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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 CR805664. 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 PerformanceReliability.......................... 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 31 Rainfall/Runoff Event Parameters and Statistics 71 41 Geometric and Hydraulic Data for Hypothetical Reservoir 109 51 Coefficients for the Transcendental Equation Representing the Percent Volume Control Produc tion Function Produced by STORM 174 52 Coefficients for the Transcendental Equation Representing the Percent Pollutant (BOD) Control Production Function Produced by STORM 175 53 Coefficients for the CobbDouglas Equation Repre senting the Percent Pollutant (BOD) Control Pro duction Function Produced by STORM 178 71 Coefficients for the CobbDouglas, Mitscherlich, and Quadratic Equations Representing the Production Functions Produced by the S/T Block 272 72 Optimal Storage/Release Combinations for Volume Control 285 73 Optimal Storage/Release Combinations for TSS Control 287 74 Optimal Storage/Release Combinations for TSS Removal 289 LIST OF FIGURES Figure Page 11 Design Event Minimizing the Sum of Control Costs and Damages. 6 12 Design Event Maximizing Net Benefits. 7 21 Basic Storage/Release System Configurations. 14 22 Batch Reactor. 19 23 FirstOrder Reactions in a Batch Reactor. 21 24 Results of Hydrocarbon Settling Test. 24 25 Integral Method of Estimating "Reaction" Order for Hydrocarbon Settling. 25 26 Tracer Response, SteadyState Flow Reactor. 28 27 PlugFlow Reactor. 34 28 Completely Mixed Flow Reactor. 35 29 Brune's Sediment Trap Efficiency Curves. 45 210 Discrete Inputs of the Inflow Rate and Pollutant Concentration at Equal Time Intervals. 52 211 Storage/Release System, Howard's Statistical Method. 54 212 Storage/Release System, STORM. 63 31 Simplified Representation of Independent Rain fall of Runoff Events. 70 32 Cumulative Gamma Distributions. 74 33 Autocorrelation Function for the Hourly Precipi tation Record of Des Moines, Iowa, 1968. 76 34 Autocorrelation Function for the Hourly Runoff Record (Generated by STORM) of Des Moines, Iowa, 1968. 78 vii Figure Page 35 Number of Events per Year Versus the Minimum Dry Period. 79 36 Determination of Effective Storage Capacity, V 86 37 Determination of Mean Effective Storage Capacity, VE. 89 38 Determination of the LongTerm Fraction of the Total Pollutant Load or Runoff Volume Not Cap tured by the Storage Basin. 91 39 Determination of the LongTerm Fraction of the Total Pollutant Load or Runoff Volume Bypassing the Interceptor or Mainsteam. 94 41 SWMM Storage/Treatment Block. 100 42 Storage/Treatment Unit. 102 43 Completely Mixed, VariableVolume Detention Unit. 112 44 PlugFlow Detention Unit. 114 45 Camp's Sediment Trap Efficiency Curves. 120 46 Design Details, Humboldt Avenue Detention Tank, Milwaukee, Wisconsin. 124 47 Effluent Flow Rate, September 12 (11:00 p.m.) to September 21 (10:00 a.m.), 1972, Humboldt Avenue Detention Tank, Milwaukee, Wisconsin. 125 48 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 51 OneInput, OneOutput Production Function. 131 52 TwoInput, One Output Production Function. 133 53 OneInput, TwoOutput Production Function. 135 54 Classical OneInput, OneOutput Production Function and the Average and Marginal Products. 137 55 Elimination of the Region of Increasing Average Product. 139 56 Graphical Representation of the Marginal Rate of Substitution and an Isocline. 142 viii Figure Page 57 Determination of the Area of Substitution. 143 58 OneInput, OneOutput CobbDouglas Production Function and the Average and Marginal Products. 148 59 TwoInput, OneOutput CobbDouglas Production Function with Isoclines and Ridge Lines. 151 510 OneInput, OneOutput Mitscherlich Production Function and the Average and Marginal Products. 152 511 TwoInput, OneOutput Mitscherlich Production Function with Isoclines and Ridge Lines. 154 512 OneInput, OneOutput Quadratic Production Function and the Average and Marginal Products. 157 513 TwoInput, OneOutput Quadratic Production Function with Isoclines and Ridge Lines. 158 514 Piecewise Linearization of a Production Function. 161 515 Factors Affecting the Quality Control Performance of Stormwater Storage/Release Systems. 165 516 Storage/Treatment (or Release) Isoquant Produced by a Transcendental Production Function. 171 517 Storage/Treatment (or Release) Production Function for BOD Control Produced by STORM, Graphical and Transcendental Representations, Minneapolis, Minne sota. 176 518 Storage/Treatment (or Release) Production Function for BOD Control Produced by STORM, Graphical and CobbDouglas Representations, Minneapolis, Minne sota. 179 519 OffLine Storage/Release Production Function for BOD Control Produced by Hydroscience Statistical Method, Compared with STORMGenerated Production Function, Denver, Colorado. 180 520 Derivation of Production Functions for InLine Storage/Release Systems From Hydroscience Sta tistical Method. 182 521 InLine Storage/Release Production Function for Pollutant Load or Volume Control Produced by Hydroscience Statistical Method, VvR = 1.73. 183 Figure Page 522 InLine Storage/Release Production Function for Additional Open Beach Days Produced by Hydroscience Statistical Method, Kingston, New York. 185 523 InLine Storage/Treatment (or Release) Production Function for Suspended Solids Control Produced by Howard's Statistical Method, Minneapolis, Minnesota. 187 524 InLine Storage/Release Production Function for BOD Control Produced by SWMM Storage/Treatment Block, Atlanta, Georgia. 189 61 Annual Cost Verus Additional Open Beach Days as a Result of Fecal Coliform Control With a Storage/Release System, Kingston, New York. 194 62 Determination of Maximum Net Benefits. 195 63 General Input Cost Function Types. 200 64 Costs for Earthen Storage Basins, September, 1976. 202 65 Costs for Covered Concrete Storage Basins, Sep tember, 1976. 203 66 Costs for Uncovered Concrete Storage Basins, Sep tember, 1976. 204 67 Convex and Concave Functions. 209 68 Terms of Equations 622. 213 69 Optimization in Production Space. 223 610 Optimal Costs for Any Storage/Release System Per formance Level. 225 611 Possible Isocost Curves. 226 612 Optimization in Cost Space. 228 71 Hypothetical Scenario. 233 72 Percent Volume Control Production Function, SWMM Level I. 239 73 Percent Pollutant (BOD) Control Production Function, SWMM Level I. 240 Figure Page 74 Cumulative Frequency Plots for Event Flows, Dura tions, Volumes, Interevent Times and Several Cumulative Gamma Distributions. 243 75 Percent Volume and Pollutant Control Production Function, Hydroscience Statistical Method. 245 76 Percent Volume Control Production Function, Results of S/T Block Simulation. 247 77 Percent TSS Control Production Function, Results of S/T Block Simulation. 248 78 Percent TSS ControltoPercent Volume Control Ratios. 249 79 Results of Settleability Tests for Total Suspended Solids. 252 710 Percent TSS Removal Production Function, Results of S/T Block Simulation. 253 711 Percent TSS Removal Production Function, Results of S/T Block Simulation (Enlarged Range). 254 712 Percent TSS RemovaltoPercent TSS Control Ratios. 256 713 Percent TSS RemovaltoPercent Volume Control Ratios. 257 714 Percent of the Time that the Hourly TSS Load from System Exceeds 50, 100, or 500 lbs/hour. 259 715 Percent TSS Removal Production Function, Completely Mixed Storage Basin with Bypass. 261 716 Percent TSS Removal Production Function, PlugFlow Storage Basin without Bypass. 262 717 Percent TSS Removal Production Function, Completely Mixed Storage Basin without Bypass. 263 718 FirstOrder TSS Removal Equations. 266 719 Percent TSS Removal Production Function, 1 K = 0.0003 sec. 267 720 Percent TSS Removal Production Function, 1 K = 0.00003 sec 268 721 Percent TSS Removal Production Function, K = 0.000003 sec1. 269 K = 0.000003 sec .269 Figure Page 722 Percent Volume Control Production Function, Rep resented by the CobbDouglas Equation. 273 723 Percent Volume Control Production Function, Rep resented by the Mitscherlich Equation. 274 724 Percent Volume Control Production Function, Rep presented by the Quadratic Equation. 275 725 Percent TSS Control Production Function, Repre sented by the CobbDouglas Equation. 276 726 Percent TSS Control Production Function, Repre sented by the Mitscherlich Equation. 277 727 Percent TSS Control Production Function, Repre sented by the Quadratic Equation. 278 728 Percent TSS Removal Production Function, Repre sented by the CobbDouglas Equation. 279 729 Percent TSS Removal Production Function, Repre sented by the Mitscherlich Equation. 280 730 Percent TSS Removal Production Function, Repre sented by the Quadratic Equation. 281 731 Expansion Paths for the Percent Volume Control Production Function, Graphical Representation. 284 732 Expansion Paths for the Percent TSS Control Pro duction Function, Graphical Representation. 286 733 Expansion Paths for the Percent TSS Removal Pro duction Function, Graphical Representation. 288 734 Application of CostSpace Optimization Procedure. 291 735 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 longterm performance of storage/ release systems, the use of the production function to summarize system performance, and several techniques to determine the most costeffective designs. The focus is on longterm analyses rather than "design storm" or singleevent approaches. Several available computer simulation and xiii statistical models capable of evaluating longterm 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 longterm 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 costeffective 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 nonurban 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 89234) 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 92500), 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 92500 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 92500 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. 362, U.S. Environmental Protection Agency, 1976). The tone and goals of PL 92500 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 95217) 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 twentyeight 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 longterm data records (i.e., many years). Longterm 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 11 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 11 yields to the benefitcost curves shown in Figure 12. 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 PRECONTROL 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 11. 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 12. 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., precomputer age), it has per formed admirably in quantity control applications. Nevertheless, there are serious drawbacks in quantity and quality control applications: 1) Preevent 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 desktop 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 longterm 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 longterm 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 costeffective 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 costeffective 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 timehonored 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 longterm 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 quasisteady 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) inline and 2) offline. The inline configuration is shown in Figure 21(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 RZ (a) InLine System RUNOFF BYPASS , _RECEIVING WTE RECEIVING WATER (b) OffLine System Figure 21. 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 offline configuration, shown in Figure 21(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) (21) dt where V(t) = volume of water in storage, L , 3 1 I(t) = inflow rate to storage, L T1 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)] (22) Equation 21 may then be written as dV I(t) f[V(t)] (23) dt Assume that a power function governs the relationship between outflow and volume, i.e., 0(t) = a[V(t)V ]b (24) where a, b = coefficients, and V = volume of stored water at which outflow begins, L3 Substituting equation 24 into equation 23 yields dV b dV I(t) a[v(t)Vo ] (25) Equation 25 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 25 becomes dV + aV(t) = I(t) + aV (26) Equation 26 is a simple linear firstorder 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)} (27) or V(t) = exp(at){ft[I(t) + aV ]exp(at)dt + V(0)} (28) 0 0 where V(O) = initial volume of stored water, L3 The obvious possible drawback in equation 28 is the inflow function, I(t). Only rather trivial functions allow the complete solution of equation 28. When b = 0, the outflow is constant, i.e., 0(t) a, and equation 25 becomes dV d = I(t) a (29) for which the solution is V(t) = ft(I(t) a)dt (210) 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 22. The storage equation for a batch reactor is dV d 0 (211) 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 22. Batch Reactor. Isothermal, irreversible reactions in a completely mixed, constant volume batch reactor are governed by the following equation (Levenspiel, 1972): V = KVCn(t) (212) dt or KCn(t) (213) dt 3 where C(t) = reactant concentration in the reactor, ML3 V = constant fluid volume in the reactor, L , K = rate constant, T1 and n = reaction order. The term dC/dt represents the reaction rate. Under the assumption of firstorder kinetics (i.e., n=l), equation 213 becomes dC = KC(t) (214) dt for which the solution is C(t)= exp(Kt) (215) 3 where C(0) = initial reactant concentration in the reactor, ML3 Equation 215 produces the family of curves shown in Figure 23. Firstorder reactions are typified by those that are unaffected by the initial concentration, C(0). The firstorder 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 213 by the natural logarithm yields dC (216) In( d_ = n In[C(t)] + ln(K) (216) 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 23. FirstOrder 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 213, i.e., [ C() ]n = { Kt (217) C(0) 1n [c(o)]ln or Ct ln 1ln C.F. = {[ t ]1n 1 C(n) = Kt for n # 1 (218) C(0) nl 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 213 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 firstorder "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 6ft settling column (essentially a batch reactor without continuous mixing). The results for hydrocarbons are shown in Figure 24. Tests of several rate orders, using the inte gral method of analysis (equations 217 and 218), are shown in Figure 25. From these plots it is clear that the settling of hydrocarbons (at least for the data collected) is approximately a fourthorder "reaction." Of course, a more accurate representation might have been obtained at some noninteger 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 24. 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 25. 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 21 governs the continuity of fluid mass, i.e., dV I(t) 0(t) (219) 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 219 does not provide this information. The two idealflow 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 arbitraryflow reactors also have unique age distributions. These reactors experience mixing levels between the extremes of the completely mixed and plugflow reactors. Consider a steadystate 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 26. Figure 2 6(a) shows the step function response associated with plugflow reactors. Figure 26(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' (220) 0 where t' = dummy variable of integration. Also, by definition, r (t)dt = rP(t)dt = 1 (221) 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 (222) Still in reactor: V ft *(t')dt' (223) 0 Left reactor: Q ftft"~(t')dt' dt" (224) 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 26. Tracer Response, SteadyState 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 (225) Q ft ft"/(t')dt'dt", 0 0 ft ft"(t')dt'dt" 0 0 (226) Differentiating with respect to t and recalling equation 220 yields = 9(t) + F(t) K(t) = (1 F(t)) (227) (228) From this result, the useful fact that the area between F(t) 1 and F(t) (the shaded area in Figure 26(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 26(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 (229) leaving the (230) Differentiating equation 220 and substituting the result for >(t) in equation 230 produces 0 = t dFt dt = flt dF(t) (231) 0 d0 30 Multiplying equation 231 by Q/V yields 6 = fl ; t dF(t) (232) 0 The integral in equation 232 also defines the shade area shown in Figure 26(c); thus, S= 1 (233) or 9 = V (234) Q Equation 234 represents the average or nominal detention timeit 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 steadystate 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 steadystate 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 228 can be written as p(t) = R Q F(t) (235) Differentiating equation 220 and substituting the result for #(t) in equation 235 produces the following: dF(t+ q F(t) = Q (236) dt V V Solving for F(t) yields F(t) = 1 exp( t) (237) (2V7 Equation 237 is represented by the curve shown in Figure 26(b). The external age distribution is P(t) = dt exp( t) (238) When the steadystate reactor is operated in the plugflow mode, the age of all fluid elements leaving at any time t is V/Q. Thus, F(t) = 0 for t < 6 (240) F(t) = 1 for t > 8 Equation 239 describes the step function shown in Figure 26(a). For unsteadystate 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) = (240) 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 240 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 (241) 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 steadystate units this is true but equation 240 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 reactorsmathematical 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 plugflow and completely mixed reactors are derivable from the onedimensional advectivedispersive equation (Medina, 1976; Medina et al., 1981a), i.e., S [E(x,t) U(x,t)C(x,t)] (So + Si) (242) at ax ax 0 3 where C(x,t) = reactant concentration in the reactor, ML3 x = distance along flow axis, L, 2 1 EL(x,t) = longitudinal dispersion coefficient, L T1 U(x,t) = longitudinal flow velocity, LT1 3 1 3 1 S. = sink of reactant, ML T . 1 A plugflow 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 242 becomes a = [ U(x,t)C(x,t)] KC (x,t) (243) Equation 243 is the pure advective form of equation 242. For prac tical purposes (e.g., computer simulation), equation 243 can be viewed as a series of discrete plugs, each acting as a batch reactor, moving along the flow axis. This representation of the plugflow reactor is shown in Figure 27. 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 28. 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 27. PlugFlow 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 28. completely mixed flow reactor with an input source, an outflow sink, and a reactive sink, equation 242 becomes d(Vl = c(t)I(t) C(t)O(t) KCn(t)V(t) (244) where C(t) = reactant concentration in the reactor and the outflow, ML3 and C (t) = reactant concentration in the inflow, ML3 Equation 244 can also be viewed as a version of the storage equation written in terms of reactant mass. Unfortunately, the nth order reaction provision in equation 244 makes the derivation of an analytical solution difficult. However, for firstorder reactions (i.e., n = 1), equation 244 becomes d( C (t)I(t) C(t)0(t) KC(t)V(t) (245) dt or C(t)d + V(t)d CI(t)I(t) C(t)0(t) KC(t)V(t) (246) Rearranging terms yields dC + 1 dV + + K] C(t) t) CI(t) (247) dt V(t) dt V(t) V(t) In terms of C(t), equation 247 is a simple firstorder 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)] (248) V(t) V(t) where C(O) = initial reactant concentration in the reactor and the 3 outflow, ML3 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 248 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 advectivedispersive equation given by equation 242 (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 plugflow reactor; in other words, queued quiescent flows. Obviously, these are more desirable conditions for settling than those found in completelymixed 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 (249) 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, ML3, 2 g = gravitational constant, LT2 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 (250) where FD = drag force of the fluid, MLT , CD = coefficient of drag, 2 A = crosssectional area of the particle, L and v = terminal settling velocity of the particle, LT1 When the particle is no longer accelerating, the drag force and the effective weight are equal, i.e., (P )gV CpAc /2 (251) and the terminal settling velocity can be calculated as (P P) V vs = [& 1/2 (252) CD p Ac 3 2 For spherical particles of diameter d, V = (n/6)d and A = (w/4)d. p c Equation 252 can be written as 4 (P p) 1/2 vs [ CD d] (253) 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; (253) D NR or 24 3 4 CD = 1+ / + 0.34 for 0.5 < NR < 104; (255) D NR 1 or CD = 0.4 for NR > 104 (256) The value of NR is calculated as NR = Vsd/v (257) where v = kinematic viscosity of the fluid, L2 T Kinematic viscosity is a function of fluid temperature. When CD is specified by equation 255, 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 steadystate, plugflow 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 (258) 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 unsteadystate 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 crosssection of data from a number of systems to develop relationships between longterm 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 longterm response of a variety of system sizes or capacities can be rapidly evaluatedwithout 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 timeseries 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 longterm 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 manmade 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 crosssection of data from a variety of storage/release systems to develop relationships between design and operating parameters and longterm 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, singlesystem relation ships are not as applicable to other proposed or inplace systems as their crosssectional counterparts. Nevertheless, the crosssectional relationships should only be used for very preliminary estimates. Summaries of crosssectional 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 crosssectional 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 29. Each curve represents a different type of sediment (i.e., fine, coarse, and "median") and each is a function of the basin capacityto 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, initialvalued 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) RungeKutta formulae, and 3) predictorcorrector 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 secondorder Adams formula approximates the differential equation dy g(yx) (259) dx by Ax (260) 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)] (261) 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 261 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 (262) 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)] (263) With this storageoutflow or storagedischarge relationship in hand, the righthand side of equation 262 is evaluated and the actual values of Vj+l and 0j+1 determined by substituting equation 263 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 RungeKutta family is the "fourthorder RungeKutta formula" (Hornbeck, 1975). The formula approximates equa tion 259 by 1 1.) Yj+I Yj + Ax[T g(yj' xj) + g(Yj+1/2' xj+1/2) (264) + 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 RungeKutta formula: dV d I(t) f[V(t)] (265) The resulting RungeKutta approximation is 1 1 j+ J Jv,/2' J+1/2) (266) 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 RungeKutta 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 266 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) (267) 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 (268) 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] (269) n1 n1 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 210 (Medina, 1976; Medina et al., 1981a). This technique completes the integralladen 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), constantvolume, 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 longterm 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 longterm 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 210. Discrete Inputs of the Inflow Rate and Pollutant Concentration at Equal Time Intervals (Medina, 1976). u TIME, t Figure 210. Discrete Inputs of the Inflow Rate and Pollutant Concentration at Equal Time Intervals (Medina, 1976). loads in a constantvolume, 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 firstorder 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 constantvolume, 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 inline configuration. A schematic is shown in Figure 2 11. The principal input to the model is a longterm hourly rainfall record. The rainfall record is transformed into a runoff record with a runoff coefficient after considering the effects of natural and manmade 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 211. 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 singleparameter exponential function; p(x) = (l/x)exp(x/x) (270) 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 rainfallrunoff 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 211): 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 wetweather treatment facility or the excess capacity available in a dryweather 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 (271) 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 271 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 = (272) 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, wetweather 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 wetweather 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 twoparameter 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 longterm runoff record. Unfortunately, such longterm 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 21) 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 longterm 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 longterm 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 plugflow reactor and assumes a constant volume once the basin is filled to the overflow depth. Pollu tants are assumed to be removed under firstorder 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 dryweather 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 variablevolume, plugflow 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 multistorm 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 firstorder reactants. The differential equations governing this system are evaluated using the discreteinput 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 longterm runoff and 62 pollutant load records with a continuous longterm hourly rainfall record. The hourly runoff record is, in turn, routed to a storage/ treatment (or release) system. The system is depicted in Figure 212. 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 plugflow 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 offline 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 212. 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 dryweather 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 plugflow or com pletely mixed reactors; 5) the ability to use a wide variety of pollutant removal equa tions in the plugflow mode; and 6) the modeling of pollutants as firstorder 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 longterm production functions. The details of the model algorithms are discussed in Chapter 4. Measure of PerformanceReliability 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 (273) a N where Nf = number of failure years, and N = total number of years. This measure of reliability is also known as "occurrencebased" relia bility and this term is preferred here. The noncatastrophic nature of stormwater quality problems and economic considerations require a differ ent definition. Thus, occurrencebased 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 (274) t T where Td time period that demand was satisfied, T, and Tt = total operating period, T. Equation 274 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 concentrationfrequency 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 (275) 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 longterm 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 longterm 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 31(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 31. 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 , kDURATION, d _I TIME (b) Simplified Representation Figure 31. Simplified Representation of Independent Rain fall or Runoff Events (Hydroscience, Inc., 1979). EVENT Table 31. 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 (31) 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 (32) 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 twoparameter 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 K1 p(x) = (/x) xl exp(Kx/x) (33) 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) (34) The cumulative gamma distribution for several values of v is shown in Figure 32. 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 32. 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 openseries 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 33 (Medina, 1976). From this analysis it was determined that the appropriate minimum dry period is 10 hours. The tolerance limit (defined at the 95percent probability level) was HOURLY PRECIPITATION RECORD DES MOINES, IOWA 1968 95% T.L. 0 20 40 60 80 100 LAG, hours Figure 33. 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 34 (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 35. 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 35, a welldefined 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 34. 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 ProgramSYNOP A computer program, named SYNOP, has been developed to analyze longterm rainfall records (Municipal Environmental Research Laboratory, 1976). Its primary function is to generate the array of statistics listed in Table 31. 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 longterm runoff records. These records may be generated by a computer simulator, by a monitoring program, or both. Runoff Statistics from Rainfall Statistics Longterm 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 (35) 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 runofftorainfall 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 (36) 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 (37) where MR mean runoff event pollutant load, M, and CR mean runoff event pollutant concentration, ML3. 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 (38) 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) (39) WR = CRQR (1 + Vc qRPcq) (310) 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 37 and 38, it is also assumed that v = R (311) and w = qR (312) 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 longterm simulator (calibrated against any available local data) or an actual longterm record. Analysis of Storage/Release Systems Direct analysis of the longterm 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 (InLine 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 longterm performance. Unfortunately, the effective storage capacity is a stochastic process with a memory, i.e., it is at least a firstorder 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 36. 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 (313) e B = E S v + Ve E (314) V = V for v < V, 6 +B E (314) e B = E' 6 >  V = 60 for v > V, 6 < VB e E =  (315)    VE EVENT 2 V EVENT I I isS 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 36. Determination of Effective Storage Capacity, V (DiToro and Small, 1979; Hydroscience, Inc., 1979) 1979). 