5EPA
United States
Environmental Protection
Agency
Environmental Research
Laboratory
Athens GA 3061 3
EPA/600/3-85/040
June 1985
Research and Development
Rates, Constants, and
Kinetics
Formulations in
Surface Water Quality
Modeling
(Second Edition)
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EPA/600/3-85/040
June 1985
RATES, CONSTANTS, AND KINETICS FORMULATIONS
IN SURFACE WATER QUALITY MODELING
(SECOND EDITION)
By
George L. Bowie, William B. Mills, Donald B. Porcella,
Carrie L. Campbell, James R. Pagenkopf, Gretchen L. Rif-pp,
Kay M. Johnson, Peter W.H. Chan, Steven A. Gherini
Tetra Tech, Incorporated
Lafayette, California 94549
and
Charles E. Chamberlin
Humboldt State University
Arcata, California 95521
Contract 68-03-3131
Project Officer
Thomas 0. Barnwell, Jr.
Technology Development and Applications Branch
Environmental Research Laboratory
Athens, Georgia 30613
ENVIRONMENTAL RESEARCH LABORATORY
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
ATHENS, GEORGIA 30613
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DISCLAIMER
The information in this document has been funded wholly or in part by
the United States Environmental Protection Agency under Contract No. 68-03-
3131 to Tetra Tech, Incorporated. It has been subject to the Agency's peer
and administrative review, and it has been approved for publication as an
EPA document. Mention of trade names or commercial products does not con-
stitute endorsement or recommendation for use by the U.S. Environmental
Protection Agency.
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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
control officials achieve water quality goals.
Basin planning requires a set of analysis procedures that can provide
an assessment on the current state of the environment and a means of predic-
ting the effectiveness of alternative pollution control strategies. This
report contains a revised and updated compilation and discussion of rates,
constants, and kinetics formulations that have been used to accomplish these
tasks. It is directed, toward all water quality planners who must interpret
technical information from many sources and recommend the most prudent course
of action that will minimize the cost of implementation of a pollutant con-
trol activity and maximize the environmental benefits to the community.
Rosemarie C. Russo
Director
Environmental Research Laboratory
Athens, Georgia
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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: dispersion, heat budgets, dissolved oxygen saturation, reaeration,
CBOD decay, NBOD decay, sediment oxygen demand, photosynthesis, pH and
alkalinity, nutrients, algae, zooplankton, and coliform bacteria. Factors
affecting the specific phenomena and methods of measurement are discussed in
addition to data on rate constants.
This report was submitted in fulfillment of Contract No. 68-03-3131 by
Tetra Tech, Incorporated, under the sponsorship of the U.S. Environmental
Protection Agency. The report covers the period June 1983 to April 1985t
and work was completed as of April 1985.
IV
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CONTENTS
Fdreword 111
Abstract |v
Figures V1<1'
Tables x]
Acknowledgments XV1'
1. Introduction 1
1.1 Background 1
1.2 Purpose and Use of This Manual 2
1.3 Scope and Arrangement of Manual 2
1.4 General Observations on Model Formulations, Rate Constants,
and Coefficients 3
1.5 References. 4
2. Physical Processes 6
2.1 Introduction 6
2.2 Advective Transport , 11
2.3 Dispersive Transport 17
2.4 Surface Heat Budget 61
2.5 References 76
3. Dissolved Oxygen 90
3.1 Dissolved Oxygen Saturation 90
3.2 Reaeration 101
3.3 Carbonaceous Deoxygenation . 135
3.4 Nitrogenous Biochemical Oxygen Demand 158
3.5 Sediment Oxygen Demand (SOD) 173
3.6 Photosynthesis and Respiration 188
3.7 References ........ 205
4. pH and Alkalinity 231
4.1 Introduction 231
4.2 Carbonate Alkalinity System 231
4.3 Extended Alkalinity Approach 236
4.4 Summary 241
4.5 References 242
5. Nutrients 244
5.1 Introduction 244
5.2 Nutrient Cycles 245
5.3 General Modeling Approach for All Nutrients 247
5.4 Temperature Effects 253
5.5 Carbon Transformations 255
5.6 Nitrogen Transformations 257
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5.7 Phosphorus Transformations ,. 265
5.8 Silicon Transformations 265
5.9 Algal Uptake 268
5.10 Excretion 271
5.11 Sediment Release 272
5.12 Summary 273
5.13 References 274
6. Algae 279
6.1 Introduction 279
6.2 Modeling Approaches 281
6.3 Cell Composition 285
6.4 Growth 287
6.5 Respiration and Excretion . 342
6.6 Settling 345
6.7 Nonpredatory Mortality 351
6.8 Grazing 357
6.9 Summary 363
6.10 References 365
7. Zooplankton 375
7.1 Introduction 375
7.2 Temperature Effects 376
7.3 Growth 378
7.4 Respiration and Mortality 400
7.5 Predatory Mortality 409
7.6 Summary 416
7.7 References 418
8. Coliform Bacteria 424
8.1 Introduction 424
8.2 Composition and Assay 426
8.3 Modeling Coliforms 428
8.4 Summary 449
8.5 References . 450
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FIGURES
Number Page
2-1 One-dimensional geometric representation for river
systems. .... ................... 9
2-2 Two-dimensional geometric representation for lake
systems. ...
2-3 Oscillation of velocity component about a mean value . . 18
2-4 Diffusion coefficients characteristic of various
environments ............. . ........ 20
2-5 Dependence of the horizontal diffusion coefficient
on the scale of the phenomenon ............ 31
2-6 Okubo's diffusion data and 4/3 power lines ....... 32
2-7 Factors contributing to tidal ly averaged dispersion
coefficients in the estuarine environment ....... 42
2-8 Dispersion coefficients in streams ........... 54
2-9 Clear sky solar radiation according to Hamon, Weiss
and Wilson (1954) ................... 65
3-1 Predicte'd reaeration coefficients as a function of
depth from thirteen predictive equations ....... HI
3-2 Comparisons of predicted and observed reaeration
coefficients for the formula of Dobbins (1965) (a)
and Parkhurst-Pomeroy (1972) (b) ........... 112
3-3 Formula of Bennett and Rathbun (1972) compared against
observed reaeration coefficients . .......... 113
3-4 Calculated versus experimental reaeration coefficients
for equations of (a) Tsivoglou and Wallace (1972),
(b) Padden and Gloyna (1971), and (c) Parkhurst and
Pomeroy (1972) .................... 114
Vll
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Number
FIGURES (continued)
Page
3-5 Reaeration coefficient versus energy dissipation
(a) for flow rates between 10 and 280 cfs and
(b) for flow rates less than 10 cfs .......... 115
3-6 Field data considered by three different investigations. H6
3-7 Reaeration coefficient (I/day) vs. depth and velocity
using the suggested method of Covar (1976) ...... H7
3-8 Ratio of reaeration coefficient under windy conditions
to reaeration coefficient without wind, as a
function of wind speed ....... . ........ 123
3-9 Division of head loss structures by dam type ...... 126
3-10 Sources and sinks of carbonaceous BOD in the aquatic
environment ...................... 137
3-11 Deoxygenation coefficient (k ,) as a function of depth. . 147
3-12 Example computation of kR based on BOD measurements of
stream water ..................... 156
3-13 Effect of reduced nitrogen concentration on
nitrification rate as reported by Borchardt (1966) . . 162
3-14 Effect of temperature on nitrification as reported
by Borchardt (1966) .................. 166
3-15 pH dependence of nitrification ............. 167
3-16 Nitrogenous biochemical oxygen demand versus travel
time in Shenandoah River ............... 171
3-17 Diurnal variation of (P-R) in Truckee River near
Station 2B ...................... 198
3-18 Concept of Stokes total time derivative ......... 199
3-19 Algal productivity and chlorophyll relationships for
streams ..................... ... 202
4-1 [SCn - SC.J plotted against reported alkalinity ..... 239
5-1 Nutrient interactions for carbon, nitrogen, and
phosphorus ...................... 246
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FIGURES (continued)
Number Page
5-2 Nutrient interactions for silica 247
5-3 Nitrogen cycle 248
5-4 Phosphorus cycle 249
5-5 Effect of pH and temperature on unionized ammonia. . . . 264
6-1 Major types of temperature response curves for algal
growth 294
6-2 Envelope curve of algal growth rate versus temperature
for data compiled from many studies involving many
different species 298
6-3 Temperature-growth curves for major algal groups .... 304
6-4 Comparison of light response curves for algal growth . . 312
6-5 Michaelis-Menten saturation kinetics for algal growth
limitation by a single nutrient 324
7-1 Growth rate and grazing rate as a function of food
supply for zooplankton with constant filtration rates
and assimilation efficiencies 380
7-2 Comparison of the Ivlev and Michaelis-Menten functions
with the same half-saturation value 388
7-3 Comparison of reverse Michaelis-Menten formulation
(a) and Canale et aJL 's, (1975, 1976) formulation (b) for
filtration rate as a function of food concentration. . 394
7-4 Frequency histograms for zooplankton assimilation
efficiencies 403
7-5 Frequency histograms showing variations in zooplankton
assimilation efficiencies with different food types. . 404
7-6 Frequency histograms of zooplankton respiration rates. . 415
7-7 Frequency histogram of nonpredatory mortality rates
for zooplankton 416
8-1 Relationship between pathogen or virus decay rates and
coliform decay rates based on figure presented by
Chamberlin (1982) 425
ix
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FIGURES (continued)
Number
8-2 Typical mortality curves for coliforms as a function
of time
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TABLES
Number Page
2-1 Major Reviews of Modeling State-of-the-Art 7
2-2 Values for Empirical Coefficients a, and a? 28
2-3 Tidally Averaged Dispersion Coefficients for Selected
Estuaries 45
2-4 Tidally Averaged Dispersion Coefficients 46
2-5 References Related to Longitudinal Dispersion. ..... 49
2-6 Summary of Studies of Transverse Mixing in Streams ... 57
2-7 Transverse Mixing Coefficients in Natural Streams and
Channels 58
2-8 Summary of Field Data for Transverse Dispersion
Coefficients 59
2-9 Summary of Nondimensional Diffusion Factors in Natural
Streams 60
2-10 Values for Short Wave Radiation Coefficients A and B . . 66
2-11 Values for Empirical Coefficients 69
2-12 Evaporation Formula for Lakes and Reservoirs 71
3-1 Methods Used by Selected Models to Predict Dissolved
Oxygen Saturation 91
3-2 Solubility of Oxygen in Water Exposed to Water-
Saturated Air at 1.000 Atmosphere Pressure 93
3-3 Values for the Bracketed Quantity Shown in Equation 3-11
to be Used with the Corresponding Temperatures and
Pressures 95
3-4 Comparison of Dissolved Oxygen Saturation Values
from Ten Equations at 0.0 mg/1 Salinity and
1 atm Pressure 97
xi
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TABLES (continued)
Number
3-5 Comparison of Dissolved Oxygen Saturation Values from
Selected Equations at a Chloride Concentration of
20,000 mg/1 (36.1 ppt Salinity) and 1 atm Pressure . . 98
3-6 Reaeration Coefficients for Rivers and Streams 103
3-7 Summary of Studies which Reviewed Stream Reaeration
Coefficients ..... 108
3-8 Equations that Predict the Effects of Small Dams on
Stream Reaeration. . '25
3-9 Reported Values of Temperature Coefficient 127
3-10 Sources of Stram Reaeration Data 128
3-11 Reaeration Coefficients for Lakes 130
3-12 Reaeration Coefficients for Estuaries 132
3-13 Values of the Temperature Compensation Coefficient
Used for Carbonaceous BOD Decay. 142
3-14 Coefficient of Bed Activ.ity as a Function of Stream
Slope 143
3-15 Deoxygenation Rates for Selected U.S. Rivers 144
3-16 Expressions for Carbonaceous Oxygen Demand Used in
Water Quality Models ..... 150
3-17 Values of Kinetic Coefficients for Decay of
Carbonaceous BOD 152
3-18 Expressions for Nitrogenous Biochemical Oxidation
Rates Used in a Variety of Water Quality Models ... 160
3-19 Summary of Factors that Influence Nitrification 164
3-20 Temperature Correction Factor, #, for Nitrification. . . 165
3-21 Case Studies of Nitrification in Natural Waters 169
3-22 Summary of Nitrification Rates 172
xi i
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TABLES (continued)
Number Page
3-23 Some Typical Values of the Temperature Coefficient for
SOD Rate Coefficients Used in Water Quality Models . . 179
3-24 Model Formulations Commonly Used in SOD Computations . .
3-25 Average Values of Oxygen Uptake Rates of River Bottoms .
3-26 Measured Values of Sediment Oxygen Demand in Rivers
and Streams,
189
3-27 Measured Values of Sediment Oxygen Demand in Lakes
and Reservoirs .................... 190
3-28 Measured Values of Sediment Oxygen Demand in
Estuaries and Marine Systems ............. 19"!
3-29 Oxygen Produced per Mass of Algae ............ 192
3-30 Oxygen Consumed per Mass of Algae ............ 193
3-31 Summary of Methods to Predict Photosynthetic Oxygen
Production and Respiration without Simulating Algal
Growth and Death ................... 194
3-32 Photosynthetic Oxygen Production and Respiration
Rates in Rivers .................... 204
4-1 Options and their Required Input Parameters for PHCALC . 240
5-1 Comparison of Nutrient Models .............. 254
5-2 Rate Coefficients for Carbon Transformations ...... 256
5-3 Rate Coefficients for Nitrogen Transformations ..... 259
5-4 Rate Coefficients for Denitrification .......... 262
5-5 Rate Coefficients for Phosphorus Transformations .... 266
5-6 Rate Coefficients for Silica Transformations ...... 267
6-1 General Comparison of Algal Models ........... 283
6-2 Nutrient Composition of Algal Cells - Percent of
Dry Weight Biomass .................. 286
6-3 Nutrient Composition of Algal Cells - Ratio to Carbon. . 288
xiii
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TABLES (continued)
Number
6-4
6-5
6-6
6-7
6-8
6-9
6-10
6-11
6-12
6-13
6-14
6-15
6-16
6-17
6-18
6-19
6-20
7-1
7-2
7-3
7-4
Nutrient Composition of Algal Cells - Ratio to
Chlorophyll a
Algal Maximum Growth Rates
Comparison of Temperature Adjustment Functions for
Algal Growth
Comparison of Light Limitation Formulations
Algal Saturating Light Intensities
Half-saturation Constants for Light Limitation
Half-Saturation Constants for Michael is-Menten
Growth Formulations
Comparison of Algal Growth Formulations
Half-Saturation Constants for Variable Stoichiometry
Formulations
Minimum Cell Quotas
Maximum Internal Nutrient Concentrations
Maximum Nutrient Uptake Rates
Half-Saturation Constants for Nutrient Uptake
Model-Specific Nutrient Uptake Parameters
Algal Respiration Rates
Phytoplankton Settling Velocities
Algal Nonpredatory Mortality Rates
General Comparison of Zooplankton Models
Zooplankton Maximum Consumption Rates
Zooplankton Maximum Filtration Rates
Zooplankton Maximum Growth Rates
289
291
3n5
319
320
321
327
329
332
333
334
339
340
341
346
352
358
377
382
383
384
xiv
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TABLES (continued)
Number Page
7-5 Comparison of Temperature Adjustment Functions for
Zooplankton Growth and Consumption .......... 385
7-6 Michael is-Menten Half-Saturation Constants for
Zooplankton Consumption and Growth .... ......
7-7 Threshold Feeding Levels for Zooplankton . . ......
7-8 Comparison of Zooplankton Growth Formulations ...... 396
7-9 Zooplankton Assimilation Efficiencies .......... 401
7-10 Zooplankton Respiration Rates .............. 410
7-11 Zooplankton Mortality Rates ............... 413
8-1 Factors Affecting Coliform Disappearance Rates ..... 435
8-2 Coliform Bacteria Freshwater Disappearance Rates
Measured Iji situ ................... 436
8-3 Values for Coliform-Specific Disappearance Rates Used
in Several Modeling Studies ......... ...... 437
8-4 Nutrient KS Values for Escherichia Coli ......... 438
8-5 Values of Co, C', k, and k1 from the Ohio River ..... 441
8-6 Summary of Decay Rates of Tc, Fc, and Fs ........ 442
8-7 Comparison of kg Estimates Based on Chamberlin and
Mitchell (1978) with Additional Values ........ 444
8-8 Parameter Estimates for Lantrip (1983) Multi-Factor
Decay Models ..................... 446
8-9 Experimental Hourly T-90 Values ............ 449
xv
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ACKNOWLEDGMENTS
Special thanks are due to the participants in the Rates Manual Workshop
held at Tetra Tech, Lafayette during November 29-30, 1984 to review the
first draft of the report. These include Ray Whittemore (National Council
of the Paper Industry for Air and Stream Improvement, Inc.)» Steve
McCutcheon (U.S. Geological Survey), Kent Thornton (Ford, Thornton, and
Norton, Inc.), Vic Bierman (U.S. Environmental Protection Agency), Tom
Barnwell (U.S. Environmental Protection Agency), Don Scavia (National
Oceanic and Atmospheric Administration), Tom Gallagher (HydroQual, Inc.),
Carl Chen (Systech, Inc.), Jerry Orlob (University of California, Davis),
Lam Lau (National Water Research Institute, Ontario, Canada), Bill Walker
(private consultant), and Peter Shanahan (Environmental Research and
Technology, Inc.). Betsy Southerland (U.S. Environmental Protection Agency)
was unable to attend but also participated in the review of the first draft.
The above individuals provided many useful comments and references which
were incorporated in the final report.
Numerous other individuals also provided reference materials during the
preparation of this report. Although they are too numerous to mention here,
their input is greatly appreciated.
We would also like to thank Trudy Rokas, Susan Madson, Gloria Sellers,
Belinda Hamm, and Faye Connaway for typing and preparing the report, and
Marilyn Davies for providing most of the graphics.
Finally, thanks are due most to Tom Barnwell and the U.S. Environmental
Protection Agency, Environmental Research Laboratory, Athens, Georgia for
both funding the project and providing technical input and guidance.
xvi
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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 two
decades. Simulation techniques offer an integrated and relatively sound
course for evaluating wasteload abatement alternatives. Predictions of
system behavior based upon mathematical simulation techniques may be
misleading, however, particularly if the physical mechanisms involved are
not accurately represented 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 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. There is, however, a need for
a document that summarizes the rate constants and coefficients
(e.g., nitrification rates and reaeration rates) needed in the models. This
document is intended to satisfy that need.
The first edition of this document was published seven years ago (Zison
e_t jil_., 1978). Because an extensive body of literature on rate constants
and modeling formulations has emerged since that time, the United States
Environmental Protection Agency has sponsored an updating of the manual. In
addition, a workshop was held to evaluate the manual, to review the
1
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formulations and associated coefficients and rate values, and to provide
further data for the final document. As a result of the literature review
and workshop, a substantially new manual has been produced.
1.2 PURPOSE AND USE OF THIS MANUAL
This manual is intended for use by practitioners as a handbook—a
convenient reference on modeling formulations, constants, and rates commonly
used in surface water quality simulations. Guidance is provided in
selecting appropriate formulations or values of rate constants for specific
applications. The manual also provides a range of coefficient values that
can be used to perform sensitivity analyses. Where appropriate, measurement
techniques for rate constants are also discussed.
It was impossible, however, to encompass all formulations or to examine
all 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 a sampling from a larger set of
data. It should also be noted that there are often very real limitations
involved in using literature values for rates rather than observed system
values. It is hoped 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.
1.3 SCOPE AND ARRANGEMENT OF MANUAL
In preparing this manual, an attempt was made to present a
comprehensive set of formulations and associated constants. In contrast to
the first edition (Zison e_t a_L,1978), the manual has been divided into
sections containing specific topics. Following this introduction, chapters
are presented that discuss the following topics:
i Physical processes of dispersion and temperature
® Dissolved oxygen
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• pH and alkalinity
c Nutrients
• Algae
• Zooplankton
• Coliforms
The parameters that are addressed in this manual are those that
traditionally have been the focus of water quality management and the focus
for control of conventional pollutants. These include temperature,
dissolved oxygen, nutrients and eutrophication, and coliform bacteria.
Higher organisms (fish, benthos) are not discussed, nor are the details of
higher trophic levels in ecosystem models. Also, hydrodynamic processes,
although important, are not dealt with in detail.
1.4 GENERAL OBSERVATIONS ON MODEL FORMULATIONS, RATE CONSTANTS, AND
COEFFICIENTS
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 formulations 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 value's for parameters. Even where the parameter is to be
chosen by calibration, it is clearly important to establish whether the
calibrated value is within a reasonable range or not. Recent references on
model calibration include Thomann (1983), National Council on Air and
Stream Improvement (1982), and Beck (1983). Users should be aware that an
acceptable model calibration does not imply that the model has predictive
capability. The model may contain incorrect mechanisms, and agreement
between model predictions and observations could have been obtained through
an unrealistic choice of parameter values. Further, the future status of
the prototype may be controlled by processes not even simulated in the
model.
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Values of many constants and coefficients are dependent upon the way
they are used in modeling formulations. For example, while pollutant
dispersion is an observable physical process, modeling this process is
partly a mathematical construct. Therefore, constants that are used to
represent the process (i.e., dispersion coefficients) cannot be chosen
purely on the basis of physics since they also depend on the modeling
approach. For example, to determine the dispersion coefficients in a
model application to an estuary, both the time and length scales of the
model must be considered. Whether the model is tidally averaged or
simulates intra-tidal variations, and whether the model is 1-, 2-, or 3-
dimensional, all influence the value of the appropriate dispersion
coefficient for that model. Ford and Thornton (1979) discuss scale effects
in ecological models, and conclude that inconsistent scales for the
hydrodynamics, chemistry, and biology may produce erroneous model
predictions.
Since coefficient values are never known with certainty, modelers are
constantly faced with the question of how accurately rate constants should
be known. The relationship between uncertainty in coefficient values and
model predictions can be evaluated by performing sensitivity analyses. For
models with few parameters, sensitivity analyses are generally
straightforward. However, for complex models, sensitivity analyses are no
longer straightforward since many dynamic interactions are involved.
Sensitivity analyses are discussed in detail in Thornton and Lessem (1976),
Thornton (1983), and Beck (1983).
1.5 REFERENCES
Beck, M.B. 1983. Sensitivity Analysis, Calibration, and Validation.
Iji: Mathematical Modeling of Water Quality: Streams, Lakes, and Reservoirs.
International Institute for Applied Systems Analysis. Editor: G.T. Orlob.
Ford, D.E. and K.S. Thornton. 1979. Time and Length Scales for the One-
Dimensional Assumption and its Relation to Ecological Models. Water
Resources Research. Vol. 15, No. 1, pp. 113-120.
National Council of the Paper Industry for Air and Stream Improvement, Inc.
1982. A Study of the Selection, Calibration and Verification of
Mathematical Water Quality Models. NCASI Tech. Bull. 367, New York.
4
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Thomann, R.V. 1982. Verification of Water Quality Models. Journal of
Environmental Engineering Division, ASCE. Vol. 108, No. EE5, October,
pp. 923-940.
Thornton, K.W. and A.S. Lessem. 1976. Sensitivity Analysis of the Water
Quality for River-Reservoir Systems Model. U.S. Army Waterways Experiment
Station. Misc. Paper Y-76-4.
Thornton, K.W. 1983. Sensitivity Analysis in Simulation Experimentation.
Encyclopedia of Systems and Control. Pergamon Press.
Zison, S.W., W.B. Mills, D. Deimer, and C.W. Chen. 1978. Rates Constants
and Kinetics Formulations in Surface Water Quality Modeling. Prepared by
Tetra Tech, Inc., Lafayette, CA, for Environmental Research Laboratory,
USEPA, Athens, GA. EPA-600/3-78-105. 335 pp.
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Chapter 2
PHYSICAL PROCESSES
2.1 INTRODUCTION
The purpose of this chapter is to give the reader an overview of how
the major physical processes are incorporated into water quality and
ecosystem simulations. Since a detailed review is beyond the scope of this
report, the reader is encouraged to review the articles listed in Table 2-1
which represent several of the more complete and recent reviews of the
state-of-the-art in physical process modeling.
Physical processes often simulated in water quality models include flow
and circulation patterns, mixing and dispersion, water temperature, and the
density distribution (which is a function of temperature, salinity, and
suspended solids concentrations) over the water column. It is stressed that
quality predictions are very dependent upon the physical processes and how
well these are represented in the water quality simulations. Despite this
dependence, the modeler often is forced to make a trade-off between
acceptable degree of detail in water quality vs. physical process simulation
due to cost or other restrictions. It is desirable from the standpoint of
both the engineer and ecosystem analyst, therefore, to select the simplest
model which satisfies the temporal and spatial resolution required for water
quality and/or ecosystem simulation. For example, the optimum time step for
dynamic simulation of a fully-mixed impoundment would be 3-6 hours for
capturing diurnal fluctuations, and daily or weekly for strongly stratified
impoundments which normally exhibit slowly varying conditions. In terms of
spatial resolution required, the analyst should take advantage of the
possible simplifications of dominate physical characteristics (i.e.,
physical shape, stratified layers, mixing zones, etc.).
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TABLE 2-1. MAJOR REVIEWS OF MODELING STATE-OF-THE-ART
French, R.H. 1983. Lake Modeling: State-of-the-Art. In: CRC Critical
Reviews in Environmental Control, Vol. 13, Issue 4, pgs. 311-357.
Harleman, D.R.F. 1982. Hydrothermal Analysis of Lakes and Reservoirs.
Journal of the Hydraulics Division, ASCE. Vol. 108, No. HY3, pp. 302-325.
Johnson, B. 1982. A Review of Multi-Dimensional Numerical Modeling of
Reservoir Hydrodynamics. U.S. Army Corps of Engineers, Waterways Experiment
Station.
Fischer, H.B., List, E.J., Koh, R.C.Y. Imberger, J., and Brooks, N.H. 1979.
Mixing in Inland and Coastal Waters. Academic Press, New York.
Hinwood, J.B., and Wall is, I.G. 1975. Review of Models of Tidal Waters.
Journal of the Hydraulics Division, ASCE, Vol. 101, No. HY11, Proc. Paper
11693, November, 1975.
Orlob, G.T., ed. 1984. Mathematical Modeling of Water Quality: Streams,
Lakes, and Reservoirs. John Wiley and Sons, Wiley-Interscience, N.Y., N.Y.
Elhadi, N. , A. Harrington, I. Hill, Y.L. Lau, B.G. Krishnappan. 1984.
River Mixing: A State-of-the-Art Report. Canadian Journal of Civil
Engineering. Vol. 11, No. 11, pp. 585-609,
2.1.1 Geometric Representation
2.1,1.1 Zero-Dimensional Models
Zero-di me nsiona 1 models are used to estimate spatially averaged
pollutant 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).
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2.1.1.2 One-Dimensional Models
Most river models use a one-dimensional representation, where the
system geometry is formulated conceptually as a linear network of segments
or volume sections (see Figure 2-1). Variation of water quality
parameters occur longitudinally (in the x-direction) as the water is
transported out of one segment and into the next. The one-dimensional
approach is also a popular method for simulation of small, deep lakes, where
the vertical variation of temperature and other quality parameters is
represented by a network of vertically stacked horizontal slices or volume
segments.
2.1.1.3 Multi-Dimensional Models
Water quality models of lakes and estuaries are often two- or three-
dimensional in order to represent the spatial heterogeneity of the water
bodies. Depending on the system, two-dimensional representations include a
vertical dimension with longitudinal segmentation for deep and narrow lakes,
reservoirs, or estuaries (Figure 2-2).
Three-dimensional spatial representations have been used to model
overall lake circulation patterns. Part of the reason for this need is the
concern with the water quality of the near-shore zone as well as deep zones
of lakes. In addition, the different water quality interactions in these
zones can lead to changes in the overall lake quality that cannot be
predicted without this spatial definition.
2.1.2 Temporal Variation
Ecological models are 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 conditions
which are unaffected by internal conditions of the system, have constant
values. Inflows and outflows are discharged to and drawn from the system at
8
-------
I-
Control Volume, Xj
X.| * volume element
QO = water withdrawals from element X,
xi 1
QI = water discharged to element X.
x. i
E = evaporation
F = precipitation
Q.j+1 = advective flow to downstream element
Q. -| = advective flow from upstream element
AX = longitudinal dimension of element
QI
Figure 2-1. One-dimensional geometric representation for river systems (Chen and Wells, 1975).
-------
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 represent
mean conditions observed in a system, and therefore the model cannot
simulate cyclic phenomena.
A wide variety of planning problems can be analyzed by use of steady-
state or quasi-steady (slowly varying) 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 or quasi-steady
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 distribution and recycle.
tributary
inflow
tributary
inflow
horizontal
segmentation
vertical
segmentation
outflow
Figure 2-2. Two-dimensional geometric representation for lake
systems (Baca and Arnett, 1976).
10
-------
Many water quality or ecological models for rivers and lakes are
concerned with the simulation of water quality variables that have
substantial temporal variation and are linked to processes and variables
that vary considerably. For example, the seasonal distribution of certain
biological species and related abiotic substances may be of major
importance. In these instances, dynamic models are required.
The process of selecting the correct time and length scales and then
matching these with an appropriate model demands both an a priori
understanding of the dominate physical, chemical and biological processes
occurring within the system, as well as an understanding of a given model's
theoretical basis and practical application limits. Proper guidance for
model selection and application best comes from a thorough review of the
relevant literature appropriate to the specific problem at hand. Ford and
Thornton (1979), for example, present a detailed discussion of the time and
length scales appropriate for the vertical one-dimensional modeling approach
for reservoirs and lakes. The references presented in Table 2-1 as well as
several others cited throughout this chapter discuss model compatibility
requirements for various water body types and applications.
The remainder of this chapter focuses on advective transport,
dispersive transport, and the surface heat budget.
2.2 ADVECTIVE TRANSPORT
The concentration of a substance at a particular site within a system
is continually modified by the physical processes of advection and
dispersion 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:
V«9C W5C d IV <9Cx d ,y 6>CN 9 ,y <9Cv _
~^ + ~dl ~ at (Kx di' ~ w ( y dy> " dl (Kz dl> ~
11
-------
where c = mean concentration of constituent, mass/volume
u = mean velocity in x-direction, length/time
v = mean velocity in y-direction, length/time
w = mean velocity in z-direction, length/time
2
K ,K ,K = eddy diffusion coefficients, length /time
i/
2S = sum of source/sink rates, mass/(volume-time)
t = time
Difficulties exist in trying to correctly quantify the terms in this
equation. The unsteady velocity field (u,v,w) is usually evaluated
separately from Equation (2-1) so that the pollutant concentration, c, can
be prescribed. The complete evaluation of the velocity field involves the
simultaneous solution of the momentum, continuity, hydrostatic, and state
equations in three dimensions (see Leendertse and Liu, 1975; Hinwood and
Wallis, 1975). Although sophisticated hydrodynamic models are available,
the detail and expense of applying such models are often not justified in
water quality computations, especially for long term or steady-state
simulations where only average flow values are required. For example, the
annual thermal cycle for a strongly stratified reservoir with a relatively
low inflow to volume ratio has been successfully simulated with only a crude
one-dimensional, steady-state application of mass and energy conservation
principles. On the other hand, simulation of large, weakly stratified
impoundments dominated by wind driven circulation may require the ultimate,
full representation of the unsteady velocity field in three dimensions.
The purpose of this section is to briefly familiarize the reader with
the various types of approaches used to evaluate the velocity field in water
quality models. Most hydrodynamic models internally calculate hydrodynamics
with relatively little user control except for specification of forcing
conditions such as wind, tides, inflows, outflows and bottom friction.
Thus, the following paragraphs present only a summary discussion of the
approaches used, organized according to the dimensional treatment of the
model.
12
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2.2.1 Empirical Specification of Advection
This is the crudest approach, in that the advective terms of the
advection-diffusion equation (Equation 2-1) are directly specified from
field data. Empirical specification is quite common in water quality models
for rivers, but is also often used in steady-state or slowly-varying estuary
water quality models (e.g., O'Connor et: aj_. (1973)). In these types of
estuary models, specification of the dispersion coefficients is critical
since dispersion must account for the mixing which in reality is caused by
the oscillatory tidal action. Due to the highly empirical treatment of the
physical processes in such models, the model "predictions" remain valid for
only those conditions measured in the field. These models cannot predict
water quality variations under other conditions, thus increasing the demand
on field data requirements. Examples of models representative of the above
approach include .0' Connor et al_. (1973) and Tetra Tech (1977).
2.2.2 Zero Dimensional Models for Lakes
A coarse representation of the water system as a continuously stirred
tank reactor (CSTR) is often sufficient 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.
Examples of zero-dimensional models include lake models by Anderson et al .
(1976).
2.2.3 One-Dimensional Models for Lakes
For lake systems with long residence times and stratification in the
vertical direction, vertical one-dimensional representations are common.
Horizontal layers are imposed and advective transport is assumed to occur
only in the vertical direction.
13
-------
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 vertical advective flow
(via the continuity equation) between all elements above the level of entry.
The elements below this level, containing higher density water, are assumed
to be unaffected. Examples of one-dimensional lake models include Lombardo
(1972, 1973), Baca et _al_. (1974), Chen and Orlob (1975), Thomann et aJL
(1975), Imberger et al_. (1977), HEC (1974), Markofsky and Harleman (1973),
and CE-QUAL-R1 (1982).
For lake or reservoir systems exhibiting complex horizontal interflows,
inflows, and outflows, semi-empirical formulations have been developed to
distribute inflows and to determine the vertical location from which
outflows arise, depending on stratification conditions. Examples include
models by U.S. Army Corps of Engineers (1974), Baca e_t aj_. (1976), and Tetra
Tech (1976).
2.2.4 One-Dimensional Models for Rivers
Most river models represent river 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. In general, there are two approaches to
treat the advection in river models. One approach requires field
calibration of the river flow properties by measuring flows and cross
sectional geometry at each model segment over a range of flow magnitudes. A
power series can then be developed for each cross section to interpolate or
extrapolate for other flow events. Such dn approach is especially
appropriate for rivers exhibiting complex hydraulic properties (i.e.,
supercritical flows, cascades, etc.) and when steady state solutions are of
interest. Examples of such models include Tetra Tech (1977).
A second, more rigorous approach for simulating river advection
involves the simultaneous solution of the continuity and momentum equations
for the portion of the river under study. This approach is considered more
14
-------
"predictive" than the former since empirical flow data are required only for
model calibration and verification. It is also more accurate and
appropriate for use in transient water quality simulations. In either case,
however, geometrical data on the cross-sectional shapes of the river are
required. Examples of models representative of the latter approach include
Brocard and Harleman (1976), and Peterson et al_. (1973).
2.2.5 One-Dimensional and Pseudo-Two-Dimensional Models for Estuaries
A natural extension of the one-dimensional river model has been to
estuary systems, either as a one-dimensional representation for narrow
estuaries or as a system of multiple interconnecting one-di rnens i onal
channels for pseudo-representation of wider or multi-channeled estuaries.
In either case, advection is determined through the simultaneous solution of
the continuity and momentum equations together with appropriate tidal
boundary conditions. These types of models are generally quite flexible in
their ability to handle multiple inflows, transient boundary conditions, and
complex geometrical configurations. Two primary approaches include the
"link-node" network models by Water Resources Engineers (WRE) (1972), and
the finite element model (Galerkin Method) by Harleman et aj_. (1977).
2.2.6 Two-Dimensional Vertically Averaged Models for Lakes and Estuaries
Vertically averaged, two-dimensional models have proven to be quite
useful, especially in modeling the hydrodynamics and water quality of
relatively shallow estuaries and wind-driven lakes. The crucial assumption
of these models is the vertically well-mixed layer that allows for vertical
integration of the continuity, momentum, and mass-transport equations. Such
models are frequently employed to provide the horizontal advection for water
quality models since they are relatively inexpensive to operate compared to
the alternatives of large scale field measurement programs or fully three-
dimensional model treatments. There exist well over fifty models which
would fit into the two-dimensional, vertically averaged classif'ication.
Examples of models that have been widely used and publicized include Wang
and Connor (1975), Leendertse (1970), Taylor and Pagenkopf (1980), and
Simons (1976).
15
-------
2.2.7 Two-Dimensional Laterally Averaged Models for Reservoirs and
Estuaries
In recent years, laterally averaged models have become standard
simulation techniques for reservoirs or estuaries which exhibit significant
vertical and longitudinal variations in density and water quality
conditions. The two-dimensional laterally averaged models require the
assumption of uniform lateral mixing in the cross channel direction.
Although this simplification eliminates one horizontal dimension, the
solution of the motion equations in the remaining longitudinal and vertical
dimensions requires a much more rigorous approach than for the two-
dimensional vertically averaged models. In order to correctly simulate the
vertical effects of density gradients on the hydrodynamics and mass
transport, both the motion (continuity and momentum) and advective-diffusion
equations must be solved simultaneously. In addition, such models must also
treat the vertical eddy viscosity (momentum transfer due to velocity
gradients) and eddy diffusivity (mass transfer due to concentration
gradients) coefficients, which are directly related to the degree of
internal mixing and the density structure over the water column.
Mathematical treatment of vertical diffusion and vertical momentum transfer
varies greatly between models, and will be discussed further in this
document. Examples of laterally averaged reservoir models include Edinger
and Buchak (1979) and Norton e_t aK (1973). Examples of laterally averaged
models developed for estuaries include Blumberg (1977), Najarian et al.
(1982) and Wang (1979).
2.2.8 Three-dimensional Models for Lakes and Estuaries
Fully three-dimensional and layered models have been the subject of
considerable attention over the last decade. Although still a developing
field, there are a number of models which have been applied to estuary,
ocean, and lake systems with moderate success. As with laterally
averaged two-dimensional models, the main technical difficulty in this
approach is in the specification of the internal turbulent momentum transfer
and mass diffusivities, which are ideally calibrated with field
16
-------
observations, thus making availability of adequate prototype data an
important consideration. An additional factor of great importance is the
relatively large computation cost of running three-dimensional models,
especially for long-term water quality simulations. In many cases, the
effort and cost of running such models is difficult to justify from purely a
water quality standpoint. However, as computational costs continue to
decrease and sophistication of numerical techniques increases, such models
will eventually play an important role in supplying the large scale
hydrodynamic regimes in water quality simulations. Examples of the more
prominent three-dimensional models include Blumberg and Mel lor (1978),
Leendertse and Liu (1975), Sheng and Butler (1982), Simons (1976) and King
(1982).
2.3 DISPERSIVE TRANSPORT
2.3.1 Introduction
The purpose of this section is to show how dispersive transport terms
are incorporated into the equations of motion and continuity by temporal and
spatial averaging (a detailed discussion of this subject is also given by
Fischer e^t al. (1979)). A consequence of temporal averaging of either
instantaneous velocity or concentration is to produce a smoothed velocity or.
concentration response curve over time. Figure 2-3 illustrates both
instantaneous velocity and time-smoothed curves. The velocities V and ¥ are
related by
V = V + V (2-2)
where V = instantaneous velocity, length/time
V = time-smoothed velocity, length/time
V = velocity deviation from the time-smoothed velocity,
length/time
The velocity component V is a random component of velocity which vanishes
when averaged over the appropriate time interval (i.e., V = 0).
17
-------
By averaging, the stochastic components are removed from the momentum
and mass conservation equations. However, cross product terms appear in the
equations, such as V'V and V'V in the case of the momentum equation, and
XX X j
V'C' and V'C' in the case of the mass conservation equation (where C1 is the
instantaneous concentration fluctuation, and V^ and V^ are the random
velocity deviations in the x and y directions, respectively). In the case
of the momentum equation these terms are called turbulent momentum fluxes,
and in the case of the mass conservation equation they are called turbulent
mass fluxes. It is through these terms that eddy viscosity and eddy
diffusivity enter into the momentum and mass conservation equations.
To solve the time-smoothed equations, the time averaged cross product
terms are expressed as functions of time averaged variables. Numerous
empirical expressions have been developed to do this. The expressions most
often applied are analogous to Newton's law of viscosity in the case of
turbulent momentum transport and Pick's law of diffusion in the case of
turbulent mass transfer. Expressed quantitatively these relationships are
of the form:
o
3
01
V= time smoothed velocity
V-instantaneous velocity
V
V'-V-V
TIME
Figure 2-3.
Oscillation of velocity component about a mean
value (redrawn after Bird et ji]_., 1960).
18
-------
9MV
- (2-3)
V^CT = K g (2-4)
where E = eddy viscosity, mass/( length-time)
2
K = eddy diffusivity, length /time
V = time smoothed velocity in the x direction, length/time
/\
£ = time smoothed concentration, mass/volume
P = mass density, mass/ volume
In natural water bodies the turbulent viscosity and diffusivity given
by Equations (2-3) and (2-4) swamp their counterparts on the molecular
level. The relative magnitude between eddy di ffusivities and molecular
diffusion coefficients is depicted graphically in Figure 2-4.
In addition to temporal averaging, spatial averaging is often used to
simplify three dimensional models to two or one dimensions. As an
illustration consider the vertically averaged mass transport equation.
Before averaging, the governing three dimensional mass transport equation is
typically written as:
c c)
.(V . (V . (QZ) (2.5)
where c = the local (time smoothed) concentration, mass/volume
u,v,w = the local (time smoothed) water velocities,
length/time
2
Q ,Q ,Q, = the local diffusive fluxes, mass/(length -time)
x y z
Before spatial averaging, the local concentration and velocities can be
expressed by a vertically averaged term and a deviation term:
19
-------
10
6-
104-
102-
o
CD
CO
\
CN
E
o
UJ
o
LU
O
o
Z
O
00
10°
10-2-
io-4-
10-6-
10-8-
10-10.
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 H2O
•Proteins in H20
THERMAL DIFFUSION
Salts in
• Ionic Solutes in
Porus Media
(Sediments,Soils)
Figure 2-4. Diffusion coefficients characteristic of various
environments (redrawn after Lerman, 1971).
20
-------
c = ch + ch
u = uh + uh
v = vh + v^ (2-6)
where c,u,v = previously defined
ch = vertically averaged concentration, mass/volume
h
o
c/ = deviation from c, at any point in the water column,
mass/volume
u. ,v. = vertically averaged water velocities, length/time
h h
= /udz, /vdz (2-8)
0 0
u/jV/ = deviation from u. , v. at any point in the water column,
length/time
h = local water depth, length
Equation (2-5) can now be written in its vertically averaged form:
9ch d(u.c.) <9(v.c- ) r-
- + -*- * - - - J "x^h)*
It is noted that when vertical integration is performed on the three-
dimensional mass conservation equation, cross product terms appear in the
resulting two-dimensional equation, just as they do when temporal averaging
is done because vertical gradients generally exist in both the concentration
and velocity fields. The horizontal turbulent diffusion fluxes Q r Q are
usually expressed in terms of the gradients of the vertically averaged
concentration and the turbulent diffusion coefficient, which in general form
are written:
21
-------
dC. dC.
«x • -xx -5? - Sy -5? <2-10a)
dC. dC.
Q = -e —- - e —- (2-10b)
y yx <9x yy ay
where e , e , e , e = turbulent eddy diffusion coefficients
xx xy yx yy
By analogy, the horizontal transport terms, u^c^ and v^c^, associated with
vertical velocity variations (i.e., differential advection), are expressed
by means of the shear dispersion coefficients:
i -
- Exy I*
. 9C. . dC.
,, i „, _ rd n ra n /0
Vh - -Eyx ~ai - Eyy ~gy
where E ,E ,E ,E = shear dispersion coefficients
xx xy yx yy
By combining Equations (2-10) and (2-11), the final form of the vertically
averaged mass transport equation can be written as:
h + D
dt dx. ay dx \ xx dx xy dy
-2.1 n h —Li + n h
9y \ xy <9x yy
where D ,D ,D = dispersion coefficients which account for mass transport
xx xy yy
due to both concentration and velocity gradients over the vertical.
One-dimensional mass conservation equations result when a second
spatial averaging is performed. The one-dimensional equations express
changes along the main flow axis. As expected, the diffusion terms are
again different from their two-dimensional counterparts. Consequently, the
type and magnitude of the diffusion terms appearing in the simulation
22
-------
equations depends not only on the water body characteristics, but the model
used to simulate the water body.
2.3.2 Vertical Dispersive Transport
Vertical dispersive transport of momentum and mass becomes important in
lakes or estuaries characterized by moderate to great depths. In a lake
environment, vertical mixing is generally caused by wind action on the
surface through which eddy turbulence is transmitted to the deeper layers by
the action of shear stresses. In estuaries, typically the vertical mixing
is induced by the internal turbulence driven by the tidal flows, in addition
to surface wind effects. Similarly, the internal mixing in deep reservoirs
is primarily caused by the flow-through action. In each environment,
however, the amount of vertical mixing is controlled, to a large extent, by
the degree of density stratification in the water body.
Treatment of vertical mixing processes in mathematical models is
generally achieved through the specification of vertical eddy viscosity (E )
and eddy diffusivity (K ) terms, as previously discussed. As observed by
McCutcheon (1983), however, there is little consensus on what values the
vertical eddy coefficients should have and how eddy viscosity and eddy
diffusivity are related. At present, the procedure for estimating these
coefficients is generally limited to empirical techniques that range from
specifying a constant E and K to relating to some measure of stability,
i.e., the Richardson number Ri . In this approach, the ratio of the
coefficients for stratified flow to the coefficients for unstratified flow
is expressed as a function of stability f(s):
= f2(s), (2-14)
and EvQ=PrKvo (2-15)
23
-------
where E = kzu+(l - z/h) for shear layers and Pr = the Prandtl or Schmidt
vo *
number, which is generally close to unity for open-channel shear flow
(Watanabe et a_l_., 1984).
In addition
k = von Karman's constant (-0.4), dimension!ess
z = distance above the bottom, length
u* = shear velocity, length/time
h = depth of flow, length
A widely used formula which relates E /E to stability involves the
Munk and Anderson (1948) formulation (as reported by McCutcheon (1983)):
Ev/Evo = (1 + 10 Ri)'1/2 (2-16)
and
Kv/KvQ = (1 + 3.33 RiT3/2 (2-17)
O
where Ri = ^ _ /f|M j s dimension! ess (2-18)
P = density, mass/volume
u = the mean horizontal velocity at a point z above the bottom,
1ength/time
2
g = acceleration of gravity, length/time
As reported by McCutcheon (1983), in a recent review of available data
for stratified water flows (Delft, 1979) Equations (2-16) and (2-17) were
found to fit the data better than several other similar formulations.
Models by Waldrop (1978), Harper and Waldrop (1981), Edinger and Buchak
(1979), O'Connor and Lung (1981), Najarian et al_. (1982), and Heinrich, Lick
and Paul (1981) use this scheme. In some models, the coefficients and
exponents in Equation (2-16) and (2-17) are not adjusted, and any
discrepancies between field measurements and model predictions are
attributed to the inexactness of the model. In other models, the
coefficients and exponents are calibrated on a site specific basis.
24
-------
For model simulations of mixing through and below the thermocline, the
Munk and Anderson type formulas appear to be less adequate (McCutcheon,
1983). Odd and Rodger (1978) developed site specific eddy viscosity
formulations for the Great Ouse Estuary in Britain:
Ev/Evo = (1 + bRi)" for Ri£l (2
and
EV/EVQ = (1 + b)"n for Ri>l (2-20)
where b and n are coefficients. The depth varying Ri is used if Ri
increases continuously starting at the bed and extending over 75 percent or
more of the depth. Where a significant peak in Ri occurs in' the vertical
gradient, that peak Ri is used for all depths in the equation above.
McCormick and Scavia (1981) make a correction for K in Lake Ontario and
Lake Washington studies that is similar to corrections of E by Odd and
Rodger. Above the hypo! imnion , they apply a modification of the Kent and
Pritchard (1959) equation:
Kv - u,/0RQ (2-21)
where RQ = -kz2 £ f£ / u^ (2-22)
/3 = constant
Below the thermocline a constant K was specified for Lake Ontario. In
Lake Washington, Equation (2-21) and (2-22) were applied throughout- the
depth. In Lake Washington bottom shear was important for mixing as opposed
to deeper Lake Ontario where surface wind shear dominated the mixing
process.
Several other formulations for E and K have been developed which are
not based on the Munk and Anderson equations. For example, Blumberg (1977),
in his laterally averaged model of the Potomac River Estuary, employed an
expression for K which uses a ratio of Ri to a critical Ri to relate K to
stabil ity, where:
25
-------
Kv = Klz2(¥l2|"l'1-^-^' (2-23)
where K.. is a turbulence constant which must be determined by calibration,
and Ri is the critical Richardson number, which is the value of Ri at which
mixing ceases due to strong stratification. Blumberg also related E to K
through the following formulation:
E = K (1 + Ri) for 0 < Ri < Ri (2-24a)
V V t*
EV = KV = 0 for Ri > Ric (2-24b)
Using the above formulations, Blumberg obtained reasonably good
comparisons for salinity distributions in the Potomac River.
Simons (1973) based his formulation for K for Lake Ontario on the
results of dye diffusion experiments performed by Kullenberg j?t al_. (1973)
where K is expressed as:
K = C | | (2-25)
where C = an empirical constant, (2~8)10~
W = wind speed, length/time
N = Brunt-Vaisala frequency, ^ — » time"
an " ^Z -I
sp- = vertical shear of the current, time
OL
Simons also assumed that the vertical eddy viscosity coefficient was based
on a similar relationship.
The above formulation is a result of experiments performed in fjords,
coastal and open sea areas, as well as from Lake Ontario, and is generally
valid for expressing the vertical mixing in the upper 20 m for persistent
winds above 4-5 m/sec. The lower value of the numerical constant refers to
the lake data and the higher value to the oceanic data.
26
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For low and varying wind speeds Equation (2-25) will not be valid
(Murthy and Okubo, 1975). In these cases the internal mixing is considered
to be governed by local processes, i.e., the energy source is the kinetic
energy fluctuations. Kullenberg (as reported by Murthy and Okubo (1975))
proposed the following relation for weak local winds:
10 10 10
where q L = V/ + V/
x y
V' ,V' = velocity fluctuations in the x and y directions,
A y
respectively, length/time
Equation (2-26) is representative of the vertical mixing both above and
below the thermocline under conditions of low wind speeds.
Tetra Tech (1975) has used the following empirical expressions for
utation of the vertical eddy thermal diffusiv
dimensional hydrodynamic simulation of Lake Ontario.
computation of the vertical eddy thermal diffusivity, K , in their three-
(2'27)
where p = density of fresh water at 4°C, mass/volume
2
TS = surface wind stress, mass/(length-time ).
Lake systems that are represented geometrically as a series of
completely 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 coefficient:
Kv = al + a2 Vw d-' (2-28)
27
-------
p
where K = vertical dispersion coefficient, m /sec
z = depth, m
Vw = wind speed, m/sec
d = depth of thermocline, m
2
a,,a? = empirical constants, m /sec and m respectively
The following table of values (Table 2-2) for a, and a2, as given by
Baca and Arnett (1976), were obtained from previous model applications.
TABLE 2-2. VALUES FOR EMPIRICAL COEFFICIENTS a1 and
Lake
Max.
Description Depth (m)
•j
(m /sec)
(m)
American Falls
Lake Washington
Lake Mendota
Lake Wingra
Long Lake
well-mixed
stratified
stratified
well-mixed
linearly
stratified
18
65
24
5
54
1 x 10"5
1 x 10'6
5 x 10"7
5 x 10"5
5 x 10"6
1 x 10"4
1 x 10"5
5 x 10"5
2 x 10"4
5 x 10"5
The vertical eddy viscosity and eddy diffusivity concepts continue to
be practical and are a popular means for simplifications of the momentum and
mass conservation equations. As pointed out by Sheng and Butler (1982) and
McCutcheon (1983), however, a wide variety exists among the various forms of
the vertical turbulence stability functions determined empirically by
various investigators, and suggest that the appropriate stability function
is dependent on the type of numerical scheme used and the nature of the
water body under study. The wide variation in formulations is, in part, due
to the attempt to fit empirical functions determined under specific field
conditions to a wider range of water body types and internal mixing
phenomena. Due to the possibility of applying an empirical relationship
28
-------
beyond its valid limits, site-specific testing of formulations for E and K
will probably be required in most model applications.
The above discussion has concentrated on the eddy diffusion concept on
which many models are based. However, an alternative to this approach is
the mixed layer concept which has been successfully applied by numerous
investigators to predict the vertical temperature regime of lakes and
reservoirs. As summarized by Harleman (1982), the mixed layer or integral
energy concept involves the following: the turbulent kinetic energy (TKE)
generated by the surface wind stress is transported downward and acts to mix
the upper water column layer. At the interface between the upper mixed
layer and the lower quiescent layer, the remaining TKE, plus any that may be
locally generated by interfacial shear (minus dissipation effects), is
transferred into potential energy by entraining quiescent fluid from below
the interface into the mixed layer. This entrainment, in addition to any
vertical advective flows, determines the thickness of the mixed layer. TKE
is also produced by convective currents which occur during periods of
cooling, and can contribute to the mixing process. Also, the total vertical
heat balance due to surface heat flux and internal absorption must be
considered in evaluating the vertical density distribution and potential
energy of the water column. A discussion of the mixed layer model approach
can be found in Harleman (1982), French (1983) and Ford and Stefan (1980).
Models based on this approach include those by Stefan and Ford (1980),
Hurley-Octavio et _§]_. (1977), Imberger et aj_. (1977) and CE-QUAL-R1 (1982).
2.3.3 Horizontal Eddy Diffusive Transport
Generally, horizontal eddy diffusivity is several orders of magnitude
greater than the vertical eddy diffusivity (see Figure 2-4). 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
p
cm /sec. Unlike diffusive transport in open-channel type flows, diffusion
in open water, such as in lakes and oceanic regimes, cannot be effectively
related to the mean flow characteristics (Watanabe §t aj_., 1984). Oceanic
or lake turbulence represents a spectrum of different eddies resulting from
29
-------
the breakdown of large-scale circulations in shore zones and by wind and
wave induced circulations. Attempts to analyze this phenomenon have
demonstrated that the horizontal diffusive transport, D, depends on the
length scale L of the phenomenon. The most widely used formula is the four-
thirds power law:
Dh = ADL4/3 (2-29)
where AD is the dissipation parameter (of the order .005, with Dh in
cm2/sec). The length scale L is loosely defined depending on the nature of
the diffusion phenomenon. For a waste discharge in the ocean, for example,
L is often estimated based on the diffuser length, which is typically the
order of a kilometer. Another example is to estimate L based on the length
of the tidal excursion in estuaries or coastal areas. When using Equation
(2-29) to estimate the diffusion coefficient in two or three-dimensional
numerical models, the length scale is often taken as the size of the
horizontal grid spacing, since this approximates the minimum scale of eddies
which can be reproduced in the model.
Useful summaries of lake and ocean diffusion data are given by Yudelson
(1967), Okubo (1968) and Osmidov (1968). Okubo and Osmidov (1970) have
graphed the empirical relationship between the horizontal eddy diffusivity
and the length scale, as shown in Figure 2-5. According to Figure 2-5:
Dh^2 x 10"3 L4/3 for L < 105cm
Dh~104 for 105 < L < 5 x 105 cm
Dh = 10"3 L4/3 for L > 5 x 105 cm (2-30)
2
where Dh is in cm /sec and L in cm. Based on these empirical observations,
it is seen that the dissipation parameter of the four-thirds law decreases
at larger length scales.
A comprehensive collection of diffusion data in the ocean was presented
by Okubo (1971), who proposed as best fit to all the data the relation:
30
-------
= 0.01L
1.15
for 10 < L < 10
cm
(2-31)
which is graphed in Figure 2-6. According to Christodoulou et aj_. (1976),
the four-thirds law seems theoretically and experimentally acceptable for
expressing the horizontal eddy diffusivity in large lakes and in the ocean,
providing the length scales of interest are not of the order of the size of
the energy containing eddies. In addition, the four-thirds law is not fully
acceptable near the shore, due to the shoreline and bottom interference.
Two examples of the use of Equation (2-29) in lake models are in Lam
and Jacquet (1976) and Lick et aj_. (1976). Lam and Jacquet obtained the
following formulation for the horizontal eddy diffusivity for lakes, based
on experimental results:
109
108-
107-
o 106-
0)
CO
CN
e io5-
o
si
21 104'
103<
102'
10 •
10s
10'
10C
LENGTH,cm
Figure 2-5. Dependence of the horizontal diffusion
coefficient on the scale of the phenom-
enon (after Okubo and Osmidov (1970)).
31
-------
10M
o
0)
CO
E
o
10
c I
5-l
A North Sea, 1964
€> North Sea, 1962
D New York Bight
• Off Cape Kennedy
O Off California
+ Banana River
x Other
103
Figure 2-6.
104
io5 io6
LENGTH, cm
107
IO8
Okubo's diffusion data and 4/3 power lines
(after Okubo (1971)).
= .0056L
1.3 *
(2-32)
2
where D, = horizontal eddy diffusivity, cm /sec
L = length scale of grid, cm
As reported by Lam and Jacquet, for a grid size larger than 20 km, the
fi ?
diffusivity is expected to be essentially constant (10 cm /sec).
Lick (1976) used a similar formulation after Osmidov (1968), Stommel
(1949), Orlob (1959), Okubo (1971) and Csanady (1973):
32
-------
Dh = a E1/3 L4/3 (2-33)
where a = constant of proportionality, of the order 0.1
E = rate of energy dissipation per unit mass
n r o
Observations by Lick indicated values of 10 to 10 cm /sec for D, for the
overall circulation in the Great Lakes with smaller values indicated in the
near-shore regions.
The above relationships can be used as a general guide to evaluate the
horizontal diffusivities in a numerical model, where the grid size may be
regarded as the approximate length scale of diffusion. However, as pointed
out by Murthy and Okubo (1977): (1) the data upon which these empirical
relations are obtained do not represent diffusion under severe weather
conditions, and thus may include a bias towards relatively mild conditions;
(2) the horizontal diffusivity can vary (depending primarily upon the
environmental conditions) by an order of magnitude for the same length scale
of diffusion; (3) the definition of the length scale of diffusion for the
horizontal diffusivity is somewhat arbitrary; and (4) the horizontal
diffusivity varies by an order of magnitude between the upper and lower
layers of oceans and deep lakes. Thus, to develop reliable three-
dimensional models the scale and stability dependence of eddy diffusivities
and the large variability of the magnitude of the eddy diffusivity with
depth and environmental factors (wind, waves, inflows, etc.) must somehow be
incorporated into the models.
The formulations for horizontal eddy diffusivity discussed above are
generally representative of empirical (physical) diffusion behavior and are
most compatible with a three-dimensional approach. As previously discussed,
horizontal dispersion is the "effective diffusion" that occurs in two-
dimensional mass transport equations that have been integrated over the
depth. Thus the horizontal dispersion must account for both horizontal eddy
diffusivity due to horizontal turbulence and concentration gradients, as
well as the effective spreading caused by velocity and concentration
variations over the vertical. In addition, any simplifications in the
33
-------
velocity field used in modeling must also be accounted for in the dispersion
coefficients. The less detailed the flow field is modeled, the larger the
dispersion coefficient needs to be to provide for the spreading that would
occur under the actual circulation (Christodoulou and Pearce, 1975).
Therefore, the dispersion coefficients are characteristic not only of the
flow conditions to be simulated, but more significantly of the way the
process is modeled. Hence these coefficients are model-dependent and
difficult to quantify in any general, theoretical manner. For example, many
two-dimensional models use a constant dispersion coefficient over the
whole model domain as well as over time despite the fact that dispersion
changes both spatially and temporally as the circulation features change.
An example of a model that uses constant dispersion coefficients is
Chri stodoul ou et al_. (1976).
One two-dimensional model which utilizes variable dispersion
coefficients (velocity dependent) in time and space is the finite difference
model by Taylor and Pagenkopf (1981). They utilize Elder's (1959)
relationship for anisotropic flow where the dispersion of a substance is
proportional to the friction velocity, u*, and the water depth, h, as
follows:
Dr = 5.9 u*h (2-34)
Dn = 0.2 u^h (2-35)
p
where D = dispersion coefficient along the flow axis, length /time
D = dispersion coefficient normal to the flow axis,
2
length /time
u* = V8 I U I ' length/time
f = Darcy-Weisbach friction factor, dimensionless
I "*"!
IUI = absolute value of mean velocity along flow axis,
length/time
The above relationship is incorporated into the two dimensional mass
conservation equation resulting in an anisotropic mixing process which
calculates a dispersion coefficient at each time step and node as a function
34
-------
of the instantaneous flow conditions. The expressions used for the
dispersion coefficients in the model are as follows:
Dxx = ( + q) (5'9 - 5'7 Sl
Dxy = 11.4 \ (qx2 + qy2) sin & cos 6 (2-37)
Dyy = (q + q) (5'9 ' 5'7 cos^ (2'38)
where D ,D Y..,D = dispersion coefficients
•*•* *y yy _-.
e = tan" (q /q )
j *
q = flow in x direction
q = flow in y direction
The above model has been successfully tested agai nst dye diffusion
experiments in Flushing Bay, New York, and in Community Harbor, Sau di Arabia
(Pagenkopf and Taylor (1985); Taylor and Pagenkopf (1981)).
A two-dimensional, finite element water quality model was developed by
Chen jet _al_. (1979), based on the earlier model by Chri stodoulou et a! .
(1976). They provided for flow-dependent anisotropic dispersion
coefficients by using the following relationships:
D = ^V.4 + .-** (2-39)
x ul/6 -x ;
n
Dy -
* * ** **
where -~ and ; ar-e user-defined constants as are e and £ , the latter
x y x y
being provided for additional dispersion effects such as wind and marine
traffic.
Whether the two-dimensional model in question utilizes constant or
flow-dependent dispersion coefficients, the dispersion mechanism is usually
somewhat dependent on factors typically beyond user control, such as
numerical i nstabi 1 i'ti es and grid sire averaging effects. It is therefore
-------
stressed that any application of a two-dimensional water quality model be
verified either through site-specific salinity or dye tracer data.
Naturally, when performing field tracer experiments the time and length
scales of the field phenomenon should be compatible with the time and length
scales to be represented in the model. For example, a dye study lasting
only a few hours is not valid for verification of a model using a daily
computational time step. Similarly, a dye study confined to a small portion
of a large lake or estuary will not allow for verification of the model over
the entire system.
2.3.4 Longitudinal Dispersive Transport in Estuaries
As previously discussed, longitudinal dispersion is the "effective
diffusion" that occurs in one-dimensional mass transport equations that have
been integrated over the cross sectional area perpendicular to flow. This
one-dimensional approach to modeling has often been applied to tidal and
nontidal rivers, and to estuaries.
The magnitude of the one-dimensional dispersion coefficient in
estuaries and tidal rivers is determined in part by the time saale over
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 therefore water quality relationships are resolved in greater detail and
hence, in such models, smaller dispersion coefficients are needed than in
those which, for example, average hydrodynamics 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-
dimensional 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
constituent 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
36
-------
interface between 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
time varying 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
location 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 conservative
constituent (salinity), empirical measurements are easily performed. 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. Empirical measurements can alternatively be
based, however, on dye release experiments.
2.3.4.1 Time Varying Longitudinal Dispersion
A model which is not averaged over the tidal cycle is more capable of
representing the mixing phenomena since it represents the time varying
advection in greater detail. However, the averaging effects of spatial
velocity gradients (shear) and density gradients must still be accounted
for. The specification of longitudinal dispersion coefficients is closely
associated with the type of mathematical techniques used in a given model.
Most of the model developments for one-dimensional representation of
estuaries has occurred in the early 1970's, and the most prominent
techniques are summarized below.
37
-------
The "link-node" or network model developed originally by WRE (1972) and
commonly known as the Dynamic Estuary Model (DEM) used the basic work of
Feigner and Harris (1970) to describe the numerical dispersion in the
constant density region of an estuary:
DL = Cl E1/3Le4/3 (2-41)
2
where D. = longitudinal dispersion coefficient, length /time
E = rate of energy dissipation per unit mass
L = mean size of eddies participating i'n the mixing process
C-, = function of relative channel roughness
For computational purposes, Feigner used the following simplification:
DL = 0.042 |u| R (2-42)
where R = hydraulic radius, ft
|u| = absolute value of velocity, ft/sec
There exists no corresponding formulation for the longitudinal
dispersion coefficient in the salinity intrusion regions of estuaries.
Rather, a careful calibration procedure is required using available salinity
data to prescribe the appropriate dispersion coefficients. Obviously, this
approach somewhat restricts the predictive nature of such models since a
substantial amount of empirical data is necessary for proper model
appl ication.
Similar versions of the DEM exist in one form or another. Not all
versions, however, include the option for specification of longitudinal
dispersion. This stems from the fact that considerable numerical dispersion
occurs in the DEM from the first order, explicit, finite difference
treatment of the advective transport terms. Feigner and Harris (1970) gave
some comparisons of different weightings of the first order differencing in
terms of trade-offs between numerical mixing, accuracy, and stability. Work
on this problem has been done by Bella and Grenney (1970) and a numerical
38
-------
estimate'of this dispersion can be given by the following equation:
num
\ \(l-2v) Ax - VAt (2-43)
where v represents the weighting coefficient assigned to the concentrations
of two adjacent nodes.
This equation shows that the numerical dispersion is a function of
Ax, At, and the velocity, V, which is a function of location and time. This
equation is useful for estimating the magnitude of numerical dispersion. It
illustrates the lack of control that the modeler has over this phenomena in
the DEM.
Daily and Harleman (1972) developed a network water quality model for
estuaries which uses a finite element numerical technique. The hydraulics
are coupled to the salinity through the density-gradient terms in the manner
formulated by Thatcher and Harleman (1972). The high accuracy finite element
Galerkin weighted residuals technique is relatively free of artificial
numerical dispersion. The longitudinal dispersion formulation combines both
the vertical shear effect and the vertical density-induced circulation
effect through the following expression:
0
D(x,t) = K|^|+ m DT (2-44)
d^ T
where D(x,t) = temporally and spatially varying dispersion coefficient,
ft2/sec.
§ = s/s where s(x,t) is the spatial and temporal
distribution of salinity, ppm
s = ocean salinity, ppm
8 = x/L
L = length of estuary, ft (to head of tide)
25/6
DT = Taylor's dispersion coefficient in ft /sec = 77 u nRh
u = u(x,t) tidal velocity, ft/sec
39
-------
n = Manning's friction coefficient
R, = hydraulic radius, ft.
2
K = estuary dispersion parameter in ft /sec = u L/1000
u = maximum ocean velocity at the ocean entrance, ft/sec
m =a multiplying factor for bends and channel
irregularities
One-dimensional, time varying modeling using this expression has been
performed for several estuaries, a recent example being an application
(Thatcher and Harleman, 1978) to the Delaware Estuary wherein the time-
varying calculations were made for a period'of an entire year in order to
provide a model for testing different water management policies.
For real time simulations in the constant density region of estuaries
and tidal rivers, the following expression has been proposed (TRACOR, 1971):
D, = 100 n U RU5/6 (2-45)
L max H v
where D. = longitudinal dispersion coefficient in the constant
2
density region, ft /sec
n = Manning's roughness coefficient, ft
Um,« = maximum tidal velocity, ft/sec
ffla X
RU = hydraulic radius, ft.
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
coefficients can be found by fitting the solution of the salinity mass
transport equation to the observed data. As reported in TRACOR (1971), this
technique 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:
40
-------
DL = 13000 (l - f ) (2-46)
where D. = real time longitudinal dispersion coefficient in salinity
L 2
intrusion region, ft /sec
L = length of entire tidal region of the estuary.
2 2
At the estuary mouth, D. was found to be 13,000 ft /sec or 40 mi /day
72
(1.2 x 10 cm /sec) by using the technique described above. Under the same
conditions in a constant density region, Equation (2-38) predicts D. =
2252
175 ft /sec, or 0.5 mi /day (1.6 x 10 cm /sec). This illustrates the large
difference that can be expected between the real time dispersion coefficient
in the salinity intrusion region of an estuary and in the constant density
region. For more detailed discussions of real time longitudinal dispersion
in estuaries, see Holley et al . (1970) and Fischer et al . (1979).
2.3.4.2 Steady State Longitudinal Dispersion
For tidal ly averaged or net nontidal flow simulations, the dispersion
coefficients must somehow include the effects of oscillatory tidal mixing
which has been averaged out of the hydrodynamics representation. No known
general analytical expressions exist for this coefficient. Hence, it is
cautioned and emphasized that steady-state dispersion coefficients 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
et jfl_. (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 2-7.
41
-------
Salinity Gradient Mixing
MOUTH
Figure 2-7.
HEAD
Factors contributing to tidally averaged dispersion
coefficients in the estuarine environment (modified
after Zison et al., 1977).
The expression used is:
(2-47)
where DL = tidally averaged dispersion coefficient, ft /sec
Cj = tidally-induced mixing coefficient (dimensionless)
y = tidally averaged depth, ft
|u| = tidally averaged absolute value of velocity, ft/sec
-------
The first term on the right side of Equation'(2-47) represents mixing
brought about by the oscillatory flows associated with the ebbing and
flooding of the tide. The second term represents additional mixing when
longitudinal salinity gradients are present. It is noted that, in practice
the above formulation requires careful calibration using field salinity data
due to the high empirical dependency of this relationship.
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:
D = Rs = rc(f-l) t (2-48)
L A(ds/dx) A(df/dx) { !
2
where D. = 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 = —jj—, unitless
a = normal ocean salinity of the coastal water into which the
estuary empties, mg/1
x = distance along estuary axis, ft.
D. can be calculated at any location within the estuary if the river
flow, cross-sectional area, and salinity or freshwater fraction
distributions are known.
The above method has certain pitfalls which are pointed out by Ward and
Fischer (1971) in their analysis of such an application to the Delaware
Estuary. They point out that the use of a dispersion coefficient
relationship, i.e., a functional relationship of dispersion to distance,
which is also directly related to the measured upstream freshwater inflow.
neglects entirely the basic response of the waterbody to variations in
freshwater inflow. Ward and Fischer show, for example, that it may take a
period of months for the estuary to adjust to a short period change in
43
-------
freshwater discharge and that any dispersion coefficient relationship based
on a simple correlation analyses may be seriously in error.
Hydroscience (1971) has collected values of tidally averaged dispersion
coefficients for numerous estuaries, and these values are shown in
Table 2-3.
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 2-4. Many values were developed using the fraction of
freshwater method just discussed. Additional values for the longitudinal
dispersion coefficient have been summarized in Fischer et a_l_. (1979).
2.3.4.3 The Lagrangian Method
The models discussed in previous sections of this chapter have all been
based on the Eulerian concept of assigning velocities and concentrations to
fixed points on a spatial grid. As previously discussed, the fixed grid
approach tends to introduce a fictitious "numerical" dispersion into the
mass transport solution since the length scale of the diffusion process is
somewhat artificially imposed depending on the grid detail. To avoid such a
problem, an alternative approach termed the Lagrangian method has been used
by Fischer (1972), Wallis (1974), and Spaulding and Ravish (1984) for models
of estuaries and tidal waters. Briefly, the Lagrangian method establishes
marked volumes of water, distributed along the channel axis, which are moved
along the channel at the mean flow velocity. Numerical diffusion is almost
entirely eliminated, since there is no allocation of concentrations to
specific grid points; rather, the "grid" is a set of moving points which
represent the centers of the marked volumes. Longitudinal dispersion
between marked volumes can be set according to appropriate empirical or
theoretical diffusion behavior (Fischer ert _al_., 1979). The Lagrangian
method has been primarily applied to channelized estuaries such as the
Suisun Marsh (Fischer, 1977) and Bolinas Lagoon (Fischer, 1972), and more
recently has been extended by Spaulding and Ravish (1984) to simulate
particulate transport in three dimensions.
44
-------
TABLE 2-3. TIDALLY AVERAGED DISPERSION COEFFICIENTS FOR SELECTED ESTUARIES
(from Hydroscience, 1971)
Estuary
Delaware River
Hudson River (NY)
East River (NY)
Cooper River (SC)
Savannah River (GA, SC)
Lower Raritan River (NJ)
South River (NJ)
Houston Ship Channel (TX)
Cape Fear River (NC)
Potomac River (VA)
Compton Creek (NJ)
Wapplnger and
Flshkin Creek (NY)
Freshwater
Inflow
(cfs)
2,500
5,000
0
10,000
7,000
150
23
900
1,000
550
10
2
Low Flow
Net Nontldal
Velocity (fps)
Head - Mouth
0.12-1.000
0.037
0.0
0.25
0.7-0.17
0.047-0.029
0.01
0.05
0.48-0.03
0.006-0.003
0.10-0.013
0.004-0.001
Dispersion
Coefficient
(mlVday)*
5
20
10
30
10-20
5
5
27
2-10
1-10
1
0.5-1
(ft2/sec)
1610
6450
3230
9680
3230-6450
1610
1610
8710
645-3230
320-3230
320
160-320
V
*1 m12/day = 322.67 ft2/sec.
2.3.5 Dispersive Transport in Rivers
2.3.5.1 Introduction
Dispersive transport in rivers is typically, but not always, modeled
using a one-dimensional equation such as:
dc + uac
9t <9x
where C = concentration of solute, mass/length
U = cross-sectional averaged velocity, length/time
2
D, = longitudinal dispersion coefficient, length /time
x = longitudinal coordinate, length
t = time
45
(2-49)
-------
TABLE 2-4. TIDALLY AVERAGED DISPERSION COEFFICIENTS
(FROM OFFICER, 1976)
Estuary
Dispersion
Coefficient Range
(ft2/sec)
Comments
San Francisco Bay, CA
Southern Arm
Northern Arm
Hudson River, NY
Narrows of Mercey, UK 1,430-4,000
Potomac River, MD
65-650
Severn Estuary, UK
Tay Estuary, UK
Thames Estuary, UK
Yaquina Estuary
Measurements were made at slack
200-2,000 water over a period of one to a
500-20,000 few days. The fraction of
freshwater method was used.
Measurements were taken over
three tidal cycles at 25 loca-
tions.
4,800-16,000 The dispersion coefficient was
derived by assuming D. to be
constant for the reach studied,
and that it varied only with flow.
A good relationship resulted be-
tween D. and flow, substantiating
the assbraption.
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 D. .
Bowden recalculated D, values
originally determined by Stommel,
who had used the fraction of
freshwater method. Bowden in-
cluded the freshwater inflows from
tributaries, which produced the
larger estimates of D. .
530-1,600 The fraction of freshwater method
was used. At a given location, D.
was found to vary with freshwater
inflow rate.
3,640 Calculations were performed using
(high flow) the fraction of freshwater method,
600-1000 between 10 and 30 miles below
(low flow) London Bridge.
650-9,200 The dispersion coefficients for
(high flow) high flow conditions were substan-
140-1,060 tially higher than for low flow
(low flow) conditions, at the same locations.
The fraction of freshwater method
was used.
75-750
46
-------
Because of the difficulty of accurately solving Equation (2-49)
numerically, some researchers (e.g., Jobson, 1980a; Jobson and Rathbun,
1985) have chosen a Lagrangian approach, where the coordinate system is
allowed to move with the local stream velocity. Using this approach,
Equation (2-49) become:
^L = 9_ /D 9
dt 9£ ( L 9
t
where £ = x- J* Udr
o
The numerically troublesome advective term does not appear in Equation (2-
50). In general , the equation can be solved more easily and with more
accuracy than Equation (2-49).
A second method used to simulate dispersive transport in rivers is to
consider lateral mixing in addition to longitudinal mixing. A typical form
of the two-dimensional equation is:
9L , i \ 9C 9 / 9C\ , 9 i 9C\ /o n \
di + u(y) ft " ^ (Sc w} + ey (ey 9^ (2^l]
where u(y) = depth averaged velocity of water, which is a function of
y, and is no longer the cross-sectional averaged
velocity, length/time
= depth averaged longitudinal diffusion coefficient,
x 2
length /time
e = depth averaged lateral diffusion coefficient,
y 2
length /time
y = lateral coordinate, length
Note that longitudinal dispersion coefficient, DL, in Equation (2-49) is not
the same as the longitudinal diffusion coefficient, e , in Equation
A
Typically, D»E.
47
-------
2.3.5.2 Longitudinal Dispersion in Rivers
Fischer (1966, 1967a, 1967b, 1968) has performed much of the earlier
research on longitudinal dispersion in natural channels. Prior to Fischer,
Taylor (1954) studied dispersion in straight pipes and Elder (1959) studied
dispersion in an infinitely wide open channel. More recently Fischer crt al .
(1979) and Elhadi ejt al_. (1984) have provided a comprehensive review of
dispersion processes.
Researchers have shown that Equation (2-49) is valid only after some
initial mixing length, often called the Taylor length or convective period.
While the convective period has been a topic of active research in the
literature (e.g., Fischer, 1967a and b; McQuivey and Keefer, 1976a; Chatwin,
1980), this concept is not embodied in one-dimensional" water quality models
in general use.
Table 2-5 summarizes references on stream dispersion. • The references
include information from at least one of the following areas:
• methods to predict D. , typically for model applications
t methods to measure DL from field data
• data summaries of dispersion coefficients
t approaches used to simulate dispersion in,a non-Fickian
manner.
Bansal (1971), Elhadi and Davar (1976), Elhadi jet jK (1984) also provide
reviews of stream dispersion.
To date, the predictive capabilities of expressions for dispersion
coefficients have not been thoroughly tested. However, it is known that the
Taylor (1954) or Elder (1959) formulas do not accurately predict dispersion
coefficients for natural streams. Glover (1964) found that dispersion
coefficients in natural streams were likely to be 10 to 40 times higher than
predicted by the Taylor or Elder equations. The lateral variation in stream
velocity is the primary reason for the increased dispersion not accounted
for by Taylor and Elder. Fischer (1967a) quantified the contribution of the
48
-------
TABLE 2-5. REFERENCES RELATED TO LONGITUDINAL DISPERSION
Reference
Comments
Taylor (1954)
Elder (1959)
Glover (1964)
Krenkel (1960)
Parker (1961)
Fischer (1967a, 1967b)
Elhadi and Davar (1976)
Fischer (1968)
Bansal (1971)
Godfrey and Frederick (1970)
Thackston (1966)
D. =10.1R u*; pipe flow.
D. = 5.93Hu*; lateral velocity variation
not considered.
D, = SOORu*; natural streams.
DL G^H1'2^0'3; two-dimensional channel.
(E = USg)
DL 14.3R
; open channel flow.
; concentration variances
are measured after an initial period.
Long tails may introduce some error.
DL =— / q'(y)dz/
A o
/d(y)
(U(y,z)-u)dz
o
This formula considers the effects of
lateral velocity changes.
2
DL=-fdT
q'(y)dy.
D. = Q.3u'l~ -^- ; a simplification of the
integral equation above
Fischer also discusses another method for
determining D. called the routing procedure.
Reviewed many methods to predict D. . Found
D,/(Hu*) is not a constant as reported by
many researchers.
Field measurements of D, were made in
the Green and Duwamish Rivers.
-------
TABLE 2-5. (continued)
Reference
Comments
Thackston and Krenkel (1967)
Miller and Richardson (1974)
McQuivey and Keefer (1974)
The limitations of dispersion equations
which do not consider lateral velocity
variations are discussed. Site specific
measurements of D. are recommended.
In laboratory experiments, D. varied from
0.6 ft /sec to 66 ftVsec.
Dispersion coefficient data were reviewed,
including hydraulic data, for 17 rivers.
ir
D, = 0.66 ^.
3
'
McQuivey and Keefer (1976b)
Liu (1977)
0.058
Dispersion tests performed in the
Mississippi River are summarized.
Fischer (1975)
Hays et _al.. (1966)
Thackston and Schelle (1970)
Day (1975)
Day and Wood (1976)
Liu and Cheng (1980)
B *• 0.18/VqRsV'5
I U/
Summary of D. values also reported.
D,
_ O.OlluV
Liu (1977) shows this is a special case of
his formula when /3= 0.011.
Several conceptual models of mass exchange
with dead zones are presented and the
Fickian Equation is modified to include mass
transfer to and from dead zones.
Application of Hays et ^1.
model to TVA stream (Tata.
(1966) dead zone
Longitudinal dispersion of fluid particles
in small mountain streams 1n New Zealand was
investigated. It was shown that the
dispersion coefficient increased with
distance and never approached an asymptotic
value.
Longitudinal dispersion of fluid particles
in the Missouri River and in a small moun-
tain stream was Investigated. The dis-
persing particles were shown to behave
differently from the Taylor type model. A
method to predict dispersion was developed.
A non-Fickian model 1s presented to predict
stream dispersion.
(continued)
50
-------
TABLE 2-5. (continued)
Reference
Comments
Sabol and Nordin (1978)
Valentine and Wood (1977)
Valentine and Wood (1979)
Rutherford, Taylor, Davies (1980)
Beltaos (1980a)
A modified model of stream dispersion is
presented that includes the effects of
storage along the bed and banks.
Effects of dead zones on stream dispersion
are addressed
Experimental results are provided to show
how dead zones modify longitudinal dis-
persion.
A hybrid method is discussed to predict
dispersion in the Waikato River, New
Zealand.
Dispersion processes in streams are
reviewed and it is shown that many
experimental results do not comply with
Fickian dispersion theory. A non-Fickian
dispersion model is proposed.
Beltaos (1982)
Bajraktarevic - Dobran (1982)
Beer and Young (1983)
Jobson (1980a)
Jobson (1985) and
McBride and Rutherford (1984)
Jobson and Rathbun (1985)
Dispersion in steep mountain
streams is examined.
Fischer's methods are successfully applied
to predict dispersion in mountainous
streams.
Methods are developed to predict dispersion
in rivers including the effects of dead
zones, using a (j.n.m) model.
The Fickian Equation is solved with a
Lagrangian scheme to avoid lumping numerical
dispersion with actual physical dispersion.
See Jobson (1980b).
Determined that D. and coefficients for
nonconservative water quality constituents
could be determined simultaneously during
calibration. D. determined by this method
is in good agreement with literature values
(Jobson) or match D, values determined from
dye studies (McBride and Rutherford).
Numerical dispersion minimized with a
Lagrangian routing procedure that provides
more consistent estimates of D. than the
method of moments for pool and riffle
streams. Applying this procedure to peak
dye concentrations yielded D. to within 10X
of estimates based on the entire
concentration-time curves.
(continued)
51
-------
TABLE 2-5. (continued)
Footnotes:
A = cross-sectional area
b = channel width
£ = wave velocity
d(y) - depth of water at y
E = rate of energy dissipation per unit mass of fluid
E = lateral turbulent mixing coefficient
H = stream depth
K = regional dispersion factor
1 lateral distance from location of maximum velocity
2
a - variance of distance - concentration curves
? ?
°t2' °tl = variance °f time concentration curves
ty, "L = mean times of passage
p = mass density of water
Q = discharge at steady base flow
q'(y) integral of velocity deviation on depth
R = hydraulic radius
Rp - pipe radius
S - slope of energy gradient at steady base flow
U = mean velocity of flow in reach
i
u = deviation of velocity from cross-sectional mean
U = mean velocity of flow at sampling point
u* = shear velocity
ft - coefficient of viscosity of water
W = channel width at steady base flow
52
-------
lateral velocity variation on stream dispersion.
A number of the formulas in Table 2-5 are of the type DL/(U*H) =
constant. However, several researchers, including Bansal (1971), Elhadi and
Davar (1976), and Beltaos (1978a) have shown that the ratio DL/(U*H) is not
a constant. Figure 2-8 shows this ratio can vary by several orders of
magnitude.
Two widely used methods of predicting the longitudinal dispersion
coefficients were developed by Liu (1977) and Fischer (1975) and are shown
in Table 2-5. Liu showed that Fischer's method is identical to his own
when j3 = 0.011.
Although numerous researchers (e.g., Sabol and Nordin, 1978) have shown
how to include the effects of dead zones on dispersive transport, this
refinement does not yet appear to be in general use in water quality models
today. In fact, some water quality models do not include dispersion at all
(at least physical dispersion; numerical dispersion may be present,
depending on the solution technique used).
Dispersion can be neglected in certain circumstances with very little
effect on the predicted concentration distributions. Thomann (1973), Li
(1972), and Ruthven (1971) have investigated the influence of dispersion.
Ruthven gave a particularly simple expression for a pollutant which decays
at a rate k. If
kD 1
-T < -b = -04
u
then the concentration profile will be affected by no more than 10 percent
if dispersion is ignored. Consider, for example, a decaying pollutant with
2
k = 0.5/day in a stream where U = 1 fps and D, = 500 ft /sec. The ratio
p L
kD,/U =.003, which indicates that dispersion can be ignored. This guideline
assumes that the pollutant is being continuously released and conditions are
at steady state. The basic presumption is that if the concentration
gradient is small enough, the dispersive transport is also small, and
53
-------
o Godfrey and Frederick (1970)
A Glover (1964), rectangular flume
A Glover (1964), triangular flume
D Glover (1964),South Platte River
• Glover (1964), Mohawk River
• Yotsukura et. al.(1970), Missouri River
• Fischer, Sacramento River (see Sooky,1969)
v Fischer (1968),Green-Duwamish River
+ Fischer (1967),trapezoidal flume
T Smooth .meandering flume
$ Rough, meandering flume1
» Smooth,meandering flume
* Thackston and Schnelle (1969)
° Hou and Christensen (1976)
Width/radius of curvature -0.28
Width/radius of curvature -0.14
10,000 q
1,000 -
100 -
10 -
10 100
WIDTH/DEPTH
1000
Figure 2-8. Dispersion coefficients in streams (Beltaos, 1978a),
54
-------
perhaps negligible. On the other hand when pollutants are spilled,
concentration gradients are large and dispersion is not negligible.
Thomann (1973) investigated the importance of longitudinal dispersion
in rivers that received time variable waste loadings, and therefore produced
concentration gradients in the rivers. His results showed that for small
rivers, dispersion may be important when the waste loads vary with periods
of 7 days or less. For large rivers, dispersion was found to be important
whenever the waste load was time-variable.
2.3.5.3 Lateral Dispersion in Rivers
Although two-dimensional water quality models are less widely used in
rivers than one-dimensional models, lateral mixing has been the topic of
considerable research. Models that simulate lateral mixing are particularly
useful in wide rivers where the one dimensional approach may not be
applicable. Vertical mixing is rarely simulated in river modeling because
the time required for vertical mixing is usually very rapid compared to the
time required for lateral mixing. Thermal plumes are an exception.
An example of a model that simulates lateral mixing in rivers is the
RIVMIX model of Krishnappan and Lau (1982). The model is particularly
useful for delineating mixing zones or regulating the rate of pollutant
discharge so that concentrations outside of the mixing zones are limited to
allowable values.
When lateral and longitudinal mixing are both simulated, the x and y
coordinates are generally assumed to continuously change to be oriented in
the longitudinal and transverse directions. Although Equation (2-51)
should rigorously contain metric factors (Fukuoka and Sayre, 1973) to
account for these continuous changes, modelers typically assume the metric
factors are unity.
Lateral mixing coefficients are usually presented in one of the
following two forms:
55
-------
y (2-52)
or 2
Dy=£f (2-53)
2
where e = lateral mixing coefficient, length /time
y 52
D = lateral diffusion factor, length /time
H = water depth, length
a,jS = coefficients that vary from river to river
u+ = friction velocity, length/time
3
Q = stream flow, length /time
W = width of river, length
D and e are related by, the following formula:
J J
Dy = HUmx£y (2-54)
where m = average metric value in x- direction (~1)
A
Equation (2-52) is generally the most widely used of the two formulas.
Equation (2-53) is used when the two-dimensional convective-diffusion
equation is expressed in terms of cumulative discharge (Yotsukura and Cobb,
1972).
Table 2-6 summarizes studies of transverse mixing in streams. Data
from the literature are summarized in Tables 2-7 through 2-9. Table 2-9
contains values of (3 for use in Equation (2-53).
Elhadi et al . (1984) have recently provided a detailed review of
lateral mixing in rivers. They concluded that lateral mixing coefficients
can be predicted with accuracy only in relatively straight channels.
56
-------
TABLE 2-6. SUMMARY OF STUDIES OF TRANSVERSE MIXING IN STREAMS
Reference
Comments
Okoye (1970)
Prych (1970)
Yotsukura, Fischer, Sayre (1970)
Yotsukura and Cobb (1972)
Hoi ley (1975)
Hoi ley and Abraham (1973)
Yotsukura and Sayre (1976)
Shen (1978)
Lau and Krishnappan (1981)
Somlyody (1982)
Gowda (1978)
Mescal and Warnock (1978)
Benedict (1978)
Henry and Foree (1979)
Beltaos (1980)
Cotton and West (1980)
Hoi ley and Nerat (1983)
Demetracopoulous and Stefan (1983)
Webel and Schatzmann (1984)
This study presented a detailed analysis of laboratory experiments
of lateral mixing.
This study detailed the effects of density differences on lateral
mixing.
A lateral dispersion coefficient of 1.3 ft /sec was determined for
the Missouri River.
Studies of lateral mixing were performed on the South River,
Atrisco Feeder Canal, Bernardo Conveyance Channel, and the Missouri
River.
A two-dimensional model of contaminant transport in rivers was
developed and applied to the Missouri and Clinch Rivers, p was
experimentally determined using •*
dx
Transverse dispersion measurements were made in the Waal and Ussel
Rivers, Holland. The change of moments method was used.
Transverse cumulative discharge was used as an independent variable
replacing transverse distance in the 2-D mass transport equation.
The approach of Yotsukura and Sayre (1976) was extended to include
transient mixing.
Field data for transverse mixing coefficients were summarized. A
further extension of the approach of Yotsukura and Sayre was made.
Values of e /(u^H) were found to depend on depth/width ratios.
Tracer studies were performed in five streams to predict lateral
mixing coefficients. A numerical model used in the study was an
extension of the work of Yotsukura and Sayre (1976).
Transverse mixing coefficients were measured in the Grand River.
A study of lateral mixing 1n the Ottawa River produced the
expression S = 0.043HU.
This study reviewed various mixing expressions.
An approximate method of two-dimensional dispersion modeling was
presented.
Transverse mixing characteristics of three rivers 1n Alberta,
Canada were documented by tracer tests for open water and ice
covered flow conditions.
Rhodamine WT dye was used to determine the transverse diffusion
coefficient on a straight reach of an open channel.
Inclusion of secondary mixing as part of a lateral diffusion
coefficient was concluded to have a limited physical basis.
Transverse mixing was studied in wide and shallow rivers using
heated discharge as a tracer. A modified method of moments was
developed to compute transverse mixing coefficients.
An experimental study was conducted to Investigate variations 1n
transverse mixing coefficients 1n straight, rectangular channels.
E-/(u*H) was found to be constant.
E = lateral mixing coefficient
U = cross-sectional average velocity
ff = variance of concentration in y-direct1on
u4 » shear velocity
H = depth
57
-------
TABLE 2-7. TRANSVERSE MIXING COEFFICIENTS IN NATURAL STREAMS AND CHANNELS
(FROM BELTAOS, 1978a)
Source
Channel
and
Description
W
(m)
W
U
(ra/s)
/(Hu*)
Comments
Glover 1964
Yotsukura
et _aJK, 1970
Yotsukura and
Cobb, 1972
Sayre and Yeh,
1973
Engmann and
Kellerhais,
1974
Meyer, 1977
Krlshnappan &
Lau, 1977
Beltaos, 1978b
Columbia River
Missouri River, two mild
alternating bends
South River, few mild
bends
Missouri River, sinuous,
severe bends
Lesser Slave River, ir-
regular, almost contorted
meander, no bars; sinu-
osity = 2.0
Mobile River, mostly
straight, one mild curve
Meandering laboratory
flume with "equilibrium
bed". Planview sinu-
soidal. Meander wave-
length^ irW=1.88m
Athabasca River below Fort
McMurray, straight with
occasional islands, bars;
s1nous1ty=l.O
305 100
183 68.7
18.2 46.2
234 59.1
43.0 17.0
430 87.2
.30 10.5
.30 15.9
.30 7.6
.30 10.2
.30 9.0
.30 11.6
.30 10.0
373
170
1.35
1.74
.21
1.98
.65
.30
.26
.27
.31
.30
.28
.23
.32
.95
.034
.014
.284
.015
.045
.028
.162
.105
.163
.208
.271
.156
.101
.028
.74 Test results and analysis
approximate
.60 Flow distribution available at only
two cross sections
.30 Analysis by streamtube method
3.30 Analysis by numerical and
analytical methods. Periodical
variation of E detected;
average value indicated here
.33 Effects of transverse advection
lumped together with transverse
dispersion. Reanalysis of ice
covered data' by streamtube method
gave e /Ru* .16
7.20 Steady-state condition unlikely
Evaluation of Ey by a numerical
simulation method. Use of constant
Cyqave more consistent results
than laterally variable values of
.75 Slug-injection tests; analysis by
streamtube method applied to dosage
(see also Beltaos 1975)
Beltaos, 1978b
Beltaos, 1978b
Beltaos
(unpublished)
Beltaos
(unpublished)
Athabasca River below
Athabasca, Irregular
meanders with occa-
sional bars, islands;
sinuosity=1.2
Beaver River near Cold
Lake, regular meanders,
point bars and large
dunes, sinuosity=1.3
North Saskatchewan River
below Edmonton, nearly
straight, few, very mild
bends with occasional
bars, islands; sinuo-
sity=1.0
Bow River at Calgary,
sinuous with frequent
islands; mid-channel bars
diagonal bars, sinu-
osity^. 1
320 156
42.7 44.6
213 137
104 104
.86
.50
.58
1.05
.067 .41
.062 1.0
.152 .25
.143 .61
Steady-state concentration tests.
Analysis by stream-tube method.
By steady-state concentration and
slug-injection tests. Analysis
by streamtube and numerical
methods respectively
A amplitude of meanders
f fraction factor
R hydraulic radius
U cross-sectionally averaged velocity
W = width
H depth
EV lateral mixing coefficient
58
-------
TABLE 2-8. SUMMARY OF FIELD DATA FOR TRANSVERSE DISPERSION COEFFICIENTS
(LAU AND KRISHNAPPEN, 1981)
Width,
Data Source in meters
Yotsukura and Cobb (1972)
Missouri River near
Blair
Yotsukura and Cobb (1972)
South River
Yotsukura and Cobb (1972)
Aristo Feeder Canal
Yotsukura and Cobb (1972)
Bernado Conveyance
Channel
Beltaos (1978a), Athabasca
below Fort McMurray
Beltaos (1978a), Athabasca
River below Athabasca
Beltaos (1978a), North
Saskatchewan River
below Edmonton
Beltaos (1978b), Bow River
at Calgary
Beltaos (1978b), Beaver
River near Cold Lake
Sayre and Yeh (1975)
Missouri River below
Cooper Generation
Station
Lau and Krishanppan (1977)
Grand River below
Kitchener
183.0
18.3
18.3
20.1
373.0
320.0
213.0
104.0
42.7
234.0
59.2
Average velocity
in meters
W/H per second
66.7
46.2
27.3
28.7
170.0
156.0
137.0
104.0
44.6
59.1
117.0
1.74
0.18
0.67
1.25
0.95
0.86
0.58
1.05
0.50
1.98
0.35
Shear Velocity
1n meters
per second
0.073
0.040
0.062
0.061
0.056
0.079
0.080
0.139
0.044
0.085
0.069
Dispersion Coefficient,
Friction £y, 1n meters squared
factor per second £y/u*W ey/utH
0.014
0.220
0.069
0.020
0.028
0.067
0.152
0.143
0.062
0.015
0.314
0
0
0
0
0
0
0
0
0
1
0
.101
.0046
.0093
.013
.092
.066
.031
.085
.042
.110
.009
7.5 x
6.3 x
8.2 x
10.6 x
4.4 x
2.6 x
1.8 K
5.9 x
22.4 x
55.8 x
2.2 x
•5
10'3
_Q
10 J
0
10 3
0
10 3
-^
10 J
•1
10 3
ID'3
•5
10 3
o
10'3
•3
10 3
10'3
0.50
0.29
0.22
0.30
0.75
0.41
0.25
0.61
1.00
3.30
0.26
Sinuosity
S
1.1
1.0a
1.0a
1.0a
1.0a
1.2
1.0a
1.1
1.3
2.1
1.1
-------
TABLE 2-9. SUMMARY OF NONDIMENSIONAL DIFFUSION FACTORS IN NATURAL STREAMS
(FROM GOWDA, 1984)
Source of data
Hamdy and Klnkead
(1979) St. Clair River
Glover (1964) Columbia
River near
Rlchland
Hoi ley and Abrahan
(1973)Haal River
Yotsukura and Cobb
(1972) Missouri River
near Blair
Beltaos (1980b) Athabasca
River below Fort
McMurray
Beltaos (1980b) Athabasca
River below
Athabasca
Holly and Abrahan
(1973) Ijssel River
Beltaos (1980b) Beaver
River near Cold Lake
Yotsukura and Cobb
(1972) Bernardo Conve-
yance Channel
Gowda (1980) Grand
River below Waterloo
Yotsukura and Cobb
(1972) Atrlsco Feeder
Canal near Bernallllo
Yotsukura and Cobb
(1972) South River
near the Town of
Wayresboro
Gowda (1980) Boyne
River below Al listen
Notes:
D W
*••*
Discharge,
1n cubic Mean
meters width, 1n
Salient features per second meters
12.0 km straight 6,800.00 819.3
stretch with
an Island
0.11 km stretch 1,235.30 304.8
with a gradual
S-curve
10.0 km straight 1,027.75 266.1
stretch
10.0 km stretch 965.60 183.0
with mild alter-
nating curve
17.6 km stretch 776.00 373.0
with occasional
bars and Islands
17.0 km stretch 566.00 320.0
with Irregular
meanders, occa-
sional bars and
Islands
8.6 km stretch 269.75 69.5
with three al-
ternating bends
1.5 km stretch 20.5 42.7
with regular
meanders, point
bars and large
dunes
2.0 km straight 17.75 20.1
stretch
3.4 km stretch 12.54 57.3
with two alter-
nating curves
2.0 km straight 7.42 18.3
stretch with a
channel of nearly
uniform cross-
section
0.4 km stretch 1.53 18.2
with a few very
slight bends
0.2 km straight 0.82 8.85
stretch
Mean
Mean velocity Nond1mens1onal
depth, 1n In meters diffusion
meters per second factor, /3
10.00 0.83 5.9 x 10 "4
3.00 1.35 4.7 x 10'4
4.70 0.82 5.3 x 10"4
2.74 1.74 6.6 x 10"4
2.20 0.95 7.8 x 10'4
2.05 0.86 8.4 x 10"4
4.00 0.97 23.0 x 10"4
0.96 0.50 41.0 x 10"4
0.70 1.25 81.0 x ID"4
0.56 0.39 10.0 x 10'4
0.67 0.67 13.0 x 10"4
0.38 0.21 25.0 x 10'4
0.43 0.22 25.0 x 10"4
°y ' ^ Vy
W channel width
Q - flow rate
H = depth
U * velocity
mx » average value of matrix (=1) 1n x- direction
60
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2.3.6 Summary
The previous sections have provided a brief review on the treatment of
dispersive transport in water quality models. This has included a
discussion of vertical dispersion in lakes and estuaries, and horizontal
(lateral and longitudinal) dispersion in lakes, estuaries, and rivers. It
is readily seen that a wide variety of numerical formulations for dispersion
exist in the literature. Formulations for dispersion coefficients tend to
be model-dependent and are all based to some extent on general lack of a
complete understanding of the highly complex turbulence induced mixing
processes which exist in natural water bodies. In all cases, due to this
model and empirical dependence, it is -desirable to include a careful
calibration and/or verification exercise using on-site field data for any
water quality modeling application.
2.4 SURFACE HEAT BUDGET
The total heat budget for a water body includes the effects of inflows
(rivers, discharges), outflows, heat generated by chemical-biological
reactions, heat exchange with the stream bed, and atmospheric heat exchange
at the water surface. In all practicality, however, the dominant process
controlling the heat budget is the atmospheric heat exchange, which is the
focus of the following paragraphs. In addition, however, it is also
important to include the proper boundary conditions for advective exchange
(e.g., rivers, thermal discharges, or tidal flows) when the relative source
temperature and rate of advective exchange is great enough to affect the
temperature distribution of the water body.
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 can be
readily measured. Alternatively, and quite conveniently, the various energy
transfer phenomena which occur at the air-water interface can be considered
in a heat budget formulation.
61
-------
In a complete atmospheric heat budget formulation, the net external
heat flux, H, is most often formulated as an algebraic sum of several
component energy fluxes (e.g., Baca and Arnett, 1976; U.S. Army Corps of
Engineers, 1974; Thomann et aj_., 1975; Edinger and Buchak, 1978; Ryan and
Harleman, 1973; TVA, 1972). A typical expression is given as:
H =
where H = net surface heat flux
iQ = shortwave radiation incident to water surface,
30 to 300 kcal/m 2/hr
2
tQ = reflected short wave radiation, 5 to 25 kcal/m /hr
4-Q = incoming long wave radiation from the atmosphere, 225 to
360 Kcal/m2/hr
2
+Q = reflected long wave radiation, 5 to 15 kcal/m /hr
ar
fQ, = back radiation emitted by the body of water,
220 to 345 kcal/m2/hr
2
•t-Q = energy utilized by evaporation, 25 to 900 kcal/m /hr
|Q_ = energy convected to or from the body of water, -35 to 50
2
kcal/m /hr at the surface
NOTE: The magnitudes are typical for middle latitudes of the
United States. The arrows indicate if energy is coming
into the system (+), out of the system (t), or both ($).
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, short wave solar radiation, relative humidity, water
temperature, latitude, and longitude.
Estimation of the various heat flux components has been the subject of
many theoretical and experimental studies in the late 1960's and early
62
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1970's. Most of the derived equations rely heavily on empirical
coefficients. These formulations have been reviewed extensively by the
Tennessee Valley Authority (1972), Ryan and Harleman (1973), Edinger «rt aK
(1974), and Paily ^t jjK (1974). A summary of the most commonly used
formulations in water quality models is given in the following sections.
2.4.1 Measurement Units
The measurement units in surface heat transfer calculations do not
follow any consistent units system. For heat flux, the English system units
2 2
are BTU/ft /day. In the metric system, the units are either Kcal/m /hr or
2
watt/m (1 watt = 1 joule/sec). The Langley (abbreviated Ly), equal to 1
2
cal/cm , also persists in usage. The following conversions are useful in
this section:
1 BTU/ftVday
1 watt/m2
1 Ly/day
1 kcal/m2/hr
1 kilopascal
1 mb
1 mm Hg
1 in Hg
= 0.131 watt/m
= 7.61 BTU/ft2/day
= 0.483 watt/m2
=1.16 watt/m2
= 10 mb
= 0.1 kilopascal
= 1.3 mb
= 33.0 mb
0.271 Ly/day
2.07 Ly/day
3.69 BTU/ft2/day
2.40 Ly/day
7.69 mm Hg
0.769 mm Hg
0.13 kilopascal
25.4 mm Hg
= 0.113 kcal/rrr/hr
= 0.86 kcal/m2/hr
= 0.42 kcal/m2/hr
= 8.85 BTU/ft2/day
= 0.303 in Hg
= 0.03 in Hg
= 0.039 in Hg
= 3.3 kilopascal
2.4.2 Net short wave Solar Radiation, Q
sn
Net short wave solar radiation is the difference between the incident
and reflected solar radiations (Qs - Qsr). Techniques are available and
described in the aforementioned references to estimate these fluxes as a
function of meteorological data. However, in order to account for the
reflection, scattering, and absorption incurred by the radiation through
interaction with gases, water vapor, clouds, and dust particles, a great
deal of empiricism is involved and the necessary data are relatively
extensive if precision is desired.
63
-------
One of the most common simplified formulations for net short wave solar
radiation (Anderson, 1954; Ryan and Harleman, 1973) is expressed as:
Qsn = Qs - Qsr = 0.94 Qsc (1-0.65C2) (2-56)
2
where Q = clear sky solar radiation, kcal/m /hr
j \+
C = fraction of sky covered by clouds
As reported by Shanahan (1984), Equation (2-56) is an approximation in that
it assumes average reflectance at the water surface and employs clear sky
solar radiation. In certain circumstances atmospheric attenuation
mechanisms are much greater than normal, even under cloudless conditions.
For such situations, the more complex formulae described by TVA (1972) are
required.
A number of methods are available for estimating the clear sky solar
radiation. TVA (1972) presents a formula for Q as a function of the
O v*
geographical location, time of year, and hour of the day. Thackston (1974)
and Thompson (1975) report methods for calculating daily average values of
solar radiation as a function of latitude, longitude, month, and sky cover.
Hamon jjt jjl_. (1954) have graphed the daily average insolation as a function
of latitude, day of year and percent of possible hours of sunshine, and is
given in Figure 2-9.
Lombardo (1972) represents the net short wave solar radiation,.Q
(langleys/day), with the following expression:
Qsn = (1-R) Qs (2-57)
where Qs = short wave radiation at the surface (langleys/day)
R = reflectivity of water = 0.03, or alternately:
R = AaB (A,B given below in Table 2-10)
= sun's altitude in degrees
64
-------
3000
CTl
tn
2500
>
o
\
»»
u.
\
m
c
o
'£
T)
oc
o
0)
i
(A
JS
O
2000
1600
1000
BOO
0 10 20 30 10 20 28 10 20 30 10 20 30 10 20 30 10 20 3O 10 20 30 10 20 30 10 20 30 10 20 30 10 20 30 10 20 30
Jan F«b Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Figure 2-9. Clear sky solar radiation according to Hamon, Weiss and Wilson (1954)
-------
TABLE 2-10. VALUES FOR SHORT WAVE 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
The WQRRS model by the U.S. Army Corps of Engineers (1974) considers
the net short wave 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 short wave
solar radiation by considering absorption and scattering in the atmosphere.
A final important note on calculation of the net short wave solar
radiation regards the effects of shading from trees and banks primarily on
stream systems or rivers with steep banks. Shading can significantly reduce
the incoming solar radiation to the water surface, resulting in~ water
temperatures much lower than those occurring in unobstructed areas. Jobson
and Keefer (1979) present a method to account for the reduction of incoming
solar radiation by prescribing geometric relations of vertical obstruction
heights and stream widths for each subreach of their model of the
Chattahoochee River.
2.4.3 Net Atmospheric Radiation, Q
The atmospheric radiation is characterized by much longer wavelengths
than solar radiation since the major emitting elements are water vapor,
carbon dioxide, and ozone. The approach generally adopted to compute this
flux involves the empirical determination of an overall atmospheric
emissivity and the use of the Stephan-Boltzman law (Ryan and Harleman,
1973). The formula by Swinbank (1963) has been adopted by many
investigators for use in various water quality models (e.g., U.S. Army Corps
of Engineers, 1974; Chen and Orlob, 1975; Brocard and Harleman, 1976). This
66
-------
formula was believed to give reliable values of the atmospheric radiation
within a probable error to +_5 percent. Swinbank's formula is:
Qan = Qa - Q,v, =1-16 x 10"13 U + 0.17C2) (Ta +460)6 (2-58)
an a ar
2
where Q = net long wave atmospheric radiation, BTU/ft /day
C = cloud cover, fraction
T = dry bulb air temperature, °F
a
A recent investigation by Hatfield jrt al. (1983) has found that the formula
by Brunt (1932) gives more accurate results over a range of latitudes of
26°13'N to 47°45'N and an elevation range of -30m to + 3,342m. Brunt's
formula is:
Qari = 2.05xlO~8(l+0.17C2)(T +460) 4( 1+0.149 v^) (2-59)
an a £.
2
where () = net long wave atmospheric radiation, BTU/ft /day
an
e~ = the air vapor pressure 2 meters above the water surface, mm
Hg
T = air temperature 2 meters above the water surface, °F
a
2.4.4 Long Wave Back Radiation, Q.
The long wave back radiation from the water surface is usually the
largest of all the fluxes in the heat budget (Ryan and Harleman, 1973).
Since the emissivity of a water surface (0.97) is known with good precision,
this flux can be determined with accuracy as a function of the water surface
temperature:
Qbr = 0.97 <7T$4 (2-60)
2
where Qbr = long wave back radiation, cal/m /sec
T = surface water temperature, °K
o = Stefan-Boltzman constant =1.357 x 10"8, cal/m2/sec/°K4
67
-------
The U.S. Army Corps of Engineers (1974) uses the following
linearization of Equation (2-60) to express the back radiation emitted by
the water body:
Qbr = 73.6 + 1.17 T (2-61)
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 percent relative to Equation (2-60).
2.4.5 Evaporative Heat Flux, Q
Evaporative heat loss occurs as a result of the change of state of
water from a liquid to vapor, requiring sacrifice of the latent heat of
vapori zati.on. The basic formulation used in all heat budget formulations
(e.g., Ryan and Harleman, 1973; U.S. Army Corps of Engineers, 1974; Chen and
Orlob, 1975; Lombardo, 1972) is:
Qe=PLwE <2-62)
p
where Q = heat loss due to evaporation, kcal/m /sec
3
p = fluid density, kg/m
LW = latent heat of vaporization, kcal/kg
or L, = 597 - 0.57 T
w s
E = evaporation rate, m/sec
T = surface water temperature, °C
The general expression for evaporation from a natural water surface is
usually written as:
E = (a + bW) (es - ea) (2-63)
where a,b = empirical coefficients
58
-------
W = wind speed at some specified elevation above water
surface, m/sec
es = saturation vapor pressure at the surface water
temperature, mb
e = vapor pressure of the overlying atmosphere, mb
Various approaches have been used to evaluate the above expression.
In a very simplified approach, the empirical coefficient, a, has often been
-9 -9
taken to be zero, while b ranges from 1 x 10 to 5 x 10 (U.S. Army Corps
of Engineers, 1974). The value of e is a nonlinear function of the surface
water temperature. However e can be estimated in a piecewise linear
fashion as follows:
es = a. + /3n. Ts (2-64)
where a-,/3. = empirical coefficients with values as given in
Table 2-11.
T = surface water temperature, C
TABLE 2-11. VALUES FOR EMPIRICAL COEFFICIENTS
Temperature Range, C
0-1
5-10
10-15
15-20
20-25
25-30
30-35
35-40
ai
6.05
5.10
2.65
-2.04
-9.94
-22.29
-40.63
-66.90
01
0.522
0.710
0.954
1.265
1.659
2.151
2.761
3.511
A more convenient formula for the saturation vapor pressure, e , is
presented by Thackston (1974) as follows:
es = exp [l7.62 - 9501/(T$ + 460)] (2-65)
69
-------
where e = saturation vapor pressure at the surface water temperature,
in Hg
T = water temperature, °F
The standard error of prediction of Equation (2-55) is reported by Thackston
(1974) to be 0.00335.
A large number of evaporation formula exist for a natural water
surface, as demonstrated in Table 2-12 (Ryan and Harleman, 1973). Detailed
comparisons of these formulae by the above authors showed that the
discrepancies between these formulae were not significant. Both Ryan and
Harleman (1973), and TVA (1968) recommend the use of the Lake Hefner
evaporation formula developed by Marciano and Harbeck (1954), which has the
best data base, and has been shown to perform satisfactorily for other water
bodies. The Lake Hefner formula is written as:
Qe = 17 W2 (es - e2) (2-66)
2
where Q = heat loss due to evaporation, BTU/ft /day
W2 = wind speed at 2 meters above surface, mph
e = saturated vapor pressure at the surface water temperature,
mm Hg
e^ = vapor pressure at 2 meters above surface, mm Hg
It is important to note that the Lake Hefner formula was developed for lakes
and may not be universally valid for streams or open channels due to
physical blockage of the wind by trees, banks, etc.; and due to differences
in the surface turbulence which affects the liquid film aspects of
evaporation (McCutcheon, 1982). Jobson developed a modified evaporation
formula which was used in temperature modeling of the San Diego Aqueduct
(Jobson, 1980) and the Chattahoochee River (Jobson and Keefer, 1981). This
formula is written as:
E = 3.01 + 1.13 W (ec - e_) (2-67)
S a
70
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TABLE 2-12. EVAPORATION FORMULA FOR LAKES AND RESERVOIRS
(RYAN AND HARLEMAN, 1973)
Name
Lake
Hefner
Kohler
Zaykov
Meyer
•^j
Morton
Rohwer
Formula in
Original Form
E=6.25-10~4W8(es-e8)
E=.00304W4(es-e2)
E=[,15+.108W2](es-e2)
E=10{l+.lW8)(es-e8)
E=(30CH-50W)(es-ea)/p
E=.77l[l.465-.0186B]x
[.44+.118W](es-ea)
where B=atmos. press.
Units*
cm/3 hr
knots
mb
in. /day
miles/day
in. Hg
mm/ day
m/s
mb
in. /month
mph
in. Hg
i n . /month
mph
in. Hg
in. /day
mph
in. Hg
Observation
Levels
Sm-wind
Bm-e,
a
4m-wind
2m-e,
a
2m-wind
2m- e,
a
25 ft-wind
25 ft-e,
a
8m-wind
2m- e
a
0.5-1 ft-wind
1 inch-e,
a
Time
Increments
3 hrs
Day
Day
Monthly
Monthly
Daily
Water Body
Lake Hefner
Oklahoma
2587 acres
Lake Hefner
Oklahoma
2587 acres
Ponds and
small reservoirs
Small lakes
and reservoirs
Class A pan
Pans
85 ft
diameter tank
1300 acre
Reservoir
Formula at sea-level
Meas. Ht. Spec. Units
BTU/ft2/day mph, mm Hq
12.4W8(es-e8)
17.2W2(es-e2)
15.9W4(es-e2)
17.5W2(es-e2)
(43+14W2)(es-e2)
(73+7.3W3)(es-e8)
(80+10W2)(es-e2)
(73.5-H2.2Wg)(es-e2)
(73.5+14.7W2(es-e2)
(67+10W2)(es-e2)
Remarks
Good agreement with Lake
Mead, Lake Eucumbene,
Russian Lakes.
Essentially the same as
the Lake Hefner Formula.
Based on Russian
experience. Recommended
by Shulyakovskiy
e is obtained daily from
mean morning and evening
measurements of T , R...
Increase constant! by 10% if
average of maximum and
minimum used.
Data from meteorological
stations. Measurement
heights assumed.
Extensive pan measurements
using several types of pans.
Correlated with tank
reservoir data:
*For each formula, the units are for evaporation rate, wind speed, and vapor pressure.
-------
where E is in mm/day
W = wind speed at some specified elevation above the water
surface, m/sec
e = vapor pressure at the same elevation as the wind,
a
kilopascals
e = saturation vapor pressure at the water surface temperature,
kilopascals
It is noted that the wind speed function of Equation (2-67) was reduced by
30 percent during calibration of the temperature model for the Chattahoochee
River (McCutcheon, 1982). The original Equation (2-67) was developed for
the San Diego Aqueduct which represented substantially different climactic
and exposure conditions than for the Chattahoochee River. McCutcheon (1982)
notes that the wind speed function is a catchall term that must compensate
for a number of difficulties which include, in part:
• Numerical dispersion in some models.
• Inaccuracies in the measurement and/or calculation of wind
speed, solar and long-wave radiation, air temperature, cloud
cover, and relative humidity.
• Effects of wind direction, fetch, channel width, sinuosity,
bank and tree height.
• Effects of depth, turbulence, and lateral velocity
distribution.
• Stability of the air moving over the stream.
2.4.6 Convective Heat Flux, Q
Convective heat is transferred between air and water by conduction and
transported away from (or toward) the air-water interface by convection
72
-------
associated with the moving air mass. The convective heat flux is related to
the evaporative heat flux, Q , through the Bowen ratio:
R =3|= (6.19 x 10"4) p
Ts-Ta
es - ea
(2-68)
where R = Bowen Ratio
p = atmospheric pressure, mb
T = dry bulb air temperature, °C
a
T = surface water temperature, C
e = saturation vapor pressure at the surface water temperature,
mb
e, = vapor pressure of the overlying atmosphere, mb
a
The above formulation is used in the surface heat transfer budget of
several models (e.g., U.S. Army Corps of Engineers, 1974; Brocard and
Harleman, 1976).
2.4.7 Equilibrium Temperature and Linearization
The preceding paragraphs present methods for estimating the magnitudes
of the various components of heat transfer through the water surface.
Several of these components are nonlinear functions of the surface water
temperature, T . Thus, they are most appropriately used in transient water
quality simulations where the need to predict temperature variations is on
the time scale of minutes or hours. However, for long term water quality
simulations or for steady state simulations, it is more economical to use a
linearized approach to heat transfer. As developed by Edinger and Geyer
(1965), and reported by Ryan and Harleman (1973), this approach involves two
concepts, that of equilibrium temperature, T£, and surface heat exchange, K,
where H can now be written as:
H = K (Ts - TE) (2-69)
73
-------
The equilibrium temperature, TE, is defined as that water surface
temperature which, for a given set of meteorological conditions, causes the
surface heat flux H, to equal zero. The surface heat exchange coefficient,
K, is defined to give the incremental change of net heat exchange induced by
an incremental change of water surface temperature. It varies with the
surface temperature and thus should be recalculated as the water temperature
changes.
2.4.7.1 Equilibrium temperature, Tr-
The equilibrium temperature Tr is the temperature toward which every
water body at the site will tend, and is useful because i1^ is dependent
solely upon meteorological variables at a given site. A water body at a
surface temperature, T , less than TF, will have a net heat input and thus
W L.
will tend to increase its temperature. The opposite is true if T > TF.
W L.
Thus, the equilibrium temperature embodies all the external influences upon
ambient temperatures.
Certain formulations for the equilibrium temperature have been
developed which require an iterative or trial and error solution approach
(Ryan and Harleman, 1973). An approximate formula for obtaining Tr has been
developed by Brady ^t al. (1969) which has been shown to yield fairly
accurate results:
TF = ^ + T, (2-70)
L 23 + f(W) (0+ .255) a
where Qgn = net short wave solar radiation, BTU/ft2/day
T = dew point temperature of air, °F
f(w) = empirical wind speed relationship
= 17W2 (based on Lake Hefner data), BTU/ft2/day/mm Hg
fB = proportional i ty factor which is a fuirction of
temperature, mm Hg/°F
W~ = wind speed at 2 meters above surface, mph
74
-------
The expression for /3 is written as:
/3= .255 - .0085 T* + .000204 T*2 (2-71)
where
T* = \
2.4.7.2 Surface Heat Exchange Coefficient, K
The surface heat exchange coefficient, K, relates the net heat transfer
rate to changes in water surface temperature. An expression for K developed
by Brady et a]_. (1969), (and reported by Ryan and Harleman, 1973) is:
K = 23 + (13 + .255) 17W, (2-73)
W L.
where W? = wind speed at 2 meters, mph
and ft is evaluated at T based on Equation (2-62):
W W
/3w = .255 - .0085 TW + .000204 T2 (2-74)
Charts giving K as a function of water surface temperature and wind
speed are given by Ryan and Stolzenbach (1972), assuming an average relative
humidity of 75 percent. Shanahan (1984) presents a calculation procedure to
determine T£ and K from average meteorological data.
2.4.8 Heat Exchange with the Stream Bed
For most lakes, estuaries, and deep rivers, the thermal flux through
the bottom is insignificant. However, as reported by Jobson (1980) and
Jobson and Keefer (1979), the bed conduction term may be significant in
determining the diurnal variation of temperatures in water bodies with
depths of 10 ft (3m) or less. Jobson (1977) presents a procedure for
accounting for bed conduction which does not require temperature
measurements within the bed. Rather, the procedure estimates the heat
75
-------
exchange based on the gross thermal properties of the bed, including the
thermal diffusivity and heat storage capacity. The inclusion of this method
improved dynamic temperature simulation on the San Diego Aqueduct and the
Chattahoochee River.
2.4.9 Summary
The previous section has presented a brief summary of the most
frequently used formulations for surface heat exchange in numerical water
quality models. These formulations are widely used and have been shown to
work quite well within the normal range of meteorological and surface water
conditions, provided a reasonably complete data base is available on
meteorological conditions at the site of interest. Meteorological data
requirements include atmospheric pressure, cloud cover, and at a known
surface elevation: wind speed and direction, relative humidity, and wet and
dry bulb air temperatures. Shanahan (1984) presents a useful summary of
meteorological data requirements for surface heat exchange computations.
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Valentine, E.M. and I.R. Wood. 1977. Longitudinal Dispersion with Dead
Zones, Journal of the Hydraulic Division, ASCE, Vol. 103, No. HY9.
Valentine, E.M. and I.R. Wood. 1979. Dispersion in Rough Rectangular
Channels. Journal of the Hydraulics Division, ASCE, Vol. 105, No. HY12,
pp. 1537-1553.
88
-------
Waldrop, W.R. 1978. Hydrothermal Analyses Using Computer .Modeling and
Field Studies. Verification of Mathematical and Physical Models in
Hydraulic Engineering, Proceedings of the 26th Annual Hydraulics Specialty
Conference. ASCE, pp. 38-43.
Wang, J.D. 1979. Finite Element Model of 2-D Stratified Flow. Journal of
the Hydraulics Division, ASCE, Vol. 105, No. HY12, Proc. Paper 15049,
pp. 1473-1485.
Wang, J.D. and J.J. Connor. 1975. Mathematical Modeling of Near Coastal
Circulation, Technical Report No. 200, R.M. Parsons Laboratory, MIT.
Wallis, I.G. 1974. Lagrangian Models of Waste Transport for Estuaries and
Tidal Inlets. Geophysical Fluid Dynamics Laboratory, Monash University,
Clayton, Victoria, Australia.
Ward, P.R.B. and H.B. Fischer. 1971. Some Limitations on Use of the One-
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Reservoirs. John Wiley and Sons, Wiley-Interscience, N.Y., N.Y.
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89
-------
Chapter 3
DISSOLVED OXYGEN
3.1 DISSOLVED OXYGEN SATURATION
3.1.1 Introduction
Dissolved oxygen saturation, commonly symbolized as C and expressed in
mg/1 , is a basic parameter used in a great many water quality models. Since
dissolved oxygen predictions are often primary reasons for developing water
quality models, accurate values for C are needed.
Table 3-1 illustrates the equations used to calculate saturation
dissolved oxygen values in a number of water quality models. The most
frequently used equation is the polynomial equation developed by Elmore and
Hayes (1960) for distilled water (Equation (3-1) in Table 3-1). In this
equation, neither pressure nor salinity effects are considered (pressure is
assumed to be 1 atm and salinity is 0 ppt).
Effects of pressure on saturation values are expressed as a ratio of
site pressure to sea level (Equation (3-5)) or as a function of elevation
(Equation (3-6)). Effects of salinity (relevant to estuaries and oceanic
systems) are considered in the last two model equations (Equations (3-7) and
(3-8)). When used in fresh water applications, the sections of the
equations in which the saline term appears reduce to zero and have no effect
on the dissolved oxygen saturation. Every saturation equation, whether or
not modified to include non-standard pressure or salinity, evaluates
dissolved oxygen saturation as a function of temperature.
90
-------
TABLE 3-1. METHODS USED BY SELECTED MODELS TO PREDICT DISSOLVED OXYGEN SATURATION
Equation
Number
3-1
3-1
3-1
3-1
3-2
3-3
3-4
3-5
3-5
3-6
3-7
3-8
Model Name
(or Description)
Limnological Model
for Eutrophic Lakes
and Impoundments
EXPLORE-1
Level Ill-Receive
Water Quality Model
for Large Lakes:
Part 2: Lake Erie
WRECEV
QUAL-II
CE-QUAL-R1
One Dimensional Steady
State Stream Water
Quality Model
HSPF (Release 7.0)
DOSAG and DOSAG3
Pearl Harbor Version
of Dynamic Estuary
Model (DEM)
RECEIV-II
Model
Reference
Baca and Arnett,
1976
Battelle, 1973
Medina, 1979
Di Toro and
Connolly, 1980
Johnson and
Duke, 1976
Roesner, et al.,
1981
U.S. Army COE,
1982
Bauer, et al.,
1979
Imhoff, et al.,
1981
Duke and Masch,
1973
Genet et al.,
1974
Raytheon Co. ,
1974, and
Weiss, 1970
Equation or Method for Dissolved Oxygen
Saturation C (mg/1)
Cs = 14.652 - (0.41022 T) + (0.007991 T2) - (7.7774xlO~5 T3)
T = °C
Same as above
Same as above
Same as above
Cs = 14.62 - 0.3898 T + 0.006969 T2 - 5.897xlO"5 T3
T =°C
Cs = 24.89-0.4259 T + 0.003734 T2 - 1.328xlO~5 T3
T = °F
C = (14 6)e'~(°'027767 " °-00027 T + 0.000002 T2) T)
C = (14. 652-. 41022 T + 0.007910 T2 -7.7774xlO~5 T3) (BP/29.92)
T = °C
BP = Barometric pressure (in.Hg)
Same as above
(14.62 - (0.3898 T) + (0.006969 T2) - (5.897xlO*5 T3))
[l.O - (6.97xlO'6 E)]5'167
T = °C
E = Elevation, ft.
Cs = 14.5532 - .38217 T + .0054258 T2-CL(1.665xlO"4-5.866xlO"6T + 9.796xlO"8 T2)
T = °C
CL = Chloride concentration (ppm)
Cc - 1.4277 exp[-173.492 + 24963. 39/T + 143.3483 ln(T/100.) ,
s -0.218492 T + S(-0. 033096 + 0.00014259 T - 0.00000017 r)]
T = °K = °C + 273.15
S = Salinity (ppt)
-------
3.1.2 Dissolved Oxygen Saturation As Determined by the APHA
The APHA (1985) presents a tabulation of oxygen solubility in water as
a function of both chlorinity and water temperature (see Table 3-2). This
table is the work of Benson and Krause (1984) who collected the data and
developed the equations for conditions in which the water was in contact
with water saturated air at standard pressure (1.000 atm).
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, expressed here three ways, is:
Salinity (ppt or O/OQ) = 0.03 + 0.001805 Chlorinity (mg/1) (3-9a)
or
Salinity (ppt or 0/OQ) = 5.572 x 10"4(SC) + 2.02 x 10"9(SC)2 (3-9b)
where SC = specific conductance in micromhos/cm
or
Salinity = 1.80655 (chlorinity as ppt) (3-9c)
where chlorinity and salinity are as defined in the footnote to Table 3-2.
Equation (3-9b) is from USGS (1981) and Equation (3-9c) is from APHA
(1985).
The APHA (1985) recommends that the concentration of oxygen in water
(at different temperatures and salinity) at equilibrium with water saturated
air be calculated using the equation below (Benson and Krause, 1984):
In Cs = -139.34411 + (1.575701 x 105/T) (3-10)
-(6.642308 x 107/T2) + (1.243800 x 1010/T3)
-(8.621949 x 10U/T4)
-Chl[(3.1929 x 10"2) - (1.9428 x 10/T)
+ (3.8673 x 103/T2)]
92
-------
TABLE 3-2. SOLUBILITY OF OXYGEN IN WATER EXPOSED
TO WATER-SATURATED AIR AT 1.000 ATMOSPHERIC
PRESSURE (APHA, 1985)
Tanp.
1n°C
ChloHnity. ppt
0.0 5,0 10.0 15.0 20.0 25.0
Dissolved Oxygen, mq/1
Difference
per 0.1 ppt
ChloHnity
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
11.0
12.0
13.0
14.0
15.0
16.0
17.0
18.0
19.0.
20.0
21.0
22.0
23.0
24.0
25.0
26.0
27.0
28.0
29.0
30.0
31.0
32.0
33.0
34.0
35.0
36.0
37.0
38.0
39.0
40.0
14.621
14.216
13.829
13.460
13.107
12.770
12.447
12.139
11.843
11.559
11.288
11.027
10.777
10.537
10.306
10.084
9.870
9.665
9.467
9.276
9.092
8.915
8.743
8.578
8.418
8.263
8.113
7.968
7.827
7.691
7.559
7.430
7.305
7.183
7.065
6.950
6.837
6.727
6.620
6.515
6.412
13.728
13.356
13.000
12.660
12.335
12.024
11.727
11.442
11.169
10.907
10.656
10.415
10.183
9.961
9.747
9.541
9.344
9.153
8.969
8.792
8.621
8.456
8.297
8.143
7.994
7.850
7.711
"7.575
7.444
7.317
7.194
7.073
6.957
6.843
6.732
6.624
6.519
6.416
6.316
6.217
6.121
12.888
12.545
12.218
11.906
11.607
11.320
11.046
10.783
10.531
10.290
10.058
9.835
9.621
9.416
9.218
9.027
8.844
8.667
8.497
8.333
8.174
8.021
7.873
7.730
7.591
7.457
7.327
7.201
7.079
6.961
6.845
6.733
6.624
6.518
6.415
6.314
6.215
6.119
6.025
5.932
5.842
12.097
11.783
11.483
11.195
10.920
10.656
10.404
10.162
9.930
9.707
9.493.
9.287
9.089
8.899
8.716
8.540
8.370
8.207
8.049
7.896
7.749
7.607
7.470
7.337
7.208
7.083
6.962
6.845
6.731
6.621
6.513
6.409
6.307
6.208
6.111
6.017
5.925
5.835
5.747
5.660
5.576
11.355
11.066
10.790
10.526
10.273
10.031
9.799
9.576
9.362
9.156
8.959
8.769
8.586
8.411
8.242
8.079
7.922
7.770
7.624
7.483
7.346
7.214
7.087
6.963
6.844
6.728
6.615
6.506
6.400
6.297
6.197
6.100
6.005
5.912
5.822
5.734
5.648
5.564
5.481
5.400
5.321
10.657
10.392
10.139
9.897
9.664
9.441
9.228
9.023
8.826
8.636
8.454
8.279
8.111
7.949
7.792
7.642
7.496
7.356
7.221
7.090
6.964
6.842
6.723
6.609
6.498
6.390
6.285
6.184
6.085
5.990
5.896
5.806
5.717
5.631
5.546
5.464
5.384
5.305
5.228
5.152
5.078
0.016
0.015
0.015
0.014
0.014
0.013
0.013
0.012
0.012
0.012
0.011
0.011
0.011
0.010
0.010
0.010
0.009
0.009
0.009
0.009
0.009
0.008
0.008
0.008
0.008
0.007
0.007
0.007
0.007
0.007
0.007
0.006
0.006
0.006
0.006
0.006
0.006
0.006
0.006
0.005
0.005
DEFINITION OF SALINITY
Although salinity has been traditionally defined as the total solids in water after all carbonates
have been converted to oxides, all bromide and iodide have been replaced by chloride, and all
organic matter has been oxidized, the new scale used to define salinity 1s based on the electrical
conductivity of seawater relative to a specified solution of KC1 and Had (UNESCO, 1981). The scale
is dimensionless and the traditional dimensions of parts per thousand (I.e., mg/g of solution) no
longer applies.
DEFINITION OF CHLORINITY
Chlorinity 1s now defined in relation to salinity as.follows:
Salinity- 1.80655 (ChloHnity)
Although chlorinity is not equivalent to chloride concentration, the factor for translating a
chloride determination in seawater to include bromide, for example, is only 1.0045 based on the
molecular weights and the relative amounts of the two ions. Therefore, for practical purposes,
chloride (in mg/g of solution) is nearly equal to chlorinity in seawater. For wastewater, a
knowledge of the ions responsible for the solution's electrical conductivity 1s necessary to correct
for the ions impact on oxygen solubility and use of the tabular value or the equation is
Inappropriate unless the relative composition of the wastewater is'Similar to seawater.
93
-------
where C = equilibrium oxygen concentration, mg/1 , at 1.000 atm
(standard pressure)
= temperature (°K) = °C + 273.150 and °C is within 0.0 to
T
Chi = chlorinity within 0.0 to 28.0, ppt
40.0°C
Table 3-2 replaces the older table of previous APHA Standard Methods
editions. The USGS (1981) has replaced older tables based on calculations
of Whipple and Whipple (1911) with tables generated from an equation by
Weiss (1970) (Equation 3-8).
The APHA (1985) recommends that saturation dissolved oxygen
concentration at non-standard pressure be calculated using the following
equation:
r = r P
s s
(l-Pwv/P)
(3-11)
where C,
P
P
wv
In P.
6
T
WV
= equilibrium oxygen concentration at non-standard
pressure, mg/1
= equilibrium oxygen concentration at 1.000 atm, mg/1
= pressure, atm, and is within 0.000 to 2.000 atm
= partial pressure of water vapor, atm, which may be
computed
= 11.8571- (3840.70/Tk) - (216961/Tk2)
= temperature in °K
= 0.000975 - (1.426 x 10~5TJ + (6.436 x 10~8T 2)
o
= temperature in C
The expressions for P and 6 are also from APHA (1985).
For altitudes less than approximately 4000 feet the bracketed quantity
is very nearly 1 and at these altitudes multiplying C by P(atm) alone
i S i
results in a good approximation of C . A more accurate calculation of C
94
-------
can be made by using Table 3-3. The quantity in brackets from Equation
(3-11) is tabulated for temperatures between 0-40°C and for pressures from
1.1 to 0.5 atm (Benson and Krause, 1980). As an approximation of the
influence of altitude, C decreases about 7 percent per 2,000 feet of
elevation increase.
TABLE 3-3 VALUES FOR THE BRACKETED QUANTITY SHOWN IN EQUATION 3-11
TO BE USED WITH THE CORRESPONDING TEMPERATURES AND PRESSURES
(BENSON AND KRAUSE, 1980)
T(°C)
1.1
1.0
0.9
P atm
0.8
0.7
0.6
0.5
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
1.0005
1 .0007
1.0010
1.0015
1.0021
1.0029
1.0039
1.0053
1.0071
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
0.9994
0.9991
0.9987
0.9982
0.9974
0.9965
0.9952
(0.9935)
(0.9913
0.9987
0.9980
0.9971
0.9959
0.9942
0.9921
(0.9892)
(0.9854)
(0.9805)
0.9977
0.9966
0.9950
0.9929
(0.9901)
(0.9864)
(0.9814)
(0.9750)
[0.9665]
0.9963
0.9946
0.9922
(0.9889)
(0.9845)
(0.9787)
[0.9711]
[0.9610]
[0.9479]
0.9944
0.9918
0.9882
(0.9833)
[0.9767]
[0.9680]
[0.9566]
[0.9415]
[0.9217]
Explanation of Interpolation Procedure:
Linear interpolation in P and T will introduce an error <0.02% in the upper and left
sections of table. Interpolation using numbers in parentheses will lead to errors
<0.05%. With the numbers in brackets, interpolation errors become larger. Either
temperature or presssure may be interpolated first, as illustrated for T - 3.00 C and
P = 0.67 atm by the two arrays shown below.
Temperature Interpolated First Pressure Interpolated First
0.7 0.67 0.6 0.7 0.67 0.6
0
3
5
0
0
0
.9977
.99704
.9956
0.9965
answer
0.
0.
0.
9963
99528
9946
0
3
5
0
0
.9977
.9966
0
0
0
.99728
.9965,
.99600
0.9963
answer
0.9946
95
-------
Earlier the APHA (1980) calculated the effects of barometric pressure
on dissolved oxygen saturation as:
' P - P.
"s "s
C = C f WV
This is equivalent to Equation (3-11) when 6=0.
3.1.3 Comparison of Methods
Table 3-4 compares the dissolved oxygen saturation values for Equations
(3-1) through (3-8) and APHA (1971) against the values in Table 3-2 from
the APHA (1985), Equation (3-10). The comparisons are performed at 0.0 mg/1
salinity and sea level. When the values from the equations are compared
o *
with the APHA (1985) values within the temperature range 10-30 C and the
maximum differences examined, four "groups^" of differences appear. Values
from Equation (3-8) are in the group that shows the least difference from
APHA (1985): 0.03 mg/1 higher than the APHA (1985) predictions. Values
from Equations (3-2), (3-4), (3-6) and APHA (1971) are in the second group
with differences of .07 to .11 mg/1 higher than APHA (1985). Values from
Equations (3-1), (3-3) and (3-5) are in the third group with differences of
.11 to .13 mg/1 lower than APHA (1985). Equation (3-7) produced differences
that comprise the fourth group with some values >0.4 mg/1 higher than APHA
(1985). Generally, the maximum differences with each equation occur at
higher temperatures, when dissolved oxygen depletion may contribute to
serious water quality problems.
In Table 3-5 Equations (3-7), (3-8), (3-13) and APHA (1971) (those
including salinity factors) are evaluated at a chloride concentration of
20,000 mg/1 at 1 atm pressure and compared to APHA (1985) values.
* Typically, the temperature range in which most freshwater water quality
analyses take place.
96
-------
TABLE 3-4. COMPARISON OF DISSOLVED OXYGEN SATURATION VALUES FROM TEN
EQUATIONS AT 0.0 mg/1 SALINITY AND 1 ATM PRESSURE
Temperatur
°C
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
11.0
12.0
13.0
14.0
15.0
16.0
17.0
18.0
19.0
20.0
21.0
22.0
23.0
24.0
25.0
26.0
27.0
28.0
29.0
30.0
31.0
32.0
33.0
34.0
35.0
36.0
37.0
38.0
39.0
40.0
e
(3-1)
14.652
14.250
13.863
13.491
13.134
12.791
12.462
12.145
11.842
11.551
11.271
11.003
10.746
10.499
10.262
10.034
9.816
9.606
9.404
9.209
9.022
8.841
8.667
8.498
8.334
8.176
8.021
7.871
7.723
7.579
7.437
7.298
7.159
7.022
6.885
6.749
6.612
6.474
6.335
6.194
6.051
Equation Number From Table
(3-2)
14.620
14.237
13.868
13.512
13.169
12.838
12.519
12.213
11.917
11.633
11.360
11.097
10.844
10.601
10.367
10.142
9.926
9.718
9.518
9.3251
9.140
8.961
8.789
8.624
8.464
8.309
8.160
8.015
7.875
7.739
7.606
7.477
7.350
7.227
7.105
6.986
6.868
6.751
6.635
6.520
6.404
(3-3)
14.650
14.248
13.861
13.490
13.133
12.790
12.460
12.144
11.841
11.550
11.270
11.002
10.744
10.497
10.260
10.033
9.814
9.604
9.401
9.207
9.019
8.838
8.664
8.495
8.331
8.172
8.017
7.866
7.719
7.574
7.432
7.292
7.154
7.016
6.880
6.743
6.606
6.468
6.329
6.188
6.045
(3-4)
14.600
14.204
13.826
13.465
13.120
12.790
12.475
12.173
11.883
11.606
11.340
11.085
10.840
10.605
10.378
10.161
9.951
9.749
9.555
9.367
9.186
9.011
8.842
8.679
8.521
8.367
8.219
8.075
7.935
7.800
7.668
7.539
7.414
7.293
7.174
7.058
6.945
6.834
6.726
6.620
6.517
(3-5)
14.652
14.250
13.863
13.491
13.134
12.791
12.462
12.145
11.842
11.551
11.271
11.003
10.746
10.499
10.262
10.034
9.816
9.606
9.404
9.209
9.022
8.841
8.667
8.498
8.334
8.176
8.021
7.871
7.723
7.579
7.437
7.298
7.159
7.022
6.885
6.749
6.612
6.474
6.335
6.194
6.051
3-1
(3-6)
14.620
14.237
13.868
13.512
13.169
12.838
12.519
12.213
11.917
11.633
11.360
11.097
10.844
10.601
10.367
10.142
9.926
9.718
9.518
9.325
9.140
8.961
8.789
8.624
8.464
8.309
8.160
8.015
7.875
7.739
7.606
7.477
7.350
7.227
7.105
6.986
6.868
6.751
6.635
6.520
6.404
(3-7)
14.553
14.176
13.811
13.456
13.111
12.778
12.456
12.144
11.843
11.553
11.274
11.006
10.748
10.502
10.266
10.041
9.827
9.624
9.432
9.251
9.080
8.920
8.772
8.634
8.506
8.390
8.285
8.190
8.106
8.033
7.971
7.920
7.880
7.850
7.832
7.824
7.827
7.841
7.866
7.901
7.948
(3-8)
14.591
14.188
13.803
13.435
13.084
12.748
12.426
12.118
11.823
11.540
11.268
11.008
10.758
10.517
10.286
10.064
9.850
9.644
9.446
9.254
9.070
8.891
8.720
8.554
8.393
8.238
8.088
7.943
7.802
7.666
7.533
7.405
7.281
7.161
7.043
6.930
6.819
6.711
6.606
6.505
6.405
APHA
(1971)
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
-
-
_
-
APHA(1985)
3-10
14.621
14.216
13.829
13.460
13.107
12.770
12.447
12.139
11.843
11.559
11.288
11.027
10.777
10.537
10.306
10.084
9.870
9.665
9.467
9.276
9.092
8.915
8.743
8.578
8.418
8.263
8.113
7.968
7.827
7.691
7.559
7.430
7.305
7.183
7.065
6.950
6.837
6.727
6.620
6.315
6.412
Equation (3-13) is based on the data of Green and Carritt (1967). From
their data Hyer j3t aj_. (1971) developed an expression relating C to both
temperature and salinity.
GS is given by:
= 14.6244 - 0.367134T + 0.0044972r
0.0966S + 0.00205ST + 0.0002739S'
97
(3-13)
-------
TABLE 3-5. COMPARISON OF DISSOLVED OXYGEN SATURATION VALUES FROM
SELECTED EQUATIONS AT A CHLORIDE CONCENTRATION OF
20,000 mg/1 (36.1 ppt SALINITY) AND 1 ATM PRESSURE
Temperature
°C
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
11.0
12.0
13.0
14.0
15.0
16.0
17.0
18.0
19.0
20.0
21.0
22.0
23.0
24.0
25.0
26.0
27.0
28.0
29.0
30.0
31.0
32.0
33.0
34.0
35.0
36.0
37.0
38.0
39.0
40.0
Equation
(3-7)
11.215
10.953
10.699
10.452
10.212
9.978
9.752
9.532
9.320
9.114
8.915
8.723
8.538
8.360
8.189
8.025
7.868
7.718
7.574
7.438
7.308
7.186
7.070
6.961
6.859
6.764
6.676
6.595
6.521
6.454
6.394
6.340
6.294
6.254
6.221
6.196
6.177
6.165
6.160
6.162
6.171
Number from
(3-8)
11.400
11.105
10.823
10.553
10.295
10.048
9.811
9.585
9.367
9.158
8.958
8.765
8.580
8.402
8.231
8.067
7.908
7.755
7.607
7.465
7.327
7.194
7.066
6.942
6.822
6.594
6.594
6.485
6.379
6.277
6.177
6.081
5.987
5.896
5.808
5.722
5.638
5.557
5.477
5.400
5.325
Table 3-1
(3-13)
11.492
11.203
10.924
10.653
10.391
10.139
9.895
9.661
9.435
9.218
9.011
8.812
8.623
8.442
8.270
8.108
7.954
7.809
7.674
7.547
7.429
7.321
7.221
7.130
7.049
6.976
6.912
6.857
6.812
6.775
6.747
6.729
6.719
6.718
6.726
6.743
6.770
6.805
6.849
6.902
6.965
APHA
(1971)
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
_
^
—
_
_
_
_
_
-
APHA (1985)
(3-10)
11.354
11.067
10.790
10.527
10.273
10.031
9.801
9.575
9.362
9.156
8.957
8.769
8.586
8.411
8.241
8.077
7.922
7.770
7.624
7.482
7.347
7.215
7.087
6.964
6.844
6.727
6.616
6.507
6.401
6.297
6.197
6.100
6.005
5.912
5.822
5.734
5.648
5.564
5.481
5.400
5.322
98
-------
where T = temperature, C
S = salinity, ppt.
The values were compared over a temperature range of 5-30 C.
Equation (3-8), as before, agreed closely-with APHA (1985) throughout the
5-30°C temperature range with a maximum difference of .022 mg/1 less than
APHA (1985). Equation (3-7) had differences of .08 less than and .04 mg/1
greater than APHA (1985) from 5-25°C and near .2 mg/1 higher than APHA
(1985) at 30°C. The values from the APHA (1971) (reported to the nearest
tenth mg/1) had a maximum difference range of 0 to .1 mg/1 higher than APHA
(1985) and the fourth equation, Equation (3-13), varied the most from APHA
(1985) with differences in the range of approximately .03 to 0.5 mg/1
higher.
3.1.4 Methods of Measurement
Elmore and Hayes (1960) have summarized the work of numerous
researchers who have measured dissolved oxygen saturation. According to
Elmore and Hayes, Fox in 1909 used a gasometric technique in which a known
volume of 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
percent oxygen.
From Fox's expression, Whipple and Whipple (1911) converted their
results from milliliters per liter to parts per million. These results were
tabularized, circulated and used as standards by water agencies for years,
and are only now being gradually replaced with tables developed from more
elaborate equations.
Benson and Krause (1984) determined the solubility of oxygen in fresh
and seawater over a temperature of 0-60°C using an equilibrator different
from the Jacobsen Worthington-type equilibrator used in previous
investigations. They felt the new apparatus minimized the uncertainties
associated with methods involving thin films of liquids (Benson, et al.,
99
-------
1979). The dissolved gas values were determined with use of a mercury
manometric system. The resulting data and equations were compared to
previous sets of values from Carpenter (1966), Green (1965), and Murray and
Riley (1969). The APHA (1985) subsequently adopted the Benson and Krause
concentrations as tabulated in. Table 3-2. In earlier work involving fresh
water only (Benson and Krause, 1980) the new concentration values were
recommended by Mortimer (1981) for use in fresh water systems.
To date there is no "standard method" recommended by APHA to measure
saturated dissolved oxygen. The laboratory methods noted in the preceeding
paragraphs are sophisticated methods developed and/or modified for each
research effort and are not conducive to simplier laboratory environments
nor are they adaptable for field use.
Calibration of popular dissolved oxygen probes is carried out under
saturation conditions by methods recommended by the instrument manufacturers
in conjunction with a table such as Table 3-2. The values obtained may be
verified with one of the several wet chemistry iodometric methods (or
"Winkler" titrations) (APHA, 1985).
3.1.5 Summary
Notable differences exist among the results obtained by various
methods used to determine saturated dissolved oxygen values under specified
conditions of temperature, salinity and pressure. These discrepancies may
be as high as 11 percent for high saline conditions (Table 3-5). Under
conditions of zero salinity observed differences are generally less than
2 percent (Table 3-4). The accuracy of the Elmore and Hayes expression, one
of the most frequently used formulas, rapidly deteriorates at water
temperatures exceeding 25°C. The algorithm, Equation (3-8), used in ttte
RECEIV-II model (Weiss, 1970 and USGS, 1981) matches the APHA (1985) data
better than any formula reviewed, for both saline and freshwater conditions.
The algorithm, Equation (3-10), presented in APHA (1985) and its
corresponding table of saturation values, Table 3-2, are based on latest
research and provide the most accurate values of Cg to date. Knowing the
100
-------
possible sources of error using any other particular formulation for C
permits the user to decide whether they are significant in a particular
study.
3.2 REAERATION
3.2.1 Introduction
Reaeration is the process of oxygen exchange between the atmosphere
and a water body in contact with the atmosphere. Typically, the net
transfer of oxygen is from the atmosphere and into the water, since
dissolved oxygen levels in most natural waters are below saturation.
However, when photosynthesis produces supersaturated dissolved oxygen
levels, the net transfer is back into the atmosphere.
The reaeration process is modeled as the product of a mass-transfer
coefficient multiplied by the difference between dissolved oxygen saturation
and the actual dissolved oxygen concentration, that is:
Fc = kL(Cs-C) (3-14)
where F = flux of dissolved oxygen across the water surface, mass/
area/time
C = dissolved oxygen concentration, mass/volume
C = saturation dissolved oxygen concentration, mass/volume
k. = surface transfer coefficient, length/time
For practically all river modeling applications and for
vertically mixed estuaries a depth averaged flux (F'), is used:
F1 = — = — (^c~^' H
c H H s ^
where H = water depth,-length
101
-------
In Equation 3-15 the surface transfer rate and depth are typically
combined into a single term, called the reaeration rate coefficient or
reaeration coefficient, denoted in the literature by k0 or k :
(- a
kL
k2 = £- (3-16)
3.2.2 Reaeration in Rivers
3.2.2.1 Overview
Rivers have been the focus of the majority of reaeration research in
natural waters. Some of the equations that have been developed for rivers
have been successfully applied to estuaries, and is indicative of the lack
of estuarine reaeration research.
Table 3-6 summarizes reaeration coefficient expressions (k~ values) for
rivers. All formulas for reaeration in Table 3-6 are depth averaged values
and are in units of I/day. The table also shows the units required for the
parameters in each formula, and when possible the range of conditions used
in the development of the formulas. All values of k^ are base e, and are
referenced to 20 C, unless otherwise noted. Although base e values are used
directly in most modeling formulations, in the earlier days of reaeration
research, k~ values were often expressed in base 10. The relationship
between base e and base 10 reaeration coefficients is:
k? = In (10)k? = 2.303 k? (3-17)
base e base 10 base 10
Stream reaeration research began in earnest in the late 1950's, and
continues today. The formulas that are shown in Table 3-6 are based on
theory, empiricism, or a combination of the two. In the late 1960's the
radioactive tracer method was introduced by Tsivoglou and Wallace (1972).
The tracer method, or a modification of it, forms the basis for much pf the
research being conducted on reaeration today.
102
-------
TABLE 3-6. REAERATION COEFFICIENTS FOR RIVERS AND STREAMS
o
CO
Author(s) k2, base e(l/day at 20°C)
O'Connor and Dobbins (1958) 12.9U°'5
H1'5
Churchill et al. (1962) 11.6U0'969
H1.673
Owens etiL- <1964) 21.7U°'67
Owens .et.aK (1964) 23. 3U^'73
langbeln and Durum (1967) 7.6U
^753
Isaacs and Gaudy (1968) 8.62U
Parkhurst and Poraeroy (1972) 48. 4(1+0. 17F2)(SU)3/8
H
AA0-85
Negulescu and Rojanskl (1969) 10-9(ff)
Thackston and Krenkel (1969) 24.9(l+F°-5)u*
H
Lau (1972b) 2515/lilj^
Units
U-fps
H-feet
U-fps
H-feet
U-fps
H-feet
U-fps
H-feet
U-fps
H-feet
U-fps
H-feet
U-m/s
S-m/m
H-meters
U-fps
H-feet
u.-fps
H-feet
U*-fps
H-feet
Applicability
Moderately deep to deep channels; 1-
-------
TABLE 3-6. (continued)
Author(s)
Krenkel and Orlob (1962)
Krenkel and Orlob (1962)
Padden and Gloyna (1971)
Cadwallader and
McDonnell (1969)
Bansal (1973)
Bennett and Rathbun (1972)
Dobbins (1964)
Ice and Brown (1978)
McCutcheon and Jennings (1982)
k2> base e(l/day at 20°C)
234(US>'408
8.4 D^"321
H2'32
6.9U0-703
336(US)°'5
H
4.67U0'6
H1'4
1Q6U0.413S0.273
H1.408
20.2U0'607
H1'689
117(l+F2(US)°-375)cot|r4.10(US)°-125l
(0.9+F)U5H L (0.9+F)°'5J
37H2/3S1/2U7/V/2
Q2/3
[ / a I 24 \ * 1
'ln L1'2 V(30.48H)2/ J
T
Units
U-fps
S-ft/ft
H-feet
0, -ft2/m1n
H=feet
U-fps
H-feet
U-fps
S-ft/ft
H-feet
U-fps
H-feet
U-fps
S-ft/ft
H-feet
U-fps
H-feet
S-ft/ft
H-feet
S-ft/ft
U-fps
q-ft3/sec
H-feet _ ,„
a=1.42 (1.1)T'ZO
T °r
Applicability
Based on I1 wide flume data. 0.08'< H<0.2'
Experiments performed 1n a 1' wide flume by deoxygenatlng
the water. Other similar formulas are also reoorted. The
flume dispersion coefficient, OL, was below the range
expected 1n natural systems.
Regression analysis performed on data where 9. 82^28. 8/day.
Based on multlvarlate analysis of reaeratlon data.
Based on reanalysls of reaeratlon data 1n numerous rivers.
These two equations are based on a reanalysls of historical
data, with the second equation being al most as good a
predictor as the first, but not having the slope term.
Theory combined with measurements 1n natural streams, and
flume data of Krenkel and. Orlob (1963).
Based on data collected 1n several small Oregon streams.
Based on the Velz method (1970) and replaces the Iterative
technique. The expressions for the mix Internal I are basec
0.0016 + 0.0005 H H<2.26 ft
0.0097 ln(H) - 0.0052 H>2.26 ft
(continued)
-------
TABLE 3-6. (continued)
Author(s)
Long (1984)
Foree (1976)
Foree (1977)
k2. base e( I/day at 20°C)
1.923U0'273
H0.894
0.30+0.19S1/Z at 25°C
• 0.888 (O.eS+O^S1'15^0-25 at 25°C
for 0.05 < q < 1
Units
U -meters/ sec
H-meters
S-feet/mile
S-feet/mile
q-cfs/ml2
Applicability
Known as the 'Texas* equation. Based on data collected on
streams in Texas.
Radioactive tracer technique used on small streams in
Kentucky. 0l
• 0.42 (0.63+0.4S1'15) at 25°C
for q<0.05
en
Tsivoglou and Wallace (1972)
Tslvoglou and Neal (1976)
0.054 - at 25°C
• 0.11 /-^JJ
f or 1 < Q < 10 cf s
. 0.054
for 25 < Q < 3000 cf s
Ah-feet
t-days
Ah-feet
t-days
Based on sunmary of radioactive tracer applications
to 5 rivers.
Based on data collected on 24 different streams using
radioactive tracer method.
Grant (1976)
Grant (1978)
O.OoY-^}
at 25°C
at 25°C
Ah-feet
t-days
Ah-feet
t-days
Based on data from 10 small streams in Wisconsin using
radioactive tracer techniques:
2.1 < k,<55/day
1.2 i S<:<70 ft/mile
0.3 < Q < 37 cfs
Based on radioactive tracer data developed on Rock River.
Wisconsin and Illinois:
0.01< k,S0.8/day
0.25
-------
TABLE 3-6. (continued)
o
CTi
Author(s)
SMndala and Truax (1980)
k2, base e(l/day at 20°C)
• O.OS/-^ at 25°C
for Q < 10 cf s
• 0.06^ at 25°C
for 10 < Q < 280 cfs
Units
Ah-feet
t-days
Applicability
Based on statistical analysis of reaeratlon coefficients
for rivers 1n 7 states, where the radioactive tracer Method
was used to find the reaeratlon rates.
Eloubaldy and Plate (1972)
Haltingly (1977)
Gulliver and Stefan (1981)
Frexes et a].. (1984)
Wind effects analyzed. See text for discussion.
Wind effects analyzed. See text for discussion.
Wind effects analyzed.
Wind effects analyzed.
Definitions of Symbols:
DL = longitudinal dispersion coefficient
f * Froude number
U
(gh)0'5
g * acceleration due to gravity
Ah = change 1n stream bed elevation between two points
q « stream discharge divided by drainage area
R » hydraulic radius
S • slope
t * travel time between two points where Ah measured
U » stream velocity ,
u* « shear velocity - VgRS
W = width
-------
3.2.2.2 Reviews of Stream Reaeration
Over the past decade, several researchers have reviewed reaeration
formulas, and have tried to evaluate the performance of the formulas. One
of the earlier reviews, Bennett and Rathbun (1972), is also an excellent
source for reaeration theory. They describe the theories behind various
conceptual models of reaeration (including film, renewal, penetration, film-
penetration, and two-film theory models), semi-empirical models, and
empirical models. They also discuss methods to determine the reaeration
coefficient that include dissolved oxygen balances in natural streams,
dissolved oxygen balances in recircu1ating flumes, the distributed
equilibrium technique (where sodium sulfite is usually added to the water to
deoxygenate it), and the radioactive tracer technique.
Table 3-7 summarizes the Bennett and Rathbun review in addition to
other studies that have compared reaeration coefficients. The studies
conclude that no single formula is best for all rivers. For one set of data
one formula may be best, while for another set of data another formula may
appear to be best.
Figure 3-1 compares 13 reaeration coefficient expressions for a range
of depths (from Bennett and Rathbun, 1972). The figure illustrates the
variability between predictions for a velocity of 1.0 fps and slope of
0.0001. The range of differences between predicted values spans one to two
orders of magnitude. The formulas agree with each other best within the
depth range of 1 to 10 feet, typical of many rivers.
Figure 3-2 compares calculated and observed reaeration coefficients for
the formulas of Dobbins (1965) and Parkhurst and Pomeroy (1972). These
formula were found by Wilson and MacLeod (1974) to best fit the observed
data. Notice that the spread of data is slightly less than one order of
magnitude.
The data of Wilson and Macleod also show that the depth - velocity
model of Bennet and Rathbun (1972) does not fit the experimental data nearly
107
-------
TABLE 3-7. SUMMARY OF STUDIES WHICH REVIEWED
STREAM REAERATION COEFFICIENTS
•- • Bennett and Rathbun (1972)
• Thirteen equations were evaluated.
• The standard error of the estimate was used as a measure of the difference between
predicted values and data.
t The equation which provided the best fit to their original data set was Krenkel (1960).
• The equations which best fit the entire range of data were: O'Connor and Dobbins (1958),
Dobbins (1965), Thackston and Krenkel (1969).
• Of the thirteen equations the Churchill e_t al. (1962) formula provided the best fit to
natural stream data.
t The Bennett and Rathbun formula, developed from the data evaluated during their review,
provided a smaller standard error for natural streams than the other 13 equations.
• There was a significant difference between predictions from equations derived from flume
data and equations derived from natural stream data.
• The expected root-mean-square error from different measurement techniques is: 15 percent
using the radioactive tracer technique; 65 percent using the dissolved oxygen mass balance,
and 115 percent using the disturbed equilibrium method.
Lau (1972b)
• Both'conceptual and empirical models were reviewed.
• Conclusions reported were similar to those of Bennett and Rathbun.
• It was found that no completely satisfactory method exists to predict reaeration.
Wilson and MacLeod (1974)
• Nearly 400 data points were used in the analysis.
• Sixteen equations were reviewed.
t The standard error of estimate and graphical results were both used in error analysis.
• It was concluded that equations which use only depth and velocity are not accurate over the
entire range of data investigated.
• The methods of Dobbins (1965) and Parkhurst and Pomeroy (1972) gave the best fits to the
data investigated.
Rathbun (1977)
• Nineteen equations were reviewed.
• Equation predictions were compared against radioactive tracer measurements on 5 rivers
(Chattahoochee, Jackson, Flint, South, Patuxent).
t The best equations in terms of the smallest standard error estimates was Tsivoglou- Wallace
t 1 |?'05?8)? Parkhur>st-Pomeroy (1972) (0.0818), Padden-Gloyna (1971) (0.0712) and Owens
• No one formula was best for all five rivers.
108
-------
TABLE 3-7. (continued)
Rathbun and Grant (1978)
• Compared the radioactive and modified tracer techniques for Black Earth Creek and Madison
Effluent Channel 1n Wisconsin.
• Differences 1n Black Earth Creek were -9% to 4% in one reach and 16% to 32% on another reach
attributable to Increased wind during the latter part of the test.
• Unsteady flow during the Madison Effluent Channel tests led to differences of as much as 25
to 58% 1n one case and -5% to 3% in another.
Shlndala and Truax (1980)
• Reaeration measurements for streams in Mississippi, Wisconsin, Texas, Georgia, North
Carolina, Kentucky, and New York were made using the radioactive tracer technique.
• The energy dissipation model resulted in the best correlation for reaeration coefficient
prediction for small streams. The following escape coefficients (defined as the coefficients
of ^r in energy dissipation models for reaeration coefficients) were recommended:
0.0802/ft .for Q <10 cfs
0.0597/ft ,for 10 < Q < 280 cfs
NCASI Bulletin (1982b)
• Six reaeration formulas were compared against measurements made using radioactive tracer
techniques and hydrocarbon tracer techniques for a reach of the Ouachita River, Arkansas.
t The hydrocarbon tracer technique produced reaeration rates higher than both the radioactive
tracer and empirical formulas.
• The O'Connor - Dobbins (1958) equation was chosen as the best empirical equation.
Kwasnik and Feng (1979)
• Thirteen reaeration formulas were reviewed and compared against values measured using the
modified tracer technique for two streams in Massachusetts.
• The equations of Tsivoglou-Wallace (1972) and Bennett-Rathbun (1972) gave the closest
predictions to the field values.
• The study indicates that results using the modified tracer technique are reproducible.
Grant and Skavroneck (1980)
• Four modified tracer methods and 20 predictive equations were compared against the
radioactive tracer methods for 3 small streams in Wisconsin.
• Compared to the radioactive tracer method the errors in the modified tracer techniques were:
11% for the propane-area method
18% for the propane-peak method
21% for the ethylene-peak method
26% for the ethylene-area method
• Compared to the radioactive tracer method, the equations with the smallest errors were:
18% for Ts1voglou-Neal (1976)
21% for Negulescu-Rojanski (1969)
23% for Padden-Gloyna (1971)
29% for Thackston-Krenkel (1969)
32% for Bansal (1973)
109
-------
TABLE 3-7. (continued)
House and Skavroneck (1981)
• Reaeration coefficients were determined on two creeks 1n Wisconsin using the propane area
modified tracer technique and compared against 20 predictive formulas.
• The top five predictive formulas were:
Tsivoglou - Neal (1976), 34% mean error
Foree (1977), 35% mean error
Cadwallader and McDonnell (1969), 45X, mean error
Isaacs-Gaudy (1968), 45X, mean error
Langbein-Durum (1967), 495!, mean error.
Zison et al_. (1978)
• Thirteen reaeration formulas were reviewed, but none were compared against historical data.
• Covar's method (1976) was discussed which shows how stream reaeration can be simulated by
using three formulas (O'Connor-Dobbins (1958), Churchill et al_. (1962), and Owens et &]_.
(1964)), each applicable in a different depth and velocity regime.
Yotsukura et al_. (1983)
• Developed a steady injection method to avoid uncertainty in dispersion corrections.
• Determined reproducibility to be 4%.
• Found negligible effect of wind where stream banks are high.
Ohio Environmental Protection Agency (1983)
t Eighteen reaeration coefficient equations were compared against data collected in 28 Ohio
streams.
• The streams were divided into four groups based on slope and velocity. The best predictive
equations for each group are shown below:
Group Slope (ft/mile)
1 <3
2 3-10
3 3-10
4 >10
Flow (cfs)
All data
<30
>30
All data
Preferred Equation
Negelescu-Royanski (1969)
Krenkel-Orlob (1962)
Parkhurst-Pomeroy (1972)
Thackston-Krenkel (1969)
Parkhurst-Pomeroy (1972)
Tsivoglou-Neal (1976)
as well (see Figure 3-3). This was the formula which Bennett and Rathbun
(1972) found produced the smallest error of the formulas they reviewed.
Figure 3-4 shows the three reaeration formulas found by Rathbun (1977)
to best predict observed values for the Chattahoochee, Jackson, Flint,
110
-------
1000
100-
10-
>-
<
1-
UJ
o
o
ce:
LU
<
UJ
0.1-
0.01-
0.001
Mean Velocity=1.0 feet per second
Slope =.0.0001
Range of Experimental Data
—,. ___________—,
Equation
Identification
-1
Key:
1
2
3
4
5
6
7
8
9
10
11
12
13
Dobbins (1965)
Krenkel (I960)
Thackston (1966)
Negulescu and Rojansk! (1969)
Thackston (1966)
Fortescue and Pearson (1967)
O'Connor and Dobbins (1956)
O'Connor and Dobbins (1956)
Issacs and Gaudy (1968)
Owens et al. (1964)
Isaacs ancTGaudy (1968)
Churchill et al. (1962)
Owens et aTT TT964)
NOTE: References repeated in the key
indicate that the authors developed
more than one formula for reaeration
rate.
0.01 0.1 1 10 100
DEPTH, IN FEET
Figure 3-1. Predicted reaeration coefficients as a function of depth from
thirteen predictive equations (from Bennett and Rathbun, 1972)
111
-------
1000
Q"
LU
100-
o 10-
LU
O
LU
DC
LU
<
LU
DC
1 -
z
Q 0.1-
0.01
0.01 0.1 1 10 100 1000
REAERATION COEFFICIENT OBSERVED, DAY'1
1
Q
LU
1000'
z
UJ
o
U.
LL
LU
O
O
DC
LU
<
LU
DC
100-
o 10-
1 -
0.01
I
0.1
0.01 0.1 1 10 100 1000
REAERATION COEFFICIENT OBSERVED, DAY"1
(b)
Figure 3-2. Comparisons of predicted and observed reaeration coefficients
for the formula of Dobbins (1965) (a) and Parkhurst-Pomeroy
(1972) (b).
112
-------
South, and Patuxent Rivers. The range of reaeration coefficients analyzed
here is considerably smaller than analyzed by Wilson and Macleod. The
Tsivoglou - Wallace method is noticeably better than either the Padden-
Gloyna or Parkhurst-Pomeroy methods. However, the Tsivoglou-Wallace method
was originally developed using this data set, so it is not surprising that
the fit is best.
Figure 3-5 shows the energy dissipation model of Shindala and Truax
(1980) applied to streams with flow rates less than 280 cfs. They found
that the best fit to the data was achieved when the flow rate was divided
into two groups: less than 10 cfs and greater than 10 cfs.
Covar (1976), as discussed by Zison _et _al_. (1978) found that the
research of O'Connor-Dobbins (1958), Churchill et _al_. (1962), and Owens et
al. (1964) could be used jointly to predict stream reaeration coefficients
1000
C 0.01
0.01 0.1 1 10 100 1000
REAERATION COEFFICIENT OBSERVED, DAY''
Figure 3-3. Formula of Bennett and Rathbun (1972) compared
against observed reaeration coefficients.
113
-------
for a range of depth and velocity combinations. Figure 3-6 shows the data
points collected by each investigator and the regions Covar choose to divide
the applicable formulas. Figure 3-7 shows the plots of reaeration
prediction. Note that the predictions approximately match at the boundaries
of each region.
9.6
z
IU
f 7.2H
UJ
O
o
UJ
fi
a.
S
_
2.4 4.8 7.2 9.6 0 2.4 4.8
EXPERIMENTAL REAERATION COEFFICIENT, DAY'1
(a) (b)
7.2
96
I
9.6
UJ
i 7-2-
LL
UJ
O
o
2 4.8 H
UJ
-------
40
DATA FOR FLOWRATES
BETWEEN 10.0 AND 280.0 CFS
120 180 240 300 360 420
ENERGY DISSIPATION.SU (FEET/DAY)
(a)
460
50
DATA FOR FLOWRATES
LESS THAN 10.0 CFS
160
240
320
400
480
560
640
Figure 3-5.
ENERGY DISSIPATION, SU (FEET/DAY)
(b)
Reaeration coefficient versus energy dissipation (a) for flow rates
between 10 and 280 cfs and (b) for flow rates less than 10 cfs.
(Note: Curves for predicted reaeation coefficients are forced
through the origin).
115
-------
3.2.2.3 Measurement Techniques
Methods to determine reaeration rates based on instream data include
the dissolved oxygen balance, deoxygenation by sodium sulfite, productivity
measurements, and tracer techniques (both radioactive tracers and hydocarbon
tracers). Today, use of tracers is the most widely accepted method.
Productivity measurements are sometimes used, but because of their indirect
approach could be subject to considerable error. Some of these methods are
discussed in Kelly et a/L (1975), Hornberger and Kelly (1975), and Waldon
(1983). Only the tracer methods are discussed here.
Q.
UJ
0
50
40
30
20-
10
8
6
4
3
.6-
.4-
.3-
A O'Connor-Dobbins
o Churchill,et.al.
D Owens,et.aj.
A
.1
A
A
A
A
DA
D
D DD
D
"A" Line
•"B"Line
.2 .3 A .6 .8 1 23456
VELOCITY, ft./sec.
Figure 3-6. Field data considered by three different investigations.
116
-------
The tracer method which appears to produce the most accurate results is
the radioactive tracer technique developed and reported by Tsivoglou et^ al.
(1965), Tsivoglou (1967), Tsivoglou ei_ a]_. (1968), Tsivoglou and Wallace
(1972), and Tsivoglou and Neal (1976). The method involves the
instantaneous and simultaneous release of three tracers: krypton-85,
tritium, and a fluorescent dye. The fluorescent dye indicates when to
sample the invisible radioactive tracers and provides travel time
information as well. The tritium acts as a surrogate for dispersion: the
t
LU
Q
.3-
• ^^^z
.1 .2 .3 A .6 .8 1 2 3456
VELOCITY, ft./sec.
Figure 3-7. Reaeration coefficient (I/day) vs. depth and velocity
using the suggested method of Covar (1976).
117
-------
tritiated water disperses in the same manner as the natural water. The
krypton-85 is lost to the atmosphere in a constant, known ratio compared
with dissolved oxygen. The formula used is:
= exp (-kkpt) (3-18)
/C. \
where 7^- = concentration ratios of krypton and tritium at
\°tr/A,B
locations A and B when the dye peaks at each location
t = travel time between A and B
k. = atmospheric exchange rate of krypton
kkr
Since -r^- = 0.83±0.04, the dissolved oxygen reaeration rate, k?, can be
K2
found directly from k. . The ratio 0.83 was found in the laboratory and has
not been proven to be constant for all conditions.
Wilhelms (1980) has applied the radioactive tracer technique to flow
through a hydraulic model. The results compared favorably with results from
disturbed-equilibrium tests.
Because of the costs and potential hazards of using this method, other
tracer techniques have been developed which do not use radioactive tracers.
These methods have been discussed by Rathbun .et _al_. (1975), R,athbun ^t al.
(1978), Rathbun and Grant (1978), Kwasnik and Feng (1979), Bauer et jTL
(1979), Rathbun (1979), Jobson and Rathbun (undated), Grant and Skavroneck
(1980), House and Skavroneck (1981), Rainwater and Holley (1984), Wilcock
(1984a). and Wilcock (1984b). Not all researchers agree on the accuracy to
the modified tracer techniques. Kwasnik and Feng (1979), Grant and
Skavroneck (1980), House and Skavroneck (1981) all reported successful
applications of the method. However, NCASI (1982b) reported that the
hydrocarbon tracer technique produced results higher than both the
radioactive tracer and empirical methods. The application was on a large
113
-------
sluggish stream. Rainwater and Holley (1984) have investigated two
assumptions of the hydrocarbon tracer technique (constant ratios between
mass transfer coefficients and negligible absorptive losses) and found both
assumptions to be valid for that particular study.
The modified tracer techniques use a hydrocarbon gas tracer and a
fluorescent dye (e.g., rhodamine-WT) as the dispersion-dilution tracer.
Sometimes two different tracer gases (e.g., ethylene and propane) can be
used simultaneously to yield two estimates of reaeration rate. Two methods
can be used: the peak concentration method and the total weight method.
Using the total-weight method the exchange rate of the tracer with the
atmosphere, k is computed as follows:
(3-19)
where AU and Ad = areas under the gas concentration-versus-time curves
at the upstream and downstream ends of the reach,
respectively, and
Q and Q. = stream discharge at each end of the reach.
The reaeration coefficient k? is computed as:
(3-20a)
(3-20b)
Recently Wilcox (1984a, b) has proposed methyl chloride as a gas tracer. At
20°C,
kT
k2 = "757 ' for meth^1 chloride (3-20c)
119
kT
.87
\c
kT
[.72
, ethylene
, propane
-------
The methyl chloride transfer coefficient kT was found to exhibit a
temperature dependence.
The peak concentration method is similar in form to the radioactive
tracer equation:
(3-21)
where k-r = the base e desorption coefficient for the tracer
gas;
t.-t = the time of travel between the peak concentrations;
CT and Cn = the peak concentrations of the tracer gas and
rhodamine-WT dye, respectively
(Ap) ., (AD) = area under dye versus time curve downstream and
uptream, respectively
More recently Yotsukura et_ al. (1983) have conducted tests to assess
the feasibility of a steady-state propane gas tracer method as a means of
estimating reaeration coefficients. The tests were conducted on Cowaselon
Creek, New York. It was concluded that the steady state method, which also
includes an instantaneous injection of dye tracer, is feasible and provides
a reliable method of determining the reaeration coefficient.
3.2.2.4 Special Influences on Reaeration
In addition to hydraulic variables which typically appear in the
expressions in Table 3-6, the reaeration coefficient can be influenced by
certain special factors which include:
• surfactants
• suspended particles
120
-------
• wind
• hydraulic structures, and
• water temperature
While surfactants, suspended solids, and wind can influence reaeration
in rivers, in practice the effects of these factors are rarely if ever
included in water quality models. Discussion of the influence of
surfactants is given in Zison et a±. (1978), Poon and Campbell (1967), and
Tsivoglou and Wallace (1972). The influence of suspended solids is
discussed by Holley (1975) and Alonso et _§]_. (1975).
3.2.2.4.1 Wind Effects
While wind effects are typically not included in reaeration predictions
in rivers, there is evidence that at high wind speeds, the reaeration rates
can be significantly increased. These effects are occasionally alluded to
in the literature when experimental measurements are abnormally high.
Eloubaidy and Plate (1972) performed experiments in the wind-wave
facility at Colorado State University. They arrived at the following
expression for the surface transfer coefficient, k, , in feet per day:
cu* h u*
k, = s c (3-22)
v
where C = a constant of proportionality
2
v = kinematic viscosity of water, m /sec
U* = surface shear velocity due to wind, m/sec = 0.0185 Vw
s
V,, = wind speed, m/sec
I
IL = water shear velocity defined as y9nSr » m/sec
•x v v*
h •= normal depth (i.e., depth with uniform flow), m
Sc = pressure-adjusted channel slope, unitless, SQ +
P = mass density of water, kg/m
2
g = gravitational constant, m/sec
S = slope of energy gradient (channel slope for uniform flow),
unitless
121
-------
o
4^- = air pressure gradient in the longitudinal direction, kg/m -
QX n
sec
From their experiments Eloubaidy and Plate found that C = .0027.
The variables comprising Equation 3-22 are readily obtainable, with the
exception of the pressure gradient. The authors determined that an error on
the order of 2 percent was obtained in k2 (= k^/h) by neglecting the
pressure gradient.
A summary of the conditions under which Equation 3-22 was developed is:
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, 0.63, 0.75 cfs at 0.0043 slope
water depth: 0.385 feet
Note the extremely high wind velocities used in the experiments
(greater than 22 fps). Hence the validity of the approach to lesser wind
speeds typically encountered in the natural environment has not been
demonstrated.
Mattingly (1977) also performed laboratory studies of the effects of
wind on channel reaeration. He obtained this empirical expression:
- 1 = 0.2395 V 1'643 (3-23)
o w
where k2 = reaeration coefficient under windy conditions, I/day
(k2)Q = reaeration coefficient without wind, I/day
VM = wind velocity in meters per second in the free stream
above the boundary layer near the water surface
122
-------
A plot of the experimental data is shown in Figure 3-8. Note the
importance of wind induced reaeration at moderate to high wind speeds.
Further discussion of the effects of wind are found in Gulliver and Stefan
(1981) and Frexes et al_. (1984).
Because wind effects are typically neglected in river and stream
reaeration modeling, this approach is equivalent to assuming a zero wind
velocity. For many water quality model ing applications, such as wasteload
allocation, this approach is reasonable.
100-
10—
JC
CN
D
WATER VELOCITY
O = 18.0 cm/sec
A= 9.0 cm/sec
D = 4.5cm/sec
i i i I i
10
WIND SPEED, m/sec
100
Figure 3-8. Ratio of reaeration coefficient under windy conditions
to reaeration coefficient without wind, as a function
of wind speed (based on laboratory studies).
123
-------
3.2.2.4.2 Small Dams
On many rivers and streams small to moderate sized dams are present.
Dams can influence reaeration by changing the dissolved oxygen deficit from
1 to 3 mg/1 (typically) in a very short reach of the river. Table 3-8
summarizes various predictive equations that have been used to simulate the
effects of small dams. Avery and Novak (1978) discuss limitations of these
equations and aspects of oxygen transfer at hydraulic structures.
Butts and Evans (1983) have reviewed various approaches that predict
the effects of small dams on channel reaeration and further collected field
data on 54 small dams located in Illinois to determine their reaeration
characteristics. They identified 9 classes of structures, and quantified
the aeration coefficient b for use in the following formula:
C -C
r = r^- = 1 + 0.38abh (1 - O.llh) (1 + 0.046T) (3-24)
"
where a = water quality factor (0.65 for grossly polluted streams; 1.8 for
clean streams)
b = weir dam aeration coefficient
h = static head loss in meters
T = water temperature, °C
Figure 3-9 shows the general structural classification and the aeration
coefficient, b, for each class.
The present review does not include influences of large dams,
artificial reaeration, or other hydraulic structures. Cain and Wood (1981)
discuss aeration over Aviemore Dam, 40 m (130 ft) in height, Banks et _al_.
(1983) and NCASI (1969) discuss effects of artificial reaeration, and
Wilhelms _et aK (1981), Wilhelms (1980), and Wilhelms and Smith (1981)
further discuss reaeration related to hydraulic structures.
124
-------
TABLE 3-8. EQUATIONS THAT PREDICT THE EFFECTS OF SMALL DAMS
ON STREAM REAERATION
Reference
Gameson (1957)
Gameson et a±. (1958)
Jarvis (1970)
Holler (1971)
Holler (1971)
Department of the
Environment (1973)
Department of the
Environment (1973)
Nakasome (1975)
Foree (1976)
r =
r -
r!5
r20
r20
r =
r =
log
r =
Predictive Equation
l+0.5abh
l+0.11ab(l+0.046T)h
= 1.05 h°'434
=1+0. 91h
- 1+0. 21h
l+0.69h(l-0.11h)(l+0.0464T)
l+0.38abh(l-0.11h)(l+0.046T)
, \ n AC-7C1.1-28 0.62 ,0.439
e(r20) = 0.0675h q d
exp(O.lbh)
Units
h,
h,
h,
h,
h,
h,
h,
d,
q,
h,
in
in
in
in
in
in
in
h,
in
in
meters
feet
meters
meters
meters
meters
meters
in meters
m2/hr
feet
Source
field
model
model
model
survey
prototype
model
model
model
field
survey
Symbols: r-,-
Cs-Cu
= dissolved oxygen saturation
C , C . = concentration of dissolved oxygen upstream and downstream of dam,
u respectively
a = measure of water quality (0.65 for grossly polluted; 1.8 for clear)
b = function of weir type
h - water level difference
d = tailwater depth below weir
q = specific discharge.
T = water temperature, °C
3.2.2.4.3 Temperature Effects on Reaeration
The influence of temperature on reaeration is typically simulated using
the following type of temperature dependence:
k2(T)=k2(200C)0'
where T = water temperature, °C
B = temperature adjustment factor
125
(3-25)
-------
Table 3-9 summarizes values of 0 from the literature. Typically values
of 1.022 to 1.024 are used in most modeling applications.
Schneiter and Grenney (1983) developed a different approach to simulate
temperature corrections over the ranges 4°C to 30°C. Their approach
effectively allows 0 to vary as a function of temperature. However, the
approach is not widely used.
Head Loss Structure
I I
Dan or Heirs Rock Barriers
I I
Sharp Crested Broad Crested
Vertical Face Sloping Face Round Crest Flat Crest
|| (Straight) (Curved Sloping Face) II
Gates Heir Sloping Face Vertical Face
Curved Straight Straight Step
Irregular Regular
Reference Hunter
5432
Reference Dam Type
Numbers (2)
1 Flat broad-crested regular step 0.70
2 Flat broad-crested irregular step 0.80
3 Flat broad-crested vertical face 0.80
4 Flat broad-crested straight slope face 0.90
5 Flat broad-crested curved face 0.75
6 Round broad-crested curved face 0.60
7 Sharp-crested straight slope face 1.05
8 Sharp-crested"vertical face 0.80
9 Sluice gates with submerged discharge 0.05
Figure 3-9. Division of head loss structures by dam type.
126
-------
TABLE 3-9. REPORTED VALUES OF TEMPERATURE COEFFICIENT
Temperature
Coefficient, 6 Reference
1.047 Streeter, et al. (1926)
1.0241 Elmore and West (1961)
1.0226 Elmore and West (1961)
1.020 Downing and Truesdale (1955)
1.024 Downing and Truesdale (1955)
1.016 Dowining and Truesdale (1955)
1.016 Streeter (1926)
1.018 Truesdale and Van Dyke (1958)
1.015 Truesdale and Van Dyke (1958)
1.008 Truesdale and Van Dyke (1958)
1.024 Churchill ^t _al_ (1962)
1.022 Tsivoglou (1967)
1.024 Committee on Sanitary Engineering Research (1960)
3.2.2.5 Sources of Data
Many sources of stream reaeration rates exist in the literature.
Table 3-10 summarizes a number of the major sources. Many state agencies
are also repositories of reaeration data.
3.2.3 Reaeration in Lakes
Simulation of reaeration in lakes is normally accomplished using the
surface transfer coefficient kL rather than the depth averaged k2- Most
often in lake simulations the surface transfer coefficient kL is assumed to
be a function of wind speed.
127
-------
TABLE 3-10. SOURCES OF STREAM REAERATION DATA
Source
Contents
Owens et al., (1964)
O'Connor and Dobbins (1958)
Churchill et al. (1962)
Tsivoglou and Wallace (1972)
Bennett and Rathbun (1972)
Foree (1976)
Grant (1976)
Grant (1978)
Zison et al. (1978)
Kwasnik and Feng (1979)
Grant and Skavroneck (1980)
House and Skavroneck (1981)
Shindala and Truax (1980)
Terry et ah (1984)
Bauer et al. (1979)
Goddard (1980)
Reaeration coefficients using disturbed equilibrium
technique for six rivers in England (Ivel, Lark, Derwent,
Black Beck, Saint Sunday's Beck, Yewdale Beck), and
associated hydraulic data.
Reaeration data for Clarion River, Brandywine Creek,
Illinois River, Ohio River, and Tennessee River.
Reaeration data using dissolved oxygen balance downstream
from deep impoundments for Clinch River, Holston River,
French Broad River, Watauga River, Hiwassee River.
Hydraulic properties and radioactive tracer measured
reaeration coefficients for Flint, South, Patuxent, Jackson,
and Chattahoochee Rivers.
Summaries of data from Churchill et al_., (1962), Owens et
al., (1964), Gameson ert al. (1958), O'Connor and Dobbins
TT958), Tsivoglou et 71., (1967,1968), Negulescu and
Rojanski (1969), ThackTEonU966), Krenkel (I960).
Radioactive tracer measurements and reaeration hydraulic
characteristics for small streams in Kentucky, and
reaeration measurements for small dams in Kentucky.
Reaeration measurements and hydraulic characteristics for 10
small streams in Wisconsin.
Reaeration measurements and hydraulic characteristics for
Rock River, Wisconsin.
Summary of reaeration coefficients and hydraulic
characteristics for rivers throughout the United States.
Reaeration data using the modified tracer technique on
selected streams in Massachusetts.
Reaeration data from three small streams 1n Wisconsin.
Reaeration data for two small streams 1n Wisconsin.
Radioactive tracer measurements of reaeration rates and
escape coefficients, plus hydraulic data, for rivers in
Mississippi, Wisconsin, Texas, Georgia, North Carolina,
Kentucky and New York.
Hydrocarbon tracer measurements of k, and hydraulic data for
Spring Creek, Osage Creek, and Illinois River, Arkansas.
Bennett-Rathbun (1972) best fit all three streams. Eight
equations were tested.
Hydrocarbon tracer measurements of k- and hydraulic data for
the Yampa River, Colorado best matcned the Tsivoglou Neal
and Thackston and Krenkel energy dissipation type equations.
Lau's equation was extremely error prone. Nineteen
equations were tested.
Hydrocarbon tracer measurements of k- and hydraulic data
from the Arkansas River 1n Coloradcr were used to test 19
equations. The best fitting equations were those by
Dobbins, Padden and Gloyna, Langbein and Durum, and
Parkhurst and Pomeroy.
128
-------
TABLE 3-10. (continued)
Source
Contents
Hren (1983)
Rathbun et al_. (1975)
NCASI (1982c)
farkhurst and Pomeroy (1972)
Ice and Brown (1978)
Ohio Environmental Protection
Agency (1983)
Long (1984)
Radioactive tracer measurements for the North Fork Licking
River, Ohio.
Hydrocarbon tracer measurements for West Hobolochitts Creek,
Mississippi.
Radioactive tracer measurements for Ouachita River,
Arkansas, and Dugdemona River, Louisiana.
Reaeration coefficients were determined by a deoxygenation
method in 12 sewers in the Los Angeles County Sanitation
District.
Reaeration coefficients were determined using sodium sulfite
to deoxygenate the water in small streams in Oregon.
Reaeration coefficients were determined for 28 different
streams in Ohio using predominantly the modified tracer
technique, and in one case the radioactive tracer technique.
Reaeration coefficients, hydraulic data, and time of travel
data collected on 18 streams in Texas.
Since many lakes are not vertically well-mixed, multiple layers are
often used to simulate dissolved oxygen dynamics. Atmospheric reaeration
occurs only through the surface layer, and then dissolved oxygen is
dispersed and advected to layers lower in the water body.
Table 3-11 summarizes various methods that have been used to simulate
reaeration in lakes. With the exception of the method of Di Toro and
Connolly (1980), all formulas include a wind speed term. Di Toro and
Connolly applied a constant surface transfer coefficient to Lake Erie. They
found that the surface layer of the lake remained near saturation so that
the value of k, used was not important as long as it was sufficiently high
to maintain saturated dissolved oxygen levels in the surface layer.
All the surface transfer coefficients shown in Table 3-11 should be
viewed as empirical; the researchers have simply hypothesized that the
suggested formulas are adequate to simulate reaeration. The coefficients (a
and b) are of limited validity, and should be treated as calibration
parameters. O'Connor (1983) has analyzed from a more theoretical point of
view the effects of wind on the surface transfer coefficient.
129
-------
TABLE 3-11 . REAERATION COEFFICIENTS FOR LAKES
Author(s)
Surface Transfer Rate, k, (m/day)
Di Toro and Connolly (1980)
Chen et al., (1976)
Banks (1975)
Baca and Arnett (1976)
Smith (1978)
Liss (1973)
Downing and Truesdale (1955)
Kanwisher (1963)
Broecker et a]_. (1978)
Yu et _al_. (1977)
Broecker and Peng (1974)
2.0
86400D
D
V
(200-605.5 m/sec
kL a + bV
a 0.005 - 0.01 m/day
b = 10"6 - 10"5 m"1
V = wind speed, m/day
kL a + br
a = 0.64 m/day
b - 0.128 secV^ay
V = wind speed, m/s
kL 0.156 V
k 0.0269V
,0.63
1.9
V <4.1 m/sec
V > 4.1 m/sec
V wind speed, m/sec
kL = 0.0276V2'0
V = wind speed, m/sec
kL 0.0432V2
V = wind speed, m/sec
kL 0.864V
V = wind speed, m/sec
kL - 0.319V
V wind speed, m/sec
kL = 0.0449V
V - wind speed, m/sec
130
-------
TABLE 3-11. (Cont'd)
Author(s) Surface Transfer Rate, k. (m/day)
Weiler (1975) k,_ » 0.398 V <1.6 m/sec
kL = 0.155V2 V >1.6 m/sec
V = wind speed, m/sec
Notes:
1. Elevation of wind speed measurements is not always reported.
2. a and b are empirically determined.
Some limited research has addressed the influence of rainfall on
reaeration (Banks _et _al_., 1984; Banks and Herrera, 1977). Rainfall effects
are more of theoretical interest rather of practical concern.
3.2.4 Reaeration in Estuaries
The present state of reaeration simulation in estuaries combines
concepts used in river and lake approaches. Very little original research
on estuarine reaeration has been completed to date.
Table 3-12 summarizes different formulations that have been used to
predict reaeration in estuaries. The different approaches include both kL
(surface transfer) and k,, (depth averaged) reaeration terms. In some
models, k? can be specified (e.g., Genet et ^1_., 1974 and MacDonald and
Weismann, 1977). O'Connor et _§].. (1981) specified the surface transfer rate
to be 1 m/day in their two-layered model of the New York Bight. One of the
more widely used approaches is the O'Connor (1960) formula, which has
subsequently been modified to include wind speed terms (Thomann and
Fitzpatrick, 1982).
Few field studies have been performed for the purpose of directly
measuring reaeration in estuaries. Baumgartner _et _al_., (1970) used Krypton-
85 to measure the range of reaeration in the Yaquina River Estuary. However,
no predictive formulas were developed.
131
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TABLE 3-12. REAERATION COEFFICIENTS FOR ESTUARIES
Reference
Reaeration Rate
O'Connor (1960)
(DLU0)
1/2
H
372
(I/day)
U = mean tidal velocity over a complete
cycle, m/day
D, - molecular diffusivity of oxygen, m /day
H - average depth, m
Genet et _§!_., (1974) kg = user specified
O'Connor et a],., (1981) kL = 1 m/day
MacDonald and Weisman (1977) k, * user specified
Harleman et al_., (1977)
V°-6HW,
= 10.86
(I/day)
V = tidal velocity, ft/sec
H = depth, ft
WT = top width, ft
A = cross-sectional area, ft
Thomann and Fitzpatrick (1982) k2 = ȣ! + 1^81 (0.728W0'5 - 0.317W + 0.0372W2) (I/day)
V - depth averaged velocity, fps
H - depth, ft
W = wind speed, m/sec
Ozturk (1979)
4.56V
4/3
H
(I/day)
V - mean tidal velocity, m/sec
H * mean depth, m
aThe coefficient 10.86 1s the recommended value, but can be changed as discussed by Harleman
et al. (1977).
Tsivoglou (1980) has discussed the application of radioactive tracer
techniques to small estuaries within the Chesapeake Bay. Special discussion
was given to the Ware River Est.uary.
132
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3.2.5 Summary
The most common method of simulating reaeration in rivers is to use the
depth averaged k~ approach, while in lakes the surface transfer rate k, is
typically used. In estuaries either k~ or k, is used, depending on the
importance of stratification. Very little research on reaeration has been
done in either lakes or estuaries. In lakes, reaeration is typically
specified to be a constant or to be a function of wind speed. Little
information is available on how to select parameters in the wind speed
functions. Site specific calibration of the. parameters may be required.
In contrast to lakes and estuaries much research has been conducted on
reaeration in rivers. Thirty-one formulas were shown earlier in Table 3-6.
The formulas have been developed based on hydraulic parameters, most often
depth and velocity. Consequently, the variables in reaeration expressions
are generally not of concern in distinguishing among the utility of the
formulas. One exception is formulas that contain longitudinal dispersion
coefficients, which are difficult to quantify.
Considerable evidence shows that reaeration formulas are most
applicable over the range of variables for which they were developed, and
outside of that range, errors might be quite large. This suggests that
reaeration rates developed from laboratory flume data' may be quite limited
for natural stream applications. Some research supports this supposition
(Bennett and Rathbun, 1972).
Previous reviews of stream reaeration (see Table 3-7) have shown that
no one formula is best under all conditions, and depending on the data set
used, the range of the reaeration coefficients in the data set, and the
error measurement selected, the "best" formula may change. Some of the
reaeration rate expressions which have been judged "best" during past
reviews are:
• The O'Connor and Dobbins (1958), Dobbins (1964), and
Thackston and Krenkel (1969) formulas best fit the entire
range of data reviewed by Bennett and Rathbun (1972).
133
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t The Churchill et !]_. (1962) formula provided the best fit to
natural stream data in the Bennett and Rathbun review.
• The methads of Dobbins (1964) and Parkhurst and Pomeroy
(1972) gave the best fits to the data reviewed by Wilson and
MacLeod (1974).
0 The Tsivoglou-Wallace and Parkhurst-Pomeroy methods were best
in the review by Rathbun (1977).
• The energy dissipation model produced the best correlation
for small streams based on the study of Shindala and Truax
(1980).
From previous reviews, one of the more popular and more accurate
methods for reaeration rates prediction is the energy dissipation method of
Tsivoglou. The method requires knowledge of the escape coefficient, which
appears to depend on streamflow. Typical values of the escape coefficient
are 0.08/ft for flow rates less than 10 cfs, and 0.06/ft for flow rates
between 10 and 280 cfs.
The method of Covar (1976), which combines the O'Connor-Dobbins,
Churchill et _§]_., and Owens et _§_]_., formulas, has merit in that it attempts
to limit the use of the three formulas to within the depth-velocity range
for which they were developed. However, for relatively small and shallow
streams, the method of Owens et_ j/L, tends to overestimate reaeration, so
that the energy dissipation method, which appears to perform well in small
streams, could be used to supplement the method.
The radioactive tracer method appears to be the best method for
measuring stream reaeration coefficients. Even so, the coefficients that
are predicted are valid only for the particular flow condition existing at
the time of sampling. Thus to completely characterize the range of values
of the reaeration coefficient would require numerous sampling events or use
of an acceptable predictive equation.
134
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Sampling methods which require indirect knowledge of parameters that
are difficult to quantify should be avoided. The gas tracer method has been
used with at least partial success, but applications do not yet appear
widespread. When stream reaeration rates are being measured the wind should
be light or calm; otherwise wind effects can produce atypical reaeration
rates.
In deep, slowly moving backwater regions of rivers reaeration can
either be simulated using a river formula or lake formula. The O'Connor-
Dobbins method is probably the most appropriate stream formula to use,
although for very slowly moving backwater regions the predicted reaeration
coefficient can be between 0.01 to 0.05/day, which is below the range of k2
values used in the development of the formula. If a lake reaeration formula
is used, the reaeration rate coefficient can exceed the range predicted
using the O'Connor-Dobbins formula. Under these conditions, wind and not
depth and velocity can control the rate of reaeration.
3.3 CARBONACEOUS DEOXYGENATION
3.3.1 Introduction
Biochemical oxygen demand (BOD) is the utilization of dissolved oxygen
by aquatic microbes to metabolize organic matter, oxidize reduced nitrogen,
and oxidize reduced mineral species such as ferrous iron. The term BOD is
also applied to the substrate itself. Concentrations of reduced minerals in
waste streams are usually inconsequential, and so BOD is commonly divided
into two fractions: that exerted by carbonaceous -matter (CBOD) and that
exerted by nitrogenous matter (NBOD). In domestic wastewaters, CBOD is
typically exerted before NBOD, giving rise to the well-known two-stage BOD
curve (although the processes can be simultaneous in natural systems and
certain industrial effluents). Because wastewaters are potentially high in
BOD, and because dissolved oxygen concentration is used as a principal
determinant of the health of an aquatic system, BOD is a widely applied
measure of aquatic pollution. This section discusses dissolved and
135
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suspended CBOD; Section 3.4 deals with NBOD and Section 3.5 treats benthic
oxygen demand or sediment oxygen demand (SOD). All are related processes.
Figure 3-10 shows the major sources and sinks of carbonaceous BOD in
natural waters. Anthropogenic inputs include point sources and nonpoint
sources such as urban runoff and feedlot runoff. Autochthonous sources
derived from the aquatic biota (particularly algae) can be important in some
systems. Also, re-entrainment of oxygen-demanding material from benthic
deposits may occur. Removal of CBOD from the water column occurs through
sedimentation, microbial degradation and the sorption to or uptake by the
benthic flora. Some components of BOD may also volatilize from the water
column. Carbonaceous material which has settled or been sorbed becomes part
of the benthic oxygen demand.
It is important that the analyst distinguish in the modeling process
between both the sources of BOD and the instream removal mechamisms. Waste
load allocation decisions based upon models which consider CBOD as a
"lumped" quantity may not accurately or fairly assess the water quality
impact of the point sources.
Efforts to characterize CBOD kinetics have focused chiefly on
water-column decay processes, and that is the major emphasis of this
section. A general expression for BOD decay is:
BOD + BACTERIA + 02 + GROWTH FACTORS (NUTRIENTS)
—+- C02 + H20 + MORE BACTERIA + ENERGY
3.3.2 Water Quality Modeling Needs
Nearly all water quality models characterize CBOD decay with first
order kinetics represented by:
136
-------
where L = ultimate CBOD, mg/1
k , = first order rate coefficient, I/day, base e
t = time, days
This equation when coupled with stream dissolved oxygen kinetics becomes the
classic Streeter-Phelps equation:
D =
kdl
-k .t
e d -e
kf
+ D e
o
(3-27)
POINT AND NON-POINT
SOURCE INPUTS
AUTOCHTHONOUS SOURCES
Dead invertebrates, Fecal Algal Exudates
algae,fish .microbes Pellets
CARBONACEOUS BOD
DISSOLVED AND
SUSPENDED
SCOURING AND LEACHING
FROM BENTHIC DEPOSITS
MICROBIAL
DEGRADATION
SETTLING FROM
WATER COLUMN
ADSORPTION/ABSORPTION BY
BENTHIC BIOTA
Figure 3-10.
Sources and sinks of carbonaceous BOD in the
aquatic environment.
137
-------
where D = dissolved oxygen deficit, mg/1
k? = stream reaeration rate, I/day, base e
D = initial stream deficit, mg/1
This equation in principle is similar in nearly all state-of-the-art water
qua!ity models.
In using this representation of BOD/DO for waste load allocation
modeling, the analyst may require measurement or estimation of three
independent factors which include:
(a) the magnitude of ultimate CBOD of the point sources and the
resulting instream spatial distribution,
(b) the magnitude and spatial distribution of the instream CBOD
removal rate, and
(c) the ratio of point source ultimate CBOD to 5-day CBOD (if
compliance is to be based upon CBOD,-).
It is important to note that the water quality model is based upon
ultimate CBOD and not CBODr. Some models internally convert from 5 day to
ultimate using an assumed ratio. In the case of the QUAL-II model (NCASI,
1982a), this ratio is 1.46 and is not user specified. This assumption has
significant implications to water quality modeling because recent experience
has shown that this ratio is both wasteload and receiving water specific.
Ultimate to 5 day ratios as high as 30 have been reported for some paper
industry wastewaters (NCASI, 1982d). Since first order kinetics are assumed
in most models, the ultimate to 5 day ratio is not independent of the decay
rate, k,. Consequently, analysts should be certain that the river water
ultimate to 5 day BOD is not as-signed independently of the rate, k ,.
3.3.3 Nomenclature
Since microbial degradation is not the only process contributing to the
observed depletion of CBOD in a water body (see Figure 3-10), laboratory
rates of carbonacous deoxygenation must be distinguished from those which
138
-------
occur in the field. The following terms are used herein to maintain these
distinctions:
k-| = laboratory-derived CBOD decay rate,
k^ = CBOD decay rate in natural waters
ks = CBOD settling rate
kR = overall rate of CBOD removal from water column
By these definitions,
kR = kd + ks (3-28)
kd ^ kp typically (3-29)
Note that uptake/sorption by the benthic biota is not explicitly dealt
with. In practice, the effects of instream deoxygenation and benthic
biological CBOD removal are difficult to distinguish. Thus reported k.
values may incorporate both processes. Unless otherwise specified, all rate
coefficients discussed in this section are corrected to 20 C, are to the
base e, and are in units of inverse days.
3.3.4 Factors Affecting CBOD Removal
A number of factors are known to influence the rate at which CBOD is
removed from the water column. Chief among these are water temperature,
hydraulic factors, stream geometry and the nature of the carbonaceous
material. The influence of these factors has been described by both
theoretical and empirical formulations.
Like all biochemical processes, CBOD decay occurs at a rate which
increases with increasing temperature up to the point where protein
denaturation begins. This temperature dependence is generally formulated
for a limited range of temperature as:
(3-30)
139
-------
where kT = rate constant at temperature T
k^g = rate constant at 20°C
0 = an empirical coefficient.
This formulation is based on the Arrhenius equation which incorporates
the energy of activation of the overall decay reaction. Arrhenius proposed
the relationship:
d Ink _ _ E /, -n
dT - -2 <331>
where T = absolute temperature, °K
R = universal gas constant
E = activation energy of the reaction
k = rate constant
Integrating Equation (3-31) results in
k -E
ln = (3"32)
where T = arbitarily chosen reference temperature, °K
k = rate constant at temperature T
Equation (3-32) can be rewritten as
-E (T-T V
exp p T T° ) (3-33)
/-MT^
VToT
Equations (3-30) and (3-33) are identical if 6 is defined as
0 = exp (p^r) (3-34)
Note that whether T-TQ is in units of °C or °K is of no concern. Thus 6,
which is assumed to be independent of temperature in Equation (3-30), really
has some slight temperature dependence.
140
-------
Table 3-13 shows values of 6 which have been used for CBOD decay. The
value 1.047 is very widely used and corresponds to an energy of activation
of 7900 calories per mole measured by Fair _et aj_. (1968). There are limits
to the applicability of this approach because the activation energy is not
actually constant. Studies by Schroepfer _et _a]_. (1964) indicate that the
value of 1.047 for 6 is valid between 20°C and 30°C, but higher values are
appropriate at lower temperatures. Fair _et aJL (1968) suggest 6 values of
1.11 and 1.15 for 10°C and 5°C, respectively. Few water quality models
incorporate a varying temperature dependence for CBOD degradation. Some
impose temperature limits, generally 5-30°C, outside of which the reaction
is considered not to occur. The model SSAM-IV (Grenney and Kraszewski,
1981) adjusts the BOD decay rate for temperature via the expression:
kT = rk2Q (3-35)
wnere r = 0.1393 exp (0.174(1-2))
0.9 + 0.1 exp (0.174(1-2))
This is equivalent to varying the value of 6 with temperature.
The 1.047 value originated from the work of Phelps and Theriault
(Phelps, 1927, Theriault, 1927). The 6 value of 1.047 was an average value
obtained from three separate studies with a reported standard deviation of
0.005. Moore noted in 1941 that the correlation of the CBOD decay rate with
temperature using the Arrenhius model was not strong, since correlation
coefficients of 0.56 to 0.78 were obtained (Moore, 1941).
Water turbulence is hypothesized to influence the rate of BOD depletion
in a receiving water in several ways. It influences kg by controlling such
processes as scour and sedimentation. Increased turbulence may enhance
contact between BOD and the benthic biological community. It also
influences the carbonacous deoxygenation rate, so that laboratory samples
which are agitated during incubation yield higher k-j values than quiescent
samples (see Morrissette and Mavinic, 1978, for example). This confounds
the use of k, values from static laboratory tests in place of field values
141
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TABLE 3-13
VALUES OF THE TEMPERATURE COMPENSATION COEFFICIENT
USED FOR CARBONACEOUS BOD DECAY
0, TemperatureTemperature
Correction Factor Limits (°C) Reference
1.047
1.05
Chen(l970)
Harleman et al_. (1977)
Medina (1979)
Genet _et _al_. (1974)
Bauer et _§]_. (1979)
ORB (1983)
Bedford et a]_. (1983)
Thomahn and Fitzpatrick (1982)
Velz (1984)
Roesner _et al_. (1981)
Crim and Lovelace (1973)
Rich (1973)
1.03-1.06
1.075
1.024
(0-5)-(30-35)
Smith (1978)
Imhoff et al. (1981)
Metropolitan Washington Area
Council of Governments (1982)
1.02-1.06
Baca and Arnett (1976)
Baca et al. (1973)
1.04
1.05-1.15
5-30
Di Toro and Connolly (1980)
Fair et al_. (1968)
of kd. To more closely duplicate natural conditions, some investigators
used stirring during laboratory incubations (NCASI, 1982a). This particular
experiment showed nj3 effect of stirring on the reaction kinetics.
142
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Adjustment factors based on stream characteristics have also been used
in BOD calculations. Bosko (1966) expressed kd in terms of k] for streams
by the expression:
kd = k1 + n(V/D) (3-36)
where V = stream velocity, length/time
D = stream depth, length
n = coefficient of bed activity, dimensionless
The coefficient of bed activity is a step function of stream gradient;
values are given in Table 3-14. This expression has been used in a version
of QUAL-II applied to rivers in New England (ORB, 1983; Van Benschoten and
Walker, 1984; Walker, 1983), by Terry et aj_. (1984) on the Illinois River,
Arkansas, and by Chen and Goh (1981).
TABLE 3-14. COEFFICIENT OF BED ACTIVITY AS A FUNCTION OF STREAM SLOPE
(from BOSKO, 1966)
Stream
Slope (ft/mi)
2.5 .1
5.0 .15
10.0 .25
25.0 .4
50.0 .6
Stream hydraulic factors may also account for differences between the
deoxygenation rate k. and the overall BOD removal rate kR. Table 3-15 shows
examples of such differences in six U.S. rivers. Higher values of kR are
attributable to settling of particulate BOD. Bhargava (1983) observed rapid
settling of particulate BOD just downstream from sewage outfalls in two
143
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Indian rivers, where kR was several times greater than farther downstream.
He modeled this effect by considering the BOD to be composed of two
fractions, using the expression:
Vc
Lt = 4(1- ^t) + L2exp(-kdt) (3-37)
where L. = BOD remaining at downstream travel time t
U
L, = portion of original BOD removed by settling
L2 = portion of original BOD subject to in-stream degradation
V = settling velocity of particulate BOD
D = average stream depth
TABLE 3-15. DEOXYGENATION RATES FOR SELECTED U.S. 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 1
(day *)
3.0
0.15
0.3
0.2
.14-. 13
0.05
*
kR ,
(day'1)
3.0
1.7
0.75
1.0
2.5
0.5
*
Note: These data are over 20 years old. It is likely that advances in
waste treatment have altered the BOD kinetics in these waterways.
Some modelers distinguish between benthic and water-column CBOD
removal, and assign rate coefficients to each type. For example, the sum of
144
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settling and benthic biological CBOD uptake is widely portrayed as a first-
order process (Baca and Arnett, 1976; Grenney and Kraszewski, 1981; Duke and
Masch, 1973; Orlob, 1974):
--(k + k)L (3-38)
where k . = water-column deoxygenation rate
ko = total removal rate to the benthos by settling and sorption
The settling rate alone may be derived from the particle settling
velocity and mean depth of the water column:
k = — (3-39)
s D
The effects of scour are often incorporated into the benthic removal
coefficient k,. This may be done implicitly, or by calculating k^ as the
sum of two first-order coefficients having opposite sign (Bauer et al . ,
1979). Scour of benthic BOD is also treated as a zero-order process
(e.g., Baca _et _a]_. , 1973):
aL = -kDL + L (3-40)
et R a
where L = rate of BOD re-entrainment by scour, mg/(l-day).
a
The nature of the oxygen-demanding material also affects the rate of
its removal from a receiving water. Particulate BOD, while it may be
susceptable to settling, is more refractory than soluble BOD. Also two
waters having the same ultimate BOD may show very different BOD depletion
profiles. For in-stream BOD arising from a wastewater inflow, the degree of
treatment of the wastewater is important. In general, the higher the degree
of treatment, the greater the degree of waste stabilization, and the lower
the deoxygenation rate will be. Fair ^t a]_. (1968) cite deoxygenation rates
of 0.39, 0.35 and 0.12-0.23 per day for raw wastewater, primary and
secondary effluent, respectively.
145
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Martone (1976) observed a similar trend with paper industry
wastewaters. Following biological treatment, rates as low as 0.02 per day,
base e were observed. This low rate was attributed to the refractory humic
material remaining in the wastewater. Similar low rates were also noted in
receiving streams (NCASI, 1982a).
The U.S. Environmental Protection Agency (1983), using the data of
Hydroscience (1971) and Wright-McDonnell (1979) has derived a relationship
between stream depth and CBOD removal. This is shown in Figure 3-11. Note
that the predicted decay rate corresponds to the sum of water column and
benthic deoxygenation. Should SOD data be available, modelers are cautioned
when using this figure to avoid double counting of SOD in the oxygen
balance.
To this point, depletion of dissolved oxygen caused by CBOD decay has
been implicitly considered to depend only on the concentration of substrate,
i.e., CBOD. However, at low dissolved oxygen concentrations, oxygen may be
limiting to the reaction. Provision for this "oxygen inhibition" is
incorporated into many water quality models as discussed below.
Autochthonous sources may be a major influence on BOD dynamics. In
lakes, carbon fixed by phytoplankton may become the predominant source of
CBOD. Investigators have dealt with the input of autochthonous CBOD in
several ways. Modeling Onondaga Lake in New York, Freedman et_ al . (1980)
considered the biological contribution to water-column CBOD to be equivalent
to the mass rate of phytoplankton production of organic material. Baca and
Arnett (1976) considered the death rates of phytoplankton and zooplankton
separately. These affected BOD according to the expression:
= -kd L + a (FpP + FzZ) (3-41)
where a = stoichiometric coefficient, mg02/mgC
FZ = death rate of zooplankton from fish predation, I/day
F = death rate of phytoplankton from zooplankton grazing, I/day
146
-------
P = phytoplankton concentration, mg-C/1
Z = zooplankton concentration, mg-C/1
A Potomac Estuary Model by Thomann and Fitzpatrick (1982) considers
the "non-predatory" death rate of phytoplankton to augment water-column
CBOD:
(3-42)
10
5 -
1
QC
LU
OL
0.5 -
0.1 -
KEY
• Hydroscience Data (1971)
o Wright-McDonnell Data(1979)
0.3 0.5 1 1.5
5 10
DEPTH (feet)
50 100
'NOTE: kd includes a
Benthic Deoxygenation
Component
Figure 3-11. Deoxygenation coefficient (k^) as a function of depth
147
-------
where k,p = death rate of phytoplankton other than from grazing,
I/days
P = phytoplankton carbon, mg/1
BOD ,- = ratio of ultimate to 5-day CBOD, taken as 1.85 for
uo
phytoplankton
3.3.5 Predictive Expressions for Deoxygenation
The carbonaceous deoxygenation rate is determined in two general ways.
Most investigators base their measures of kd on the results of field or
laboratory experiments that monitor dissolved oxygen or ultimate CBOD. In
stream modeling, this traditional approach has recently been augmented by
efforts to quantify k, as a function of hydraulic parameters.
It is important to note that these correlations relied upon published
values of k . (such as Figure 3-11). No distinction was made as to how k.
was obtained; and in these correlations, observed instream values have equal
weight with measured laboratory values. Thus, considerable ambiguity exists
in the published literature with regard to the meaning of k , and the
resulting correlation may be of limited value.
Bansal (1975) attempted to predict deoxygenation rates based on the
Reynolds number and the Froude number. This approach was found to have
limited applicability (Novotny and Krenkel, 1975). More commonly, k. is
found as a function of flow rate, hydraulic radius or average stream depth.
Wright and McDonnell (1979) utilized data from 36 stream reaches in the U.S.
to derive the expression:
= (10.3)Q~°'49 (3-43)
where Q = flow rate, ft /sec
They found that above flow rates of about 800 ft3/sec, kd is not a
function of flow rate. The lower limit of the applicability of this
expression is approximately 10 ft3/sec. Below this flow rate, deoxygenation
148
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rates were noted to consistently fall in the range 2.5-3.5 per day,
independent of streamflow. For this same range of flowrates (between 10 and
800 cfs), an expression based on channel wetted perimeter was also found
successful in predicting k.:
kd = 39.6P~°'84 (3-44)
where P = wetted perimeter, feet
The deoxygenation rate coefficient has also been expressed as an
exponential function of stream depth (Hydroscience, 1971; Medina, 1979) and
hydraulic radius (Grenney and Kraszewski, 1981).
Regardless of how carbonaceous deoxygenation rate coefficients are
derived, they are widely applied in only two ways: first-order decay and
simultaneous first-order decay. In the latter case, the CBOD is partitioned
into more than one fraction; each fraction is degraded at a specific rate
according to first-order kinetics. The first-order approximation for CBOD
decay has been widely criticized, and multi-order or logarithmic models have
been used by individual investigators (see Hunter, 1977 for a review).
Martone (1976), in a study of BOD kinetic models, observed that first
order kinetics did not universally describe observed BOD data. In a few
cases, a two-stage carbonaceous BOD model resulted in a better statistical
fit (McKeown et^ aj_., 1981). The Wisconsin Department of Natural Resources
included this alternative formulation in its QUAL-III model (Wisconsin DNR,
1979). However, no alternative formulation has been shown to be universally
superior, and oxygen-sag computations are comparatively easily performed for
first-order decay. Hence, this is the pre-eminent model in use today.
Table 3-16 shows the expressions used by water quality modelers to
describe the consumption of oxygen as a function of water column CBOD decay.
Note that nonoxidative processes such as settling, where CBOD is removed
from or added to the water column, do not contribute to dissolved oxygen
149
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TABLE 3-16. EXPRESSIONS FOR CARBONACEOUS OXYGEN DEMAND
USED IN WATER QUALITY MODELS
Depletion Rate of Dissolved
Oxygen by CBOD Decay, |^ Model and Reference
-k L MIT-DNM (Harleman et ah, 1977)
Dynamic Estuary Model (DEM) (Genet et^K, 1974)
EXPLORE-I (Baca et a±., 1973)
USGS river modelTBauer et _aj.., 1979)
HSPF (Inhoff et a\_., 19817
DOSAS3 (Duke and Masch, 1973)
DIURNAL (Deb and Bowers. 1983)
QUAL-II (Roesner et al., 1981)
O'Connor et al. (T58TT*
Lake Erie Model* (Di Toro and Connolly, 1980)
Potomac Estuary Model (PEM) (Thomann and Fitzpatrick, 1982)
Level Ill-Receiving (Medina, 1979)
Wright and McDonnell (1979)
Rinaldi (1979)
Bedford et al. (1983)
-k.L . - k,L . . WQRRS (Smith, 1978)
1 so1 i aet CE-QUAL-R1* (Corps of Engineers, 1982)
Chen et _§!.* (1974)
-kl (depth, D> 2.44m) 1 RECEIV-II (Raytheon, 1974)
° I WRECEV (Johnson and Duke, 1976)
-0.434
-Cj(Rh C2)L SSAM-IV (Grenney and Kraszewski, 1981)
-k^L Freedman e^ _§]_. (1980)
*"L" represents a fraction of organic carbon, soluble and/or detrital, rather than CBOD.
Definition of symbols:
kd field CBOD oxidation rate
L carbonaceous BOD concentration
Oj concentration of dissolved oxygen
kn half-saturation constant for oxygen
2
a,b,Cj,C2 empirically-determined coefficients
D water depth
Q stream flow rate
k,, k, oxidation rates for two CBOD fractions
I... soluble CBOD (or. dissolved organic carbon)
Ldet particulate CBOD (particulate organic carbon)
Rh steam hydraulic radius
. nonlinear 0« inhibition coefficient
150
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depletion, and are not included in the expressions. In cases where k . is
d
calculated within the model using a hydraulic expression, that expression is
included in the table. As shown, the rate expressions do not include
temperature correction coefficients. Some of the models listed (starred
references) do not treat CBOD per se, but organic carbon or carbonaceous
detritus. The effect of low dissolved oxygen concentration is generally
handled through a Michaelis-Menten formulation. A representative value of
kQ2, the half-saturation coefficient for oxygen uptake, is 0.5 mg/1. Some
models partition oxygen-demanding matter into soluble and particulate
fractions, with different rate coefficients. In limnological models, the
particulate or detrital fraction may be determined as a function of the
death of phytoplankton and zooplankton, with no additional particulate CBOD
present.
3.3.6 Values of Kinetic Coefficients
Table 3-17 is a compilation of deoxygenation rate coefficients and the
methods by which they were determined. Unless otherwise specified, the
coefficient is kd. In some cases, investigators reported kR values as such;
in other cases, rates reported as deoxygenation were actually observations
of total removal (kn) and they are cited as such. Most of the data are from
rivers, although some lake and estuary values have been reported. The range
of values reported as in-stream deoxygenation rates is wide, spanning more
than two orders of magnitude.
3.3.7 Measurement of Ultimate BOD Decay Rate
In laboratory studies using BOD bottles, BOD exertion is found as the
difference between sample and control dissolved oxygen depletion.
Respirometry studies and reaerated stirred-reactor studies involve
essentially continuous monitoring of oxygen usage. The results of these
laboratory experiments produce cumulative oxygen demand-vs-time
relationships.
A number of methods have been used to derive k-, from these curves.
Among these are:
151
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TABLE 3-17- VALUES OF KINETIC COEFFICIENTS FOR DECAY
OF CARBONACEOUS BOD
kd
Location (I/days 9 20 C, base e)
Potomac Estuary 1977
1978
Willamette River, OR
Chattahoochee River, GA
Ganga River, India
Yamuna River, India
S. Fork,
Shenandoah River
Merrimack River, Mass
Gray's Creek, Loulsana
Onondaga Lake, New York
Yampa River, Colorado
Skravad River, Denmark
Seneca Creek
Kansas (6 rivers)
Michigan (3 rivers)
Truckee River, Nevada
Virginia (3 rivers)
N. Branch, Potamac, WV
South Carolina (3 rivers)
New York (2 rivers)
New Jersey (3 rivers)
Houston Ship Channel, TX
Cape Fear R. Estuary, NC
Holston River, Tenn
New York Bight
White River, Arkansas
N. Fork Kings River, CA
Lake Washington, WA
Ouachita River, Arkansas
36 U.S. river reaches
plus laboratory flume
San Francisco Bay
Estuary
Boise River, ID
W. Fork, Trinity
River, TX
0.14 ± 0.023
0.16 ± 0.05
0.1-0.3
0.16
3.5-5.6 (kp)
1.4 K
0.4(kR)
0.01-0.1
1.44 (kR)
K
0.10
0.40
0.15
0.90 (kR)
0.008
0.02-0.60
0.56-3.37
0.36-0.96
0.30-1.25
0.4
0.3-0.35
0.125-0.4
0.2-0.23
0.25
0.23
0.4-1.5
0.05-0.25
0.004-0.66 (k.,)
0.2
0.2
0.15
0.17 (kR)
0.02 (k")
0.08-4.24
0.2
0.75
0.06-0.30
Method of
Determining
Coefficient
field study
field study
field study
field study
model
calibration
model
cal ibration
model
calibration
field study
various
methods
model
calibration
laboratory
study
calibration
laboratory
study
field studies
laboratory
study
Reference
US EPA (1979a)
US EPA (1979b)
Baca et a±. (1973)
Bauer et .al.. (1979)
Bhargava (1983)
Deb and Bowers (1983)
Camp (1965)
Crane and Malone (1982)
Freedman et aK (1980)
Grenney and Kraszewskl (1981)
Hvitved-Jacobsen (1982)
Metropolitan Washington
Council of Governments (1982)
Reported by Bansal (1975)
Novotny and Krenkel (1975)
O'Connor et 3\_. (1981)
Terry et al_. (1983)
Tetra Tech (1976)
Chen and Orlob (1975)
Hydrosdence (1979)
NCASI (1982a)
Wright and McDonnell (1979)
Chen (1970)
Chen and Wells (1975)
Jennings et ^1- (1982)
152
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TABLE 3-17. (Cont'd)
Location
Mil lamette River, OR
Arkansas River, CO
k!l
(I/days 0 20°C, base e)
0.07-0.14
1.5
Method of
Determining
Coefficient
lab and field
field study
Reference
McCutcheon (1983)
Lower Sacramento 0.41
River, CA
Delaware River Estuary 0.31
Wappinger Creek 0.31
Estuary, NY
Potomac Estuary 0.16,0.21
Speed River, Ontario 1.0
field study
Hydroscience (1972)
Thomann and Fitzpatrick (1982)
Gowda (1983)
1. The linear least-squares technique of Reed and Theriault
2. Thomas' graphical slope method
3. The moment method of Moore (1941)
4. Orford and Ingram1s logarithmic method
5. Rhame's two-point method
6. Nemerow's general laboratory method (graphical)
7. The daily difference method of Tsivoglou (1958)
8. The rapid ratio method of Sheehy (1960)
9. Nonlinear regression method of NCASI (1982d).
The first six methods are discussed by Nemerow (1974). Gaudy et_ _al_. (1967)
review and compare a number of calculation methods. Some of the techniques
assume a particular kinetic model for the data, while others do not. The
linear least-squares method can be used with a first or second-order BOD
dependency, with somewhat different calculations. Orford and Ingram1s
method assumes that cumulative BOD exertion varies with the logarithm of
elapsed time, and no limiting value is approached. The nonlinear regression
technique has the advantage of flexibility in evaluating alternative BOD
models.
Barnwell (1980) developed a nonlinear least-squares technique for
fitting laboratory CBOD progressions. It is based upon the first-order
decay model, and is suitable for implementation on programmable calculators
or microcomputers. It allows computation of confidence contours for the
estimates of k, and ultimate CBOD. The nonlinear regression technique also
153
-------
provides estimates of the confidence contours. Further discussion of BOD
measurement techniques are contained in Stover and McCartney (1984) and
Stamer et al_. (1983).
Estimates of the length of time necessary to evaluate the BOD
parameters have been provided by Berthouex and Hunter (1971). They
determined, using statistical arguments, that this length of time is a
function of the anticipated decay rate, k-,. The time computed from 4/k, is
suggested as the maximum value. Barnwell (1980) and NCASI (1982d) have
shown that the estimate of the confidence contours is directly related to
the length of time the BOD experiment was conducted. As the length of time
increases, the confidence contours get smaller.
In field estimation of deoxygenation rates, water samples from along
the stream reach are collected, and their ultimate CBOD values are
determined in the laboratory. Graphical methods are then used to find the
CBOD decay rate. These techniques are based on a mass balance for BOD in
the stream. Note that if unfiltered water samples are used, the rate
calculated is k^, not k.. It may be that the two rates are essentially
equivalent. An unvarying profile of suspended solids along the reach may
indicate the validity of these measurements to estimate k.. Alternatively,
filtered samples may be incubated, and the contribution of particulate
matter to BOD assumed to be insignificant.
The calculation methods described herein are based upon simplified
forms of the BOD mass-balance equations. The user should assess carefully
whether the necessary simplifying assumptions can reasonably be applied to
the study system.
One simple and commonly used technique is for streams influenced by
continuous point sources. The stream reach under study should have a
relatively constant cross section, constant flow rate, and a single point-
source BOD loading. The BOD concentration downstream from the source is
given by:
154
-------
-v
L = LQexp \ J (3-45)
where X = distance downstream from source, length
L = BOD concentration immediately downstream from source, at
X = 0, mass/volume
V = average stream velocity, length/time
A graph of the logarithm of BOD concentration versus distance
downstream should show a straight-line relationship with a slope of -kR/V if
decay is first order. Sometimes the slope may be more steep for the first
few miles below a point source, where settling of BOD as well as decay is
occurring (Deb and Bower, 1983). The slope may be found graphically or by
linear regression. Figure 3-12 is an example of this type of computation.
If the slope is determined by regression, the natural log of BOD should be
regressed on distance. If the slope is found graphically from a semi-log
plot, it must be multiplied by 2.3 (to convert from base-10 to base-e) for
model applications.
The same approach is possible for tidally influenced rivers, as
discussed in Zison ^t ^1_. (1978). However, the tidally averaged dispersion
coefficient is required as an additional piece of information and will add
some degree of uncertainty to the predicted kd value.
3.3.8 Summary and Recommendations
Although its shortcomings have been widely discussed, the first-order
model is still the common method for simulating instream CBOD kinetics.
Relative ease of computation, a long history of use and the absence of
alternative formulations which are superior over a range of conditions are
probably responsible for this precedent.
In estimating k., there is increasing use of various stream hydraulic
parameters. Estimates based on flow rate seem to be most successful,
155
-------
15
=• 10-
O)
E
Q
o
CO
O
5-
4-
3-
2-
u
V-4 mi/day
KR--Slope xV
-2.3(0.6)
-1.4/day
Slope =2.3 pft
-0.6/mile
8
MILE
10
12
Point Source
14
Figure 3-12. Example computation of kR based on BOD measurements
of stream water.
156
-------
although stream geometric parameters such as hydraulic radius and depth are
also used. The use of hydraulic characteristics for kd prediction has
limits, since deoxygenation is independent of flow rate at both high and low
flow. These predictive equations should be used with caution.
To assess CBOD fluxes based on site-specific data, it is essential to
have some familiarity with the water body under study. A reconnaissance
survey can help elucidate the possible importance of CBOD sedimentation or
resuspension, as well as the magnitude of aquatic biological processes. The
survey is also an opportunity to assess what assumptions can reasonably be
made about the system to simplify calculations.
For those river waters and effluents which contain significant
concentrations of NBOD, the analyst must consider an appropriate procedure
for the separation of NBOD from CBOD in the ultimate BOD test. Currently,
two techniques are used which include: the use of nitrification inhibitors
such as TCMP and others, and the monitoring of nitrogen series with time
during the test to define the NBOD. There is currently no consensus as to
which technique is best. Nitrification inhibitors have been observed to
have an unpredictable inhibition effect on the CBOD kinetics as well
(Martone, 1976). For large modeling projects, the monitoring of nitrogen
species in the BOD bottle tests can create significant additional laboratory
expense. Though likely to be more expensive, the latter technique provides
more information regarding the CBOD and NBOD kinetics and is recommended by
NCASI (1982b).
The investigator should exercise caution in using deoxygenation
coefficients obtained for other water bodies. The wide range of values in
Table 3-17 indicates substantial variation in rate estimation and reporting
procedures. Unfortunately, many investigators automatically equate k^ or kR
with k., and do not fully consider the different meanings of these rates.
Some report k. and kR values without stating whether these apply to total
BOD or CBOD, are temperature-corrected, are to base e or base 10, etc.
157
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One way to handle these uncertainities is to conduct sensitivity
analyses of model predictions. Such analyses are beyond the scope of many
projects; however, results are available for many widely-used models either
in the model documentation or in the final reports of large-scale projects.
Examples of sensitivity analyses for deoxygenation rate coefficients are
Crane and Malone (1982), Thomann and Fitzpatrick (1982) and NCASI (1982a).
In addition, it is possible to quantitatively evaluate the uncertainly
associated with an estimated coefficient. Barnwell's (1980) and NCASI's
(1982b) calculation techniques allow computation of confidence limits for an
estimated k, value. Jaffe and Parker (1984) provide a procedure for
estimating the uncertainty of kd values as influenced by the field sampling
scheme. Chadderton et_ aj_. (1982) evaluate the relative contributions to
uncertainty of the parameters of the Streeter-Phelps equation.
3.4 NITROGENOUS BIOCHEMICAL OXYGEN DEMAND
3.4.1 Introduction
The transformation of reduced forms of nitrogen to more oxidized forms
(nitrification) consumes oxygen. Although nitrification is also a nutrient
transformation process, this section addresses the oxygen consumption
aspects, since numerous models simulate nitrogenous biochemical oxygen
demand (NBOD) without detailing nitrogen transformations.
Nitrification is a two-stage process. The first stage is the oxidation
of ammonia to nitrite by Nitrosomononas bacteria:
NH* + 1.5 02—»-NO~ + H20 + 2H+ (3-46)
(14 gm) (48 gm)
Stoichiometrically 48/14 or 3.43 gm of oxygen are consumed for each gram of
ammonia-nitrogen oxidized to nitrite-nitrogen. During the second stage of
nitrification Nitrobacter bacteria oxidize nitrite to nitrate:
153
-------
NO;; + 1/2 02—a-NO^ (3-47)
(14 gm) (16 gm)
Stoichiometrically 16/14 = 1.14 gm of oxygen are consumed per gram of
nitrite-nitrogen oxidized. If the two reactions are combined, the complete
oxidation of ammonia can be represented by:
NH* + 2 02—*-NO~ + H20 + 2H+ (3-48)
(14 gm) (64 gm)
As expected, 64/14 = 4.57 gm of oxygen are required for the complete
oxidation of one gram of ammonia.
In the reactions above, the organic-nitrogen form does not appear,
since organic-nitrogen is hydrolyzed to ammonia, and does not consume oxygen
in the process. However, organic nitrogen will eventually contribute to the
NBOD, as the following equation shows:
NBOD
= 4.57 (NQ + NX) + 1.14 N2 (3-49)
where N~ = organic-nitrogen concentration, mass/volume
N, = ammonia-nitrogen concentration, mass/volume
N2 = nitrite-nitrogen concentration, mass/volume
The stoichiometric coefficients of 3.43, 1.14, and 4.57 in the
equations above are actually somewhat higher than the total oxygen
requirements because of cell synthesis. Some researchers (e.g., Wezernak
and Gannon, 1967 and Adams and Eckenfelder, 1977) have suggested that the
three coefficients be reduced to 3.22, 1.11, and 4.33, respectively.
3.4.2 Modeling Approaches
Modelers use both the two-stage and one-stage approach to simulate NBOD
decay, as shown by Table 3-18. First order kinetics is the predominant
159
-------
method used to simulate the process. Oxygen limitation is used by some
modelers (e.g., O'Connor ^t _§_]_., 1981; Thomann and Fitzpatrick, 1982; and
Bedford _et aj.., 1983).
Relatively few modelers explicitly simulate the effects of benthic
nitrification (exceptions are Williams and Lewis, 1984 and Mills, 1976).
The models of Williams and Lewis, and Mills were developed for relatively
shallow streams where bottom effects could be important. Of these two, only
Mills looks at the details of oxygen and nitrogen transfer from the water
column into an attached nitrifying biofilm. Several studies (Kreutzberger
and Francisco, 1977; Koltz, 1982) have confirmed that nitrifying bacteria
can thrive in the beds of shallow streams, and that, in the streams they
investigated, nitrification occurred primarily in the bed, and not in the
water column. Denitrification has been shown to occur in stream sediments
TABLE 3-18. EXPRESSIONS FOR NITROGENOUS BIOCHEMICAL OXIDATION RATES
USED IN A VARIETY OF WATER QUALITY MODELS
Expression for Nitrogenous Oxidation
Rate. eDO/at
Model and/or Reference
- a k N. - a k N,
1 n. 1 2 n, 2
-knLn
WQRRS (Smith, 1978)
Bauer et _aj_. (1979)
QUAL-II (Roesner et jH., 1981)
SSAM IV (Grenney and Kraszewski, 1981)
CE-QUAL-R1 (U.S. Army COE, 1982)
RECEIV II (Raytheon, 1974)
NCASI (1982d)
Baca and Arnett (1976)
MIT Transient Water Quality Model (Harleman et al., 1977)
DOSAG3 (Duke and Masch, 1973)
HSPF (Imhoff et al., 1981)
Genet _et a]_. (1974)
DIURNAL (Deb and Bowers, 1983)
Gowda (1983)
EXPLORE-1 (Baca et al., 1973)
Bauer et al. (1979)
Di Toro and Matystik, 1980
160
-------
TABLE 3-18. (Cont'd)
Expression for Nitrogenous Oxidation
Rate. gDO/at
Model and/or Reference
- a- k
3 n °2 + Knit
N,
O'Connor ^t _§]_. (1981)
Thomann and Fitzpatrick (1982)
" a3 a4 Q Q + K Nl
J 4 U2 nit L
Time Shifted First Order (time delayed)
Lagged First Order (nonoxidative step
followed by an oxidative step)
Benthic Nitrification:
- a, S (zero order kinetics)
- Jc (Monod kinetics)
Bedford et &\_. (1983)
NCASI (1982d)
NCASI (1982d)
Williams and Lewis (1984)
Bauer ^t _al. (1979)
Mills (1976)
Definition of Symbols:
k
dissolved oxygen concentration
half-saturation constant
zero order benthlc nitrification rate
benthic oxygen flux rate by nitrifying
organisms growing in an attached
biofllm
amnonla to nitrite oxidation rate
nitrite to nitrate oxidation rate
NBOD decay rate
3.43, typically
1.14, typically
4.57, typically
b unspecified
N0~ -N
nitrogenous BOD
as well (Wyer and Hill, 1984). Denitrification is discussed in more detail
in Chapter 5.
The most straightforward method of including the effects of organic
nitrogen on the potential depletion of dissolved oxygen is to simulate the
conversion of organic nitrogen to ammonium nitrogen (a rate of O.I/day is
typically used). The increased ammonia concentration is then available to
exert an oxygen demand. However, it is not clear that all the models in
161
-------
Table 3-18 simulate the organic nitrogen to ammonia conversion. Some models
appear to combine ammonia and organic nitrogen together into a single term.
While first order kinetics is the most popular approach for simulating
nitrification in natural systems, Monod and zero-order kinetics are often
used to simulate nitrification in wastewater treatment processes (Hall and
Murphy, 1980; Charley _et _al_. , 1980; Rittmann and McCarty, 1978).
'Figure 3-13 shows how nitrification is simulated using Monod kinetics. "At
the high level of reduced nitrogen compounds found in wastewater,
nitrification can proceed at its maximum rate, and thus is zero order
(independent of substrate concentration). At lower reduced nitrogen
concentrations, first order kinetics are applicable.
Vmax
LU
fe
cc
o
5
£ O.SVmax
£
ZERO ORDER
FIRST
ORDER
Kc
REDUCED NITROGEN CONCENTRATION
Figure 3-13.
Effect of Reduced Nitrogen Concentration on Nitrification
Rate as Reported by Borchardt (1966).
162
-------
Several researchers (e.g., Wi Id _et _§_]_. , 1971; Kiff, 1972; Huang and
Hopson, 1974) have established concentration ranges of ammonia nitrogen when
zero order kinetics appear to be followed. The range is quite wide, from
1.6 mg/1 to 673 mg/1. Concentrations of ammonia-nitrogen in natural waters
can exceed the lower end of the scale reported, and indicate that zero order
or Monod kinetics may be appropriate in these circumstances (e.g., see
Wilber et _al.. , 1979).
3.4.3 Factors That Affect Nitrification
Table 3-19 summaries studies that have investigated factors that
influence the rate of nitrification. The factors include pH, temperature,
ammonia and nitrite concentrations, dissolved oxygen, suspended solids, and
organic and inorganic compounds. Sharma and Ahlert (1977) also prpvide a
review of previous studies.
Many of the studies have been carried out in controlled environments,
and not in natural waters. Also, the concentration of organic substances
which have inhibitory effects on nitrification are often, but not always,
well above 1 mg/1 (Wood _et al . , (1981)), so that the compounds are not
likely to be inhibitory in natural waters.
Modelers typically consider only the temperature effect on
nitrification, although a few do model dissolved oxygen limitations (see
Table 3-18). Other inhibitory or stimulatory effects are assumed to be
included in the "reference" rate (typically at 20°C) measured or otherwise
selected for the modeling applications.
Researchers have found that within the temperature range of 10°C to
30°C temperature effects can be simulated by the following expression:
T-20 f 3-501
(6 buj
where k on = nitrification rate coefficient at 20°C
n^U
0 = temperature correction factor
163-
-------
TABLE 3-19. SUMMARY OF FACTORS THAT INFLUENCE NITRIFICATION
Reference
Factors Investigated
Comments
Sharma and Ahlert (1977)
Temperature, pH, Nitrogen
Concentrations, Dissolved
Oxygen, Organic Compounds
In reviews of previous studies found: 12 studies for dissolved
oxygen, 15 studies for pH, 14 studies for the effect of ammonia
levels on nitrification, 11 studies of effects of nitrate levels
on nitrification, 34 studies on substances that are required or
stimulate nitrification; 47 studies on substances that inhibit
nitrification.
Stenstrom and Poduska (1980)
Wild, Sawyer, and McMahon (1971)
Dissolved Oxygen
pH, Temperature, Ammonla-
nltrogen
01
Kholdebarin and Oertli (1977a) pH, Ammonia-nitrogen
Kholdebarin and OertH (1977b) Suspended Solids
Bridle, CHmenhage, Stelzlg
(1979)
Qulnlan (1980)
Hood, Hurley, Matthews (1981)
Hockenbury and Grady (1977)
pH, Temperature, Amnonla-
nitrogen, Copper
Temperature
Organic Compounds
Organic Compounds
In this literature review of the effects of dissolved oxygen
concentrations on nitrification, the lowest concentration where
nitrification occurred is approximately 0.3 mg/1. However, the
dissolved oxygen level required for no oxygen inhibition varied to
as high as 4.0 mg/1, while other researchers found only 0.5 mg/1
1s required.
Studies were conducted in a pilot nitrification unit receiving
trickling filter effluent. Ammonia nitrogen did not Inhibit
nitrification at concentrations less than 60 mg/1. Optimum pH for
nitrification was found to be 8.4. The rate af nitrification
Increased with temperature in the range 5 C to 30 C.
For water samples collected from the Whitewater River, California,
the optimum pH for nitrification of ammonia and nitrite was 8.5.
Nitrite oxidation was stimulated by the addition of 3 mg/1
ammonium.
In water from the Whitewater River in California, suspended solids
were found to have a stimulatory effect on nitrification,
presumably caused by the physical support provided by the sol Ids.
In batch reactors ammonia nitrification was not inhibited for TKN
levels up to 340 mg/1. The optimum pH for nitrification was 8.5.
The nitrification rate Increased approximately 2.5 fold for each
10 C Increase. Copper concentrations of 3000 mg/1 produced no
adverse effect; concentrations of 6000 mg/1 were Inhibitory.
Temperature for optimal ammonia and nitrite oxidation was found to
depend on nitrogen concentrations. At low nitrogen concentra-
tions the optimum temperatures were 35.4 C for ammonia oxidation
and 15.4°C for nitrite oxidation.
Laboratory studies were conducted using filtered liquor from
return activated sludge. Approximately 20 compounds were tested
in concentrations from 10 to 330 mg/1. Approximately half the
compounds had no Inhibitory effects.
This study reviewed previous work on the influence of organic
compounds on nitrification. Additionally, they found that many
compounds did not Inhibit nitrification at concentrations as high
as 100 mg/1, while other compounds inhibited nitrification at
concentrations less than 1 mg/1.
-------
Values of the temperature correction factor are reported in Table 3-20.
Temperature correction values are slightly higher for ammonia oxidation than
for nitrite oxidation. The mean temperature correction values are 1.0850
for ammonia oxidation and 1.0586 for nitrite oxidation. Many models use
temperature correction factors slightly lower than these values. Typically
modelers use only one temperature correction coefficient, and do not
distinguish between temperature corrections for ammonia and nitrite
oxidation. Example of temperature correction factors used in selected
models are:
• 1.05, EXPLORE-1 (Baca etaK, 1973)
0 1.065, MIT Nitrogen model (Harleman _et _a_L, 1977)
• 1.08, New York Bight model (O'Connor _et a/L, 1981)
• 1.047, QUAL-II (Roesner et _al_., 1981), USGS Steady State Model
(Bauer et _al_., 1979)
• 1.045, Potomac Estuary Model (Thomann and Fitzpatrick, 1982)
TABLE 3-20
TEMPERATURE CORRECTION FACTOR, 6, FOR NITRIFICATION
Reference Ammonia Oxidation Nitrite Oxidation
Stratton (1966); Stratton and
McCarty (1967)
Knowles et _aj_. (1965)
Buswell et _al_. (1957)
Wild _et _al_. (1971)
Bridle et _aj_. (1979)
Sharma and Ahlert (1977)
Laudelout and Van Tichelen (1960)
Mean
1.0876
1.0997
1.0757
1.0548
1.1030
1.069
-
1.0850
1.0576
1.0608
-
-
-
1.0470
1.0689
1.0586
165
-------
• 1.02-1.03, WQRRS (Smith, 1978)
• 1.08, Lake Erie model (Di Toro and Connolly, 1980)
While Equation (3-44) can provide adequate temperature correction up to
approximately 30°C, beyond this temperature the nitrification rate is
inhibited by the high temperature, so the relationship no longer holds.
Figure 3-14 illustrates the effect of temperature on nitrification and shows
that the rate rapidly decreases at temperatures beyond 30 C.
100
90-
80-
70
60
50-
o
b
UJ
CC
li.
O
Z 40
111
o
IT
UJ 30
20-
10-
10 20 30 40 50
TEMPERATURE,°C
60 70
Figure 3-14. Effect of Temperature on Nitrification as Reported
by Borchardt (1966).
The influence of pH on rates of nitrification is also quite important.
If pH is outside of the range 7.0 to 9.8, significant reduction in
nitrification rates can occur. Table 3-19 indicated that the optimal pH for
nitrification is approximately 8.5 and at pH values below about 6.0,
nitrification is not expected to occur. Figure 3-15 shows the effect of pH
166
-------
on ammonia and nitrite oxidation. A more thorough review of pH effects is
contained in Sharma and Ahlert (1977).
Effects of solid surfaces have frequently been documented as being
important for nitrification (e.g., Kholdebarin and Oertli, 1977). The
following section discusses this effect more fully through a number of case
studies.
UJ
cc
2
ID
2
X
<
LU
O
DC
UJ
CL
100
90-
80-
70-
60-
50-
40-
30-
20-
10-
0
6.0
LLJ
£
cc
5
D
2
X
UJ
O
QC
UJ
0.
70
8.0
ph
9.0
10.0
(a) AMMONIA OXIDATION (Wild et.al.,1971)
100
90-
80-
70-
60-
50-
40-
30-
20-
10-
0
0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5
Ph
(b) NITRITE OXIDATION (Myerhof,l9l6)
Figure 3-15. pH Dependence of Nitrification.
167
-------
3.4.4 Case Studies and Nitrification Rates
Table 3-21 summaries case studies of nitrification in natural waters.
These studies are intended to show how various researchers have determined
nitrification rates in natural waters, some of the complications that can
occur in doing so, and what the rates are.
Except for Slayton and Trovato (1978, 1979) all "the case studies are
for streams or rivers. Note the high variability in nitrification" rates
from study to study. For rivers, documented first order nitrification rates
varied from 0.0/day to 9.0/day. For the two Potomac estuary studies, the
nitrification rates were fairly small and constant (0.1 to 0.14/day). The
nitrification rate was often determined from plots of TKN or NBOD versus
distance or travel time. Figure 3-16 shows an example. A number of the
studies (e.g., Koltz (1982) and Ruane and Krenkel (1978)) emphasized that
algal uptake of ammonia can be an important transformation and should be
accounted for in the rate determination. The increase of nitrate nitrogen
can be monitored, as well as the decrease in ammonia nitrogen for more
conclusive evidence that nitrification is occurring. Bingham ejt a±. (1984)
show how the nitrification rate constant is changed in a QUAL-II application
when algae is simulated compared to when algae is not simulated.
Several of the case studies have enumerated nitrifying bacteria present
in the water column and in the sediments (e.g., Kreutzberger and Francisco
(1977)). Far more nitrifying organisms are typically present in the
sediments than in the water column. Case studies on the following rivers
have reached the same conclusion:
t Kanawha River, West Virginia (U.S. EPA, 1975)
• Tame and Trent Rivers, England (Curtis et ^1_., 1975)
• North Buffalo Creek, North Carolina (Williams and Lewis, 1984)
• Willamette River, Oregon (Rinella et_ _al_., 1981)
t Chattahoochee River, Georgia (Jobson, undated)
16C
-------
TABLE 3-21. CASE STUDIES OF NITRIFICATION IN NATURAL WATERS
Reference
Study Area
Purpose of Study
Reported
Nitrification Rates
Methods of Determining
Nitrification Rates
Comments
en
Uezernak and Gannon
(1968)
Stratton and HcCarty
(1969)
Blain (1969)
Gowda (1983)
Curtis (1983)
Deb and Bowers
(1983)
Deb, Klafter-Snyder,
and Richards (1983)
Ruane and Krenkel (1978)
Koltz (1982)
Clinton River,
Michigan, a
shallow stream
with velocities
of 1-2 fps
Speed River,
Canada, a
relatively
shallow river
with velocities
from 0.3 to
1.5 fps
Still River,
Connecticut
South Fork of
Shenandoah River
Leatherwood,
Creek, Arkansas
Holston River,
Tennessee
Iowa and Cedar
Rivers, Iowa
To mathematically
model nitrifica-
tion 1n a stream
(This was one of
the earlier
modeling attempts)
To determine the
affects of nitrifica-
tion on dissolved oxygen
levels within the river
To determine the fate
of ammonia in the
river by simulating
oxidative and non-
oxidative transforma-
tions
To simulate the
dissolved oxygen of the
river using the
DIURNAL model
To simulate the
dissolved oxygen
dynamics of a small
surface-active stream
for wasteload allocation
purposes
To examine the various
nitrogen transformations
that occur 1n the river
To determine the
locations and rates
of nitrification down-
stream from two waste-
water treatment plants
ammonia oxidation:
3.1-6.2/day
nitrite oxidation:
4.3-6.6/day
0.2-4.41/day
0.0-0.4/day
0.2-1.25/day
1.1-7. I/day
0.15-0.3/day
0.5-9.0/day
(continued)
Measurements of
ammonia, nitrite, and
nitrate at three
locations within the
stream
Plots of TKN versus
travel time
Comparison of total
ammonia decrease to
nitrate increase
Plots of NBOD versus
travel time
Plots of TKN versus
travel time
Rate of ammonia
reduction and rate
of nitrate increase
Rate of ammonia
reduction and rate
of nitrate increase
The nitrogen balance developed
Indicated that nitrification was
primary mechanism responsible for
observed nitrogen transformations.
NBOD predicted to be much
more important on the dissolved
oxygen deficit than CBOD.
The complexity of the nitrogen
cycle 1n the Holston River is
discussed Including the effects
of ammonia transformations other
than caused by nitrification.
Algal assimilation of ammonia
appeared to be an important
transformation process. Labora-
tory rates of nitrification varied
from 0.02-0.35/day.
-------
TABLE 3-21. (continued)
Reference
Study Area
Purpose of Study
Reported
Nitrification Rates
Methods of Determining
Nitrification Rates
Comments
Kreutzberger and
Francisco (1977)
Morgan Creek,
Ruin Creek, and
Little L1ck
Creek; three
shallow streams
In North Carolina
To determine the
distributions of
nitrifying organisms,
and to examine the
nitrogen transformation
occurring in the streams
Counts of nitrifying organisms
were enumerated in the water
column and in the top 1 cm of
sediments. The populations were
much larger in the sediments,
which indicated that nitrifica-
was occurring predominantly in
the sediments and not In water
column.
Cirello et aK (1979)
o
Finstein and
Matulewich (1974)
Passaic River,
New Jersey
Passaic River,
New Jersey
To determine whether
nitrification was a
significant process in
the Passaic River
To determine the
distribution of
nitrifying bacteria
In the river
There were high ammonia nitrogen
concentrations in the river with
relatively little nitrification
occurring. The potential for
nitrification appeared high, and
was expected to be exerted if
water quality within the river
improved.
Nitrifying bacteria were found to
be from 21 to 140,000 times more
abundant voluraetrically in sedi-
ments than in the water column.
SI ayton and Trovato
(1978. 1979)
Potomac Estuary
To determine factors
important in the
oxygen balance within
the estuary
0.10-0.14/day
Thomas Graphical
Method
-------
O)
CD
4.0
3.0-
2.0-
1.0
0.9-
0.8
0.5
Kn"1.25/day
Kn=0.2/day
0 0.5
Figure 3-16.
1.0 1.5
TIME OF TRAVEL,DAYS
2.0
2.5
Nitrogenous biochemical oxygen demand
versus travel time in Shenandoah River
(Deb and Bowers, 1983).
Additional nitrification rates are shown in Table 3-22. Bansal (1976)
has documented nitrification rates in numerous rivers throughout the United
States, and developed a method to predict nitrification rate based on
hydraulic data. His method has been criticized by Gujer (1977) and Brosman
(1977) and is not reported.
Relatively few nitrification rates were found for lakes or estuaries.
The few data in Table 3-22 for lakes and estuaries are generally in the
range O.I/day to 0.5/day.
171
-------
TABLE 3-22. SUMMARY OF NITRIFICATION RATES
River
Grand River,
Michigan
Clinton River,
Michigan
Truckee River,
Nevada
South Chickamaugo Creek,
Tennessee
Oostanaula Creek,
Tennessee
Town Branch,
Alabama
Chattahoochee River,
Georgia
Willamette River,
Oregon
Flint River,
Michigan
Upper Mohawk River,
New York
Lower Mohawk River,
New York
Bange Canal near
Upper Mohawk River,
New York
Ohio River
Big Blue River,
Nebraska
Delaware River
Estuary
Willamette River,
Oregon
Ouachlta River,
Arkansas and Louisiana
Potomac Estuary
Lake Huron and
Saginaw Bay
New York Bight
Maximum
3.9
15.8
4.0
2.4
1.9
0.8
—
--
0.7
2.5
0.3
0.3
0.25
0.25
0.25
0.54
—
~
—
~
~~
Average
2.6
5.7
1.9
1.9
—
—
0.7
0.44
—
1.4
0.25
0.3
0.25
0.25
0.11
0.3
0.75*
1.05**
0.1*
0.5**
0.09-0.13
0.20
0.025
Minimum
1.9
2.2
0.4
—
1.1
0.1
--
—
0.4
0.1
0.25
0.3
0.25
0.25
0.03
0.09
~
—
--
~
•~
Reference
Courchaine (1968)
Wezernak and Gannon (1968)
O'Connell and Thomas (1965)
Tennessee Valley Authority
Ruane and Krenkel (1978)
Tennessee Valley Authority
Ruane and Krenkel (1978)
Tennessee Valley Authority
Ruane and Krenkel (1978)
Stamer et al_. (1979)
Rinellaet al_. (1981)
Bansal (1976)
Bansal (1976)
Bansal (1976)
Bansal (1976)
Bansal (1976)
Bansal (1976)
Bansal (1976)
Alvarez-Montalvo, et aT_.
undated
NCASI (1982c)
Thomann and Fltzpatrick, 1982
D1 Toro and Matystik, 1980
O'Connor et _§],. 1981
Note: Nitrification rates are 1n units of I/day.
* Ammonia Oxidation
** Nitrite Oxidation
172
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3.4.5 Summary
Typically modelers simulate nitrification by first order kinetics,
either the single stage or two stage approach. Most nitrification rate data
have been collected in streams and rivers, where the rates can be quite
variable due to bottom effects. Instream rates can differ significantly
from laboratory or bottle rates. However, for large bodies of water
(typically lakes or estuaries) the relative importance of the bottom is
diminished, and nitrification rates tend to approach bottle rates.
Available data suggest nitrification rates between 0.1 to 0.3/day are often
appropriate for large lakes, large rivers, or estuaries.
In flowing waters, instream nitrification rates are often determined
based on TKN versus travel time. Care should be taken that the assumptions
of the approach are met, and that processes that transform nitrogen other
than nitrification are assessed (i.e., the other components of the nitrogen
cycle).
Because benthic nitrification can be important in small streams, it is
important not to "doubly count" oxygen sinks in modeling applications. A
component of the sediment oxygen demand would include benthic nitrification,
so the two processes need to be accounted for in a mutually exclusive way
for modeling applications.
Very few studies actually try to measure populations of nitrifiers in
natural systems. This, however, is the most conclusive method to confirm
that nitrification is occurring.
3.5 SEDIMENT OXYGEN DEMAND (SOD)
3.5.1 Concept of SOD
Oxygen demand by benthic sediments and organisms can represent a large
fraction of oxygen consumption in surface waters. Benthal deposits at any
given location in an aquatic system are the result of the transportation and
173
-------
deposition of organic material. The material may be from a source outside
the system such as leaf litter or wastewater particulate BOD (allochthonous
material), or it may be generated inside the system as occurs with plant
growth (autochthonous material). In either case, such organic matter can
exert a high oxygen demand under some circumstances. In addition to oxygen
demand caused by decay of organic matter, resident invertebrates can
generate significant oxygen demand through respiration (Walker and
Snodgrass, 1984). The importance of this process to water quality modeling
is reflected in a recent symposium (Hatcher and Hicks, 1984). This same
symposium also reviewed measurement techniques and a concensus favoring j_n
situ measurement was reached.
It is generally agreed (e.g., Martin and Bella, 1971) that the organic
matter 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 are performed jm 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 (e.g., see Di Toro, 1984), then additional modeling
complexity is necessary.
The process is usually referred to as sediment oxygen demand (SOD)
because of the typical mode of measurement: enclosing the sediments in a
chamber and measuring the change in dissolved oxygen concentration at
several time increments. This technique is used in the laboratory or
2
in situ. The oxygen utilized per unit area and time (gO^/m -day) is the
SOD. The technique measures oxygen consumption by all of the processes
enclosed in the chamber: chemical reactions, bacterially mediated redox
reactions, and respiration by higher organisms (e.g., benthic worms,
insects, and molluscs). Background water column respiration is then
subtracted from this rate to compute the component due solely to the
174
-------
sediment interface. SOD is usually assumed to encompass the flux of
dissolved constituents such as DO to sediment and reduced chemicals to the
water column. However, solid particle flux as BOD or sediment entrainment
or settling is modeled separately.
The major factors affecting SOD are: temperature, oxygen concentration
at the sediment water interface (available oxygen), makeup of the biological
community, organic and physical characteristics of the sediment, current
velocity over the sediments, and chemistry of the interstitial water. Each
of these factors is a resultant of other interacting processes occurring
elsewhere in the aquatic system. For example, temperature and available
oxygen can be changed as a result of transport and biochemical processes in
the water column or system boundaries. Temperature and oxygen are usually
modeled explicitly, and can be used as input variables to the SOD process
equations. Another important linkage is that the biological community will
change with the water quality (e.g., oxygen and nutrient concentrations) and
productivity of the system. The organic characteristics will change over
the long term due to settling of organic matter (detritus, fecal matter,
phytoplankton) and its subsequent degradation and/or burial by continued
sedimentation. The biological community and the organic and physical
characteristics of the bottom sediments are usually treated as a composite
characteristic of the particular system. Recently, techniques have been
developed for investigating these factors; however, the usual technique is
to measure the SOD directly rather than the underlying factors that control
the processes of SOD.
At least two major factors affecting SOD are usual ly'neglected in SOD
modeling. Current velocity is often neglected despite the fact that it has
a major effect on the diffusive gradient of oxygen beginning just below the
sediment-water interface. Most measurement techniques provide mixing,by
internal mixing or by recirculating or flow-through systems to minimize the
effect of concentration gradients. However, the velocity of such systems
may be insufficient (WlYlttemore, 1984a) or may be so vigorous as to cause
scour and resuspension. Interstitial water chemistry affects substrates for
biochemical and non-biochemical oxidation-reduction reactions and their
175
-------
reaction rates. This factor is also usually neglected in SOD measurements
and kinetic formulations.
3.5.2 Kinetics
The generalized equation for sediment oxygen demand is:
= f(d1sso1ved oxygen,
at n
temperature, organisms, substrate) (3-51)
where H = water depth, m
2
SOD = sediment oxygen demand (as measured), gO?/m -day
t = time
C = oxygen concentration in the overlying water, mg/1
3.5.2.1 Dissolved Oxygen
The benthic oxygen consumption has been hypothesized to depend on the
dissolved oxygen concentration in the overlying waters (e.g., Edwards and
Rolley, 1965; McDonnell and Hall, 1969):
SOD = a Cb (3-52)
where 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, primarily as a function of the population density of
benthic invertebrates.
Lam j2t jil_. (1984) use a Michael is-Menten relationship to express the
effects of oxygen on SOD:
dc - ks AS c
176
-------
2
where k = rate constant for SOD in Lake Erie, 0.1 g 0-,/m -day
s 2 L
A_ = area of the sediment, m
3
V = volume of water layer, m
Kn = oxygen half saturation constant (1.4 mg/1)
U2
C = oxygen concentration, mg/1
Walker and Snodgrass (1984) divided SOD in Hamilton Bay in Lake Ontario
into two fractions: chemical-microbial (CSOD) and biological (BSOD). The
chemical fraction was defined as a first-order function of oxygen:
CSOD = kt(T) C (3-54)
where k^(T) = temperature-adjusted rate constant for biochemical SOD,
I/day
The biological fraction was estimated to be 20-40 percent due to
macroinvertebrates in Hamilton Bay sediments but still followed a Michael is-
Menten relationship:
BSOD = u(T) ., x r (3-55)
where u(T) = temperature-adjusted rate constant for biological SOD
2
(obtained by measurement: range = 0.58 to 5.52 g Op/m -
day) , I/day
Kn = oxygen half-saturation constant (1.4 mg/1)
It is interesting to note the similarity between the two estimates of Kn
U2
(Lam e^al_. , 1984; Walker and Snodgrass, 1984).
The direct effects of dissolved oxygen on the rate constant are
generally neglected except in a few models. For example, in the HSPF model
(Johansen j?t jil_. , 1981), dissolved oxygen concentration affects the rate of
sediment oxygen utilization exponentially:
177
-------
dC
f - - r <3-56'
2
where kT = the temperature adjusted rate constant, mg/m -day
3.5.2.2 Temperature
Temperature effects on SOD are most commonly modeled using the
van't Hoff form of the Arrhenius relationship:
(3-57)
where kT = the rate at ambient temperature T
k-p = the rate at a reference temperature (usually Tr=20 C)
6 = the temperature coefficient for adjusting the rate
(Table 3-23)
Although this form of the relationship is the most common and gives
equivalent results to the Arrhenius equation, it is not preferred in
standard nomenclature (Grau et_ _al_., 1982).
The exceptions to use of Equation (3-57) are RECEIV-II (Raytheon,
1974), HSPF (Johanson et _al_., 1981), and SSAM-IV (Grenney and Kraszewski,
1981). RECEIV-II apparently does not provide a temperature correction for
the SOD rate coefficient although other rate coefficients in the model are
adjusted according to Equation (3-57) with 6= 1.047 for CBOD. HSPF uses a
linear function for adjusting the SOD for temperature:
kT = 0.05 Twk2Q (3-58)
where kj = the temperature adjusted coefficient
k2g = the rate constant at 20°C
TW = water temperature, °C
178
-------
TABLE 3-23. SOME TYPICAL VALUES OF THE TEMPERATURE COEFFICIENT
FOR SOD RATE COEFFICIENTS USED IN WATER QUALITY MODELS
Model
DOSAG-3
QUAL-II
Vermont QUALII
Lake Erie Model
WASP
WASP
LAKECO
WQRRS
ESTECO
DEM
EAM
EAM
USGS-Steady
AQUA- IV
EXPLORE- I
Laboratory/Field Studies
e
1.047
1.047
1.047
1.08
1.08
1.1
1.02
1.02-1.04
1.02-1.04
1.04
1.02
1.047
1.065
1.02-1.09
1.05
1.040-1.130
Q10(20°C)*
1.58
1.58
1.58
2.16
2.16
2.59
1.22
1.22-1.48
1.22-1.48
1.48
1.22
1.58
1.88
1.22
1.63
1.5-3.4
Reference
Duke & Masch (1973)
Roesner e_t aK (1977)
JRB (1983)
Di Toro & Connolly (1980)
Thomann & Fitzpatrick (1982)
O'Connor e_t a_K (1981)
Chen & Orlob (1972, 1975)
Smith (1978)
Brandes (1976)
Genet e_t ^L (1974)
Bowie ejt al_. (1980)
Tetra Tech (1980), Porcella ejt al.. (1983)
Bauer et ai- (1979)
Baca & Arnett (1976)
Baca et ai. (1973)
Zisone_t al. (1978); Whittemore (1984b)
* Q10(20°C) = ratio of
at
=6
10
-------
Grenney and Kraszewski (1981) used a modification of the Thornton and
Lessem (1978) equation for SSAM-IV to provide, essentially, a continuously
variable adjustment coefficient (0) for the rate constants in biological
processes. The equation adjusts over a temperature range of 5 to 30°C which
is similar to using Equation (3-51) with a variable 6 coefficient:
K, ey
-------
Additional field experience and the use of divers to place the respirometers
should measurably improve these results.
3.5.2.3 Biological Effects on SOD
The biological component is usually neglected when modeling SOD,
because of the complexity of modeling benthic microorganisms and
macroinvertebrates. The spatial and seasonal variability in SOD caused by
sediment biological processes and communities results in variation in SOD
that modelers appear to account for by varying the temperature coefficient.
Some investigators have attempted to incorporate this variation directly in
the model (Grenney and Kraszewski, 1981), or have suggested that the value
of the temperature coefficient changes with season (e.g., Bradshaw et al.,
1984) or with location downstream (e.g., Mancini ^t ail_. ,1984) . Other models
(LAKECO, ESTECO, WQRRS, EAM) incorporate a benthic organisms compartment and
may be able to evaluate the effects of benthos on SOD directly. However, no
verification studies have been discovered that demonstrate this to be a
useful technique.
3.5.2.4 Substrate Variability
The process describing the substrate utilized is where most models
differ (Table 3-24). In the first water quality models that were widely
used (DOSAG-3, QUAL-II), the decay of substrate is assumed to balance
continued settling resulting in a steady-state sediment concentration of
oxygen-demanding substrate. The resulting equation is:
(3'60)
p
where k-p = temperature adjusted rate constant SOD, gO^/m -day
H = mean water depth, m
As shown in Table 3-24, most models have followed this approach.
181
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TABLE 3-24. MODEL FORMULATIONS COMMONLY USED IN SOD COMPUTATIONS
Formulation
k/A
k/H
Units
k,mg00/m day
2
A.ni
2
k .mgO-Xm day
Description
SOD rate normalized
by bottom area
SOD rate normalized
Model (Reference)
DOSAG-3 (Duke & Masch, 1973)
QUAL-II (Roesner e_t aU (1977)
Vermont QUAL-II (JRB, 1983)
IICCC Ctn=.^w ID*, .or- 0+ 5,1 1Q7QA
H>m AQUA-IV (Baca & Arnett, 1976)
WASP (O'Connor et aK 1981)
RECEIV-II (RaytTieon, 1974)
OEM (Genet e_t a_K 1974)
HSPF (Johanson et aK 1981)
r§ a k SED a,mgO,/mg Sed Conversion factor LAKECO* (Chen & Orlob, 1972, 1975)
kj/day Decay rate W™*
-------
Substrate has been incorporated directly into ESTECO, LAKECO, WQRRS,
EAM, and EXPLORE-I. Different settling rates of oxygen-demanding organic
materials can lead to different amounts of sediment materials, and
consequently different SOD rates calculated according to:
^ = - a k SED (3-61)
where a = stoichiometric conversion factor relating oxygen to organic
sediment, mg CL/mg sediment
k = sediment decay rate constant, I/day
SED = sediment substrate that is subject to decay
In .EXPLORE-I, only carbonaceous BOD is simulated as the substrate (SED),
which in turn is affected by scour or settling from the water column. In
the other models, all of the nutrient elements (C, N, P) are transformed
according to a first-order reaction (k SED) but sediment oxygen demand is
exerted only by carbon. Values of the conversion factor for sedimented
organic carbon to 0? lie in the range of 1.2 to 2.0 mgO^/mg sediment.
Nitrogen decays to ammonium and is released to the overlying waters where
nitrification can take place (see Section 3.4). Other nutrients also enter
the overlying waters as a result of similar transformations.
In some versions of the WASP model (Di Toro and Connolly, 1980; Thomann
and Fitzpatrick, 1982), the oxygen-demanding materials in the sediment are
divided into multiple compartments. First, the decay processes of sediment
organic matter generate concentrations of CBOD and NBOD constituents in
interstitial waters. Then both CBOD and NBOD are released to the water
column where they subsequently decay in the appropriate compartments. In
addition to CBOD release, oxygen utilization in the interstitial water is
computed as oxygen equivalents, and diffusion into the interstitial water
compartment is determined. If oxidation in excess of the amount available
from diffusion occurs, these excess "oxygen equivalents" continue to
represent a potential demand on the dissolved oxygen system. Finally, a
deep oxygen demand has been hypothesized in an attempt to account for the
measured oxygen demand. These concepts are described Di Toro and Connolly,
183
-------
1980. More recently, Di Toro (1984) has provided an additional correction
to SOD from denitrification of nitrate, although he suggests that this
correction is usually negligible.
3.5.3 Measurement Techniques
Essentially three types of measurement techniques have been used to
estimate SOD rates: model calibration to estimate SOD, in situ measurements
using respiration chambers, and laboratory respiration chamber measurement
using cores or dredged samples. However, all three methods have severe
disadvantages and the uncertainty of calculating SOD rates is so great that
the simple formulations in the model equations (Table 3-24) are very
appealing to model users. Unfortunately, these simple formulations will not
result in credible models with good predictive capability when single values
are used for rates and coefficients.
It would be expected that considerable spatial and temporal variation
would occur in SOD. Spatially, the bed sediments of streams, lakes , and
estuaries vary in their physical and chemical characteristics, rates of
deposition, and other factors. For example, a stream may have .fine
sediments in low velocity areas and coarse cobble or boulders in steep
gradient-high velocity reaches. Depth and velocity can vary significantly
in any one cross-section. Reservoirs have deposition zones near inlets and
at dam structures. Estuaries like streams and lakes vary considerably in
substrate type and water velocity but are influenced by the salinity
gradient and an added factor of coagulation and rapid settling in zones
where fresh and saline waters mix.
Another source of variation is temperature. Temperature varies
seasonally but that is accounted for in use of the van't Hoff or similar
relationships. However, temperature and season both cause a shift in
benthic community composition. Macroinvertebrate populations, especially
emergent insects, change dramatically with life stage. Also, it would be
expected that considerable variation in microbial community characteristics
would occur in response to temperature changes.
184
-------
These spatial and seasonal characteristics suggest that a large number
of SOD measurements would be required to estimate and obtain sufficient
variation in rate coefficients. This has led to the development of in situ
•> •-- •———
and laboratory methods for measuring SOD that will be site-specific and
seasonal for SOD. SOD mapping strategies may be necessary. Ideally,
in situ methods would provide the best approach, but considerable variation
in results occurs because of problems associated with field sampling:
t Horizontal and longitudinal non-homogeneity of stream bottom
materials. Areas of cobble, soft sediments, logs, and
bedrock, increase the cost of measurement because more
samples are needed. Soft, flocculent sediments are very
difficult to evaluate with in situ methods. In some streams,
an inaccurate characterization of reach-averaged SOD will be
obtained.
• Difficulties in placement of respiration chamber. For
example, obtaining a complete seal in cobbled and bouldered
areas or where significant interaction with the ground water
system occurs is essentially impossible.
• Mixing in the respiration chamber may not be modeled
correctly nor simulate natural conditions and this is
reflected in the wide variance in results from measurements.
For example, the Institute of Paper Chemistry reported on a
comparison of 5 j_n s i tu samplers of two basic types
(recirculating and internally mixed) and found the results to
be markedly different (Parker, 1977).
Laboratory measurements suffer from similar problems. They would
appear to work reasonably well for aquatic systems of relatively uniform
sediment characteristics, but heterogeneous sediments often lead to
measurement variability.
135
-------
Some practices improve laboratory measurement: correcting of results
for varying sediment depth is usually unnecessary when depths exceed
5-10 cm; undisturbed core samples are preferred over dredge samples even
though they are more costly to collect; storage of samples and acclimation
of samples to laboratory temperatures is discouraged because of potential
changes in benthos or substrate; divers may help to improve precision.
In regard to the effect of variability in oxygen-demanding materials,
there appears to be no strong relationship between SOD and various measures
of organic matter (NCASI, 1978), but this may have been due to inaccurate
measurement techniques. Improper mixing (i.e, velocity too high or too
low), inadequate oxygen supp-ly, storage or improper pretreatment of samples
in the laboratory, and inappropriate laboratory temperatures may lead to
errors that prevent the derivation of SOD/substrate relationships. However,
Gardiner _et al. (1984), using a laboratory chamber, showed that SOD was
related to chemical oxygen demand (COD) of the sediments in Green Bay, a
large gulf in the northwest corner of Lake Michigan, according to the
following equation:
SOD = 7.66 COD/(156.5 + COD) (3-62)
As further evidence, the higher SOD values coincided with areas of summer
dissolved oxygen depletion in Green Bay.
Given the many sources of measurement error, it is not surprising that
Whittemore (1984b) was unable to correlate literature SOD values obtained in
o
simultaneous field and laboratory measurements. He obtained a low r value
of 0.58. But even more significant, the j_n situ SOD values were
consistently higher than laboratory derived values at low SOD concentrations
and the reverse observed at high SOD concentrations. This systematic error
indicates the need for better methods of estimating SOD as well as
developing a better understanding of the component SOD mechanisms.
The model calibration approach to estimating SOD is essentially a
determination of the SOD rate by calibration subject to the constraint of a
186
-------
reasonable range of SOD values. Thomann (1972) used literature SOD rates
and modeling experience to suggest SOD ranges for certain environments
(Table 3-25). The model approach (e.g., Terry and Morris, 1984; Draper
^t jj_., 1984), by itself, contains considerable variance because there are
uncertainties in the other processes (reaeration, nitrification,
respiration, photosynthesis, flow) as well as the considerable spatial and
temporal variation expected in most aquatic environments. Lam ert _al_. (1984)
suggest that variation in dissolved oxygen load to Lake Erie owing to
TABLE 3-25. AVERAGE VALUES OF OXYGEN UPTAKE RATES OF
RIVER BOTTOMS (AFTER THOMANN, 1972)
o
Uptake (g 02/m -day)
Bottom Type and Location
Sphaerotilus - (10 gm dry wt/m)
Municipal Sewage Sludge-
Outfall Vicinity
Municipal Sewage Sludge-
"Aged" Downstream of Outfall
Estuarine mud
Sandy bottom
Mineral soils
9
Range
-
2-10.0
1-2
1-2
0.2-1.0
0.05-0.1
20°C
Average
7
4
1.5
1.5
0.5
0.07
hydrologic fluctuations could easily mask the effects of SOD on water column
oxygen.
3.5.4 Summary
There is a diversity of modeling and measurement techniques used for
predicting oxygen consumption by sediments. This diversity reflects the
need for better process descriptions and measurement techniques. Simple
zero-order model formulations have been used, but first-order multi-
component reactions with a separate benthic organism component may be needed
to accurately model sediment oxygen demand (SOD).
187
-------
Consequently, it is suggested that modelers use site-specific SOD
rates. In situ methods such as described in Whittemore (1984a) and Markert
_et al . (1983) are more useful and credible than laboratory methods at this
time.
As an aid to estimating SOD rates and establishing reasonable ranges
for calibration, the SOD literature values in Tables 3-26, 3-27, and 3-28
are presented for rivers and streams, lakes and reservoirs, and estuaries
and marine environments, respectively. These should be considered only as
order of magnitude estimates.
3.6 PHOTOSYNTHESIS AND RESPIRATION
3.6.1 Introduction
Photosynthetic oxygen production (P) and respiration (R) can be
important sources and sinks of dissolved oxygen in natural waters. Many
models simulate these processes directly in terms of algal growth and
respiration. For example, net algal growth is simulated with the QUAL-II
model (Roesner j5t jj_. , 1981) using:
)A (3-63)
where A = algal concentration, mass/volume
M = specific growth rate of algae, I/time
P = algal respiration rate, I/time
a = algal settling rate, I/time
The net algal oxygen production minus consumption is simulated by
QUAL-II as:
(3-64)
where C = dissolved oxygen concentration, mass/volume
188
-------
al ~ oxygen production per unit of algal mass, mass oxygen/mass
algae
a,, = oxygen uptake per unit of algal mass, mass oxygen/mass algae
The stoichiometric coefficients a., and a. relate algal growth and death to
oxygen production and consumption. Tables 3-29 and 3-30 summarize values of
these coefficients used in different models.
TABLE 3-26. MEASURED VALUES OF SEDIMENT OXYGEN DEMAND
IN RIVERS AND STREAMS
SOD, g02/m day
Environment
Experimental Conditions
References
0.022-0.92
0.09±0.02 (312"C
0.15±0.04 (6>20°C
0.20+0.03 (P28°C
0.29±0.07 (?36°C
0.18+0.05 (912"C
0.55±0.22 (@20"C
0.60±0.28 (I?28"C)
0.87+0.23 ((?36°C)
3.2-5.7
0.52-3.6
2-33
0.9-14.1
<0.1-1.4(@20°C)
0.27-9.8
0.10-5.30
(920°C)
1.1-12.8
0.3-1.4
0.20-1.2
1.7-6.0
1.5-9.8
4.6-44.
Upper Wisconsin River
Eastern U.S. River
Southeastern U.S. River
Fresh shredded tree bark
Aged shredded tree bark
Four eastern U.S. rivers
downstream of paper mill
discharges
Eastern U.S. river
downstream of paper
mill discharge
Northern Illinois rivers
(N = 89 stations)
Six stations in
eastern Michigan rivers
New Jersey rivers
(10 stations)
Swedish rivers
Swedish rivers
Spring Creek, PA
74 samples from
from 21 English rivers
Streams
60-hour laboratory core incubation,
periodic mixing, 4°C, dark
45 day incubation of
0.6 liters sediment in
3.85 liters BOD
dilution water, light
10-liter incubations in
aged tap water, room
temperature, light
In-situ chamber
respTrometers, 22-27°C, light,
stirred at varying rates;
open-ended tunnel respirometer,
in-situ. 22-27°C, dark
In-situ respirometer
stirred at various rates
9-16 C, dark, 6 = 1.08
In-situ respirometry, dark,
T"=~5°~- 31°C
time l>s-3 hours
respirometry 1n
stirred chambers, 15-27 hours
dark, 19-25°C, 6 = 1.08
In-situ respirometer, dark,
30 minutes-8 hours, stirred.
Temperature unknown
In-situ respirometer, light,
stirred, 0-10°
Laboratory Incubations,
stirred, dark, 5-10°C
Laboratory incubators 1n
dark, stirred, 20 C
Laboratory incubation of
cores; 15DC
Oxygen mass balance
Sullivan et^ a\_. (1978)
NCASI (1981)
NCASI (1971)
NCASI (1978)
NCASI (1979)
Butts & Evans (1978)
Chiaro & Burke (1980)
Hunter e_t a]_. (1973)
Edberg & Hofsten (1973)
Edberg & Hofsten (1973)
McDonnell & Hall (1969)
Rolley & Owens (1967)
James (1974)
189
-------
TABLE 3-27. MEASURED VALUES OF SEDIMENT OXYGEN DEMAND
IN LAKES AND RESERVOIRS
SOD, g02/m day
Environment
Experimental Conditions
References
1-7
0-2.2
0.4-2.6
0.21-1.5
5.5 (31-32.5°C)
5.1 (22.5-25.6°C
2.1 (13.2-16.1°C
0.84-3.3
0.4-3.6
0.40-0.45
0.27
0.12-0.22
0.47-0.92
0.72-8.40
0.6-3.6
1.7-8.9
0.17-0.5
0.54-0.71
0.3-1.0
0.076-0.48
0.004-0.012
Green Bay, Lake Michigan
Fish culture ponds
Swedish lakes
Swedish lakes
Horseshoe Lake, IL
Lake Apopka, FL
Lake Apopka, FL
Hyrum Reservoir, UT
Lake Powell
Shagawa Lake
Swedish Lakes
Lakes
Hamilton Harbor,
Lake Ontario
Lake Mohegan, NY
Swedish lakes
Swedish lakes
Lake Hartwell, SC
Marion Lake, BC
Lake Superior
Lab incubation in darkness, 20 C
In situ respirometry with 100-cm
long plexiglass columns (dark pvc),
over 47 days. Temperature unknown
In situ resplrometer. light
stirred", 5-18
Laboratory incubations,
stirred, dark, 10-13°C
In situ respirometry, dark,.
stirred about 1 hour
Laboratory incubation of
cores at room temperature,
2-3 hours, light. No stirring.
Laboratory flow-through system
(closed, 100 1 volume)
3-phase microcosms,
25DC, dark
In-situ chambers (1m ), at
7-inrTdepths; 12-14 C (est.)
Laboratory measurement with
undisturbed cores; used in situ
temperatures
Oxygen mass balance
In situ chambers, 11-16°C
Measurement based on mass
balance, continuous flow
lab chamber, 22-32 C
In situ & laboratory measurements,
winter temperatures
Laboratory Incubation of
undisturbed cores, 8 C
Laboratory chambers, 18°C
Laboratory incubation of
undisturbed cores, no mixing, 15°C
Laboratory incubation of
undisturbed cores, 4UC
Gardiner et al_. (1984)
Shapiro S Zur (1981)
Edberg & Hofsten (1973)
Edberg 4 Hofsten (1973)
Butts & Evans (1979)
Bel anger (1981)
Medine et aK (1980)
Sonzogni et al_. (1977)
Graneli (1977)
James (1974)
Polak & Haffner (1978)
Fillos (1977)
Edberg (1977)
Andersen (1977)
Brewer e_t al_. (1977)
Hargrave (1969)
Glass & Podolski (1975)
In addition to algal respiration, respiration from zooplankton and
nekton can contribute to oxygen depletion, and would be included in Equation
(3-64), along with additional equations to describe their growth and death.
Models that simulate algae and zooplankton (such as those in Tables 3-29 and
3-30) are discussed in detail in Chapters 6 and 7 of this report. This
section describes methods to predict P-R without simulating algal growth or
respiration. The methods pertain largely to streams and rivers, and are
useful in that they simplify the modeling approach.
190
-------
It should be mentioned that some water quality models do not simulate
photosynthesis and algal respiration. This approach is valid where P=0 and
R=0. Other models simulate only daily average photosynthetic oxygen
production ("P) and daily average respiration (R). If, on a daily average
basis, P-R — 0, these models would predict little effect of algal activity on
dissolved oxygen. However, if P and R are both large numbers, then actual
dissolved oxygen levels will be higher during the day and lower at night
than predicted by the models.
3.6.2 Methods
Table 3-31 summarizes the methods reviewed to predict photosynthetic
oxygen production and respiration without simulating algal growth. The
methods consist of either single station methods or two-station methods.
Odum (1956) appears to be one of the first researchers to use this approach.
TABLE 3-28. MEASURED VALUES OF SEDIMENT OXYGEN DEMAND
IN ESTUARIES AND MARINE SYSTEMS
SOD, g02/m day Environment
0.10±0.03 (@12°C) A North Carolinian estuary
0.20±0.05
0.22±0.09
0.37±0.15
2.32±0.16
1.88±0.018
0.14-0.68
(320 C)
028°C)
@36°C)
Buzzards Bay near raw
sevage outfall
Buzzards Bay control
5°C) Puget Sound
Experimental Conditions
45 day incubation of 0.6 liters
sediment in 3.85 liters BOD dilution
water, light
In-situ dark respirometers, stirred,
1^3 days. Temperature unknown
Laboratory incubations
References
NCASI (1981)
Smith e^aK (1973)
Pamatmat et al. (1973)
0.20-0.76 (10"C) sediment cores
0.30-1.52 (15°C)
0.05-0.10
1.25-3.9
0.02-0.49
0.9-3.0
0.4-0.71
0-10.7
0.3-3.0
San Diego Trough
(deep marine sediments)
Yaquina River estuary,
Oregon
Eastern tropical Pacific
The Baltic Sea
The Baltic Sea
Delaware Estuary
(22 stations)
Fresh & brackish waters,
In-situ respirometry for 5-13
hours, 4°C, light
Dark laboratory incubators,
stirred, 20°C
Shipboard incubations, 15°C
stirred, dark
In-situ light respirometer,
stirred, 10°C
Laboratory incubations, stirred,
dark, 10°C
In-situ dark respirometry,
13-14°C
In-situ respirometry, 0-18°C
Smith (1974)
Martin & Bella (1971)
Pamatmat (1971)
Edberg & Hofsten (1973)
Edberg & Hofsten (1973)
Albert (1983)
Edberg & Hofsten (1973)
Sweden
Laboratory cores, 5-13 C
191
-------
TABLE 3-29. OXYGEN PRODUCED PER MASS OF ALGAE
Model
Value
Reference
DOSAG3
QUAL-II
WASP
WASP
WASP
LAKE ECO
WQRRS
AQUA-IV
ESTECO
EAM
EAM
EAM
DEM
Vermont-
QUAL-II
1.4 - 1.
1.4 1.
°
2
mg algae (D.W. )
mg Op
mg algae (D.W.)
2.67 mg 02/mg C
2.66 mg 02/mg C
.133 mg 02/mg Chi-a
2.67 mg 02/mg C
1.6 mg 02/mg algae (D.W.)
1.6 mg 02/mg algae (D.W.)
1.6 - 2.66 mg 02/mg C
1.6 - 1.8 mg 02/mg algae (D.W.)
1.24 mg 02/mg algae (D.W.)
1.6 mg 02/mg algae (D.W.)
1.24mg • 02/mg algae (D.W.)
1.6 mg 02/mg algae (D.W.)
Duke & Masch (1973)
Roesner et al_. (1977)
Di Toro & Connolly (1980)
O'Connor et al_. (1981)
O'Connor et al_. (1981)
Thomann & Fitzpatrick (1982)
Chen & Orlob (1975)
Smith (1978)
Baca & Arnett (1976)
Brandes (1976)
Porcella et. a].. (1983)
Bowie et a]_. (1980)
Tetra Tech (1980)
Feigner & Harris (1970)
1.4 1.8 mg 02/mg algae (D.W.) JRB (1983)
Note:
D.W. dry weight
Both numerical and analytical techniques have since been developed. The
light-dark bottle technique and benthic chamber method are also included in
the table.
As shown in Table 3-31, O'Connell and Thomas (1965) applied a total
derivative approach for P-R calculation, and compared the results against a
192
-------
second procedure using a submerged algal chamber. Respiration was corrected
for oxygen consumption by bacterial oxidation. Figure 3-17 compares the two
methods for a station on the Truckee River, and shows good agreement.
O'Connor and Di Toro (1970) use a half cycle sine wave or a Fourier
series to find the time varying photosynthetic oxygen production rate. In
TABLE 3-30. OXYGEN CONSUMED PER MASS OF ALGAE
Model
DOSAG 3
QUAL-II
WASP
WASP
WASP
LAKE ECO
WQRRS
AQUA-IV
ESTECO
EAM
EAM
EAM
DEM
Vermont
QUAL-II
Value
Reference
1.6 - 2.3 mg 02/mg algae (D.W.)
1.6 - 2.3 mg 02/mg algae (D.W.)
1.87 mg 02/mg C
2.0 mg 02/mg C
2.0 mg 02/mg C
.10 mg 02/mg Chl-a^
1.6 mg 02/mg algae (D.W.)
1.6 2.0 mg 02/mg algae (D.W.)
1.6 2.66 mg 02/mg C
1.6 - 1.8 mg 02/mg algae (D.W.)
.95 mg 02/mg algae (D.W.)
1.6 mg 02/mg algae (D.W.)
.95 mg 02/mg algae (D.W.)
1.6 mg 02/mg algae (D.W.)
1.6 - 2.3 mg 02/mg algae (D.W.)
Duke & Masch (1973)
Roesner et al_. (1977)
Di Toro & Connolly (1980)
Thomann & Fitzpatrick (1982)
O'Connor e_t aj_. (1981)
O'Connor et a1_. (1981)
Chen & Orlob (1975)
Smith (1978)
Baca & Arnett (1976)
Brandes (1976)
Porcella et £]_. (1983)
Bowie et_ a]_. (1980)
Tetra Tech (1980)
Feigner & Harris (1970)
JRB (1983)
This is multiplied by an oxygen limitation factor,
saturation constant equal to 0.1 mg/1.
0,
2 , where K is a half-
Note:
D.W. = dry weight
193
-------
TABLE 3-31. SUMMARY OF METHODS TO PREDICT PHOTOSYNTHETIC OXYGEN PRODUCTION
AND RESPIRATION WITHOUT SIMULATING ALGAL GROWTH AND DEATH
Source
Equations
Symbols
Comments
Odum (1956)
see comments
D'Connell and Thomas (1965) P - R = Q + U |£ - k,(C -C) +
9\. Of, C 5
knN
UD
see comments
U = stream velocity
k- = reaeratlon rate
C = dissolved oxygen
Cs = dissolved oxygen saturation
k1 = CBOD decay rate
L = CBOD
k =• nitrification rate
n
N = NBOD
1. Photosynthetic oxygen production was based
on a graphical procedure. Either two
stations or single station approaches
could be used. A method was also
presented to find the reaeratlon
coefficient.
1. P-R was found in two Independent ways. In
the first, all terms in the dissolved
oxygen mass-balance were found
independently and then P-R was found as
the only remaining term in the oxygen
balance. In the second method, an algal
chamber was placed on the river bed.
2. The two methods gave comparable results.
3. The approach was used on the Truckee
River, where attached algae were abundant.
O'Connor and Oi Toro (1970) Half cycle sine wave:
{-
\p
p-/ni (p s/s s
I 0 when t$ + p < t < t + 1
Fourier series extension:
P =
where
bn = cos (ni7p)
= rate of photosynthetic
oxygen production,
mg/(l-day)
n(x)=max1mum rate of
photosynthetic oxygen
production, mg/(l-day)
» time of day when source
begins
» fraction of day when
source is active
(continued)
1. This approach is found In DIURNAL, a
stream model developed by O'Connor and
D1 Toro.
2. The approach is potentially applicable to
any vertically mixed water body.
3. The method of Erdmann (1979a) was used to
evaluate P and R for a wasteload
allocation application on the Shenondoah
River (Deb and Bowers, 1983) and on
Leatherwood Creek, Arkansas (Deb et al..
1983).
4. O'Connor and Di Toro (1970) applied the
method to the Grand, Clinton, and Flint
rivers in Michigan, the Truckee River in
Nevada, and the Ivel River in Great
Britain. They used a trial and error
procedure to determine Pm, ts, P and R to
best fit observed diurnally varying
dissolved oxygen data.
-------
TABLE 3-31. (continued)
Source
Equations
Symbols
Comments
Kelly, Hornberger, Cosby
(1975)
P - R
Hornberger and Kelly (1972)
tn
A = unknown coefficients 1.
w « 2 77/48
The An are determined based on 2.
measurements of dissolved oxygen
at either one of two locations
1n a stream. They are chosen to 3.
give a "best fit" between
predicted and observed dissolved
oxygen values.
dissolved
concentration
oxygen
U - stream velocity
k, = reaeratlon rate
G =• dissolved oxygen saturation
A 48-hour cycle was used so that values at
the beginning and end of a day are not
constrained to be Identical.
R 1s total respiration, Including both
algal respiration and bacterial decay.
The single station analysis can be used
when the dissolved oxygen concentrations
at the upstream and downstream stations
are approximately the same.
Three methods were examined to predict
P-R: a finite difference method, an
analytical solution assuming P-R remains
constant over the time Interval, and a
second analytical method assuming P-R
varies linearly over a time step.
The analytical methods were preferred over
the numerical approach from a conceptual
point of view, and because time steps
smaller than the residence time through
the stream reach could be used.
Erdmann (1979a)
P - R - k2(cs-c) -
where:
C =• concentration of dissolved
nm oxygen at station m and
time n
Dt
C2rCll+C22~C12 C12~C11+C22 C21)U = t 1 m e of sample at
downstream station
:, = time of sample at upstream
1 station
t * travel time between two
r stations
k- = reaeratlon rate
C ^dissolved oxygen
s saturation
C = dissolved oxygen
(continued)
Respiration 1s first computed at night
when P = 0. Then P 1s computed during the
day using known R.
The method was applied to Charles River,
Massachusetts.
R 1s total oxygen consumption rate by both
algae and bacteria.
-------
TABLE 3-31. (continued)
Source
Erdmann (1979b)
Equations
F- (ACu+ACd)
D + D .
R - k, ( „ ) + (AC +
Symbols
P = d a i 1 y average
photosynthesis
Comments
1. The method is a simplification of Erdmann
(1979a) and is used to predict dally
average values of P-R from data at two
stations .
10
cr>
4Cu.4Cd
respiration
reaeration rate
travel time between
two stations
diurnal range of
dissolved oxygen at
upstream stations
dally average
dissolved oxygen
deficit at upstream
and downstream
stations
•dally average
d1s s o 1 ved oxygen
concentratlon at
upstream and
downstream stations
2. The method was applied to the Charles
River, Massachusetts.
3. Some Important assumptions Include
constant temperature and symmetrical
diurnal curves.
Gulliver, Mattke, Stefan
(1982)
kzlcs-c)
U = stream velocity 1.
k, = reaeration rate
C = dissolved oxygen
C « dissolved oxygen saturation 2.
0, = longitudinal disperson
coefficient
3.
(continued)
A finite difference computer model DORM
was used to route dissolved oxygen changes
between two stations and includes the
effects of temperature variations and
dissolved oxygen levels on respiration.
The model was applied to experimental
stream reaches in the U.S. EPA's
Monticello Ecological Research Station,
Minnesota.
For the channels analyzed, 1t was found
that affects of longitudinal dispersion
were negligible. However the results were
sensitive to reaeration, residence time
between the two stations, and temperature
dependent processes (saturation and
respiration rates).
-------
TABLE 3-31. (continued)
Source
Equations
Symbols
Comments
U.S. EPA (1983)
light and dark bottle technique
U.S. EPA (1983)
benthic chamber
1. Light and dark bottles are suspended at
various depths in water and dissolved
oxygen measurements are made at regular
Intervals to determine P-R.
2. This method suffers from numerous
limitations which include:
• only photosynthetic activity of algae
1n water column is measured
* the estimate of R includes algal and
bacterial respiration
• the P-R is a point estimate, rather
than representative of a reach.
1. P-R of attached algae 1s measured using a
clear benthic chamber and a covered (dark)
chamber.
-------
their applications, they used a trial and error procedure to determine P-R
that best fit diurnally varying dissolved oxygen data. In the Deb and
Bowers (1983) application of the same method, Erdmann's approach (1979a) was
used to evaluate P-R. The method of Erdmann combines all terms which
contribute to deoxygenation (algal respiration, CBOD decay and NBOD decay)
into a single respiration term. To find algal respiration, CBOD and NBOD
are subtracted from total community respiration.
Kelly et jfL, (1975), also shown in Table 3-31, use a Fourier series,
but with a 48 hour period. The coefficients A are not true Fourier
coefficients but are based on a best fit between predicted and observed
dissolved oxygen values. Cohen and Church (1981) have more recently applied
these methods to measure productivity of algae in cultures open to the
atmosphere.
+1.5
+1.0
+0.5
£ 0
o>
E
cr
-0.5
-1.0
-1.5
0000
i i i i i i i i i i i i i i i i i i i i i i i
0400 0800 1200 1600
TIME OF DAY
2000
2400
FINITE DIFFERENCE DATA
ALGAE CHAMBER DATA
Figure 3-17.
Diurnal variation of (P-R) in Truckee River near
Station 2B (O'Connell and Thomas, 1965).
198
-------
Erdmann (1979a, 19795) has developed methods to predict time-varying
P-R values and daily average values. In the time varying case the concept
of the Stokes total time derivative is used (see Figure 3-18). The total
derivative is the sum of the time derivative (dC/dt) and the advective
derivative (U<9C/<9x). The time derivative is evaluated as the average of two
times, and the advective derivative is the average between two stations.
Figure 3-18. Concept of Stokes total time derivative. Here
DC/Dt = 0.43 mg Ci'h (from Erdmann, 1979a).
Gulliver _et _§]_., (1982) provide a literature review of the various
methods used to predict P-R in streams. They also developed a computerized
model to determine P-R that includes dispersion. However, they found that
effects of dispersion were negligible for their applications. Several
applications of diurnal curve analyses not reported in Table 3-31 include
the work of Schurr and Ruchts (1977) who used a single station method to
predict monthly average P-R values, and the work of Simonsen and Harremoes
(1978) who used a two station approach to predict P-R on a river in Denmark.
199
-------
The final two methods shown in Table 3-31 are the light-dark bottle
method and the benthic chamber method. These methods measure P-R of algae
in the water column (light-dark bottles) and of attached algae (benthic
chamber). The methods provide single point estimates that may not be
representative of the water body as a whole.
Some models simulate daily average photosynthetic oxygen production
rather than time-varing production. Erdmann (1979b) shows that, the daily
average photosynthesis oxygen products rates, P, can be approximated by:
P" = 2ADO (mg/i/nr) (3_65)
24
where ADO = daily maximum dissolved oxygen concentration minus daily
minimum dissolved oxygen concentration, mg/1
This approximation appears to be valid only for reaeration rates less than
0.2/day (Manhattan College, 1983).
A second method of estimating P is to integrate a sinusoidal curve that
represents the instantaneous photosynthetic oxygen production rate. The
result is:
- 2 Pmp
P = -^r- (3-66)
where Pm = maximum daily photosynthetic oxygen production rate,
mg/l/day
p = fraction of day when algae are producing oxygen, decimal
fraction
The U.S. EPA (1983) describes a third method to estimate daily average
production based on light-dark bottle measurements:
(3-67)
COS (rrtj/f) - COS(n-t2f)
200
-------
where P = observed average production rate between times t~ and t,
AT = (t2 - t^/24
f = number of hours in day when oxygen is being produced
Relationships between photosynthet i c oxygen production and
chlorophyll-a have been developed by a number of researchers. While a
detailed review of these methods is outside of the scope of this section,
several of the more commonly used formulations are summarized here.
Megard jrt al . (1979) developed the following expression for daily average
photosynthetic oxygen production:
(3-68)
where I = light intensity at the water surface
I = light intensity at depth z
C, = chlorophyll-a concentration
a
£c = specific attenuation of light by chlorophyll-a
EW = specific attenuation of light by all causes other than
chlorophyll-a
Pm = maximum daily photosynthetic oxygen production rate,
mg/l/day
Demetracopoulos and Stefan (1983) modified this expression to predict
hourly photosynthetic oxygen production, and used the expression in a model
of the Mississippi River.
In experiments on the Sacramento-San Joaquin Estuary, Bailey (1970)
correlated the daily photosynthetic oxygen production rate to a number of
factors. The resulting expression was:
T0.677
P = 3.16 C ^-i + 0.16T - 0.56H (3-69)
av a k
201
-------
where P
I
k,
T
H
C
av
average dally gross photosynthetic rate, mg/l-day
mean daily solar intensity, cal/sq.cm-day
light extinction coefficient, I/meter
mean temperature, C
mean water depth, m
mean chlorophyll, mg/1
Finally, simple relationships between chlorophyll-a and, Pm have been
proposed (U.S. EPA, 1983). Figure 3-19 shows how Pm/Ca ratios are
a
O
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0-
Probable Range
\ „,--'"
Carbon
Ca
65
50
35
No nutrient limitation
T
T
10
12
14
16 18 20 22 24
WATER TEMPERATURE (°C)
1 1 1 1—
26 28 30 32
Figure 3-19. Algal productivity and chlorophyll relationships
for streams (U.S. EPA, 1983).
influenced by water temperature and algal carbon/Ca ratios. For a typical
water temperature (20°C) and a typical carbon/Ca ratio (50), Pm/Ca = 0.25.
However, this ratio is likely to vary between 0.1 to 0.6 for the range of
conditions present in streams.
202
-------
3.6.3 Data
Table 3-32 summarizes data reviewed on photosynthetic oxygen production
and respiration. Respiration is sometimes reported as total community
respiration and at other times as algal respiration. As shown by the data,
photosynthetic oxygen production can be quite variable, both over distance
and time. In the Havelse River, for example, average photosynthetic oxygen
o
production rates varied from 0.2 to 25.9 g/(m -day). One of the primary
reasons for the variability was because solar radiation intensity changed by
more than an order of magnitude between measurement periods.
3.6.4 Summary
Most water quality models that simulate photosynthetic oxygen
production and algal respiration simulate algal growth and respiration.
Stoichiometric coefficients are used to convert growth and respiration to
oxygen production and consumption. Tables 3-29 and 3-30 summarize these
coefficients.
Some river water quality models use the approach that photosynthetic
oxygen production and respiration can be modeled without the necessity of
simulating algal activity. Rather, some type of curve, such as a sine curve
or more generally a Fourier series, is used instead, where certain
parameters must be delineated to characterize the curve.
Typically instream dissolved oxygen measurements at two stations are
used to generate P-R data. Either finite difference or continuous solutions
to dissolved oxygen mass balance equations are used. Mhile light-dark
bottles or benthic chambers can in principal be used to find the required
information, these approaches are limited in a number of ways. The two
station methods are better in that they provide an integrated estimate of
algal activity.
However, two station methods should also be used cautiously. In a
sense, the methods are curve fitting techniques: they are used to fit a
203
-------
curve based on dissolved oxygen variation between two stations. Typically
other rate constants such as reaeration rates, carbonaceous BOD decay,
nitrogenous BOD decay are needed to fit the curves. Thus errors in these
coefficients are propagated into P-R calculations. Also care should be
taken if results are extrapolated to other situations (e.g., different
temperatures, different solar intensities, and different nutrient loadings).
TABLE 3-32. PHOTOSYNTHETIC OXYGEN PRODUCTION AND RESPIRATION RATES IN RIVERS
Reference
O'Connor and Di Toro (1970)
O'Connor and Di Toro (1970)
O'Connor and Di Toro (1970)
O'Connor and Di Toro (1970)
O'Connor and Di Toro (1970)
Thomas and O'Connell (1966)
Thomas and O'Connell (1977)
T Pm
River °C g/m -day
Grand, Michigan 28 12.7 37.6
Clinton, Michigan 21 13.2 22.9
Truckee, Nevada 28 12.9 26.
Ivel , Great Britain 16 24.
Flint, Michigan 28 4. 40.
North Carolina Streams
Laboratory Streams
Pav R
2 2
g/m -day g/m -day
4.4 13.0 9.3 12. 7a
4.2 7.3 9.3a
4.8 9.6 3.6 6.2a
9.0 4.6a
1.3 18. 4. 20a
9.8 21. 5b
3.4 4.0 2.4 2.9b
Erdmann (1979a,b)
Deb and Bowers (1983)
Charles, Massachusetts 19-25
Shenandoah, Virginia
23
4.8 17.
Algal respiration only
Total community respiration
Measurements were made over the
0.0 12.
0.0 36.l
0.9 5.9°
Kelly
Kelly
Kelly
Kelly
Kelly
Kelly
et
et
1*
et
et
et
Simonsen
Gul 1 iver
li-
al.
al_.
al_.
al-
ll-
and
ejt
(1975)
(1975)
(1975)
(1975)
(1975)
(1975)
Harremoes (1978)
al_. (1982)
Baker,
Virginia
Rappahannock, Virginia
S. Fork
Rivanna
South,
Mechums
Havelse
Rivanna, Virginia
, Virginia
Virginia
, Virginia
, Denmark
Experimental Channels 9-24 5. 45.
0.
6
2
2
45
.1
.1
.3
2.0
1
0.2
1.5
.3
25. 9C
14.8
1.
7,
3.
5.
5.
2.
4.8
2.6
.9
.3
,4
.4
,3
,6
b
b
b
b
b
b
22.
10.
9b
,7b
period of one year, and solar radiation varied by more than a factor of 10.
204
-------
In cases where diurnal water temperature changes are great, diurnal curve
analyses should include temperature correction effects.
All of the approaches reviewed in Table 3-31 have apparently been
successfully applied. However, no comprehensive comparison of the
approaches against the same data set were found. In cases where a
significant amount of data is available for analysis, a computerized
approach such as Kelly jrt aj_. (1975) or Gulliver et _al_. (1982) appears to be
better than trial and error procedures. The method that has been most
rigorously tested is the DORM model of Gulliver et al_. (1982). Also these
methods can be used when the distance between stream stations is great,
because the models do not assume that P-R remains constant over the travel
time between the stations.
Under the appropriate conditions the simpler approach of Erdmann
(1979a,b) can be used. One restriction on using approaches where P-R is
assumed constant over the time increment is that the travel time between
stations must be short (i.e., 1 to 3 hours) so that the constant P-R
assumption is not violated.
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230
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Chapter 4
pH AND ALKALINITY
4.1 INTRODUCTION
The subjects of pH and alkalinity are becoming increasingly important
as society begins to deal with acidic precipitation. New models developed
to analyze effects of alternative controls on inputs of acidity to sensitive
aquatic environments use alkalinity as a state variable, then predict pH
from alkalinity (Gherini _et _§_]_., 1984). Earlier models did not contain many
of the processes that affect pH, and their predictive capability was
adequate for some, but not all, environments (e.g. Henriksen, 1979). More
elaborate models now exist which take into account a more complete picture
of the constituents that comprise alkalinity in the dilute systems that are
at risk from acidic precipitation (organic acids, other non-carbonate weak
acids, etc.) and which compute other source-sinks of alkalinity and factors
that affect pH (Chen^t aj_., 1984).
4.2 CARBONATE ALKALINITY SYSTEM
The carbonate system is of great importance in lakes, rivers, and
estuaries. Carbonate chemistry of natural waters has been described in
detail elsewhere (Stumm and Morgan, 1970, 1981; Trussell and Thomas, 1971;
Park, 1969; Butler, 1982; Chen and Orlob, 1972, 1975). The carbon dioxide
p
(CO,,) - bicarbonate (HCOl) - carbonate (CO- ) equilibrium is the major
buffer system in aquatic environments. This equilibrium directly affects
the pH, which in turn can affect the biological and chemical constituents of
the system. For example, it may become necessary to simulate pH and
alkalinity in order to compute the toxicant, un-ionized ammonia (see
Chapter 5), or to determine available concentrations of metals
(e.g., Gherini et _al_., 1984).
231
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Since algae use carbon dioxide as a carbon source during
photosynthesis, this is a nutrient which can reduce the growth rate when
alkalinity is low and other nutrients are high (Goldman, ^t _al_., 1972).
Most models 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
consumed by algal growth. The major source is atmospheric exchange.
The major chemical species considered to constitute alkalinity are
dissolved carbon dioxide, bicarbonate, and carbonate ion, together with the
hydrogen and hydroxyl ions. Mass balance equations assume that ionic
equilibrium exists and calculate carbon inputs and outputs from a pool of
total inorganic carbon (TIC). Conversions between different carbon forms
are based on stoichiometric equivalents. The carbon dioxide form is
involved in most of the important processes, including surface reaeration,
respiration, excretion, algal uptake, and organic decay reactions. However,
dissolved carbon dioxide combines with water to form carbonic acid, which,
in turn, dissociates to bicarbonate ion, carbonate ion, and hydrogen ion.
Since the dissociation reactions occur very rapidly in comparison to the
other biological and chemical processes, dissolved carbon is modeled as the
2-
sum of CO,., + HCC> + C0_ , and is referred to as total inorganic carbon
(TIC).
Dissolved inorganic carbon is derived from several sources. These
include surface reaeration; respiration by fish, zooplankton, benthic
animals, and algae; soluble excretion by fish, zooplankton, and benthic
animals; and the decay of organic matter in the form of detritus, sediment,
and sewage BOD. Dissolved carbon is removed by assimilation during algal
photosynthesis.
Conceptually, the mass balance equation defining these relationships
for the EAM model (Tetra Tech, 1980) is expressed as follows:
232
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^ A
sed) I sed
1=1 j=l \
n_^ /
• Zr • • C.
zoo) + "f^ (aV Ari' Calg)
\ (zooi ' Zdexi ' Czoo) + (ben) (Bdex) (Cben)
+ BOD KBOD CBOD
^ ' \ / \
+ /Rrn \ /CO - CO \ /Area) (4-1)
\ LU2/ V sat V \ /
= detritus decay + sediment decay + fish respiration
+ benthic animal respiration + zooplankton respiration
+ algal respiration + fish excretion + zooplankton excretion
+ benthic animal excretion - algal assimilation
+ BOD decay + surface reaeration.
Although Equation (4-1) is a substantially complete picture of TIC
dynamics in an aquatic system, most models do not contain the same degree of
complexity. However, whether multi-compartmented or few compartments, the
general aspects of the process are modeled similarly. Also, the inputs and
outputs can be based on C0? with suitable stoichiometric conversions (e.g.,
Di Toro and Connolly, 1980) rather than TIC.
Surface reaeration of CO^ from atmospheric sources is done in a way
similar to oxygen (Section 3.2). However, only minimal effort to measure
233
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C02 reaeration is necessary and literature values have been used (Emerson,
1975; Liss, 1973). Reaeration occurs only at the surface of the water body,
and is a function of the carbon dioxide saturation level. The saturation
concentration is a function of the water temperature as it affects the
Henry's law constant (KH) for computing COp .:
C02sat = KH PC02 (4-2)
where pC02 is the partial pressure of C02 in the atmosphere (generally
0.00033 atmospheres is used) and
KH =
2385.73 - 14.0184 + 0.0152642 TK
T
K J (4-3)
where MCQ = 44,000 mg/mole, C02
TK = temperature in K = 273.15 + °C
KM = Henry's law constant, mg/(liter-atm)
After computing the total inorganic carbon according to the mass
balance in Equation (4-1), the dissolved carbon dioxide concentration is
calculated using relationships derived from the equilibrium constants of the
dissociation reactions. The reactions involved are:
H2C03^± HCO~ + H+ KI (4-4)
HCO~;=±C03 + H+ K2 (4-5)
H00=± H+ + OH" K (4-6)
f. ^ w
where the equilibrium constants are defined as
KI =[_n^3JL" J (4_7)
[H2C°3]
-------
(4-8)
[HCO-]
= [H+] [OH'] (4-9)
The equilibrium constants K^ K2, and KW vary with temperature according to
the following relationships (Tetra Tech, 1979):
[l4.8435 - 0.032786 TK - (3404.71/TK)]
Kx = 10 (4-10)
[6.498 - 0.02379 TK - (2902.39/TK)1
K2 = 10 (4-11)
["35.3944 - 0.00835 TK - (5242. 4/TK - 11.826 log (TK)J
Kw = 10 (4-12)
In a carbonate system, the alkalinity (alk) is calculated according to
the mass balance equation:
alk = alkalinity = |~HCO~] + 2 [cOg] + [oH'J - [H+] (4-13)
Other processes can affect alkalinity in aquatic systems. Addition of
acids and nitrification reduce alkalinity, and uptake of nitrate by algae
increases alkalinity. Because of the magnitude of the ammonia concentration
in waters receiving municipal effluents, nitrification can affect alkalinity
substantially, generating 2 equivalents of acid (H ) per equivalent of
ammonia oxidized (see Section 3.4). Similarly in eutrophic waters, nitrate
uptake can increase alkalinity by the production of approximately 1
equivalent of base (OH") per equivalent of nitrate taken up by plant cells.
These corrections would be of consequence in low alkalinity waters (less
than 200^eq/l), and would be applied to Equation (4-13).
Once the total inorganic carbon and alkalinity have been determined
using the mass balance equations (4-1, 4-13), the hydrogen ion concentration
235
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can be calculated by trial and error solution of the following relationship:
After [H ]is determined, it is substituted into the expression for CO,,,
which can then be solved directly for the dissolved COp concentration:
(4-15)
Ki K-l • l\n
1 + — +
V] [H+l2
Not all models compute inorganic carbon species or pH. Generally these
computations have been made primarily in lake systems where they are of
significance in acid precipitation or are used for additional model
verification as in Di Toro and Connolly (1930). In all cases, the
formulations are based on the above derivations, although the computation
details may differ from model to model. Water quality models that contain
the COp, alkalinity, pH formulations include those discussed in the
following references:
Smith, 1978 WQRRS
Thomann et jfl_. , 1974 LAKE-3
Di Toro and Connolly, 1980 Lake Erie Model
Scavia, £t _al_. , 1976 Lake Ontario Model
Tetra Tech, 1980 EAM
WES, 1982 CE-QUAL-R1
4.3 EXTENDED ALKALINITY APPROACH
4.3.1 Definition of Extended Alkalinity
The mass balance equation (4-13) has ignored several H -ion acceptors,
and is appropriate in many instances. In very low alkalinity waters,
236
-------
however, the concentration of these neglected H -ion acceptors can be
significantly large. The neglected H -ion acceptors include organic
substances with carboxyl and phenolic hydroxyl groups, for example:
R-COO" +
R-COOH (organic acids)
(4-16)
and the monomeric aluminum species and their complexes, for example,
A1
3+
and
(4-17)
Al-R
A1
3+
(4-18)
An extended alkalinity relationship would include the alkalinity associated
with water itself, the carbonate system, the monomeric aluminum system and
its organic complexes, and dissolved organic acid anions. The dissolved
organic carbon alkalinity can be represented by a triprotic (H^R-i) and/or
monoprotic (HR, ) model organic acid with fixed dissociation constants and a
fixed number of acid-base functional groups per unit mass of carbon
( eq/mgC). The components of the total alkalinity, as represented by the
H -ion acceptors, are given below:
Alk = AlkH 0 + Alkc + AlkR + AlkR + AlkA1
water carbonate organic
system acids
aluminum system
(4-19)
where
AlkH n = [OH"] - [H+]
(4-20)
Alkc = [HCO~]
Alk
3[R3']
237
(4-21)
(4-22)
-------
AlkR = [R-
AlkA1 = [A1(OH)2+] + 2[Al(OH)p.+ 3[A1(OH)°] + 4[A1(OH)'] (4-24)
AlkA1.Q - 3[A1 Rj + [A1R2+] + 2[Al(R2)+] + 3[A1(R2)3] (4-25)
An alternative representation of solution-phase alkalinity, which is
mathematically equivalent to the above is given as follows,
Alk= 2k ZkNk = ICB - 2CA (4-26)
where SCD = the sum of the base cations
K+] + [wj] (4-27)
2C. = the sum of the strong acid anions
= 2 [so42'] + [NO-] + [cr] (4-28)
The derivation is based on the mass balance equation and the solution
electroneutral ity condition. Figure (4-1) shows the equivalence for
lakes in the State of Washington.
4.3.2 Modeling Extended Alkalinity
The concept of extended alkalinity has been incorporated in a model
called PHCALC. This model was developed primarily for the ILWAS model
(Tetra Tech, 1983), and was later modified into an interactive FORTRAN
program to compute any one of the following options: pH, alkalinity, total
inorganic carbon (TIC) and "solution equilibration". The solution
equilibration approach is similar to the approach for pH, except that
alkalinity can be adjusted for gibbsite precipitation or dissolution.
Table 4-1 shows the list of required parameters for any given option.
All the concentrations on the left-hand-side of Equations (4-20)
through (4-25) can be expressed in terms of ionization fractions and
238
-------
320
KEY:
LAKES IN STATE
OF WASHINGTON
-40
-40
120 180 200 240
REPORTED ALKALINITY (/ieq/D
280 320
Figure 4-1. [£CR-SO. ] plotted against reported alkalinity
(from Gherini et al_., 1984).
temperature-dependent dissociation constants. Fluoride and sulfate
concentrations are required for the determination of their complexations
with aluminum.
4.3.3 Equilibrium Constants and Solubility Products
The equilibrium constants used in PHCALC are obtained by first
expressing a thermodynamic temperature dependence for a related constant,
1C,:
239
-------
TABLE 4-1. OPTIONS AND THEIR REQUIRED INPUT PARAMETERS FOR PHCALC
Options* Parameters Required to be Specified
pH Alk, TIC or EQp Aly, OACp OAC2, F, S042", T
Alk pH, TIC or EQp Aly or EQ2, OACp OAC2, F, S042", T
TIC pH, Alk, A1T, or EQ2, OACp OAC2, F, S042", T
EQ Alk, TIC or EQp Aly, EQ2, OACp OAC2, F, S042", T
Definition of Parameters:
Alk alkalinity
TIC total inorganic carbon
EQ equilibration of a solution with A1(OH)_
A1-, total aluminum
OAC, total organic acid (1)
OAC2 total organic acid (2)
F fluoride concentration
p_
SO, sulfate concentration
T temperature, °C
EQ, ratio of TIC to air-equilibrated TIC (specified for open
system)
EQ2 -log (Ksp) for A1(OH)3 mineral or one of the following minerals
for the equilibration with gibbsite
AG - amorphous gibbsite (pKsD = 31-19)
MG - microcrystalline gibbsite (pKg = 32.64)
NG - natural gibbsite (pK$ = 33.22)
SG - synthetic gibbsite (pK$ = 33.88)
*0ptions are the parameters to be computed
240
-------
Iog10 K. = a + y + cT + dlog1QT
(4-29)
The constants a, b, c and d are given as follows
w
6.0875
545.56
-17052
-4470.99 -0.01706
0.12675
d Reference
0 Stumm & Morgan, 1981
-215.21 Loewenthal & Marais,
1978
-6.498
2902.39 0.02379
Loewenthal & Marais,
1978
K
'H
-14.0184
2385.73 0.0152642
0
Stumm & Morgan, 1981
K , K,, and.Kp are dimensionless while KM is in moles liter" atm~ .
KM has to be multiplied by RT to convert to a dimensionless form. R is the
universal gas constant and T is the absolute temperature in degrees Kelvin
in the range of 273 K to 313 K.
The solubility products used in the equilibration with gibbsite were
shown earlier in Table 4-1.
4.4 SUMMARY
Two approaches have been presented for the relationship of total
inorganic carbon, alkalinity and pH. For waters with low dissolved organic
carbon (with little color) and high alkalinity (al k2l200^eq/l) , the
conventional alkalinity definition is recommended. For waters with high
dissolved organic carbon and waters with alk <200 ^ieq/1 where the
alkalinities contributed by aluminum and organic acids are no longer
negligible, the extended alkalinity approach is recommended. The
equivalence between the expression Alk = SC_ - 2C. and the extended
alkalinity definition provides a convenient tool in alkalinity evaluation.
241
-------
4.5 REFERENCES
Butler, J.N. 1982. Carbon Dioxide Equilibria and their Applications.
Addison-Wesley Pub. London. 259 p.
Chen, C.W., and G.T. Orlob. 1972. Ecological Simulation for Aquatic
Environments, Report to Office of Water Resources Research OWRR C-2044,
Water Resources Engineers Inc., Walnut Creek, California.
Chen, C.W., and G.T. Orlob. 1975. Ecologic Simulation of Aquatic
Environments. Systems Analysis and Simulation in Ecology, Vol. 3, B.C.
Patten, (ed.). Academic Press, New York, N.Y. pp. 476-588.
Chen, C.W., S.A. Gherini, J.D. Dean, R.J.M. Hudson, and R.A. Goldstein.
1984. Development and Calibration of the Integrated Lake-Watershed
Acidification Study Model. In Modeling of Total Acid Precipitation Impacts.
J.L. Schnoor (Ed.), Butterworth, Boston, Mass.
Di Toro, D.M., and J.P. Connolly. 1980. Mathematical Models of Water
Quality in Large Lakes Part 2: Lake Erie, EPA-600/3-80-065, U.S.
Environmental Protection Agency, Duluth, Minnesota.
Emerson, S. 1975. Gas Exchange Rates in Small Canadian Shield Lakes.
Limnol. Oceanogr. 20:754-761.
Gherini, S.A., C.W. Chen, L. Mok, R.A. Goldstein, R.J.M. Hudson, and G.F.
Davis. 1984. The ILWAS Model: Formulation and Application. In the
Integrated Lake-Watershed Acidification Study. 4: Summary of Major Results.
EPRI EA-3221. p. 7-1 to 7-46.
Goldman, J.C., D.B. Porcella, E.J. Middlebrooke, and D.F. Toerien. 1972.
Review Paper: The Effect of Carbon on Algal Growth: Its Relationship to
Eutrophication. Water Res. 6:637-679.
Henriksen, A. 1979. A Simple approach For Identifying and Measuring
Acidification in Freshwater. Nature, 278, 542.
Liss, P.S. 1973. Processes of Gas Exchange Across an Air-Water Interface.
Deep-Sea Res. 20: 221-238.
Loewenthal, R.E., and G.V.R. Marais. 1978. Carbonate Chemistry of Aquatic
Systems: Theory and Application. Volume 1. Ann Arbor Science, Michigan.
Parks, P.K. 1969. Oceanic C0? System: An Evaluation of Ten Methods of
Investigation: 179-186. c
Scavia, D., B.J. Eadie, and A. Robertson. 1976. An Ecological Model for
Lake Ontario Model Formulation, Calibration, and Preliminary Evaluation.
NOAA Technical Report ERL 371-GLERL 12. NOAA. Boulder Colorado. 63 p.
Smith, D.I. 1978. WQRRS, Generalized Computer Program for River-Reservoir
Systems. User's Manual 401-100, 100A: U.S. Army Corps of Engineers,
Hydrologic Engineering Center (HEC), Davis, California. 210 pp.
242
-------
Snoeyink, V.L., and D. Jenkins. 1980. Water Chemistry. New York. John
Wiley & Sons.
Stumm, W., and J.J. Morgan. 1970. Aquatic Chemistry. (New York: Wiley-
Interscience).
Stumm, W., and J.J. Morgan. 1981. Aquattc Chemistry, 2nd Ed., Wiley, New
York.
Tetra Tech, Inc. 1979. Methodology for Evaluation of Multiple Power Plant
Cooling System Effects, Volume II: Technical Basis for Computations.
Electric Power Research Institute, Report EPRI EA-1111.
Tetra Tech, Inc. 1980. Methodology for Evaluation of.Multiple Power Plant
Cooling System Effects, Volume V. Methodology Appl ication to Prototype-
Cayuga Lake. Electric Power Research Institute, Report EPRI EA-1111.
Tetra Tech, Inc. 1983. The Integrated Lake-Watershed Acidification Study,
Volume 1: Model Principles and Application Procedures. Electric Power
Research Institute. Report EPRI EA-3221.
Thomann, R.V., D.M. Di Toro, R.P. Winfield, and D.J. O'Connor. 1975.
Mathematical Modeling of Phytoplankton in Lake Ontario. I. Model
Development and Application. EPA-660/3-75-005. USEPA, Corvallis, Oregon,
97330. 177 p.
Trussell, R.R., and J.F. Thomas. 1971. A Discussion of the Chemical
Character of Water Mixtures. J. American Water Works Assoc. 63(1), 49.
243
-------
Chapter 5
NUTRIENTS
5.1 INTRODUCTION
Certain elements are referred to as nutrients because they are
essential to the life processes of aquatic organisms. The major nutrients
of concern are carbon, nitrogen, phosphorus, and silicon. Silicon is
important only for diatoms, one of the major components of the algal
community. Other micronutrients such as iron, manganese, sulphur, zinc,
copper, cobalt, and molybdenum are also important. However, these latter
nutrients are not considered in water quality models because they are
required only in trace amounts and they are usually present in quantities
adequate to meet the biochemical requirements of the organisms.
Nutrients are important in water quality modeling for several reasons.
For example, nutrient dynamics are critical components of eutrophication
models since nutrient availablility is usually the main factor controlling
algal blooms. Algal growth is typically limited by either phosphorus or
nitrogen, with the exception of diatoms which are often silicon Limited.
Some blue-green algae can fix nitrogen and are therefore not limited by
nitrogen. Carbon is usually available in excess although in some cases it
may also be limiting. Carbon is also important because of its role in the
pH-carbonate system, as discussed in Chapter 4.
Nitrogen is important in water quality modeling for reasons other than
its role as a nutrient. For example, the oxidation of ammonia to nitrate
during the nitrification process consumes oxygen and may represent a
significant portion of the total BOD. Also, high concentrations of
unionized ammonia can be toxic to fish and other aquatic organisms.
244
-------
5.2 NUTRIENT CYCLES
Nutrients are present in several different forms in aquatic systems:
• dissolved inorganic nutrients
• dissolved organic nutrients
• particulate organic (detrital) nutrients
• sediment nutrients
• biotic nutrients (algae, aquatic plants, zooplankton, fish,
benthic organisms)
Only the dissolved inorganic forms are available for algal growth. These
include dissolved C0?, ammonia, nitrite, and nitrate nitrogen,
orthophosphate, and dissolved silica.
Each nutrient undergoes continuous recycling between the major forms
listed above. For example, dissolved inorganic nutrients are removed from
the water column by algae and aquatic plants during photosynthesis. These
nutrients are distributed to the other aquatic organisms through the food
web. Dissolved inorganic nutrients are returned to the water through the
soluble excretions of all organisms, the decomposition of organic detritus
and sediments, and the hydrolysis of dissolved organic nutrients. In
addition, dissolved CCL and N2 gases exchange with the atmosphere.
Suspended particulate nutrients are generated through the particulate
excretions of aquatic animals and the death of planktonic organisms.
Organic detritus and phytoplankton which settle to the bottom contribute to
the sediment nutrients. Decomposition of suspended organic detritus and
organic sediment releases both dissolved organic and dissolved inorganic
nutrients to the water.
Many of the above interactions are shown in Figure 5-1 for carbon,
nitrogen, and phosphorus and in Figure 5-2 for silicon. Figures 5-3 and 5-4
present more detailed descriptions of the nitrogen and phosphorus cycles.
In. addition to the internal recycling of nutrients within the
waterbody, nutrients are also introduced through wasteloads (both point and
nonpoint sources), river or tributary inflows, runoff, and atmospheric
precipitation.
i-T1 D
-------
TOTAL
INORGANIC
CARBON
Figure 5-1. Nutrient interactions for carbon, nitrogen, and phosphorus
(from Tetra Tech, 1979).
246
-------
5.3 GENERAL MODELING APPROACH FOR ALL NUTRIENTS
Nutrient dynamics are governed by the following processes;
• dissolved inorganic nutrients
- photosynthetic uptake
- excretion
- chemical transformations (e.g., oxidation of NH.J
- hydrolysis of dissolved organic nutrients
- detritus decomposition
- sediment decomposition and release
- external loading
SETTLING SETTLING
IN OUT
^RESPIRATION,
Figure 5-2. Nutrient interactions for silica (from Tetra Tech, 1979).
247
-------
co
VaATIZATION
OFNHj
ATMOSPHERIC
EXCHANGE
I
LIVING PARTICULAR N
I
FIXATION
DISSOLUTION
f
UPTAKE j
RELEASE
1
MOLECULAR
NITROGEN (N?)
RESPI
AMMONIFICATION
DENITRIFICATION
NITRIFICATION
DENITRIFICATION
LJL
RELEASE
N03-N
NITRIFICATION
DENITRIFICATION
N02-N
UPTAKE
FIXATION
NATION
RESPIRATION
SOLUBLE
ORGANIC N
SORPTION
DESORPTION
I
NON-LIVING
PARTICULATE N
T
I
| SETTLING I
~
SEDIMENT PROCESSES
ASSIMILATION
FIXATION
NITRIFICATION
DENITRIFICATION
DECOMPOSITION
AMMONIFICATION
LOSS (SEDIMENT TRAPPING)
Figure 5-3. Nitrogen cycle (from Baca and Arnett, 1976).
-------
UPTAKE
1X1
-pi
1
1
1
EASE |
.IVING PARTICULATE P UPTAKE
i |
1
1
RELE
i
ASE UP
t
At
n I /
/
RELEASE ^
(E SOLUBLE COMPLEX P
SOLUBLE
ORGANIC P
HYDROLYSISi
\J
J
r
DISSOLVED REACTIVE
PHOSPHORUS (P0
-------
• dissolved organic nutrients
- excretion
- hydrolysis
- detritus decomposition
- sediment decomposition and release
- external loading
• particulate organic nutrients
- particulate excretions
- plankton mortality
- decomposition
- settling
- zooplankton grazing
- external loading
• sediment nutrients
- detritus settling
- algal settling
- sediment decomposition and release
Only processes affecting the abiotic forms of nutrients are discussed in
this chapter since the biotic components of water quality models are
discussed in Chapters 6 (Algae) and 7 (Zooplankton).
Nutrients are modeled by using a system of coupled mass balance
equations describing each nutrient compartment and each process listed
above, plus the transport processes of advection and dispersion discussed in
Chapter 2. The general equations for each nutrient, omitting the transport
and external loading terms, can be expressed as follows:
dissolved inorganic nutrients:
at = - Vs * fl es * Kl S' - K2 S * Korg Sorg * f2 Kdet Sdet
f3 Ksed Ssed
250
-------
dissolved organic nutrients:
dS
es - Korg Sorg + ^ ' V Kdet Sdet
~ f3> Ksed Ssed
particulate organic nutrients:
dS
- ep * MP - Kdet Sdet - Ks Sdet ' Gz
sediment nutrients:
' Ks Sde«t * \ - Ksed Ssed
where S = dissolved inorganic nutrient concentration, mass/volume
S1 = another inorganic form of the nutrient which decays to the
form S (e.g., NH, NO,) , mass/ volume
S = dissolved organic nutrient concentration, mass/volume
Sdet = susPendecl particulate organic nutrient concentration,
mass/ volume
S . = organic sediment nutrient concentration, mass/volume
K, = transformation rate of S1 into S, I/time
K~ = transformation rate of S into some other dissolved
inorganic form of the nutrient, I/time
K = hydrolysis rate of dissolved organic nutrient, I/time
Kdet = decomP°s"itlon rate °f particulate organic nutrient, I/time
K , = decomposition rate of organic sediment nutrient, I/time
K = settling rate for particulate organic nutrient, I/time
V = photosynthetic uptake rate for nutrient S, mass/volume-
time
e = soluble excretion rate of nutrient by all organisms,
mass/ volume-time
f, = fraction of soluble excretions which are inorganic
251
-------
f = fraction of detritus decomposition products which are
immediately available for algal uptake
f, = fraction of sediment decomposition products which are
immediately available for algal uptake
e = particulate excretion rate of nutrient by all animals,
mass/volume-time
M = total rate of plankton mortality, mass/volume-time
G = detritus grazing rate by zooplankton, mass/volume-time
A = algal settling rate to sediment, mass/volume-time
Note that all of the transformations between the various abiotic
nutrient compartments are described by first-order kinetics. This approach
is used in almost all water quality models. Nutrient models differ
primarily in the specific nutrients simulated (i.e., C, N, P, and Si) and in
the number of compartments used to describe each nutrient cycle (i.e.,
dissolved inorganic forms such as NH,, NCL, and NO.,; dissolved organic
components; particulate organic components; sediments; and biotic components
such as algae and zooplankton).
For example, many models omit carbon since it does not limit algal
growth in most situations. Silicon is generally modeled only when diatoms
are simulated as a separate phytoplankton group.
The nutrient cycles are often simplified by combining or omitting some
of the forms described above. For example, many models do not simulate
sediment nutrients explictly with a mass balance equation such as
Equation (5-4). Instead, user-specified sediment fluxes are specified in
Equations (5-1) and (5-2). Dissolved organic nutrients are also left out of
most models. In these cases, the decomposition products of the detritus and
sediments as well as all soluble excretions go directly to the dissolved
inorganic nutrient compartments. This in effect combines the suspended
particulate and dissolved organic compartments into a- single "unavailable"
nutrient compartment which decays to produce available inorganic forms.
252
-------
Nitrogen models also differ in the forms of inorganic nitrogen which
are included, as well as in some of the processes modeled. For example,
some models include only ammonia and nitrate, rather than the full oxidation
sequence of ammonia to nitrite to nitrate. While most models include the
nitrification reactions, only a few include denitrification. Also, only a
few models include nitrogen-fixation by blue-green algae.
Sediments and particulate organic detritus are often modeled as single
compartments, rather than having a separate compartment for each nutrient.
In this case, the corresponding compartments for each nutrient are
determined from the product of the total sediment and detritus
concentrations and the stoichiometric ratios for each nutrient. The
stoichiometric ratios are generally the same as those used for algae (see
Section 6.3 of Chapter 6) so that mass is conserved during nutrient
recycling.
Table 5-1 presents a comparison of the various nutrient forms included
in several models. Transformation processes and the corresponding rate
coefficients for each specific nutrient are discussed below, along with
model formulations for nutrient uptake, excretion, and sediment release.
Formulations for plankton mortality and zooplankton grazing are discussed in
Chapters 6 and 7. Settling formulations for particulate organic detritus
are essentially the same as the simplest formulations used for phytoplankton
settling described in Chapter 6 (i.e., the settling rate equals the user-
specified settling velocity divided by the depth of the model segment).
5.4 TEMPERATURE EFFECTS
Temperature influences the rates of all of the nutrient transformation
processes discussed above. All of the first-order rate coefficients in
Equations (5-1) through (5-4) are therefore temperature dependent. Almost
all models use the exponential Arrhenius or van't Hoff relationship to
describe these effects. A reference temperature of 20°C is usually assumed
when specifying each rate coefficient, resulting in the following equation:
253
-------
TABLE 5-1. COMPARISON OF NUTRIENT MODELS
Model
(Author)
AqUA-IV
CE-QUAL-R1
CLEAN
CLEANER
MS. CLEANER
DEM
DOSAG3
EAM
ESTECO
EXPLORE-1
HSPF
LAKECO
MIT Network
QUAL-I1
RECEIV-II
SSAM IV
WASP
WQRRS
Bierman
Canale
Jorgensen
Lehman
Nyholm
Scavia
Nutrients
C N
X
X X
X X
X X
X X
X
X
X X
X X
X X
X X
X X
X
X
X
X
X X
X X
X
X
X
X X
X
X X
Modeled
F Si
X
X
X
X
X X
X
X
X X
X
X
X
X
X
X
X
X X
X
X X
X X
X
X
X
X
Dislvd.
Inorg.
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Dislvd.
Organic
N
X
X
X
N
X
1
X
N
Partic.
Organic
X
X
X
X
X
X
X
X
X
X
X
X-
X
X
X
X
X
X
Nutrient
Sedi-
ments
X
X
X
X
X
X*
*
X
X
X
p
it
X
X
*
X
X
X
X
X
X
Forms
Algae
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Zoo- Other
plankton Organisms
X
X X
X X
X X
X X
X X
X X
X
X
X X
X
X
X X
X
X
X X
X
Inorganic
NH3 N02
X X
X X
X X
X X
X X
X X
X X
X X
X X
X X
X X
X X
X
X
X X
X
X
Nitrogen Forms
Total
N03 Avail
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
References
Baca i Arnett (1976)
WES (EWqOS) (1982)
Bloomfield et aT_. (1973)
Scavia & Park (1976)
Park et al.. (1980)
Feigner & Harris (1970)
Duke & Masch (1973)
Tetra Tech (1979, 1980)
Brandes & Masch (1977)
Baca et al_. (1973)
Johanson et jil_. (1980)
Chen & Orlob (1975)
Harleraan et al_. (1977)
Roesner et a±. (1981)
Raytheon (1974)
Grenney & Kraszewski (1981)
Di Toro et al_. (1981)
Smith (1978)
Bierman et al.. (1980)
Canale et al.. (1975, 1976)
Jorgensen (1976)
Lehman et aj.. (1975)
Nyholm (1978)
Scavia et al_. (1976)
ro
01
Specify flux.
-------
K - K - 5 R)
KT - K2Q b-b)
where KT = rate coefficient at temperature T, I/time
T = temperature, C
K2Q = rate coefficient at 20°C, I/time
6 = temperature adjustment coefficient
This relationship is derived in Section 3.3 of Chapter 3.
A few models use different temperature adjustment formulations. For
example, Canale (1976) uses a linear relationship and Grenney and Kraszweski
(1981) use a logistic equation as a temperature adjustment function.
5.5 CARBON TRANSFORMATIONS
Table 5-2 presents rate coefficients for carbon decay processes along
with the corresponding temperature adjustment factors. As shown in the
table, these coefficients have a broad range, indicating a lack of detailed
process characterization. Process characterization has been neglected in
carbon models since the relationship of carbon dynamics to water quality
modeling has not been considered essential. In fact, most water quality
models do not include carbon since it is not usually a limiting nutrient.
In the Lake Erie version of WASP (Di Toro and Connolly, 1980), the rate of
decay of particulate organic carbon to C02 has been further reduced by using
a saturation relationship (Di Toro and Connolly, 1980). However, the decay
rates in all other models are computed according to the first-order kinetics
discussed above.
Most of the temperature adjustment factors in Table 5-2 range from 1.02
to 1.047, corresponding to Q,Q values ranging between 1.2 and 1.6. The
exception is the Lake Erie WASP model (Di Toro and Connolly, 1980), which
uses a temperature correction factor of 1.08 (Q,n = 2.16) for decay of
settled algae and sediment organic matter. Also, the decay rate constants
for these compartments are generally higher than those used in other models.
255
-------
TABLE 5-2. RATE COEFFICIENTS FOR CARBON TRANSFORMATIONS
POC -
K
0.1**
0.05
0.001
0.003
0.02
0.1
0.005-0.05***
0.001-0.02***
- co2
e
1.04
1.045
1.02
1.020
1.020
1.047
1.02-1.04***
1 .040***
SOC *
K
0.00025
0.001
0.0015
0.001
0..0015
0.001-0.01***
0.001-0.02***
C02 SA -•• SOC
Ff If f\
"* \J
1.08 0.02 1.08
1.02
1.047
1.020
1.047
1.02-1.04***
1.040***
SA •* C02 References
K e
0.02 1.08 Di Toro & Connolly (1980)
O'Connor et. aK (1981)
Chen & Orlob (1972, 1975)
Tetra Tech (1980)
Bowie et a]_. (1980)
Porcella et aK (1983)
Smith (1978)
Brandes (1976)
*Abbreviations are defined as follows:
POC - Participate Organic Carbon
C02 - Carbon Dioxide
SOC - Sediment Organic Carbon
SA - Settled Algae n
U2
**This rate is multiplied by an oxygen limitation factor, 'RT+OT, where K, is a half-saturation constant for oxygen.
***Model documentation values.
-------
5.6 NITROGEN TRANSFORMATIONS
Nitrogen dynamics are modeled in a considerably more complex manner
than carbon because of their substantial biogeochemical role, important
oxidation-reduction reactions, and because other important water quality
variables such as oxygen are affected by nitrogen. The processes that are
simulated in water quality models include:
• Ammonification - release of ammonia due to decay processes
(deamination, hydrolysis).
• Nitrification - oxidation of ammonia to nitrate (NOZ)
directly (one-stage process) or to nitrite (NOl) and then to
nitrate (two-stage process). Nitrification is discussed in
detail in Section 3.4 of Chapter 3 in reference to its
^effects on dissolved oxygen.
• Denitrification - reduction of nitrate to N~ under anaerobic
conditions. This process also produces N?0 ( 10 percent of
total reduced), but since N?0 has not been shown to have an
appreciable effect on water quality, N^O production has not
been modeled.
• Uptake - accumulation of inorganic nitrogen by plants during
photosynthetic growth. Both ammonia and nitrate are
accumulated, with preference for ammonia over oxidized forms,
although not all models include this preference.
• Nitrogen fixation - reduction of N2 to ammoniated compounds.
Nitrogen fixation by blue-green algae is an important
external input of nitrogen accumulation in waterbodies that
materially affects nitrogen dynamics. However, uptake of
inorganic ions takes precedence over nitrogen fixation.
257
-------
In addition to the above processes, unionized ammonia can play a significant
role as a toxicant depending on the ammonia concentration, pH, and
temperature.
Table 5-3 presents rate coefficients for the major nitrogen decay and
abiotic transformation processes along with the corresponding temperature
ajdustment factors. The decay processes shown include breakdown of complex
organic compounds (particulate organic nitrogen, PON) to simpler organics
(dissolved organic nitrogen, DON) or to ammonia, the breakdown of sediment
nitrogen to ammonia, and the oxidation of ammonia to nitrate. Rate
constants for ammonia decay to nitrite and then to nitrate or from ammonia
to nitrate directly are approximately commensurate as an overall rate
process. The rate coefficients for some of the decay processes in some
versions of WASP are further reduced by saturation kinetics (Di Toro and
Connolly, 1980; Di Toro and Matystik, 1980; Thomann and Fitzpatrick, 1982;
O'Connor et al. , 1981). For example, the decay of particulate organic
nitrogen to ammonia is reduced as chlorophyll a decreases, and the
nitrification rate is reduced as dissolved oxygen decreases, according to
saturation kinetics.
The temperature adjustment factors have a wide range of values,
indicating some uncertainty in this coefficient. The Q.Q values generally
range from 1.2 to 2.4, but with one value as high as 3.7.
5.6.1 Denitrification and Nitrogen Fixation
Both of these processes affect the mass balance of nitrogen because
nitrogen is transported to (denitrification) or from (nitrogen fixation) the
atmosphere rather than recycling within the water. Although both processes
have been shown to be important in certain aquatic environments,
denitrification is not commonly included in models. HSPF (Johanson ert al.,
1980), CE-QUAL-R1 (WES, 1982), Jorgensen (1976), AQUA-IV (Baca and Arnett,
1976), and some versions of WASP (Di Toro and Connolly, 1980; Thomann and
Fitzpatrick, 1982; O'Connor et a/L, 1981) include denitrification.
258
-------
TABLE 5-3. RATE COEFFICIENTS FOR NITROGEN TRANSFORMATIONS
PON 'DON DON - NH3 PON -
K 6 K 6 K
0.035
0.03**
0.03***
0.03***
0.075
0.14
0.001
0.020 (linear) 0.020 (linear)
0.020 (linear) 0.020 (linear)
0.02 1.020 0.02 1.020
0.02 (linear) 0.024 (linear)
0.003
0.1
0.01**
0.005**
0.1**
0.2**
>NH NH *
6 K
(linear)
i.o'a
1.08
1.08
1.08
(linear)
1.02 0.003-0.03
1.020 0.02
1.047 0.02
NI
1.08
1.02
1.072
N02 NH3 -- N03 N02 - N03 SEDN •*• NHj References
6 K 6 K 6 K 8
Calibration Values
0.04 (linear) Thonann et ah (1975)
Thomann et ah (1979)
0.12*** 1.08 0.0025 1.08 Di Toro & Connolly (1980)
0.20 1.06 Di Toro & Hatystik (1980)
0.09-0.13*** 1.08 0.0004 1.08 Thomann & Fitzpatrick (1982)
0.025*** 1.08 O'Connor et al_. (1981)
Salas & Thoraann (1978)
1.02 0.09 1.02 0.001 1.02 Chen & Orlob (1972, 1975)
0.060 (linear) Scavia j^t aK (1976)
0.1 (linear) Scavia (1980)
0.1 1.020 Bowie et ah (1980)
0.16 (linear) Canale et^ al_. (1976)
1.047 0.25 1.047 0.0015 1.047 Tetra Tech (1980)
1.047 0.25 1.047 0.0015 1.047 Porcella et al_. (1983)
0.95-1.8*** 1.14 Nyholm (1978)
Bierman et al_. (1980)
Jorgensen -(1976)
Jorgensen et aj_. (1978)
(continued)
-------
TABLE 5-3. (continued)
ro
CT>
o
PON » DON DON -» NH3 PON - NH3 NH3 * NI
K 6 K 6 K 6 K
0.1-0.4 NI 0.1-0.5
0.02-0.04 1.02-1.09 0.1-0.5
0.1-0.5
0.1-0.5
0.005-0.05 1.02-1.04 0.05-0.2
0.001-0.02 1.040 0.05-0.2
D2 NH3 -> N03
8KB
Model Documentation Values
NI
1.02-1.09
1.047
1.047
1.02-1.03
1.02
0.04-3.0 (logistic)
0.001-1.3**** NI
N02 + NOj SEDN •* NH
KB K
5. -10. NI
3. -10. 1.02-1.09 0.01-0.1
0.5-2.0 1.047
0.5-20 1.047
0.2-0.5 1.02-1.03 0.001-0.01
0.2-0.5 1.02 0.001-0.02
,, References
6
Baca et a]_. (1973)
1.02-1.09 Baca 4 Arnett (1976)
Duke & Hasch (1973)
Roesner e_t aK (1978)
1.02-1.04 Smith (1978)
1.040 Brandes (1976)
Grenney 4 Kraszewski (1981)
Collins 1 Wlosinski (1983)
*Abbreviations are defined as follows:
NI - No Information
PON - Particulate Organic Nitrogen
DON - Dissolved Organic Nitrogen
SEDN - Sediment Organic Nitrogen
"Unavailable nitrogen decaying to algal-available nitrogen.
Chi a
i~n I a
*"Di Toro & Connolly (1980) and Di Toro & Matystik (1980) multiply the PON NhL rate by a chlorophyll limitation factor, "K.+CH1 a^,
where K. is half-saturation constant = 5.0 »ig CHa a/1. '
2
Di Toro & Connolly (1980) and Thomann & Fitzpatrick (1982) multiply the NH, NO, rate by an oxygen limitation factor, K_+0?) where
K2 is a half-saturation constant - 2.0 mg02/l. 0
O'Connor et al. (1981) multiply the NH, NO, rate by an oxygen limitation factor, K,-KL, where K, is a half-saturation constant
= 0.5 mg02/l. J J J i J
Nyholm (1978) uses a sediment release constant which is multiplied by the total sedimentation rate of algae and detritus.
""Literature value.
-------
Denitrification rates and the corresponding temperature adjustment
coefficients are listed in Table 5-4. The decay rates for the WASP model
are further modified according to a saturation type relationship based on
the dissolved oxygen concentration. The rate decreases rapidly as 0^
increases above 0.01 mg/1. This rate would be equivalent to that of
Jorgensen (1976) when 0? = 5 mg/1. This indicates disagreement in
conceptualization of the process or in its quantitative response between the
two models. Sediment nitrate denitrification helps decrease the gradient of
the sediment oxygen demand (SOD) and may lead to a reduced requirement for
SOD (see Chapter 3.5; also, Di Toro, 1984).
Nitrogen fixation by blue-green algae is modeled by assuming that
growth is not limited by nitrogen and that nitrogen fixation makes up for
all nitrogen requirements which cannot be satisfied by ammonia and nitrate.
Some type of saturation relationship is typically used to partition the
nitrogen requirements between nitrogen fixation and uptake of ammonia and
nitrate. The major features of these relationships are as follows: 1) no
fixation occurs when ammonia plus nitrate are above some critical threshold
concentration; 2) for concentrations below the threshold, nitrogen fixation
increases as ammonia and nitrate decrease; and 3) when ammonia and nitrate
become very low, all of the nitrogen requirements are supplied by fixation.
Nitrogen fixation is included in the EAM (Tetra Tech, 1979), Scavia e_t al.
(1976), Canale et_ a]_. (1976), and Bierman e_t al_._ (1980) models.
5.6.2 Unionized Ammonia
Although nitrogen is an important nutrient required by microorganisms,
plants, and animals, certain forms such as unionized ammonia (NFL) can be
toxic. Unionized ammonia is toxic to fish at fairly low concentrations.
For example, water quality criteria ranging from 0.0015 to 0.12 mg N/l for
the 30-day average concentration have been suggested (USEPA, 1984). This
range exists because the biological response varies at different temperature
and pH values.
Both analytical measurement techniques and most model formulations for
ammonia are based on total ammonia:
261
-------
TABLE 5-4. RATE COEFFICIENTS FOR DENITRIFICATION
Nitrate
K
0.1*
0.1**
0.09*
0.1*
0.002
0.02-0.03
o.-r.o***
-* Nitroaen Gas
0
1.045
1.045
1.045
1.045
No Information
No Information
1.02-1.09***
References
Di Toro & Connolly (1980)
Di Toro & Connolly (1980)
Thomann & Fitzpatrick (1982)
O'Connor et al_. (1981)
Jorgensen (1976)
Jorgensen et al. (1978)
Baca & Arnett (1976)
*This rate is multiplied by an oxygen limitation factor, K,+0^,
is a half-saturation constant = 0.1mgO?/l.
**The same rate applies to sediment NO, dentrification.
***Model documentation values.
where K
1
X = total ammonia = NH-, + NH.
(5-6)
The concentrations of NH3 and NH* vary considerably over the range of pH and
temperature found in natural waters, but each can be readily calculated
assuming that equilibrium conditions exist (Stumm and Morgan, 1981).
Unionized ammonia exists in equilibrium with ammonia ion and hydroxide ion
(Emerson, et aj_., 1975):
NH(g)
= NH
OH
(5-7)
The reaction occurs rapidly and is controlled largely by pH and temperature.
Thus, unionized ammonia is calculated from the equilibrium expression:
262
-------
(NH*)(OH") (NH*)Kw
Ki = (NH3)(H20) = --—
Rearranging and taking the negative logarithm:
(NH!)
log Tfjyj-y = pKw - pK.j - pH (5-9)
The quantity pKh is called the hydrolysis constant. Substituting and taking
the inverse logarithms,
'X - NH,\ (pKh-pH)
= 10 n = R (5-10)
and solving for NH.,.
3 1 + R
(5-11)
Thurston ei^ a! . , (1974) determined the temperature correction for the
hydrolysis constant as follows:
pKh = 0.09018 + 2729. 92/T (5-12)
where T = absolute temperature, K
Substituting this relationship into Equation 5-7, unionized ammonia in
moles/liter becomes a function of measured ammonia, temperature, and pH.
Most water quality models predict the concentration of measured ammonia (X)
in units of weight/volume as a resultant of processes of nitrification,
ammoni f ication, respiration, and assimilation. For NH^-N, there are
14,000 mg/mole and
NH -H. mg/1 . (5-13)
263
-------
Although more cumbersome, a table of equilibrium values for unionized
ammonia can be used in a model (e.g., USEPA, 1984). Figure 5-5 illustrates
the relationship between pH, water temperature, and unionized ammonia.
Q
LU
N
O
I
111
O
DC
ai
Q.
0.01
0
Figure 5-5.
TEMPERATURE (°C)
Effect of pH and temperature on unionized ammonia
(from Willingham, 1976).
264
-------
5.7 PHOSPHORUS TRANSFORMATIONS
Table 5-5 presents rate coefficients and temperature correction factors
for the various phosphorus transformation processes included in water
quality models. The transformations include the decay of particulate
organic phosphorus (POP), sediment phosphorus (SEDP), and settled algae (SA)
directly to PO.-P or into intermediate forms (dissolved organic phosphorus,
OOP) before decaying to PO.-P. The decay rates have a broad range,
indicating some uncertainty in quantifying these processes. Similarly,
there is a broad range in temperature coefficients, with a Q,0 range from
1.2 to 2.4, except for a QIQ value of 3.7 for Nyholm (1978). Several of
the WASP models adjust the phosphorus decay rates using a saturation
equation based on algal biomass (Di Toro and Connolly, 1980; Di Toro and
Matystik, 1980; Salisbury £t a/L_, 1983; Thomann and Fitzpatrick, 1982). In
the case where chlorophyll a is used to estimate algal biomass, the half-
saturation constant is 5.0 g/1, and where carbon is used to estimate algal
biomass, the value is 1.0 mgC/1.
5.8 SILICON TRANSFORMATIONS
Silicon can be limiting only for diatoms, so its biogeochemical cycle
is simulated only when diatoms are modeled as a separate algal group.
Diatoms are important because of their role in phytoplankton succession,
their role in aquatic food chains, and their potential effects on water
treatment plants. Table 5-6 presents decay rates and temperature adjustment
coefficients for silicon. In contrast to the other nutrients, particulate
and sediment silicon decay directly to dissolved inorganic silicon rather
than passing through a dissolved organic phase. The range of the first-
order decay rates for particulate silica decay is 0.003-0.1 (I/day). The
temperature adjustment factor varies between 1.02 and 1.08, corresponding to
a Q1Q range of 1.2 to 2.2.
265
-------
TABLE 5-5. RATE COEFFICIENTS FOR PHOSPHORUS TRANSFORMATIONS
r-o
CTi
CTv
POP -OOP POP -
K () K
0.14
0.03
0.03"
0.22" 1.08
0.14
0.001
0.02
0.2 (linear)
0.003
0.02
0.1
0.1
0.005
0.1
0.5-0.8
0.1-0.7***
0.1-0.7***
0.005-0.05***
0.001-0.02***
Sediment Sediment Sedirent
• P04 OOP - P04 StOP - OOP SEOP ., po OOP - P04 SA - OOP SA - PO
fl K0K0 K 9 K 0 f. A K «
(linear)
l.OB
l.OB
0.22" 1.08 0.0004 1.08 0.0004 1.08 0.0004 1.08 0.02 1.08 0.02 1.08
(linear)
1.02 0.001 1.02
(linear)
0.2 (linear)
1.020 0.0015 1.047
1.020 0.001 1.020
1.047 0.0015 1.047
1.14 1.0-1.7 1.14**
1.08
1.02 0.0018 1.02
1.072
1.02-1.09*** 0.1-0.7*" 1.02-1.09*"
1.02-1.09*"
1.02-1.04*** 0.001-0.01*" 1.02-1.04*"
1 .040"*
References
Thomann et aj.. (1975)
Thomann et •]_. (1979)
01 Toro I Connolly (1980)
01 Toro 4 Matyslik (1980)
Salisbury et a_L (1983)
Thomann I Fltzpatrlck (1982)
Salas 1 Thomann (1978)
Chen » Or lob (1972. 1975)
Scavia et al. (1976)
Scavla 11980)
Canale el aK (1976)
Tetra Tech (1980)
Bowie et t\_. (1980)
Porcella et, a_K (1983)
Nyholm (1978)
Blernan et i\_. (1980)
Joroensen (1976)
Jorgensen ejt a_[. (1978)
Baca et aj. (1973)
Baca I Arnett (1976)
Smith (1978)
Brandes (1976)
Abbreviations are defined as
fol lows:
POP - Participate Organic Phosphorus SEOP - Sediment Organic Phosphorus
OOP - Dissolved Organic Phosphorus SA - Settled Algae
P04 - Phosphate Chl a
•*Di Toro t Connolly (1980), Ot Toro i Matystlk (1980) and Salisbury et a^. (1983) multiply this rate by a chlorophyll limitation factor, C.»CH1 a^,
•here Kj Is a half-saturation constant - 5.0 fig Chl a/1. Al al C
ThoMnn S Fitzpatrlck (1982) •ultlply this rate by an algal carbon limitation factor, ic^TAlgal^, where 1^ is a half-saturation constant - l.(tagC/l.
Nyhola (1978) utilizes a sediment release constant which is multiplied by total sedimentation of algae and detritus.
•••Model documentation values.
-------
TABLE 5-6. RATE COEFFICIENTS FOR SILICA TRANSFORMATIONS
ro
en
Particulate _
Silica
K
0.0175
0.1
0.04
0.03
0.003
0.01
0.04
0.005
^ Dissolved
Silica
8
1.08
1.08
(linear)
(linear)
1.020
1.020
1.047
1.08
Sediment + Dissolved
Silica Silica References
K 8
Thomann et^ al_. (1979)
Di Toro & Connolly (1980)
Scavia (1980)
Canale ert aj_. (1976)
0.005 1.047 Tetra Tech (1980)
0.001 1.020 Bowie et al_. (1980)
0.0015 1.047 Porcella et. al. (1983)
Bierman et al_. (1980)
-------
5.9 ALGAL UPTAKE
Two major approaches are used to simulate nutrient uptake by algae in
water quality models. The most common method is the fixed stoichiometry
approach in which the nutrient composition of the algae is assumed to remain
constant. Under this assumption, the nutrient uptake rates are equal to the
algal gross growth rate times the corresponding nutrient fractions of the
algal cells:
Vs = «s M A (5-14)
where V = uptake rate for nutrient S, mass/volume-time
a = nutrient fraction of algal cells, mass nutrient/mass algae
fji = gross growth rate of algae, I/time
A = algal concentration, mass/volume
This formulation is used in all fixed stoichiometry models. Typical values
of the nutrient compositions of algae are given in Tables 6-2 to 6-4 of
Chapter 6. Algal growth formulations and the corresponding model
coefficients are discussed in Section 6.4 of Chapter 6.
The second approach to modeling nutrient uptake is the variable
stoichiometry approach. In this method, the internal nutrient composition
of the algal cells varies with time depending on the external nutrient
concentrations in the water column and the relative rates of nutrient uptake
and algal growth. The uptake rate depends on the difference between the
internal nutrient concentration in the algal cells and the external
concentration in the water. The internal concentration of each nutrient is
assumed to range between a minimum stoichiometric requirement (called the
minimum cell quota or subsistence quota) and some maximum internal
concentration. In general, the uptake rate increases both as the external
nutrient concentration increases and as the internal nutrient concentration
decreases toward the minimum cell quota. However, the uptake rate decreases
as the internal concentration approaches the maximum internal level,
regardless of the external concentration in the water.
268
-------
In contrast to fixed stoichiometry models, the uptake formulations
used in variable stoichiometry models vary from model to model. Some models
even use different formulations for different nutrients. Variable
stoichiometry formulations for nutrient uptake are discussed in
Section 6.4.4.3 of Chapter 6, since nutrient uptake is an integral part of
the algal growth formulations in variable stoichiometry models. The major
formulations are given in Equations (6-63) to (6-67).
5.9.1 Ammonia Preference Factors
Since algae use two forms of nitrogen, ammonia and nitrate, during
uptake and growth, many models use ammonia preference factors in the uptake
formulations to account for the fact that algae tend to preferentially
uptake ammonia over nitrate. Ammonia preference factors are generally used
in fixed stoichiometry models when both ammonia and nitrate are simulated.
In this case, the 'uptake equations for ammonia and nitrate become:
and
where VMM = ammonia uptake rates mass/volume-time
NH3
VMH = n"itrate uptake rate, mass/volume-time
M3
/3N,, = ammonia preference factor
O
a.. = nitrogen fraction of algal cells
Ammonia preference factors are generally not needed in variable
stoichiometry models since separate formulations with different coefficients
can be used to distinguish between ammonia and nitrate uptake rates.
The ammonia preference factor .,„ partitions the nitrogen uptake
w
required for a given amount of algal growth between ammonia and nitrate.
The preference factor can range from 0 to 1, with 1 corresponding to a
269
-------
situation in which all the nitrogen requirements are obtained from ammonia
uptake, and 0 corresponding to a situation in which all the nitrogen is
obtained from nitrate. The value of the preference factor is generally a
function of the ammonia and nitrate concentrations in the water.
The simplest form of the ammonia preference factor assumes there is
no preference for either form of nitrogen and partitions the uptake
according to the relative proportions of ammonia and nitrate in the water:
NH3
= (5-17)
where NH.-, = ammonia concentration, mass/ volume
NO., = nitrate concentration, mass/volume
This approach is used in EXPLORE-1 (Baca et al. , 1973), LAKECO (Chen and
Orlob, 1975), WQRRS (Smith, 1978), CE-QUAL-R1 (WES, 1982), EAM (Tetra Tech,
1979), ESTECO (Brandes, 1976), and earlier versions of WASP (Thomann et a!.,
1975).
Other models which assume there is a preference for ammonia uptake have
used the following formulations for the preference factor:
yi NH3
(5-18)
3 + N03
NH.,
3 (5-19)
NH3
(5-20)
VNfl+M- V I NO
/•) Mil .3 T (i I •)] MU-j
o o o o
(5-21)
where y,, y2, y3, y. = coefficients in ammonia preference factor
formulations
270
-------
Equation (5-18) is used in SSAM IV (Grenney and Kraszewski, 1981) and Scavia
e_t al. (1976), Equation (5-19) in an early Lake Erie WASP model by Di Toro
et al. (1975), Equation (5-20) in AQUA-IV (Baca and Arnett, 1976) and Canale
e;t al. (1976), and Equation (5-21) in more recent versions of WASP by
Thomann and Fitzpatrick (1982) and O'Connor et al. (1981).
5.10 EXCRETION
Nutrient excretion by algae and zooplankton is one of the major
components of nutrient recycling. In almost all models, nutrient excretion
is modeled as the product of the respiration mass flux and the nutrient
stoichiometry of the organisms. The equations for algal excretion and
zooplankton excretion are:
and
esa ' "sa ra
esz - asz rz Z (5-23)
where e = algal excretion rate of nutrient S, mass/volume-time
sa
e = zooplankton excretion rate of nutrient S9 mass/volume-time
a = nutrient fraction of algal cells, mass nutrient/mass algae
sa
a = nutrient fraction of zooplankton, mass nutrient/mass
zooplankton
r = algal respiration rate, I/time
a
r = zooplankton respiration rate, I/time
A = algal concentration, mass/volume
Z = zooplankton concentration, mass/volume
The excretion formulations for other organisms such as fish or benthic
animals is the same as for zooplankton. Respiration rate formulations for
algae and zooplankton are discussed in Section 6.5 (Chapter 6) and 7.4
(Chapter 7), respectively. The nutrient compositions of algae are presented
in Tables 6-2 to 6-4 of Chapter 6. The nutrient compositions of zooplankton
are typically assumed to be the same as for algae in fixed stoichiometry
models so that nutrient mass is conserved as biomass cycles through the food
web.
-------
5.11 SEDIMENT RELEASE
Three major approaches have been used to simulate nutrient release from
the sediments in water quality models. The simplest approach is to specify
an areal flux from the bottom in the mass balance equations for dissolved
nutrients. This technique is commonly used in river models and in models
which do not dynamically simulate sediments as a separate constituent (e.g.,
QUAL-II (Roesner et al., 1981), DOSAG3 (Duke and Masch, 1973), and HSPF
(Johanson et, aj_._, 1980)). Sediment release rates are highly site-specific,
and are determined largely by model calibration of the dissolved nutrients.
The second approach is to model sediment nutrients as a dynamic pool
using a mass balance equation such as Equation (5-4). In this method,
nutrients are released according to a first-order decay rate:
Sed (5-24)
where R = sediment release rate of nutrient S, mass/volume-time
a = stoichiometric ratio of nutrient per mass organic sediment
K . = organic sediment decay rate, I/time
Sed = concentration of organic sediment, mass/volume
The organic sediment pool increases as algae and suspended organic detritus
settle to the bottom, and decreases as the sediment decomposes. This
approach is used in LAKECO (Chen and Orlob, 1975), Chen et al. (1975), WQRRS
(Smith, 1978), CE-QUAL-R1 (WES, 1982), EAM (Tetra Tech, 1979), and ESTECO
(Brandes, 1976). In some models, a fraction of the settled particulates is
assumed to be refractory and unavailable for mineralization.
The third approach to modeling sediment release uses a more complex
mechanistic approach in which: 1) organic sediments undergo the same decay
sequences as particulate organics in the water column but with the decay
products going to the interstitial water rather than the overlying water,
and 2) the nutrients in the interstitial waters diffuse to the overlying
water at a rate depending on the concentration gradient between the
272
-------
interstitial water and overlying water. This approach is used in some
versions of WASP (e.g., Di Toro and Connolly, 1980; Thomann and Fitzpatrick,
1982). A few models also include denitrification in the transformation
reactions.
Nyholm (1978) simulates sediment release dynamically without actually
modeling sediments by assuming the release rates equal the product of a
temperature dependent coefficient times the sedimentation rates of algal and
detrital nutrients to the bottom.
5.12 SUMMARY
Carbon, nitrogen, phosphorus, and silicon are the major growth limiting
nutrients included in water qua'lity models. Nitrogen is also important
because of the effects of nitrification on dissolved oxygen dynamics and
because of ammonia toxicity. All nutrients recycle continuously in the
water column between particulate and sediment forms, dissolved organic
forms, dissolved inorganic forms, and biotic forms. The important processes
are decomposition of organic particulates and sediments, decay of dissolved
organic to inorganic forms, chemical transformations such as nitrification,
photosynthetic uptake of dissolved inorganic forms, and soluble and
particulate excretion by aquatic organisms. Denitrif ication and nitrogen
fixation are also important in some situations.
First-order kinetics are used in almost all models to describe the
various decay processes and transformations. The exponential Arrhenius or
van't Hoff relationship is used to adjust the rate coefficients for
temperature effects. Some of the processes are modified by Michaelis-Menten
type saturation kinetics in a few models. Uptake and excretion are based on
algal growth rates and algal and zooplankton respiration rates combined with
the nutrient stoichiometries of the organisms. More complex formulations
are used for nutrient uptake in variable stoichiometry models. Sediment
release rates are usually modeled either by specifying a nutrient flux or
modeling sediments as a nutrient pool subject to first-order decay. A few
models use more complex formulations which include decay reactions in the
273
-------
Interstitial waters and diffusion between the interstitial waters in the
sediment and the overlying water column.
5.13 REFERENCES
Baca, R.G., W.W. Waddel, C.R. Cole, A. Brandstetter, and D.B. Clearlock.
1973. EXPLORE-I: A River Basin Water Quality Model. Battelle, Inc.,
Pacific Northwest Laboratories, Richland, Washington.
Baca, R.G. and R.C. Arnett. 1976. A Limnological Model for Eutrophic Lakes
and Impoundments. Battelle, Inc., Pacific Northwest Laboratories, Richland,
Washington.
Bierman, V.J., Jr. 1976. Mathematical Model of the Selective Enhancement
of Blue-Green Algae by Nutrient Enrichment. In: Modeling Biochemical
Processes in Aquatic Ecosystems. R.P. Canale (¥3.), Ann Arbor Science
Publishers, Ann Arbor, Michigan, pp. 1-31.
Bierman, V.J., Jr., D.M. Dolan, E.F. Stoermer, J.E. Gannon, and V.E. Smith.
1980. The Development and Calibration of a Multi-Class Phytoplankton Model
for Saginaw Bay, Lake Huron. Great Lakes Environmental Planning Study.
Contribution No. 33. Great Lakes Basin Commission, Ann Arbor, Michigan.
Bloomfield, J.A., R. A. Park, D. Scavia, and C.S. Zahorcak, 1973. Aquatic
Modeling in the Eastern Deciduous Forest Biome. U.S. International
Biological Program. ln_: Modeling the Eut^rophi cation Process.
E.J. Middlebrook, D.H. Talkenborg, and I.E. Maloney, (eds.). Utah State
University, Logan, Utah. pp. 139-158.
Bowie, G.L., C.W. Chen, and D.H. Dykstra. 1980. Lake Ontario Ecological
Modeling, Phase III, Tetra Tech, Inc., Lafayette, California. For National
Oceanic and Atmospheric Administration, Great Lakes Environmental Research
Laboratory, Ann Arbor, Michigan.
Brandes, R.J. 1976. An Aquatic Ecologic Model for Texas Bays and
Estuaries. Water Resources Engineers, Inc., Austin, Texas. For the Texas
Water Development Board, Austin, Texas.
"%
Brandes, R.J. and F.D. Masch. 1977. ESTECO--Estuarine Aquatic Ecologic
Model: Program Documentation and User's Manual. Water Resources Engineers,
Inc., Austin, Texas. For the Texas Water Development Board, Austin, Texas.
Canale, R.P., L.M. Depalma, and A.H. Vogel. 1975. A Food Web Model for
Lake Michigan. Part 2 - Model Formulation and Preliminary Verification.
Tech. Report 43, Michigan Sea Grant Program, MICHU-SG-75-201.
Canale, R.P., L.M. Depalma, and A.H. Vogel. 1976. A Plankton-Based Food
Web Model for Lake Michigan. In: Modeling Biochemical Processes in Aquatic
Ecosystems. R.P. Canale (ed.). Ann Arbor Science Publishers, Ann Arbor,
Michigan, pp. 33-74.
274
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Chen, C.W. and G.T. Orlob. 1972. Ecologic Simulations of Aquatic
Environments. Water Resources Engineers, Inc., Walnut Creek, California.
For the Office of Water Resources Research.
Chen, C.W. and C.T. Orlob. 1975. Ecological Simulation for Aquatic
Environments. In: Systems Analysis and Simulation in Ecology, Vol. 3.
B.C. Patten (ed.). Academic Press, Inc., New York, New York. pp. 476-588.
Chen, C.W., M. Lorenzen, and D.J. Smith. 1975. A Comprehensive Water
Quality-Ecological Model for Lake Ontario. Tetra Tech, Inc., Lafayette,
California. For National Oceanic and Atmospheric Administration, Great
Lakes Environmental Research Laboratory, Ann Arbor, Michigan.
Collins, C.D. and J.H. Wlosinski. 1983. Coefficients for Use in the U.S.
Army Corps of Engineers Reservoir Model, CE-QUAL-R1. U.S. Army Corps of
Engineers, Waterways Experiment Station, Vicksburg, Mississippi.
Di Toro, D.M., D.J. O'Connor, R.V. Thomann, and J.L. Mancini. 1975.
Phytoplankton-Zooplankton Nutrient Interaction Model for Western Lake Erie.
In: Systems Analysis and Simulation in Ecology, Vol. III. B.C. Patton
(ed.). Academic Press, Inc., New York, New York. 423 pp.
Di> Toro, D.M. and J.P. Connolly. 1980. Mathematical Models of Water
Quality in Large Lakes. Part II: Lake Erie. U.S. Environmental Protection
Agency, Ecological Research Series. EPA-600/3-3-80-065.
Di Toro, D.M. and W.F. Matystik, Jr. 1980. Mathematical Models of Water
Quality in Large Lakes. Part I: Lake Huron and Saginaw Bay.
U.S. Environmental Protection Agency, Ecological Research Series. EPA-
600/ 3-80-056.
Duke, J.H., Jr. and F.D. Masch. 1973. Computer Program Documentation for
the Stream Quality Model DOSAG3, Vol. I. Water Resources Engineers, Inc.,
Austin, Texas. For U.S. Environmental Protection Agency, Systems
Development Branch, Washington, D.C.
Emerson, K., R.C. Russo, R.E. Lund, and R.V. Thurston. 1975. Aqueous
Ammonia Equilibrium Calculations: Effect of pH and Temperature. J. Fish.
Res. Board Can., 32(12):2379-2383.
Feigner, K.D. and H. Harris. 1970. FWQA Dynamic Estuary Model,
Documentation Report. U.S.D.I., F.W.Q.A., Washington, D.C.
Grenney, W.J. and A.K. Kraszewski. 1981. Description and Application of
the Stream Simulation and Assessment Model: Version IV (SSAM IV).
Instream Flow Information Paper. U.S. Fish and Wildlife Service, Fort
Collins, Colorado, Cooperative Instream Flow Service Group.
Harleman, D.R.F., J.E. Dailey, M.L. Thatcher, T.O. Najarian, D.N. Brocard,
and R.A. Ferrara. 1977. User's Manual for the M.I.T. Transient Water
Quality Network Model — Including Nitrogen-Cycle Dynamics for Rivers and
Estuaries. R.M. Parsons Laboratory for Water Resources and Hydrodynamics,
275
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Massachusetts Institute of Technology, Cambridge, Massachusetts. For U.S.
Environmental Protection Agency, Corvallis, Oregon. EPA-600/3-77-010.
Johanson, R.C., J.C. Imhoff, and H.H. Davis. 1980. User's Manual for
Hydrological Simulation Program - Fortran (HSPF). Hydrocomp, Inc., Mountain
View, California. For U.S. Environmental Protection Agency, Athens.
Georgia. EPA-600/9-80-015.
Jorgensen, S.E. 1976. A Eutrophication Model for a Lake. Ecol. Modeling,
2:147-165.
Jorgensen, S.E., H. Mejer, and M. Friis. 1978. Examination of a Lake
Model. Ecol. Modeling, 4:253-278.
Lehman, J.T., D.B. Botkin, and G.E. Likens. 1975. The Assumptions and
Rationales of a Computer Model of Phytoplankton Population Dynamics.
Limnol. and Oceanogr., 20(3):343-364.
Nyholm, N. 1978. A Simulation Model for Phytoplankton Growth and Nutrient
Cycling in Eutrophic, Shallow Lc :s. Ecol. Modeling, 4:279-310.
O'Connor, D.J., J.L. Mancini, and J.R. Guerriero. 1981. Evaluation of
Factors Influencing the Temporal Variation of Dissolved Oxygen in the
New York Bight, Phase II. Manhattan College, Bronx, New York.
Park, R.A., C.D. Collins, C.I. Connolly, J.R. Albanese, and B.B. MacLeod.
1980. Documentation of the Aquatic Ecosystem Model MS.CLEANER. Rensselaer
Polytechnic Institute, Center for Ecological Modeling, Troy, New York. For
U.S. Environmental Protection Agency, Environmental Research Laboratory,
Office of Research and Development, Athens, Georgia.
Porcella, D.B., T.M. Grieb, G.L. Bowie, T.C..Ginn, and M.W. Lorenzen. 1983.
Assessment Methodology for New Cooling Lakes, Vol. 1: Methodology to Assess
Multiple Uses for New Cooling Lakes. Tetra Tech, Inc., Lafayette,
California. For Electric Power Research Institute. Report EPRI EA-2059.
Raytheon Company, Oceanographic & Environmental Services. 1974. New England
River Basins Modeling Project, Vol. Ill - Documentation Report, Part 1 -
RECEIV-II Water Quantity and Quality Model. For Office of Water Programs,
U.S. Environmental Protection Agency, Washington, D.C.
Roesner, L.A., P.R. Giguere, and D.E. Evenson. 1981. Computer Program
Documentation for the Stream Quality Model QUAL-II. U.S. Environmental
Protection Agency, Athens, Georgia. EPA 600/9-81-014.
Salas, H.J. and R.V. Thomann. 1978. A Steady-State Phytoplankton Model of
Chesapeake Bay. Journal WPCF, 50(12):2752-2770.
Salisbury, D.K., J.V. DePinto, and T.C. Young. 1983. Impact of Algal-
Available Phosphorus on Lake Erie Water Quality: Mathematical Modeling.
For U.S. Environmental Protection Agency, Environmental Research Laboratory,
Duluth, Minnesota.
276
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Scavia, D., B.J. Eadie, and A. Robertson. 1976. An Ecological Model for
Lake Ontario - Model Formulation, Calibration, and Preliminary Evaluation.
Natl. Ocean, and Atmos. Admin., Boulder, Colorado. NOAA Tech. Rept. ERL
371-GLERL 12.
Scavia, D. and R.A. Park. 1976. Documentation of Selected Constructs and
Parameter Values in the Aquatic Model CLEANER. Ecol. Modeling, 2:33-58.
Scavia, D. 1980. An Ecological Model of Lake Ontario. Ecol. Modeling,
8:49-78.
Smith, D.I. 1978. Water Quality for River-Reservoir Systems. Resource
Management Associates, Inc., Lafayette, California. For U.S. Army Corps of
Engineers, Hydrologic Engineering Center (HEC), Davis, California, pp 210.
Stumm, W. and J.J. Morgan. 1970 (First Edition).
Wiley-Interscience. New York, New York. 583 pp.,
1981 (Second Edition).
780 pp.
Tetra Tech, Inc. 1979. Methodology for Evaluation of Multiple Power Plant
Cooling System Effects, Volume II. Technical Basis for Computations.
Tetra Tech, Inc., Lafayette, California. For Electric Power Research
Institute. Report EPRI EA-1111.
Tetra Tech, Inc. 1980. Methodology for Evaluation of Multiple Power Plant
Cooling System Effects, Volume V. Methodology Application to Prototype -
Cayuga Lake. Tetra Tech, Inc., Lafayette, California. For Electric Power
Research Institute. Report EPRI EA-1111.
Thomann, R.V., D.M. Di Toro, R.P. Winfield, and D.J. O'Connor. 1975.
Mathematical Modeling of Phytoplankton in Lake Ontario, Part 1. Model
Development and Verification. Manhattan College, Bronx, New York. For U.S.
Environmental Protection Agency, Corvallis, Oregon. EPA-600/3-75-005.
Thomann, R.V., J. Segna, and R. Winfield. 1979. Verification Analysis of
Lake Ontario and Rochester Embayment Three-Dimensional Eutrophication
Models. Manhattan College, Bronx, New York. For U.S. Environmental
Protection Agency, Office of Research and Development.
Thomann, R.V. and J.J. Fitzpatrick. 1982. Calibration and Verification of
a Mathematical Model of the Eutrophication of the Potomac Estuary.
Government of the District of Columbia, Washington, D.C.
Thurston, R.V., R.C. Russo, and K. Emerson. 1974. Aqueous Ammonia
Equilibrium Calculations. Fisheries Bioassay Laboratory, Montana State
Univ., Boseman, Montana. Technical Report No. 74-1.
U.S. Environmental Protection Agency. 1984.
Protection of Aquatic Life and Its Uses -
Standards. Washington, D.C.
Water Quality Criteria for the
Ammonia. USEPA, Criteria and
WES (Waterways Experiment Station). 1982. CE-QUAL-R1: A Numerical One-
Dimensional Model of Reservoir Water Quality, Users Manual. Environmental
277
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and Water Quality Operational Studies (EWQOS), U.S. Army Corps of Engineers,
Waterways Experiment Station, Vicksburg, Mississippi.
Willingham, W.T. 1976. Ammonia Toxicity. EPA 908/3-76-001.
278
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Chapter 6
ALGAE
6.1 INTRODUCTION
Algae are important components of water quality models for several
reasons. For example:
• Algal dynamics and nutrient dynamics are closely linked
together since nutrient uptake during algal growth is the
main process which removes dissolved nutrients from the
water, and algal respiration and decay are major components
of nutrient recycling.
• Algal processes can cause diurnal variations in dissolved
oxygen due to photosynthetic oxygen production during the
daylight combined with oxygen consumption due to algal
respiration during the night. Seasonal oxygen dynamics may
also be closely tied to algal dynamics, particularly in
highly productive stratified systems, since the respiration
and decomposition of algae which settles below the photic
zone is often a major source of oxygen depletion.
• Algae can affect pH through the uptake of dissolved CO,,
during photosynthesis and the recycling of C0_ during
respiration.
• Algae are the dominant component of the primary producers in
many systems, particularly in lakes and estuaries. Since
279
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they form the base of the food chain, they play a major role
in the dynamics of all successive trophic levels.
• Suspended algae are often a major component of turbidity.
• Algal blooms can restrict recreational uses of water,
sometimes resulting in fish kills under severe conditions.
• Algae can cause taste and odor problems in water supplies,
and filter clogging problems at water treatment facilities.
Two general approaches have been used to simulate algae in water
quality models: 1) aggregating all algae into a single constituent (for
example, total algae or chlorophyll _a), or 2) aggregating the algae into a
few dominant functional groups (for example, green algae, diatoms, blue-
greens, dinoflagellates, etc.).
The first approach is commonly used in river models since the major
focus is on short term simulations (days to weeks) where the primary
interest is the effects of algae on general water quality parameters such as
dissolved oxygen, nutrients, and turbidity. Typical examples include
QUAL-II (Roesner et .aJL, 1981; NCASI, 1982, 1983), DOSAG3 (Duke and Masch,
1973), and RECEIV-II (Raytheon, 1974). In contrast, lake and reservior
models tend to use the second approach since the focus is on long term
simulations (months to years) of eutrophication problems where seasonal
variations in different types of algae are important (Bierman, 1976; Bierman
£t aj_., 1973, 1980; Canal e et aj_. , 1975, 1976; Chen et al_. , 1975; Tetra
Tech, 1979, 1980; Park et aK, 1974, 1975, 1979, 1980; Scavia et al_., 1976;
Scavia, 1980; Lehman ^t a]_., 1975). Species-specific differences in
nutrient requirements, nutrient uptake rates, growth rates, and temperature
preference ranges result in a"seasonal succession of dominance by different
phytopl ankton groups. It is often important to distinguish these
differences in order to realistically model both nutrient dynamics and
phytopl ankton dynamics, and to predict the occurrence of specific problems
such as blue-green algal blooms. Multi-group models typically use the same
280
-------
general model formulations for all groups, but provide different coefficient
values to characterize the differences between groups.
6.2 MODELING APPROACHES
Phytoplankton dynamics are governed by the following processes:
growth, respiration and excretion, settling, grazing losses, and
nonpredatory mortality (or decomposition). A general equation which
includes all of these processes and forms the basis for almost all
phytoplankton models can be expressed as:
$ = (/i- r - ex - s - m) A - G (6-1)
where A = phytopl ankton biomass or concentration (dry weight biomass,
chlorophyll _a, or equivalent mass of carbon, nitrogen, or
phosphorus), mass or mass/volume
^ = gross growth rate, I/time
r = respiration rate, I/time
e = excretion rate, I/time
A
s = "sett!ing rate, I/time
m =, nonpredatory mortality (or decomposition) rate, I/time
G = loss rate due to grazing, mass/time or mass/volume-time
This equation is appropriate when phytopl ankton are modeled in terms of
either biomass or nutrient equivalents (carbon, nitrogen, phosphorous,
etc.). However, if phytoplankton are expressed in terms of cell numbers,
the growth rate is replaced with the cell division rate, and the respiration
and excretion terms are omitted since they pertain to changes in biomass
rather than cell numbers. The resulting equation is:
dA
dT ' for, - S - m) An - Gn ^
where A = phytoplankton cell numbers, numbers or numbers/volume
fi = cell division rate, I/time
281
-------
G = 1 o s s rate due to grazing, numbers/time or
numbers/volume-time
The cell division rate in Equation .(6-2) is assumed to be a continuous
process although in reality cell division is a discrete event which is often
expressed in terms of the number of divisions per day, n.. The continuous
division rate u. is related to the discrete rate n , by u = n . In2.
r n a n Q
Most models express phytoplankton in terms of biomass (or nutrient or
chlorophyll a equivalents) rather than cell numbers. This facilitates the
modeling of both nutrient cycles and food web dynamics since it allows a
more direct linkage between the phytoplankton equations and the mass balance
equations for both nutrients and higher trophic levels such as zooplankton
and fish. Phytoplankton cell numbers are used in a few models whose focus
is restricted to phytoplankton dynamics (e.g., Lehman et aJL, 1975; Cloern,
1978).
The major differences between different phytoplankton models are:
1) the number of phytoplankton groups modeled, 2) the specific formulations
used for each process, and 3) the manner in which the various processes and
corresponding terms in Equations (6-1) or (6-2) are combined. Some of the
basic features of different phytoplankton models are compared in Table 6-1.
The specific process formulations are discussed in later sections.
Many models combine several of the processes in Equation (6-1) into a
single term, thereby simplifying the equation. For example, respiration and
excretion are usually combined into a single respiration term. Respiration
is often combined with growth so that the growth rate n represents the net
growth rate, rather than the gross growth rate as in Equation (6-1). This
is consistent with net growth rates typically reported in the literature
from laboratory cultures. Some models combine respiration with the other
loss terms to give a net loss rate which includes respiration and mortality.
Other models combine grazing and nonpredatory mortality into a single
mortality term, particularly when algal grazers are not modeled explicitly.
282
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TABLE 6-1. GENERAL COMPARISON OF ALGAL MODELS
Model
(Author)
AQUA- IV
CE-QUAL-R1
CLEAN
CLEANER
MS. CLEANER
DEM
OOSAG3
EAM
ESTECO
EXPLORE-1
HSPF
LAKECO
MIT Network
QUAL-II
RECEIV-II
SSAM IV
WASP
WQRRS
Bierman
Canal e
Jorgensen
Lehman
Nyholm
Scavia
Number of Groups
Phyto-
ilankton
1
2
2
3
4
1
1
4
2
1
1
2
1
1
1
1
2
2
5
4
1
5
1
5
Attached Zoo-
Algae plankton
1
1
1 3
1 3
1 5
1 3
1
1
1 1
1
1
1
2
2 1
2
9
1
6
Growth
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
"recesses
Respir-
ation
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Computed
Settling
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Separately in
Nonpredator^
Mortality
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Model
Predatory
Mortality
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Algal Units
Dry Wt. Other Cell
Biomass Chi a Carbon Nutrient Numbers
X
X
X
X
X
X
X
X
X
X
X
X
N
X
X
X
X X
X
X
X
X
X
X
X
Reference
Baca & Arnett (1976)
WES (EWQOS) (1982)
Bloomfield et al_. (1973)
Scavia & Park (1976)
Park et al_. (1980)
Feigner & Harris (1970)
Duke & Masch (1973)
Tetra Tech (1979, 1980)
Brandes & Masch (1977)
Baca et al_. (1973)
Johanson et al . (1980)
Chen & Orlob (1975)
Harlerean et ah (1977)
Roesner et ah (1981)
Raytheon (1974)
Grenney & Kraszewski (1981
Di Toro et aK (1981)
Smith (1978)
Bierman et a]_. (1980)
Canale et al.. (1975. 1976)
Jorgensen (1976)
Lehman et^ ah (1975)
Nyholm (1978)
Scavia et a]_. (1976)
ro
co
CO
-------
Because of these variations, it is very important to understand the
assumptions of a particular model when selecting coefficients. Care must be
taken both when extracting values from one model and applying them to
another, or when using experimental measurements reported in the literature.
For the latter case, the experimental conditions should be checked to make
sure they are consistent with the assumptions of the model. If they are
different, the appropriate adjustments should be made.
Attached algae (periphyton) and aquatic macrophytes have the same
growth requirements as phytoplankton (light and nutrients) and are subject
to the same basic processes of growth, respiration and excretion,
grazing,and nonpredatory mortal ity. Therefore, they are usually modeled
using the same general approach and process formulations as phytoplankton,
although the specific values of the model coefficients will vary. The major
differences are: 1} periphyton and macrophytes are associated with the
bottom substrate and are expressed in terms of area! densities rather than
volumetric densities or concentrations; 2) periphyton and macrophytes do not
have settling losses, but instead they have additional losses due to
sloughing or scouring from the bottom substrate; 3) periphyton and
macrophytes are not subject to hydrodynamic transport; and 4) macrophytes
use nutrients from the sediments and interstitial waters rather than
nutrients in the water column. The general model equation for attached
algae and macrophytes can be expressed as:
dA
^- (,-r-ex-Sl -m)Ab-Gb (6-3)
where Ab = periphyton or macrophyte biomass (dry weight biomass,
chlorophyll a, or equivalent mass of carbon, nitrogen, or
phosphorus), mass or mass/area
S.j = sloughing or scouring rate, I/time
Gb = loss rate due to grazing, mass/time or mass/area-time
Benthic algae or macrophytes are included in only a few models such as CLEAN
(Park £_t aU, 1974), CLEANER (Park et aH.. , 1975), MS.CLEANER
284
-------
(Park et a].., 1980), EAM (Tetra Tech, 1979, 1980), WQRRS (Smith, 1978), HSPF
(Johanson ^t aj_. , 1980), SSAM IV (Grenney and Kraszewski, 1981), and in
Canale and Auer (1982) and Scavia et aj_. (1975).
6.3 CELL COMPOSITION
The majority of models express algae and other biological constituents
as either dry weight biomass (Chen and Orlob, 1972; Chen _et .al_., 1975; Park
et a_L, 1974, 1975, 1979, 1980; Tetra Tech, 1979, 1980; Brandes and Masch,
1977; Smith, 1978; Johanson et _al_., 1980; Grenney and Kraszewski, 1981;
Bierman jst _al_., 1973, 1980; Jorgensen, 1976; Jorgensen et aj_., 1978; Nyholm,
1977, 1978) or carbon (Baca and Arnett, 1976; Baca _et a]_. , 1973, 1974;
Canale jrt aj_., 1975, 1976; Scavia _et _al_., 1976; Scavia, 1980). Nitrogen or
phosphorus have also been used in a few models, which focus on a single
nutrient cycle and assume that particular nutrient always limits algal
growth (Najarian and Harleman, 1975; Harleman et_ £]_., 1977). Some models
express phytoplankton as chlorophyll a since both field measurements and
water quality standards are often' reported in these units (Roesner et al.,
1981; Duke and Masch, 1973; Raytheon, 1974; Di Toro et al_., 1971, 1977;
Di Toro and Matystik, 1980; Di Toro and Connolly, 1980; O'Connor et al_.,
1975; Thomann et al_., 1975., 1979).
Dry weight biomass is related to the major nutrients (carbon, nitrogen,
and phosphorus) and chlorophyll d. through stoichiometric ratios which give
the ratios of each nutrient to the total biomass. Typical algal nutrient
compositions are summarized in Tables 6-2 to 6-4. Algae expressed as
carbon, nitrogen, phosphorus, or chlorophyll a can be converted to dry
weight biomass or any of the other units by using the stoichiometric ratios
presented in the tables.
Most conventional water quality models assume the nutrient compositions
of the cells and the resulting stoichiometric ratios are constant. In
reality, cell stoichiometry varies with species, cell size, physiological
condition, and recent environmental conditions (external nutrient
concentrations, light, and temperature), although it is often assumed
285
-------
Diatoms
Green Algae
TABLE 6-2. NUTRIENT COMPOSITION OF ALGAL CELLS
- PERCENT OF DRY WEIGHT BIOMASS
Percent of Dry Weight Biomass
Algal Type
Total
Phytoplankton
C
40. -50.
N P
8. -9. 1.5
Si Chi a_ References
Tetra Tech (1976)
40.
60.
40.-50.*
7.2
1.0
6.1
7.-9.*
8.-9.*
0.88
1.-1.2*
1.2-1.5*
50.* 9.* 1.2*
42.9-70.2** 0.6-16.** 0.16-5.**
1.5-9.3** 0.08-1.17**
40.
40.
7.2
7.2
1.0
19.-50.** 2.7-5.9** 0.4--2.0**
20.-53**
1.0
35.-48.** 6.6-9.1** 2.4-3.3**
15.-74.**
20. -24.
50.
Chen & Wells (1975, 1976)
Tetra Tech (1980)
Bowie et al. (1980)
PorcelTa et al_. (1983)
2. Bierman (1976)
Nyholm (1978)
Jorgensen (1976)
Smith (1978)
5. -10.* Roesner et al . (1980)
Duke & MascTT(1973)
Brandes (1976)
Baca & Arnett (1976)
Jorgensen (1979)
Tetra Tech (1980)
Bowie et al. (1980)
PorcelTa ef al. (1983)
Bierman e_t *\_. ("1976)
Di Toro et al. (1971)
Bierman et al_. (1980)
Tetra Tech (1980)
Bowie et al. (1980)
Porcella et al_. (1983)
Di Toro et ah (1971)
Bierman et al . (1980)
Blue-green
Algae
40.
7.2
1.0
28.-45.** 4.5-5.8** 0.8-1.4**
38.-39.**
1.-3.**
0.25**
Tetra Tech (1980)
Bowie et al. (1980)
PorcelTa et. al. (1983)
Di Toro et al_. (1971)
Bierman e_t a]_. (1980)
Baca & Arnett (1976)
Jorgensen (1979)
236
-------
TABLE 6-2. (continued)
Percent of Dry Weight Biomass
Algal Type
N
Si
Chl-a
References
Dinoflagel lates
Flagellates
Benthlc Algae
37.-47** 3.3-5.0** 0.6-1.1**
10.-43.**
7.2
40.
7.2
1.0
40.
29.-67.**
Chrysophytes 35.-45.** 7.8-9.0** 1.2-3.0
1.0
40.-50.* 7.-9.* 1.-1.2*
275. O'Connor et a]_. (1981}
Di Toro et al. (1971)
Bierman et al. (1980)
Tetra Tech (1980)
Bowie et al. (1980)
Porcella et ah (1983)
Bierman et al. (1980)
Jorgensen (1979)
Tetra Tech (1980)
Bowie et al. (1980)
PorcelTa e_t al- (1983)
Smith (1978)
*Hode1 documentation values.
**Literature values.
constant for modeling purposes. Several of the more recent algal models,
however, have included variable cell stoichiometry in their formulations to
simulate processes such as luxury uptake and storage of nutrients (Bierman
et_al_., 1973, 1980; .Bierman, 1976; Lehman et _aj_., 1975; Jorgensen, 1976;
Jorgensen et al_., 1978; Nyholm, 1977, 1978; Park et _§_]_., 1979, 1980; Canale
and Auer, 1982). These models are discussed later with reference to
phytoplankton growth and nutrient uptake formulations.
6.4 GROWTH
Algal growth is a function of temperature, light, and nutrients. The
major growth limiting nutrients are assumed to be phosphorus, nitrogen, and
carbon, with the addition of silicon for diatoms. Other essential
micronutrients such as iron, manganese, sulfur, zinc, copper, cobalt,
287
-------
molybdenum, and vitamin B,2 may also limit growth under conditions of
restricted availability (particularly in oligotrophic systems). However,
these effects are generally not included in models since micronutrients
are usually not simulated. The algal growth rate formulations used in
almost all models can be expressed in general functional form as:
=/zmax(T
ref
) f(T)
where
^ ref'
f(T)
T
f(L,P,N,C,Si)
L
P
algal growth rate, I/time
maximum growth rate at a particular reference
temperature Tref under optimal conditions of
saturated light intensity and excess nutrients,
I/time
temperature function for growth
temperature, C
growth limiting function for light and nutrients
light intensity
available inorganic phosphorus concentration,
mass/volume
TABLE 6-3. NUTRIENT COMPOSITION OF ALGAL CELLS
- RATIO TO CARBON
Algal Type
Total
Phytoplankton
N
C
0.17 - 0.25
0.18
P
C
0.025
0.024
Si
C References
Thomann & Fitzpatrick (1982)
Di Toro et al_. (1971)
Scavia et al . (1976)
Diatoms
0.2
0.05 - 0.17** 0.024 - 0.24**
0.05 - 0.43** 0.025 - 0.05**
0.18 0.024 0.6
0.067 - 0.21** 0.003 - 0.14** 0.06-0.77**
Scavia (1980)
Canale et a]_. (1976)
Baca & Arnett (1976)
Jorgensen (1979)
Scavia (1980)
Jorgensen (1979)
**Literature Values.
280
-------
TABLE 6-4. NUTRIENT COMPOSITION OF ALGAL CELLS
- RATIO TO CHLOROPHYLL a
Algal Type
N
CFTa
P
ChTa
Si
WTa
References
Total
Phytoplankton
Diatoms
Green Algae
Blue-green
Algae
Dinoflagellates
50.-100. 7.-15. 0.5-1.0
7.2
25.-112.** 7.-29.**
10.-100.** 2.7-9.1**
0.5
0.63
1.0**
100. 10.-15. 0.5-1.0 40.-50.
0.5
50.-200.*
18.-500** 2.2-74.6** 0.27-19.2** 2.4-50.7**
25.-100.*
14.-67.*
275.
19.3
Thomann et al. (1975, 1979)
O'Connor et al. (1981)
Di Toro & Matystik (1980)
Di Toro & Connolly (1980)
Salas & Thomann (1978)
Salisbury et al_. (1983)
Larsen et aj_. (1973)
Jorgensen (1979)
O'Connor ejt al_. (1981)
Di Toro & Connolly (1980)
Di Toro & Matystik (1980)
Thomann e^ al. (1979)
Salisbury e^ al_. (1983)
Baca & Arnett (1976)
Di Toro et al- (1971)
Baca & Arnett (1976)
Baca & Arnett (1976)
O'Connor et al. (1981)
*Model documentation values.
**Literature values.
N
C
Si
= available inorganic nitrogen concentration,
mass/volume
= available inorganic carbon concentration,
mass/volume
= available inorganic silicon concentration,
mass/volume
Note that the growth limiting function f(L,P,N,C,Si) is simplified in
many models by excluding some of the nutrients. For example, silicon is
289
-------
included only in models which simulate diatoms as a separate algal group
(Bierman _et _§]_., 1973, 1980; Bierman, 1976; Canale e_t a]_., 1975, 1976;
Scavia £t al_., 1976; Scavia, 1980; Chen et_ _al_., 1975; Tetra Tech, 1979,
1980; Lehman _et _a]_, 1975; Park et _aj., 1979, 1980; Di Toro and Connolly,
1980). Carbon is frequently omitted since it is often available in excess
relative to phosphorus and nitrogen (Bierman e_t jj_., 1980; Scavia et al.,
1976; Nyholm, 1978; Canale* et aj_., 1975, 1976; Baca and Arnett, 1976;
Di Toro and Matystik, 1980). Some models include only one nutrient,
phosphorus or nitrogen, and assume that nutrient,is limiting at all times
for the particular system under consideration (Najarian and Harleman, 1975;
Canale and Auer, 1982).
It should also be noted that the nutrient concentrations in the growth
limiting function f(L,P,N,C,Si) correspond to the "external" nutrient
concentrations in the water for some models, and to the "internal" nutrient
concentrations in the algal cells for other models. These distinctions will
be discussed in more detail below.
6.4.1 Temperature Effects On Maximum Growth Rates
The quantity f* ^(T^ J f(T) in Equation (6-4) represents the effects
fflaX i GT
of temperature variations on maximum algal growth rates under conditions of
optimum light and nutrients. The maximum growth rate u_av must be specified
tHaX
at a reference temperature T - which is consistent with the particular
temperature function f(T) used in the model. The reference temperature may
correspond to 20°C, optimum temperature conditions, or some other
temperature, depending on the form of the temperature function. Therefore,
maximum growth rate coefficients obtained from one model may have to be
adjusted before using the coefficients in another model which has a
different temperature adjustment function. Maximum growth rates for algae
are tabulated in Table 6-5, along with the corresponding reference
temperatures.
Although numerous temperature adjustment functions have been used to
model a.lgae, most of them fall into one of three major categories
290
-------
TABLE 6-5. ALGAL MAXIMUM GROWTH RATES
Algal Type
Maximum Growth
Reference,
Rate (I/day) Temperature ( C)
References
Total
Phytoplankton
1.3 2.5
1. 2.5
1. 2.
1.5
1. - 2.7
1.5
1.8 2.53
2.4
0.2 8.*
1. 3.*
1. 3.*
0.2 8.*
1.5 2.*
0.58 3.**
20°C
20°C
20°C
20°C
'opt
20°C
opt
'opt
20°C
'opt
20°C
20°C
20°C
20°C
O'Connor et al_. (1975, 1981)
Thomann et al_. (1974, 1975, 1979)
Thomann & Fitzpatrick (1982)
Di Toro & Connolly (1980)
Di Toro & Matystik (1980)
Di Toro et al_. (1971, 1977)
Salas & Thomann (1978)
Salisbury et al_. (1983)
Chen (1970)
Chen & Orlob (1975)
Chen & Wells (1975, 1976)
Tetra Tech (1976)
Battelle (1974)
Grenney & Kraszewski (1981)
Scavia & Park (1976)
Younjgberg (1977)
Nyholm (1978)
Jorgensen (1976)
Jorgensen ejt al_. (1978)
Larsen et al_. (1973)
Baca & Arnett (1976)
Smith (1978)
Roesner et. al_. (1980)
Duke & Masch (1973)
Grenney & Kraszewski (1981)
Brandes (1976)
Jorgensen (1979)
Diatoms
2.1
2.0 2.5
2.0 - 2.1
2.1
1.6
1.8 - 2.5
3.0
20°C
'opt
20°C
25°C
10° - 14°C
'opt
Topt
29.1
Di Toro & Connolly (1980)
Thomann et al_. (1979)
Salisbury et aJL. (1983)
Tetra Tech (1980)
Bowie et al_. (1980)
PorcelTa et a]_. (1983)
Canale et al_. (1976)
Bierman (1976)
Bierman et^ al_. (1980)
Scavia et al_. (1976)
Scavia TT980)
Lehman et al. (1975)
-------
TABLE 6-5. (continued)
Maximum Growth Reference
Algal Type Rate (I/day) Temperature ( C)
1.75**
0.55 3.4**
1.1 5.0**
Green Algae 1.9
1.4
2.0 - 2.5
1.9
1.8 - 2.5
1.6
3.0
1.5 3.9**
0.7 - 2.1**
0.9 4.1**
9.0 9.2**
1.4 2.4**
1.5 3.9**
1.3 4.3**
5.65**
Blue-green Algae 0.8
0.7 - 1.0
1.6
1.4 - 1.9
1.1 - 2.0
1.1
2.5
1.6 2.5
0.41 - 0.86**
0.2 4.9**
2.0 3.9**
0.5 11.**
27°C**
20°C**
20°C**
25°C
20°C
Topt
20°C
Topt
25°C
Topt
25°C**
20° C
25°C**
39°C**
20°C**
25°C**
35°C**
40°C**
25°C
20° - 25°C
20°C
Topt
Topt
25°C
Topt
Topt
20°C**
25°C**
35°C**
40°C**
References
Di Toro et al. (1971)
Collins & Wlosinski (1983)
Jorgensen (1979)
Bierman (1976)
Bierman_£t al. (1980)
Tetra Tech (1980)
Bowie et al. (1980)
PorcelTa et al_. (1983)
Canale et a]_. (1976)
Scavia et al. (1976)
Scavia "(T980)
DePinto ejt al_. (1976)
Lehman et al_. (1975)
Di Toro et al. (1971)
Collins & Wlosinski (1983)
Jorgensen (1979)
Bierman (1976)
Bierman ejt &\_. (1980)
Canale et al. (1976)
Youngberg (1977)
Scavia & Park (1976)
Scavia (1980)
DePinto et al- (1976)
Lehman et a^ (1975)
Tetra Tech (1980)
Bowie et al. (1980)
PorcelTa et ah (1983)
Jorgensen (1979)
Collins & Wlosinski (1983)
292
-------
TABLE 6-5. (continued)
Algal Type
Dinoflagellates
Flagellates
Chrysophytes
Maximum Growth Reference
Rate (I/day) Temperature ( C)
0.2 - 0.28
2.16**
0.2 2.1**
1.6
1.2
1.5
1.5
0.4 2.9**
20°C
20°C
20°C
Topt
20°C
Topt
Topt
25°C**
References
O'Connor e_t aj_. (1981)
Di Toro et af[. (1971)
Collins & Wlosinski (1983)
Tetra Tech (1980)
Porcella et_ aj_. (1983)
Bierman et aj_. (1980)
Lehman et aj_. (1975)
Lehman et_ al. (1975)
Collins & Wlosinski (1983)
Coccolithophores
Benthic Algae
1.75 2.16"
0.5 1.5
25°C**
opt
Jorgensen (1979)
Tetra Tech (1980)
Porcella et al. (1983)
1
1
0.2
0.5
.08
.5
0.8*
1.5*
Topt
20°C
20°C
Topt
Auer and Canale (1982)
Grenney & Kraszewski (1981)
Grenney & Kraszewski (1981)
Smith (1978)
*Model documentation values.
**Literature values.
(Figure 6-1): 1) linear increases in growth rate with-temperature, 2)
exponential increases in growth rate with temperature, and 3) temperature
optimum curves in which the growth rate increases with temperature up to the
optimum temperature and then decreases with higher temperatures.
The simplest type of temperature adjustment function assumes a linear
temperature response curve above some minimum temperature T^. This
relationship can be expressed in general form as:
293
-------
f(T) =
T - Tmin
ref ~ min
(6-5)
1.
'Tref " Tmin
T -
min
ref min
T + j8
where T . = lower temperature limit at which the growth rate is zero,
°C
Exponential /
Curve .'
Temperature
Optimum
Curve
TEMPERATURE,°C
Figure 6-1. Major types of temperature response curves for
algal growth.
294
-------
T - = reference temperature corresponding to the value of the
maximum growth rate u (T f), C
ITlaX i cl
y = TJ j-^ r = slope of growth vs. temperature curve
ref min
j3 = T Y = y-intercept of growth vs. temperature
ref " min curve
This equation is typically used in simplified form by choosing a lower
temperature limit Tmi. equal to zero so that Equation (6-5) becomes:
f(T) = J— (6-6)
ref
Reference temperatures of either 20°C or 1°C are usually used which results
in:
f(T) = o (6"7)
or f(T) = T (6-8)
This approach is used in EXPLORE-I (Baca _et a]_., 1973) and RECEIV-II
(Raytheon, 1974) and by Di Toro _et al. (1971) in an early version of WASP.
Some models use piecewise linear functions for algal growth with
different slopes over- different temperature ranges (Bierman _et TQ (6-9b)
295
-------
with u (T _) = n (T J (6-9c)
^ ref ^maxv opt' v
where T = optimum temperature at which the growth rate is maximum,
°P °c
This assumes growth increases linearly with temperature until the maximum
growth rate is attained, and then remains at the maximum rate as temperature
increases further.
The most commonly used exponential temperature adjustment functions are
based on the Arrhenius or van t Hoff equation:
10
where K, = reaction rate at temperature T,
K~ = reaction rate at temperature T,,
Q1Q = ratio of reaction rates at 10°C temperature increments
This equation can be rearranged into a more useful form as:
K2 = Kj Q10\~TO~7 (6-11)
f'M
°r K(T) = K(Tref) Q10\ 10 / (6-12)
= K(Tref) f(T)
where f(T) is the temperature adjustment function:
f(T) = Q10V^^~/ (6-13)
The temperature adjustment function (Equation (6-13)) is generally expressed
in a more simplified form as:
296
-------
f(T) - Q-ref (6-14)
- fl
where 0 = Q-,^ ' = temperature adjustment coefficient
The temperature adjustment coefficient 8 typically has a value between 1.01
and 1.2, with a value of 1.072 corresponding to a doubling of the growth
rate for every 10°C increase in temperature. Eppley (1972) found that 6
equals 1.066 for an exponential envelope curve of growth rate versus
temperature data compiled from a large number of studies involving many
different species (Figure 6-2).
Most models which use exponential temperature functions assume a
reference temperature of 20°C which gives the familiar equation (Chen and
Orlob, 1975; Baca and Arnett, 1976; Roesner et_ _a]_. , 1981; Brandes and Masch,
1977; Duke and Masch, 1973; Thomann et _al_. , 1979; Thomann and Fitzpatrick,
1982; Di Toro and Matystik, 1980; Di Toro and Connolly, 1980; O'Connor
et_al., 1981):
f(T) = 0 (T'20 C) (6-15a)
with Mmax(Tref) = /VX(20°C) (6-15b)
However, Thomann _et jil_. (1975) and Eppley (1972) use a reference temperature
of 0°C which results in:
f (T) = #T (6-16a)
With "max^ref > * "C> (6-16b)
The above equations assume that the temperature adjustment coefficient
8 has the same value regardless of the reference temperature. However, a
few models have applied Equation (6-14) in a piecewise manner assuming that
the value of 0 varies over different temperature intervals.
297
-------
Many formulations have been used to generate temperature optimum curves
for algal growth. The reference temperature is generally set at the optimum
temperature for maximum growth, and the temperature adjustment function is
normalized so it has a maximum value of 1.0 at the optimum temperature and
smaller values elsewhere. Most curves begin with a zero value at the lower
temperature tolerance limit, increase to a maximum value of 1.0 at the
optimum temperature, and then decrease back to a value of zero at the upper
temperature tolerance limit. These types of curves are typically based on
growth vs. temperature data for a single species. These data generally show
no growth at very low temperatures followed by an exponential increase in
10
20 30
TEMPERATURE °C
Figure 6-2.
Envelope curve of algal growth rate versus temperature
for data compiled from many studies involving many
different species (adapted from Eppley, 1972; Goldman,
1981).
298
-------
growth with temperature over a large part of the temperature range.
However, the growth rate eventually levels off to some maximum value at the
optimum temperature, and then begins to decline at very high temperatures
until growth finally ceases at some upper temperature limit.
Lehman et_ _al_. (1975) use a skewed normal distribution as a temperature
optimum curve for phytoplankton growth. The equation is:
with
f(T) = exp
M (T *
^maxv ref
[ 2 3 ( T ' Ml
[ • (\ - v' J
,) = a (T . )
f ^max opt'
(6-17a)
(6-17b)
where T = optimum temperature at which the growth rate is maximum,
°P °c
Tx = Tmin
T <-
= Tmax for T > Topt
Ti = lower temperature limit at which the growth rate is zero,
T = upper temperature tolerance limit at which growth ceases,
Ilia A
Jorgensen (1976) and Jorgensen _et al_. (1978) use a modified form of Equation
(6-17a) which is expressed as:
f(T) = exp (-2.3
T - T
opt
opt ~ min
(6-18)
Several models including CLEAN (Bloomfleld et _al., 1973), CLEANER
(Scavia and Park, 1976), MS.CLEANER (Park et ^1, 1979, 1980), and Scavia
^t _al_. (1976) use a temperature optimum function originally developed by
Shugart e_t jTL (1974). This formulation can be expressed as:
f(T) = Vx ex(1'V)
(6-19a)
299
-------
V = ma* " (6-19b)
max" opt
x =
(6_19c)
W '
-------
where K = a scaling constant used in the original equation from which
Equation (6-21a) was derived,
df(T]
dt
= K
/ Tmax - T \
\ max ~ opt/
(6-21c)
These equations result in a temperature optimum curve which is always skewed
to the right.
Thornton and Lessem (1978) developed a temperature optimum curve by
combining two logistic equations, one describing the rising limb of the
curve below the optimum temperature and one describing the falling limb of
the curve above the optimum temperature. The second curve is rotated about
the y-axis and shifted to the right along the x-axis until the approximate
peaks of both curves coincide. The left side of the temperature curve is
expressed as:
VT> Vy.(T-T_.
1 + 1<
(6-22a)
1
1 (Topt(l)
., 1n
- T '. ) '"
min'
~K2 (1 - Kj)"
Kj (1 -
K2)
(6-22b)
and the right side is expressed as:
max
-T)
'
•T)- l]
(6-23a)
(6-23b)
301
-------
where T tm= lower limit of optimum temperature range, C
opt { i j
T t(p\= upper limit of optimum temperature range, C
y, = rate coefficient for left side of curve
y~ = rate coefficient for right side of curve
K, = rate multiplier near the lower temperature limit
K. = rate multiplier near the upper temperature limit
K2 = 0.98
K3 = 0.98
The temperature curve is defined as the product of Equations (6-22a)
and (6-23a):
f(T) = KA(T) Kgd) (6-24a)
with
By using different values of the logistic equation parameters for each side,
an assymmetric growth curve can be generated. The values of K» and K, are
set equal to 0.98 rather than 1.0 so that the peak of the combined logistic
equation is close to 1.0 (since the logistic equation would otherwise only
approach 1.0 assymptoti cal ly) . Two values of the optimum temperature,
^oot(l) anc' ^oDt(2)' are usec^ to a^ow an opt"1'"111111 temperature range, rather
than a single optimum temperature value. This formulation is used in CE-
QUAL-R1 (WES, 1982), WQRRS (Smith, 1978), and EAM (Tetra Tech, 1979, 1980).
The left side of the curve (the basic logistic equation, Equation (6-22a))
is also used as a temperature adjustment curve in SSAM IV (Grenney and
Kraszewski, 1981).
The MIT one-dimensional network model (Najarian and Harleman, 1975;
Harleman ^t aj_. , 1977) uses a temperature optimum curve which is defined as:
forT< V
302
-------
rT - T
T _ °pt 1 for T > T
max opt '
with maxref maxopt (6-25c)
The values of the exponents n and m are 2.5 and 2.0, respectively (Najarian
and Harleman, 1975) .
Some type of temperature optimum curve is generally more appropriate
than a linear or exponential formulation when considering a single algal
species or functional group, since growth usually slows down and eventually
ceases above some upper temperature limit for any given species. This
approach is used in most models which simulate several algal groups (e.g.s
Chen^t aj_., 1975; Tetra Tech, 1979, 1980; Park et a]_. , 1979, 1980; Canal e
_et aj_., 1975, 1976; Scavia _et aj_. , 1976; Lehman et ^K , 1975; Smith, 1978;
WES, 1982), since seasonal variation in temperature is one of the major
factors causing seasonal succession in the dominance of different groups
(diatoms, greens, blue-greens, etc.). However, since many species are
lumped into a few functional groups, the temperature optimum curves and
maximum growth rates should be defined so that they encompass the
temperature-growth curves of all dominant species in the defined groups.
Canale and Vogel (1974) developed a set of temperature-growth curves for
diatoms, green algae, blue-green algae, and flagellates based on a
literature review of growth data for many species (Figure 6-3).
Since the temperature function includes both the effects of increasing
temperature on the growth rates of many individual species as well as shifts
in the species composition toward dominance by warmer water species, some
modelers have preferred to use exponential or linear formulations
over the whole temperature range, particularly when only one or two groups
are simulated (Chen and Orlob, 1975; Thomann _et al . , 1979; Di Toro and
Matystik, 1980; Di Toro and Connolly, 1980; Nyholm, 1978). This assumes
that as temperature increases, the species composition changes so that
species with optimum temperatures near the ambient temperature (and with
303
-------
Mixed Population .
TEMPERATURE,°C
Figure 6-3. Temperature-growth curves for major algal groups
(from Canale and Vogel, 1974).
higher maximum growth rates) tend to dominate the phytoplankton assemblage.
Eppley (1972) showed that an exponential relationship describes the envelope
curve of growth rate versus temperature data from a large number of studies
with many different species (Figure 6-2). However, this approach may
overestimate the net growth of the assemblage if the growth rates are based
on the maximum growth rate of the species assumed to be dominant at any
given instant, since much of the biomass will include species which
predominated earlier under different temperature conditions (Swartzman and
Bentley, 1979). Exponential or linear functions which increase indefinitely
with temperature can also be justified in situations where the maximum water
temperatures are always below the optimum temperatures for the species
present. For example, Canale and Vogel (1974) assumed a linear relationship
below the temperature optimum for each algal group in Figure 6-3.
The temperature formulations used in different models are compared in
Table 6-6.
304
-------
TABLE 6-6. COMPARISON OF TEMPERATURE ADJUSTMENT FUNCTIONS FOR ALGAL GROWTH
Temperature Formulation (Equation No.)
Model Optimum Other
(Author) Linear Exponential Curve Curve
AQUA-IV 6-14
CE-QUAL-R1 6-24
CLEAN 6-19
CLEANER 6-19
MS. CLEANER 6-19
DEM 6-14
DOSAG3 6-14
EAM 6-24
ESTECO 6-14
EXPLORE-1 6-6
HSPF piecewise
linear
saturation
LAKECO 6-14
MIT Network 6-25
QUAL-II 6-14
RECEIV-II 6-6
SSAM IV logistic
equation
WASP 6-14
WQRRS 6-24
Bierman piecewise piecewise
linear linear
saturation
Canal e piecewise
1 inear
Jorgensen 6-18
Lehman 6-17
Nyholm 6-14
Scavia 6-19
Reference
Temperature
20°C
Topt
Topt
Topt
Topt
20°C
20°C
Topt
20°C
1°C
20°C
Topt
20°C
1°C
20°C
20°C
Topt
1°C
Topt
rtn-f
opt
20°C
Topt
Reference
Baca & Arnett (1976)
WES (EWQOS) (1982)
Bloomfield et al. (1973)
Scavia & Park (1976)
Park et a]_. (1980)
Feigner & Harris (1970)
Duke & Masch (1973)
Tetra Tech (1979, 1980)
Brandes & Masch (1977)
Baca et al. (1973)
Johanson et aK (1980)
Chen & Orlob (1975)
Harleman et al. (1977)
Roesner et a]_. (1981)
Raytheon (1974)
Grenney & Kraszewski (1981)
Di Toro et al. (1981)
Smith (1978)
Bierman e_t al. (1980)
Canale et al- (1975, 1976)
Jorgensen (1976)
Lehman et a^ (1975)
Nyholm (1978)
Scavia et al. (1976)
305
-------
6.4.2 Algal Growth Limitation
In addition to temperature effects, algal growth rates are limited by
both light and nutrient availability. As mentioned above, only
macronutrients (phosphorous, nitrogen, carbon, and silicon) are generally
included in models. Growth limitation was expressed previously as the
factor f(L,P,N,C,Si) in the algal growth equation:
H = n (T ,) f(T) f(L,P,N,C,Si) (6-4)
r pmax ref v ' v ' ' ' ' ' '
Separate growth limiting factors are typically computed for light and each
potentially limiting nutrient. The number of nutrients considered will vary
between models depending on the particular system under consideration. Each
growth limitation factor can range from a value of 0 to 1. A value of 1
means the factor does not limit growth (i.e., light is at optimum intensity,
nutrients are available in excess, etc.) and a value of 0 means the factor
is so severely limiting that growth is stopped entirely.
Four major approaches have been used to combine the limiting factors
for light and each limiting nutrient:
1) a multiplicative formulation in which all factors are multiplied
together:
f(L,P,N,C,Si) = f(L) f(P) f(N) f(C) f(S1) (6-26)
where f(L) = light limitation factor
f(P) - nutrient limitation factor for phosphorous
f(N) = nutrient limitation factor for nitrogen
f(C) = nutrient limitation factor for carbon
f(Si) = nutrient limitation factor for silicon (for
diatoms)
2) a minimum formulation in which the most severely limiting factor
alone is assumed to limit growth:
306
-------
f(L,P,N,C,Si) = min [f (L) ,f (P) ,f (N) ,f (C) ,f (Si )] (6-27)
where min l~x, jX^.x^ , .. .1 = minimum of each factor x.
.
3) a harmonic mean formulation which combines the reciprocal of each
limiting factor in the following manner:
f(L,P,N,C,Si) = 1 - - - - — (6-28)
1TL7 +
where n = number of limiting factors (5 in this case)
4) an arithmetic mean formulation which uses the average of each
1 i mi ting factor:
f(L,P,N,C,Si) = f(L) * f(P) * f(N) * f(C) * f(S1) (6_2g)
The multiplicative formulation has been used in many models (Chen and
Orlob, 1972, 1975; Di Toro et al_. , 1971, 1977; Di Toro and Matystik, 1980;
Di Toro and Connolly, 1980; Thomann et a_K , 1975, 1979; O'Connor et _al_. ,
1975; Jorgensen, 1976; Jorgensen et al_. , 1978; Canale el ll- , 1975, 1976;
Lehman _et ci]_. , 1975; Roesner e_t aj_., 1981; Baca et aj_. , 1973; Duke and
Masch, 1973; Brandes and Masch, 1977). This approach assumes that several
nutrients in short supply will more severely limit growth than a single
nutrient in short supply. The major criticism of this approach is that the
computed growth rates may be excessively low when several nutrients are
limiting. Also, the severity of the reduction increases with the number of
limiting nutrients considered in the model, making comparison between models
difficult. Many models assume that light limitation is multiplicative, but
use one of the other approaches for nutrient limitation (e.g., Bierman
et jj]_., 1980; Bierman, 1976; Baca and Arnett, 1976; Nyholm, 1978; Raytheon,
1974).
The minimum formulation is based on "Liebig's law of the minimum" which
states that the factor in shortest supply will control the growth of algae.
307
-------
This approach has been popular in many recent algal models (Bierman et a!. ,
1980; Park et aj[. , 1979, 1980; Scavia, 1980; Smith 1978; Tetra Tech, 1979,
1980; WES, 1982; Johanson et a\_., 1980; Grenney and Kraszewski , 1981; Chen
j?t _al_., 1975; Baca and Arnett, 1976). The minimum formulation is often used
only for nutrient limitation, with a multiplicative formulation for the
light limitation factor.
The harmonic mean formulation is based on an electronic analogy of
several resistors in series. The rationale for this formulation is that it
includes some interaction between multiple limiting nutrients, but it is not
as severely limiting as the multiplicative formulation. This approach has
been used in only a few models, for example, the original CLEAN (Bloomfield
et a_K, 1973) and CLEANER (Scavia and Park, 1976) models and Nyholm (1978).
The current version of MS.CLEANER (Park _et ^1_. , 1980) has abandoned this
formulation in favor of the minimum formulation. In fact, the harmonic mean
formulation and minimum formulation produce similar growth response curves
under a wide range of conditions (Swartzman and Bentley, 1979).
The rationale for the arithmetic mean formulation is the same as for
the harmonic mean formulation (i.e., it considers the effects of multiple
nutrient limitation, but is not as severely limiting as the multiplicative
formulation). However, this formulation (e.g., Patten, 1975; Patten et aj_.,
1975) is rarely used since it does not restrict growth enough. For example,
the arithmetic mean formulation allows growth even if a critical nutrient
such as phosphorus is totally absent, as long as other nutrients are
available.
These and other formulations for combining multiple growth limitation
factors are reviewed in De Groot (1983),
6.4.3 Light Limitation
Light limitation formulations consist of two components: 1) a
relationship describing the attenuation of light with depth and the effect
308
-------
of algae on light attenuation, and 2) a relationship defining the effect of
the resulting light levels on algal growth and photosynthesis.
The attenuation of light with depth is defined in essentially all
models by the Beer-Lambert law:
I(z) = IQ e'yz (6-30)
where I(z) = light intensity at depth z below the surface
z = depth, length
I = light intensity at the surface
y = light extinction coefficient, I/length
The light intensity at the surface I is a function of location, time of
year, time of day, meterological conditions, and shading from topographic
features or riparian vegetation. The surface light intensity used in the
algal growth formulations corresponds only to the visible range, which is
typically about 50 percent of the total surface solar radiation used in the
heat budget computations. Almost all radiation outside of the visible range
is absorbed within the first meter below the surface (Orlob, 1977). In
addition, some models (for example, MS. CLEANER) assume that only a portion
of the visible radiation (about 50%) is available for photosynthesis (Park
tst a].., 1980; Strickland, 1958).
Light attenuation in models differs primarily in the way the light
extinction coefficient X is formulated. The simplest approach is to assume
a constant value of y. This approach is reasonable for short term
simulations or over periods when turbidity does not change significantly.
However, in long term simulations, y should be computed dynamically to
account for seasonal variations in turbidity due to algal shading or
variations in suspended solids loads.
The light extinction coefficient is most commonly defined as the linear
sum of several extinction coefficients representing each component of light
absorption. The components include all suspended particulates
309
-------
(phytoplankton, zooplankton, organic and inorganic particulates) as well as
dissolved organic matter. The general equation is:
y = y + i>, (6~31)
0 1=1 1
= y + Eajci (6~32)
i=l
where y = base light extinction coefficient for water without
particulates or dissolved organic matter, I/length
y. = light extinction coefficient corresponding to each component
of light absorption i, I/length
n = total number of absorption components considered in the
formulation
C. = concentration of absorption component i, mass/volume
a. = coefficient for absorption component i relating the
concentration C. to the light extinction coefficient y^
Many models include the effects of all components except phytoplankton
in the base extinction coefficient y (by assigning a higher value), and
then compute the temporal variations in y as a function of the algal
densities only. This assumes phytoplankton blooms are the major cause of
turbidity changes. Equation (6-32) then becomes:
y = yQ + aj_ A (6-33)
where y = light extinction coefficient for all absorption components
but phytoplankton, I/length
a, = coefficient relating the phytoplankton concentration A to
the corresponding light extinction coefficient for
phytoplankton (also called the self-shading factor),
I/(length-mass/volume)
A = phytoplankton concentration, mass/volume
310
-------
This provides a way of incorporating self-shading effects in the light
limitation portion of the algal growth formulation. Some models which use
this approach use a nonlinear formulation to describe the relationship
between the phytop 1 ankton concentration and the light extinction
coefficient. The general expression is:
b?
Y = yQ + a1 A + a2 A * (6-34)
where apa^ = coefficients of the equation relating phytop 1 ankton
concentrations to the light extinction coefficient
^2 ~ exponent of the equation relating phytop 1 ankt on
concentrations to the light extinction coefficient
The second component of the light limitation formulation represents the
light limitation factor f(L) in Equations (6-26) through (6-29). f(L)
defines the relationship between ambient light levels and algal growth rates
or rates of photosynthesis. Essentially all formulations fall into one of
two major categories (Figure 6-4): 1) saturation type relationships in
which the growth rate increases linearly with light at low intensities, but
gradually levels off at high intensities to reach a maximum value at the
optimum (or saturating) light intensity, or 2) photoinhibition relationships
which are similar to the above curves below the optimum light intensity, but
which predict decreases in growth rates above the optimum intensity due to
photoinhibition effects.
Saturation type responses are typically described by either a
Michael is-Menten (1913) type relationship (Chen and Orlob, 1975; Jorgensen,
1976; Duke and Masch, 1973; Tetra Tech, 1979; Roesner et al_. , 1981; Johanson
et_a]_., 1980; Smith, 1978; WES, 1982):
where f(L) = light limitation function for algal growth
I = light intensity
311
-------
= half-saturation constant defining the light level at which
growth is one-half the maximum rate
or a Smith (1936) formulation (Park et _a_L, 1980):
f(L) =
(6-36)
where a. = constant in the Smith formulation (1/a^ is the slope of the
linear portion of the photosynthesis vs. light curve),
I/light
LU
CC
X
o
oc
Photoinhibition
Curve
LIGHT INTENSITY
Figure 6-4. Comparison of light response curves for algal growth.
312
-------
Vollenweider (1965) modified the Smith formulation to give a more
general relationship of which the Smith equation is a special case. The
Vollenweider form includes photoinhibition effects, and is expressed as:
f(L) = / ^=W . L=\ (6-37)
where a^ = photoinhibition factor, I/light
n = exponent
Baca and Arnett (1976) use this formulation in AQUA-IV with the exponent n
equal to 1.
The most commonly used photoinhibition relationship is the Steele
(1965) formulation:
(6-38)
where I = optimum (saturating) light intensity
This formulation is used in many models including Di Toro et aj_. (1971,
1977), Di Toro and Matystik (1980), Di Toro and Connolly (1980), Thomann
et £]_. (1975, 1979), Thomann and Fitzpatrick (1982), O'Connor et a].. (1981),
Bloomfield _et al.. (1973), Park jet aj_. (1974, 1975, 1979, 1980), Scavia
jet aj_. (1976), Najarian and Harleman (1975), Bierman eit ail_. (1980), Canale
et il. (1975, 1976), Lehman et a\_. (1975), and Baca et al. (1973).
Park et _aj_. (1980) use the Steele formulation above the saturating
light intensity I and the Smith formulation below I They feel that the
Steele formulation is not accurate below the inhibition threshold since the
predicted photosynthesis response is partially dependent on the response
above the threshold (Park jet jiT_. , 1979). Under non-inhibiting light
313
-------
conditions, this may result in a light limitation factor which is too low
(Groden, 1977).
Walker (1975) found that the Steele formulation underpredi cts
photosynthesis rates at high light intensities (above saturation) for some
algae, so he modified it by adding an additional parameter n:
(6-39)
where n = parameter for modified Steele formulation
This parameter adjusts the rate of decline of the photosynthesis vs. light
curve for light intensities above and below the optimum. The original
Steele formulation assumes n=l, while Walker used n values of 0.67, 0.80,
and 1.0 for three different algal groups.
A few models include light adaptation algorithms in their light
limitation formulations to account for the fact that algae adapted to low
Vight levels have a more rapid response to changing light conditions
(steeper slope of photosynthesis vs. light curve) than algae adapted to high
light levels. Algae adapt to changing light conditions by varying the
chlorophyll content of their cells, with algae adapted to lower light
intensities having more chlorophyll.
Nyho.lm (1978) simulates this effect by varying the value of the
saturating light intensity at different times of the year to shift the peak
of the light limitation function f(L). The I values are maximum during
summer and minimum during winter. This shifts the slope of the light
response curve so it is steepest during the winter when the algae are
adapted to low light levels.
Groden (1977) developed a more complicated formulation for the
MS.CLEANER model which dynamically computes the slope of the photosynthesis
314
-------
vs. light curve as a function of light intensity, and then uses this
information to compute the saturating light intensity as a function of both
light and temperature. The equation for the slope of the photosynthesis vs.
light curve in the light inhibited range is:
a = Kj_ ln(I) - K2 (6-40)
where a = slope of photosynthesis vs. light curve
K«,K? = constants
This is based on the assumptions that 1) the slope a is a linear function of
the chlorophyll content of the cells and 2) chlorophyll decreases
exponentially with light intensity until it reaches some minimum value
(Groden, 1977). The values of ^ and K2 used in MS.CLEANER are 0.1088 and
0.0704, respectively (Groden, 1977; Park et a]_., 1980). The equation for
the saturating light intensity is:
Smith (1980) developed a formulation for computing the saturating light
intensity as a function of the maximum photosynthetic quantum yield, maximum
growth rate, temperature, light extinction coefficient per unit chlorophyll,
and the carbon to chlorophyll ratio of the algae. The equation is:
Mmax(Tr ef) f(T) C e
I = max reT - H— (6-42)
s ac
where C = carbon to chlorophyll ratio
^mT.v, = maximum photosynthetic quantum yield, moles carbon
max
fixed/mole photons absorbed
315
-------
a = coefficient for light extintion per unit chlorophyll,
1/Oength-mass chlorophyll/volume)
The effects of light adaptation are included in the carbon to chlorophyll
ratio C . This ratio typically ranges from 20 to 100, with 20 corresponding
to low-light, high-temperature conditions, and 100 corresponding to high-
light, low-temperature conditions (Smith, 1980; Eppley, 1972). Based on
observations that the maximum photosynthesis rate typically occurs at the
depth where the light intensity is about 30 percent of the surface value (I
=0.3 I ), Smith (1980) suggested the following relationship for estimating
C as a function of the ambient light levels:
°-3Ta ,6
(6~
where I = daily average light intensity at the surface
These formulations are used by Thomann and Fitzpatrick (1982) in the Potomac
Estuary version of WASP. One advantage of this approach is that I and C
are defined in terms of parameters which are well documented in the
literature (
-------
Since light also varies continuously with time, most models integrate
the light limitation function f(L) over 24 hours to get a daily average
value for a given time of the year and set of meteorological conditions.
This is generally approximated by multiplying the light limitation function
by the photoperiod (expressed as the fraction of the day in which the sun is
out) and by using the average light intensity during the daylight hours as
I in the formulation. This approach is used in steady-state models and
dynamic models which use daily time steps. The alternative approach when
short time steps (minutes to hours) are used is to compute the light
limitation and algal growth formulations dynamically throughout the day
using instantaneous values of IQ. The latter method simulates the diurnal
variations in algal photosynthesis.
The depth and time integrated Michael is-Menten formulation for light
limitation (Equation (6-35)) is expressed as:
(6'44)
where f = photoperiod (expressed as a fraction of the day)
d = water depth, length
I = average light intensity at the surface during the daylight
hours
when averaged over the whole water depth or as:
f(L) = y (z P2 i In -^ ^H (6-45>
where z, = depth at top of layer, length
z2 = depth at bottom of layer, length
when averaged over a single layer (for example, in a vertically segmented
lake model).
317
-------
The analogous expressions for the Smith formulation (Equation (6-36))
are:
f(L) =/n
and
f(L) =
(6-46)
In
(6-47)
For the Steele formulation (Equation (6-38)), the depth and time
integrated expressions are:
'
f(L) =
2.718 f
y d
(6-48)
and
f(L) =
2.718 f
(Z9 - zi)
(6-49)
Light limitation factors are compared for several models in Table 6-7.
Saturating light intensities and half-saturation constants for light
limitation are presented in Tables 6-8 and 6-9.
6.4.4 Nutrient Limitation
Two major approaches have been used to compute nutrient limitation
factors in algal models. The first approach is based on Monod (1949) or
Michaelis-Menten (1913) kinetics and assumes that the growth rates are
determined by the external concentrations of available nutrients. External
here refers to the nutrient concentrations in the water column as opposed to
the internal concentrations in the algal cells. This approach assumes the
nutrient composition of the algal cells remains constant, and is generally
referred to as fixed stoichiometry models.
313
-------
TABLE 6-7. COMPARISON OF LIGHT LIMITATION FORMULATIONS
Model
(Author)
AQUA- IV
CE-QUAL-R1
CLEAN
CLEANER
MS. CLEANER
DEM
DOSAG3
EAM
ESTECO
EXPLORE-1
HSPF
LAKECO
MIT Network
QUAL-II
RECEIV-II
SSAM IV
WASP
WQRRS
Bierman
Canale
Jorgensen
Lehman
Nyholm
Scavia
Light Limitation Formulation
Michael is-
Steele Smith Menten Vollenweider Other Reference
X Baca & Arnett (1976)
X WES (EWQOS) (1982)
X Bloomfield ejt a]_. (1973)
X Scavia & Park (1976)
X* X* Park et a]_. (1980)
X Feigner & Harris (1970)
X Duke & Masch (1973)
X Tetra Tech (1979, 1980)
X Brandes & Masch (1977)
X Baca et a]_. (1973)
X Johanson et al_. (1980)
X Chen & Orlob (1975)
X Harleman et a_l_. (1977)
X Roesner et al_. (1981)
X Raytheon (1974)
none Grenney & Kraszewski (1981)
X Di Toro et al_. (1981)
X Smith (1978)
X Bierman e_t a]_. (1980)
X Canale et a\_. (1975, 1976)
X Jorgensen (1976)
X Lehman et al_. (1975)
piecewise Nyholm (1978)
linear
saturation
X Scavia et aK (1976)
*Smith formulation used below light saturation, Steele formulation used above light saturation.
319
-------
TABLE 6-8. ALGAL SATURATING LIGHT INTENSITIES
Algal Type
Saturating Light Intensity
(1 angleys/day)
References
Total
Phytoplankton
300 350
250 350
200 300
216
288
Thomann et a]_. (1975, 1979)
Salas & Thomann (1978)
Di Toro et al_. (1971)
Di Toro & Connolly (1980)
Di Toro & Matystik (1980)
O'Connor et al. (1975)
Scavia et al_. (1976)
Scavia & Park (1976)
Scavia (1980)
Youngberg (1977)
Desormeau (1978)
Larsen et al. (1973)
Diatoms
225
300
8 - 100
225
144
Thomann et al_. (1979)
Di Toro & Connolly (1980)
Scavia et jj]_. (1976)
Scavia TJ980)
Bierman (1976)
Bierman e_t jj]_. (1980)
Canale et .a_l_. (1976)
Lehman et al_. (1975)
Green Algae
88 100
160
65
Bierman (1976)
Bierman et al_. (1980)
Canale et al_. (1976)
Lehman et al. (1975)
Blue-green Algae
44 50
43
600
300 350
250
Bierman (1976)
Bierman et a]_. (1980)
Lehman et al_. (1975)
Canale et a\_. (1976)
Youngberg (1977)
Scavia (1980)
Flagellates
Chrysophytes
288
100
86
Lehman et al_. (1975)
Bierman et ^1_. (1980)
Lehman et al. (1975)
320
-------
TABLE 6-9. HALF-SATURATION CONSTANTS FOR LIGHT LIMITATION
Half-Saturation Constant
Algal Type (Kcal/m2/sec)
References
Total
Phytoplankton
0.002 - 0.006
0.0046
0.002 - 0.006*
0.005*
0.003 0.005*
0.004 0.006**
0.0044**
Chen (1970)
Chen & Orlob (1975)
Chen & Wells (1975, 1976)
U.S. Army Corps of Engineers (1974)
Tetra Tech (1976)
Jorgensen (1976)
Jorgensen ejt aj_. (1978)
Smith (1978)
Roesner et al_. (1980)
Duke & Masch (1973)
Brandes (1976)
Jorgensen (1979)
Collins & Wlosinski (1983)
Diatoms
Green Algae
0.003
0.002*
0.00005 0.0012**
0.00005 0.0026**
0.002 - 0.004
0.002*
0.0003 0.0011**
0.0003 - 0.0106**
Tetra Tech (1980)
Bowie et al. (1980)
PorcelTa et aj_. (1983)
Tetra Tech (1979)
Jorgensen (1979)
Collins & Wlosinski (1983)
Tetra Tech (1980)
Bowie et aj_. (1980)
Porcella et aj_. (1983)
Tetra Tech (1979)
Jorgensen (1979)
Collins & Wlosinski (1983)
Blue-green Algae
0.002 - 0.004
0.002*
Tetra Tech (1980)
Bowie et al_. (1930)
PorcelTa et al. (1983)
Tetra Tech (1979)
Dinoflagellates
0.002*
0.0043 - 0.0053**
Tetra Tech (1979)
Collins & Wlosinski (1983)
(continued)
321
-------
TABLE 6-9. (continued)
Algal Type
Half-Saturation Constant
(Kcal/m2/sec)
References
Flagellates
Chrysophytes
Benthic Algae
0.002 0.004
0.0044**
0.002*
0.0014 0.0017**
Coccolithophores 0.0003 0.0016**
0.01 0.005
0.002 0.006*
Tetra Tech (1980)
Porcella et_ al_. (1983)
Collins & Wlosinski (1983)
Tetra Tech (1979)
Collins & Wlosinski (1983)
Collins & Wlosinski (1983)
Tetra Tech (1980)
Bowie et aj_. (1980)
Porcella et al_ (1983)
Smith (1978)
*Model documentation values.
**Literature values.
The second approach assumes that algal growth is a two-step process,
the first step being nutrient uptake and the second step being cell growth
or division. Cell growth depends on the internal concentrations of
nutrients within the cells, rather than external concentrations in the
water. The uptake rates are dependent on both the external and internal
concentrations. Since uptake and growth are modeled separately, the
nutrient composition of the cell may change with time, resulting in variable
stoichiometry or internal pool models. These models simulate processes
such as luxury uptake of nutrients which allows growth even when external
nutrients are depleted.
6.4.4.1 Nutrient Limitation in Fixed Stoichiometry Models
The majority of water quality models are of the fixed stoichiometry
type. These models are generally based on conventional Monod or Michael is-
Menten kinetics. The algal growth equation for a single limiting nutrient
under conditions of optimum temperature and light can be expressed as:
322
-------
<6-50>
where s = concentration of the limiting nutrient in the
water, mass/volume
KS = half-saturation constant for the limiting nutrient,
mass/volume
The quantity f(s) = (v s, ) is the growth limitation factor for the
Ks + s
nutrient s. The half-saturation constant refers to the concentration of the
nutrient at which the growth rate is one half of its maximum value. The
above equation results in a hyperbolic growth curve (Figure 6-5) in which
growth increases approximately linearly with nutrients at very low nutrient
concentrations, but gradually levels off to a maximum growth rate at high
nutrient levels (growth saturation). At this point, the nutrient is no
longer limiting, so further increases in the external nutrient supply do not
affect growth.
Fixed stoichiometry models typically compute a separate growth
limitation factor f(s) for each nutrient modeled, and then combine the
factors using any one of the four methods discussed above in Equations
(6-26) to (6-29) (i.e., multiplicative formulation, minimum formulation,
harmonic mean formulation, or arithmetic mean formulation). The specific
nutrient limitation factors are:
PO
(NH. + NO,)
+ (NH3 * N03) <6
C0y
(6-53)
323
-------
(6-54)
where PO,
(NH3+N03)
CO,
Si
KSi
available dissolved inorganic phosphorus
concentration (orthophosphate), mass/volume
available dissolved inorganic nitrogen concentration
(ammonia plus nitrate), mass/volume
available dissolved inorganic carbon concentration
(carbon dioxide), mass/volume
available dissolved silicon concentration,
mass/volume
half-saturation constant for phosphorus, mass/volume
half-saturation constant for nitrogen, mass/volume
half-saturation constant for carbon, mass/volume
half-saturation constant for silicon, mass/volume
LU
h-
cr
i
o
cr
O
Figure 6-5.
NUTRIENT CONCENTRATION
Michaelis-Menten saturation kinetics for algal
growth limitation by a single nutrient.
324
-------
The number of growth limiting factors included in a given model depends
on both the particular algal species present and the chemistry of the water
body under consideration. For example, silicon limitation is only
appropriate for diatoms. Nitrogen limitation can generally be omitted for
nitrogen-fixing blue-green algae (although nitrogen kinetics for blue-greens
must still be included to correctly describe the nitrogen cycle). Carbon
limitation is frequently excluded from algal models since carbon is often
assumed to be available in excess and is therefore not modeled as a state
variable. Lake models often assume phosphorus is the only limiting
nutrient, while estuary models often assume nitrogen is limiting at all
times.
The way in which nitrogen limitation is computed also varies from model
to model. For example, some models simulate available nitrogen as a single
constituent (Bierman ej^ aj_., 1980; Jorgensen et^ _§_]_., 1978; Nyholm, 1978;
Thomann _et _al_., 1979), while other models simulate ammonia, nitrite, and
nitrate separately and assume both ammonia and nitrate are available for
algal growth (Chen and Orlob, 1975; Baca and Arnett, 1976; Baca et al.,
1973; Smith, 1978; Najarian and Harleman, 1975; Duke and Masch, 1973).
QUAL-II simulates the various forms of nitrogen, but assumes algal growth is
only limited by nitrate (Roesner et_ aj_., 1981). Some models include factors
to account for ammonia preference by algae in their nutrient uptake
formulations (Scavia _e_t al., 1976; Canale e_t al., 1976; Grenney and
Kraszewski, 1981; Thomann and Fitzpatrick, 1982; O'Connor et al_., 1981; JRB,
1983). Ammonia preference factors are discussed in Chapter 5.
Values of the Michaelis-Menten half-saturation constants for each
limiting nutrient are available from many sources, including both the
modeling literature and the experimental literature. However, care must be
taken when using this information since the values reported will depend on
the particular model formulations used for the modeling literature, and on
the experimental conditions for the scientific literature. For example, if
a multiplicative formulation is used to compute algal growth
(Equation(6-26) ), the half-saturation constants should be smaller than the
corresponding constants where a minimum formulation is used (Equation
325
-------
(6-27)). In general, the more limiting nutrients that are considered with a
multiplicative formulation, the smaller the value of each half-saturation
constant. This is necessary in order to get the same growth response with
both formulations when more than one nutrient is limiting simultaneously.
This is true of both the modeling literature and the experimental
literature. When the harmonic mean formulation is used (Equation (6-28)),
the half-saturation constants should generally be somewhere between the
values of the minimum and multiplicative formulations. Half-saturation
constants for each limiting nutrient are tabulated in Table 6-10.
Table 6-11 compares the algal growth formulations used in several
models, including the growth limiting factors used, the specific
formulations for nutrient limitation, and the methods for combining multiple
limiting factors.
6.4.4.2 Nutrient Limitation In Variable Stoichiometry Models
Variable Stoichiometry models assume that the growth limiting factor
for nutrients, f(P,N,C,Si) in Equation (6-4), is a function of the internal
levels of the nutrients in the algal cells rather than the external
concentrations in the water column. The internal concentrations are
generally defined as:
_ internal mass of nutrient in cells /fi cc\
H dry weight biomass of cellsID-OD;
where q = internal nutrient concentration, mass nutrient/biomass algae
Internal nutrient levels depend on the relative magnitudes of the nutrient
uptake rates and the algal growth rates. The uptake rates are functions of
both the internal and external nutrient concentrations, while the growth
rates depend primarily on the internal concentrations.
Variable Stoichiometry models differ in 1) the specific process
formulations used to simulate uptake and growth, 2) the number of nutrients
considered, and 3) the ways in which multiple limiting factors are combined.
326
-------
TABLE 6-10. HALF-SATURATION CONSTANTS FOR MICHAELIS-MENTEN GROWTH FORMULATIONS
Half-Saturation Constant
Algal Type
Nitrogen
(mg/1 )
Phosphorus
(mg/1 )
Carbon
(mg/1 )
Silicon
(mg/1)
References
Total Phytoplankton
Diatoms
0.025
0.0005 0.03
0.01 0.4
0.2
0.025
0.06 0.08
0.015
0.014
0.025 0.3*
0.04 0.10*
0.2 0.4*
0.004 0.08
0.02 0.03
0.006 0.025
0.02
0.0025
0.001
0.006 0.03*
0.02 0.05*
0.03 0.05*
0.015 0.3* 0.0025 0.08*
0.10 0.4* 0.03 0.05*
0.0014 0.018 0.006**
0.025 0.2** 0.002 0.08**
0.0015 0.15**
0.02 0.075**
0.015 0.03
0.025
0.002
0.001 0.002
0.03 0.8
0.5
0.02 0.04*
0.15*
0.03
0.025 0.030 0.004 0.009
0.015 0.0025
0.015* 0.03* 0.03*
0.0063 0.12** 0.01 0.025**
0.025**
0.003 0.923** 0.001 0.163**
0.08
0.030 0.1
0.03
0.03
0.1
0.08*
O'Connor et al. (1975, 1985)
Thomann et aTT (1974, 1975, 1979)
Thomann & Fltzpatrick (1982)
Di Toro & Hatystik (1980)
Di Toro & Connolly (1980)
Di Toro et al_. (1971, 1977)
Sal as & Thomann (1978)
Salisbury et al_. (1983)
Chen (1970)
Chen & Orlob (1975)
Chen & Wells (1975, 1976)
U.S. Army Corps of Engineers (1974)
Tetra Tech (1976)
Jorgensen (1976)
Jorgensen et al_. (1978)
Battelle (1974)
Grenney 8 Kraszewski (1981)
Canale et al_. (1976)
Larsen et §1- (1973)
Baca & Arnett (1976)
Smith (1978)
Roesner e_t al_. (1980)
Duke & Masch (1973)
Grenney & Kraszewski (1981)
Brandes (1976)
Di Toro et aj_. (1971)
Jorgensen (1979)
O'Connor et al_. (1981)
Collins & Wlosinski (1983)
Tetra Tech (1980)
Bowie et aj_. (1980)
Porcella et. al_. (1983)
Thomann et al_. (1979)
Di Toro & Connolly (1980)
Salisbury et al_. (1983)
Scavia et al_. (1976)
Scavia (T980)
Canale et aj_. (1976)
Bierman (1976)
Tetra Tech (1979)
Di Toro et al_. (1971)
Jorgensen (1979)
Collins & Wlosinski (1983)
327
-------
TABLE 6-10. (continued)
Half-Saturation Constant
Algal Type
Nitrogen
(mg/1)
Phosphorus Carbon
(mg/1) (mg/1)
Silicon
(mg/1)
References
Green Algae
Blue-green Algae
Dinoflagellates
Flagellates
Chrysophytes
Coccolithophores
Benthic Algae
0.03 0.035
0.004
0.03
0.15 0.01
0.001 0.035 0.005 0.024
0.15
0.03*
0.005 0.15**
0.006 1.236**
0.
0.001
0.015
0.*
0.062 4.34**
0.005
0.08*
0.007 0.13**
0.019 0.589**
0.08
0.0084 0.13**
0.001 0.052**
0.015
0.006**
0.0025
0.03* 0.03*
0.01**
0.002 0.475** 0.068 - 1.5**
0.010.- 0.02 0.03
0.01 0.015
0.01
0.0025
0.06* 0.03*
0.006** 0.031 0.088**
0.06*
0.012
0.03*
0.03
0.02*
0.047 0.076**
0.03*
0.006 0.019**
0.05 0.1 0.004 0.008 0.03 0.1
0.06 0.08 0.02
0.04 0.10* 0.02 0.05* 0.02 0.04*
0.015 0.3* 0.0025 0.08*
*Model documentation values.
"Literature values.
Tetra Tech (1980)
Bowie et al. (1980)
PorcelTa el al. (1983)
Di Toro et al. (1971)
Scavia ejt al_. (1976)
Scavia & Park (1976)
Scavia (1980)
Canale et al_. <1976)
Tetra Tech (1979)
Jorgensen (1979)
Collins & Wlosinski (1983)
Tetra Tech (1980)
Bowie et al_. (1980)
Porcella et al_. (1983)
Scavia & Park (1976)
Scavia (1980)
Di Toro et aj_. (1971)
Canale et al.. (1976)
Tetra Tech (1979)
Collins & Wlosinski (1983)
O'Connor ei_ al_. (1981)
Tetra Tech (1979)
Di Toro et al. (1971)
Collins & Wlosinski (1983)
Tetra Tech (1980)
Porcella et al_. (1983)
Jorgensen (1979)
Collins & Wlosinski (1983)
Tetra Tech (1979)
Collins & Wlosinski (1983)
Collins & Wlosinski (1983)
Tetra Tech (1980)
Bowie et al. (1980)
Porcella et al.. (1983)
Grenney & Kraszewski (1981)
Smith (1978)
Grenney & Kraszewski (1981)
328
-------
TABLE 6-11. COMPARISON OF ALGAL GROWTH FORMULATIONS
Model
(Author)
AQUA- IV
CE-QUAL-R1
CLEAN
CLEANER
MS. CLEANER
DEM
DOSAG3
EAM
ESTECO
EXPLORE-1
HSPF
LAKECO
MIT Network
QUAL-II
RECEIV-II
SSAM IV
WASP
WQRRS
Bierman
Canale
Jorgensen
Lehman
Nyholm
Scavia
Growth Limiting Factors
Light P04 N03 NH3 C02 Si
X X X X
X X X X X
X X X X X
X X X X X
X X X X X X
XXX
XXX
X X X X X X
X X X X X
X X X X
X X X X X
X X X X X
X XX
XXX
X XXX
XXX
X X X X X
X X X X X
X X X X X
X X X X X
X X X X X
X X X X X X
X X X X
X X X X X
Stoichiometry
Fixed Variable
X
X
X
X
C & Si N & f
X
X
X
X
X
X
X
X
X
X
X
X
X
Si N & F
X
X
X
X
X
Nutrient Limitation
Formulation
Michaelis-
Menten Other
X
X
X
X
X* 6-51*
X
X
X
X
X
X
X
X
X
X
X
X
X
X** 6-52**
X
6-53
6-53
6-54, 55
X
Method for Combining Factors
Multipl- Harmonic
icative Minimum Mean
light nutrients
X
X
X
X
X
X
X
X
X
X
X
light
X
light nutrients
X
X
X
light nutrients
X
X
X
light nutrients
X
Reference
Baca & Arnett (1976)
WES (EWQOS) (1982)
Bloomfield et_ aT_. (1973)
Scavia & Park (1976)
Park et a]_. (1980)
Feigner & Harris (1970)
Duke & Masch (1973)
Tetra Tech (1979, 1980)
Brandes & Masch (1977)
Baca et al_. (1973)
Johanson et al . (1980)
Chen & Orlob (1975)
Harleman et al . (1977)
Roesner et al. (1981)
Raytheon (1974)
Grenney & Kraszewski (1981)
Di Toro et a_K (1981)
Smith (1978)
Bierman et al . (1980)
Canale et. aj_. (1975, 1976)
Jorgensen (1976)
Lehman et. al_. (1975)
Nyholm (1978)
Scavia et al. (1976)
oo
ro
*Fixed Stoichiometry Michaelis-Menten formulation used for carbon and silicon, with variable Stoichiometry formulations for nitrogen and phosphorus.
**Fixed Stoichiometry Michaelis-Menten formulation used for silicon, with variable Stoichiometry formulations for nitrogen and phosphorus.
-------
Several different formulations have been used to compute nutrient
limitation factors in variable stoichiometry models. As with fixed
stoichiometry models, the limitation factors may range from 0 to 1. Most
models assume a minimum internal stoichiometric nutrient requirement at
which growth is zero. This minimum level is often called the minimum cell
quota or subsistence quota. Algal growth (and the nutrient limitation
factors) are assumed to increase with increasing internal nutrient levels
above the minimum cell quota until the maximum growth rate is attained.
Some type of hyperbolic function is typica-lly used to express this
saturation type relationship.
The following expressions have been used to determine growth limitation
factors in variable stoichiometry models:
f(q) =
(6-56)
where f(q)
q
q .
q
f(q)
f(q)
f(q)
f(q) H
1 -
"•mm
q - q
'mm
iax
K, + (q - q . )
VM Hmin;
q - q
+ (
^
- q • )
H'
(6-57)
(6-58)
(6-59)
(6-60)
nutrient limitation factor
internal nutrient concentration, mass nutrient/biomass
al gae
minimum internal stoichiometric requirement (cell
quota), mass nutrient/biomass algae
maximum internal nutrient concentration, mass
nutrient/biomass algae
^'Saturation constants for growth limitation
330
-------
Equation (6-56) is equivalent in form to the Michaelis-Menten relationship
except that the internal rather than the external nutrient concentration is
the independent variable. This equation is used in MS.CLEANER for both
nitrogen and phosphorus limitation (Park e_t al., 1980). Equation (6-57)
also has the same form as the Michaelis-Menten relationship, but the
independent variable is the internal nutrient concentration in excess of the
minimum cell quota. This equation is used by Bierman (1976) and
Bierman et al_. (1973, 1980) for nitrogen and phosphorus. Equation (6-58)
was originally developed by Droop (1968), and it is used in several models
including Lehman et aj_. (1975), Jorgensen (1976), Jorgensen e_t al. (1978,
1981), and Canale and Auer (1982) for all nutrients simulated in these
models. Equation (6-58) can be derived from Equation (6-57) by assuming K~
= q . , as was demonstrated by Rhee (1973, 1978) for phosphorus and nitrogen
(Bierman, 1981). Equations (6-59) and (6-60) are used by Nyholm (1978) for
nitrogen and phosphorus, respectively. Note that Equation (6-59) is a
linear rather than hyperbolic relationship. Also, Equation (6-60) is
similar to Equation (6-57) since the second factor in Equation (6-60) is a
constant once qm. , qm_, and K., are defined.
m I n max j
Since variable stoic hiometry formulations have not been widely used,
data for the model parameters are limited. Values for the various half-
saturation constants are presented in Table 6-12. Note that the half-
saturation constants (K..,K2, and K.) have different values since the
corresponding equations are different. Minimum cell quotas and maximum
internal nutrient concentrations are tabulated in Tables 6-13 and 6-14.
The ways in which variable stoichiometry formulations are used varies
between different models. Some models use variable stoichiometry
formulations only for phosphorus and nitrogen, combining them with
conventional Michael is-Menten kinetics for carbon and silica (Park et a*1 . ,
1980; Bierman ^t jaj., 1980). while other models use variable stoichiometry
formulations for all nutrients modeled (Lehman j^t a_[., 1975; Jorgensen,
1976). In a few cases, different internal nutrient formulations are used
for different nutrients in the same model (Nyholm, 1978). In some models.
331
-------
TABLE 6-12. HALF-SATURATION CONSTANTS FOR VARIABLE STOICHIOMETRY FORMULATIONS
Half-Saturation Constant
Nutrient Type Value
Phosphorus Kj 0.005 g/m
K, 0.724xlO"7 /imole/cell
0.0005 mg/mg (D.W.)
0.312xlO"8 ^mole/cell
0.0005 mg/mg (D.W.)
0.148xlO"7 (/mole/cell
0.0005 mg/mg (D.W.)
0.488xlO"8 /^mole/cell
0.0007 mg/mg (D.W.)
0.566xlO~8 /^mole/cell
Algal Type
Total Phytoplankton
Diatoms
Green Algae
Flagellates
Blue-greens (N-fixing)
Blue-greens (non N-fixing)
Reference
Desormeau (1978)
Bierman et aj_. (1980)
Nitrogen
0.0007 mg/mg (D.W.)
0.003 mg/mg (D.W.)
0.05 g/m
O.SOlxlO'5 ^mole/cell
0.025 mg/mg (D.W.)
0.345xlO"6 (/mole/cell
0.025 mg/mg (D.W.)
0.163xlO"5 ^mole/cell
0.025 mg/mg (D.W.)
0.377xlO"6 ^mole/cell
0.025 mg/mg (D.W.)
0.438xlO"6 /^mole/cell
0.025 mg/mg (D.W.)
O.HxlO"7 ^mole/cell
0.14x10
0. 23x10" 7 //mole/cell
0.14x10" jimole/cell
Total Phytoplankton
Total Phytoplankton
Diatoms
Green Algae
Flagellates
Blue-greens (N-fixing)
Blue-greens (non N-fixing)
Diatoms
Green Algae
Blue-greens (N-fixing)
Blue-greens (non N-fixing)
Nyholm (1978)
Desormeau (1978)
Bierman et al_. (1980)
Bierman (1976)
carbon and silica are not included as potentially limiting nutrients
(Nyholm, 1978).
The combined effects of multiple limiting nutrients in variable
stoichiometry models are dealt with in the same basic ways as in fixed
stoichiometry models (i.e., multiplicative formulation (Equation (6-26)),
minimum formulation (Equation (6-27)), or harmonic mean formulation
(Equation (6-28)). However, when a minimum (or threshold) formulation is
used, the limiting nutrient is often determined by comparing the internal
332
-------
TABLE 6-13. MINIMUM CELL QUOTAS
Algal -Type
Total
Phytoplankton
Diatoms
co Green Algae
CO
CO
Blue-green Algae
Dinoflagellates
Flagel lates
Chrysophytes
Benthic Algae
Nitrogen
0.015-0.02
0.015
0.04
0.520xlO"7
O.SOlxlO"5
0.025
-7**
6-xlO '
0.520xlO"7
0.345xlO"6
0.025
0. 520-0. 853xlO"7
0. 377-0. 438xlO"6
0.025
-7**
1.1x10 '
_7**
3.9x10
0.163xlO"5
0.025
0.18-0.3xlO"7**
Minimum Cell Concentration
Phosphorus Carbon Silicon
0.001-0.003 0.15-0.18
0.001 0.15-0.4
0.00146
0.3-0.7**
0.20xlO"8
0.724xlO"7
0.0005
0.9-30.X10"9** 0.2-40.X10"7**
0.45-0.6**
0.20xlO~8
0.312xlO"8
0.0005
1.7-4.5xlO~9**
>0.5**
0. 583-1. 34xlO~9
0. 488-0. 566xlO"8
0.0007
_Q**
2.5x10
>0.5**
-Q**
ll.xlO
0.148xlO"7
0.0005
0**
0.5x10
0.0005
Units
mg/mg (D.W.)
mg/mg (D.W.)
mg/mg (D.W.)
mg/mg (D.W.)
jimoles/cell
^moles/eel 1
mg/mg (D.W.)
^moles/eel 1
(ig/mm cell
volume
^moles/cell
fjmoles/cel 1
mg/mg (D.W.)
jjmoles/cell
//g/mm cell
volume
/^moles/cell
^moles/cell
mg/mg (D.W.)
//moles/cell
fig/mm cell
volume
^moles/cell
/imoles/cell
mg/mg (D.W.)
^moles/cell
mg/mg (D.W.)
References
Jorgensen (1976, 1983)
Jorgensen et aj_. (1978, 1981)
Nyholm (1978)
Jorgensen (1981)
Bierman (1976)
Bierman et al . (1980)
Lehman et al . (1975)
Jorgensen (1979)
Bierman (1976)
Bierman et al . (1980)
Lehman et al_. (1975)
Jorgensen (1979)
Bierman (1976)
Bierman et al . ( 1980)
Lehman et al_. (1975)
Jorgensen (1979)
Lehman et !]_. (1975)
Bierman et al . (1980)
Lehman et al_. (1975)
Auer and Canale (1982)
"Literature values.
-------
TABLE 6-14. MAXIMUM INTERNAL NUTRIENT CONCENTRATIONS
CO
-F*
Maximum Cell Concentration
Algal Type
Nitrogen
Phosphorus
Carbon Silicon
Units
References
Total
Phytoplankton 0.08-0.12
0.013-0.03
0.6
0.1 0.02
0.08-0.12** 0.013-0.035**
mg/mg (D.W.) Jorgensen (1976, 1983)
Jorgensen et al_. (1978, 1981)
mg/mg (D.W.) Nyholm (1978)
mg/mg (D.W.) Jorgensen et al_. (1981)
**Literature values.
-------
phosphorus to internal nitrogen ratio with a threshold ratio, rather than
computing the growth limitation factor for each nutrient and using the
smallest value.
Table 6-11 compares the growth formulations used in several variable
stoichiometry and fixed stoichiometry models. The comparisons show which
limiting factors are included, which formulations are used to compute
nutrient limitation, and how multiple limiting factors are combined.
6.4.4.3 Nutrient Uptake In Variable Stoichiometry Models
In fixed stoichiometry models, the nutrient composition of the algal
cells is assumed to remain constant, so nutrient uptake is directly related
to the algal growth rate by the stoichiometric ratio of nutrient mass to
cell biomass. The nutrient uptake rate can then be expressed as:
v = fz q (6-61)
Vrf
where v = nutrient uptake rate, mass nutrient/mass algae-time
f* = algal growth rate, I/time
q = constant internal nutrient concentration, mass
nutrient/biomass algae
The growth rates are assumed to be functions of the external nutrient
supplies (plus temperature and light) as computed by Michaelis-Menten type
relationships (Equation (6-50)).
In contrast, nutrient uptake rates in variable stoichiometry models are
functions of both internal nutrient levels in the cells and external
nutrient concentrations in the water. The general relationship is typically
of the form:
v = vmax(Tref} f(T) f(q»s) f(L) (6
where vm (T .) = maximum nutrient uptake rate at reference
max rer
335
-------
temperature T ,, mass nutrient/mass algae-time
f(T) = temperature function for uptake
f(q,s) = nutrient uptake limitation function
q = internal nutrient concentration, nutrient mass/cell
biomass
s = external nutrient concentration, mass/water volume
f(L) = light limitation function for uptake
The temperature and light functions for uptake are essentially the same as
those used for algal growth.
Variable stoichiometry models are distinguished primarily by the
specific formulations used for the uptake limitation function f(q,s). These
functions define the feedback between uptake rates and both internal
and external nutrient levels. Some formulations attempt a more mechanistic
approach, while others tend to be empirically based. In general, the uptake
rates increase with the external nutrient supplies but at the same time
decrease as the internal nutrient levels approach their saturation values.
Uptake rates approach zero when either external nutrients are depleted or
when internal nutrients reach their maximum saturated levels. However,
neither of these conditions can persist since nutrients are continually
recycled and since phytoplankton growth increases the algal biomass relative
to the internal nutrient mass which in effect reduces the internal nutrient
concentrations under conditions of restricted uptake.
The following formulations have been used to express internal and
external nutrient effects on uptake rates in variable stoichiometry models:
(6-64)
336
-------
f(q»s) =
(6-65)
with q = q. e\min / (6-67b)
where qm= = maximum internal nutrient concentration, mass
fflaX
nutrient/biomass algae
q • = minimum internal stoichiometric requirement (cell
quota), mass nutrient/biomass algae
q, = internal available nutrient concentration, mass
nutrient/ volume
q . . = minimum internal available nutrient concentration,
mass nutrient/volume
C. = internal concentration of uptake inhibitor, mass
nutrient/biomass algae
f . = fraction of total internal nutrient concentration
which acts as an inhibitor to nutrient uptake (this
corresponds to the acid-soluble polyphosphate
fraction of total internal phosphorus, or the
cellular free amino acid fraction of total internal
nitrogen)
Kul'Ku2'Ku3 = nalf~saturation constants for nutrient uptake,
mass nutrient/volume water
337
-------
K. = half-saturation constant for inhibition of nutrient
uptake, mass nutrient/biomass algae
K = affinity coefficient, volume/mass nutrient
cl
Equation (6-63) is used by Koonce and Hasler (1972), Equation (6-64) by
Lehman et al_. (1975) and Jorgensen (1976), Equation (6-65) by Rhee (1973)
and Park e_t a]_. (1980), Equation (6-66) by Di Toro (1980), Auer and Canale
(1982), and Canale and Auer (1982), and Equations (6-67a) and (6-67b) by
Bierman et al_. (1973, 1980).
Maximum nutrient uptake rates and half-saturation constants for uptake
are presented in Tables 6-15 and 6-16. Minimum cell quotas and maximum
internal nutrient concentrations were presented previously in Tables 6-13
and 6-14. Some of the more model specific parameters are presented in
Table 6-17.
Although variable stoichiometry models more realistically represent
nutrient uptake and cell growth than fixed stoichiometry models, they do it
at the expense of additional model complexity and computational costs.
Algal growth computations in variable stoichiometry models require shorter
time steps since the time scale for nutrient uptake is on the order of hours
while the time scale for algal growth is on the order of days. Also,
spatial variability in external and internal nutrient concentrations
complicates transport since algae with different internal stoichiometries
will be transported into the same model segment, requiring some type of
averaging procedure at each time step.
Another criticism of variable stoichiometry models is that more model
coefficients are required than in fixed stoichiometry models. Several
coefficients are required for both the uptake and growth formulations.
Since these coefficients must describe the response of species assemblages
rather than the single species evaluated in laboratory experiments, they
must be determined largely by model calibration. This introduces additional
uncertainty in the model results. Also, the data base for variable
stoichiometry coefficients is much smaller than for conventional Michaelis-
Menten parameters.
338
-------
TABLE 6-15. MAXIMUM NUTRIENT UPTAKE RATES
GO
OJ
UD
Maximum Uptake Rate
Algal Type
Total
Phytoplankton
Diatoms
Green Algae
Blue-green Algae
Flagellates
Chrysophytes
Coccolithophores
Benthic Algae
Nitrogen
0.15
0.012-0.03
0.14
0.01-0.035**
0.01-0.035**
0.0024**
0.015
0.125
0.72-4.32**
_n**
0. 3-120. xlO
1.52-8.33xlO~6**
0.060
0.125
-ft**
2.2-10.6x10 H
2.14-5.56xlO"6**
0.040
0.125
0.042xlO"6**
0.125
ft**
1.4-3.8x10 B
-i n**
4.-9.X10
Phosphorus
0.0014
0.0014-0.008
0.1
0.003-0.01**
0.003-0.01**
0.02-2.95**
0.024
0.500
q**
0.7-8. xlO y
0.133
0.500
ft**
1.2-4. xlO B
0.042-0.059
0.500
0.500
_7**
2.4x10 '
2.01-13.9xlO~9**
0.045
Carbon Silicon Units
0.55 I/day
0.40-1.21 I/day
I/day
0.2-0.7** I/day
0.2-1.4** I/day
pmoles/hr
I/ day
I/ day
I/day
2.6-950.X10"9** pmoles/cell-hr
0. 073-26. 6xlO"6** ymoles/cel 1-hr
I/day
I/ day
umoles/cell-hr
ymoles/cell-hr
I/day
I/day
u moles/eel 1-hr
I/day
pmoles/cell-hr
pmoles/cell-hr
pmoles/cell-hr
I/day
References
Jorgensen (1983)
Jorgensen et aj_. (1978, 1981)
Desorraeau (1978)
Jorgensen e^ aj_. (1978)
Jorgensen (1981)
Jorgensen (1979)
Bier-man (1976)
Bierman et aj_. (1980)
Jorgensen (1979)
Lehman et al. (1975)
Jorgensen (1979)
Bierman (1976)
Bierman et aj_. (1980)
Lehman e;t aj_. (1975)
Jorgensen (1979)
Bierman (1976)
Bierman et al . (1980)
Jorgensen (1979)
Bierman et al_. (1980)
Lehman e_t aj_. (1975)
Jorgensen (1979)
Lehman et_ a]_. (1975)
Auer and Canale (1982)
"Literature values.
-------
TABLE 6-16. HALF-SATURATION CONSTANTS FOR NUTRIENT UPTAKE
Phytoplankton
Group
Total Phytoplankton
Diatoms
Green Algae
Blue-green Algae
Dinoflagellates
Flagel lates
Chrysophytes
Coccol ithophores
Bacillariophyceae
Benthic Algae
Nitrogen
(mg/1)
0.2
0.2
0.05
0.0014-0.007**
0.030*
0.0028-0.105**
0.0014-0.130**
0.0042-0.105**
0.030*
0.0024-0.02**
0.0014-0.02**
0.0024-0.02**
0.030*
0.980**
0.0067-0.980**
0.0015-0.133**
0.0015-0.144**
0.0014-0.133*
0.030*
0.007-0.077**
0.0014-0.0084**
0.0014-0.0084**
0.0014-0.0084**
0.0014**
0.0014-0.0028**
0.0014-0.0043**
0.0063-0.120**
Half-Saturation Constant
Phosphorus Carbon Silicon
(mg/1) (mg/1) (mg/1)
0.02-0.03 0.5
0.02 0.5-0.6
0.07
0.0028-0.053**
0.060*
0.18-0.053 0.022-0.098**
0.0002-0.053** 0.0053-0.098**
0.020*
0.019-0.155**
0.0009-1.500**
0.015-0.060*
0.060*
0.016-0.496**
0.009-0.496**
0.125
References
Jorgensen (1976, 1983)
Jorgensen et_ al_. (1978)
Desormeau (1978)
Jorgensen (1979)
Bierman e_t al_. (1980)
Lehman et al_. (1975)
Eppley et al_. (1969)
Jorgensen (1979)
Bierman et al_. (1980)
Lehman et a]_. (1975)
Eppley et al_. (1969)
Jorgensen (1979)
Bierman et. al_. (1980)
Lehman et al_. (1975)
Jorgensen (1979)
Lehman et al_. (1975)
Eppley et al_. (1969)
Jorgensen (1979)
Bierman et, aj_. (1980)
Jorgensen (1979)
Lehman et al_. (1975)
Eppley e_t al_. (1969)
Jorgensen (1979)
Lehman et. aj_. (1975)
Eppley et aj_. (1969)
Jorgensen (1979)
Jorgensen (1979)
Auer and Canale (1982)
*Apparent half-saturation values under nutrient-starved conditions.
**Literature values.
340
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TABLE 6-17. MODEL-SPECIFIC NUTRIENT UPTAKE PARAMETERS
Model Parameter
Nutrient Type Value
Phosphorus K. 0.0001 g/m3
Ki 0.0007 mg/mg (D.W.)
fi 0.01%
K O.SlSxlO^l/mol
0.167xlO/l/mol
0. 518-2. OxlOn/mol
0.518 x 10°l/inol
K O.SOxlO^l/mol
a 0.50xlO°l/mol
0.90-1.0x10 1/mol
tt S/'i
0.5 yg/1
0.5 yg/1
qdmin 0.215xlO~ynol/l cell vol.
0.215x10 "imol/l cell vol.
0.107xlO"/mol/l cell vol.
Nitrogen KI- 0.0005 g/m3
f. 0.05%
K O.lOOxlO^l/mol
a O.lOOxlo'l/mol
O.lOOxlo'l/mol
0.100x10 1/mol
K O.lOxlO^l/mol
a O.lOxlo'l/mol
0.10x10 1/mol
I Is/'!
3. yg/1
3. yg/1
^Hm-in 0.267xlO"^mol/l cell vol.
0.267xlO~V>l/l cell vol.
0.267xlO~V>l/l cell vol.
Algal Type
Total Phytoplankton
Benthic Algae
Total Phytoplankton
Diatoms
Green Algae
Blue-green Algae
Flagellates
Di atoms
Green Algae
Blue-green Algae
Diatoms
Green Algae
Blue-green Algae
Flagellates
Diatoms
Green Algae
Blue-green Algae
Total Phytoplankton
Total Phytoplankton
Diatoms
Green Algae
Blue-green Algae
Flagellates
Diatoms
Green Algae
Blue-green Algae
Diatoms
Green Algae
Blue-green Algae
Flagellates
Diatoms
Green Algal
Blue-green Algae
Reference
Desormeau (1978)
Auer and Canale (1982)
Desormeau (1978)
Bierman et_ al_. (1980)
Bierman (1976)
Bierman et aj_. (1980)
Bierman (1976)
Desormeau (1978)
Desormeau (1978)
Bierman e;t aj_. (1980)
Bierman (1976)
Bierman et al_. (1980)
Bierman (1976)
1
Di Toro (1980) and Di Toro and Connolly (1980) have shown that since
the time scale for nutrient uptake is a fraction of the time scale for algal
growth and is usually much smaller than the time scale for changes in
external nutrient concentrations, many of the complexities of variable
stoichiometry models can be avoided by assuming cellular equilibrium with
341
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external nutrient concentrations at each time step. This allows algal
growth to be computed using conventional Michaelis-Menten kinetics, but at
the same time allows the internal stoichiometry of the algae to vary. Since
the cells are assumed to equilibrate immediately with the external nutrient
concentrations during transport, both the computational difficulties
associated with the rapid uptake dynamics and the problem of algae with
different internal stoichiometries being transported into the same model
segment are eliminated. Variable stoichiometry formulations are more
important to accurately simulating nutrient re-cycling than to computing
algal growth, so this scheme may be a reasonable compromise between the
variable stoichiometry formulations discussed above and conventional fixed
stoichiometry formulations.
6.5 RESPIRATION AND EXCRETION
Respiration and excretion are generally combined and modeled as a
single term which includes all metabolic losses and excretory processes.
These losses represent the difference between gross growth and net growth.
Since net growth (rather than gross growth) is typically reported in the
literature, some models lump respiration, excretion, and gross growth into a
single net growth term, rather than simulating each process separately.
However, it is generally more appropriate to compute growth and respiration
separately since growth rates are sensitive to nutrient supplies while
respiration rates depend primarily on temperature. Also, respiration and
excretion are important components of nutrient recycling, so these processes
are usually computed separately for use in the nutrient dynamic equations.
Most models express respiration (plus excretion) as either a constant
loss term or as a function of temperature. The general expression is:
r = r(Tref} VT) (6'68)
where r = rate of respiration plus excretion, I/time
i"(T *) = respiration rate at a particular reference temperature
Tref, I/time
f (T) = temperature function for respiration
342
-------
The temperature functions for respiration use the same formulations
discussed above for growth (Equations (6-5)" through (6-25)). Most models
use the same temperature function and coefficients for both processes. The
major approaches are 1) linear increases in respiration with temperature, 2)
exponential increases in respiration with temperature, and 3) temperature
optimum curves in which respiration increases with temperature up to the
optimum temperature and then decreases with higher temperatures. The most
commonly used-exponential formulation is the Arrhenius relationship with a
reference temperature of 20°C (Equation (6-15a)). Some models, for example
CE-QUAL-R1 (WES, 1982), use the left hand side of a temperature optimum
curve or a logistic equation (Equation (6-22a)) to define temperature
effects on respiration. This approach assumes respiration increases
exponentially at low temperatures, but eventually levels off to some maximum
value at higher temperatures.
A few models use formulations which relate the respiration rate to the
physiological condition of the algal cells. For example, Scavia (1980)
represents respiration as the sum of two components, 1) a low maintenance
rate representing periods of minimal growth, and 2) a rate which is directly
proportional to the photosynthesis rate (as defined by the growth limitation
factor):
?,K,C,Si} (6-69)
where r . (T .) = base respiration rate under conditions of minimal
minv ref
growth (poor physiological condition) at reference
temperature T ,., I/time
K (T f) = maximum incremental increase in respiration under
conditions of maximum growth (optimum physiological
condition) at reference temperature T -, I/time
Both rates are multiplied by a temperature adjustment function.
The MS.CLEANER model uses a similar formulation which expresses
respiration as the sum of endogenous respiration and photorespiration
343
-------
(Groden, 1977; Park et _al_. , 1980). The endogenous respiration is defined
as:
re = .0175 e'069T (6-70)
where r = endogenous respiration rate, I/time
T = temperature, C
Photorespiration is defined as a constant fraction of the temperature
adjusted maximum photosynthesis rate in early versions of MS. CLEANER
(Groden, 1977):
rP = Kpl
where r = photorespiration rate, I/time
K , = fraction of maximum photosynthesis rate which is oxidized
by photorespiration (typically 5 to 15%)
and as a fraction of the actual photosynthesis rate (including temperature,
light, and nutrient limitati'on effects) in later versions (Park et a! . ,
1980):
rp - Kp2 M (6-72)
where K « = fraction of actual photosynthesis rate which is oxidized by
photorespiration
MS. CLEANER also considers excretion as a separate loss term, in contrast
to most models which lump respiration and excretion together. Excretion is
formulated similar to photorespiration. However, since the excretion of
photosynthate and photorespiratory compounds relative to carbon assimilation
(photosynthesis) is highest at both low light levels and inhibitory high
light levels, the excretion rate is expressed as (Desormeau, 1978; Collins,
1980):
344
-------
e = K 1 - f(L) 11 (6-73)
x e \ /
where e = excretion rate, I/time
X
Kg = fraction of photosynthesis excreted
f(L) = light limitation factor
fj. = growth (photosynthesis) rate, including effects of
temperature, light, and nutrient limitation, I/time
Lehman et _al_. (1975), Jorgensen (1976), and Jorgensen^taj_. (1978,
1981) use variable stoichiometry formulations which relate the respiration
rate to the internal carbon levels of the cells. The ratio of the internal
carbon level to the maximum internal carbon level is used to define the
physiological state of the cells. The respiration rate increases with the
internal carbon level according to the equation:
- WTref> li^l (6-74>
where r (T f) = maximum respiration rate at reference temperature
max ret
Tref, I/time
C. = internal carbon level, mass carbon/biomass algae
C = maximum internal carbon level, mass carbon/biomass
max
al gae
Algal respiration rates are tabulated in Table 6-18.
6.6 SETTLING
Phytoplankton settling rates depend on the density, size, shape, and
physiological state of the phytoplankton cells, the viscosity and density of
the water, and the turbulence and velocities of the flow field. The
settling velocities for spherical particles in still water can be computed
from S.toke's law. Stoke1 s law can be modified to account for non-spherical
phytoplankton cells by using an "equivalent radius" and "shape factor" in
the formulation (Scavia, 1980):
345
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TABLE 6-18. ALGAL RESPIRATION RATES
Algal Type
Respiration
Rate (I/day)
•Reference
Temperature ( C)
References
Total
Phytoplankton
0.05 0.15
20°C
0.05 - 0.10
0.08
0.10
0.088 0.6
0.051
0.05
0.005 0.12*
0.05 0.2*
0.05 -.0.5*
0.02 0.8*
0.05 0.10**
0.05 - 0.20**
20°C
20°C
20°C
'opt
20°C
20°C
20°C
20°C
20°C
20°C
20°C
20°C
Di Toro et al. (1971, 1977)
O'Connor etai. (1975, 1981)
Thomann etal. (1974, 1975, 1979)
Di Toro & Matystik (1980)
Di Toro & Connolly (1980)
Thomann & Fitzpatrick (1982)
Salisbury et al_. (1983)
Chen & Orlob (1975)
Chen & Wells (1975, 1976)
Tetra Tech (1976)
Canale et al_. (1976)
Lombardo (1972)
Jorgensen (1976)
Jorgensen et^ a\_. (1978)
Brandes (1976)
Grenney & Kraszewski (1981)
Baca & Arnett (1976)
Smith (1978)
Roesner et jH_. (1980)
Duke & Masch (1973)
Grenney & Kraszewski (1981)
Collins & Wlosinski (1983)
Jorgensen (1979)
Diatoms
0.04 0.08
0.07 0.08
0.03 0.05
0.05 0.25
0.05 - 0.59**
20°C
20°C
20°C
opt
20°C
Thomann et^ al. (1979)
Di Toro & Connolly (1980)
Salisbury et al_. (1983)
Di Toro et al. (1971)
Porcella et_ al_. (1983)
Tetra Tech (1980)
Bierman (1976)
Bierman ejt al_. (1980)
Scavia et a].. (1976)
Scavia TJ980)
Bowie et al. (1980)
Collins & Wlosinski (1983)
(continued)
346
-------
TABLE 6-18. (continued)
Algal Type
Green Algae
Respiration
Rate (I/day)
0.05 0.07
0.05 0.25
Reference
Temperature ( C)
20°C
References
Tetra Tech (1980)
Porcella e_t al_. (1983)
Scavia et al . (1976)
0.03 0.05
0.01 0.46**
Blue-green Algae 0.05 0.065
0.05 0.25
0.03 0.05
0.10 0.92**
Dinoflagellates 0.047
opt
20°C
20°C
20°C
opt
20°C
20°C
20°C
Scavia TT980)
Bowie et al_. (1980)
Bierman (1976)
Bierman e_t al_. (1980)
Collins & Wlosinski (1983)
Tetra Tech (1980)
Porcella et al_. (1983)
Scavia et aj_. (1976)
Scavia T1980)
Bowie et_ al. (1980)
Bierman (1976)
Bierman et^ al_. (1980)
Collins & Wlosinski (1983)
O'Connor et al. (1981]
Flagellates 0.05
0.05 - 0.06
Chrysophytes 0.15 0.32**
Benthic Algae 0.02 0.1
0.44
0.1
0.02 0.8*
0.05 - 0.2*
20UC
20°C
20°C
20°C
Topt
20°C
20°C
20°C
Bierman et_ aj_. (1980)
Tetra Tech (1980)
Porcella et al. (1983)
Collins & Wlosinski (1983)
Tetra Tech (1980)
Bowie et al . (1980)
Porcella e_t al- (1983)
Auer and Canale (1982)
Grenney & Kraszewski (1981)
Grenney & Kraszewski (1981)
Smith (1978)
*Model documentation values.
**Literature values.
347
-------
(P - P )
where V = settling velocity, length/time
s 2
g = acceleration of gravity, length/time
R = equivalent radius (based on a sphere of equivalent volume),
length
3
p = density of the cell, mass/length
P = water density, mass/length
v = kinematic viscosity
F = shape factor
The shape factor has a value >1.0 and accounts for all factors which
reduce the settling velocities below that of an equivalent spherical
particle, for example increased drag due to diatom spicules, flat or
elongated cells, clusters or colonies of cells, etc. In a model of Lake
Ontario, Scavia (1980) used a shape correction factor of 1.3 for small
diatoms, 2.0 for large diatoms, and 1.0 for all other algal groups.
In practice, very few model s use Stoke's law as a model formulation
(Scavia _et £]_., 1976; Scavia, 1980; Park et a]_. , 1980) . Most model s lump
many species into a few algal groups, so representative values of the cell
radius, shape factor, and cell density are difficult to define, making this
level of detail unnecessary. Since the shape factor is really a calibration
parameter, it is more direct to simply use the settling velocity as a
calibration parameter. Also, Stoke's law does not account for turbulence
and flow velocities which tend to keep algae in suspension or resuspend
settled algae. Additional factors which further complicate settling include
the production of gas vacuoles or gelatinous sheaths which make some species
buoyant, and the fact that settling velocities may vary with the nutritional
state or physiological condition of the cells.
Settling rates are also partly dependent on the structure of the model.
For example, one-dimensional layered lake models typically use settling
velocities which are an order of magnitude lower than measured values or
348
-------
values used in two-or three-dimensional models which simulate hydrodynamic
processes (Scavia and Bennett, 1980). This is probably because one-
dimensional models do not adequately represent vertical transport process
such as upwelling or entrainment of phytoplankton in large-scale
circulations which effectively reduce the net settling rates (Scavia and
Bennett, 1980).
Because of the above factors, most models specify phytoplankton
settling velocities directly as model coefficients. The settling rate in
Equations (6-1) or (6-2) is generally expressed as:
V.
s = -r- (6-76)
d
V = settling velocity, length/time
where s = settling rate, I/time
V = settling velocity,
d = water depth, length
In layered models, algae settling in from the above layer, as well as
algae settling out of the layer, must be included in the formulation. This
also requires consideration of the bottom topography, since a fraction of
the algae will settle onto the bottom area associated with each layer.
Equation (6-76) is refined in some models by including a temperature
function which accounts for changes in settling velocities due to
temperature effects on the density and viscosity of water. The settling
rate is then expressed as:
V(T )
s = fs(T) (6-77)
where V (T ,) = settling velocity at reference temperature T -,
length/time
.f (T) = temperature adjustment function for the settling
velocity
349
-------
Typical examples of temperature adjustment functions include (Tetra Tech,
1980):
f,(T) =
where T = temperature in C.
or (Scavia and Park, 1976):
157.5
0.069T - 5.3T + 177.6
(6-78)
fs(T) = 1 + as T
(6-79)
where a = slope of settling velocity vs. temperature curve
Scavia _et al_. (1976) have also expanded the settling rate formulation to
account for variations in settling velocities due to the physiological
condition of the phytopl ankton cells. The basic assumption is that the
cells are healthiest and the settling rates smallest when neither light nor
nutrients are limiting growth. The settling rates are therefore expressed
as a function of the growth limitation factor f (L,P,N,C,Si ) . Potential
formulations include (Scavia et al_. , 1976; Scavia, 1980):
. _
' \f(L,P,N,C,STT+ K
set.
(6-80)
or
V (T }
, smaxv ref; f /-,-%
5 H T<-V' 1
f(L,P,N,C,Si)
(6-81)
where vsmax(Tref) = maximum settling velocity at reference temperature
Tref under P°°i" physiological condition,
length/time
Kset 'Kset = constants of tne settling formulations
A few models require specification of the settling rate s rather than
the settling velocity Vg as a model calibration coefficient. When used in
350
-------
this way, the settling rate may take on a wide range of values since it
depends as much on the water depth as the settling velocities of the algae.
Phytoplankton settling velocities are presented in Table 6-19.
Additional data are available in a review by Smayda (1970).
6.7 NONPREDATORY MORTALITY
Nonpredatory mortal ity accounts for all algal losses which are not
explicitly accounted for by the grazing term or other loss processes in the
model (for example, settling and respiration if they are not computed
explicitly). Nonpredatory mortality includes processes such as senescence,
bacterial decomposition of cells (parasitism), and stress-induced mortality
due to severe nutrient deficiencies, extreme environmental conditions, or
toxic substances. The nonpredatory mortality rate in Equations (6-1),
(6-2), or (6-3) is generally specified as a constant model coefficient.
This is in contrast to the predatory mortality or grazing rate which is
computed dynamically to reflect changes in the predator densities.
In some models, a temperature adjustment function is used with
nonpredatory mortality which results in:
m = m(Tref) fm(T) (6-82)
where m = nonpredatory mortality rate, I/time
m(T -) = nonpredatory mortal ity rate at reference temperatur*e
Tref» I/time
fm(T) = temperature function for mortality
The temperature functions for mortality generally use the same formulations
used for growth and respiration (Equations (6-5) through (6-25)). However,
if a temperature optimum curve is used for growth, the temperature function
for mortality will often use only the left hand portion of the curve to
produce a temperature response curve in which mortality increases with
temperature until some maximum mortality rate is reached.
351
-------
TABLE 6-19. PHYTOPLANKTON SETTLING VELOCITIES
Algal Type
Settling Velocity (m/day)
References
Total
Phytoplankton
Diatoms
0.05 0.5
0.05 0.2
0.02 0.05
0.4
0.03 0.05
0.05
0.2 0.25
0.04 0.6
0.01 - 4.0*
0. 2.0*
0.15 2.0*
0. - 0.2*
0. - 30.**
0.05 - 0.4
0.1 - 0.2
0.1 - 0.25
0.03 - 0.05
0.3 - 0.5
2.5
0.02 14.7**
0.08 - 17.1**
Chen & Orlob (1975)
Tetra Tech (1976)
Chen (1970)
Chen & Wells (1975, 1976)
O'Connor etjj]_. (1975, 1981)
Thomann etal_. (1974, 1975, 1979)
Di Toro & Matystik (1980)
Di Toro & Connolly (1980)
Thomann & Fitzpatrick (1982)
Canale et aj_. (1976)
Lombardo (1972)
Scavia (1980)
Bierman et^ al. (1980)
Youngberg (1977)
Jorgensen (1976)
Jorgensen j^t a^. (1978, 1981)
Baca & Arnett (1976)
Chen & Orlob (1975)
Smith (1978)
Duke & Masch (1973)
Roesner et^ al_. (1977)
Brandes (1976)
Jorgensen (1979)
Bierman (1976)
Bierman et^ a\_. (1980)
Thomann et a]_. (1979)
Di Toro & Connolly (1980)
Tetra Tech (1980)
Porcella et al_. (1983)
Canale £t al_. (1976)
Smayda & Boleyn (1965)
Lehman et al_. (1975)
Collins & Wlosinski (1983)
Jorgensen (1979)
(continued)
352
-------
TABLE 6-19. (continued)
Algal Type
Settling Velocity (m/day)
References
Green Algae
Blue-green Algae
Flagellates
Dinoflagellates
Chrysophytes
0.05 - 0.19
0.05 0.4
0.02
0.8
0.1 0.25
0.3
0.08 0.18**
0.27 - 0.89**
0.05 0.15
0.
0.2
0.1
0.08 0.2
0.10 0.11**
0.5
0.05
0.09 - 0.2
0.07 0.39**
8.0
2.8 6.0**
0.5
Coccolithophores 0.25 13.6
0.3 1.5**
Jorgensen ejt aj_. (1978)
Bierman (1976)
Bierman et_ al. (1980)
Canale et a]_. (1976)
Lehman et_ al_. (1975)
Tetra Tech (1980)
Porcella et. ah (1983)
DePinto e_t al. (1976)
Collins & Wlosinski (1983)
Jorgensen (1979)
Bierman (1976)
Bierman e_t al. (1980)
Canale et. al. (1976)
Lehman et al- (1975)
DePinto e_t al. (1976)
Tetra Tech (1980)
Porcella et ah (1983)
Collins & Wlosinski (1983)
Lehman et al. (1975)
Bierman ejt al. (1980)
Tetra Tech (1980)
Porcella et ah (1983)
Collins & Wlosinski (1983)
O'Connor et al. (1981)
Collins & Wlosinski (1983)
Lehman e_t al. (1975)
Collins & Wlosinski (1983)
Jorgensen (1979)
*Model documentation values.
**Literature values.
353
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A few models use more sophisticated formulations for nonpredatory
mortality which try to relate the mortality rate to the physiological
condition of the algal cells or to the size of the decomposer population
(De Pinto, 1979). For example, Scavia _et _aj_. (1976) use the value of the
growth limitation factor f (L ,P ,N ,C ,Si ) as a measure of cell health and
express the mortality rate as:
where m (T ,.) = maximum nonpredatory mortal i ty under poor
max ref K J J
physiological conditions at reference temperature
Tref, I/time
This assumes minimal mortality and algal decomposition when growth
conditions are optimal, and maximum mortality when conditions are severely
limiting.
Lehman ^t al . (1975) use a similar approach, but also include the
duration of growth limiting conditions in the formulation. They define the
mortal ity rate as:
m - WTref>
where T = number of days of suboptimal conditions (defined as ..
50 ma X
.05) , time
KSQ = coefficient defined as In2 divided by the number of days at
suboptimal conditions until m increases to % m
MS. CLEANER expresses nonpredatory mortality as a function of both the
internal nutrient concentrations and temperature such that the mortality
rate increases exponentially under conditions of either nutrient starvation
or critical ly high temperatures. The equation is (Desormeau, 1978; Park
et al., 1980):
Kn(Ncrit-f(P'N'C'S1))
-------
where K = nonpredatory mortality rate coefficient, I/time
K = exponent for nutrient starvation
f(P,N,C,Si) = variable stoichiometry nutrient limitation factor
for algal growth
N .. = critical value of f(P,N,C,Si) for starvation
O I I L
mortality
T .. = critical temperature for nonpredatory mortality
C i I L
This assumes that when the internal nutrient levels drop below the
subsistence quota, increased senescence, bacterial colonization, and cell
lysis occur.
Bierman _e_t jaj_. (1980) use a nonpredatory mortality function which
indirectly includes the size of the decomposer bacteria population in the
formulation. Although the bacteria are not modeled explicitly, they are
assumed to increase in proportion to the total algal concentration (the sum
of all algal groups in the model). Therefore, increases in the bacteria
associated with the bloom of one algal group will result in higher mortality
rates for all other groups since a higher decomposer population is
established. The equation is:
where K (T f) = nonpredatory mortality rate coefficient at reference
temperature T f, 1/time-algae
A. = concentration of algal group i, mass/volume
n = total number of algal groups
Nyholm (1978) uses a Michael is-Menten type saturation function of the
algal concentrations in his formulation for algal mortality:
355
-------
where m (T r) = maximum nonpredatory mortality rate at reference
max ref
temperature T f> I/time
A = algal concentration, mass/volume
K , = half-saturation constant for algal nonpredatory
ml
mortality, mass/volume
At high algal concentrations, this is equivalent to the basic first order
formulation (Equation (6-82)), while at very low algal levels, the mortality
rate is essentially a second order relationship analogous to Equation
(6-86). However, even though the mortality rate is second order at low
algal densities, the Michaelis-Menten term reduces the net rate at low
densities below the maximum first-order rate at high algal densities.
The Michaelis-Menten formulation is also used by Di Toro and Matystik
(1980), Di Toro and Connolly (1980), and Thomann and Fitzpatrick (1982) in
their formulation for the decomposition of organic matter (dead algal
cells), although a basic first-order formulation is used for algal
nonpredatory mortality. These models use the Michaelis-Menten formulation
to account for the effects of the bacterial population on decomposition
rates, assuming that decomposers (and the resulting decomposition rates)
increase in proportion to the algal densities at low concentrations, but
that other factors limit decomposition rates at high algal densities
(Di Toro and Matystik, 1980; Di Toro and Connolly, 1980). These mechanisms
could also be assumed for nonpredatory mortality.
Rodgers and Salisbury (1981) use a modified Michaelis-Menten
formulation for nonpredatory mortality which includes the effects of both
bacterial activity and the physiological condition of the algal cells on
algal decomposition:
" ' Vx
where M = algal growth rate, I/time
= half-saturation constant for algal nonpredatory mortality,
mass-time/volume
356
-------
The mortality rate is directly proportional to the algal biomass (an
indicator of bacterial activity) and inversely proportional to the algal
growth rate (an indicator of the physiological condition of the cells), both
through a saturation type relationship which limits the maximum rate.
Some models include formulations to account for stress-induced
mortality due to factors such as extreme temperatures or toxic substances.
Stress related mortality is typically modeled by expanding the nonpredatory
mortality term to include additional terms for these effects, for example:
m
= m(Tref) fm(T) + mT(Tref} fT(T) + mx fx(X) (6'89)
where mj(T e^) = thermal mortality rate at reference temperature
Tpef, I/time
fy(T) = thermal mortality response curve
m = toxic mortality rate, I/time
A
f (X) = dose-response curve for toxic mortality
A
X = concentration of toxicant, mass/volume
Toxic effects can also be included in the growth and respiration
formulations.
Algal nonpredatory mortality rates are presented in Table 6-20.
6.8 GRAZING
Algal grazing losses can be modeled in several ways, depending on 1)
whether predator populations are simulated in the model, and 2) whether
alternate food items are available for the predators.
When predators are not explicitly modeled, predator-prey dynamics
cannot be simulated, so grazing effects are typically handled by either
assuming a constant grazing loss which is specified by the user as a model
input parameter:
357
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TABLE 6-20. ALGAL NONPREDATORY MORTALITY RATES
Nonpredatory Mortality
Algal Type Rate (I/day)
References
Total
Phytoplankton
Diatoms
Benthic Algae
0.02
0.003 0.17
0.03
0.005 0.10
0.01 0.1
0.03
0.- 0.8
Thomann & Fitzpatrick (1982)
Baca & Arnett (1976)
Scavia et al_. (1976)
Salas & Thomann (1978)
Jorgensen (1976)
Jorgensen et al_. (1978)
Scavia et al. (1976)
Tetra Tech (1980)
Bowie et a]_. (1980)
Porcella et al. (1983)
G = constant
(6-90)
where G = loss rate due to grazing, mass algae/time
or by assuming a loss rate which is directly proportional to the algal
densities (e.g., RECEIV-II (Raytheon, 1974)):
G = e A
or
G =
(6-91)
(6-92)
where e
(T .)
v ref;
VT)
= grazing rate coefficient, I/time
= algal biomass or density, mass or mass/volume
grazing rate coefficient at reference temperature
Tref, I/time
temperature function for grazing
358
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The second formulation is equivalent to that often used for non-predatory
mortality (Equation (6-82)), so both nonpredatory mortality and grazing
losses are typically combined into a single total mortality term when
predator populations are not directly simulated:
mtot
where m. . = total mortality rate, I/time
mtot^ref^ ~ total mortality rate at reference temperature T ,,
I/time
fm(T) = temperature function for mortality
The temperature functions used for grazing are the same as those discussed
previously for algal growth, respiration, and mortality (Equations (6-5) to
(6-25)).
Many general water quality models include a single zooplankton group
to provide a more realistic grazing formulation for algae (Baca et al . ,
1973; Johanson _et ^]_. , 1980; Najarian and Harleman, 1975). The zooplankton
are often added only to obtain better simulations of algal dynamics, rather
than to evaluate the zooplankton dynamics of the system. The coupled algae
and zooplankton equations provide the major features of predator-prey
interactions since the algal grazing rate is defined as a function of the
zooplankton density which in turn varies dynamically with the food supply
(algal concentration). The algal grazing rate in these models is typically
expressed either in terms of a zooplankton filtration rate:
G = Cf A Z (6-94)
or G = Cf(Tref) fg(T) A Z (6-95)
where Cf = zooplankton filtration rate, water volume/mass
zooplankton-time
359
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Z = zooplankton biomass or concentration, mass or
mass/ volume
Cf(T f) = filtration rate at reference temperature T ,.,
water volume/mass zooplankton-time
or in terms of a zooplankton ingestion rate:
G = C Z (6-96)
or G = Cg(Tref) fg(T) Z (6-97)
where C = zooplankton ingestion rate, mass algae/mass
zooplankton-time
C (T ,.) = ingestion rate at reference temperature T f)
gv ref s ref
mass algae/mass zooplankton-time
Ingestion rates are often back-calculated from computed zooplankton
growth rates based on the equation (Chen and Orlob, 1975; Smith, 1978; Tetra
Tech, 1979; WES, 1982):
Cg = (6-98)
where g = zooplankton growth rate, I/time
E = zooplankton assimilation efficiency
In this approach, zooplankton growth rates are first computed as a function
of food supply and temperature, and then the amount of algae which would
have to be consumed to produce the growth is computed from Equation (6-98).
The alternative approach is to specify or compute the ingestion rates
directly, and then calculate the zooplankton growth rates based on the
amount of food consumed and the assimilation efficiencies. Specific
formulations for zooplankton filtration rates, ingestion rates, growth
rates, and assimilation efficiencies are discussed in detail in Chapter 7.
360
-------
Models which simulate only a single algal and zooplankton group tend
to oversimplify predator-prey dynamics since a single constituent represents
all primary producers and another single constituent represents all
consumers. This ignores the complexities of the food web, as well as
differences in foraging strategies, grazing rates, and food preferences
between different types of predator organisms. This approach may be
adequate in short term simulations where one group of phytopl ankton and
zooplankton are dominant. However, in long term simulations, more than one
group of algae and zooplankton should be used to adequately simulate
predator-prey interactions and population dynamics.
Algal grazing rates in multi-group models are functions of alternative
food sources and food preferences, as well as predator densities, algal
densities, and temperature. The basic grazing formulations are essentially
the same as those mentioned above for a single zooplankton group, except
that 1) grazing losses must be considered for each potential predator which
grazes the algae, and 2) total grazing rates calculated for a given predator
must be partitioned among the various food items which it consumes. Some
models also consider differences in the ingest ion or assimilation
efficiencies between different food items (Scavia _et aTL , 1976; Park et al . ,
1980), and differences in the feeding behavior of different zooplankton
groups (e.g., non-selective filterers, selective filterers, carnivorous
raptors, omnivores, etc.) ( Canal e _et aj_. , 1975, 1976; Park _et aj_. , 1980).
Grazing losses for non-selective feeders can be partitioned between
different algal groups by distributing them in proportion to the algal
concentrations:
(6-99)
361
-------
where G.. = loss rate of algal group i due to grazing by zooplankton
group j, mass algae/time
C. = total ingestion rate of zooplankton group j on all food
J
items, mass food/mass zooplankton-time
A. = biomass or concentration of algal group i, mass or
mass/volume
F, = biomass or concentration of potential food item k consumed
by zooplankton group j, mass or mass/volume
n = number of potential food items
Z. = biomass or concentration of zooplankton group j, mass or
J
mass/volume
When grazing is expressed in terms of a filtration rate this partitioning is
done automatically since the grazing losses are simply the algal
concentrations times the volumetric filtration rates.
The above expression can be modified to account for selective feeding
behavior by using food preference factors. These are weighting factors
which reflect the probability that a given food will be consumed relative to
the others when all foods are present in equal concentrations. The
preference factors account for feeding differences due to factors like food
particle size and shape, desirability and quality of food, and zooplankton
feeding behavior. The grazing losses for each algal group subject to
selective feeding can be expressed as:
P. . A.
G..= C. —U—1— Z. (6-100)
ij J n j v
k=l kj' k
where P.. = preference factor for zooplankton j grazing on algal group
i
P.. = preference factor for zooplankton j grazing on food item k
The total ingestion rates C. for each predator are the same as discussed
J
above for a single zooplankton group (Equations (6-94) through (6-98)).
362
-------
When several predators are modeled, the total grazing loss for a given
algal group is the sum of the grazing losses from each predator:
6. = £G... (6-101)
where G. = loss rate for algal group i due to grazing by all predators,
mass algae/time
n = total number of predators grazing on algal group i
Any of the previous formulations can be used to define the incremental
grazing rates G. . associated with each predator.
Zooplankton grazing rates are tabulated in Chapter 7, along with more
detailed descriptions of the grazing formulations for zooplankton.
6.9 SUMMARY
Phytopl ankton and attached algae are generally modeled as a biomass
pool using the same mass balance approach used for nutrients and other
constituents. The simpler models lump all algae into a single group, while
more complex models distinguish between different functional groups such as
green algae, diatoms, and blue-green algae. Single-group models are
commonly used in rivers, while multi-group models are more common in lakes
where long-term simulations of the seasonal succession of different types of
phytoplankton are important.
Algal dynamics depend on growth, respiration, excretion, settling,
nonpredatory mortality, and predation. Although some of these processes can
be measured in the field or laboratory, most of the coefficients defining
the process rates are usually determined by model calibration. This is
necessary since the rates will vary with environmental conditions such as
temperature, light, nutrient concentrations, and predator densities as well
as with the species composition of the algae, all of which change
continually with time. Literature values from laboratory experiments are
useful for establishing reasonable ranges for the coefficients. However,
363
-------
specific experimental results are difficult to apply directly since
experiments typically use a single species rather than the species
assemblages represented in models, and since experimental conditions may not
represent conditions in the field. Model constructs must be relied upon to
describe the effects of changing environmental and ecological conditions on
the process rates.
Most processes in algal models are assumed to be temperature
dependent. Three major approaches have been used to describe these effects:
linear temperature response curves, exponential curves, and temperature
optimum curves. The exponential Arrhenius relationship is commonly used
when only one algal group is simulated, while temperature optimum curves are
more common in multi-group models.
The most important and complex formulations in algal models are the
growth formulations. Growth is a function of temperature, light, and
nutrients. Light limitation is typically defined by either a saturation
type relationship or a photoinhibition relationship in which growth
decreases at light intensities above the optimum. Most models use
Michaelis-Menten kinetics to describe nutrient limitation effects and assume
the nutrient composition of the algal cells rema.ins constant. More
sophisticated models allow the internal stoichiometry of the algae to vary
with changes in the external nutrient concentrations. These models simulate
nutrient uptake and algal growth as two separate steps. Nutrient uptake is
first computed as a function of both the internal nutrient levels in the
cells and the external concentrations in the water. Algal growth is then
computed based on the internal nutrient concentrations in the cells.
Various formulations have been used to describe uptake and growth kinetics
in variable stoichiometry models. These formulations are more complex and
involve more model coefficients than fixed stoichiometry models.
Most models use simple temperature-dependent first-order relationships
to describe respiration, settling, and nonpredatory mortality. A few models
include the effects of the physiological condition of the algae on these
processes by making them a function of the growth rate, growth limitation
factor, or internal nutrient level (in variable stoichiometry models). Some
364
-------
models also include the effects of the decomposer bacteria population on
nonpredatory mortality. These' latter effects are modeled indirectly by
assuming the decomposers increase in proportion to the algal densities and
using algal concentrations as an indicator of the bacterial population,
rather than by simulating the decomposers directly. Both second-order
mortality formulations and Michaelis-Menten type saturation relationships
have been used to describe these effects.
Algal grazing is usually modeled as a first-order loss when
zooplankton are not simulated. When zooplankton are modeled, algal grazing
is a function of the algal densities, zooplankton densities, and the
zooplankton filtration rates or consumption rates. In multi-group models
which include several algal and zooplankton groups, selective feeding
behavior can be simulated by including food preference factors in the
grazing formulations.
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J.E. Titus, P.R. Weiler, J.W.
Bloomfield, H.H. Shugart, Jr., R.S. Booth,
J.F. Koonce, D. Scavia, M.S. Adams,
Dettmann, J.A. Hoopes, D.D. Huff, S. Katz,
J. LaRow, D.C. McNaught, J.L. Peterson,
Wilkinson, and C.S. Zahorcak. 1974. A
Generalized Model for Simulating Lake Ecosystems. Simulation, 23(2): 33-50.
Park, R.A. , C.D. Collins, O.K. Leung, C.W. Boylen, J. Albanese, P.
deCaprariis, and H. Forstner. 1979. The Aquatic Ecosystem Model
MS.CLEANER. Proc. of First International Conf. on State of the Art of Ecol.
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Park, R.A., T.W. Groden, and C.J. Desormeau. 1979. Modifications to the
Model CLEANER Requiring Further Research. Jji: Perspectives on Lake
Ecosystem Modeling. D. Scavia and A. Robertson (eds.). Ann Arbor Science
Publishers, Ann Arbor, Michigan, pp. 87-108.
Park, R.A., C.D. Collins, C.I. Connolly, J.R. Albanese, and B.B. MacLeod.
1980. Documentation of the Aquatic Ecosystem Model MS.CLEANER. Rensselaer
Polytechnic Institute, Center for Ecological Modeling, Troy, New York. For
U.S. Environmental Protection Agency, Environmental Research Laboratory,
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Patten, B.C. 1975.
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A Reservoir Cove Ecosystem Model. Trans. Am. Fish
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Jji: Systems~Analysis and Simulation in Ecology. Patten, B.C. (ed.).
Academic Press, New York, New York. pp. 206-415.
Porcella, D.B., T.M. Grieb, G.L. Bowie, T.C. Ginn, and M.W. Lorenzen. 1983.
Assessment Methodology for New Cooling Lakes, Vol. 1: Methodology to Assess
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California. For Electric Power Research Institute. Report EPRI EA-2059.
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Raytheon Company, Oceanographic & Environmental Services. 1974. New England
River Basins Modeling Project, Vol. Ill - Documentation Report, Part 1 -
RECEIV-II Water Quantity and' Quality Model. For Office of Water Programs,
U.S. Environmental Protection Agency, Washington, D.C.
Rhee, 6.Y. 1973. A Continuous Culture Study of Phosphate Uptake, Growth
Rate and Polyphosphate in Scenedesmus sp. J. Phycol., 9:495-506.
Rhee, G.Y. 1978. Effects of N:P Atomic Ratios and Nitrate
Algal Growth, Cell Composition and Nitrate Uptake. Limnol,
23:10-25.
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Rodgers, P. and D. Salisbury. 1981. Water Quality Modeling of Lake
Michigan and Consideration of the Anomalous Ice Cover of 1976-1977.
J. Great Lakes Res.9 7(4):467^480.
Roesner, L.A., P.R. Giguere, and D.E. Evenson. 1981.
Documentation for the Stream Quality Model QUAL-II. U,
Protection Agency, Athens, Georgia. EPA 600/9-81-014.
Computer Program
S. Environmental
Roesner, L.A., P.R. Giguere, and D.E. Evenson. 1981. User's Manual for the
Stream Water Quality Model QUAL-II. U.S. Environmental Protection Agency,
Athens, Georgia. EPA-600/9-81-015.
Salas, H.J. and R.V. Thomann. 1978. A Steady-State Phytoplankton Model of
Chesapeake Bay. Journal WPCF, 50(12)=2752-2770.
Salisbury, O.K., J.V. DePinto, and T.C. Young. 1983. Impact of Algal-
Available Phosphorus on Lake Erie Water Quality: Mathematical Modeling.
For U.S. Environmental Protection Agency, Environmental Research Laboratory,
Duluth, Minnesota.
Scavia, D. 1979. Examination of Phosphorus Cycling and Control of
Phytoplankton Dynamics in Lake Ontario with an Ecological Model. J. F,ish.
Res. Board Can., 36:1336-1346.
Scavia, D.
8:49-78.
1980. An Ecological Model of Lake Ontario. Ecol. Modeling,
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Formulation of a Generalized Model for Simu
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Deciduous Forest Biome Memo Report'75-4.
R.A. Park. 1975. The
ating Aquatic Macrophyte
Troy, New York. Eastern
Scavia, D., B.J. Eadie, and A. Robertson. 1976. An Ecological Model for
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Scavia, D. and R.A. Park. 1976. Documentation of Selected Constructs and
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Shugart, H.H., R.A. Goldstein, R.V.
A Terrestrial Ecosystem Energy Model
264.
O'Neill, and
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J.B. Mankin. 1974. TEEM:
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373
-------
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374
-------
Chapter 7
ZOOPLANKTON
7.1 INTRODUCTION
Zooplankton are included in water quality models primarily because of
their effects on algae and nutrients. Algal dynamics and zooplankton
dynamics are closely tied through predator-prey interactions. Nutrient
dynamics are also influenced by zooplankton since zooplankton excretion is
an important component of nutrient recycling, and because of the effects
zooplankton have on algal dynamics. These interrelationships are
particularly important for long-term simulations in lakes and estuaries
since both zooplankton and algal densities may change by orders of magnitude
over periods of several months.
As with phytoplankton, zooplankton have been modeled both as a single
constituent representing total zooplankton and as several functional groups.
The functional groups may represent different feeding types (e.g.,
herbivores, carnivores, omnivores, non-selective filter feeders, selective
filter feeders, etc.) or different taxonomic groups (cladocerans, copepods,
rotifers, etc.). While many models use only one group, multiple-group
models more realistically represent trophic interactions since, for example,
herbivorous zooplankton can be distinguished from carnivorous species.
However, multi-group models require more coefficients and model parameters,
as well as more detailed information for model calibration.
Zooplankton dynamics are governed by the same general processes as
.phytoplankton: growth, respiration and excretion, predation, and
nonpredatory mortality. The major difference is that zooplankton are not
subject to settling losses since they are motile and migrate vertically in
the water column, typically in a diurnal pattern. As a result, zooplankton
375
-------
are usually simulated using the same types of equations and formulations as
used for phytoplankton. The general zooplankton equation which forms the
basis of almost all models is:
§ •
-------
TABLE 7-1. GENERAL COMPARISON OF ZOOPLANKTON MODELS
Model
(Author)
AQUA- IV
CE-QUAL-R1
CLEAN
CLEANER
MS. CLEANER
EAM
ESTECO
EXPLORE-1
HSPF
LAKECO
MIT Network
WASP
WQRRS
Blerman
Canale
Jorgensen
Scavia
Number of Groups
Zoo-
lankton
1
1
3
3
5
3
1
1
1
1
1
2
1
2
9
1
6
Phyto-
plankton Fish
1
2 3
2 3
3 3
4 8
4 20
2 3
1
1
2 3
1
2
2 3
5
4
1 I
5
Zooplankton
Growth
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Processes
Respiration
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Computed Separately In Model
Nonpredatory
Mortality
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Predatory
Mortality
X
X
X
X
X
X
X
X
X
X
X
X
Zooplankton Units
Dry Wt. Other
Blomass Carbon Nutrient
X
X
X
X
X
X
X
X
X
X
N
X
X
X
X
X
X
Reference
Baca & Arnett (1976)
WES (EWQOS) (1982)
Bloomfleld e_t al_. (1973)
Scavia 4 Park (1976)
Park e_t al_. (1980)
Tetra Tech (1979, 1980)
Brandes 8, Masch (1977)
Baca e_t al_. (1973)
Johanson et al . (1980)
Chen & Orlob (1975)
Harleman et al_. (1977)
D1 Toro e_t al_. (1981)
Smith (1978)
Bierman £t aj_. (1980)
Canale e_t al_. (1975, 1976)
Jorgensen (1976)
Scavia et^ aj[. (1976)
CO
-------
nonpredatory mortality are generally direct functions of temperature, and
predation is indirectly related through temperature effects on the
consumption rates of zooplankton predators. In most models, the temperature
response formulations used for zooplankton are identical to those used for
phytoplankton, and the same temperature function is generally used for all
processes affecting a given zooplankton group. The major differences in the
response functions between different organisms are the particular
coefficient values used to define the shapes and slopes of the response
curves, the optimum temperatures, and the upper-and lower lethal limits. A
few models use different formulations for each process. For example, CE-
QUAL-R1 (WES, 1982) uses an optimum curve for growth, a logistic equation
for respiration, and a reverse logistic equation for nonpredatory mortality.
The various formulations used to define temperature effects are
described in detail in the algal growth section of the report
(Section 6.3.1), and they will not be repeated here. In general, all
formulations can be classified as either linear response curves, exponential
response curves, or temperature optimum curves which exhibit maximum process
rates at the optimum temperature and decreasing rates as the temperature
moves away from the optimum.
7.3 GROWTH
Zooplankton growth formulations represent increases in the biomass of
the population due to both reproduction and the growth of individuals. The
growth rate depends on the amount of food which is ingested and assimilated,
and is therefore a function of food densities, ingestion rates, and
assimilation efficiencies. Part of the assimilated food goes into
individual growth and metabolic losses, and part goes into reproduction.
Both ingestion rates and assimilation efficiencies vary according to
many factors, including (Leidy and Ploskey, 1980):
• Zooplankton factors such as species, age, size, feeding type,
sex, reproductive state, and physiological or nutritional
state
378
-------
• Food related factors such as food concentration, type,
particle size, quality, and desirability of the food
• Temperature
Ingestion rates also vary on a diurnal basis, with maximum feeding rates
typically occurring at night. Peak nighttime grazing rates have been shown
to range from 2 to 27 times the minimum daytime rates (Leidy and Ploskey,
1980).
Almost all zooplankton growth formulations are based on the following
fundamental relationship:
9Z = Cg E (7-2)
where g = zooplankton growth rate, I/time
C = ingestion or grazing rate, mass food/mass zooplankton-time
E = assimilation efficiency, fraction
Since most zooplankton are filter feeders, the ingestion rate is often
expressed in terms of a volumetric filtration rate multiplied times the
total available food concentration. In this case, the above equation
becomes:
gz = Cf FT E (7-3)
where Cf = zooplankton filtration rate, water volume/mass zooplankton-
time
FT = total available food concentration, mass/volume
For raptorial feeders, the previous equation (Equation (7-2)) is generally
used.
The simplest growth formulations assume constant filtration rates and
assimilation efficiencies (Figure 7-1). For this situation, the growth rate
379
-------
LU
fe
CC
FOOD CONCENTRATION, FT
Figure 7-1. Growth rate and grazing rate as a function of food supply
for zooplankton with constant filtration rates and
assimilation efficiencies (adapted from Leidy and
Ploskey, 1980).
is directly proportional to the food supply. More sophisticated models
include more complex formulations for the ingestion (or filtration) rates
and the assimilation efficiencies to account for variability due to factors
like food densities, food types, different feeding methods, and temperature
effects on feeding and growth (Canale eit al., 1975, 1976; Scavia et al.,
1976; Scavia, 1980; Scavia and Park, 1976; Park et al., 1975, 1979, 1980).
The effects of food density and temperature on zooplankton growth rates can
be expressed in general functional form as:
or
Emax(Tref) f(T)
WTref>
(7-4)
ff(FpF2,...Fn)
(7-5)
where C (T ,.)
gmaxv ref
maximum ingestion rate at reference temperature
Tref under conditions of saturated feeding
380
-------
(excess food supply), mass food/mass
zooplankton-time
= maximum filtration rate at reference
temperature T f, water volume/mass
zooplankton-time
E (T .p) = maximum assimilation efficiency at reference
temperature T f, fraction
f(T) = temperature function for ingest ion or
filtration and assimilation
f (FpF2,...Fn) = growth limitation factor for ingestion
j
formulation (Equation (7-2)) accounting for
food density effects on ingestion rates and/or
assimilation rates (where [ri»F2''"|rn are the
concentrations of the potential food items)
ff(F,,Fp,...F ) = growth limitation factor for filtration
formulation (Equation (7-3)) accounting for
food density effects on filtration rates and/or
assimilation rates
In some models, the maximum ingestion rate and the maximum
assimilation efficiency are combined into a single parameter representing
the maximum growth (or assimilation) rate (Chen and Orlob, 1972, 1975; Chen
et^jfL, 1975; Jorgensen, 1976; Jorgensen et aU, 1978, 1981, 1983; Najarian
and Harleman, 1975; Smith, 1978; WES, 1982; Tetra Tech, 1979). In this
case, Equation (7-4) becomes:
9z ' 3max f fg (?-6)
where ci (T f) = maximum zooplankton growth rate at reference
~rnax rer
temperature T f, I/time
Maximum consumption rates, filtration rates, and growth rates are
presented in Tables 7-2, 7-3, and 7-4, respectively.
331
-------
TABLE 7-2. ZOOPLANKTON MAXIMUM CONSUMPTION RATES
Zooplankton Maximum
Group Consumption Rate (I/day)
Total
Zooplankton 0.8
0.35 - 0.50
0.24 1.2**
Omnivores 1.4
0.43
Carnivores 1.6
0.7
Fast Ingesters 0.7
Slow Ingesters 0.1
Cladocerans 1.6 1.9
0.045 13.8**
0.045 2.3**
References
Scavia & Park (1976)
Bierman (1976)
Collins & Wlosinski (1983)
Scavia (1980)
Bowie et al. (1980)
Canale et al_. (1976)
Scavia et al_. (1976)
Canale et al_. (1976)
Bierman ejt al_. (1980)
Bierman ejt al_. (1980)
Scavia et al. (1976)
Scavia TJ980)
Bowie et al. (1980)
Leidy & Ploskey (1980)
Collins & Wlosinski (1983)
Copeods
Rotifers
Mysids
1.7 1.8
0.10 0.47**
1.8 2.2
3.44**
3.44**
1.0 - 1.2
Scavia et al_. (1976)
Scavia TT980)
Bowie et al- (1980)
Collins & Wlosinski (1983)
Scavia et al. (1976)
Scavia TJ980)
Bowie ejt al. (1980)
Leidy & Ploskey (1980)
Collins & Wlosinski (1983)
Scavia et al- (1976)
Scavia TT.980)
Bowie et a]_. (1980)
**Literature values.
382
-------
TABLE 7-3. ZOOPLANKTON MAXIMUM FILTRATION RATES
Zooplankton Group
Total Zooplankton
Herbivores
Carnivores
Cladocerans
Copepods
Rotifers
Maximum
Filtration Rate
0.13
0.05
0.8
0.7
1.0
3.5
0.2
0.192
0.2
0.009
0.18
0.18
1.0
0.05
0.161
0.05
0.02
0.02
0.006
0.6
0.6
0.007
1.2
- 0.2*
1.10**
- 1.4
3.9
4.0
1.6**
0.682**
1.6**
177**
9.4**
9.4**
6.5
- 2.2**
2.21**
2.2**
- 4.1**
- 5.28**
35.**
1.5**
- 1.5**
0.576**
Units
1/mgC-day
1 /mg C- day
l/mg(D.W.)-day
1/mgC-day
1/mgC-day
l/mg(D.W.)-day
l/mg(D.W.)-day
l/mg(D.W.)-day
l/mg(D.W.)-day
ml/animal-day
ml/animal-day
ml/animal-day
l/mg(D.W.)-day
l/mg(D.W.)-day
l/mg(D.W.)-day
l/mg(D.W.)-day
ml/animal-day
ml/animal-day
ml/animal-day
l/mg(D.LJ.)-day
ml/animal-day
ml/animal-day
References
Di Toro et al . (1971)
O'Connor et al_. (1975, 1981)
Baca & Arnett (1976)
Di Toro et al_. (1971)
Thomann et al . (1975, 1979)
Di Toro & Connolly (1980)
Di Toro & Matystik (i960)
Salisbury et a]_. (1983)
Thomann et al . (1975, 1979)
Di Toro & Connolly (1980)
Di Toro & Matystik (1980)
Salisbury et jH_. (1983)
Canale et_ al_. (1976)
Di Toro et al. (1971)
Lombardo (1972)
•Jorgensen (1979)
Leidy & Ploskey (1980)
Wetzel (1975)
Jorgensen (1979)
Canale et al_. (1976)
Di Toro et al_. (1971)
Lombardo (1972)
Jorgensen (1979)
Wetzel (1975)
Jorgensen (1979)
Leidy & Ploskey (1980)
Di Toro £t al_. (1971)
Jorgensen (1979)
Leidy & Ploskey (1980)
*Model documentation values.
**Literature values.
383
-------
The temperature function f(T) in the above equations uses the same
types of formulations discussed previously for phytoplankton. Experimental
results suggest optimum type response curves for short term changes in
temperature, but more of a linear response curve when acclimation has time
to occur (Leidy and Ploskey, 1980). Work by Geller (1975) indicates
acclimation times may range from 4 to 6 weeks, which is short enough for
zooplankton to acclimate to the typical seasonal variations in temperature,
but not to rapid changes (for example, thermal plume effects). However,
since feeding is expected to slow down or cease as the temperature
approaches the upper lethal limit, an optimum type response curve is
appropriate if it is skewed so that the optimum occurs near the upper lethal
limit. Table 7-5 presents a comparison of the temperature' adjustment
functions used in several zooplankton models.
TABLE 7-4. ZOOPLANKTON MAXIMUM GROWTH RATES
Zooplankton Group
Maximum
Growth Rate (I/day)
References
Total Zooplankton
0.15 0.25
0.175 0.2
0.1 0.3*
Chen (1970)
Chen & Orlob (1975)
Chen & Wells (1975, 1976)
Jorgensen (1976)
Jorgensen et jil_. (1978)
U.S. Army Corps of Engineers (1974)
Brandes (1976)
Smith (1978)
Cladocerans
Copepods
Rotifers
Mysids
0.15 0.30** Jorgensen (1979)
0.35 0.5 Tetra Tech (1980)
Porcella et a]_. (1983)
0.27 0.74** Jorgensen (1979)
0.5 Tetra Tech (1980)
0.44 0.45 Porcella et al_. (1983)
0.24 - 0.76** Jorgensen (1979)
0.14 Tetra Tech (1980)
*Model documentation values.
**Literature values.
384
-------
TABLE 7-5. COMPARISON OF TEMPERATURE ADJUSTMENT
FUNCTIONS FOR ZOOPLANKTON GROWTH AND CONSUMPTION
Temperature Formulation
Model
(Author) Linear Exponential
AQUA- IV
CE-QUAL-R1
CLEAN
CLEANER
MS. CLEANER
EAM
ESTECO 6-14
EXPLORE-1 X
HSPF 6-14
LAKECO 6-14
MIT Network
WASP X
WQRRS
Bierman X
Canale piecewise
linear
Jorgensen
Scavia
(Equation No. )
Optimum Other
Curve Curve
none
6-24
6-19
6-19
6-19
6-24
6-25
6-24
6-18
6-19
Reference
Temperature
Topt
Topt
Topt
Topt
Topt
20°C
1°C
20°C
20°C
Topt
1°C
Topt
20°C
1°C
Topt
Topt
Reference
Baca & Arnett (1976)
WES (EWQOS) (1982)
Bloomfield e_t al_. (1973)
Scavia & Park (1976)
Park et aj_. (1983)
Tetra Tech (1979, 1980)
Brandes & Masch (1977)
Baca et. al. (1973)
Johanson jrt al_. (1980)
Chen & Orlob (1975)
Harleman et aJL (1977)
Di Toro et al. (1981)
Smith (1978)
Bierman et. aj_. (1980)
Canale et. aj_. (1975, 1976)
Jorgensen (1976)
Scavia e_t al. (1976)
7.3.1 Growth Limitation
The growth limitation functions used in the above equations,
f (FpF2,...F ) and ff (FpF2,.. .Fn), are somewhat different since the latter
function is multiplied times the total available food concentration FT to
give the net grazing rate. Therefore:
fg(F1,F2,...Fn)^ff(F1,F2,...Fn)
(7-7)
385
-------
Both functions typically represent some type of saturation response to
feeding, assimilation, and growth. Experimental observations show that at
low food concentrations, zooplankton ingestion rates increase with increases
in the food supply. For filter feeders which are filtering water at a
constant rate, the grazing rate is directly proportional to th'e food
concentration (Figure 7-1). Grazing rates for predatory zooplankton also
increase with the food supply at low food concentrations since less energy
and time are required to find and capture prey items as the prey density
increases. However, as food becomes more abundant, the grazing rates
eventually become saturated and level off at some maximum value after which
the grazing rate becomes independent of the food supply. Filter feeders can
regulate their ingestion rates at high food levels by reducing their
filtering rates as the food concentration increases. At low concentrations,
the feeding rate is limited by the ability of the zooplankton to filter
water, while at high concentrations, it is limited by the ability to ingest
and digest the food (Leidy and Ploskey, 1980). Similarly, the feeding rates
for carnivorous zooplankton are limited at low prey densities by the ability
of the zooplankton to find and capture prey items, while at high prey
densities, they are limited by the ability to process, ingest, and digest
the prey. Also, at very high ingestion rates, zooplankton growth may be
limited by assimilation rates since ingested food remains in the gut for
less time, resulting in only partial digestion and reduced assimilation
efficiencies.
While the saturation type feeding response has been demonstrated in
numerous studies, work by Mayzaud and Poulet (1978) indicates that
zooplankton may be able to acclimate to changing food concentrations by
adjusting their digestive enzyme levels, allowing them to filter at maximum
rates over a much wider range than suggested by the saturation response
curves of short term experiments (Leidy and Ploskey, 1980). This results in
a linear response curve with ingestion rates directly proportional to the
food supply. However, some upper limit on feeding and growth must exist
based on theoretical arguments, so a saturation response curve is probably
appropriate, even though the saturating food levels may be much higher than
386
-------
typically experienced in the field except perhaps during phytoplankton
blooms.
Two major approaches are used to simulate saturation responses in
zooplankton models, the Michaelis-Menten (1913) formulation and the Ivlev
(1966) formulation. The Michaelis-Menten formulation is a hyperbolic
function analagous to that used in phytoplankton growth calculations, and is
probably the most common approach used in water quality models (Chen and
Orlob, 1972, 1975; Di Toro and Connolly, 1980; Di Toro and Matystik, 1980;
Bloomfield et af[., 1973; Park et _al_., 1974, 1975, 1979, 1980; Scavia et a]_.,
1976; Scavia, 1980; Canale jjt jfL, 1975, 1976; Bierman, 1976; Bierman
et _al_., 1980; Baca et _§]_., 1973, 1974; Baca and Arnett, 1976; Najarian and
Harleman, 1975). The basic equation is:
where F = total available food supply, mass/volume
half-saturation c<
growth, mass/volume
K = half-saturation constant for zooplankton feeding and
The Ivlev formulation is an exponential function which is more popular in
biologically oriented models (Kremer and Nixon, 1978). The general equation
is:
-K F
VFl'F2""Fn) = * - e (7~9)
where K = proportionality constant for Ivlev formulation
Figure 7-2 shows a comparison of the Michaelis-Menten and Ivlev
functions where both functions have the same half-saturation value (i.e.,
K = -ln(%)/K ). Both response functions range from minimum values of 0 at
very low food concentrations to maximum values of 1 at food saturation.
However, for food concentrations below the half-saturation constant (K ),
the Ivlev function is slightly lower than the Michaelis-Menten function.
387
-------
For food concentrations above K the Ivlev function is higher and
approaches saturation sooner than the Michaelis-Menten function. Note that
both functions are used with the total ingestion form of the growth equation
(Equation (7-4)) rather than with the filtration form (Equation (7-5)),
since the growth limitation function in the filtration form must always be
multiplied times the total food supply to get the net response.
Both the Michael is-Menten and Ivlev formulations can be modified to
allow for threshold food concentrations below which zooplankton do not feed.
This provides a refuge for prey organisms when they are present in very low
concentrations. The resulting equations are:
LU
*
DC
o
z
N
<
DC
O
DC
O
LU
!Sc
DC
O
DC
O
LLJ
LU
DC
MICHAELIS-MENTEN
F
Figure 7-2.
Kz 2K;
FOOD CONCENTRATION, FT
Comparison of the Ivlev and Michael is-Menten functions with the
same half-saturation value (i.e., K = -ln(%)/Kz) (adapted from
Swartzman and Bentley, 1977, and Leidy and Ploskey, 1980).
383
-------
and VFl'F2"--Fn) = 1 - e T ° (7'12)
where F = threshold food concentration below which feeding does not
occur, mass/volume
Zooplankton half-saturation constants and threshold feeding levels are
presented in Tables 7-6 and 7-7.
A few models, for example CLEAN, CLEANER, and MS.CLEANER (Bloomfield
et_al_., 1973; Park et _§]_., 1974, 1975, 1979, 1980; Scavia and Park, 1976),
use a modified Michael is-Menten formulation in which the half-saturation
constant varies as a function of zooplankton densities to account for
competition and feeding interference effects. The equation is:
Kz = Kzl + Kz2 Z (7-13)
where K , = feeding area coefficient, mass/volume
K 2 = competition or interference coefficient
Other saturation response functions besides the Michaelis-Menten and
Ivlev formulations have been used in some models. For example, rectilinear
saturation curves have been constructed by assuming feeding increases
linearly with food concentration until a critical food density is reached,
and then levels off at a maximum rate for all concentrations above the
critical density. This is expressed as:
_*• /1- r- i- \ _ i
for FT > Fsat
389
-------
TABLE 7-6. MICHAELIS-MENTEN HALF-SATURATION CONSTANTS FOR ZOOPLANKTON
CONSUMPTION AND GROWTH
Zooplankton
Group
Total
Zooplankton
Half- Saturation
Constant*** Units
0.010 0.060 mg(Chl a_)/l
0.5 (growth) mg/1
References
Di Toro et al . (1971)
O'Connor et al_. (1975,
Chen (1970)
1981)
0.5 - 2.0
mg/1
1.0 mg/1
0.2 0.6* (growth) mg/1
0.06 0.6*
mg/1
Chen & Orlob (1975)
Chen & Wells (1975, 1976)
Jorgensen (1976)
Jorgensen et aj_. (1978)
Bierman et^ jil_. (1980)
U.S. Army Corps of Engineers (1974)
Brandes (1976)
Smith (1978)
Baca & Arnett (1976)
Herbivores 0.010 0.015 mg(Chl a_)/l Thomann et al_. (1975,1979)
Di Toro & Connolly (1980)
Di Toro & Matystik (1980)
Salisbury et al. (1983)
Carnivores
Omnivores
Cladocerans
Copepods
0.010
0.02
0.2
0.2
0.15
0.375
0.16 0.2
0.5
0.8 (growth)
1.8 (growth)
0.16 0.4
1.0
1.2 (growth)
mg(Chl aj/l
mgC/1
mgC/1
mgC/1
mgC/1
mg/1
mgC/1
mg/1
mg/1
mg/1
mgC/1
mg/1
mg/1
Thomann et^ al. (1975)
Scavia et al_. (1976)
Canale et al_ (1976)
Canale et^ al. (1976)
Scavia (1980)
Bowie et_ al. (1980)
Scavia e_t al. (1976)
Scavia "(1980)
Bowie et_ al- (1980)
Tetra Tech (1980)
Porcella et al. (1983)
Scavia et a]_. (1976)
Scavia "(1980)
Bowie et_ al. (1980)
Tetra Tech (1980)
Rotifers
0.2 0.6
mgC/1 Scavia et al. (1976)
Scavia TJ980)
390
-------
TABLE 7-6. (continued)
ZoopTankton
Group
Mysids
Half-Saturation
Constant***
0.5
2.0 (growth)
0.10 0.20
0.5
2.0 (growth)
Units
mg/1
mg/1
mgC/1
mg/1
mg/1
References
Bowie et_ al_. (1980)
Porcella et al_. (1983)
Scavia et al. (1976)
Scavia TT980)
Bowie et aj_. (1980)
Tetra Tech (1980)
*Model documentation values.
***Half-saturation constants are for consumption unless specified for growth.
where F . = food concentration when feeding saturation occurs,
mass/volume
VFl'F2"--Fn' '
for FT > Fsat
when a threshold feeding concentration F is used.
The growth limitation functions used with the filtration form of the
growth equation (Equation (7-5)) are different than the saturation response
functions discussed above since they must be multiplied by the available
food concentration to get the total response. In contrast to the previous
functions, these functions generally decrease with increases in the food
supply to account for factors like reduced filtering rates, adjustments in
particle size selectivity, and reduced assimilation efficiencies which occur
at high food concentrations. These types of functions generally have
maximum values of 1 at low food densities and decrease assymptotical ly
toward some minimum value as the food density increases.
391
-------
Di Toro and Matystik (1980) and Di Toro and Connolly (1980) use a
reverse Michaelis-Menten formulation to simulate reductions in filtration
rates as food concentration increases:
ff(FlfF2,...Fn) =
(7-16)
where Kf = food concentration at which the filtration rate is 1/2 of
its maximum value, mass/volume
TABLE 7-7. THRESHOLD FEEDING'LEVELS FOR ZOOPLANKTON
Zooplankton Group
Total Zooplankton
Carnivores
Omnivores
Threshold Feeding Level
0.028 mg/1
0.01 mg/1
0.20 mg/1
0.01 mgC/1
0.001 mgC/1
0.025 mg/1
References
Scavia & Park (1976)
Youngberg (1977)
Bierman et al_. (1980)
Scavia et aj_. (1976)
Scavia (1980)
Bowie et al_. (1980)
Cladocerans
Copepods
Rotifers
Mysids
0.02 0.05 mgC/1
0.05 mg/1
0.02 0.05 mgC/1
0.05 mg/1
0.02 - 0.05 mgC/1
0.05 mg/1
0.02 0.05 mgC/1
0.05 mg/1
Scavia et a]_. (1976)
Scavia TJ980)
Bowie et_ al_. (1980)
Scavia et aj_. (1976)
Scavia TT980)
Bowie et al. (1980)
Scavia et al_. (1976)
Scavia TT980)
Bowie et al. (1980)
Scavia et al_. (1976)
Scavia TT980)
Bowie et al. (1980)
392
-------
This function approaches 0 assymptotically at high food densities, resulting
in a saturation response for total consumption (Figure 7-3a).
Canale et _al_. (1975, 1976) use a slightly different formulation to
account for reductions in filtering rates and changes in particle size
selectivity at high food levels:
K1 FT + K2
ff = FT I K, <7-17)
where K, = multiplier for minimum filtering rate (minimum value of f-r)
Kp = food concentration at which the filtering rate is half way
between its minimum and maximum value,
ff = 1/2 (K1 + 1), mass/volume
This function approaches K, assymptotically at high food levels rather than
0. As a result, the total consumption rate continues to increase in
proportion to the food supply at high food concentrations since the
volumetric filtration rate remains at a constant minimum level
(Figure 7-3b). However, a saturation type response can be generated by
setting the minimum multiplier K, equal to 0, in which case this formulation
is identical to Equation (7-16).
A reverse Michael is-Menten formulation has also been used to simulate
reductions in the assimilation efficiencies of filter feeders at high food
concentrations (Di Toro et a^., 1971, 1977; Di Toro and Matystik, 1980; Di
Toro and Connolly, 1980; Thomann et _al_. , 1975, 1979; Canale et _al_. , 1975,
1976). The equation is:
where K = food concentration at which the assimilation efficiency is
a
1/2 of its maximum value, mass/volume
393
-------
LJJ
tc
QC
Consumption Rate
FOOD CONCENTRATION, FT
(a)
LU
fe
QC
Consumption Rate
FOOD CONCENTRATION, FT
(b)
Figure 7-3. Comparison of reverse Michaelis-Menten formulation (a) and
Canale et. aj_. 's (1975, 1976) formulation (b) for filtration
rate as a function of food concentration.
394
-------
If a constant volumetric filtration rate is used (Di Toro et aj_., 1971,
1977; Canals _§t _§_]_., 1975, 1976), this results in a Michael is-Menten type
relationship for total consumption in which the maximum assimilation rate
(growth rate) equals the product of the constant filtration rate, maximum
assimilation efficiency, and the food concentration at half-maximum
assimilation efficiency K (ignoring temperature effects):
9z fmax max T
However, Di Toro and Matystik (1980) and Di Toro and Connolly (1980) also
use this formulation with a reverse Michael is-Menten formulation for the
filtration rate, which results in a more complicated expression for total
consumption involving the product of a Michael is-Menten term and a reverse
Michaelis-Menten term:
9z-cf..xE»axl(8lr4T-ilT-T-r) (7-?°>
Zooplankton growth and consumption formulations are compared for
several models in Table 7-8.
7.3.2 Food Supply
The total available food concentration FT used in all of the above
growth formulations can be defined in several ways. The simplest approach
is to assume all potential food items can be consumed with equal efficiency
and define FT as the sum of the available food concentrations:
FT - FK (7-21)
395
-------
TABLE 7-8. COMPARISON OF ZOOPLANKTON GROWTH FORMULATIONS
Model
(Author)
AQUA- IV
CE-QUAL-R1
CLEAN
CLEANER
MS. CLEANER
EAH
ESTECO
EXPLORE-1
HSPF
LAKECO
Phyto-
plankton
1
2
2
3
4
4
2
1
1
2
Food Sources
Preference
Zoo- FKtors
Detritus plankton Used
1 X
1 3 X
1 3 X
2 5 X
1 3 X
1 X
1 X
MIT Network 1
HASP
HQRRS
merman
Canals
Jorgensen
Scavlt
2
2
5
4
1
5
1
1 X
X
9 X
1 6 X
Basic Approach
Growth Total Filtration
Computed Ingestlon Rate
Directly Computed Computed
X
X
X
X
raptorial filter
feeders feeders
X
X
X
X
X
X
X
carnivores filter
feeders
X
X
Growth Limitation Formulation
Variable Variable Threshold
M1chael1s- Assimilation Filtration Feeding
Menten Ivlev Efficiency Rate Included
X
X
X X
X X
raptors » X* X
saturation
fllterers
X
X
X
X
X
X
X X
X
X X
carnivores nonselecttve selective
fllterers fllterers
X X
X X
Assimilation Efficiency
Varies Varies
with with
Constant Food Type Food Cone.
X
X
X
X
X
X
X
X
X
X
X
X
X
X
carnivores S selec- nonselectlve
tlve fllterers fllterers
X
X
GO
*Hax1muB assimilation rate used for constant rate filters, with excess consumption egested as pseudofeces.
-------
where F, = concentration of potential food item k, mass/vol ume
n = total number of potential food items
A more realistic approach recognizes that food items vary in the
efficiency and frequency at which they are utilized by zooplankton, even if
all food items are present in equal concentrations. This is due to factors
such as food particle size and shape, desirability and quality of different
types of food, ease of capture, and zooplankton feeding behavior. For
example, many filter feeders are able to selectively filter different food
items with different efficiencies, varying their selectivity according to
the abundance and desirability of the various food items present. Food
particle shape and size are important distinguishing features since, for
example, filamentous algae are often actively rejected or avoided while
individual cells of the same species in suspension may be consumed (Leidy
and Ploskey, 1980). However, the quality and desirability of the food are
also important, since senescent cells are less likely to be consumed than
healthy cells of the same species. For raptorial feeders, particle size and
shape are not quite as critical since they are able to tear large prey items
into smaller pieces before consuming them. Prey desirabil ity and ease of
capture then become more important.
The above factors are accounted for in models by assigning feeding
preference factors to each potential food item. Preference factors can have
values ranging from 1 to 0, with 1 corresponding to a food item which is
desirable and easily captured and consumed (or filtered), and 0
corresponding to a food item which is never consumed. Food preference
factors have been called selectivity coefficients, electivities, ingestion
efficiencies, and several other names in different models, but they all
basically represent the same thing--weighting factors which reflect the
probability that a given food item will be consumed relative to the others
when all foods are present in equal concentrations. They account for the
fact that some food items may be less available for consumption than
indicated by their concentrations alone. When food preference factors are
specified, the total available food concentration FT is defined as:
397
-------
FT= E Pk F (7-22)
I k=1 K k
where P. = food preference factor for food item k
K
F, = concentration of food item K, mass/volume
K
n = total number of potential food items
Vanderploeg and Scavia (1979) show how preference factors can be derived
from the different forms of data reported in zooplankton feeding
experiments. Infield situations, preference'factors may change as the
composition of the food supply changes. However, this level of
sophistication is generally not included in current ecological models.
7.3.3 Assimilation Efficiencies
In addition to differences in food preferences or ingest ion
efficiencies for different food types, food items may also differ in their
assimilation efficiency by zooplankton. The assimilation efficiencies for
different food types varies with the energy content, digestibility, and
quality of the food (Leidy and Ploskey, 1980). For example, the
assimilation efficiencies for algae are typically higher than for detritus
and bacteria, although the assimilation efficiencies for blue-green algae
are also generally low. Algae with gelatinous sheaths or resistant cell
walls and masses of colonial cells may pass through a zooplankton gut intact
and in viable condition (Wetzel , 1975), indicating minimal assimilation
efficiencies for these food items. The animal foods of raptorial feeders
are assimilated more efficiently than plant foods. Also, since the energy
content and digestibility of algae and detritus vary much more widely than
animal foods, the assimilation efficiencies for herbivores and
detritivores typically cover a much wider range than for carniv.orous
zooplankton (Leidy and Ploskey, 1980).
Variations in the assimilation efficiencies of different food items can
be modeled in several ways. One approach is to incorporate these effects in
the food preference factors, for example, by assigning a low value to the
preference factor for blue-green algae relative to the other algal groups.
398
-------
This in effect lowers the amount of blue-green algae available for
zooplankton assimilation and growth. Another approach is to define
different maximum assimilation efficiencies for different food items, to
compute net assimilation separately for each food item, and then to sum the
individual assimilation terms to get the total zooplankton growth rate
(Scavia _et al . , 1976; Scavia, 1980). This can be expressed for the total
consumption formulation (Equation (7-4)) as (ignoring temperature effects):
Com,v Z-, Em=,v ^r, (FijF,,...!
gmax (<1maxk gkl'2
n'
(7-23)
where C = maximum total consumption rate, mass food/mass
zoopl ankton-time
E = maximum assimilation efficiency for food item
III QAi
k
f (F,,F9, ...F ) = growth limitation factor for food item k
gk i ^ n
n = total number of potential food items
and for the filtration formulation (Equation (7-5)) as:
raLE-k P* F>] (7-24)
where C,, = maximum volumetric filtration, volume/mass
zoopl ankton-time
ff (F, ,F?,.. .F ) = gr owth 1 i mi tat i on f uncti on for f il trati on
formul ation
food prefere
F. = concentration of food item k, mass/vol une
P. = food preference factor for food item k
Note that growth limitation factors must be computed separately for each
food item in the total consumption formulation since the quantities which
are summed must reflect both the assimilation efficiencies and the amounts
of food consumed for each different food item.
399
-------
For the Michael is-Menten formulation, the individual growth limitation
factor may be defined as:
f I? F F)=
: 2"""
PI F,
^ - (7-25)
k=l
This is equivalent to the total Michael is-Menten factor when summed over all
food items:
n
n n P. F. APiA
•^ ,• / r- r- r \ _ V"1 K K ' _ K~i K K
rTTn(7-26)
Analogous expressions for the Ivlev formulation are more difficult to
formulate, since the individual terms are not consistent with the total
growth limitation function, even under conditions of equal assimilation
efficiencies.
As discussed previously, assimilation efficiencies may decrease with
increases in ingestion rate at high food concentrations since the retention
time in the gut decreases resulting in incomplete digestion and reduced
assimilation. Model formulations to describe these effects have already
been discussed in the growth limitation section (Equation (7-18)).
Zooplankton average assimilation efficiencies are presented in
Table 7-9. Figures 7-4 and 7-5 present frequency histograms of assimilation
efficiency data compiled by Leidy and Plosky (1980).
7.4 RESPIRATION AND MORTALITY
Zooplankton respiration and mortality are modeled using the same
general formulations as phytopl ankton. Almost all models represent both
respiration and nonpredatory mortal ity rates as either constant coefficients
or simple functions of temperature. The basic equations are:
400
-------
TABLE 7-9. ZOOPLANKTON ASSIMILATION EFFICIENCIES
Zooplankton
Group
Assimilation Efficiency
References
Total
Zooplankton
Herbivores
0.60 - 0.75
0.63
0.7
0.6
0.5 0.8*
0.5 0.7*
0.6 (max.)
Carnivores 0.6 (max.)
0.5
0.4 (Cladocerans)
Omnivores 0.5
(0.2 for detritus, blue-green algae)
0.4
Cladocerans
0.5
Copepods
(0.2 for detritus, blue-green algae)
0.5
0.8 (max/
0.5
(0.2 for detritus, blue-green algae)
0.7
Di Toro et al_. (1971)
O'Connor etal_. (1975, 1981)
Jorgensen (1976)
Jorgensen et_ a]_. (1978)
Tetra Tech (1976)
Chen & Wells (1975, 1976)
Bierman e_t aj_. (1980)
Brandes (1976)
Smith (1978)
Baca & Arnett (1976)
Thomann et al_. (1975, 1979)
Di Toro F~Connolly (1980)
Di Toro & Matystik (1980)
Salisbury et ^1_. (1983)
Thomann et al_. (1975, 1979)
Di Toro S~~Connolly (1980)
Di Toro & Matystik (1980)
Salisbury et al_. (1983)
Scavia et al_. (1976)
Canale et_ al_. (1976)
Scavia (1980)
Bowie et_ al_. (1980)
Canale et al. (1976)
Scavia et al. (1976)
Scavia TTgM)
Bowie et al. (1980)
Tetra Tech (1980)
Porcella et al. (1983)
Canale et al. (1976)
Scavia et al- (1976)
Scavia TT980)
Bowie et al. (1980)
Canale et al. (1976)
401
-------
TABLE 7-9. (continued)
Zooplankton
Group
Assimilation Efficiency
References
Rotifers
Mysids
0.5
(0.2 for detritus, blue-green algae)
0.5
0.5
(0.2 for detritus, blue-green algae)
0.5
Scavia et a]_. (1976)
Scavia TT980)
Bowie et al_. (1980)
Tetra Tech (1980)
Porcella et al. (1983)
Scavia et al_. (1976)
Scavia TT980)
Bowie et a]_. (1980)
Tetra Tech (1980)
*Model documentation values.
and
where r
r!(T.
fr(T)
m
f (T)
nr '
r = r (T f) f (T)
z zv ref rv '
m = m (T f) f (T)
z zv ref nr '
(7-27)
(7-28)
ref)
zooplankton respiration rate, I/time
respiration rate at reference temperature T f, I/time
temperature function for respiration
zooplankton nonpredatory mortal ity rate, I/time
nonpredatory mortal ity at reference temperature
Tref, I/time
temperature function for nonpredatory mortality
Since the respiration and nonpredatory mortal ity rate equations have the
same basic form and typically use the same temperature functions, many
models combine both processes into a single loss term:
(7-29)
where dz(Trgf) = total loss rate due to both respiration and
nonpredatory mortality at reference temperature T f>
1/time
402
-------
16 -
12 •
4 •
n •
—
—
—
TOTAL
ZOOPLANKTON
•"^
.20
.40 .60 .80
ASSIMILATION EFFICIENCY
.1.0
16
12
e •'•
i 8-.
4- •
CLADOCERANS
.20
M .60
ASSIMILATION EFFICIENCY
.80
1.0
O
GO
6 •
ROTIFERS
nn
AO .60
ASSIMILATION EFFICIENCY
uo
8-
6-
2-
COPEPODS
.20
M £0
ASSIMILATION EFFICIENCY
UO
Figure 7-4. Frequency histograms for zooplankton assimilation efficiencies (from Leidy and Ploskey, 1980)
-------
In a few models, the respiration rate is partitioned into two
components, 1) the standard respiration rate representing the combined basal
metabolism and digestion energetics and 2) the active respiration rate which
represents the additional respiration associated with zooplankton activity.
These two components can be distinguished by using different temperature
response functions for each component. For example, standard respiration is
FREQOEHCY
*••
> «• 00 M
(
3REEN ALGAE AS FOOD
.20 ,40 ,CO ,80
ASSIMILATION EFFICIENCY
1.0
12-
8.
4- •
BUTE-GREEN ALGAE AND/OR
DETRITUS AS FOOD
.20 >0 ,60 .80
ASSIMILATION EFFICIENCY
1.0
Figure 7-5. Frequency histograms showing variations in zooplankton
assimilation efficiencies with different food types
(from Leidy and Ploskey, 1980)
404
-------
typically associated with an exponential temperature curve which increases
until the upper lethal limit is approached, while the active respiration
rate may be associated with a temperature optimum curve:
fa(T) (7-30)
where rstcj(Tref) = standard respiration rate at reference temperature
Tref, I/day
fs(T) = temperature function for standard respiration
ract(Tref) = active respiration rate at reference temperature
Tref, I/day
fa(T) = temperature function for active respiration
Another approach is to assume that the activity level (and active
respiration) is proportional to the feeding level by using a Michaelis-
Menten or Ivlev function:
f (F1'F2' • • -Fn) ^7
where f (F,, F?,.. .F ) = growth limitation factor as a function of food
supply
This approach is used by Scavia et_ _a_K (1976) and Scavia (1980) where the
first term represents the minimum endogenous respiration rate under
starvation conditions and the second term represents the increase in
respiration associated with feeding.
A similar formulation is used in CLEANER (Scavia and Park, 1976) and
MS.CLEANER (Park et _al_., 1979, 1980) where the active respiration rate is
expressed as a fraction of the total consumption rate:
r =\r . (T f) + K C 1 f(T) (7-32)
|_rmir ref r gj v ' v '
where r . (T ^) = minimum endogenous respiration under starvation
m i n r et'
conditions at reference temperature T ., 1/tii
405
i me
-------
K = fraction of ingested food which is respired
C = ingestion rate, I/time
The CLEANER and MS. CLEANER models also include additional factors to
account for crowding effects and population age effects on both respiration
and nonpredatory mortal ity rates. The crowding factor is expressed as:
(7-33)
where f , = crowding factor
crd 3
K = crowding coefficient
Z = zooplankton carrying capacity, mass or mass/volume
C dp
This factor increases the respiration and mortal ity rates as zooplankton
density increases. The age factor accounts for the effects of the
population age structure on the net respiration and mortal ity rates since
these rates generally vary with age. The basic assumption is that the
population consists primarily of immature individuals at low zooplankton
densities and of adults at high population densities (Scavia and Park,
1976). The age factor represents the difference between adult and juvenile
rates. The age factor for respiration is expressed as:
W • 1 + Kl -^ - 1 '7-34>
where f aae = age factor for respiration
Krx = fractional increase in respiration rate between young
zooplankton and adults
and the age factor for mortality is expressed as:
where fmage = age factor for nonpredatory mortality
406
-------
K = fractional decrease in mortality rate between young
III X
zooplankton and adults
Both the crowding and age structure factors are multiplied with the
respiration and no npredatory mortal ity rates defined in Equations (7-32) and
(7-28) to incorporate these effects into the rates.
Some versions of CLEANER (Youngberg, 1977) also include an oxygen
reduction factor in the respiration equation to account for decreases in
respiration at low dissolved oxygen levels. The equation is:
00 - 0 .
_ 2 mm -
ox 2 mi n
where f = oxygen reduction factor
\J A
Op = ambient oxygen concentration, mg/1
0 . = minimum oxygen requirement, mg/1
K = half-saturation constant for oxygen limitation (set at
0.9 mg/1)
Bierman et _al_. (1980) use a second order formulation for zooplankton
mortality when the zooplankton density exceeds a critical level. This
accounts for density dependent effects on both natural mortality and
predatory mortality (which is not directly simulated in this model) at high
densities. The equation is:
m= [-"l+Km Z] f f7'37'
where m, (T J = mortality rate below the critical zooplankton density
at reference temperature T -, I/time
K (T f) = density dependent mortality coefficient for increased
mortality above the critical zooplankton density at
reference temperature Tref, I/mass zooplankton-time
407
-------
The nonpredatory mortal ity rate can also be partitioned into several
components which account for specific types of mortality such as natural
senescence, thermal ly- induced mortal ity, toxic mortal ity, and stress-induced
mortality due to low dissolved oxygen, pH extremes, starvation, etc. The
general equation is:
fl
f2(T) f(02,PH,...) + mf(Tref) f3(T) + ff(Fy) (7-38)
where m (T f) = mortality rate due to senescence at reference
temperature Trgf , I/time
f (T) = temperature function for senescent mortality
mT(T f) = thermal mortality rate at reference temperature
Tref, I/time
fy(T) = thermal mortality response curve
m (T f) = toxic mortality rate at reference temperature
Tpef, I/time
f-i(T) = temperature function for toxic mortality
f (X) = dose-response curve for toxic mortality
A
X = concentration of toxicant, mass/ volume
m (T _) = stress- induced mortality rate for low dissolved
oxygen, pH extranes, etc., at reference temperature
Tref 1/time
f2(T) = temperature function for stress-induced mortality
f(0,>,pH...) = stress- induced mortal ity function for low dissolved
oxygen, pH extremes, etc.
mf(Tref) = starvation-induced mortality rate at reference
temperature Tr f, I/time
f^CO = temperature function for starvation mortality
f-f(Fy) = starvation mortality function
Various formulations could be used to define these effects, although most
current models deal only with natural mortality and sometimes thermal
effects.
408
-------
Zooplankton respiration rates and mortality rates are presented in
Tables 7-10 and 7-11. Figures 7-6 and 7-7 present frequency histograms of
respiration rates and nonpredatory mortality rates from data compiled by
Leidy and Ploskey (1980).
7.5 PREDATORY MORTALITY
Zooplankton predatory mortality is modeled using the same formulations
described previously for phytopl ankton. However, since Zooplankton are
often the highest trophic level included in water qual ity model s, predator-
prey dynamics between Zooplankton and higher trophic levels cannot usually
be simulated. Therefore, predation by fish and carnivorous Zooplankton is
modeled by either assuming a constant predation loss which is specified as a
model input parameter:
GZ = constant (7-39)
where G = total predatory mortality rate by all zoopl ankton
consuners, mass zoopl ankton/time
or by assuming a loss rate which is directly proportional to the zoopl ankton
densities:
Gz = ez Z (7-40)
or G, = e,(T^rf) fg(T) Z (7-41)
where e = predatory mortality rate coefficient, I/time
Z = zooplankton biomass or concentration, mass or
mass/volume
e (T ,,) = predatory mortality rate coefficient at reference
temperature T f, I/time
f (T) = temperature function for predatory mortality
Since these formulations are essentially the same as those used for
nonpredatory mortality, nonpredatory mortality and predation losses are
409
-------
TABLE 7-10. ZOOPLANKTON RESPIRATION RATES
Zooplankton
Group
Total
Zooplankton
Herbivores
Carnivores
Omnivores
Cladocerans
Respiration Rate
0.01
0.02 - 0.035
0.36
0.02 0.16
0.005 0.02
0.001 - 0.11*
0.005 0.3*
0.02 - 0.03
0.007 - 0.02
0.30
0.04 - 0.06
0.08 0.33
0.04 - 0.06
0.1 0.36
0.017 - 0.10
0.04 - 0.06
0.157 - 0.413**
0.090 0.216**
0.006 0.772**
8.5 - 14.2**
Units
I/day
I/day
I/day
I/day
I/day
I/ day
I/day
I/day
I/day
I/ day
I/day
I/day
I/day
I/day
I/day
I/day
I/day
I/day
I/day
ml 02
Temperature
20°C
20°C
20°C
20°C
20°C
20°C
20°C
20°C
20°C
Topt
20°C
Topt
20°C
Topt
20°C
20°C
20°C
20°C
Topt
18°C
References
Chen (1970)
Chen & Orlob (1975)
Chen & Wells (1975, 1976)
Jorgensen (1976)
Jorgensen et al . (1978)
Lombardo (1972)
O'Connor et ah (1975)
Tetra Tech (1976)
U.S. Army Corps of Engineers (1974)
Brandes (1976)
Smith (1978)
Baca & Arnett (1976)
Thomann et al. (1975, 1979)
Di Toro & Connolly (1980)
Di Toro & Matystik (1980)
Salisbury .et al. (1983)
Thomann et al . (1975, 1979)
Di Toro 3TConnolly (1980)
Di Toro & Matystik (1980)
Salisbury et al- (1983)
Scavia e_t al. (1976)
Canale et al. (1976)
Scavia (1980)
Bowie et. al. (1980)
Canale et. 3K (1976)'
Scavia et al. (1976)
Scavia TT9f50~)
Bowie e_t aj_. (1980)
Tetra Tech (1980)
Porcella et. al- (1983)
Canale et al- (1976)
Lombardo (1972)
Leidy & Ploskey (1980)
Collins & Wlosinski (1983)
Di Toro et al . (1971)
mg(D.W.)-day
410
-------
TABLE 7-10. (continued)
Zooplankton
Group
Copepods
Rotifers
Mysids
Respiration Rate Units
5.4 14.2** ml °2
mg(D.W.)-day
14.2** ml °2
mg(D.W.)-day
0.1 0.35 I/day
0.04 0.06 I/day
0.017 I/day
0.085 - 0.550** I/day
0.064 - 0.738** I/day
0.043 - 0.695** I/day
3.0 12.2** ml °2
mg(D.W.)-day
2.93 - 18.9** m1 °2
mg(D.W.)-day
3.0 - 13.5** ml °2
mg(D.W.)-day
0.12 0.40 I/day
0.15 I/day
0.163 0.677** I/day
0.05 0.28 I/day
0.022 I/day
Temperature
20°C
20°C
Topt
20°C
20°C
20°C
20°C
Topt
20°C
20°C
20°C
Topt
20°C
20°C
Topt
20°C
References
Lombardo (1972)
Jorgensen (1979)
Scavia et al. (1976)
Scavia TT980)
Bowie et al.. (1980)
Canale et a]_. (1976)
Tetra Tech (1980)
Lombardo (1972)
Leidy & Ploskey (1980)
Collins & Wlosinski (1983)
Di Toro et al . (1971)
Lombardo (1972)
Jorgensen (1979)
Scavia et al. (1976)
Scavia TT980)
Bowie et al- (1980)
Porcella et ah (1983)
Leidy & Ploskey (1980)
Scavia et al. (1976)
Scavia TT980)
Bowie et al. (1980)
Tetra Tech (1980)
*Model documentation values.
**Literature values.
often combined into a single total mortality term when higher trophic levels
are not directly simulated:
"tot
= m,(T rf) + e (T
'zv'ref
zvref
UT)
(7-42)
411
-------
where m, = total mortality rate, I/time
m
t(T f) = total mortality rate at reference temperature
Tref, I/time
In ecologically oriented model s where long term seasonal changes in
population dynamics are important, zooplankton are often separated into
several functional groups based on general feeding types (filter feeders,
carnivorous raptors, omnivores, etc.) or on major taxonomic groups
(cl adocerans, copepods, rotifers) (Canal e et_ al_., 1975, 1976; Scavi a et
aj_., 1976; Scavia, 1980; Parkejt^l_., 1974, 1975, 1979, 1980; Chen et _§]_.,
1975; Tetra Tech, 1979). Although several species must be lumped into each
functional group, this approach recognizes the importance of complexities in
the food web, different foraging strategies, and predator population
dynamics in evaluating both zooplankton and phytoplankton dynamics. Several
planktivorous fish groups are also sometimes provided for the same reasons.
(Chen et a].., 1975; Tetra Tech, 1979; Park et aj_., 1979, 1980).
In these situations, zooplankton predation rates are computed as the
sum of the consumption rates by all potential predators, including
carnivorous or omnivorous zooplankton and planktivorous fish. The general
relationship for predatory mortality can be expressed as:
n
G
Z . f-4
1 J = -
Cy
* A •
Fkj
(7-43)
where G = total predatory mortality rate for zooplankton group i,
mass zooplankton/time
n = total number of zooplankton consumers
C- = total consumption rate by predator group j, I/time
X. = biomass or concentration of predator group j, mass or
mass/volume
P.. = food preference factor for predator group j feeding on
zooplankton group i
412
-------
TABLE 7-11. ZOOPLANKTON MORTALITY RATES
Zooplankton
Group
Total
Zooplankton
Carnivores
Omnivores
Fast Ingesters
Slow Ingesters
Cladocerans
Copepods
Mortality Rate (I/day)
0.075
0.125
0.025 0.033
0.005
0.02
0.015
0.005*
0.001 - 0.005*
0.005 - 0.02*
0.003 - 0.075**
0.01
0.01
0.005
0.05
0.01
0.01
0.04 - 0.05
0.001 - 0.005
0.01
0.1
0.0007 - 0.027**
0.001 - 0.027**
0.01
0.05
0.002
0.003 - 0.005
0.01
Mortality Type
total
nonpredatory
nonpredatory
nonpredatory
nonpredatory
total
nonpredatory
nonpredatory
nonpredatory
total
nonpredatory
fish grazing
fish grazing
nonpredatory
nonpredatory
nonpredatory
fish grazing
fish grazing
nonpredatory
nonpredatory
nonpredatory
nonpredatory
nonpredatory
fish grazing
fish grazing
nonpredatory
nonpredatory
References
Di Toro et al_. (1971)
Jorgensen (1976)
Jorgensen et al_. (1978)
Chen and Wells (1975, 1976)
Tetra Tech (1980)
O'Connor e_t al_. (1981)
U.S. Army Corps of Engineers (1974)
Brandes (1976)
Smith (1978)
Jorgensen (1979)
Scavia et_ aK (1976)
Scavia et a]_. (1976)
Scavia (1980)
Bierman ejt al_. (1980)
Bierman et_ al_. (1980)
Scavia et a]_. (1976)
Scavia ejt a1_« (1976)
Scavia (1980)
Tetra Tech (1980)
Porcella e_t a]_. (1983)
Leidy & Ploskey (1980)
Collins & Wlosinski (1983)
Scavia ejt al_. (1976)
Scavia et al_. (1976)
Scavia (1980)
Canal e et aj_. (1976)
Tetra Tech (1980)
413
-------
TABLE 7-11. (continued)
Zooplankton
Group
Rotifers
Mysids
Mortality Rate (I/day)
0.0005 0.153**
0.003 - 0.155**
0.01
0.12
0.01
0.1
0.08
0.01
Mortality Type
nonpredatory
nonpredatory
nonpredatory
nonpredatory
nonpredatory
fish grazing
fish grazing
nonpredatory
References
Leidy & Ploskey (1980)
Collins & Wlosinski (1983)
Scavia et al_. (1976)
Porcella et_ al_. (1983)
Scavia et al_. (1976)
Scavia et aj_. (1976)
Scavia (1980)
Tetra Tech (1980)
*Model documentation values.
**Literature values.
Z. = biomass or concentration of zooplankton group i, mass or
mass/volume
n. = total number of potential food items for predator group j
J
P^ • = food preference factor for predator group j feeding on food
item k
F.. = biomass or concentration of potential food item k consumed
by predator group j, mass or mass/volume
A
The quantity (P... I./ £ P^. p ) in Equation (7-43) represents the
fraction of the total food consumption by predator group j which is provided
by zooplankton group i. The quantity C.X. represents the total rate of food
J v
ingestion by predator group j. Ingestion rate formulations for carnivorous
zooplankton were discussed in the previous section. Consumption rates for
planktivorous fish are generally modeled in the same way. As discussed in
the algae chapter, consumption rates are sometimes back-calculated from
computed growth rates and known assimilation efficiencies using the
equation:
414
-------
12 --
10 -•
8 .. r-
6 ••
4 • •
2 ••
CLADOCERANS
n n
rfl
m
.25 .50 .75 1.0
RESPIRATION RATE (I/day)
1.25
1.50
1?
10 • •
8 ••
'3
I 6
£
* 4
7
COPEPODS
n
.25 .50 .75
RESPIRATION RATE (I/day)
1.0
12
10
4 ••
ROTIFERS
m
r
.25 .50 .75
RESPIRATION RATE (I/day)
1.0
FIGURE 7-6. Frequency histograms of zooplankton respiration
rates (from Leidy and Ploskey, 1980).
415
-------
c -
EJ
(7-44)
where C.
vJ
EJ
total consumption rate for predator group j, I/time
growth rate for predator group j, I/time
assimilation efficiency for predator group j
When different assimilation efficiencies are used for different food items,
consumption rates are generally calcul ated directly for each food item and
combined with the food specific assimilation efficiencies to determine net
growth (as discussed in Section 7.3.3).
7.6 SUMMARY
Zooplankton are typically modeled as a biomass pool using the same mass
balance approach used for nutrients, phytoplankton, and other constituents.
24 -
20 -
16 -
5
z
o- 1? '
8 -
4 '
0
TOTAL ZOOPLANKTON
1 1
11 III
.01 .02 .03 .04 o05
NONPREDATORY MORTALITY RATE (I/day)
Figure 7-7- Frequency histogram of nonpredatory mortality rates
for zooplankton (from Leidy and Ploskey, 1980).
416
-------
The simplest models lump all zooplankton into a single group, while more
complex models distinguish between different feeding types or different
taxonomic groups.
Zooplankton dynamics depend on growth, reproduction, respiration,
excretion, predation, and nonpredatory mortal ity. However, these processes
are not generally measured in the field for a specific model application
since: 1) many of them are difficult or impossible to measure directly; 2)
the rates depend on environmental conditions (e.g., temperature), ecological
conditions (e.g., food supply and predator densities), and the species
composition of the zooplankton, all of which change continually with time;
«•
and 3) the fluxes depend largely on the zooplankton densities, which may
vary by orders of magnitude over a seasonal cycle.
As a result, many of the model coefficients must be determined by model
calibration rather than by measurement. Model constructs must be relied
upon to describe the effects of different factors on these processes.
Literature values from 1 aboratory experiments are useful for establishing
reasonable ranges of the process rates and coefficients. However, specific
experimental results are difficult to apply directly since experiments
typically use a single species rather than the species assemblages
represented in models, and since experimental conditions may not represent
conditions in the field.
Most models include formulations to describe the effects of temperature
on all process rates. Food density effects on growth and consumption are
typically modeled using saturation kinetics similar to those used for
phytoplankton. Respiration and mortal ity rates are most commonly modeled as
first-order losses, although a few models use more complicated formulations
which include the effects of other factors, for example, crowding effects.
Since few models include higher trophic levels such as fish, predatory
mortality is typically treated in a simplistic manner.
417
-------
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Lombardo, P.S. 1972. Mathematical Model of Water Quality in Rivers and
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Mayzaud, P. and S.A. Poulet. 1978. The Importance of the Time Factor in
the Response of Zooplankton to Varying Concentrations of Naturally Occurring
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Michael is, L. and M.L. Menten. 1913. Biochemische Zeitschrift., 49:333.
Naja*rian, T.O. and D.R.F. Harleman. 1975. A Nitrogen Cycle Water Quality
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Hydrodynamics, Massachusetts Institute of Technology, Cambridge,
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O'Connor, D.J., D.M. Di Toro, and R.V. Thomann. 1975. Phytoplankton Models
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O'Connor, D.J., J.L. Mancini, and J.R. Guerriero. 1981. Evaluation of
Factors Influencing the Temporal Variation of Dissolved Oxygen in the New
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Park, R.A., R.V. O'Neill, J.A. Bloomfield, H.H. Shugart, Jr., R.S. Booth,
R.A. Goldstein, J.B. Mankin, J.F. Koonce, D. Scavia, M.S. Adams,
L.S. Clesceri, E.M. Colon, E.H. Dettmann, J.A. Hoopes, D.D. Huff, S. Katz,
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CHAPTER 8
COLIFORM BACTERIA
8.1 INTRODUCTION
Coliform concentrations in natural waters have been used as an
indicator of potential pathogen contamination since at least the 1890's
(Whipple, 1917). Until recently, coliforms have been considered to be less
sensitive to environmental stresses than enteric pathogens. Accordingly,
coliforms were believed to be more persistent in natural waters and,
therefore, a "safe" or conservative index of potential pathogen levels.
However, recent evidence about enteric viruses, opportunistic
pathogens, and pathogenic Escherichia coli have raised doubts that coliforms
are the "ideal indicator" (Sobsey and Olson, 1983). First, enteric viruses
appear to generally have both lower decay rates than coliforms and also a
lower ID-50 (i.e., the dose required to infect 50 percent of the persons
exposed) than most bacterial enteric pathogens. Second, opportunistic
pathogens (e.g., Pseudomonas aeruginosa, Aeromonas hydrophila, and
Legionella pneumophila) often have major non-fecal sources and are able to
grow in natural waters. These pathogens generally have a high ID-50,
threatening primarily immunologically compromised persons such as hospital
patients who are being given immunological suppressants. Finally, some
strains of _§_. coli produce an enteric toxin that results in gastroenteritis.
In the context of drinking water, Olivieri (1983) has recommended that
different indicators be used when different aspects of pathogen behavior are
of interest, e.g., indicator of feces, treatment efficiency, or post-
treatment contamination. Chamber!in (1982) has compared coliform (combining
Total Coliform, Fecal Coliform, and £. coli) decay rates with pathogen and
424
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virus decay rates measured simultaneously and has found that the respective
2
decay rates were highly correlated (r = 0.73) and that within-species
variability was as great as pathogen-to-coliform variation (see Figure 8-1).
At low decay rates, coliform decay rates were approximately equal to
pathogen decay rates while at the highest decay rates, pathogen decay was
slower.
0.001
0.001
0.01
COLIFORM DECAY RATE (1/HR)
Figure 8-1.
Relationship between pathogen or virus decay rates and coliform
decay rates based on figure presented by Chamber! in (1982).
Decay rates were estimated by Chamber! in based on data from
Baross et _al_. (1975) (A), Morita (1980) (X), McFeters et al .
(1974) (T), McCambridge and McMeekin (1981) (O), LantrTp
(1983) (•), and Kapuscinski and Mitchell (1981) (D). The
line shown represents coliform decay rates equal to pathogen
decay rates.
425
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In addition, epidemiological studies have revealed that enterococci
levels are more closely associated with enteric disease than are coliforms
(Cabelli et_ a_K 1982). This work has in part motivated a proposed revision
of the contact recreation bacterial water quality criteria: switching from
fecal coliforms to IE. coli and/or enterococci (U.S. Environmental Protection
Agency, 1984).
Taken as a whole, these issues may serve to motivate modelers to
include additional indicators as state variables and to use coliforms as jin
indicator rather than as the indicator.
8.2 COMPOSITION AND ASSAY
The coliform group consists of both fecal and non-fecal components.
The fecal component includes mainly the Escherichia and Klebsiella genera
while the non-fecal component includes mainly the Enterobacter and
Citrobacter genera commonly associated with soils and plants (Dufour, 1977).
Neither the multiple tube (MPN) nor the membrane filter (MF) techniques
for Total Coliforms (TC) effectively differentiates between the fecal and
non-fecal components. The Fecal Coliform (FC) tests (either MPN or MF)
provide a better differentiation at the cost of additional labor and time
plus more exacting equipment requirements. The tests require either
supplemental tests run on TC or incubation at elevated temperatures within
precise limits (i.e., 44.5°C ± 0.2°C). These more stringent conditions
eliminate most of the non-fecal component while still permitting the fecal
component to survive. FC represents from 15 to 90 percent of the TC,
depending on sample source. Unfortunately, there are major non-fecal
sources of FC, most commonly of Klebsiella species (Hendry et aj_. 1982).
Pulp mill wastewater provides a frequent example. Tests for E_. col 1 are
even more specific to fecal sources, but again incur further costs for labor
and time.
The non-fecal components of the coliforms, especially the Enterobacter
and Citrobacter genera, are of limited use in indicating fecal contamination
426
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but do indicate prior contact with soil or plant material. In addition,
these genera are capable of regrowth in nutrient-rich natural waters or
where surfaces are available for growth.
Fecal streptococci (FS) provide another common indicator of fecal
contamination (Clausen et^ jil_. 1977). Although all FS belong to the single
genus Streptococcus, there are again fecal and non-fecal components.
Enterococci and SL faecal is are more specific to fecal sources than the non-
enterococcal streptococci. FS and particularly the enterococci are often
considered to be able to survive longer in natural waters than either TC or
FC. Chamberlin (1982) compared FS (combining TC, FC, and E. coli results)
decay rates in cases where the rates were measured in the same experiments
2
and found a high correlation (r = 0.80) between the logarithm of the
respective rates. In addition, the relationship between the logarithms of
the rates had a slope estimated by linear regression that was not
significantly different (p = 0.05) from 1.0. The intercept was marginally
distinguishable from 0.0 at p = 0.01 and was estimated as -0.31. This
suggests that coliform decay rates were generally twice as large as FS decay
rates but that the rates changed generally by equal amounts from one
environment to another. According to Geldreich and Kenner (1969), the FC/FS
ratio is useful in discriminating between recent human versus animal fecal
contamination. If the ratio exceeds approximately 1 (although 4 is often
cited as the cut-off value), the source is presumptively human fecal
material while if the ratio is less than 1, the source is assumed to be
animal feces. But as Dutka and Kwan (1980) have observed, the ratio can
change dramatically once the material enters natural waters. They monitored
changes from an initial ratio of 2.7 to a low of 0.07 and a high of 22.5 in
a single experimental run.
Other proposed fecal indicators have been discussed by Olivieri (1983)
and include Clostridium perfringens, yeasts, and RNA coliphages. None of
these novel indicators has become generally accepted.
Beyond the selections of a particular indicator or set of indicators,
recent work has shown the importance of sublethal stress or injury of
influencing observed concentrations in decay studies (Rose .ejt _al_. 1975;
427
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Bissonette et_ aj_. 1977). Rhodes and Kator (1982) and Kapuscinski and
Mitchell (1981) have, among others, substantiated these results and have
suggested particular mechanisms of injury. Consequently, the decision to
use or not use a resuscitation step (e.g., incubation at 35°C in less
selective medium for two hours) can have a major impact on the observed
decay rates.
8.3 MODELING COLIFORMS
Modeling of coliforms is done for one main reason—establ ishing the
level of fecal and/or soil pollution and potential pathogen contamination.
The usual approach is simply to simulate disappearance and' to estim-ate
coliform levels as a function of initial loading and the disappearance rate
which, in turn, is a function of time or distance of travel from the source
and of environmental conditions such as temperatures, salinity, and light
intensity.
8.3.1 Factors Affecting Disappearance 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 organisms
may, in fact, increase in numbers in natural waters where conditions are
adequate (Lombardo, 1972; Mitchell and Chamberlin, 1978).
Factors can be conveniently classified into three categories:
physical, physicochemical, and biochemical-biological. However, note that
synergisms (e.g., osmotic effects and photo-oxidation) and interferences
(e.g., sedimentation versus photo-oxidation) may exist. Kapuscinski and
Mitchell (1980) and Bitton (1980) have reviewed factors that govern virus
inactivation in natural waters and present essentially a parallel list to
the one given below.
428
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Physical factors that can affect the coliform population in natural
waters, resulting in an apparent increase or decrease in the coliform
disappearance rate include:
• Photo-oxidation
• Adsorption
• Flocculation
• Coagulation
• Sedimentation
• Temperature
Physicochemical factors include
• Osmotic effects
• pH
• Chemical toxicity
• Redox potential
Biochemical-biological factors include:
0 Nutrient levels
t Presence of organic substances
• Predators
• Bacteriophages (viruses)
t Algae
• Presence of fecal matter
8.3.1.1 Physical Factors
Chamber!in and Mitchell (1978) have noted that, although many data have
been collected on coliform disappearance rates, mechanisms mediating the
rates have historically been poorly understood. According to Chamberlin and
Mitchell, however, light is one of the most important factors. They observe
that it is difficult to show statistically significant relationships between
coliform disappearance rates and many factors usually hypothesized as
429
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influencing those rates. In contrast, significant relationships between
light intensity and coliform disappearance rates can be demonstrated.
Chamberlin and Mitchell (1978) have shown that field data statistically
support the photo-oxidation model (to be discussed), and data presented by
Wallis _e_t _al_. (1977) also appear to implicate incident light. Subsequent
work by Sieracki (1980), Kapuscinski and Mitchell (1983), Lantrip (1983),
and others has demonstrated that viruses and enteric bacterial pathogens are
also sensitive to light but that viruses are generally less sensitive than
coliforms.
Chamberlin and Mitchell (1978) have elaborated upon possible mechanisms
by which light may increase coliform disappearance rates. They point out
that although in many cases of light induced mortality, one or more
photosensitizing substances are involved, visible and near ultraviolet (UV)
light can kill IE. col i in the absence of exogenous photosensi ti zers.
Grigsby and Calkins (1980) have confirmed the significance of the near UV.
One suggested mechanism is that light quanta drive some exogenous or
endogenous chromophore to an electronically excited state. The chromophore,
in the process of returning to the ground state, transfers its absorbed
light energy to another substance to form superoxides (Op), which, in turn,
cause damage to cellular components. Alternatively, the activated
chromophore may cause damage directly, without the agency of a
superoxygenated intermediate. Kapuscinski and Mitchell (1981) observed that
injury to the catalase system is the most likely site of damage in £. coli
and that the damage can be repaired if the coliforms are transferred to an
appropriate recovery medium. Krinsky (1977) has, on the other hand, argued
that the "cause of death" may be division-inhibition, mutation, and/or
membrane damage.
Substances within coliform and other bacterial cells are effective,
near-UV chromophores, including ubiquinones, porphyrins, and tryptophan
(Krinsky 1977). Exogenous sources of photo-oxidants include algal pigments,
lignins, and humic and fulvic acids. More highly colored and turbid waters
have been shown to produce peroxides, singlet oxygen, and hydroxide radicals
430
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at greater rates than well waters (for example, Zepp ejt aj_. 1977; Cooper and
Zika, 1983).
Adsorption, coagulation, and flocculation may affect coliform
disappearance 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 (1978), early investigations by
several workers have demonstrated that clays tend to adsorb conforms 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. Gannon
ejt al. (1983) found that 90 to 96 percent of the coliforms entering a lake
from upland watersheds were associated with 0.45 to 5 fzm particles.
Sedimentation involves the settling out of bacterial particles and
aggregates. The rate of disappearance may be materially influenced by
aggregation and sedimentation, but the magnitude and direction of the change
in rate is not well understood. The mechanism of apparent disappearance 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). However simple, sedimentation may sometimes be
the predominant mechanism of removal as Gannon ejt a]_. (1983) demonstrated in
a field study of coliform survival in a lake. Accordingly, modeling
coliform disappearance in the water column may give misleading results,
particularly where shellfish are harvested for human consumption. Reduction
in coliform levels in the water column may simply mean increased numbers in
the benthos.
Temperature influences most, if not all, of the the other factors.
Bitton (1980) and Lantrip (1983) argue that temperature is the single most
important modifier of decay rates, especially in freshwater and in the dark.
431
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8.3.1.2 Physicochemical Factors
Mitchell and Chamberlin (1978) report that physicochemical factors may
have significant effects on disappearance 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
addition, Sieracki (1980) has observed a synergism with light effects. Work
by Zaf iriou and True (1979) suggest that nitrite photolysis in seawater may
be a partial cause. 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
"oliogodynamic action" of silver and copper salts. According to Mitchell
and Chamberlin (1978), heavy metals have been implicated as important
mediators of E,. coli disappearance 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 disappearance rates. In
addition to this, redox may influence disappearance rates in other ways,
although data on this are not extensive.
Finally, Kott (1982) has presented evidence that when coliforms undergo
the transition from the generally low oxygen environment of sewage to the
higher oxygen levels found in seawater, the oxygen shock promotes rapid
decay.
8.3.1.3 Biochemical and Biological Factors
Nutrient concentrations may be important in determining disappearance
rates under some conditions. In many nutrient studies, the apparent impact
of nutrient addition to the coliform culture is due to chelation of heavy
metal ions (Mitchell and Chamberlin,^1978). Thus, the apparent decrease in
disappearance rate in many cases may not be due to the additional nutrient,
but instead to reduce toxicity of the culture medium. Mitchell and
Chamberlin (1978) cite the work of Jones (1964) who found that E. coli would
432
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not grow at 37°C in either filter-sterilized natural or synthetic seawater
supplemented with glucose, ammonium chloride, and potassium phosphate.
Inhibition could be reversed by autoclaving, by addition of very small
amount of organic matter, or by addition of metal chelating or complex ing
agents. Jones demonstrated that two levels of toxic metals would produce
the inhibitory effect, and concluded that the apparent influence on
disappearance rates was due to naturally occurring trace heavy metals in
solution. Furthermore, as Mitchell and Chamberlin (1978) note, other
researchers have obtained experimental results implicating heavy metals, and
their chelation upon addition of nutrients, in apparent changes in
disappearance rates.
In some situations, it appears that nutrient levels influence
disappearance rates in ways unrelated to toxic metals availability. Savage
and Hanes (1971). and Chamberlin (1977), for example, have reported growth-
limiting effects of available BOD or organic matter. Recent work by Dutka
and Kwan (1983) indicates that after-growth and long-term persistence is
particularly sensitive to nutrient levels. Further, it is possible that the
level of nutrients affects coliform predators, thereby influencing rates of
grazing on coliforms. Mitchell and Chamberlin (1978) report that predators
in natural waters may be significant in reducing coliform populations given
high predator levels. They cite three groups of micro-organisms which may
be importantly 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 disappearance of coliforms in both
fresh and seawater, although Lantrip (1983) did not observe a significant
predator influence in chamber experiments using freshwater. 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 (1978), 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.
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Some forms of phytoplankton produce antibacterial agents which are
excreted into the water column. These substances are heat-liable macro-
molecules, and according to Mitchell and Chamberlin (1978) 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 disappearance 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 8-1 is a summary of factors influencing coliform disappearance
rates.
8.3.2 Modeling Formulations
Traditionally, coliform modeling has only taken into account
disappearance, and a simple first-order kinetics approach has been used
(Baca and Arnett, 1976; Chen, et_ _al_., 1975; Chen ejt _al_., 1976; U.S. Army
Corps of Engineers,1974; Chen and Orlob, 1975; Lombardo, 1973; Lombardo,
1972; Smith, 1978; Anderson et aj_. 1976; Huber, et _al_. 1972; Hydroscience,
1971; Chen and Wells, 1975; Tetra Tech, 1976b):
dC _ ,c (8-1)
dt KU
or
Ct = CQe"kt (8-2)
where C = coliform concentration, MPN or count/100 ml
CQ = initial coliform concentration, MPN or count/100 ml
C. = coliform concentration at time t, MPN or count/100 ml
-1 -1
k = disappearance rate constant, day or hr~
t = exposure time, days or hours.
434
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A summarized listing of values for k is presented in Table 8-2. The
data summarize 30 studies of rates measured in situ. Table 8-3 shows values
for k from a number of modeling studies. The median rate for the jji situ
studies is .04 hr" with 60 percent of the values less than .05 hr"1 and 90
percent less than .22 hr"1.
TABLE 8-1. FACTORS AFFECTING COLIFORM DISAPPEARANCE RATES
Factor
Effects
Sedimentation
Temperature
Adsorption, Coagulation, Flocculation
Solar Radiation
Nutrient Deficiencies
Predation
Bacteriophages
Algae
Bacterial Toxins
Physiochemical Factors
Important with regard to water column coliform
levels, particularly where untreated or primary
sewage effluent or stormwater is involved, and
under low vertical mixing conditions. May
adversely affect shellfish beds by depositing
coliforms and fecal matter into benthos.
Probably the most generally influential factor
modifying all other factors.
Inconclusive.
Important; high levels may cause more than 10-fold
increase in disappearance rate over corresponding
rate in the dark in seawater. Rates also
materially increased in freshwater.
Appear to accelerate disappearance. Numerous
studies have indicated that increasing nutrient
levels of seawater decrease disappearance rates.
Several species of organisms (bacteria, amoebae)
have been shown to attack and destroy £. coli.
Importance of predation depends strongly on the
concentration of predators.
Apparently not important.
Bactericidal substances are known to be produced
by planktonic algae. Substances may be
photoactivators, mediating the influence of light
on coliform disappearance. This might account for
variability of data in studies of light-induced
disappearance rates. Another hypothesis is that
algal predators with blooms concomitant with algal
blooms may produce substances toxic to £. col i or
may prey upon them.
Antibiotic substances produced by indigenous
bacteria are not believed important in coliform
disappearance.
Apparently, pH, heavy metals content, and the
presence of organic chelating substances mediate
coliform disappearance rates. Importance of each,
however, is poorly understood at present.
Salinity strongly enhances the effect of solar
radiation.
435
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A number of researchers have determined values for the half saturation
constant (K ) for E. coli growth, using the Monod expression:
b
TABLE 8-2. COLIFORM BACTERIA FRESHWATER DISAPPEARANCE RATES MEASURED
IN SITU (AFTER MITCHELL AND CHAMBERLIN, 1978)
System
Ohio River
Upper Illinois River
Lower Illinois River
"Shallow Turbulent Stream"
Missouri River
Tennessee River
(Knoxville)
Tennessee River
(Chattanooga)
Sacramento River
Cumberland River
Glatt River
Groundwater Stream
Leaf River
(Mississippi)
Wastewater Lagoon
Maturation Ponds
Oxidation Ponds
Lake Michigan
Ford Lake
(Ypsilanti, Michigan)
DeGray Reservoir
(Arkansas)
Temperature
Summer (20°C)
Winter (5°C
June-September
October and May
December-March
April and November
June-September
October and May
December-March
April and November
Winter
Summer
Summer
Summer
Summer
10°C
7.9-25.5°C
19°C
UTII
10-17°C
August
October 1976 (15°C)
March 1977 (10°C)
June 1977 (20°C)
k(l/hr)
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
0.005
0.072
0.23
1.1
0.021
0.017
0.00833-0.029
0.083
0.07
k = 0.108
•(1.19)T-20
0.36
0.4
0.052
0.109 and 0.016
0.138 and 0.114
Reference
Frost and Streeter (1924)
Hoskins et a].. (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 et a].. (1934)
Wuhrmann (1972)
Mahloch (1974)
Klock (1971)
Marais (1974)
Marais (1974)
Zanoni et al_. (1978)
Gannon et &\_. (1983)
Thornton et al_. (1980)
Modified from Mitchell and Chamberlin (1978).
436
-------
(8-3)
where n = growth rate at nutrient concentrations, day"1
S =
PM ~
K =
concentration of growth limiting nutrient, mg/1
maximum growth rate, day"1
half-saturation constant producing the half-maximal value
of ^, mg/1
Table 8-4 shows some reported values for K .
However, Gaudy et_ aj_. (1971) have shown that the Monod expression
(Equation 8-3) is not adequate to describe transient coliform growth
behavior. Accordingly, as suggested by Mitchell and Chamberlin (1978), the
utility of the KS value is in evaluating which nutrient may be growth
limiting rather than in estimating a growth rate, ^.
TABLE 8-3. VALUES FOR COLIFORM-SPECIFIC DISAPPEARANCE RATES
USED IN SEVERAL MODELING STUDIES
System
k @20 C,
1/hr
North Fork Kings River, .042
Cal i form'a
Various Streams .0004-.146
Lake Ontario .02-.083
Lake Washington .02
Various Streams .042-.125
Boise River, Idaho .02
San Francisco Bay Estuary .02
Long Island Estuaries, .02-.333
New York
Reference
Chen, et al_. (1976)
Baca and Arnett (1976)
U.S. Army Corps of Engineers (1974)
Chen and Orlob (1975)
Hydroscience (1971)
Chen and Wells (1975)
Chen (1970)
Tetra Tech (1976)
437
-------
TABLE 8-4. NUTRIENT K VALUES FOR ESCHERICHIA COLI (AFTER MITCHELL AND CHAMBERLIN, 1978)
Nutrient Medium
Glucose minimal medium
seawater
OJ
co
Lactose seawater
minimal medium
Phosphate minimal medium
minimal medium
T
oc
30
30
20
20
30
Ks
Micromoles
22.
19.4
41.7
405.
550.
44.
50.
111.
0.7
17.35
Remarks Reference
Monod (1942)
Moser (1958)
Schultz and Lipe (1964)
Jannasch (1968)
Jannasch (1968)
Monod (1942)
uptake study Medveczky and Rosenberg (1970)
Shehata and Marr (1971)
Glucose
30
0.378
Shehata and Marr (1971)
-------
Work on coliforms in the Ohio River by Frost and Streeter (1924)
revealed that the log decay rate for coliforms is nonlinear with time.
Accordingly, use of a simple decay expression such as Equation (8-1) with a
single value of k is only an approximation to the actual disappearance
process. Such an approach must, to some extent as a function of time,
overestimate 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
8-2. 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:
-let ' L' t
Ct = Coe + coe
where C^ = coliform concentration at time t, MPN or count/100 ml
C ,C' = concentrations of each of the two hypothetical organism
types, MPN or count/100 ml
k,k' = decay rates for the two organism types, day
Table 8-5 shows values for C , C' k, and k1 for E_. col i as estimated by
Phelps (1944).
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:
CT = CT e' (8-5)
't 'o
CF = Cp e"kf* (8-6)
rt ro
C. = C, e~kst (8-7)
439
-------
0.01
Figure 8-2. 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).
-------
TABLE 8-5. VALUES OF Co, C1, k, AND k1 FROM THE OHIO RIVER
PHELPS (1944)
Parameter
C (percent)
k (I/day)
Half-life (day)
Cg (percent)
k1 (I/day)
Half- life (day)
Warm Weather
99.51
1.075
.64
.49
.1338
5.16
Cold Weather
97
1.165
.59
3.0
.0599
11.5
where C, = organism concentration at time t, MPN or count/100 ml
C = organism concentration at time zero, MPN or count/100 ml
Table 8-6 provides data for kj, k<- and kp as summarized from Lombardo
(1972).
As discussed earlier, recent studies have suggested that incident light
levels strongly affect coliform disappearance rates. Chamberlin and
Mitchell (1978) have defined a light level-dependent disappearance rate
coefficient as
k' = yoe'az (8-8)
where k1 = the light dependent coliform disappearance rate, 1/hr.
2
kn = proportionality constant for the specific organism, cm /cal
2
¥ = incident light energy at the surface, cal/cm -hr
a = light attenuation coefficient per unit depth
z = depth in units consistent with a.
441
-------
TABLE 8-6. SUMMARY OF DECAY RATES OF TC, FC, AND FS,
REPORTED BY LOMBARDO (1972)
Indicator
TC
FC
FS
n
16
13
5
Median
k (1/hr)
0.038
0.048
0.007
Minimum
k (1/hr)
0.010
0.008
0.002
Maximum
k (1/hr)
0.105
0.130
0.063
Then, incorporating the vertical dispersion of bacterial cells,
-V7 - - E7 - k'C(z.t) (8-9)
7. OL. 7- O L.
2
where E = the vertical dispersion coefficient, cm /hr
V = the vertical settling velocity, cm/hr
An expression of this kind is useful where the vertical distribution of
coliforms is nonuniform over depth and where disappearance is assumed to be
solely a function of light intensity. Chamberlin (1977) has presented
solutions of Equation 8-2 for various ranges of V , E , kn, a, and H (depth
of water column) using dimensionless variables.
According to an independent development by Mancini (1978) and
Chamberlin and Mitchell (1978), 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:
and
442
-------
k = k (8-11)
where f = the depth-averaged light intensity, cal/cm2/hr
H = the depth of the water column in units consistent with a
k = the depth-averaged light-dependent disappearance rate, hr"
The depth-averaged, light-dependent, disappearance rate, k, may be used
in the first order disappearance expression for a vertically mixed water
body so that:
f = -kC (8-12)
It is clear that the use of such a model (Equation (8-12)) 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 8-7 presents some values for k«.
Since coliforms and other indicators are known to decay in the dark,
Mancini (1978) and Lantrip (1983) have developed decay rate models combining
light-dependent and light-independent (i.e., dark) components. The model
proposed by Mancini expresses k1 as a function of temperature, percent
seawater, and depth-averaged light intensity:
k. = (0.8+0.006(%SH))1>07T-20 + ^ (8
where T = water temperature in C.L
The model coefficients were estimated based on a combination of
laboratory, chamber, and field studies. Note that k^ is not expressed
as a function of either salinity or temperature.
443
-------
TABLE 8-7. COMPARISON OF k« ESTIMATES BASED ON CHAMBERLIN AND MITCHELL
(1978rWlTH ADDITIONAL VALUES
Organism
Conform Group
Study
14 field studies
Mean
5' percentlle
95 percentile
24 field studies
Mean
5 percentile
95 percentile
61 laboratory studies
Mean
5 percentlle
95 percentile
k£
(cm2/cal)
0.481
0.163
1.25
0.168
0.068
0.352
0.136
0.062
0.244
Data Source
Gameson and Gould (1975)
Foxworthy and Kneeling (1969)
Gameson and Gould (1975)
Fecal Conform
Total Conforms
Fecal Conforms
Escherlchia coli
Seratla marcescens
Bacillus subtilis
var. nlger
Fecal Streptococci
Estimated from diurnal
field experiments 1n SW
Estimated from compilation of
field and laboratory studies,
both SW and FW.
22 chamber studies 1n FW
Mean
Minimum
Maximum
22 chamber studies 1n FW
Mean
Minimum
Maximum
4 field studies
Mean
Minimum
Maximum
4 laboratory studies
Mean
4 field studies
Mean
Minimum
Maximum
1 laboratory study
3 laboratory studies
Minimum
Maximum
3 field studies
1 field study
12 field studies, Initial rates
Mean
Minimum
Maximum
23 chamber studies in FW
Mean
Minimum
Maximum
0.18 at I = 1.0 cal/cm?hr Bellalr et al. (1977)
0.07 at I 0.1 cal/ci/hr
Manc1n1 (1978)
Salmonella typhimurlum 2 laboratory studies
0.042
0.004
0.000
0.013
0.005
0.000
0.011
0.362
0.321
0.385
0.354
0.192
0.093
0.360
0.002
0.048
0.123
0.000
0.007
0.091
0.004
0.184
0.008
0.001
0.028
1.48
6.40
Lantrlp (1982)
Lantrlp 91982)
Gameson and Gould (1975)
Gameson and Gould (1975)
Gameson and Gould (1975)
Gameson and Gould (1975)
Gameson and Gould (1975)
Gameson and Gould (1975)
Foxworthy and Kneeling (1969)
Lantrlp (1982)
Elsenstark (1970)
444
-------
lantrip (1983) developed a set of temperature and light-dependent
models based on a series of chamber studies conducted in freshwater.
Separate models were determined for TC, FC, and FS. He used nonlinear
regression methods to determine the "best" coefficient values and reported
both the "best" estimates and associated standard deviations. • The three
models have the same form:
k1 = kdj2(y-20 + kf/ (8-14)
where kd 2Q = "dark" decay rate at 20°C (l/h\rl
6 = temperature correction term
The coefficients for the three models are summarized in Table 8-8. Note
that Lantrip also considers kf to be independent of temperature.
Finally, many investigators have noted an initially very low decay rate
in laboratory and field studies. For example, see Mitchell and Chamberlin
(1978), Mancini (1978), and others. Kapuscinski and Mitchell (1983) and
Severin et _al_. (1978) have argued that this "shoulder" in the decay curve is
not the consequence of growth or particle breakup but is instead due to the
nature of the photo-oxidation process. Severin _e_t a±. present two
mechanistic models that would produce a "shoulder":
• Multi-target model based on assumption that several targets
or sites in the organism must be hit before the organism will
be killed:
Ct-C0ll-|
-------
- c
"
where n = event threshold for inactivation
(8-16)
Such models are still novel in engineering applications and have not yet
been incorporated into water quality models.
8.3.3 Methods of Measurement
Estimates of the coliform disappearance rate, k, may be obtained in a
number of ways in the laboratory chamber studies, or, preferably, j_n 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 disappearance may be determined
for various combinations of conditions. Unfortunately "bottle effects"
often distort laboratory results as shown by Zanoni and Fleissner (1982),
TABLE 8-8. PARAMETER ESTIMATES FOR LANTRIP (1983)
MULTI-FACTOR DECAY MODELS
Indicator n
TC
FC
FS
Estimate 38
Standard
Error
Estimate 41
Standard
Error
Estimate 38
Standard
Error
Standard Error kd,20
Regression (1/hr)
0.0151 0
0
0.020 0
0
0.0183 0
0
.0301
.0044
.0305
.0057
.0294
.0050
1
0
1
0
1
0
6
.0893
.0208
.0978
.0280
.0859
.0234
(
0
0
0
0
0
0
z*
cm /cal )
.0022
.00065
.00377
.00081
.00502
.00076
446
-------
since enteric bacterial growth is promoted by availability of surfaces for
attachment.
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 (e.g., Zanoni et aj_. 1978 and Gannon et aj_.
1983 provide recent examples) is to add a slug of a conservative tracer
substance (a dyes rare element, or radioisotope) to the steady-state
discharge. 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
sufficiently 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
concentration 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" or 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
447
-------
has a mean velocity of 2 ft per second, then the first sampling site should
be sampled:
5280 ft v 1 second 1 hr _ 7, hr
mile x 2 ft 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
corresponding to the peak loading. Where possible, dye studies or other
techniques should be used to characterize stream dispersion at the sampling
location. Then, by integrating under the curve, total surviving coliforms
can be estimated. If, on the other hand, discharge and stream conditions
are clearly at steady-state, sampling times are of no consequence.
Equation (8-17) may be used to estimate k where a slug dose of tracer
has been introduced into the discharge (assuming first-order decay):
k = -In (CtF0/FtC0)/t (8-17)
where F = discharge concentration of tracer, mg/1
Ft = observed concentration of tracer, mg/1
If no tracer is used and conditions approximating plug flow exist, then:
k = -ln(Ct/C0)/t (8-18)
where CQ = concentration of coliforms at the base sampling site, MPN
or count/100 ml
Regardless of the technique used for estimating k, it is important to
concurrently quantify, to the extent possible, those variables which
influence k. For example, light levels should be measured or at least
estimated 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 8-9 shows how serious
such errors can be. The data show T-90 values for coliforms as a function
448
-------
TABLE 8-9. EXPERIMENTAL HOURLY T-90 VALUES
(AFTER WALLIS, ET AL., 1977)
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
of incident light. T-90 values are the times required for 90 percent
mortality. The associated k values are .058 hr~ in the dark and .1 hr~ at
midday. It is clear that estimating a single value for a k could result in
greater than order-of-magnitude errors.
8.4 SUMMARY
The coliform group is of interest as an index of potential pathogen
contamination in surface waters and has become one of the more commonly
modeled water quality parameters. Modeling coliforms usually involves the
use of a simple first-order decay expression to describe disappearance.
Since regrowth is generally neglected, no growth terms are normally included
in the model.
The disappearance 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 near-UV and visible range) is important as are a number of
449
-------
physicochemical factors. Rates of disappearance are also sensitive to the
salinity of the water which also affects the influence of light on
disappearance rates.
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