Analysis of storage/release systems in urban stormwater quality management

MISSING IMAGE

Material Information

Title:
Analysis of storage/release systems in urban stormwater quality management
Physical Description:
xiv, 309 leaves : ill. ; 28 cm.
Language:
English
Creator:
Nix, Stephan J
Publication Date:

Subjects

Subjects / Keywords:
Storm water retention basins   ( lcsh )
Storm sewers   ( lcsh )
Water -- Pollution   ( lcsh )
Water quality   ( lcsh )
Genre:
bibliography   ( marcgt )
non-fiction   ( marcgt )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1982.
Bibliography:
Includes bibliographical references (leaves 300-307).
Statement of Responsibility:
by Stephan Jack Nix.
General Note:
Typescript.
General Note:
Vita.

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
oclc - 09588764
ocm09588764
System ID:
AA00011129:00001

Full Text














ANALYSIS OF STORAGE/RELEASE SYSTEMS
IN URBAN STORMWATER QUALITY MANAGEMENT
















BY

STEPHAN JACK NIX


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY



UNIVERSITY OF FLORIDA


1982
























To My Mothert, FatheA,
and Btothes, David, Michae~,
RichaLrd, Kenneth, and Patrick,

and

My Wife, Autaire,
and Childrien, Jed rtey and Stephanie















ACKNOWLEDGEMENTS


I have benefited from the advice, knowledge, and friendship of

many individuals during the course of this study and my academic career

at the University of Florida. I am especially indebted to my doctoral

committee chairman and mentor, Dr. James P. Heaney, who expended con-

siderable effort to direct this dissertation and my general course of

study. Dr. Heaney's unique outlook on environmental problems has cer-

tainly influenced my own. Equally appreciated is the assistance, advice,

and encouragement provided by Dr. Wayne C. Huber. Dr. Huber's mastery

of many areas in mathematics and engineering has been particularly

valuable.

Special thanks are due to Dr. Herbert A. Bevis, Dr. Bryan E. Melton,

Dr. Gary D. Lynne, Dr. Barry A. Benedict, and Dr. W. Brian Arbuckle for

their assistance and review of this dissertation. I am also grateful to

Mr. Robert Dickinson and the many other colleagues and friends who aided

and encouraged me over the last several years.

The persistence and professional typing of Mrs. Dibbie Dunnam,

Mrs. Peggy Paschall, and Mrs. Wendy Stafford contributed greatly to the

completion of this study. I am deeply grateful to them for their efforts

under difficult circumstances.

Some of the work presented here was supported by the Municipal

Environmental Research Laboratory, United States Environmental Protection

Agency through Project CR-805664.


iii















TABLE OF CONTENTS


Page

ACKNOWLEDGEMENTS............................................... iii


LIST OF TABLES ................... ........ .... ...... ..... ... ..... vi


LIST OF FIGURES.................................................. vii


ABSTRACT ................. .... ........ ........................... .. xiii


CHAPTER 1: INTRODUCTION.......................................... 1


CHAPTER 2:


STORAGE/RELEASE SYSTEMS IN URBAN STORMWATER QUALITY
MANAGEMENT....... ............................ ........


Introduction....... ............. ................... .........
Basic Configurations .........................................
Theoretical Representations..............................
Evaluation Techniques......................................
Measure of Performance--Reliability..........................
Summary.......................................... ............


CHAPTER 3: STATISTICAL ANALYSIS OF STORAGE/RELEASE SYSTEMS.......

Introduction ........... ......................... ...........
System Conceptualization ...................................
Characterization of Rainfall and Runoff Events..............
Analysis of Storage/Release Systems......................
Summary and Critique......................... ................


CHAPTER 4:


COMPUTER SIMULATION OF STORAGE/RELEASE SYSTEMS:
THE STORM WATER MANAGEMENT MODEL STORAGE/TREATMENT
BLOCK ............. ....... ..................... ... ......


Introduction................................................
Model Structure.. ............. ............................
Modeling Techniques ........................................
Application to a Storage/Release System....................
Summary................ .................................









Page


CHAPTER 5: PRODUCTION THEORY AND STORAGE/RELEASE SYSTEMS.......... 128

Introduction................................................. 128
Production Theory and Production Functions................... 129
Mathematical Representations of Production Functions......... 146
Production Functions for Storage/Release Systems........... 164
Summary...................................................... 188


CHAPTER 6: STORAGE/RELEASE SYSTEM OPTIMIZATION.................. 190

Introduction............................................. 190
Design Optimization Problem................................. 191
Cost Functions........................ ........................ 196
Optimization Techniques..................................... 201
Summary... .......................... .. ........ ............ .. 230


CHAPTER 7: CASE STUDY............................................ 231

Introduction................................................. 231
Problem Setting.............................................. 232
Developing the Production Functions......................... 236
Mathematical Representations of the Production Functions..... 271
System Optimization......................................... 282
Summary .............. ............ ........ ... ... ... .... 292


CHAPTER 8: SUMMARY AND CONCLUSIONS.............................. 295


REFERENCES................. ....................... ................ 300


BIOGRAPHICAL SKETCH................................................ 308















LIST OF TABLES


Table Page


3-1 Rainfall/Runoff Event Parameters and Statistics 71

4-1 Geometric and Hydraulic Data for Hypothetical
Reservoir 109

5-1 Coefficients for the Transcendental Equation
Representing the Percent Volume Control Produc-
tion Function Produced by STORM 174

5-2 Coefficients for the Transcendental Equation
Representing the Percent Pollutant (BOD) Control
Production Function Produced by STORM 175

5-3 Coefficients for the Cobb-Douglas Equation Repre-
senting the Percent Pollutant (BOD) Control Pro-
duction Function Produced by STORM 178

7-1 Coefficients for the Cobb-Douglas, Mitscherlich,
and Quadratic Equations Representing the Production
Functions Produced by the S/T Block 272

7-2 Optimal Storage/Release Combinations for Volume
Control 285

7-3 Optimal Storage/Release Combinations for TSS Control 287

7-4 Optimal Storage/Release Combinations for TSS Removal 289














LIST OF FIGURES


Figure Page


1-1 Design Event Minimizing the Sum of Control
Costs and Damages. 6

1-2 Design Event Maximizing Net Benefits. 7

2-1 Basic Storage/Release System Configurations. 14

2-2 Batch Reactor. 19

2-3 First-Order Reactions in a Batch Reactor. 21

2-4 Results of Hydrocarbon Settling Test. 24

2-5 Integral Method of Estimating "Reaction" Order
for Hydrocarbon Settling. 25

2-6 Tracer Response, Steady-State Flow Reactor. 28

2-7 Plug-Flow Reactor. 34

2-8 Completely Mixed Flow Reactor. 35

2-9 Brune's Sediment Trap Efficiency Curves. 45

2-10 Discrete Inputs of the Inflow Rate and Pollutant
Concentration at Equal Time Intervals. 52

2-11 Storage/Release System, Howard's Statistical
Method. 54

2-12 Storage/Release System, STORM. 63

3-1 Simplified Representation of Independent Rain-
fall of Runoff Events. 70

3-2 Cumulative Gamma Distributions. 74

3-3 Autocorrelation Function for the Hourly Precipi-
tation Record of Des Moines, Iowa, 1968. 76

3-4 Autocorrelation Function for the Hourly Runoff
Record (Generated by STORM) of Des Moines, Iowa,
1968. 78


vii








Figure Page


3-5 Number of Events per Year Versus the Minimum
Dry Period. 79

3-6 Determination of Effective Storage Capacity, V 86

3-7 Determination of Mean Effective Storage Capacity,
VE. 89

3-8 Determination of the Long-Term Fraction of the
Total Pollutant Load or Runoff Volume Not Cap-
tured by the Storage Basin. 91

3-9 Determination of the Long-Term Fraction of the
Total Pollutant Load or Runoff Volume Bypassing
the Interceptor or Mainsteam. 94

4-1 SWMM Storage/Treatment Block. 100

4-2 Storage/Treatment Unit. 102

4-3 Completely Mixed, Variable-Volume Detention Unit. 112

4-4 Plug-Flow Detention Unit. 114

4-5 Camp's Sediment Trap Efficiency Curves. 120

4-6 Design Details, Humboldt Avenue Detention Tank,
Milwaukee, Wisconsin. 124

4-7 Effluent Flow Rate, September 12 (11:00 p.m.) to
September 21 (10:00 a.m.), 1972, Humboldt Avenue
Detention Tank, Milwaukee, Wisconsin. 125

4-8 Effluent Suspended Solids Mass Rate, September 12
(11:00 p.m.) to September 21 (10:00 a.m.), 1972,
Humboldt Avenue Detention Tank, Milwaukee, Wisconsin. 126

5-1 One-Input, One-Output Production Function. 131

5-2 Two-Input, One Output Production Function. 133

5-3 One-Input, Two-Output Production Function. 135

5-4 Classical One-Input, One-Output Production Function
and the Average and Marginal Products. 137

5-5 Elimination of the Region of Increasing Average
Product. 139

5-6 Graphical Representation of the Marginal Rate of
Substitution and an Isocline. 142


viii








Figure Page


5-7 Determination of the Area of Substitution. 143

5-8 One-Input, One-Output Cobb-Douglas Production
Function and the Average and Marginal Products. 148

5-9 Two-Input, One-Output Cobb-Douglas Production
Function with Isoclines and Ridge Lines. 151

5-10 One-Input, One-Output Mitscherlich Production
Function and the Average and Marginal Products. 152

5-11 Two-Input, One-Output Mitscherlich Production
Function with Isoclines and Ridge Lines. 154

5-12 One-Input, One-Output Quadratic Production
Function and the Average and Marginal Products. 157

5-13 Two-Input, One-Output Quadratic Production
Function with Isoclines and Ridge Lines. 158

5-14 Piecewise Linearization of a Production Function. 161

5-15 Factors Affecting the Quality Control Performance
of Stormwater Storage/Release Systems. 165

5-16 Storage/Treatment (or Release) Isoquant Produced
by a Transcendental Production Function. 171

5-17 Storage/Treatment (or Release) Production Function
for BOD Control Produced by STORM, Graphical and
Transcendental Representations, Minneapolis, Minne-
sota. 176

5-18 Storage/Treatment (or Release) Production Function
for BOD Control Produced by STORM, Graphical and
Cobb-Douglas Representations, Minneapolis, Minne-
sota. 179

5-19 Off-Line Storage/Release Production Function for
BOD Control Produced by Hydroscience Statistical
Method, Compared with STORM-Generated Production
Function, Denver, Colorado. 180

5-20 Derivation of Production Functions for In-Line
Storage/Release Systems From Hydroscience Sta-
tistical Method. 182

5-21 In-Line Storage/Release Production Function for
Pollutant Load or Volume Control Produced by
Hydroscience Statistical Method, VvR = 1.73. 183









Figure Page

5-22 In-Line Storage/Release Production Function
for Additional Open Beach Days Produced by
Hydroscience Statistical Method, Kingston,
New York. 185

5-23 In-Line Storage/Treatment (or Release) Production
Function for Suspended Solids Control Produced
by Howard's Statistical Method, Minneapolis,
Minnesota. 187

5-24 In-Line Storage/Release Production Function for
BOD Control Produced by SWMM Storage/Treatment
Block, Atlanta, Georgia. 189

6-1 Annual Cost Verus Additional Open Beach Days
as a Result of Fecal Coliform Control With a
Storage/Release System, Kingston, New York. 194

6-2 Determination of Maximum Net Benefits. 195

6-3 General Input Cost Function Types. 200

6-4 Costs for Earthen Storage Basins, September, 1976. 202

6-5 Costs for Covered Concrete Storage Basins, Sep-
tember, 1976. 203

6-6 Costs for Uncovered Concrete Storage Basins, Sep-
tember, 1976. 204

6-7 Convex and Concave Functions. 209

6-8 Terms of Equations 6-22. 213

6-9 Optimization in Production Space. 223

6-10 Optimal Costs for Any Storage/Release System Per-
formance Level. 225

6-11 Possible Isocost Curves. 226

6-12 Optimization in Cost Space. 228

7-1 Hypothetical Scenario. 233

7-2 Percent Volume Control Production Function,
SWMM Level I. 239

7-3 Percent Pollutant (BOD) Control Production Function,
SWMM Level I. 240








Figure Page


7-4 Cumulative Frequency Plots for Event Flows, Dura-
tions, Volumes, Interevent Times and Several
Cumulative Gamma Distributions. 243

7-5 Percent Volume and Pollutant Control Production
Function, Hydroscience Statistical Method. 245

7-6 Percent Volume Control Production Function,
Results of S/T Block Simulation. 247

7-7 Percent TSS Control Production Function, Results
of S/T Block Simulation. 248

7-8 Percent TSS Control-to-Percent Volume Control
Ratios. 249

7-9 Results of Settleability Tests for Total Suspended
Solids. 252

7-10 Percent TSS Removal Production Function, Results
of S/T Block Simulation. 253

7-11 Percent TSS Removal Production Function, Results
of S/T Block Simulation (Enlarged Range). 254

7-12 Percent TSS Removal-to-Percent TSS Control Ratios. 256

7-13 Percent TSS Removal-to-Percent Volume Control Ratios. 257

7-14 Percent of the Time that the Hourly TSS Load from
System Exceeds 50, 100, or 500 lbs/hour. 259

7-15 Percent TSS Removal Production Function, Completely
Mixed Storage Basin with Bypass. 261

7-16 Percent TSS Removal Production Function, Plug-Flow
Storage Basin without Bypass. 262

7-17 Percent TSS Removal Production Function, Completely
Mixed Storage Basin without Bypass. 263

7-18 First-Order TSS Removal Equations. 266

7-19 Percent TSS Removal Production Function,
-1
K = 0.0003 sec. 267

7-20 Percent TSS Removal Production Function,
-1
K = 0.00003 sec 268

7-21 Percent TSS Removal Production Function,
K = 0.000003 sec-1. 269
K = 0.000003 sec .269








Figure Page

7-22 Percent Volume Control Production Function, Rep-
resented by the Cobb-Douglas Equation. 273

7-23 Percent Volume Control Production Function, Rep-
resented by the Mitscherlich Equation. 274

7-24 Percent Volume Control Production Function, Rep-
presented by the Quadratic Equation. 275

7-25 Percent TSS Control Production Function, Repre-
sented by the Cobb-Douglas Equation. 276

7-26 Percent TSS Control Production Function, Repre-
sented by the Mitscherlich Equation. 277

7-27 Percent TSS Control Production Function, Repre-
sented by the Quadratic Equation. 278

7-28 Percent TSS Removal Production Function, Repre-
sented by the Cobb-Douglas Equation. 279

7-29 Percent TSS Removal Production Function, Repre-
sented by the Mitscherlich Equation. 280

7-30 Percent TSS Removal Production Function, Repre-
sented by the Quadratic Equation. 281

7-31 Expansion Paths for the Percent Volume Control
Production Function, Graphical Representation. 284

7-32 Expansion Paths for the Percent TSS Control Pro-
duction Function, Graphical Representation. 286

7-33 Expansion Paths for the Percent TSS Removal Pro-
duction Function, Graphical Representation. 288

7-34 Application of Cost-Space Optimization Procedure. 291

7-35 Final Cost Curve for Hypothetical Scenario. 293















Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy



ANALYSIS OF STORAGE/RELEASE SYSTEMS
IN URBAN STORMWATER QUALITY MANAGEMENT



By

Stephan Jack Nix

December, 1982

Chairman: James P. Heaney
Major Department: Environmental Engineering Sciences

Urban stormwater runoff acts as a transport medium for a variety of

wastes. This phenomenon can cause water quality problems and, as a re-

sult, corrective measures sometimes are required. Storage/release sys-

tems, consisting of a storage basin and a release mechanism, are often

used for this purpose. Unfortunately, there are few data to assess their

performance in this role and the techniques currently used to design and

analyze these systems are generally inadequate. A comprehensive and us-

able approach, reflecting the dynamic aspects of this problem, is pre-

sented in order to fill this need.

This dissertation develops and presents the necessary theories and

evaluation techniques to assess the long-term performance of storage/

release systems, the use of the production function to summarize system

performance, and several techniques to determine the most cost-effective

designs. The focus is on long-term analyses rather than "design storm"

or single-event approaches. Several available computer simulation and

xiii









statistical models capable of evaluating long-term performance are dis-

cussed. However, these models are either too inflexible or simple to be

generally applicable. To meet the need for a more sophisticated and

flexible model, a computer simulator known as the Storm Water Management

Model Storage/Treatment Block is introduced. The production function, a

concept taken from economic theory, is used to summarize the relationship

between long-term performance and the characteristics of the system. The

properties of production functions, as well as several possible mathemat-

ical representations, are also discussed. Simple analytical and graphi-

cal optimization techniques, which require the information contained in

the production function, are presented and developed for use in the de-

termination of cost-effective designs.

A hypothetical case study is used to demonstrate many of the tech-

niques and procedures, illustrate the generally expected results, and

suggest an overall methodology for analyzing storage/release systems.

Emphasis is also placed on the results produced by the Storage/Treatment

Block and its sensitivity to changes in the assumptions governing system

behavior.


xiv














CHAPTER 1

INTRODUCTION



Traditionally, urban stormwater has been viewed as a flooding and

drainage problem. Urban areas, by their nature, produce more runoff

than non-urban areas for a given storm event. To compound the problem,

urban areas more expeditiously remove stormwater because the roughness

of the land surface has been reduced and natural storage areas have been

eliminated. The damages incurred by uncontrolled stormwater runoff can

be impressive, dramatic, and disastrous. The need for solutions to

urban drainage and flooding problems is and has been apparent.

Not so apparent is the role urban stormwater plays as a transport

medium for pollutants. Undoubtedly, urban stormwater runoff is con-

taminated; but is it a water quality problem? It was not until the

early 1960's that this question was addressed at all. The first concern

on a national scale came with the Water Quality Act of 1965 (PL 89-234)

which provided some assistance to state and local authorities in develop-

ing controls for stormwater discharges (including combined sewer over-

flows). The Federal Water Pollution Control Act Amendments of 1972 (PL

92-500), however, marked the beginning of intense national concern over

the quality of the nation's waters and, as part of that concern, the

following goals were established:

1) The discharge of pollutants into navigable waters is to be

eliminated by 1985.






2
2) Navigable waters should be of sufficient quality by July 1,

1983, to protect aquatic life, wildlife, and recreation.

3) Areawide water quality management planning processes should be

developed and utilized in order to meet and maintain the

previously stated goals.

Other provisions included funds for research and aid for the imple-

mentation of management plans. The commitment mandated by PL 92-500 is

immense and was probably a product of the general thinking of the early

1970's embodied in the phrase "if we can send men to the moon, we ought

to be able to. ." The goals of PL 92-500 may be more difficult and

costly to achieve. This act specifically recognizes nonpoint pollution

sources as potentially damaging and Environmental Protection Agency

(EPA) guidelines called for "an analysis of the magnitude of existing

and anticipated urban stormwater problems" (p. 3-62, U.S. Environmental

Protection Agency, 1976).

The tone and goals of PL 92-500 had a dramatic effect on the way in

which urban stormwater was studied and analyzed. This act and sub-

sequent EPA guidelines on its interpretation tended to define urban

stormwater as a "problem" before enough field data and analysis had been

compiled to categorize it as such. This dampened the importance of

subsequent monitoring work and delayed answering the essential question:

Is urban stormwater a water quality problem? The Clean Water Act of

1977 (PL 95-217) responded to the unanswered questions and overwhelming

control cost estimates by suspending Federal funding for the treatment

and control of stormwater discharges from separate sewer systems. The

EPA also responded to the dearth of solid information and data by

creating the Nationwide Urban Runoff Program (NURP) in 1978. This





3

program is collecting data from twenty-eight planning efforts around the

United States and providing a coordinating body through which adminis-

trative and technical support is provided to assure consistency and

maximize the effectiveness of the effort.

When is urban stormwater a water quality problem? There are no

obvious guidelines. It can probably be said that urban stormwater is a

problem when it impairs or denies beneficial uses of a receiving water.

However, all this statement accomplishes is to shift the burden of

definition from "problem" to "beneficial use." Beneficial to whom or

what? By whose perception? There are no absolutes. Regulatory agen-

cies have assumed some of the specification of beneficial uses through

water quality criteria, effluent standards, and the classification of

receiving waters. Almost invariably, though, the substantiation for

these regulations is weak or nonexistent. Often, legislation is written

in such a way to make it very difficult to interpret.

The best way to determine the existence of an urban stormwater

quality problem is through a combination of intense local monitoring and

analysis and informed public forums to determine local perceptions. The

process must be localized because the potential for a problem is highly

site specific. Local factors influencing this determination are

1) climatology and hydrology;

2) characteristics of the receiving water;

3) the level of beneficial use desired; and

4) attitudes, resources, and values.

The process is partially subjective; but in a democratic society, the

need to address this question must ultimately come from the populace and

its ranking of urban stormwater problems in national and local priorities.






4

After an urban stormwater quality problem is identified, concern is

directed towards providing an effective means of control. The use of

storage/release systems has been the prominent means of control (Lager

et al., 1977). Regulatory pressures are further solidifying the use of

storage/release systems for this purpose. Storage/release systems

provide storage to capture a portion of the highly variable stormwater

flows and pollutant loads, detention capability to allow pollutant

removal to proceed, and a release mechanism through which the storage

capacity can be recharged and the contents released in a more controlled

fashion. Storage/release systems are not limited to structures such as

detention basins. In fact, this type of system is encountered by storm-

water in the urban environment at several points and in several forms.

Many have some potential to abate a stormwater quality problem. Among

the storage/release systems prevalent in the urban environment are

1) rooftops,

2) parking lots (porous and nonporous),

3) catch basins (including the percolating type),

4) natural depressions and impoundments,

5) soil storage and percolation,

6) sewer systems, and/or

7) manmade detention basins.

The focus of this study is on the basic theory and analytical

techniques useful to the analysis and design of storage/release systems

employed in urban stormwater quality management. Nevertheless, it

should be emphasized that the techniques and theory reviewed and formu-

lated herein are applicable to a wide range of water quality management

problems in which storage/release systems can play a role. This thesis








emphasizes urban stormwater, primarily because the developmental work

was so imbued; however, the generality of the work should not be lost.

The analysis of urban stormwater storage/release systems relies on

long-term data records (i.e., many years). Long-term data can be in

time series form or summarized by statistical parameters. Quantity

control facilities have long been designed by a statistically based

method known as the "design event." A design event is defined as a

runoff event or storm with a specific duration, depth, and return period

by which a facility design is based. The use of a design event is a

widely accepted method of designing storage/release facilities and it is

often mandated by statutory requirements. Unfortunately, the underlying

reasoning behind the selection of a design event has been forgotten by

many in the field of urban stormwater management. Figure 1-1 depicts

the costs of constructing and operating a hypothetical storage/release

system to handle specific design events and the damages incurred by any

subsequent runoff events. The damages could be attributed to stormwater

quality and/or quantity. Assuming the damages could be quantified in

monetary terms, the rational analyst would select the event inflicting

the minimum total cost (damages plus the control costs) on society

(James and Lee, 1971). An alternative approach presents the reduction

in damages as a "benefit" and Figure 1-1 yields to the benefit-cost

curves shown in Figure 1-2. In this case, the point where the net

benefits (total benefits less costs) are maximized is selected as the

design event. In either case, the answer is the same. Of course, there

is considerable analysis behind the development of the cost and damage

or benefit information over such a wide variety of conditions and, in

fact, it is usually very difficult to place a monetary value on damages















PRE-CONTROL DAMAGES









/ TOTAL


0
0



t,




m
I-.
V)
o
o
0=


Qt

(0'
U
0
C
I
4



4
4
C
-J


DESIGN EVENT RETURN PERIOD, years


Figure 1-1.


Design Event
and Damages.


Minimizing the Sum of Control Costs


CONTROL COSTS





























BENEFITS (REDUCTION
OF DAMAGES)


NET
BENEFITS


CONTROL COSTS


DESIGN EVENT RETURN PERIOD, years

Figure 1-2. Design Event Maximizing Net Benefits.






8

or benefits (see Chapter 6). Nevertheless, the simple logic behind the

selection of a design event appears to have been subdued by regulatory

and technical dogma. A review of several manuals intended for prac-

ticing engineers and analysts produces very few references to this

fundamental principle (Lager and Smith, 1974; Lager et al., 1977;

Municipal Environmental Research Laboratory, 1976; American Public

Works Association, 1981).

The design event method is widely used and attractive because of

the relative ease with which it is applied once the design event is

established. The fact that most regulations and statutes require storage/

release designs based on particular design storms solidifies its popu-

larity. The method is fundamentally sound and, given the technical

atmosphere from which it evolved (i.e., pre-computer age), it has per-

formed admirably in quantity control applications. Nevertheless, there

are serious drawbacks in quantity and quality control applications:

1) Pre-event design conditions (e.g., antecedent dry period,

previous storm characteristics, storage/release system status)

are difficult to assess because of the nearly random nature of

of storm events.

2) Additional uncertainties exist when developing a design storm

for water quality purposes. This is because of the lack of

historical data, the inability to measure benefits (or dam-

ages) associated with different levels of water quality, the

unreliability of water quality measurements, and the unclear

relationship between stormwater flows and pollutant loads (Nix

et al., 1981).

3) "Design" conditions in the receiving water are also necessary.








The advent of computers has opened the door to the analysis of the

behavior of storage/release systems over a long period and, thus, re-

duces the need for the design event. The proliferation of efficient

computers with large memories allows a return to the original intent of

the design event concept which is the analysis of a wide range of

storage/release system designs in order to locate the most effective

alternative.

Several mathematical models of the urban hydrologic cycle are

available. These include nomographic or desk-top procedures based on

computer results, statistical methods, or computer simulators. Unfortu-

nately, mathematical models are not perfect representations of the

physical world and this is why stormwater quantity and quality monitoring

is important. Monitoring is vital to our understanding of the physical,

chemical, and biological processes involved in the urban hydrologic

cycle, and, more specifically, the behavior of storage/release systems.

However, monitoring alone cannot provide the long-term information

needed to analyze and design these systems under the highly variable

conditions presented by stormwater flows and pollutant loads. Besides

being very expensive, monitoring does not directly provide, a priori,

the information needed to characterize the behavior of a wide range of

designs. Properly constructed models can provide this predictive informa-

tion. Analyses of future monitoring data will improve the models and,

concurrently, the models might be able to provide guidance for the

monitoring programs.

Unfortunately, many analysts are distrustful of or intimidated by

models, especially computer simulators. The distrust probably comes

from the extensive use of models to analyze stormwater problems in






10

recent years at the expense of adequate monitoring. Models alone are

not up to the task. Modeling and monitoring should be considered

complementary, not mutually exclusive. Consider the statement by

Sonnen (1980):

Mathematical models are relevant to a consideration
of data or information needs, because ostensibly
the intent of these models is to provide a means
by which quality prediction could obviate the need
for monitoring. (p. 33)

With this commonly held view of mathematical models, it is little wonder

that their use so often ends in disappointment. Mathematical models are

imperfect because our knowledge of the underlying principles is imper-

fect. While every effort should be made to increase this understanding,

properly calibrated and verified models are certainly useful and capable

of providing valuable insights.

The overall objective of this thesis is to construct a comprehensive

framework for analyzing storage/release systems used in urban stormwater

quality management in light of the need for long-term analyses and the

present condition of the data base. The specific objectives are to

1) review basic theory and available analytical techniques;

2) develop and present a flexible storage/release computer

simulator;

3) explore the use of production theory and the production function

in characterizing the performance of storage/release systems;

4) review simple analytical optimization techniques and develop

a graphical optimization technique whereby the production

information can be used to design cost-effective systems; and








5) synthesize the previous steps through the use of a case

study.



To this end, Chapter 2 discusses the basic theory and available ana-

lytical techniques. Chapter 3 explores a recently developed statistical

technique for analyzing storage/release systems (and urban stormwater

in general). This technique serves as an analytical prelude to the more

sophisticated computer simulator presented in Chapter 4. This simulator

was developed for this thesis and as part of a larger package of urban

stormwater simulation modules. Chapter 5 investigates the use of pro-

duction functions to represent the performance of storage/release systems

as urban stormwater quality control devices. Chapter 6 reviews simple

analytical optimization techniques and presents a flexible, effective

graphical method for guiding the cost-effective design and operation of

storage/release systems. Chapter 7 provides a case study through which

some observations can be made and the merits or demerits of many

of the techniques explored. This application will also point out the

integrated nature of the techniques and suggest a general methodology.

Chapter 8 summarizes the main points and presents concluding statements.















CHAPTER 2

STORAGE/RELEASE SYSTEMS IN URBAN STORMWATER QUALITY MANAGEMENT



Introduction

Storage/release systems play an important role in urban stormwater

quality management and, in fact, are the most commonly used control

technology (Lager et al., 1977; Finnemore, 1982). The systems are used

in a variety of situations in the urban environment. They are also

predominant in a number of agricultural applications (Loehr, 1974; Meta

Systems, Inc., 1979; Nix and Melton, 1979). Among potential urban

stormwater applications are the following:

1) control of combined sewer overflows;

2) sediment control from disturbed sites (e.g., construction);

and/or

3) control of pollutants and debris from industrial,

commercial, residential, and other urban land areas.

These systems are constructed in several different manners. Some are

covered or uncovered concrete basins; others are earthen (lined or

unlined) basins. Several are designed to serve aesthetic purposes as

well as to provide pollution control. In some cases, existing ponds or

lakes are used to provide the system. The actual form is dependent on

site conditions, the consideration of all system purposes, economic

factors, and public desires.








Urban stormwater storage/release systems have primarily been de-

signed through the use of the design storm. As revealed in Chapter 1,

this time-honored method is probably not equipped to handle the design

of stormwater quality management systems. This chapter explores the

basic configurations and theories needed to examine the problem in the

context of long-term evaluation. The theoretical exposition provides

the necessary foundation for the review and development of a series of

evaluation techniques.



Basic Configurations

The basic storage/release system in the urban stormwater flow

scheme receives a highly variable input with a large random component.

This fact makes the analysis of these systems much more complex than

those operating under steady state or quasi-steady state conditions

(e.g., sewage treatment facilities). Not only are these systems re-

ceiving fluctuating stormwater flows but also they accept highly vari-

able pollutant loads which further complicate the situation.

There are essentially two storage/release system configurations:

1) in-line and 2) off-line. The in-line configuration is shown in Figure

2-1(a). In this arrangement the flow is first directed to the storage

unit, which either accepts the flow or bypasses all or part of it.

Flows entering the storage unit are drained or released in some re-

strained fashion (unrestrained release implies no storage). The deci-

sion to accept or bypass flows is a function of the role of the system.

It is sometimes advantageous to capture the early portions of flow

events and bypass all other flows after the unit is filled. This will

be called the bypass mode. In some cases, it may be useful to allow









RUNOFF


BYPASS


RELEASE




- -RECEIVING WATER --R-Z-


(a) In-Line System


RUNOFF


BYPASS


, _RECEIVING WTE
RECEIVING WATER-


(b) Off-Line System


Figure 2-1. Basic Storage/Release System Configurations.








all flows to enter the unit and design a release mechanism to handle a

wide range of flows and provide the necessary relief.

The off-line configuration, shown in Figure 2-1(b), only accepts

flows from the mainstream after a predetermined capacity has been ex-

ceeded. The term "stream" is loosely defined to mean any movement of

stormwater. The excess is routed to the storage unit which may accept

or bypass part or all of the flow in the manner discussed above. The

flows entering storage are released in some restrained manner but the

flows returned to the mainstream must not exceed its capacity.



Theoretical Representations

Regardless of the configuration, the manner in which flows and/or

pollutants are handled by the storage/release system can be concep-

tualized with the same basic theoretical framework. This framework will

be used to establish a background for the performance evaluation tech-

niques discussed later in this chapter and in Chapters 3 and 4.



Storage Equation

Storage/release systems attenuate input flows and pollutant loads

by providing a repository from which controlled releases are made. The

flow continuity relationship for a storage/release system is given by

(Linsley et al., 1975)

dt = I(t) 0(t) (2-1)
dt

where V(t) = volume of water in storage, L ,

3 -1
I(t) = inflow rate to storage, L T-1

3 -1
0(t) = outflow rate from storage, L T and

t = time, T.






16

Theoretically, a storage unit could become so large that it would cap-

ture all inflows and release them at a constant rate. Such a unit would

totally "equalize" the flows. Obviously, physical and economic con-

siderations often prohibit such systems. However, some level of equali-

zation is often desirable in a number of applications. Some examples

include

1) the improvement of stormwater treatment unit efficiency by

minimizing peak flows;

2) the construction of smaller treatment facilities by reducing

the need to design for peak flows; and

3) the attenuation of shock loads from combined sewer overflows

or urban runoff on receiving waters.

Naturally, as the storage unit becomes smaller, the equalization effect

is reduced.

The storage equation is a simple, elegant representation of the

physical system. However, the equation is difficult to solve for all

but the most trivial functions of I(t) and 0(t). For example, assume

that the outflow is a function of the fluid volume in the storage basin,

i.e.,

0(t) = f[V(t)] (2-2)

Equation 2-1 may then be written as

dV I(t) f[V(t)] (2-3)
dt

Assume that a power function governs the relationship between outflow

and volume, i.e.,

0(t) = a[V(t)-V ]b (2-4)

where a, b = coefficients, and








V = volume of stored water at which outflow begins, L3

Substituting equation 2-4 into equation 2-3 yields

dV b
dV I(t) a[v(t)-Vo ] (2-5)

Equation 2-5 is very difficult to solve except under very restrictive

situations (Dooge, 1973). A few of the simpler cases are discussed

below.

When b = 1, the relationship between 0(t) and V(t) is linear and

equation 2-5 becomes

dV
+ aV(t) = I(t) + aV (2-6)

Equation 2-6 is a simple linear first-order differential equation for

which the solution is (Ross, 1964)

V(t) = exp(-fa dt){f[I(t) + aV exp(fta dt)dt
0 0 0 0

+ V(0)} (2-7)

or V(t) = exp(-at){ft[I(t) + aV ]exp(at)dt + V(0)} (2-8)
0 0

where V(O) = initial volume of stored water, L3

The obvious possible drawback in equation 2-8 is the inflow function,

I(t). Only rather trivial functions allow the complete solution of

equation 2-8.

When b = 0, the outflow is constant, i.e., 0(t) a, and equation

2-5 becomes

dV
d = I(t) a (2-9)


for which the solution is

V(t) = ft(I(t) a)dt (2-10)
0

This solution is useful for pumped outflow. The only restriction,

again, is the funciton I(t) and its ability to be integrated.






18

Reactor Theory and Reaction Kinetics

Reactor theory is a useful tool in the fields of chemistry and

chemical engineering. This theoretical framework also provides a good

structure for analyzing the behavior of pollutants (i.e., reactants) in

storage units (Metcalf and Eddy, Inc., 1972; Rich, 1973; Medina, 1976;

Medina et al., 1981b). The application is not necessarily limited to

chemical reactions as several physical and biological processes can

also be characterized by the mathematical representations found in

reaction kinetics. Reactors are often segregated into three ideal

reactor types: the batch reactor and two flow reactors, completely

mixed and plug flow. Each type of reactor essentially routes fluid

elements (along with any associated reactants, pollutants, etc.) in a

different manner. Whereas the storage equation describes the continuity

of mass for a storage unit, it does not provide this routing information.

The flow regime is particularly important in determining the ability of

the storage unit to carry out the desired reaction (i.e., pollutant

removal).



Batch reactors. Batch reactors are normally charged with reactants,

completely mixed, and allowed to react over time without inflow to or

outflow from the reactor. The system is shown in Figure 2-2. The

storage equation for a batch reactor is

dV
d 0 (2-11)
dt

The ideal batch reactor provides a convenient point to investigate some

of the basics of reaction kinetics. However, the concepts are appli-

cable to the other reactor types.






19















dV
dt


Figure 2-2. Batch Reactor.








Isothermal, irreversible reactions in a completely mixed, constant-

volume batch reactor are governed by the following equation (Levenspiel,

1972):

V = -KVCn(t) (2-12)
dt

or KCn(t) (2-13)
dt
-3
where C(t) = reactant concentration in the reactor, ML-3

V = constant fluid volume in the reactor, L ,

K = rate constant, T-1 and

n = reaction order.

The term -dC/dt represents the reaction rate. Under the assumption of

first-order kinetics (i.e., n=l), equation 2-13 becomes

dC = KC(t) (2-14)
dt

for which the solution is

C(t)= exp(-Kt) (2-15)

-3
where C(0) = initial reactant concentration in the reactor, ML-3

Equation 2-15 produces the family of curves shown in Figure 2-3.

First-order reactions are typified by those that are unaffected by the

initial concentration, C(0). The first-order assumption is commonly

made because of the simplicity of the governing equation and the fact

that it appears to perform adequately for many purposes. However, it

does not always fit the situation and, thus, it is sometimes necessary

to determine the correct reaction order.

Transforming equation 2-13 by the natural logarithm yields

dC (2-16)
In(- d_ = n In[C(t)] + ln(K) (2-16)
TEP





21








1.0

K= 0.01 hr





O

05-

- K= 0. I hr





K= 1.0 hr1



0 5 10 15 20
TIME, t, hours
Figure 2-3. First-Order Reactions in a Batch Reactor.






22

This transformation allows the estimation of the reaction order, n, and

the rate constant, K, through a simple linear regression of ln(-dC/dt)

against ln[C(t)]. The value of dC/dt is taken from a concentration

versus time curve produced by representing the batch reactor experimental

data; its value at any particular point in time is the slope of the

curve at that point. This is known as the differential method of analy-

sis (Levenspiel, 1972). The principal drawback of this technique is

that the differentiation of experimental data via graphical means intro-

duces extra "noise" to the analysis due to the visual error in deter-

mining the slopes (Butt, 1980).

An alternative approach, known as the integral method of analysis

(Levenspiel, 1972), tests individual rate equations of any order (except

n = 1) until a suitable one is found. The analysis is based on the

integration of equation 2-13, i.e.,

[ C() ]n = { Kt (2-17)
C(0) 1-n
[c(o)]ln
or
Ct l-n 1l-n
C.F. = {[ t ]1-n 1 C(n) = Kt for n # 1 (2-18)
C(0) n-l

where C.F. is known as the concentration function. By plotting values

of C.F. against values of t for the experimental data and a particular

value of n, a determination can be made as to whether the proper value

of n, the reaction order, has been selected. This is done by noting how

well the values of C.F. and t are represented by a straight line. If

they are, the value of n is correct; if not, another value must be

selected and tested. Standard linear regression techniques can be used

to determine the appropriateness of each value of n. An example of this

method is shown later.






23

Reaction kinetics and environmental processes. Although reaction

kinetics is primarily applied to chemical processes, it is also useful

in environmental and sanitary engineering. The basic mathematical

representation given by equation 2-13 is applicable to biological and

physical phenomena as well as chemical reactions. For example, first-

order kinetics is often used to represent the oxidation of substrate by

microorganisms in the analysis and design of biological treatment units

in sanitary waste facilities (Fair et al., 1968; Rich, 1973). The

destruction of coliforms by disinfectants is also commonly treated as a

first-order "reaction" (Chick, 1908; Collins et al., 1971). It is also

useful in the analysis of storage/release systems where the settling of

particles is often treated as a "reaction" (City of Milwaukee et al.,

1975; Smith, 1975; Medina, 1976; Medina et al., 1981b).

An excellent example of how reaction kinetics can be applied to an

environmental engineering problem is in settleability testing. Whipple

and Hunter (1981) analyzed the settleability of several pollutants in

urban runoff through the use of a 6-ft settling column (essentially a

batch reactor without continuous mixing). The results for hydrocarbons

are shown in Figure 2-4. Tests of several rate orders, using the inte-

gral method of analysis (equations 2-17 and 2-18), are shown in Figure

2-5. From these plots it is clear that the settling of hydrocarbons (at

least for the data collected) is approximately a fourth-order "reaction."

Of course, a more accurate representation might have been obtained at

some non-integer value near 4. It is important to note that a settling

column should be designed with a depth closely conforming to expected

field conditions (i.e., the actual basin) in order for the rate constant

and "reaction" order to be transferable.










































10 20 3C
TIME, t, hours


Figure 2-4.


Results of Hydrocarbon Settling Test (Whipple
and Hunter, 1981).


0 4.0-
-
E0

2IC
z




Z
w
0
z
0
2.0-
z
0



O 1.0.
Q
x-


















LL


0.-
2F




Z
z 0.8-


=o
0
U.





z
S0.4-
z
0








O
0.2-




0
0


Figure 2-5.


n = 1.5



n=2



n=3

n=4
n=5


10 20 30 40
TIME, t, hours

Integral Method of Estimating "Reaction" Order
for Hydrocarbon Settling.






26

Flow reactors general characteristics. Flow reactors recieve

inflows and release outflows, whereas batch reactors do not. The fluid

storage equation given by equation 2-1 governs the continuity of fluid

mass, i.e.,

dV I(t) 0(t) (2-19)
dt

However, the behavior of the system as a reactor depends on how individ-

ual fluid parcels or elements are mixed and routed through the system.

Unfortunately, equation 2-19 does not provide this information.

The two ideal-flow reactors represent the extreme levels of mixing.

In the completely mixed reactor, all inflow elements are immediately,

uniformly, and completely dispersed throughout the reactor. The plug-

flow reactor queues the flow such that all fluid elements leave the

reactor in the same order they entered. The extremes of mixing repre-

sented by these ideal reactors are better understood by investigating

the age or detention time distributions of the fluid elements in the

reactor and in the outflow. Intermediately mixed or arbitrary-flow

reactors also have unique age distributions. These reactors experience

mixing levels between the extremes of the completely mixed and plug-flow

reactors.

Consider a steady-state reactor with constant and equal inflow and

outflow rates, i.e., I(t) = 0(t) = Q, and a constant fluid volume, i.e.,

V(t) = V. The mixing level is arbitrary. Assume that a conservative

tracer is continuously injected with the inflow at a constant concentra-

tion, C., beginning at an arbitrary starting point, t = 0. The system

response, as measured by the ratio of the outflow concentration, C(t),

to the inflow concentration, Ci, or F(t) = C(t)/Ci, has several possible






27

forms. These are shown as functions of (-t) in Figure 2-6. Figure 2-

6(a) shows the step function response associated with plug-flow reactors.

Figure 2-6(b) shows the response of a completely mixed system. These

responses will be demonstrated later. The response shown in Figure 2-

6(c) is what might be expected for a reactor with an intermediate level

of mixing.

The ages or detention times of the tracer elements in the reactor

have a statistical distribution (Danckwerts, 1953; Butt, 1980). This

distribution is defined by p(t). The distribution of tracer ages in the

outflow is defined by #(t). In other words, the fraction of all tracer

material in the reactor having ages between t and t + dt is p(t)dt.

Similarly, for the outflow, the fraction is 4(t)dt. Thus, the term F(t)

can also be defined as the fraction of the fluid leaving the reactor

that has resided in the reactor for less than t, i.e.,

F(t) = ftc(t')dt' (2-20)
0

where t' = dummy variable of integration.

Also, by definition,

r (t)dt = rP(t)dt = 1 (2-21)
0 0

Assuming C. = 1 (the specific units are irrelevant), the "balance

sheet" for the tracer at time t is as follows:

Entered reactor: Qt (2-22)

Still in reactor: V ft *(t')dt' (2-23)
0



Left reactor: Q ftft"~(t')dt' dt" (2-24)
0 0











1.0-






ZoA-
u.,


U 1.0

t


(a) Plug Flow


















(b) Completely
Mixed

















(c) Intermedi-
ately Mixed


0 1.0
O to S t
*


Figure 2-6. Tracer Response, Steady-State Flow
Reactor (Butt, 1980).


---- -








The variables t' and t" are dummy variables of integration. The term

Q ft"((t')dt' is the mass rate of tracer leaving the reactor at time t".
he material balance at time t is
The material balance at time t is


Qt = V ft*(t')dt' +
0



t ft(t')dt' +
V 0 V


(2-25)


Q ft ft"/(t')dt'dt",
0 0


ft ft"(t')dt'dt"
0 0


(2-26)


Differentiating with respect to t and recalling equation 2-20 yields


= 9(t) + F(t)



K(t) = (1 F(t))


(2-27)



(2-28)


From this result, the useful fact that the area between F(t) 1 and

F(t) (the shaded area in Figure 2-6(c)) is 1.0 emerges, i.e.,


1 E fJ(t)dt f. (1 F(t))dt
0 0
The last integral is the shaded area shown in Figure 2-6(c).

The average age or detention time of all fluid elements

reactor at any time t, 0, is given as follows:


Srt4(t)dt
O = 0 = f t4(t)dt since J<(t)dt = 1.
fJ (t)dt 0 0
0


(2-29)


leaving the





(2-30)


Differentiating equation 2-20 and substituting the result for >(t) in

equation 2-30 produces

0 = t dFt- dt = flt dF(t) (2-31)
0 d0






30

Multiplying equation 2-31 by Q/V yields

6 = fl ; t dF(t) (2-32)
0
The integral in equation 2-32 also defines the shade area shown in

Figure 2-6(c); thus,

S= 1 (2-33)

or

9 = V (2-34)
Q

Equation 2-34 represents the average or nominal detention time--it is

not the detention time of all fluid elements passing through the system.

This calculation is a familiar and useful tool, but it is limited to

steady-state reactors or storage units (for any level of mixing). The

average age of all fluid elements leaving the reactor up to time t, e,

is also V/Q.

When the steady-state reactor is completely mixed, the internal and

external age distributions are equal and the tracer concentration in the

outflow is equal to that in the tank. Thus, equation 2-28 can be written

as

p(t) = R Q F(t) (2-35)

Differentiating equation 2-20 and substituting the result for #(t) in

equation 2-35 produces the following:

dF(t+ q F(t) = Q (2-36)
dt V V

Solving for F(t) yields

F(t) = 1 exp(- t) (2-37)
(2V7








Equation 2-37 is represented by the curve shown in Figure 2-6(b). The

external age distribution is


P(t) = dt exp(- t) (2-38)

When the steady-state reactor is operated in the plug-flow mode,

the age of all fluid elements leaving at any time t is V/Q. Thus,

F(t) = 0 for t < 6
(2-40)
F(t) = 1 for t > 8

Equation 2-39 describes the step function shown in Figure 2-6(a).

For unsteady-state reactors (i.e., I(t), 0(t), and V(t) are time

variable), the average detention time of all fluid elements leaving the

reactor at time t is

ft' (t,tldt')
0
8(t) = (2-40)
J'o(t,t')dt'
0

This equation is valid for any level of mixing. However, the dis-

tribution of fluid element ages, 4(t,t'), is complicated by the varia-

ble nature of the inflows and outflows and, thus, it changes with time

t. As a result, equation 2-40 may be difficult or impossible to inte-

grate directly. The average detention time for all flows leaving the

reactor, up to time t, is

fto(t')e(t')dt'
() = 0 (2-41)
ftO(t')dt'
0
The concept of detention time is useful in understanding the

behavior of various reactor types. Unfortunately, the term has been








misunderstood by practitioners in sanitary and environmental engineering,

especially those investigating stormwater storage/release systems. A

common error is to assume that the flow leaving a storage unit has an

average detention time of V(t)/0(t). In steady-state units this is true

but equation 2-40 clearly indicates that this is not so for unsteady-

state conditions. In fact, it is possible for the outflow to have a

variety of age distributions for the same value of V(t)/0(t). The true

average detention time at any time, t, must account for the history of

the unit. This erroneous assumption has often been used to calculate

the "detention time" (or related value, such as the overflow velocity)

at various points in time. Another common error is to calculate the

average detention time over a given period by dividing the average

values of V(t) and 0(t) or averaging the values of V(t)/0(t). (At

times, [O(t) + I(t)]/2 is substituted for 0(t).) Relationships between

these parameters and the removal of particular pollutants have also been

developed (Lager et al., 1977). These relationships might be of some

predictive value but the term "detention time" should not be used. It

would be preferable to use a term such as "detention parameter" in order

to make it clear that such calculations are only indications of the

detention ability of the storage basin.



Flow reactors--mathematical models. The previous subsection

described the general mixing and detention characteristics of flow

reactors. This subsection will describe mathematical models for plug-

flow and completely mixed reactors. These models will be particularly

useful in the development of a computer simulator in Chapter 4.








The mathematical models for plug-flow and completely mixed reactors

are derivable from the one-dimensional advective-dispersive equation

(Medina, 1976; Medina et al., 1981a), i.e.,

S [E(x,t) U(x,t)C(x,t)] (So + Si) (2-42)
at ax ax 0
-3
where C(x,t) = reactant concentration in the reactor, ML-3

x = distance along flow axis, L,
2 -1
EL(x,t) = longitudinal dispersion coefficient, L T-1

U(x,t) = longitudinal flow velocity, LT-1
-3 -1

-3 -1
S. = sink of reactant, ML T .
1
A plug-flow reactor is characterized by a regime in which flow through

the reactor is queued; i.e., the fluid elements (or plugs) exit the

reactor in the same order that they arrived. In other words, there is

no dispersion (EL = 0). Thus, with a reactive sink, equation 2-42

becomes

a = [- U(x,t)C(x,t)] KC (x,t) (2-43)

Equation 2-43 is the pure advective form of equation 2-42. For prac-

tical purposes (e.g., computer simulation), equation 2-43 can be viewed

as a series of discrete plugs, each acting as a batch reactor, moving

along the flow axis. This representation of the plug-flow reactor is

shown in Figure 2-7.

Completely mixed reactors immediately disperse all inflows (in-

cluding reactants, tracers, pollutants, etc.) completely and uniformly

throughout the reactor; i.e., 3C/3x = 0. Therefore, the concentration

of the reactant is the same in both the reactor and the outflow. A

schematic of this type of reactor is shown in Figure 2-8. For a






















V(t)

(t) ,C(t) C(t) 0(t),C(t)


IPLUGj
I I


NOTE: j and Cj(t) are the volume and reactant concentration
of plug j, respectively.


Figure 2-7. Plug-Flow Reactor.














I(t), C(t)
-n1


V(t), C(t)



o<=>


0(t),C(t)
, I -I >


Completely Mixed Flow Reactor (Rich, 1973).


0-


--


Figure 2-8.








completely mixed flow reactor with an input source, an outflow sink, and

a reactive sink, equation 2-42 becomes

d(Vl = c(t)I(t) C(t)O(t) KCn(t)V(t) (2-44)

where C(t) = reactant concentration in the reactor and the outflow,

ML3 and

C (t) = reactant concentration in the inflow, ML-3

Equation 2-44 can also be viewed as a version of the storage equation

written in terms of reactant mass.

Unfortunately, the n-th order reaction provision in equation 2-44

makes the derivation of an analytical solution difficult. However, for

first-order reactions (i.e., n = 1), equation 2-44 becomes

d( C (t)I(t) C(t)0(t) KC(t)V(t) (2-45)
dt

or

C(t)d + V(t)d CI(t)I(t) C(t)0(t) KC(t)V(t) (2-46)

Rearranging terms yields

dC + 1 dV + + K] C(t) t) CI(t) (2-47)
dt V(t) dt V(t) V(t)

In terms of C(t), equation 2-47 is a simple first-order differential

equation with the following solution (Ross, 1964):

C(t) = exp(t 1 dV ot) + K]dt) [ftexp{ft 1 dV
0 V(t) dt V(t) 0v(0 ) dt


+ (t) + K]dt} (t) CI(t) dt + C(0)] (2-48)
V(t) V(t)

where C(O) = initial reactant concentration in the reactor and the
-3
outflow, ML-3

The integrals in this solution make it difficult to work with for most

functions of V(t), 0(t), and C (t). In fact, it is only possible to








evaluate equation 2-48 with the most trivial forcing functions (Medina,

1976; Medina et al., 1981a). Fortunately, numerical techniques can be

used to approximate and evaluate equations that are not directly solvable.

Intermediately mixed flow is defined as any degree of fluid element

mixing between plug and completely mixed flow (Metcalf and Eddy, Inc.,

1972). Reactors with this type of mixing regime can be analyzed with

the full advective-dispersive equation given by equation 2-42 (Medina,

1976; Butt, 1980; Medina et al., 1981a). Although most real reactors

are, to some degree, arbitrary flow units, an attempt is usually made to

achieve one of the extremes. For this reason, intermediately mixed flow

reactors are not investigated in this study.



Particle Settling

The settling of particles is one of the most important mechanisms

contributing to the removal of pollutants in storage units. Storage

units used to settle particles are designed to closely resemble the

ideal plug-flow reactor; in other words, queued quiescent flows.

Obviously, these are more desirable conditions for settling than those

found in completely-mixed reactors. Four types of settling are generally

recognized (Fair et al., 1968; Clark et al., 1977):

1) discrete particles in a dilute suspension,

2) settling of flocculent suspensions,

3) hindered settling of discrete particles, and

4) compression.

The settling of discrete particles in a dilute suspension is assumed to

be unhindered and a function only of fluid and particle properties.








The settling of flocculent suspensions is different from discrete parti-

cle settling because the particles are coalescing and, thus, changing

their properties as they settle. Concentrated suspensions of discrete

particles cause significant displacement of the supporting fluid and

hinder settling. Compression occurs when the suspension becomes so

concentrated that particles are in contact with each other and supported

somewhat by the compacted mass. Unhindered discrete particle settling

can be described in terms of simple dynamics. The other forms of set-

tling are generally not adaptable to direct analysis by fundamental

physical principles, although the hindered settling of discrete particles

can be analyzed through theories developed for the backwashing of filter

media (Fair et al., 1968). All forms of settling can be analyzed experi-

mentally through the use of settling column or settleability tests

(Clark et al., 1977). Such tests should always be conducted in situa-

tions where settling is a major pollutant removal mechanism.

Theoretical settling relationships, when available, can provide

useful tools for understanding the behavior of storage/release systems

and developing mathematical models. The theory governing the unhindered

settling of discrete particles is briefly discussed below. Fortunately,

this concept of settling probably mirrors the settling of particles in

stormwater more closely than the other types of settling.

A discrete particle is one that does not change its shape, weight,

or size while settling (Fair et al., 1968). In a dilute suspension,

such a particle will accelerate until the drag or frictional resistance

of the fluid equals the effective weight of the particle. The effective

weight is given by








Fp = (Ps P)gVp (2-49)
-2
where F = effective weight of the particle, MLT ,
p
-3
p = mass density of the particle, ML ,
-3
p = mass density of the fluid, ML-3,
-2
g = gravitational constant, LT-2 and
3
V = volume of the particle, L.
P
A dimensionally derived relationship for the frictional drag exerted by

the fluid is given by Fair et al. (1968):

FD f CDAcPs2/2 (2-50)

where FD = drag force of the fluid, MLT ,

CD = coefficient of drag,
2
A = cross-sectional area of the particle, L and

v = terminal settling velocity of the particle, LT-1

When the particle is no longer accelerating, the drag force and the

effective weight are equal, i.e.,

(P )gV CpAc /2 (2-51)

and the terminal settling velocity can be calculated as

(P P) V
vs = [& 1/2 (2-52)
CD p Ac
3 2
For spherical particles of diameter d, V = (n/6)d and A = (w/4)d.
p c
Equation 2-52 can be written as
4 (P p) 1/2
vs [ -CD d] (2-53)


The coefficient of drag, CD, varies with the Reynolds number, NR. The

Reynolds number is a dimensionless measure of the effect of fluid

viscosity in fluid systems. For spherical particles, CD, is approximated

by







24
C for NR < 0.5; (2-53)
D NR

or

24 3 4
CD = 1+ /- + 0.34 for 0.5 < NR < 104; (2-55)
D NR 1

or

CD = 0.4 for NR > 104 (2-56)
The value of NR is calculated as

NR = Vsd/v (2-57)

where v = kinematic viscosity of the fluid, L2 T-

Kinematic viscosity is a function of fluid temperature. When CD is

specified by equation 2-55, an iterative technique is required to solve

for v (Sonnen, 1977).

The vertical travel length of a discrete particle (or any particle)

during the particle's residence time in the storage unit is the deciding

factor in the removal of the particle from the fluid stream. If resus-

pension is ignored it can be assumed that a particle reaching the bottom

of a storage unit is removed. The ability of a particle to reach the

bottom is, of course, a function of the settling velocity, v In a

steady-state, plug-flow system in which the incoming particles are uni-

formly distributed along the depth of a rectangular basin with a horizon-

tal bottom, the removal fraction of particles with velocity v is v /v .

The time it takes the particles to reach their terminal velocity is

ignored. The variable vc is the settling velocity required for a parti-

cle entering at the top of the settling zone to reach the bottom, i.e.,

d
S(258)
v c = 6- (2-58)


where vc = critical particle settling velocity, LT-





41

de = depth of fluid in the storage basin, L, and

6 = detention time of all fluid elements leaving the basin, T.

Obviously, if v > v c, all particles with a velocity v are removed.

When unsteady-state conditions are encountered, the analysis is not as

straightforward. One approach to this problem is discussed in Chapter

4.



Evaluation Techniques

The theoretical presentation given in the previous section serves

as a backdrop to a series of practical tools for analyzing the response

of urban stormwater storage/release systems. It was readily apparent

that the basic differential equations governing the behavior of storage

units are not directly solvable with anything but the most trivial input

functions. Thus, it becomes necessary to develop and use alternative

techniques to evaluate the system's response in some reasonable manner.

The basic techniques are empirical methods, numerical analysis,

statistical analysis, and computer simulation. Empirical methods use a

cross-section of data from a number of systems to develop relationships

between long-term pollutant removal and design and operating parameters.

Numerical analysis essentially approximates the original differential

equation with a simpler representation. These methods are sometimes

clumsy and, by definition, subject to some level of error. However,

they are generally well adapted to computerization. Statistical anal-

ysis, as the name implies, relies on a set of statistical parameters

describing the inflows to the system and a relatively simple repre-

sentation of the storage/release system to perform the analysis.

Statistical techniques can be powerful tools, especially in preliminary






42

analyses. They do suffer from several limiting assumptions and they

lack the ability to generate a time series of system responses to a time

series of incoming flows and pollutant loads. However, once the re-

quired influent statistics are obtained, the long-term response of a

variety of system sizes or capacities can be rapidly evaluated-without

computer aids. This is an impressive advantage but the price it extracts

is a lack of flexibility in specific system designs and operating

policies. Computer simulators, for the most part, automate the numerical

solution techniques discussed above. The power of a detailed simulator

is flexibility and its ability to produce time-series responses to

serial stimuli. These are important features; just as the ease of

application of a statistical technique is a potentially valuable trait.

Some critics of computer simulators have implied that the use of a

computer is a liability. This is not as compelling an argument as it

once was because of the recent proliferation of a new generation of

powerful and relatively inexpensive computers.

Computer simulators and statistical techniques have complementary

traits. This fact will be exploited in Chapter 7 to develop long-term

performance information at different levels of detail. One obvious but

necessary caveat is that neither class of techniques is without flaws--

both are nothing but mathematical models of a physical system. Such

representations must be calibrated and verified with measured data in

order to insure their accuracy. The need for calibration is an indict-

ment of model assumptions and implies, a priori, imperfection. It is

sometimes useful to remember that all man-made systems designed to

measure or analyze natural systems are imperfect models. Measuring

devices (e.g., dissolved oxygen meters) are nothing more than electrical





43

models coupled to a measurable, and often indirectly related, quantity.

Laboratory methods are also models which make certain assumptions about

the quantities to be measured. Thus, before discounting computer simu-

lators or statistical techniques, it is wise to recall the ubiquitous

nature of models. The only real evil of simulators and statistical

methods is that they are one more step removed from the "real" world,

but they can open up a broader understanding of the system without

requiring extensive (and expensive) monitoring programs. Mathematical

models and monitoring programs should be viewed as mutually necessary

and beneficial.



Empirical Techniques

Empirical techniques use a cross-section of data from a variety of

storage/release systems to develop relationships between design and

operating parameters and long-term pollutant removal. These relation-

ships are developed from a wide range of system designs and are gener-

ally based on the storage basin volume, annual inflows and/or the

drainage area. Relationships for single systems are also possible but

these are usually developed for individual storm characteristics (Davis

et al., 1978; McCuen, 1980). By their nature, single-system relation-

ships are not as applicable to other proposed or in-place systems as

their cross-sectional counterparts. Nevertheless, the cross-sectional

relationships should only be used for very preliminary estimates.

Summaries of cross-sectional relationships have been prepared by

Chen (1975), Ward et al. (1977), and Nix et al. (1981). Most of the

cited work is taken from the literature dealing with the sediment re-

tention capability of reservoirs. Most of these cross-sectional





44

relationships are limited to normally ponded reservoirs (i.e., wet

basins). A prominent example is provided by Brune's sediment

trap efficiency curves (Brune, 1953). These curves are shown in Figure

2-9. Each curve represents a different type of sediment (i.e., fine,

coarse, and "median") and each is a function of the basin capacity-to-

annual inflow ratio. Brune's curves are widely used in reservoir design

and, because of their ease of application, they have been used in the

analysis of small sediment retention basins (Chen, 1975). Meta Systems,

Inc. (1979) presents a summary of additional relationships for the

removal of sediment, nitrogen, phosphorus, and other pollutants in

agricultural detention basins.



Numerical Solution of Governing Equations

The differential equations defining the continuity of mass in

storage/release systems are not amenable to direct analytical solutions

except under very restrictive conditions. Fortunately, numerical

techniques are available to circumvent this problem. Numerical solu-

tions are generally obtained by (Hornbeck, 1975)

1) the direct numerical solution of the differential equation

in question; or

2) numerically evaluating the integrals found in analytical

solutions where the integrand is not readily integrable.

The primary advantage of numerical techniques is that they can be

readily applied to a wide range of differential equations or integral-

laden analytical solutions. This is not to say, however, that the

solution is always easy to attain. Numerical techniques are





45








-00







o o





eH
m






U








--4
u >
8 C


3 "








z E-
U .

-. 0





O m

0 -



z
00







SNO
N -
%vw. NJI
r x -
4 \. )\ "<:
8o IX ^^ ^^ :

iV N





46

approximators and, by definition, are flawed and subject to convergence

and stability problems. Nevertheless, many problems can be solved with

the judicious selection of a solution technique.



Numerical solution of ordinary differential equations. A numerical

solution to an ordinary, initial-valued differential equation can be

obtained through one of several techniques. Some of the more important

classes of techniques are (Hornbeck, 1975)

1) multistep methods,

2) Runge-Kutta formulae, and

3) predictor-corrector methods.

Each of these classes has its own advantages and disadvantages. A

specific group of multistep methods and a specific member of the Runge-

Kutta family will be discussed below to illustrate their applicability

to the differential equations describing storage units.

The closed Adams formulae are a group of multistep methods in which

truncated, backward Taylor series expansions are used to approximate the

differential equation. For example, the second-order Adams formula

approximates the differential equation

dy g(yx) (2-59)
dx

by

Ax (2-60)
Yj+I f Yj +-2 [gj+l + gj]
where Ax = constant interval of x,

y.,x. = values of y and x at the beginning of interval j;

yj+l,xj+l values of y and x at the end of interval j;
gj = g(yj, x.); and

gj+l = g(Yj+l' Xj+l).







This numerical method can be used to approximate the storage equa-

tion in the following manner:

V V + [(I j j ) + (I O)] (2-61)

where At = constant time interval or step, T;

V.,I.,0. = values of V(t), I(t), and 0(t) at the beginning of time
J step j; and

V. + I, +,0 -= values of V(t), I(t), and 0(t) at the end of time
step j.

For any time step, it is assumed that the values of Ij, Oj, Vj, and I j+

are known. Thus, the terms in equation 2-61 are rearranged to write the

unknown variables, 0j+. and V+l, in terms of the known quantities,

i.e.,

it hAt
Vj + O. = V + (I + I. ) -A (2-62)
2j+1 +1 j j+1 3 2

Unfortunately, Vj+1 and Oj+0 cannot be determined until a relationship

is developed between them, i.e.,

0(t) = f[V(t)] (2-63)

With this storage-outflow or storage-discharge relationship in hand, the

right-hand side of equation 2-62 is evaluated and the actual values of

Vj+l and 0j+1 determined by substituting equation 2-63 for 0j+1 and

solving for V j+ (and in turn, Oj+ ). The method is repeated for the

next time step with the j+l values of this time step becoming the j-

subscripted values of the next time step. This method (also known as

the Puls method) is presented in more detail in Chapter 4.

The differential equation representing the routing of reactants or

pollutants in a completely mixed reactor or storage basin can also be

analyzed in a manner similar to that shown above. This application is

also discussed in Chapter 4.








A well known method of the Runge-Kutta family is the "fourth-order

Runge-Kutta formula" (Hornbeck, 1975). The formula approximates equa-

tion 2-59 by


1 1.)
Yj+I Yj + Ax[T g(yj' xj) + g(Yj+1/2' xj+1/2)


(2-64)


+ 1
3 g(Yj+1/2' xj+1/2)

where Ax = constant interval of x;

y ,x. = the values of y and x at
j;


1
6 g(j+1i j+li



the beginning of interval


yj+lxj+l = the values of y and x at the end of interval j;
Ax
x. =x + --
j+1/2 j 2'
Ax
Yj+1/2 Yj + I g(yj xj);
S Ax
j+1/2= j +- +1/2' xj+1/2) and

j+- Yj + Ax g(y +12' xj+112).
As an example, the following form of the storage equation is set up with

the fourth order Runge-Kutta formula:

dV
d I(t) f[V(t)] (2-65)

The resulting Runge-Kutta approximation is


1 1
j+ J Jv,/2' J+1/2)


(2-66)


where g(V,t)

At

V., t
J J


Vj+1' tj+l


1 1
+ (+1/2' tj+l/2 + jj+' j+l

= I(t) f(V(t));

= constant time interval or step, T;

= values of V(t) and t at the beginning of time step
J;

i values of V(t) and t at the end of time step j;








tj+/2 = t + --*
j+1/2 2+

At
VtI = V. +-- [I(t.) -f(V)];
J3+1/2 j 2j

V1/2 V + [I(t /2) f(V )/2]; and
j+1/2 j 2 j+1/2 j+1/2

V1 = V. + At [I(t.j 2) f(V / )].
3+1 3 j+1/2 3+1/2
Runge-Kutta formulae are probably the most commonly used of the

numerical methods (Hornbeck, 1975). They are easily programmed and

offer good stability. It can be seen in equation 2-66 that the form of

I(t) or f[V(t)] poses no obvious problem for the technique, but stabili-

ty problems are always possible.



Numerical integration. Numerical integration can be used to com-

plete the solution of analytical solutions containing integrals that are

not readily evaluated. Essentially, numerical integration approximates

the complicated integrand with a simple function over relatively short

intervals. There are several techniques, including the following

(Hornbeck, 1975):

1) trapezoidal rule,

2) Simpson's rule,

3) Romberg integration, and

4) Gauss quadrature.

The trapezoidal rule approximates the function to be integrated,

f(x), with a series of straight lines over a finite number of equal

intervals, Ax. This piecewise linear approximation results in a series

of trapezoids for which the area is easily evaluated and, thus, the

approximate value of the integral obtained. Simpson's rule achieves the








same goal with parabolic arcs instead of straight lines. Romberg inte-

gration uses the trapezoidal rule with improved error analysis. Gauss

quadrature employs unequally spaced intervals and approximates f(x) with

orthogonal polynomials.

As an example, consider the following form of the storage equation:

dV
d I(t) aV(t) (2-67)

where the outflow, O(t), is a linear function of V(t). The solution to

the equation at time t is (Ross, 1964)

V(t) = exp(-at) ftI(t)exp(at)dt (2-68)
0

when the initial condition, V(0), is zero. Obviously, the inflow (or

forcing) function can take on a complex form and be impossible to solve

analytically. However, V(t) can be estimated by using the trapezoidal

rule:

At At
V(t) = exp(-at)[I(0) + 2I(t1)exp(atl)-I+


+ 21(t )exp(at )- + I(t )exp(atn)A] (2-69)
n-1 n-1 2 n np 2
where At is the constant time step or interval, and the terms in brack-

ets represent the trapezoidal approximation of I(t)eat from t = 0 to t
n
(=t). The approximation is made over n equally spaced time steps or

intervals. In general, the accuracy is dictated by the size of the time

step. For highly fluctuating functions, smaller time steps are desir-

able. For smoother functions, a straight line approximation is often

reasonably accurate with larger time steps.



Discrete inputs. A third approach, not listed earlier, approxi-

mates the input or forcing function (e.g., I(t)) by a series of discrete,








equally spaced inputs such as that shown in Figure 2-10 (Medina, 1976;

Medina et al., 1981a). This technique completes the integral-laden

solutions for simpler versions of the storage equation and equations

describing the various reactor models by substituting discrete inputs

and evaluating the integrals over each time interval. Unfortunately,

the fact that the technique requires an analytical solution limits its

usefulness. In a general sense, the direct numerical evaluation of the

differential equation is probably more useful.



Statistical Techniques

Statistical techniques have been developed to analyze stormwater

runoff and the response of general storage/release systems (Howard,

1976; DiToro and Small, 1979; Hydroscience, Inc., 1979; Charles Howard

and Assoc., 1979; Howard et al., 1979), constant-volume, completely

mixed equalization basins (DiToro, 1975) and the effects of pollutant

loads on receiving waters (Hydroscience, Inc., 1979; E.D. Driscoll and

Assoc., 1981). Each method requires and/or develops a set of statistics

describing the long-term characteristics of the incoming flows and is

simple to apply to the analysis of storage/release systems once these

statistics are developed. The result is a set of statistics describing

the long-term performance of the system. However, these methods are

limited by their inability to produce the transient response of a

storage/release system.



Equalization basins. Equalization basins are designed to smooth

out variations in the influent flow rate and/or pollutant loads. DiToro

(1975) developed a statistical approach to the analysis of pollutant

















4-











o -4tk
-I



















O
0 .





-J
U
z
0









TIME, t

Figure 2-10. Discrete Inputs of the Inflow Rate and Pollutant
Concentration at Equal Time Intervals (Medina, 1976).
u










TIME, t

Figure 2-10. Discrete Inputs of the Inflow Rate and Pollutant
Concentration at Equal Time Intervals (Medina, 1976).








loads in a constant-volume, completely mixed basin. Several parameters

were used to describe the probabilistic nature of the input pollutant

loads and flow. These parameters were then used to develop parameters

describing the effluent pollutant loads. A first-order reaction charac-

terized the behavior of pollutants within the basin. The procedure was

compared with a simple simulator to verify the statistical derivations.

Medina (1976) also compared DiToro's method with his constant-volume,

completely mixed basin simulator (see later discussion). In both cases,

the results were very similar.



Howard's statistical method. Howard (1976) developed and subse-

quently modified (Charles Howard and Assoc., 1979; Howard et al., 1979)

a statistical method to obtain expressions for the probability distri-

butions of runoff from a watershed and analyze the performance of storage/

release or storage/treatment systems. The storage/release system is

modeled as an in-line configuration. A schematic is shown in Figure 2-

11.

The principal input to the model is a long-term hourly rainfall

record. The rainfall record is transformed into a runoff record with a

runoff coefficient after considering the effects of natural and man-made

depression storage. Rainfall or storm events are defined by specifying

the minimum dry period which separates independent storms or events. An

autocorrelation analysis is suggested to determine this value. A more

complete discussion of the definition of an independent event is present-

ed in Chapter 3. The duration of each event is easily obtained from the

record, as is the depth or volume. The average intensity of each event

is its volume divided by the duration. Thus, the rainfall record becomes












RAINFALL








URBAN AREA






RUNOFF



S BYPASS








STORAGE

RELEASE



TREATMENT
PLANT OVERFLOW
PLANT





-- RECEIVING WATER --


Figure 2-11. Storage/Release System, Howard's
Statistical Method (Howard et al., 1979).








a series of rectangular hydrographs. With the selected event definition,

the time between individual events is readily calculated.

With the rainfall record segregated into independent events, the

average values for event duration, intensity, volume, and time between

events times can be determined. It is assumed that each of these rain-

fall characteristics has a probability distribution which can be speci-

fied by the mean through a single-parameter exponential function;

p(x) = (l/x)exp(-x/x) (2-70)

where x random variable, x > 0,

p(x) = probability density or distribution function of x, and

x = mean value of x.

It is also assumed that event intensities and durations are independent

of each other. A similar assumption is made between event volumes and

the times between events.

Based on several simplifying assumptions, exponential distributions

are derived for runoff event intensities, depths, durations and the

times between events with parameters that are derived from the rainfall

parameters (i.e., mean values of flow, volume, duration and the time

between events). An alternative approach would be to derive these statis-

tics from the results generated by a rainfall-runoff simulator or from

actual data. Pollutant concentrations in runoff are assumed to be

constant and independent of all event parameters.

The storage/release or storage/treatment analysis is based on the

following assertions (see Figure 2-11):

1) The treatment plant operates at a constant flow rate, 0, as

long as water is in the storage basin. This rate can be

interpreted in several ways. It can be the flow capacity of a








wet-weather treatment facility or the excess capacity available

in a dry-weather or sanitary sewage facility. It can also be

defined as a constant storage basin release rate.

2) For the analysis of pollution control, the efficiency of the

treatment unit, n., is assumed to be constant. The storage

basin also "treats" flows passing through it by settling or

some other mechanism related to detention time. The effect of

various storage capacities and release rates is approximated

by

TV = a logl0(DT) + b (2-71)

where nV = average removal efficiency of the storage
basin, 0 5 n < 1.0,

DT = detention parameter, T, DT < DTMIN,

DTMIN = minimum value of DT for which equation
2-71 is valid, T, and

a,b = coefficients.

The detention parameter is so named to distinguish it from

detention time. It is defined as

VB
DT = (2-72)

where VB = storage basin capacity, L3

Howard et al. (1979) refer to DT as the average detention time

over the period of interest. This is a misnomer and should be

discouraged; it is only an indicator of the detention ability

of the basin (see earlier discussion). Basins with higher

values of DT tend to hold water longer. The variable DT

provides a convenient parameter upon which to estimate nV but

the equation must be calibrated either to actual field data or





57

simulation results. It is not correct to equate DT with holding

times in settleability or treatability tests conducted in the

laboratory. This would falsely assume that all water parcels

passing through storage have the same detention time.

3) The bypassed flows receive no treatment, and, thus, enter the

receiving water with the original pollutant concentration.

4) The storage basin is assumed to be full at the end of each

storm. This requirement is reasonable for small basins but

can be a problem for larger basins.

With these assumptions and the runoff statistics and distributions, it

becomes a relatively simple task to estimate the expected number of

overflow events (events exceeding the storage capacity) per year, the

average annual volume of overflows, and the average annual runoff and

pollutant control efficiency.



Hydroscience statistical method. Hydroscience, Inc. (1979) has

presented a statistical method capable of analyzing urban stormwater

runoff, storage/release systems, wet-weather treatment devices, and

receiving water impacts. Much of the runoff analysis work appears in

the Areawide Assessment Procedures Manual (Municipal Environmental

Research Laboratory, 1976). The analyses of storage/release systems and

wet-weather treatment devices were developed by DiToro and Small (1979)

and Small and DiToro (1979), respectively, and were included in the

Hydroscience report.

The Hydroscience method develops a set of rainfall statistics with

a method similar to that used by Howard (1976). However, in this








case, the minimum dry period that drives the coefficient of variation

(the standard deviation divided by the mean) of the time between events

(i.e., interevent times) to unity is used to separate events. The

reason for this criterion revolves around the use of the exponential

distribution to characterize the time between events. A fundamental

characteristic of the exponential distribution is that the coefficient

of variation is unity.

The Hydroscience method assumes that event durations and inten-

sities are gamma distributed. The gamma distribution is a two-parameter

distribution requiring the mean and coefficient of variation. In

general, because the gamma distribution accounts for different varia-

bility patterns (through the coefficient of variation), it is more adept

at representing these characteristics than the exponential distribution

(Howard et al., 1979).

The Hydroscience method includes a simple method to obtain runoff

event statistics from rainfall statistics. Additionally, it is assumed

that the gamma distribution describes the runoff event flows and volumes.

An alternative, and superior, way to derive runoff event statistics is

to directly analyze a long-term runoff record. Unfortunately, such

long-term records are usually not available. However, a simulator could

provide the necessary data.

The Hydroscience method is capable of analyzing the two basic

system configurations (shown in Figure 2-1) with three restrictions.

One restriction is that the storage unit is operated with the bypass

mode in place. Secondly, all releases from the storage unit are at a

constant rate. The third restriction is that pollutant removal is

handled by assuming constant removal efficiencies. Based on several





59

simplifying assumptions, several sets of nomographic curves depicting

the long-term performance of the system were developed. Performance is

defined in terms of what portion of the runoff volume or pollutant load

does not appear as a bypassed quantity.

Receiving water impacts are also analyzed by the Hydroscience

method. This feature is potentially useful in relating the storage/

release system design and operation to impacts. Unfortunately, this

application is limited because the issue of how the storage/release

system affects runoff statistics is not sufficiently addressed.

The Hydroscience method will be discussed in more detail in Chapter

3. In general, this method is more flexible and has a longer history of

use and application (Municipal Environmental Research Laboratory, 1976;

Hydroscience, Inc., 1979, E.D. Driscoll and Assoc., 1981) than Howard's

method. Thus, it will be used, along with a simulator developed in

Chapter 4, to analyze the long-term performance of storage/release

systems.



Computer Simulation

Computer simulators have become common tools in scientific and

engineering endeavors. The advent of computers with large memories and

data handling capacities has allowed the development of sophisticated

environmental system simulators capable of detailed analyses. The

recent revolution in computer accessibility has brought this capacity to

most analysts. The need to analyze urban stormwater storage/release

systems has spawned a small family of computer simulators over the last

decade, ranging from very simple models with limited capabilities to

more complex models capable of simulating a wide range of designs and








operating policies. Simulators have the advantage (over the statistical

methods) of being able to produce the transient response of these systems.

Several of these models are briefly reviewed below.



City of Milwaukee detention tank model. In a study done for the

City of Milwaukee concerning the design and operation of the Humboldt

Avenue combined sewer overflow detention basin, a simple model was

developed to aid in the analysis (City of Milwaukee et al., 1975). This

model treats the detention basin as a plug-flow reactor and assumes a

constant volume once the basin is filled to the overflow depth. Pollu-

tants are assumed to be removed under first-order conditions. The model

also has a simple surface runoff module which develops a series of

hourly runoff values from an hourly rainfall record. The volume remain-

ing in the basin at the end of each storm is drained at a constant rate

and all retained pollutants, including the removed quantities, are

thoroughly mixed for release with this flow. This option was included

to simulate the return of the basin contents to an interceptor sewer

and, in turn, to a sanitary sewage treatment facility. The model was

used to study the effectiveness of varying basin sizes and dry-weather

drawdown rates. The results were also coupled with a receiving water

model to estimate the effect of the basin on water quality in local

waterways. The program was written in FORTRAN and is displayed in the

report cited above.



University of Kentucky sediment detention basin model (DEPOSITS).

DEPOSITS is a moderately sophisticated model developed by Ward et al.

(1977) to simulate sediment detention facilities. It models these







storage/release systems as variable-volume, plug-flow reactors. However,

in this case, the pollutants (i.e., sediment loads) are removed by

simulating the settling of particles. Because it is designed to simulate

sediment detention basins, the model also calculates the effect of

sediment deposition on the storage capacity of the basin. The model was

verified on a number of surface mine sedimentation ponds. DEPOSITS

requires particle size distributions and a complete hydraulic descrip-

tion of the basin and outlet structure. The model is primarily designed

to be run on single storm event hydrographs but it can be adapted for

multi-storm simulations.



Medina's storage/treatment model. Medina (1976; 1981a with others)

constructed a model that treats urban stormwater detention basins as

completely or intermediately mixed reactors. Pollutants are treated as

first-order reactants. The differential equations governing this system

are evaluated using the discrete-input method described in an earlier

section. Unfortunately, the model is limited to a linear relationship

between outflow and basin volume. This model was also successfully

applied to the Humboldt Avenue detention basin (Medina et al., 1981b)

and linked with a receiving water model to study the impact of the

stormwater detention basins on water quality (Medina, 1976; Medina et

al., 1981b).



Corps of Engineers Storage, Treatment, Overflow, Runoff Model (STORM).

The Corps of Engineers STORM model is a widely used model in the field

of urban and rural stormwater runoff modeling (Hydrologic Engineering

Center, 1977). The model was designed to generate long-term runoff and





62

pollutant load records with a continuous long-term hourly rainfall

record. The hourly runoff record is, in turn, routed to a storage/

treatment (or release) system. The system is depicted in Figure 2-12.

Runoff exceeding the maximum treatment rate is stored for release at a

later time. If the storage capacity is exceeded, the excess overflows

directly to the receiving water. When runoff eases to the point where

the treatment rate is no longer exceeded, the storage unit is drained at

the rate equivalent to the difference between the maximum treatment rate

and the runoff rate. The STORM model is written in FORTRAN and is

specifically designed to process decades of hourly rainfall data. The

model also provides simple statistics for runoff quantity and quality

before and after encountering the storage/release system. In addition,

the program accounts for the age of runoff held in the storage unit (on

a plug-flow basis). However, it does not simulate pollutant reduction

due to particle settling or any other mechanism. The term "treatment"

is used in the STORM literature but no pollutant removal is simulated.

The term is more precisely linked with the mainstream capacity associated

with the off-line storage/release configuration. Flows remaining in the

mainstream or returning to it are often routed to a treatment facility

of some type and, thus, the term "treatment" is commonly applied.



EPA Storm Water Management Model (SWMM), Storage/Treatment Block.

The author has developed the Storage/Treatment Block as part of this

study and the comprehensive Storm Water Management Model (Huber et al.,

1981). The S/T Block is a flexible, FORTRAN simulator capable of model-

ing several types of wastewater storage and/or treatment units and

system configurations, including storage/release systems. The S/T Block












RAINFALL


RUNOFF


- --. RECEIVING WATER-


Figure 2-12.


Storage/Release System, STORM
(Hydrologic Engineering Center, 1977).


URBAN AREA





64

is readily coupled with the other portions of the SWMM model. The other

modules simulate the urban runoff process, the routing of flows and

pollutants through collection systems, and the impacts of urban runoff

on receiving waters. Among the more relevant and interesting features

of the S/T Block are

1) the ability to model a wide variety of storage basin geom-

etries and outlet structures;

2) the capability to simulate the dry-weather release of the

basin contents;

3) the ability to characterize pollutants by particle size/

specific gravity or settling velocity distributions and to

simulate particle settling;

4) a provision for modeling storage units as plug-flow or com-

pletely mixed reactors;

5) the ability to use a wide variety of pollutant removal equa-

tions in the plug-flow mode; and

6) the modeling of pollutants as first-order reactants in the

completely mixed mode.

The Storage/Treatment Block is probably the most versatile model of its

kind. It will be used in later chapters of this study to provide the

necessary simulation capacity to develop long-term production functions.

The details of the model algorithms are discussed in Chapter 4.



Measure of Performance--Reliability

The information provided by the various evaluation techniques can

be expressed in terms of reliability. Reliability is a widely used

engineering tool that provides a performance measure for systems operating







on stochastic processes. Kritskiy and Menkel (1952, indirect ref.

from Klemes, 1981) recognized three reliability measures for use in the

design analysis of water supply reservoirs. These measures are discussed

below and modified for application to urban stormwater quality problems.

Annual reliability, Ra is defined in water supply parlance as the

probability that no failure in water supply will occur within a year.

Failure is defined as not being able to meet a predetermined draft from

the reservoir. The value of R is estimated by

N N
R = i (2-73)
a N

where Nf = number of failure years, and

N = total number of years.

This measure of reliability is also known as "occurrence-based" relia-

bility and this term is preferred here. The non-catastrophic nature of

stormwater quality problems and economic considerations require a differ-

ent definition. Thus, occurrence-based reliability may be redefined to

denote the probability that a failure will not occur more than a speci-

fied number of times per year (or other period). Alternatively, annual

reliability could be defined as the average number of occurrences or

failures per year (or other period). This measure of reliability has

been used in stormwater quality management to indicate the annual number

of times a storage/release system overflows or is bypassed (Roesner et

al., 1974; Labadie and Grigg, 1976; Charles Howard and Assoc., 1979;

Howard et al., 1979) and the reduction in the annual number of beach

closings due to the control of combined sewer overflows with a storage/

release system (E.D. Driscoll and Assoc., 1981).






66

Time reliability, Rt, is defined in the water supply context as the

portion of the time that the water demand or specified draft was satis-

fied, i.e.,

T
R = d- (2-74)
t T

where Td time period that demand was satisfied, T, and

Tt = total operating period, T.

Equation 2-74 is useful for the purposes herein when the term "demand"

is replaced with "water quality objective." This modified definition is

the essence of the pollutant load- or concentration-frequency rela-

tionships found in several studies on the effects of stormwater pollution

control on receiving water quality (Medina, 1976, 1979; Heaney et al.,

1977; Scholl and Wycoff, 1981; E.D. Driscoll and Assoc., 1981; Medina et

al., 1981b).

Volume reliability, R is defined for water supply reservoirs as

the portion of the total demand volume actually supplied during the

operating period, i.e.,

f(qd qs) dt
qs< d (2-75)
R 1--
v t (qd) dt
0


where qd = water demand rate, L3 T, and

qs = water supply rate, L T.

This measure of reliability can be applied to stormwater storage/release

systems by allowing the numerator to represent the total volume and/or

pollutant load not captured by the system. Correspondingly, the denomi-

nator represents the total volume and/or load entering the system. This

is a popular measure of performance in stormwater quality studies and it





67

is often reported as the average performance level (Heaney et al., 1976,

1977, 1979; Hydroscience, Inc., 1979; DiToro and Small, 1979; Nix et

al., 1981).

The use of reliability in the analysis of urban stormwater quality

management and, specifically, storage/release systems is not new, but

the term is rarely used. This is unfortunate because it is a descriptive

term and entirely appropriate to storage/release systems operating under

such conditions. The classification scheme described above allows a

broad framework with which to evaluate stormwater quality problems. Not

all of the evaluation techniques are capable of providing the information

needed to estimate each form of reliability. Obviously, the selection

of a technique must fit the nuances of the problem.



Summary

This chapter has established the basic theoretical framework for

analyzing storage/release systems and briefly reviewed several useful

evaluation techniques. These techniques range from simple empirical

relationships to sophisticated computer simulations. Examples of two of

the more useful techniques will be investigated in detail in the next

two chapters. The concept of reliability was introduced as a unifying

framework for quantifying the long-term performance of storage/release

systems as stormwater quality control devices.














CHAPTER 3

STATISTICAL ANALYSIS OF STORAGE/RELEASE SYSTEMS



Introduction

Knowledge of the probability distributions governing the inflow

stream to a storage/release system provides the basis for a set of

powerful statistical analysis procedures. Hydroscience, Inc. (1979)

developed a methodology to analyze long-term hourly rainfall records and

to develop a set of statistics describing the resulting stormwater

runoff characteristics of urban areas. Given the set of runoff statis-

tics and governing distributions, the control effectiveness of various

treatment, interception and storage/release devices can be determined

through techniques proposed by Small and DiToro (1979) and DiToro and

Small (1979). This material was also included in the Hydroscience work

(1979). These techniques were later used by the author and others in a

report describing a methodology to analyze combined sewer overflows,

their control, and impact in receiving waters (E. D. Driscoll and Assoc.,

1981). The techniques are described in some detail in the remainder of

this chapter.



System Conceptualization

The basic storage/release system configurations discussed in

Chapter 2 are amenable to the techniques forming the Hydroscience

statistical method. However, some implications were necessary to








avoid overburdening the methodology with theoretical details that would

probably prove to be debilitating. Foremost among the simplifications

is the assumption that the storage basin is operated with the bypass

mode in place. The second simplification requires the interevent (i.e.

dry weather) releases from the storage unit to be at a constant rate.

The third simplification assumes that pollutant removal is determined by

constant removal efficiencies. This particular restriction is probably

the most untenable of the system simplifications.



Characterization of Rainfall and Runoff Events

The stochastic rainfall or runoff process is segregated into a

series of independent, randomly occurring events as shown in Figure 3-

l(a). This representation is further simplified by characterizing each

event by a uniform, rectangular hydrograph (see Figure 3-1(b)). Each

event is described by its duration, volume, average flow rate or inten-

sity, and the elapsed time since the last event or interevent time. The

interevent time is measured between event midpoints. The discussion

that follows is valid for rainfall and runoff events.



Statistical Properties of Event Characteristics

The characteristics describing each rainfall or runoff event are

random variables. Statistics describing these characteristics can be

estimated from historical records. The complete set of statistics

required for this method is given in Table 3-1. The mean and coefficient

of variation are used to describe each random variable. The sample mean

is given by












-~ MINIMUM DRY PERIOD

I I I
I-- EVENT -. EVENT.
I I I
I I I '
I I I i


0 TIME


(a) Actual Record and Event Delineation


TIME BETWEEN
EVENT MIDPOINTS, -
D ,

k-DURATION, d -_I


TIME


(b) Simplified Representation
Figure 3-1. Simplified Representation of Independent Rain-
fall or Runoff Events (Hydroscience, Inc., 1979).


-EVENT

















Table 3-1. Rainfall/Runoff Event Parameters and Statistics


Parameter For each event Mean Coefficient of
variation

Intensity or
flow rate, L T q Q v

Duration, T d D vd

Volume, L3 v V v

Time between 6 A v
event midpoints
(interevent
time), T


a. Event intensities/flow rates and volumes are often normalized
ovey the catchment area. When this is done, the units become
LT and L, respectively.
b. The subscripts "P" and "R" are used to denote a rainfall or
runoff event parameter or statistic, respectively.







n
x l x (3-1)
i=l
where x = mean of the sample

x ith sample, and

n = total sample size.

The sample coefficient of variation is

v = s/x (3-2)

where v = sample coefficient of variation, dimensionless, and
n
s = (xi x)n sample standard deviation.
i=l
In effect, the first two moments of the data are used to describe the

event characteristics. The mean describes the central tendency and the

coefficient of variation measures the variance around the mean. The

coefficient of variation is used in place of the standard deviation

because it frees the measure of variability from dependence on specific

dimensions. High values of v reflect greater variability in the random

sample; low values reflect lesser variability. Of course, sample statis-

tics only approximate the true statistics; they become more accurate as

the sample population increases.

The sample statistics allow the assignment of one- or two-parameter

probability density functions to each of the event characteristics.

Rainfall and runoff event flows and durations appear to be well repre-

sented by the gamma distribution (Hydroscience, Inc., 1979). Such a

distribution requires an estimate of the mean and coefficient of varia-

tion and is given as
K K-1
p(x) = (/x) xl exp(-Kx/x) (3-3)
r(K)

where p(x) = gamma probability distribution function of x,

K = 1/v and







r(K) = gamma function with argument K.

The probability distribution function for the time between rainfall and

runoff event midpoints is well represented by an exponential distri-

bution (Hydroscience, Inc., 1979). However, the exponential distribu-

tion is simply a special case of the gamma distribution in which the

coefficient of variation is 1.0, i.e.,

p(x) = (1/x) exp(-x/x) (3-4)

The cumulative gamma distribution for several values of v is shown in

Figure 3-2. The cumulative plots allow the extraction of valuable

frequency information. For example, if the mean runoff event flow was

determined to be 0.10 inches/hour and the coefficient of variation found

to be 1.5, it could be stated that 8 percent of all events exceed 0.30

inches/hour. If the average number of events is known to be 100 per

year, one could conclude that 8 events per year exceed 0.30 inches/hour

and the return period for these events is 1.50 months.

Throughout the methodology, it is assumed that event flows and

durations and interevent times are independent. This is probably not

a valid assumption. For example, areas that experience short intense

storm cells in the summer and long, less intense storms in the winter

would show some level of dependence between flows and durations. However,

this assumption allows some interesting and relatively simple analytical

procedures to be developed and it appears to be adequate for the objec-

tives of the methodology.

The gamma and exponential distributions have often been used in the

analyses of rainfall data (Thor, 1951, 1958; Eagleson, 1970; Chow and

Yen, 1976; Howard, 1976; Yen, 1977; Howard et al., 1979; Hydroscience,

Inc., 1979). However, it may not always be the most appropriate.





74























95-


90
9





w-













0 2 4 6 8 10

MULTIPLES OF THE MEAN, x/T

Figure 3-2. Cumulative Gamma Distributions
(Hydroscience, Inc., 1979).
0V


I-

U)



I-


LU0
0 68t
MUTPE FTEMAxi

Fiue32 uuaie am itiuin
(HdocecIc,17)







Distributions of rainfall event characteristics are skewed; i.e., there

are many small observations and few large ones. Other distributions,

including the lognormal and Weibull functions, are also capable of

representing such populations. The advantage of the gamma distribution

is that it appears to be reasonably accurate and simple enough to deter-

mine analytical solutions for the performance of storage/release systems.



Definition of an Independent Event

The statistical method is based on the analysis of independent or

uncorrelated events. Up to this point, no definition of an independent

event has been given. The most widely accepted approach is to select

the minimum amount of "dry" time which must occur between two separate

independent events (Heaney et al., 1977; Hydroscience, Inc., 1979). Any

activity not preceded by this minimum is not a separate, independent

event. The "dry" time is not the interevent time; the latter is calcula-

ted between event midpoints after the events are defined with the former.

The appropriate value will usually be different for rainfall and runoff

records because of the effects of the catchment system.

Several methods have been proposed to determine the minimum dry

period. Medina (1976), Howard (1976), and Heaney et al. (1977) recom-

mend that an open-series autocorrelation analysis be performed on hourly

rainfall records to determine the time lag at which no significant

autocorrelation is present (as measured by the autocorrelation coeffi-

cient). An autocorrelogram for the 1968 hourly rainfall record in Des

Moines, Iowa, is shown in Figure 3-3 (Medina, 1976). From this analysis

it was determined that the appropriate minimum dry period is 10 hours.

The tolerance limit (defined at the 95-percent probability level) was

















HOURLY PRECIPITATION RECORD
DES MOINES, IOWA 1968


95% T.L.


0 20 40 60 80


100


LAG, hours


Figure 3-3.


Autocorrelation Function for the Hourly
Precipitation Record of Des Moines, Iowa,
1968 (Medina, 1976).


-02








first reached at 7 hours but it was recommended that the point where the

autocorrelation coefficient first reaches zero be selected (i.e., 10

hours). This rainfall record was used to generate a runoff record with

the computer simulator STORM. The generated data set was analyzed in

the same manner; the resulting autocorrelogram is shown in Figure 3-4

(Medina, 1976). In this case, the dry period time separating independent

events is 9 hours. The differences between the two autocorrelograms

are due to the fact that STORM first satisfies depression storage and

evaporation before producing runoff as a simple product of a runoff

coefficient and the rainfall rate. Hydroscience, Inc. (1979) has noted

that this technique is biased at shorter lag times because the autocorre-

lation coefficient is inflated due to high correlations within events.

They suggest that it is better to perform the autocorrelation analysis

on the events resulting from different minimum dry periods.

A second approach investigates the relationship between the minimum

dry period and the number of events per year (Medina, 1976; Heaney et

al., 1977). Several such relationships are plotted in Figure 3-5. The

point where an increase in the minimum dry time no longer causes a

significant decrease in the number of events determines the proper

value. Such an approach is somewhat more subjective than the auto-

correlation technique; as one can see in Figure 3-5, a well-defined

breakpoint is not always evident.

The third approach, advocated by Hydroscience, Inc. (1979), assumes

that the time between independent rainfall or runoff events is exponen-

tially distributed. The exponential distribution and its discrete

counterpart, the Poisson distribution, are commonly used to describe

times between independent, probabilistic events of several kinds (Benjamin

















HOURLY RUNOFF RECORD
DES MOINES, IOWA 1968


95% T.L.


95% T.L


0 20 40 60 80
LAG, hours


100


Figure 3-4.


Autocorrelation Function for the Hourly Runoff
Record (Generated by STORM) of Des Moines, Iowa,
1968 (Medina, 1976).













/
Lo
: -0

.*_












,. / w
I o

) Z :

CD
t m I
Z 0 ( I o
0 w44

0


8 : i I
f o 0* 0S






Io w

i 3 I SI N3 : -8n
I '


/ / o
1. .


I EN


I 000
S/






80

and Cornell, 1970). Given this assumption, the minimum dry period is

adjusted until the coefficient of variation of interevent times (deter-

mined from a sufficiently long record) is 1.0. Recall that an exponen-

tial distribution is a special case of the gamma distribution in which v

= 1.0. This is certainly not an absolute event definition but it pro-

vides a convenient framework for characterizing the distribution of

interevent times. Based on this approach, analyses performed by Hydro-

science, Inc. (1979) and E. D. Driscoll and Assoc. (1981) indicate that

most areas of the United States have minimum dry periods of 3 to 15

hours.

An alternative to the selection of a minimum dry period to separate

independent events investigates the local meteorological patterns to

determine independent events based on the storm structure or scale

(Eagleson, 1970, Heaney et al., 1977). For example, two periods of

rainfall or runoff may come from two separate, independent convective

cells or the same frontal system. This approach involves considerable

study and is probably not warranted for the purposes of the Hydroscience

method.



Synoptic Analysis Program-SYNOP

A computer program, named SYNOP, has been developed to analyze

long-term rainfall records (Municipal Environmental Research Laboratory,

1976). Its primary function is to generate the array of statistics

listed in Table 3-1. These statistics are provided on a monthly basis,

an annual basis, and for all storms over the entire record. The monthly

values are particularly useful where wide seasonal differences occur.

Although designed to analyze rainfall records (specifically in the form








provided by the National Climatic Center, NOAA), the program can also

analyze long-term runoff records. These records may be generated by a

computer simulator, by a monitoring program, or both.



Runoff Statistics from Rainfall Statistics

Long-term hourly rainfall records, extending over many years, are

more commonly available than equally detailed runoff records. A large

network of rainfall gauges has long been maintained by the National

Weather Service, whereas runoff records are somewhat sporadic and rela-

tively short. The Hydroscience statistical method includes techniques

to estimate runoff statistics from rainfall event statistics. The tech-

niques were derived to allow preliminary analyses from little more than

the rainfall record.



Quantity statistics. Two extremely simple relationships are used

to derive the mean runoff event volume and flow from the corresponding

rainfall statistics. For volume,

V = RcVP (3-5)

where VR = mean runoff event volume, L3,

R = average runoff coefficient, and

Vp mean rainfall event volume, L3

The runoff coefficient, Rc, represents the average runoff-to-rainfall

ratio. Of course, this ratio varies from storm to storm, but for prelim-

inary analyses the estimate is probably adequate. The value of R can

be estimated by an analysis of local rainfall/runoff data or estimated

from one of several simple techniques (Miller and Viessman, 1972; Hydro-

logical Engineering Center, 1977; Hydroscience, Inc., 1979).





82

The mean runoff flow is calculated as

D
QR = RcQ P D (3-6)
R RP DR

where QR = mean runoff event flow rate, L T-,
3 -1
Qp = mean rainfall event intensity, L T-,

Dp = mean rainfall event duration, T, and

DR = mean runoff event duration, T.

The ratio Dp/DR is included to account for runoff continuing after the

rainfall event has subsided. This is particularly useful in large

catchments where travel times are lengthy. The value of DR is estimated

by a unit hydrograph procedure developed by Brater and Sherill (1975).

The details are presented in the Hydroscience report (1979).

The mean interevent time for runoff events, AR, is assumed to equal

the rainfall value, Ap. Given the simple coefficient method of esti-

mating VR, this is reasonable. Actually, many small rainfall events

produce no runoff because initial storage or abstractions are not

satisfied. This would cause the value of AR to be greater than A.

The coefficients of variation for runoff event flows, vqR, and volumes,

vvR' are also assumed to equal their rainfall counterparts.

The above measures are crude. It would be preferable to drive a

runoff simulator with the hourly rainfall record and analyze the results

with SYNOP. Any available runoff data could be used to calibrate the

model. Obviously, the best method would analyze a long series of

observed runoff flows but such records are scarce.



Quality statistics. The mean pollutant load for all runoff events

is determined by the mean pollutant concentration and the mean runoff

event volume:








MR = CRVR (3-7)

where MR mean runoff event pollutant load, M, and

CR mean runoff event pollutant concentration, ML-3.

This simple equation assumes that the pollutant concentrations are

independent of runoff volumes. This is probably not true, but is assumed

to be adequate for most preliminary analyses. A similar relationship

was developed for the mean pollutant load rate, i.e.,

WR = CRQR (3-8)

where WR = mean runoff event pollutant load rate, MT-.

Again, the assumption of independence between pollutant concentrations

and runoff flows is made. If the independence assumptions are inade-

quate, the following corrections can be made:


R = CRVR (1 + Vc vRPcv) (3-9)


WR = CRQR (1 + Vc qRPcq) (3-10)

where vc = coefficient of variation for runoff event pollutant
concentrations,

Pc = linear correlation coefficient between pollutant
concentrations and runoff volumes, -1 < p < 1
and = cv *

Pcq linear correlation coefficient between pollutant concen-
trations and runoff flow rates, -1 < p < 1.
= cq =
A positive value of pcv or pcq would indicate that higher flows or

volumes produce higher concentrations. A negative value would indicate

that the dilution effect of large runoff events is dominant. Unfor-

tunately, deriving statistically significant estimates for p and
cv
Pq requires a large amount of data.








From the assumptions associated with equations 3-7 and 3-8, it is

also assumed that

v = R (3-11)

and w = qR (3-12)

where v = coefficient of variation of runoff pollutant loads, and

v = coefficient of variation of runoff pollutant load rates.

It also seems reasonable to expect the distribution of loads and load

rates to be similar to the probability distributions for runoff volumes

and flows (i.e., gamma distributed).

Once again, it should be emphasized that it would be more appropri-

ate to analyze the results of a long-term simulator (calibrated against

any available local data) or an actual long-term record.



Analysis of Storage/Release Systems

Direct analysis of the long-term behavior of storage/release

systems under the stimulus of a series of random runoff events is possi-

ble if the probability distributions governing the event characteristics

are known (DiToro and Small, 1979; Hydroscience, Inc., 1979). The gamma

and exponential distributions were established in the pertinent litera-

ture and in the previous section as applicable to the characteristics of

independent rainfall or runoff events. This knowledge and the system

conceptualization, presented earlier, form the basis of the analytical

procedures developed by DiToro and Small (1979), and presented by Hydro-

science, Inc. (1979). Their work is summarized below. The subscript

"R" is dropped from most of the runoff event parameters and statistics

to avoid unnecessary clutter.







Storage (In-Line Configuration)

A storage basin captures, from a particular event, up to its current-

ly available or effective capacity, Ve. If the volume of the event

exceeds V the excess is bypassed. The only time the entire storage

basin capacity, VB, is available is when the basin is totally emptied of

water captured from previous storms. The basin is emptied between

events at a constant release rate, S. It is readily seen that the

average effective storage capacity, VE, and the release rate, 0, deter-

mine the basin's long-term performance.

Unfortunately, the effective storage capacity is a stochastic

process with a memory, i.e., it is at least a first-order Markov or

autoregressive process. In other words, the current value of Ve is a

function of previous events. The analysis of processes such as these

can be difficult and unwieldy. In order to facilitate the development

and application of a representative model of the storage/release system,

the interaction between any two events is assumed to be as shown in

Figure 3-6. Event 1 begins with the mean effective storage capacity,

VE, available to store runoff. The volume associated with an arbitrary

runoff event (denoted here as event 1), v, fills the basin to (VB VE)

+ v. Between events 1 and 2, the basin is emptied at a constant release

rate, 2. At the beginning of event 2, the basin has an available capaci-

ty of V The value of V is a function of v and the time between
e e
events, 6:

V = VB for v > VE, 6> B (3-13)
e B = E S-

v + Ve E (3-14)
V = V for v < V, 6 +B E (3-14)
e B = E' 6 > -


V = 60 for v > V, 6 < VB
e E = -- (3-15)















-- ------------ ---
VE EVENT 2 V
EVENT I I

i-s-S


TIME


LEGEND


V 8
VEa
vR
Ve:


maximum storage capacity, L3
mean effective storage capacity, L3
event I runoff volume, L3
release rate, L3T'
effective storage capacity, L3
time between runoff event midpoints,T


Figure 3-6.


Determination of Effective Storage Capacity,
V (DiToro and Small, 1979; Hydroscience, Inc.,
1979)
1979).