&EPA
           United States
           Environmental Protection
           Agency


           Research and Development
            Environmental Research
            Laboratory
            Athens GA 30613
EPA/600 3-85 065
August 1985
Computer Program
Documentation for the
Enhanced Stream
Water Quality
Model QUAL2E
                                     >  I

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                                       EPA/600/3-85/065
                                       August 1985
COMPUTER PROGRAM DOCUMENTATION FOR THE ENHANCED
       STREAM WATER QUALITY MODEL QUAL2E

                      by

Linfield C.  Brown* and Thomas 0.  Barnwell,  Jr.**

       *Department of Civil Engineering
               Tufts University
              MedfoYd, MA  02155

     **Environmental Research Laboratory
     U.S. Environmental  Protection Agency
              Athens, GA  30613
       Cooperative Agreement No.  811883
           t^-rtrt pro
        ENVIRONMENTAL RESEARCH LABORATORY
        OFFICE OF RESEARCH AND DEVELOPMENT
       U.S.  ENVIRONMENTAL PROTECTION  AGENCY
                ATHENS,  GEORGIA 30613

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                                 DISCLAIMER

      The information in this document has been funded wholly or in part by
the United States Environmental Protection Agency under Cooperative Agreement
No. 811883 to Tufts University.  It has been subject to the Agency's peer and
administrative review, and it has been approved for publication as an EPA
document.  Mention of trade names or commercial products does not constitute
endorsement or recommendation for use by the U.S. Environmental Protection
Agency.
                                     ii

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                                  FOREWORD

      As environmental controls become more costly to implement and the
penalties of judgment errors become more severe, environmental  quality
management requires more efficient management tools based on greater
knowledge of the environmental  phenomena to be managed.   As part of this
Laboratory's research on the occurrence, movement, transformation,  impact
and control of environmental contaminants, the Assessment Branch develops
management or engineering tools to help pollution control officials achieve
water quality goals.

      The  stream water quality  model QUAL-II  is  widely used  for waste  load
allocations, discharge  permit  determinations,  and other  conventional
pollutant  evaluations in the United States.   Since its introduction in
1970,  several  different versions  of the model  have evolved.  This  manual
presents  the most  recent modifications  in the  form of an enhanced  state-of-
the-art model  called  QUAL2E, which was  developed  by the  National Council
for  Air and Stream Improvement (NCASI).   Distribution and maintenance of
the  QUAL2E computer program, and  training and  assistance to model  users,
will  be provided by EPA's  Center  for  Water Quality Modeling  at this Labora-
tory.
                                      Rosemarie C.  Russo,  Ph.D.
                                      Director
                                      Environmental  Research Laboratory
                                      Athens,  Georgia
                                    iii

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                                  ABSTRACT

     Presented in this manual  are recent modifications  and  improvements  to
the widely used stream water quality model  QUAL-II.   Called QUAL2E,  the
enhanced model  incorporates improvements in  eight areas:   (1)  algal,
nitrogen, phosphorus, and dissolved oxygen interactions;  (2)  algal  growth
rate; (3) temperature; (4) dissolved oxygen;  (5)  arbitrary  non-conservative
constituents; (6) hydraulics;  (7) downstream  boundary concentrations;  and
(8) input/output modifications.

     QUAL2E, which can be operated either as  a  steady-state or  as  a  dynamic
model, is intended for use as  a  water quality planning  tool.  The  model,
for example, can be used to study the impact  of waste loads on  instream
water quality or to identify the magnitude and  quality  characteristics of
nonpoint waste loads as part of  a field sampling  program.   The  user  also
can model effects of diurnal variations in meteorological data  on  water
quality  (primarily dissolved oxygen and temperature)  or examine diurnal
dissolved oxygen variations caused by algal growth and  respiration.

     This report was submitted in partial fulfillment of Cooperative Agree-
ment No. 811883 by Tufts University under the partial sponsorship  of the
U.S.  Environmental Protection Agency.  This  report  covers  a period  from
August 1984 to June 1985, and work was completed  as  of  June 1985.
                                    IV

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                                CONTENTS
FOREWORD	ill
ABSTRACT	   iv

1.  INTRODUCTION 	    1
    1.1  QUAL2E Development  	    2
         1.1.1  History	    2
         1.1.2  Enhancements to QUAL-II  	    3
         1.1.3  Information Sources  	    4
         1.1.4  Organization of This Report	    5
    1.2  QUAL2E  Computer Model  	    5
         1.2.1  Prototype Representation 	    5
         1.2.2  Model Limitations  	    6
         1.2.3  Model Structure and Subroutines  	    6
         1.2.4  Program Language and Operating Requirements  	    6

2.  GENERAL MODEL FORMULATION  	    9
    2.1  Introduction  	    9
    2.2  Conceptual  Representation 	   10
    2.3  Functional  Representation 	   10
    2.4  Hydraulic Characteristics 	   14
         2.4.1  Discharge Coefficients 	   14
         2.4.2  Trapezoidal Cross Sections 	   15
         2.4.3  Longitudinal Dispersion  	   15
    2.5  Flow Augmentation 	   18

3.  CONSTITUENT REACTIONS AND INTERRELATIONSHIPS 	   21
    3.1  General Considerations  	   21
    3.2  Chlorophyll a_ (Phytoplankton Algae)	   21
         3.2.1  Algal Respiration Rate	   23
         3.2.2  Algal Specific Growth Rate	   23
         3.2.3  Algal-Light Relationships  	   25
         3.2.4  Algal-Nutrient Relationships 	   33
         3.2.5  Temperature Dependence in Algae Simulation 	   34
    3.3  Nitrogen Cycle	   34
         3.3.1  Organic Nitrogen 	   34
         3.3.2  Ammonia Nitrogen	   35
         3.3.3  Nitrite Nitrogen	   35
         3.3.4  Nitrate Nitrogen 	   36
         3.3.5  Inhibition of Nitrification at Low DO	   36

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                            CONTENTS (cont'd)
                                                                        Page

     3.4  Phosphorus Cycle  	   37
          3.4.1  Organic Phosphorus 	   37
          3.4.2  Dissolved Phosphorus 	   38
     3.5  Carbonaceous BOD	   38
     3.6  Dissolved Oxygen  	   39
          3.6.1  Dissolved Oxygen Saturation Coefficient  	   40
          3.6.2  Atmospheric Reaeration Coefficient Estimation  ....   41
          3.6.3  Ice Cover	   47
          3.6.4  K2 Default Values	   47
          3.6.5  Dam Reaeration	   48
     3.7  Coliforms   	   48
     3.8  Arbitrary Nonconservative Constituent ... 	   49
     3.9  Temperature	   49
     3.10 Temperature Dependence of Rate Coefficients 	   50
     3.11 Reaction Rates and Physical Constants ... 	   52

4.   FUNCTIONAL REPRESENTATION OF TEMPERATURE 	   55
     4.1  Basic Temperature Equation  	 	   55
     4.2  Definition of HN	   56
     4.3  Net Short-Wave Solar Radiation  	   58
          4.3.1  Extraterrestrial Radiation   	   59
          4.3.2  Radiation Scattering and Absorption  	   61
          4.3.3  Cloudiness	   63
          4.3.4  Reflectivity	   63
     4.4  Long-Wave Atmospheric Radiation 	   64
     4.5  Water Surface Back Radiation  	   64
     4.6  Evaporation	   65
     4.7  Conduction	   67

5.   COMPUTATIONAL REPRESENTATION 	   68
     5.1  Prototype Representation  	   68
     5.2  Forcing Functions 	   69
     5.3  Model Limitations 	   70
     5.4  Numeric Solution Technique  	   71
          5.4.1  Formulation of the Finite Difference Schmem  	   71
          5.4.2  Method of Solution   	   74
          5.4.3  Boundary Conditions  	   76

6.   REFERENCES     	   78

APPENDIX A.  QUAL2E USERS MANUAL  	   82
APPENDIX B.  EXAMPLE INPUT/OUTPUT DATA SETS   	  118

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                            LIST OF FIGURES
No.                                                                   Page
1-1    General Structure of QUAL2E  	    7
II-l   Discretized Stream System  	   11
II-2   Stream Network of Computational Elements and Reaches 	   12
III-l  Major Constituent Interactions in QUAL2E .-	   22
III-2  QUAL2E Light Functions   	   28
IV-1   Heat Transfer Terms Associated with Interfacial Heat Transfer    57
V-l    Classical Implicit Nodal Scheme  	   71

                              LIST OF TABLES
No.                                                                   Page
II-l   Values of Manning's "n" Roughness Coefficient  	   17
II-2   Experimental Measurements of Longitudinal Dispersion
       in Open Channels	   19
III-l  Comparison of Dissolved .Oxygen Saturation Concentrations   . .   42
111-2  Default Temperature Correction Values for QUAL2E	   51
111-3  Typical Ranges for QUAL2E Reaction Coefficients	   52
IV-1   Definition of Heat Transfer Terms Illustrated in Figure IV-1     58
IV-2   Empirical Coefficients for Determing Rs  	   64
                                   vii

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                              ACKNOWLEDGMENT

     Over the years, many investigators  have contributed  to  the development
of what has become QUAL2E.  The foundation  upon  which  the model has  been
built was laid by the Texas Water Development Board  in the late 1960s  in
the QUAL-I model.  Many versions of the  model  emerged  in  the 1970s.  The
lineage of QUAL2E can be traced to work  done for the Southeast Michigan
Council of Governments (SEMCOG) by Water Resources Engineers,  Inc.  (now
Camp, Dresser, McKee Inc.).  QUAL-II/SEMCOG was  chosen for distribution
by the Center for Water Quality Modeling (CWQM)  in the late  1970s and  began
to receive wide use in water quality modeling and wasteload  allocation
programs.

     QUAL-II/SEMCOG was throughly reviewed, tested,  and documented  by  the
National Council  of the Paper Industry for  Air and Stream Improvement, Inc.,
(NCASI) as discussed in NCASI Technical  Bulletin No. 391.  Changes  arising
from this review were incorporated in a  model  called QUAL-II/NCASI,  which
was adopted for distribution by the Center  for Water Quality Modeling.
Because of a mutual interest in the program, CWQM partially  sponsored  an
NCASI review of other versions of the QUAL-II computer program and  incor-
porated useful features of these versions in the program  called QUAL2E.

     Appendix A of this documentation report, the QUAL2E  users manual, is
taken directly from NCASI Technical  Bulletin No.  457,   "Modifications  to
the QUAL-2 Water Quality Model and User  Manual for QUAL2E Version 2.2."

     We express our appreciation to NCASI,  for permission to reprint this
material in the QUAL2E documentation report.

     The QUAL2E program also has been made  available for  IBM PC-compatible
microcomputer.  The microcomputer installation of this program was  performed
by Mr. Bruce Bartell of Computer Sciences Corporation, Inc.  and was  made
possible through the support of Mr. King Boynton of  the US EPA's Office
of Water and through an agreement with the  US-Spain  Joint Committee  for
Scientific and Technical Cooperation.
                                    vm

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                              1.   INTRODUCTION

     QUAL2E is a comprehensive and versatile  stream water  quality model.
It can simulate up to 15 water quality constituents in any combination
desired by the user.  Constituents which  can  be  simulated  are:
          1.  Dissolved Oxygen
          2.  Biochemical Oxygen  Demand
          3.  Temperature
          4.  Algae as Chlorophyll a_
          5.  Organic Nitrogen as N
          6.  Ammonia as N
          7.  Nitrite as N
          8.  Nitrate as N
          9.  Organic Phosphorus  as P
          10. Dissolved Phosphorus as P
          11. Coliforms
          12. Arbitrary Nonconservative Constituent
          13. Three Conservative  Constituents
The model is applicable to dendritic streams  that are well  mixed.   It
assumes that the major transport  mechanisms,  advection and dispersion,  are
significant only along the main direction of  flow (longitudinal  axis  of the
stream or canal).  It allows for  multiple waste  discharges, withdrawals,
tributary flows, and incremental  inflow and outflow.  It also has the
capability to compute required dilution flows for flow augmentation to
meet any prespecified dissolved oxygen level.
     Hydraulically, QUAL2E is limited to  the  simulation of time  periods
during which both the stream flow in river basins and input waste loads are
essentially constant.  QUAL2E can operate either as a steady-state  or as  a

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dynamic model, making it a very helpful water  quality  planning tool.  When
operated as a steady-state model,  it  can  be  used  to  study the impact of
waste loads (magnitude, quality and location)  on  instream water  quality
and also can be used in conjunction with  a field  sampling program to
identify the magnitude and quality chracteristics of nonpoint source
waste loads.  By operating the model  dynamically, the  user  can study the
effects of diurnal variations in meteorological data on  water quality
(primarily dissolved oxygen and temperature) and  also  can study  diurnal
dissolved oxygen variations due to algal  growth and  respiration.  However,
the effects of dynamic forcing functions, such as headwater flows or point
loads, cannot be modeled in QUAL2E.
1.1  QUAL2E DEVELOPMENT

1.1.1  History

     QUAL2E, a new release of QUAL-II,  was  developed  jointly  by  the
National Council  for Air and Stream Improvement,  Inc.  (NCASI) and the  EPA
Center for Water Quality Modeling (CWQM),  Environmental Research Labora-
tory, Athens, GA.  It includes many modifications made  in the model since
the revisions to the SEMCOG version of  QUAL-II  (NCASI,  1980).  QUAL2E  is
intended to supersede all  prior releases  of QUAL-II.

     QUAL-II is an extension of the stream  water  quality model QUAL-I
developed by F. D. Masch and Associates and the Texas  Water Development
Board (1971) and the Texas Water Development Board  (1970).  In 1972, Water
Resources Engineers, Inc.  (WRE) under contract  to the  U.S. Environmental
Protection Agency, modified and extended  QUAL-I to  produce the first ver-
sion of QUAL-II.  Over the next 3 years,  several  different versions of
the model evolved in response to specific  user  needs.   In March  1976,  the
Southeast Michigan Council  of Governments  (SEMCOG)  contracted with WRE to
make further modifications and to combine  the best  features of the exist-
ing versions of QUAL-II into a single model. The significant modifica-
tions made in the SEMCOG version by WRE (Roesner  et al_., 1981a and b)
were:

     •  Option of English  or metric units  on input  data

     •  Option for English or metric output—choice is  independent of
        input units

     t  Option to specify channel hydraulic properties  in terms  of
        trapezoidal  channels or stage-discharge and velocity  discharge
        curves

     t  Option to use Tsivoglou's computational method  for stream
        reaeration

     •  Improvement in output display routines

     t  Improvement in steady-state temperature computation routines

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     The SEMCOG version of QUAL-II was later reviewed,  documented,  and
revised (NCASI, 1982).  The revised SEMCOG version  has  since  been main-
tained and supported by the EPA Center for Water Quality Modeling  (CWQM).
In 1983, EPA, through the CWQM, contracted with NCASI to continue the
process of modifying QUAL-II to reflect state-of-the-art water quality
modeling.  Extensive use of QUAL-II/SEMCOG had uncovered difficulties that
required corrections in the algal-nutrient-light interactions.  In  addition,
a number of modifications to the program input and  output had been  suggested
by users.  This report describes the most recent modifications made to
QUAL-II and fully documents the enhanced model, now named QUAL2E.
1.1.2  Enhancements to QUAL-II

     QUAL2E, a modified version of QUAL-II/SEMCOG,  incorporates  improvements
in eight areas:

     1.   Algal, nitrogen, phosphorus,  dissolved oxygen interactions

          •    Organic nitrogen state variable
          •    Organic phosphorus state variable
          •    Nitrification inhibition at  low DO
          •    Algal preference factor for  NH3

     2.   Algal growth rate

               Growth rate dependent upon  both NH3  and  N03  concentrations
               Algal self-shading
               Three light functions for growth rate attenuation
               Three growth rate attenuation  options
               Four diurnal averaging options for light

          Temperature

          •    Link to algal growth via solar radiation
          •    Default temperature correction factors

          Dissolved Oxygen (DO)

          •    New Standard Methods DO saturation function
          •    Traditional SOD  units (g/m2-day or g/ft2-day)
          •    Dam reaeration option

          Arbitrary non-conservative constituent

          •    First order decay
          •    Removal (settling) term
          •    Benthal source term

          Hydraulics

          t    Input factor for longitudinal  dispersion

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          t    Test  for  negative  flow  (i.e., withdrawal greater than flow)
          t    Capability for incremental outflow along reach

     7.   Downstream boundary

          •    Option for specifying downstream boundary  water quality
                 constituent concentrations

     8.   Input/output modifications

          •    Detailed summary of hydraulic calculations

          •    New coding forms

          •    Local climatological  data echo  printed

          •    Enhanced steady-state convergence

          •    Five part final  summary including components  of DO
                 deficit and plot of DO and BOD
1.1.3  Information Sources

     Major sources of information for this  revised  documentation  were:

     1.   Roesner, L. A., Giguere, P. R.  and  Evenson,  D«  E.   Computer
          Program Documentation for Stream  Quality  Modeling  (QUAL-II).
          U.S. Environmental  Protection Aghens,  GA. EPA-600/9-81-014,
          February 1981.

     2.   'JRB Associates.  Users Manual for Vermont QUAL-II  Model.
          Prepared for U.S. Environmental Protection Agency,  Washington,
          DC.  June 1983.

     3.   'National Council for Air and Stream Improvement.   A Review of
          the Mathematical Water Quality  Model QUAL-II  arid Guidance  for
          its Use. NCASI. Newg York, NY.  Technical  Bulletin  No. 391,
          Decemberr 1982.

     This documentation of QUAL2E consolidates material  from these and
other sources into a single volume.  The  basic theory  and mechanics  behind
the development of QUAL2E are described in  this  volume,  which is  intended
to supplement the users manual (NCASI, 1985). The  QUAL2E user manual con-
tains a detailed description of data requirements,  as  well as the input
coding forms and an example input/output  data file. Both reports and a
copy of the QUAL2E computer code are available from the Center for Water
Quality Modeling, U.S. Environmental Protection  Agency,  Athens, GA 30613.

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 1.1.4  Organization of this Report

     The  general program structure, specifications, and limitations of
 QUAL2E are discussed in the remainder of this chapter.  Chapter 2 des-
 cribes the conceptual and functional representation of QUAL2E as well as
 the hydraulic characteristics of the model.  The mathematical basis of
 the water quality constituent formulations is presented in Chapter 3.
 Chapter 4 presents the frame work for modeling temperature.  It is ex-
 tracted verbatim from Roesner jjt aj_., 1981.  Chapter 5 describes the com-
 putational representation of the model and the numerical solution algorithm,
 A  list of references used in this report is found in Chapter 6.

     For  the convenience of the majority of users, all of the units speci-
 fications are given in the English system of measurement.  QUAL2E, however,
 will recognize either English or metric units.


 1.2  QUAL2E COMPUTER MODEL

 1.2.1  Prototype Representation

     QUAL2E permits simulation of any branching, one-dimensional stream
 system.   The first step in modeling a system is to subdivide the stream
 system into reaches, which are stretches of stream that have uniform hy-
 draulic characteristics.  Each reach is then divided into computational
 elements  of equal length.  Thus, all reaches must consist of an integer
 number of computational  elements.

     There are seven different types of computational  elements:

          1.   Headwater element

          2.   Standard element

          3.   Element just upstream from a junction

          4.   Junction element

          5.   Last element in system

          6.   Input element

          7.   Withdrawal  element

 Headwater elements begin every tributary as well  as the main river system,
and as such, they must always  be the first  element in  a headwater reach.
A standard element is one that does not qualify as one of the remaining
six element  types.  Because incremental  flow is permitted in all  element
types, the only input permitted in a standard element  is incremental
flow.  A type 3 element  is used to designate an element on the main  stem
just upstream of a junction.   A junction element (type 4)  has a  simulated
tributary entering it.  Element type 5 identifies the  last computational

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element in the river system;  there should  be  only  one  type  5 element.
Element types 6 and 7 represent  inputs (waste loads  and  unsimulated
tributaries) and water withdrawals, respectively.  River reaches,  which
are aggregates of computational  elements,  are the  basis  of  most  data
input.  Hydraulic data, reaction rate coefficients,  initial conditions,
and incremental flows data are constant for all  computational  elements
within a reach.

1.2.2  Model Limitations

     QUAL2E has been designed to be a relatively general program;  however,
certain dimensional limitations  have been  imposed during program develop-
ment.  These limitations are:

     t    Reaches:  a maximum of 50

     •    Computational elements:  no more than 20 per reach  or a total
          of 500

     •    Headwater elements:  a maximum of 10

     •    Junction elements:   a  maximum of 9

     t    Input and withdrawal elements:  a maximum  of 50

QUAL2E incorporates features  of ANSI FORTRAN 77 that allow  these limita-
tions to be easily changed.


1.2.3  Model Structure and Subroutines

     QUAL2E is structured as  one main program supported by  49  different
subroutines.   Figure 1-1 illustrates the functional  relationships  between
the main program and the subroutines.  New state variables  can be added
or modifications to existing  relationships can be made with a  minimum of
model restructuring through the  simple addition of appropriate subroutines.

     The structural framework of QUAL2E has  been modified from prior  ver-
sions.  The large MAIN program and subroutine INDATA have been divided
into smaller groups of subroutines, each with a more narrowly  defined
task.  The new subroutines in QUAL2E include  the algal  light  functions
(GROW/LIGHT), the steady state algal output summary  (MRPT1),  the organic
nitrogen and phosphorus state variables (NH2S, PORG),  and the  line printer
plot routine (PRPLOT).  This  reorganization of QUAL2E  into  smaller pro-
grammatic units is the first  step in adapting the model  to  micro and
minicomputers, which have limited space for memory.


1.2.4  Program Language and Operating Requirements

     QUAL2E is written in ANSI FORTRAN 77  and is compatible with mainframe
and personal computer systems that support this language.  QUAL2E typically

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requires 256K bytes of memory and  uses  a  single  system  input device
(cards or disk file) and the  system's line  printer  (or  disk file) as the
output device.

     If the system's normal FORTRAN  input device unit is  not unit 5 or
the output unit is not unit 6,  then  the variables "NI"  and "NJ" in the
subroutine INDATA should be changed  to  reflect the  system's I/O unit
identifiers.

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                       2.  GENERAL MODEL FORMULATION
2.1  INTRODUCTION

     The primary objective of any stream water quality model  development
is to produce a tool  that has the capability for simulating the behavior
of the hydrologic and water quality components of a stream system.  The
development of this tool to simulate prototype behavior by applying
mathematical model on a digital computer proceeds through three general
phases (Water Resources Engineers, Inc., 1967):

          1.   Conceptual representation

          2.   Functional representation

          3.   Computational representation

     Conceptual representation involves a graphic idealization of the
prototype by description of the geometric properties that are to be
modeled and by identification of boundary conditions and interrelation-
ships between various parts of the prototype.  Usually, this process
entails dividing the prototype into discrete "elements" of a size com-
patible with the objectives that the model  must serve, defining these
elements according to some simple geometric rules, and designating the
mode by which they are connected, either physically or functionally, as
integral parts of the whole.  A part of this conceptual structuring is
the designation of those boundary conditions to be considered in the
simulation.

     Functional representation entails formulation of the physical fea-
tures, processes, and boundary conditions into sets of algebraic equations.
It involves precise definition of each variable and its relationship to
all other parameters that characterize the model  or its input-output
relationships.

     Computational representation is the process whereby the functional
model is translated into the mathematical forms and computational proce-
dures required for solution of the problem over the desired time and
space continuum.  It is concerned with development of a specific solution
technique that can be accommodated by the computer and with codification
of the technique in computer language.

     In the remainder of this section the Conceptual Representation of
QUAL2E will be described together with its general Functional  Representa-

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tion for mass transport, hydraulic characteristics,  and longitudinal
dispersion.  Chapter 3 will  discuss specific constituent reactions  and
interactions.  Chapter 4 will  develop the Functional  Representation of
stream temperature as simulated in QUAL2E.
2.2  CONCEPTUAL REPRESENTATION

     Figure II-l shows a stream reach (n)  that  Tias  been  divided  into  a
number of subreaches or computational  elements, each  of  length AX.  For
each of these computational  elements, the  hydrologic  balance  can be
written in terms of flows into the upstream face of the  element  (Q-j-i),
external  sources or withdrawals (Qxi), and the  outflow (Q-j) through the
downstream face of the element.  Similarly, a materials  balance  for any
constituent C can be written for the element.   In the materials  balance,
we consider both transport (Q'C) and dispersion (A  £L aC)  as  the movers
                                                   AX" 3x
of mass along the stream axis.  Mass can be added to  or  removed  from  the
system via external sources  and withdrawals (QxCx)-j and  added or removed
via internal  sources or sinks (Sj) such as benthic  sources and biological
transformation.  Each computational  element is  considered  to  be  completely
mixed.

     Thus, the stream can be conceptualized as  a string  of completely
mixed reactors—computational  elements—that are linked  sequentially  to
one another via the mechanisms of transport and dispersion.   Sequential
groups of these reactors can be defined as reaches  in which the  computa-
tional elements have the same hydrogeometric properties—stream  slope,
channel cross section, roughness, etc.--and biological  rate constants—
BOD decay rate, benthic source rates, algae settling  rates, etc.--so  that
the stream shown at the left of Figure 11-2 can be  conceptually  represented
by the grouping of reaches and computational elements shown on the  right
of Figure 11-2.
2.3  FUNCTIONAL REPRESENTATION

2.3.1  Mass Transport Equation

     The basic equation solved by QUAL2E  is  the one  dimensional  advection-
dispersion mass transport equation,  which is numerically  integrated  over
space and time for each water quality constituent.   This  equation  includes
the effects of advection, dispersion, dilution, constituent  reactions and
interactions, and sources and sinks.  For any constituent, C,  this equation
can be written as:
      3M     3(AyDi  3x)          3(AX u  C)                  dC
      _  =  JJLi - 1  dx   -   — • -   dx  +   (Axdx) ~  +  s     II-l
      at        8x                 3x                      dt

                                  10

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                               Computational
                                 Element i
(QC)S
                                                FLOW
                                                BALANCE
                                     MASS
                                     BALANCE
  Figure II-l.  Discretized Stream System

-------
                                                               Number
                                       Coflipwlotlonol
                                              Numbtr
Figure 11-2.   Stream Network of Computational Elements  and Reaches
                                12

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where

     M    =    mass (M)

     x    =    distance (L)

     t    =    time (T)

     C    =    concentration (M L~3)

     Ax   =    cross-sectional  area

     D|_   =    dispersion coefficient (L2 T'1)

     "u    =    mean velocity (L T-1)

     s    =    external source or sinks (M T~l)

Because M = VC, we can write

          3M   a(vc)      ac     3V
          _ = __ = v  — + C —                                   II-2a
          3t     3t        3t     3t
where
          V = AX dx = incremental volume
If we assume that the flow in the stream is steady, i.e., 8Q/3t = 0, then

the term 3V/3t = 0 and equation II-2a becomes

           3M     3C
          -- = V --                                                  II-2b
           at     at

Combining equations II-l and II-2b and rearranging,

                   3T
     8C     a(AxDL lx)     8(AX "u C)  dC   s

     at       Ax ax         AX  ax    dt   v


     The terms on the right-hand side of the equation represent, respec-
tively, dispersion, advection, constituent changes, external sources/sinks,
and dilution.  The dC term refers only to constituent changes such as
                   dt                                       3C
growth and decay, and should not be confused with the term — , the local
                                                            at
concentration gradient.  The latter term includes the effect of constituent
changes as well as dispersion, advection, sources/sinks, and dilutions.

                                    13

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     Under steady-state conditions, the local  derivative becomes  equal  to
zero; in other words:

               8C
               — = 0                                               II-4
               at

Changes that occur to individual  constituents  or particles  independent  of
advection, dispersion, and waste  inputs are  defined  by  the  term

          dC
          — = individual  constituents changes                       11-5
          dt

These changes include the  physical, chemical,  and biological  reactions  and
interactions that occur in the stream.  Examples of  these changes  are
reaeration, algal respiration and photosynthesis, and coliform die-off.
2.4  HYDRAULIC CHARACTERISTICS

     QUAL2E assumes that the stream hydraulic  regime is  steady-state;
i.e., 3Q/^t = 0, therefore, the hydrologic balance for a computa-
tional  element can be written simply as (see Figure II-l):
          (     - (QX).                                               .1-6


where (Qx). is the sum of the external  inflows and/or withdrawals  to that
element,  i
2.4.1  Discharge Coefficients

     Once equation II-6 has been solved for Q,  the other hydraulic
characteristics of the stream segments can be determined by  equations  of
the form:


          u  =  aQb                                                  II-7

          Ax =  Q/U                                                  1 1-8

and

          d  =  aQB                                                  1 1-9

where a, b, a and e are empirical  constants, and d is  the stream depth.
These constants usually can be determined from  stage-discharge  rating
curves.


                                    14

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2.4.2  Trapezoidal Cross Sections
     Alternatively, if the cross-sectional properties of the stream segment
are available as a function of the depth d, u can be obtained as a function
of discharge by the trial and error solution of Mannings equation:

              1>486      J>n    i/?
          Q = _ Ax Rx2/3 Se !/2                                  11-10
                n
where
     AX = cross-section area of the channel or canal, ft2
     Rx = mean effective hydraulic radius, ft
     n  = Manning roughness factor (usual range 0.010 to 0.10)
     Se = slope of the energy grade line (unit!ess)
     Q  = discharge, ft^/sec
The value for "u is then determined from equation II-8.

2.4.3  Longitudinal Dispersion
     Dispersion is basically a convective transport mechanism.   The term
"dispersion" is generally used for transport associated with spatially
averaged velocity variation, as opposed to "diffusion," which is reserved
for transport that is associated primarily with  time-averaged velocity
fluctuations.
     Taylor (1956) was able to derive a predictive equation for the
longitudinal dispersion coefficient, DL, in long straight pipes, as
          DL = 10 r0 u*,  ft2/sec                                    11-11
where r0 is the pipe radius and u* is the average shear velocity defined
as
          u* =  /T0/p,  ft/sec                                       11-12
where
          TO = boundary shear stress, lb/ft2, and
          p  = mass fluid density, Ib-sec2/ft4
Some investigators have attempted to apply Taylor's expression  to stream-
flow.  Such applications are only approximate, however,  because of the
difference between the geometry or velocity distributions in streamflow
and those in a pipe.
                                   15

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     Elder (1959).assumed that only the vertical  velocity gradient  was
important in streamflow and developed  an expression  analogous to  Taylor's
expression:

          DL = Kdu*                                                 11-13-

where d is the mean depth in feet of the stream.   Elder  used a  value of
5.93 for K in this equation.

     Other investigators have derived  similar expressions for D|_  and found
it to be extremely sensitive to lateral  velocity  profiles.  Elder's
expression, however, seems adequate in one-dimensional situations where
the channel is not too wide.  For very wide  channels,  Fisher (1964) has
shown that half-width rather than depth is the dominant  scale and there-
fore is important to the definition of the longitudinal  dispersion  coeffi-
cient.  Equations 11-11 and 11-13 can  be written  in  terms of the  Manning
Equation and other variables characteristic  of stream  channels.

     As an example, for steady-state open-channel  flow.

          u* = C ^~RSe~                                              11-14

where

          C  = Chezy's coefficient

          R  = the hydraulic radius

          Se = the slope of the energy grade line

Chezy's coefficient is given by:

                   Rl/6
               C = -—                                             11-15
where n is the Manning roughness coefficient  tabulated  for different  types
of channels in Table II-l.

     Se, the slope of the energy gradient,  is given by

                         u" n    9
               S  = (	O2                                   H-16
                6    1.486 R2/3

where "u is the mean velocity.  Substituting equations  11-14,  11-15 and II-
16 into equation 11-13 and letting R = d for  a wide channel yields the
expression

               DL = 3.82 K n u" d5/6                                 11-17
                                   16

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                                TABLE II-l

              VALUES OF MANNING'S "n" ROUGHNESS  COEFFICIENT

                          After Henderson  (1966)
Artificial Channels                                            n


Glass, plastic, machined metal                                0.010

Dressed timber, joints flush                                 0.011

Sawn timber, joints uneven                                   0.014

Cement plaster                                               0.011

Concrete, steel troweled                                     0.012

Concrete, timber forms, unfinished                           0.014

Untreated gunite                                          0.015-0.017

Brickwork or dressed masonry                                 0.014

Rubble set in cement                                         0.017

Earth, smooth, no weeds                                      0.020

Earth, some stones, and weeds                                0.025


Natural River Channels                                         n


Clean and straight                                        0.025-0.030

Winding with pools and shoals                             0.033-0.040

Very weedy, winding and overgrown                         0.075-0.150

Clean straight alluvial channels                              0.031 d!/6

                                                  (d  =  D-75 size  in ft.
                                                     =  diameter that 75
                                                       percent  of parti-
                                                       cles are smaller
                                                       than)
                                 17

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where

          DL = longitudinal  dispersion coefficient,  ft2/sec

          K  = dispersion constant (dimensionless)

          n  = Manning's roughness coefficient  (dimensionless)

          u  = mean velocity, ft/sec

          d  = mean depth, ft

     Typical values for dispersion coefficients,  D|_,  and  values  of  the
dispersion constant, K, cited by Fisher et  al.  (1979),  are given  in  Table
11-2.  Note that the dispersion constant,  K,  shown  in this table is one
to three orders of magnitude greater than  that  used  by Elder.


2.5  Flow Augmentation

     When the DO concentration in a stream drops  below some  required target
level, such as the state water quality standard for  DO, it may be desirable
to raise this DO concentration by augmenting  the  flow of  the stream.
According to the originators of the flow augmentation  routine in QUAL2E,
Frank D. Masch and Associates and the Texas Water Development Board
(1971), the amount of flow necessary to bring the DO concentrations up to
required standards cannot be calculated by an exact  functional relationship.
A good approximation of the  relationship is used  in  QUAL2E and has  the
following quadratic form:
              - DOT - D0min                                           11-18

and
                              DOR
          Q  = Q [ — + 0.15 ( — )2]                                  11-19
                  DOT         DOT
where,
              = dissolved xoygen concentration  required  to meet  target
                conditions, mg/L

          DOj = required target level  of DO, mg/L

        D0min = minimum DO concentration (critical  level) in  the oxygen
                sag curve, mg/L

          QR  = amount of flow augmentation  required,  ft3/sec

          QQ  = flow at the critical  point in the  oxygen sag  curve,
                ft3/sec


                                    18

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                             TABLE  II-2



EXPERIMENTAL MEASUREMENTS  OF  LONGITUDINAL DISPERSION  IN OPEN CHANNELS



                (After Table  5.3,  Fisher et al., 1979)
Channel
Chicago Ship Channel
Sacramento River
River Derwent
South Platte River
Yuma Mesa A Canal
Trapezoidal Laboratory
Channel with roughened
sides



Green-Duwarmish River
Missouri River
Copper Creek (below gage)


Clinch River


Copper Creek (above gage)
Powell River
Cinch River
Coachella Canal
Bayon Anacoco

Nooksack River
Wind/Bighorn Rivers

John Day River

Comite River
Sabine River

Yadkin River

Depth
d
(ft)
26.5
13.1
0.82
1.5
11.3
0.115
0.154
0.115
0.115
0.069
0.069
3.61
8.86
1.61
2.79
1.61
2.79
6.89
6.89
1.31
2.79
1.90
5.12
3.08
2.98
2.49
3.61
7.09
1.90
8.10
1.41
6.69
15.6
7.71
12.6
Width
W
(ft)
160
--
--
—
--
1.31
1.41
1.31
1.12
1.08
0.62
66
66
52
59
52
154
197
174
62
112
118
79
85
121
210
194
226
82
112
52
341
417
230
236
Mean
Velocity
u
(ft/sec)
0.89
1.74
1.25
2.17
2.23
0.82
1.48
1.48
1.44
1.48
1.51
—
5.09
0.89
1.97
0.85
1.05
3.08
2.62
0.52
0.49
0.69
2.33-
1.12
1.31
2.20
2.89
5.09
3.31
2.69
1.21
1.90
2.10
1.41
2.49
Shear
Velocity
u*
(ft/sec)
0.063
0.17
0.46
0.23
1.13
0.066
0.118
0.115
0.114
0.108
0.127
0.16
0.24
0.26
0.33
0.26
022
034
0.35
0.38
0.18
0.16
0.14
0.22
0.22
0.89
0.39
0.56
0.46
0.59
0.16
0.16
0.26
0.33
0.43
Dispersion
Coefficient
(ft2/Sec)
32
161
50
174
8.2
1.3
2.7
4.5
0.8
4.3
2.4
70-92
16,000
215
226
102
151
581
506
97
102
87
103
355
420
377
452
1722
151
700
151
3390
7200
1200
2800
Dispersion
Constant
K
20
74
131
510
8.6
174
150
338
205
392
270
120-160
75000
500
250
245
235
245
210
220
200
280
140
524
640
170
318
436
172
146
650
3100
1800
470
520
                                    19

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     The model  augments the stream  flow  by  first  comparing, after  steady-
state conditions have been  reached,  the  simulated DO concentration with
the prespecified target level  of DO in each reach.  If the calculated DO
is below the target level,  the program finds those upstream sources that
the user has specified for  dilution purposes, and adds water equally from
all these sources.  The DO  calculations  are then  repeated.  This process
continues until the DO target  level  is satisfied.  (NOTE:  The flow
augmentation subroutine can be used for  DO  only.)
                                    20

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             3.  CONSTITUENT REACTIONS AND INTERRELATIONSHIPS
3.1  GENERAL CONSIDERATIONS

     One of the most important considerations in determining the waste-
assimilative capacity of a stream is its ability to maintain an adequate
dissolved oxygen concentration.  Dissolved oxygen concentrations in streams
are controlled by atmospheric reaeration, photosynthesis,  plant and animal
respiration, benthal demand, biochemical oxygen demand,  nitrification,
salinity, and temperature, among other factors.

     The most accurate oxygen balance would consider all  significant fac-
tors.  The QUAL2E model  includes the major interactions  of the nutrient
cycles, algae production, benthic oxygen demand, carbonaceous oxygen
uptake, atmospheric aeration and their effect on the behavior of dissolved
oxygen.  Figure III-l illustrates the conceptualization  of these interac-
tions.  The arrows on the figure indicate the direction  of normal  system
progression in a moderately polluted environment; the directions may be
reversed in some circumstances for some constituents.  For example, under
conditions of oxygen supersaturation, which might occur  as a result of
algal photosynthesis, oxygen might be driven from solution, opposite to
the indicated direction of the flow path.

     Coliforms and the arbitrary nonconservative constituent are modeled
as nonconservative decaying constituents and do not interact with other
constituents.  The conservative constituents, of course,  neither decay  nor
interact in any way with other constituents.

     The mathematical relationships that describe the individual  reactions
and interactions are presented in the following paragraphs.


3.2  CHLOROPHYLL A (PHYTOPLANKTONIC ALGAE)

     Chlorophyll ^ is considered to be directly proportional to the
concentration of phytoplanktonic algal  biomass.  For the  purposes of this
model algal biomass is converted to chlorophyll ^ by the  simple relation-
ship:
                    Chi  1  = a0 A                                   III-l
                                    21

-------
where
          Chi ji = chlorophyll ^concentration, ug-Chl a/L
          A     = algal biomass concentration, mg-A/L
          a0    = a conversion factor, ug Chi ^/mg A
The differential equation that governs the growth and production of algae
(chlorophyll aj is formulated according to the following relationship.
                                      Atmospheric
                                      Reaeration
           Figure III-l.  Major Constituent Interactions in QUAL2E
                                    22

-------
                    dA             oj
                    — = yA - pA - — A                            III-2
                    dt              d

where

          A  =  algal biomass concentration, mg-A/L

          t  =  time, day

          v  =  the local specific growth rate of algae as defined  below,
                which is temperature dependent, day*

          p  =  the local respiration rate of algae,  which is  temperature
                dependent, dayl

          <*1 =  the local settling rate for algae,  which is temperature
                dependent, ft/day

          d  =  average depth, ft


3.2.1  Algal Respiration Rate

     In QUAL2E, the single respiration rate parameter,  p,  is used to
approximate three processes:  (a) the endogenous respiration of  algae,  (b)
the conversion of algal phosphorus to organic phosphorus,  and  (C) the con-
version of algal  nitrogen to organic nitrogen.  No  attempt is  made  to use
separate rate coefficients for these three processes, as is done in the
State of Vermont, revised Meta Systems version of QUAL-II  (JRB Associates,
1983; and Walker, 1981).


3.2.2  Algal Specific Growth Rate

     The local specific growth rate of algae, y, is known  to be  coupled
to the availability of required nutrients (nitrogen and phosphorus) and
light.  A variety of mathematical expressions for expressing multiple
nutrient-light limitations on algal  growth, rate have  been  reported  (De
Groot, 1983; Scavia and Park, 1976;  and Swartzman and Bentley, 1979).
QUAL2E has the capability of modeling the interaction among these limiting
factors in three different ways.

     Growth Rate Option 1.  Multiplicative.   The kinetic expressions used
to represent the effects of nitrogen, phosphorus, and light are  multiplied
together to determine their net effect on the local algal  growth rate.
This option has as its biological basis the  multiplicative effects  of
enzymatic processes involved in photosynthesis:
                    Umax (FD  (™)  (Fp)                           ni-3a
                                    23

-------
where

          Mmax  = maxlnium specific algal  growth rate,  day"-*-

          FL    = algal  growth limitation factor for light

          FN    = algal  growth limitation factor for nitrogen

          FP    = algal  growth limitation factor for phosphorus

This formulation is used in the SEMCOG version  of QUAL-II.

     Growth Rate Option  2.  Limiting  Nutrient.   This option  represents  the
local algal  growth rate  as limited by light  and either nitrogen  or. phosphorus,
but not both.  Thus, the nutrient/light effects are multiplicative, but the
nutrient/nutrient effects are alternate.   This  formulation mimics  Liebig's
law of the minimum:

                v = Umax (FL) Min (FN.FP)                          III-3b

Thus, the algal growth rate is controlled by the nutrient  (N or  P)  with the
smaller growth limitation factor.  This option  is used in the  State of
Vermont version of QUAL-II.

     Growth Rate Option  3.  Inverse Additive.   This option,  a  compromise
between options 1 and 2, is a modification of an intuitive form  suggested
by Scavia and Park (1976) and is mathematically analogous to the total
resistance of two resistors in series.  In this option, an effective nutrient
limitation factor is computed as the  average of the inverse  reciprocals of
the individual nitrogen  and phosphorus growth limitation factors,  i.e.,
                               1/FN + 1/FP

Thus, the algal  growth rate is controlled by a multiplicative relation
between light and nutrients, but the nutrient/nutrient interactions are
represented by an inverse average.   This option has  been  used by  Water
Resources Engineers in the application of a QUAL-II-like  model, WREDUN, to
Lake Dunlap (Brandes and Stein, no  date).

     Walker (1983) has cautioned against using the inverse-additive option
in systems where one nutrient is in excess  (say nitrogen, so that FN -»• 1.0)
and  the other is extremely  limiting  (say phosphorus, so that FP -»• 0.0).
In this case the value of the nutrient attenuation factor approaches 2
FP,  rather than FP, as expected.
                                     24

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3.2.3  Algal -Light Relationships

3.2.3.1  Light Functions

     A variety of mathematical relationships between photosynthesis and
light have been reported in the literature (Jassby and Platt,  1976;
Field and Effler, 1982).  Although they differ in mathematical  form,  the
relationships exhibit similar characteristics.  All  show an increasing
rate of photosynthesis with increasing light intensity up to a maximum or
saturation value.  At high light intensities, some of the expressions
exhibit photoinhibition, whereas others show photosynthetic activity
remaining at the maximum rate.

     QUAL2E recognizes three options for computing the algal growth limi-
tation factor for light, FL in Equations III-3a,b,c.  Light attenuation
effects on the algal growth rate may be simulated using a Monod half-
saturation method, Smith's function (Smith, 1936), or Steele's equation
(Steele, 1962).

     Light Function Option 1.  Half Saturation.   In this option, the  algal
growth limitation factor for light is defined by a Monod expression:


               FL          Z
                 Z     ......  ..                                       -
                       KL + iz
where

          FLZ = algal growth attenuation factor for light at intensity I2

          Iz  = light intensity at a given depth (z),  Btu/ft2-hr

          KL  = half saturation coefficient for light, Btu/ft2-hr

          z   = depth variable, ft


     Light Function Option 2.   -Smith's Function.  In this option, the  algal
growth limitation factor for light is formulated to include  second order
effects of light intensity:
          FL   =  _ - -—                                 III-4b
                  (KL2 + I,2)1/2

where

          K|_ = light intensity corresponding to 71% of the maximum  growth
               rate, Btu/ft2-hr

          with the other terms as defined in Equation III-4a.
                                     25

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     Light Function Option 3. .Steel's Equation.   This  option  incorporates
an exponential function to model  the effect  of photoinhibition on  the  algal
growth rate:

                   Jz           Iz
          FLZ  =  ( —)  exp (1 - «-)                                    III-4c
                   KL           KL

where

          KL = saturation light intensity at which the  algal growth  rate  is
               a maximum, Btu/ft^-hr

          with the other terms as defined in Equation III-4a.

Note:  The parameter KL, which appears in all  three light function equations
is defined differently  in each.

     All of the light functions in Equations III-4a,b,c  express the  value
of FL for an optically  thin layer.  In QUAL2E photosynthesis occurs  throughout
the depth of the water  column.  Light intensity varies  with depth  according
to Beer's law:

               Iz = I exp (-x z)                                    III-5

where

          Iz = light intensity at a  given depth (z), Btu/ft2-hr

          I  = surface  light intensity, Btu/ft2-hr

          \  = light extinction coefficient, ft~l

          z  = depth variable, ft


     When Equation III-5 is substituted into Equations  III-4a,b,c  and
integrated over the depth of flow, the depth-averaged light attenuation
factor is obtained.  The resulting expressions for the  three options are:

     Option 1:  Half Saturation

                             KL + I
          FL = (I/Ad) In [	i]                           III-6a
                          KL + Ie'xd

                    KL  = light intensity at  which  growth rate  is 50%
                         of the maximum growth rate.
                                    26

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     Option 2:  Smith's Function
                                  + (1 +
          FL =  (1/Xd) ln[ - - -      .  .]          III-6b
                         I/KLe-xd + (1 +
                    K|_ = light intensity at which growth rate is
                         of the maximum growth rate.
     Option 3:  Steel's Equation


                                        -I/K
- e
                                            ._
                                            L]
                    KL = light intensity at which growth rate is
                         equal to the maximum growth rate.

where

          FL = depth-averaged algal growth attenuation factor for light

          KL = light saturation coefficient, Btu/ft2-hr

          A  = light extinction coefficient, ft~l

          d  = depth of flow, ft

          I  = surface light intensity, Btu/ft2-hr

     The relative merits of these light functions are discussed by various
authors (Bannister, 1974; Platt et al_. , 1981; Swartzmann and Bentley, 1979;
and Field and Effler, 1982).  The half saturation method is the form used
in the SEMCOG version of QUAL-II.  Evidence shows that the use of Smith's
function is preferrable over the half saturation method if photoinhibition
effects are unimportant (Jassby and Platt, 1976).  The mathematical  forms
of Equations III-4a,b,c are compared graphically in Figure 1 1 1-2.  All
three equations have a single parameter, Ki_; however, it is defined differ-
ently in each equation.  In Figure II 1-2 trie values of KL are  selected  so
that each curve passes through a common point, namely FL = 0.5 at I  = 5
intensity units (i.e., a half saturation rate equal  to 5 light intensity
units).


3.2.3.2  Light Averaging Options

     Steady state algal simulations require computation of an  average value
of FL, the growth attenuation factor for light, over the diurnal  cycle.
There are four options in QUAL2E for computing this average.  The options

                                    27

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arise from combinations  of situations  regarding two factors:

     •    The source  of  the solar radiation data used in the computation,
          i.e., whether  it is  supplied externally by the user or calcu-
          lated internally in  the temperature heat balance.

     •    The nature  of  the averaging  process, i.e., whether hourly values
          of FL are averaged,  or a  single daylight average value of solar
          radiation  is used to estimate the mean value of FL.

     The four daily light  averaging options are defined below.  In
each case, the half saturation light function is used as an example; in
practice any of the three  light functions may be employed.

     Option 1:  FL is computed from one daylight average solar radiation
value calculated in the  steady state temperature heat balance:
FL
                = AFACT * f *
          FL
                   Ad
                                  Ia1
     1.0-r
             Saturation
                   Half Saturation
                                   1 = Half Saturation  ; KL =  5.0

                                   2 = Smith's Function ;  KL = 8.66

                                   3 = Steele's Equation ; KL = 21.55
o
     0.0
                     Light  Intensity, I  (arbitrary  units)

                  Figure 111-2.  QUAL2E Light Functions

                                    28

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          Ialg  = TFACT* Iternp                                  III-7C

where

          FL    = algae growth attenuation factor for light,  adjusted  for
                  daylight hours and averaging  method

          AFACT = a light averaging factor,  used to provide similarity
                  between calculations using a  single average value  of
                  solar radiation and computations using the  average of
                  hourly values of FL

          f     = fraction of daylight hours

          Fl_i   = growth attenuation factor_for light, based  on  daylight
                  average light intensity (Iaig

          X     = light extinction coefficient, ft'1

          d     = mean depth of stream,  ft

          KL    = half saturation coefficient for light, Btu/ft2-hr

          Taig  = daylight average, photosynthetically active, light
                  intensity, Btu/ft2-hr

          TFACT = fraction  of  solar radiation computed  in the temperature
                  heat balance that is photosynthetically active

          Itemp = daylight  average light intensity as computed in the
                  temperature  heat balance, Btu/ft2-hr

     Option 2:  FL is computed from one daylight average solar radiation
 value  supplied externally by the user.  The calculations required to ob-
 tain FL iji option 2 are the same as those for option 1, except that  the
 value  of  Ia]g is computed directly from user input of photosynthetically
 active solar radiation:

          Talg = kot/N                                           n 1-8

 where

          Itot = total daily photosynthetically active solar radiation,
                  Btu/ft2

          N    = number of daylight hours per day, hr

     Both Itot and N are supplied by the user as input information.
 Equations III-8, III-7b, and III-7a are used to compute the value of FL.
 Because the user input value of It0t is assumed to be the photosyntheti-
 cally active radiation, the factor TFACT is not used in option 2.
                                   29

-------
     Option 3:   FL is obtained  by  averaging the hourly daylight values of
FL that are computed from the hourly daylight values of solar radiation
calculated in the steady  state  temperature heat balance:


               FL    = f *  FL2                                 III-9a


                       1 N   1      KL + lalg.i
               FL2   = - I —   [— -  9'   i]             III-9b
                       Ni=l   xd   K  + Ie-*d
                  ,   =TFACT*

  where

          FL2       =    average  of N hourly values of FL, based on
                         hourly values  of  light intensity (Iaig,i)

          Ia]q -j     =    hourly value of photosynthetical ly active light
                         intensity, Btu/ft2-hr

          Hemp i    =    hourly value of light intensity as computed in
                         the steady state  temperature heat balance, Btu/
                         ft2-hr

          with other terms  are defined  in  Equations III-7a,b,c, and III-8.

 Because the average FL computed  in option 3 (and 4) is an average of
 diurnal ly varying values of FL,  the factor AFACT is not used in the
 calculations.

      Option 4:  FL is obtained by averaging the hourly daylight values of
 FL that are computed from  the hourly daylight values of solar radiation
 calculated from a single value of total daily, photosynthetical ly active,
 solar radiation and an assumed cosine  function.  The calculations required
 to obtain FL are the same  as those for option 3, except that the values of
 Jalg.i are computed from an internally specified cosine function:


                              COS 2 wi
                           (1 -- ) ,    1 = 1,N               IH-10
                                 N + 1
 As in the case of option  2,  both  I^0t and N are supplied by the user.
 Equations 111-10, III-9b, and  III-9a are then used to compute the value
 of FL.  Because the user  specified  value of Itot is assumed to be photo-
 synthetical ly active,  the factor  TFACT is not used with option 4.
                                   30

-------
     Three empirical factors—diurnal  cosine  function, AFACT, and TFACT--
are used in the formulations of the  four  light  averaging options.

     Two diurnal cosine functions were evaluated  for  use in QUAL2E :   (1)
a modified form of the one in the SEMCOG  version  of QUAL-II, and  (2) the
form used in QUAL-TX (Texas Water Development Board,  1984).  The  function
in SEMCOG was modified to produce non-zero  solar  radiation values for each
daylight hour, as given in Equation  111-10.   The  form used in QUAL-TX is:
                   Itot      *(i-l)         *1
                   - ' [COS(~— )  -  COS(-)]  .     1-l.N         IH-H
                    2N          N          N
     Equations 111-10 and III-ll were  evaluated  by comparing simulated
values of FL from modeling options  2 and  4  (i.e., in effect computing
values of AFACT).  Simulations  were performed over a range of values of
Ki_, A, d, I^ot, anc' N» as we^  as f°r  eacn  °f the three  light functions.
The values of AFACT averaged 0.92 and  0.94  for the SEMCOG and Texas
equations, respectively.  There was no compelling reason to include both
functions (the user specified the one  to  be used).  The  diurnal cosine
function used in QUAL2E, therefore, is the  modified SEMCOG version given
in Equation 111-10.

     AFACT is the adjustment factor accounting for the nonlinear averaging
inherent in computing a daily average  value of FL.  From the simulations
just described, a resonable value of AFACT  is 0.92, with a range from 0.85
to 0.98.  Zison et al_. (1978) report ah implied  value of 1.0 (Eq. 3.33),
and Walker (1983j~suggests using a  value  of 0.85.

     TFACT is  the  photosynthetically active fraction  of  total  solar  radia-
tion.  When performing algae simulations, it is  important that  the value
of.light  intensity and light saturation coefficient,  Ki, be  in  units of
photosynthetically active radiation,  PAR  (Bannister,  1974; Field and
Effler, 1983;  and Stefan et aU, 1983).  Because the  temperature heat
balance computes total radiation over  a wide spectrum, this  value must  be
adjusted  to PAR if it is to be used in the  algae simulation.  The ratio
of energy in the visible band  (PAR) to energy  in the  complete  (standard)
spectrum  is approximately 0.43 to 0.45 (Bannister,  1974  and  Stefan et
al., 1983).  TFACT is a user input  variable; thus a value to meet site
specific  conditions may be used.

     Summary of Daily Averaging Options:  The  selection  of a light averag-
ing option depends largely on the extent  to which the user wishes to
account for the diurnal variation in  light  intensity.  Options  1 and 2
use a single calculation of FL based on an  "average"  daily solar radiation
value.  Options 3 and 4 calculate hourly  values  of  FL from hourly values
of solar  radiation and then average the hourly  FL values to obtain the
daily average  value.  Options 1 and 3  use the  solar radiation from the
temperature heat balance routines.   (Thus both algae  and temperature
simulations draw on the same source for solar  radiation.)  Options 2 and


                                    31

-------
4 use the solar radiation value provided by the user for algae simulation.
Thus, either option 2 or 4 must be selected when algae are simulated and
temperature is not.  The light averaging factor (AFACT) is used to provide
similarity in FL calculations between options 1 and 2 versus options 3
and 4.  The solar radiation factor (TFACT)  specifies the fraction  of the
solar radiation computed in the heat balance, which is photosynthetically
active.  It is used only with options 1 or  3.

     In dynamic algae simulations, photosynthetically active radiation is
computed hourly using Equation III-9c unless temperature is not simulated,
in which case photosynthetically active solar radiation data must  be
supplied with the local  climatology data.


3.2.3.3  Algal Self Shading

     The light extinction coefficient, x, in Equations III-6a,b,c  is
coupled to the algal  density using the nonlinear equation


                X = XQ + A1 agA + X2(aQA)2/3                    111-12

where

      XQ = non-algal  portion of the light extinction coefficient,  ft"1

      \l = linear algal  self shading coefficient, ft"1 (ug-Chlai/L)"1

      \2 - nonlinear algal self shading coefficient, ft"1 (ug-Chla/L)~2/3

      ofj = conversion factor, ug-Chl ji /mg  A

      A  = algal biomasis concentration, mg-A/L

     Appropriate selection of the values of  X^ and  Xg allows modeling of
a variety of algal self-shading, light-extinction relationships:

     t  No algal self shading  (QUAL-II  SEMCOG)

           \l =  X2 = 0

     a  Linear algal self  shading (Meta Systems QUAL-II)

           Xi i  0  ,  X2 = 0

     •  Nonlinear  algal  self  shading  (Riley  Eq., in Zison  et^ al_., 1978)

           \! =  0.00268,  ft"1  (ug-ChU/L)"1

           X2 =  0.0165,  ft'1  (ug-ChU/L)"2/3
                                     32

-------
     or

           \l = 0.0088, m'1 (ug-ChU/L)'1

           \2 = 0.054, m'1 (ug-ChU/L)"2/3


3.2.4  Algal  Nutrient Relationships

     The algal growth limitation factors for nitrogen (FN)  and for phos-
phorus (FP) are defined by the Monod expressions:
                       Ne
               FN =       .                                      111-13
                    N  +K
and
                       P2
               FP = 	                                     111-14
                    P2 + Kp

where

          Ne = the effective local  concentration of available inorganic
               nitrogen, mg-N/L

          KN = the Michaelis-Menton half-saturation constant  for nitrogen,
               mg-N/L

          ?2 = the local concentration of dissolved phosphorus,  mg-P/L

          Kp = the Michaelis-Menton half-saturation constant  for
               phosphorus, mg-P/L

     Algae are assumed to use ammonia and/or nitrate as a source of in-
organic nitrogen.  The effective concentration of available nitrogen  is
given by:


          Ne = N! + N3                                           111-15

where

     HI = concentration of ammonia nitrogen, mg-N/L

     N3 = concentration of nitrate nitrogen, mg-N/L


     The empirical half-saturation constants for nitrogen,  K^,  and phos-
phorus, Kp, are used to adjust the algal  growth rate to account  for those


                                     33

-------
factors that can potentially 1imit algal  growth.   Each  constant  is  actually
the level  at which that particular factor limits  algal  growth  to  half  the
maximal or "saturated" rate (Zison et jil_., 1978).   Table III-3 at the  end
of this chapter lists typical  values of the half-saturation  constants  for
nitrogen and phosphorus.  If algal concentrations  are simulated  and
either nitrogen, phorphorus, or both are not simulated,  the  program
assumes that the parameter not simulated is not limiting.


3.2.5  Temperature Dependence in Algae Simulation

     The algal growth rate and death rates are temperature dependent.
They are corrected within the model, as are all  other temperature dependent
systems variables, according to the procedure explained in Section  3.10.
3.3  NITROGEN CYCLE

     In natural aerobic waters, there is a stepwise transformation from
organic nitrogen to ammonia, to nitrite, and finally to  nitrate.   The
nitrogen cycle in QUAL2E contains all four of these components,  as shown
in Figure III-l.  The incorporation of organic nitrogen  as  a  state variable,
an organic nitrogen settling term, and an algal  nitrogen uptake  preference
factor are the primary enhancements to the nitrogen cycle in  QUAL2E com-
pared to the SEMCOG version of QUAL-II.  The differential equations
governing transformations of nitrogen from one form to another are shown
below.
3.3*1  Organic Nitrogen


          dN4
          —- = 01 P A - 33 N4 - o4 N4                           111-16
          dt

where

     N4  =  concentration  of organic nitrogen, mg-N/L

     33  =  rate  constant  for hydrolysis of organic nitrogen to
           ammonia  nitrogen, temperature dependent, day*

     a}  =  fraction of  algal biomass that is nitrogen, mg-N/mg-A

     p   =  Algal  respiration rate, day-1

     A   =  algal  biomass  concentration, mg-A/L

     04  =  rate  coefficient for  organic nitrogen settling, temperature
           dependent, dayl


                                    34

-------
3.3.2  Ammonia Nitrogen
          —•» = foNA - BiNi + °3/d - FI onjiA                     111-17
          dt
where
          Fi = PNNI/(PNNI + (i ~ PN)NS)                           ni-18
     NI = the concentration of ammonia nitrogen, mg-N/L
     N3 = the concentration of nitrate nitrogen, mg-N/L
     N4 = the concentration of organic nitrogen, mg-N/L
     B! = rate constant for the biological oxidation of ammonia nitrogen,
          temperature dependent, day*
     B3 = organic nitrogen hydrolysis rate, day'1
     ai = fraction of algal biomass which is nitrogen, mg-N/mg-A
     
-------
where

     HI = the concentration of ammonia nitrogen, mg-N/L

     N2 = the concentration of nitrite nitrogen, mg-N/L

     3} = rate constant for the oxidation of ammonia  nitrogen,
          temperature dependent, dayl

     32 = fate constant for the oxidation of nitrite  nitrogen,
          temperature dependent, dayl


3.3.4  Nitrate Nitrogen


          dN3
                     - (1 - F)cqyA                             111-20
           dt
where
     F  = fraction of algal  nitrogen taken from ammonia  pool,  as
          defined in Section 3.3.2

     cq = fraction of algal  biomass that is nitrogen,  mg-N/mg-A

     y  = local specific growth rate of algae,  dayl


3.3.5  Inhibition of Nitrification at Low Dissolved Oxygen

     QUAL2E has the capability of inhibiting (retarding)  the rate of
nitrification at low values  of dissolved oxygen.  This inhibition effect
has been reported by others  (Department of Scientific  and Industrial
Research, 1964; Texas Water  Development Board,  1984).

     Nitrification rates are modified in QUAL2E by computing an inhibition
correction factor (having a  value between zero  and one)  and  then  applying
this factor to the values of the nitrification  rate coefficients, P]_,  and
32-  The nitrification  rate correction factor  is computed according to
a  first order  equation:


               CORDO =  1.0 - EXP(-KNITRF * DO)                    111-21

where

     CORDO  =  nitrification rate correction factor

     EXP    =  exponential function
                                    36

-------
     KNITRF = first order nitrification inhibition  coefficient, mg/L-1

     DO     = dissolved oxygen concentration, mg/L

     The correction factor is applied to the ammonia  and  nitrite oxida-
tion rates by:
     Ammonia:  (3i)1nhib. = CORDO * Wlnput                    HI-22

     Nitrite:  (32)inhib. = C0RDO * (^input                    in~23
     A value of 0.6 for KNITRF closely matches  the  inhibition  formula-
tion in QUAL-TX, the Texas Water Development Board  version  of  QUAL-II,
whereas, a value of 0.7 closely simulates the data  for the  Thames  Estuary
3.4  PHOSPHORUS CYCLE

     The phosphorus cycle operates like the nitrogen  cycle  in  many  respects.
Organic forms of phosphorus are generated  by the  death  of algae, which
then converts to the dissolved inorganic state, where it  is available to
algae for primary production.  Phosphorus  discharged  from sewage treatment
plants is generally in the dissolved inorganic  form and is  readily  taken up
by algae (Zison et aU , 1978).  QUAL2E revises  the SEMCOG version of QUAL-
II, which included only dissolved phosphorus, to  simulate the  interactions
between organic and dissolved phosphorus.   Below  are  the differential
equations governing transformations of phosphorus from  one  form to  another.


3.4.1  Organic Phosphorus
                   =  a? p A - 34?!  - acPi                         111-24
                dt

 where

          P! = the  concentration of  organic phosphorus, mg-P/L

          a£ = phosphorus  content of algae, mg P/mg-A

          p  = algal  respiration rate, day-1

          A  = algal  biomass  concentration, mg-A/L

          34 = organic phosphorus decay rate, temperature dependent,
               day-1

          05 = organic phorphorus settling rate, temperature dependent ,
               day-1
                                    37

-------
3.4.2  Dissolved Phosphorus


               dP2
               «— = 34?! + 02/d -  «2yA                           111-25
                dt

where

          ?2 ~ concentration of inorganic  or  dissolved  phosphorus, mg-P/L

          02 = benthos source rate  for dissolved  phosphorus, temperature
               dependent, mg-P/ft^-day

          d  = mean stream depth, ft

          p  = algal growth rate, dayl

          A  = algal biomass concentration, mg-A/L



3.5  CARBONACEOUS BOD

     The QUAL2E model assumes a first order  reaction  to describe  deoxygen-
ation of ultimate carbonaceous BOD  in the  stream.   The  BOD function as
expressed in the model also takes  into account  additional BOD  removal due
to sedimentation, scour and flocculation,  which do  not  exert an oxygen
demand (Thomas, 1948):


               dL
               — = - iqi - K3L                                  111-26
               dt

 where

           L  =  the concentration of  ultimate carbonaceous BOD, mg/L

           KI  =  deoxygenation  rate coefficient, temperature dependent, day"-*-

           l<3  =  the rate  of loss of carbonaceous BOD due to settling,
                temperature dependent, dayl

     QUAL2E simulates  ultimate BOD in the general case; however, the user
 may choose to use 5-day  BOD values for input and output.  In this case,
 the model  will  make the  necessary coversions from 5-day to ultimate BOD.
 The conversion  equation  is:


                BOD5 =  BODU (1.0 - EXP(5 * KBOD))                 111-27
                                    38

-------
where

          BOD5 = 5-day BOD, mg/L

          BODU = ultimate BOD, mg/L

          KBOD = BOD conversion rate coefficient, day1


     The SEMCOG version of QUAL-II uses a value of 0.23 day1 for KBOD.
With QUAL2E, the user may specify the appropriate value for this conver-
sion.  Note:  when modeling 5-day BOD, the conversion coefficient is
applied to all input BODs forcing functions (headwaters, incremental
flows, point loads, and the downstream boundary condition).
3.6  DISSOLVED OXYGEN

     The oxygen balance in a stream system depends on the capacity of the
stream to reaerate itself.  This capacity is a function of the advection
and diffusion processes occurring within the system and the internal
sources and sinks of oxygen.  The major sources of oxygen, in addition to
atmospheric reaeration, are the oxygen produced by photosynthesis and the
oxygen contained in the incoming flow.  The sinks of dissolved oxygen
include biochemical oxidation of carbonaceous and nitrogenous organic
matter, benthic oxygen demand and the oxygen utilized by respiration
(Zison et aj_., 1978).

     The differential equation used in QUAL2E to describe the rate of
change of oxygen is shown below.  Each term represents a major source or
sink of oxygen.

dO
— = K2(0*-0) + (03 y - 04?) A - K! L - K4/d - 05 Bj Nj. - 05  &2 N2   IH-28
dt

where

     0    =    the concentration of dissolved oxygen, mg/L

     0*   =    the saturation concentration of dissolved oxygen at the
               local  temperature and pressure, mg/L
          »
     03   =    the rate of oxygen production per unit of algal photo-
               synthesis, mg-0/mg-A

     04   =    the rate of oxygen uptake per unit of algae respired,
               mg-0/mg-A

     05   =    the rate of oxygen uptake per unit of ammonia nitrogen
               oxidation, mg.->0/mg-N

                                    39

-------
     «5   =    the rate of oxygen uptake per unit  of  nitrite  nitrogen
               oxidation,  mg-0/mg-N

     v    =    algal  growth rate, temperature dependent, day"1

     p    =    algal  respiration  rate,  temperature  dependent, day"1

     A    =    algal  biomass concentration,  mg-A/L

     L    =    concentration of ultimate carbonaceous  BOD, mg/L

     d    =    mean stream depth, ft

     K!   =    carbonaceous BOD deoxygenation rate, temperature  dependent,
               day1

     l<2   =    the reaeration rate in  accordance with  the Fickian
               diffusion analogy, temperature dependent, day-1

     l<4   =    sediment oxygen  demand  rate,  temperature dependent,
               g/ft2-day

     B!   =    ammonia oxidation  rate  coefficient,  temperature dependent,
               day1

     &2   =    nitrite oxidation  rate  coefficient,  temperature dependent,
               day1

     NI   =    ammonia nitrogen concentration, mg-N/L

     N£   =    nitrite nitrogen concentration, mg-N/L


3.6.1  Dissolved Oxygen Saturation Concentration

     The solubility of dissolved  oxygen in water decreases with  increasing
temperature, increasing dissolved solids concentration, and decreasing
atmospheric pressure  (Zison et  al_., 1978).  QUAL2E  uses a predictive
equation for the saturation "(equilibrium) concentration of dissolved
oxygen (APHA, 1985).


     InO* = -139.34410 + (1.575701 x  1Q5/T)  -  (6.642308 x 10?/T2)

            + (1.243800 x 1Q10/T3) -  (8.621949 x lO11/!^)          111-29

where:

          0* = equilibrium oxygen concentration  at 1.000 atm, mg/L

          T  = temperature (°K) = (°C+273.150) and °C is within  the
               range 0.0 to 40.0°C


                                  40

-------
     For non-standard conditions of pressure, the equilibrium concentra-
tion of dissolved oxygen is corrected by the equation 1 1 1-30:


                    (1-Pwv/P) (HP)
          Op = 0*P [..             ..]                             111-30
                      (1-Pwv) (1-*)
where
          Op  = equilibrium oxygen concentration at non-standard pressure,
                mg/L

          0*  = equilibrium oxygen concentration at 1.000 atm, mg/L

          P   = pressure (atm) and is within 0.000 to 2.000 atm

          Pwv = partial pressure of water vapor (atm), which may be
                computed from:


                          = 11.8571 - (3840. 70/T)  - 216961/T2)    111-31
and


          $  = 0.000975 - (1.426 x 10-5t) + (6.436 x 10-8t2)       111-32

where

          t  = temperature, °C

     The equations in Standard Methods (1985)  for computing dissolved oxy-
gen saturation concentrations also include corrections  for salinity  and
chloride.  Because neither salinity nor chloride is explicitly  modeled,
QUAL2E does not correct 0* for chloride or salinity.  Furthermore, the
pressure correction to 0* (Equation 1 11-30)  is made only  when temperature
is modeled, because barometric pressure data are a primary requirement of
the heat balance equations.

     The dissolved oxygen saturation concentrations computed from the
Texas and SEMCOG versions of QUAL-II are compared to those from the
Standard Methods formulations of QUAL2E in Table III-l.


3.6.2  Atmospheric Reaeration Coefficient Estimation

     The reaeration coefficient (Ko) is most often expressed as a function
of stream depth and velocity.  QUAL2E  provides eight options for estimating
or reading in K£ values, which are discussed in the sections below.   A
comparative study of reaeration prediction equation performance has  been
reported by St. John et al_. (1984).

                                   41

-------
                      TABLE III-l
COMPARISON OF DISSOLVED  OXYGEN  SATURATION CONCENTRATIONS
   (Barometric Pressure  =  1 atm,  Chloride =  0.0 mg/L,
    Equilibrium with Air Saturated  with  Water  Vapor)
Temperature,
°C
0.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
QUAL-II
SEMCOG
14.632
14.227
13.837
13.461
13.100
12.752
12.418
12.096
11.787
11.489
11.203
10.927
10.661
10.406
10.159
9.922
9.692
9.471
9.257
9.050
8.849
8.655
8.465
8.281
8.101
7.925
7.753
7.584
7.417
7.252
7.089
6.927
6.765
6.604
6.442
6.280
6.116
5.950
5.782
5.612
5.438
QUAL-TX
Texas
14.584
14.187
13.806
13.441
13.091
12.755
12.433
12.124
11.828
11.544
11.271
11.009
10.758
10.517
10.285
10.062
9.848
9.642
9.444
9.253
9.069
8.891
8.720
8.555
8.396
8.241
8.092
7.948
7.807
7.672
7.540
7.412
7.288
7.167
7.049
6.935
6.823
6.715
6.609
6.506
6.406
QUAL2E
Std. Meth.
14.621
14.217
13.830
13.461
13.108
12.771
12.448
12.139
11.843
11.560
11.288
11.027
10.777
10.537
10.306
10.084
9.870
9.665
9.467
9.276
9.093
8.915
8.744
8.578
8.418
8.264
8.114
7.969
7.828
7.691
7.559
7.430
7.305
7.183
7.065
6.949
6.837
6.727
6.620
6.515
6.413
                           42

-------
.K2 Option 1

     Option 1 allows the user to read in K2 values that have been pre-
viously selected by the modeler.  This option is useful in modeling
unusual situations such as ice cover (see Section 3.6.3).


K? Option 2

     Using data collected in field measurements of stream reaeration,
Churchill, Elmore, and Buckingham (1962) developed the following expression
for K2 at 20°C.


          K220 = 5.026 IT °-969 d-1-673 x 2.31                    111-33

where

          u  =  average velocity in the stream, ft/sec.

          d  =  average depth of the stream, ft

          K2 =  reaeration coefficient, day"1


Kp Option 3

     O'Connor and Dobbins (1958) proposed equations based on the turbulence
characteristics of a stream.  For streams displaying low velocities  and
isotropic conditions, Equation 111-34 was developed:


                 20 = (Dm ") ' |                                  m_34
                *       dl.50

For  streams with high velocities and nonisotropic conditions, the rela-
tionship  is:


                      4800^*5 S^'25
               Ko20  =    I    ..."  ,.  x  2.31                     111-35
                L          dl-25
where

          S0 = slope of the  streambed, ft/ft

          d  = mean  stream depth, ft

          ~\\  = mean  velocity, ft/day

          K2 = reaeration coefficient, day"1


                                   43

-------
and Dm is the molecular diffusion coefficient (ft2/day),  which is given
by:


          Dm = 1.91 x 103 (1.037)1"-20                            111-36


Equation 111-34 has been found to be generally applicable for most cases
and is the equation used in QUAL2E for Option 3.   Equation II1-35 can
be used to calculate l<2 outside the model  and input it directly under
Option 1.


K? Option 4
     Based on the monitoring of six streams  in England,  Owens et al.  (1964)
developed an equation for shallow,  fast moving streams.   Equation 111-37
can be used with streams that exhibit depths of 0.4 to 11.0  feet and  velo-
cities of 0.1 to 5.0 ft/sec:


          K220 = 9.4 Tj0.67/d1.85 x  2.31                          111-37

where

          u = mean velocity, ft/sec

          d = mean depth, ft


K? Option 5 .

     Thackston and Krenkel (1966) proposed the following equation based
on their investigation of several rivers in  the Tennessee Valley Authority
system.

                                      u*
                K2,0 = 10.8 (1 + F0-5) «. x 2.31                 111-38
                                      d

 where F is the Froude number, which is given by:

                        u*
                F  =  —.                                        111-39


 and u* is the shear velocity, ft/sec.:
                      	      u  n / g
                u* = / d Seg = i                                   111-40
                          6     1.49 dl-167

                                    44

-------
where
          d  =  mean depth, ft
          g  =  acceleration of gravity, ft/sec2
          Se =  slope of the energy gradient
          u  =  mean velocity, ft/sec
          n  =  Manning's coefficient

K  Option 6
     Langbien and Durum (1967) developed a formula for l<2 at 20°C:

               K220 = 3.3 "u/d1-33 x 2.31                         111-41

where
          IF = mean velocity, ft/sec
          d = mean depth,  ft

K;? Option 7
     This option computes the reaeration coefficient from a power function
of flow.  This empirical  relationship is similar to the velocity and  depth
correlations with flow used in the hydraulics section of QUAL2E, i.e.,

               K2 = aQb                                          1 1 1-42
 where
           a  =  coefficient  of flow for K2
           Q  =  flow,  ft3/sec
           b  =  exponent on  fl ow for  K2

 Kg  Option  8
     The method of Tsivoglou and Wallace  (1972)  assumes  that the  reaera-
 tion coefficient  for a reach is proportional to  the  change  in elevation
 of  the  water surface in  the  reach and inversely  proportional to the  flow
 time through the  reach.  The equation is:
                                    45

-------
                 20
               K220 = c —                                      111-43
                        tf

where

          c  =  escape coefficient, ft-1

          Ah =  change in water surface elevation in reach,  ft

          tf =  flow time within reach, days


Assuming uniform flow, the change in water surface elevation is


                Ah = Se AX                                       111-44

where

          Se = slope of the energy gradient, ft/ft

          AX = reach length, ft

and the time of passage through a reach is


                      AX
               tf =  «--                                         111-45
                      u

where

          IT = mean velocity in reach, ft/sec


Substituting the above in equation 111-43 gives


               K220 =  (3600 x 24) cSe u                            111-46


     Equation  111-46 is the form of Option 8 used in QUAL2E.  The constants
3600 and 24 convert velocity to units of feet per day.  The slope may be
input directly for computing K2 with this option, or it can be calculated
by:

                         _2
                         u   n2
               Se =  ___                                  111-47
                     (1.49)2 d4/3
                                   46

-------
where

          d = mean depth, ft

          n = Manning's coefficient


     The escape coefficient is usually treated as a variable and determined
empirically.  TenEch (1978) recommends the following guideline in deter-
mining c values, analogous to that recommended for uncalibrated stream
segments by Tsivoglou and Neal (1976):


          c = 0.054 ft-1 (at 20°C) for 15 _< Q <_ 3000 ft3/sec

          c = 0.110 ft-1 (at 20°C) for 1 ^ Q ^ 15 ft3/sec


3.6.3  Ice Cover

     Ice cover on streams during winter low flow conditions may signifi-
cantly affect reaeration.  Reaeration rates are decreased because ice
cover reduces the surface area of the air-water interface through which
reaeration occurs (TenEch, 1978).  Approaches recommended by TenEch
(1978) for estimating the extent of ice cover include:

     o    Statistical analyses of past records

     o    Steady state heat budget analysis (including  the U.S. Army
          Corps of Engineers differential  equations)

     o    Extensive field observations
     To adjust the reaeration rate for winter ice cover conditions  in  the
QUAL2E model, the calculated reaeration rate must be multiplied  by  an  "ice
cover factor" and input under Option 1.  TenEch recommends  factors  ranging
from 0.05 for complete ice cover to 1.0 for no ice cover.  Depending on
the extent of cover, reaeration values can be greatly reduced.


3.6.4  K? Default Values

     There are no default Kg values in QUAL2E.  In some versions of QUAL-
II, a default value of K£ is computed, accounting for the influences of
wind-induced turbulence and diffusion under low-velocity conditions.  In
those models, when the calculated values of Kg are less than two divided
by the depth of the reach (2/d), Kg is set equal to 2/d.  This feature
has not always proved useful, particularly when simulating the very low
reaeration rates; thus it is not included in QUAL2E.
                                   47

-------
3.6.5  Dam Reaeration
     QUAL2E has the capability of modeling oxygen input to the system from
reaeration over dams.  The following equation  described by Zison et a1.
(1978) and attributable to Gameson is used to  estimate oxygen  input from
dam reaeration.
               Da - Db = [1 -                     	] Da         111-48
                              1 + O.llab(l  + 0.046T)H
where
          Da = oxygen deficit above dam,  mg/L
          £>b = oxygen deficit below dam,  mg/L
          T  = temperature, °C
          H  = height through which water falls,  ft
          a  = empirical parameter
             = 1.25 in clear to slightly  polluted water
             = 1.0 in polluted water
          b  = empirical parameter
           . = 1.0 for weir with free fall
             = 1.3 for step weir or cascades
The factors H, a and b are input for each dam.   The model  includes  a
provision for bypassing some or all of the flow around the dams  (e.g.,
through generators).  The fraction of the total  flow that  spills over the
dam is supplied as an input variable.

 3.7  COLIFORMS
      Coliforms are used as an indicator  of pathogen contamination  in sur-
 face waters.   Expressions  for estimating coliform  concentrations are
 usually  first order decay functions, which only  take  into account  coliform
 die-off  (Zison et aj_.,  1978).  The QUAL2E  model  uses  such an expression:

                     dE
                     —  = - K5 E                                  111-49
                     dt       b
                                    48

-------
where

          E  = concentration of coliforms, colonies/100 ml

          K5 = coliform die-off rate, temperature dependent, day"1



3.8  ARBITRARY NONCONSERVATIVE CONSTITUENT

     QUAL2E has the provision for modeling an arbitrary nonconservative
constituent (ANC).  In addition to a first order decay mechanism, there
are source and sink terms in the mass balance.  The differential  equation
describing the interactions for an arbitrary nonconservative constituent
is:

               dR
               — = -K6 R - o6R + a7/d                       111-50
               dt

where

       R  =  concentration of the nonconservative constituent,  mg-ANC/L

       Kg =  decay rate for the constituent, temperature dependent, day"1

       ag =  rate coefficient for constituent settling, temperature
             dependent, day-1

       07 =  benthal source for constituent, temperature dependent,
             mg-ANC/ft2-day

       d  =  mean stream depth, ft
3.9  TEMPERATURE

     Temperature is modeled by performing a heat balance on each computa-
tional  element in the system.  The heat balance accounts for temperature
inputs and losses from the forcing functions as well  as the heat exchanged
between the water surface and the atmosphere.   The air-water heat balance
terms include long and short wave radiation, convection, and evaporation
using:


               Hn = Hsn + Han - Hb - Hc - He                     111-51

where

      Hn  = net heat flux passing the air water surface, Btu/ft^-day
                                   49

-------
      Hsn = net short wave solar radiation after losses  from
            absorption and scattering in the atmosphere  and  by
            reflection at the interface, Btu/ft2-day

      Han = net long wave atmosphere radiation  after reflection,
            Btu/ft2-day

      Hb  = outgoing long wave back radiation,  Btu/ft2-day

      HC  = convective heat flux,  Btu/ft2-day

      He  = heat loss by evaporation, excluding sensible heat loss,
            Btu/ft2-day
     In order for QUAL2E to perform the heat  balance  computations,  the
user must supply a variety of data, including the  longitude  and  latitude
of the basin, the time of year,  evaporation coefficients,  and  a  dust
attenuation coefficient.  Local  climatological  information in  the form  of
time of day, wet and dry bulb air temperatures,  atmospheric  pressure,
cloud cover and wind velocity also must be  provided.   These  data are
applied uniformly over the entire river basin.

     In the dynamic mode, local  climatological  data must be  supplied at
regular (typically 3 hour) intervals.   In this  manner the  source/sink
term for the heat balance is updated in time  to simulate the diurnal
response of the steady hydraulic system to  changing temperature  condi-
tions.

     In the steady state mode, average local  climatological  data must
be supplied by the user.  The program uses  linear  approximations for the
longwave back radiation and evaporation terms for  solution of  the  steady
state heat balance.  The reader  is referred to Chapter 4 of  this report
for a detailed treatment of the  temperature simulation.
3.10 TEMPERATURE DEPENDENCE OF RATE  COEFFICIENTS

     The temperature values computed in QUAL2E  are used to correct  the rate
coefficients in the source/sink terms for the other water quality variables.
These coefficients are input at 20°C and are  then  corrected to temperature
using a Streeter-Phelps type formulation:

               XT = X20 0  (T-20°)                              111-52

where

     Xj  = the value of the coefficient at the local temperature (T)

     X2Q = the value of the coefficient at the standard temperature
            (20°C)


                                     50

-------
                          TABLE  111-2
      DEFAULT TEMPERATURE CORRECTION, 9, VALUES FOR QUAL2E
Rate Coefficient
BOD Decay
BOD Settling
Reaeration
SOD Uptake
Organic N Decay
Organic N Settling
Ammoni a Decay
Ammonia Source
Nitrite Decay
Organic P Decay
Organic P Settling
Dissol ved P Source
Algal Growth
Algal Respiration
Algal Settling
Coli form Decay
ANC
ANC
ANC
Symbol
K!
KS
K2
K4
S3
°4
31
°3
^2
34
°5
°2
y
P
°1
KS
<6
°6
°7
Default
SEMCOG
1.047
-
1.0159
-
-
-
1.047
-
1.047
-
-
-
1.047
1.047
-
1.047
1.047
-
-
Values
QUAL2E
1.047
1.024
1.024
1.060
1.047
1.024
1.083
1.074
1.047
1.047
1.024
1.074
1.047
1.047
1.024
1.047
1.000
1.024
1.000
Note:  - = not temperature dependent in QUAL-II SEMCOG.

ANC = Arbitrary Nonconservative Constituent
                             51

-------
       0   = an empirical constant for each reaction coefficient
  The values of the temperature correction factors, 9, may be specified by
  the user.  In the absence of user specified values, the default values
  shown in Table III-2 are employed.  For comparison purposes, the 9 values
  used in the SEMCOG version of QUAL-II are also listed in Table III-2.

       If temperature is not simulated, the temperature value specified for
  the initial condition is assumed to be the temperature for the simulation.
  3.11 REACTION RATES AND PHYSICAL CONSTANTS

       The chemical and biological reations that are simulated by QUAL2E
  are represented by a complex set of equations that contain many system
  parameters; some are constant, some are spatially variable, and some are
  temperaturedependent.  Table 111-3 lists these system parameters and
  gives the usual range of values, units, and types of variation.  Kramer
  (1970), Chen and Orlob (1972), and Zison et al_. (1978) give detailed
  discussions of the basic sources of data, ranges and reliabilities of
  each of these parameters.  Final selection of the values for many of
  these system parameters or measurement of sensitive ones should be made
  during model calibration and verification.
                                TABLE 111-3
              TYPICAL RANGES FOR QUAL2E REACTION  COEFFICIENTS
             •


                                               Range  of   Variable  Temperature
Variable  Description                  Units    Values    by Reach   Dependent
ao
al
«2
a3
04
a5
Ratio of chlorophyl 1-a
to algal biomass
Fraction of algal biomass
that is Nitrogen
Fraction of algal biomass
that is Phosphorus
02 production per unit of
algal growth
02 uptake per unit of
algae respired
02 uptake per unit of
NH3 oxidation
ug-Chl a
mg A
mg-N
mg A
mg-P
mg A
mg-Q
mg A
mg-0
mg A
mg-0
mg N
10-100
0.07-0.09
0.01-0.02
1.4-1.8
1.6-2.3
3.0-4.0
No
No
No
No
No
No
No
No
No
No
No
No
                                      52

-------
              TABLE III-3 (cont'd)
TYPICAL RANGES FOR QUAL2E REACTION  COEFFICIENTS
Variable Description
«6
^max
P
KL
KN
Kp
x°
M
*2
PN
°i
CT2
°3
04
°5

Q£ uptake per unit of
N02 oxidation
Maximum algal growth rate
Algal respiration rate
Michaelis-Menton half-
saturation constant
for light (Option 1)
Michaelis-Mention half-
saturation constant
for nitrogen
Michaelis-Menton half-
saturation constant
for phosphorus
Non-algal light extinc-
tion coefficient
Linear algal self-shading
coefficient
Nonlinear algal self-
shading coefficient
Algal preference factor
for ammonia
Algal settling rate
Benthos source rate for
dissolved phosphorus
Benthos source rate for
ammonia nitrogen
Organic nitrogen
settling rate
Organic phosphorus
settling rate

Units
mg-0
mg N
day'1
day1
Btu/ft2-
min
mg-N/L
mg-P/L
ft'1
I/ft
ug Chl-a/L
I/ft
Range of Variable Temperature
Values by Reach Dependent
1.0-1.14
1.0-3.0
0.05-0.5
0.02-0.10
0.01-0.20
0.01-0.05
Variable
0.002-0.02
0.0165
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
(ug Chl-a/L)Z/J (Riley)
-
ft/day
mg-P
ft^-day
mg-0
ft^-day
day'1
day'1
53
0.0-1.0
0.5-6.0
Variable
Variable
0.001-0.10
0.001-0.10

No
Yes
Yes
Yes
Yes
Yes

No
Yes
Yes
Yes
Yes
Yes


-------
              TABLE III-3 (cont'd)
TYPICAL RANGES FOR QUAL2E REACTION  COEFFICIENTS
Variabl
°6
°7
*1
K2
K3
K4
K5
<6
Pi
e Description
Arbitrary non-conserva-
tive settling rate
Benthal source rate for
arbitrary non-conserva-
tive settling rate
Carbonaceous deoxygenera-
tion rate constant
Reaeration rate constant
Rate of loss of BOD due
to settling
Benthic oxygen uptake
Col i form die-off rate
Arbitrary non-conserva-
tive decay coefficient
Rate constant for the
Units
day'1
mq-ANC
ft2-day
day"1
day'1
day'1
m-0
ft* -day
day -1
day'1
day -1
Range of Variable Temperature
Values by Reach Dependent
Variable
Variable
0.02-3.4
0.0-100
-0.36-0.36
Variable
0.05-4.0
Variable
0.10-1.00
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
B4
  biological  oxidation
  of NH3 to N02

Rate constant for the
  biological  oxidation
  of N02 to N03

Rate constant for the
  hydrolysis of organic-
  N to ammonia

Rate constant for the
  decay of organic-P
  to dissolved-P
                      day
                         '1
        0.20-2.0
                                                      Yes
day-1   0.02-0.4     Yes
day1   0.01-0.7     Yes
Yes
                                                     Yes
                                                     Yes
                       54

-------
               4.  FUNCTIONAL REPRESENTATION OF TEMPERATURE
4.1  BASIC TEMPERATURE EQUATION
     The basic mass transport equation for QUAL2E was given in Section
II as (see equation II-3):
                             _§C
               8C     3(AXDL 3x)     3(AX u C)   dC   s
               --  =  ,            -        „   , + - + -           iv-l
               3t        Ax  3x         Ax  3x     dt   V

In temperature modeling, C is taken as the concentration of heat (HL~3)
and can be equated to temperature through the relationship
                         C = p c (T - T0)                         IV-2

where
          p  =  the density of water (M L~3)
          c  =  the heat capacity of water (HM~1 D-1)
          T  =  the water temperature
          T0 =  an arbitrary base temperature
          M  =  mass
          H  =  heat energy flux
          D  =  degrees
The parameters p and c can be considered constant for practical  purposes.
Also, the internal  heat generation dC, which results from viscous dissi-
                                   dt
pation of energy and boundary friction, is generally small  enough to be
                                    55

-------
considered negligible.  Thus setting ^C = 0 in equation IV-1  and substituting
                                     dt
equation IV-2 for C gives us (after some simplification):

                             *L
               3T     3(AXDL 3x)     9(AX u T)   1   s
               —  =  _—  -         'i   + —  -            IV-3
               3t        Ax 9x         Ax 3x     pC  V


     The source term s/V (with units of HL-3T-1) accounts  for all  heat
transferred across the system boundaries, i.e., heat transferred across
the air-water interface and heat conducted across mud-water interface.
Heat transfer across the mud-water interface is generally  insignificant;
hence, s/V takes on the identify of the net rate of heat input per unit
volume of stream through the air-water interface.

     It is most convenient to represent the interfacial heat  transfer
rate as a flux (HN) having units of HL'M"1.  For a stream element of
length dx and mean surface width W, H|\j is related to s/V as follows.

     The total rate of heat input across the air-water interface is H^ dx
W. _This heat isjdistributed uniformly throughout the underlying volume
of Ax dx, where Ax is the mean cross-sectional area of the element.  Thus
the rate of heat gain per unit volume of water, s/V, is computed as:


               s     s     HN (Wdx)   HN

               V   Ax dx    Ax dx     d


where d = AX/W is the hydraulic depth of the stream.  Substituting equation
IV-4 into equation IV-3 gives the generalized form of the temperature
equation:


                             JL
               9T     3(AXD[_ 3x)     8(AX u T)   HN
               _  =  ,            _           | + «_.„              iv-5
               3t        Ax 3x         Ax  3x     pcd
4.2  DEFINITION OF HN

     Heat is transferred across the air-water interface of a surface
water body by three difference processes:  radiation exchange, evaporation,
and conduction.  The individual heat terms associated with these processes
are shown in Figure IV-1 and are defined in Table IV-1 with the typical
ranges of their magnitudes in northern latitudes also listed.
                                     56

-------
     The expression  that  results from the sumnation of these various
energy fluxes is:
where
          HN

          H
           sn
               HN  =  Hsn + Han - (Hb ± Hc + He)
                                              IV-6
net energy flux passing the air-water interface,
Btu/ft2-day

net short-wave solar radiation flux  passing
through the interface after losses due to
absorption and scattering in the  atmosphere
and by reflection at the interface,  Btu/ft2-day

net long-wave atmospheric radiation  flux passing
through the interface after reflection, Btu/ft^-day

outgoing long-wave back radiation flux, Btu/ft2-day

convective energy flux passing back  and forth
between the interface and the atmosphere,  Btu/ft2-day

energy loss by evaporation, Btu/ft2-day
 These mechanisms by which heat is exchanged between the water surface  and
 the atmosphere are fairly well understood and are adequately documented  in
 the literature by Edinger and Geyer (1965).  The functional  representation
            H.
        Hb
         4
                                     4
                         nc
                                                     AIR-WATER
                                                    'INTERFACE
            Figure  IV-1.
     Heat Transfer Terms Associated with
     Interfacial Heat Transfer
                                   57

-------
                                TABLE IV-1
                    DEFINITION OF HEAT TRANSFER  TERMS
                         ILLUSTRATED IN FIGURE  IV-1
               Heat Term                     Units           Magnitude
                                                         (BTU/ft2-dayl)


Hs   =    total incoming solar or            HL^T'1          400-2800
          short-wave radiation


H$r  =    reflected short-wave radiation     HL"2T'1           40-200


Ha   =    total incoming atmospheric         HL'2?"1         2400-3200
          ratiation


Har  =    reflected atmospheric radiation    HL"2^1           70-120


Hb   =    back radiation from the water      HL"2^1         2400-3600
          surface


He   =    heat loss by evaporation           HL^T"1          150-3000

     •
Hc   =    heat loss by conduction to         HL^r1         -320  to +400
          atmosphere
of these terms has been defined  by  Water Resources  Engineers,  Inc.  (1967).
The formulations reported here were extracted  from  that more detailed work
by Frank D. Masch and Associates and the Texas Water  Development  Board
(1971).
4.3  NET SHORT-WAVE SOLAR RADIATION

     The net incoming solar radiation is  short-wave  radiation which  passes
directly from the sun to the earth's  surface.   Its magnitude depends on:
the altitude of the sun, which varies daily  as  well  as  seasonally  for  a
fixed location on the earth; -the dampening  effect of scattering and
absorption in the atmosphere due to cloud cover,  and the  reflection  from
the water surface.

                                     58

-------
     The net amount of solar radiation  which  reaches  the  surface  of  the
reach may be represented functionally on  an hourly  basis  by:
                    Hsn  = ^Jo  jj^   (^-  RSJ    (1  -  0.6JSC)       IV-7
                           (i)  (ii)    (111)           (W)~~
 where
           Hsp  =    net short-wave solar radiation flux, Btu/ft2-hr
           H0   =    amount of radiation flux reaching the earth's
                     atmosphere, Btu/ft2-hr
           at   =    atmospheric transmission term
           Rs   =    Albedo or reflection coefficient
           CL   =    cloudiness as a fraction of sky covered
      It is appropriate for purposes of this discussion to identify and
 treat separately the four components in equation IV-7 as (i)  extra-
 terrestrial solar radiation, (ii)  radiation scattering and absorption,
 (iii) reflectivity, and (iv) cloudiness.
 4.3.1  Extraterrestrial Radiation
      The short-wave solar radiation flux that strikes the earth's  outer
 atmosphere over a given period of time is given by Water Resources
 Engineers, Inc. (1967) as:
                     Hsc       ^
                HQ = ,—- { sin — sin 6 (te -
                     r2        180
                     12     ir<|>              Trte
                     M. cos —* cos 6 [sin (—••)  -  sin (•—•)]}  r     IV-8
                     IT      180             12           12
 where
           Hsc  =    solar constant =  438.0  Btu/ft2-hr
           r    =    normalized radius of the earth's  orbit
           $    =    latitude of the site, degrees
                                   59

-------
           >te   =     hour  angles corresponding to the beginning and end,
                     respectively, of any time interval between sunrise
                     and sunset

           r     =     a  correction factor for diurnal exposure to
                     radiation flux

     Listed below are several  parameters  in equation  IV-8 requiring
further definition as described  by  Water  Resources  Engineers,  Inc.  (1967).
a.  Relative Earth-Sun Distance--

                                         2*
               r    =    1.0 + 0.17 cos [— (186-Dy)]               IV-9
                                         365

where Dy is the number of the day of the year (beginning January 1)


b.  Declination--

                         23.45        2ir
               6    =    —1— TT cos [— (173-Dy)]                  IV-10
                          180         365


c,  Hour Angles--
           •

               tb   =    STb - Ats + ET - 12                         IV-11


and

               te   =    STe - Ats + ET - 12                         IV-12


where ST|j, STe are the standard times at the beginning and end of the
time interval selected
          ET   =    an expression for time from a solar ephemeris that
                    represents the difference in hours between "true
                    solar time" and that computed on the basis of a yearly
                    average.  It is given for each day of the year, Dy, by
                                   60

-------
                                                    2lT
                  ET   =    0.000121  -  0.12319 sin [— (Dy-1)  -  0.0714]
                                                    365
                                         4*
                            0.16549 sin [---  (Dy-1)  + 0.3088]       IV-13
                                         365

        Ats  =    difference between  standard  and  local  civil time
                  in hours as determined from:

                            e
                  Ats  =    ~ (LSm - Mm)                          IV-14
                            J. 
-------
in which a" is the mean atmospheric transmission coefficient after
scattering and absorption, given by:


               a" = exp { - [0.465 + 0.0408 Pwc]

                    [0.179 + 0.421 exp (-0.721 9am)] 9am}           IV-20


where 9am is the optical  air mass given by the expression:

                                exp (-Z/2531)
               9am =	             IV-21
                     sin a + 0.15 (180a + 3.885)-l-253
                                    IT

in which

          Z    =    elevation of the site in feet

          a    =    sun's altitude in radians, given by:

                                          1T(j)              TT(J>
                         a = arc sin [sin *•— sin 6 + cos <^—
                                          180             180

                                       Trt
                             cos <5 cos --]                          IV-22
                                       12

    •
in which t is the hour angle, described by an equation similar to equation
IV-11 and IV-12.

     Pwc in equation IV-20 is the mean daily precipitable water content
in the atmosphere, given by the expression:


               PWC  =    0.00614 exp (0.0489Td)                     IV-23


where T^ is the dewpoint in °F, which can be obtained from the expression:


               Td   =    In [(ea + 0.0837)/0.1001]/0.03             IV-24


where ea is the water vapor pressure of the air.

     The mean atmospheric coefficient, a1, can also be represented by an
equation of the form of equation IV-20 as:
                                    62

-------
               a1   =    exp { - [0.465 + 0.0408 Pwc]

                         [0.129 + 0.171 exp (-0.880 Oam)] 0am}     IV-25


     Dust attenuation of the solar radiation flux, which is represented
in equation IV-19 by the quantity d, varies with optical air mass, season
of the year, and geographic location.  Water Resources Engineers, Inc.
(1967) gives a range of 0-0.13 for several locations.


4.3.3  Cloudiness

     The dampening effect on the solar radiation flux is given by Water
Resources Engineers, Inc. (1967) as


               Cs   =    1.0 - 0.65 c£                            IV-26


where  C|_  is the  decimal  fraction of the sky covered.  Water Resources
Engineers,  Inc.  (1967) reports that equation IV-26 gives satisfactory
results except for heavy overcast conditions, i.e., when C|_ approaches
1.0.
 4.3.4   Reflectivity

     The  reflection coefficient, Rs, can be approximately computed as a
 function  of the  solar altitude, a, by Anderson's (1954) empirical formula:


               Rs   =    AaB                                       IV-27


 where a is in  degrees, and A and B are functions of cloudiness, CL.
 Values  for A and B given by Anderson (1954) are shown in Table IV-2.
                                TABLE IV-2
                EMPIRICAL COEFFICIENTS FOR DETERMINING R<
                          After Anderson (1954)
Cloudiness
CL
0
Clear
0.1 - 0.5
Scattered
0.6 - 0.9
Broken
1.0
Overcast
Coefficients    A       BAB       AB       AB

               1.18   -0.77   2.20  -0.97     0.95   -0.75     0.35   -0.45
                                    63

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4.4  LONG-WAVE ATMOSPHERIC RADIATION

     The long-wave radiation emitted by the atmosphere varies  directly
with the moisture content of the atmosphere.   Although it  is primarily
dependent on air temperature and humidity,  it can also be  affected by
ozone, carbon dioxide, and possibly other materials  in the atmosphere.
Anderson (1954) indicated that the amount of atmospheric radiation is
also significantly affected by cloud height.   The amount of long-wave
atmospheric radiation that is reflected is  approximately a constant
fraction of the incoming radiation, found by Anderson (1954) to  be
approximately 0.03.

      The net  atmospheric radiation flux  can  be expressed  as:


           Han =  [2.89 x 10"6]  o (Tfl +  460)6  (1.0 + 0.17Cf)(l-RL)     IV-28

 where

           Han =  net  long-wave  atmospheric  radiation  flux,  Btu/ft^-hr

           o   =  Stefan-Boltzman constant,  1.73 x 10-9  Btu/ft2/hr/
                 °Rankine4

           Ta   =  air  temperature at a level 6  feet above the water
                 surface, °F

           RL   =  reflectivity of the water  surface for atmospheric
                 radiation =  0.03

           C[   =  cloudiness,  fraction of  cloud cover
 4.5  WATER SURFACE BACK  RADIATION

      The third source of radiation  transfer through the air-water  interface
 is long-wave  back  radiation  from the water surface, H^, which represents
 a loss  of heat from the  water.   It  can  be seen from Table  IV-1 that back
 radiation accounts for a substantial portion  of  the heat loss from a  body
 of water.  This loss is  expressed by the Stefan-Boltzman Fourth  Power
 Radiation Law for  a blackbody  as:


                Hb    =    0.97  a (Ts + 460)4                        IV-29


 where

           Hb    =    water surface back  radiation flux, Btu/ft2-hr

           Ts    =    water surface temperature, °F

                                   64

-------
     Equation IV-29 can be linearized over a given temperature range as


               Hb   =    
-------
          W    =    wind speed,  in  mph, measured 6 feet above the water
                    surface

          es   =    saturation  vapor pressure  of the  air, in. of Hg,
                    at the temperature of the  water surface, as
                    given by

          es   =    0.1001 exp  (0.03 Ts) -  0.0837

and

          ea   =    water vapor pressure, in.  of Hg,  at a height of
                    6 feet above the water  surface, given as

          ea   =    ewb - 0.000367  Pa (Ta - Twb)


                                                                 IV-34
                             1571

where

                    saturation  vapor pressure, in. of Hg, at the
                    wet bulb temperature from  the expression
          ewb  =    0.1001 exp (0.03 T^)  -  0.0837                 IV-35

          Pa   =    local  barometric pressure,  in. of Hg

          Twb  =    wet bulb air temperature,  °F

          Ta   =    dry bulb air temperature,  °F


The literature contains a wide range of values  for the  evaporation
constants a and b.  Roesner (1969)  reports that a good  average  value of  a
would be 6.8 x 10-4 ft/hr-in.  of Hg, while b would best be  represented by
2.7 x l.O'4 ft/hr-in. of Hg.-mph.

     To linearize the variation of  evaporation  rate  with  surface  water
temperature Ts, equation IV-34 is approximated  over  5°F intervals as:


               es   =    01 + BI Ts                               IV-36


Sets of 01, Bi are specified for twenty-one  5°F intervals between 35°F
and 135°F.  The linearized evaporation expression  is used in  the  steady-
state temperature solution.
                                    66

-------
     The sensible evaporative heat loss can be expressed simply as:


               Hv   =    c y E (Ts - T0)                          IV-37


where

          c    =    heat capacity of water = 1 Btu/lb-°F

          T0   =    reference temperature, °F


Sensible heat loss is very small compared to the other heat loss components
in the energy budget and thus is not included in the QUAL2E temperature
computation.



4.7  CONDUCTION

     Heat that is transferred between the water and the atmosphere due to
a temperature difference between the two phases is normally called
conduction.  .Using the fact that transfer by conduction is a function of
the same variables as evaporation, it is possible to arrive at a propor-
tionality between heat conduction and heat loss by evaporation.  This
proportionality, known as Bowen's ratio, is expressed as:


               B    =    - = CB [ $    «] —~                   IV-38
                         He       es - ea  29.92


where Cg is a coefficient = 0.01.

     By using Bowen's ratio, the rate of heat loss to the atmosphere by
heat conduction, Hc, can be defined as:

                                             Pa
               Hc   =    y HL (a+bW) (0.01 ——) (Ts - Ta)         IV-39
                                           .29.92


For practical purposes, the ratio (Pa/29.92) can be taken as unity.
                                   67

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                      5.  COMPUTATIONAL REPRESENTATION
5.1  PROTOTYPE REPRESENTATION

     To expand upon the basic conceptual  representation presented in Sec-
tions 1 and 2, QUAL2E permits any branching,  one-dimensional  stream system
to be simulated.  The first step involved in  approximating the  prototype
is to subdivide the stream system into reaches,  which  are  stretches of
stream that have uniform hydraulic characteristics.   Each  reach is then
divided into computational  elements of equal  length  so that all  computa-
tional elements in all reaches are the same length.   Thus, all  reaches
must consist of an integer number of computational elements.

     There are seven different types of computational  elements:

     1.   Headwater element

     2.   Standard element

     3.   Element just upstream from a junction

     4.   Junction element

     5.   Last element in system

     6.   Input element

     7.   Withdrawal  element

Headwater elements begin every tributary as well as  the main river system,
and as such, they must always be the first element  in a headwater reach.
A standard element is one that does not qualify  as  one of  the remaining
six element types.  Because incremental flow is  permitted  in all  element
types, the only input permitted in a standard element is incremental
flow.  A type 3 element is used to designate  an  element on the  mainstem
that is just upstream of a junction.  A junction element (type  4), has a
simulated tributary entering it.  Element type 5 identifies the last com-
putational element in the river system (downstream  boundary); there
should be only one element type 5.  Element types 6  and 7  represent
elements which have inputs (waste loads and unsimulated tributaries) and
water withdrawals, respectively.
                                    68

-------
     River reaches, which are aggregates of computational  elements, are
the basis of most data input.  Hydraulic data, reaction rate coefficients,
initial conditions, and incremental flow data are constant for all  computa-
tional elements within a reach.
5.2  FORCING FUNCTIONS

     Forcing functions are the user specified inputs that drive the
system being modeled.  These inputs are specified in terms of flow, water
quality characteristics, and local climatology.  QUAL2E accommodates four
types of hydraulic and mass load forcing functions in addition to local
climatological factors—headwater inputs, point sources or withdrawals,
incremental inflow/outflow along a reach, and the (optional) downstream
boundary concentration.

     1.  Headwater Inputs - Headwater inputs are typically the upstream
boundary conditions at the beginning of the system.  They are the condi-
tions required to generate the solution of the mass balance equations for
the first computational element in each headwater reach.  Headwaters are
also the source of water for flow augmentation.

     2.  Point Sources and/or Withdrawals - These loads are used to repre-
sent point source discharges into the system (i.e., sewage and industrial
waste, or storm water runoff) and losses from the system resulting from
diversions.  In QUAL2E point source discharges may represent either raw
or treated waste loads.  If raw waste loads are used, the effect of
treatment can be simulated by applying a specific fractional  removal  for
carbonaceous BOD to each point source load.

     3.  Incremental  Inflow - QUAL2E has the capability to handle flow
uniformly added or removed along a reach.  The total flow increment along
a reach is apportioned equally to all  computational  elements in the
reach.  This feature can be used to simulate the effects of non-point
source inputs to the system, or the effect of loss of stream flow to the
groundwater.

     4.  Downstream Boundary Concentration (optional) - QUAL2E has the
capability of incorporating known downstream boundary concentrations of
the water quality constituents into the solution algorithm.   This feature
is useful  in modeling systems with large dispersion in the lower reaches
(e.g., estuaries). -When downstream boundary concentrations are supplied,
the solution generated by QUAL2E will  be constrained by this boundary
condition.  If the concentrations are not provided, the constituent
concentrations in the most downstream element will  be computed in the
normal fashion using the zero gradient assumption (see Section 5.4.3).

     Local  climatological  data are required for the dynamic  simulation of
algae and temperature.  The temperature simulation uses a heat balance
across the air-water interface and thus requires values of wet and dry
bulb air temperatures, atmospheric pressure, wind velocity,  and cloud


                                   69

-------
cover.  The algal  simulation requires values of net solar radiation.
These climatological data must be input at regular time intervals over
the course of the simulation.  For modeling steady-state temperature  and
algae, average daily local climatological  data are required.   All clima-
data are applied uniformly over the entire river basin.
5.3  MODEL LIMITATIONS

     QUAL2E has been developed to be a relatively general  program;  however,
certain dimensional  limitations have been imposed upon it  during program
development.  These limitations are as follows:

     Reaches:  a maximum of 50

     Computational elements:  no more than 20 per reach or 500 in total

     Headwater elements:  a maximum of 10

     Junction elements:  a maximum of 9

     Input and withdrawal elements:  a maximum of 50 in total

(Note:  These limitations may be modified, if necessary, by the user by
altering the PARAMETER statement specifications  in the common  blocks of
the program.)

     QUAL2E can be used to simulate any combination of the following
parameters or groups of parameters.

     1.   Conservative minerals (up. to three at  a time)

     2.   Temperature

     3.   BOD

     4.   Chlorophyll a^

     5.   Phosphorus cycle (organic and dissolved)

     6.   Nitrogen cycle (organic, ammonia, nitrite, and nitrate)

     7.   Dissolved oxygen

     8.   Coliforms

     9.   An arbitrary nonconservative constituent

All parameters can be simulated under either steady-state  or dynamic
conditions.  If either the phosphorus cycle or the nitrogen cycle are
not being simulated, the model presumes they will not limit algal growth.


                                    70

-------
5.4  NUMERICAL SOLUTION TECHNIQUE

     At each time step and for each  constituent, Equation II-3 can be
written I times, once for each of  the  I  computational elements in the
network,  because it is not possible to  obtain  analytical solutions to
these equations under most prototype situations, a finite difference
method is used—more specifically, the classical implicit backward differ-
ence method (Arden and Astill, 1970; Smith,  1966; and Stone and Brian,
1963).

     The general basis of a finite difference scheme is to find the value
of a variable (e.g., constituent concentration) as a function of space at
a time step n+1 when its spatial distribution at the nth time step is
known.  Time step zero corresponds to  the  initial condition.  Backward
difference or implicit schemes are characterized by the fact that all
spatial derivatives (9/3x) are approximated  in  difference form at
time step n+1.
5.4.1  Formulation of the Finite  Difference  Scheme

     The finite difference scheme is  formulated by considering the consti-
tuent concentration,  C, at four points  in the mnemonic scheme as shown in
Figure V-l.

     Three points are required at time  n+1 to approximate the spatial
derivatives.  The temporal  derivative is approximated at distance step i.

     Equation II-3 can be written in  finite  difference form in two steps.
First, the advection  and diffusion terms are differentiated once with
respect to x, giving:
          DOWNSTREAM4-
UPSTREAM
element i-H
element i
                            I
                           •o-
                                                    N  :  t -
                                                             At
                           i
                Figure  V-l.  Classical Implicit Nodal Scheme

                                   71

-------
           (ADL -) -  (ADL ~)
      aci        a* i        3x i_!      (A u
      at
          + __ + _                                                v-i
            dt    Vi
where
          Vi = Ai A Xi

Secondly, expressing the spatial derivative of the diffusion terms in finite
difference and thence the time derivative of C in finite difference, there
results:
                                 T] c"+} - [(ADL)J] cf *
               At                      V-j  A Xj
                                                                 s
                                              .       rn+1
                                              )  + ri  Cn+1 + Pi + —
In equation V-2, the term dC/dt is expressed as:

               dC,-
               ^ = r. cn+l + D.
               «—•   r1 v,1   T PI
                dt
where
          ^ = rate constant
          Pi = internal constituent sources and sinks (e.g., nutrient
               loss from algal growth, benthos sources, etc.)

Note  that  the  dC/dt for every  constituent modeled by QUAL2E can be expressed
in  this  form.
                                    72

-------
     If equation V-2 is rearranged in terms  of the coefficients of
Cn+1, and C^}, we obtain the equation:
                    b1
where
                        At     Qi_i At
                                                          V-3
        = 1.0 + [(ADL)i +
                                At        At
                                     + Qi - - r. At
     Ci = -[(ADL)i
7.  = r.n .
*i    ui
                     At
                    At
                            At
 The values of ai,  b.j,  Ci,  and 1^  are  all  known  at  time  n,  and  the C^
 terms are the unknowns at  time step n+1.

      In  the case of  a  junction element  with  a tributary upstream element,
 the basic equation becomes:
     a1  Cnt} + bi  Cf1  + Ci  Cn!j  +  dj  Cn+1  =  Zi
                                                          V-4
 where
                  At
                                  At
      j     =  the element  upstream  of junction element  i

      Cn+1  =  concentration  of constituent in element j at time n+1


      It  can  be  seen  that the dj term  is analogous to  the a^ term.  Both
 terms account for  mass  inputs  from upstream due to dispersion and advec-
 tion.
                                   73

-------
                                    oU-j
     Under steady-state conditions, •—• = 0 in equation  V-l.   Working
                                    3t
through the finite difference approximations and  rearranging  terms  as
before, the steady-state version of equation V-3  is  derived:
                   n+l + h. rO+l + c-  rn+l  =
                         Di h    + ci  H-l
where
                (ADL)l
               Si
     Note that equation V-5 is the same as equation V-3,  with  three
changes:

          o    At = 1.0

          o    the constant 1.0 in b-j  = 0.0

          o    the initial  concentration Cn in Z^  = 0.0


5.4.2  Method of Solution

     Equations V-3 and V-5 each represent a set of simultaneous  linear
equations whose solution provides the  values of Cn+1 for  all i's.
Expressed in matrix form, this set of  equations appears as:
                                    74

-------
bl cl
3o bo C2
a3 b3 C3
• • •

ai bi ci
• • •
al-l bl-l CI-1
al bl




X




V "
c||+1
cfi
•

T1
•
'PI
cf1





=



zl
Z2
Z3
•

Zi
•
ZM
zi
                                                                       V-6
The left matrix is  a tri-diagonal  matrix.   An  efficient  method  that  readily
lends itself to a computer solution of such a  set of equations  is:
Diyide through the first equation in V-6 by b^ to obtain:

          Cn+1 + Wx
                                                                  V-7
 where
                      and  G^  =  I\l\>i.
      Combine  the  expression  for  b^  (see V-3) and the  second equation in
 V-6 to eliminate  ag and the  result  is:
             +1
                      W2  c§+  = G2
                                                             V-8
where
          W2 =
                       wl
                       and 62 =
                                  - 82
     Combine equation V-8 and the third equation in V-6 to eliminate 33
and the result is:
                                   75

-------
               Cf 1 + W3 Cf 1 = G3                              V-9

where
                   c3               Z3 - a3 G2
          W3 = i            and 63 = •      .   •
               b3 - a3 W2           b3 - a3 W2

     Proceed through the equations, eliminating a-j  and storing the values
of W-j and G-j given by:
                         ci
               Wi = .              i  = 2, 3,  .... I                V-10
                    bi - ai  Wi_i

and
                    zi  - ai
               Gi = - ,  i  = 2,  3,  .... I                 V-ll
                    bi - ^i  wi-l

     The last equation is solved for Cn+1 by


               Cn+1 = Gj                                            V-12


     Solve for Cn^J, Cn+J, . .  . , C}+1  by back substitution.


               Cn+1 = Gi - W,. Cnlj,  i =  1-1, 1-2,  . .  .  ,  1          V-13


 5.4.3  Boundary Conditions

      In most situations of interest, transport is unidirectional  in nature,
 i.e., there is no significant  transport  upstream.  Therefore,  the concen-
 tration at some point just  upstream from the beginning  or end  of the
 stream reach of interest can be used as  the boundary  condition.


 5.4.3.1  Upstream Boundary  (Headwater Elements)

      For headwater elements there is no  upstream, i-1,  element.   Thus,
 the headwater driving force is substituted  in Equation  V-3  for the upstream
 concentration G-J_I.  Because the  headwater  concentrations are  fixed, they
 are incorporated on the right  hand  side  of  Equation V-3 in  the known term Z-j ,
 for headwater elements as follows.

                           SiAt
                Z, = Cin + — + pi At  - ai Cn                       V-14
                                   76

-------
where C0 is the upstream boundary condition (headwater concentration).


5.4.3.1  Downstream Boundary (Last Element in the System)

     QUAL2E has two options for modeling the downstream boundary.   One
uses a zero gradient assumption; the other incorporates fixed downstream
constituent concentrations into the solution algorithm.

     Zero Gradient Assumption (Arden and Astill,  1970)--For the last
computational  element in the system, there is no  downstream,  i+1,  element.
At this boundary, a zero gradient assumption is made that  replaces C-j+i
with C-j_i.  In this manner, the downstream boundary acts as a mirror to
produce a zero gradient for the concentration of  the constituent variable.
The coefficient a-j, therefore, is modified to include the  dispersion effect
normally found in the coefficient c-j for the last element  in the system.
Thus, the equation for a-j in V-3 becomes:

                                            At    Qi_iAt
               ai = -[((ADL)M + (ADL)I) 	+	]           .v-is
                                          VjAxj     V!

and

               G! = 0                                               V-16

where I = index of the downstream boundary element

     Fixed Downstream Constituent Concentrations—For this boundary
option, the user supplies known downstream boundary concentrations C[_B
for each water quality constituent.  Thus, the value of Cj+i  in Equation
V-3 becomes


                CI+1  = CLB                                        V-17


 Because the boundary  concentrations are  known  in  this  option,  they  are
 incorporated  on  the  right  hand  side of  Equation V-3  in  the known term Z-j
 for  the  downstream boundary  element then  results  as


                           SjAt
                zi = c! +  — + PiAt - ci CLB                       v-is
                           VI
                                    77

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                          6.   REFERENCES
American Public Health Association,  Inc.,  Standard Methods for the
     Examination of Water and  Wastewater,  American Public Health Associa-
     tion, 1965 (12th edition),  1985 (16th edition).

Anderson, E.R., Energy Budget  Studies  in Water Loss  Investigations—Lake
     Hefner Studies, Technical Report, U.S.  Geological Survey, Washington,
     DC, Prof. Paper 269, 1954.

Arden, B.W. and K.N. Astill, Numerical  Algorithms:   Origins and Applica-
     tions, Addison-Wesley,  Reading, MA, 1970.

Bannister, T.T., "Production Equations in  Terms  of Chlorophyll Concentra-
     tion, Quantun Yield, and  Upper  Limit  to Production," Limnology and
     Oceanography, Vol. 19,  No.  1, pp  1-12,  January  1974.

Brandes, R.J. and A.B. Stein,  WREDUN Model Documentation Report, Water
     Resources Engineers, Inc.,  prepared for Texas Department of Water
     Resources, Construction Grants  and Water Quality  Planning Division,
     no date.

Chen, C.W. and G.T. Orlob, Final  Report, Ecologic Simulation of Aquatic
     Environments, Water Resources Engineers, Inc.,  prepared for the
     Office of Water Resources Research, U.S. Department of the Interior,
     Washington, DC, October 1972.

Churchill, M.A., H.L. Elmore and R.A.  Buckingham, "The Prediction of
     Stream Reaeration Rates," International  Journal of Air and Water
     Pollution, Vol 6, pp 467-504, 1962.

DeGroot, W.T., "Modelling the  Multiple Nutrient  Limitation of Algal
     Growth," Ecological Modelling,  Vol. 18,  pp  99-119, 1983.
Duke, James H., Jr., Provision of a  Steady-State  Version of the Stream
     Model, QUAL, Water Resources Engineers,  Inc.,  Austin, TX, prepared
     for the U.S.  Environmental  Protection Agency,  Washington, DC,
     November 1973.

Edinger, J.E. and J.C. Geyer,  Heat Exchange in the  Environment, Johns
     Hopkins Univ., Baltimore, MD, 1965.
                                 78

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Elder, J.W., "The Dispersion of a Marked Fluid in Turbulent  Shear Flow,"
     Jour. Fluid Mech.. Vol. 5, Part 4,  pp 544-560,  May 1959.

Field, S.D. and S.W. Effler, "Photosynthesis-Light Mathematical  Formula-
     tions," Journal of Environmental  Engineering Division,  ASCE, Vol.
     108, No.  EE1, pp 199-203, February 1982.

Field, S.D. and S.W. Effler, "Light-Productivity  Model  for Onondaga,  N.Y.,"
     Journal Environmental  Engineering Division,  ASCE,  Vol.  109,  No.  EE4,
     pp 830-844, August 1983.

Fisher, H.B., Discussion to "Time of Travel  of Soluble  Contaminants  in
     Streams," by T.J. Buchanan, Proc. Sanitary Eng. Div., ASCE,  v. 6,
     1964.

Fisher, H.B., E.J. List, R.C.Y. Koh, J.  Imberger, N.H.  Brooks, Mixing in
     Inland and Coastal Waters, Academic Press, New York,  NY, 1979.

Frank D. Masch and Associates and the Texas  Water Development Board,
     Simulation of Water Quality in Streams  and Canals, Theory and
     Description of the QUAL-I Mathematical  Modeling System, Report 128,
     the Texas Water Development Board,  Austin, TX,  May 1971.

Henderson, F.M;, Open Channel  Flow, Macmillan Co., New  York, NY,  1966.

JRB Associates, "Users Manual  for Vermont QUAL-II Model,"  prepared for
     U.S. Environmental Protection Agency, Washington,  DC, June  1983.

Jassby, A.D., and T. Platt, "Mathematical  Formulation of the Relationship
     Between Photosynthesis and Light for Phytoplankton,"  Limnology and
     Oceanography. Vol. 21, No. 4, July  1976, pp  540-547.

Kramer, R.H., A Search of the Literature for Data Pertaining to
     Bioenergetics and Population Dynamics of Freshwater Fishes,  Desert
     Biome Aquatic Program, Utah State University, Logan, UT, August
     1970.

Langbien, W.B. and W.H. Durum, The Aeration  Capacity of Streams,  U.S.
     Geological Survey, Washington, DC,  Circ. 542, 1967.

National Council  for Air and Stream Improvement,  Inc.,  A Review of
     the Mathematical  Water Quality Model  QUAL-II and Guidance for its
     Use. NCASI,  New York,  NY, Technical  Bulletin No. 391, December 1982.

National Council  for Air and Stream Improvement,  Inc.,  QUAL2E User Manual,
     NCASI, New York,  NY,  Technical  Bulletin No.  457, April  1985.

O'Connor, D.J. and W.E. Dobbins, "Mechanism  of Reaeration in Natural Streams,"
     Trans. ASCE, Vol. 123, pp 641-684,  1958.
                                   79

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Owens, M., R.W. Edwards and J.W. Gibbs,  "Some Reaeration Studies in Streams,"
     International  Journal  of  Air  and Water Pollution, Vol. 8, No. 8/9,
     pp 469-486, September 1964.

Platt, T., K.H. Mann,  and  R.E. Ulanowicz  (eds.), Mathematical Models in
     Biological Oceanography,  Unesco Press, Paris, 1981.

Roesner, L.A., Temperature Modeling in Streams, Lecture notes, water
     quality workshop, Tennessee Valley  Authority, Knoxville, TN, 1969.

Roesner, L.A., P.R. Giguere, and D.E. Evenson, Computer Program Documentation
     for Stream Quality Modeling (QUAL-II).  U.S. Environmental Protection
     Agency, Athens, GA,  EPA-600/9-81-014, February 1981b.

Roesner, L.A., P.R. Giguere, and D.E. Evenson, User's Manual for Stream
     Quality Model  (QUAL-II).  U.S. Environmental Protection Agency,
     Athens, GA, EPA-600/9-81-015, February 1981b.

Scavia, D. and R.A. Park,  "Documentation  of Selected Constructs and
     Parameter Values  in the Aquatic Model CLEANER," Ecological Modeling,
     Vol. 2, pp 33-58, 1976.

Smith, E.L., "Photosynthesis in Relation  to Light and Carbon Dioxide,"
     Proceedings, National  Academy of Sciences, Vol. 22, pp 504-510, 1936.

Smith, J.D., Solutions to  Partial  Differential Equations, Macmillan Co.,
     New York, NY,  1966.

St. John, J.R., T.W. Gallagher, and P.R.  Paquin, "The Sensitivity of the
     Dissolved Oxygen  Balance  to Predictive Reaeration Equations," in Gas
     Transfer at Water Surfaces, W. Brutsaert and G. Jirka, eds., D. Reidl
     Publishing Company, Dordrecht, Holland, 1984.

Steele, J.H., "Environmental Control of  Photosynthesis in the Sea,"
     Limnology and  Oceanography, Vol. 7,  pp 137-150, 1962.

Stefan, H.G., J.J.  Cardoni, F.R. Schiebe, and C.M. Cooper, "Model of Light
     Penetration in a  Turbid Lake," Water Resources Research, Vol. 19,
     No. 1, pp 109-120, February 19BT.

Stone, H.L. and P.O.T. Brian,  "Numerical  Solution of Convective Transport
     Problems," Journal American Institute of Chemical Engineers, Vol. 9,
     No. 5, pp 681-688, 1963.

Streeter, H.W. and  E.B. Phelps, Study of  the Pollution and Natural Purifi-
     cation of the  Ohio River. U.S. Public Health Service, Washington, DC,
     Bulletin No. 146  (reprinted 1958),  1925.

Swartzman, G.L. and R. Bentley, "A Review and Comparison of Phytoplankton
     Simulation Models," Journal of the  International Society for Ecological
     Modelling. Vol. 1, Nos. 1-2,  pp 30-81, 1979.
                                     80

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Taylor, G.I., "The Dispersion of Matter in Turbulent  Flow Through  a Pipe,"
     Proceedings. Royal  Society of London, Vol. 234A,  No. 1199, pp 456-475,
     March 6, 1954.

Texas Water Development  Board, Simulation of  Water  Quality  in Streams
     and Canals, Program Documentation and User's Manual, Austin,  TX,
     September 1970.

Texas Water Development  Board, QUAL-TX Users  Manual,  Version 2.5,  Water
     Quality Management  Section, Austin,  TX,  November 1984.

TenEch Environmental  Consultants, Inc.  Waste Load  Allocation
     Verification Study:  Final Report.  Prepared for Iowa  Department
     of Environmental Quality, July 1978.

Thackston, E.L. and P.A. Krenkel, Reaeration  Prediction  in  Natural Streams,
     Journal of the Sanitary Engineering  Division,  ASCE, Vol. 95,  No.  SA1,
     pp 65-94, February  1969.

Thomas, H.A., Jr., "Pollution Load Capacity o.f Streams." Water and Sewage
     Works. Vol. 95,  No. 11, pp 409-413,  November 1948.

Tsivoglou, E.C. and J.R. Wallace, Characterization  of Stream Reaeration
     Capacity, Prepared  for U.S. Environmental Protection Agency,
     Washington, DC,  1972.

Tsivoglou, E.C. and L.A. Neal, "Tracer Measurement  of Reaeration:  III.
     Predicting the Reaeration Capacity of  Inland Streams," Jour.
     WPCF, Vol. 48, No.  12, pp 2669-2689, December  1976.

Walker, W.W., QUAL2 Enhancements and Calibration to the  Lower Winooski.
     Prepared for the Vermont Agency of Environmental  Conservation,
     Montpelier, VT,  December 1981.

Walker, W.W.  Personal Communication,  1983.

Water Resources Engineers, Inc., Prediction of Thermal Energy Distribution
     in Streams and Reservoirs, Prepared  for  the California Dept.  of Fish
     and Game, 1967.

Water Resources Engineers, Inc. Progress  Report on  Contract No. 68-01-0713,
     Upper Mississippi River Basin Model  Project, Sponsored by the
     Environmental Protection Agency,  submitted to  Environmental Protection
     Agency, September 21, 1972.

Wunderlich, W.O., The Fully-Mixed Stream  Temperature  Regime. ASCE  Specialty
     Conf., Utah State Univ., Logan, Utah, 1969.

Zison, S.W., D.B. Mills, D. Deimer, and C.W.  Chen,  Rates, Constants, and
     Kinetics Formulations in Surface  Water Quality Modeling.  Prepared
     for USEPA Environmental Research  Lab., Athens, Georgia, September 1978,
     revised 1985, EPA-600/3-85-040.


                                   81

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                           APPENDIX A

                       QUAL2E User Manual*
    The following sections illustrate the coding of input data
forms.
A.  Title Data

    All 16 cards are required in the order shown.  The first
two are title cards, and columns 22 through 80 may be used to
describe the base, date of simulation, etc.  Title cards 3
through 15 require either a "yes" or "no" in columns 10 through
12 and are right justified.  Note that each of the nitrogen and
phosphorus series must be simulated as a group.

    For each conservative substance (up to three) and the
arbitrary non-conservative, the constituent name must be
entered in columns 49 through 52.  Corresponding input data
units are entered in columns 57 through 60 (e.g.. mg/L).

    QUAL2E simulates ultimate BOD in the general case.  If the
user wishes to use 5-day BOD for input and output, the program
will internally make the conversions to ultimate BOD.  This
conversion is based upon first order kinetics and a decay rate
that can be specified by the user (Type 1 Data, line 8).  If no
value is specified, the program uses a default value of 0.23
per day, base e.   NCASI recommends that users work only with
ultimate BOD unless they have detailed knowledge of the river
water and point source BOD kinetics.  To use the 5-day BOD
input/output option, write "5-Day Biochemical Demand" in mg/L
on the title 7 card beginning in column 22.

    Card 16 must read ENDTITLE beginning in column 1.
 *From:   Modifications  to  the QUAL-2 Water Quality Model  and  User
  Manual  for  QUAL-2E Version 2.2.  National Council of  the  Paper
  Industry  for  Air  and  Stream Improvement, Inc., New York,  NY.
  NCASI Tech. Bulletin  No.  457. April  1985.  Used by permission.


                               82

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B.  Data Type 1 - Program Control

    Type 1 Data define the program control options and the
characteristics of the stream system configuration, as well as
some of the geographical/meteorological conditions for modeling
temperature.  There are a maximum of 17 Data 1 cards.  The
first 13 are required, while the last four are necessary only
if temperature is being simulated.

    The QUAL2E program recognizes Type 1 Data by comparing the
first four characters (columns 1-4) of each data card with a set
of internally fixed codes.  If a match between the code and
characters occurs, then the data are accepted as supplied on
the card by the user.  If a match does not occur, then the
program control options will revert to default values and the
system variables for the unmatched codes will be assigned the
value zero (0.0).

    The first seven cards control program options.  If any
characteristics other than those shown below are inserted in
the columns 1 through 4, the actions described will not occur.

LIST - Card T.  list the input data.

WRIT - Card 2.  write the intermediate output report,
       WRPT2 (see SUBROUTINE WRPT2 in the documentation
       manual,  NCASI Technical Bulletin No. 391).

FLOW - Card 3,  use flow augmentation.

STEA - Card 4 shows this is a steady-state simulation.  If it
       is not to be a steady-state, write DYNAMIC SIMULATION,
       or NO STEADY STATE, and it is automatically a dynamic
       simulation.

TRAP - Card 5,  cross-sectional data will be specified for each
       reach.  If discharge coefficients are to be used for
       velocity and depth computations,  write DISCHARGE
       COEFFICIENTS,  or NO TRAPEZOIDAL CHANNELS,  beginning in
       column 1.

PRIN - Card 6.  local climatological data specified globally for
       the basin simulation will appear in the final output
       listing.

PLOT - Card 7.  dissolved oxygen and BOD will be plotted in final
       output listing.

    The next two cards provide further program flags and coeffi-
cients.   This information is supplied in two data fields per
card;  columns 26-35.  and 71-80.   Note that the character codes
                               83

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in columns 1-4 must occur as shown in order for the data to be
accepted by the program.

FIXE - Card 8. specifies: (a) whether the downstream boundary
       water quality constituent concentrations are fixed (user
       specified), and  (b) the value of the rate coefficient
       for converting input 5-day BOD to ultimate BOD.  A value
       of 1.0 (or larger) in columns 26-35 specifies that the
       downstream boundary water quality constituent
       concentrations will be supplied in Data Types 13 and
       13A.  A value less than 1.0 (usually 0.0 or blank) in
       these columns means that the downstream boundary
       concentrations are not user specified.  In this case,
       the concentrations in the most downstream element (Type
       5) will be computed in the normal fashion using the zero
       gradient assumption (1, 11).  The second value on this
       card, columns 71-80, is the rate coefficient for
       converting 5-day to ultimate BOD.  It is used only when
       5-day BOD is being modeled (Title Card 7).  If the
       columns are left blank, the model uses a default value
       of 0.23 per day, base e.  Note that this conversion
       factor is applied to all input BODg forcing functions
       (headwaters, incremental flows, point loads, and the
       downstream boundary condition).

INPU - Card 9, specifies whether the user will input and/or
       output in metric or English units.  The value of 1.0 (or
       larger) in card  columns 26-35 specifies metric input.
       The value of 1.0  (or larger) in card column 71-80
       specifies metric units for output.  Any value less than
       1.0 (usually 0.0 or blank) will specify English units.

    The next four cards describe the stream system.  There are
two data fields per card, columns 26-35 and 71-80.  The program
restrictions on the maximum number of headwaters, junctions,
point loads, and reaches are defined by PARAMETER statements in
the Fortran code.  These statements may be modified by the user
to accommodate a particular computer system or QUAL2E
simulation application.  The values of the maximum constraints
are as follows:

       Maximum number of headwaters                  10
       Maximum number of junctions                    9
       Maximum number of point loads                 50
       Maximum number of reaches                     50
       Maximum number of computational elements     500

NUMB - Card 10, defines the number of reaches into which the
       stream is segmented and the number of stream junctions
       (confluences) within the system.
                                84

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NUM_ - Card 11. shows the number of headwater sources and the
       number of inputs or withdrawals within the system.  The
       inputs can be small streams, wasteloads, etc.
       Withdrawals can be municipal water supplies, canals.
       etc.  NOTE:  Withdrawals must have a minus sign ahead of
       the flow in Data Type 11 and must be specified as
       withdrawals in Data Type 4 by setting IFLAG = 7 for that
       element.  Note, the code for Card 11 is 'NUM_' (read:
       NUM space) to distinguish it from the code for Card 10.
       NUMB.

TIME - Card 12. contains the time step interval in hours and the
       length of the computational element in miles
       (kilometers).  The time step interval is used only for a >
       dynamic simulation, thus it may be omitted if the
       simulation is steady-state.

MAXI - Card 13. provides information with different meanings de-
       pending on whether a dynamic or steady-state simulation
       is being performed.  For a dynamic simulation, the
       maximum route time is specified in columns 26-35.  This
       value represents the approximate time in hours required
       for a particle of water to travel from the most upstream
       point in the system to the most downstream point.  The
       time increment in hours for intermediate summary reports
       of concentration profiles is specified in columns
       71-80.  For a steady-state simulati-on. the maximum
       number of iterations allowed for solution convergence is
       entered in columns 26-35.  The value in columns 71-80
       may be left blank because it is not required in the
       steady-state solution.

    The next four cards provide geographical and meteorological
information and are required only if temperature is being
simulated.  There are two data fields per card, columns 26-35
and 71-80.  Note:  the character codes in columns 1-4 must
occur as shown in order for the data to be accepted by the
program.

LATI - Card 14, contains the basin latitude and longitude and
       represent mean values in degrees for the basin.

STAN - Card 15. shows the standard meridian in degrees, and the
       day of the year the simulation is to begin.

EVAP - Card 16. specifies the evaporation coefficients.  Typical
       values are AE = 6.8 x 10~4 ft/hr-in Hg and BE = 2.7 x
       10~4ft/hr-in Hg-mph of wind for English units input.
       or AE = 6.2 xlO~6 m/hr-mbar and BE = 5.5 x 10~6
       m/hr-mbar-m/sec of wind for metric units input.
                               85

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ELEV - Card 17. contains the mean basin elevation in feet
       (meters) above mean sea level, and the dust attenuation
       coefficient (unitless) for solar radiation.  The dust
       attenuation coefficient generally ranges between zero
       and 0.13.  Users may want to consult with local
       meteorologists for more appropriate valiies.

       Data Type 1 must end with an ENDATA1 card.


C.  Data Type 1A - Global Algal. Nitrogen, Phosphorus, and Light
    Parameters

    These parameters and constants apply to the entire
simulation and represent the kinetics of the algal, nutrient.
light interactions.  It is important to note that proper use of
all options in QUAL2E requires detailed knowledge of the algal
growth kinetics appropriate for the water body being simulated.

    These data cards are required only if algae, the nitrogen
series (organic, ammonia, nitrite, and nitrate), or the
phosphorus series (organic and dissolved) are to be simulated.
Otherwise they may be omitted, except for the ENDATA1A card).
Information is supplied in two data fields per card, columns
33-39 and 74-80.  As with Type 1 Data, the QUAL2E program
recognizes Type 1A Data by comparing the first four characters
(columns 1-4) of each card with a set of internally fixed
codes.  If a match between the codes and the characters occurs.
then the data are accepted as supplied on the card by the
user.  If a match does not occur, then the system variables for
the unmatched codes will be assigned the value zero (0.0).
Note: the spaces (under bars) are an integral (necessary) part
of the four character code.

O_UP - Card 1, specifies the oxygen uptake per \rnit of ammonia
       oxidation, and oxygen uptake per unit of nitrite
       oxidation.

O_PR - Card 2. contains data on oxygen production per unit of
       algae growth, usually 1.6 mg O/mg A. with a range of 1.4
       to 1.8.  It also contains data on oxygen uptake per unit
       of algae, usually 2.0 mg O/mg A respired, with a range
       of 1.6 to 2.3.

N_CO  - Card  3.  concerns  the  nitrogen content and  phosphorus
       content  of  algae  in mg  per mg of  algae.   The fraction  of
       algae  biomass which is  nitrogen  is  about  0.08  to  0.09.
       and the  fraction  of algae  biomass which  is  phosphorus  is
       about  0.012  to  0.015.
                               86

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ALG_ - Card 4. specifies the growth and respiration rates of
       algae.  The maximum specific growth rate has a range of
       1.0 to 3.0 per day.  The respiration value of 0.05 is
       for clean streams, while 0.2 is used where the NE and
       P2 concentrations are greater than twice the half
       saturation constants.

N_HA - Card 5. contains the nitrogen and phosphorus half
       saturation coefficients.  The range of values for
       nitrogen is from 0.2 to 0.4 mg/L and for phosphorus the
       value normally used is 0.04 mg/L.

LIN_ - Card 6, contains the linear and nonlinear algal self
       shading light extinction coefficients.  The coefficients
       \! and X-2 are defined below.

       \1 = linear algae self-shading coefficient
            (l/ft)/(ug chla/L), or (l/m)/(ug chla/L)

       \2 = nonlinear algae self-shading coefficient
            (l/ft)/ug chla/L)2/3. or (l/m)/(ug chla/L)2/3

       These two self-shading coefficients are used with
       \o. the non-algal light extinction coefficient (Type
       6B Data) in the general light extinction eguation shown
       below:

                  \  = \0  +  X^A +

       where X is the total light extinction coefficient and
       A is the algae biomass concentration in mg A/L and
       
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LIGH - Card 7. contains the solar light function option for
       computing the effects of light attenuation on the algal
       growth rate,  and the light saturation coefficient.
       QUAL2E recognizes three different solar light function
       options.  The light saturation coefficient is coupled to
       the selection of a light function,  thus care must be
       exercised in specifying a consistent pair of values.
          The depth integrated form of the three light
       functions and the corresponding definitions of the light
       saturation coefficient are given in Figure A-l and
       outlined in the following table.
       Light Function Option
         (Columns 33 - 39)

       1  (Half Saturation)

       2  (Smith's Function)


       3  (Steele's Function)
Light Saturation Coefficient*
	(Columns 74 - 80)	

Half Saturation Coefficient

Light intensity corresponding
to 71% of maximum growth rate

Saturation Light Intensity
       * Units of the Light Saturation Coefficient are as
         follows:
         English:  BTU/ft2-min     and   Metric:   Langleys/min
          Light Function Option 1 utilizes a Michaelis-Menton
       half saturation formulation for modeling the algal
       growth limiting effects of light (FL).   It is the method
       used in the SEMCOG version of QUAL-2.  Option 2 is
       similar to Michaelis-Menton, but utilizes a second order
       rather than first order light effect.  Both options 1
       and 2 are monotonically increasing functions of light
       intensity.  Option 3 includes a photo-inhibition effect
       at high light intensities and has been reported in
       (9.13).

DAIL - Card 8. contains the light averaging option (columns
       33-39) and the light averaging factor (columns 74-80).
       These values are used only in a steady-state simulation.
       The light averaging option allows the user to specify the
       mannerin which the light attenuation factor (FL in Figure
       A-l) is computed, from the available values of solar
       radiation.  A summary of these options is given below.
                                88

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         Option      	Description	

          1          FL is computed from one daily average solar
                     radiation value calculated in the steady-
                     state temperature subroutine (HEATER).

          2          FL is computed from one daily average solar
                     radiation read from Data Type 1A.

          3          FL is obtained by averaging the 24 hourly
                     values of FL. that are computed from the
                     24 hourly values of solar radiation calcu-
                     lated in the steady-state temperature sub-
                     routine (HEATER).

          4          FL is obtained by averaging the 24 hourly
                     values of FL. that are computed from the
                     24 hourly values of solar radiation
                     computed from the total daily solar
                     radiation (Data Type 1A) and an assumed
                     cosine function.
       Note: that if options 1 or 3 are selected, temperature
             must be simulated.
          The light averaging factor (columns 74-80) is used to
       make a single calculation using daylight average solar
       radiation (Option 1 or 2) agree with average of
       calculations using hourly solar radiation values (Option
       3 or 4).  The factor has been reported to vary from 0.85
       to 1.00.

    The selection of a daily (diurnal) light averaging option
depends largely on the detail to which the user wishes to
account for the diurnal variation in light intensity.  Options
1 and 2 utilize a single calculation of FL based on an average
daylight solar radiation value.  Options 3 and 4 calculate
hourly values of FL from hourly values of solar radiation and
then average the hourly FL values to obtain the average
daylight value.  Options 1 and 3 use the solar radiation from
the temperature heat balance routines (thus both algae and
temperature simulations draw on the same source for solar
radiation).  Options 2 and 4 use the solar radiation value in
Data Type 1A for the algae simulation.  Thus either option 2 or
4 must be selected when algae are simulated and temperature is
not.  The light averaging factor is used to provide similarity
in FL calculations between options 1 and 2 versus options 3 and
4.  The solar radiation factor (Data Type 1A. card 11)
specifies the fraction of the solar radiation computed in the
heat balance which is photosynthetically active.  It is used
only with options 1 or 3.
                               89

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Light  Functions    (LFNOPT)

    •   Option 1:   Half saturation (SEMCOG)

        FL = (-i)  In  [   KL + \ ,]
              Ad       K L + I e

        KL =  light intensity  at which growth rate  is  50% of  the
              maximum growth rate.

    •   Option 2:   Smith's Function (14)

               1            T /If T  -L /I  J. t T /VT \  \ ^f ^"
        FL = (-i)  In  [	  ,H                  H , iV,]
              Ad       I/KLe-Xd + (1 + (I/KLe-Xd)2)1/2

        KL =  light intensity  at which growth rate  is  71% of  the
              maximum growth rate.

    •   Option 3:  Steele's Equation  (9)

        FL = 2.718  [e-(e"Xd
              Ad

        KL =  light intensity at which growth rate  is  equal to the
              maximum growth rate.

    Notation:   FL  =   light attention factor
                 X  =   extinction coefficient *
                 d  =   depth
                 I  =   surface  light  intensity
                KL  =   light saturation  coefficient
               * Algal Self-shading

                    •  General Equation         X . Xo f Xx  oo A  +  X2 (a0A)2/3

                    where: X »  light extinction coefficient
                          X9 »  non-algal extinction
                          \l -  linear algal self shading coefficient
                          \2 »  non-linear algal self shading coefficient
                          A  «  algal biomass concentration (mg/L)
                          °0 -  chlorophyll a to algae biomass ratio (ug chla/ragA)
                    •  Special Cases
                          •  No Self-shading (SEMCOG)


                          •  Linear Self-shading (META)


                          •  Non-linear Self-shading (TetraTech)

                            xl f  X2  f  °

                    e.g. X » XQ + 0.0088^ + 0.054(aQA)2/3 (Riley Eq.,
                             metric units)
            FIGURE A-l.   ALGAL GROWTH RATE


                                90

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    In dynamic algae simulations option 3 is used (default)
unless temperature is not simulated in which case solar
radiation data are read in with the local climatology data.

NUMB - Card 9. contains the number of daylight hours (columns
       33-39). and the total daily radiation (BTU/ft2,  or
       Langleys) (columns 74-80).  This information is used if
       light averageing options 2 or 4 are specified for the
       simulation.

ALGY - Card 10, contains the light-nutrient option for computing
       the algae growth rate (columns 33-39). and the algal
       preference factor for ammonia nitrogen (columns 74-80).
       The light-nutrient interactions for computing algae
       growth rate are as follows (see also Figure A-l):
       Option
          Description
                   Multiplicative:  (FL) * (FN) * (FP)
                   Limiting Nutrient:  FL * [minimum (FN. FP)]
                   Inverse Additive:     FL * 2
                                      1/FN + 1/FP
     Option 1 is the form used in QUAL-2 SEMCOG,  while option 2
     is used in the META Systems Version of QUAL-2.   Option 3 is
     described by Scavia and Park (13).

     The algal preference factor for ammonia (columns 74-80)
     defines the relative preference of  algae for ammonia and
     nitrate nitrogen (see also Figure A-2).  The user defines
     this preference by specifying a decimal value between 0
     and 1.0,  for example:
     Algal  Preference
         factor
       for  Ammonia
          Interpretation
          0.0

          0.5

          1.0
Algae will use only nitrate for growth.

Algae will have equal preference for
ammonia and nitrate.
Algae will use only ammonia for growth,
                               91

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    Nutrient Attenuation Factors

        •  Nitrogen:  NE = NH3 + NC>3  (mg-N/L)
                     FN = NE/(KN + NE)
        where:  FN = Nitrogen attenuation factor
               KN = Nitrogen half saturation coefficient  (mg-N/L)

        •  Phosphorus:   FP = ?2/(KP + P£)
                       ?2 = Dissolved Phosphorus (mg-P/L)
        where:  FP = Phosphorus attenuation factor
               KP = Phosphorus half saturation coefficient (mg-P/L)

    Algal Preference foe Ammonia
             F  =
                        (PN)
                   (PN)(NH3) + (1-PN)N03
        where:  PN = Algal preference for ammonia nitrogen (0-1.0)
                  PN=  0.0; algae will use only nitrate for growth
                  PN=  0.5; algae will have equal preference for
                          ammonia and nitrate for growth
                  PN=  1.0; algae will use only ammonia for growth
             F  =  Fraction of algal nitrogen uptake from ammonia pool.


                    FIGURE A-2.  ALGAL GROWTH  RATE
ALG/ - Card  11.  contains  the  factor for converting the solar
     radiation value from the heat balance  to  the solar
     radiation value appropriate for the algae simulation
     (columns 33-39) and  the  value of the first order
     nitrification inhibition coefficient (columns 74-80).
           The solar radiation factor specifies  the fraction  of
     the  solar radiation computed in the heat  balance
     (subroutine HEATER) that is photosynthetically active
     (i.e..  used by algal  cells for growth).   It is required
     only in steady-state  simulations when  light averaging
     options 1 or 3 (Data  Type 1A, card 8)  are  selected.  A
     decimal value between 0  and 1.0 specifies  the value  of
     this fraction.  Typically the value of  this fraction is
     about 0.45 (14).

           The first order  nitrification inhibition coefficient
     is the value of KNITRF in the following equation (see
     Figure A-3).

             CORDO = 1.0 -  EXP (-KNITRF *DO)
                                 92

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where:
        DO = dissolved  oxygen  concentration  (mg/L), and
   CORDO = correction  factor  applied  to ammonia  and nitrite
               oxidation rate coefficients.
    Nitrification Rate Correction Factor  (CORDO)

       •  CORDO = 1.0 - EXP(-KNITRF *  DO)
       •  The value  of KNITRF is supplied by the user,
          the default value in QUAL-2E is 10.0
Applied to Ammonia and Nitrite Oxidation Rates

      Ammonia:     ^l^inhib   =  ^l^ino t * (CORD°)

      Nitrite:     (Sn),_u,u   =  (B0).__..1. * (CORDO)
                       'Vinhib.
    Magnitude of  Correction Factor
                      'Vinput
       The following table contains  values of CORDO as a  function
    of DO (row)  and KNITRF (column).
DO
(mg/L)
0.1
0.2
0.3
0.4
0.5
0.7
1.0
1.5
2.0
3.0
4.0
5.0
7.0
10.0
0.5
.05
.10
.14
.18
.22
.30
.39
.53
.63
.78
.86
.92
.97
.99
0.7
.07
.13
.19
.24
.30
.39
.50
.65
.75
.88
.94
.97
.99
1.00
KNITRF
1.0
.10
.18
.26
.33
.39
.50
.63
.78
.86
.95 1
.98 1
.99 1
1.00 1
1.00 1
2.0
.18
.33
.45
.55
.63
.75
.86
.95
.98
.00
.00
.00
.00
.00
5.0
.39
.63
.78
.86
.92
.97
.99
1.00
1.00
1.00
1.00
1.00
1.00
1.00
10.0
.63
.86
.95
.98
.99
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
       KNITRF = 0.6, closely matches the nitrification inhibition
                 function  in QUAL-2 Texas (4).
      KNITRF


    FIGURE  A-3.
0.7,  closely matches the nitrification inhibition
  data  of the Thames Estuary  (12).

   NITRIFICATION INHIBITION AT LOW DO

               93

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          A value of 0.6 for KNITRF closely matches the
     inhibition formulation in QUAL-Z Texas, while a value of
     0.7 closely matches the data for the Thames Estuary (12).
     The default value of KNITRF is 10.0, i.e., no inhibition
     of nitrification at low dissolved oxygen.  A table of
     CORDO values as a function of KNITRF and DO is given in
     Figure A-3.

ENDA - The last card in Data Type 1A must be an ENDATA1A card,
     regardless of whether algae, nitrogen, or phosphorus are
     simulated.
D. Data Type IB - Temperature Correction Factors

   Several of the processes represented in QUAL2E are affected
by temperature.  The user may elect to input specific
temperature correction factors.  In the absence of such
information, default values are used as noted in Figure A-4.
The user need supply only those values that are to be changed.

   Data Type IB information is supplied as follows:

     Alphanumeric code for each temperature
     coefficient as noted in Figure A-4;         Columns 10-17

     User specific temperature coefficient       Columns 19-26
The last card in Data Type IB must be an ENDATA1B card,
regardless of whether any of the default values are modified.
E. Data Type 2 - Reach Identification and River Mile/Kilometer
   Data

   The cards of this group identify the stream reach system by
name and river mile/kilometer by listing the stream reaches
from the most upstream point in the system to the most
downstream point.  When a junction is reached, the order is
continued from the upstream point of the tributary.  There is
one card per reach.  The following information is on each card:

    Reach Order or Number                      Columns 16-20

    Reach Identification or Name               Columns 26-40

    River Mile/Kilometer at Head of Reach      Columns 51-60

    River Mile/Kilometer at End of Reach       Columns 71-80
                                94

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                                   DEFAULT VALUES
       INDEX   RATE COEFFICIENT
        1      BOD Decay
        2      BOD Settling
        3      Reaeration
        4      SOD Uptake
        5      Organic N Decay
        6      Organic N Settling
        7      Ammonia Decay
        8      Ammonia Source
        9      Nitrite Decay
       10      Organic P Decay
       11      Organic P Settling
       12      Dissolved P Source
       13      Algae  Growth
       14      Algae  Respiration
       15      Algae  Settling
       16      Coliform Decay
       17      Non-cons Decay
       18      Non-cons Settling
       19      Non-cons Source

           FIGURE A-4.   DEFAULT THETA VALUES FOR QUAL2E
SEMCOG
1.047
-
1.0159
-
-
-
1.047
-
1.047
-
-
-
1.047
1.047
-
1.047
1.047
-
-
QUAL-2E
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
.047
.024
.024
.060
.047
.024
.083
.074
.047
.047
.024
.074
.047
.047
.024
.047
.000
.024
.000
CODE
BOD DECA
BOD SETT
OXY TRAN
SOD RATE
ORGN DEC
ORGN SET
NH3 DECA
NH3 SRCE
N02 DECA
PORG DEC
PORG SET
DISP SRC
ALG GROW
ALG RESP
ALG SETT
COLI DEC
ANC DECA
ANC SETT
ANC SRCE
    A very useful  feature of QUAL2E pertaining to modifications
of reach  identification once the system has  been coded is that
existing  reaches may be subdivided (or new reaches added) with-
out renumbering the reaches for the whole system.   If. for
example,  it  is desired to divide the river reach originally
designated as REACH 3 into two reaches, the  division is made by
calling the  upstream portion REACH 3 and the "new reach"
downstream REACH 3.1.  Up to nine such divisions can be made
per reach (3.1-3.9);  thus REACH 3 (or any other reach) can be
divided into as many as 10 reaches numbered  3,  3.1-3.9.  This
option of dividing a reach is useful particularly when new
field data indicate a previously unknown or  a change in
geomorphology, or  when the addition of a new or proposed load
alters the biochemistry in the downstream portion of the
reach.  If this option is invoked, the number of reaches
specified in Data  Type 1 must be changed to  the new total
number of reaches.
    Note:  It  is  important to realize that  this  option cannot
be used to subdivide  a reach into more  (and  thus smaller)
                                95

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computational elements, in an attempt to provide greater detail
to the simulation.  All computational elements must have the
same length (as specified in Type 1 Data).

    This option also will allow the user to add a new reach to
the system; for example, taking a tributary that was initially
modeled as a point source and changing it to a modeled reach
(or reaches) in the basin.  This type of modification adds a
junction to the system and thus the junction information in
Data Types i.  4. and 9 must be modified accordingly.


    This group of cards must end with ENDATA2.
F.  Data Type 3 - Flow Augmentation Data

    These cards, except ENDATA 3. are required only if flow
augmentation is to be used.  The cards in this group contain
data associated with determining flow augmentation requirements
and available sources of flow augmentation.  There must be as
many cards in this group as in the reach identification group.
The following information is on each card.


    Reach Order or Number                        Columns 26-30

    Augmentation Sources (the number of          Columns 36-40
    headwater sources which are avail-
    able for flow augmentation)

    Target Level (minimum allowable              Columns 41-50
    dissolved oxygen concentration (mg/L)
    in  this reach)

    Order of Sources (order of avail-            Columns 51-80
    able headwaters, starting at most
    upstream points


    This card group must end with ENDATA3, even if no flow
augmentation is desired.


G.  Data Type 4 - Computational Elements Flag Field Data

    This group  of cards identifies each type of computational
element in each reach.  These data allow the proper form of  the
routing equations to be used by the program.  There are seven
element types allowed, they are listed below.
                                96

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          I FLAG     	Type	

           1        Headwater source element.

           2        Standard element,  incremental inflow/
                    outflow only.

           3        Element on mainstream immediately upstream
                    of a junction.

           4        Junction element.

           5        Most downstream element.

           6        Input (point source) element.

           7        Withdrawal element.


    Each card.in this group (one for each reach), contains- the
following information:

    Reach Order <5r Number                          Columns 16-20

    Number of Elements  in the Reach                Columns 26-30

    Element Type  (these are numbers                Columns 41-80
    of a set, identifying each
    element by type).
    Remember that once a system has been coded, reaches can be
divided or new ones added without necessitating the renumbering
of the entire system (see Data Type 2 - Reach Identification
and River Mile/Kilometer Data for application and constraints).
When this option is invoked, the element types and number of
elements per reach for the affected reaches must be adjusted in
Data Type 4 to reflect the changes.

    This card group must end with ENDATA4.


H.  Data Type 5 - Hydraulics Data

    Two options are available to describe the hydraulic
characteristics of the system.  The first option utilizes a
functional representation while the second option utilizes a
geometric representation.  The option desired is specified in
Data Type 1. card 5.  The code "TRAPEZOIDAL" specifically
denotes the geometric representation.  Any other code, such as
"NO TRAPEZOIDAL", or "DISCHARGE COEFFICIENTS", specifies the
functional representation.

                               97

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    Note: With either option, the effect is global (for the
entire system).  This option is not reach variable.

    If the first option is selected, velocity is calculated as
V = aQb and depth is found by D =aQ,3 .   Each card repre-
sents one reach and contains the values of a. b, a, and 6.
as described below.
    Reach Order or Number                         Columns 16-20

    Dispersion Constant                           Columns 23-30

    a. coefficient for velocity                   Columns 31-40

    b. exponent for velocity                      Columns 41-50

    a. coefficient for depth                      Columns 51-60

    |3. exponent for depth                         Columns 61-70

    Mannings "n" for reach (if                    Columns 71-80
    not specified, the program
    default value is 0.02)
    The dispersion constant is the value of K in the general
expression relating the longitudinal dispersion coefficient to
the depth of flow and shear velocity (10).
                             DL  =  Kdu*
where:
       DL = longitudinal dispersion coefficient.
            (ft2/day. m2/day)

       K  =  dispersion constant, dimensionless

       d  =  mean depth of flow, (ft.m)

       u* =  shear velocity, (ft/sec, m/sec) = (gdS)-1/2

       g  =  gravitational constant (ft/sec2, m/sec2)

       S  =  slope of the energy grade line (ft/ft, m/m)
Substitution of the Manning equation for S, leads to the
following expression for the longitudinal dispersion
coefficient. DL.


                               98

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                         DL = 3.82 KnVd5/6

where:
       n = Mannings roughness coefficient, and

       V = Mean stream velocity (ft/sec, m/sec).


Typical values of K range from 6  to 6000.  A value of 5.93 leads
to the Elder equation for longitudinal dispersion, which is the
one used in the SEMCOG version of QUAL-2.

    The coefficients a, b. a. and p should be expressed to
relate velocity, depth and discharge units as follows:
         System
0
m3/sec
ft3/sec
V
m/sec
ft/sec
D
m
ft
        Metric

        English
  If the second option is selected, each reach is represented as
a trapezoidal channel.  This data form is also used to specify
the trapezoidal cross-section (bottom width and side slope), the
channel slope and the Manning's "n" corresponding to the reach.
The program computes the velocity and depth from this data using
Manning's Equation and the Newton-Raphson (iteration) method.
One card must be prepared for each reach as follows:


  Reach Order or Number                          Columns 16-20

  Dispersion Constant, K                         Columns 23-30

  Side Slope 1 (run/rise; ft/ft, m/m)            Columns 31-40

  Side Slope 2 (run/rise; ft/ft, m/m)            Columns 41-50

  Bottom Width of Channel.                       Columns 51-60
  (feet, meters)

  Channel Slope (ft/ft, m/m)                     Columns 61-70

  Mannings "n" (Default = 0.020)                 Columns 71-80

  This group of data must end with an ENDATA5 card.
                               99

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I.     Data Type 6 - BOD and DO Reaction Rate Constants Data

  This group of cards includes reach information on the BOD
decay rate coefficient and settling rate, sediment oxygen
demand, as well as the method of computing the reaeration
coefficient.  Eight options for reaeration coefficient
calculation are available.  These are listed below.
         K2 OPT     	Method (9)	

           1        Read in values of K2.

           2        Churchill.

           3        O'Connor and Dobbins.

           4        Owens and Gibbs.

           5        Thackston and Krenkel.

           6        Langbien and Durum.

           7        Use equation K2 = aQb

           8        Tsivoglou-Wallace.


  One card is necessary for each reach,  and contains the
following information:

  Reach Order or Number                          Columns 16-20

  BOD Decay Rate Coefficient (I/day)              Columns 21-28

  BOD Removal Rate by Settling (I/day)            Columns 29-36

  Sediment Oxygen Demand                         Columns 37-44
  (gm/ft2-day, gm/m2-day)

  Option for K2 (1-8, as above)                  Columns 45-48

  K2 (Option 1 only) Reaeration                  Columns 49-56
  Coefficient, per day, base e. 20C

  a. Coefficient for K2 (Option 7)               Columns 57-64
  or Coefficient for Tsivoglou
  (Option 8)

  b. Exponent for K2 (Option 7) or               Columns 65-72
  Slope of the Energy Gradient, Se,
  (Option 8)

                              100

-------
    The units of a and b vary depending on whether option 7 or
8 is used and on whether the input data are in English or
Metric units, as follows:

    Units of a:                  English             Metric
    Option 7 (Coefficient)   Consistent with     Consistent with
                             flow in cfs         flow in cms

    Option 8 (Coefficient)      I/ft                  1/m

    Units of b:                 English              Metric	

    Option 7 (Exponent)      Consistent with     Consistent with
                             flow in cfs         flow in cms

    Option 8 (Se)            Dimensionless       Dimensionless


    For option 8 (Tsivoglou's option), the energy gradient.
Se need not be specified if a Manning "n" value was assigned
under Hydraulic Data Type 5.  Se will be calculated from
Manning's Equation using the wide channel approximation for
hydraulic radius.

    This group of cards must end with ENDATA6.


J.  Data Type 6A - N and P Coefficients

    This group of cards is required if algae, the nitrogen
series (organic nitrogen, ammonia, nitrite, and nitrate), or
the phosphorus series (organic and dissolved) are to be
simulated.  Otherwise, they may be omitted.  Each card of this
group, one for each reach, contains the following information:


    Reach Order or Number                          Columns 20-24

    Rate Coefficient for Organic-N                 Columns 25-31
    Hydrolysis (I/day)

    Rate Coefficient for Organic-N                 Columns 32-38
    Settling (I/day")

    Rate Coefficient for Ammonia                   Columns 39-45
    Oxidation (I/day)

    Benthos Source Rate for Ammonia                Columns 46-52
    (mg/ft2-day, mg/m2-day)
                               101

-------
    Rate Coefficient for Nitrite                   Columns 53-59
    Oxidation (I/day)

    Rate Coefficient for Organic                   Columns 60-66
    Phosphorus Decay (l/day)

    Rate Coefficient for Organic                   Columns 67-73
    Phosphorus Settling (I/day)

    Benthos Source Rate for Dissolved              Columns 74-80
    Phosphorus (mg/ft2-day. mg/m2-day)
    Note that the benthos source rates are expressed per unit
of bottom area.  Other versions of QUAL-2 use values per length
of stream.  To convert to the areal rate, divide the length
value by the appropriate stream width.

    This group of cards must end with ENDATA6A, even if algae.
nitrogen, or phosphorus are not simulated.
K.  Data Type 6B - Algae/Other Coefficients

    This group of cards is required if algae, the nitrogen
series, the phosphorus series, coliform, or the arbitrary
non-conservative is to be simulated.  Otherwise, they may be
deleted.  Each card of the group, one per reach, contains the
following information:
    Reach Order or Number                          Columns 20-24

    Chlorophyll a. to Algae Ratio                   Columns 25-31
    (ug chla/mg algae)

    Algal Settling Rate (ft/day, m/day)            Columns 32-38

    Non-Algal Light Extinction                     Columns 39-45
    Coefficient (I/ft, 1/m)

    Coliform Decay Coefficient (I/day)             Columns 46-52

    Arbitrary Non-Conservative Decay               Columns 53-59
    Coefficient (I/day)

    Arbitrary Non-Conservative Settling            Columns 60-66
    Coefficient (I/day)

    Benthos Source Rate for Arbitrary              Columns 67-73
    Non-Conservative  (mg/ft2-day. mg/m2-day)
                              102

-------
    This group of cards must end with ENDATA6B. even if algae.
nitrogen, phosphorus, coliform. or the arbitrary
non-conservative are not simulated.
L.  Data Type 7 - Initial Conditions - 1
    This card group, one card per reach, establishes the initial
conditions of the system, with respect to temperature, dissolved
oxygen concentration, BOD concentrations, and conservative
minerals.  Initial conditions for temperature must always be
specified whether it is simulated or not.  The reasons for this
requirement are: (a) when temperature is not simulated, the
initial condition values are used to set the value of the
temperature dependent rate constants; (b) for dynamic
simulations the initial condition for temperature, and every
other quality constituent to be simulated, defines the state of
the system at time zero; and (c) for steady state simulations
of temperature, an initial estimate of the temperature between
freezing and boiling is required to properly initiate the heat
balance computations.  Specifying 68°F or 20°C for all
reaches is a sufficient initial condition for the steady state
temperature simulation case.  The information contained is as
follows:
    Reach Order or Number                          Columns 20-24

    Temperature (F or C)                           Columns 25-31

    Dissolved Oxygen (mg/L)                        Columns 32-38

    BOD (mg/L)                                     Columns 39-45

    Conservative Mineral I*                        Columns 46-52

    Conservative Mineral II*                       Columns 53-59

    Conservative Mineral III*                      Columns 60-66

    Arbitrary Non-Conservative*                    Columns 67-73

    Coliform  (No./lOO ml)                          Columns 74-80

    * - Units are those specified on the Title Card.

    This group of cards must end with ENDATA7.
                              103

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M.  Data Type 7A - Initial Conditions - 2

    This group of cards is required if algae, the nitrogen
series, or the phosphorus series are to be simulated.  The
information is coded as follows:
    Reach Order or Number                          Columns 20-24

    Chlorophyll a. (ug/L)                           Columns 25-31

    Organic Nitrogen as N (mg/L)                   Columns 32-38

    Ammonia as N (mg/L)                            Columns 39-45

    Nitrite as N (mg/L)                            Columns 46-52

    Nitrate as N (mg/L)                            Columns 53-59

    Organic Phosphorus as P (mg/L)                 Columns 60-66

    Dissolved Phosphorus as P (mg/L)               Columns 67-73


    This group of cards must end with ENDATA7A. even if algae,
nitrogen, or phosphorus are not simulated.


N.  Pita Type 8 - Incremental Inflow - 1

    This group of cards, one per reach,  accounts for the
additional flows into the system not represented by point
source inflows or headwaters.  These inflows, which are assumed
to be uniformly distributed over the reach, are basically
groundwater inflows and/or distributed surface runoff that can
be assumed to be approximately constant through time.

    An important new feature to QUAL2E is that incremental
outflow along a reach may be modeled.  This option is useful
when field data show a decreasing flow rate in the downstream
direction indicating a surface flow contribution to groundwater.


    Each card, one for  each reach,  contains  the following
information:

    Reach Order or Number                          Columns 20-24

    Incremental Inflow  (cfs.                       Columns 25-31
       m3/sec) outflows are indicated
       with a minus  "-" sign.

    Temperature (F.  C)                             Columns 32-38

                              104

-------
    Dissolved Oxygen (mg/L)                        Columns 39-44

    BOD (mg/L)                                     Columns 45-50

    Conservative Mineral I                         Columns 51-56

    Conservative Mineral II                        Columns 57-62

    Conservative Mineral III                       Columns 63-68

    Arbitrary Non-Conservative                     Columns 69-74

    Coliform (No./lOO ml)                          Columns 75-80

    This group of cards must end with ENDATA8.


O.  Data Type 8A - Incremental Inflow - 2

    This group of cards is a continuation of Data Type 8. and
is required only if algae, the nitrogen series or the
phosphorus series are to be simulated.  Each card, one per
reach, contains the following information:


    Reach Order or Number                          Columns 20-24

    Chlorophyll 
-------
indicated in Figure A-5;  that is.  the junctions must be ordered
so that the element numbers lust downstream of the junction are
specified in ascending order.  In Figure A-5.  the downstream
element numbers for Junction 1.  2 and 3 are 29, 56.  and 64,
respectively.  There is one card per junction, and the
following information is  on each card:
    Junction Order or Number                       Columns 21-25

    Junction Names or Identification               Columns 35-50

    Order Number of the Last Element               Columns 56-60
    in the reach immediately
    upstream of the junction
    (see Figure A-5).  In the
    example, for Junction 1.
    the order number of the
    last element immediately
    upstream of the junction
    is number 17.  For
    Junction 2, it is number
    49.  For Junction 3.  it
    is number 43.

    Order Number of the First Element              Columns 66-70
    in the reach immediately down-
    stream from the junction.
    It is these numbers that must
    be arranged in ascending
    order.  Thus, for Figure A-5
    these order numbers are as
    follows:

                                 Downstream
               Junction          Element No.

                  1                  29
                  2                  56
                  3                  64
    Order Number of the Last Element               Columns 76-80
    in the last reach of the tribu-
    tary entering the junction.
    For Figure A-5 these order
    numbers for Junctions 1, 2,
    and 3 are 28, 55, and 63,
    respectively.

    This group of cards must end with ENDATA9. even if there
are no junctions in the system.

                               106

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                       Most Upstream
                           Point
                                          Reach
                                          Number
                   Computational
                   Element Number
FIGURE A-5.   STREAM NETWORK EXAMPLE  TO ILLUSTRATE DATA INPUT


                                 107

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Q.  Data Type 10 - Headwater Sources Data - 1
    This group of cards,  one per headwater,  defines the flow.
temperature,  dissolved oxygen.  BOD.  conservative mineral.
algae, nutrient,  coliform,  and  arbitrary nonconservative
concentrations of the headwater.  The following information is
on each card:
    Headwater Order or number
    Starting at Most Upstream Point

    Headwater Name or Identification

    Flow (cfs, m3/sec)

    Temperature (F. C)

    Dissolved Oxygen Concentration (mg/L)

    BOD Concentration (mg/L)

    Conservative Mineral I

    Conservative Mineral II

    Conservative Mineral III

    This group of cards must end with ENDATA10.


R.  Data Type 10A - Headwater Sources Data - 2

    This group of cards supplements the information in Data
Type 10. and is required if algae, the nitrogen series, the
phosphorus series, coliform. or arbitrary non-conservative are
to be simulated.  Each card, one per headwater, contains the
following data:
Columns 15-19


Columns 20-35

Columns 36-44

Columns 45-50

Columns 51-56

Columns 57-62

Columns 63-68

Columns 69-74

Columns 75-80
    Headwater Order or Number

    Arbitrary Non-Conservative

    Coliform. (No./lOO ml)

    Chlorophyll a. (ug/L)

    Organic Nitrogen as N (mg/L)

    Ammonia as N (mg/L)

    Nitrite as N (mg/L)
Columns 16-20

Columns 21-26

Columns 27-32

Columns 33-38

Columns 39-44

Columns 45-50

Columns 51-56
                              108

-------
    Nitrate as N (mg/L)

    Organic Phosphorus as P (mg/L)

    Dissolved Phosphorus as P (mg/L)
Columns 57-62

Columns 63-68

Columns 69-74
    This group of cards must end with ENDATA10A, even if algae.
nitrogen, phosphorus, coliform. or arbitrary non-conservative
are not simulated.
S.  Data Type 11 - Point Load - 1

    This group of cards is used to define point source inputs
and point withdrawals from the stream system.  Point sources
include both wasteloads and unsimulated tributary inflows.  One
card is required per inflow or withdrawal.  Each card describes
the percent of treatment (for wastewater treatment), inflow or
withdrawal, temperature, and dissolved oxygen, BOD,  and
conservative mineral concentrations.  They must be ordered
starting at the most upstream point.  The following information
is on each card:
    Point Load Order or Number

    Point Load Identification or Name

    Percent Treatment (use only if in-
    fluent BOD values are used)

    Point Load Inflow or Withdrawal
       (cfs. m3/sec) (a withdrawal must
       have a minus {"-") sign

    Temperature (F. C)

    Dissolved Oxygen Concentration (mg/L)

    BOD Concentration (mg/L)

    Conservative Mineral I

    Conservative Mineral II

    Conservative Mineral III
Columns 15-19

Columns 20-31

Columns 32-36


Columns 37-44



Columns 45-50

Columns 51-56

Columns 57-62

Columns 63-68

Columns 69-74

Columns 75-80
    This group of cards must end with ENDATA11,
                              109

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T.  Data Type 11A - Point Load - 2

    This group of cards supplements Data Type 11 and contains
the algal, nutrient, coliform and arbitrary non-conservative
concentrations of the point source loads.  This information is
necessary only if algae, the nitrogen series, the phosphorus
series, coliform, or the arbitrary non-conservative are to be
simulated.  Each card, one per waste load (withdrawal) contains
the following information:

    Point Load Order or Number                     Columns 16-20

    Arbitrary Non-Conservative                     Columns 21-26

    Coliform (No./lOO ml)                          Columns 27-32

    Chlorophyll a. (ug/L)                           Columns 33-38

    Organic Nitrate as N (mg/L)                    Columns 39-44

    Ammonia as N (mg/L)                            Columns 45-50

    Nitrite as N (mg/L)                            Columns 51-56

    Nitrate as N (mg/L)                            Columns 57-62

    Organic Phosphorus as P (mg/L)                 Columns 63-68

    Dissolved Phosphorus as P (mg/L)               Columns 69-74


    This group of cards must end with ENDATA11A, even if algae,
nitrogen, phosphorus, coliform, or arbitrary non-conservative
are not simulated.


U.  Data Type 12 - Dam Reaeration

    This group of cards  is required if oxygen  input from re-
aeration over dams  is to be modeled as a component of the dis-
solved  oxygen simulation.  Dam reaeration effects are estimated
from the empirical  equation attributed to Gameson (9).  The
following  inputs are  required:


    Dam Order or Number                            Columns 20-24

    Reach Number of Dam                            Columns 25-30

    Element Number Below Dam                       Columns 31-36
                              110

-------
    ADAM Coefficient:                              Columns 37-42
      ADAM = 1.25 for clear to
             slightly polluted waters
      ADAM = 1.00 for polluted water

    BDAM Coefficient:                              Columns 43-48
      BDAM = 1.00 for weir with
             free fall
      BDAM = 1.30 for step weirs or
             cascades

    Percent of Flow Over Dam                       Columns 49-54
    (as a fraction 0.0-1.0)

    Height of Dam (ft, m)                          Columns 55-60
    This group of cards must end with ENDATA12. even if oxygen
input from dam reaeration is not to be modeled.
V.  Data Type 13 - Downstream Boundary - 1
    This data card supplies the constituent concentrations at
the downstream boundary of the system.  It is required only if
specified in Data Type 1. card 8.  This feature of QUAL2E is
useful in modeling systems with large dispersion in the lower
reaches (e.g.. estuaries).  When downstream boundary
concentrations are supplied the solution generated by QUAL2E
will be constrained by this boundary condition.  If the
concentrations are not provided, the constituent concentrations
in the most downstream element will be computed in the normal
fashion using the zero gradient assumption (I. 11).

    Downstream boundary values for temperature, dissolved
oxygen. BOD, conservative mineral, coliform, and arbitrary
non-conservative are required as follows:
    Temperature (F, C)                             Columns 25-31

    Dissolved Oxygen (mg/L)                        Columns 32-38

    BOD Concentration (mg/L)                       Columns 39-45

    Conservative Mineral I                         Columns 46-52

    Conservative Mineral II                        Columns 53-59

    Conservative Mineral III                       Columns 60-66
                              111

-------
    Arbitrary Non-Conservative                     Columns 67-73

    Coliform (No./lOO ml)                          Columns 74-80
    This data group must end with an ENDATA13 card, even if the
fixed downstream boundary concentration option is not used in
the simulation.
W.  Data Type 13A - Downstream Boundary - 2

    This group of data (one card) is a continuation of Data
Type 13.  It is required only if the fixed downstream boundary
condition is used and if algae, the nitrogen series, the
phosphorus series are to be simulated.  This card contains the
downstream boundary concentrations for algae, nitrogen, and
phosphorus as follows:
    Chlorophyll a. (ug/L)                           Columns 25-31

    Organic Nitrogen as N (mg/L)                   Columns 32-38

    Ammonia as N (mg/L)                            Columns 39-45

    Nitrite as N (mg/L)                            Columns 46-52

    Nitrate as N (mg/L)                            Columns 53-59

    Organic Phosphorus as P (mg/L)                 Columns 60-66

    Dissolved Phosphorus as P (mg/L)               Columns 67-73


    This data group must end with an ENDATA13A card, even if
the fixed downstream boundary condition is not used, and if
algae, nitrogen, or phosphorus are not simulated.


X.  Climatological Data

    Climatological data are required for the following cases:

    1.  Temperature simulations, both steady-state and dynamic,
    2.  Dynamic simulations where algae is being simulated, and
        temperature is not.


If neither temperature nor dynamic algae are being simulated,
these cards may be omitted.
                               112

-------
    For steady-state temperature simulation, only one card is
required which gives average values of the climatological
data.  For dynamic simulation, each card represents readings at
three hour intervals, chronologically ordered.  There must be a
sufficient number of cards to cover the time period specified
for the simulation (Data Type 1, card 13, maximum route time).
The following information is on each card.
    Month                                          Columns 18-19

    Day                                            Columns 21-22

    Year (last two digits)                         Columns 24-25

    Hour of Day                                    Columns 26-30

    Net Solar Radiation*                           Columns 31-40
       (BTU/ft2-hr. Langleys/hour)

    Cloudiness**, fraction in                      Columns 41-48
       tenths of cloud cover

    Dry Bulb Temperature** (F, C)                  Columns 49-56

    Wet Bulb Temperature** (F, C)                  Columns 57-64

    Barometric pressure                            Columns 65-72
       (inches Hg. millibars)

    Wind speed**   (ft/sec, m/sec)                  Columns 73-80
          Required only if dynamic algae is simulated and
          temperature is not.

          Required if temperature is simulated.
    There is no end card for the climatological data.
Y.  Plot Reach Data

    This data type is required if the plotting option for
DO/BOD is selected (Data Type 1, card 7. PLOT DO/BOD).  The
following information is required for QUAL2E to produce a line
printer plot.
                              113

-------
    1. Card I - BEGIN RCH
       Reach number at which plot                  Columns 11-15
       is to begin
    2. Card 2 - PLOT RCH

       a. Reach numbers in their                   Columns 11-15
          input order (1,  2.  3..NREACH)            Columns 16-20
                                                           21-26
       b. If a reach is not to be                           etc.
          plotted,  (i.e..  a tributary)                     76-80
          replace the reach number
          with a zero.

       c. Use additional PLOT RCH cards
          if there  are more than 14
          reaches in the system.
    3.  Additional plots can be obtained by repeating the
       seguence of BEGIN RCH and PLOT RCH cards.
    As an example of the plotting option, suppose that for the
river system shown in Figure A-5, one.wishes to obtain two
DO/BOD plots: one for the main stream (Reaches 1, 2, 5. 6. 10,
and 11) and one for the second tributary (Reaches 7 and 9).
The plot data would appear in the following order.

       BEGIN RCH 1
       PLOT RCH 120056000  10  11
       BEGIN RCH 7
       PLOT RCH 00000070900

    No ENDATA card is required for the PLOT information.
Z.  Summary

    Constructing a consistent and correct input data set for a
QUAL2E simulation must be done with care.  This user's guide is
designed to assist the user in this process.  It has been
NCASI's and EPA's experience that two of the most frequently
made errors in constructing a QUAL2E input data set are:


    (a)   Using a numerical value that is inconsistent with the
          units option selected, and


    (b)   Not adhering to the 4 character input codes for Data
          Types 1 and 1A.

                              114

-------
     As  an  aid  to  the  units  problem.  Table  A-l  is  included in
 this report.   It  provides a complete summary of all  the input
 variables  whose dimensions  are  dependent  on whether  English or
 metric  units are  selected.   Finally,  the  user  is  encouraged to
 check and  recheck the input codes  in Data  Types 1 and 1A for
 accuracy,  especially  the  codes  for cards  10 and 11 of Data Type
 1  (i.e..  "NUMB" and  "NUM_").
                     REFERENCES  -  APPENDIX A
 1.    "A Review of the Mathematical Water Quality Model QUAL-2
       and Guidance for its Use", NCASI Technical Bulletin No.
       338. October 1980 (Revised No. 391. December. 1982).

 2.    "Computer Program Documentation for the Stream Quality
       Model QUAL-2". prepared for Southeast Michigan Council
       of Governments (SEMCOG). Detroit, Michigan (July, 1977).

 3.    "Users Manual for the Stream Quality Model QUAL-2".
       prepared for Southeast Michigan Council of Governments
       (SEMCOG). Detroit.  Michigan (July. 1977).

 4.    "QUAL-TX User's Manual (Draft)". Texas Water Development
       Board. Austin. Texas (June. 1981).

 5.    "Calibration and Application of QUAL-2 to the Lower Win-
       ooski River:  Preliminary Studies", prepared for state
       of Vermont by Meta  Systems. Inc. (July. 1979).

 6.    Norton. W.R.. et al. "Computer Program Documentation for
       the Stream Quality Model-QUAL-2". for EPA Contract 68-
       01-1869 (August.  1974).

 7.    Patterson. D.. e_t a_l, "Water Pollution Investigation
       Lower Green Bay and Lower Fox River". Wisconsin DNR.
       (EPA 68-01-1572). (June. 1975).

 8.    "A Study of the Selection, Calibration, and Verification
       of Mathematical Water Quality Models". NCASI Technical
       Bulletin No. 367. (March 1982).

 9.    "Rates, Constants,  and Kinetic Formulations in Surface
       Water Quality Modeling". USEPA. Athens. Georgia. NTIS
       PB-290938 (March. 1978).

10.    Fisher, H.B.. E.J.  List. R.C.Y. Koh,  J. Imberger. N.H.
       Brooks, Mixing in Inland and Coastal  Waters. Academic
       Press, 1979.
                               115

-------
TABLE A-l.
LIST OF QUAL2E INPUT VARIABLES THAT  ARE
    ENGLISH/METRIC UNIT DEPENDENT

CARD
DATA OR
TYPB klM
1 8
8
1 11
1 IS
15
1 16
1A 6
6
1A 7
1A 9
2 all
• 11
5 all
(Discharge
Coefficient)
S all
(Trapezoidal)
6 all
6 all
6 all
6A all
CB all
7 all
B all
10 all
11 all

12 all
13 1
LCD all
VARIABLE DESCRIPTION
Input Units specification
Output Units Specification
Length of Computational Element
Evaporation coeftlcient
Evaporation Coefficient
Ban In Elevation
Linear Algal Extinction Coeff.
Non-linear Algal Extinction
Coefficient
Light Saturation Coefficient
Total Daily Solar Radiation
River Mile/km to Head of Reach
River Mile/da to End of Reach
Coefficient Flow for Velocity
Exponent Flow tot Velocity
Coeftlcient on Flow for Depth
Exponent on Flow for Depth
Bottom Width of Channel
SOD Rate
Option 7 for kj
Coefficient on flow for Kj
Exponent on flow for k2
Option 8 for Kj
Coefficient for Tsivoglou Eq.
Slope of Energy Gradient
Benthal Source Rate for
An>monla-N
Benthal Source Rate for
Phosphorus
Algal settling Rate
Non-algal Extinction Coefficient
Arbitrary Nonconeervative
Benthal Source Rate
Initial Condition-Temperature
Incremental Inflow
Flow Rate
Temperature
Headwater Conditions
Flow Rate
Temperature
Point Source/Withdrawal
Flow Rate
Temperature
Height of Dam
Downstream Boundary-Temperature
Solar Radiation
Dry Bulb Temperature
Het Bulb Temperature
Barometric pressure
Wind Speed
FORTRAN
CODE NAH8
METRIC
METOUT
DELX
AE
BE
ELEV
EXALG1
EXALG2
CKL
SONET
RMTHOR
RMTEOR
COEFQV
Exrogv
COEFOH
EXPOQH
WIDTH
CM
COEQK2
EXPQK2
COEQK2
EXPQK2
GNH3
SPHOS
ALOSET
EXCOEF
8RCANC
TINIT
01
TI
HWKLOH
HHTEHP

HSFLOH
HFTEHP
HDAH
LBTEHP
6OI.HR
DRYBI.B
METBLB
ATMPR
HIND
UNITS
ENGLISH
O
0
mile
ft/hr-ln Hq
ft/hr-in Hg-mph
ft
1/ft-ug chla/L
l/ft-(ug chla/L2/3
BTU/tt2
BTU/ft2
mile
mile
Consistent with
flow, velocity
and depth in
cts. fps. ft
respectively
ft
gm/ft2-day
Consistent with
flow in cfs
I/ft
ft/ft
Mg/ft2-day
mg/ft2-day
ft/day
I/day
»g/ft2-day
F
cfs
F
cfs
F

cfs
F
ft
F
BTO/ft2-hr
F
F
in Hg
ft/sec

METRIC
1
1
kilometer
m/hr-mbar
m/hr-mbar-m/sec
meters
1/m-ug chla/L
l/m-(ug chla/L2/3)
langley/mln
langleys
kilometer
kilometer
Consistent with
flow, velocity, and
depth in cms, mps.
• respectively
meters
gm/m2-day
Consistent with
flow in cms
I/meter
meter/mater
mg/m2-day
mg/mz-day
m/day
I/meter
•g/m2-day
C
cms
C
cms
C

cms
C
meters
C
langleys/hr
C
C
mbar
•/sec
                         116

-------
11.     Arden.  B.W.  and K.N.  Astill.  Numerical Algorithms: Ori-
       gins and Applications,  Addison-Wesley. 1970.

12.     Department of Scientific and Industrial Research. Ef-
       fects of Polluting Discharge on the Thames Estuary. Wa-
       ter Pollution Research Technical Paper No. 11. Her Maj-
       esty's Stationery Office. London. 1964.

13.     Scavia. D. and R.A. Park, "Documentation at Selected
       Constructs and Parameter Values in the Aquatic Model
       CLEANER". Ecological Modelling. Vol. 2. pp. 33-58, 1976.

14.     Bannister. T.T., "Production Equations in Terms of Chlo-
       rophyll Concentration.  Quantum Yield, and Upper Limit to
       Production". Limnology and Oceanography.  Vol. 19, No. 1.
       pp. 1-12 (Jan., 1974).
                             117

-------
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