolved
Matter
Sodium Chloride
Glycine
Glucose
Soluble Starch
Peptone
Nutrient Broth
Concentration
mg/1
111!
1MI
1©@©
1MI
1101
Rati« a
i.M-
9.92
®.9»
®.87
S.51
1.47
134
-------
Artificial reaeration by means of mechanical aerators will not be dis-
cussed here (see, for example, Whipple, et al_. (1969), and Weber (1972).
However, it 'iould be noted that reaeration due to ship mixing may be of
importance -^ certain situations. In the Houston Ship Channel, Kramer
(1974) attempted to predict reaeration based on several predictive formulas,
such as are shown in Table 3-1. In all cases the predicted values were
considerably less than observed. The author felt that this discrepancy
was due largely to artificial mixing induced by ship traffic and wind
effects.
3.2.4 Tabulated Values of the Reaeration Coefficient
Tables 3-4 and 3-5 below show reaeration coefficients and other
characteristics of streams and rivers as taken from Hydroscience (1971)
and Elmi (1975). Table 3-6 shows reaeration coefficients observed in
tidal rivers and estuaries. Included in the tables are data on hydraulic
characteristics of the systems, as well as original references. Additional
values for stream reaeration rates are included in Table 3-17 in Section
3.6 on "Nitrogenous Deoxygenation." Some duplication exists between that
table and Tables 3-4 and 3-5.
Because they are based upon hydraulic type variables which are readily
quantifiable, reaeration rates for streams, rivers, and estuaries have
been extensively computed and tabulated (see, e.g., Table 3-1). Since these
values generally do not change substantially over time, the values in the
tables may give reasonable estimates of the reaeration rate for the systems
tabulated. For lakes, however, reaeration has generally been expressed as
a function of wind speed, and is not closely related to lake characteristics
Hence there are no "typical" values of K[_ or k2 for a particular lake, and
no tabulations will be given here. Referral to Section 3.2.2, however,
will show expressions based on wind speed for K|_,
3.2.5 Techniques Used to Estimate Reaeration in Situ
Many methods have historically been used to estimate the reaeration
rate in receiving water bodies. One approach consists of using a mass-
balance equation for dissolved oxygen where all the parameters except kg are
measured, leaving kg to be calculated. Churchill, et_ aj_. (1962) have used
this approach and expressed kg in terms of the dissolved oxygen deficit
observed at two points in a stream:
In D - In D,
k2 = *-£ ' (3-21)
where DQ = deficit at upstream location, mg/1
D-i = deficit at downstream location, mg/1
t = time of travel between the two locations, days
135
-------
TABLE 3-4. REAERATION COEFFICIENTS AND OTHER CHARACTERISTICS
OF STREAMS AND RIVERS (AFTER HYDROSCIENCE, 1971)
CO
River Name
Grand River
(Michigan)
Clinton R.
(Michigan)
Truckee R.
(Nevada)
Flint River
(Michigan)
Jackson R.
(Virginia)
N. Branch
Potomac R.
(Md. , W.Va.)
Clarion R.
(Penna. )
South River
Ivel River
(England)
Depth
Category (ft. )
Shallow 1.9
Shallow 1.58
Shallow 1.67
1.67
1.67
Shallow 2.1
2.6
2.6
1.7
1.9
Shallow 3.
Shallow 2.
Shallow 1.
1.9
1-2
Shallow 1.21
1.51
1.09
1.50
1.08
0.38
1.12
1.46
1.31
2.44
2.03
Area
(ft2)
320.0
44.6
150.
150.
150.
210.
200.
400.
290.
400.
365.
100.
Flow
(cfs)
295.
33.
180.
195.
271.
134.
174.
174.
204.
204.
100.
100.
1-10.
35.
4.86
4.15
3.87
15.40
4.86
4.15
3.87
15.40
15.40
10.07
10.07
Velocity
(fps)
0.92
0.72
1.20
1.30
1 .81
0.64
0.83
0.44
0.73
0.51
0.27
1.0
0.55
0.14
0.14
0.13
0.37
0.16
0.15
0.13
0.38
0.47
0.23
0.22
KL
(ft/day)
9.
5.58
4.64
6.83
6.45
9.75
4.00
2.55
8.65
14.70
4.16
6.35
P20°C
(I/day)
4.5
5.9
5.6
5.7
6.6
3.5
3.9'
3.1
5.0
2.2
4.1
9.0
2.26
2.35
2.06
3.20
2.37
4.57
2.09
1.18
3.18
6.18
0.90
1.66
(?20"C
(I/day) Reference
0.59 O'Connor and Di Toro (1970)
3.37 O'Connor and Di Toro (1970)
0.36 O'Connor and Di Toro (1970)
0.36
0.96
0.56 O'Connor and Di Toro (1970)
0.63
0.69
0.69
0.69
1.25 O'Connor with Hydroscience (19C2a),
(1962b), Hydroscience (1967b)
0.40 Hydroscience (1969a)
3. O'Connor and Dobbins (1958)
2.
Owens, et al . (1964)
k , is the deoxygenation coefficient. See Section 3.5.
(continued)
-------
TABLE 3-4 (continued)
River Name
Lark River
(England)
Derwent R.
(England)
Black Beck
River
(England)
St. Sunday's
Beck (Eng.)
Yewdale Beck
(England)
Elk River
(Penna. )
Mohawk River
Mohawk River
(New York)
North Branch
Susquehanna
New River
(Virginia)
Depth Arf
Category (ft.) (ft )
Shallow 1.
1.
1.
2.
Shallow 0.
0.
0.
Shallow 0.
0.
0.
0.
0.
1.
Shallow o.
0.
Shallow 0.
0.
0.
0.
0.
0.
Shallow 0.
Shallow 3.
Intermediate 15.
Intermediate 4.
Intermediate 5.
74
47
82
41
72
89
85
40
40
39
60
69
00
82
78
64
48
72
66
67
69
9
143.
3800.
1700.
1720.
Flow
(cfs)
10.94
10.
36.
36.
21.
21.
21 .
2,
2.
2.
17.
17.
17.
19.
19.
5.
5.
17.
17.
17.
17.
.94
20
20
.60
60
.60
.70
70
70
70
70
.70
10
10
10
10
30
30
30
.30
Velocity
(fps)
0.
0.
0.
0.
1.
1.
1.
0.
0.
0.
1.
1.
1.
1.
1.
0.
0.
1.
1.
1.
1.
,28
.37
.50
.43
,37
.19
.07
44
,56
63
83
81
54
07
27
46
60
16
31
,30
25
0.97
KL
(ft/ day)
2.
6.
5.
1.
41.
39.
37.
19.
20.
21.
54.
39.
34.
32.
23.
14.
19.
25.
24.
21.
24.
12.
.48
04
,10
,48
90
70
80
30
90
00
80
40
20
40
70
40
20
30
80
10
,60
k2 kd*
020°C @20rjC
(I/day) (I/day) Reference
0.
2
1.
0.
.78
.12
.41
31
31.80 Owens, et al . (1964)
24.53
34.
25.
28.
22.
49.
30.
18.
21.
16.
12.
30.
18.
20.
17.
19.
5.
57
59 Owens, et al . (1964)
34
,80
17
77
46
05 Owens, et al. (1964)
06
04 Owens, et al. (1964)
32
90
25
09
16
84 O'Connor and Dobbins (1958)
.07-4.0 .23 O'Connor & Hydroscience (1967)
800.
1000.
1200.
0.
0.
21
60
70
.07-4.0 .40 O'Connor & Hydroscience (1968)
1.
1.
5 0.35 Hydroscience (1965a)
04 2.5 Hydroscience, (1966a)
0.5 Hydroscience, (1967a)
k, is the deoxygenation coefficient.
See Section 3.5.
(continued)
-------
TABLE 3-4 (continued)
CO
oo
Ri ver Name
Wabash
(Indiana)
Clinch R.
(T.V.)
Holston
(T.V.)
Fr. Broad
(T.V.)
Depth Ar
Category (ft.) (ft
~a Flow
2) (cfs)
Intermediate 5-7
Intermediate 3
5.
4
6
5
7
Intermediate 11.
2,
2
4.
9.
6,
7,
7
5
8,
3,
Intermediate 9.
10.
3.
4.
.27
.09
.42
.14
.65
.17
.41
,12
.93
.54
.50
.29
,52
.07
.44
.06
.98
.38
.19
.29
,74
5.72
Wautaga R.
(T.V.)
Hiwassee R.
(T.V.)
6.
4.
6.
7.
9.
Intermediate 3.
Intermediate 3.
2.
98
29
01
16
49
42
02
83
10
14
10
10
10
11
12
17
44
12
17
12
17
1000-
5000
3300
4500
3190
5C90
5910
5930
,385
3230
6400
,085
,440
6540
,500
,500
5590
,930
952
,010
,120
,105
8775
,455
,270
4150
8775
,455
,270
3112
1145
1145
Velocity
(fps)
3
•3
3
2
2
2
2
2
3
4
3,
2
3
3,
3
4
2
2
3,
2
3.
4,
52
85,
3.
5.
3,
3.
.07
.69
.10
.68
.78
.64
.92
.47
.44
.65
.94
.51
.15
.30
. ! 1
.28
.73
.41
.06
.40
,46
,02
^75
,23
.71
.0
.05
,91
KL
(ft/day)
17.
16.
1
2
020°C
-------
TABLE 3-4 (continued)
CO
i-D
River Name Category
Ohio River Deep
Depth
(ft.)
32.
Area
(ft2)
43,000
Flow
(cfs)
6000
i/
Velocity L
(fps) (ft/day)
,14
020°C
(I/day)
.06
(?20°C
(I/day)
0.25
Reference
Streeter and Phelps
Hydroscience (l%fh)
(1925)
Malcolm Pirnie (1969)
Upper Hudson Deep
(Troy, N.Y.-
Saugerties)
Lower Sac- Deep
ramento R.
Upper James Deep
River (Va.)
Illinois R. Deep
17.5
21.0
15-20
15.5
10-12
9.2
9.0
8.9
6000
6750
8000
11,500
14,500
14,500
14,000
14,500
13,500
15,000
3000
4500
10,000
1800
2600
9000
7500
4500
3800
1350
8000
0.
1 .
1.
5
.5
.5
0.16
0.
0.
0.
0.
0.
0.
1.
1.
1,
18
.63
.53
.31
.28
.13
,37 6.
.57 7.
.63 7.
.34
.34
.28
.15
.24
.15
.14
.12
.13
.22
.24
.22
.225
.269
.224
.125
.165
.40
.48
.30
.31
.41
.39
.38
.43
.07
Hydroscience (1965a)
Hydroscience (1964e)
O'Connor (1963b)
Hydroscience (1964d)
O'Connor and Dobbins
(1958)
k . is the deoxygenation coefficient. See Section 3.5.
-------
TABLE 3-5, HYDRAULIC CHARACTERISTICS AND REAERATION RATES
OBSERVED BY TSIVOGLOU-WALLACE FOR FIVE RIVERS (FROM ELMI, 1975)
Length Flow Velocity Depth Elevation ko Observed by
River Studied (cfs) (ft/sec) (ft) Change (ft) Gas Tracer
Flint 9.9 mi. 5-27 .31-.88 .82-1.96 8.1-57.8 .101-.698/hr @ 25°C
South 18.3 mi. 47-207 .82-1.27 1.05-2.4 2.1-47.7 .125-.324/hr @ 25°C
Chatahoo- 120.0 mi. 1070-3300 1.72-2.45 3.62-7.66 1.2-17.8 .029-.061/hr @ 25°C
chee
Patuxent 7.0 mi. 9.8-19.5 .22-.39 .8-1.1 4.8-29.6 .101-.199/hr @ 25°C
Jackson -- 90-130 .321-.673 1.67-3.14 9.2-47.16 .07 -.364/hr G> 25°C
-------
TABLE 3-6. REAERATION COEFFICIENTS AND OTHER CHARACTERISTICS
OF TIDAL RIVERS AND ESTUARIES (AFTER HYDROSCIENCE, 1971)
Name
Delaware R.
Estuary
Depth
(ft.)
25.
ttrea
(ft2)
20
75
150
,000
,000
,000
Net Non-
Tidal
Flow
(cfs)
2,500
270,000
Hudson River
East River
Cooper River
(S. Carolina)
Savannah River
(Ga. , S.C.)
Lower Raritan
River (N.J.)
South River
(N.J)
Houston Ship
Channel (Tex)
Cape Fear R.
Estuary (N.C.)
35.
40.
40.
10.
-1
28.
14.
17.5
12.
25.
9.7
20.
13.0
135
80
40
10
40
3
5
2
17
25
2
9
33
,000
,000
,000
,000
-i
,000
,200
,000
,500
,500
,000
,100
,700
,000
5,000
0
10,000
7,000
150
23
900
2,600
1,000
Advecti ve
Velocity
(fps)
.12
.033
.016
.009
0.037
0.
0.25
0.7
0.17
.047
.029
0.01
.05
.10
.48
.10
.03
Disper-
sion
Coef.
\ day /
5.
020nC P20"C
/ 1 \ /_L\
\dayj \day/
0.3 0.
17
Reference
Hydroscience
Hydroscience
(1966c)
(1969b)
O'Connor (1963a)
20.
10.
30.
50.
10.
f.
20.
5.
5.
27.
40.
2.
10.
0.25 0.
0.23 0.
0.30 0.
0.30 0.
*
09
08
08
1
Hydroscience
Eckenfelder
Hydroscience
Hydroscience
Hydroscience
(1964a,1964b,1964c,1968a)
and Hydroscience (1968)
(1970b>
(1965b)
(1970C)
0.65
0.20 0.32
0.
0.20 0.
0.25 0.
0.23 0.
0.
0.
18
40
10
3
1
37
Hydroscience
Hydroscience
Hydroscience
Hydroscience
(1964a, 1965c)
(1964a)
(1970d)
(1966b)
(continued)
-------
TABLE 3-6 (continued)
Name
Potomac River
Compton Creek
(New Jersey)
Wappinger &
Fishkill Cks.
(New York)
River Foyle
Estuary
(N. Ireland)
Depth
(ft.)
10.
4
25.
14.5
10.5
9.
4.
10.
15.
25.
5.
Ar69
o
(ft2)
100,000
t
1 ,700,000
1,000
790
500
2,000
10,000
15,000
20,000
40,000
Net Non-
Tidal Advective
Flow Velocity
(r.fs) (fps)
550 .006
.0003
10 .010
.013
2 .004
.001
250 .025
.017
.013
.006
Disper-
sion
1.0
^
10.
1.0
0.5
1.0
5.
5.
5.
5.
kd
P20"C
/ 1 \
\~toy)
0.47
0.23
0.30
0.30
k2c
/ 1 \
Vdiy/
0.38
0.3
0.48
0.25
0.29
0.22
0.10
1.16
Reference
Kydroscience (1970a)
Hydroscience (1964b)
-------
In order for Equation (3-21) to be valid, all processes affecting the
dissolved oxygen level other than atmospheric reaeration (such as dilution,
dispersion, deoxygenation processes, and photosynthesis), must be negli-
gible. In the Churchill, e_t aj_. study the average BOD5 in the stream reach
under consideration was only 0.8 mg/1, and they deemed the deoxygenation
process to be negligible.
A consideration of importance equal to the assumptions discussed above
for Equation (3-21) is an accurate determination of the saturation value of
dissolved oxygen, which is needed to compute the deficits D0 and D]. Isaacs
and Gaudy (1968) have shown that errors from 5 to 50 percent can occur in
the prediction of V.2 due to poor determination of the deficits.
Additional methods can be developed to determine V.2 based on other
versions of the dissolved oxygen mass balance equation. When this is done,
care should be taken to assure that the assumptions underlying the particular
mass balance expression are met for the system to which it is being applied.
An equally important consideration is that increasing complexity of the mass
balance equation means more parameter estimates, and potentially, a larger
error can be incorporated into the estimation of V.2-
Hornberger and Kelly (1975) have adapted the work of Odum (1956) in
developing two productivity-based methods for prediction of reaeration in
streams. The first method uses observed nighttime changes in oxygen con-
centration, and the second compares daytime net productivity with incident
radiation. These methods require that several assumptions be made:
1. productivity is linearly related to incident light,
2. nighttime respiration is constant, and
3. the reaeration coefficient is constant over a single day (this
implies steady-state flow and constant water temperature).
The authors suggest that one advantage of these methods over many others
is their amenability for use with automatic recording equipment. Odum's
method (unmodified) has been applied successfully to estuaries (Juliano,
1969).
A radioactive tracer technique originally developed and applied by
Tsivoglou (1967, 1972), Tsivoglou, et al_. (1965, 1968), and Tsivoglou and
Wallace (1972), and later applied by Foree (1976) has been found useful
for predicting reaeration in small streams. This method does not require
measurement of dissolved oxygen saturation, as many other methods do.
The application of the tracer technique involves the use of tritiated
water, dissolved krypton-85, and a fluorescent dye (Foree, 1976). The
tracer study provides for evaluation of gas exchange capacity, time of
flow, and longitudinal and total dispersion.
143
-------
The reaeration coefficient is calculated from:
l
ln FT
Q
_
2 0.83 tf
where Ru»R.j - measured ratios of krypton-85 activity to
tritium activity at the upstream and
downstream ends of the reach, respectively
tf = time of passage of the dye peak through
the reach, in days
0.83 = ratio of exchange coefficient for krypton
to exchange coefficient for oxygen
The sulfite technique (Owens, ejt a]_. , 1964) is also applicable to
reaeration measurement in flowing waters. The water is first deoxygenated,
and the rate of oxygenation is observed. For lakes or estuaries the dome
method has been developed (Copeland and Duffer, 1964). Juliano (1969)
later applied this technique to a location in the Sacramento-San Joaquin
Estuary, California.
3.2.6 Evaluating Reaeration for a Particular System
In contrast to some other parameters, such as the deoxygenation rate,
where choosing a value for the parameter is the basic issue of concern, the
issue involved here is choosing an appropriate formulation for the system
to be simulated. The traditional approach chosen by most modelers has been
to neglect the contribution of wind to the reaeration rate in streams and
to use a formulation based. only on hydraulic parameters. However, for many
estuary and lake simulations, especially where net advective flow is small,
wind effects should be included'.
Certainly there is no lack of expressions available from which to
predict reaeration. However, there has been a lack of generalized approaches
with which to select appropriate expressions in a given situation. One
exception to this is the work done by Covar (1976) that is applicable to
streams. He plotted the data used by three groups of researchers (O'Connor
and Dobbins (1958), Churchill, et. al_. (1962), and Owens, et al_. (1964)) for
the reaeration expressions they developed (see Figure 3-277 These expres-
sions were based on stream velocity and depth. The "A" line which divides
the data of O'Connor and Dobbins from that of Churchill, et. al_. is also
the line where the two equations formulated by these authors yield identical
results. That this line separates the two data sets so discretely lends
credence to the work of both groups. Cpvar arbitrarily set the "B" line
at a depth of two feet to define the region containing most of the data of
Owens, et_ al_. For each of the three areas Covar calculated the reaeration
equation, and plotted families of curves (Figure 3-3).
144
-------
50
40
30
20-
10
8
T 6
M—
*-~ 4
.c
^. 3
o>
Q 2
1.0
.8
.6
.4
.3
A O'Connor-Dobbins (1958)
o Churchill, et gl, (1962)
a Owens, et o[. (1964)
A
\^
A
a PD a
Q
"B" Line
.1 .2 .3 .4 .6 .8 I 2 3456
Velocity (ft./sec.)
Figure 3-2. Field data considered by three
different investigators (Covar, 1976)
145
-------
.2 .3 .4 .6 .8 I 2 3456
Velocity (ft./sec.)
Figure 3-3. k2 vs. depth and velocity using the
suggested method of Covar (1976).
146
-------
While the match-up of reaeration values along the "B" line is not exact,
along the "A" line it is. Thus, Covar has developed a method of predicting
stream aeration as a function of velocity and depth for streams with
velocities of up to six feet per second and depths of up to 50 feet.
For non-stratified estuaries, and especially tidally influenced rivers,
O'Connor's (I960) formulation has probably been the most widely applied
expression containing the variables velocity and depth. This expression,
however, does not account for reaeration due to wind.
Since the temperature correction factor, as previously discussed, can
result in significant variations in the prediction of k2, some consideration
should be given to its choice. The guidelines of Metzger (1968) can be
used to make this estimate. If KL (as needed in Metzger1s method) cannot
be estimated, then it may be expedient to use a 0 in the "middle range"
(i.e., 1.01 to 1.025) rather than at either extreme. More accurate deter-
minations of 9 can be obtained by making measurements of k2 at different
temperatures and plotting the logarithm (base 10) of k2 versus temperature.
The slope of the resulting line is the logarithm of 6, or
6 = exp (2.3 slope) (3-23)
where the "slope" line should be the line of best fit through the plotted
points.
Since each predictive formula has a limited number of variables, the
user should consider whether any other effects could be important for the
system being modeled. In particular, wind effects (for formulas depending
only on hydraulic variables), suspended solids, and surfactants may be
extraneous variables whose effect on reaeration is not quantifiable (to
date) but which are, nevertheless, important determinants of reaeration.
3.2.7 References
Alonso, C.V., J.R. McHenry, and J.C.S. Hong, 1975. "The Influence of
Suspended Sediment on the Reaeration of Uniform Streams," Water Research,
Vol. 9, 695.
Baca, R.G. and R.C. Arnett, 1976. A Limnological Model for Eutrophic Lakes
and Impoundments^ Battelle Pacific Northwest Laboratories.
Banks, R.B., 1975. "Some Features of Wind Action on Shallow Lakes,"
Journal of Environmental Engineering Division, ASCE, Vol. 101.
Barrett, M.J., A.L. Gameson, and C.G. Ogden, 1960. "Aeration Study of
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U.S. Geological Survey, Professional Paper 737.
147
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Bird, R.B., W,E. Stewart, and E.N. Lightfoot, 1960. Transport Phenomena,
John Wiley & Sons.
Chen, C.W., D.J. Smith, and S. Lee, 1976. Documentation of Hater Quality
Models for the Helms Pumped Storage Project, Tetra Tech, Incorporated.
Churchill, M., H. Elmore, and R. Buckingham, 1962. "The Prediction of
Stream Reaeration Rates." Journal of Sanitary Engineering Division,
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Copeland, B.J. and W.R. Duffer, 1964. "Use of a Clear Plastic Dome to
Measure Gaseous Diffusion Rates in Natural Waters," Limnology and Ocean-
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Covar, A.P., 1976. "Selecting the Proper Reaeration Coefficient for Use
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Downing, A.L., and G.A. Truesdale, 1955. "Some Factors Affecting the Rate
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Quality and Wastewater Treatment Alternatives - City of Montreal, Canada,"
City of Montreal, Province of Quebec, Canada.
Elmi, H., 1975. "A Study of Assimilative Capacities for Receiving Streams."
Research Report, Florida Technological University.
Elmore, H.L. and W.F. West, 1961. "Effects of Water Temperature on Stream
Reaeration." Journal of Sanitary Engineering Division, ASCE, Vol. 87,
No. SA6, 59.
Eloubaidy, A.F. and E.J. Plate, 1972. "Wind-Shear Turbulence and Reaeration
Coefficient." Journal of Hydraulic Division, ASCE, HY1, 153.
Foree, E.G., 1976. "Reaeration and Velocity Prediction for Small Streams,"
Journal of the Environmental Engineering Division, ASCE, EE5, 937.
Gameson, A.L., K.G. Van Dyke, and C.G. Oger, 1958. "The Effect of
Temperature on Aeration at Weirs," Water and Water Engineering, London.
Gloyna, E.A., 1971. "Transport of Organic and Inorganic Materials in
Small Scale Ecosystems." Advances in Chemistry Series, Vol. 106.
Holley, E.R., 1975. "Oxygen Transfer at the Air-Water Interface," in
Interfacial Transfer Processes in Water Resources, Report No. 75-1, State
University of New York at Buffalo.
Hornberger, G.M. and M.G. Kelly, 1975. "Atmospheric Reaeration in a River
Using Productivity Analysis," Journal of Environmental Engineering Division,
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148
-------
Hydrologic Engineering Center, 1974. Water Quality for River-Reservoir
Systems, U.S. Army Corps of Engineers.
Hydroscience, Inc., 1964a. "Preliminary Report Pollution Analysis of the
South and Lower Raritan Rivers," Hercules Powder Company, with E.I. duPont
de Nemours and Company.
Hydroscience, Inc., 1964b. "Preliminary Estimate of the Effects of the
Proposed Sewage Discharge to Compton Creek," Middletown Township Board of
Health, Middletown, New Jersey.
Hydroscience, Inc., 1964c. "Analysis of Model Dye Dispersion Tests for
La Guardia Airport Runway Extensions," Port of New York Authority.
Hydroscience, Inc., 1964d, "Assimilation Capacity of the Upper James
River," State Water Control Board Commonwealth of Virginia.
Hydroscience, Inc., 1964e. "Pollution Assimilation Capacity of the Lower
Sacramento River," County of Sacramento, California.
Hydroscience, Inc., 1965a. "Pollution Analysis of the Upper Hudson River
Estuary," Malcolm Pirnie Engineers.
Hydroscience, Inc., 1965b. "Preliminary Evaluation of the Assimilation
Capacity of the Cooper Estuary," West Virginia Pulp and Paper Company
(Westvaco) Charleston, South Carolina.
Hydroscience, Inc., 1965c. "Pollution Analysis of the South and Lower
Raritan Rivers," E.I. duPont de Nemours and Company.
Hydroscience, Inc., 1966a. "Water Quality Studies - New River," Celanese
Corp. of America, Celco Plant, Narrows, Virginia.
Hydroscience, Inc., 1966b. "Water Quality Analysis of the Cape Fear River
Estuary," E.I. duPont de Nemours and Company.
Hydroscience, Inc., 1966c. "Water Quality Analysis of the Delaware Estuary,1
Industrial Participants, Consultant Project Technical Advisory Committee,
Delaware Estuary Comprehensive Study^, Philadelphia, Pennsylvania.
Hydroscience, Inc., 1967a. "Water Quality Studies II, New River," Celanese
Corp. of America, Celco Plant, Narrows, Virginia.
Hydroscience, Inc., 1967b. "Water Quality Analysis of the Jackson River,"
West Virginia Pulp and Paper Company (Westvaco) Covington.
Hydroscience, Inc., 1968a. "Mathematical Models for Water Quality for the
Hudson-Champlain and Metropolitan Coastal Water Pollution Control Project,"
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149
-------
Hydroscience, Inc., 1968b. "Water Quality Analysis for the Markland Pool
of the Ohio River, Malcolm Pirnie Engineers.
Hydroscience, Inc., 1969a. "Water Quality Analysis of the North Branch,
Potomac River, near Luke, Maryland, West Virginia Pulp and Paper Company
(Westvaco) Fine Papers Division.
Hydroscience, Inc., 1969b. "Nitrification in the Delaware Estuary,"
Delaware River Basin Commission, Trenton, New Jersey.
Hydroscience, Inc., 1970a. "Preliminary Report - Feasibility of the Potomac
Estuary as a Supplemental Water Supply Source," U.S. Army Corps of Engineers,
Hydroscience, Inc., 1970b. "Analysis of the Effects on Water Quality of a
Proposed Fill Program in a Portion of Bowery Bay within La Guardia Airport
Boundaries," Port of New York Authority.
Hydroscience, Inc., 1970c. "Water Quality Analysis of the Savannah River
Estuary," American Cyanamide Co., Wayne, N.J.
Hydroscience, Inc., 1970d. "Mathematical Model and Water Quality Analyses -
Houston Ship Channel," Galveston Bay Study, Texas Water Quality Board,
TRACOR, Inc.
Hydroscience, Inc., 1971. Simplified Mathematical Modeling of Water
Quality, U.S. Environmental Protection Agency, Washington, D.C.
Hyer, P.V., C.S. Fang, E.P. Ruzecki, and W.J. Hargis, 1971. Hydrography
and Hydrodynamics of Virginia Estuaries, Studies of the Distribution of
Salinity and Dissolved Oxygen in the Upper York System, Virginia Institute
of Marine Science.
Isaacs, W.P. and A.F. Gaudy, 1968. "Atmospheric Oxygenation in a Simulated
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Report for the Estuary Water Quality Models. Tetra Tech, Inc7, technical
report.
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150
-------
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for Reaeration in Open Channel Flow." Technical Bulletin No. 61, Department
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Industrial and Chemical Engineering, Vol. 16, No. 12, 1215.
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River Requirements for Control of Water Pollution and Expansion of Mill
Creek Wastewater Treatment Plant, Volume II," January.
Mancy, K.H. and W.E. Barlage, undated. "Mechanism of Interference of
Surface Active Agents with Gas Transfer in Aeration Systems," New Concepts
in Biological Waste Treatment.
Mastropietro, M.A., 1968. "Effects of Dam Reaeration on Waste Assimilation
Capacities of the Mohawk River," Proceedings of the 23rd Industrial Waste
Conference, Purdue University.
Metzger, I., 1968. "Effects of temperature on Stream Aeration," Journal
of Sanitary Engineering Division, ASCE, Vol. 94, SA6, 1153.
Negulescu, M. and V. Rojanski, 1969. "Recent Research to Determine
Reaeration Coefficient," Water Research, Vol. 3, No. 3, 189.
O'Connor, D.J., 1960. "Oxygen Balance of an Estuary," journal of Sanitary
Engineering Division, ASCE, SA3, 35.
O'Connor, D.J., 1963a. "Report on Analysis of the Dye Diffusion Data in
the Delaware River Estuary, Evaluation of Diffusion Coefficients by Analog
Computation," United States Public Health Service Regional Office,
Philadelphia, Pennsylvania.
O'Connor, D.J., 1963b. "Analysis of the Upper James River by Analog
Computation Techniques," State Water Control Board, Commonwealth of
Virginia.
O'Connor, D.J. and W.E. Dobbins, 1958. "Mechanism of Reaeration in
Natural Streams," American Society of Civil Engineers Transactions,
V. 123, 641.
O'Connor, D.J. with Hydroscience, Inc., 1962a. "Preliminary Report -
Assimilation Capacity of the Jackson River," West Virginia Pulp and Paper
Company (Westv"aco).
151
-------
O'Connor, D.J. with Hydroscience, Inc., 1962b. "Preliminary Report -
Mechanical Aeration of the Jackson River," West Virginia Pulp and Paper
Company (Westvaco).
O'Connor, D.J. and Hydroscience, Inc., 1967. "Preliminary Report - Water
Quality Analysis of the Mohawk River-Barge Canal," New York State Department
of Health.
O'Connor, D.J. and Hydroscience, Inc., 1968. "Water Quality Analysis of
the Mohawk River-Barge Canal," New York State Department of Health.
O'Connor, D.J. and D.M. Di Toro, 1970. "Photosynthesis and Oxygen Balance
in Streams," Journal of the Sanitary Engineering Division, ASCE, April.
Odum, H.T., 1956. "Primary Production in Flowing Waters," Limnology and
Oceanography, Vol. 1, 102.
Owens, M., R.W. Edwards, and J.W. Gibbs, 1964. "Some Reaeration Studies
in Streams," International Journal of Air and Water Pollution, Vol. 8, 469.
Poon, C.P.C. and H. Campbell, 1967. "Diffused Aeration in Polluted Water,"
Water and Sewage Works, Vol. 114, 461.
Rood, O.E. and E.R. Holley, 1974. "Critical Oxygen Deficit for a Bank
Outfall," Journal of Environmental Engineering Division, ASCE, Vol. 100, EE3.
Streeter, H.W., 1926. "The Rate of Atmospheric Reaeration of Sewage
Polluted Streams," in Transactions ASCE, Vol. 89, 1351.
Streeter, H.W. and E.B. Phelps, 1925. "A Study of the Pollution and
Natural Purification of the Ohio River," Public Health Bulletin #146,
Washington, D.C.
Streeter, H.W., C.T. Wright, and R.W. Kehr, 1936. "Measures of Natural
Oxidants in Polluted Streams II," Sewage Works Journal, Vol. 8, 282.
Thackston, E.L. and P.A. Krenkel, 1969. "Reaeration Prediction in Natural
Streams," Journal of Sanitary Engineering Division, ASCE, Vol. 95, SA1, 65.
Truesdale, G.A. and K.G. Van Dyke, 1958. "The Effect of Temperature on
the Aeration of Flowing Waters," Water and Waste Treatment Journal, Vol. 7, 9.
Tsivoglou, E.G., e_t aj_., 1965. "Tracer Measurement of Stream Reaeration I.
Laboratory Studies," Journal of the Water Pollution Control Federation,
Vol. 37, No. 10, 1343.
Tsivoglou, E.G., 1967. "Tracer Measurement of Stream Reaeration,"
Federal Water Pollution Control Administration.
152
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Tsivoglou, E.G., et al_., 1968. "Tracer Measurement of Stream Reaeration II.
Field Studies," Journal of the Water Pollution Control Federation, Vol. 40,
No. 2, Part 1, 285.
Tsivoglou, E.G., 1972. "Direct Tracer Measurement of the Reaeration
Capacity of Streams and Estuaries," Water Pollution Control Research
Series Report 16050, U.S. Environmental Protection Agency.
Tsivoglou, E.G. and L.A. Neal, 1976. "Tracer Measurement of Reaeration III.
Predicting the Reaeration Capacity of Inland Streams," Journal of Wate_r
Pollution Control Federation, Vol. 48, No. 12, 2669.
Tsivoglou, E.G. and J.R. Wallace, 1972. "Characterization of Stream
Reaeration Capacity," Ecological Research Series Report No. EPA-R3-72-012,
U.S. Environmental Protection Agency.
Weber, W.J., 1972. Physicochemical Processes for Water Quality Control,
John Wiley & Sons.
Whipple, W., J.V. Hunter, B. Davidson, F- Dittman, and S. Yu, 1969.
"Instream Aeration of Polluted Rivers," published by Water Resources
Institute, Rutgers University, New Jersey.
3.3 DISSOLVED OXYGEN SATURATION
3.3.1 Introduction
Dissolved oxygen saturation, commonly symbolized as Cs and expressed
in mg/1, is a basic parameter used in a great many surface water quality
models. Since dissolved oxygen prediction is often the reason for develop-
ing a water quality model, accurate values of Cs are needed. Additionally,
values of dissolved oxygen saturation are often used in the estimation of
other parameters, such as the reaeration coefficient as discussed previously
in Section 3.2.5.
Use of inaccurate Cs values can lead to poor values of other para-
meters. Isaacs and Gaudy (1968) showed, for example, that predicted k2
values could be in error by as much as 50 percent based on incorrect
estimates of Cs-
3.3.2 Factors Affecting Dissolved Oxygen Solubility
The solubility of dissolved oxygen in water decreases with increasing
temperature, increasing chloride concentration (or concentration of ionic
impurities in general), and decreasing atmospheric pressure (i.e., partial
pressure of oxygen in the atmosphere). Each of these factors has been
quantified (although not always with consistent results) and will be dis-
cussed here.
153
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The APHA (1971) presents a tabulation of oxygen solubility in water
as a function of both chloride concentration (chlorinity) and water tempera-
ture (see Table 3-7). These calculations were made by Whipple and Whipple
(1911), for conditions in which the water was in contact with dry air con-
taining 20.90 percent oxygen. Since chlorinity is related to salinity, and
salinity is more often measured than chlorinity, the relationship between
the two quantities is of interest. The relationship is:
Salinity (ppt or °/oo) = 0.03 + 0.001805 Chlorinity (mg/1) (3-24)
In 1960, Elmore and Hayes prepared a report for the American Society
of Civil Engineers concluding that the values in APHA (1971) tend to be
high within a temperature range of 10°C to 30°C while being substantially
correct in the temperature range 0°C to 5°C (Table 3-8). The authors con-
cluded that since their own results were corroborated by several other
workers in the field (e.g., Winkler (1891) and Morris (1959)), the values
listed in Table 3-8 are more satisfactory for application to natural stream
conditions. Unfortunately there is no concensus among researchers on which
set of data is the better. Holley, et al_. (1970), for example, found dis-
solved oxygen saturation values to differ from those in Table 3-8, by a
substantial amount in some cases (1-1.5 mg/1).
Elmore and Hayes (1960) developed a predictive formula for Cs from the
data in Table 3-8:
Cs = 14.652 - 0.41022T + 0.0079910T2 - 0.000077774T3 (3-25)
where C = dissolved oxygen saturation, mg/1
T = water temperature, °C
Equation (3-25) is valid at standard pressure and zero salinity.
Several investigators have quantified the solubility of dissolved
oxygen as a function of salt content, as well as temperature. Fair, et al.
(1968) found that oxygen solubility in saline waters could be approximated
by multiplying the solubility in freshwater by a reduction factor. The
following relationship results:
= r
L
so
- (?xlO-5)
(3-26)
where C = solubility at zero salinity, mg/1
s" = salinity of water expressed as chloride, mg/1
If values of Cso are used from APHA (1971) as shown in Table 3-7, Equation
(3-26) predicts the remainder of the values in that table within 0.1 mg/1,
154
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TABLE 3-7. SOLUBILITY OF OXYGEN IN WATER
(APHA, 1971)
Temp.
in
°C
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
Chloride Concentration
0
5,000
10,000
in Water mg/1
15,000
20,000
Di fference
per 100 mg
Chloride
Dissolved Oxygen mg/1
14.6
14.2
13.8
13.5
13.1
12.8
12.5
12.2
11.9
11.6
11.3
11.1
10.8
10.6
10.4
10.2
10.0
9.7
9.5
9.4
9.2
9.0
8.8
8.7
8.5
8.4
8.2
8.1
7.9
7.8
7.6
7.5
7.4
7.3
7.2
7.1
13.8
13.4
13.1
12.7
12.4
12.1
11.8
11.5
11.2
11.0
10.7
10.5
10.3
10.1
9.9
9.7
9.5
9.3
9.1
8.9
8.7
8.6
8.4
8.3
8.1
8.0
7.8
7.7
7.5
7,4
7.3
13.0
12.6
12.3
12.0
11.7
11.4
11.1
10.9
10.6
10.4
10.1
9.9
9.7
9.5
9.3
9.1
9.0
8.8
8.6
8.5
8.3
8.1
8.0
7.9
7.7
7.6
7.4
7.3
7.1
7.0
6.9
12.1
11.8
11.5
11.2
11.0
10.7
10.5
10.2
10.0
9.8
9.6
9.4
9.2
9.0
8.8
8.6
8.5
8.3
8.2
8.0
7.9
7.7
7.6
7.4
7.3
7.2
7.0
6.9
6.8
6.6
6.5
11.3
11.0
10.8
10.5
10.3
10.0
9.8
9.6
9.4
9.2
9.0
8.8
8.6
8.5
8.3
8.1
8.0
7.8
7.7
7.6
7.4
7.3
7.1
7.0
6.9
6.7
6.6
6.5
6.4
6.3
6.1
0.017
0.016
0.015
0.015
0.014
0.014
0.014
0.013
0.013
0.012
0.012
0.011
0.011
0.011
0.010
0.010
0.010
0.010
0.009
0.009
0.009
0.009
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
155
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TABLE 3-8. DISSOLVED OXYGEN SATURATION VALUES
IN DISTILLED WATER (ELMORE AND HAYES, 1960)
Temperatu
°C
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
re
0.0
14.65
14.25
13.86
13.49
13.13
12.79
12.46
12.14
11.84
11 .55
11.27
11.00
10.75
10.50
10.26
10.03
9.82
9.61
9.40
9.21
9,02
8.84
8.67
8.50
8.33
8.18
8.02
7.87
7.72
7.58
7.44
Di ssol ved Oxygen in
0.1
14.61
14.21
13.82
13.46
13.10
12.76
12.43
12.11
11 .81
11 .52
11 .24
10.98
10.72
10.48
10.24
10..01
9.79
9.58
9.38
9.19
9.00
8.82
8.65
8.48
8.32
8.16
3.01
7.86
7.71
7.56
7.42
0.2
14.57
14.17
13.79
13.42
13.06
12.72
12.40
12.08
11.78
11 .49
11.22
10.95
10.70
10.45
10.22
9.99
9.77
9.56
9.36
9.17
8.98
8.81
8.63
8.46
8.30
8.14
7.99
7.84
7.69
7.55
7.41
0.3
14.53
14.13
13.75
13.38
13.03
12.69
12.36
12.05
11.75
11 .47
11.19
11 .93
10.67
10.43
10.19
9.97
9.75
9.54
9.34
9.15
8.97
8.79
8.62
8.45
8.29
8.13
7.98
7.83
7.63
7.54
7.40
0.4
14.49
14.09
13.71
13.35
13.00
12.66
12.33
12.02
11.72
11.44
11.16
10.90
10.65
10.40
10.17
9.95
9.73
9.52
9.32
9.13
8.95
8.77
8.60
8.43
8.27
8.11
7.96
7.81
7.66
7,52
7.38
Hi 1 1 iqranis
0.5
14.45
14.05
13.68
13.31
12.96
12.62
12.30
11.99
11 .70
11.41
11.14
10.87
10.62
10.33
10.15
9.92
9.71
9.50
9.30
9.12
8.93
8.75
8.58
8.42
8.25
8.10
7.95
7.80
7.65
7.51
7.37
Per Liter
0.6
14.41
14.02
13.64
13.28
12.93
12.59
12.27
11.96
11.67
11.38
11.11
10.85
10.60
10.36
10.12
9.90
9.69
9.48
9.29
9.10
8.91
8.74
8.56
8.40
8.24
8.03
7.93
7.78
7.64
7.49
7.35
0.7
14.37
13.98
13.60
13.24
12.89
12.56
12.24
11.93
11 .64
11.35
11 .08
10.82
10.57
10.33
10.10
9.88
9.67
9.46
9.27
9.08
8.90
8.72
8.55
8.38
8.22
8.07
7.92
7.77
7.62
7.48
7.34
0.3
14.33
13.94
13.56
13.20
12.86
12.53
12.21
11.90
11 .61
11.33
11.06
10.80
10.55
10.31
10.08
9.86
9.65
9. J4
9.25
. 9.06
8.88
8.70
8.53
8.37
8.21
8.05
7.90
7.75
7.61
7.47
7.32
0.9
14.29
13.90
13.53
13.17
12.82
12.49
12.18
11.87
11.58
11 .30
11 .03
10.77
10.52
10.23
10.06
9.84
9.63
9.42
9.23
9.04
S.86
3.68
8.52
S.35
8.19
8.04
7.89
7.74
7.59
7.45
7.31
156
-------
except for within the temperature range of 0°C to 5°C at a chloride concen-
tration of 15,000 to 20,000 mg/1. Within this region, discrepancies of up
to 0.4 mg/1 occur.
From the data of Green and Carritt (1967), Hyer, et, aj_. (1971) developed
an expression relating Cs to both temperature and salinity. Cs is given by:
C - 14.6244 - 0.367134T + 0.0044972T2
S o (3-27)
- 0.0966S + 0.00205ST + 0.00027395^
where T = temperature, °C
S = salinity, ppt.
At zero salinity, the values predicted for Cs by Equation (3-27) agree with
the values in Table 3-7 to within 0.1 mg/1, and with the values in Table 3-8
to within 0.2 mg/1. For nonzero salinity, Equation (3-27) agrees with the
values in Table 3-7 to within 0.1 to 0.2 mg/1 except for temperatures between
21 to 30°C with a chloride concentration of 15,000 to 20,000 mg/1. In this
region a departure of from 0.2 to 0.7 mg/1 occurs.
Barometric pressure also affects Cs, and the correction expression is
(APHA, 1971):
where C = saturation^ value at sea level, at the temperature
of the water, mg/1
C ' = corrected value at the altitude of the river, mg/1
P, = barometric pressure at river altitude, mm Hg
P = saturation vapor pressure of water at the river
v temperature, mm Hg
As an approximation of the influence of altitude, C decreases about 7% per
2,000 feet of elevation increase.
Henry's law can also be used to estimate the solubility of dissolved
oxygen, usually to within 1-3% of the correct value (Daniels and Alberty,
1967). Henry's law is expressed as:
Cs = kH p (3-29)
157
-------
where kH = coefficient of absorption, ml/1
P02 = partial pressure of oxygen, atmospheres
Values for the absorption coefficient are shown in Table 3-9.
TABLE 3-9. SOLUBILITIES OF OXYGEN IN WATER FOR
DIFFERENT TEMPERATURES (CRC, 1967)
Oxygen
Absorption
Temperature Coefficient,
°c
0
5
10
15
20
25
30
35
40
°F
32
41
50
59
68
77
86
95
104
(ml/liter)
48.9
42.9
38.0
34.2
31.0
28.3
26.1
24.4
23.1
The absorption coefficient expresses the volume (in ml at standard
pressure) of oxygen dissolved in a liter of water. To convert this volume
to mass (as mg) the ideal gas law is used:
P0? kH M
Cs = -|-f- (3-30)
where M = molecular weight of oxygen (=32)
RU = universal gas constant, 0.082055 l-atm/gm°K
T = absolute temperature (°K), at which k,, is
calculated M
3.3.3 Methods of Measurement
Elmore and Hayes (1960) have summarized the work of numerous research-
ers who have measured dissolved oxygen saturation. According to Elmore and
Hayes, Fox in 1909 used a gasometric technique in which a known volume of
158
-------
pure oxygen was exposed to a known volume of water. After equilibrium had
been established the volume of oxygen above the water was determined, and
the solubility calculated assuming air contained 20.90% oxygen. From the
tests an expression was developed for the oxygen absorption coefficient as
a function of temperature.
From Fox's expression, Whipple and Whipple (1911) converted their
results from mi Hi liters per liter to parts per million. These results were
given in Table 3-7.
Truesdale, et_ aj_. (1955) used gaseous nitrogen to deoxygenate water
samples, and then allowed the water to return to equilibrium conditions.
At different time intervals, subsamples were removed and analyzed by an
adaptation of the Winkler (1891) procedure.
Morris (1959) conducted a series of 12 experiments under conditions
similar to those of Truesdale, ejt al_. (1955). Morris1 experiments were
conducted under controlled atmospheric conditions and using titrimetric
methods. Following that, thirteen more experiments were conducted using
a gasometric technique. The results tended to support the findings of Fox.
Then, in 1960, Elmore and Hayes undertook to determine dissolved
oxygen saturation values in water. The work was done because of disagree-
ment remaining among researchers. The results of Elmore and Hayes (1960)
were summarized earlier in Table 3-8. The procedures they used were simi-
lar to the Truesdale group.
The determination of the solubility of oxygen in seawater from 0°C to
35°C and for various chlorinities was done by Green (1965), and discussed by
Green and Carritt (1967). Green used a Jocobsen-Worthington equilibrator
and a titration method similar to that described by Carpenter (1965).
3.3.4 Summary
Today, significant differences exist among the results obtained by
various researchers regarding the "true" value of dissolved oxygen satura-
tion under specified conditions. These discrepancies can be as large as
11 percent for high salinity conditions (compare Table 3-7 with Equation
(3-27)). Under conditions of zero salinity, the discrepancies observed are
generally less than two percent (compare Table 3-7 and Table 3-8). Knowing
the possible ranges of errors in using a particular formulation for Cs
permits the user to decide whether they are significant in his study. In
instances where the outcome is sensitive to Cs, and only a few values of Cs
are needed, it may be justifiable to measure Cs directly rather than to use
tabulated values.
3.3.5 References
APHA (American Public Health Association), 1971. Standard Methods for the
Examination of Water and Wastewater, 13th Edition, APHA, Washington, D.C.
159
-------
Carpenter, J.H., 1965,, "The Accuracy of the Winkler Method for Dissolved
Oxygen Analysis," Limnology and Oceanography, X, 135-140,
CRC (Chemical Rubber Company), 1967. Handbook of Chemistry and Physics.
Daniels, F. and R.A. Alberty, 1967, Physical Chemistry. 3rd Edition,
John Wiley & Sons, New York.
Elmore, H.L. and T.W. Hayes, 1960. "Solubility of Atmospheric Oxygen in
Water," Twenty-Ninth Progress Report of the Committee on Sanitary Engineer-
ing Research, Journal Sanitary Engineering Division, ASCE, Vol. 86, SA4, 41.
Fair, G.M., J.C. Geyer, and D.A. Okun, 1968. Water and Wastewater
Engineering, Vol. 2, Water Purification and Wastewater Treatment and
Disposal, John Wiley & Sons, New York.
Green, E.J., 1965. A Redetermination of the Solubility of Oxygen in Sea-
water and Some Thermodynamic Implications of the Solubility Relations.
Ph.D. Thesis, Mass. Inst. of Technology.
Green, E.J. and D.E. Carritt, 1967. "New Tables for Oxygen Saturation
of Seawater," J. Marine Research, 25(2).
Holley, E.R., et aj_., 1970. "Effects of Oxygen Demand on Surface Reaeration,"
Research Report 46, Water Resources Center, University of Illinois, 80 p.
Holley, E.R., 1975. "Oxygen Transfer at the Air-Water Interface, in
Interfacial Transfer Processes in Water Resources^ State University of
New York at Buffalo. Report 75-1.
Hyer, P.V., C.S. Fang, E.P. Ruzecki, and W.J. Hargis, 1971. Hydrography
and Hydrodynamics of Virginia Estuaries, Studies of the Distribution of
Salinity and Dissolved Oxygen in the Upper York System, Virginia Institute
of Marine Science.
Isaacs, W.P. and A.F. Gaudy, 1968. "Atmospheric Oxidation in a Simulated
Stream," Journal Sanitary Engineering Division, ASCE, Vol. 94, SA2, 319.
Morris, J.C., 1959. Final Report, Contract SAph 69705, Dept. of Health,
Education, and Welfare with Harvard University.
Sawyer, C.N. and P.L. McCarty, 1967. Chemistry for Sanitary Engineers,
McGraw-Hill Book Company.
Truesdale, G.A., A.L. Downing, and G.F. Lowden, 1955. Journal of Applied
Chemistry, Vol. 5, No. 53.
Whipple, G.C. andM.C. Whipple, 1911. Journal American Chemical Society,
33:362.
Winkler, L.W., 1891= Ber d. Deutschen Chem. Ges., Vol. 24, No. 89.
160
-------
3.4 PHOTOSYNTHESIS
3.4.1 Introduction
In simulating the dissolved oxygen resources of aquatic ecosystems,
the need frequently arises to include the process of photosynthesis. Simu-
lation of photosynthetic oxygen production can be accomplished in one of
two ways. One method is to first model algal growth and then relate oxygen
production to the predicted algal growth rate. The second method is simply
to include a term for time averaged or instantaneous photosynthetic oxygen
production in the dissolved oxygen mass balance equation without simulating
algal growth. The latter method, although easier from a modeling stand-
point, still requires the estimation of a number of parameters, either by
direct measurement from the water body to be simulated, or by the use of
previously tabulated values.
3.4.2 Simulating Photosynthetic Oxygen Production
Many investigators have simulated photosynthetic oxygen production
rates without modeling algal growth. This section will be limited to a
case-by-case discussion of these methods, and will not include algal growth
kinetics. For a discussion of algal growth kinetics see Section 3.9.
O'Connell and Thomas (1965) examined photosynthetic oxygen production
by benthic algae and rooted aquatic plants in the Truckee River during the
months of July, 1962, and August, 1963. O'Connell and Thomas used two
methods both of which involved determining net photosynthesis (P-R). In
their first method, all terms (other than P-R) were evaluated in an un-
steady dissolved oxygen mass balance equation at two hour time increments
over a two-day period. Data were collected at a number of locations along
the stream. In their mass balance equation, BOD concentrations, BOD decay
rate, dissolved oxygen deficits, the reaeration coefficient, and the sub-
stantial concentration derivative (defined below) required evaluation.
Additionally the researchers included oxygen consumption due to nitrifi-
cation by finding the stoichiometric quantity of oxygen necessary to oxi-
dize the ammonia and nitrite found to have either been removed from solution
or converted to nitrate. The substantial derivative reflects the concen-
tration change that is observed moving downstream at the local stream
velocity.
By using a tracer dye, O'Connell and Thomas were able to determine the
time of passage of a "slug" of water as it passed from station to station.
Their results were such that net P-R for each of a number of locations and
at all hours throughout the day could be found. Figure 3-4 illustrates the
diurnal P-R variation at one of their stations.
In their second method, O'Connell and Thomas used algal chamber
measurements to obtain an independent estimate of the net oxygen change in
the stream due to photosynthesis. This method consisted of using plexiglass
chambers containing submersible pumps. The chambers were sealed with a
161
-------
sheeting that was oxygen impermeable. The chambers were then placed in the
stream and petri dishes, upon which Oscillatoria had been grown, were placed
in the chamber. Tests were conducted at various times during the day and
night as well as at different locations. Because the test chambers were
filled with river water, oxygen was consumed by both carbonaceous and
nitrogenous oxygen demanding material as well as plant respiration. Net
P-R values were corrected for this sink of oxygen. The outcomes of the two
approaches were quite similar, as illustrated by Figure 3-4.
o
.c
+ 15
+ 10
+05
01
I
Q. -
II
-15
0000
0400
0800 1200 1600
Time of Day
2000
2400
Finite Difference Data
Algae Chamber Data
Figure 3-4. Diurnal variation of (P-R) in Truckee River near
Station 2B (O'Connell and Thomas, 1965).
O'Connor and Di Toro (1970) have hypothesized what the functional forms
might be for photosynthetic oxygen production and for oxygen depletion by
respiration. From collected data, they were able to evaluate proportionality
constants needed in their expressions. The photosynthetic expression was
assumed to be time-dependent, while that for respiration was assumed time-
independent. Based upon the results of earlier studies (Westlake, un-
dated) O'Connor and Di Toro assumed that the photosynthetic oxygen produc-
tion rate, averaged over depth, would resemble the diurnal incident solar
162
-------
radiation curve. For the sake of simplicity, O'Connor and Di Toro also
assumed that the photosynthetic oxygen production curve, P(t), could be
represented by a half cycle sine wave:
P(t) =
Pm sin
i K)
when t < t < t + p
(3-31)
0 when t <_ t or t >_ t + p
where P =
maximum rate of photosynthetic oxygen production,
mg/l-day
t = current time, expressed as a fraction of a day
tg = the time of day representing sunup, expressed
as a fraction of a day
p = daylight fraction of the day
Since this function was assumed to have a period of one day, its periodic
extension could be expressed as a Fourier Series (not discussed here).
In Equation (3-31) both ts and p are established using the times of
sunrise and sunset. O'Connor and Di Toro found that the best fits to their
data were obtained if ts was slightly later than sunrise and p was slightly
less than the total time between sunrise and sunset. It remained then to
quantify Pm and R (respiration). O'Connor and Di Toro adopted a heuristic
approach, based primarily on the diurnal fluctuation at various stations
downstream and guided by whatever qualitative information was available
about the algal populations involved. Calculations were carried out on the
Grand, Clinton, Truckee, Ivel, and Flint Rivers. The resulting formula-
tions agreed reasonably well with observations on these streams. Weeter
and Bell (1971) have used a modification of O'Connor and Di Toro's model.
Their approach was verified on the Wabash River, Indiana, and found to
agree well with observed data,
Both of the methods described above for finding instantaneous production
rates (O'Connell and Thomas, 1965; and O'Connor and Di Toro, 1970) can be
used to find the net average photosynthetic oxygen production over a day.
In the first method net P-R is determined by finding the area under the P-R
versus time curve, such as the curve that was shown in Figure 3-4. The net
production is not zero since the total daytime production is greater than
the total nighttime consumption.
163
-------
From O'Connor and Di Toro's methods, average daily photosynthesis
(Pav, mg/l-day) is given by integrating Equation (3-31) over 24 hours, giving:
Pav = 2p P/TT (3-32)
Kelly, e_t a]_. (1975) have described a method also applicable to streams
where the net photosynthetic oxygen production rate (P-R) is expressed as a
Fourier cosine series. A period of 48 hours was used so that values at the
beginning and end of a day were not constrained to be identical. Their
problem reduced to determining enough Fourier coefficients so that a finite
number of terms could accurately represent P-R.
Their method used digitally recorded diurnal oxygen concentration
changes to solve for the coefficients in an oxygen mass-balance equation.
Data were recorded at two stations on a river and the unknown coefficients
were selected so that the downstream values were accurately predicted from
the upstream values. A unique feature of this work is that Kelly, et al.
developed an error analysis procedure applicable to their method.
In the work just described on stream photosynthesis, neither researcher
explicitly included light intensity in their expressions. For deeper
bodies of water, such as impoundments, inclusion of this term along with
an extinction coefficient is necessary. Such an expression is given by
(U.S. EPA, 1976):
2 718 f
-KeH
where a = e
P = average daily algal oxygen production rate,
av mg/l-day
P = the light saturated rate of oxygen
production, mg/l-day
I = the average light intensity during the daylight
portion of the day (any units consistent with
those chosen for I )
I = the light saturated intensity (optimum light
intensity for growth)
f = the number of hours of daylight, per day
164
-------
T = twenty-four hours
KS = the extinction coefficient, 1/m
H = depth, m
The following range was suggested for the extinction coefficient:
0.1-0.5/m for very clear impoundments,
0.5-2.5/m for moderately turbid waters, and
2.5/m or greater for very turbid waters
The U.S. EPA suggested that Ps was related to the chlorophyll-a concentra-
tion as follows:
Pc = 0.25 Chi (3-34)
s a
where Chi = chlorophyll-a concentration, yg/1
a
The U.S. EPA also suggested an expression for predicting respiration, R, as a
function of chlorophyll-a_ concentration:
R = r • Chi (3-35)
a
where R = mg oxygen utilized/1-day
Chi = chlorophyll-a concentration in ug/1
a —
r = constant ranging from 0.005 to 0.030 with
0.025 a common value. This corresponds to
a 10 to 1 ratio of PS to R
The respiration rate is known to vary considerably and depends on the
nutrient concentration and age of the culture. Hence the average daily
algal respiration calculated from equation (3-35) is subject to some degree
of uncertainty.
Bailey (1970) also developed an expression for gross photosynthesis
(in the Sacramento-San Joaquin estuary) that included a light limitation
term. It is:
T0.667
Pay = 3.16 chl |p + 0.16T - 0.56H (3-36)
165
-------
av
where Pau = average daily gross photosynthetic rate,
mg/l-day
= mean daily solar intensity, cal/sq.cm-day
I
ke
T
H
= light extinction coefficient, m
= mean' temperature, °C
= mean water depth, m
-1
chl = mean chlorophyll, mg/1
The multiple correlation coefficient obtained in developing Equation (3-36)
was 0.96. This means that 96^ or 92 percent of observed variability of Pav
could be accounted for by I, ke, T, H, and chlorophyll concentration with
this form of equation.
3.4.3 Tabulated Values for Photosynthesis
Table 3-10 shows values of gross average photosynthetic oxygen produc-
tion and average respiration for several waterbodies. These values are
expressed on an areal basis. To convert from an area! basis to mg/l-day in
vertically well-mixed systems, divide the areal rate (gm/sq.m-day) by the
average depth (in meters)-
TABLE 3-10. SOME AVERAGE VALUES OF GROSS PHOTOSYNTHETIC
PRODUCTION OF DISSOLVED OXYGEN AND AVERAGE RESPIRATION
(AFTER THOMANN, 1972, AND THOMAS AND O'CONNELL, 1966)
Water Type
Average Gross
Production
(gm/m2-day)
Average
Respiration
(gm/m^-day)
Truckee River Bottom
attached algae
Tidal Creek Diatom Bloom
(62-ir<)Xi06 diatoms/1)
Delaware Estuary summer
Duwamish River Estuary
Seattle, Washington
Neuse River System
North Carolina
River Ivel
North Carolina Streams
Laboratory Streams
3-7
0.5-2.0
0.3-2.4
3.2-17.6
9.8
3.4-4.0
11.4
6.7-15.4
21.5
2.4-2.9
166
-------
The use of Equations (3-34) and (3-36) depend on estimates of chloro-
phyll-^ concentrations. Table 3-11 presents chlorophyll-a values on an
areal basis for various communities. Also presented in that table are
assimilation numbers, defined as grams of oxygen produced per hour per gram
of chlorophyll-a_. Hence by estimating the chlorophyll-a_ concentration the
maximum oxygen production rate can also be estimated. Equation (3-34) is
an example of an application of this concept.
TABLE 3-11. CHLOROPHYLL-a AND ASSIMILATION NUMBER OF
VARIOUS COMMUNITIES (AFTER ODUM, et al_., 1958)
Assimilation
Euphotic Zone* Number, gm Oxygen
Chlorophyll-a_ Per Hour Per gm
Plankton Communities gm/m^ Chlorophyll-a_
Not including bottom plants:
Long Island Sound 0.1-0.6 1.3
Diatom bloom, Moriches Bay 0.20 4.5
one meter
Sewage Pond, Kadoka, S.D. 1.5 2.
Shallow Aquatic Communities
with Bottom Plants:
Rocky Mountain Stream, Utah 0.3-1.6 0.7-2.0
Blue-green algal mat, polluted 2.5
stream, Mission River, Texas,
August, 1957
*From the surface to the depth where light intensity 1» of surface light.
3.4.4 Summary
Of the methods presented here for simulating photosynthetic oxygen
production, all but one (O'Connell and Thomas, 1965) provide quantitative
relationships. All of the relationships require the user to make estimates
of several parameters, such as the maximum rate of photosynthesis. Many of
the measurement techniques used by the researchers described depend on
determining, in the dissolved oxygen mass balance equation, all terms other
than P-R, from which P-R can then be determined. Inherent in this procedure
is the hazard of incorporating errors due to errors in estimating values of
the required parameters, such as the deoxygenation rate, k
-------
the light and dark chambers method, can offer an advantage in this respect
(see Thomas and O'Connell (1966) and Mclntyre, ejt a]_. (1964) for applica-
tions). However, this method has a disadvantage in that it only samples at
one point, and also disturbs the algal population and surrounding flow
field.
Several of the equations presented here (Equations (3-34), (3-35), and
(3-36) require chlorophyll concentration measurements. Table 3-11 has
presented only a limited amount of data on chlorophyll-a_ concentrations.
Also chlorophyll-a_ concentrations are not constant, although over a rela-
tively short time period, chlorophyll may be modeled as such. Each of the
expressions for photosynthesis that contain terms for chlorophyll predict
daily average rates, and not instantaneous rates. It would generally be
inappropriate to use either of these expressions in a model with a time
step shorter than one day, except to approximate diurnal dissolved oxygen
ranges that might be expected given typical chlorophyll concentrations.
3.4.5 References
Bailey, T.F., 1970. "Estuarine Oxygen Resources-Photosynthesis and
Reaeration," Journal Sanitary Engineering Division, ASCE. Vol. 96, SA2, 279.
Bella, D.A., 1970. "Dissolved Oxygen Variations in Stratified Lakes,"
Proceedings, American Society of Civil Engineers, Journal Sanitary
Engineering, SA5, October.
Kelly, M.G., G.M. Hornberger, and B.J. Cosky, 1975. "A Method for Monitoring
Eutrophication in Rivers," Department of Environmental Sciences, University
of Virginia. Prepared for Office of Water Research and Technology.
Mclntyre, C.D., R.L. Garrison, H.K. Phinney, and C.F. Warren, 1964. "Primary
Production in Laboratory Streams," Limnology and Oceanography, Vol. 9, 92.
O'Connell, R.L. and N.A. Thomas, 1965. "Effects of Benthic Algae on Stream
Dissolved Oxygen," Proceedings of the American Society of Civil Engineers,
Journal of Sanitary Engineering Division, SA3, June, 1965.
O'Connor, D.J. and D.M. Di Toro, 1970. "Photosynthesis and Oxygen Balance
in Streams," Journal Sanitary Engineering Division, ASCE. Vol. 96, SA2.
Odum, H.T., W. McConnell, and W. Abbot, 1958. "The Chlorophyll-a_ of
Communities," Institute of Marine Science, Vol. I, 65-69.
Thomann, R.V., 1972. Systems Analysis and Water Quality Management.
Environmental Research and Applications, Inc.; New .York.
Thomas, N.A. and R.-L. O'Connell, 1966. "A Method for Measuring Primary
Production by Stream Benthos," Limnology and Oceanography. Vol. II,
No. 3, pp 386-392.
168
-------
U.S. Environmental Protection Agency, 1976. Areawide Assessment Procedures
Manual. Office of Research and Development, Cincinnati, Ohio, Report
EPA-600/9-76-014.
Weeter, D.W. and J.M. Bell, 1971. A Systems Analysis Investigation for
Water Quality Management, Presented at the Indiana Water Pollution Control
Association Annual Conference, November 16, Indianapolis, Indiana.
Westlake, D.F. (undated). "A Model for Quantitative Studies of Photo-
synthesis by Higher Plants in Streams," Air and Hater Pollution Journal,
Vol. 10, 883.
3.5 CARBONACEOUS DEOXYGENATION
3.5.1 Introduction
Biochemical oxygen demand (BOD) is the oxygen consumed by micro-
organisms utilizing organic matter as food and breaking down the complex
compounds into simpler products. This process can be divided into two
stages: the first being due to oxidation of carbonaceous matter and the
second due to nitrogenous substance oxidation. Only carbonaceous oxidation
is described here; oxidation of nitrogenous matter is discussed in
Section 3.6.
The amount of BOD present in an aquatic system has traditionally been
measured by applying the standard BODs test to a water sample. Appropriate
factors are available which can then be applied to the 5-day BOD values in
order to obtain the ultimate first stage oxygen demand. These factors vary
from 1.10 to 2.40 with 1.45 being the most common (i.e., ultimate BOD =
1.45 BOD5) (Crim and Lovelace, 1973).
In addition to externally applied loads of BOD such as those entering
the system from a sewage treatment facility, the amount of BOD present in
a system may also be affected by the following internal sources: 1) the
death of algae, zooplankton, fish and bacteria, 2) excretions by algae,
zooplankton, fish and bacteria, and 3) benthal demands.
The BOD decay process can be described by the following expression:
SUBSTRATE + BACTERIA + 0? + GROWTH FACTORS *•
(NUTRIENTS)
C02 + H20 + MORE BACTERIA + ENERGY
At present virtually all investigators assume some form of first order
reaction to describe the kinetics of BOD exertion. Typically the BOD
expression is represented as:
169
-------
T - kd L (
where L = BOD remaining at time t, mg/1
k . = deoxygenation coefficient, day"
In the discussion that follows, it will be necessary to distinguish
between different but related BOD decay rates. The following convention
will be followed:
k , = receiving water deoxygenation rate due to BOD exertion,
k-, = laboratory deoxygenation rate due to BOD exertion, and
k = receiving water exertion rate of BOD (includes all
sinks of BOD, including deoxygenation and settling).
Some attempts have been made to develop relationships between these rates,
as will be discussed later.
3.5.2 Factors Affecting Deoxygenation Rates
Many factors are known to influence both the deoxygenation rate, kd,
and the disappearance rate of BOD, kr. Those to be discussed are:
• temperature
• hydraulic parameters
• degree of waste treatment, and
• other in situ physicochemical processes.
The most commonly used temperature correction expression for k(j is of
the following form:
Ho e?'
where k, = deoxygenation rate at temperature T (°C), day"
(k ,) = deoxygenation rate at 20°C, day"
a 20
0 = temperature correction coefficient
Values reported by modelers for the temperature coefficient are shown in
Table 3-12. Also included in that table are the deoxygenation coefficients
used in the studies.
170
-------
TABLE 3-12. VALUES OF TEMPERATURE CORRECTION FACTOR, 6
AND DEOXYGENATION COEFFICIENT, krf
Location
Deoxygena tion
Coefficient, kj at
20nC, day"1(base e)
Temperature
Coefficient, 0
Reference
N. Fork Kings
River, California
Lake Washington
ilerrimack River
Boise River, Idaho
San Francisco Bay
Estuary
o.:
2.5
0.2
0.1 1.5
0.2
0.06 0.36
0.01 0.1
0.2 2.0
0.25
0.2
0.2
0.1 0.3
0.23
1.02 1.09
1.075
1.047
1 .047
Baca and Arnett, 1976
Tetra Tech, 1976a
Hydrologic Engineering
Center, 1974
Chen and Orlob, 1975
Lombardo, 1972
Camp, 1965
Hydroscience, 1971
Chen and Wells, 1975
Chen, 1970
Thomann, e_t al_., 1974
Tetra Tech, 1976b
Crim and Lovelace, 1973
Bansal (1975) has investigated the effects of various hydraulic param-
eters on kd by combining these parameters into dimension!ess groups and
then, by using historical data, employing regression methods to develop
relationships between the groups. He chose various combinations of param-
eters, and that combination yielding the best results is shown in Figure
3-5. The data points from which the line of best fit was developed are not
included on the figure; however the predictive formulation is, including the
correlation coefficient and the standard error of estimate.
The data Bansal (1975) used in his work are presented in Table 3-13.
From his work Bansal concluded that the ratio of Reynolds number to Froude
number offered the best method (with respect to those parameter combinations
he evaluated) of predicting k^ independently of stream location, pollution
load, and channel configuration. The effect of temperature is implicitly
included in the Reynolds number. This temperature influence is that affect-
ing water density and viscosity, and not microbial activity, as expressed
by Equation (3-38). It is more meaningful to incorporate the effects of
171
-------
IO8
IO7
io6
5
IO4
io3
io2
Log (kdH2/v) = -3.606+1.383 Log ((pUH/y)/(uVgH) )
Correlation Coefficient = .958
Std. Error of Estimate = .342
/
/
io4
/
/
io5
/
io6
/
/
io7
'
io8
ior
•}
Figure 3-5. Deoxygenation parameter (k^H /v) versus the
Reynolds Number and Froude Number (pUH/p)/(UV/gTT)
(from Bansal, 1975).
172
-------
TABLE 3-13. DEOXVGENATION RATE CONSTANTS (FROM BANSAL, 1975)
Cross-
sectional
Discharge Area
cfs sq.ft.
Kansas
Kansas
Kansas
Republ
River at
15,200
2,160
2,090
2,440
1,300
828
632
1,080
Bonner Springs, Kansas
4,300
1,200
1,170
1,300
850
550
425
710
Top
Width
ft.
770 % ^
505 2.-M-
500 2-3^
525 2. "5"
450 \-°(
415 \-3
405 \-o£T
432 \-W>
Temp .
°C
25
28
25
24
9
5
9
14
kd
Observed
.02
.12
.12
.24
.02
.16
.26
.17
River at Lecompton, Kansas
1,750
1,360
2,060
2,300
1,040
793
1 ,170
River at
3,040
1,460
1,800
2,690
1,900
764
631
608
ic River
258
657
609
201
15
36
249
750
590
880
1 ,000
450
350
500
Topeka, Kansas
1,450
700
865
1,285
910
365
310
290
below Milford, Kansas
184
412
392
140
10
26
177
725 \-O£>
660 cS^
774 \-\H
757 \-3"2-
592 . ~l\e>
538 ,tS
620 , Si
468
437
447
466
450
405
368
364
196
251
249
166
68 ,V^»
77 .34
192
27 •
32 —
28 --
10 -
6
0
16
22
31
27
18
7
7
11
15
24
28
24
14
1
0
16
.19
15
.35
.30
47
.23
.05
.08
.07
.1
.37
.14
.06
.23
.10
.18
.19
.07
.25
.14
.23
.29
(continued)
173
-------
TABLE 3-13 (continued)
Smoky
Smoky
Discharge
cfs
Cross-
sectional
Area
sq.ft.
Top
Width
ft.
Temp .
°C
kd
Observed
Hill River at Enterprise, Kansas
215
100
373
146
113
210
157
Hill River at New
185
67
734
Solomon River at Miles,
Kansas
£1
35
117
Ri ver at Wamego,
890
1,540
2,530
1,470
680
535
483
Big Bl ue River at Tuttle
1,060
162
90
810
961
232
50
50
122
57
211
83
65
120
88
Cambridge, Kansas
87
35
252
Kansas
65
40
125
Kansas
390
670
1,080
730
300
225
190
Creek, Kansas
1,050
70
42
490
1,000
108
30
30
(continued)
119
69
131
91
75
118
57
86
79 .If 4
88
53 V2.
49 . &2
61
413
468
540
462
395
381
375
194
76
62 •(&*&
191
194
92
52 uS^
52 . 5Hs
27
32
24
6
0
15
14
27
31
29
28
30
21
27
26
27
15
0
2
7
22
24
27
23
8
1
6
13
.09
.26
.16
.32
.14
.17
.24
.09
.19
.27
.21
.19
.06
.23
.13
.30
.26
.28
.11
.06
.21
.14
.23
.28
.6
.37
.15
.2
174
-------
TABLE 3-13 (continued)
Cross-
sectional
Discharge Area
cfs sq.ft.
Kansas River at Manhattan (Fort Riley),
1 ,250
559
1,200
511
Solomon River at Glen Elder
44
47
79
30
48
33
30
4,250
1 ,750
4,050
367
, Kansas
38
40
58
30
41
31
30
Top
Width
ft.
Kansas
533
493
530
247
49 0~t'?:,
49 .I? 2-
53 VV
33 . '-\ \
49 , SH-
36 , 1?t (s>
33 ,C{(
Temp.
°C
26
32
26
25
28
28
24
7
0
0
5
kd
Observed
.26
.15
.10
.09
.20
.23
.10
.35
.34
.37
.35
Saline River at Tescott, Kansas
Smoky
Smoky
Grand
8.3
5.8
132
Mil 1 River at Mentor,
138
35
675
Hill River at Langley
77
147
493
249
14
21
18
81
River, Michigan
295
14
10
75
Kansas
88
20
288
, Kansas
60
85
210
122
15
2]
19
60
320
19.4
17.8
30
83
81.5
92.5
57
67
90
75
20
23
22
58
168.4
2.7
28
21
24
26
26
23
27
24
23
7
1
6
11
20
.37
.25
.26
.10
.15
.42
.42
.51
.14
, .14
.20
.28
.29
.33
.59
(continued)
175
-------
TABLE 3-13 (continued)
Cross-
sectional Top
Discharge Area Width
cfs sq.ft. ft.
Temp.
°C
kd
Observed
Clinton River, Michigan
33
44.6
28.22
20
3.37 ^^
Truckee River, Nevada
Flint
180
195
271
River. Michigan
134
174
174
204
204
150
150
150
210
200
400
296
400
89.8
89.8
89.8
100
76.9
153.8
170.6
210.5
20
20
20
--
20
20
20
20
.36
.36
.96
.56
.63
.69
.69
.69
Jackson River, Virginia
North
North
100
Branch Potomac River
100
Branch Susquehanna
1,000
365
122
20
1.25 ^-^
(Maryland, West Virginia)
100
1,700
50
425
20
20
.4
.35
New River, Virginia
Upper
Lower
1,200
1 ,720
344
20
.5
Hudson, Troy, New York
3,000
4,500
Sacramento River
10,000
6,000
6,750
8,000
(continued)
343
321
457
20
20
20
.125
165
.4
176
-------
TABLE 3-13 (continued)
Cross-
sectional
Discharge Area
cfs sq.ft.
Top
Width Temp.
ft. °C
L-
kd
Observed
Upper Jan.es River. Virginia
1,800 8,000
2,600 11,500
9,000 14,500
7,500 14,000
4,500 14,500
3,800 13,500
1,350 15,000
742
935.5
935.5
903.2
935.2
871
967.7
20
20
20
20
20
20
20
48
.30
.31
.41
.39
.38
.43
Cooper River, South Carolina
10,000 40,000
1,000
20
.3
Savannah River, Georgia and South Carolina
7,000 10,000 1,000
6,800 40,000 1,428.6
20
20
.3
.3
South New Jersey
23
2,500
208.3
20
.2
Compton Creek, New Jersey
10 1,000
10 790
69
75
20
20
.23
.23
177
-------
temperature by using Bansal's expression to evaluate k^ at some base
temperature, and then use Equation (3-38) to correct for temperature
variations from the base temperature.
Tables 3-4 and 3-6, previously presented in Section 3.2.4 on reaera-
tion, also contain a number of deoxygenation coefficients. Table 3-17 in
Section 3.6 on nitrogenous oxidation contains additional kd values.
Hydroscience (1971) has developed a relationship for predicting kj
in streams and rivers that takes into account both stream depth and degree
of waste treatment. This relationship, developed from data surveys con-
ducted by Hydroscience, is graphically shown in Figure 3-6. Data points
are not included in the figure. The dependency of k
-------
10.0
1.0
o
°0
OJ
0.05
0.3
Stable, Rocky Bed
Moderate Treatment
Some Ammonia
MEAN
Unstable, Sandy Channel
Highly Treated Effluent
with Nitrification
1.0
10.0
100.0
DEPTH (FT.)
Figure 3-6. Deoxygenation coefficient as a function of depth (after Hydroscience, 1971)
-------
TABLE 3-14. DEOXYGENATION RATES FOR SOME SELECTED RIVERS
(ECKENFELDER AND O'CONNOR, 1961)
River
Elk
Hudson
Wabash
Willamette
Clinton
Tittabawassee
Flow
(cfs)
5
620
2800
3800
33
--
Temp.
(°C)
12
22
25
22
--
--
BOD5
(mg/1)
52
13
14
4
—
--
kd
(day'1 )
3.0
0.15
0.3
0.2
.14-. 23
0.05
kr
(day-1 )
3.0
1.7
0.75
1.0
2.5
0.5
kd = k1 + n (V/D) (3-39)
where k, = stream deoxygenation rate coefficient, base e, day"
k, = laboratory deoxygenation rate coefficient,
base e, day1
V = stream velocity, ft/sec
D = stream depth, feet
n = coefficient of bed activity
The second term on the right side of Equation (3-39) and the values
of n in Table 3-15 reflect the importance of organisms in the stream bed
that utilize BOD.
The coefficient n is taken as a step function of stream slope, as
shown in Table-3-15 below.
Several investigators have modified the traditional BOD decay formula-
tion, Equation (3-37) to account for several other phenomena. In an
attempt to better represent internal sources of BOD in addition to the
externally applied loads, Baca and Arnett (1976) proposed the following
formulation:
180
-------
TABLE 3-15. CHANGES IN COEFFICIENT OF
BED ACTIVITIES BY STREAM SLOPE
Stream
Slope (ft/mi)
2.5
5.0
10.0
25.0
50.0
n
.1
.15
.25
.4
.6
3L/3t = -^L + a/F P + FzZJ
(3-40)
\ r <- i
where L = concentration of BOD, mg/1
a = stoichiometric constant, mg-O^/mg-C
F = death rate of zooplankton due to fish predation, day~
F = death rate of phytoplankton due to zooplankton predation, day~
P = phytoplankton concentration, mg/1
Z - zooplankton concentration, mg/1
Thomas (1948) introduced the concept that the river removal rate,
kr, could be defined as the sum of a laboratory coefficient, ki, and an
additional coefficient, k3, that would account for additional BOD removal
due to factors such as sedimentation, scour, flocculation, and volatilization
occurring in benthal deposits.
Baca and Arnett (1976) incorporated a similar concept into their BOD
formulation for the Gray's Harbor/Chehalis River study. Their expression
has the following form:
3L/3t = -( k1 + k3 JL + P (3-41)
where L = ultimate BOD, mg/1
k-, = laboratory deoxygenation coefficient; typically 0.1-0.8/day
181
-------
ko = "sedimentation" coefficient; typicall-y 0.0-3.5/day
P = scour coefficient; typically 0.0-0.8 mg/l-day
Orlob (1974) is another investigator who has used the sedimentation coeffi-
cient in his work.
3.5.3 Measurement of BOD Decay Rate
The removal rate of BOD from receiving water can be determined indirectly
from in situ BOD measurements by using the BOD mass-balance equation to solve
for the removal rate. There are three aspects of this approach that should
be considered:
1. The removal rate so obtained may approximate either kr
or k,j. If filtered BOD samples are taken and the
settleable fraction of BOD is not assumed to represent
a significant demand on oxygen resources, then kj is
obtained. If total BOD is sampled then kr is obtained.
2. This method is limited to cases where sufficient data
are available, or can be generated, to perform the
necessary calculations in the mass-balance equation used.
3. The method is usually applicable in "idealized"
situations where the mass-balance equation can be
explicitly solved for kr (or kd). In many situations,
the assumptions required to reduce the mass-balance
equation to such a simplified form may not be justifiable.
Three illustrations of this technique will be shown here, two applicable
to streams, and one to ti dally influenced rivers. Each method entails the
use of a plot.
For the first method, the stream should have a relatively constant
cross section, constant flow rate, and a BOD loading located at a position
that will be called x = 0. The BOD concentration can then be expressed as:
(3-42)
where U = mean stream velocity
x = distance downstream from BOD source
LQ = BOD concentration at x=o (just below source)
132
-------
Plotting the log of BOD concentration (BODs or BODult.) versus distance
should produce a generally straight line with slope of -kr/U. An example
plot is shown in Figure 3-7. Care must be exercised to convert the slope
from base 10 logarithms as given in the semi-log plot to base e logarithms.
This conversion is accomplished by multiplying the kr value in base 10
logarithms by 2.3, as shown in the figure.
The second illustration is similar to the first, except that a uniform
inflow of water occurs along the reach. Otherwise the assumptions for the
two illustrations are the same. The uniform inflow is assumed to be free
of BOD. The mass-balance equation then is:
L = L.
o\T
(3-43)
where 0 = flow rate at x = 0
Q = Qo + (AQ) x, flow rate at a distance x
a =
A_ =
krAo
AQ
stream cross-sectional area = Q /U
DO = stream velocity at x = 0
AQ = incremental flow per unit of distance (assumed
constant over the reach under consideration)
For this case a log-log plot of BOD and flow rate, Q, should be developed.
The slope of the plot will be -a. From the known A0 and AQ, kr can be
obtained directly. Note that kr need not be converted from base 10 loga-
rithms to base e, as it already is to the base e.
For tidally influenced rivers where the same assumptions hold as in
the first illustration above, BOD concentration can be expressed as a
function of distance, x, from the discharge point by:
L = L exp ( j,x j, x < 0 (in upstream direction) (3-44)
L = L exp I J2x ], x > 0 (in downstream direction) (3-45)
Jl 2E
<
1 + Jl +
(3-46)
183
-------
10.0
5.0
in
Q
O
co 1.0
U = 4 Miles/Day
kr = - Slope x U
Slope =
Miles
= 0.16/Day
( 0.0'7\ / 4 Miles v
\ M i I n c / \ nn« /
Day
8
DISTANCE (MILES)
INPUT
16
24
32
36
Figure 3-7. Example of computation of kr from stream data
(from Hydroscience, 1971).
184
-------
~w L] - V1" ~[?~ J (3'47)
where E = tidally averaged dispersion coefficient
U = net tidal velocity
By plotting the log of BOD versus distance upstream or downstream, the slope
of the plot becomes j-j or J2 respectively, and kr can then be obtained. If
the abscissa is distance upstream, then x should be negative and j-j positive
If the abscissa is distance downstream, then x is positive and J2 negative.
From laboratory measurements of BOD exerted versus time, various
techniques can be used to obtain the BOD decay rate, k], where this decay
rate is the laboratory decay rate. From the collected data, ki can be
obtained by one of the five methods described in Nemerow (1974):
1. least-squares technique,
2. slope-method,
3. moments method,
4. logarithmic method, and
5. Rhame's two-point method.
Nemerow also describes an additional laboratory procedure which he uses.
Gaudy, e_t aj_. (1967) provides further background on these methods.
Using data from the Clinton and Tittabawassee Rivers, Gannon (1966)
has compared the many methods of k] prediction and has suggested that the
Reed-Theriault method (Theriault, 1927) may be superior.
Standard Methods (APHA, 1971) describes an empirical bottle test for
determining BOD. Care must be exercised in comparing these BOD values to
those which might be observed in surface waters, however. As stated in
Standard Methods:
"The test is of limited value in measuring
the actual oxygen demand of surface waters, and the
extrapolation of test results to actual stream
oxygen demands is highly questionable, since the
laboratory environment does not reproduce stream
conditions, particularly as related to temperature,
sunlight, biological population, water movement
and oxygen concentration."
185
-------
As a corollary to this comment, the decay rates (laboratory versus instream)
would not be expected to be identical. Tierney and Young (1974) have
addressed this question, as previously discussed, and derived an expression
relating Iq and k^ (Equation (3-39)) applicable to streams.
3.5.4 Summary
As yet, no unified theory has evolved for predicting deoxygenation
rates of receiving waters. There are even arguments that the first order
decay rate is invalid (Regal and Schroeder, 1976). Further, the literature,
at times, fails to state whether measured BOD rates are k
-------
3.5.5 References
APHA (American Public Health Association), 1971. Standard Methods for the
Examination of Water and Waste Water, 13th Edition"] American Public Health
Assn., Washington, D.C.
Baca, R.G. and R.C. Arnett, 1976. A Limnological Model for Eutrophic Lakes^
and Impoundments, Battelle Northwest Laboratories.
Bansal, M.K., 1975. "Deoxygenation in Natural Streams," Water Resources
Bulletin, Vol. 11, No. 3, pp 491-504.
Bosko, K., 1966. "Advances in Water Pollution Research," International
Association on Water Pollution Research, Munich.
Camp, T.R., 1965. "Field Estimates of Oxygen Balance Parameters," Journal
Sanitary Engineering Division, ASCE, Vol. 91, SA5, 1.
Chen, C.W. and G.T. Orlob, 1975. "Ecological Simulation for Aquatic
Environments" from Systems Analysis and Simulation in Ecology, Vol. Ill,
Academic Press.
Chen, C.W. and J. Wells, 1975. Boise River Water Quality-Ecological Model
for Urban Planning Study. Prepared for U.S. Army Corps of Engineers, Walla
Walla District and Idaho Water Resources Board.
Chen, C.W., 1970. "Concepts and Utilities of Ecological Model," Journal
Sanitary Engineering Division, ASCE, Vol. 96, SA5.
Crim, R.L. and N.L. Lovelace, 1973. Auto-Qual Modeling System, U.S. EPA-
440/9-73-003; Office of Air and Water Programs, Monitoring and Data Support
Divis ion.
Eckenfelder, W.W. and D.J. O'Connor, 1961. Biological Waste Treatment,
Pergamon Press, New York.
Flegal, T.M. and E.D. Schroeder, 1976. "Temperature Effects on BOD
Stoichiometry and Oxygen Uptake Rate," Jour. Water Pollution Control Fed.,
Vol. 48, No. 12, p 2700
Gannon, 1966. Reference lost.
Gaudy, A.F., e_t a\_., 1967. Methods for Evaluating the First Order Constants
k and L for BOD Exertion. M-l Center for Water Research in Engineering,
Oklahoma State University.
Hydrologic Engineering Center, 1974. Water Quality for River-Reservoir
Systems, U.S. Army Corp of Engineers.
Hydroscience, Inc., 1971. Simplified Mathematical Modeling of Water
Quality for EPA Water Programs.
187
-------
Lombardo, P.S., 1972. Mathematical Model of Water Quality in Rivers and
Impoundments, Hydrocomp, Inc.
Nemerow, N.L., 1974. Scientific Stream Pollution Analysis. Scripta Book
Company, Washington, D.C.
Orlob, G.T., 1974. Mathematical Modeling of Estuarial Ecosystems.
Proceedings of the International Conference on Transport of Persistent
Chemicals in Aquatic Ecosystems, Ottawa, Canada.
Tetra Tech, Inc., 1976a. Documentation of Hater Quality Models for the
Helms Pumped Storage Project. Prepared for Pacific Gas and Electric
Company.
Tetra Tech, Inc., 1976b. Estuary Water Quality Models, Long Island, New
York - User's Guide. Prepared for Nassau-Suffolk Regional Planning Board,
New York.
Theriault, E.J., 1927. The Oxygen Demand of Polluted Waters. United States
Public Health Service, Washington, D.C., Public Health Service Bulletin 173.
Thomann, R.V., D.M. Di Toro, and D.J. O'Connor, 1974. Journal Environmental
Engineering Division, ASCE, Vol. 100, EE3.
Thomas, H.A., Jr., 1948. "Pollution Load Capacity of Streams," Water and
Sewage Works Journal, Vol. 95, 409.
Tierney, G.F. and G.K. Young, 1974. "Relationship of Biological Decay to
Stream Morphology." Prepared by Meta Systems, Inc., Springfield, Virginia.
3.6 NITROGENOUS DEOXYGENATION
3.6.1 Introduction
The rate parameter that is used in simulating nitrification (the oxi-
dation of reduced forms of nitrogen) as a first order, single stage process
is commonly denoted as kn and most often is expressed in units of day"'.
This representation of the nitrification process is shown in Equation (3-48)
f - -v <3-48'
where N = total oxidizable nitrogen, mg/1
-1
k = first order decay rate, day
188
-------
Many other methods of simulating the nitrification process have been
used and were briefly discussed in Section 2.5.1. The present section will
deal with the rate constant kn and will discuss variables that affect
nitrification as well as methods that have been used to measure kn. Finally,
values of kn, as found in the literature, will be presented.
As shown in Equation (3-48), using the single stage formulation for
simulating nitrification requires values of total oxidizable nitrogen (TON)
as well as kn. Generally TON is taken as the sum of ammonia-nitrogen and
organic-nitrogen, although some researchers have chosen to use only ammonia-
nitrogen. To compute the oxygen demand resulting from nitrification, the
TON is typically multiplied by 4.57, the number of grams of oxygen required
to oxidize one gram of ammonia-nitrogen to nitrate. This figure is com-
puted by using the half reaction oxidation-reduction equations for nitrifi-
cation (presented earlier in Section 2.5.1). In reality, however, some
oxygen required in the nitrification process can be obtained through in-
organic carbon sources, reducing the total oxygen demand slightly from that
estimated by using the half reaction equations. The amount of reduction
depends upon the biochemical and nutritional status of the cell. Nitrifica-
tion synthesis-oxidation equations used by Adams and Eckenfelder (1977)
resulted in a value of 4.33 grams of oxygen for the complete oxidation of
one gram of ammonia nitrogen.
3.6.2 Factors Which Affect Nitrification
Several environmental factors have been shown to influence the rate
at which nitrification occurs. Among them are pH, temperature, suspended
particle concentrations, hydraulic parameters, and the benthos of the
receiving waters. These factors will be discussed in this section.
A number of researchers (Meyerhof (1917), Engel and Alexander (1958),
and Wild, e_t aj_. (1971)) have studied the effects of pH on nitrification.
Meyerhof found the optimum pH for ammonia nitrification to be about 8.6.
He also found that at pH values of 7.9 and 9.3 the rate decreases to 50
percent of the maximum. Although Wild, et^ al_. found nearly the same
optimum pH (8.4), the pH values at which the rate had decreased to 50 per-
cent represented a wider pH range, specifically, 7.1 to 9.8. Engel and
Alexander found, contrary to the results of Meyerhof and of Wild, et al.,
that the nitrification rate formed a plateau from pH 7.0 to 9.0 where the
rate everywhere within this range was at least 95 percent of the maximum.
Outside of this range the rate of nitrification rapidly decreased.
Meyerhof (1916) also investigated the rate of nitrite oxidation and
found optimum pH values to be between 8.5 and 9.0. This result is sub-
stantiated by Kholdebarin and Oertl i (1977b) who found the optimum rate of
nitrite oxidation in the Whitewater River to be at pH 8.5.
Researchers have generally found temperature to strongly affect
nitrification rates. Although most investigators have found that at a
nominal temperature of 20°C, a 10°C temperature change produces a large
change in the rate of nitrification, at least one researcher (Borchardt,
189
-------
1966) found that within the range of 15°C to 35°C temperature had little
influence. Most researchers, however, have found that within the tempera-
ture range of 10°C to 30°C the temperature effects can be simulated by the
following expression:
kn = kn 9T-20 (3-49)
n n20
where k = nitrification rate parameter at 20°C
n20
8 = temperature correction factor
Since the oxidation rate of ammonia is typically much slower than the rate
of nitrite oxidation to nitrate, the rate of the first reaction controls
the overall reaction rate. Hence, values of 6 for the ammonia oxidation
reaction are the appropriate ones to use in the single stage formulation.
Table 3-16 summarizes 0 values found by different researchers for that
reaction. The values are generally valid within a temperature range of
10°C to 30°C.
TABLE 3-16. TEMPERATURE CORRECTION FACTOR 6
FOR AMMONIA OXIDATION (McCARTY, UNPUBLISHED NOTES)
Reference
Stratton
Knowles,
Garrett
Buswell ,
Wild, et.
(1966)
ejt
ejt
al-
ii-
(1965)
(1957)
al_. (1971)
1
1
1
1
1
e
.0876
.0997
.0953
.0757
.0548
AT
Doubl
8.
7.
7.
9.
13.
for
ing kn
3°
3°
6°
5°
0°
C
C
C
C
C
Equation (3-49) indicates that nitrification rates monotonically
increase with increasing temperature. Wild, et al. (1971) found this to
be true for the temperature range of 5°C to 30°C. For temperatures above
30°C this may not be true, however, and there is clearly some maximum
temperature above which the rate decreases rapidly. Outside of the range
5°C to 30°C Equation (3-49) should not be used. The optimum temperature
for nitrification is generally accepted to be between 25°C to 28°C.
190
-------
In surface waters, a stimulating effect of suspended particles on the
nitrification rate has been reported by a number of workers (Kholdebarin and
Oertli, 1977a). In their studies, Kholdebarin and Oertli collected water
samples from the Whitewater River in the Coachella Valley in Southern Cali-
fornia. Their findings indicated that although nitrifying organisms can
function well in media lacking any measurable suspended solids, suspended
particulates stimulated their activity. The authors felt that this effect
was caused by the particles acting as sites for the proliferation of the
nitrifying bacteria.
Many workers have reported that the bottoms of streams can also offer
sites for the growth of nitrifying organisms (e.g., Matulewich and Finstein,
1975 and Blain (1969)). This is particularly true for rocky bottoms of
shallow streams. Mills (1976) has documented evidence supporting this
contention, as well as developing a mathematical model to simulate the
process in natural streams.
Nitrification has also been shown to occur extensively in the top few
centimeters of the sediments of the Rivers Trent and Tame in England. At
least 80 percent of the observed ammonia nitrification in those rivers
occurred there (Curtis, ejt al_., 1975).
Bansal (1976) performed extensive analysis in relating kn to different
characteristics of natural streams by using dimensionless groups. The best
correlations betV/een predicted and observed values occurred when using
the stream Reynolds and Froude numbers to predict kn. These two dimension-
less numbers depend primarily on temperature, depth, and velocity parameters
which ordinarily are readily available or easily measurable. BansaTs work
is somewhat controversial, however, and has been criticized by at least two
workers (Gujer (1977) and Brosman (1977)). The reason for the criticisms
is the fact that Bansal chose to ignore the biochemical aspects of nitrifi-
cation and considered only hydraulic stream parameters.
3.6.3 Summary of Rate Values
Bansal (1976) has compiled kn data for numerous rivers around the
country. These values are shown in Table 3-17 which also contains k
-------
TABLE 3-17. NITRIFICATION RATE CONSTANTS (FROM BANSAL, 1976)
Discharge
(cfs)
Cross-
sectional
Area
(sq ft)
Top
Width
(ft)
Temperature
°C
Reported
kj Base e
day"1
Reported
k2 Base e
day1
kn
Reported
(295.0)
320.0
Grand River (Mich.)
168.4 28.0
3.996
1.9
(180.0)
(195.0)
(271.0)
150.0
150.0
150.0
Truckee River (Nev.)
89.82
89.82
89.82
27.8
27.8
27.8
0.09
0.49
1.30
5.625
5.902
7.190
2.4
2.4
2.4
Flint River (Mich.)
134.0
174.0
174.0
204.0
204.0
210.0
200.0
400.0
290.0
400.0
100.0
76.92
153.85
170.59
210.5
28.0
28.0
28.0
28.0
28.0
0.76
0.95
0.95
0.95
0.95
2.786
2.487
1.641
3.971
2.801
2.0
0.1
0.1
2.5
2.5
Upper Mohawk River (N.Y.)
143.0
143.0
147.0
305.0
176.0
185.0
318.0
195.0
205.0
353.0
301.0
340.0
729.0
331.0
374.0
801.0
392.0
451.0
,018.0
392.0
451.0
,018.0
557.0
656.0
,525.0
,088.0
,272.0
3,655.0
,108.0
,286.0
,727.0
,128.0
,311.0
3,799.0
400.0
400.0
,400.0
,400.0
120.0
120.0
120.0
120.0
120.0
120.0
200.0
200.0
200.0
200.0
200.0
200.0
700.0
700.0
700.0
500.0
500.0
500.0
650.0
650.0
650.0
900.0
900.0
900.0
800.0
800.0
800.0
700.0
700.0
700.0
200.0
200.0
200.0
200.0
48.0
48.0
48.0
40.0
40.0
40.0
64.5
64.5
64.5
57.1
57.1
57.1
189.2
189.2
189.2
375.0
375.0
375.0
100.0
100.0
100.0
120.0
120.0
120.0
422.2
422.2
422.2
70.0
70.0
70.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
0.23
0.23
0.23
0.23
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.23
0.23
0.23
0.23
0.23
0.23
0.23
0.23
0*23
0.23
0.23
0.23
0.063
0.063
0.064
0.099
.875
.993
.526
.193
.290
.558
.911
.132
.950
.599
.796
4.416
1.256
1.366
2.227
0.713
0.775
1.264
0.737
0.813
1.348
0.741
0.814
1.534
0.245
0.267
0.506
0.589
0.644
1.220
0.25
0.25
0.25
0.25
0.30
0.30
0.30
0.30
0.30
0.30
0.30
0.30
0.30
0.30
0.30
0.30
0.30
0.30
0.30
0.30
0.30
0.30
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
(continued)
192
-------
TABLE 3-17 (continued)
Discharge
(cfs)
Cross-
sectional
Area
(sq ft)
Top
Width
(ft)
Temperature k
-c
Reported
day"1
Reported
k>2 lia Lie e
day1
Reported
Middle Mohawk River (H.Y. )
1,300.0
1,300.0
1,600.0
1 ,610.0
1,620.0
1,640.0
1,640.0
1,640.0
1,690.0
1 ,715.0
1,715.0
1,715.0
1,715.0
1 ,735.0
1,735.0
1,735.0
1,735.0
16.0
34.0
16.0
26.0
16.0
20.0
5,000.0
1 ,500.0
1,500.0
4,600.0
5,200.0
6,200.0
6,200.0
6,200.0
7,500.0
7,500.0
7,500.0
7,500.0
6,000.0
6,000.0
6,000.0
9,500.0
19,000.0
2,200.0
2,200.0
2,200.0
3,000.0
3,000.0
3,000.0
385.0
136.4
125.0
354.0
400.0
477.0
Lower Mohawk
477.0
459.0
555.5
555.5
535.7
535.7
428.6
428.6
428.6
633.6
1,266.6
Barge Canal (parallel
176.0
176.0
176.0
230.8
230.8
230.8
23.5
23.5
23.5
23.5
23.5
23.5
River (N.Y.)
23.5
23.5
23.5
23.5
23.5
23.5
23.5
23.5
23.5
23.5
23.5
to upper Mohawk
19.0
19.0
19.0
19.0
19.0
19.0
0.30
0.30
0.30
0.30
0.30
0.30
0.30
0.30
0.30
0.30
0.30
0.30
0.30
0.30
0.30
0.30
0.30
River)
0.23
0.23
0.23
0.23
0.23
0.23
0.132
0.344
0.345
0.158
0.147
0.133
0.134
0.127
0.115
0.116
0.110
0.110
0.126
0.127
0.127
0.087
0.058
0.017
0.027
0.017
0.018
0.013
0.015
0.30
0.30
0.30
0.30
0.30
0.30
0.30
0.30
0.30
0.30
0.30
0.30
0.30
0.30
0.30
0.30
0.30
0.25
0.25
0.25
0.25
0.25
0.25
Ohio River-Markland Pool
26,000.0
29,000.0
32,000.0
23,000.0
17,000.0
14,000.0
25,000.0
30,000.0
10,000.0
20,000.0
17,500.0
14,500.0
15,000.0
36,000.0
36,000.0
44,000.0
44,000.0
50,000.0
50,000.0
56,000.0
56,000.0
30,000.0
35,000.0
70,000.0
70,000.0
35,000.0
1 ,333.3
1,333.3
1,375.0
,375.0
,562.5
,562.5
,555.5
,555.5
12,500.0
14,000.0
1 ,707.3
1 ,707.3
14,000.0
27.0
26.0
28.0
24.0
28.0
24.0
27.0
26.0
24.0
16.0
28.0
24.0
28.0
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.241*
0.25*
0.25
0.25
0.224*
0.085
0.091
0.067
0.057
0.042
0.039
0.042
0.048
1.624
1.719
0.052
0.023
2.291
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
Big Blue River (Nebr.)
25.5
25.6
24.2
59.0
36.0
38.0
25.0
25.0
0.160
0.100
19.067
3.468
0.164
0.164
*Estimated values of k,.
(continued)
193
-------
TABLE 3-17 (continued)
Discharge
(cfs)
22.1
32.3
20.94
132.0
122.0
152.0
194.0
210.0
192.0
295.0
333.0
8,900.0
8,900.0
8,900.0
5,600.0
5,600.0
5,600.0
2,000.0
2,000.0
2,000.0
3,000.0
3,000.0
3,000.0
Cross-
sectional
Area
(sq ft)
36.1
27.1
22.01
102.6
77.0
85.2
161.2
96.4
112.3
190.0
148.0
45,000.0
145,000.0
220,000.0
45,000.0
175,000.0
220,000.0
45,000.0
175,000.0
220,000.0
45,000.0
175,000.0
220,000.0
Top
Width
(ft)
Big Blue
32.0
26.0
24.0
85.0
55.0
50.0
110.0
79.0
127.0
126.0
86.0
Delaware
1 ,800.0
7,000.0
8,800.0
1 ,800.0
7,000.0
8,800.0
1,800.0
7,000.0
8,800.0
1,800.0
7,000.0
8,800.0
Temperature
°C
River (Nebr.)
23.5
23.5
23.5
23.5
27.0
26.0
27.7
28.0
29.0
29.0
27.8
River Estuary
10.0
10.0
10.0
21.0
21.0
21.0
20.0
20.0
20.0
26.0
26.0
26.0
Reported
kj Base e
day"1
0.200
0.170
0.170
0.110
0.238*
0.239*
0.240*
0.237*
0.236*
0.243*
0.243*
0.30**
0.30**
0.30**
0.30**
0.30**
0.30**
0.30**
0.30**
0.30**
0.30**
0.30**
0.30**
Reported
k£ Base e
day1
6.731
10.825
11.716
9.471
8.552
7.076
6.774
12.444
16.740
7.484
7.840
0.051
0.023
0.020
0.035
0.015
0.013
0.019
0.008
0.007
0.023
0.010
0.008
kn
Reported
0.164
0.253
0.253
0.046
0.051
0.051
0.051
0.051
0.108
0.108
0.032
0.085
0.085
0.085
0.385
0.385
0.385
0.325
0.325
0.325
0.535
0.535
0.535
*Estimated values of
These values of k, are referenced to 20°C
194
-------
10
8
10'
CM
X
IOJ
10'
Log -'i- I -3.421+1.36 Log ^ / v
Correlation coefficient .936
Standard error of estimate - .360
10'
10'
10
8
I09
Figure 3-8. Nitrification parameter knH /v versus the ratio of the
Reynolds number pVH/y and the Froude number (Bansal, 1976)
Courchaine (1968) has plotted nitrogenous BOD on a logarithmic scale,
and determined the decay rate from the slope of the line. This is a proce-
dure similar to that described in Section 3.5 for measuring carbonaceous BOD
decay rates. Other mass-balance methods in that section, such as measuring
kd in a tidally influenced river, can be applied to determine kn if the
appropriate conditions are met. Nelder and Mead (1965) and Bard (1967) have
also used data measured at two or more locations to estimate nitrification
rates.
195
-------
Thomann (1963) has determined kn by plotting long-term BOD data on a
semi-log plot. The slope of the line provides an estimate of the decay
coefficient during the second (nitrogenous) stage of decay.
Thomann, et aK (1971) has taken a different approach to estimating
nitrification rates. He used a finite-difference approximation to solve a
set of simultaneous linear equations. The nitrification rates calculated
from this set of equations were adjusted so that observed dissolved oxygen
values throughout the stream were accurately reflected by the equations.
Although Thomann simulated nitrification in two stages, his procedure is
applicable to a single stage simulation approach.
Das and Cibulka (1976) have used the two-point method of Rhame (de-
scribed in Nemerow (1974)) to determine kn. To do this, they first inhib-
ited nitrification using 2-chloro-6-( trichloromethyl) pyridine, as
recommended in the U.S. EPA Training Manual (1974). The inhibited BOD
curve was subtracted from the uninhibited curve resulting in a curve from
which kn could be determined.
3.6.5 Summary
The single stage representation of nitrification is certainly a
simplification of the process, and some (Gujer (1977) and Brosman (1977))
would argue that it is an oversimplification. Nevertheless it is a widely
used approach, and will probably continue to be so.
Methods for measuring kn are varied, as discussed previously. There
is no concensus as to which nitrogen forms are actually represented in
in situ kn observations. Some monitor only the disappearance of ammonia-
nitrogen, and others, the disappearance of ammonia plus organic nitrogen.
If the disappearance of ammonia-nitrogen is monitored, and no organic-
nitrogen is present, the two methods will produce the same result; other-
wise they will not.
The method most widely used to determine the stoichiometric equiva-
lence between oxygen utilization and nitrification is to use the oxidation-
reduction half reactions for nitrification, which produces a stoichiometric
equivalence of 4.57. Using this approach yields a value which is a few
percentage points higher than the true equivalence. However, because the
actual value may change depending on environmental conditions, because a
considerable effort would be required to improve upon the value, and
because the 4.57 figure produces conservative results as far as dissolved
oxygen mass-balance calculations are concerned, the higher figure can
justifiably be used.
Any method for measuring kn using in^ situ data based on an NBOD mass-
balance equation should be studied to determine if the formulation assump-
tions are valid for the receiving water under consideration. Since the
aquatic nitrogen cycle is complex (see Brezonik (1973)) somewhat obscure
sources or sinks of NBOD may exist, and if significant, could affect the
estimated kn.
196
-------
Laboratory measurements of k^ can produce results that differ signifi-
cantly from what might be measured in situ. As discussed previously,
benthic effects, pH, and suspended particle levels can affect nitrification
rates, and all these factors cannot be simulated accurately in the labora-
tory. The methods previously discussed that deal with simply determining
whether or not nitrification is occurring have taken the approach of eval-
uating kn. Another method exists which, in a qualitative sense at least,
can be used to determine whether or not nitrification occurs. This method
involves determining directly whether a viable population of nitrifiers
exists in situ. Relatively few researchers have done this, either as a
supplement or as an integral part of their work in receiving waters. How-
ever, two cases should be mentioned. The first dealt with measuring
Nitrosomonas and Nitrobacter in sediments of the Rivers Trent and Tame
(Curtis, et al. (1975)), where large concentrations of these organisms were
found. The second dealt with the evaluation of the same organisms in the
Passaic River (Finstein, ejt aK, 1977) where disagreement existed as to
whether or not nitrification was occurring.
Low concentrations of nitrifiers (on the order of hundreds per milli-
liter) suggest that nitrification is unimportant, while high concentrations
(on the order of millions per milliliter) indicate that nitrification is
probably important.
3.6.6 References
Adams, C.E. and W.W. Eckenfelder, March, 1977. "Nitrification Design
Approach for High Strength Ammonia Wastewaters," Journal of Water Pollution
Control Federation, Vol. 49, No. 3, pp 413-421.
Bansal, M.K., 1976. "Nitrification in Natural Streams," Journal of Water
Pollution Control Federation, Vol. 48, No. 10, pp 2380-2393.
Bard, Y., 1967. "Nonlinear Parameter Estimation and Programming," IBM
New York Sci. Center, Report 320-2902.
Blain, W.A., 1969. Discussion of "Evaluation of Nitrification in Streams,"
Journal of the Sanitary Engineering Division, ASCE, SA5, pp 956-958.
Borchardt, J.A., 1966. "Nitrification in the Activated Sludge Process."
In: The Activated Sludge Process, Div. of Sanitary and Water Resources Eng.,
Univ. of Michigan, Ann Arbor.
Brezonik, P.L., 1973. "Nitrogen Sources and Cycling in Natural Waters,"
EPA-660/3-73-002.
Brosman, D.R., 1977. Discussion of "Nitrification in Natural Streams,"
Journal of Water Pollution Control Federation, Vol. 49, No. 5, pp 876-877.
197
-------
Buswell, A.M., e_t al_., 1957. "Laboratory Studies on the Kinetics of the
Growth of Nitrosomonas with Relation to the Nitrification Phase of the BOD
Test," Applied Microbiology, 2., 21-25.
Courchaine, R.J., 1968. "Significance of Nitrification in Stream Analysis--
Effects on the Oxygen Balance," Journal Mater Pollution Control Fed.,
40, 835.
Curtis, E.J.C., K. Durrant, and M.M.I. Harman, 1975. "Nitrification in
Rivers in the Trent Basin," Water Research, Vol. 9, pp 255 to 268.
Das, K.C. and J.J. Cibulka, 1976. "A Case History of Water Quality
Considerations in the James River at Richmond," Presented at the 49th
Annual Water Pollution Control Federation Conference, Minneapolis,
Minnesota, October 3-8.
Engel, M.S. and Alexander, M., 1958. "Growth and Autotrophic Metabolism
of Nitrosomonas Europaea," Jour. Bacterio., 76, 217.
Finstein, M.S., J. Cirello, P.F. Strom, Mil. Morris, R.A. Rapaport, and
S. Goetz, April, 1977. "Evaluation of Nitrification in the Water Column
of the Passaic River," Water Resources Research Institute, Rutgers
University, New Jersey.
Gujer, W., 1977. Discussion of "Nitrification in Natural Streams,"
Journal of Water Pollution Control Federation, Vol. 49, No. 5, pp 873-875.
Kholdebarin, B. and J.J. Oertli, 1977a. "Effect of Suspended Particles and
Their Sizes on Nitrification in Surface Waters," Journal of Water Pollution
Control Federation, Vol. 49, No. 7, pp 1693-1697.
Kholdebarin, B. and J.J. Oertli, 1977b. "Effect of pH and Ammonia on the
Rate of Nitrification of Surface Water," Journal of the Water Pollution
Control Federation, Vol. 49, No. 7, pp 1688-1692.
Knowles, C., A.L. Downing, and M.J. Barrett, 1965. "Determination of
Kinetic Constants for Nitrifying Bacteria in Mixed Culture, with the Aid
of an Electronic Computer," Journal of General Microbiology, 38, 263-278.
Matulewich, V.A. and M.S. Finstein, 1975. "Water Phase and Rock Surfaces
as the Site of Nitrification," Abstr. Annual Meeting American Society of
Microbiology, N 30.
Meyerhof, 0., 1916. Arch, f. die ges Physiologie, 164, 416.
Meyerhof, 0., 1917. Arch, f. die ges Physiologie, 166, 255.
Mills, W.B., 1976. A Computational Model for Predicting Biofilm Nitrifi-
cation in Streams. Engineer's Thesis, Stanford University.
198
-------
Nelder, J.A. and R. Mead, 1965. "A Simplex Method for Function Minimiza-
tion," Computer Journal, 7, 308.
Nemerow, N.L., 1974. Scientific Stream Pollution Analysis, Scripta Book
Company.
Stratton, F.E. and P.L. McCarty, 1969. Discussion of "Evaluation of
Nitrification in Streams," Journal of the Sanitary Engineering Division,
ASCE, SA5, pp 952-955.
Stratton, Frank, 1966. "Nitrification Effects on Oxygen Resources in
Streams," Ph.D. Thesis, Stanford University.
Thomann, R.V., et al., 1971. "The Effect of Nitrification on the Dissolved
Oxygen of Streams and Estuaries," Environ. Eng. and Sci. Program, Manhattan
College, Bronx, N.Y.
Thomann, R.V., 1963. "Mathematical Model for DO," Journal Sanitary Engineer-
ing Division, ASCE, Vol. 89, SA5, 1.
United States Environmental Protection Agency Training Manual, 1974.
Chemical Analysis for Water Quality Treatment.
Wild, H.E., C.N. Sawyer, and T.C. McMahon, 1971. "Factors Affecting
Nitrification Kinetics," Journal of Water Pollution Control Federation,
Vol. 43, No. 9, pp 1845-1854.
3.7 BENTHIC OXYGEN DEMAND
3.7-1 Introduction
Often, it is necessary to consider benthic oxygen demand when simula-
ting dissolved oxygen in receiving waters, regardless of the water body
type. Benthal deposits at any given location in an aquatic system are the
result of the transportation and deposition of organic material. The
material may be from an outside source (allochthonous material), or it may
have local origins (autochthonous material). In either case such organic
matter can exert a high oxygen demand under some circumstances.
Many investigators (e.g., Martin and Bella, 1971) have shown that
the rate of oxygen demand is influenced by two different phenomena. The
first is the rate at which oxygen diffuses into the bottom sediments and
is then consumed. The second is essentially the rate at which reduced
organic substances are conveyed into the water column, and are then
oxidized. Traditional measurement techniques, whether they be performed
in situ or in the laboratory, do not differentiate between the two processes
but measure, either directly or indirectly, the gross oxygen uptake. Hence,
in modeling dissolved oxygen, a single term in the dissolved oxygen mass
balance formulation is normally used for both processes. If the two
phenomena are modeled separately, then an additional BOD source is included
199
-------
in the BOD mass balance to account for the organic material being added to
the water column from the benthos. Concurrently, a lesser benthic demand
is used in the dissolved oxygen mass balance, Ogunrombi and Dobbins (1970)
investigated these two phenomena, and developed expressions for the BOD and
DO mass balances. In the experiments they performed (using a synthetic
mixture approximating heavily polluted stream water), they found that the
BOD which diffused back into the overlying water was 28 percent as large as
the oxygen demand caused by oxygen diffusing into the benthic deposits.
In the discussion that follows, the gross benthic oxygen demand will
be treated as a single term.
3.7.2 Factors Affecting Benthic Oxygen Demand
Values of benthal oxygen uptake can vary considerably as will be shown
in Section 3.7.3. Hunter, e_t aj_. (1973), and others, have undertaken studies
to determine exactly what characteristics of the benthic deposits were
significant in affecting the uptake rates. Hunter, et^ aj_. first looked at
the chemical constituents comprising the benthic material. He examined per-
cent moisture, percent volatile, kjeldahl nitrogen, COD, BODs, BOD2Q, TOC
and hexane-benzene extractables. Hunter then used multiple regression
analyses to find out how these constituents correlated with observed benthic
oxygen demands. The one constituent that explained most of the variance of
uptake rates was COD. However, it explained only 37% of the variance, and
the relationship was found to be not significant at the 95% confidence level.
Hunter, e_t al_. then continued the analyses by examining the influence
of biological constituents, in particular the benthic macro invertebrate
population. They found strong correlations between benthic demand rate
and the presence of tubificial worms, although the respiration of the
worms comprised only a small fraction of the benthic demand. Hunter, et al.
hypothesized several possible mechanisms for the worms' influence on oxygen
demand. These included: 1) an increase in the depth to which oxygen
diffuses in the deposit, 2) an increase in the effective surface area over
which uptake can occur, and 3) the transport of organic substances to the
benthic surface by the worms.
The findings of Hunter, e_t al_. concerning the importance of macro-
invertebrates are supported by the earlier work of Rolley and Owens (1967).
Like Hunter, e_t al_., Edeberg and Hofsten (1973) also found that no simple
correlations existed between the organic content in the sediment and the
oxygen uptake rate.
The effects of sediment depth on oxygen consumption have been studied.
For sewage sludge, Baity (1938) developed the following equation:
s = a D°'485 0-50)
b
200
-------
where D = depth of sediment, cm
a = an empirical constant (found to
be 2.7 at 22°C)
SB = benthic uptake rate of oxygen,
gm/m^-day
The maximum depth that was used in Equation (3-50) was 20 cm. Not all
researchers have agreed with the equation and its constants, however.
Fillos and Molof (1972) found oxygen uptake to be independent of sediment
depths greater than 10 cm. Stein and Denison (1967) found no evidence of
increased oxygen uptake at increasing sediment depths, while Davidson and
Hanes (1968) found that deeper deposits of freshly deposited cellulosic
material exerted a significantly greater oxygen uptake than shallower
deposits. Here, however, once consolidation of the bottom deposit was
essentially complete, the oxygen uptake rate became independent of depth.
The benthic oxygen consumption rate also has been hypothesized to
depend on the dissolved oxygen concentration in the overlying waters.
Many researchers (e.g., Edwards and Rolley, 1965; McDonnell and Hall, 1969)
have hypothesized the relationship to be:
SB = a Cb (3-51)
where C = dissolved oxygen concentration
a,b = empirically determined constants
In the McDonnell and Hall (1969) study b was found to be 0.30 and a to vary
from 0.09 to 0.16. The dissolved oxygen concentrations ranged from 0.5
mg/1 to 6.0 mg/1.
Lombardo (1972) also computes the benthic oxygen demand as a function
of overlying dissolved oxygen concentration:
SB = a] -e- (3-52)
where a = constant depending on nature of deposits (0.96
to 8.52 for Sn in gm/m^-day)
That the overlying dissolved oxygen concentration is important in
influencing the benthal demand remains in dispute. Martin and Bella (1971)
showed that benthal oxygen consumption appears independent of the oxygen
concentration in the water for dissolved oxygen values above 2 mg/1. They
201
-------
reasoned that the benthal oxygen uptake was primarily caused by release
of oxygen-demanding substances, and not the diffusion of oxygen into the
benthos.
Martin and Bella (1971) have also investigated the effects of over-
lying water mixing rates on the benthic oxygen demand. Samples were
collected from the Yaquina Estuary near Newport, Oregon. The mixing speed
in the apparatus used for testing was adjusted so that the suspended solids
in the apparatus compared with the suspended solids measurements taken in
the estuary. Results showed the average unmixed oxygen uptake to be 1.9
gm/m^-day, while the uptake rate under mixed conditions was 80 percent
higher.
Perhaps the best documented and most agreed upon factor affecting
benthic oxygen demand is temperature. Many investigators who incorporate
benthic demand into their models prefer to estimate the benthic demand at
a reference temperature (usually 20°C) and then incorporate a temperature
correction factor for other temperatures. The relationship can be expressed
as:
J-20
320
(3-53)
where the reference temperature is 20°C. Many different values of e have
been determined. These are summarized in Table 3-18.
TABLE 3-18. THE EFFECT OF TEMPERATURE
ON BENTHIC OXYGEN UPTAKE
Researcher
Edeberg and Hofsten (1973)
Edeberg and Hofsten (1973)
Edeberg and Hofsten (1973)
Edwards and Rolley (1965)
Karlgren (1968)
McDonnell and Hall (1969)
Pamatmat (1971)
Pamatmat (1971)
Thomann (1972)
Range
(°c)
5-15
10-20
15-25
10-20
2-22
5-25
5-10
5-15
10-30
Temperature
Coeffi-
cient, 9
1.130
1.080
1.040
1.077
1.090
1.067
1.088
1.041
1.065
SR
B(T+10)
SR
BT
3.4
2.1
1.5
2.1
2.4
1.9
2.7
1.5
1.9
202
-------
The last column in the table indicates how a 10°C temperature increase
affects the uptake rate. A temperature coefficient of 1.0718 results in a
doubling of the uptake rate for a 10°C increase. Many researchers have
found that the temperature coefficient decreases somewhat with increasing
temperature (see Table 3-18).
3.7.3 Values of Benthic Oxygen Uptake
Table 3-19 summarizes values obtained by Hunter, e_t aj_. (1973) from
which they tried to develop correlations between chemical characteristics
of the deposits and benthic oxygen demand. They also collected additional
samples within the Passaic River to investigate the relationship of benthic
demand to invertebrate population. These results are shown in Table 3-20.
The distance measurements in the table are with respect to Pine Brook, where
a heavy waste load enters the river. Below that point there are no known
point sources. Thus, the river in the region under investigation represents
a polluted zone followed by a zone of gradual improvement. All of Hunter's
measurements were carried out in situ using a benthic respirometer.
TABLE 3-19. SOURCE AND GENERAL CHARACTERISTICS
OF BENTHAL DEPOSITS (HUNTER, ET AL. 1973)
j»
1
2
3
4
5
6
7
8
9
10
Source
(River)
Passaic
Passaic
Passaic
Passaic
Passaic
Passaic
Passaic
Elizabeth
Mi 11 stone
Mile Run
Benthal
Oxygen
Demand
g/m2/d
10.141
12.76
2.423
2.284
1 .981
1.888
1.726
9.81
2.012
1.083
Moisture
7
52.2
42.7
51.5
52.7
52.4
48.1
40.2
38.2
40.8
24.3
Volatile
Solids
I
12.4
10.8
14.8
12.0
12.6
11.4
11.2
11.7
7.4
5.6
TOC
mg/g
297
326
278
209
584
750
311
289
280
144
COD
mg/g
2245
4064
3272
2639
2640
2563
2482
3724
1281
713
BOD5
mg/g
75
82
190
80
70
125
162
190
23
5
Kjel.N
mg/g
3.8
3.2
3.3
3.5
4.4
3.6
4.0
1.6
0.6
1.1
203
-------
TABLE 3-20. RELATIONSHIP BETWEEN BENTHAL OXYGEN DEMAND
AND TUBIFEX WORM POPULATION (HUNTER, ET AL.., 1973)
Distance
Station ft.
I 100
III +1 ,800
IV +15,000
V +30,000
VI +50,000
Benthal Oxygen
Demand, grams/
meter2/day
11.45
6.42
2.42
1.89
1.73
Tub if ex
Worms
Numbers/ft^
26,400
1 1 , 1 60
3,800
3,000
400
Oxygen
Demand
mg/worm/day
0.035
0.041
0.022
0.012
0.049
Edeberg and Hofsten (1973) conducted both in situ and laboratory
experiments for receiving water bodies in Sweden. Their results are shown
in Table 3-21. The in situ values were generally found to be considerably
higher than the laboratory values. The authors felt these differences
were caused by the disruption of sediment structure and biological condi-
tions during sampling, and the difference between natural and contrived
laboratory conditions.
Rolley and Owens (1967) investigated the sediments from 74 sites in
12 river systems in southern England. The rates of oxygen consumption
varied from 0.144 gm/m2-day to 9.84 gm/m2-day. A summary of their findings
is presented in Table 3-22. No significant difference was found between
rates of oxygen consumption in winter and summer in any of the first four
groups. The fifth group (Group E) was not analyzed because of the hetero-
geneous nature of the sites. Also, it was found the demands from Groups A
and B were similar for any given parameter and equal to approximately half
the values found in Groups C and D. It should be noted that the volumetric
discharge from the activated sludge treatment plants was considerably
higher than that from the trickling filter plants.
Thomann (1972) has also compiled data from various sources and this
information is presented in Table 3-23.
3.7.4 Measurement Techniques
There are two major categories of techniques for measuring benthic
oxygen demand: TJT_ sj_tu techniques and laboratory analysis techniques.
204
-------
TABLE 3-21. OXYGEN CONSUMPTION OF SEDIMENTS IN THE LABORATORY
AND IN SITU (FROM EDEBERG AND HOFSTEN, 1973)
Site
Date
Org.
Hatter
( :. Dry
Weight)
Temp.
Oxygen Con-
sumption in
Laboratory
(g 02 nrz
Karl ho 1m 3
Lakes
Ekoln**
Erken 1
Erken 2
Morrviken 1**
llorrviken 2**
Ramsen
Running Waters
Arbogaan 1
Venaviken
Arbogaan 2
Sjomosjon
Arbogaan 3
Jaders Bruk
7/15/71
9/ 1/71
8/31/71
6/29/72
6/29/72
7/28/71"
7/29/71
7/30/71
8/18/71
3/22/72
7/ 9/71
5/ 4/72
5/ 4/72
8/26/71
3/ 8/72
10/ 7/71
3/ 8/72
10/ 8/71
3/ 8/72
12.8
12.8
7.9
17.7
11 .4
19.0
60.5
2.6
0.6
1 .3
40.
30.
12.2
10.3
11 .
8.5
10
10
13
10
10
10
10
5
10
5
0.71
0.40
0.58-1.2
0.32-0.36
1.1
1 .5
0.21-1.08
0.26-1.2
0.31-0.61
0.42-0.63
15
15
16
13
15
15
17
17
18
4
14
5
7
17
2
10
2.5
1.1
1 .2
3.0
0.92
0.93
1.3
1.7
2.6
0.43
0.50
1 .8
2.4
2.3
1 .44
0.68
0.31
Sediment contains fibrous matter from a papennill where sulphite arid groundwood pulp is produced.
Sediment contains fibrous matter from a hard board mill.
Eutrophic lake with newly deposited algae.
205
-------
ro
o
en
TABLE 3-22. RATES OF OXYGEN CONSUMPTION AND CHEMICAL PROPERTIES OF MUD DEPOSITS
SAMPLED FROM VARIOUS SITES (AFTER ROLLEY AND OWENS, 1967)
Group
(12)
Oxygen consumption
(g/m2 per day at 15°C
at oxygen level of 7 mg/1)
Permanganate value
(g 02/100 g* in 4 hr. at 27°C)
Humic acid (g/100 g)
Loss on ignition (g/100 g)
Organic carbon (C) (g/100 g)
Kjeldahl nitrogen (N) (g/100 g)
C/N ratio
Humic acid/C ratio
Winter
Summer
5
2
10
6
0
10
0
1 .2
1.44
4+2.
4±1 .
7+4.
H3.
5±0.
3±3.
4±0.
A**
6
4
3
6
2
9
2
B E L
Group B
(21)
5
2
10
5
0
10
0
1 .2
1.44
7+2.7
5±1.3
7i4.4
7±2.7
6+0.3
2±3.2
5i0.2
0 W E F F L U E
Group C
(10)
7
3
15
7
0
9
0
2.4
3.36
6+3.7
0±2.1
4±7.8
5±4.8
8±0.5
3+2.2
5-h0.2
NTS
Group D
(5)
10
4
20
11
1
10
0
2.4
2.4
7+2.9
7+2.1
9+7.7
8+3.7
2+0.6
9+2.9
4+0.2
Group E
(21-23)
1
1
6
3
13
6
0
9
0
68
2
2
2
1
3
5
8
5
* All chemical properties are for 100 g dried mud.
** Group definitions:
(A) Sites above effluent outfalls that are free from polluting influences (12 sites).
(B) Sites affected by effluents from percolating-filter plants operating to nominal standards of
30 mg/1 suspended solids and 20 mg/1 biochemical oxygen demand (BOD) (21 sites).
(C) Sites affected by effluents from percolating-filter plants operating to nominal standards of
50 mg/1 suspended solids and 30 mg/1 BOD (10 sites).
(D) Sites affected by effluents from activated-sludge plants (5 sites).
(E) Sites not suitable for inclusion in the above groups (23 sites). These include sites receiving
effluents from mixed treatment works and also sites where the upstream flow or discharge pattern
was complex.
-------
TABLE 3-23. AVERAGE VALUES OF OXYGEN UPTAKE RATES OF
RIVER BOTTOMS (AFTER THOMANN, 1972)
p
Uptake (gms 02/m -day)
@ 20°C
Approximate
Bottom Type and Location Range Average
p
Sphaerotilus - (10 gm dry wt/m ) - 7
Municipal Sewage Sludge-
Outfall Vicinity 2-10.0 4
Municipal Sewage Sludge-
"Aged" Downstream of Outfall
Cellulosic Fiber Sludge
Estuarine mud
Sandy bottom
Mineral soils
1-2
4-10
1-2
0.2-1.0
0.05-0.1
1.5
7
1.5
0.5
0.07
Hunter, et al. (1973) used a benthal respirometer of the type described
by Stein and Denison (1967) to make in situ measurements. The respirometer
is an opaque chamber (to prevent photosynthesis) that is lowered to the
stream bottom where its open face is in contact with the mud surface. The
oxygen level of the water in the chamber is determined for as long a period
as possible, while not allowing the dissolved oxygen to fall below 0.5 mg/1.
The oxygen consumed is then calculated from the difference between the
initial and final DO concentrations, and corrected using a BOD determina-
tion of the river water at the same temperature over the same area. Edeberg
and Hofsten (1973) also used the method of Stein and Denison, but with some
modifications.
The kajak bottom sampler (Edmonson and Winberg, 1971) has been used
to obtain sediment cores for laboratory benthic oxygen demand analysis.
Edeberg and Hofsten (1973) modified this technique slightly. They used a
plexiglas sediment sampling tube which was 500 mm in length and 70 mm in
diameter. The laboratory experiments were carried out directly in these
tubes in the dark, and at constant temperature. A magnetic stirrer was
used to keep the water above the sediment homogeneous, and an oxygen
electrode was used for oxygen measurements.
207
-------
Martin and Bella (1971) used an apparatus that mixed the mud surface
with the overlying water. Mixing blades were immersed in each sampling
tube, the height above the mud and rotation rate being adjustable. Samples
were collected in water below the low tide mark so that no sample was
taken in an area normally exposed at low tide. After the tubes were
pushed onto the mud and the samples collected., each tube was reaerated to
saturation and a 4-inch layer of mineral oil was floated on the surface
to prevent reaeration. McDonnell and Hall (1969) used similar procedures
in their work at Spring Creek, a highly eutrophic stream in central
Pennsylvania. Other laboratory techniques have been described in Hanes
and Irvine (1966), Edwards and Rolley (1965), and Knowles, ejt aj_. (1962).
3.7.5 Summary
The numerical values of benthic oxygen demand that have been presented
here vary considerably, ranging from approximately 0.1 to 10.0 gm/nr-day.
There is no universal agreement on the factors causing this variation.
Hence there is no one expression which has achieved general acceptance for
computing Sg in receiving waters.
In contrast, the temperature correction factor has received more
universal acceptance. Substantial variation in values for this factor,
though, has also been observed by numerous researchers. Several researchers
recommend a temperature correction factor, 6, such that the benthic uptake
rate is doubled for every 10°C (which would correspond to 6 = 1.0718).
If it is undesirable to use a predictive formula to calculate SB>
there remain the alternatives of either using tabulated values, or directly
measuring the uptake rate. For systems such as small lakes, where the up-
take rate can be assumed spatially constant, only one measurement or esti-
mate need be made. However for systems with large spatial variations,
there is evidence (e.g., Hunter, ejt aj_. (1973) and Rolley and Owens (1967))
that the benthic uptake can significantly change over distance within the
water body. Accordingly, one measurement or estimate may not be sufficient
to accurately characterize benthic uptake.
If direct measurement is not possible or is impractical, the selection
of literature values should be guided by the following considerations as a
minimum:
• similarity of water bodies
• temperature of measurement and temperature
correction needed to correct rate to water body
being modeled
t location with respect to discharges, and the source
rate and uniformity of benthic material
208
-------
3.7.6 References
Baity, H.G., 1938. "Some Factors Affecting the Aerobic Decomposition of
Sewage Sludge Deposits," Journal Sewage Works, Vol. 10, 539.
Davidson, R. and N. Hanes, 1968. "Effects of Sludge Depth on the Oxygen
Uptake of a Benthal System." Proceedings of the 23rd Int. Wastes Confer-
ence, Purdue University.
Edeberg, N. and B.V. Hofsten, 1973. "Oxygen Uptake of Bottom Sediments
Studies and in Laboratory," Water Research, Vol. 7, pp 1285-1294.
Edmonson, W.T. and G.G. Winberg, 1971. "A Manual on Methods for the
Assessment of Secondary Productivity in Fresh Waters," IBPS Handbook,
Number 17, Blackwell, Oxford.
Edwards, R.W. and H.L.J. Rolley, 1965. "Oxygen Consumption of River Muds,"
Journal of Ecology, Vol. 53, 1.
Fillos, J. and A.H. Molof, 1972. "Effects of Benthal Deposits on Oxygen
and Nutrient Economy of Flowing Waters," Journal Water Pollution Control
Federation, Vol. 44, 644.
Hanes, N.B. and R.L. Irvine, 1966. "Oxygen Uptake Rates of Benthal Systems
by a New Technique." Proceedings of the 21st International Waste Congress,
Purdue University Extension Series 121, Lafayette, Ind.
Hunter, J.V., M.A. Hartnett, and A.P. Cryan, 1973. "A Study of the Factors
Determining the Oxygen Uptake of Benthal Stream Deposits," Department of
Environmental Sciences, Rutgers University.
Karlgren, L., 1968. Fibersediment och vattendragens syrebalans.--IVL-
konferensen, 1967. Institutet for Vatten-och Luftvardsforskning. B 28.
Stockholm, 1968.
Knowles, G., R.W. Edwards, and R. Briggs, 1962. "Polarographic Measure-
ment of the Rate of Respiration of Natural Sediments," Limnology and
Oceanography, Vol. 7, 481.
Lombardo, P.S., 1972. Mathematical Model of Water Quality in Rivers and
Impoundments, Hydrocomp., Inc.
Martin, D.C. and D.A. Bella, 1971. "Effect of Mixing on Oxygen Uptake
Rate of Estuarine Bottom Deposits," Journal of the Water Pollution Control
Federation, Vol. 43, No. 9.
McDonnell, A.J. and S.D. Hall, 1969. "Effect of Environmental Factors on
Benthal Oxygen Uptake," Journal of Water Pollution Control Federation,
Vol. 41, No. 8, Part 2, R353.
209
-------
Ogunrombi, J.A. and W.E. Dobbins, 1970. "The Effects of Benthal Deposits
on the Oxygen Resources of Natural Streams," Journal of the Hater Pollution
Control Federation. Vol. 42, No. 4, 538.
Pamatmat, M.M., 1971. "Oxygen Consumption by the Seabed-VI Seasonal Cycle
of Chemical Oxidation and Respiration in Puget Sound," Int. Revue Ges.
Hydrobiol. Hydrogr.. Vol. 56, 769.
Rolley, H. and M. Owens, 1967. "Oxygen Consumption Rates and Some Chemical
Properties of River Muds," Water Research. Vol. 1, p 759.
Stein, J. and J. Denison, 1967. "In Situ Benthal Oxygen Demand of
Cellulosic Fibers." Proceedings 3rd Int. Conference on Water Pollution
Resources, Water Pollution Control Federation, Washington, D.C., Vol. 3, 181
Thomann, R.V., 1972. Systems Analysis and Water Quality Measurement.
Environmental Research and Applications, Inc., New York.
3.8 COLIFORM BACTERIA
3.8.1 Introduction
The abundance of coliforms in natural waters has traditionally been
used as an indicator of pathogen contamination. In the past, it was gener-
ally believed that coliform bacteria are not as sensitive to environmental
stresses as organisms more commonly considered as pathogenic*. Accordingly,
it was felt that coliforms are more persistent in natural waters and are,
therefore, a "safe," and conservative index of pathogen contamination.
Lombardo (1972) reports, however, that this may not be true under all
circumstances. Data presented by Gallagher and Sino (in Lombardo, 1972)
show that the numbers of the pathogen Salmonella typhimurium decline more
slowly than numbers of coliforms, and it is clear that there is no reason
to assume that coliform die-off rates are lower than those of all other
pathogenic forms.
In discussing the impact of light on die-off rates of coliforms,
Chamberlin and Mitchell (in press) have noted that fecal streptococci also
do not necessarily die off at the same rate as do coliforms. As a result
of reported findings such as these, the widespread practice of using numbers
of coliform bacteria as an indicator of pathogen contamination is now sub-
ject to question.
*It should be noted first that coliforms themselves can be pathogenic
under certain conditions as in the mammalian urinary tract, and second,
that the pathogenic organisms once believed more susceptible do not
include most viruses, about which little is known regarding persistence,
210
-------
3.8.2 Composition and Assay
The coliform group consists of fecal and non-fecal components. The
fecal component is comprised of the enteric Escherichia group, while the
non-fecal component represents the genus Aerobacter commonly found in soils
and on leaves and grain.
The usual method of assaying for coliforms using the multiple tube
fermentation technique does not differentiate between coliforms of fecal
and non-fecal origin. To differentiate between Escherichia and Aerobacter,
other, more time-consuming procedures are required, such as incubation at
elevated temperature, and inoculation onto selective media such as eosin-
methylene blue (EMB) agar. High temperatures inhibit non-fecal coliforms,
and fecal coliforms respond in characteristic ways on the selective media,
thus permitting assay of the fecal component of total coliforms.
The non-fecal coliform component is of limited sanitary significance
and is capable of regrowth in nutrient-rich waters. Fecal coliforms,
usually representing about 15 percent of the total (Lombardo, 1972) gener-
ally require conditions different from those in unpolluted or moderately
polluted streams, and whether regrowth actually occurs in natural waters
is still in doubt.
The fecal streptococcus is another measure of the bacteriological
quality of a water body. The presence of "fecal streps" in streams is
generally an indication of fecal contamination. The ratio of fecal coli-
forms to fecal streptococci can be used as an indication of the source of
fecal pollution. According to Goldreich and Kenner as reported by Lombardo
(1972), human feces would have a ratio not greater than 0.7. Clearly,
however, this concept would only be useful where fecal contamination is
fresh or where the relative loss of each species is known so that different
die-off rates can be accounted for.
3.8.3 Modeling Coll forms
Modeling of coliforms is done for one main reason—establishing the
level of fecal and/or soil pollution and potential pathogen contamination.
The usual approach is simply to simulate die-off and to estimate coliform
levels as a function of initial loading and the die-off rate which, in
turn, is a function of time or distance of travel from the source.
3.8.3.1 Factors Affecting Die-Off Rates
Upon discharge to a water body, environmental conditions determine
the extent to which coliform regrowth and death occur. Fecal coliforms
and streptococci are occasionally observed to increase in numbers, although
this may be due to disaggregation of clumps of organisms. Non-fecal organ-
isms may, in fact, increase in numbers in natural waters where conditions
are adequate (Lombardo, 1972; Mitchell and Chamberlin, in press).
211
-------
Several physical factors can affect the coliform population in
natural waters, resulting in an apparent increase or decrease in the coli-
form die-off rate. These factors include:
t Photo-oxidation •
• Adsorption •
• Flocculation
Physicochemical factors include:
• Osmotic effects •
• pH •
Biochemical and biological factors include:
Coagulation
Sedimentation
Chemical toxicity
Redox potential
Bacteriophages (viruses)
Algae
Presence of fecal matter
t Nutrient levels •
t Presence of organic t
substances •
• Predators
3.8,3,1.1 Physical Factors
Chamberlin and Mitchell (in press) have noted that although much data
has been collected on coliform die-off rates, mechanisms mediating the
rates have historically been poorly understood. According to Chamberlin
and Mitchell, however, light is one of the important factors. They
observe that it is difficult to show statistically significant relation-
ships between coliform die-off rates and factors usually hypothesized as
influencing those rates. In contrast, significant relationships between
light intensity and coliform die-off rates can be demonstrated. Chamberlin
and Mitchell (in press) have shown that field data statistically support
the photo-oxidation model (to be discussed), and data presented by Wallis,
et_ a]_. (1977) also appear to implicate incident light. Chamberlin and
Mitchell (in press) have elaborated upon possible mechanisms by which
light may increase coliform die-off rates. They point out that although
in many cases of light induced mortality, one or more photosensitizing
substances are involved, visible and near ultraviolet light can kill
E. coli in the absence of exogenous photosensitizers.
One suggested mechanism is that light quanta drive some exogenous or
endogenous chromophore to an electronically excited state. The chromo-
phore, in the process of returning to the ground state, transfers its
absorbed light energy to another substance to form superoxides (02«)>
which in turn cause damage to cellular components. Alternatively, the
activated chromophore may cause damage directly, without the agency of a
superoxygenated intermediate.
Several substances present in bacterial cells are effective photo-
sensitizers. These include tryptophan (an amino acid) and porphyrins.
Also, the photosensitizing substance may originate outside the coliform
cell—for example, as a result of algal biosynthesis.
212
-------
Adsorption, coagulation, and flocculation may affect coliform die-off
rates, although few quantitative data are available. Adsorption refers to
the attachment of coliform organisms to suspended particles. Coagulation
refers to the coalescence of bacteria into clumps, and flocculation refers
to the formation of soft, loose aggregates incorporating much water.
According to Mitchell and Chamberlin (in press), early investigations
by several workers have demonstrated that clays tend to adsorb coliforms
more than do silts or sands. This is, of course, commonly the case with
sorbed substances. As Mitchell and Chamberlin point out, the nature and
stability of coliform aggregates incorporating other particulate matter
depends to a very large extent upon the physicochemical nature of the
particles.
Sedimentation involves the settling out of bacterial particles and
aggregates. The rate of die-off may be materially influenced by aggre-
gation and sedimentation, but the magnitude and direction of the change
in rate is not well understood. The mechanism of apparent die off due
to sedimentation is actually simple removal of cells from the water
column—that is, transfer of matter from one physical compartment (the
water column) to another (the benthos). Accordingly, modeling coliform
die off in the water column may give misleading results particularly
where shellfish are harvested for human consumption. Reduction in coli-
form levels in the water column may simply mean increased numbers in the
benthos.
3.8.3.1.2 Physicochemical Factors
Mitchell and Chamberlin (in press) report that physicochemical
factors may have significant effects on die-off rates. Survival rates
of E. coli, for example, are inversely proportional to salinity both in
natural seawater (due to osmotic and other effects) and in artificial
salt solutions. In general, E. coli have been found to survive longer
in lower pH salt solutions (pH < 8) than under alkaline conditions.
Heavy metal toxicity toward microorganisms has been known since the
late nineteenth century. A great number of studies have been done on
the "oligodynamic action" of silver and copper salts. According to
Mitchell and Chamberlin (in press), heavy metals have been implicated as
important mediators of E. coli die-off rates, and the heavy metal effects
may be reduced by addition of chelating agents. Redox potential, through
its effect on heavy metals solubilities, also affects die-off rates. In
addition to this, redox may influence die-off rates in other ways, al-
though data on this are not extensive.
3.8.3.1.3 Biochemical and Biological Factors
Nutrient concentrations may be important in determining die-off
rates under some conditions. In many nutrient studies, the apparent
impact of nutrient addition to the coliform culture is due to chelation
213
-------
of heavy metal ions (Mitchell and Chamberlin, in press). Thus the apparent
decrease in die-off rate in many cases may not be due to the additional
nutrient, but instead, to reduced toxicity of the culture medium. Mitchell
and Chamberlin (in press) cite the work of Jones (1964) who found that
E. coli would not grow at 37°C in either filter-sterilized natural or syn-
thetic seawater supplemented with glucose, ammonium chloride, and potassium
phosphate. Inhibition could be reversed by autoclaving, by addition of
very small amounts of organic matter, or by addition of metal chelating or
complexing agents. Jones demonstrated that low levels of toxic metals
would produce the inhibitory effect, and concluded that the apparent influ-
ence on die-off rates was due to naturally occurring trace heavy metals in
solution. Furthermore, as Mitchell and Chamberlin (in press) note, other
researchers have obtained experimental results implicating heavy metals,
and their chelation upon addition of nutrients, in apparent changes in die-
off rates.
In some situations, it appears that nutrient levels influence die-off
rates in ways unrelated to toxic metals availability. Savage and Hanes
(1971), for example, have reported growth-limiting effects of available BOD.
Finally, it is possible that the level of nutrients affects coliform pred-
ators, thereby influencing rates of grazing on coliforms. Mitchell and
Chamberlin (inr press) report that predators in natural waters may be signifi
cant in reducing coliform populations. They cite three groups of micro-
organisms which may be important in seawater. These are cell wall-lytic
marine bacteria, certain marine amoebae, and marine bacterial parasites
similar to Bdellovibrio bacteriovorus. Experiments performed by a number
of researchers have implicated predators in die-off of coliforms in both
fresh and seawater. Bacteriophages, on the other hand, are apparently of
minor importance, despite their demonstrated presence in sea water. The
relative insignificance of phages, according to Mitchell and Chamberlin
(in press), stems from their ineffectiveness in killing E. coli where the
bacterial cells are not actively growing and multiplying, and the rapid
inactivation of the phages by seawater.
Some forms of phytoplankton produce antibacterial agents which are
excreted into the water column. These substances are heat-labile macro-
molecules, and according to Mitchell and Chamberlin (in press), at least
one, a chlorophyllide, is active only if the system is illuminated. The
fact that at least one antibacterial agent is activated by light suggests
that algae may play a mediating role in the effect of light on die-off
rates.
Other mechanisms of algal anti-coliform activity have been suggested.
One is that during algal blooms, other organisms which prey on both algae
and coliforms may also increase in numbers.
Table 3-24 is a summary of factors influencing coliform die-off rates.
214
-------
TABLE 3-24, FACTORS AFFECTING COLIFORM DIE-OFF RATE
(ABSTRACTED FROM MITCHELL AND CHAMBERLIN, IN PRESS)
Factor
Effects
Sedimentation
Adsorption, Coagulation,
Flocculation
Solar Radiation
Nutrient Deficiencies
Predation
Bacteriophages
Important with regard to water column
coliform levels, particularly where
untreated or primary sewage effluent
is involved, and under low vertical
mixing conditions. May adversely
affect shellfish beds by depositing
coliforms and fecal matter into benthos.
Inconclusive.
Important; high levels may cause more
than 10-fold increase in die-off rate
over corresponding rate in the dark in
seawater. Rates also materially
increased in freshwater.
Appears to accelerate die-off. Numerous
studies have indicated that increasing
nutrient levels of seawater decrease
die-off rates. Some studies have shown
that substantial nutrient additions
appear to stimulate growth but actually
may simply cause clump disaggregation.
Several species of organisms (bacteria,
amoebae) have been shown to attack and
destroy E. coli. Importance of preda-
tion is well established, but general
guidelines for estimating extent of
predation are difficult to provide
without field data.
Apparently not important.
(continued)
215
-------
TABLE 3-24. (Continued)
Factor
Effects
Algae
Bacterial Toxins
Physicochemical Factors
Bactericidal substances are known to be
produced by planktonic algae. Substances
may be photoactivators, mediating the
influence of light on coliform die-off.
This might account for variability of
data in studies of light-induced die-off
rates. Another hypothesis is that algal
predators with blooms concomitant with
algal blooms may produce substances toxic
to E. coli or may prey upon them.
Antibiotic substances produced by indig-
enous bacteria are not believed important
in coliform die-off.
Apparently, pH, heavy metals content,
and the presence of organic chelating
substances mediate coliform die-off
rates. Importance of each, however,
is poorly understood at present.
216
-------
3.8.3.2 Modeling Formulations
Traditionally, coliform modeling has only taken into account die-off,
and a simple first-order kinetics approach has been used (Baca and Arnett,
1976; Tetra Tech, 1975; Tetra Tech, 1976a; U.S. Army Corps of Engineers,
1974; Chen and Orlob, 1975; Lombardo, 1973; Lombardo, 1972; Anderson, et_ al_.,
1976; Huber, et al., 1972; Hydroscience, 1971; Chen and Wells, 1975; Tetra
Tech, 1976b):
— = -kC (3-54)
_1 j_ IX « V ^ *J /
dt
or
Ct = Coe'kt (3-55)
where C = coliform concentration, mg/1
C = initial coliform concentration, mg/1
C.,. - coliform concentration at time t, mg/1
k = die-off rate constant, day"
A summarized listing of values for k is presented in Table 3-25. The
data summarize 28 studies of rates measured in situ. Table 3-26 shows
values for k from a number of modeling studies. The median rate for the
in situ studies is .04 hr~^ with 60 percent of the values less than .05 hr~'
and 90 percent less than .11 hr" .
A number of researchers have determined values for the half saturation
constant (Ks) for E. coli growth, using the Monod expression:
M - -TT-, (3-56)
where y = growth rate at nutrient concentration s, day
s = concentration of growth limiting nutrients mg/1
y... = maximum growth rate, day
K = half saturation constant producing the half
maximal value of y, mg/1
Table 3-27 shows some reported values for Ks.
217
-------
TABLE 3-25. COLIFORM BACTERIA FRESHWATER DIE-OFF RATES MEASURED IN SITU
ro
oo
(AFTER
System
Ohio River
Upper Illinois River
Lower Illinois River
"Shallow Turbulent Stream"
Missouri River
Tennessee R. (Knoxville)
Tennessee R. (Chattanooga)
Sacramento River
Cumberland River
Glatt River
Groundwater Stream
Leaf River (Miss.)
Wastewater Lagoon
Maturation Ponds
Oxidation Ponds
MITCHELL AND
T (°C)
Summer (20°C)
Winter (5°C)
June-Sept.
Oct. and May
Dec. -Mar.
Apr. and Nov.
June-Sept.
Oct. and May
Dec. -Mar.
Apr. and Nov.
Winter
Summer
Summer
Summer
Summer
10°
7.9-25.5
19°C
11 T"
CHAMBERLIN, IN PRESS)
kthr"1,
Base e)
0.049
0.045
0.085
0.105
0.024
0.043
0.085
0.037
0.026
0.029
0.63
0.020
0.043
0.055
0.072
0.23
1.1
0.021
0.017
0.00833-0.029
0.083
0.07
k = 0.108
. n -mJ-20
Reference
Frost and Streeter (1924)
Hoskins, e_t al_. (1927)
Hoskins, et_ al_. (1927)
Kittrell and Kochtitzky (1947)
Kittrell and Furfari (1963)
Kittrell and Furfari (1963)
Kittrell and Furfari (1963)
Kittrell and Furfari (1963)
Kittrell and Furfari (1963)
Wasser, e_t a]_. (1934)
Wuhrmann (1972)
Mahloch (1974)
Klock (1971)
Marais (1974)
Marais (1974)
-------
TABLE 3-26. VALUES FOR COLIFORM SPECIFIC DIE-OFF RATES
USED IN SEVERAL MODELING STUDIES
System
k @20°C,
Reference
North Fork Kings River,
Cal ifornia
Various Streams
Lake Ontario
Lake Washington
Various Streams
Boise River, Idaho
San Francisco Bay Estuary
.042
.0004-. 146
.02-. 083
.02
.042-. 125
.02
.02
Tetra Tech (1976a)
Baca and Arnett (1976)
U.S. Army Corps of
Engineers (1974)
Chen and Orlob (1975)
Hydroscience (1971)
Chen and Wells (1975)
Chen (1970)
Long Island Estuaries,
New York
.02-.333
Tetra Tech (1976b)
219
-------
TABLE 3-27. NUTRIENT Kc VALUES FOR E. COLI (AFTER MITCHELL AND CHAMBERLIN, IN PRESS)
o
Nutrient
Glucose
Lactose
Phosphate
Medium °C Micromoles Remarks
minimal medium 22.
19.4
41.7
30 405.
30 550.
seawater 20 44.
seawater 20 50.
minimal medium 111.
minimal medium 0.7 uptake study
minimal medium 30 17.35
Reference
Monod (1942)
Moser (1958)
Schultz and Lipe
(1964)
Jannasch (1968)
Jannasch (1968)
Monod (1942)
Medveczky and
Rosenberg (1970)
Shehata and Marr
(1971)
Glucose
30
0.378
Shehata and Marr
(1971)
-------
However, Gaudy, et al_. (1971) have shown that the Monod expression
(Equation (3-56)) is not adequate to describe transient coliform growth
behavior. Accordingly, as suggested by Mitchell and Chamberlin (in press),
the^utility of the Ks value is in evaluating which nutrient may be growth
limiting rather than in estimating a growth rate, y.
Work on coliforms in the Ohio River by Frost and Streeter (1924) re-
vealed that the log decay rate for coliforms is nonlinear with time. Accord
ingly, use of a simple decay expression such as Equation (3-54) with a
single value of k is only an approximation to the actual die-off process.
Such an approach must, to some extent and as a function of time, over-
estimate and/or underestimate dC/dt. One approach to solving the problem
of a time-variable decay rate is to decompose the death curve into two
components, each having its own decay rate (Velz, 1970). This approach is
predicated upon typical death rate curves such as those shown in Figure 3-9.
These curves have essentially two regions, each with its own characteristic
slope, and the coliform concentration as a function of time may be defined
as:
Ct = CQe-kt + C^e-^ (3-57)
where C^ = coliform concentration at time t, mg/1
C,Cg = concentrations of each of the two
hypothetical organism types, mg/1
k,k' = decay rates for the two organism
types, day-'
Table 3-28 shows values for C0, CQ, k, and k" for E. coli as estimated
by Phelps (1944).
TABLE 3-28. VALUES OF C0> Co", k, AND k'
FROM THE OHIO RIVER (PHELPS, 1944)
Parameter Warm Weather Cold Weather
CQ (percent)
k (day), base e
Half-life (day)
CQ (percent)
k" (day), base e
Half-life (day)
99.51
1.075
.64
.49
.1338
5.16
97
1.165
.59
3.0
.0599
11.5
221
-------
100
0.01
0
Figure 3-9. Typical mortality curves for coliforms as a function
of time. Curve A is for cool weather while curve B
represents warm weather decay (redrawn from Velz, 1970)
222
-------
Lombardo (1972), in an effort to more meaningfully model coliforms,
has formulated the dynamics of the coliform population plus streptococci
with three separate first order expressions:
-kTt
CT = CT e ' (3-58)
't 'o
-kft
C, = Cf e (3-59)
Tt To
-k t
Cs = C e s (3-60)
st so
where Ct = organism concentration at time t, mg/1
CQ = organism concentration at time zero, mg/1
Subscript T represents total coliforms,
f represents fecal coliforms, and
s represents fecal streptococci
Table 3-29 provides data for kT, k. and kf as provided by Lombardo
(1972). ' s T
As discussed earlier, recent studies have suggested that incident light
levels strongly affect coliform die-off rates. Chamberlin and Mitchell (in
press) have defined a light level-dependent die-off rate coefficient as
k' = k£ yTaz (3-61)
where k' = the light dependent coliform die off
rate, 1/hr
k = proportionality constant for the specific
organism, cm2/cal
£ = incident light energy at the surface,
0 cal/cm2-hr
a = light attenuation coefficient per unit
depth (see Section 2.3.5)
z = depth in units consistent with a
223
-------
Fecal Col iform
Fecal Streptococci
TABLE 3-29. RATE OF DISAPPEARANCE OF COLIFORM AND
FECAL STREPTOCOCCUS BACTERIA (FROM LOMBARDO, 1972)
Bacteria
Fecal
Fecal
Fecal
Fecal
Fecal
Fecal
ro
ro
Total
Total
Total
Total
Fecal
Fecal
Col
Col
Col
iform
iform
i f o rm
Col iform
Col
Col
Col
Col
Col
Col
Col
Col
i f o rm
iform
i f o rm
iform
i f o rm
iform
i form
i f o rm
Reference
Geldreich, et
Geldreich, et
Geldreich, et
Geldreich, et
Geldreich, et
Geldreich, et
Klock (1971)
Klock (1971)
Evans, et al .
Evans, et al .
Evans, et al .
Evans , et al .
il-
il-
al-
ii-
il-
il-
(1968)*
(1968)*
(1968)*
(1968)*
(1968)*
(1968)*
K
0.
0.
0.
0.
0.
0.
1/hr
,0105
0654
0413
0259
0134
.0076
0.0288-0.0096
0.0384-0.0144
(1968)
(1968)
(1968)
(1968)
0.
0.
0.
0.
.0384
.0211
0577
0481
Remarks
Average value for stormwater runoff
stored at 10''C for 14 days
Average value for stormwater runoff
stored at 20"C, 1st day
Average value for stormwater runoff
stored at 20° C, 2nd day
Average value for stormwater runoff
stored at 20"C, 3rd day
Average value for stormwater runoff
stored at 20" C, 4th-7th day
Average value for stormwater runoff
stored at 20°C, 8th-14th day
Clean rivers and lakes
Polluted rivers and lakes
Stormwater runoff, initial regrowth
Stormwater runoff, initial die-away
Stormwater runoff, initial die-away
Stormwater runoff, 24 hours after
Evans, et al. (1968
Evans, et al. (1968)
0.0962
0.0625
initial die-away
Stormwater runoff, 48 hours after
initial die-away
Stormwater runoff, initial die-away
(continued)
-------
TABLE 3-29 (continued)
ro
ro
en
Bacteria Reference
Fecal Streptococci Geldreich, et al . (1968)*
Fecal Streptococci Geldreich, et_ al . (1968)*
Fecal Streptococci Geldreich, et al . (1968)*
Fecal Streptococci Geldreich, et al . (1968)*
Fecal Col i form CDWR (1962)
Total Col i form CDWR (1962)
Total Coll form CDWR (1962)
Fecal Col i form CDWR (1962)
Total Col i form CDWR (1962)
Total Coliform CDWR (1962)
Fecal Coliform CDWR (1962)
(continued)
K 1/hr
0.0019
0.0125
0.0065
0.0044
0.13
0.1053
0.04
0.0668
0.0223
0.0182
0.0419
Remarks
Average value for stormwater runoff
stored at 10"C for 14 days
Average value for storniwater runoff
stored at 20"C, 2 days
Average value for stormwater runoff
20°C, 3rd day
Average value for stormwater runoff
20°C, 4th-14th day
Unchlorinated primary domestic
effluent discharged nearby,
average rate in June
Unchlorinated primary domestic
effluent discharged nearby,
average rate in June
Unchlorinated primary domestic
effluent discharged nearby,
average rate in October
Industrial and primary treatment
discharges average rate in June
Industrial and primary treatment
discharges average rate in June
Industrial and primary treatment
discharges average rate in October
Many small secondary treatment
effluents, large chlorinated
primary treatment effluent and
sugar beet wastewater from
August to December, initial rate
in June
-------
TABLE 3-29 (continued)
Bacteria
Re ference
K 1/hr
Remarks
Total Coliform
CDWR (1962)
0.0425
Total Coliform
CDWR (1962)
0.051
Fecal Coliform
CDWR (1962)
0.0542
ro
ro
Total Coliform
CDWR (1962)
0.0293
Total Coliform
Total Coliform
Total Coli form
CDWR (1962)
CDWR (1962)
CDWR (1962)
0.0889
0.0761
0.0374
Many small secondary treatment
effluents, large chlorinated
primary treatment effluent and
sugar beet wastewater from
August to December Initial rate
in June
Many small secondary treatment
effluents, large chlorinated
primary treatment effluent and
sugar beet wastewater from
August to December. Average rate
in August
Many small secondary treatment
effluents, large chlorinated
primary treatment effluent and
sugar beet wastewater from
August to December. Average rate
in August
Many small secondary treatment
effluents, large chlorinated
primary treatment effluent and
sugar beet wastewater from
August to December. Initial rate
in October
Cited results of lower Illinois
River, summer, first 20 hours
travel from peak population
Cited results of lower Illinois
River, summer, 21st to 30th hours
of travel from peak population
Cited results of lower Illinois
River, summer, 31st to 50th hour
of travel from peak population
''Note that Lombardo (1972) does not reference Geldreich, ^et. a]_. 1968.
to Lombardo where several references to Geldreich are listed.
The reader is referred
-------
Then, incorporating the vertical dispersion of bacterial cells,
^- = Ez 8 C^at)- - k'C (z,t) (3-62)
9
where EZ = the vertical dispersion coefficient, cm /hr
An expression of this kind is useful where the vertical distribution of
coliforms is nonuniform over depth and where die-off is assumed to be solely
a function of light intensity.
According to Chamberlin and Mitchell (in press), if the bacterial cells
can be assumed uniform over depth (i.e., the water column is vertically
mixed), then the depth-averaged light intensity and the depth-averaged decay
rate, respectively, may be computed:
(3-63)
\J . N_™I > m
and
k"= k£I (3-64)
_ 2
where 1 - the depth-averaged light intensity, cal/cm -hr
H = the depth of the water column in units consistent
with a
k - the depth-averaged light-dependent die-off rate, hr~
The depth-averaged, light-dependent die-off rate, k, may be used in
the first order die-off expression for a vertically mixed water body so
that
a| = kC (3-65)
It is clear that the usejof such a model (Equation (3-65)) might be
further refined by computing k using a sinusoidal function to estimate
light levels and incorporating the influence of such factors as latitude,
day of the year, time of day, and atmospheric conditions including cloud
cover and dust effects. Table 3-30 presents some values for k^.
227
-------
TABLE 3-30. COMPARISON. OF kA ESTIMATES
(AFTER CHAMBERLIN AND MITCHELL, IN PRESS)
Organism
oo
Escherichia coli
Seratia marcescens
Study
24 Field Studies
Mean
5%'tile
95%'tile
61 Laboratory Studies
Mean
5%'tile
95%'tile
4 Field Studies
Mean
Min
Max
4 Laboratory Studies
Mean
4 Field Studies
Mean
Min
Max
£
(cm2/cal)
Data Source
Coli form group
14 Field Studies
Mean
5%'tile
95% 'tile
0.481
0.163
1.25
Gameson and Gould (1975)
Bacillus subtil is var. niger 1 Laboratory Study
0.168
0.068
0.352
0.136
0.062
0.244
0.362
0.321
0.385
0.354
0.192
0.093
0.360
0.002
Foxworthy and Kneeling (1969)
Gameson and Gould (1975)
Gameson and Gould (1975)
Gameson and Gould (1975)
Gameson and Gould (1975)
Gameson and Gould (1975)
(continued)
-------
TABLE 3-30 (continued)
Organism
ro
ro
Study
(cm/cal)
Data Source
Fecal Streptococci
Salmonella typhimurium
3 Laboratory Studies
Min
Max
3 Field Studies
1 Field Study
12 Field Studies, initial rates
Mean
Min
Max
2 Laboratory Studies
0.048
0.123
0.000
0.007
0.091
0.004
0.184
1.48
6.40
Gameson and Gould
Gameson and Gould
(1975)
(1975)
Foxworthy and Kneeling (1969)
Eisenstark (1970)
-------
3.8.3.3 Methods of Measurement
Estimates of the coliform die-off rate, k, may be obtained in a number
of ways in the laboratory or, preferably, in situ. For laboratory estimates,
samples of effluent may be taken along with samples of receiving water.
Then, under controlled conditions of light, temperature, and dilution, the
time rate of die-off may be determined for various combinations of condi-
tions.
In situ, k values can be determined whether the flow regime is well
defined or not, although there are inherent errors involved in each method.
Where there are no flow regime data, or where flows are of a transient
nature, a commonly used method is to add a slug of a conservative tracer
substance (a dye, rare element, or radioisotope) to the steady-state dis-
charge. Then the discharge plume is sampled, dilution is estimated from
concentrations of tracer, and the dilution corrected coliform counts permit
k to be estimated. It should be recognized that this technique may give
misleading results where the dilution of the tracer is due to mixing with
water heavily contaminated with the same discharge. Since the tracer had
been introduced as a slug, there is no way to know how much of the surviving
coliforms originated in the tracer-dosed effluent and how much came from pre-
dosing or post-dosing effluent. However, where the flow regime is suffi-
ciently predictable and stable to assure that dilution occurs essentially
with ambient water, and where coliform levels in the ambient water are
known, this should'not be a problem.
Another method, which is particularly useful where discharge is to a
channel, is as follows. First, a base sampling site is established below
the discharge where the water column is fully mixed normal to the direction
of flow. Then samples are taken at the base site and at several points
downstream. Based upon known velocities and the change in coliform concen-
tration with distance (time), k values may be estimated. Clearly, errors
will be introduced to the extent that there is incomplete lateral mixing of
the stream, nonuniform longitudinal velocities laterally and vertically
across the channel, and unknown inflows causing dilution or introducing
additional coliforms between sampling sites. Also, sampling can be done
so that the same "parcel" of water is sampled, in case the discharge is
not at steady-state. For example, if the first sampling site is one mile
below the base site, and the channel flow has a mean velocity of 2 feet
per second, then the first sampling site should be sampled
5280 ft Y 1 second 1 hr = 7q ,
mile x 2 feet x 3600 seconds -/J nr
or 44 minutes after sampling at the base site. Clearly, however, this does
not account for dispersion, and the 44 minutes is an average figure corre-
sponding to the peak loading. Where possible, dye studies or other
technique should be used to characterize stream dispersion at the sampling
location. Then, by integrating under the curve, total surviving coliforms
230
-------
can be estimated. If, on the other hand, discharge and stream conditions
are clearly at steady-state, sampling times are of no consequence.
Equation (3-66) may be used to estimate k where a slug dose of tracer
has been introduced into the discharge (assuming first order decay).
k = -£n (Ct Fo/Ft CJA (3-66)
where FQ = discharge concentration of tracer, mg/1
F£ = observed concentration of tracer, mg/1
If no tracer is used and conditions exist approximating plug flow, then:
K ~ i
where CQ = concentration of coliforms at the base
sampling site, mg/1
Regardless of the technique used for estimating k, it is important to
concurrently quantify, to the extent possible, those variables which in-
fluence k. For example, light levels should be measured or at least esti-
mated over the period for which k is estimated. If this is not done, and
if the effects of the important parameters are not taken into account in
modeling coliforms, serious errors will result. Table 3-31 shows how
serious such errors can be. The data show T-90 values for coliforms as a
function of incident light. T-90 values are the times required for 90 per-
cent mortality. The associated k values are .058 hr~' in the dark and
.1 hr-1 at midday. It is clear that estimating a single value for k could
result in greater than order-of-magnitude errors.
3.8.4 Summary
Coliforms are of interest as an index of pathogen contamination in
surface waters, and the group of organisms has become one of the most
commonly modeled water quality parameters. Modeling coliforms usually
encompasses the use of a simple first order decay expression to describe
die-off. Since regrowth is generally conceded as unimportant, no growth
terms are normally included in the model.
The die-off rate, k, is a function of a number of variables, the
effects of all of which are not well understood. It now appears that light
(in the visible range) is important as are a number of physicochemical
factors and the presence of heavy metals. Rates of die-off are also sensi-
tive to the salinity of the water, this also affecting the influence of
light on die-off rates.
231
-------
TABLE 3-31. EXPERIMENTAL HOURLY T-90 VALUES (AFTER WALLIS, ET AL., 1977)
ro
CO
ro
Time of Day
0100
0200
0300
0400
0500
0600
0700
0800
T-90 (hours)
40
40
40
40
40
19
8.0
4.6
Time of Day
0900
1000
1100
1200
1300
1400
1500
1600
T-90 (hours)
3.2
2.5
2.3
2.5
2.9
3.3
3.9
4.6
Time of Day
1700
1800
1900
2000
2100
2200
2300
2400
T-90 (hours)
5.3
6.7
8.5
11
14
20
27
34
-------
3.8.5 References
Anderson, D.R., J.A. Dracup, T.J. Fogarty, and R. Willis, 1976. "Water
Quality Modeling of Deep Reservoirs," Journal Water Pollution Control
Federation, Vol. 48, No. 1.
Baca, R.G. and R.C. Arnett, 1976. A Limnological Model for Eutrophic Lakes
and Impoundments. Battelle Pacific Northwest Laboratories.
CDWR (California Department of Water Resources, Sacramento), 1962.
Sacramento River Water Pollution Survey, Appendix C: Public Health Aspects,
Bulletin No. Ill.
Chamber!in, C. and R. Mitchell (in press). "A Decay Model for Enteric
Bacteria in Natural Waters," to appear in Water Pollution Microbiology,
Vol. 2, R. Mitchell, ed.
Chen, C.W., 1970. "Concepts and Utilities of Ecological Model," ASCE,
Journal of the Sanitary Engineering Division, Vol. 96, No. SA5.
Chen, C.W. and G.T. Orlob, 1975. "Ecological Simulation for Aquatic
Environments," in Systems Analysis and Simulation in Ecology, Vol. Ill
(Academic Press).
Chen, C.W. and J. Wells, 1975. Boise River Water Quality-Ecological Model
for Urban Planning Study, Tetra Tech technical report prepared for U.S. Army
Engineering District, Walla Walla, Wash., Idaho Water Resources Board, and
Idaho Dept. of Environmental and Community Services.
Eisenstark, A., 1970. Mutation Res., Vol. 10, No. 1.
Evans, F.L., E.E. Goldreich, S.R. Weibel, and G.G. Robeck, 1968. "Treatment
of Urban Stormwater Runoff," Journal Water Pollution Control Federation,
Vol. 40, No. 5, R162-170.
Foxworthy, J.E. and H.R. Kneeling, 1969. Eddy Diffusion and Bacterial
Reduction in Waste Fields in the Ocean, Univ. Southern Calif., Los Angeles.
Frost, W.H. and H.W. Streeter, 1924. Public Health Bulletin 143, U.S.
Public Health Service, Wash., D.C.
Gameson, A.L.H. and D.J.Gould, 1975. In Proc. Int. Symp. on Discharge of
Sewage from Sea Outfalls, Pergamon Press, London.
Gaudy, A.F., Jr., A. Obayashi, and E.T. Gaudy, 1971. "Control of Growth
Rate by Initial Substrate Concentration at Values Below Maximum Rate,"
Applied Microbiology, Vol. 22, pp 1041-1047.
233
-------
Geldreich, E.E. and B.A. Kenner, 1969. "Concepts in Fecal Streptococci in
Stream Pollution," Journal Water Pollution Control Federation, Vol. 41,
No. 8, R336-352.
Hoskins, J.K., C.C. Ruchhoft, and L.G. Williams, 1927. "A Study of the
Pollution and Natural Purification of the Illinois River. I. Surveys and
Laboratory Studies," Public Health Bulletin No. 171.
Huber, W.C., D.R.F. Harleman, and P.J.Ryan, 1972. "Temperature Prediction
in Stratified Reservoirs," ASCE, Journal of the Hydraulics Division,
Vol. 98, No. HY4.
Hydroscience, Inc., 1971. Simplified Mathematical Modeling of Water Quality,
EPA-Water Programs.
Jannasch, H.W., 1968. "Competitive Elimination of Enterobacteriaceae
from Seawater," Appl. Microb., Vol. 16, pp 1616-1618.
Jones, 6.E., 1964. "Effect of Chelating Agents on the Growth of Escherichia
coli in Seawater," J. Bact.. 87:483-99.
Kittrell, F.W. and O.W. Kochtitzky, Jr., 1947. "Natural Purification
Characteristics of a Shallow Turbulent Stream," Sew. Works J., Vol. 19,
pp 1032-1048.
Kittrell, F.W. and S.A. Furfari, 1963. "Observations of Coliform Bacteria
in Streams," Journal Water Pollution Control Federation, Vol. 35, p 1361.
Klock, J.W., 1971. "Survival of Coliform Bacteria in Wastewater Treatment
Lagoons," Journal Water Pollution Control Federation, Vol. 43, pp 2071-2083.
Lombardo, P.S., 1972. "Mathematical Model of Water Quality in Rivers
and Impoundments," technical report, Hydrocomp, Inc.
Lombardo, P.S., 1973. Critical Review of Currently Available Water Quality
Models, Hydrocomp, Inc., technical report, contract number 14-31-0001-3751.
Mahloch, J.L., 1974. "Comparative Analysis of Modeling Techniques for
Coliform Organisms in Streams," Appl. Microb., Vol. 27, pp 340-345.
Marais, GerritV.R,, 1974. "Faecal Bacterial Kinetics in Stabilization
Ponds," Journal of the Environmental Engineering Division, ASCE,
Vol. lOO(EEI), pp 119-139.
Medveczky, N. and H. Rosenberg, 1970. "The Phosphate Binding Protein of
Escherichia coli," Nature, Vol. 211, pp 158-168.
Mitchell, R. and C. Chamberlin (in press). "Factors Affecting the Survival
of Indicator Organisms in the Aquatic Environment." To appear in Indicators
of Enteric Contamination in Natural Waters, G. Berg, ed.
234
-------
Monod, J., 1942. Recherches sur la Croissance des Cultures Bacteriennes,
Hermann, Paris.
Moser, H., 1958. "Dynamics of Cell Populations," Publications of the
Carnegie Instn., No. 614.
Phelps, E.B., 1944. Stream Sanitation, Wiley, New York, p 209.
Savage, H.P. and N.B. Hanes, 1971. "Toxicity of Seawater to Coliform
Bacteria," Journal of the Mater Pollution Control Federation, Vol. 43,
pp 854-861 .
Schulze, K.L. and R.S. Lipe, 1964. "Relationship Between Substrate
Concentration, Growth Rate, and Respiration Rate of E. coli in Continuous
Culture," Archiv fiir Mikrobiologie, Vol. 48, pp 1-20.
Shehata, I.E. and A.G. Marr, 1971. "Effect of Nutrient Concentration on
the Growth of Escherichia coli," J. Bact., Vol. 107, pp 210-216.
Tetra Tech, Inc., 1975. A Comprehensive Water Quality Ecological Model
for Lake Ontario (final report), prepared for National Oceanic and Atmos-
pheric Administration.
Tetra Tech, Inc., 1976a. Documentation of Water Quality Models for the
Helms Pumped Storage Project, prepared for Pacific Gas and Electric Company.
Tetra Tech, Inc., 1976b. Estuary Water Quality Models, Long Island,
New York-User's Guide, technical report prepared for Nassau Suffolk
Regional Planning Board, Hauppauge, New York 11787.
U.S. Army Corps of Engineers (Hydrologic Engineering Center), 1974. Water
Quality for River-Reservoir Systems (technical report).
Velz, Clarence J., 1970. Applied Stream Sanitation, Wiley Interscience,
New York, pp 242-246.
Wallis, I.G., J.T. Bellair, and G.A. Parr-Smith, 1977. "T-90 Values for
Faecal Coliforms in Seawater," prepared for Symposium on Rate Constants,
Coefficients, and Kinetics Formulations in Surface Water Modeling" held in
Concord, California, February 23-25.
Waser, E., W. Husmann, and G. Blochliger, 1934. Ber. Schweiz. Bot. Ges.,
Vol. 43, p 253.
Wuhrmann, K., 1972. "Stream Purification," Chapter 6 in R. Mitchell (ed.),
Water Pollution Microbiology, Wiley-Interscience, N.Y., 1972.
235
-------
3.9 ALGAE
3.9.1 Introduction
Water column algal populations have historically been among the most
commonly simulated water quality parameters. This has been the case for a
number of reasons. These include:
• Algal metabolism can strongly affect other water quality
parameters such as:
- Dissolved oxygen
- BOD
- Nutrient levels
- PH
- Turbidity
In some cases, particularly with regard to DO, phytoplankton can
cause great diurnal fluctuations.
• Algae are an important component of the primary
producers, particularly in lake ecosystems. They
are an important factor in understanding
eutrophication status and processes.
t Algae affect tastes and odors in potable water supply.
Under some conditions, they may cause problems in
use of water for industrial processes.
• Significant algal blooms can affect use of waters
for recreation.
This section is devoted to the formulations and rate constants used in
simulating algal population dynamics. Although many different approaches
have been taken in the past, a commonly used general approach has developed.
This approach is emphasized in this section, along with information relating
to the algal specific growth rate, the algal endogenous respiration rate,
algal settling, and the Michaelis-Menton half-saturation constants as they
relate to algal population dynamics.
It is very important to recognize that many models, which purport to
simulate "algae" actually seek to predict some composite behavior. Invari-
ably, although there may be relatively few dominant algal species present in
the water column at a point in time, this dominance, and indeed the entire
plankton species profile is capricious. Accordingly, not only is it a gross
approximation to simulate the algal assemblage as a single alga, but perhaps
236
-------
even more important is the error inherently introduced by using constant
rates in long time-period simulations. The various rates are highly species
and temperature dependent. Accordingly, while selected rate constants may
be satisfactory over some short period of time, they may introduce serious
simulation errors where prototype conditions are changing substantially.
It should also be noted that no distinction is made in many models as
to the physical characteristics of the "alga." In general, relationships
to be discussed are for phytoplanktonic forms.
The basic approach to simulating phytoplankton in many popular models
is founded upon the expression:
(3-68)
where Gp = specific growth rate, day"
Dp = specific death rate, day"
P = phytoplankton concentration, mg/1
The fundamental equation is usually expanded to account for the effects of
growth-limiting conditions, respiration, predation, and settling. In order
to incorporate the effects of growth limiting conditions, the maximum spe-
cific growth rate is modified. Commonly this is done in a fashion analogous
to:
Gp = MKp(T) (3-69)
where M = Monod type half-saturation expression
for the growth limitation terms for
nutrients, light, and temperature,
unitless.
= specific growth rate, day
-1
Kp(T) = Temperature-dependent maximum-specific growth rate, day
The specific death rate (Dp) may be separated into a respiration term, a
predation term, and a settling term:
Dp = Rp + Sp + Fp (3-70)
where Rp = respiration rate, day"
Sp = settling rate, day"
FD = predation rate, day
237
-------
The equation becomes:
- RP - sp - FPP (3-71)
3.9.2 Phytoplankton Specific Growth Rate (Gp)
3.9.2.1 Factors Affecting Gp
As discussed above in general terms, there are several factors which
can limit or at least mediate the rate of phytoplankton growth. These are:
• Availability of nutrients • Temperature
t Light levels • Availability of silica (diatoms)
It is very clear that some factor or factors must provide a limitation
on the rate of growth and replication of algal cells. Obviously, in theory,
environmental factors may be optimal and may not provide such limitation.
In this case the biochemistry of the organism determines the rate of growth
and replication, and the rate is the maximum or saturated growth rate, Kp.
In natural systems, however, non-optimal levels of one or more factors or
substances required for growth are likely to cause the rate to be less than
V
A large body of research and literature has been developed over recent
years regarding growth limitation in natural water bodies. There is some
evidence that more than half of the lakes in the U.S. east of the Rockies
are phosphorus limited (Gakstatter, e_t a]_., 1975)*. Studies performed by
Maloney, et_ al_. (1972) on waters from nine Oregon lakes (in-vitro studies)
have shown that in four lake samples, phosphorus addition alone greatly
stimulated algal growth. The addition of nitrogen alone stimulated algal
growth slightly in two lakes. In one lake, which was highly oligotrophic
(severe deficiency of one or more nutrients), the addition of nitrogen,
phosphorus, and carbon had no effect on algal growth rates.
Powers, et_ al_. (1972) reported in situ studies on natural algal popula-
tions in Minnesota and Oregon. In Shagawa Lake in Minnesota, additions of
phosphorus, of nitrogen, and of both nitrogen and phosphorus elicited in-
creases in algal growth rates, thus suggesting that the concept of a single
growth limiting substance must be expanded to accommodate limitation by a
combination of substances or factors. In Burntside River in Minnesota,
Powers, e_t al_. (1972) found that algal growth was stimulated only when
According to Lorenzen (pers. comm.), the concept of limitation of growth
is often misinterpreted, and results of studies commonly reflect laboratory
/->i'iin,-l-i-l--ir\v-»<~ v» 3 •(- L-i i"» vi 4-l-iTiM "I^L-a ^Tn^T rl\/nam-i/"«c-
conditions rather than lake algal dynamics.
238
-------
phosphorus and nitrogen were added in combination. Algal growth in moder-
ately productive Triangle Lake was stimulated by the addition of phosphorus
and further stimulated by the addition of phosphorus plus nitrogen.
Schelske and Stoermer (1972) have found significant correlations
between particulate phosphorus and the rate of carbon fixation in Lake
Michigan. Based upon enrichment experiments conducted in 1970, phosphorus
was found to be limiting, while nitrogen was not limiting.
Welch, ejt al_. (1973) have discussed the trophic status of Lake
Washington and Lake Sammamish in the State of Washington. Results of
nutrient diversion from the lakes have shown that while diversion has led
to a decrease in productivity in Lake Washington, and suggest that this
lake is phosphorus limited, diversion of phosphorus has not substantially
affected productivity in Lake Sammamish. Welch, et al_. suggest that the
phosphorus change in Lake Sammamish has occurred on the asymptotic part of
the response curve. Although this might appear tantamount to saying that
Lake Sammamish is not phosphorus limited, Welch, e_t a]_. also suggest that
the availability of phosphorus is mediated by iron released from sediments
below an anaerobic hypolimnion. The iron reacts with phosphorus, making it
unavailable as a nutrient, and deposits the phosphorus in the sediment. If
the level of phosphorus were increased significantly, the authors argue,
the production rate would also increase. This means that the lake is
phosphorus limited to an extent, but the effect on production of short-
term, small scale changes in phosphorus is masked by the dynamic behavior
of sediment phosphorus exchange.
Much of the literature has emphasized phosphorus and nitrogen studies,
and research done in recent years has suggested that phosphorus is more
often limiting than nitrogen. Very much less often, studies are conducted
to determine the extent to which carbon is limiting. Kerr, e_t al_. (1972),
for example, investigated the role of nitrogen, phosphorus, and carbon in
Shriner's Pond, a small fishpond in Georgia. Their findings suggest that
while nitrogen, phosphorus and potassium additions to the pond caused in-
creased algal production, the carbon content also influenced the rate of
production. Powers, et al_. (1972) found that carbon increased productivity
in one instance when added with nitrogen and phosphorus to Burntside River.
In another case, moderately productive Triangle Lake exhibited increased
productivity through the addition of carbon. Overall, it appears that while
carbon may not be as important as either phosphorus or nitrogen, it probably
plays a significant role in algal growth regulation in a limited number of
surface waters.
In addition to the macronutrients (i.e., phosphorus, nitrogen,
potassium, carbon), micronutrients (i.e., essential metals, sulfur) may be
limiting under certain conditions, particularly in oligotrophy. Goldman
(1972), for example, found that significant trace element deficiencies
existed in 82 percent of the 28 oligotrophic lakes he studied. In ten
lakes in New Zealand, the addition of cobalt, iron, zinc, manganese and
molybdenum elicited phytoplanktonic responses ranging from maxima of less
than a ten percent increase in productivity to one case of more than an 80
239
-------
percent increase. Goldman also obtained positive responses from trace
element additions to lakes in Alaska (molybdenum, cobalt, manganese, zinc
individually), Africa (iron and various combinations of nitrogen, sulfur,
potassium, and phosphorus in eutrophic Lake George) and California (Lake
Tahoe and others, addition of iron, zinc, and manganese individually).
Algal growth limitation by non-optimal light levels has been extensively
investigated both in situ and in vitro. Light provides the energy in the
carbon fixation process which uses carbon dioxide and water in the green
plant cell biosynthesis of glucose, glucose polymers (cellulose, starches,
various sugars), and other energy storing substances along with some amino-
acids. The energy stored in the photosynthesis process, of course, provides
the energy for cellular active transport, respiration, biosynthesis, and
other energy consuming processes vital to the algal cell.
In view of the role of light in cellular energetics and biochemical
synthesis, substantial changes in light levels must have a very major effect
on algal cell function in general and growth rates in particular. In a
study by Stepanek (1969), a close statistical relationship between the
duration of sunshine and the numbers of nannoplankton produced in Sedlice
Reservoir was found (r = .929). Stepanek also examined statistical rela-
tionships between numerical productivity and solar azimuth and blue-violet,
and total visible components of incident solar radiation. Generally, the
correlation coefficients were reasonably high with r^ > .64 in all but the
correlation between total visible radiation and numerical productivity
(r2 - .42).
Jorgensen and Steeman-Nielsen (1969) have discussed rates of photo-
synthesis in Skeletonema costatum (a diatom) as a function of light level
and adaptation period.According to their data, at 8°, 14°, and 20°C, the
photosynthetic rate (carbon fixation per cell per hour) is essentially
constant in the range 15K to 30K lux. The rate is greatest at 20 degrees.
In the range from about 8 to 15 degrees, the rate of photosynthesis in-
creases, but at a decreasing rate. In the range 0 degrees to about 8
degrees, the photosynthetic rate increases linearly, with a slope of about
2.3 x 10~7 i_igC/ cell -hour-lux. The daily growth rate at 10K lux was 3.7
for cells adapted to 8°C compared with 5.4 for 20°C adapted cells.
Jorgensen and Steeman-Nielsen also suggest that S_. costatum cells grown
with a short light period synthesize and maintain a higher level of photo-
synthetic pigments than cells grown with a short dark period. The short
light period grown cells also show by far the highest photosynthetic rate
(by about a factor of 1.5 in the range 10K to 30K lux).
Thomas (1966) has studied the effects of illuminance on three species
of oceanic phytoplankton. Figure 3-10 shows the effects of light levels
on two forms of Gymnodinium. Figure 3-11 shows similar curves for
Nannochloris and Chaetoceros species. Table 3-32 shows optimal growth
illumination using various measures as criteria (Brown and Richardson,
1968). Figures 3-10 and 3-11 show that up to about 400 foot candles,
the change in phytoplankton growth rate is approximately linear. Between
240
-------
CO '-5
1 Z>
UJ O
1- X
2^ L0
*% r
QL CO
O
GYMNODINIUM 581
x
X .X
X t' ' '
x/
/
/ 1 1 1 1 1 1 1 1
500 1000 1500 2000
ILLUMINANCE ft-c
CO 1-5
o:
I 13
UJ O
I- X
O " 0.5
o: co
GYMNODINIUM 582
x x
500 1000 1500 2000
ILLUMINANCE ft-c
Figure 3-10. Effect of light levels on growth rates
of Gymnodinium (Thomas, 1966).
241
-------
4.0
CO
, §
£0
or sf
3.0
2.0
00
Si
Q
> 1.0
NANNOCHLORIS 582
500 1000 1500
ILLUMINANCE ft-c
CO
o:
i ^
CO
1-0
o
05
U-D
CHAETOCEROS 581
250 500 750
ILLUMINANCE ft-c
Figure 3-11.
Effects of light levels on growth rate
of Nannochloris and Chaetoceros
(Thomas, 1966).
242
-------
TABLE 3-32. OPTIMAL GROWTH ILLUMINATION (ALL VALUES IN FT-C)
FOR VARIOUS MEASUREMENTS (BROWN AND RICHARDSON, 1968)
oo
Cyanidium caldarum
GU>_epcajDs_a_ ajjjicola
Phormidium luridum
Phormidium persicinum
Porphyridium aerugineum
Porphyridium cruentum
-Astasia long_a
Amphidinium sp.
Cryptomonas ovata
Nitzschia closterium
Ochromonas danica
Sphacelari_a sp.
Chlorel la pyrenoidosa
Chlorococcum wimmeri
Euglena graci'1 is
Tribonema aequale
Growth
400
400
500
200
400
800
200-1000
600
>1000
400->1100
700
200-600
>2750
>1000
>1100
800
DT hr*
77
29
18
24
37
21
21
48
22
48
12
30 days
13
67
26
18
Ps
<50
>1150
--50
400
<50
50
-
>1200
250
>1100
>1350
-50
400-1300
>1000
400
>1300
Res
700
150
>600
200
-1200
>1200
500
>1100
>1350
300
800
>1000
>1100
>1300
Q-Hili
<50
100-400
<75->1300
150
400
-
<50
<25
<50
400
<50
400
400
100
100
Pigmentation
<50
<50
500
200
<50
150
-
90
250
<50
400
<50
400
Chloro.^50
Asta.>1000
175
no max.
Cell
Max.
>1000
800
-
600
400
-50
<50
<50
<25
250
400
>1100
_
Vol uine
% Max. Inc.
>100
150
-
210
190
30
59
120
47
1800
110
80
_
*0oubling time (DT) refers to the time required for the population of 1-liter cultures to increase 100%. Lower DT periods are readily
obtained for short periods with small volumes of dilute cell suspensions. Doubling time is measured at the initial point of greatest growth.
P = photosynthetic rate, Res = respiration, Q-Hill - quinone-hill reaction.
-------
about 400 and 700 foot candles, growth increases at a decreasing rate with
increased illuminance, and above about 700 foot candles, increased light
has little or no effect on growth rate. It should be noted, however, that
illuminance levels can be substantially higher than those represented in
the figures. At higher levels photoinhibition may occur. In impoundment
depth profiles of chlorophyll-a_ concentrations, in conjunction with
decreased growth rates caused by low light levels at greater depth, this
phenomenon gives rise to characteristic subsurface bell-shaped curves.
The literature is replete with studies of photosynthetic rates as a
function of depth below the water surface. Such studies show the effects
of stratification, light attenuation, and/or temperature. Temperature
effects on algal growth rates are essentially due to the effect of tempera-
ture on chemical reaction rates. Since biochemical reactions are enzyme
catalyzed, effects of temperature on algal growth rates reflect the effects
of temperature on enzyme molecule characteristics and integrity. According
to data from West and Todd (1966), given temperature correction
Equation (3-72),
6
(T-Tr)
(3-72)
where T = ambient temperature, °C
Tr = reference temperature, °C
Kj = enzyme catalyzed reaction rate at temperature T, day"
Ky = enzyme catalyzed reaction rate at the reference
r temperature, T , day"'
0 = characteristic constant
8 can range from 1.01 to 1.18, for T and Tr within the normal enzyme tem-
perature range. A temperature increase of 15°C, which commonly about
doubles reaction rates in uncatalyzed organic reactions, can result in a
1.2 to 12-fold increase in enzymatic reaction rate. Accordingly, algal
growth rates are quite sensitive to temperature.
Smayda (1969a) studied the effect of temperature (and light levels and
salinity) on a marine diatom, Detonula confervacea. Figure 3-12 shows some
of Smayda's results.
244
-------
ro
-p.
en
_ 1.6-
-' 1.4-
-------
3.9.2.2 Applications and Modeling Approaches
3.9.2.2.1 General Discussion of Gp and Use of the Michaelis-Menton
Formulation
Several approaches are commonly used to describe the specific
growth rate Gp (or y). This rate is known to be a function of temperature,
light, and nutrient concentration as discussed above. Numerous investigators
(see, for example, Baca and Arnett, 1976; U.S. Army Corps of Engineers,
1974; Canale, e_t al_., 1976; and O'Connor, ert aj_., 1975) have used a specific
growth rate formulation of the following general form:
(3-73)
where K (T)
= saturated growth rate, a function of
temperature, day"'
r(I ,K ) = light reduction term due to non-optimal
incident light, a function of the saturated
light intensity, Is, and the extinction
coefficient, K > unitless
i = subscript representing each of the n
specific nutrients with potential to
limit growth (primarily nitrogen and
phosphorus, and sometimes carbon and
silicate)
Km - = the ith Michaelis-Menton or half-saturation
' constant, mg/1
Ci
= the ith nutrient concentration, mg/1
Tetra Tech (1975, 1976) and the U.S. Army Corps of Engineers (1974) have
used the following growth expression:
where K,
K, 0(T-2°)
" LI
L2+LI
r P04 i
[P2+P04_
- co2 -
c2+co2
N
_ N2+N _
= maximum specific growth rate at 20°C, days"
= temperature coefficient, ranging from
1.02-1.06
= half-saturation constant for algae
utilizing phosphorus, mg/1
(3-74)
246
-------
N2 = half-saturation constant for algae
utilizing nitrogen, rng/1
£>2 = half-saturation constant for algae
utilizing carbon, mg/1
L£ = half-saturation constant for algae
utilizing light energy, kcal/m^-sec
PO^jNjCOp = concentrations of phosphorus, nitrogen,
and carbon dioxide, respectively, mg/1
(or consistent units with ?£, N2, and 02)
LI = incident light level in kcal/m^-sec
The variable LI refers to the available light intensity, which is
known to vary as a function of both depth and time. This parameter is
discussed in detail elsewhere in this report (Section 2.3.5).
Chen and Orlob (1975) and Chen and Wells (1975) used this same formu-
lation for the specific growth rate in model applications to Lake Washington,
San Francisco Bay Estuary, and Boise River, Idaho.
The Michaelis-Menton half-saturation constants are commonly used to
adjust the growth rate of phytoplankton (or other forms) to suboptimal
levels of potentially growth limiting factors. Such factors include
nutrients and light. Each constant is actually the level at which the
particular factor limits growth to half the maximal or "saturated" rate.
Thomann, Di Toro, and O'Connor (1974) formulated the phytoplankton
growth expression in a similar fashion as follows:
(3-75)
where G = growth rate averaged over depth
Pj
KT = maximum growth rate, days" °C~
T = temperature, °C
f = photoperiod; ranging from 0.3-0.7
(fraction of a day, unit!ess)
K = extinction coefficient; ranging
from 0.1-3.0 m'1
247
-------
= depth, m
al
% - ip
Is = saturation light intensity, foot candles
Ia = incident light, foot candles
N = nitrogen concentration, mg-N/1
D = phosphorus concentration, mg-P/1
Kn,Kp = half-saturation constant for total inorganic
nitrogen and total phosphorus, respectively, mg/1
Thomann, Di Toro, and O'Connor (1975) incorporated a nonlinear temperature
maximum growth rate where the temperature correction is made as in
Equation (3-72). Here, 6 was set equal to 1.066 with a reference tempera-
ture of 0°C.
Baca and Arnett (1976) have applied an algal growth rate related to the
concentration of the principal nutrients, nitrogen and phosphorus, using a
modified Michaelis-Menton expression. The nutrient term formulation is
given as:
C_ D,
MIN
KP+D1
(3-76)
where C = effective nitrogen concentration, mg/1
K ,K = half-saturation constants for nitrogen and
p phosphorus, mg/1
D-i - soluble phosphorus concentration, mg/1
MIN = the minimum value function, MIN (A-i ,A?,'-',A )=
Ai 1 A-| ,A2, • • •,A where i = 1, • • • n n
248
-------
The effective nitrogen concentration, C , is given by:
Cn =
p.C + (1 - p)C, C and
1
Cl + C3' Cl or C3 <£
(3-77)
where p = preference factor
e = critical nitrogen level
C-| ,C3 = ammonia and nitrate concentration, mg-N/1
Baca and Arnett (1976) also included a multiplicative term to account
for the effect of toxic substances as follows:
where G-, = Michael is-Menton growth limitation
term due to toxic substance
T = concentration of toxic substance, mg/1
Kt = toxicity constant, mg/1
Baca and Arnett (1976) included this toxic substances term in the
specific growth expression because they felt that toxic substances generally
inhibit growth rates rather than increase respiration (death).
Canale, ejt al_. (1976) is yet another investigator who has employed
this type of specific growth formulation. Canale 's formulation includes a
maximum growth term which is temperature-dependent, a light reduction
factor, and a modified Michael is-Menton expression for nutrient reduction
effects. Their nutrient reduction term is formulated much the same way as
Baca and Arnett's (1976) expression and, in addition, he includes a multi-
plicative term for silicate growth limitation (see Sections 2.5.1-2.5.3).
In addition to these more common approaches, other investigators have
suggested quite different formulations for specific algal growth, Gp-
Bierman's (1976) complex model of phytoplankton-growth kinetics includes
four phytoplankton groups: 1) diatoms, 2) greens, 3) non-nitrogen fixing
blue-greens, and 4) nitrogen fixing blue-greens. Table 3-33 is a qualita-
tive listing of the operational differences among the four phytoplankton
types, as given by Bierman (1976).
249
-------
TABLE 3-33. QUALITATIVE DIFFERENCES
AMONG PHYTOPLANKTON TYPES MODELED BY BIERMAN (1976)
P-HYTOPLANKTON TYPE
Characteristic
Property
Nutrient Requirements
Relative Growth Rates
Diatoms
P, N, Si
High
Greens
P, N
High
Bl ue-Green
Blue-Green Non N-
(N-Fixing) Fixing)
P, N P
Low Low
(optimum at 25°C)
Phosphorus Uptake Affinity Low
Sinking Rate High
Grazing Pressure High
Low
High
High
High
Low
None
High
Low
None
A unique feature of Bierman's model is that cell growth is considered
to be a two-step process involving separate nutrient uptake and cell syn-
thesis mechanisms. Motivation for this variable stoichiometry approach is
that an increasingly large body of experimental evidence indicates that
the mechanisms of nutrient uptake and cell growth are actually quite
distinct (Bierman, 1976). The model includes carrier mediated transport
of phosphorus and nitrogen using a reaction-diffusion mechanism, and
possible intermediate storage in excess of the immediate metabolic needs
of the cell. Specific cell growth rates are assumed to be directly depend-
ent on the intracellular levels of these nutrients, in contrast to the
classical Michael is-Menton approach which relates these rates directly to
extracellular dissolved phosphorus.
Bierman (1976) simulated the specific growth rate of phytoplankton,
G, as the minimum value of the following three functions:
G.. = MIN <
"Gmax * f(T) * f(I) ' [l-exp(-0.693(P/PQ-l))]
f(T) ' f(I) • [(N-No)/(KNCELL + N-NQ)]
f(T) ' f(I) - [sCM/(KSCM + SCM)]
max
max
(3-79)
(3-80)
(3-81)
where KNCELL = intracellular half-saturation constant
nitrogen-dependent growth, moles - N/l
KSCM = intracellular half-saturation constant
silicon-dependent growth, moles - Si/1
for
for
250
-------
P = moles phosphorus per phytoplankton cell
P = minimum stoichiometric level of phosphorus
per phytoplankton cell (mol/cell)
N = moles nitrogen per phytoplankton cell
N = minimum stoichiometric level of
nitrogen per phytoplankton cell (mol/cell)
SCM = silicon concentration in solution (mol/1)
f(T) = e(T-2°) Where 6 = 1.07 for diatoms,
1.08 for green algae, and 1.1 for
blue-green algae
T = temperature °C, given by
T = TMAX
value
0 5-0 5 sin pn • TIME + «j>-[ j
u.a u.o sin
of sin argument in radians
•I •
TMAX = maximal water temperature during
summer, °C
TIME = day of year, 30 day months
$ = -1890. Value gives sin (arg) = 0 for
Nov. 1. Nov. 1 = day 301.
f(I) = [l/(ke • DEPTH)] • [e'a1-e ~a°]
al = (la/Is) e -(ke ' DEPTH), unitless
ao = la/Is, unitless
ke = (1.9/secchi depth) + .17 • TCROP *
.633 + .17 TCROP, nT1
la = 2000 ft-C, surface incident light level
Is = saturation light level, ft-C
DEPTH = water column depth, m
TCROP = total phytoplankton biomass,
mg dry wt/1
251
-------
Since the traditional Michael is-Menton approach to nutrient uptake
kinetics does not include a feedback mechanism, Michael is-Menton kinetics
are actually a special case of this two part uptake kinetic theory in which
the cell's nutritional state is assumed to be constant.
A major problem in attempting to simulate a complex chemical-bio-
logical process is that the models usually contain coefficients for
which direct measurements do not exist. It is possible that more than
one set of model coefficients could produce an acceptable "fit." In the
transition from single-class to multi-class models, this problem becomes
particularly acute because it is no longer sufficient to ascertain a
range of literature values for a given coefficient. Multi -class models
necessitate the definition of class distinctions within this range.
Given such circumstances, Bierman (1976) concludes that many of the coeffi-
cients in multi -class models simply must be estimated.
The procedure used in Hydrocomp's (Lombardo, 1972) model is to treat
the algal uptake of each nutrient separately and then determine which
uptake rate causes the smallest phytoplankton growth rate during each
time period. Hydrocomp's method assumes a nutrient is limiting for each
time period but that the identity of the limiting nutrient may change over
successive time periods. This concept, also considered by Bierman (1976)
as described previously, has been considered by Orlob (1974) to be a
plausible alternative to the single multiplicative Michael is-Menton
formulation.
Hydrocomp's (Lombardo, 1972) expression for phosphate uptake is
formulated as:
/ P04 NOs
= VMAXP
VMAXF
_ _
pp \ CMMP + P04 / CMMN + NO,
where G = P04 uptake rate, hr~
VMAXP = maximal P04 uptake rate, hr"1 ;
typically 0.3
CMMP = Michael is constant for phosphorus;
typically 0.0303 mg/1
CMMN = Michael is constant for nitrogen;
typically 0.0284 mg/1
P0« = PO* concentration, mg-P/1
N03 = N03 concentration, mg-N/1
252
-------
The P04 uptake rate, Gpp, may presumably be converted to algal bio-
mass production using chemical equivalence information. Hydrocomp (Lombardo,
1972) assumes a constant chlorophyll-a_ to phosphorus ratio of 0.6.
The expression used for nitrogen limited growth is given as:
Gpn=VMAXN (3~83)
where G = N07 uptake rate, hr
PI I O
VMAXN = maximal NOs uptake rate, hr" ;
typically 0.7/hr
The following expression is proposed for nitrate uptake under light-
limited growth conditions (Lombardo, 1972):
V = VMAXL (3-84)
where G . = nitrate uptake rate, hr~
CLT = Michael is constant; typically 0.033
langleys/min
ZI = light intensity, langleys/min
VMAXL = maximal NOs uptake rate under light-
limiting conditions, hr~^;
typically 0.3/hr
3.9.2.2.2 Temperature and the Algal Specific Growth Rate
Two notable differences in the growth expression used by Thomann,
Di Toro, and O'Connor (1974) and the one used by Chen and Orlob (1975) are
the terms for light effects and temperature effects. Thomann, et al.,
assume a linear relationship between growth rate and temperature while
Chen and Orlob apply the temperature correction expression commonly used
to adjust temperature-dependent rates:
253
-------
where T = temperature, °C
6 = characteristic temperature correction constant
Baca and Arnett (1976) incorporate a temperature dependent maximum
specific growth rate term similar to that of Tetra Tech (1975).
Assuming neither light nor nutrients are in short supply, Lombardo
(1972) defines the growth rate to be directly dependent on temperature.
The following expression, as suggested by McCombie (Lombardo, 1972), is used:
G = 0.006T - 0.035 for: 28>T
-1
(3-86)
where G = specific growth rate, hr
T = water temperature, °C
Lehman, et_ a]_. (1975) report another approach to temperature correc-
tion of Gn. In their model, which is useful for lakes having ice cover in
winter, they assume temperature to be constant over a 24 hour period. For
each day during ice-free months, temperature is computed as:
min ' max
[l-Cos(2,(D-Dm )/(De-Drtl))]i DmT
opt
(3-88)
(3-89)
254
-------
where exp(x) = the exponential function, ex
T . = optimal temperature, °C
T££ = lower limit of T, °C
TU£ = upper limit of T, °C
Lehman, ejb al_. (1975) suggest that this is an inexact approach to the
Arrhenius equation of enzyme activity that Johnson, et^ aj_. (1954) suggested
may represent the exponential growth phase of microorganism populations.
Lassiter (1975) notes that the approach of Equation (3-85) assumes a
monotonically increasing exponential response of the maximal temperature-
dependent growth rate to temperature, although this is a reasonable approxi-
mation only over a part of the normal temperature range that an algal cell
will tolerate. Actually, growth rates have some time-variable optimal
temperature remote from which the growth rate drops off very substantially.
Lassiter cites a complex set of formulations reported by Bloomfield, et a!.
(1973) to compute k, a general biotic growth rate (I/day):
(3-90)
/ -> r , ]/2i2\
X = (IT [l + (1 + 40/W) J j/400 (3-91)
(3-92)
(3-93)
where T. = upper limiting temperature at which the
growth rate is zero, °C
T = optimum temperature for growth, °C
T = ambient temperature, °C
K, = reference rate constant, day"
T-, = reference temperature, °C
Qin = ratio of rate constants at two temperatures,
1 T1 and TZ
255
-------
T2 = temperature at which K2 is to be determined, °C
= computed rate constant at T2, day
This model, as noted by Lassiter, is satisfactory at near optimum
temperatures, but is not very satisfactory remote from optimum T. He cites
a formulation by Johnson, e_t al_. (1954) which is based upon thermodynamic
and physical chemistry precepts and is capable of simulating algal response
to temperature over a wide range:
_
(AS/R - AH/RT)
(3-94)
U W;
where C = a scaling constant
= the heat of activation for transition
state intermediates
AH = the heat of activation for the reaction
for which K is the rate coefficient
AS = the entropy of activation of the reaction
R = the gas law constant
As Lassiter points out, the thermodynamic constants in Equation (3-94)
are difficult to estimate. Lassiter and Kearns (1973) developed a model
accounting for the following (quoting Lassiter, 1975):
• "Rate coefficients are always positive, approaching
zero asymptotically with respect to values of an
external stimulus; thus a change in the rate in
response to an external stimulus must be proportional
to the magnitude of the rate itself."
t "A biological rate constant reaches maximum at an
optimum temperature; therefore, the rate of change
of the constant is hypothesized to be proportional
to the deviation of temperature from optimum."
t "The rate, which diminishes when the optimum tempera-
ture is exceeded (its rate of change becomes
negative), decreases more rapidly as the upper limit
is approached. An inverse relationship between the
rate of change and deviation from the maximum
temperature (T£) is hypothesized."
256
-------
Based upon these three assumptions, the formulation is given as:
dk _ ak(Tm
Integrating (3-95) with k = km when T =
k =
(VTm)
(3-96)
T>T£
where a = a scaling constant
k = maximal growth rate, day'
Lassiter notes that the model (Equations (3-95) and (3-96)) has been
applied to a number of biological processes including the temperature-
dependence of growth rates of Chlorella species, egg production of several
wood fungus species, growth rates of four species of aquatic snails, and
luminescence from certain luciferin/luciferase reactions. In all cases,
using the method of least squares, the fit of the model to the data over a
wide range of temperatures has been good.
3.9.2.2.3 Light Levels and the Algal Specific Growth Rate
In addition to the use of a Michaelis-Menton formulation for sub-
optimal light levels (discussed in Section 3.9.2.2.1), a number of other
formulations to compute the algal specific growth rate as a function of
light have been used. Some of the formulations estimate photosynthetic
rate, but this can be easily and fairly accurately related to growth rate.
Baca and Arnett (1976) have used a light reduction term developed by
Vollenwider as a direct multiplier of y (the maximal growth rate) as
follows:
AI 1
+ (AI)2 (1 + (al)2)n (3-97)
257
-------
where A = low light constant; typically 0.00054
a = photo inhibit!on factor, 1ux~
I = light intensity, lux
n = 1 (from studies by Battelle)
Di Toro, e_t al_. (1971) report that by averaging the expression (over
the euphotic depth and over 24 hours):
F [i(Zit)3 = Ip^j eL S J (3-98)
s
the following expression may be obtained:
, F_
u = u
(3-99)
where y = growth rate
I = radiation at depth z at time t
I = radiation intensity at which the
maximum specific growth rate is
achieved (saturation intensity)
z = depth
t = time
F[I(z,t)] = relative photosynthesis
F = maximum fractional reduction in
daily specific growth rate over
euphotic depth
R(t) = total daily radiation
K = total daily radiation at which half of
r Fm.v is attained
lllaA
The euphotic depth is taken to be the depth to which one percent of the
surface radiation penetrates.
258
-------
Lehman, ejt aj_. (1975) cite a function for photosynthesis reported by
Steele:
P(I) = P
x ' max
where Pmax = maximal photosynthetic rate, any
productivity units
P(I) = the photosynthetic rate at light
intensity I, units consistent with PmaN,
max
I = the ambient light intensity,
cal cm"2 min"'
I . = light intensity for saturated
p photosynthetic rate, cal cm"^ min"1
Lehman, et_ al_. (1975) further modify the expression to account for end-
product inhibition:
where C = cellular maximum carbon store
capacity, mol/cell
C = cellular carbon storage inactive
organisms, mol/cell
C = cellular growth limiting carbon
content, mol/cell
Lassiter (1975) cites a model used by Steele:
P - a pm I e1"31 (3-102)
where a = a scaling constant
p = maximum photosynthetic rate (same
variable as Pmax in Equation (3-100)),
productivity units
2 1
I = ambient light level, cal cm min
p = photosynthetic rate at light level I,
productivity units
259
-------
The derivation of the equation was not described by Steele, but Lassiter
points out that the formulation fits several sets of photosynthesis-light
curves, and suggests a derivation consistent with the following assumptions
(quoting Lassiter (1975)):
"1. The rate of change of p with a unit change in I
depends both on the value of p and on the deviation
of I from the optimum I Um);
2. The rate of change of p with a unit change in I is
inversely proportional to I, i.e., the photo-
inhibition effect."
The formulation is:
d£- kP ( '''"»; (3-103)
dl I
Integrating Equation (3-103),
/ , \k k[ HI/L.) I
(3-104)
According to Lassiter, Steele had set k = 1 and I/I = a to get Equation
(3-102). m
Substituting the exponential light attenuation equation,
-1.19nTz
Iz = I0 e (3-105)
where I = light intensity at depth z,
cal cm~2 min'1
I = incident light (at the surface),
r»a 1 rm~t- min~l
cal
n-r = overall extinction rate per meter
of depth, accounting for water,
color, turbidity
z = depth, m
into Equation (3-102), and integrating over depth (z ) of the euphotic zone,
260
-------
P = Pm^ ! 1Q. . "6X l (3-106)
where p = the average photosynthetic rate over depth
Substituting for pm an equation attributed to Bannister (Lassiter, 1975),
Pm = 12 *m Im hc C/e (3-107
where 12 = the atomic weight of carbon
4>m = maximum quantum yield
C = chlorophyll-a_ concentration
h = the rate of light absorption by chlorophyll-a
\+ —
and substituting
b = C/B
where B = biomass
b = conversion constant, chlorophyll per
unit biomass
into Equation (3-107) and dividing through by B, equation (3-106) becomes:
12 cL Im h_ b ,
(3-108)
where TT = the rate of carbon fixation per unit
biomass
Then the specific growth rate subject only to light limitation, y" is given
by:
£' = ^a ^ (3-109)
261
-------
where na = the stoichiometric coefficient relating
total biomass to carbon (fixation)
3.9.2.2.4 Michaelis-Menton Constants - Deficiencies of the Approach
One of the commonly used formulations for correcting y (saturated
growth rate) for limiting nutrients and suboptimal light is the Michaelis-
Menton expression (see Section 3.9.2.2.1). The theory is based upon the
work of Michaelis and Menton in 1913 and of Monod (see Sykes, 1973), and
was originally applied to enzyme kinetics. Monod applied it to continuously
stirred chemostat cultures.
In order to relate the ambient concentration of some limiting nutrient,
S, to the growth rate, y, Monod proposed a relationship of the form
(3-110)
l\ T O
where y = specific growth rate, day~
-1
y = maximum specific growth rate, day
K = a constant, same units as S
This is the form widely adopted and used in many models which simulate
algal population dynamics. Equations (3-73), (3-74), (3-75), and (3-78)
all incorporate this formulation.
The constant K in Equation (3-110) is called the Michaelis-Menton
half-saturation constant. The form of Equation (3-110) is hyperbolic, and
is as shown in Figure 3-13, which represents the responses of three differ-
ent algae to changes in concentration of a hypothetical factor, S. The
plot also shows the significance of K, the half-saturation constant, and
order of growth kinetics.
The Michaelis-Menton formulation has several problems when used as in
Equation (3-74). First, the application implicitly assumes that the effect
of several suboptimal factors is simply multiplicative, although there is
certainly no compelling reason to assume such a relationship. Equation
(3-76) goes to the other extreme, assuming that only one nutrient can be
limiting. This is known to be untrue. Furthermore, none of the expres-
sions presented earlier account for luxury uptake. Luxury uptake occurs
under circumstances where there is an excess of a particular nutrient. The
nutrient is then stored and used when there is a deficiency of that nutrient,
This phenomenon is likely to result in the Michaelis-Menton formulation
overestimating the impact of short-term, transient nutrient deficiencies,
and a lag in long-term deficiency impacts being manifested physically in
the prototype.
262
-------
VMAX3
Alga3
Zero'th Order Kinetics Region
— First Order Kinetics Region
VMAX represents the maximum growth rate (G) for each algal species.
KS is the substrate concentration (S) at which the growth rate is
half maximal (V). Within the region labeled "first order kinetics,
the growth rate is a function of S and might be represented as
dC/dt kS while in the zero'th order kinetics region, the growth
rate is independent of substrate concentration. Here, other
factors 1 imit G.
Figure 3-13.
Growth kinetics of three hypothetical algal species
as a function of substrate concentration.
In a discussion by Mar (1976), the very use of the Monod theory is
brought into question. Mar cites observed variations in values for the
half-saturation constants, suggesting that the variability may be due more
to an improper formulation in the Monod theory than to true variability of
algal nutrient requirements. If the change in biomass per unit time is
given as:
dB_
dt
B,
(3-111)
where B = biomass concentration
K = Michaelis-Menton half-saturation constant
S = substrate concentration
263
-------
Mar points out that under conditions of large S, greater biomass concen-
trations result in greater productivity predictions (reasonable). How-
ever, where S becomes very small, Equation (3-111) approaches:
which states that an increase in biomass will still result in increased
productivity. In fact, regardless of how small S becomes, and how clearly
limiting the substrate represented actually is in the prototype, if B is
very large, the model will predict substantial increases in biomass. Mar
points out, further, that the introduction of a decay term into the expres
sion as:
or
dB
dt ~ |Ms I K + S I " Kd
B (3-114)
where K , = decay rate
does not solve the problem since under the condition:'
1 »
the same anomalous growth dependence upon biomass is described. That is,
for constant and small S, productivity is a function of biomass.
Mar has suggested an alternative expression which does not depend
upon Michaelis-Menton half-saturation constants. The expression is of
the same form as the Monod formulation:
(3-115)
where a = conversion of substrate to biomass.
264
-------
It should be noted that as S becomes small, and
(3-116)
- Bys aS/B = y$ aS (3-117)
Thus for small S, productivity is not materially affected by the value of B.
3.9.2.2.5 Half-Saturation Constant Values
Table 3-34 lists reported values of the half-saturation constants for
nitrogen, phosphorus, silicate, carbon, and light. Other data are also
presented in the table including saturated growth rates and saturated light
intensities. Table 3-35 presents Michaelis-Menton half-saturation constants
as reported by Di Toro, ejt aj_. (1971). Table 3-36, taken from Eppley, e_t
al. (1969), provides nitrogen half-saturation constants for marine phyto-
plankton, while Table 3-37 provides values from Lehman, ejb al_. (1975).
Table 3-38 shows minimum cell nutrient quotas for some marine and fresh-
water phytoplankton.
3.9.2.2.6 Temperature Correction Constants
A commonly used temperature correction formulation for algal growth
rates is given above as Equation (3-85). The same formulation was cited in
Equation (3-72). The expression for the algal specific growth rate is:
(T-Tr)
y = ye (3-118)
Values of 6 are presented in Table 3-39 (from Di Toro, 1971).
3.9.3 Phytoplankton Specific Death Rate, Dp
3.9.3.1 Factors Affecting Dp
The algal specific death rate, Dp in Equation (3-70) represents all
sinks of phytoplankton biomass. Sinks include the endogenous respiration
rate, grazing by zooplankton and other herbivores, settling, parasitization,
and stress-induced death.
3.9.3.1.1 Endogenous Respiration Rate
The endogenous respiration rate is the time rate of algal biomass con-
version back to carbon dioxide per unit weight of organic carbon content
(O'Connor, e_t al_., 1973). As a process, it can be considered as the inverse
of photosynthesis. Endogenous respiration, while providing for the mainten-
ance of the algal cell and its direct energy requirements, may be considered
as partial death of the cell since biomass is consumed.
265
-------
TABLE 3-34. VALUES FOR THE HALF-SATURATION CONSTANT IN MICHAELIS-MENTON GROWTH FORMULATIONS
Phytoplankton
Description
Total Phytoplankton
Total Phytoplankton
Total Phytoplankton
Total Phytoplankton
Total Phytoplankton
Total Phytoplankton
Total Phytoplankton
Warm Water
Uarm Uater
Cold Water
Cold Uater
Diatoms
Small Diatoms
Large Diatoms
Green
Green
Blue-Green
Blue-Green (N-Fixing)
Blue-Green (Non N-Fixing)
Small Cells Favoring
Low Nutrients
Small Cells Favoring
Low Nutrients
Large Cells Favoring
High Nutrients
Large Cells Favoring
High Nutrients
Readily Grazed
Fast Settl ing
Not Readily Grazed
Not Fast Settl ing
Maximum
Speci fie
Growth
Rate, (Days'1 )
0.2-8.0
2.0
2.5
2.0
1.3
2.1
1.0-2.0
2.0
1.-2
2.5
1.-3.
2.1 (25"C)
2.1
2.0
1.9 (25°C)
1.9
1.6
0.8 (25°C)
0.8 (25"C)
1.0
1.5
2.0
2.0
1.5
2.0
Nitrogen
(mg/1)
0.025-0.3
0.025
0.025
0.025
0.025
0.025
0.025
0.07
0.05-0.3
0.01
0.1-0.4
-
-
-
-
0.015
0.015
-
-
0.3
0.3
0.4
0.4
0.02
0.4
HALF-SATURATION CONSTA
Phosphorus Silicate Carbon
(mg/1) (mg/1) fmq/1)
0.006-0.03
.
.
0.005
0.010
0.002
0.006-0.025
0.015 - 0.03
0.02-0.05 - 0.4-0.6
0.0? - 0.04
O.OU4-O.UJ - 0.5-0.8
-
0.03
0.03
-
0.0025
0.0025
-
-
0.03 - 0.5
0.03 - 0.5
0.05 - 0.6
0.05 - 0.6
0.02 - Q.05
0.05 - 0.8
(continued)
NTS
Light
(Kcal/m'Vsec)
-
-
-
-
-
-
-
0.002
0.002-0.004
0.003
0.004-0.006
-
-
-
-
-
-
-
-
0.003
0.002
0.006
0.004
0.003
0.006
Reference
Baca and Arnett (1976)
O'Connor, ejt a_l . (1975)
O'Connor, et al . (1975)
O'Connor, et al. (1975)
O'Connor, et a\_. (1975)
O'Connor, et aj_- (1975)
Battelle (1974)
Tetra Tech (1976)
U.S. Army Corps of
Engineers (1974)
Tetra Tech (1976)
U.S. Army Corps of
Engineers (1974)
Bierman (1976)
Canale, et al 1. (1976)
Canale, et al . (1976)
Bierman (1976)
Canale, e_t al_. (1976)
Canale, e_t_ al_. (1976)
Bierman (1976)
Bierman (1976)
Chen and Orlob (1975)
Chen (1970)
Chen and Orlob (1975)
Chen (1970)
Chen and Wei Is (1975)
Chen and Wells (1975)
-------
TABLE 3-34 (continued)
ro
Phytoplankton
Description
Total Phytoplankton
Total Phytoplankton
Total Phytoplankton
Total Phytoplankton
Total Phytoplankton
Total Phytoplankton
Total Phytoplankton
Warm Water
Warm Water
Cold Water
Cold Water
Saturated
Light
Intensity
(Ft-Candles)
-
300
300
300
350
350
-
-
-
-
-
Chemical
Composition
(fraction by weight) Temperature
_> * Tolerance
C N f Limits (°C)
-
-
-
-
-
.
-
0.4 0.08 0.015 10-30
10-30
0.4 0.08 0.015 5-25
5-25
Location
of Study
-
San Joaquin River
San Joaquin Delta Estuary
Potomac Estuary
Lake Erie
Lake Ontario
Grays Harbor/Chehal is
River, Washington
N. Fork Kings River, Calif.
-
N. Fork Kings River, Calif.
-
Reference
Baca and Arnett (1976)
O'Connor, et al . (1975)
O'Connor, et aK (1975)
O'Connor, et al_. (1975)
O'Connor, et aj. (1975)
O'Connor, e_t aj_. (1975)
Battelle (1974)
Tetra Tech (1976)
U.S. Army Corps of
Engineers (1974)
Tetra Tech (1976)
U.S . Army Corps of
Engineers (1974)
Diatoms
Small Diatoms
Large Diatoms
Green
Green
Blue-Green
Blue-Green (N-fixtng)
Blue-Green
(Non N-Fixing)
Small Cells Favoring
Low Nutrients
Small Cells Favoring
Low Nutrients
Large Cells Favoring
High Nutrients
Large Cells Favoring
High Nutrients
Readily Grazed
Fast Settling
Not Readily Grazed
Not Fast Settling
0.5 0.09 0.015
0.5 0.09 0.015
Saginaw Bay, Lake Huron
Lake Michigan
Lake Michigan
Saginaw Bay, Lake Huron
Lake Michigan
Lake Michigan
Saginaw Bay, Lake Huron
Saginaw Bay, Lake Huron
Lake Washington
San Francisco Bay Estuary
Lake Washington
San Francisco Bay Estuary
Boise River, Idaho
Boise River, Idaho
Biennan (1976)
Canale, et al_. (1976)
Canale, et aj_. (1976)
Bierman (1976)
Canale, et aj[. (1976)
Canale, et al_. (1976)
Bierman (1976)
Bierman (1976)
Chen and Orlob (1975)
Chen (1970)
Chen and Orlob (1975)
Chen (1970)
Chen and Wells (1975)
Chen and Wells (1975)
-------
TABLE 3-35. MICHAELIS-MENTON HALF-SATURATION CONSTANTS FOR
NITROGEN AND PHOSPHORUS (FROM DI TORO, ET AL_., 1971)
Organism
Chaetoceros gracilis
(marine diatom)
Scenedesmus gracile
Natural association
Microcystis aeruginosa
(blue-green)
Phaeodactylum tricornutum
Oceanic species
Oceanic species
Neritic diatoms
Neritic diatoms
Neritic or littoral
Flagellates
Natural association
Oligotrophic
Natural association
Eutrophic
Nutrient
P04
*+
Total N
Total P
P04
P04
P04
N03
NH3
N03
NH3
N03
NH3
N03
NH3
N03
NH3
Michael is
Constant,
yg/Liter
as N or P
25
150
10
6a
10a
10
1.4-7.0
1.4-5.6
6.3-28
7.0-120
8.4-130
7.0-77
2.8
1.4-8.4
14
18
Estimated.
268
-------
TABLE 3-36. MICHAELIS-MENTON HALF-SATURATION CONSTANTS (Ks) FOR UPTAKE OF NITRATE AND AMMONIUM
BY CULTURED MARINE PHYTOPLANKTON AT 18°C. Ks UNITS ARE yMOLES/LITER (AFTER EPPLEY, EJ_ AL_., 1969)
ro
CTl
UD
Organism
Oceanic species
Coccolithus huxleyi BT-6
C . huxl eyi F-5
Chaetoceros gracil is
Cyclotella nana 13-1
Neritic diatoms
Skeletonema costatunr
Le_ptocyl indrus danicus
Rhizosolenia stol terfothi i
R. robusta'i
Ditylum brightwellii
Coscinodiscus lineatus
C. wailesii
Asterionella japonica
Neritic or littoral flagellates
Gonyaulax polyedra
Gymnodinium splendens
Monochrysis lutheri
Isochrysis galbana
Dunaliella tertiolecta
Natural marine communities (from
01 igotrophic
Eutrophic
N I T R
ATE
±95%
KS Conf. Limit
0.1
0.1
0.3,0.1
0.3,0.7
0.5,0.4
1.3,1.2
1.7
3.5,2.5
0.6
2.4,2.8
2.1,5.1
0.7,1.3
8.6,10.3
3.8
0.6
0.1,0.1
1.4
Maclsaac and Dugdale,
<0.2(6 expts)
>.! .0(3 expts)
0.3+
1.6
0.5,0.2
0.4,0.5
0.4,0.1
0.5,0.1
0.4
1.0,1.0
1.7
0.5,0.6
0.3,1.8
0.3,0.5
--,2.4
0.9
0.3
0.2,0.2
1.1
1969)
A M
K
s
0.1
0.2
0.5,0.3
0.4
3.6,0.8,0.
3.4,0.9,0.
0.5,0.5
5.6,9.3
1.1
2.8,1.2
4.3,5.5
1.5,0.6
5.7,5.3
1.1
0.5
0.1
0.1-0.6(3
1.3 (1
M 0 N I U M
±95%
Conf. Limit
0.7
0.9
0.5,0.3
0.3
8 0.8,0.7,0.5
5 1.4,0.2,0.4
0.9,0.4
2.0,1.5
0.6
2.6,1.0
5.4,2.0
1.2,0.8
0.6,1.1
1.0
0.4
0.6
expts)
expt)
Cell
Di am*
(u)
5
5
5
5
8
21
20
85
30
50
210
10
45
47
5
5
8
Geometric mean diameter rounded off to the nearest micron.
"'"This notation means that -0.2 < Ks < 0.4. Negative Ks values have no physical interpretations.
*At 28°C, Ks for nitrate uptake was 1.0 ± 0.5; at 8°C, it was 0.0 ± 0.5.
An oceanic species according to Cupp (1943J.
-------
TABLE 3-37. HALF-SATURATION CONSTANTS FOR N, P, AND Si UPTAKE (yM) REPORTED FOR MARINE
AND FRESHWATER PLANKTON ALGAE (AFTER LEHMAN, ET AL., 1975)
Cyclotella
nana
Dunal iel la
tertiol ecta
Asterionella
japonica
Honochrysis
lutheri
Fragilaria
N03 0.4-1.9
1.8
0.35
0.5
NH4 0.4
N03 0.21
NH4 0.17
N03 1.4
NH4 0.6
N03 0.7-1.3
1.0
NH4 1.0
N03 0.42
NH4 0.29
N03 0.6
NH4 0.4
NO, 0.6-1.6
Carpenter and Guillard (1971)
Maclsaac and Dugdale (1969)
Caperon and Meyer (1972)
Eppley, e_t al_. (1969)
Caperon and Meyer (1972)
Eppley, e_t al_. (1969)
Eppley and Thomas (1969)
Eppley, et al. (1969)
Caperon and Meyer (1972)
Eppley, ejt al_. (1969)
Carpenter and Guillard (1971)
Leptocyl indricus
danicus
Rhizosolenia
stol terfothii
Rhi zosolenia
robiista
Dityl urn
brightwel 1 ii
Coscinodiscus
1 ineatus
Coscinodiscus
wailesii
Eucjlena
grac i 1 i s
Cyclotella
nana
Thalassiosira
N03
N03
N03
N03
N03
NH4
N03
P0l
P04
f°A
1.25 Eppley, Rogers and McCarthy
0.7 (1969)
1.7
0.5
3.0
7.5
0.6
1.1
2.6
2.0
3.6
4.9
16. Blum (1966)
0.58 Funs, et al_. (1972)
1.72
pinnata
Bellochia sp.
Coccochloris
stagnina
Phaeodactylum
tricornutum
Anabaena
cylTndrica
Chlorella
pyrenoidosa
Chaetoceros
gracilis
Gonyaulax
polyedra
Gymnodinium
splendens
Coccolithus
huxleyi
Skeletonema
co s ta tum
Isochrysis
galbana
N03
N0
N0
N03
NH4
N0
N03
NH4
N0
N0
0.1-0.9
0.31
2.6
70.
40.
25.
0.2
0.4
9.5
5.5
3.8
1.1
0.1
0.1
0.45
0.8
0.1
Caperon and Meyer (1972)
Ketchum (1939)
Hattori (1962)
Knudsen (1965)
Eppley, e_t al_. (1969)
f'Tuvia'tTlis
Chlorel la
pyrenoidosa
Mitzschia
actinastreoides
Scenedesmus sp.
Pediastrum
0. soci_al_e var.
americanum
Nitaschia
actinastreoidcs
Thai n_s_s joj_i_ra_
pseudonana'
Thalass_ioji_ra_
decipien's
Skeletonema
costati™
L_icmpphpra_ sp.
Di_tyluni
brightwellii
P04
P°4
P°4
P°4
P04
P04
Si-
Si
Si
Si-
Si
Si
Si
0.6
1.1
0.8
0.5
3.5
1.4-2.9
1.39
3.37
0.80
2.58
2.96
Jeanjean (1969)
Muller (1972)
Rhee (1973)
Lehman unpublished
Muller (1972)
Paasche (1973a)
Paasche (1973b)
-------
TABLE 3-38. MINIMUM CELL NUTRIENT QUOTAS (yMOLES CELL"1)
OF P, Si, AND N FOR SdME MARINE AND FRESHWATER
PHYTOPLANKTON (AFTER LEHMAN, EJ_ AL_., 1975)
Phosphorus:
Asterionel la
formosa
Asterionel la
japonlca
Cyclotella
nana
Nltzschia
actinastreoides
Phaeodactylum
tricornutum
Chlorella
pyrenoidosa
Scenedesmus
quadricauda
Scenedesmus sp.
Thalassiosira
fluviatil is
A. formosa
Gymnodinium
Pi nobryon
Anabaena
Si 1 icon:
Navicula
pell iculosa
Ni tzschia
al ba
Asterionel la
formosa
Fragilaria
crotonensis
Thalassiosira
pseudonana
Nitrogen:
Isochrysis
galbana
Asterionella
formosa
Gymnodinium
Dinobryon
Anabaena
2 x ICf9
1 .5-3. x 10~9
1 .5 x 10"9
0.9 x 10"9
3 x 10'9 (35u)**
4 x 10"9 (50-55p)**
2 x 10"9
3 x 10"9
4.5 x 10~9
1 .7 x 10"9
12.5 x 10"9
3 x 1(T8
1.1 x 10"8
0.5 x 10"9
2.5 x 10"9
0.5 x 10"7
3 x 10"7
2 x 10'6
1.8 x 10"6
4 x 10"6
2 x 10"8
3 x 10'8
6 X 10"7
3.9 x 10~7
1.8 X 10'8
1 x 10"7
Mackereth (1953)
Mu'ller (1972)
Fuhs (1969)
Muller (1972)
Rhee (1973)
Fuhs, et al . (1972)
Grim (1939)*
Busby and Lewin (1967)
Lewin and Chen (1963)
Hughes and Lund (1962)
Grim (1939)*
Paasche (1973a)
Droop (1973)
Grim (1939)*
*These quantities are not necessarily the minimum.
"cell length.
271
-------
TABLE 3-39. MAXIMUM (SATURATED) GROWTH RATES AS A FUNCTION OF
TEMPERATURE (FROM DI TORO, ET AU , 1971)
Organism
Chi orel la ellipse idea
(green alga)
Nannochloris atomus
(marine flagellate)
Nitzschia closterium
(marine diatom)
Natural association
Chi orel la pyrenoidosa
Scenedesmus quadricauda
Chlorella pyrenoidosa
Chlorella vulgaris
Scenedesmus obliquus
Chlamydomonas reinhardti
Chlorella pyrenoidosa
(synchronized culture)
(high-temperature strain)
Saturated Growth
Rate, K1
Temperature Base e, Day~l
25
15
20
10
27
19
15.5
10
4
2.6
25
25
25
25
25
25
10
15
20
3.14
1.2
2.16
1.54
1.75
1.55
1.19
0.67
0.63
0.51
1.96
2.02
2.15
1.8
1.52
2.64
0.2
1.1
2.4
272
-------
According to O'Connor, et. al_. (1973), the temperature dependence of
endogenous respiration has been investigated, and a straight line fit seems
reasonable. Lund (1965) has noted that under nutrient depletion, some algae
go into a kind of morphological or physiological resting stage. O'Connor,
ejt afl_. suggest that, whereas endogenous respiration is nutrient dependent,
in some cases an expression such as the Michaelis-Menton formulation might
be useful for simulating endogenous respiration.
3.9.3.1.2 Grazing
Herbivorous zooplankton reduce water column phytoplankton through
grazing. According to O'Connor, ejt al_. (1973), the interaction between phyto-
plankton and herbivorous zooplankton is quite complex, and only a first
approximation to the process can be given. Herbivorous zooplankton feed by
straining the water and removing whatever phytoplankton and detritus are
present. The phytoplankton represent a food source for the zooplankton, and
phytoplankton blooms are commonly associated with succeeding blooms of her-
bivorous zooplankton.
The zooplankton grazing rate, according to O'Connor, ejt ^1_., is commonly
expressed in terms of volume of water filtered per unit of zooplankton bio-
mass per unit time. This representation, of course, is very convenient from
a simulation standpoint. The filtering rate of herbivorous zooplankton has
been observed to vary with temperature, the degree of variation being zoo-
plankton species dependent. O'Connor, et_ al_. note that the filtering rate
is also dependent upon particulate concentration (Burns and Rigler, 1967),
phytoplankton cell size (Mullin, 1963), and concentration of phytoplankton
(McMahon and Rigler, 1965). Further, selective grazing according to phyto-
plankton cell size has also been observed (Burns, 1969).
3.9.3.1.3 Settling, Parasitization, and Stress
Settling out of phytoplankton is known to involve a complex set of
phenomena including vertical turbulence effects, vertical density distribu-
tion, and the physiological state of the different species of phytoplankton
(Thomann, §t aj_., 1975). With regard to species, the generation of gelat-
inous sheaths by phytoplankton, for example, has been shown to be of impor-
tance in settling, and apparently the settling velocity of nutrient rich
cells is somewhat lower than that of cells that are nutritionally deficient.
In some instances the phytoplankton settling velocity may be zero or the
cells may be sufficiently buoyant to move upward in the water column
(Thomann, et_ a]_., 1975). Thomann, et_ al_. (1975) report on work by Burns
and Pashley which shows a general decrease in settling velocity with depth
and a marked seasonal variation of settling velocity.
Lombardo (1972) reports on work done by Eppley, e^t a_L which suggests
that there are three discrete levels of phytoplankton sinking rates. These
may be differentiated as follows:
1. rate during logarithmic growth
273
-------
2. rate during the stationary period
3. rate found in neutrally buoyant cells
In those rivers and estuaries where the primary transport is one-dimen-
sional along the longitudinal axis of flow, settling may not be significant.
The mixing and turbulence associated with both freshwater flow and tidal
action are frequently of sufficient magnitude to minimize settling effects
(O'Connor, et al_. , 1975).
Parasitization and stress are two other factors which may contribute to
the death of algal cells. Parasitization represents the infection of the
algal cell by other microorganisms. Stress can result from nonoptimal condi-
tions (distinct from changes in growth rates) and from the presence of toxic
substances, although the modeling of toxic substances is commonly done by
altering the respiration or growth rate. In reality, of course, toxic sub-
stances might well affect any or all of the rates of respiration, growth,
settl ing, and death.
3.9.3.2 Modeling the Phytoplankton Specific Death Rate
Di Toro, et al_. (1971), O'Connor, et al . (1975), Thomann, et al_. (1975),
ahd the Fisheries Research Board of Canada~TSimons, 1976) have proposed a
general expression describing the phytoplankton death rate of the following
type:
where K^ = endogenous respiration rate of phytoplankton,
a function of temperature, day"'
C = grazing rate of herbivorous zooplankton,
y 1/day-mg zooplankton carbon
Z = zooplankton carbon concentration, mg/1
K = Michael is-Menton half-saturation constant for
p zooplankton grazing on phytoplankton, mg/1
W = settling velocity, m/day
P = phytoplankton concentration, mg/1
H = depth for settling out, m
The same investigators and others (Thomann, et_ aj_. , 1974; and Lombardo,
1973; in modeling studies of the Potomac and San Joaquin Delta Estuaries),
represented the temperature corrected endogenous respiration rate, R (in
274
-------
), as a linear function of temperature as follows:
R = K2 T (3-120)
where K~ = endogenous respiration rate, day~ °C~
T = temperature, °C
In a modeling study for Lake Ontario, Thomann, et^ a]_. (1975), proposed
the following expression for endogenous respiration:
R = K2 0(T-20) (3-121)
where Ko = endogenous respiration rate, day"
9 = temperature coefficient; typically 1.08
As discussed earlier, the filtering rate, Cg, in formulations like
Equation (3-119) is known to vary as a function of the size of the phyto-
plankton cell being ingested, the concentration of the phytoplankton, the
amount of particulate matter present, and temperature. However, as an approx-
imation, a single constant grazing coefficient, Cg, is used.
Baca and Arnett (1976) used the following representation for the phyto-
plankton death rate (day"') in their water quality model:
Dp = F1 + F2 (3-122)
where F-, = algal respiration rate, day~
F2 = algal decomposition rate, day~ ; typically 0.003-0.17
Tetra Tech (1975, 1976) and the U.S. Army Corps of Engineers (1974)
consider the phytoplankton death rate, Dp, to be a function of endogenous
respiration, zooplankton grazing, and sinking losses. Generally, two values
of respiration are available in their formulations. When the temperature is
below the lower bound of a temperature tolerance limit, the standard respira-
tion rate is used; otherwise the active respiration rate is used. Tetra
Tech (1975, 1976) and Chen and Wells (1975) consider the amount of phyto-
plankton biomass consumed by the zooplankton to be a function of both the
amount of algae grazed and the digestive efficiency of the zooplankton.
A formulation used by Bierman (1976) for the phytoplankton death rate,
Dp, also considers losses to be due to respiration (and cell death), grazing,
and sinking. His death term is formulated as follows:
R = RLYS • T • TCROP (3-123)
275
-------
where R = rate of phytoplankton biomass loss, day
RLYS - algal death rate (day-°C-mg/l)-1
T = temperature, °C
TCROP = total phytoplankton biomass, mg/1
Lombardo (1972) reports an expression for the phytoplankton death rate
which is essentially the same as that given by Thomann, O'Connor and Di Toro,
Lehman (1975) considers the algal specific death rate to be due to respira-
tion and physiological mortality. The formulation for the loss of algal
cells due to algal death is
^- =(" .693 y -(V/D) - M] N (3-124)
and
M =
- exp(k-SG)] (3-125)
where N - algal cell concentration, numbers/ml
y = cell division rate, day
V = sinking rate, m/day
D - mean epi limnetic depth, m
M = fraction of population dying per day
Mmax = maxi'mal faction dying per day at
suboptimal condition
SG = number of suboptimal days
k = .693/d
d = number of suboptimal days required for
M = W2
Lassiter (1975) uses the formulation:
'T>Tm
(3-126)
T
-------
where D = the specific death rate due to excessive
temperature, day~l
T = ambient temperature, °C
Tm = optimal temperature, °C
a = scaling coefficient, day~
'1
k = k in Equation (3-96), day
m = "1
k = km in Equation (3-96), day
3.9.3.3 Vakies for Algal Specific Death Rates and Related Constants
Table 3-40 provides rate values for algal respiration, zooplankton
filtering, and algal sinking, and values of the Michaelis grazing constant
and zooplankton digestive efficiency. Values given are for 20°C unless
noted otherwise. Tables 3-41 and 3-42 are from Di Toro, ejt a]_. (1971) and
show rates of phytoplankton endogenous respiration and zooplankton grazing
rates.
3.9.4 Benthic Algae
Through photosynthesis, benthic algae and rooted aquatic plants affect
some of the major water quality parameters, e.g., dissolved oxygen and
nutrient concentrations. In a stream, benthic algae and rooted aquatic
plants can be a major influence on the dissolved oxygen resources due to
photosynthetic processes taking place during the day and respiration proc-
esses occurring at night. Benthic algae and rooted aquatic plants also may
influence the nutrient content of flowing water by their extraction of
nutrients. A stream which supports benthic algae may exhibit a decrease in
eutrophying influence on a receiving lake by the fixing of organic nutrients
in the stream and resulting reduction in nutrient discharge.
Benthic algal dynamics, when considered in a water quality model, are
usually treated in the same way as phytoplankton dynamics, except that ben-
thic algae are stationary within a reach and are not allowed to exceed a
maximum concentration (Lombardo, 1972; Lombardo, 1973).
3.9.5 Methods for Measuring Algal Growth and Death Rates and Coefficients
Methods for measuring the various rates and coefficients described
above are generally much too complex and organism-specific to discuss here.
The reader is referred to the substantial body of technical literature on
the various specific subjects. Table 3-43 provides references on various
topics which may be helpful in initiating a literature review on parameters
discussed in this section.
277
-------
TABLE 3-40. VALUES FOR COEFFICIENTS IN PHYTOPLANKTON SPECIFIC DEATH RATE EXPRESSIONS
Phytoplankton
Description
Phytoplankton
Phytoplankton
Phytoplankton
Phytoplankton
Phytoplankton
Phytoplankton
PJiytopJankton
Phytoplarvkton
ro
^-J Phytoplankton
CO
Phytoplankton
PhytopJankttsn
Phytoplankton
-Warm Water
Warm Water
Cold Water
Cold Water
Diatoms/Greens
Blue-Greens
Algal Zooplankton
Respiration Filtering
Rate g20°C Rate (?20°C
(day-1) (1/mg-C-day)
0.005-0.12
-
0.08
0.10 0.13
0.08 0.25
O.LD 1.2
0,1-0 0.18
0.10
0.10
0.10±0.02 0.13
0.1
-
0.1 0.01
(active) (inactive)
0.05
0.1 0.01
{active) (inactive)
0.05
0.0015* 0.35-0.50**
0.0015* 0.35-0.50**
Zooplankton Michael is
Digestive Sinking Grazing
Efficiency Rate Constant
(mg/mg) (m/day) (ug chlor/1)
-
0.5-0.8 0.-2.
0.00-0.005
0.6 - 50
0.65 - 50
"0.60 0.1 10
0.-6Q - 50
0.4
0.5
0.5
0.7 0.2
0.2
0.7 0.05
0.05
0.4
0.15
Location
of Study
-
Lake Washington
Lake Michigan
San Joaquin River
Lake Erie
take Ontario
San Joaquin DeJta Estuary
Potomac Estuary
Boise River, Idaho
San Francisco E-ay Estuary
N. -Fork Kings River,
Cal ifornia
lake Washington
N. Fork Kings River,
Cal ifornia
Lake Washington
Saginaw Bay, Lake Huron
Saginaw Bay, lake Huron
Reference
Baca and Arnett (1976)
Chen and Orlob (1975)
Canale, e_t a_K M976)
O'Connor, e_t al_. (1975)
O'Connor, et al_. (1975)
O'Connor, et_ll- (1975)
O'Connor, e^ al_. {1975)
O'Connor, et -a], {1975)
Lowbardo (1972)
T3i T-oro, et -al_. {1371)
Chen and Wells (1975)
Chen (1970)
Tetra Tech 11976)
Chen and Orlob (1975)
Tetra Tech (1976)
Chen and Orlob (1975)
Bierraan (1976)
Bierman (1976)
* - 0.0015 (day-'C-mg/ir1
** - 0.35-0.50 mg algae/ing Zooplankton -day
-------
TABLE 3-41. VALUES FOR ENDOGENOUS RESPIRATION RATES
OF PHYTOPLANKTON (DI TORO, ET AL., 1971)
Organism
Nitzschia closterium
Nitzschia closterium
Coscinodiscus excentricus
Natural association
Temper-
ature ,
13 C
6
35
20
16
16
2
18
2.0
17.9
Endogenous
Respiration Rate,
Day~l (Basee
0.035
0.170
0.08
0.075
0.11
0.03
0.12
0.024=0.
O.llOzO.
)
012
007
TABLE 3-42. VALUES FOR GRAZING RATES OF ZOOPLANKTON
(DI TORO, EJ_AL_., 1971)
Organism
Reported
Grazing Rate
Grazing Rate,
Liter/fig Dry
Wt.-Day
Rotifer
Brachionus calyciflorus
Copepod
Calanus sp.
Calanus finmarchicus
Rhincalamus nasutus
Centropages hamatus
Cladocera
Daphnia sp.
Daphnia sp.
Daphnia magna
Daphnia magna
Natural association
Georges Bank
0.05-0.12C
67-208
27a
98-6703
81 a
57-82a
D-110U
0.6-1.5
0.67-2.0
0.05
0.3-2.2
0.67-1.6
1.06
0.2-1.6
0.74
0.2-0.3
0.8-1.10
ml/animal-day
ml/mg wet wt-day
279
-------
TABLE 3-43. REFERENCES FOR METHODS OF MEASUREMENT
OF VARIOUS ALGAL MODELING PARAMETERS
Reference
Subject
Braarud (1961)
Cassie (1963)
Eppley, et al_. (1966)
Eppley (1968)
Eppley, et aj_. (1969)
Fitzgerald and Nelson
(1966)
James and Birge (1938)
Megard (1972)
Powers, et_aK (1972)
Smayda (19695)
Strickland (1960)
Cultivation of organisms
Multivariate analysis of phytoplankton
data
Marine phytoplankter culture
Carbon content of phytoplankton
Half-saturation constants
Limiting and surplus phosphorus in
algae
Light absorption
Rates of photosynthesis and phyto-
plankton growth
Nutrient limitation of algae
Settling of algae
Marine phytoplankton production
3.9.6 Summary
Phytoplankton can dramatically affect surface water quality through
photosynthesis, respiration, nutrient uptake, and death. In general, model-
ing approaches consider rates of growth (a source of phytoplankton biomass)
and respiration, settling, and zooplankton grazing (sinks of phytoplankton
biomass). Factors influencing the growth rate are phytoplankton type,
nutrient availability, light intensity and duration, and temperature. The
impacts of nutrient and light growth limitation are generally modeled using
a Monod or Michaelis-Menton type formulation, although this has a number of
deficiencies.
280
-------
Factors affecting the death rate include phytoplankton species, abun-
dance of predators, temperature, presence of toxic substances, and osmotic
conditions. Respiration, which may be conceptualized as the inverse of
photosynthesis, consumes biomass carbon and is simulated as one component
of death. Toxic substance effects are sometimes simulated as growth-limiting
factors rather than as factors affecting the death rate.
3.9.7 References
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and Impoundments. Battelle Pacific Northwest Laboratories.
Battelle, Inc., 1974. Development of a Mathematical Water Quality Model for
Grays Harbor and the Chehalis River, Washington: Documentation Report,
Pacific Northwest Laboratories, Richland, Washington.
Bierman, V.J., Jr., 1976. "Mathematical Model of the Selective Enhancement
of Blue-Green Algae by Nutrient Enrichment," in R.P, Canale, ed,, Modeling
Biochemical Processes in Aquatic Ecosystems, Ann Arbor Science Press.
Bloomfield, J.A. , R.A. Park, D. Scavia, and C.S. Zahorcak, 1973. "Aquatic
Modeling in the Eastern Deciduous Forest Biome," in Middlebrooks, Falkenborg,
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Blum, J.J., 1966. "Phosphate Uptake by Phosphate Starved Euglena." J^_Gen_i_
Physiol., Vol. 49, pp 1125-1137.
Braarud, T. , 1961. "Cultivation of Marine Organisms as a Means of Under-
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Brown, Thomas E. and F.L. Richardson, 1968. "The Effect of Growth Environ-
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Burns, C.W., 1969. "Relation Between Filtering Rate, Temperature, and Body
Size in Four Species of Daphnia," Limnol . Oceanog. , Vol. 14, No. 5,
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Daphnia in Lake Water and in Suspensions of Yeast," Limnol . Oceanog. ,
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Busby, W.F. and J. Lewin, 1967. "Silicate Uptake and Silica Shell Forma-
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Canals, R.P., L.M. DePalma, and A.M. Vogel, 1976. "A Plankton-Based Food
Web Model for Lake Michigan," in R.P. Canale, ed., Modeling Biochemical
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Caperon, J. and J. Meyer, 1972. "Nitrogen-limited Growth of Marine Phyto-
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Carpenter, E.J. and R.R.L. Guillard, 1971. "Intraspecific Differences in
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Cassie, R.M., 1963. "Multivariate Analysis in the Interpretation of
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Chen, C.W. and G.T. Orlob, 1975. "Ecological Simulation for Aquatic
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Cupp, E.E., 1943. "Marine Phytoplankton Diatoms of the West Coast of North
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Eppley, R.W., 1968. "An Incubation Method for Estimating the Carbon
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Constants for Uptake of Nitrate and Ammonium by Marine Phytoplankton,"
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Fuhs, G.W., 1969. "Phosphorus Content and Rate of Growth in the Diatoms
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Annual A1BS Meeting, Corvallis, Oregon, August, 17-22.
Goldman, C.R., 1972. "The Role of Minor Nutrients in Limiting the
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Grim, J., 1939. "Beobachtungen am Phytoplankton des Bodensees (Obersee)
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Hattori, A., 1962. "Light-Induced Reduction of Nitrate, Nitrite, and
Hydroxylamine in a Blue-Green Alga, Anabaena cylindrica," Plant Cell
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Hughes, J.C. and J.W.G. Lund, 1962. "The Rate of Growth of Asterionella
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James, H.R. and E.A. Birge, 1938. "A Laboratory Study of the Absorption of
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Jeanjean, R., 1969. "Influence de la carence en phosphore sur les vitesses
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Johnson, F.H., H. Eyring, and M.J. Polissar, 1954. The Kinetic Basis of
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Jorgensen, E.G. and E. Steeman-Nielsen, 1969. "Adaptation in Plankton
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Kerr, P.O., D.L. Brockway, D.F. Paris, and J.T. Barnett, Jr., 1972. "The
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Ketchum, B.H., 1939. "The Absorption of Phosphate and Nitrate by Illuminated
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Knudsen, 6., 1965. "Induction of Nitrate Reductase in Synchronized Cultures
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Lassiter, R. and O.K. Kearns, 1973. "Phytoplankton Population Changes
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Lassiter, R.P., 1975. Modeling Dynamics of Biological and Chemical
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Lehman, J.T., D.B. Botkin, and Gene E. Likens, 1975. "The Assumptions and
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3.10 ZOOPLANKTON
Attempts to understand algal population dynamics and nutrient cycling
in surface waters generally require a parallel and simultaneous under-
standing of herbivorous zooplankton population dynamics. The term "zoo-
plankton," however, encompasses a large variety of forms of animal plank-
ton, not all of which are herbivorous. Some have the ability to photo-
synthesize. Some are predatory, while some are not. Some forms are
seasonal with a dormant period, and some overwinter by forming cysts. Some
are microscopic, and some are easily visible to the naked eye. The zoo-
plankton species composition in any water body is highly mutable, being
dependent to a large extent upon such factors as phytoplankton population
dynamics, availability of organic particulate matter, bacterial popula-
tions, temperature, and salinity. Prior to discussing population dynamics,
models, and constants, it is important to understand the extreme complexity
of zooplankton-biotic community relationships. Such an understanding should
lead to an appreciation for the degree of simplification necessary in
developing a practical model as well as an understanding of what such a
model can and cannot do.
3.10.1 Important Zooplankton Organisms
Freshwater zooplankton populations are generally dominated by the
cladocera and copepoda (two crustaceans) and by the rotifers, with a number
of other forms being important under some circumstances. The latter in-
clude the protozoans and some forms having only part of their life cycles as
zooplankton. Organisms which are zooplanktonic only as part of their life
cycles include some kinds of flatworms, aquatic insects, mites, gastrotrichs,
and coelenterates. Some fish also spend part of their life cycles as organ-
isms which can truly be termed zooplankton.
According to Wetzel (1975), the protozoans can sometimes represent a
significant proportion of the total numbers and biomass of zooplankton.
Wetzel cites data from Lake Dal nee, a large lake in Kamchatka, Siberia
(U.S.S.R.), where the midsummer pelagial zooplankton population had a large
flagellate and ciliate protozoan component. Their maximal biomass was ob-
served to coincide with a decline in a summer algal bloom and with a bloom
of bacterial forms. The protozoans were also observed to occur in distinct
layers in the water column.
287
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According to Wetzel, the Lobosan Rhizopod Difflugia occurs commonly in
both oligotrophic and eutrophic lakes. A number of ciliate protozoans may
represent important components of zooplankton numbers, but usually only in
shallow lakes or in anaerobic hypolimnia. The rotifers, in contrast, are
generally an important component of freshwater zooplankton populations, are
commonly sessile (attached) and tend to be associated with littoral sub-
strates (Wetzel, 1975). Numbers tend to be quite high where associated
with small submersed macrophytes, as opposed to situations where the organ-
isms are more exposed and subject to predation. The sessile organisms
undergo a number of morphological changes in going from the sessile to
planktonic form. For example, there is commonly a weight reduction with
simultaneous volume increase. Swimming apparatus are developed along with
other swimming adaptations such as those which reduce the sinking rate.
Among rotifers which are sessile during part of their life cycles and those
which are entirely planktonic, the free-swimming forms commonly are ciliated,
the cilia providing for locomotion as well as feeding functions.
The seasonal behavior of rotifer populations is complex. Some forms
are perennial while others (the stenotherms) bloom in winter and early
spring. Still others bloom in summer in response to food availability.
Wetzel cites data for Vorderer Finstertaler See, a deep alpine drainage lake
in Austria, which has two rotifers constituting 99 percent of the zooplank-
ton population. One of the two rotifers (Keratella humalis) is a steno-
therm, blooming in midwinter and declining in warmer weather. The other
rotifer (Polyarthra do!ichoptera) is also a coldwater form which tolerates
anoxic conditions. It was found to dominate in midsummer.
According to Wetzel, the seasonal distribution of rotifer populations
is likely to vary considerably from lake to lake in a given geographical
area, and the populations are likely to be only very generally synchronous
among such lakes.
In fresh water, the crustacean zooplankters are dominated by the
Cladocerans and Copepoda. The freshwater Branchiopoda (fairy and clam
shrimps) are common planktonic forms in shallow saline inland waters,
particularly playas in semiarid regions.
The Cladocerans are generally very small (microzooplankton) having a
highly variable seasonality. Some are perennial, while some survive cold
periods through a resting egg stage. Some Cladocerans are coldwater forms
living in northerly lakes and commonly tolerating low dissolved oxygen
concentrations.
Wetzel cites population dynamics data for Base Line Lake, Michigan,
as a typical case. The zooplankton population consisted mainly of Daphnia.
Two population maxima were observed in Daphnia galeata mendotae. The first
occurred in the spring, and the second occurred late in autumn. This
species overwintered in the free-swimming form rather than as a resting
stage. The birth rates in spring and autumn were nearly equal. In winter,
reproduction ceased and a slow death rate determined populations over time.
During the summer, predation by the large Cladoceran Leptodora was found
to be important.
288
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In late summer, Daphnia retrocurva bloomed, greatly exceeding
JX_ galeata. EK retrocurva and JD. catawba were found to have late summer
maxima. Both overwinter in the egg stage. The presence of D. catawba in
late summer and resulting competition with CL galeata was deemed responsible
for delaying the autumnal maximum of the latter.
In a shallow, eutrophic lake in northwestern Pennsylvania (Sanctuary
Lake), D. galeata exhibited maxima similar to those observed in Michigan.
Again D. retrocurva was codominant in late summer, and the predaceous
Leptodora kindtii grazed the Daphnia down in summer. The population dynamics
of Leptodora were correlated with loss rates of prey. The predators were
found to shift seasonally among Daphnia, Ceriodaphnia, Bosmina, and Chydorus
and the copepods Cyclops and Diaptomus (Wetzel, 1975).
Like the rotifers, the Cladocera exhibit seasonal polymorphism. Al-
though cyclomorphic growth patterns in these organisms do not mean the
difference between sessile and planktonic habit, they do, as was true with
rotifers, influence vulnerability to predation. In rotifers, the free-
swimming form is likely to be more vulnerable than the sessile form. In
Daphnia, changes appear to make the organism less conspicuous, and therefore
less likely to be consumed by fish.
The Harpacticoid copepods are almost entirely associated with macro-
scopic vegetation and particulate organic matter in the littoral zone. The
planktonic Cyclopoid copepods form major components of the copepod zoo-
plankton, particularly in small shallow lakes. The Cyclopoid copepods are
primarily littoral benthic species. The Calanoid copepods, on the other
hand, are almost exclusively pelagial zone plankton.
Some Cyclopoid copepods are known to have diapause periods at either
the egg or copepodite stage, with or without encystment (Uetzel, 1975). In
some, the diapause occurs in winter while in some, it is in summer. In
Cyclops strenuus strenuus, the reproduction characteristics result in a
bimodal population over the year. Cyclops strenuus abyssorum, on the other
hand, has only a single reproductive period per year.
Coexisting Cyclopoid copepod species exhibit asynchronous maxima and
diapause periods, which is believed to minimize competition. The coexist-
ence mechanisms include seasonal separation, vertical separation, and
differences in terms of food particle sizes taken.
3.10.2 Modeling Zooplankton
A popular method for representing zooplankton assemblages is through
use of Michaelis-Menton formulations analogous to those used in modeling
phytoplankton. The zooplankton are assumed to grow in accord with food
availability. The food, in turn, is presumed to be algae, despite the very
much more complex trophic relationships which usually exist in the prototype
(see Section 3.10.2.1.1).
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The zooplankton, in turn, are preyed upon by higher levels in the food
chain, and undergo endogenous respiration and death. It is clear, based
upon known predator-prey relationships that the monotonic food chain is
artificial, and represents a gross oversimplification of reality. The
defense of the models, however, lies in practicality and the fact that for
most purposes, simplified models lead to satisfactory estimates.
Zooplankton biomass concentrations are generally assumed to be influ-
enced by the following processes: a) growth, b) respiration, c) grazing
by higher trophic levels, and d) death. As with typical phytoplankton
representations, these processes are usually combined in a source/sink
formulation which is then included in a complete mass balance expression.
The source/sink formulation accounting for changes in zooplankton
concentration is of the following general form in the majority of water
quality models:
dZ - In \n (3-127)
where G = gross specific growth rate, day"
DZ = death rate, day
Z = zooplankton concentration, mg/1
3.10.2.1 Gross Specific Growth Rate
There are a variety of formulations commonly used to compute the gross
specific growth rate. The formulations incorporate varying degrees of
complexity, and the variation is often the major reason for differences
among formulations reported by various researchers.
Generally, the rate, Gz, is considered to be a function of algal
density, ingestion or grazing rate, and assimilation efficiency. Food
material is typically assumed to flow through a zooplankton system according
to the following schematic (Lombardo, 1972):
+ ASSIMILATION (GROWTH)
FOOD -> GRAZING + INGESTION
(FILTERING) "* EXCRETION
where INGESTION = GRAZING (FILTERING) RATE X FOOD CONCENTRATION
ASSIMILATION = ASSIMILATION EFFICIENCY X INGESTION
290
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3.10.2.1.1 Factors Affecting Gz
Feeding mechanisms and food availability are a primary determinant of
growth rates. Some models which incorporate phytoplankton and zooplankton
assume a single filtering rate of the phytoplankton by the zooplankton.
Others account for more than one feeding mechanism. Regardless of the
assumed model, the mechanism of grazing is an important consideration in
computing Gz, as is the type of food preferentially sought.
Most protozoan forms of zooplankton feed on bacteria, algae, partic-
ulate detritus and other protozoans. Some are carnivorous, feeding on small
metazoans (Wetzel, 1975). Mechanisms of ingestion are generally through
pinocytosis and formation of food vacuoles. Some protozoans are known to
feed actively on algae. Amoeboid forms are also known to consume diatoms.
Rotifers feed by sedimenting seston particles into the mouth orifice
by means of the pulsating action of the coronal cilia. There is a consider-
able variation in the size of food particles consumed, with most particles
being less than about 12 microns in diameter. Larger cells, some being up
to 50 microns in diameter, however, are sometimes ruptured to release cell
particulate matter (Wetzel, 1975). Asplanchna is a raptorial rotifer which
seizes and ingests whole cells or at least cell contents of algae, other
rotifers, and small planktonic crustaceans. According to Wetzel (1975),
some algae, such as Chlorella, are not eaten as readily as others, possibly
due to some negative physiological response elicited by the algal cell
content.
Feeding among the crustacean zooplankters is usually through filtering
of particles by the setae. The food collects in the ventral food groove,
and is moved forward to the mouth.
Some forms, particularly species of Polyphemus and Leptodora, are
predaceous and feed on protozoa, rotifers, and small crustaceans. Recent
research has suggested that some zooplankton forms have the ability to
select food for consumption and that such selection can well affect which
phytoplankton will bloom and which will not (Wetzel, 1975).
The mouth parts of the Harpacticoids are adapted for seizing large
particles and scraping food from them. No filtering mechanism occurs in
free-living Cyclopoida. Feeding is raptorial. The carnivorous Cyclopoida
include the Macrocyclops, Acanthocyclops, Cyclops, and Mesocyclops. Her-
bivorous Cyclopoida include Microcyclops, Acanthocyclops, and Eucyclops,
which feed upon a variety of algae (Wetzel, 1975JT
The quantity of food consumed by zooplankton is not quantitatively
converted to biomass. Most models account for this through an assimilation
factor. However, assimilation efficiencies vary as a function of ambient
conditions. The assimilation efficiency has been found, for example, to
decrease with increasing food concentration. The assimilation efficiency
291
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has also been found to decrease with decreasing caloric food value, repre-
sentative of detritus and non-nutritious particles in the water. Lombardo
(1972) reported that healthy algae have a caloric value of about 5 cal-
ories/mg dry weight.
At high phytoplankton concentrations, the zooplankton do not metabolize
all the phytoplankton grazed, but rather, they excrete a portion in undi-
gested form. According to Wetzel (1975), ingestion of algae does not always
mean that the cells will be assimilated. Algae that have especially resist-
ant cell walls, gelatinous sheaths, or masses of colonial cells can pass
through the gastric tract of some zooplankton in viable form. In some
cases, individual cells may be assimilated while colonies of the same algal
species pass through relatively unaltered.
Another factor affecting 6Z is water temperature. As was true for
phytoplankton, rates of growth of zooplankton are influenced by biochemical
reaction rates. These rates determine the speed of swimming, the rate of
feeding and filtering, the rate of assimilation, and the rate of excretion.
3.10.2.1.2 Formulations for Computing Gz
Many investigators (see, for example, Tetra Tech, 1975, U.S. Army
Corps of Engineers, 1974, Bierman, 1976, O'Connor, et_ aj_. , 1975, and
Orlob, 1974) have used Michaelis-Menton terms in their growth formulations.
This type of representation may be illustrated by considering the following
general form of the Michaelis-Menton expression as given by Thomann, et al . ,
(1975):
G = ACK (3-128)
where A = assimilation efficiency relating phytoplankton
^ biomass ingested to zooplankton biomass produced, unitless
C = filtering rate @ 20°C, £/mg-ch1-day
K = phytoplankton biomass concentration at which the
growth rate, Gz, is half the maximum possible
growth rate, mg-chl/£
P = phytoplankton biomass concentration, mg-chl/£
The filtering rate of zooplankton which use a filtering mechanism
varies according to a number of factors. These include:
food particle size
food particle density
acceptability of food
temperature
species of zooplankton
292
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In addition, some zooplankton are known to filter intermittently, and,
possibly, only during darkness (Lombardo, 1972). Lombardo also reported
that when filter feeding crustaceans encounter low food concentrations, the
feeding rate is limited by the ability of the animal to filter water.
The phenomenon of variable assimilation efficiency is accommodated with
the following functional relationship:
zp \ K + P /
By multiplying this term by the biomass term, CqP, the specific growth
rate expression given in Equation (3-128) is obtained.
Under conditions of nonlimiting food sources, i.e., when P»Kmp, the
growth expression (Equation (3-128)) reduces to:
AzP Cg Kmp
which is the "zooplankton maximum growth rate" for saturating phytoplankton
concentrations.
In Equation (3-128), phytoplankton biomass is measured as mg-chl/£
while zooplankton biomass is measured as mg-C/1. Therefore a carbon to
chlorophyll ratio (Azp) must be specified. In model applications to five
different locations (San Joaquin River, San Joaquin Delta and Potomac
Estuaries, and Lakes Erie and Ontario), a value of 50 was used.
Baca and Arnett (1976) use the following expression for zooplankton
specific growth rate based on the formulations given by Thomann, O'Connor,
and Di Toro:
_
where C - zooplankton grazing rate, day
The specific growth rate expression proposed by Bierman (1976) has a
similar form:
Gz = RZMAX - A . I { - ZEFF (nK'L) ' A(L) - } (3-131)
2 zp L = l < KZSAT (K,L) + I ZEFF (K,L) * A(L) '
where n = number of phytoplankton species
RZMAX = zooplankton maximum ingestion rate, day"
293
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A = zooplankton assimilation efficiency, unit! ess
ZEFF(K,L) = ingestion efficiency of zooplankton species K
for phytoplankton species L, unitless
KZSAT(K,L) = half-saturation concentration of phytoplankton L
for grazing by zooplankton K, mg/1
A(L) = phytoplankton L concentration, mg/1
Bierman's model considers two separate groups of zooplankton which can
graze upon two distinct groups of phytoplankton. The two zooplankton types
are differentiated on the basis of maximum ingestion rates. The formula-
tion for gross specific growth rate is actually a form of the expression
used by Baca and Arnett (1976) (Equation (3-130)) with two modifications.
First, the term ZEFF(K,L) in the Bierman formulation is used to accommodate
zooplankton preference for one type of phytoplankton over another. Second,
the summation of a number of Michael is-Menton terms is incorporated to allow
consideration of the limiting influence on total growth of the availability
of specific phytoplanktonic groups,
Tetra Tech (1975, 1976), Chen and Wells (1975), and the U.S. Army Corps
of Engineers (1974) use the following formulation for the zooplankton
specific growth rate:
Gz = ZMAX ULG/(ALG2 + ALG) j (3-132)
where ZMAX = maximum specific growth rate @ 20°C, day"
AL&2 = half-saturation constant for zoorlankton
grazing on algae, mg/1
ALG - effective algal concentration, mg/1
- ALGl + ALG2
ALG1 = concentration of type 1 algae, mg/1
ALG2 = concentration of type 2 algae, mg/1
PREF 1 = preference for zooplankton grazing on
type 1 algae, decimal
PREF 2 = preference for zooplankton grazing on
type 2 algae, decimal
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Equation (3-132) can be made analogous to Equation (3-128) if ZMAX
is set equal to the quantity Azp Cg Kmp. As previously illustrated, the
quantity AZp Cg Kmp from Thomann, O'Connor and Di Toro's expression repre-
sents a maximum growth rate under saturated phytoplankton conditions.
Tetra Tech (1975, 1976), Bierman (1976), and the U.S. Army Corps of
Engineers (1974) have included phytoplankton species preferences in their
formulations. This addition was necessary because more than one species of
phytoplankton were considered. In Tetra Tech's model, the zooplankton
feeding preference is used to establish a single effective algal concentra-
tion. This simplification allows input of only one half-saturation con-
stant for zooplankton growth and adjustment of the concentration of algae
accordingly.
Canale, e_t a_L (1976) propose a complex zooplankton model which con-
siders the following nine different types of zooplankton:
1. Leptodora and Polyphemus (predators)
2. Cyclops (small omnivores)
3. Cyclops
4. Diaptomus
5. Limnocalanus and Epischura
naupli i
6. Diaptomus (small herbivorous copepods)
7. Limnocalajuis and Epischura (large omnivores)
8. Daphnia (large herbivorous cladocerans)
9. ^c^mina^ and Holopedium (small herbivorous
cladocerans)
RAPTORS
SELECTIVE
FILTERERS
J-SELECTIVE
FILTERERS
Conceptually, the model is similar to those previously described. The
specific growth rate formulation, in qualitative terms, can be described as
the product of an ingestion rate, or "eating rate" as it is called by
Canale, and an assimilation efficiency. In equation form, it becomes:
/assimilation j /eating)
\ efficiency / \ rate /
(3-133)
where C = zooplankton concentration for zooplankton
z type z, mg/1
295
-------
The assimilation efficiency represents milligrams zooplankton carbon devel
oped per milligram of food carbon consumed and the eating rate represents
milligrams of food carbon consumed per milligram of zooplankton carbon per
day.
Much of the complexity of the formulation is due to the inclusion of
three different types of eating (grazing) behavior - raptorial, selective
filtering, and non-selective filtering. For raptors, the eating rate
expression is given as:
(EC.
— J
E C. + KFOOD
i
where Z C^ = sum of concentrations of all states
i (i) that can serve as food for raptor
state z
KFOOD = food level half-saturation constant for
state z (raptors), mg food C/l ; typically 0.2
A?z = snatching rate at 20°C, mg food C/mg zoopl . C-day
(0.70 - Leptodora and Polyphemus, 0.43 - Cyclops)
4>Z(T) = temperature correction term, unitless
The eating expression for selective filterers is:
eating „-,
rate ~~ A7z
where A9 = minimum filtering rate multiplier,
unitless; typically 0.1
A10 = food level where multiplier is 1/2 (1+A9)
m9 f°od C'; typically 0.2
A7 - maximum filtering rate, 1/mg zoopl. C-day:
2.6 - Cyclops nauplii
6.5 - Diaptomus naupl ii
5.2 - Limnocalanus and Epischura nauplii
1 .0 - Diaptomus
1 .25 - Limnocalanus and Epischura
296
-------
And for non-selective filterers, the expression is:
("ale") = A7Z *2(T) E C, (3-136)
Z 1
Nonselective filterers cannot lower their filtering rate when the
plankton content of the water increases. Therefore, they operate below
the maximum possible efficiency. The assimilation efficiency for non-
selective filterers is given by the following expression:
assimilation \ m-,,,/ A24 \ /, 107\
efficiency) s A11N Z C- + A24 (3-137)
/z \i "" /
where A11N = maximum efficiency possible, mg
zoopl. C/mg food C; typically 0.8
A24 = half-maximum efficiency food level for
nonselectives, mg food C/l; typically 0.2
Species preference is accommodated by means of the following
expression:
Z r
preference of species z ak k
,_ , ,„,
( for species k /
The electivity, a|, is defined as the fraction of species z diet that would
be composed of food species k if all food species were present in equal con
centrations. The preference of z for k is proportional to the product of
the electivity of species z for species k and the concentration of species
k(i.e., C|<). Electivities used in Canale's model are unverified.
3.10.2.1.3 Growth Rate Data
Table 3-44 provides values for the zooplankton maximum specific growth
rate, conversion efficiency, grazing rate, Michael is-Menton half-saturation
constant, and chemical composition. Tables 3-45 through 3-48 show filter-
ing rates for various zooplankton forms. Tables 3-49 and 3-50 give produc-
tion data for both herbivorous and predatory zooplankton. Table 3-51 gives
food utilization efficiencies for several zooplankton forms.
297
-------
TABLE 3-44. VALUES FOR COEFFICIENTS IN MICHAEL IS-MENTON ZOOPLANKTON GROWTH FORMULATIONS
Zooplankton
Description
Zooplankton
Zooplankton
Zooplankton
Zooplankton
IV>
U3 Zooplankton
oo
Raptor Feeders
Selective
Filtering
Feeders
Zooplankton
Zooplankton
Zooplankton
Zooplankton
Zooplankton
Zooplankton
Zooplankton
Zooplankton
Zooplankton , , , .
Maximum Zooplankton
~ . r- Lonver-
r !£ D * sion Effi- Zooplankton
Growth Rate Clencyifl 6lrazing
(day ) (decimal; Rate
0.5-0.7 0.05-0.2 day"1
0.30
0.15-0.30
0.15
0.6 ' 0.35-0.50 day"1
0.4
0.7
0.6 0.13 1/mg-C-day
0.6 0.18 1/mg-C-day
-
0.65 0.25 1/mg-C-day
0.60 1.2 1/mg-C-day
0.25
0.25'
0.6 0.5 1/mg-dry wt-day
Michael is-
Menton
Hal f-Saturation
Constant
0.06-0.6 mg-C/1
0.4 mg/1
0.4-0.6 mg/1
0.5 mg/1
:
-
50 ug chlor/1
50 ug chlor/1
-
50 pg chlor/1
10 \ig chlor/1
0.5 mg/1
0.5 mg/1
0.06 ug chlor/1
Chemical
Composi tion
(fraction
by weijjhj;) Location
C " ~ N " P of Study
-
0.4 0.08 0.015 North Fork Kings
River, California
-
Lake Washington
: :
-
San Joaquin River,
Cal i form' a
San Ooaquin Delta
Estuary, California
Potomac Estuary
Lake Erie
Lake Ontario
San Francisco Bay
Estuary
0.5 0.09 0.015 Boise River, Idaho
-
Reference
Baca and Arnett (1976)
Tetra Tech, Inc. (1976)
U.S. Army Corps of
Engineers (1974)
Chen and Orlob (1975)
Bier-ran (1976)
Canal e e_t aj . (1976)
Canale et al . (1976)
O'Connor e_t a_l_. (1975)
O'Connor et al . (1975)
O'Connor _et aj_. (1975)
O'Connor et aj_. (1975)
O'Connor et a^. (1975)
Chen (1970)
Chen and Wells (1975)
Oi Toro et aU (1971)
-------
TABLE 3-45. VALUES FOR ZOOPLANKTON FILTERING RATES
(AFTER LOMBARDO, 1972)
Feeding Organism
Acartia tonsa
Centropages hamatus
C. typicus
Daphnia niagna
0. schodleri
D. pulex
D. galeata
D. rosea^
D, rosea
Tood Onjatiisin
Thai ass iosira
fl uvia tiiis
¥00-100" cell /ml
and Arteinia
Thalassiosira
f 1 uviatius
800~-l""o"o"cell/ml
and Arteinia
Thai assiosira
Fl uviatius
SOO-ToT'cell/ml
and Arteinia
Rhodotorula
cjl utinis
.'25 "x" 105 cells/ml
Rhodotorula
gj utinis
. 25~TT65 cells/ml
Rhodotorula
gl utinis
.25" x'105 cells/ml
Rhodotorula
gj uti nis
".25 x" 105 cells/ml
Rhodotorula
gj utinis
.25~x VO5
Mixture of
Scenedesmus,
other algae and
bacteria
Temperature (°C)
Filtering Rate (ml/mg dry wt-hr
2" 8"
17.0 33.0
8.3 9.0
6.7 13.2
15° 20°
8.0 16.6
12.6 15.9
13.6 15.9
10.3 24.3
5" 10°
13.8 18.0
5° 12°
18.4 26.0
15" 22.5"
58.0 92.0
10.0 35.0
11.2 20.0
25°
19.0
11.3
1-2.8
27.9
15" 20° 25°
25.0 28.4 26.2
14° 20° 25°
27.0 22.2 21.0
0.040 ing/animal was used.
299
-------
TABLE 3-46. COMPARISON OF FILTERING RATES OF VARIOUS
CLADOCERAN ZOOPLANKTERS (AFTER WETZEL, 1975)
GO
O
O
Species
Daphnia
D. rosea
D. galeata
D. parvula
D. longjj;pina
Ceriodaphnia
C. quadrangula
Diaphanosoma
D. brachyurum
Bosmina
B. longi_rostris
Chydorus
C. sphaericus
Animal Size
Range (Length
Type of Food in mm)
In situ phytoplankton 1.3-1.6
1.5-1.7
0.7-1.2
0.7-0.9
0.9-1.4
0.4-0.6
0.1-0.2
Average
Filtering Rate
(ml animal "1
day-1)
5.5
3.6
6.4
3.7
3.8
2.3
4.6
1.6
0.44
0.18
Source
Haney (1973)
Burns and Rigler
Haney (1973)
Burns and Rigler
Haney (1973)
Nauwerck (1963)
Haney (T973)
Haney (1973)
Haney (1973)
Haney (1973)
(1967)
(1967)
-------
TABLE 3-47. FILTERING RATES AND CONTRIBUTION TO TOTAL GRAZING OF SPECIES.
DOMINANT ZOOPLANKTON OF ACIDIC DROWNED BOG LAKE, ONTARIO, IN EARLY SEPTEMBER'
(AFTER WETZEL, 1975)
Species
Bosmina longirostris
Holopedium gibberum
Daphnia parvula
Diaptomus oregonensis
Diaphanosoma brachyurum
Filtering Rates
(ml animal' day"1)
Sept. 1968 Sept. 1969
0.46 0.45
9.4
1.6
2.1
1.2
Species
to Total
Sept. 1968
85
12
0.1
2.0-
0.1
Contribution
Grazing (%)
Sept. 1969
44.8
46.2
7.5
0.9
0.4
*Data extracted from Haney (1973).
301
-------
TABLE 3-48. COMPARISON OF FILTERING RATES OF VARIOUS COPEPODS (AFTER WETZEL, 1975)
Species
OJ
O
r-o
Type of Food
Particle
Concentration
(cells x 10-3 mi-l)
Filtering Rate
(ml animal"' day"')
Source
Diaptomus
Er.~grac"i1oides Natural phytoplankton
Scenedesmus
D. siciloides Pandorina and
Chlamydomonas
D. oregonensis Chlamydomonas
Chlorella
In situ phytoplankton
D. gracilis Melosira and
Asterionel la
Chlorella
Scenedesmus
Diplosphaeria
Ankistrodesmus
Carteria
Ni tzschia
Pediastrum
Haematococcus
Bacteria
Limnocalarius
L. inacrurus Scenedesmus
Chlanydomonas
Rhodo tu ruTa~~[y ea s t )
13.6
1.5-25.0
25.0-52.0
52.0-198.0
24.2-52.0
198.0
<30.0
<30.0
<30.0
<30.0
<30.0
<30.0
<30.0
<30.0
<30.0
0.3-2.8
4.1
2.0
2.5
2.5-1.4
1.4-0.3
1.4-0.00
1.92-1.96
0.68
0.61 (5°C)
1.51 (12°C)
2.40 (20°C)
0.94 (12°C)
1.32 (20°C)
1.76 (12°C)
2.54 (20°C)
1.61 (12°C)
2.45 (20°C)
0.87 (20°C)
1.96 (20°C)
0.02 J20°C)
2.16 (20°C)
0.19 (20°C)
2.45 (<5°C)
1.24 (<5°C)
0.1 (<5°C)
Nauwerck (1959)
Malovitskaia and
Sorokin (1961)
Comita (1964)
Richman (1966)
Haney (19/3)
Malovitskaia and
Sorokin (1961)
Kibby (1971)
Kibby and Riqler
(1975)
-------
TABLE 3-49. EXAMPLES OF THE PRODUCTION OF HERBIVOROUS AND PREDATORY
ZOOPLANKTON COMMUNITIES (AFTER WETZEL, 1975)
CO
o
CO
Lake Type
OLIGOTROPHIC
Lake Baikal , USSR
Clear Lake, Ontario
Lake 239, Ontario
Lake Krugloe, USSR
MESOTROPHIC
Taltowisko Lake,
Poland
Herbivores
Predators
Lake Naroch, USSR
Lake Krasnoe, USSR
EUTROPHIC
Lake Mikolajskie, Poland
Herbivores
Predators
Lake Sniardwy, Poland
Herbivores
Predators
Kiev Reservoir, USSR
Herbivores
Predators
Severson Lake, Minnesota
Herbivores
Predators
DYSTROPHIC (high dissolved
organic matterjb
Lake Flosek, Poland
Herbivores
Predators
a Estimated using the mean caloric
' See discussion in Chapter 18 of
Bioinass (B)a Production (P)a
Period of 3311
Investigation g m kcal m g m kcal in
June-July 0.136
Sept. 0.43
Annual (0-50 m depth)
Annual 0.20
May-Nov.
Annual 0
May-Oct. 0.12
Annual 0.07 0
Annual 0.14 0
May-Oct.
May-Oct.
0.35 1
0.022 0
May-Oct.
May-Oct.
value for microconsumers (Cummins
Wetzel (1975).
3.44
3.02 16.435
0.61 3.331
.405 0.94 5.116
3.04
4.68 25.43
0.46 2.50
,38 1.12 6.11
.76 3.09 16.82
6.45 35.09
1.32 7.18
3.08 16.78
0.50 2.71
.9 9.15 49.8
.12 1.16 6.3
2.51 13.66
0.11 0.60
25.68 139.7
1.16 6.3
and Wuycheck, 1971 ).
Biomass
Turnover Time (days)
Average Range Remarks and Source
Primarily E_pischura; Moskalenko
and Votinsev 11970)
25 12-333 Herbviorous zooplankton (roti-
fers, Ho 1 oped i urn, Dajihnia,
Bosmina, DiaptomusT"; Schindler
TT970T
22 10-91
29 Arctic lake; Winberg (1970)
14.3 29",' of total lake area in
littoral zone; Kajak, et. al .
10.2 (1970); Kajak (1970) "' ""
9.1
22.4 Winberg (1970)
16.5
Kajak, et al . (1970); Hillbricht-
9.2 4.0-12.5 Ilkowska.'et al . (1970)
25.0
14.9 8.3-33
14.3
13.9 Uinberg (1970)
Comita (1972)
Sphagnum bog; high littoral and
6.3 allochthonous orqanic inputs;
25.0 Kajak, et al. (1970); Hillbricht-
Ilkowska, et al_. (1970)
-------
TABLE 3-50.
EXAMPLES OF PRODUCTIVITY OF HERBIVOROUS AND PREDATORY FORMS OF ZOOPLANKTON
(AFTER WETZEL, 1975)
CO
o
-pa
Type/Species
FILTER FEEDERS:
Cladocera:
Daphnia hyal ina
0. parvula
0. galeata mendota
D. schodleri
D. longis£_ina
Bosinina longirostris
B. longirostris and
B. corejoni
Cerioda_ghnia reticulata
Chydorus sphaericus
Cladocera
Cpj)epods:
Cyclops strenuus
Eudiaptonius graciloides
Mesocyclops edax
Diaphanosoma leuchten-
bergiarium
Diciptomus siciloids
Arc thodiaptomus
T"2 speciesT"
Lake; General Productivity
Eglwys Nynydd Reservoir, Wales;
eutrophic
Severson Lake, Minn.; eutrophic
Sanctuary Lake, Pa. ;
eutrophic reservoir
Canyon Ferry Reservoir,
Mont. ; eutropinc
Canyon Ferry Reservoir,
Mont. ; eutrophic
Lake Sevan, southern USSR
Severson Lake, Minn.
Sanctuary Lake, Pa.
Naroch Lake, USSR
Myastro Lake, USSR
Batorin Lake, USSR
Duttennere , Enqland; oligolrophic
Rydal Hater, England; eutrophic
Grasmere, England; eutrophic
Esthwaite Water, England;
eutrophic
Lake Sevan, southern USSR
Naroch Lake, USSR
Myastro Lake, USSR
Balorin Lake, USSR
Severson Lake, Minn.
Lake Sevan, southern USSR
Period of
Investigation
Annual , 1970
Annual , 1971
Annual
May-Nov. , 1966
May-Nov. , 1967
Apri 1-Sept.
April -Sept.
Annual
Annual
May-Nov. , 1966
Kay -Nov. , 1967
July-Nov.
July-Auq. , 1966
July-Aug. , 1967
May-Oct.
May-Oct.
May-Oct.
Annual
Annual
Annual
Annual
Annual
May-Oct.
May-Get.
May-Oct.
Annual
Annual
(continued)
Production Bio"iass
Estimates3 Turnover Time
g m-3 kcal m~2 (Days)
day~^ day~' Average Range Source
0.57
0.32
0.010
0.407
0.030
0.114
0.227
0.006
0.007
0.183
0.067
0.031
0.004
0.047
0.0026
0.015
0.033
0.0004
0.0005
0.0006
0.0017
0.0007
0.0010
0.0065
0.0070
0.0046
0.0067
0.0342
0.0014
21.3 3.8
15.9
0.102
3.026
0.223
10.0
6.7
58.9
0.071
1.361
0. 498
0.154
0.020
0.233
0.117 13.7
0.403 10.9
0.484 10.5
79.3
0.044 24.7
0.174 20.2
0.104 15.4
0.045
0.067
0.341
162
-333 Georgp and Edwards
(19/4)
Comita (1972)
Cummins, ei a]_. (1 969)
Wright (1965)
Wright (1965)
Mcshkova (1952) in
Winberg (1971 )
Comita (1972)
Cummins, et aj..(1969)
Winberg, et al . (1970)
Smyly (1973)
Meshkova (1952) in
Winberg (1971)
Winberg, §1 a_l_. (1970)
Comita (1972)
Meshkova t'952) i--.
Winberg (1971 )
-------
TABLE 3-50 (continued)
GO
O
en
Type/Species
FILTER FEEDERS: (Cont.)
Rotifers:
Kcratollo ^uadrata
K . r.nchlearis
H 1 inia loncjiseta
Bra thin nu 5 sp.
Polyarthra sp.
Rotifers" "
PREDATORY FEEDERS:
Cl adorera :
Leptodora kindth
Cl adocera
Copepods:
Cyclops sp.
Rotifers:
Aspl anchna priodonta
Asplanchna sp.
Synchaeta sp.
Insect larvae:
Chaborus punctipennis
Lake; General Productivity
Severson Lake
N a ro c h
Myas tro
Batorin
Lake,
Lake,
Lake,
, I'l inn.
USSR
USSR
USSR
Sanctuary Lake, Pa.
Naroch
My a s t ro
Batorin
Naroch
My a s t ro
Batorin
Naroch
My as tro
Batorin
Lake,
Lake,
Lake,
Lake,
Lake,
Lake,
Lake,
Lake,
Lake,
Sevorson Lake
USSR
USSR
USSR
USSR
USSR
USSR
USSR
USSR
USSR
, Minn.
Period of
Investigation
Annual
May-Oct.
May-Oct.
Hay-Cct.
May-Nov. , 1966
M.iy-Nov. , 1967
May-Oct.
May-Oct.
May-Oct.
Annual
Production
Est (mates'1
g m"' kcal m-2
day' dayl
0.
0.
0.
0.
0.
0.
0.
c.
0.
0.
0.
0.
0.
0.
0.
0.
Q
0.
0.
0.
0.
0.
on?i
0(107
0011
0075
0010
0027
0024
0105
003
013
0003
0009
0002
0008
0023
0094
0014
0061
0105
0031
00009
0001
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
l"\
0.
0.
0.
0.
0.
.021
0071
0112
0752
0103
120
065
.156
022
097
013
023
003
034
062
140
061
163
156
031
0009
001
I'. iOllldSS
Turnover Tin's
(Days)
Average Range Source
1
3
2
11
10
9
19
14
2
2
3
_
_
_
_
_
.7
.9
.6
_
_
.3
. 9
.3
.7
.4
9
.9
.5
.2
„
_
_
Con.ita (1972)
Winberg, et al . (1970)
Cummins, et al . (1 969)
Winberg, et a 1 . (1973)
Comita (1972)
Conversions estimated using the caloric mean of microconsumers (Cummins and Wuycheck (1971)), when mean depths available.
-------
TABLE 3-51. EXEMPLARY ESTIMATES OF EFFICIENCIES OF FOOD UTILIZATION
BY VARIOUS ANIMALS3 (AFTER WETZEL, 1975)
Organisms Egestion
ZOOPLANKTON
Daphnia pulex 69-86
Ceriodaphnia reticulata
Simocephalus vetulus
Juveniles (?) 27.6
Reproducing (+) 68.3
Leptodora kindtii 60.0
(jG
O
Mesocyclops albidus
10 Herbivores 52.4
BENTHIC ANIMALS
Asel lus aquaticus 69.7
Lestes sjxinsa 63,4
FISH
Phytophagous carp,
Cteno_phary_rujodon 86.0
Predatory perch,
Perca fluviatil is 64.2
% of Ingested
Assimilation
14-31
10.6
72.4
31.7
40.0
20-75
47.6
30.3
36.6
14.0
35.8
Food Utilized in:
Respiration
4-14
1.8
19.5
11.2
„
ca 20
40.1
24.7
13.2
12.2
16.2
% of Assimilated Energy
Expended in:
Growth and Growth and
Reproduction Respiration Reproduction Source
10-17 27-44 56-73 Richman (1958)
Czeczuga and Bobiatynska-Ksok (1970)
Klekowski (1970)
52.9 26.9 73.1
20.5 35.3 64.7
92.7 7.3 Cummins, et al. (1969);
Moshiri, et al . (1969)
ca 25 ca 50 ca 50 Klekowski and Shushkina (1966)
7.5 - - Comita (1972)
5.6 81.8 18,2 Klekowski (1970)
23.5 36.0 64.0 Klekowski , et al . (1970)
1 .9 86.0 14.0 Fischer (1970)
19.5 45.5 54.5 Klekowski , et al . (1970)
Methods of analysis and experimental conditions vary greatly and are comparable only approximately.
-------
3.10.3 Death Rate, Dz
3.10.3.1 Factors Affecting Dz
The second term in Equation (3-127) involves the zooplankton death
rate, Dz. Several factors mediate Dz. These include predation by other
zooplanRton, predation by organisms in higher trophic levels (especially
fish), endogenous respiration, mortality due to nonoptimal ambient condi-
tions, and natural mortality. All of these are, in turn, influenced by
temperature.
Predation by other zooplankton is very important in zooplankton popula-
tion dynamics. Several common types are carnivorous and consume other
zooplankton. The predator-prey relationships among the zooplankton are
very important in determining which species predominate over time. Carniv-
orous zooplankton which prey on other zooplankton include species of the
following genera:
• Asplanchna (rotifer)
Polyphemus \
Leptodora f
Macrocyclops
Acanthocyclops
Cyclops
Mesocyclops
(cyclopoid copepods)
The degree of predation on any zooplankton prey depends upon the degree
of selectivity of the predator (willingness to take other forms) antecedent
abundance of the prey species and/or other food sources, and in general,
conditions affecting the growth of the predator which are exactly analogous
to those affecting the prey species itself.
Predation by fish is another important sink to zooplankton biomass.
Recent studies have implicated planktivorous fishes in determining, to a
large extent, the dominant species of zooplankton. Often, larger zooplank-
ton are favored by the fish, and intense grazing can lead to dominance of
small zooplankton species. Studies have shown, further, that the dominance
cannot be attributed to food supply (Wetzel, 1975). Wetzel also notes that
the planktivorous rainbow trout (Salmo gaerdneri) and the yellow perch
(Perca flavescens) cannot remove from the water zooplankton smaller than
1.3 mm. Apparently, planktivorous fishes are selective feeders, seeking
out desirable food forms.
Some species of fish are obligate planktivores (e.g., the alewife,
AJosa pseudoharengus) while some can prey on plankton or other types of
food (trout and perch) (Wetzel, 1975).
307
-------
The remaining sinks to zooplankton biomass are endogenous respiration
and mortality. Endogenous respiration is the autoconsumption of zooplankton
biomass in providing for organism energy needs. Mortality may be simply the
result of organism aging. Nonoptimal conditions may make mortality a
significant sink of zooplankton biomass. Stresses such as toxic substances,
nonoptimal temperatures, and insufficient oxygen supply may cause a rapid
die-off of zooplankters.
Temperature also influences rates of predation by influencing such
factors as prey swimming speeds and avoidance ability, abundance of pred-
ators, physical condition of both predators and prey, and availability of
alternate food supplies for predators.
3.10.3.2 Formulations for Computing Dz
The zooplankton death rate, Dz, is generally considered to be a function
of the following processes: 1) endogenous respiration, 2) grazing by higher
trophic levels, and 3) natural mortality. The mortality rate is that
fraction.of the biomass which is converted to detritus by death of the
particular zooplankton type.
Thomann, et_ al_. (1975) express the death rate as a function of two
primary factors: 1) endogenous respiration, and 2) predation by higher
trophic levels. They proposed the following general expression to describe
the zooplankton death rate (Dz, I/day):
Dz = K3(T) + K4 (3-139)
where Ko(T) = endogenous respiration rate, a function
of temperature (day-°C)~^
K, = death rate attributed to higher trophic levels, day"
The death term due to predation by higher trophic levels, 1(4, is con-
cerned primarily with carnivorous zooplankton. The authors report that
there is much uncertainty about the mechanisms involved in this process
and the specific term used is empirically determined.
The death rate formulation used by Baca and Arnett (1976) is of the
following form:
Dz - Rz + Fz (3-140)
where R = endogenous respiration rate, day"
F - death rate due to fish predation, day"
308
-------
Chen and Wells (1975) and the U.S. Army Corps of Engineers (1974)
propose a zooplankton death rate expression of the following general form:
where ZM - zooplankton mortality rate, day"1
ZR = zooplankton respiration rate, day'1
ZGRAZ = rate of grazing on zooplankton, mg/l-day
PEFF = digestive efficiency of zooplankton grazer
ZOO = zooplankton concentration, mg/1
Canale, et al_. (1976) proposed the following expression for zooplankton:
where D^ = zooplankton biomass loss due to
respiration and death, mg/l-day
A14z(t) = zooplankton natural death rate, a
function of temperature,
GZ = zooplankton concentration, mg/1
Similarly, Kelly (1975), in his model application to the Delaware
Estuary, considered the zooplankton specific death rate to be a function
of endogenous respiration, zooplankton death, and predation. In the
Lombardo and Franz model (Lombardo, 1973), the zooplankton specific death
rate is considered to be a function of respiration and natural death.
Respiration is taken as a linear function of temperature, and death is
considered to occur at a constant rate.
3.10.3.3 Death Rates Data
Table 3-52 provides zooplankton respiration and mortality data from a
variety of studies. Tables 3-53 and 3-54 contain respiration rate data.
3.10.3.4 Measuring Zooplankton Rate Constants
Because of the tremendous diversity of ambient conditions and complex
ities of zooplankton population dynamics, no specific recommendations can
be made here for measuring rate constants. The reader is referred to
references cited in appropriate portions of the preceding text.
309
-------
TABLE 3-52. VALUES FOR COEFFICIENTS IN ZOOPLANKTON SPECIFIC DEATH RATE EXPRESSIONS
CJ
Zooplankton
Description
Zooplankton
Zooplankton
Zocplankton
Zooplankton
Adult
Zooplankton
Nauplii
Zooplankton
Copepods
Zooplankton
Zooplankton
Zooplankton
Zooplankton
Zooplankton
Zooplankton
Zooplankton
Zooplankton
Zooplankton Zooplankton
Respiration Mortality
Rate Rate
(days'1) (days'1)
0.005-0.3
0.005-0.02 0.02
0.001-0.10 0.005
0.01
0.06
0.04
0.003-0.005
0.075
0.10
-
0.16
0.02
0.36
0.01
0.01
Location
of Study
-
North Fork Kings River,
California
-
Lake Washington
Lake Michigan
Lake Michigan
Lake Michigan
San Joaquin River,
California
San Ooaquin River,
Delta Estuary, California
Potomac Estuary
Lake Erie
Lake Ontario
-
Boise River, Idaho
San Francisco Bay Estuary
Reference
Baca and Arnett (1976)
Tetra Tech, Inc. (1976)
U.S. Army Corps of Engineers (1974)
Chen and Orlob (1975)
Canale, et a]_. (1976)
Canale, e_t aj_. (1976)
Canale, et al_. (1976)
O'Connor, _e_t al_. (1975)
O'Connor, et_ a_i_. (1975)
O'Connor, e_t al . (1975)
O'Connor, et al . (1975)
O'Connor, e_t aj_. (1975)
Lombardo (1972)
Chen and Wells (1975)
Chen (1970)
-------
TABLE 3-53. ENDOGENOUS RESPIRATION RATES OF ZOOPLANKTON
(AFTER DI TORO, ET AL., 1971)
Temperature
Organism °C
Cladocerans 18
4
Copepods 18
4
Copepods 18
16
12
8
4
Calanus 20
finmarchicus 15
10
4
Diaptomus 25
leptopus 20
15
10
5
D. clavipes 25
20
15
10
5
D. siciloides 25
20
15
10
5
Diaptomus sp. 25
f _ 2Q
15
10
5
Respiration Rate,
ml 02 / Mg Dry
Wt-Day
14-2
2.7
12.2
3.8
8.2
6.5
5.2
4.1
3.4
4.2
2.3
1.4
1.3
12.1
7.4
5.3
2.8
2.5
12.5
8.5
5.1
2.4
1.8
21
13.5
7.8
5.5
4.8
4.3
3.0
2.1
1.7
1.1
311
-------
TABLE 3-54. VALUES FOR ENDOGENOUS RESPIRATION RATES OF ZOOPLANKTON
(AFTER LOMBARDO, 1972)
Organism
Diaptomus siciloides
D. oregonensis
D_. leptopus
D. clavipes
D. articus
Acartia clausi
2
Centropages hamatus
Temora longicornis
Daphnia magna
Copepods
(Diaptomus and Cyclops)
Copepods
(Diaptomus and Cyclops)
Cladocerans
(Ceriodaphnia and Daphnia)
Temperature, °C
02 Consumed, ul/mg-hr
Respiration rate, per hour
5°
4.42
.00563
3.88
.00495
2.33
.00297
2.14
.00273
.982
.00125
2.27
.00288
\
4°
3.3
.00422
3.8
.00487
2.7
.00345
10°
5.26
.00658
4.71
.0059
3.06
.00382
2.85
.00356
1 .48
.00185
10°
13.0
.0162
4.54
.00568
10°
5.4
.00675
3.4
.00425
8°
4.3
.00541
15°
9.92
.0122
8.65
.0106
5.07
.00625
4.64
.00571
2.06
.00254
13°
12.4
.0154
6.34
.00786
13°
5.4
.0067
12°
5.7
.00707
20°
16.0
.0193
13.3
.0167
7.76
.00938
7.05
.00852
2.93
.00354
17°
15.8
.0193
7.56
.00922
17°
7.2
.00878
16°
6.6
.00811
25°
24.4
.0288
21 .0
.0248
11.7
.0138
10.6
.0125
4.38
.00577
20°
18.9
.0229
20°
10.3
.0125
5.4
.00654
19°
8.07
.00977
12.2
.0148
14.2
.0172
Mean length 0.85 mm; assumed weight 0.005 mg/animal
Mean length 0.89 mm; assumed weight 0.015 mg/animal
Mean length 0.787 mm; assumed weight 0.015 mg/animal
312
-------
3.10.4 Summary
Zooplankton populations in surface water bodies tend to be ephemeral
over time, with population dynamics being dependent upon food availability,
temperature, and abundance of predators. Because of the importance of zoo-
plankton in phytoplankton population dynamics, zooplankton are normally
simulated in surface water ecomodels. Modeling is usually of a form similar
to that for phytoplankton - that is, zooplankters grow, respire, and die.
Michaelis-Menton formulations are commonly used in computing growth rates
with primary application in terms of phytoplankton availability. Despite
the complex population dynamics in the plankton communities, a simple repre-
sentation of fish grazing on zooplankton and zooplankton grazing on phyto-
plankton is commonly used.
3.10.5 References
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and Impoundments. Battelle Pacific Northwest Laboratories.
Bierman, V.J., Jr., 1976. "Mathematical Model of the Selective Enhancement
of Blue-Green Algae by Nutrient Enrichment," in R.P. Canale, ed., Mathe-
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Burns, C.W. and F.H. Rigler, 1967. "Comparison of Filtering Rates of
Daphnia rosea in Lake Water and in Suspensions of Yeast," Limnol. Oceanogr.,
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Canale, R.P., L.M. DePalma, and A.H. Vogel, 1976. "A Phytoplankton-Based
Food Web Model for Lake Michigan," in R.P. Canale, ed., Mathematical
Modeling of Biochemical Processes in Aquatic Ecosystems, Ann Arbor Science
Press.
Chen, C.W., 1970. "Concepts and Utilities of Ecological Model," ASCE,
Journal of the Sanitary Engineering Division, Vol. 96, No. SA5.
Chen, C.W. and G.T. Orlob, 1975. "Ecological Simulation for Aquatic
Environments," in Systems Analysis and Simulation in Ecology, Vol. Ill
(Academic Press).
Chen, C.W. and J. Wells, 1975. Boise River Water Quality-Ecological Model
for Urban Planning Study, Tetra Tech technical report prepared for U.S.
Army Engineering District, Walla Walla, Wash., Idaho Water Resources Board,
and Idaho Dept. of Environmental and Community Services.
Comita, G.W., 1964. "The Energy Budget of Diaptomus siciloides, Lilljeborg,"
Verh. Int. Ver. Limnol., Vol. 15, pp 646-653.
Comita, G.W., 1972. "The Seasonal Zooplankton Cycles, Production and
Transformations of Energy in Severson Lake, Minnesota," Arch. Hydrobiol.,
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Cummins, K.W., R.R. Costa, R.E. Rowe, G.A. Moshiri, R.M. Scanlon, and
R.K. Zajdel, 1969. "Ecological Energetics of a Natural Population of the
Predaceous Zooplankter Leptodora kindtii Focke (Cladocera)," Oikos,
Vol. 20, pp 189-223.
Cummins, K.W. and J,C. Waycheck, 1971. "Caloric Equivalents for Investi-
gations in Ecological Energetics," Mitt. Int. Ver. Limnol., Vol. 18, 158 pp.
Czeczuga, B. and E.Bobiatynska-ksok, 1970. "The Extent of Consumption of
the Energy Contained in the Food Suspension by Ceriodaphnia reticulata,"
in Z. Kajak and A. Hillbricht-Ilkowska, eds., Productivity Problems of
Freshwaters, PWN Polish Scientific Publishers, Warsaw, pp 739-748.
Di Toro, D.M., D.J. O'Connor, and R.V. Thomann, 1971. A Dynamic Model of
the Phytoplankton Population in the Sacramento-San Joaquin Delta, Advances
in Chemistry Series, Nonequilibrium Systems in Natural Water Chemistry,
Vol. 106.
Fischer, Z., 1970. "Elements of Energy Balance in Grass Carp Ctenopharyn-
godon idell a Val," Pol. Arch. Hydrobiol., Vol. 17, pp 421-434.
George, D.G. and R.W. Edwards, 1974. "Population Dynamics and Production
of Daphnia hyalina in a Eutrophic Reservoir," Freshwater Biol., Vol. 4,
pp 445-465.
Haney, J.F., 1973. "An In Situ Examination of the Grazing Activities of
Natural Zooplankton Communities," Arch. Hydrobiol., Vol. 72, pp 87-132.
Hillbricht-Ilkowska, A., I. Spodniewska, T. Weglenska, and A. Karabin, 1970.
"The Seasonal Variation of Some Ecological Efficiencies and Production
Rates in the Plankton Community of Several Polish Lakes of Different
Trophy," in Z. Kajak and A. Hillbricht-Ilkowska, eds. Productivity Problems
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Kajak, Z., 1970. "Some Remarks on the Necessities and Prospects of the
Studies on Biological Production of Freshwater Ecosystems," Pol. Arch.
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Kajak, Z., A. Hillbricht-Ilkowska, and E. Piecsynska, 1970. "The Production
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Modeling in a Resource Management Framework, Resources for the Future, Inc.,
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Kibby, H.V., 1971. "Energetics and Population Dynamics of Diaptomus
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Kibby, H.V. and F.H. Rigler, 1973. "Filtering Rates of Limnocalanus,"
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Klekowski, R.Z. and E.A. Shushkina, 1966. "Ernabrung, Atmung, Wachstum and
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-------
TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing)
1. REPORT NO.
EPA-600/3-78-105
3. RECIPIENTS ACCESSION-NO.
4. TITLE AND SUBTITLE
Rates, Constants, and Kinetics Formulations
in Surface Water Quality Modeling
5. REPORT DATE
December 1978 issuing date
6. PERFORMING ORGANIZATION CODE
7. AUTHOR(S)
S.W. Zison, W.B. Mills, D. Deimer, and C.W. Chen
8. PERFORMING ORGANIZATION REPORT NO
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Tetra Tech, Incorporated
3700 Mt. Diablo Boulevard
Lafayette, California 94549
10. PROGRAM ELEMENT NO.
1BA609
11. CONTRACT/GRANT NO.
R804450-01-2
12. SPONSORING AGENCY NAME AND ADDRESS
Environmental Research Laboratory - Athens, GA
Office of Research and Development
U.S. Environmental Protection Agency
Athens, Georgia 30605
13. TYPE OF RE PORT AND PERIOD COVERED
Final, 5/76-11/78
14. SPONSORING AGENCY CODE
EPA/600/01
15. SUPPLEMENTARY NOTES
16. ABSTRACT
Recent studies are reviewed to provide a comprehensive volume on state-of-the-
art formulations used in surface water quality modeling along with accepted values for
rate constants and coefficients. Topics covered include system geometric representa-
tion (spatial and temporal), physical processes (mass transport, heat budgets, ice for.
mation, light extinction), biological systems (fish, benthic organisms), and chemical
processes (nutrient cycles, carbonate system). A detailed discussion is also pre-
sented on issues that are ordinarily of primary interest in modeling studies. These
include reaeration, dissolved oxygen saturation, photosynthesis, deoxygenation, ben-
thic oxygen demand, coliform bacteria, algae, and zooplankton. These discussions
incorporate factors affecting the specific phenomena and methods of measurement in
addition to data on rate constants.
17.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.IDENTIFIERS/OPEN ENDED TERMS
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Planning
Simulation
Water quality
Nonpoint pollution
Model studies
12A
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