WATER POLLUTION CONTROL RESEARCH SERIES • 16130 DJH 01/71
             A PREDICTIVE MODEL
         FOR THERMAL STRATIFICATION
      AND WATER QUALITY IN RESERVOIRS
ENVIRONMENTAL PROTECTION AGENCY • WATER QUALITY OFFICE

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           WATER POLLUTION CONTROL RESEARCH SERIES

The Water Pollution Control Research Series describes the
results and progress in the control and abatement of pollu-
tion of our Nation's waters.  They provide a central source
of information on the research, development, and demon-
stration activities of the Water Quality Office, Environ-
mental Protection Agency, through inhouse research and grants
and contracts with Federal, State, and local agencies, re-
search institutions, and industrial organizations.

Inquiries pertaining to the Water Pollution Control Research
Reports should be directed to the Head, Project Reports
System, Office of Research and Development, Water Quality
Office, Environmental Protection Agency, Washington, D.C. 20242.

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       A PREDICTIVE MODEL FOR  THERMAL STRATIFICATION
               AND WATER QUALITY  IN RESERVOIRS


                            by


                       Mark Markofsky

                            and

                    Donald R. F.  Harleman
                 RALPH M. PARSONS  LABORATORY
            FOR WATER RESOURCES  AND HYDRODYNAMICS
               Department of  Civil Engineering
            Massachusetts Institute of Technology
               Cambridge, Massachusetts  02139
                            for  the

                    WATER QUALITY OFFICE
               ENVIRONMENTAL  PROTECTION AGENCY

                 Research Grant No.  16130 DJH


                        January,  1971
For sale by the Superintendent ol Documents, U.S. Government Printing Office, Washington, D.C., 20402 - Price $2

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                 EPA Review Notice
This report has been reviewed by the Water Quality Office,
EPA, and approved for publication.  Approval does not signi-
fy that the contents necessarily reflect the views and poli-
cies of the Environmental Protection Agency, nor does mention
of trade names or commercial products constitute endorsement
or recommendation for use.

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                              FOREWARD
This is the third report issued in conjunction with a continuing research

program on thermal stratification and water quality in lakes and reservoirs.

The previous reports are as follows:

1.  Dake, J.M.K.  and D.R.F. Harleman,  "An Analytical and Experimental Investi-
    gation of Thermal Stratification  in Lakes and Ponds",  M.I.T.  Hydrodynamics
    Laboratory Technical Report No. 99, September 1966.   (Portions of this
    report have also been published by the same authors  under the title:
    "Thermal Stratification in Lakes:   Analytical and Laboratory Studies",
    Water Resources Research,  Vol. 5,  No.  2,  April 1969,  pp.  484-495.)

2.  Huber, W.C.  and D.R.F.  Harleman,  "Laboratory and Analytical Studies of
    the Thermal Stratification of Reservoirs",  M.I.T.  Hydrodynamics Laboratory
    Technical Report No.  112,  October  1968.
                                  —2—

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                             ABSTRACT
Previous research on thermal stratification in reservoirs has provided
analytical methods for predicting the thermal structure and internal
flow field of a reservoir characterized by horizontal isotherms.  A
one-dimensional analytical thermal stratification prediction method
developed by Huber and Harleman is reviewed and modififed to include
the time required for the inflowing water to reach the dam face.

Various "dispersion" approaches to water quality prediction, which
depend on empirically determined dispersion coefficients, are reviewed.
Application of these methods to water quality prediction in a strat-
ified reservoir is discarded because of their inability to account for
the transient nature of the internal flow pattern generated by changing
meteorological and hydrological conditions.

A one-dimensional water quality mathematical model is developed which
incorporates the internal flow pattern predicted for a stratified
reservoir from the temperature model of Huber and Harleman.   The
water quality parameters of rivers and streams entering the reser-
voir are assumed to be known.  After initial mixing, the entering
water seeks its own density level within the horizontal stratifica-
tion field of the reservoir.  The outflow of water through the reservoir
outlet is assumed to come from a withdrawal layer whose vertical thick-
ness is a function of the time-dependent vertical temperature-density
gradient.  The water quality model is designed to predict the concen-
tration of particular water quality parameters in the outflow water
as a function of time.  In the case of non-conservative pollutants,
the model incorporates generation and/or decay rates for the substance
under consideration.

The mathematical model is tested by comparisons with measurements of
outlet concentrations resulting from pulse injections of a conservative
tracer into a laboratory reservoir with time varying inflows, outflows
and insolation.  Good agreement is obtained between measured and pre-
dicted concentration values.  Pulse injection tests of a conservative
tracer in Fontana Reservoir are simulated by means of the mathematical
model in order to illustrate the flowthrough time characteristics of
a stratified reservoir.  Field data for comparison with the theory
is not available.

The application of the mathematical model to a field case of practical
interest is demonstrated by solving the coupled set of water quality
equations for B.O.D. and D.O. predictions in Fontana Reservoir.  Field
measurements of D.O. both within the reservoir and at the outlet of
Fontana are available for the year 1966; however, measurements of
incoming B.O.D. and of the long-term B.O.D decay rate were not made.
                                —3—

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Direct comparisons of the water quality model predictions with the
field measurements of dissolved oxygen are limited by the lack of
input data.   A sensitivity analysis  to various assumptions on the
input data is made in order to  illustrate the mechanics of the water
quality prediction model.   It  is  concluded that the model is capable
of predicting the effect  of reservoir  impoundments on water quality.

This report  was submitted in fulfillment of Research Grant No. 16130 DJH
between the  Water Quality Office,  Environmental Protection Agency and the
Massachusetts Institute of Technology.

Key Words:  reservoir water quality;  thermal stratification in reservoirs;
            biochemical oxygen  demand  in reservoirs,  dissolved oxygen in
            reservoirs.
                              -4-

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                           ACKNOWLEDGEMENT
        This investigation was supported by the Water Quality Office,
Environmental Protection Agency, under Research Grant No. 16130 DJH as
part of a research program entitled "Thermal Stratification and Reservoir
Water Quality".  The project officer was Mr. Frank Rainwater, Chief, National
Thermal Pollution Research Program, FWQA Pacific Northwest Water Laboratory
at Corvallis, Oregon.  The cooperation of Mr. Rainwater and of Mr. Bruce A.
Tichenor is gratefully acknowledged.
        The authors wish to express their appreciation to Mr. Rex A. Elder,
Chief of the Engineering Laboratory Branch of the T.V.A. Division of Water
Control Planning and to Dr. W. 0. Wunderlich of the same organization for
their cooperation and assistance in supplying the field data for Fontana
Reservoir.
        Mr. Patrick Ryan, Research Assistant in the Water Resources and
Hydrodynamics Laboratory, made substantial contributions in both the
analytical and experimental phases of the research program.  Appreciation
is also extended to Messrs. Edward McCaffrey and Roy Milley for assistance
in the instrumentation and construction of experimental equipment.
        The research program was administered at M.I.T. under DSR 71381 and
72325.  Numerical computations were done at the M.I.T. Information Processing
Service Center.  Our thanks to Miss Kathleen Emperor who typed most of the
report and to Mrs. Barbara Yasney for assistance with the drafting.
        The material contained in this report was submitted by Mr. Markofsky
in partial fulfillment of the requirements for the degree of Doctor of
Philosophy at M.I.T.
                                -5-

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                          TABLE OF CONTENTS










TITLE PAGE	^




FOREWARD 	 2




ABSTRACT 	 3




ACKNOWLEDGEMENT	5




C'HAPTER 1.   INTRODUCTION	H




         1.1  Introduction 	 H




CHAPTER 2.   INTRODUCTION AND  BASIC  CONCEPTS -  THE  TEMPERATURE MODEL	15




         2.1  Introduction and  Basic  Concepts	15




         2.2  The  Exact  Equations  Governing Pollutant  Concentration




              in a Stratified Reservoir	19




         2.3  Approximations  to  the Full  Set of  Equations	26




         2.3.1  Marker and Cell  Technique	26




         2.3.2  The  Boussinesq Approximation 	 27




         2.3.3  Solutions  for Various  Systems  by Means  of  a  Dispersion




                Coefficient	28




         2.3.3.1  Constant Longitudinal Dispersion Coefficient 	 31




         2.3.3.2  The Dispersion Coefficient as  a  Function of Time	32




         2.3.3.3  The Dispersion Coefficient as  an Eddy Diffusivity	35




         2.3.3.4  Evaluation  of  the Dispersion Coefficient Approach	37




         2.3.4  A  Solution Involving  the  Temperature Equation	37




         2.4   The  Temperature Model	38




         2.4.1  The  Governing Equations	38




         2.4.2  Reservoir  Schematization  and the Velocity  Field.  .  	 47




         2.4.3  Mixing at  the Reservoir Entrance	•  •  .  .  .56
                                -6-

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         2.4.4 Lag Time Determination	  59




         2.4.4.1 The Time for the Incoming Water to Reach Its Own




                 Density Level	  59




         2.4.4.2 Horizontal Travel Time	  66




         2.4.5 Surface Instabilities and Surface Mixing	  66




         2.5 The Method of Solution of the Temperature Model	  68




         2.5.1 The Finite Element Approach	  68




         2.5.2 Stability of the Explicit Scheme-Numerical




               Dispersion	  73




CHAPTER 3. THE WATER QUALITY MODEL	  77




         3.1 The Water Quality Model	  77




         3.1.1 Introduction	  77




         3. 2 Literature Review	  78




         3.3 The Governing Equation for the Water Quality Model	  85




         3. 4 Examples	  90




         3.4.1 The Dissolved Oxygen and B.O.D. Model	  90




         3.4.1.1 Governing Equations	  90




         3.4.1.2 Formulation of the Numerical Solution	  98




         3.4.1.3 Required Inputs to the D.O. and B.O.D.  Prediction




                 Model	106




         3.4.2.1 Application of the Water Quality Model to a Pulse




                 Injection of a Conservative Tracer	]_07




         3.4.2.2 Inputs to the Pulse Injection Model	]_]_0




         3.4.2.3 Discussion of the Pulse Injection Solution	]_]_Q




         3.5 Review of the Mathematical Models. 	

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CHAPTER 4.  LABORATORY EXPERIMENTS



         4. 1 Laboratory Equipment ..... • .............................. H^



         4.2 Experimental Procedures ......... • ....................... 127



         4.3 Inputs to the Mathematical Model ........................ 129



         4.3.1 Evaluation of tne Outflow Withdrawal Layer Thickness. .131



         4.3.2 Thickness of the Inflowing Layers,  Ah, for Lag Time



               Determination ......................................... 134



         4.4 Experimental Results .................................... 134



         4.4.1 Runs With Variable Insolation and Flow Rates, Constant



               Surface Elevation .................................... -134



         4.4.1.1 Sensitivity to a Cutoff Criterion for the Upper



                 Limit of the Withdrawal Layer  When No Density



                 Gradient Exists at the Outlet ...................... -143



         4.4.1.2 Sensitivity to a Gaussian vs.  Uniform Surface



                 Distribution and the Inflow Standard Deviation,  a ,



                 for Surface Inflow ................................. J.47



         4.4.1.3 Sensitivity to the Entance Mixing Ratio, r ........ .153
                                                           m


         4.4.1.4 Numerical Dispersion ............... .
         4.4.2 Discussion of the Two Regaining Sets of Experiments. . 155



         4.4.2.1 Constant Inflow and Outflow,  No Insolation ......... 157



         4.4.2,2 Variable Inflow.  Insolation and Surface Elevation. . 3.65



         4.5 Summary of Experimental Results ........ . ....... . ....... ^53



CHAPTER 5.  APPLICATION OF THE WATER QUALITY AND TEMPERATURE MODELS



           TO FONTANA RESERVOIR ..................................... 2.72

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         5.1 Introduction	172




         5.2 Temperature Prediction	174




         5.2.1 Inputs to the Temperature Model	174




         5.2.1.1 Inflow and Outflow Rates and Temperatures	174




         5.2.1.2 Solar Insolation and Related Parameters	175




         5.2.1.3 Withdrawal Layer Thickness	176




         5.2.1.4 Other Parameters...	178




         5.2.2 Temperature Predictions	179




         5.2.2.1 Results and Conclusions for the Temperature




                 Model	191




         5. 3 Water Quality Prediction	193




         5.3.1 Conservative Tracer	193




         5.3.2 Dissolved Oxygen Predictions for Fontana




               Reservoir	198




         5.3.2.1 Inputs to the Mathematical Model	198




         5.3.2.2 Comparison with D.O.  Measurements in Fontana




                 Reservoir	201




CHAPTER 6.  CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE RESEARCH.215




         6.1 The Thermal Stratification Phenomena	215




         6. 2 Temperature Predictions	215




         6.3 Concentration Predictions	216




         6.3.1 Laboratory Experiments	216




         6.3.2 Field Results	217




         6.4 Recommendations for Future Research	218

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         6.4.1 Improvement of the Mathematical Model	218




         6.4.2 Laboratory and Field Research	219





CHAPTER 7. BIBLIOGRAPHY	 .221




APPENDIX I. THE COMPUTER PROGRAM	 . 226




APPENDIX II. INPUT VARIABLES TO THE COMPUTER PROGRAM	253




APPENDIX III. SAMPLE INPUT DATA FOR FONTANA D.O.  PREDICTIONS	261




APPENDIX IV.LIST OF FIGURES AND TABLES	270




APPENDIX V.  DEFINITION OF NOTATION	276
                               -10-

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CHAPTER 1.  INTRODUCTION
1.1   Introduction
          The construction of an impoundment on a river usually leads to
   substantial changes in water quality within the reservoir and in the
   river downstream of the reservoir.  These changes reflect modifications
   of the physical, chemical, and biological regimes which are associated
   with the increase in depth,surface  area and the reduction of velocity.
   The thermal structure of the reservoir and the temperature of the out-
   let water are important as primary  water quality factors.  In addition,
   the changing thermal structure has  a dominant effect on the detention
   time which is related to the internal flow characteristics within the
   reservoir.
          Thermal stratification occurs in practically all reservoir
   impoundments. In shallow "run of the river" reservoirs the isotherms
   tend to be tilted in the downstream direction and the stratification
   is relatively weak. In deep reservoirs, having a storage volume which
   is large compared to the annual through-flow, the isotherms are
   horizontal during most of the year  and strong stratification may
   develop  during certain seasons. This investigation is concerned
   mainly with the latter type of reservoir in which temperature and
   water quality parameters are functions of depth and time.
         The thermal stratification process is governed by a heat
   balance involving solar radiation,  surface losses by evaporation and
   conduction, and convective transfer of inflows and outflows. As a
   result of research in the past few years, the stratification process
   is now understood to the extent that reasonable predictions of the
                                 -11-

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internal temperature distributions and outflow temperature can be made




for the purpose of planning new facilities or the operation of




existing reservoirs. The thermal stratification,  through the density




variation, has a predominant influence on the flow pattern and circu-




lation within a reservoir.  Vertical motions are inhibited in density-




stratified reservoirs and outflows tend to be drawn from a layer of




restricted depth near the outlet. The flow pattern may involve




numerous counterflowing currents. This complicated internal current




structure is important in the convective and dispersive processes for




any substance introduced into the reservoir.




      Many water quality factors other than temperature are important




in a reservoir. The majority of these are affected by the distribution,




dilution, and detention time in the reservoir. An understanding of




the internal flow structure of a stratified reservoir is a pre-




requisite to rational concentration predictions of various water




quality parameters. The traditional methods of analysis, in which




the concentration is assumed to depend on only the longitudinal




coordinate, is inappropriate in a stratified reservoir because the




localized horizontal currents may restrict the particular water




quality parameter to a certain level within the reservoir for a long




period of time.




      The dissolved oxygen structure of a reservoir will be a




primary consideration in water quality because the ecological balance




in a reservoir is very sensitive to dissolved oxygen levels. The




oxygen balance in a reservoir is dependent on numerous physical and





                               -12-

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biological factors which include convective transport by internal cur-
rents, atmospheric reaeration at the surface, photosynthetic oxygen
sources associated with plant life, oxygen demands of river inflows,
bottom deposits, respiration and decomposition of aquatic organisms.
Thermally stratified reservoirs exhibit oxygen stratification with an
oxygen rich surface layer which is mixed by winds and convection cur-
rents.  The lower layers of a reservoir are often deficient in oxygen
because the oxygen demand of internal organic material exceeds the
oxygen transfer from the surface layer.  In addition, the biological
and mass transfer processes are sensitive to temperature and thus the
oxygen balance will depend on the thermal structure of the reservoir.
In view of the oxygen stratification in reservoirs, the classical
Streeter-Phelps analysis for streams, which assumes vertically mixed
conditions, is not applicable in stratified reservoirs.  The oxygen
balance should include the vertical variation of dissolved oxygen as
influenced by internal currents and the vertical distribution of oxy-
gen sources and sinks.
       In the following chapters a mathematical model for predicting
the thermal stratification phenomena in a horizontally stratified
reservoir is presented.  The temperature model is based on modifica-
tions to the work of Huber and Harleman (18) in an earlier phase of
the M.I.T. reservoir research program.  The primary objective of the
present investigation is the development of a water quality mathemati-
cal model which is coupled with the thermal stratification prediction
model.  The water quality model is initially verified by comparing
the results with measurements made under controlled laboratory condi-
                              -13-

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 tions.  A series of  tests were made on  the prediction of the  transient




 reservoir outlet concentrations which resulted from pulse injections




 of a conservative tracer into a laboratory reservoir.  Predictions are




 also given for a simulated pulse injection of a conservative tracer




 into Fontana Reservoir in the TVA system.  In this context the concept




 of detention time in a stratified reservoir is discussed.   Dissolved




 oxygen predictions are also presented for Fontana Reservoir and compared




with available field data.
                                 -14-

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CHAPTER 2. INTRODUCTION AND BASIC CONCEPTS - THE TEMPERATURE MODEL
2.1  Introduction and Basic Concepts
        The problem of predicting the temporal variation of the concen-
tration of a particular water quality parameter in the outlet and at
all points within a stratified reservoir is very difficult because of
the complicated flow patterns which are generated.  Additional compli-
cations arise if one considers a parameter such as dissolved oxygen
(DO) which experiences a time dependent decay due to biological oxygen
demand (BOD) and chemical oxygen demand (COD).
        Previous work on the concentration distribution of a conserva-
tive tracer (48) and DO (54) in a stratified reservoir has attempted
to circumvent the internal flow problem.  (These papers will be discuss-
ed in detail in Sections 2.3.3.3 and 3.2 respectively.)
        As a stream enters the main body of a thermally stratified
reservoir tLere is a certain amount of mixing and entrainment which
takes place.  If the stream temperature differs from that of the reser-
voir water with which it is mixing, the effective inflow rate and temp-
erature will depend on the amount of entrainment which takes place at
the entrance.   This "mixed" incoming water will then seek its own den-
sity level within the reservoir.  If this water is warmer than the
surface water it will enter and flow along the reservoir surface.  If
it is colder than any of the water within the reservoir, it will flow
along the bottom until it reaches the deepest portion of the reservoir.
If the incoming water is at some intermediate temperature, it will flow
along the bottom until it reaches an elevation corresponding to its
own density level, at which point it will begin to move horizontally.
                             -15-

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        As the vertical density gradient due to temperature at the out-




let increases, the vertical zone of withdrawal, 6, from the reservoir




decreases.  This gives rise to a complicated series of flows and counter-




flows within the reservoir.




        These phenomena are illustrated in Figure 2.1.




        There are many time dependent factors which are involved in




altering  the thermal structure of a reservoir.  Besides the changing




temperature of the inflowing water, there are surface and internal heat




sources due to incoming solar radiation.  Evaporative cooling, back




radiation and possible losses through the reservoir perimeter are also




important contributors to the transient thermal structure.   In addition,




the operation of the reservoir discharge will control the amount of




heat advected from the reservoir.  Due to the changing temperature field,




any pollutant or water quality parameter contained in the inflowing




water will enter the reservoir at different elevations throughout the




year, depending on the temperature of the inflowing water,  and the ther-




mal structure of the reservoir at that time.




        The water which enters the reservoir in the spring and early




summer is usually warmer than the water within the lake; it tends to




enter at  the surface and remain in the reservoir for a long period of




time.  The water entering in the late summer and fall is usually colder




than the reservoir surface water, consequently, it enters at some inter-




mediate depth.   This colder water may find its way to the reservoir out-




let much earlier than the warmer water which entered before it.  This




fact is important because the majority of the water quality parameters




are affected by the length of time which the water spends in the reservoir.



                               -16-

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                                                                                             inflow
                                                                                              from
                                                                                              stream
      outflow veloci
        field
  outf lov
1  Warm Water Inflow

2  Intermediate Inflow
3  Cold Water Inflow
                   FIGURE 2.1 THE CHANGING INFLOW LEVEL AND WITHDRAWAL DISTRIBUTION

                              IN A STRATIFIED RESERVOIR

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        The main objective of this investigation is to develop a method




of predicting the temporal variation of the concentration distribution




of a particular pollutant or water quality parameter in a stratified




reservoir.  In order to do this, a mechanism for evaluating the reser-




voir entrance mixing, the internal flow field and dispersion character-




istics must be developed.  It should be clear from the previous discus-




sion that these phenomena are related to the changing temperature struc-




ture within the reservoir.  Therefore, before concentration predictions




can be made, a method of predicting the temperature field as a function




of time is needed.  A major contribution has been made by Huber and




Harleman  (13) who have developed a one-dimensional model for predicting




the transient temperature and internal flow field in a deep reservoir




having horizontal isotherms.




        This investigation is also limited to deep reservoirs with hori-




zontal isotherms.  3y means of this assumption the mass transport phen-




omena can alsp be treated in a one-dimensional approach similar to that




taken by Huber and Harleman in treating the thermal prediction problem.




In addition, the temperature and mass transport equations are coupled




in that the same velocity field used in the temperature model can be




used in the concentration prediction model.




        The temperature model was verified by Huber and Harleman using




both laboratory and field data.  The mass transport model developed here




is verified in the laboratory by means of a pulse injection of a con-




servative tracer into a laboratory reservoir.  This type of experiment




was run to  further check  the assumptions made in the  temperature model
                              -18-

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and to fully develop the method of analyzing this type of experiment



because it is a potentially valuable field technique.  The mathematical


model is also applied to DO and BOD prediction in Fontana Reservoir in


the TVA system.



        In the following section the exact equations governing the pre-


diction of the temporal and spatial distribution of conservative and


non-conservative substances in a stratified reservoir are presented.


The approximations and assumptions necessary to solve these equations


follows.  Since the prediction of the temperature field will be shown


to be most crucial, the model of Huber and Harleman, along with certain


modifications, will be discussed in detail in this chapter.  In Chapter


3, the water quality prediction model will be developed.   This is applied


to laboratory tests in Chapter 4 and field data in Chapter 5.


2.2  The Exact Equations Governing Pollutant Concentration


     Predictions in a Stratified Reservoir


        In order to solve for the concentration of a particular pollu-


tant in a stratified reservoir one must have knowledge of the flow


field, density distribution and conservation of mass for all substances


under consideration.  Mathematically, this involves the simultaneous



solution of the equation of motion:
           3u.       3u.
            i  . ~    i
+ u,
           3t     j  3x.
               PgjL -
+
2-
3 u.
1
9x.2
J


" / 1 1 N
9x. (ui V
— '
                                                                   (2-1)
continuity:
        3t
                 3u
   -i  +  u.  ^-  =  0

   ^      J  3XJ
                                                                   (2-2)
                               -19-

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The conservation of heat equation:
3T  ,  -  9T
                                , T
                                                  sources    sinks

                                                                  -


                                                                     (2-3)
the equation of state:
p =  p(T, dissolved substances)
                                                                  (2-4)
and conservation of mass:
3t
        j   3x
             J
                .   D
                       2-

                      i_£^
                    M    2
                                                  sources    sinks
                                     3x
                                       J
                                                                    (2_5)
for each pollutant under investigation where




        u. = u.  (x,y,z,t) = velocity in the i   direction (i = 1,2,3)



        at time  t.




        u. = u.  (x,y,z,t) = velocity in the j   direction (j = 1,2,3)



        at time  t.




        p = p (x,yjZ,t)  = pressure field at time t.




        P =  p(x,y,z,t)  = the density field at time t.



        g = acceleration of gravity.





        Ui'' ui'  = Ui'  (X'^'2'^'  u-'  (x>y>z»t) = turbulent velocity
              •J                     J


        fluctuation in  the i and j direction.




        T = T (x,y,z,t)  = temperature field at time t.




        T  = T  (x,y,z,t) = turbulent temperature fluctuation at time t.



        y = IJ(T)  = dynamic viscosity.
                                   -20-

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        D  = molecular diffusivity of heat.


        c  = specific heat of water.



        sources  = sources of heat per unit volume per unit time.


        sinks  = sinks of heat per unit volume per unit time.


        c = c(x,y,z,t) = concentration field of a particular pollutant


        of time t.


        c' = c'(x,y,z,t) = turbulent concentration fluctuation.


        D  = molecular diffusivity of mass.


        source   = source of mass per unit volume per unit time.
               m                  r               r


        sink  = sink of mass per unit volume per unit time.
            m                f               f


        The last term in the equations of motion and the terms involv-


ing the cross products u.'T' and u.'c' in Equations 2.3 and 2.5 should
                        J         J

be included only if the flow is turbulent.  As Koh (23) has demonstrated,


the amount of work, W, required to vertically transport a particle of


fluid of volume, V, from depth y  to y, (Figure 2.2) in a stably strat-


ified fluid (i.e.  — £- <0) is given by
        W = V  f1   [p(yQ) -p(y)] g dy                             (2-6)
        Since this work is always positive whether y  > y, ,  y  < y, , any


vertical motion requires an addition of energy, no matter how slowly the


motion is carried out.  Thus, the existence of a vertical density strat-


ification tends to inhibit vertical motion.  The ability of a density


stratification to inhibit turbulence in the vertical direction will de-


pend on the magnitude of -^- .  This is usually expressed in the  form of
                              -21-

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to
N)
         >,


         J=
         a)
         "O
                         Work Input W
Density (p)
                    FIGURE 2.2 WORK INPUT TO DISPLACE A PARTICLE  OF FLUID


                               IN A STABLY STRATIFIED FLUID

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a Richardson number.


        The question of turbulent vs. laminar flow in a stratified res-


ervoir is most crucial since this will dictate whether the turbulent


fluctuation terms in Equations 2-1, 2-3 and 2-5, which considerably


complicate the problem, should be considered.  A basic premise of this


investigation is that the existence of horizontal isotherms in a reser-


voir suppresses vertical motion to the extent that turbulent transport


of momentum, heat or mass can be neglected.  The only exceptions will


be in the case of entrance mixing and that of a surface layer instabil-


ity caused by evaporative cooling which results in an unstable density


gradient.  As will be shown in Sections 2.4.3 and 2.4.5 these two excep-


tions can be handled quite satisfactorily without specifying the exact


form of the turbulence generated in each case.  The ultimate verifica-


tion of this assumption will be the ability or inability of a theory


neglecting turbulence to match observed values.


        Orlob and Selna (36 )> in developing a thermal prediction model


for stratified reservoirs, employ a dispersion coefficient which is

               4
the order of 10  times the molecular value.  This would, at first impres-


sion, tend to indicate a high degree of turbulence and invalidate the


assumption just discussed.  However, as is shown in Section 2.3.3.3 an


apparent turbulent dispersion term may not indicate turbulence but


rather the inability of certain assumptions in a mathematical model to


account for a very complex pheonomena.


        Orlob (54) and Huber and Harleman  (]_8 ) have presented criteria


(Table 2.1) for determining when reservoirs will tend to stratify hori-
                               -23-

-------
zontally, vertically or in some intermediate stage.   Huber and Harle-



man's criterion is based on the ratio, rR,  of the yearly volume of



inflow, V , to the reservoir volume,  v .   Orlob uses this criterion,

                                             O

replacing V  by Q, the average discharge in m /sec through the reser-



voir multiplied by the ratio of the average reservoir depth,  d-, in



meters to length in meters to define a reservoir Froude number, F_:
        r   =   Q                                                 (2-7)
         H     V
                r
                        .                                          (2-8)
                  d
where   p   =  reference density



         g  =  average vertical density gradient in the reservoir
          o
                                    -3     -4       3     -3
        Orlob suggests the use of 10   Kg m   and 10  Kg m   for g and



p  respectively, reducing Equation 2-8 to:





        IF   =  320 ^  |                                           (2-9)
        These criteria are combined and typical values presented in



Table 2.1.



        Orlob' s modification, which introduces L/d into the reservoir



criterion, is an indirect way of including the pheonomena of wind in-



duced mixing and evaporative cooling.  As L/d increases, the reservoir



will be more susceptible to mixing due to either a large surface area



for surface cooling and wind forces to act upon  (large L) or the possi-



bility of the thermocline (the depth of maximum density gradient)


                              -24-

-------
                          TABLE 2.1
RESERVOIR

Hungary Horse
2
Fontana
2
Detroit
Lake-
3
Roosevelt


Priest Rapids
4
Wells

LENGTH
(m)
1
4

4

1


2


2

4

4
.7x10
4
.6x10
4
.5x10

5
.0x10


.yxiu
4
.6x10

AVERAGE
DEPTH
(m)

70

107

56


70


18

26

DISCHARGE TO
VOLUME RATIO
(sec )

1.

2.

3.


5.


4.

6.

-8
2x10
o
5x10
0
5x10

-7
0x10

— 6
6x10
,
7x10


0.

0.

0.


0.


2.

3.

Fr

0026

0029

0030


46


4

8

CLASS

Deep

Deep

Deep


Weakly-
Stratified

Completely
Mixed
Completely
Mixed
1 Montana




2 TVA System




3 Montana




4 River run dams on the Columbia River below Grand Coulee Dam
      TABLE 2.1 RESERVOIR STRATIFICATION CRITERIA
                              -25-

-------
 reaching  the  reservoir  surface  (small  d).


        However  the  inclusion of  the ratio  y^TTg? is questionable.   BQ


 is  defined  as  an average  density  gradient for which Orlob arbitrarily


 assigns the value 10~3  Kgnf4.  Firstly, in  the  case of a completely


 mixed  reservoir,   — = 0 and 6 should also  equal  zero.  Secondly,  as
                   3y

 a predictive  tool to determine the  shape of  the  reservoir isotherms,  6Q


 would  not be  a known parameter as opposed to Q,  V  , V^, L and d which


 could  be  determined  from  the proposed reservoir  geometry and inflowing


 stream hydrographs.   Since a constant value  is assumed for this ratio,


 the results obtained by Orlob are presented.  However it is felt  that

            /p

 the ratio,  /— ,  could  be omitted from the reservoir criteria to  avoid

          >/ gBo

 the unnecessary  choice  of arbitrary values.


 2.3 Approximations  to  the Full Set of Equations


        2.3.1  Marker and Cell Technique


        Daly and  Pract  do) and Slotta (43) have presented methods  for


 solving the equations of motion numerically for  the case of laminar


 flow in a density  stratified fluid.  The procedure, in two dimensions,


 consists basically of "flagging" or marking particles in rectangular


 cells  of length  6x and  height Sy according to set  schemes.  For example,


 Slotta calls EMP a cell containing no fluid particles, FULL, a cell


 containing  particles with no adjacent Fi-IP cell,  OUT, a cell defining


 an  outlet,  etc.  The Navier Stokes equa-tions are written in a finite


 difference  form and  an  algorithm is presented for  their solution.


        The fundamental problem that arises when  trying to adopt  this


method to a thermally stratified reservoir is the  complete neglect  of
                                -26-

-------
the temperature field on density variations.  New densities are calcul-




ated in the algorithm by averaging the densities of the particles in a




given cell.  Since the thermal structure of a reservoir is continuously




varying with time, the temperature field must be determined at each




successive time step in order to correctly determine the density field.




This involves the solution of the equations of motion, continuity, the




equation of state and the conservation of heat equation which poses a




formidable, if not impossible, programing and computer storage problem.




        For very simple problems, such as withdrawal from a two layered




system and flow over a submerged ridge in a two layered system, Slotta




reports a storage requirement of 65,000 locations for a grid containing




800 cells and 3,000 particles.  Using a time step near the maximum




allowable by the stability conditions, one time cycle took seven sec-




onds on a CDC 6600.  A typical run of 200 cycles took twenty-three




minutes.  Slotta felt that this size and running times were nearly




minimal.




        Considering the added complexity of solving both the complete




equations of motion, the equations of state, continuity, and the con-




servation of heat equation for a reservoir, an alternate approach,




which would simplify the governing equations, seems to be called for.




        2.3.2  The Boussinesq Approximation




        A common assumption in phenomena governed by small density dif-




ferences is that the equations of motion can be simplified by consider-




ing density variations only in the buoyancy term.  Consider the case of




a reservoir with horizontal isotherms in which the density can be
                             -27-

-------
represented as
p(y)  =
                    Ap(y)
                                                                  (2-10)
where
           « 1
                                                                 (2-11)
        Since the vertical accelerations in a reservoir will be much




less than the free fall acceleration, g, the density fluctuation, Ap,




is neglected in the vertical acceleration term but included in the buoy-




ancy term.  The Boussinesq approximation is presented in Equation 2-12.
3v , 3v
+
O L oX

v

r32v 4
2

-^ +
v 3y
32v ,
2

9v
W 3z
32v"
2
3z
                                                                 (2-12)
Unfortunately, the Boussinesq approximation does not sufficiently sim-




plify the problem since Apis a function of y and  7 nonlinear simultan-




eous partial differential equations remain to be solved.




        2.3.3  Solutions for Various Systems by Means of a




               Dispersion Coefficient




        A widely used approach in arriving at concentration predictions




for phenomena, in which the internal flow pattern is not well under-




stood,  involves a modified one-dimensional representation of the con-




servation of mass Equation 2-5.   For example, if it is assumed that  the




phenomena is basically affected by longitudinal variations, Equation




2-5 would be written for the x t-irection:
                            -28-

-------
3c
at
h U 9C
U 9x
1 3
A 3x
r> A 3c
DA
p 3x
sources
i m
I - -
P
sinks
m
P
                                                                  (2-13)
        " i-     O.K-   A ox  p   dX      p          p



where




        c = c(x,t) = average  concentration over the depth



        U = U(x,t) = average  horizontal velocity over the depth



        A = A(x,t) = cross-sectional area normal to U



        D  = longitudinal dispersion coefficient




        Two fundamental differences appear between Equation 2-13 and a



precise one-dimensional representation of Equation 2-5.  The first is



the omission of the turbulent fluctuation terms u'c' and the second is



the replacing of the molecular diffusion coefficient, D , by a disper-
                                                       m


sion coefficient, D .  The basic philosophy of this one-dimensional dis-



persion model is to assume that all the parameters in Equation 2-13 are



uniform over the depth and width (y and z directions).  A very simple



velocity field representation is assumed, i.e. U = Q/A where Q is the



volumetric rate of flow.  The longitudinal dispersion coefficient, D ,



is used to account for any non-uniformities which may exist in the



actual velocity distribution.  This method has been used extensively



by chemical engineers to treat complex flow patterns which may exist in



process equipment as is illustrated in Figure 2.3.  Levenspiel  and



Bishoff (29) present a detailed discussion of various solutions to



Equation 2-13.  In different  phenomena, the dispersion coefficient may



be considered to be a constant, a function of space or time or some



combination of these.  In all cases, D  must be empirically determined.



Three examples follow.
                               -29-

-------
                                                          Stagnant regions
o
I
                                 Channeling; especially
                                 serious in countercurrent
                                 two-phase operations
                                                                                     Extreme short-circuiting
                                                                                     and bypassing; a result
                                                                                     of poor design
                        FIGURE 2.3 FLOW IN CHEMICAL  ENGINEERING  PROCESS  EQUIPMENT

-------
               2.3.3.1  Constant Longitudinal Dispersion  Coefficient



               Consider a steady uniform  turbulent  flow in  a  long  con-



duit of constant cross-sectional area, A  (9 ).  At  time t = 0,  tracer



fluid E is injected into fluid B as a pulse  input at x =  0.   The flow



rate, Q, is constant and it is desired to determine the spatial and tem-



poral concentration distribution of the tracer.



        Since there are no external sources  or sinks of mass  and A is a



constant equation 2-13 reduces to




        3c     _  3c         3 c
                      =  D  —-                                 (2-14)
         at        3x       p
                             o^-




The initial conditon  is




         CE  (x,0)  =   (M/pA)   6(x)                                 (2-15)



where    M  =  mass of tracer  E  introduced



       6(x) =  Dirac delta function




The conservation  of mass consideration  yields  the  further  condition



that
(2-16)
The boundary condition  on  x  is  obtained  by  stating  that  the  concentra-



tion at x = + oo  remains unchanged  with  time
        c,,  (+ -,t)   =   0                                          (2-17)
         III
                            -31-
              (x,t)  dx  =  M/pA     6  (x)  dx   =  M/PA

-------
        The solution to Equation 2-14 with these initial and boundary




conditions is





            .  JL          . - <*-"t)2/4V                      (2-18)

        CE     PA  /4irD t







        This is the equation of a Gaussian curve.   The value of the dis




persion coefficient, nowever, is yet to oe determined.  Taylor (47) has




demonstrated that for uniform turbulent flow in a  straight conduit
        D   =  10.1  r
         p            o  o





where   r   =  pipe radius





        T   =  shear stress at the wall.
         o




        D  can be either calculated from Equation 2-ly by modifying r

         P                                                           °


to be the hydraulic radius of the channel or it can be determined empir-




ically by fitting experimental data.   The actual values for c  in Equa-




tion 2-18 will depend on what is assumed for D .  The larger the value




of D  the more rapidly the flow is dispersed.   This is represented sche-




matically in Figure 2-4.




               2.3.3.2  The Dispersion Coefficient as a Function of Time




               Holly (16) considers the solution to the problem of a




pollutant undergoing first order decay while flowing in a constant area




channel in which the average cross-sectional velocity is allowed to be




a function of time.   The longitudinal dispersion coefficient is assumed




to be a function of time but independent of x.  For this case Equation




2-i3 can be written as:
                                -32-

-------
    Initial Distribution of
    Diffusant at  t = 0
                                                     /C VS X
                                                     (D = D
                    Distance
 FIGURE 2.4a CONCENTRATION VARIATION AS A FUNCTION OF DISTANCE
             AT t = t  FOR VARIOUS LONGITUDINAL DISPERSION
             COEFFICIENTS
                       Time
  FIGURE 2.4b CONCENTRATION VARIATION WITH TIME AT x = x
              FOR VARIOUS LONGITUDINAL DISPERSION COEFFICIENTS

FIGURE 2.4 CONSTANT LONGITUDINAL DISPERSION COEFFICIENT MODEL
                     -33-

-------
            +  U(t)   -  =  D  (t)
                                                                  (2-20)
where   X  = first order decay constant.




        This is equivalent to the conservation of BOD equation, neglect-



ing sources, for a stream.



        Through the substitutions:
        c  =  Ae
        a  =
        0  =
              x -J  u (t) dt
                 D (t)
                        dt
(2-21a)





<2-21b)





(2-21c)
where   T =  the reference time for which C(X,T) is known



       D  =  a reference dispersion value
Equation 2-20 is reduced to
        86
                0.2
                                                                  (2-22)
        For an instantaneous release at time



the solution is
                                              = 0  (i.e.  t =  T)  atct =  0,
        A
                         4D 0
                                                                  (2-23)
where   W  =  pounds of pollutant released
         L*



        Y  =  specific weight of the fluid
                                    -34-

-------
        A  =  flow area.



        One must again turn either to a modified form of Taylor's equa-



tion or empirical data for the determination of D  and D(t).
                                                 o


               2.3.3.3  The Dispersion Coefficient as an Eddy Diffusivity



               Morris and Thackston (48) treat the problem of the spread



of a pulse injection of dye input at the inlet of a reservoir as a two-



dimensional problem governed by two dispersion equations.  In the longi-



tudinal direction:
and in the vertical direction
where D  = longitudinal dispersion coefficient
       ij



      D  = D (y,t) = vertical eddy diffusivity.





        Equation 2-24 is treated in exactly the same manner as the prob



lem discussed in Section 2. 3. 3.1; with the solution given by Equation



2-18. "D  is determined by a fit of Equation 2-18 to field data.
      J_i


        Equation 2-25 is written in finite difference form and the solu



tion for D (y,t) also arrived at by comparison with field data.



        The treatment of Equation 2-24 as the governing equation for



the horizontal spread of the incoming water may give insight  into this



complicated phenomena.  Perhaps a modification of Equation 2-24 would



be to include the variation of vertical cross-sectional area  and solve
                               -35-

-------
Equation 2-24 by finite diff ere«:e means.   In the case of surface entrance



 U could be related to the inflowing stream rate.  However, for subsur-



face entrance, care must be taken due to the superposition of the outflow



velocity field on the flow in a given layer.  By comparing Equation 2-25



with Equation 2-13 the lack of any attempt to represent the vertical



velocity field should be noted.  Also, since DV = Dv(y,t) Equation 2-25



should be written as:
        3t
D  -
 v 3y
                                                                 (2-26)
        The lack of a vertical convection term precludes any method of



vertical transport except through dispersion.   This places quite an



empirical burden on this term which can only be determined by comparison



with field data.  The order of magnitude of D  calculated by Morris and


                               -2       -1   2
Thackston varied between 5 x 10   and 10   cm /sec. whereas the value


                                  -5   2
of the molecular diffusivity is 10   cm /sec.



        In dye tests carried out in a reservoir, the investigators re-



port  that "there appeared to be very little vertical diffusion down-



ward from the dye cloud and only slight diffusion upward".  Because the



stratified reservoir flow pattern is governed by the transient density



field which is generated,  vertical velocities will always be present if



the inflow horizontal velocity profile is different from the outflow



horizontal velocity profile as is usually the case.  Thus, the large



magnitude of the vertical  eddy diffusivity does not necessarily reflect



vertical turbulence.
                                  -36-

-------
               2.3.3.4  Evaluation of  the Dispersion Coefficient Approacti




               Relying on a dispersion coefficient  to solve all but the




most simple flow problems involves lumping all ignorance of a complicated




flow field into some empirical value or function for D  .  The concept




has practical value in cases where the flow field is governed by the




geometry of the vessel in which the fluid is flowing.   In these cases,




a dispersion coefficient will uniquely describe the mixing characterist-




ics of the vessel.  When one considers the flow complexity of a thermally




stratified reservoir, the weaknesses of this procedure become clear.




The thermal structure and flow field of a reservoir are not only a func-




tion of its geometry but also a function of the yearly meteorological




cycles, the inflowing stream flow rates and temperature and the opera-




tion of the discharge through the dam.  Even if one were to empirically




determine a functional relationship for D  which satisfied one yearly




cycle of reservoir operation, it would be doubtful that this would be




of any use in calculating the next year's pollutant concentrations.




Its use on other reservoirs and as a predictive tool for future reser-




voirs would be even more suspect.  This leads to the conclusion that a




model for predicting concentration of a pollutant in a reservoir must




be linked with temperature predictions as is discussed in the next sec-




tion.




        2.3.4  A Solution Involving the Temperature Equation




        Several attempts (4), (18 ) > (54), have been made to solve the




thermal stratification prediction problem (Equations 2-1 - 2-4) in a




reservoir.   Whatever the method, a velocity field, based on certain
                               -37-

-------
assumptions, must be calculated.  This derived internal current structure




can subsequently be used in predictions of the convective and dispersive




process acting on a substance introduced into the reservoir.




        Before one can intelligently treat the problem of a non-conserva-




tive pollutant, such as DO, one should be fairly certain that a simpli-




fied form of the conservation of mass equation , 2-5, can reasonably




predict the behavior of a conservative substance.




        To follow the development of the proposed concentration predic-




tion model, the assumptions involved in the determination of the velocity




and temperature field must be completely understood.  The velocity field




used in the proposed model is a byproduct of the thermal stratification




prediction method developed by Huber and Harleman.   This is briefly




summarized in the following section and the reader is referred to (18)




for details of the development.




2.4  The Temperature Model




        2.4.1  The Governing Equations




        The basic assumption underlying the temperature model is the




existence of horizontal isotherms.   This will be a reasonable assumption




in the case of reservoirs with a low discharge to volume ratio, and res-




ervoir Froude number.





        The governing differential  equation can be derived by consider-




ing a horizontal slice through the  reservoir as schematized in Figure




2.5.   This finite control volume is of height Ay and width B(y) with




horizontal inflow and  outflow rates Q.(y) and Q (y) respectively.   The





vertical flow rate Qv(y),  through the horizontal surface area A(y), will
                               -38-

-------
                                         Elevation,  y
(a)  Reservoir and Control Volume Illustrating Mass  Continuity
       Internal Radiation Absorption
       Heat Source
                                                     Inflow  Heat
                                                     Source
                                      Diffusive Heat  Source
         Outflow
         Heat Sink
Advective Heat Source
(b)  Control Volume Illustrating Heat Conservation


     FIGURE 2.5 CONTROL VOLUMES ILLUSTRATING CONSERVATION OF

               MASS AND ENERGY IN A STRATIFIED RESERVOIR

                           -39-

-------
be  assumed  to be  uniform over  the length of the element.  The  governing

equation  for the  distribution  T(y,t) is then formulated from conserva-

tion of heat and  volume considerations for this element, and extended  to

the entire  reservoir.

        Considering  the element in Figure 2.5a to be always filled with

water  and applying the conservation of volume principle yields:


                                     3Q
        Q (y) - Q.(y)  =  Q  - (Q  + T^  Ay)                     (2-27)
        xo  y    ^j^J"     ^v   ^v   gy



        Defining  q   and q. as  the outflow and inflow rates per unit

vertical distance reduces Equation 2-27 to:


                     30
        qi  ~ qo  =   3                                             (2-28)


        Treating  the element in Figure 2.5b  in a similar manner,  the

conservation of heat equation is  derived.   Heat is advected into  the

element: by  the incoming water q.  and away from the element by the out-

flowing water q  as described in  Equation 2-29 amd 2-30.
        Heat advected in  =  pc q.T.Ay                           (2-29)
        Heat advected out =  pc q T Ay                           (2-30)


        The heat advected in at the bottom of the element is


        PCPV

where   Q^ is assumed positive upward.
                              -40-

-------
        The diffusive heat flux is
             (DT + E)
where   D  = molecular diffusivity of heat





        E  = turbulent diffusivity of heat




The heat flux per unit area due  to transmission  of  radiation  can




be represented as







        9b = - (1-6)  4>0 e ~^ys~y)                               (2-31)






where   g =  fraction of radiation absorbed at the  surface




        n =  solar radiation absorption coefficient




          =   (t) = net solar radiation reaching the water surface




        7  =  Y (t) = water surface elevation
        s     s




        Equation 2-31 is obtained from the assumption that of the solar




insolation reaching the water surface, a certain percentage,  3, is ab-




sorbed  at the surface, and the remaining heat flux  is distributed ver-




tically as an exponential decay  (Figure 2.6).




        Finally, there is the possibility of heat flux losses through




the sides of the reservoir d> , which are expressed  as
                            m







        vp Ay





where   p = P(y) = perimeter of  the control volume.




        Assuming that the density and specific heat of water  are con-




stant over the temperature ranges considered, conservation of heat






                               -41-

-------
     1.0
     0.5
     0.2L
4»(yl
     0.11—
     0.05U
     o.oi-
     0.0
                            Surface  absorpton
                     46     8     10    12    14    16    18    20
                     Depth Below Surface, y, (Meters)
          FIGURE 2.6 PENETRATION OF RADIATION  INTO  A RE?^RVOIR
                                -42-

-------
energy applied to the control volume  in  2.5a  yields
          .     3T
     pcp A Ay —
                 pc Q T
                   pV
t   (PCPQVT)
-pcp A
                 E) f
               liTi  Ay   -  pc  qQTAy  -  PAym
        +  A
                                                             (2-32)
        In Equation  2-32  it has been  assumed  that  no  solar  insolation



flux reaches the reservoir bottom.







        Simplifying  Equation  2-32  and combining  it with  the continuity



Equation 2-28 results  in:
3T
3t


A
P4>
Tm
3T
37

1 3
A 3y
x 3T
A (D + E)
M 3y _

qi
T.-T
i
A
1 b
                          3y
                                                                  (2-33)
        Since Q  has been  assumed  to  be  uniform over  the  horizontal



cross-sectional area of  the  element,  an  average vertical  velocity  can



now be defined as



                    Qv(y,t)
        v  (y,t)  =
                      A(y)
                                                             (2-34)
where v is positive upward.
                                -43-

-------
        Substituting the expression for  b and Equation  2-34  into  Equa-



tion 2-33 yields
JI  +    — = —  —
 at     V 3y   A  3y
                                       E)
                                                   T.-T
                                                    x
             pc A
                                                                  (2-35)
        This equation isbasicalty the same equation derived by Huber  and



Harleman (18).   In order to formulate a solution two boundary conditions



are needed in y, and an initial condition in t.



        The initial condition is provided by the isothermal state of a



reservoir in the spring.  Thus at t = 0 (spring):
        T = T  at t = 0 for all y
             o
                                                          (2-36)
        At the reservoir surface, the heat absorbed due to the incoming



radiation   and atmospheric radiation   minus surface losses,    ,
           o                            a                        Li


must equal the amount of heat diffused into the reservoir from the



water surface.



        Thus at the surface y = y
                      f
                  y = y.
                                                          (2-37)
        The details of the derivation of the expression for  $  and



are given in (18)  and only the results will be presented here.
        For laboratory conditions:
                                                                  (2-38)
                                 -44-

-------
where   e =  emissivity  of  the  radiating surface (e= 0.97 in the labora-



             tory)


                                                     — 11      2      4
        a =  Stephan,  Boltzman  constant = 8.132 x 10   cal/cm  min°K




       T  =  absolute  air  temperature
        o.




       *T =  *F +  4>  +   =  evaporative heat flux





         =  conductive  heat flux





         =  heat  flux due  to long wave radiation from the water



             surface to  the atmosphere.
and
                                                   (T  - T )

                                  L  + c  T  +269.1
                                                .
                                       p  s          (e   -
(2-40)
          <£p,(f>  in  cal/cm   - min






        a  =  5 x 10    cm/min  - mm  Hg





       e   =  saturated water  vapor pressure at  the water surface



              temperature  in mm Hg




       e   =  saturated water  vapor pressure at  the air temperature
        a


              in mm Hg



        ^  =  relative  humidity in  the laboratory



        L  =  heat  of vaporization  of  water = 595.9 - 0.54 T  in cal/gm
                                                             S


       T   =  water surface temperature in °C
        s


       T   =  air temperature  in  °C
        a
                               -45-

-------
                                                                  (2-u)
        For field conditions  (55)
                = 0.97 x 0.937 x 10    oT „  (1.0 + 0.17C )
               - 0.97 aT
                        S




where   C is the cloudiness, as a fraction of'the sky covered.




        T   = absolute air temperature measured 2 meters above the
         a^i


              water surface.



        Many evaporation formula exist; the majority have the form of



Equation 2-40, with different constants and an additional term to



account for the increase in the rate of evaporation with wind speed.



The two used in this study are after Rohwer (39).
         j)  +   =  (0.000308 + 0.000185w) p (e  - ^e )
         LJ    C                              S3.
L + c T  + 269.1
                     p s          e  -
                          2
where   -,,<)>  is in kcal/m  - day
         H  C
                                                                  (2-43)
        w  -  wind speed in in/sec (measured six inches above the surface)



              and all the other terms are as defined in Equation 2-41



              with centimeters replaced by meters, calories replaced by



              kilocalories etc. and Kohler's formula (84)
                               -46-

-------
-i- A n nnm ^R TT^ f v ii.r- ^
-^T m U.UUUJ.JJ Wp t£ UJE 1
E vo ^ ^ s v a
T -T
T _i_ -, T IT79 S a
L, T C 1 1 J /Z
p s e — itie
s a
                                                                  (2-44)
where   w is in m/sec  (not less than 0.05  m/sec) and measured two



        meters above the water surface



        e_, e  in millibars
         s   a



        T  is measured  two meters above the water surface.
         a



        The second boundary condition will be at the reservoir bottom



y = y, where the temperature  changes very little during the year.  There



are several ways of stating this mathematically:
        T =  T   at y  = y   for  all  t
                                                       (2-45a)
  =  0 at y =
                            for all  t
(2-45b)
         3y
V =  0 at y = y,  for all t
/              b
                                                                  (2-45c)
        The  condition  to be applied depends on  the scheme used to solve



Equation  2-35.   In  this study  Equation  2-45b was used.



        In order to  solve Equation 2-35,  the velocity field must be



determined.  This is done by first assuming a form for  the inflow and



outflow velocity distributions.  The vertical velocities are  calculated



using Equation  2-28  as described in the next section.



        2.4.2   Reservoir Schematization and the Velocity Field



        For  any reservoir, the variation  of horizontal  cross-sectional



area, with depth A(y), is assumed to be known.  Since we are  dealing





                            -47-

-------
with a one-dimensional model in y for the temperature field, it will  be




assumed that at any reservoir elevation (as illustrated in Figure  2.7)





the width 3(y) is constant and equal to






        B(y)  .  MZI                                            (2-4.)








where   L(y) is the length of the reservoir at elevation y.




        With B(y) thus defined, the inflow and outflow rates per unit




depth as a function of y can be described as







        q-LCy.t) = ILCy.t) B(y)                                   (2-47a)







        qo(y,t) = Uo(y,t) B(y)                                   (2-47b)







where   U.(y,t) = the inflow velocity at elevation y





        U (y,t) = the outflow velocity at elevation y.





        The withdrawal velocity distribution is assumed to be governed




by an equation derived by Koh (23)  for viscous, diffusive, steady flow




toward a line sink located at x = 0 (Figure 2.8).




        The assumptions underlying  his solution are:




        1. Steady, two-dimensional  flow in the infinite half plane.




           x > 0.




        2. Small stratification,  Ap/p «1 in the flow field.




        3. The fluid viscosity is p and- the molecular diffusion coeffi-




           cient for heat or dissolved mass is D.




        4. The density is a linear  function of temperature or salt




           concentration.
                              -48-

-------
    (a)  Three Dimensional View
    (b)  Control Volume  Slice
        Side Elevation
FIGURE  2.7 CONTROL VOLUME AND  SCHEMATIZATION FOR MATHEMATICAL
          MODEL OF AN IDEALIZED RESERVOIR
                     -49-

-------
                 Region of Solution
o
I  -
           x=0
                             FIGURE 2.8 LAMINAR FLOW TOWARDS A LINE SINK (23)

-------
        5. The flowing depth is small compared to x, so that the usual




           boundary layer assumptions are made.



        6. Stratification is linear far from the sink (i.e., dp/dy =



           constant).




        7. Non-linear terms are dropped and the solution is thus limited



           to laminar flow.



        The velocity field which results can be approximated by a Gauss-



ian curve
        U  = U      e       2                                    (2-48)
         o    o max      2 a
                           o



where   U      = the velocity at y = y    = the outlet centerline
         o max                        out



        a  = the standard deviation of the outflow velocity distribution.





        The thickness of the withdrawal layer, &, is given by Koh as
               7.14 x                                            ,
                                                                 (2-49)
where   x = horizontal distance from the outlet



        g = gravitational acceleration




        DT= diffusion coefficient of temperature



        v = kinematic viscosity


                               1  dp
        e = density gradient = —  -r—






        Once the thickness of the withdrawal layer is known, the stan-



dard deviation can be chosen in such a way that a certain percentage of



the flow will be contained within the withdrawal layer.  For example,



if 95% of the flow is to be contained within y    - 6/2 < y < y    + 6/2,





                                 -51-

-------
the outflow standard deviation will be
                6/2                                               (2-50)
        °o  =  1.96
This is illustrated graphically in Figure 2.9.




        It must be emphasized here that Equation 2-48 is only an approxi-




mation of what the withdrawal velocity field might look like in a strati-




fied reservoir.  Density profiles in reservoirs are not linear and velo-




city profiles are not necessarily symmetrical about the outlet.  As yet




no satisfactory theory exists for selective withdrawal under the influ-




ence of non-linear density gradients.   It is assumed that the gradient




of the density profile at the outlet is determined and Koh's theory is




applied as if this gradient were constant throughout the depth of the




reservoir.  If the withdrawal layer is thin, this assumption will be a




good one.  However, if the density gradient at the outlet is small, this




would dictate a very large withdrawal layer which could lead to serious




errors.  This will be discussed more fully with the experimental results




(Chapter 4) .




        No work similar to Koh's has been done on the inflow velocity




distributions.  Here, different assumptions will be made depending on




whether the water is entering at the surface or sinking to its own den-




sity level.  As was discussed in Section 2.2, vertical motion in a strati-




fied fluid is suppressed.   Thus, it might be reasonable to assume that




if the water which is entering from a turbulent stream of depth d  is




warmer than the reservoir surface water, it would tend to enter the
                              -52-

-------
- 6/2
                        1.96
                       -1-96
                                                  Area =0.95
Elevation Scale   Unit Norma1! Variate
         FIGURE 2.9 DETERMINATION OF THE OUTFLOW STANDARD DEVIATION
                           -53-

-------
reservoir at the surface in a layer thickness of order dg.   However, if



the entering water was cooler than the surface water it will sink to its



own density level.  In the process of sinking there will be a certain



amount of entrainment and the mixture will begin to move horizontally



at a higher elevation.  In addition,  the momentum of the incoming den-



sity current might carry some of this incoming water past the density



level it was seeking and end up being entrained in still higher density



water.  With reference to Figure 2.10 the assumptions will  be made that



if the water is sinking, it will be distributed vertically  in a Gaussian



manner (after Huber and Harleman (18) )  described as


                       (y-y,J2
                           in
        U. = U.      e      2                                     ,9
         i    i  max     2a.                                       v/
                          i



where   IT     = the maximum inflow velocity
         i max



        y.  = the depth at which the reservoir density is the same



              as that of the incoming water



        a. = the inflow standard deviation.  This will either have
         i


             to  be measured, or assumed.




If the water is  entering at  the surface,  it is assumed that it will



enter uniformly  over a thickness equal to the depth of the entering



stream.  (Huber  and harleman treated surface and subsurface inflow as



governed by Equation 2-51.)



        The determination of the maximum velocities U      and U
                                                     o max      i max


is accomplished  by equating  the total discharge to the integral of the



discharge per unit area, over the depth of the reservoir:
                              -54-

-------
o
LU
_l
LJ
                                             	TEMPERATURE  F
                                             	OTE CONCENTRATION PPB
                                               CONFLUENCE
                                             LITTLE  TENNES5EE
                                             NANTAHALA
           76

            9-16-66
           163885
                    78
                            80
9-7-66
1643.77
                                    82    80
                        82
                                 84     80
                                               82
8-31-66
164743
    84

8-25-66
164949
O
5
LJ
           58       7068       70       72     70       72       74       76     74
                                LITTLE TENNESSEE RIVER MILES
                                                                                   76       78
           FIGURE 2.10 DYE  CONCENTRATION  PROFILES  IN FONTANA RESERVOIR
                                         -55-

-------
                                            .  dy                 (2-52)
                            B(y)  e       2     ^
If the inflow water is sinking

                                           2
                        Sy           (y-y.  )
                         S  My)  ^ ~^T-  «                 <2-53a>
                        ^

and if the inflow water enters at the reservoir surface
                             B(y)  dy                              (2-53b)
                       1 y -d
                        J s  s
        once the horizontal velocity fields are known the vertical velo-

city v, and the vertical flow rate,  Q ,  can be determined from
                                     v
                                     y
        = f    B(y)  U  (y,t)  dy  - f    B(y)  UQ (y,t)  dy
          J V                        • V.
                                     yb
        = v(y,t) A(y)                                             (2-54)


        2.4.3  Mixing at the Reseirvoir Entrance

        As an inflowing stream enters a reservoir there will be  a certain

amount of mixing and entrainment of the stream and reservoir waters.   The

rate of entrainment, 0 , is specified in terms of a fraction,  r  ,  of  the

incoming water Q. and is expressed as
                                 -56-

-------
Using this definition the effective inflow rate, Q.1, is
        V " Qm + Q± = 
-------
FIGURE 2.11 SCHEMATIC REPRESENTATION OF ENTRANCE MIXING
                         -58-

-------
indicate that, for the field, a value of r  = 1 is satisfactory.



        2.4.4  Lag Time Determination



        The equation developed in the preceeding sections are sufficient



for determining the temperature distribution in reservoirs if the assump-



tion is made that the entering water immediately reaches its own density



level and spreads instantaneously along the entire length of the reser-



voir at that particular depth.  However, it is not realistic to assume



that this process takes place instantaneously.  If the water is sinking



into the reservoir, it will take a finite amount of time for it to reach



its own density level.  Once it has reached this depth, it may still be



many miles from the dam.  Huber and Harleman did not incorporate a lag



time (for the entering water to reach the dam  face) in their model.



However, they concluded that its inclusion could significantly improve



predicted outflow temperatures during the late autumn.



        A method for accounting for lag time in the temperature model



is developed in the next two sections.



               2.4.4.1  The Time for the Incoming Water to Reach



                        Its Own Density Level



               As an approximation to the actual phenomena, the first



part of the lag time will be treated as a two-layer flow problem govern-



ed by the average density difference between the mixed inflow water at



temperature T. '  and the surface water at temperature T .  With refer-



ence to Figure 2. 12 the surface water is of density p  and the sinking
                                                     s


water of density p  + Ap.  Assuming that the flow is parallel to the
                  S


reservoir bottom (the s direction), steady and of constant thickness d




                              -59-

-------
FIGURE 2.12 TWO LAYERED FLOW SCHEMATIZATION FOR SINKING FLOW
                         -60-

-------
(i.e. no entrainment after  the  initial  entrainment  at the reservoir



entrance), the equations governing  the  motion in the lower and upper



layers are
Since
               3              d2u
        0 = - -;p-  + pg  +  y  — j     (s  component)                  (2-59a)
               o o      S      ,  £-
                             dn
        0 = - —2-  + p g      (n  component)                         (2-59b)
               dn       n
where   n is normal  to  the  reservoir  bottom.
        g  = g sin  9                                              (2-60a)
         S
        g  = - g  cos  6                                            (2-60b)
and in the upper layers p = p  ,  Equation  2-59b  can  be  integrated  to
                              S


yield
        p = - p  g (cos  9)n +  c(s)                                 (2-61)
               o




At the free surface, n  =  s tan  6,  p  =  0,  therefore






        p = - (p  g(cos 6)n + p   e s tan  6)                       (2-62)
                s               s




        Differentiating Equation  2-62  with respect  to s  and making the



substitution





        S =  sin 6                                                (2-63)
                            -61-

-------
Equation 2-59a reduces  to:
                                       2
                                      d u
        0 = - p gS +  (p  +  Ap) gS +  p — 2
               03              dn
or


          d2u       .  _                                           (2-65)
         li —~  = -  ApgS

          dn



Equation 2-65 is the s equation of motion for the bottom layer.   Inte-


grating Equation 2-65 twice yields:




        u = _ ^ n2  +       +                                  (2-66)
               2y          -L      2



The boundary conditions for the bottom layer are




        u  =  0 at n  =  0                                        (2-67a)
        u  =  u. = interfacial velocity at n = d                  (2-67b)




Substituting into Equation 2-66 results in:
        u  =  -^|Si  (nd - n2) + u± ^                              (2-68)
        Equation 2-68 may be expressed in terms of  the maximum  velocity


in the lower layer u   . by observing that
                    max             6
        du     _
                                                                  (2-69)
Thus
                             -62-

-------
max   An gS  d
                      2   n
                                 n
                           max    max
                                             max
         u.
          i
            u.
             i
                                                                 (2-70)
Keulegan (22) has shown that
         max
              =  0.59
Making this substitution
                                                        (2-71)
        ApgSd
                =  G
                                                        (2-72)
and introducing Equation 2-71 into 2-70
          1
        0.59
                 u.
                  i
n
max
d
2
n
max
d2 _
n
, max
' d
                                                                 (2-73)
From Equation 2-68, 2-69 and 2-72 it can be determined that
         max
                 d.

                 2
                          u.
                                                        (2-74)
Using this and the substitution
in Equation 2-73 yields:
                                                                 (2-75)
               (1
-t  <1 +
                                                         (2-76)
Rearranging
                             -63-

-------
         (1 +  ip)   =    4                                          (2-77)

           
-------
                 d(p  +Ap)

        R  =  u	                                        (2-84)
into Equation 2-83 yields:





        u = 0.1405 (ApgSd2)  ^i2-                                (2-85)

                             u  d   (pQ +  Ap)




With the assumption  that:





        ^  « 1                                                  (2-86)
         P



and introducing the  modified gravity g'




        g'  =  ^g                                               (2-87)
                P



Equation 2-85 reduces  to:




        u  =  0.375   (g'Sd)1/2  R1/2                               (2-88)




Defining q as the discharge per unit width




        q  =  u d                                                 (2-89)




and, introducing Equation  2-89  into 2-88  and  rearranging
        d  =  1.92    r                                            (2-90)
Thus, knowing q, the  depth  of  the  density  current  and  the  average  velo-


city of the sinking  water  can be  determined  from  Equations  2-89 and


2-90.  With this velocity  known and  the distance  to be  traveled,  the


lag time for the sinking water to reach its  own density level,  t   , can


be approximated.



                                   -65-

-------
                ±l£_lil_                                         (2-91)
        CLy       S q



               2.4.4.2  Horizontal Travel Time


               Once the water has reached its own density level it will


take a finite amount of time to travel the distance to the dam face.


This time will depend on q. and the thickness of the flowing layer, Ah.


It is assumed here that there is no entrainment as the water flows hori-


zontally and that the thickness of this layer remains constant as the


water traverses the reservoir.  Thus the horizontal travel time can be


calculated as
        t    =  L' Ah—                                          (2-92)
         LH        q
where


        L1 = average horizontal length the water has to travel.




        The value of Ah can either be determined by dye tests or assigned


some typical value as the depth of the entering stream or determined in-


directly from temperature measurements.  The third method will be dis-


cussed more fully in Chapter 5.


        2.4.5  Surface Instabilities and Surface Mixing


        In the late summer, the cooling of the reservoir surface begins


a process through which the lake eventually becomes isothermal.  Due to


increased evaporative cooling, the surface water becomes denser than the


warmer wa.ter below it.  This is an unstable situation and the surface


water begins to sink.  As it sinks, it mixes with the water beneath it,


                              -66-

-------
lowering the temperature of that water.  By this process an isothermal



layer extending down from the surface is generated.  The mixing process



will continue until a stable situation has been reached.  The thickness



of the mixed, isothermal layer increases as fall turns to winter until,



at the start of spring, the stratification process begins anew.



        The mixed layer thickness and isothermal temperature can be cal-



culated through an iterative procedure since one is dependent on the



other.  If the surface water is cooler than the water beneath it a depth



of mixing, y  . , must be assumed and a mixed temperature. T . , calculated
           'mix                                 ^      ^  mix


from              y

                      T(y) A(y) dy
         T  .   =  — — -                               (2-93)
         mix      y

                      A(y) dy


                   mix




         If T  .   is less  than the  temperature immediately beneath it



y  .  has been assumed too small and a larger value must be tried.  If
'mix                                     b


T  .  is  greater  than the temperature immediately below y . , a stable
 mix     6                   f                  j       'mix'


condition has been reached and the water will stop sinking.  However,



this is  not a guarantee  that y .  is the minimum depth for which a stable
situation exists.  Therefore, y  - y  .  should be continuously decreased
                             '  s    mix                      J


until  the thickness  of  the  isothermal layer has been determined within



the desired accuracy.   In this manner the  important process by which tur-



bulent mixing gradually produces an isothermal reservoir can be accoun-



ted for without specifying  the actual form of the turbulent diffusivity.



The advantage of  this method over some empirical method involving a



vertical eddy diffusivity should be apparent.
                                -67-

-------
2.5  The Method of Solution of thejlemperature Model




        There is no analytical way of solving Equation 2-35 subject to




the prescribed initial and boundary conditions.   Huber and Harleman dis-




cuss various techniques of numerical solutions and conclude that an im-




plicit, finite difference approach based on the Stone and Brian method




is appropriate.




        Any finite difference scheme, whether explicit or implicit, is




a way of taking a continuous equation and representing the continuous




functions by numerical approximations.   It should be noted that the con-




tinuous equation was originally derived from a finite control volume




representation of the phenomena.   Therefore, it is concluded that a




finite element schematization is  a logical way to approach the problem.




        2.5.1  The Finite Element Approach




        With reference to Figure  2-5 it is seen that all of the terms in




Equation 2-35 come from considering the changes in advection, convec-




tion, and diffusion between the sides of a control volume and heat




source inside the element.   The finite element form of the equations




for calculating the temperature field derives from Equation 2-32, the




control volume equation,  and not  Equation 2-35,  the continuous equation.




For ease of understanding,  Equation 2-32 is presented below with the




terms numbered to facilitate discussion.



    [1]                          [2]
pcp AAy     =  pcp QvT -   Pc? QV T        (pCp ^ T)  Ay
                            -68-

-------
- pc  A (1)  + E) —  -
    p     M      3y
                                 [3]
           f
3y
                     Ay
            [4]
  -[5]-
     -[6]
  pc q.T. Ay - pc q T  Ay  -  pAyA   +  Ad>   -
    p^i i      M p^o o        F  Jym       \
                               Ay
                     (2-32)
Term 2 represents the net  amount  of heat  convected  into  the  control



volume of Figure 2-5.  An  equivalent  representation is
              T)
T)
                   (2-94)
vhere the point of  evaluation  of  these  terms  is  represented  schematically



in Figure 2-13.



        Since longitudinal  uniformity has  been ass.umed:
              = VA
                                   (2-95a)
              = VA                                                (2-95b)
where    v=  v(y,t) = vertical velocity



        A = A(y) = the  longitudinal  cross  section  area.






From Figure 2-13, since there will be elements  both  above and below ele-



ment I, the temperature to be assigned  to  the convective transport will



depend on the direction of the  vertical velocity.  Thus, if v  is posi-



tive, T   will be the temperature of element III.   If v  is negative,



                              -69-

-------
                   (-PCA(D+E)
                       pT

                          I
                 "S i
            center of   A(1_8)t ." n
               ma s s
                               3T
                  (-pepA(DT+E) —


                 III
                                               B(y)T..Ay
FIGURE 2.13 POINTS OF EVALUATION OF EQUATION  2-32
                 -70-

-------
T  will be the temperature of element I.   Similarly, a positive v«  is



matched with T of element I and a negative v  with T of element II.



        An analogous representation applies "to expression 3 of Equation



2-32.



        Since expressions [4] and [5] are  independent of y and are  in



fact already in a finite element representation they remain unchanged.



        Expression  [1] and [6] pertain  to  changes occurring within  the



element and should  therefore be evaluated  at the center of mass of  the



element.



        With these  modifications Equation  2-32 can be represented as:
   -  v
nc A Ay
  p   '
         3T
=  pCpVAT
- pc vAT
+  pc  (q.T. - q T) Ay - pAy  +   (1-
     p  i i    o            m
                                         a-n(ys-y)
                                 A
        -   (1-
                   3-n(ys-y),
                                                 (2-96)
        It should be explained here that Huber and Harleman's choice of



an implicit scheme was partly based on the consideration that the Stone



and Brian procedure is unconditionally stable.  However, physical instab-



ilities were noted in their results.  In order to locate the cause of the



physical instabilities the solution technique was changed to an explicit



scheme which has the advantage of being a much easier representation in



which to follow the physical processes which are occurring. It was found



                              -71-

-------
Huber and Karleraan had neglected the important point  of  assigning  the


temperature to the convective flux term based on the  direction  of  the


vertical velocity.  With this corrected, no advantage was  seen  in  return-


ing to an implicit scheme and an explicit solution was used.


        Equation  2-96 involves only first order derivatives whose  finite


difference representation is
        _§T  =  T(t + At) - T(t)                                   (2-97a)
        3t           At
        21  =  T(y + Ay) - T(y)                                   (2-97b)
        3y          Ay
        What remains to be determined is how the lag time will be  incor-


porated into the model.  Since the model is uniform with x, water  which


has entered at a certain time is assumed to have spread out over the


entire length of the reservoir.  From the lag time Equations 2-91,


2-92, the total time, t , necessary for the water to traverse the  reser-
                       Li
voir is
        Thus, if a flow entered the reservoir at time t, the  time  at



which it will have traversed the reservo-ir to the dam face is  t +  t  .
                                                                   Li


For each physical input to the reservoir at time t  (the amount of  flow



which would enter in one day for example), t  is calculated.   This flow
                                            J_i


is then input to the mathematical model a time t  past the time that  it
                                                i_i


physically entered the reservoir.  By "lagging" the inflows in this




                                -72-

-------
manner the assumption that the inflow enters uniformly, longitudinally



dispersed is consistent with the time that it is input to the mathemat-



ical model.



        2.5.2  Stability of the Explicit Scheme-Numerical Dispersion



        A difficulty caused by choosing an explicit over an implicit



method of solution is the limitation imposed on the choice of At and



Ay by stability criteria.  The first of these criteria is
                                                                 (2-99,
This expresses mathematically that the vertical distance traveled by a



particle of water in the time interval At is not greater than one length



step, Ay-  For a typical At of 1 day, and  Ay of 2 meters, the maximum



allowable vertical velocity would be 2 m/day.  It is conceivable that



vertical velocities would be greater than this.  There are two possible



ways of coping with this problem.  (1) Use a larger Ay; (2) use a smal-



ler At.  Ideally, one would like Ay and At to be as small as possible



so alternative (2) should be used.  At the beginning of the mathemati-



cal run values for Ay and At a^e assumed.  Since it is possible that



the choice of Ay and At may lead to violation of Equation 2-99 it is



first necessary to calculate the vertical velocities before the next



temperature iteration is attempted.  If condition 2-99 has not been



violated the temperature iteration is allowed to proceed with the values



of At and Ay originally chosen.  If  inequality 2-99 has not been met




the value of At necessary to satisfy this condition, At    , is calcula-
                                                       max


ted from:



                             -73-

-------
                   Ay                                            (2-100)
        At     -  	
          max     v
                   max
where
        v    = the maximum vertical velocity in time step At,
         max
        Based on Equation 2-100, the time step At is divided into an
integer, n, number of time steps,  At  so that
                                                                (2-101a)
        At  < At
          n     max




and




        nAt  =  At                                              (2-101b)
           n
Once nAt  has been completed the time step reverts back to At until con-



dition 2-99 dictates that it be reduced again.  If it is necessary to



go through this procedure too many times it is an indication that the



original choice of At or Ay was a poor one.



        A second problem inherent in the numerical scheae is that of



numerical dispersion, D .   Consider a volume of fluid at temperature T



located at elevation J at time t, in a stratified reservoir as repres-



ented schematically in Figure 2-14 a  Due to convection this slug of



fluid will be physically transferred to a new position at time t + At



as shown by the dotted rectangle in Figure 2.14b.  However, because the



finite element scheme represents values at specific points, the numeri-



cal representation of the new location of  the slug would be that of the



solid lines in Figure 2.14b.  The difference between the dotted and



                               -74-

-------
                                             J + 1
                                             J - 1
FIGURE 2.14a A VOLUME OF WATER AT TIME t
                                             J + 1
                                             J - 1
FIGURE 2.14b THE VOLUME AT TIME t + At
          FIGURE 2.14 NUMERICAL DISPERSION
                     -75-

-------
solid figure in 2.14b is termed numerical dispersion.



        Bella (2) has presented an expression for evaluating numerical



dispersion for a variable area transport equation:
V2
               [A    yMn - 1/2.t) At]                         (

               |_ fly     A(n,t + At)   J
        The effect of D  is to increase the value of the dispersion
                       P


coefficient in Equation 2-35 from D  to D  + D .   If D  is the same
                                   m     m    p       p


order of magnitude or larger than the assumed value of D , serious



problems could result unless the entire dispersion expression is insig-



nificant compared with the other terms in the heat balance equation.



This will be discussed more fully in Chapters '4 and 5.
                             -76-

-------
CHAPTER 3.  THE WATER QUALITY MODEL




3.1  The Water Quality Model




        3.1.1  Introduction




        The concentration distribution of a single water quality para-




meter within a reservoir is governed by the three-dimensional mass trans-




port Equation (2-5).   The difficulties of utilizing this equation in a




stratified reservoir are exactly the same as the difficulties of the




three-dimensional heat transport Equation (2-3).  The basic philosophy




of the temperature distribution model described in Chapter 2, which is




applicable to reservoirs maintaining horizontal isotherms, is the simpli-




fication of the governing heat transport equation to the one-dimensional




form in which temperature is a function of vertical elevation and time.




The objective of this chapter, and the primary objective of this inves-




tigation, is to develop a mathematical water quality model based on the




one-dimensional mass transport equation to be used in conjunction with




the temperature distribution model for horizontally stratified reser-




voirs.




        The temperature distribution model considers a horizontal layer




extending over the entire reservoir, of vertical thickness Ay» located




at an arbitrary elevation within the reservoir.  At any instant of time




this layer may receive, at its upstream end, a portion of the water en-




tering the reservoir and it may lose, at its downstream end, a portion




of the water being discharged through the reservoir outlet.  The pro-




portions, of the total water entering and leaving the reservoir, which




are received and lost by a given layer, depend upon the instantaneous




temperature-density structure within the reservoir and on the tempera-




                               -77-

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ture and initial mixing of the entering water.  The continuity equation




specifies the vertical convection of water through the layer which is




necessary to maintain a volumetric balance.   Finally,  the one-dimensional




heat transport equation, with appropriate heat sources and sinks, deter-





mines the instantaneous temperature of the layer.




        The above summary is given in order to emphasize that the inter-




nal flow pattern in the reservoir is governed by the assumptions of the




temperature distribution model.   The concentration distribution of any




water quality parameter such as  conservative dye tracers or non-conserva-




tive substances such as biochemical oxygen demand  or dissolved oxygen




will be governed by the same internal flow pattern.   The instantaneous




concentration of an arbitrary layer will be determined by the one-dimen-




sional mass transport equation,  with source and decay  terms appropriate




to the water quality parameter.




        The one-dimensional water quality model is developed in the




following sections.  A general method of solution  is presented,  with




specific examples given for a pulse injection of a conservative tracer




and the continuous injection of  non-conservative substances such as




B.O.D.  and D.O.




3.2  Literature Review





        Though much work has been done on predicting dissolved oxygen




concentration in streams,  very little work has been done on the develop-




ment of methods for predicting the effects of a thermally stratified




reservoir on water quality.   The earliest attempts at  D.O.  prediction





in reservoirs show the natural tendency to apply to an impoundment the
                              -78-

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methods developed for stream D.O. prediction.



        O'Connell et. al.  (33) suggested that the dynamics of dissolved



oxygen in the euphotic zone of impoundments,- when sedimentation is not



important, could be represented by
        ~  =  k^ e V - k2(D) +  (R-P)                        (3-1)
where



        D = D.O. deficit at any time



        t = time



        k  = deoxygenation rate constant



        £  = total  organic B.O.D. at t = 0
         a


        k  = reoxygenation rate constant



        R = rate of oxygen demand by algae



        P = rate of oxygen prediction by algae




        This equation is a statement that the rate of change of oxygen



in the euphotic zone is equal to the net rate of demand of oxygen by the



surface water.  Since this method does not consider any advection of



B.O.D.  by the inflow to the reservoir or any variation of D.O. in the



vertical direction  it can at best be thought of as the governing equa-



tion of a well mixed lake with no inflow or outflow or the governing



equation for a B.O.D. bottle test in the presence of sunlight  (photo-



synthesis) .



        An alternate method of D.O. prediction is a statistical approach,



Churchill and Nicholas  (8) suggested that D.O. concentration  in the
                               -79-

-------
outflow of a reservoir be expressed as a function  of  retention time



(measured from April 1), the temperature of the outflow, and  some factor



which considers reservoir operation.  The governing mathematical expres-



sion, obtained through a multiple regression analysis,  is





                                         222
        y = a + b x  + b x  + b x  + b^x1  4- b^x2  + bfix          (3-2)







in which



        y = decrease in D.O. concentration iri the outflow (mg/£)



            between April 1 and the date for which D.O. prediction



            is desired.



        a,b,,b« ,b ,b, ,b ,b, are constants developed from the regression



        analysis.







        x1 = t/10                                                 (3-3)




where



        t = the number of days  from April 1




              n

        x2 =  £   (t/10)..  ATe..                                     (3_4)

              i=l



where



        n = number of  10-day time increments  after April 1



      ATe = increase in temperature of the  outflow, in °C, between



           April 1  and the  day at which  a  D.O.  prediction is desired.



             n


        X3 =  ?_,   lOH  .                                            (3-5)
                               ^80-

-------
where




        H = distance, in feet, above the center line of the reservoir




            outlet at which  the April 1 inflow exists on the date of




            interest, assuming no mixing in the pool and that water




            is drawn from the pool at the elevation of the outlet only.




The above definition indicates that this equation might be suitable for




a reservoir already in existence for which several years data are avail-




able and no change of the B.O.D. level of the incoming waters occurs




from year to year.  However, as a predictive tool this method would be




highly questionable unless a reservoir similar to the one proposed exists




nearby.




        Wunderlich (57) developed a graphical D.O. model which, like




that of Churchill and Nicholas, considered the D.O. concentration to be




a function of residence time of the water in the reservoir.  Since any




reservoir water quality prediction model should be related to the chang-




ing temperature field in a stratified reservoir, he also developed a




graphical temperature prediction model.  This is the most recent work




on D.O. prediction in reservoirs; a detailed description of the graphi-




cal D.O. method of Wunderlich follows.




        The following assumptions were made:  (1) the inflowing water




at the upstream end of the reservoir immediately spread out along the




entire horizontal area corresponding to its own temperature level (2)




there is no mixing of the inflow at the entrance of the reservoir and




(3) the temperature in the outlet corresponds to the temperature at




the level of the outlet.  These assumptions are suspect in light of the
                               -81-

-------
discussion in Section 2.4.2, 2.4.3 and 2.4.4.




        The basic philosophy of Wunderlich's method is that the change




of D.O. concentration can be directly related to residence time of  the




water in the reservoir.  The residence time, t , is a variable, which




for a given day's input is determined from a graphical temperature  pre-




diction method as discussed below.




        Referring to Figure 3.la, mean monthly values of inflow and




reservoir surface temperatures are plotted at the middle of each month




and connected by a continuous curve.




        Wunderlich  assumes  that  the reservoir surface  temperature  can be




calculated from meteorological  data by assuming  that  the  surface  tempera-




ture is equal to the equilibrium temperature.  The  equilibrium tempera-




ture is defined as  the  temperature at which the  net rate  of heat  trans-




fer at the surface  is equal to  zero.




        As shown in Figure  3.1b  a cumulative inflow volume  curve  is




drawn with the initial  value on  January 1 being  equal  to  the volume of




the reservoir above the intake,  ¥.  .   On the same graph the  outflow vol-




ume band is plotted.  The thickness of this band corresponds  to the res-




ervoir volume between the invert and  the top of  the intake  on  January 1.




Since it has been assumed that  the withdrawal layer corresponds to the




height of the outlet opening, no water below the outlet is  ever with-




drawn.  Thus, the amount of time necessary to discharge ¥.   is shown by




the horizontal distance, t±t, in Figure 3.1b.  During  this  time the out-




flow temperature corresponds to  the initial isothermal temperature in




the reservoir on January 1.  The outflow temperature after  this time is
                              -82-

-------
 u_
0


 a.

 S
 LU
 h-


 cc
 UJ
h-
LU
LU
  1,000
     800 -
 LU
 cr
 o
 <

 o
 o
 o
LU
    INTLOY.'

TEMPERATURE
                                                          V/ATER  SUi'd ACL

                                                          TEMPERATURE
                                     MEASURED

                                        TEMPERATURE /')

                                        INTAl'E—• _      '
                                                       OUTTLO \V_j_T EMP
             FIGURE  3.la
                                            10' SURFACE LAYER
                                                   80°   75°    65V55
                                                            70   60   50
           INFLOW

           VOLUME

           CURVE
                                                        VOLUME  CURVE
            RE.SERVOIK VOLUME  BETWEEN

            VWTEH SURFACE  AND INTAKE

            INVERT ON JANUARY I
                                            RESERVOIR  VOLUME

                                            BETWEEN INVERT

                                                TOP OF INTAKE

                                         j	I    -J
    400 -
 ^  200 -
 O
 >
      0
         JAN  FEB  MAR   APR  MAY * JUN   JUL  AUG   SEP  OCT  NOV  DEC



              FIGURE 3.1b



     FIGURE 3.1 THE GRAPHICAL TEMPERATURE PREDICTION MODEL  OF  WUNDERLICH
                                  -83-

-------
assumed to correspond to the value of the isotherm intersecting  the





centerline of the outflow volume band and the bottom of an assumed 10




thick uniform surface layer.  These predicted outflow temperatures are





projected upward and plotted in Figure 3.la.




        The residence time for a selected input is defined by Wunder-




lich as the time period between which a given input temperature appears




on the inflow volume curve and the time at which this temperature ap-




pears at the center of the outflow volume band.   These are evaluated




graphically from the horizontal distances in Figure 3.1b.   The residence




time varies for different input temperatures, thus reflecting the ther-




mal characteristics of the reservoir.




        Wunderlich notes that the rate of D.O.  decay in a  reservoir is




a function of the water quality of the inflow and the complicated inter-




play of surface and bottom D.O.  and B.O.D.  production and  consumption.




Thus,  the rate of D.O.  decay cannot readily be  generalized.   However,




Wunderlich assumed that the D.O.  in the outlet  could be calculated from








        c = CQ e"k(td}                                            (3_6)







where





        Co = the initial D-°-  Concentration for  the inflow





         c = D.O. concentration in the outlet





        td = residence  time in days of that inflow






         k = k(T) = bulk depletion factor for D.O.






        The bulk D.O.  depletion factor for  a given inflow  temperature
                              -84-

-------
was determined from a plot of the measured D.O. in the layer correspond-




ing to this temperature vs. time, Figure 3.2.  With k(T) thus calcula-




ted, "predictions" are made from Equation 3-6 and shown graphically in




Figure 3.3.  The value of k is seen to vary from 5.5 x 10   day   at




60°F to 1.6 x 10~2 day"1 at 75°F.




        It would seem that "predictions" made in this manner are merely




a check that the plot of measured U.O. vs. time has been fitted correct-




ly for a given temperature.  Whether the bulk depletion coefficient is




actually reflecting the B.O.D. in the incoming water, or the assump-




tions of no mixing and a simplified withdrawal profile, is questionable.




If, for example, the B.O.D. in the inflowing water increased new bulk




depletion factors would have to be calculated for the same inflow temp-




eratures.  Therefore, this method as a predictive tool is very weak.




In addition, if the bulk depletion factor must be determined empirically




from internal reservoir measurements, the D.O. in the outlet could be




much more easily measured than predicted by the graphical method.




        In Chapter 5 the graphical model of Wunderlich will be dis-




cussed again in order to investigate the concept of detention time as




applied to a reservoir.  The remainder of this chapter is devoted to




the development of a one dimensional model for water quality prediction




and appropriate methods of solution.




3.3  The Governing Equation for the Water Quality Model




        Following the assumptions made in the temperature model, the




conservation of mass equation will be treated as a one dimensional




problem in the vertical direction, y.  Thus,  the governing equation is:
                               -85-

-------
I
CO
_J


C9
 I
2!
O

F-

a:
f-

Ld
O

O
O


LU
C9

X
O

0
LJ


O

CO
                      FIGURE 3.2 EVALUATION OF THE BULK DEPLETION FACTOR(57)
               10


                5


                0
0


5


0


5
                      APR
                                         62
                                        FONTANA RESERVOIR

                                              MILE  61.6
                     MAY
 JUN

!966
                                       JUL
AUG
SEP

-------
JAN   FEB  MARCH  APRIL   MAY   JUNE  JULY   AUG  SEPT  OCT  NOV   DEC
 FIGURE 3.3 THE GRAPHICAL D.O. PREDICTION METHOD OF WUNDERLICH
                          -87-

-------
                                           sources
        at
V
   3y   A(y)  ay
                               v(y)
                                                  m
                                                      sinks
                                                           m
ay
                                                    (3-7)
where the source and sink terms have the units of mass/volume/time.




        If the assumption is made that the changes in density caused




by various pollutants are minimal compared to those caused by the temp-




erature field, the method developed in Chapter 2 for calculating the




velocity field in the reservoir remains unchanged.




        It is possible that the concentration of one pollutant will




depend on the concentration of another pollutant present.   If this




occurs, Equation 3-7 must be written for each pollutant and the equa-




tion solved simultaneously with coupling through the source and sink




terms.  For example, if one pollutant was undergoing a first order de-




cay and the second pollutant was also undergoing a first order decay




proportional to the amount of the first pollutant present, Equation 3-7




would be written twice:







        3c,
        at
3C1
ay
sources
1
A(y)
a
ay
ac
Tl A /-,-r^
[V^ ay J
- Kc +
sinks
                                                                 (3-7a)
8C2 i a
ay
A(y) ay
" 3C2~
_ DMA(y) 3y ^
- Kc1 +
            + V
             sources    sinks
                                                                 (3-7b)

-------
where



        K = first order decay constant



        The sources and sinks of  substances 1 and 2 are due to  (1) the



advection of mass by the inflow and outflow velocity distribution  (2)



internal production or consumption not accounted for by the first  order



decay term.  The advective  sources and sinks are directly analogous to



the advective sources and sinks of temperature discussed in Section



2.4.1.  Other sources and sinks will be  discussed in Section 3.4.2.1.



        It is also possible that  the reaction rates would be tempera-



ture dependent.  In this case information  gained from  the temperature



field determination could be used in a relation expressing the  func-



tional dependence of the reaction coefficient with temperature.



        Equation 3-7 can be simplified if  the diffusion term is neg-



lected.  The Prandtl number P  for water is
                             r
        pr = rT =  10                                             (3~8)
where   v =  kinematic viscosity  of water



        DT =  molecular diffusivity of  heat
The  Schmidt  number,  S  , for  water  is
         S   = -^ : 1000                                            (3-9)

         C   DM
where
         D   = molecular diffusivity of  mass
         M
                                 -89-

-------
        Therefore,  the  ratio  of  the  molecular diffusiyity of mass and


that of heat for  water  is
        Jl  =  _L  ~ JL                                         (3-10)
        D      S    ~ 100
        Since the molecular  diffusion of  heat  has  not been found to be


significant in the temperature  prediction it is  felt that the molecular


diffusivity of mass which is two  orders of magnitude smaller, can be


neglected.


        The convective velocity field v(y,t) in  Equation 3-7 is deter-


mined from Equation 2-54.  Thus,  in order to solve Equation 3-7, init-


ial and boundary conditions  must  be stated.   In  addition to the advec-


tive sources and sinks,  a mathematical representation of the internal


and surface source and sink  terms must be made.


3.4  Examples


        3.4.1  The Dissolved Oxygen and B.O.D. Model


               3.4.1.1  Governing Equations


               The conservation of B.O.D. and  D.O. equations are exactly


the same as Equation 3-7 with c in one case representing B.O.D. and


the other case, D.O.  As discussed in Section  3.3, the diffusion term


will be neglected.  It remains  to define  the non-advective source and


sink terms and the initial and  boundary conditions.


        The usual assumption is that B.O.D. can  be represented by a


first order decay process, i.e. the rate  of change of B.O.D. is pro-
                             -90-

-------
portional to the amount of B.O.D. present.  This is represented as
                                                                  (3-11)
or
where
         £  =  the B.O.D. at  time  T
         o



         £ =  the B.O.D. at  time  t



         K =  B.O.D.  decay rate constant





         The values  of  K and £  are  traditionally  determined from 5 day



B.O.D.  tests  with K being the order of 0.1  per  day  and  a  function of



temperature.



         In dealing  with a reservoir, where  water  can be retained from



several days  to several years, five day  B.O.D.  values generally are



not  indicative of the  total B.O.D.  in the incoming  water.  It  is gener-



ally agreed that the B.O.D.  decay process is  composed of  two stages,



carbonaceous  and nitrogenous demand.  The first stage proceeds fairly



rapidly and usually starts  as soon  as waste is  introduced into a body



of water.  At first there is a small population of  aerobic bacteria.



After the waste has been input,  the population  builds up  to a  new level,



characteristic  of  the concentration of  waste and the available oxygen.



This is the carbonaceous stage.  In the  presence  of other bacteria,
                             -91-

-------
the second stage,nitrification, may occur.  Here ammonia type nitrogen




is oxidized to the nitrite ion and subsequently to nitrate.  In many




cases, nitrification occurs several days after the carbonaceous stage




and at a much slower rate.  If the initial population of a nitrifying




bacteria is small, it may be a long time before nitrification is obser-




ved.




        Churchill (8 ) presents data showing that the inflow to the




Cherokee Reservoir (in the TVA system)  in 1952 had a 5 day B.O.D.  of




about 2 mg/£ and a 30 day B.O.D. of about 8 mg/£ while the reservoir




outflow had a 5 day B.O.D. of 1 mg/£ and a 30 day B.O.D.  of about 3




mg/£.  Thus, although the 5 day B.O.D.  decreased by only 1 mg/£, the




30 day B.O.D. decreased by 5 mg/SL.   If  the water had remained in the




reservoir for 30 days, a decrease of 5  mg/fc of oxygen, neglecting sur-




face reaeration and oxygen production by photosynthesis would have




occurred.




        Churchill also reports that in  the summer of 1945 the inflow




to Douglas Reservoir (in the TVA system) contained about 7 mg/£ of




oxygen and about 2 mg/£ of 5 day B.O.D.  while the outflow contained




about 1 mg/£ of oxygen.




        Therefore, for reservoir use, long term B.O.D. studies should




be made,  yielding  values of 10, 30  and  even 50 day B.O.D.  These data




are rarely available.   The representation of the complete B.O.D. cycle




in one mathematical function has not been satisfactorily accomplished.




Dougal and Bowmann (11)  attempt to  represent this by an expression of




the form
                              -92-

-------
where




        a and b are constants



        They report, however, that this expression failed to predict



the experimental long term B.O.D. values.



        For simplicity, the complete B.O.D. cycle will be represented



as a first order decay.







        X. =  £0e~Kt                                              (3-14)






        Since the overall rate of decay will be slower than 0.1 day



for a long term process, K will be assumed to be a constant of the



order of 0.01 day   .  Also, since the ultimate B.O.D. value will be



larger than that calculated from a 5 day B.O.D. test, a larger value



for I  will be assumed.
     o


        There are also two sources of B.O.D.  within the reservoir.  The



first is the bottom demand, which is found in new reservoirs.  This is



due to the amount of oxygen needed to oxidize the organic material orig-



inally present on the reservoir bottom.  Krenkel et. al. (26) states



that "the oxygen demand due to organic deposit generally decreases with



time after the first few years as the organic matter is slowly oxidized



or leached into solution and discharged".  In this study, bottom demand



will be neglected since there is, at present, no satisfactory way  to



quantify it and, as a general rule, it is exhausted after several  years




of reservoir life.
                               -93-

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        In the surface layers of a reservoir  there  is  the possibility


of oxygen production by photosynthesis.  Also, if the  surface water  is


not saturated with oxygen, there will be a transfer of oxygen from the


atmosphere to the surface waters.  An additional surface phenomenon is


the production of B.O.D. due to algae death and oxygen consumption by


plant respiration.  Verduin (51) estimated that in  the euphotic zone


(defined as  the depth by which 99 percent of  the incident light is


observed), photosynthetic production is about equal to the respiration


of the  total biota and that mean algae respiration  is about 12 percent


of maximum photosynthesis.  Pritchard and Carpenter (37 a), however,


reported that the rate of oxygen production by photosynthesis was dou-


ble the rate of consumption in Roanoke Rapids Reservoir.



        In the absence of conclusive information, two different assump-


tions will be tested.  The first is that in the entire euphotic zone,


the rate of production of oxygen by photosynthesis and atmospheric


reaeration is sufficient to cause D.O.  saturation.   The second is that


there is oxygen saturation down to some arbitrary depth d   , above
                                                         S 3. t

the limit of the euphotic zone.   Additionally, in the euphotic zone,


the rate of B.O.D.  production  and consumption will be assumed to be


equal.



        Following the form of  Equation"3-7,  the governing equations


based  on the previous assumptions are directly analogous to the temp-


erature  Equation,  2-35,  with redefinition of the source and sink terms:

                             -94-

-------
        3c        3c          ,  ,           N  B(y)                  /o  I/-N
        —  +  v — = - K£  +  (u.c. - u.c)  . XN                  (3-16)
        3t        3y             i i    i    A(y)
where



         
-------
        The initial condition must be stated in terms of the initial




B.O.D. and D.O. in the reservoir at time t = t±, the start of the D.O.




and B.O.D. calculations.




        Since Equations 3-15 and 3-16 are solved simultaneously along




with the one-dimensional temperature Equation 2-35, it is helpful to




discuss the relationship between the time scale of the two models.




        The initial condition in the temperature equation is that







        T = T   at  t = 0                                        (3-19)
             o






In other words, time, t,  in the temperature model is measured from




t = Oy the time at which the reservoir is assumed to be isothermal.




Consequently, the velocity field v(y,t) is referred to t = 0.




        In the water quality model, times are also referred to t = 0,




the isothermal condition.   No calculations need to be made for Equations




3-15 and 3-16 until t = t. the time at which the initial B.O.D. and
                         i



D.O. profiles are known.   However, temperature calculations must be




made from t = 0 in order  to determine v(y,t) which depends on the




temperature field.




        The saturation level of D.O. in water is a function of tempera-




ture, Figure 3.4.  A least squares parabola for this relationship is
        D.O.   = 14.48 - 0.36T + O.Q043T2                         (3-20)
            s
where




        D.0.c = the saturated D.O. value (ppm)




            T = temperature in  o.



                              -96-

-------
P.
a.
0)
d
cfl
O
•H
4J

CO
S-i
a
4-1
CO
o

Q
    15
                                                           —— tabulated values

                                                             x  calculated from least squar



                                                  D.O.   =  14.48 - 0.36T + 0.0043T2
                                                       s

                                                               (ppm)
    10
                                         Temperature  (°C)


                             FIGURE 3.4 DISSOLVED OXYGEN SATURATION VS.  TEMPERATURE
       0
10
15
20
25
30
                                                              s  fit

-------
        The added restriction is that the D.O. value calculated at  a



depth y in the reservoir cannot exceed the saturated value for this



depth.  If calculations show that D.O.(y) exceeds D.O.  calculated  at
                                                      s


that depth, D.O.(y) will be replaced by D.O. .
                                            S


        The method of solution is discussed in the following section.



               3.4.1.2  Formulation of the Numerical Solution



               Mixing at the reservoir entrance and surface mixing due



to evaporative cooling will be treated in a manner similar to that used



in the temperature model.   The finite volume representation of Equa-



tions 3-15, 3-16, is derived by considering the control volume in



Figure 3.5.
           -
        ~AAy  =  v1A1£1B1 - v2A2*2B2 + (u-u £ ) B(y) Ay    (3-21)

               - K£ A Ay
        ~  A AY  =  V!A1C1B1 " V2A2°2B2 + (uiCi~UoCo) B(y) Ay
               - K£ A Ay





where



        the subscript i refers to inflow



        the subscript o refers to outflow




        The point of evaluation of CJL and c2 will depend (as in Section



2.5.1) on the sign of the convective velocities v   v .   For example,
                                -98-

-------
u cB(y)Ay
 o
               Ill
                       A A y - lUAAy
II
                                                  B(y)
       (c can be replaced by  £ with no other changes necessary)
          FIGURE 3.5 CONTROL VOLUME FOR THE WATER QUALITY MODEL
                            -99-

-------
if v  is positive, c  refers to the concentration  in  element I (Figure



3.5).  If v  is negative c  refers to the concentration  in  element II.



A similar rule applies for c .  This also applies  to  i,  and i^.



        The surface boundary condition for B.O.D.  can be  formulated



from a conservation of mass consideration.  With reference  to  Figure



3.6a, since it has been assumed that there is no net consumption  of



B.O.D. in the euphotic zone K is zero in this region.  In addition,



there can be no transfer across the free surface.  The resulting  B.O.D.



surface boundary condition is
        A 2.  T
        —  A A y = (u. SL.-u Si ) Ay B(y; + v£A
        At           i i  o o
                                         .  at y = y       (3-23a)
                                     yg-Ay        's      '
        The bottom boundary condition for B.O.D. is similarly formula-



 ted from Figure 3.6b.
A£  7       /                / x
—  AAy =  (u.£.- u £ )Ay B(y) - v£A
At           i i   o o
                                             yb+Ay
                                                   - K£          (3-23b)
        The bottom boundary condition for D.O. is arrived at in an



analogous manner and is
        Ac  -
T^-  AAy = (uici - UQCO)  Ay B(y) - vcA
                                                      - K£       (3-24a)
        The surface D.O.  boundary condition as discussed in Section



3.4.1.1 is
                             -100-

-------
 u £Ay
  o
                                                           B(y)
B(y)
                     AA
                              v£ A
                                   y -Ay

                                    O
           FIGURE 3.6a SURFACE BOUNDARY CONDITION
                      v£A
                          yfa+Ay
u £Ay  B
 o
 (y)
           FIGURE 3.6b BOTTOM BOUNDARY CONDITION





      FIGURE  3.6  BOUNDARY CONDITIONS FOR D.O.  AND  B.O.D.


                 IN  THE NUMERICAL SCHEME
                             -101-

-------
        c = c    at  y = y                                       (3-24b)
             sat          s
        In addition, as discussed in Section 3.4.1.1, it has  been assum-



 ed  that to some arbitrary depth, d  within the euphotic zone,  the  water

                                  sat


 is  saturated.  Thus
        c = c    for d    
-------
FIGURE 3.7 THE DISTRIBUTION OF AN INPUT UNDER



           STRATIFIED CONDITIONS
                 -103-

-------
amount of mixing at the reservoir entrance (calculated in a directly



analogous manner to the mixed inflow temperature, Equation 2-54) and




on the following assumptions:



        (1) If the water is cooler than the reservoir surface water,



            and entering below the euphotic zone, the incoming D.O.



            and B.O.D. begin to undergo decay immediately upon phys-



            ical entrance into the reservoir.



        (2) If the water is entering in the euphotic zone, there is no



            netB.O.D.  consumption and the consumption of oxygen is



            assumed to be balanced by reaeration and photosynthetic



            production during the time of traverse,  t , of the reser-
                                                     l_i


            voir surface.



        These assumptions are introduced into Equations 3-21 and 3-22



through ^ and c.,  the incoming B.O.D.  and D.O.  in each time step At.



For surface entrance
        c.
         i
=  c.
                    t-tT
(3-26a)
        £-•
=  £.
                    t-tT
For subsurface entrance
              =  c,
              =  £
                    t-t.
                t-t.
(3-26b)
                                                                (3-26c)
                                                                (3-26d)
                            -104-

-------
        When convective mixing occurs, due to cooling of the reservoir




surface (as was discussed in Section 2.4.5) the D.O. and B.O.D. located




in the mixing layers will be redistributed.  Since complete mixing is




assumed to occur,  the new concentration of D.O. in this mixed layer




c .   can be determined from
 mix





                J    c(y) A(y)dy





        Cmix  =  7^	                               <3-27>
                f s
                      A(y)dy
                  mix
An analogous equation applies to fc .  .
                                  mix


        One special point of interest is the reservoir outlet.  The




concentration of D.O. in the outflow D.O.   (t) at a given time t,




can be calculated from the integral over the depth of the D.O. being




advected out by the outflow velocity distribution divided by the out-




flow rate, Equation 3-28.
f5
\   p D.O.(y.t) UQ(y,
                                           B(y)dy
        D'°-out(t) = - -      -                 (3-28)
        If the B.O.D. and D.O. profiles in the reservoir are known




at some time,(the initial condition) preferably at the start of the




stratification phenomena, t = 0, Equations 3-15 and 3-16 can be simul-




taneously explicitly solved in time steps At where the B.O.D. value,




ii, used in the D.O.  calculation is £ at the beginning of the time step.
                              -105-

-------
        All that remains is to evaluate the depth of what has been




called the euphotic zone.   This has been chosen to be the depth below




which only 1% of the incoming solar radiation penetrates.  Below this




depth there is assumed to be no photosynthesis.  If the surface water




is turbid, the depth of the surface layer will be small.  Hence, any




assumption  relating to the thickness of the surface layer is not very




critical.  However, in a clear reservoir the surface layer could be




quite deep.  The depth of the euphotic zone d , can be calculated by




setting
         -  =  0.01                                             (3-29)
        j>
         o
in Equation 2-31.   This dictates  that
        ys - y = de
where




        n  =  radiation extinction coefficient (Equation 2-31).





                3-4.1.3  Required  Inputs  to the P.O.  and B.O.U.




                         Prediction Model




                Those parameters which  can be measured directly  are




        1)  Reservoir geometry




        2)  Initial  isothermal  reservoir temperature




        3)  Inflow temperatures




        4)  Air  temperatures






                               -106-

-------
        5) Relative humidities



        6) Atmospheric radiation



        7) Inflow rates



        8) Outflow rates



        9) Surface elevation



       10) Inflow D.O.



       11) Inflow B.O.D. (long term) and decay rate, K, Equation (3-14)



       12) Initial B.O.D. and D.O. profiles



Other factors which must be chosen are:



        1) Values for absorption coefficient, n> and surface



           absorption fraction, 6(Equation 2-31).



        2) Inflow standard deviation, a  (Equation 2-51) and uniform

                                       j


           surface    entrance depth d  (Equation 2-53b).



        3) Entrance mixing ratio r  and mixing depth, d  (Equation 2-55)



        4) Thickness for lag time determination, Ah  (Equation 2-92).



        5) Thickness of the saturated surface layer, d   , (Equation
                                                      sat


           3-24c)



        6) Evaluation of the withdrawal thickness (Equation 2-49).



                3.4.2.1. Application of the Water Quality Model to



                         a Pulse Injection of a Conservative Tracer



                In this section it is desired to solve Equation 3-7



for a pulse injection of dye at time t = T into a stratified reservoir.



        The governing equation is the same as 3-21, with K = 0.  The



boundary conditions are stated in terms of no transfer across the free



surface or the bottom and thus
                              -107-

-------
        At t = T a  pulse of dye is physically injected  into  the water


entering the reservoir.  As explained in Section 3.4.1.2,  the  amount


of time necessary for  the dye to reach its own density  level and to


traverse the reraaining horizontal length of the reservoir  is the lag


time t, calculated  from Equation 2-98.  To be consistent with  the
      J_j

assumption that  tne dye is uniformly dispersed horizontally, the actual


time at which the dye  is input to the mathematical model,  t  ,  is
        t   =  T +  t                                            (3-32)
         1         L
        At time  t.,  the initial mass,  M,  of  the  conservative tracer is


considered to be distributed vertically according  to  the velocity field


which exists mathematically at time t.(Figure  3.7). Therefore, the


initial condition becomes
        c(y,t)  = 0 at t < t±                                    (3-32a)



                   M        "i^'V  B(y)dy

        c(y'ti) = ^dy-    -TTTT;—    at  t =  t.        0^
                              ^i  ^ i'


where


        Q'(t^)  =  the mixed inflow rate at time t. (Eq. 2-56)

             p  =  the density of water


        In the  formulation of the numerical solution of  this problem


Equation 3-21 applies with K = 0.  The boundary conditions are Equations


                             -108-

-------
3-23a and 3-23b with K = 0.



        As a check that the initial condition  (Equation 3-32b) is  ob-



served, it can be noted in the time interval that t =  t. -  At and  t =



t..  Equation 3-21 reduces to




                   u c.B(y) AyAt

        c = Ac  =     X_ -                                 (3-33)

                       A Ay




        Ac is the change in concentration  (from a value of  zero  in this



case) in the element in time step At.



        The concentration of tracer in the incoming water at time  t =



t .  is given by
        0.' is evaluated at time t. because it is this flow which is
        xi                        i


mathematically entering the reservoir at time t = t. = T + t  .  For
                                                   i        i_i


this same reason u. is also evaluated from the inflow rate at time t..
                  i                                                 i


        Equation 3-33 and 3-34 yield




                                    u (y,t ) B(y)Ay
                                       Q,'  (t.) -




which is identical to Equation 3-32b.



        After the initial pulse has entered  the  reservoir c. will be



equal to the mass of tracer entrained by  the inflow water at time t,



divided by the total mass inflow including  entrainment.  This  is ex-



pressed as
                             -109-

-------
                              u (y)  c(y)  B(y) Ay
                               m
               -  mixing depth  -                 (3-36)
               "
where
        u   =  the backflow velocity due to mixing (Equation 2-58)
         m
         m
            =  mixing ratio (Equation 2-55)
        From Equation 3-36 it is seen that the amount of tracer entrain-

ed depends both on the mixing ratio,  r ,  and the definition of the mix-

ing depth.  As discussed in Section 2.4.3 Huber and Harleman defined the

mixing depth as an arbitrary thickness,  d ,  extending down from the

reservoir surface.  For the case of surface  inflow it is certainly pos-

sible that the entrainment is coming from beneath the surface layer

(  28 )•  This will be further discussed in Chapter 4 in connection with

the experimental results.

                3.4.2.2  Inputs to the Pulse Injection Model

                The inputs to this mathematical model are the same as

those discussed in Section 3.4.1.3 with the  obvious exception that no

B.O.D. or D.O. data is required.  In addition the time of input of one

or more pulse injections is needed.  The model is capable of handling

up to 20 pulse injection solutions simultaneously.

                3.4.2.3  Discussion of the Pulse Injection Solution

                By means of the method discussed in the previous sec-

tion, the concentration distribution of a conservative tracer c(y,t)

can be calculated.  Thus,  if attention is fixed at one particular elev-

ation within the reservoir, a concentration time curve for that depth
                            -110-

-------
can be determined.  In addition, a cumulative mass curve, defined as

the total mass of tracer which has passed a given depth at a given time,

divided by the initial mass input vs. time can be determined.

        The point of measurement of concentration in the laboratory

experiment will be the reservoir outlet.  The concentration of tracer

in the outlet, cout(t) can be calculated from an equation analogous to

Equation 3-28.
                  {ys
                  Jy  pc(y,t) UQ(y,t) B(y) dy
        cout(t) =  ->	-	                 (3-37)
        A typical plot of cout(t) vs. time is found in Figure 3.8a.

        Integrating, with respect to time, the instantaneous amount of

mass advected out of the reservoir from time t = t. to time t and divid-
                                                  i

ing by the mass of tracer input one can determine the total percentage

of tracer, tracot, which has left the reservoir:
                   t=t
                    £  PQ (t) cout(t) dt

        tracot  =  -^i	                      (3-38)
                      mass of tracer input

        This curve is shown graphically in Figure 3.8b.  This will be

referred to as the cumulative mass out curve.

        In summary, the method developed in this section gives a pre-

diction of both the time variation of the outflow concentration  and the

total mass which has passed through the reservoir as a result of a pulse

injection of a conservative tracer.  The validity of the combined temp-

erature and water quality model will be tested in a laboratory reser-

voir using pulse injections.  It should be noted that verification of
                             -111-

-------
cout(t)
                            Time
              FIGURE 3.7a CONCENTRATION IN THE OUTLET VS.  TIME (SCHEMATIC)
    100%
 tracot
                             Time




                FIGURE 3.8b CUMULATIVE MASS OUT CURVE (SCHEMATIC)






            FIGURE 3.8 SCHEMATIC CURVES PREDICTED FOR THE PULSE



                       INJECTION SOLUTION
                              -112-

-------
both temperature and tracer concentration is a much more stringent  test




of the ability to simulate the internal flow pattern in a reservoir




than is temperature alone.




3.5  Review of the Mathematical Models




        In the field case, there does not appear to be any available




data on outflow concentrations due to pulse injections of tracers.  How-




ever, recent measurements have been made on dissolved oxygen concentra-




tions in reservoirs.  Unfortunately, the data is not complete and addi-




tional assumptions must be made in order to predict D.O. concentration.




The verification of some of these assumptions is more in the hands of




biologists than engineers.  Nevertheless, the model includes the effects




of advective inflows and outflow and convective transport, selective




withdrawal, entrance mixing, lag time, and first order decay.  The




mathematical model for concentration prediction (Equations 3-15 and




3-16) is first applied to a pulse injection of a conservative substance




into a stratified laboratory flume.  In Chapter 5, the D.O. prediction




model is tested on Fontana Reservoir in the TVA system.  It is hoped




the assumptions found on the D.O. prediction model will show where




additional research in this area should be directed.
                            -113-

-------
CHAPTER 4.  LABORATORY  EXPERIMENTS




4 .1  Laboratory Equipment




         In the Hydrodynamics  Laboratory of the Massachusetts




Institute of Technology experiments  were conducted in a




laboratory flume having the  shape  of an idealized reservoir.




The flume is not intended  to be  a  physical model of an




existing or proposed reservoir,  but  rather to be a physical




system for verifying the mathematical  models developed for




temperature and concentration  predictions  in chapters 2 and




3.   Most of the basic phenomena  involved in reservoir stra-




tification ard dilution process  are  present in the laboratory




system, except wind and wave forces  and precipitation.  The




mathematical models require, as  input,  meteorological hydro-




lical and water quality data along with the reservoir geome-




try and operation scheme.  The laboratory  simulation has the




advantage of being  a controlled  system  in  which the effects




of  different variables can be  isolated  from one another




along with a time  scale measured in  minutes instead of days.




         The laboratory reservoir  is basically the same as




that used by Huber  and Harlernan  and  is  show-n in Figure 4.1.




The main section of the flume  is thirty-six feet long, one




foot wide and of rectangular cross section.  The depth varies




linearly from four  and one-half  (4-1/2) inches at the up-




stream entrance section 4  feet long, 1  foot wide and four
                          -114-

-------
FIGURE 4.1 THE LABORATORY FLUME
              -115-

-------
and one-half (4 1/2) inches deep as constructed  to  simulate




the transition from stream to reservoir flow-  The  entire




flume is constructed of plexiglass to allow  for  visualization




of the internal flow characteristics of the  stratified  re-




servoir system.




         The inflow to the reservoir was into  the upstream




end of the four foot long entrance channel through  a  three




quarter (3/4) inch hose.  The incoming flow  was  diffused




through a short section of gravel filter located near the




entrance (Figure 4.2).  The flow rate was monitored oy  a




Brooks Flow Meter (Tube 4-9M-25-3, Float 9RS-87) and varied




by means of a valve located near the flow meter.




         The incoming water temperature was  varied  by




adjusting a temperature mixing valve connected to a heat




exchanger.  The inflow temperature was measured continuously




with a thermistor located in the entrance channel.




         outflow from the reservoir was through  a one-eighth




(1/8) inch slot in the downstream end extending  the entire




width of the model.   The outlet slot was located 22.4 inches




above the reservoir bottom.  The flow, which was gravity




driven, passed through the slot into a semicircular section




(Figure 4.3) from which it was withdrawn through three




three-eighth (3/8) inch pipes.  These pipes, approximately




2  inches in length,  lead into a three quarter  (3/4) inch




rubber hose in which a thermistor was located  to monitor
                           -116-

-------
INFLOW
                        GRAVEL FILTER
                                                                                         1/2"
             FIGURE 4.2 ELEVATION VIEW OF RESERVOIR INLET

-------
           1/2"
1/8" Slot
 Reservoir
 Interior
                                              2-1/2" I.D.
                                              3" O.D.
                                                 TO FLUOROMETER
Three 3/8" Pipes
                FIGURE 4.3 THE OUTLET  SECTION
                             -118-

-------
the outlet temperature.   The  flow  then passed through a




Tuner Model 111 Fluorometer with  a flow through cell attach-




ment (VT 110-880)  to  detect the  concentration of any fluores-




cent dye, which was used  as a  tracer,  in the  outlet water.




The outflow passed  through a  Brooks  Flow Meter identical to




that at the inflow  end  and was  controlled by  a valve.




         The Fluorimeter, which  has  four different sensiti-




vity ranges was calibrated in  the  laboratory  against samples




of known dye concentration.   A  log-log plot of Fluorometer




dial readingvs. concentration  for  three of the sensitivities




(Figure 4.4) produced a straight  line  from which a calibra-




tion equation was  obtained.   The  Fluorometer  dial reading




was continuously monitored with  a  Sanborn recorder (Figure




4.5).  The Sanborn  recorder deflection was calibrated




against the Fluorometer dial  reading.   A log-log plot




(Figure 4.6) produced a straight  line  from which the rela-




tionship between the  two  was  determined.   Since the Fluo-




 rometer is also temperature sensitive  a calibration for




temperature was also  made  (Figure  4.7).




         Temperature  measurements  in  the flume were made




with thermistor probes  (Fenwal   Electronics GA51SM2).  The




thermistor has a fast response  time,  0.35 seconds to 98%




of change in temperature.  Thermistors were attached to two




movable probes shown  schematically in  Figure  4.8.  The probes
                           -119-

-------
                                         Concentration,  c,  (gm/mJi)
    100  0
          -11
                 ^-10
tl-l
•H

T3

(13

-------
                                                        ..:.
                               .•   -..
                            <**.
FIGURE 4.5 MONITORING OF FLUOROMETER READING WITH A SA.NRORN RECORDER
                         -121-

-------
 IOC
60
c
•H

"Sio
01
                                      Sanborn Gain = 20


                                      Sanborn Sensitivity = 20
01
E
o
M
O
                        Sanborn Recorder Reading  (mm)


      FIGURE 4.6 FLUOROMETER CALIBRATION-DIAL READING  VS.  SANBORN


                 DEFLECTION
                                 -122-

-------
                                           F/F20 = 1.77 e
                                                         -0.0286 (T - 20)
.5

.4
F  = Fluorometer Readings at 0°C

F   = Fluorometer Readings at 20°C
fn
tn
.2
                       FIGURE 4.7 FLUOROMETER CALIBRATION-TEMPERATURE DEPENDENCE
 .1
                           10
                                 20
                   Temperature (°C)
30

-------
 Thermistor  Leads
    Point  Gage
            Sealant
   Thermistor
                                          Leads to Control
                                          Box
Potentiometer
                                    DC Motor
                                         Leads to Control
                                          Box
                                   Plexiglass Tubing
FIGURE 4.8 MOVABLE PROBE WITH THERMISTOR FOR TEMPERATURE

           MEASUREMENTS
                   -124-

-------
were driven by a small  remote  controlled motor, geared  to the




point gage rod and  a  ten-turn  potentiometer.   The output from




the potentiometer circuit  was  connected to  the vertical axis




of an x-y plotter (Bolt,  Berenek  and Neuman).   The thermistor




was connected through a  switching box to a  Wheatstore bridge




circuit, the output of  which was  connected  to  the horizontal




axis of the x-y plotter.   As the  movable probe made a ver-




tical traverse, a direct  plot  of  depth versus  the milli-




volt output of the  thermistor  was obtained.   The  vertical




traversing rate could be  controlled  to a maximum  speed of




about one and one-half  feet per minute.




          Artificial  insolation for  the laboratory reservoir




was provided by thirty-six heat lamps (250  watt quartz




iodine lamps GE Q250-PAR-38FL) one  foot on  center mounted




on a joist suspended  from  the  ceiling.   The  joist height




could be varied by  means  of winches  connecting the sup-




porting cables to the joist.   The intensity  of the lamps




could be varied to  simulate the solar insolation  intensities




of different periods  of  the year.   The reasons for choosing




this type of lamp and the  method  of  calibration is discussed




in detail by Huber.   Only  the  results are presented here in




a plot of the average surface  intensity vs.  lamp  height and




voltage is presented  in  Figure 4.9.




          The relative  humidity,  fy , was measured with a




Bacharod Industrial Instrument Co.  #45715 Psychrometer.





                           -125-

-------
    10.0
     5.0
 CN
  e
  u
  cd
  o
  o
  -e-
  O
  •H
  4J
  CB
  •H
  0)

  i-H
  CO
  4J
  c
  •H
  o
     2.0
     1.0
     0.5
     0.2
     0.1
                                         T	T
        10
                             18 inch Lamp Height


                             ,  „ ,.1.5
                             24 inch Lamp Height
                                               30 inch Lamp Height
20                50           100

     Lamo  Voltage  (volts)
200
FIGUk£ 4.9  LABOtlATORY  INSOLATION CALIBRATION
                                -126-

-------
Surface elevations were measured  with  a point gage located on




top of the reservoir.




A.2  Experimental Procedures




          Three different  types  of  experiments were con-




ducted.




          1.  Constant  inflow  and outflow, no insolation




          2.  Variable  inflow  and outflow, variable insolation,




              constant  surface elevation




          3.  Variable  inflow  and outflow, variable insolation,




              variable  surface elevation




          All of the  tests  were  run for approximately 6




hours.  At the start  of a  run  the reservoir was isothermal




at room temperatures.   The  inflow temperature was varied




continuously in a sinusoidal manner simulating the type of




distribution found in nature  (53).   The incoming insolation was




provided by the overhead  lamps varied  in a stepwise manner




simulating the variation  of solar intensity changes through




the year.




          At a certain  time in each experiment a known




amount of Rhodatnine  B dye  was  "instantaneously" injected at




the upstream end of  the four  foot entrance channel, down-




stream of the gravel  filter.   The outlet dye concentration




was monitored continuously  by  the Fluorimeter.  No concen-




tration measurements  were  made within  the flume but visual




evidence  (Figure 4.10)  showed  that  each dye trace spread out
                            -127-

-------
••*.«,»*%
 1
    1HH' JP
    FIGURE 4.10 DYE TRACE IN A LABORATORY FLUME (3 TRACES)
                      -128-

-------
horizontally along  the  entire  length  of the flume and that



there was no visible  turbulence  in  the reservoir except at



the entrance section.   It was  possible to make more than one



dye injection during  an experiment  if the previous tracers



were seen to have passed through the  reservoir or to be at



an elevation where  they would  not  interfere with an ad-



ditional injection.



          Temperature measurements  of the inflow and outflow



were made at 5 minute intervals, while temperature profiles



in the flume were taken at  approximately half-hour intervals,



The air temperature and relative humidity were monitored



about once an hour.



          In the tests  involving variable inflow and out-



flow, flow changes  were made  in  a  stepwise manner since no



continuous means of varying the  flow  rate was  available.



4. 3  Inputs  to the  Mathematical  Model



          In addition to the  parameters discussed in the



previous section, other parameters  remain to be determined.



These are:



          1.  Side  heat loss  flux,  d>
                                    m


          2.  Evaporation constant  a  (Equation 2-40)



          3.  Values  of the absorption coefficient, n»



              and the surface  absorption fraction, g



              (Equation 2-31)
                       -129-

-------
           4.   Thickness  of  the outflow withdrawal layer,    <5 ,



               (Equation  2-49)



           5.   A cutoff criteria for the limit  of the



               withdrawal  layer when no density gradient



               exists at  the  outlet (Section  2.4.2)



           6.   Thickness  of  the inflowing  layers, Ah,



               both for surface and subsurface  entrance



               (Equation  2-92)



           7.   Inflow standard  deviation, a.    (Equation 2-51)



               and the assumption of a uniform  flow  for



               surface entrance over a thickness  d
                                                    s


               (Equation  2-53b)



           8.   Mixing ratio,  r   and the mixing "depth  d
                              m                        m


               (Equation  2-55 and 2-58)



           9-   The effect  of  numerical dispersion D
                                                    P

               (Equation  2-102)



           The  first three parameters  were  evaluated  for the



laboratory  reservoir by  Huber  and Harleman.   Only the results



are presented .
                      y   -  a                              (4-1)

where




       Tw = Tw^yjt^  = water temperature (°C)



       T  = T  (t) =  air temperature (°C)
        3-   cL

                                              -i-i       f\       t

        a = Stephan-Boltzman constant = 8.132 x 10    cal/cm - min-°K
                           -130-

-------
and    a = 0.00003


       H = 0.03 cm


       B = 0.70
          "1
                                                             (4-2a)


                                                             (4-2b)


                                                             (4-2c)
4.3.1   Evaluation  of  the Outflow Withdrawal  Layer Thickness


           The outflow withdrawal layer thickness,  <$ ,


was calculated from  Koh's Equation 2-49.



                       ~1/6
           1/3
                                    1/3
6  =
                                                             (4-3)
where

                    2
       D = 0.00144 cm /sec  (molecular diffusivity of heat)

                 2
       v = 0.01 cm /sec


       g = 980 cm/sec



           Equation  4-3 was evaluated at  an  x chosen at


about  the midpoint  of a horizontal line  between the outlet


and the  reservoir  bottom,  so  that  x=240cm.


           Substituting the above  values  in  Equation 4-3


results  in


               ~1/6

6 =  2.2 e
6 =  const
                         1/6
                                                             (4-4a)
                                                             (4-4b)
                            -131-

-------
          The  density gradient  can  be  related to  the  tem-


perature gradient    tough the expression


       dfi.  =  d£  dT                                          (4_5)
       dy    dT  dy
          A least  squares fit of  density vs. temperature for

the ranges of  T  =  4°C to T = 26°C,  Figure 4.11, yielded

       p = 1.0 - 6.63 x 10~6 (T - 4)2  gm/cm2                    (4-6)


Thus                                      _!    -i
            i  A          9fT   /\      ,_  (cm   or m  )
       f  = I  dgi  =   	2(T " 4)  ~9  dT                      (4_?)
       e   p  dy     151000 - (T-4)2  dy


          Equation  4-7 was used in  both  the laboratory  and


field s tudy.
            The  validity of Equation  4-3 is based  on  a  small

perturbation parameter, w, which  imposes the restriction

       U  =    q  2/3   «1                                   (4-7a)
            Da  x
              o
           Koh has  presented an empirical relationship to

extend Equation  4-3 when Equation  4-7ais violated

                         -1/6        -1/6
       0^-  =  3.5 	^273     =  3-5u      f°r 0.3
-------
       1.000
LO
CO
i
                                                                        II
                        True Relationship
                            p = 1.0 - 0.00000663(1-4)
         0.994
                                                   16        20


                                               Temperature (°C)


                                 FIGURE 4.11 WATER TEMPERATURE VS.  DENSITY
28
32       36

-------
3 x 10   cm     ( -T-  r  0-1 °C/cm        )  the value of  01    is



28.  However,  the effect  of  the  correction  for this higher



value of  to  is minor  and  Equation 4-3  was used.



          For  high  stratification (  ~ - 0.3 °C/cm          )
                                     dy


measured values from  dye  traces  of 6  agreed well with  -the



values of the  order of  10cm  calculated from Equations 4-4a.



Therefore, Equation 4-4a  was  assumed to be valid.



4.3.2  Thickness of the Inflowing Layers,  Ah  , for



       Lagtime Determination



          The  thicknesses  Ah  for lagtime determination



were found from observation  to be approximately 5cm for



surface flow and 4cm  for  subsurface  flow.  A typical depth



of water in the inlet  section  is  5cm  and this is an indication



that Ah can be  related to  the  depth of  the inflowing stream.



The remaining  parameters  were  evaluated from the experiments



and are discussed with  the results.



4.4  Experimental Results



4.4.1  Runs with Variable  Insolation and Flow Rates,



       Constant	Surface  Elevation



          Two  experiments  were conducted in this series.



Since the temperature model  had  been verified previously



by Huber and Harlemanin the  experiments conducted in the



same flume, the main  objective was to  investigate the



validity of the water quality  model.  Therefore, the input



temperature, insolation and  flow rates were kept as
                          -134-

-------
identical as  possible between the  two  runs.   Thus, it  is



felt that the  dye  tests taken in  the  two runs can be



directly compared.



          First,  the final predicted  results will be pre-


sented, using  an  inflow standard  deviation,  o^ =  5 cm



(for sinking  flow)  an entrance mixing  ratio  and depth,


r  =0.2  d   =5 cm, and a depth for uniform su face entrance,  d
 mm                                             s

= 5 cm.  Then the sensitivity of both the temperature and  water quality



models  to various parameters will be discussed.



          The  typical inflow temperature variation flow



rates  and insolation values for this  set of  experiments are


found  in Figure  4.12 along with measured and predicted out-


flow temperatures.   Before the peak  temperature is reached



the predicted  outlet temperatures  are  slightly higher than


those  measured.   After the peak temperature, the  predicted


values fall off  more quickly than  the  measured values.


However, the  measured and predicted  temperatures  are all


within 1°C.


          Predicted and measured  vertical temperature


profiles are  given  in Figure 4.12a.  The predicted profiles,


though generally  slightly lower than  those measured, agree



within 1°C  in  all  cases.


          Three  dye tests, with injections at 10,33 and



329 minutes after  the start of the  test were made.  In each
                            -135-

-------
  I -
O
o
  o,
c
- INSOLATION | 7QOO
LJ
J 1 § 6000
o
i in i _i ^nnn
INFLOW -OUTFLOW RATES


i i i i
0
        100   200  300
         TIME (minutes)
00
100   200  300  400
   TIME (minutes)
               ys = 24.7 cm
               2QO
-------
                                  VARIABLE INFLOW-OUTFLOW
                                  VARIABLE INSOLATION
                                  CONSTANT SURFACE ELEVATION
                                          measured
                                    	predicted
                           TEMPERATURE  (°C)
FIGURE 4.12a TEMPERATURE  PROFILES
                              -137-

-------
        _2
test, 10    gm  of  tracer  was injected.  The results  are


presented in Figures  4.13 and 4.14 in terms of concen-


tration measured  at  the  outlet divided by the mass  injected


vs. time.   In  Figure  4.15,  the results are presented  in  terms


of the total percentage  of  tracer  which had passed  through


the flume (tracot, Equation 3-38) vs. time.


          From Figures 4.13,  4.14, it can be seen that the


order of magnitude of  the concentrations  predicted  in the


outlet is in reasonably  good  agreement with the measured


values. The measured  and  predicted arrival time of  the


peak concentration and the  peak concentration divided by the


mass injected  are presented in Table 4.1.


          It is noted  that  the peak concentrations  are in


very good agreement with  measured  values, differing at most


by 2.43 x 10    gm      The time of  the peak outlet concen-


tration is  also reasonably  well predicted.


          For  the 10  and  33 minute dye injections,  the pre-


dicted start of the outlet  concentration  curve, Figures 4.13,


4.14,  is somewhat early.  This may be partially due to


frictional  affects which  are  not  accounted for in the


mathematical model.
                         -138-

-------
I
h-1
U>

        b
        X
        8
        o
            5.0
            4.0
            3.0
            2.0
             1.0
             0
                     Input at
                     33 min
                                                    T
                          T
                           50
                                       ' Input at 329 mm '
      VARIABLE INFLOW-OUTFLOW
      VARIABLE  INSOLATION
      CONSTANT SURFACE  ELEVATION

      Mass of Tracer = I0"2gm
       	measured
       	predicted
                                              Input at
                                              33 min
IOO
ISO          20O

  TIME  (minutes)
250
300
                                                               350
                                                                                                                  40O
           FIGURE 4.13 CONCENTRATION PREDICTIONS

-------
 I
[-•
-*=>
O
X
'E
en
            5.0
            4.0
            3.0
        OJ
        o
        o
        I
        15   2.0 -
        1/1
        o
        o5  1.0
             0
                Input at
                10 min
              0
                   50
                                                    T
                                                         T
                                    VARIABLE INFLOW-OUTFLOW
                                    VARIABLE INSOLATION
                                    CONSTANT SURFACE ELEVATION

                                     Mass of Tracer = IO'2 gm
       	measured
       	predicted
                                                                                  Input at
                                                                                  10 min
100
150          ZOO

  TIME  (minutes)
250
300
                                                                                              350
                                                                                                                  400
         FIGURE  4.14 CONCENTRATION  PREDICTIONS

-------
M
-P-
M
               100-
    80
6   60
o
cc
                40
                20
                   Input at Input at
                   10 mtn 33 min
                             50
                                      VARIABLE IN FLOW-OUTFLOW
                                      VARIABLE INSOLATION
                                      CONSTANT  SURFACE ELEVATION
                                           	measured
                                           	predicted
                            100
150         200         250

        TIME (minutes)
                                                                                      300
                                                                                                   350
                                                                                                              400
                      FIGURE 4.15 CUMULATIVE MASS OUT  PREDICTIONS

-------
                       TABLE 4.1
TRACE
(MIN)

10
33
329
PEAK CONCENTRATION/MASS
(
MEASURED
1.86xlO~6
2.35xlO~6
4.88xlO~6
gnT1)
PREDICTED
1.35xlO~6
1.21xlO~6
2.40xlO~6
                                                    (MIN)




                                             MEASURED  PREDICTED




                                               255       265.0




                                               273       272.5




                                               355       362.5
TABLE 4.1 PEAK CONCENTRATION AND  ARRIVAL  TIMES-VARIABLE INFLOW-




          OUTFLOW AND INSOLATION,  CONSTANT  SURFACE  ELEVATION
                             -142-

-------
This may also account for  the  lower  predicted  rate  of  fall




off from the peak concentration.   As  can  be  seen  in Figure




4.15, both of these effects  tend  to  cause the  total pre-




dicted percentage of traces  passing  through  the  reservoir




to be higher than that measured.




          A sensitivity  analysis  for  parameters  5,  7,  8  and




9 in Section 4.3 follows.




4.4.1.1  Sensitivity to  a  Cutoff  Criterion  for the  Upper




         Limit of the Withdrawal  Layer  When  No Density




         Gradient Exists at  the Outlet




          When no density  gradient exists at the  outlet,




the thickness of the withdrawal layer,  6 ,  (Equation  2-49)




is theoretically infinite.   In practice,  6  would equal  the




total depth of water in  the  reservoir.  This corresponds to




the early portion of an  experiment when the  incoming warmer




water has not yet reached  the  outlet  and  the temperature




in the vicinity of  the outlet  is  the  initial isothermal




reservoir temperature.   As  the stratification  begins to




form, although no density  gradient exists at the  outlet, a




gradient will exist near the surface.   The  depth  at which




the density gradient becomes zero increases  with  time  until




a gradient eventually exists at elevation of the  outlet




(Figure 4.l6a).  Though  the  mathematical  model would not




"sense" a density gradient  if  none existed  at  the outlet,




the physical system tends  to withdraw water mainly  from the
                         -143-

-------
              t = t
-p-
-O
I
         outlet
    D.
    0)
   T3
t = tzme
t  = time of initia
       condition
                                                                Upp_er_Liinit_ of Withdrawal. La_ye_r_t_ =
                                                            Upper Limit of Withdrawal Layer t =
                                                        outlet
                                                                   at t = t« Withdrawal Layer
                                                                             Governed by Eq. 2-49
                    Temperature
                     FIGURE 4.16a                            FIGURE 4.16b
                              FIGURE 4.16 CUT OFF  CRITERIA FOR THE WITHDRAWAL LAYER

-------
isothermal region  (Figure  4.16b).   Thus a criterion was

needed for the magnitude  of  the  temperature gradient, (AT/AY)  »

which would dictate  the upper  limit of the withdrawal layer

in the case of zero  density  gradient at the outlet.

          Two values  were  tested,  (AT/Ay)         of 0.01 and
     o
0.001  c/cm for  the  laboratory.   Temperature predictions

were minimally effected.   However, the dye tests showed that

a criteria was definitely  needed.   In Table 4.2 it is seen

that the earlier  the  dye  traces  the more sensitive to

   (AT/Ay)         the  prediction  of the start of the arrival

of the trace are.  However,  it should be also noted that

although significant  improvement was seen in the time at

which 1% of the  tracer  was predicted to have passed through

the reservoir, less  change occurred in the 5 and 10 percent

cases and the arrival time of  the peak concentration remained

unchanged.  This  is  because  the  cutoff criterion is in effect

only as  long as  there is  no  density gradient at the outlet.

When the stratification begins to effect the density

gradient at the  outlet  Equation  4-4a governs the with-

drawal layer phenomena.   Changing the cutoff criteria
                         -145-

-------
                            TABLE  4.2
                         (a. = 2.5  r  =  0.2)
                          i         m
Cut Off Criteria
(AT/Ay)  < 0.01



(AT/Ay)  > 0.001



Measured
 Flow Through Time  (min.)

  1%      5%     10%    Peak



 47.5   202.5   220.0   270.0



202.5   225.0   237.5   270.0



220.    240.    248.    255.
Trace Input
10 min.
(AT/Ay)  < 0.01



(AT/Ay)c > 0.001



Measured
200.0   227.5   242.5   277.5



225.0   245.0   255.0   277.5



260.     268.    273.    273.
33 min.
                   TABLE 4.2 CUT OFF CRITERION
                              -146-

-------
has little effect on the prediction of the peak  time.



4.4.1.2  Sensitivity to a Guassian Vs. Uniform Surface Distribution



         and the Inflow Standard Deviation, CTj,  for  Subsurface  Inflow



         The effect of assuming a Gaussian distribution with a . =



3cm vs. a uniform distribution d  = 5cm for the  surface inflow  velocity
                                s


profile was found to have no effect on the temperature prediction early



in the run.  However, in Figures 4.17 and 4.18 the Gaussian assumption



is seen to predict slightly higher temperatures.  A  Gaussian assumption



also raises the predicted outflow temperatures (Figure 4.19), but lowers



the predicted percentage mass out (Figure 4.20,  4.21).  A Gaussian assump-



tion for the surface inflow distribution inputs  water in such a way as



to add to the stratification near the surface.   Since the thicknesses



of the inflow layers are comparable (5 m for uniform inflow and 6 m for



Gaussian) the Gaussian distribution results in warmer surface tempera-



tures .



        The higher percentage mass out prediction under the uniform



surface input distribution is due to the original input being concentra-



ted uniformly in the surface layers rather than  being diluted as in the



Gaussian distribution.  This produces higher concentration in the out-



flow and consequently higher percentage mass out prediction.



        No effect of varying a . from 2.5 to 5.0  for  sinking flow with



d  = 5 cm was noted in temperature prediction.
 S
                             -147-

-------
                                   VARIABLE  INFLOW -OUTFLOW
                                   VARIABLE  INSOLATION
                                   CONSTANT SURFACE ELEVATION

                                       	 measured
                                       	rm= 0.0, a-, = 3 cm, surface  -
                                               and subsurface entrance
                                       	IV 00, crj = 2cm, dm=5cm-
                                       ---- predicted, rm= 0.2, CTJ= 5cm r
                                                     dm=5cm
  16
                                  _L
                  _L
18
20
                             22        24        26
                            TEMPERATURE  (°C)
                                     28
                                                       30
FISUF.i  4.17 TEMPERATURE PROFILE PREDICTIONS - SENSITIVITY ANALYSIS
                                -148-

-------
   20 -
VARIABLE IN FLOW-OUTFLOW
VARIABLE INSOLATION
CONSTANT SURFACE ELEVATION
           measured
           r =02, a-: = 5cm, d=5cm
                                        	rmzOO, a-= 3 cm, surface and
                                                   subsurface entrance
                                                         , dm=5cm
                                             rm=0.3, 
-------
o
I
        o
        o

        LJ
            25
            23
            21
        (5  19
        Q_
        LJ
            17 -
            15
                             SENSITIVITY  ANALYSIS
                             OUTLET TEMPERATURES

                          VARIABLE  INFLOW-OUTFLOW -
                          VARIABLE  INSOLATION
 measured                  CONSTANT SURFACE ELEVATION^
 rm=0, ->"
              0
50
100     150     2CO    250
           TIME  (minutes)
300
350    400
         FIGURE 4.19 OUTLET TEMPERATURE PREDICTIONS - SENSITIVITY ANALYSIS

-------
 I

M
Ui

h-1

 I
             o
             o
             <
             o:
             h-
                100-
                80
60
                40
                20
                    Input at

                    10 min
                                                                           Input at

                                                                           319 min
                    VARIABLE INFLOW-OUTFLOW

                    VARIABLE INSOLATION

                    CONSTANT  SURFACE ELEVATION
                         measured
                   •••-	  rm=00,o]-3.0 for surface and subsurface entrance  / /''xX^put at

                   	1^ = 00,01=20^5 cm                    ///I0 min


                         m_   '  '_oc'  m_                      / / '


                   	rm=0.2,a-|=5D,dm=5cm               //

                                                         './



                                                     //
                                                                  /. ••'

                              5O
                         100
                                     150         200         25O



                                            TIME  (minutes)
                                                                                                    350
                                                                                                               400
                FIGURE 4.20  CUMULATIVE MASS OUT  PREDICTIONS - SENSITIVITY ANALYSIS

-------
ro
                100
                80
            fe
            tr
                40
                20 -
                        Input at
                        33 min
                             50
                                   VARIABLE  INFLOW -OUTFLOW
                                   VARIABLE  INSOLATION
                                   CONSTANT  SURFACE  ELEVATION
                                                 measured
         rm=00,o-f = 20cm,dm=5cm
         rm= O.O.o] = 3.0cm for surface and
                      subsurface  flows

                                                                           1
100
150         20O         250

        TIME (minutes)
                                                                                       3OO
	L_
         350
                                                                                                              400
             FIGURS 4.21 CUMULATIVE MASS OUT PREDICTIONS  - SENSITIVITY ANALYSIS

-------
However, from Figure  A.20  it  is  seen that increasing  a



had the effect of  reducing  concentration for the input at



319 min.  This is  expected  in  light  of the larger spreading



of inflow sinking  water  with  increase  a   and the strong



stratification at  the  outlet  which  dictates a narrow with-



drawal layer at late  times.   The  very high sensitivity of



the late traces to  o.   make predictions difficult unless



can be accurately  determined.   It was found that a value of



5 cm produced much  better  results than a value of 2.5 cm.



Since 5 cm was the  order of the  depth of flow in the inlet



channel, it is believed  that  a.   can be related to this  depth



if no other information  is  available.



4.4.1.3  Sensitivity  to  the Entrance Mixing Ratio, r
         	                                 m


          From Figures  4.17 and  4.18 it can be seen that



increasing r  from  a  value  of  0  to  0.3 has the effect of
            m


raising the predicted  temperatures.   This is because the



entrained water was assumed to come  from a surface layer of



thickness d (Equation  2-58) equal to 5 cm.  At early times
           ID


(the 98 minute profile,  Figure 4.17), the water enters at



the surface and would  tend  to  be  cooled slightly through the



entrainment process.   However, this  also reduces evaporative



heat loses and the  effects  tend  to  cancel.  However, water



entering below the  surface, as would be occurring after



the peak inflow temperature  (180  minutes) would tend to  be



heated by the mixing  process.   A  similar trend, of warmer
                        -153-

-------
outlet temperatures  for  higher r  can be noted in  the  out



let temperature  predictions.



          The  effect of  increasing r  on the cumulative  mass
                                     m


out prediction (Figures  A . 20 ,  4.21) is seen to have  the



general characteristic of  increasing the amount of tracer



material that  reaches  the  outlet.   This is related to  the



earlier arrival  time of  the  traces at the outlet as  r



increases, due to  the  assumption of a .constant layer  thick-



ness,  Ah  , for  lagtime  determinations independent of  r  .



Since  increasing r   effectively  increases the amount of  flow
                   m


input  to this  layer,  it  increases  the velocity and decreases



the lagtime.   The  inputs  at  late times (329 minutes) are



most affected  because  these  flows  are sinking "and in general



tend to be withdrawn in  a  much shorter period of time  than



the earlier flows.   Typical  of these late inputs is  the



arrival of the peak  concentration  very close to the  time



that measurable  concentrations are first observed, (Figure



4.14).  Thus,  the  earlier  arrival  time of a late input



(329 min.) means that  the  peak concentration arrives



earlier along  with higher  predicted cumulative mass  out



values.



          One  advantage  of working in the laboratory is  the



possibility of making  independent  observations of the



mixing ratio.  From  dye  tests  on both surface and subsurface



entrance an average  value  of  0.2 for r  was arrived  at.
                                       m
                           -154-

-------
For the field cases  r   can  be  estimated or deduced from  the
                     m


temperature prediction  if  temperature data are available.



This will be further discussed in Chapter 5.



          The assumption  that  all of the entrance mixing



water comes  from  a  surface  layer of thickness d  whether the
                                                m


flow is entering  at  the  surface or not was investigated.



It may not be reasonable  to assume that the entrainment  is



coming from the surface  if  the flow is entering there.   The



following assumption was  tested:  if the flow enters at  the



surface in a layer  5 cm  deep,  the entrainment comes from



a 10 cm thick layer  beneath this depth.  If the flow entered



beneath the surface  the  original assumption was used.   The



results showed virtually  no change in predicted temperatures



and concentrations  under  this  new assumption.  To avoid



arbitrarily assigning more  than one mixing depth, the



original assumption  of  d   = 5  cm from the surface, for



surface and subsurface  entrance was retained.



4.4.1.4  Numerical  Dispersion



          The sensitivity  of  the numerical procedure to



numerical dispersion was  evaluated indirectly.  From



Equation 2-99 and 2-102,  neglecting the area variation,  it



is seen that the  numerical  dispersion coefficient, D ,



is limited by




                . 2
                AT
                     - vAy
(4-9)
                           -155-

-------
whe re
         <  y                                             (4-10)
           AT
Thus, if Ay  is varied while  AT   is  kept  constant, all other



parameters being equal,  the amount  of  numerical dispersion



will change.  Ay was changed  from 2.5  to  1.5 cm while AT



was kept at 2.5 min.  From Equation  4-9  and  4-10, D   <


      2                                    2
0.5 cm /min for  Ay  = 2.5 and D   <   0.2  cm /min for
                               P


   Ay    = 1.5 cm.  Under these two  conditions  insignificant



changes occurred in the  temperature  and  concentration pre-



diction.  It was concluded that doubling  the maximum



amount of numerical dispersion did  not  affect  the results



and further adjustments  were  not  attempted.



4.4.2  Discussion of the Two  Remaining  Sets  of Experiments



          In order for any analytical  method to be of



much practical use it must be free  of  many empirical con-



stants which change in some arbitrary  fashion.  Therefore,



the values of  cr ^ , r  and d  used  in  arriving at predicted



temperature and concentration curves  (Figures  4.12 through



4.15) were kept constant  in the analysis  of the three



different types of experiments performed.   The ultimate



importance of the values obtained for  various  parameters



is that they may be useful in selecting  values of these



parameters for actual reservoir.   Thus  a     and d  were
                                         i         in


chosen to be 5 cm, the depth  of water  in  the inlet channel.
                        -156-

-------
r  was set at 0.2 as determined  for  independent experiments.




 Ah  was found to be approximately  5  cm for surface entrance




and 4 cm for subsurface  entrance which is  also the order of




the depth in the inlet channel.  The results  for the ex-




periments with variable  inflow,  outflow and insolation,




constant surface elevation  using the parameters noted above




have been presented in Figures 4.12  through 4.15.




4.4.2.1  Constant Inflow  and  Outflow,  No Insolation




          Three experiments were conducted with constant




inflow and outflow rates.   As  in the first set of  experiments




discussed, the input temperature variations were kept as




identical as possible between  the  three runs.   The flow




rates for all three runs  were  constant and one dye injection




was made in each run.  In keeping  the  flow rates and tem-




perature variation similar, dye  tests  taken in each of the




three runs can be compared.




          The temperatures  of  the  inflow for  a typical




experiment in this series,  along with  the  predicted outflow




temperatures are presented  in  Figure 4.22.  Measured and




predicted vertical temperature profiles taken at different




times in the run are compared  in Figure 4.23.




          The temperature predications are in very good




agreement with measured  values.  The peak  predicted tempera-




tures, though slightly lower  than  that measured, occur at




the same time as the measured  value.  All  predicted tem-
                        -157-

-------
       FIGU&a 4.22 INFUI TO CONSTANT INFLOW-OUTFLOW, NO INSOLATION EXPERIMENTS
CO
I
              34
              32
         o  30
         o
         LU
         CC
         LJ
         Q.
         IS
         LU
             28
26
              24
             22
             20
                0
           measured
           predicted
                                 -Input Temp.
                                   CONSTANT INFLOW-OUTFLOW

                                   NO  INSOLATION
Q0 = Q j = 7260 cc/min

   = 0.25

20.5 
-------
   20
Q  -20


I
LJ
   -40
   -GO
   -80
     16
                                                   mm
                   J	I	L
                         CONSTANT INFLOW-OUTFLOW

                         NO INSOLATION




                            	 measured
                                          	predicted
                  l.    i	I	I	I	I	I	L
18
20
                  22
                                         24
                                    26
                               TEMPERATURE (°C)
      FIGURE 4.23 TEMPERATURE  PROFILES
                                                           28
                                             30
                             -159-

-------
peratures are within  1°C of those measured.


          Three  dye  tests,  with injections at  33.  92


and 300 minutes  respectively were performed.   In each  test

  _2
10   gm. of tracer was  injected.  The results  are  again


presented in terms of  concentrations measured  at the outlet


divided by the mass  injected vs. time, and the cumulative


mass out curve in Figures  A.24 through A.27.  From  the  first.


three curves, it is  again  seen that the order  of magnitude


of the concentrations  predicted in the outlet  is in reason-


ably good agreement with measured values.   The measured


arrival time and peak  concentrations are presented in  Table


4.3.


Trace         Peak Concentration/Mass In  (gm  )     Peak Arrival Time
33


92


300
asured Predicted

3.

1.
3.

1

7
6

X

X
X

10

10
10
-6

-6

-6

1

1
2

.40

.45
.75

X

X
X

10

10
10
-6

-6

-6
Measured P
(min)
238

291
320
red

255

310
321
              TABLE 4.3  PEAK CONCENTRATION CHARACTERISTICS
          The absolute  difference  between measured and


predicted peak concentration  occurred in the test input


at 33 and was 1.7 x  10    gm      The predicted peak arrival


times are in fairly  good  agreement with those measured.


          All of the  predicted  curves follow the same
                       -160-

-------
    5.0
    4.0
    3.0
    2.0
O   1.0
      0
o
X
'E
CP

8
o
             Input at
             33 min
                                             T
                          T
   CONSTANT  IN FLOW-OUTFLOW
   NO INSOLATION


      Mass of Tracer = I0~2gm
                                       	measured
                                       	predicted
                                   Input at
                                   33 min
                                             J_
                   50
100
                                             150         ZOO

                                                TIME  (minutes)
250
                                                                                   300
                                                                 350
                                                                                                             400
    FIGURE  4.24 CONCENTRATION PREDICTION

-------
NJ
I
        10
        Q
        _x
        'E
         en
CD
o
D
             5.0
            4.0
    3.0
            2.0
         in
         i/)
        O   1.0
               0
                   50
                                     Input al
                                     92 min
                                      Mass of Tracer = 10 2gm

                                       	measured
                                       	predicted
                                                                             CONSTANT INFLOW-OUTFLOW
                                                                             NO  INSOLATION
IOO
                                                                                   Input at
                                                                                   v 92 min
150          200

  TIME  (minutes)
250
300
                                                                                                         350
                                                                                                             400
              F7.GUKE  4.25 CONCENTRATION PREDICTION

-------
OJ
I
o.u




4.0
_^^
(£
|
Q
X
Tc 3.0
b
C7>
8
E
£
•fe 2.0
$
o
\
3 ,_
o 1.0


f~\
	 1 	 1 	 1 	 1 	 1 	
i 	 1 	 1
Input at
30O min
CONSTANT INFLOW-OUTFLOW
NO INSOLATION
-
Mass of Tracer = I0'2gm

	 measured
_ 	 predicted









-
i
i
i
ii

II . Input at
JR' 3DO min
,
it
\
\\
; ! ~\\
i \ \
i \ \

i i i i i
i Vs-
L I N ^
0 50 100 150 20C 250 300 350 4CX
                                                   TIME (minutes)
                  Z 4.25 CO'JCENIRATION ?2EDIC'ITON

-------
cc
    100-
    80
    60
    40
    20
      0
             Input at
             33 min
50
          Input at
          92 min
                                                 Input at
                                                 300 min
                                     CONSTANT IN FLOW-OUTFLOW
                                     NO INSOLATION
                    	measured
                    	predicted
100          150          200          250

                     '!W ! minutes)

  FIGURE 4.27 CUML'LvATIVF ''~-A5? r)UT PREDICTIONS
300
350
400

-------
general trend as those measured  in  the  laboratory.   It is



interesting to note that  the  time  interval  between  injection



of the 33 and 92 minute traces is  59  minutes  while  that  of



their peaks is 53 minutes  (measured)  and  55 minutes



(predicted).  Between the  92  minute and 300 minute  injections



the peaks were separated  by 29 minutes  (measured) and  11



minutes (predicted) although  the inputs were  208 minutes



apart.  This most important consequence of  the  internal



thermal stratification is  well predicted  by the water  quality



model.



          Discrepancies between  the cumulative  mass  out



predictions and measurements  (Figure  4.27)  for  the  input



at 33 minutes is caused by a  slightly earlier predicted



arrival time of the traces and a slightly slower predicted



fall  from the peak concentration (Figure  4.24).  Though  the



predicted and measured concentrations never differ  by  more

              _ g

than  0.75 x 10   gm and the curves  appear quite similar,



the apparent small discrepancies are  magnified  when  the



integrals of the concentration-time curves  are  taken.



4.4.2.2  Variable Inflow,  Insolation  and  Surface Elevation



          In this experiment  three  dye  injections at



10,302 and 350 minutes were made.   The  input  data and  pre-



dicted outlet temperatures are given  in Figure  4.28.   Pre-



dicted and measured temperature  profiles  are  in Figure 4.29.



Again  a. = 5.0 cm,  r  = 0.2, d  = 5  cm,  Ah    =  5 cm  for
        i           tn          ID





                          -165-

-------
FIGURE 4.28  INPUTS TO THE  VARIABLE INFLOW-OUTFLOW, VARIABLE INSOLATION, VARIABLE  SURFACE

            ELEVATION EXPERIMENTS          f               F(_ow
                                          ,~  8000 p
                                          8      	
                  INSOLATION
                 ^=0.20
                 178 
-------
   20
- VARIABLE IN FLOW-OUTFLOW
  VARIABLE INSOLATION
- VARIABLE SURFACE  ELEVATION
E
o
C  -20
I
UJ
   -40
   -60
   -80
                                 1	1	r
     16
        18
                                                       measured
                                                 	predicted
                   J	L
                           I    i     i    I	L
                       20
22
                                   24
                                                  26
                                                     28
                                    3O
                               TEMPERATURE (°C)
    FIGURE 4.29 TEMPERATURE PROFILES
                              -167-

-------
surface input  and  4  era  for  subsurface  input.   Excellent tem-




perature predictions  result.




          Dye  concentration  predictions  (Figure 4.30) are




quite representative  of  the  measured curves  for the 10 and




302 minute traces.   The  350  minute  prediction,  though




beginning at approximately  the  same  time  as  the measured




curve, and of  the  same  order of  magnitude,  is  not  very




good.  This is probably  due  to  the  large  amount of short




circuiting occurring  at  late  times  which  magnifies dis-




crepancies between assumed  and  actual  values of a. ,  r
   *                                              i    m



and    Ah




          The  cumulative mass out curves  (Figure 4.31)




show, for the  10 minute  input, the effect  of  the earlier




predicted arrival  and slower  reduction  from  the peak




concentration.  The  302  and  350  minute traces  also reflect




the slower predicted  decline  from the  peak  concentration




values.




A.5  Summary of Experimental  Results




          In general, measured  temperatures  agreed very




well with predicted values.   Concentration  predictions were




better for traces  input  early in the stratification cycle




than later when the  inflowing water was sinking.  However,




the order of magnitude  of the predicted concentration and




the general trend  of  the measured curves  could  be  predicted.
                        -168-

-------
I
h-1
VD
          FIGU.IH: 4. so CONCENTRATION PREDICTIONS

  70
  6.0
  50
  40
       Input at
       10 mm
     i
o
  20
                  T
                  50
                                              T
                                                VARIABLE  INFLOW-OUTFLOW
                                                VARIABLE  INSOLATION
                                                VARIABLE  SURFACE ELEVATION
                                  Moss of Tracer
                                                               10  gm for Input at 10 min
                                                               ,  ^_3
                                                               5 x 10  gm for Inputs at
                                                                     302 a 350 min
                                               measured
                                          --- predicted
                                                                10 mm
                                                                Input
                                                                                                    Input at
                                                                                                    3O2 mm
                                                      Inpul ot
                                                      350 mm
                                           J_
                                              100
150          200

     TIME (minutes)
                                                                                        302 mm
                                                                                        Output
                                                                     250
                                                                                                  30O
                                                                                               350
                                                                                                           4OO

-------
   100
    80
o   60

cc
I-

    40
    20
        Input at
        10 min
                       VARIABLE  INFLOW-OUTFLOW
                       VARIABLE  INSOLATION
                       VARIABLE  SURFACE ELEVATION
                    	meogured
                    	predicted
                                              Input at       Input at
                                              302 min      350 min
                 50
100
150         2OO         250

        TIME (minutes)
                                                                            300
  IIGURE 4.31  CUMULATIVE MASS OUT PREDICTIONS

-------
Though three  different  types of experiments were  run,  and



different flow  rates  used in each, an  invarient  set  of



parameters  (  a., r , d  Ah )  was sufficient for prediction.
               i  m  m
                         -171-

-------
 CHAPTER 5.  APPLICATION OF THE WATER QUALITY AND TEMPERATURE
             MODELS TO FONTANA RESERVOIR
 5.1  Introduction
          In 1966 a detailed temperature and D.O. study was conducted
on Fontana Reservoir by the T.V.A. Engineering Laboratory, Norris,
Tennessee.  The lake, formed by 400 foot high Fontana dam, is
located on the Little Tennessee River in Western North Carolina.
Three major streams, the Little Tennessee, Tuckaseegee and Nantahala,
and several smaller streams, feed the 29 mile long reservoir
(Figure 5.1).
          The meterological, hydrological and temperature data
obtained from the 1966 survey were used by Huber and Harleman to
test their temperature model.   In this chapter the same data will
be used to compare the predicitions obtained from Huber and
Harleman's r.cdel in tha modified form developed in Chapter 2.
          In addition, cunulative mass out predictions
(Section 3.4.2.3) are presented for various conservative tracer
dye injection tests even though such field tests have not as
yet been carried out.
          It is hoped that the method of analysis developed
in Chapter 3, will motivate the undertaking of dye tests which
will shed further light on the complicated flow field and
dispersion characteristics of a stratified reservoir.  The pre-
dicted curves are compared with detention times calculated by
Wunderlich (57).
                        -172-

-------
Fontana
Dam
                                  N

                                  I
                                                                    River
                                           Little Tennessee River
              Nantahala River
     FIGURE 5.1 MAP  OF FANTANA RESERVOIR AND WATERSHED
                                  -173-

-------
          Detailed measurements of the D.O. of the incoming




streams were made daily from February through December and




D.O. profiles in the lake were measured periodically from April




through December.




          No corresponding B.O.D data exists.  Therefore B.O.D.




values had to be assumed.  The D.O. data and the assumed B.O.D.




input is applied to the D.O. and B.O.D. prediction models deve-




loped in Chapter 3.




5.2  Temperature Prediction




          5.2.1  Inputs to the Temperature Model




          The necessary inputs to the temperature model are




tabulated in Section 3.A.1.3.




          The hydrological and meterologica1 data obtained




by the T.V.A. were presented either on an hourly or daily




basis.  The computer program was run with a time step of one




day and all hourly data were reduced to daily averages.  The




values for the various parameters discussed below are pre-




sented in Appendix III in the form of computer input.




          5.2.1.1  Inflow and Outflow Rates and Temperatures




          The mathematical model is designed to handle only




one input stream to the reservoir.  Inflow rates and tempera-




tures of the five sources of water for the reservoir (the




three streams previously mentioned and the runoff from the




water sheds bordering the north and south shorelines) were




available on a daily basis.  The combined flow rate and







                            -174-

-------
weighted average of their temperatures were used  as  input



to the model.



          The reservoir outflow rate and  temperature were



available on a daily basis.   Since  the power plant operates



on a peaking power production schedule these average daily



values may hide considerable  variation in flow rates and



temperatures.



          5.2.1.2  Solar Insolation and Related Parameters



          Due to the lack of  direct radiation measurements



being available the input solar radiation values  were cal-



culated from a modification of Kennedy's  (1949) method.



In this modification, developed by  Wunderlich, variation in



the surface reflection coefficient, 3cloudiness,  C,  optical



air mass, m, solar altitude   a and  the normalized radius



vector of the earth about the sun,  r, are accounted  for.



Huber and Harleman concluded  that the radiation values  cal-



culated for Fontana, compared with  unreduced pyroheliometer



readings, should be increased by 15%.  The resulting expression



is:               A  sin                     „
                   Q r*        m               /
        *  = 1.15 -^5	  a™ (1-3) (1-0.65C)                    (5-1)
         o           2.      t
                    r



    <|>   = Incoming solar radiation  flux penetrating  the



          water surface  (energy/area-time)


                                      2

    d>   = Solar constant = 1.94 cal/cm /min
     sc


     a  = Atmospheric transmission  coefficient
                             -175-

-------
     The optical air mass, m,  is defined as  the  ratio  of  the


path length of the  sun's rays  through  the atmosphere to their


path length when the sun is directly overhead.   The value of


the atmospheric transmission coefficient, a  , was determined


from measurements at nearby areas and  found  to be 0.882.


     The average surface absorbed fraction,  3 and the


absorption coefficient,  e in Equation  2-31 were  determined


from measurements taken at different times of the year as  shown


in Figure 5.2.  The value of e used was 0.7 m~   and a value of


0.5 was used for 3.


          5.2.1.3  Withdrawal Layer Thickness


          Koh's Equation 2-49 forms the basis of the with-


drawal layer calculation.  This equation had to  be extended


in order to apply to the high flow rates encountered in the


field.  As mentioned in Section 4.3.1 Koh presents an em-


pirical relationship:



         a              a     -0-133
        —  =   3.5 	VT^          for 0.3 < 	V_  
-------
    i.o
    0.5
    0.2  h
    0.1  [~
   0.05  k
   0.02  L
   0.01
  0.005
  0.002  |-
  0.001
Solid Lines Are Measurements.
At Different Times During
The Season
                  n = 0.75 m
                    = 0.5
                        8       12       16
                          Depth  (feet)
              20
FIGURE 5.2 DETERMINATION OF ABSORPTION COEFFICIENT AND SURFACE

           ABSORBED FRACTION FOR FONTANA RESERVOIR
                           -177-

-------
and
            //
       a  = 44 m
        o
For    x = 1000 m,        .   =  480

                    Da x
                      o
              it c   / on\U.
        o     (3.5 x 480)
                                =  °-385
Then


      '  '   7'14;       -  1-0 I"1/6                             (5-3)
           As discussed by Huber and Harleman, during high strati-



 fication, Koh's Formula 5-3 predicts withdrawal thicknesses on the



 order of the diameter of the penstock opening (4 maters).  This was



 felt to be unrealistic.  Hence, the coefficient in Equation 5-3 was



 doubled, yielding the final form:
       A    9/
       &  =  2/e                                                      (5_4)




           The outflow standard deviation can then be calculated



 from Equation 2-50.



           5.2.1.4  Other Parameters



           The inflow standard deviation, a. was set at 4m.  This



 value was estimated from the observed spread of a dye trace in



 the upstream region of the reservoir (Figure 2.10).
                                  -178-

-------
          Air temperatures and relative humidities were  available

on an hourly basis and averaged to obtain daily values.

          As was mentioned in Section 2.3.4, the evaporation

formulae used in the field depend on where specific quantities are

measured.  Wind values were measured at a reservoir shore location.

These were transferred to mid-lake values by an empirical correlation

provided by the T.V.A. Engineering Laboratory and Rohwer's

evaporation formula (Equation 2-43) was used.

          The reservoir width was schematized according to Equation

2-46.  The length of the reservoir at a given depth was measured

along the Little Tennessee River.  The results are tabulated in

Table 5.1.  Huber and Harleman have shown that if the width varies

exponentially with depth the evaluation of Equation 2-52 is greatly

simplified.  A semi-log plot of width vs elevation, Figure 5-3,

produced the relationship:

          B = 0.885e°-0133?                      (5-5)

          Where

              B = width in meters

              y = elevation above sea level in meters

          5.2.2  Temperature Predictions

          In Figures 5.4-5.12 predicted outlet temperatures and

temperature profiles are presented as calculated both by Huber and

Harleman and from Equation 2-96.  Five different cases are shown.

The  first two, calculated by Huber and Harleman are for:

          1.  Molecular diffusion, no entrance mixing, no lag

              time (D = Dm = 0.0124m2/day, rm = 0)
                           -179-

-------
                        TABLE  5.1






        FONTANA  RESERVOIR AREAS,  LENGTHS AND  WIDTHS










Elevation above  sea level      Area       Length     Width
(f
13
13
14
14
15
15
16
16
17
t)
00
50
00
50
00
50
00
50
00
(m)
3
4
4
4
4
4
4
9
1
2
4
5
7
8
6
1
7
2
7
2
8
503
5
1
8

1,
4,
7,
10,
14,
21,
30,
40,
(m2)
283,
700,
249,
244,
643 ,
488,
286,
028,
469,

000
000
000
000
000
000
000
000
000

1
10
16
23
28
34
41
43
45
(m)
,77

0
,863
,07
.48
,21
,55
,03
,25
,73
7
0
2
3
8
9
8
(m)
160
157
265
308
378
420
519
694
885
                          -180-

-------
                  1000
I
M
oo
h->
I
                    500   -
                 to
                 M
                 j
                 OJ
                    200  -
                    100
                        300
           400                    500

Elevation Above Sea Level*  y,   (meters)
600
             FIGURE  5.3 EXPONENTIAL  WIDTH-ELEVATION  RELATIONSHIP FOR  FONTANA RESERVOIR

-------
                 T
I
I-1
00
  24
  20
   16
UJ
cr
cr
UJ
Q.
S
UJ
   12
                  OUTLET TEMPERATURE

                  FONTANA  RESERVOIR

                           1966
                  measured (possible range)
                        	rm = 00,.
                           4m, no tagtime

                          ; = 4m,dm=6m,no  laqtime
                                      Huber a Harteman
	rm=I.O,.
                  rm=0.25,or = 4m,dm=6m,no lagtime

                  r  = I0,cr, =4 m,dm=6m, no tagtime
                             m,dm=6m,
            FIGURE  5.4 OUTLET TEMPERATURE FOR FONTAKA RESERVOIR
   0
0
                                           1
                 50
                                         100
                              150          200

                                    DAY
                                     250
                                                                           300
                                                                                            350
        Jan
                           Feb
           Mor
Apr
                                 May
                                             June
                                                                 July
                                                 Aug
Sept
                                                                            Oct
Nov
Dec

-------
                     510
00

Vs
                     490
                     470
o>
O)
                  O


                  1
                  LU
                  _l
                  UJ
   450
                           OUTLET I,
                     430
                                                                      FONTANA RESERVOIR

                                                                      JUNE 22,1966  MILE 61.6
measured

rm-0,no lagtime (Implicit-Huber a Harleman)

rmr0.25, no togtime (Implicit- Huber a Harleman)

rmr0.25, no tagfime  (Explicit)
                                              FIGURE 5.5 TEMPERATURE PROFILES FOR



                                                               FONTANA RESERVOIR
                      410
                                                                              _L
                                                   12            16           20


                                                         TEMPERATURE  (°C)
                                                                         24
                                          28

-------
                          510
I
I-1
00
4>
I
                                                                           FONTANA  RESERVOIR

                                                                           JUNE  22,1966 MILE 61.6
                                                                           measured

                                                                           rm=I.O, no lagtime (Explicit)

                                                                           rm=I.O, Ah = 8m  (Explicit)
                                                        FIGURE 5.6  TEMPERATURE PROFILES FOR



                                                                        FONTANA RESERVOIR
                         410
                                                       12            16            20

                                                           TEMPERATURE (°C)

-------
                 510
CD
Y1
                                                                             FONTANA RESERVOIR
                                                                             JULY 20,1966  MILE 616
measured
rm=0, no lagtime (Implicit-Huber a Harleman)
rm-0.25, no lagtime (Implicit-Huber a Harleman)
r=0.25, no lagtime (Explicit)
                                                          FIGURE 5.7 TEMPERATURE PROFILES  FOR

                                                                          FONTANA RESERVOIR
                 410
                                                             16            20

                                                         TEMPERATURE (°C)

-------
1
M
CO

I
                  510
                 490
                 470
              O
LoJ
_J
UJ
                 450
                 430
                 410
                                                               FONTANA RESERVOIR
                                                               JULY 20,1966  MILE  61.6
                                                       ——• measured
                                                       	 rm=I.O, no lagtime (Explicit)
                                                       .	_ rm=I.O, Ah = 8m (Explicit)
                       OUTLET
                                                            FIGURE 5.8 TEMPERATURE PROFILES  FOR


                                                                           FONTANA RESERVOIR
                                                                   JL
                                                            _L
                                               12
                                               16            20

                                             TEMPERATURE  (°C)
24
                                                                                                    28

-------
H1
00


I
    510
    490
o
    470
                       FONTANA RESERVOIR

                       SEPT. 15,1966    MILE 61.6



                          measured


                    	rm=l.0,no lagtime  (Explicit)

                    	rm=I.O, Ah = 8m (Explicit)
O


1
UJ
_i
LU
            450
         -  OUTLET 4L
    430
                                                   FIGURE 5.9 TEMPERATURE PROFILES




                                                           FOR FONTANA RESERVOIR
     410
        0
                                8       12       16       20



                                   TEMPERATURE  (°C)
24
28

-------
oo

00

I
         C/)


         0)
         o


         I
         U
         _J
         LJ
              510
             490
       1    I    I    I    I     I    I

            FONTANA  RESERVOIR

            SEPT.  15,1966   MILE  61.6



        —— measured

        	rm=0, no lagtime (Implicit-
                                                        .
             470
450
                                    /• y

              	 ,. 	    //I


	rm=0.25, no lagtime (Implicit- ///


             Huber a Harleman)


	rm=0.25, no lagtime


             (Explicit)
    - OUTLET
             430
                                        FIGURE 5.10 TEMPERATURE PROFILES




                                                 FOR FONTANA RESERVOIR
             410
                0
                    8       12       16      20



                         TEMPERATURE  (°C)
                                            24
28

-------
    510
   490
   470
o>
O
^

y
Ld
   450
   430
   410
                  FONTANA RESERVOIR
                  NOV. 10,1966  MILE 61.6
                 •measured
                 •rm=I.O, no lagtime (Explicit)
                  rm= 1.0, Ah = 8 m (Explicit)
            OUTLET
                    _L
                            I
                    8      10     12     14
                       TEMPERATURE  (°C)
                                                 16
18
20
          FIGURE  5.11 TEMPERATURE PROFILES FOR FONTANA RESERVOIR
                           -189-

-------
    510
    490
     FONTANA  RESERVOIR
     NOV. 10,1966  MILE 61.6


——measured
	1-^=0, no lagtime (Implicit- Hufcer 8
                             Hcrleman)
	rm=0.25,no lagtime (Implicit- Huber a
                             Hartemcn)
	rm=0.25, ™ lcKJtime (Explicit)
 en
 a5
    470
Q

§  450
UJ
_J
UJ
    430
OUTLET
    410
         J_
                     8      10      12      14      16

                         TEMPERATURE (°C)
                                             18     20
           FIGURE  5.12 TEMPERATURE PROFILES FOR FONTANA RESERVOIR
                            -190-

-------
          2.  Molecular diffusion, entrance mixing, no lag time

              (D - V rm = °'25> dm = 6-0m>
          The remaining three cases were calculated from the modified
explicit scheme developed in Chapter 2 and are for:
          3.  Molecular diffusion, entrance mixing, no lag time
              
-------
is reached.  However, after the peak temperature the results




become poorer with instabilities occuring in the temperature profiles




near the surface (Figures 5.9,5.11).




          The effect of mixing (r  = 0.25) was found to be




insignificant until the cooling cycle began.  The effect, similar




to the laboratory results (Section 4.4.1.3), was to raise predicted




temperatures in the region of the outlet because the mixing was




assumed to take place with the warmer surface water.




          For comparison, the explicit numerical scheme was run




with r  = 0.25 as in case 2 of Huber and Harleman.   The outlet




temperature curve yields slightly higher values than those pre-




dicted by the implicit scheme.  Though it is difficult to specify




the exact cause, it is felt that this is due to the proper assign-




ment of temperature to the convective velocity depending on the




direction of the velocity (Section 2.5.1).  Though  the results using




the explicit scheme are better before the peak temperature, they




are almost identical to the implicit solution afterwards.  As the




outlet temperature reflects an average temperature  over the with-




drawal layer, more pronounced changes can be noted  in the tempera-




ture profiles.





          The effect of increasing r^ to a value of 1.0 is seen




to generally increase predicted temperatures.  Without any lag time




consideration outlet temperatures are predicted within 1°C for the




entire year.  Vertical temperature profiles are also in excellent




agreement and no instabilities are present.






                               -192-

-------
          The effect of including a lag time, Ah = 8m, chosen to




be indicative of the total depth of the inflowing streams, is




seen to shift the entire outlet temperature curve to the right,




thus "lagging" the outflow temperatures.  Temperatures before




the peak are lower and after the peak higher than under identical




conditions not including lag time.  The same trend can be noted




in the vertical temperature profiles.




          It was found that increasing the values of the




diffusion coefficient to 100 times the molecular values did not




change the temperature predictions.  From Equation 2-99 and




2-102 the maximum value of numerical dispersion, with Ay = 2m,




 At = 1 day is found to be approximately 50 times the molecular




values.  Thus it is concluded that neither molecular diffusion




nor numerical dispersion are significant in this analysis.




5.3.  Water Quality Prediction




          5.3.1  Conservative Tracer




          No long term dye tests were made in Fontana Reservoir.




Predicted cumulative mass out curves, analogous to those for the




pulse injection solution discussed in Chapter A, were calculated.




This was done to illustrate the mechanics of stratified reservoir




flow and for comparison with the detention time predictions derived




from the graphical method of Wunderlich,(Section 3.2).




          For the cumulative mass out prediction the same parameters




that gave the best fit of the outlet temperature curves were used




(r  = 1.0, d  = 6.0m, a.  = Am, Ah = 8m).  A hypothetical instan-







                              -193-

-------
taneous injection was made every 60 days starting March 2.  The




volume of each dye injection was equal to the total volume of flow of a




particular day.  The cumulative mass out curves for a given input




(Section 3.4.2.3) thus reflect the percentage of the days inflow




which has passed through the reservoir as a function of time.  For




example, from Figure 5.13, by September 7 (250 days), 87% of the




flow which entered on March 2 (day 61) and 30.5% of the flow which




had entered on May 1 (day 121) had passed through the reservoir.




          Figure 5.13 dramatically demonstrates the short circuiting




characteristics of a stratified reservoir.   The warm inflow of




March 2 and May 1 entered at the reservoir surface 60 days apart.




The outlet cumulative mass out curves are for the most part




parallel, separated by approximately 60 days.  This in indicative of




convection being the major transport mechanism in the vertical




direction.  The cooler inflows of late summer and of the fall




(August 31 and October 29, days 241 and 301) enter beneath the




reservoir surface at their respective density levels.  Once entered,




the vertical distance to the outlet is reduced by the subsurface




entrance, these inflows tend to reach the outlet relatively sooner




than the spring inflows.  For example it is predicted that ten




per cent (10%) of the input of October 29 would have passed




through the reservoir by November 7 (day 308), i.e., nine (9)




days later.  The corresponding time for the input of March 2 is




ninety four (94) days.
                           -194-

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         FIGURE  5.13  FONTANA RESERVOIR  SIMULATION OF VARIOUS PULSE  INJECTIONS
               100
1
I-1
vo
Ln
1
               80
               60
            O
            O
                  0
   FONTANA  RESERVOIR

 SIMULATION  OF VARIOUS

 PULSE  INJECTION TESTS

          I966
              IOO
I50          200

      DAY
250
300
350
                     Jan.
Feb.     Mar.    Apr.     May     June    July     Aug.     Sept.    Ocf      Nov.     Dec.

-------
          During the cooling cycle, inflows tend to enter below




the surface of the reservoir.  Each successively-cooler input tends  to




enter lower than the input which enters before it.  This has the




effect of raising the level of the withdrawal layer and preventing




the complete withdrawal of a given day's input.  For example, by




December 21 (350 days) only seventy seven per cent (77%) of the flow




which entered on May 1 had been withdrawn from the reservoir.  Since




the gradual process of surface mixing due to surface cooling is well




advanced by late December, it is highly probable that all of the




flow which entered on May 1 would not pass through the reservoir




until the following spring or summer.  For later inputs this effect




becomes more pronounced.  For example, only 52 per cent of the




inflow of July 1 (day 181) had passed through the reservoir by




December 21.




          In view of the above discussion, it is clear that it is




extremely difficult to define precisely what is meant by a detention




time for a given reservoir input.  Wunderlich, as stated in




Chapter 3, defined the detention time, t,, as the time span




between a given input temperature and the time at which that




temperature appeared in the outlet.  In Table 5.2 the detention




times are presented for inputs of every 60 days from March 2




as calculated by the graphical method of Wunderlich (Figure 3.1).




For comparison, the corresponding percentages of these inputs




which would have passed through the reservoir at the end of their




respective "detention times" and by day 350 (December 21) from






                            -196-

-------
                            Table 5.2

                   Detention Times  (t  )    Tracot after  t   Tracot  by

                    Wunderlich  (57)            (%)           December 21

Input                   (Days)	      Equation  3.28      (%)	

March 3 (Day 61)           0                    	              96



May l(Day 120)           123                     25              77



July l(Day 181)          	                   	              47



August 31 (Day 2A1)       	                   	              39



October 29(Day 301)       20                     18              31
Table 5.2  Comparison of Predicted Cumulative Mass Out Values with
           The Detention Times of Wunderlich.
                            -197-

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Figure 5.13 are given.




          The "detention times" calculated from Wunderlich's method




do not correlate with the values calculated from Equation 3-38.




Though Wunderlich calculates no outflow from the inputs of July 1 and




August 31, Equation 3-38 predicts that 47% and 39% of these inputs,




respectively, would have passed through the reservoir by December 21.




Though the curves of Figures 5.13 have not been verified from field




measurements they are indicative of the stratified reservoir flow




through pattern since the results follow the trend verified in the




laboratory.  It can be generally concluded that the use of one




"detention time" for a given input in an attempt to describe its




flow through time in a stratified reservoir gives results which do




not reflect the complicated short circuiting characteristics of a




stratified reservoir.




     5.3.2  Dissolved Oxygen Predictions for Fontana Reservoir




          5.3.2.1  Input to the Mathematical Model




          In addition to the inputs to the temperature model already




discussed (section 5.2), several additional parameters need to be




specified in order to solve the D.O. prediction problem.  These are:




          1.  The D.O. and B.O.D. of the incoming streams and the




              long term B.O.D.  decay rate, K (Equation 3-14)




          2.  The initial conditions for B.O.D. and D.O. in the




              reservoir at time t = t,




          3.  A surface boundary condition which effectively accounts




              for the interplay between D.O. and B.O.D. production and







                                -198-

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              consumption at the reservoir surface  (this was discussed




              in Section 3.4.1.1).




          The D.O. of the incoming streams to Fontana reservoir was




monitored daily from February through December of 1966 from random




samples analyzed in the field using a simplified Winkler test kit.




Daily weighted averages of the five incoming streams were used as




inputs to the model.




          The B.O.D. of the incoming streams was sporadically




sampled in 1965. In the most polluted stream, Tuckaseegee, at most




twelve tests were made at a given monitoring station.  The results




were presented in terms of five day B.O.D. with no long term




B.O.D. reported.  Typical D.O. and B.O.D. data is presented in




Table 5.3.  The station number refer to points along the various




rivers as shown in Figure 5.1.  A weighted average of the median




values for the station closest to the reservoir produced a




five day B.O.D. of about 1.5 ppm.  As was discussed in section




3.4.1.1, long term B.O.D. values, due to nitrification, are higher




than five day B.O.D. values.  Lacking any long term data, a con-




stant input value of 8 ppm of B.O.D. was assumed.




          A value for the first order decay constant, K, also




had to be assumed.  Again, considering a slow, long term decay,




two different values,  0.01 and  0.05 day  were tested.  Though




K is probably temperature dependent there was no basis for




assuming the functional relationship.  It was also  felt that




a constant value would more clearly illustrate the  other






                             -199-

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STATION NO.
  11
  12
  14

1.

NO. Tests
Maximum
Minimum
Median
NO. Tests
Maximum
Minimum
Median
NO. Tests
Maximum
Minimum
Median
TABLE 5.3
FLOW B.O.D.5 B.O.D.1Q B.O.D.^
(cfs) (mg/JO (rag/ JO (mg/A)
0 5
2.1
0.9
1.3
0 5
1.2
0.7
0.8
721
7.0+ 7.0+ 7.4+
1.5 3.5 7.4+
1.7 3.5 7.4+
   TABLE 5.3 B.O.D. MEASUREMENTS IN FONTANA  RESERVOIR INFLOWS
                              -200-

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assumptions which had been made.

          Unfortunately, there was no  B.O.D. profile taken

at the time that the first D.O. profile  in  the lake was made,

(April 20, 1966).  In fact, the only B.O.D. measurements taken

in the reservoir were in July and August of 1965 at depths no

greater than 20 meters.  Therefore, initial conditions, B.O.D^,

for Equation 3-15 had to be assumed.

          In order to illustrate the sensitivity of the results

to the initial condition, calculations were carried out for a

uniform B.O.D.-^ of 3 ppm and for zero  B.O.D.^ in the reservoir

on March 1.   (Table 5.4)

          Since the reservoir was isothermal on March 1, it

was assumed that the D.O. in the reservoir was uniform at that

time.  An inspection of the measured outlet D.O. in February and

March indicated that a reasonable initial D.O. value would be 8 ppm

on March 1.  This differs from the saturated value of 12.2 ppm

for the isothermal reservoir temperature of 6.7°C that one might

be tempted to assume.

          5.3.2.2  Comparison with D.O. Measurements in Fontana

                   Reservoir

          Predicted outlet D.O. concentrations and profiles for

various days of the year for different initial and input B.O.D.

condition and D.O. surface assumptionsare presented in

Figures 5.14-5.23.  The same parameters  that were arrived at from

the temperature model  (rm = 1.0, d^ =  6m, c^ = 4m, Ah = 8m,

Section 5.2) were used.  The different trends which result from
                            -201-

-------
                              Table  5 A

            D.O. Initial      B.O.D. Initial      K

Run             (ppm)             (ppm)	     (Day)

180             0.01


                                                       Entire Euphotic
203             0.05
                                                       Zone Saturated


383             0.01
                                                  0.05
                  8ppm              3ppm          0.05  Top 3m
                                                         Saturated
Table 5.4  The Various Initial Condition Tested in the D.O. Analysis,
                            -202-

-------
   12
o
Q  8
    6
    0
              DISSOLVED OXYGEN  IN
          FONTANA  RESERVOIR  OUTLET
                      1966
                         Euphotic zone
                            saturated
                             	 measured
                              --  BODj=0, K = 0.01 day'1
                             	BOD| = 3, K = 0.01 day'1
     0
50
100
150
200
250
300
350
                                              Day
                          J_
       Jan.     Feb     Mar.     Apr.    May    June   July      Aug.    Sept.

  FIGURE 5.14 OUTLET D.O. CONCENTRATIONS FOR FONTANA RESERVOIR
                                                          Oct
                                                      Nov
                                                 Dec

-------
I
N>
O
E
CL
Q.
—

q
ci
            O
                12
                10 U
                 0
                             T
                          DISSOLVED OXYGEN  IN

                      FONTANA  RESERVOIR OUTLET

                                   1966
                                                  Euphotic zone

                                                     saturated

                                                  Top 3 meters
                                                     saturated
                                            —- measured

                                            	 BODj=0,K = 0.05 day

                                            	BOD) = 3, K =0.05 day'
                 50
       100
                                  150
200
250
300
                                                           Day
                        _L
350
Jan.    Feb.
Mar.
                              Apr.    May     June    July     Aug     Sept.
                                                                                        Oct.
                              Nov.
                          Dec
              FIGURE 5.15 OUTLET  D.O.  CONCENTRATIONS FOR FONTANA RESERVOIR

-------
          FONTANA RESERVOIR

          APRIL 20,1966  MILE 61.6

                             	  measured
             Entire Euphotic \   -;-•- BODj =0,K=O.OI day"1
    500

2  460
LU
_J
Ld




    420
    380
             zone saturated
	BOD; = 3,K-O.OI day"
        0
     8
                    DISSOLVED  OXYGEN (ppm)
12
     FIGURE 5.16 DISSOLVED OXYGEN PROFILES FOR FONTANA RESERVOIR
                          -205-

-------
   500
tn
£460
£
UJ
UJ
   420
   380
     I       i       i      i       i       i    '
FONTANA  RESERVOiR
APRIL 20,1966 MILE 61.6
                   — measured
  Entire Euphotic \   	- BOD;=0,K=0.05 day"1
  zone saturated  I	 BODj = 3,K=0.05 day"1
   Tap 3 meters |  ~— BOD; =3,K= 0.05 day'1
     saturated
       0            4            8            12

                   DISSOLVED  OXYGEN  (ppm)
        FIGURE 5.17 DISSOLVED OZY2EN PROFILES 3'0il FOICTAWA RESERVOIR
                         -206-

-------
    500
en
o>
    460
LU

LU
    420
    380
           FONTANA RESERVOIR
           JULY 19,1966 MILE 61.6
                                   measured
            Entire Euphotic
            zone saturated
        0
	BOD j=0,K = 0.01 day'1
	BOD; = 3, K = 0.01 day'1
     s
      8
                    DISSOLVED OXYGEN (ppm)
12
        FIGURE 5.18 DISSOLVED OXYGEN PROFILES FOR FONTANA RESERVOIR
                      -207-

-------
   500
 in
   460
LJ
_l
UJ
   420
   380
               measured
               BODj=0,K=OO5day~'  ,
               BODj=3,K= 0.05 day"1  I
               BOD;=3,K= 0.05 day"1 !
       0
       Entire Euphotic
       zone saturated
        Top 3 meters
          saturated
FONTANA RESERVOIR
JULY  19,1966  MILE 61.6
     8
                     DISSOLVED  OXYGEN (ppm)
      FIGURE 5.19 DISSOLVED OXYGEN PROFILES FOR FONTANA RESERVOIR
                       -208-

-------
   500

   460
LJ
_l
LU
   420
   380
              i       i      i       r

         FONTANA RESERVOIR

         SEPT. 7,1966  MILE 61.6
            Entire Euphotic
            zone saturated
       0
measured

BODi=0,K=O.OIday~l
    j = 3,K=O.OIday
-i
8            12
                    DISSOLVED OXYGEN (ppm)
        FIGURE 5.20 DISSOLVED OXYGEN PROFILES FOR FONTANA RESERVOIR
                   -209-

-------
   500 -
0)
O
   460
   420
   380
        FONTANA  RESERVOIR
        SEPI 7,1966 MILE 61.6
       0
Entire Euphotic
zone saturated  i
Top 3 meters   ]
  saturated
         rrrHTd —— •<.
                                measured
                                    j = 0,K=0.05
                                    j = 3,K=0.05
                           -- — BODj=3,K=0.05
         4
8
                   DISSOLVED  OXYGEN (ppm)
      FIGURE 5.21 DISSOLVED OXYGEN PROFILES FOR FONTANA RESERVOIR
                  -Ziu-

-------
o>
o>
O
I
LU
LU
    500
460
    420
    380
               i	1	1	
          FONTANA  RESERVOIR
          NOV 7, 1966 MILE 61.6
        0
              Entire Euphotic
              zone  saturated
                               measured
                         	BOD j=0,K= 0.01 day"1
                         	BOD: = 3, K= 0.01 day"1
                              8
                    DISSOLVED  OXYGEN  (ppm)
12
      FIGURE 5.22 DISSOLVED OXYGEN PROFILES FOR FONTANA RESERVOIR
                   -211-

-------
    500
CO

£
(V
O


1
LU
_l
L±J
460
   420
   380
         FONTANA RESERVOIR
         NOV. 7,1966  MILE 61.6
        Entire Euphotic
        zone saturated

        Top 3 meters
          saturated
       0
measured
BODj=0,K=0.05dayH
      3,K = 0.05day~'
      3,K = 0.05day~'

    M
    i!
    ii
    Ji
                              8
            12
                    DISSOLVED  OXYGEN (ppm)
        FIGURE 5.23 DISSOLVED OXYGEN PROFILES FOR FONTANA RESERVOIR
                   -212-

-------
 The different assumptions mentioned  above  help  to  illustrate  the




 mechanics of the D.O. prediction model  and  its sensitivity to




 the various assumptions resulting from  a  lack  of  certain data.




           From Figures 5.14 and 5.15  (considering for a moment




 the case where the entire euphotic  zone (approximately 6m for




 Fontana) has been assumed to be saturated  (Section 3.4.1.2)), it is




 seen that if K is constant, and the initial B.O.D. value is changed




 from 0 to 3.0 ppm, lower D.O. predictions result  until about day




 225 (August 13).  This corresponds  to the time at which the




 temperature in the outlet is beginning  to rise (Figure 5.4)




 indicating that the warm inflow water of  the previous months




 is reaching the outlet.  Thus, the assumption  for the initial




 B.O.D. in the reservoir tends to affect the outlet D.O. only




 as long as the major part of the water  discharged is the




 water which was initially in the reservoir.  The  same trend




 is found in the predicted D.O. profiles.





           The effect of increasing the  decay rate, K, is to increase




the rate of D.O. consumption within the  reservoir.  A value of K =




0.05 instead of 0.01 day  produces lower predicted D.O. values




in all cases.



          Changing the surface assumption for D.O. from




saturation in the entire euphotic zone (6m) to  saturation to a




depth of 3m (initial B.O.D. = 3 ppm) is  seen to result in generally




lower D.O. predictions.  This is due to  two phenomena. The first is




the obvious fact that less dissolved oxygen is  being input to the model





                             -213-

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in the surface  region.  The  second  is  involved  in  the assumption




that  the inflowing water  is  mixing  with  the water  over  the  top  6nr




of the reservoir.  Because this entire depth is not saturated under




the assumption  of surface saturation to  only 3m, less dissolved




oxygen is entrained in the incoming water through  the mixing




process.




          One curious point  about all of the profiles is the




prediction of a reversal near the bottom of the reservoir.  This




is due to the inflows of March 6-10 which were saturated with




D.O.  but colder than the initial isothermal temperature of 6.7 C.




Therefore, in the mathematical model, high oxygenated water




was brought to the bottom layer of the reservoir displacing




the water which was originally there.  Since no bottom oxygen




demand was assumed, the only mechanism of D.O.  consumption was




the B.O.D. originally present in this water.  Since a constant




value of 8 ppm B.O.D.  was assumed for all of the inflows,  the




maximum D.O. consumption was 8 ppm.   Perhaps there was some residual




B.O.D at the bottom or the incoming  B.O.D.  of the March 6-10




water was greater than 8 ppm.  In the absence of detailed  data,




it is impossible to come to a definite conclusion.

-------
CHAPTER 6.  CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE RESEARCH
6.1 The Thermal Stratification Phenomena

        In reservoirs  characterized by horizontal  isotherms, water enter-
ing at the upstream end undergoes  some initial mixing and enters the

reservoir at an elevation  corresponding to  its own  density. The water
which enters at the beginning of  the  stratification season tends to
enter at  the reservoir surface and remain in  the reservoir for a rela-
tively longer  time than the cooler water which enters late in the strat-
ification cycle.  The  thickness of  the internal withdrawal layer near
the reservoir  outlet  depends on the vertical  temperature gradient at
the outlet, decreasing as  the temperature gradient  increases. The temp-
erature and water quality  in the  outflow are  average values for the
water in  the withdrawal layer. As  summer turns to  fall, evaporative
cooling and the resulting  surface  instabilities tend to cause a mixed
isothermal layer  which eventually  returns the reservoir to a completely
isothermal state. During this mixing  process  the water quality of the
reservoir also tends  to become uniform.

6.2 Temperature Predictions
        The  temperature model  of  Huber  and  Harleman was modified to
include an  internal  lagtime  and  a proper  assignment of  temperatures
to  the vertical  convective term  depending  on the  sign  of  the  convec-
tive  velocity. The horizontal  advective velocities were determined
from  a  selective  withdrawal  theory developed by  Koh and assumptions  about
the shape of  the  inflow velocity distribution.  Vertical convection was
found to  be the major mechanism for heat transport within the reservoir.
No vertical turbulent diffusivities were introduced at any time

                              -215-

-------
 into the mathematical model.   During the initial period of the year in




 which temperature profiles are stable with regard to vertical density




 distribution, the effect of vertical diffusion does not appear to be




 important.   In later periods  of potentially unstable vertical density




 distributions  the effect of vertical mixing is accounted for indirectly




 by the development of a uniformly mixed surface layer.  In both labora-




 tory and field cases excellent temperature predictions were obtained




 with the modified temperature model during the entire yearly stratifi-




 cation cycle.  This is an improvement over the model of Huber and




 Harleman in which outlet temperature predictions after the time of the




 peak outflow temperature tended to be lower than observed values.




 6.3  Concentration Predictions




         6.3.1  Laboratory Experiments




         Outlet concentration predictions for pulse injections of a con-




.. servative tracer in a laboratory reservoir agreed well with measured




 values.   The predicted time at which measurable values of tracer first




appeared in the outlet was usually somewhat earlier than Heasured values.




 However, the time of the peak outlet concentration was fairly well




 predicted.   The concentration predictions were found to be more sensi-




 tive than temperature predictions to assumptions about the shape of




 the inflow velocity profile and the amount of mixing at the reservoir




 entrance.  However, one set of parameters was sufficient to predict




 reasonable results for the three different types of experiments con-




 ducted.   Difficulty was occasionally found with predicting outlet con-




 centrations from pulse injection toward the end of the stratification

-------
cycle.  This is attributed  to  the high degree of short circuiting which




occurs at late times and the resulting sensitivity to the choice of the




various parameters in the mathematical model.  Nevertheless, the labora-




tory tests illustrated  that many of  the parameters involved in the mathe-




matical model can be related to the  depth of the inflowing stream at the




head of the reservoir.




         6.3.2  Field Results




         The simulations of pulse injection tests for an actual reservoir




served to illustrate the flow  through time characteristics of a stratified




reservoir.  The trends  are  similar to those found in the laboratory.




Flows entering toward the latter part of the stratification cycle tend




to reach the outlet much more  quickly, relative to the time of inflow,




than flows which enter  in the  spring.  It is unfortunate that there are




no long term pulse injection dye tests available for comparison with the




predicted values.




         In 1966 detailed temperature and D.O. measurements were made in




Fontana reservoir and its inflowing  streams.  Though  long term B.O.D.




data was not available, the water quality mathematical model was tested




using assumed values for initial B.O.D. and values for the B.O.D. of




the inflowing streams.  In  addition, having no detailed information




about the complicated oxygen balance in the surface regions, two differ-




ent assumptions were tested.   The resulting sensitivity analysis to the




various assumptions made about the input B.O.D. data provided several




interesting observations.




         A first order  decay rate was assumed for the long term B.O.D.




process.  Lower D.O. prediction resulted from higher values of the decay





                                -217-

-------
constant.




         Increasing the value assumed for the initial amount of B.O.D. in




the reservoir decreases the amount of D.O. in the outlet until the tempera-




ture at the outlet begins to rise.  After this time, the initial value




assumed for the B.O.D.  in the reservoir changed D.O. profiles and outlet




D.O. concentrations very slightly.  This indicated that the quality of




the discharged water began to be determined by the quality of the water




in the inflowing streams as the warm inflow began to reach the outlet.




         Assuming the entire eupthotic zone to be saturated produced




higner D.O. predictions than the assumption of D.O.  saturation to only




3 meters.




         The lack of sufficient input water quality data made it difficult




to r.iake a direct comparison of measured and predicted values except




through a sensitivity analysis.  The combination of the information




gained from the pulse injection simulation and the D.O. predictions indi-




cates that the use of a detention time approach for water quality predic-




tion in a stratified reservoir tends to greatly oversimplify a very com-




plex problem.




6.4  Recommendations for Future Research




         6.4.1  Improvement of the Mathematical Model




         1. The present model is capable of handling only one entering




            stream at the head of the reservoir.  In the case of the




            T.V.A. Fontana data, the input temperature and water




            quality of the incoming streams were averaged to yield




            one value of T., Q., a..  This may not be representative




            of the actual inflow to the reservoir.  It is possible

-------
   that one stream could be colder than another (for example,




   if one stream was the discharge from another reservoir) and




   thus it could enter the reservoir at a different depth with-




   out interacting.  Therefore provision should be made to




   accommodate several input streams to the reservoir independ-




   ently of one another.




   Similarly provision should also be made to handle more than




   one outlet from the reservoir.  For this case the outflow




   withdrawal velocity distribution could be considered to




   be the sum of the Gaussian distribution of the individual




   outlets.




2.  At present observed water surface elevations are an input




   to the mathematical model.  These could be computed from




   a continuity equation applied to the entire reservoir




   including precipitation and evaporative mass loss in addi-




   tion to the inflow and outflow contributions.




3.  The water quality model is oriented toward treating sub-




   stances undergoing a first order decay.  The decay rate




   has been assumed to be constant and independent of tempera-




   ture.  A more general model could be developed to treat




   other types of decay rates or water quality interactions




   including decay rates which are temperature dependent.




6.4.2  Laboratory and Field Research




1.  Much work remains to be done on the determination of the




   inflow velocity distribution in a continuously stratified




   reservoir.  A theoretical prediction of the spread of the





                             -219-

-------
            inflow layers is almost imperative if multiple inflows




            are  to be incorporated into the model.  This would also




            be beneficial in determining a proper thickness, Ah, for




            the  lag time determination.




         2.  The  time for the inflow to reach its own density level




            was  based on a two-layered theory.  A method which accounts




            for  the continuous stratification in the reservoir would




            be a more rigorous approach.




         3.  Laboratory tests for a continuous injection of tracer would




            be another step toward verifying the model for conditions




            closer to those encountered in the field.   Experiments using




            radioactive tracer with known decay rates  would be a more




            striagent test of the mathematical model.




         4.  There is a need for lorg term water quality data in existing




            reservoirs.   Included in this are (1) the  initial reservoir




            water quality at the beginning of the stratification cycle,




            (2)  long term B.O.D. and chemical oxygen demand (C.O.D.)




            tests on the water in the inflowing streams and the effects




            of temperature on these processes and (3)  evaluation of the




            complicated D.O. balance in the euphotic zone.




            It is hoped that the development of a method for analyzing




D.O.  and other water quality parameters in a stratified reservoir will




provide the  incentive for field data collection programs to be used in




further tests of the mathematical model.
                                   -220-

-------
 CHAPTER 7-   BIBLIOGRAPHY

 1.   Austin, Garry H., Gray, Donald A., and Swain, Donald A., Report
     on Multilevel Outlet Works at Four Existing Reservoirs, Bureau
     of Reclamation, U. S. Department of the Interior, Denver, Colorado,
     August 1968.

 2.   Bella,  David A., Finite-Difference Modelling of River and Estuary
     Pollution, Ph.D. Thesis, New York University, New York, April 3,
     1967.

 3.   Bohan,  J.  P. and Grace, J. L.,"Mechanics of Flow from Stratified
     Reservoirs in the Interest of Water Quality", U. S. Army Engineering
     Waterways Experiment Station, Corps of Engineers, Vicksburg, Miss.,
     1969.

 4.   Burt,  W. C., "Preliminary Study of the Predicted Water Changes at
     the Lower Snake River Due to the Effects of Projected Dams and
     Reservoirs", Water Research Associates, Corvalis, Oregon, November
     1963.

 5.   Camp,  T. R., Water and Its Impurities^, Reinhold Publishing Company,
     New York, 1963.

 6.   Carslaw, H. S. and Jaeger, J. C., Operational Methods in Applied
     Mathematics, Dover Publications, Inc., New York, 1963.

 7.   Cederwall, Klas and Hansen, Jens, Tracer Studies on Dilution and
     Residence Time Distribution in Receiving Waters, Water Research,
     Vol. 2, No. 4, June 1868, pp. 297-310.

 8.   Churchill, M. A. and Nicholas, W. R., Effects of Impoundments on
     Water Quality, A.S.C.E., SA6, December 1967.

 9.   Daily,  J. W. and Harleman, D. R. F.,  Fluid Dynamics, Addison Wesley
     Publishing Company, Inc., Reading, Mass., 1966.

10.   Daly,  B. J. and Pracnt, W. E. , A_Numeric_al_Study of Density Current
     Surges, Los Alamos Scientific Laboratory of the University of
     California, Los Alamos, New Mexico, 1968.

11.   Dougal, M. D. and Baumann, E. R., Mathematical Models for Expressing
     the B.O.D. in Water Quality Studies,  Proc. 3rd Annual Am. Water
     Research Conference, San Francisco, November 1967.

12.   Fenerstein, D. L. and Selleck, R. E., Fluorescent Tracers for
     Dispersion Measurements^ A.S.C.E., SA4, August 1963.

13.   Gannon, J. J., River and Laboratory B.O.D. Rate Considerations,
     A.S.C.E., SA1, February 1966.


                              -221-

-------
14.   Harleman,  D.  R.  F.  and Abraham, G.,  One-Dimensional Analysis of
     Salinity Intrusion in the Rotterdam Waterway, Delft Hydraulics
     Laboratory,  Publication No.  44, October 1966.

15.   Harleman,  D.  R.  F.  and Stolzenbach,  K.  D.,  "A Model Study of Thermal
     Stratification Produced by Condenser Water  Discharge", M.I.T. Hydro-
     dynamics Laboratory Technical Report No.  107, October 1967.

16.   Holley, E. R., Discussion of Difference Modeling of Stream Pollution,
     by David A.  Bella and William E.  Dobbins,  Proc. A.S.C.E., Vol. 94,
     No. SA5, Paper 6192, October 1968.

17.   Holley, E. R. and Harleman,  D.  R.  F.,"Dispersion of Pollutants in
     Estuary Type Flows", M.I.T.  Hydrodynamics Laboratory Technical
     Report No. 74, January 1965.

18.   Huber, W.  C.  and Harleman, D.  R.  F., "Laboratory and Analytical
     Studies of the Thermal Stratification of Reservoirs", M.I.T. Hydro-
     dynamics Laboratory Technical Report No.  112, October 1968.

19.   Ingols, Robert S.,  Discussion of  Some Effects of Water Management
     on Biological Production in Missouri River  Main Stem Reservoirs,
     Proceedings  of the Specialty Conference on  Current Research into
     the Effects  of Reservoirs on Water  Quality, Vanderbilt University,
     Nashville, Tennessee, 1968.

20.   Jaske, R.  T.  and Spurgeon, J.  L.,  A Special Case, Thermal Digital
     Simulation of Waste Heat Discharge,  Water  Research, Vol. 2, No. 11,
     November 1968.

21.   Kennedy, R.  E. ,  "Computation of Daily Insolation Energy", Bulletin,
     American Met. Society.. Vol.  30, No.  6,  pp.208-213, June 1949.

22.   Keulegan,  G.  H., Laminar Flow at  the Interface of Two Liquids,
     Journal of Research for the National Bureau of Standards, Vol. 32,
     June 1944.

23.   Koh, R.C.Y.,  "Viscous Stratified  Flow Towards a Line Sink", W. M.
     Keck Laboratory Report, KH^R-6, California  Institute of Technology,
     1964.

24.   Kohler, M. A., "Lake and Pan Evaporation"  in Water Loss Investiga-
     tion, Lake Hefner Studies, Technical Report, U.S.G.S. Professional
     Paper 269, 1954.

25.   Krenkel, P.  A.,  Cawley, W. A. and Minch, V. A., The Effects of
     Impoundments on River Waste Assimilative Capacity, Journal of the
     Water Pollution Control Federation, 37, 9,  September 1965.
                                -222-

-------
26.  Krenkel, P. A., Thackston, E. L. and Parker, F. L. , IJhe Influence
     j3J_Impoundinents on Waste Assimilation Capacity. Proc. of the
     Specialty Conference on Current Research into  the  Effects of
     Reservoirs on Water Quality, Vanderbilt University, Nashville,
     Tennessee, TR 17, 1968.

27.  Krenkel, P. A., Thackston, E. L. and Parker, F. L., Impoundment and
     Temperature Effect on Waste Assimilation, Proc. A.S.C.E., SA1, Feb.
     1969.

28.  Lean, G. H. and Whillock, A. Z., The Behavior  of a Warm Water Layer
     Flowing Over Still Water, International Association for Hydraulic
     Research, llth International Congress, Leningrad,  1965.

29.  Levenspiel, 0. and Bischoff, K. B. , Patterns of Flow in Chemical
     Process Vessels, Advances in Chemical Engineering, Vol. 4, New
     York, 1963.

30.  Hiyauchi, T., "Residence Time Curves" Chemical  Engineering (Japan)
     Vol. 17, p. 382, 1953.

31.  Murphy, K. L. and Timpany, P. L., Design and Analysis of Mixing
     for an Aeration Tank, A.S.C.E., SA5, October 1967.

32.  O'Connell, R. L. , Thomas, N. A., Godsil, P. J. and Hearth, C. R.,
     Report of Survey of the Trucker River, U. S. Public Health, 1963.

33.  O'Connell, R. L. and Thomas, N. A., Effect of  Benthic Algae on
     Stream Dissolved Oxygen, Proc. A.S.C.E. Journal of the Sanitary
     Engineering Division, SA3, 1965.

34.  O'Connor, D. J. and DiToro, D. M., The Solution of the Continuity
     Equation in Cylindrical Coordinates with Dispersion and Advection
     for an Instantaneous Release, Symposium on Diffusion in Oceans
     and Fresh Water, August 31-September 2, 1964.

35.  O'Connor, D. J. and DiToro, D. M., The Distribution of Dissolved
     Oxygen in a Stream with Time Varying Velocity. W.R.R., Vol. 4,
     No. 3, June 1968.

36.  Orlob, G. T. and Selna, L. G., "Mathematical Simulation of Thermal
     Stratification in Deep Reservoirs", A.S.C.E. Specialty Conference
     on Water Quality, Portland, Oregon, January 1968.

37.  Posey, Frank H. and DeWitt, J. W., Effects of  Reservoir Impoundment
     on Water Quality, A.S.C.E. P01, January 1970.

37a. Pritchard, D. W. and Carpenter, J. H., Appendix to "A Study of  the
     Effects of a Submerged Weir in the Roanoke Rapids  Reservoir Upon
     Downstream Water Quality", by F.F. Fish, C.H.J. Hull, B.J. Peters
     and W.E. Knight, Special Report No. 1, Roanoke River Studies.

                               -223-

-------
38.  Purcell,  L.  T. ,  The Aging of Reservoir Waters, Journal of the
     American  Water  Works Association,  31, 10, October 1969.

39.  Rehwer, C.,  Evaporation from Free  Water Surfaces", U. S. Dept.
     of Agriculture,  Technical Bulletin No. 271,  December 1931.

40.  Scalf, M.  R. , Witherow, J.  L. and  Priesing C.  P-, IRON-59 as^
     Solids Tracer  in Aqueous Suspensions, A.S.C.E., SA6, December
     1968.

41.  Shamir, U. Y. and Harleman, D.  R.  F., "Numerical and Analytical
     Solutions of Dispersion Problems in Homogeneous and Layered
     Aquifers", M.I.T. Hydrodynamics  Laboratory Technical Report No.
     89, May 1966.

42.  Slotta, L. S. and Elwin, E. H.,  Entering Streamflow Effects on
     Currents  of a Density Stratified Model Reservoir, Bulletin No. 44,
     Engineering Experiments Station, Oregon State  University, Corvallis,
     Oregon, October  1969.

43.  Slotta, L. S. and Terry, M. D.,  The Numac Method for Non-Homogen^
     eous Unconfined  Marker Cell Calculations  Bulletin No.  44, Part II,
     Engineering Experiments Station, Oregon State  University, Cor-
     vallis, Oregon,  October 1969.

44.  Stigter,  C.and  Siemens, J., Calculations of  Longitudinal Salt-
     Distributions in Estuaries  as a  Function of  Time, Delft Hydraulics
     Laboratory,  Publication No. 52,  October 1967.

45.  Sundaram,  T. R., et.  al.,  An Investigation of  the Physical Effects
     of Thermal Discharges into Cayuga  Lake, Cornell Aeronautical Lab.,
     Inc.,  Buffalo, New York, November  1969.

46.  Symons, J. M.,  Irwin, W. H., Clark R. M. and Robeck, G. G.,
     Management and Measurement of P.O.  in Impoundments, U.  S. Dept.
     of the Interior, FWPCA, Cincinnati, Ohio, September 1966.

47.  Taylor, G. I.,  "The Dispersion  of  Matter in  Turbulent Flow Through
     a Pipe",  Proc. Royal Society (London) (A), 223 (1954).

48.  Thackston, E. L. and Morris, M.  W., Tracing  Polluted Reservoir
     Inflows with Fluorescent Dyes,  Vanderbilt University, TR 21,
     1969.

49.  Thirumurthi-Dhandapani, A Break-Through in the Tracer Studies
     of Sedimentation Tanks, Journal, Water Pollution Control Federa-
     tion,  Part 2, Vol.  41, No.  11,  November 1969.

50.  Thomas, H. A. and McKee, J. E.,  "Longitudinal  Mixing in Aeration
     Tanks", Sewage Works Journal, Vol.  14, 1942.

-------
51.  Verduin, J., Primary Predictions in Lakes_, Limnology and Oceano-
     graphy, 1, 4, April 1956.

52.  Villemonte, J. R., Rohlich, G. A. and Wallace, A. T., Hydraulic
     and Removal Efficiencies in Sedimentation Basins, Third Interna-
     tional Conference on Water Pollution Research, Munich, Germany,
     Paper 16, 1966.

53.  Ward, J. C. , Annual Variating Stream Water Temperature, Proc.
     A.S.C.E., No. SA6, December 1963.

54.  Water Resources Engineering, Inc., Mathematical Models for the
     Prediction of Thermal Energy Changes in Impoundments, Final Report
     to the FWPCA, Columbia River Thermal Effects Project, December
     1969.

55.  Wilson, James, U. S. Department of the Interior Techniques of
     Water Resources Investigation of the USGS - Fluorometric Proced-
     ures for Dye Tracing, Book 3, Chapter A12, 1968.

56.  Wunderlich, W. 0. , "Heated Mass Transfer Between a Water Surface
     and the Atmosphere", Internal Memorandum, T.V.A. Engineering
     Laboratory, Norris, Tennessee, 1968.

57.  Wunderlich, W. 0. and Elder, R., "Graphical Temperature and D.O.
     Prediction Methods", Water Resources Research, T.V.A. Division
     of Water Control  Planning, Engineering Laboratory, Norris, Tenn.,
     April 1969.
                             -225-

-------
                          APPRENDIX I




                      THE COMPUTER PROGRAM




      In this appendix the FORTRAN computer program used to solve the




finite volume representations of the temperature and water quality




equations developed in Chapters 2 and 3 is presented. The program con-




sists of a MAIN routine and sixteen subprograms. Temperatures are




referred to by T and concentrations by C.




      The MAIN routine performs all of the input and output except




writing the output for the pulse injection concentrations and cum-




ulative mass out information. This is done in subroutine SPECOT (N).




The MAIN routine initializes many variables and constants, adjusts the




surface elevation and calls for either solution to a pulse injection




of a concervative tracer or for dissolved oxygen predictions.




      At the beginning of the MAIN routine is a clock routine to indi-




cate the time required for the computations (the subroutine CLOCK is




a library program at the Massachusetts Institute of Technology Informa-




tion Processing Services Center). The time required to compute both




temperatures, D.O., B.O.D. profiles and outlet values for three hun-




dred (300) time steps and fifty (50) distance steps is approximately




three (3) minutes.





      Comment cards are included in bpth the MAIN routine and in the




subprograms to indicate points of interest and the specific function




of each of the subprograms. A listing of the necessary input variables




to the program is presented in APPENDIX II. In APPENDIX III, sample




input data for the D.O. prediction model is presented for the. case of





                               -226-

-------
initial B.O.D. = 0., initial D.O. = 8 ppm, K = 0-05 day   and satura-




tion of the entire euphotic zone.
                                 -227-

-------
C  RFSERVOTR STRATIFICATION  AND  CONCENTRATION PREDICTION PROGRAM*  1970.
             T(60,2),EL(60),XL(bO)»A(bO)»TI(310)»TA(310)»SIGH(310)
             FIN(310)»WINL>(310)»DD(310)»QI<310)»QO(310)»P(50) »NPR
             UOMAX (2) ,UIMAX<2) » DTT I , DTTA, DTSIGH* DTFIN»DTWlND»DTDD» DTQI
             UTOO, JM,JOUT,JIN,KDIF,KSUR,KOH,KQ»KLOSS»YSUR»YOUT,OT»DY
             TSTOP,EVRCON, OMEGA, BZ, SPREAD, SIGMA I, S I GMAO»ETADY»TVARI
             TVARO,EVAP,RAD»TAIR,PSI»DERiV»HAFDEL»EPSIL»GJ
             YbOT»NN»HETA,DAJM,DELCON»V ( bO»l)»UI( 60»D»DTT
             RHO.HCAP,KMIX,RMIX, JMIXti, MI XED, QMI X »KAREA.DATRAD» ATRAD ( 310)
             AR,wINOY,CO,CI,B< 60)»S( 60),EX<  60)»EXO(  60)»ARF,UO( 60»1)
             QIN<310) ,T1N(310) ,CC(20,bO,2) »CCC(20,310) »COUT (20 » 310)
             CCT (20,310) ,UQMlX<60) ,XINF(60) ,OUTF<60) ,MIXH,MM
             SURF  (310) -, GRAV » SLOPE * VISCOS»LAGT IM ( 310)
             PMASOT(20) »PMASIN(20) » ET » NTRAC ( 20 ) » ITR» ISTO» I SOI » IS02
             ISTON»IST01,THICK1»THICK2»UOXLE(60»20) » DO (306) , BOD (306)
             NLEVE< JOb) , VOL»NW»NDE F »Z»Z 1 »DDOC»NGDET»L)60D» JEUP
             NBOUNU»NoRID
      DIMENSION WH(20)'»AA(60) ,XXL(bO)
      EQLUVALtNCE (N,NN)
c H»FAD IN ALL DATA FOK PROGRAM.
                   ('VH(I) ,1 = 1,20)
                   (iVH( I) ,1=1,20)
                   (*'H( I) ,1 = 1,20)
                   JM, JOUT.KUIF,KSUR,KOH ,KQ»KLOSS»NPRINT»KAREA»KMlXt
      COMMON
      COMMON
      COMMON
      COMMON
      COMMON
      COMMON
      COMMON
      COMMON
      COMMON
      COMMON
      COMMON
      COMMON
      COMMON
      COMMON
      COMMON
      COMMON
            (5,900)
      WRITE(^,900)
      READ  (->,900)
      WEAD  (5,901)
     1MIXED
      READ  (5,902)
      READ  (5
      READ
      REAO
      READ
      READ
      READ
      READ
      READ
      REAO
      READ
                   YSUK,YOUT,OT»T STOP »TZERO»EVPCON» OMEGA »UZ
                   SPREAU,SIGMAI»ETA»BETA»RriO»HCAP,OELCON»RMlX
                   .\ITl»i\TA,NSIGH,iNKlN»NSURF,NDD»NQI,NQO
                   OT!I,OTTA,DTSIGH,DTFIN,DSURF,DTDD»DTQI»DTQO
                   (Tl ( I) ,I = 1,NTI)
                   (TA( I) , 1 = 1, NT A)
                   (SIGH(I) »I=1»NSIGH)
                   (Fl.xi(I) , I = 1,NFIN)
                   (SLIRF(I) ,I = 1,NSUHF)
                   (Dn(l) ,1=1, MOD)
                   cn ( i) »i = i»fMOi)
                   ('30( I) , l = l,iMQO)
                  SLOP£»GKAV»VISCOS
  903
C
C
C
C
C
        (5,90?)
        (^,90?)
        (5,902)
        (5,90?)
   REAO(5,903)
   FORMAT(3^12.2)
   REAO(5.401) f\lGi)tT
N«UUND=1=EUPHOTIC /ONE SATURATED
iMHOUNn=?.=v,ATUKATION OF ARHITRARY  SURFACE  LAYER THICKNESS TO BE SPECIFIED,
NROUND=l=7EvO SURFACE LAYER THICKNESS  FOR SATURATION.
Nr,OET=l=OE TENT I ON TI-iE MOOEL
NGOFT=?=00 CALCULAflOM
   GO TO  ( 1^298, 15299) ,NblJET
RFAD DATA FOR PULSE INJECTION.
15298 READ(5.927)  I TR, (,v,TKAC ( I ) , 1 = 1 , 1TR)
s*?7   FORMAT  (IS/1615)
      R£AD(5,901)  Nt)ET
1333^ NDOCA=1 ODOO
      GO TO  Ib297
C  READ DATA  F JR DO.HOU  RKEO 1C F I OMS.
                                 -228-

-------
15299
      READ(S,^n^)  UDOC«L)riOi)
      P-EAO(5.902)  (Ou( 1) » 1 = 1 »,x|OISSO>
      RE A0( 5.^02) )
C  NPHOF=l=UNlFowM INITIAL 00, HOD
c  NPf*oF=?=LiNhA* INITIAL QO.OOD PKOKILLS.
      (30 TO  ( 123S7, l?3ba)
12357 PE*r>(5.902)  001. ROD!
      GO TO  1235^
1235* PEAD  (5.902) OOH.our
12359 COMTlNUt
      NT=?AC(l)=-2

13333 READ  (5,
1563  FORMAT(2F10.b.15)
15297 CONTINUE
      REAO  (5,902)   MICK1, THICK2
      DY  =  ( YVJR-i'OiJT) /FLOAT (JM-JOuT)
      YyOT  =  YOUT-OY*KLOAT(JOUT-1)
      GO  TO (<+.?),  KAWEA
C READ  IN DATA  FOR  OTHER THAN LABORATORY  RESERVOIR IF INDICATED.
            (5.901)  NAA,NX XL»NrtI NO,NATRAD,JMP
            (5,902)  QAA,UXXL»OT*/lND»OATKAU»AAB»XXLBfAKF
      PEAO  (5.902)   (AA(1),I=1.NAA)
      PEAD  (5,902)   (XXL(I),I=1»NXXL)
      PEAf)  (5.902)   (rt 1NO ( I ) » 1 = 1 «N*IINL»)
      PEAO  (5,^02)   (ATKAOd) , I = 1,MATKAO)
      DO  3  1=1,JMP
      T (I .1)  =  T/hRO

      ELd) = YBOT*!)Y*FLOAT d-1)
      PA  =  (EL( I)-AAB)/L)AA

      Ad)  =  AA(L+i) + (RA-FLOAT (D )*(AA(L-«-2)-AA(L+i))
      Ad)  =  A< I)*AKF
      PA  =  (EL< I )-XXLH)/OXXL
      I   = RA
      XLd) = XXL(L*1)*(RA-FLOAT(L) )*(XXL(L + 2)-XXL(L*l) )
     3 b(l)  =  BZ*ARF*F*P(OM£GA*EL(I))
C THF NU^REP  0.3989423=1.0/SQRT(2*PI).
      P(3?) = 0.39^9423/BZ/ARF
      p(34)  = YOUT*CHtGA
      P(35)  = H>(32) J>€XP(
      GO  TO  5
    6. JMP =  JM+If- IX( (33.0-YSU«)/OY*0.5)
C THF NUM^tR 0
      CO  = O.O
      CI  = 0.0130R«b6
      APF =  1.0
      AR  = 0.7S8HE-10   * ( T A (1 ) +273. 16)
       CO  8 1=1. JMP
       M ( T )  = 3" .4*
       EL ( T )  = Y^OT*OYttFLOAT ( 1-1 )
       T(T.l)  = r/FK')

                                   -229-

-------
                                                 /UY
      IF  (EL(I)-22.4) 6,7,7
    6  X|_( I)  = 10.0*(EL(I) +B7.0)
      GO  TO  H
    7  XL (I)  = 1093.5
    H  A (I)  = • *L< I)*JO.<+H
    5  BB  =  OT/A (JOUT+D/DY
      FO=(A(1)+A(JM))/2.0+A(JM)*(SURF(1)-EL(JM)
      JM1=JM-1
      DO  13  I=2,JM1
   ]3  EO=EO + A( 1 )
      EO=EO*DY*TZERO*0. 1E04
      DTT=OT
            = QOUT(0)*dd
            (6.400)  (WH(I),1=1,20)
                    JM,YSUR,RHO
                    JOUr,YOJT,HCAP
                    QY,VBOT,ETA
                    OT,TZERO,BETA
                    SBEFAiSIGMAl,OMEGA
                     TSTOP,SPREAD,BZ
                    KDIF,KSUR,KOH  ,KQ,KLOS5,KAREA,EVPCON»OELCON,KMIX
            (6,906)
            (0,907)
            (h,908)
            (h,910)
    WRITE
    WRITE
    WRITE
    WRITE
    WRITE
    WRITE
    WRITE
    WRITE
INITIALIZE
    DO H50
    QIN(N)=0.0
    7IM(M)=0.0
    DO 851 M=1,ITR
    CCC(M,N)=0.0
    COUT(M.N)=0.0
    PMASOT(M)=0.u
    PMASIN(M)=0.0
851 CCT(M,M)=o .0
«50 CONTINUE
    DO 852 1=1.60
    DO «5^ M=l > IT^
    CC(M,I, 1)=0.0
             MANY VARIABLES.
             '\J=1.3in
  «53
jaooi
19487
      OUTF( I)=0.0
      OOMIX(I)=0.0
      XIMF(I)=0.0
      GO TO  <3*000,3HOO1)»NGUET
      GO TO  ( 19488, 19^6 H ,-JPPOF
      DO 87123 J=1,JM
      CC( 1, J,l)=ixm+(t-LUAT( J)^Df-OY)/(YSUR-Y80T)tt
-------
      00  PI 31
2131  VUL=VOL +
H655  COMTTMUt
      JXM=JM
      N  =  0
      JMIXH  = JM-M
      QMIX  =  0.0
      FT =  0.0
      PAO  =  0.0
      e.vA°  =  o.o
      F2=0.0
      ^3=0.0
      TAIP  =  0.0
      t> S T L  = 0 . 0
              = 0.0
      JIN  =  J
      YSU3P  =
      FTAOY  = tTA*'JY
      DO  ISO  I=1,JMH
      5(T)  =  (.'JY*FLOAr ( 1-1)
       IF
  145  FX ( I)  =
       GO  TO  IbO
  lu*  FX (I)  = 0.0
  ISO  COMTI^Ut
       P(?S)  = FLOAT MIXED) M
       IF (D(2S) -0.0) M01»ttG?»801
  Ho?  P (?S) =0.000001
      GO  TO  (^,11), KijIF
    i D I F  =  U ( N 1 )
   11   IF  (JM-50) 15. IS* 16
   IS Jp  = J.vi
      00  TO  17
   1ft jp  = so
   17 GO  TO  
-------
      GO  TO  ?b
   ?4 TS  =TTI'\i(M+l)
^5    CONTJNUt
C  LOCATE  ACTUAL  LEVEL OF OAYb  INPUT
      DO  4745  1=1.JM
      J=JM+1-1
      IF(TS-T*4746*4746
4745  COMTINUt
4746  JJN=J+1
      IF(JIN-J^)  4747.4/47.4748
4f48  JIM=JM
4747
      GO TO  ( 19000 * 19001) ,NbUET
19001 IF(JIN-J£U^)  1900^»1^00J» 19003
19001 NLF\/fc"(M*l)=l
      GO TO  19000
1900? NLEVF(N*1)=?
19000 COMTINUt
      GO TO  (45»31) »  KSUW
C COMPUTATIONS  dHEN  SURFACE ELEVATION VARIES WITH  TIME.
    31 WA =  (
      L  =  H>A
            =  SJ^F (L+l) * (^A-FLOAT (L) )*( SURF (L + 2) -SURF (L* 1 ) )
       IF
   31=  M =  1+IF IA.( ( A^S(OrS)-OY/2.0)/OY)
       JM =  JM+1FIX (SIbN(1.0»OYS) )»M
      JMIXM  =  JM-MlXtl)
      9(?^1  =  FLOAT(MIXEO)*(rt(JM)+ri(JM!Xb))
      IF  (JM-bO)  37»3?».iB
   37 JP  =  J?"
      GO  TO  34

   39 IF  (f)VS)  4S.45*40
   40 JJM =  JM-M
      no  4?  (=1*M
      J = JM+ 1 -1
   4? COMT
       MU01=N001*1
       ET  = ET+J7
       MM=O

       IF  (N-^r-
-------
       CC ("», Ji-i. 1)=£
       GO  TO  MSI
o9?    CC(M, JXM, 1 ) = O.S*  CC(M. JXNU1)*A< JXM)/U(JAM)+0.5*A(JM)
       CC(M, JM, 1 ) =CC(M. J*M, 1)
  H^l  CONTINUE
C  THIS  IS THt LAGTIMK  DETERMINATION.
HO 10  CONTINUE
       C'LTT=Q01M (N) * ( 1 . 0**MIX)/B( JM)
       IF(JM-^-JlN)  ^ 70 » 6 70. -371
  870  VELF=OL I
       XLAG=XL ( IM
       r--u TO H/^
            O = 6.iSt-06* ( ( T ( JM, 1 ) -4. 0 > **?.- ( TS-A. 0 ) **2) /2. 0
       GO TO  (b?3«
       SLOPf_=(EL ( JM) -EL ( JIN) )/(XL ( JM)-XL (JIN) )
873    COMTINUt
       [)FLOw=( 1 ,'->a      )-*(ULlT*VISCOS/GPRIME/SLOPE)*«0.33
       SLniST=FLOAT ( JM- JIM) *UY /SLOPE
       XLAG=SLL)IST/VELI" +^L (JI'N)/HVELF
   H7?  LAGTlM(i\i)=XLAG/UT
C   END OF THE LAGTI-1E UETEK-llMAT ION.
       QIM (MD =;)IN (ML) *Q«JlN U>
       TIM (ML) = (TIN(^L) *( Ji'M(ML) -QG)lN(N) ) +TTIM(N) *UQIN(N) )/QIN(ML)
                      ^t ( « . I3» • )=• » 13)
       TP=0.0
       DO 1023  J = J'-1IX-)« JM
       TP=TP/FLOAT (MI xto*i >
       TS=(Tir\: (M) +TMJ>-?^IA) / ( i.
       DO ?7  I = 1.J-I
       J = JM+l-I
       IF (TS-T ( J, 1 ) ) 11 '. 30» 30
    ^7 CON'TI'MUt
    30 JIM=J*1
       IF ( JIN-J1"1)  1691. 33 » 33
    33 JIN=JM
 1691   CONTINUE.
       QQ = QIN("M)
    HI F = T I N ( M )
       CALL
       p (?4)  =  1 S
    47 GJ =  (FLA IM(M) +FLXli-J< ^+ 1 ) ) /AKF
 c   ASSUMES  TmAT  v»ur/ur LESS  THAN UNITY EOK  STABILITY.
       VVV = AHS (V {^« 1 ) )
       DO Sol  J=3.JM
       IF (VV\/-ArtS(\/( J
   SO? VVV = AHC- (V ( J» 1 ) )
   SOI COMTTNUt
       IF ( y/vV-V'^)  S03
                                     -233-

-------
  504 DT=OY/VVV
      IDT=OTT/'JI
      GO To
  So3 IIJT = I
C  FMH STABILITY  CHECK.
  SOS 00 74 "=1.IOT
      CALL S^EED (N)
c SLJR SPEED COMMUTES  */ITriORA*AL THICKNESS AND  VELOCITIES AT EACH  TIME  STEP.
C  SUB XMIX CALCULATED  COMPOSITION OF INFLOW.
      CALL X^IMN)
C  SUM SPFCAL  CALCULATES DISTRIBUTION OF SPECIFIED INPUTS OF 00,800.
      CALL SPECAL(Ni)
      00  111<+  J=?,JMM
      DELTA=(1 ,0-BE rA)*FLXlN(N)*(EXP(-ETA*(E.L< JM) -EL < J) -UY/2.0) )*A(J)-
     HXP (-ETA*(EL( JM) -EL( J) +DY/2.0) )*A(J-1) ) /A ( J) /DY/HCAP/RHO
C  CHECKS DIRECTION  OF  VELOCITY TO ASSURE PROPER  TEMPERATURE AND CONCENTRATE
C  ASSIGNMENT.
      IF(V(J.D)  1160, 11 bO, 11 61
 1160 IF (V ( J+1. 1) ) 1 1 70, 11 70, 1 171
 1170 nFLTH=(\Mj,l)*T(J,l)MA(J)+A-2.0*T< J, 1) ) /UY/OY
      nELTO = DO( L)*(T(J-l,l)-T(J+l,l))*(A
  11^3 Dt"LTJM = OT* ( (l.i)-rttTA)-^FLXlN(N)iJ-(A(JM) -EXP ( -ET A*DY/2. 0 ) *
     1A (JM-l) ) /A( JM) /OY*2.0/riCAP/RHO + UI (JM, 1 ) * (TS-T(JM,1) )*8(JM)
     ]/A ( JM) -|j;j( l) * ( f ( JM, 1) -T ( JM-1 , 1) )
     3  /OY/DY»?.0+ ( BE TA-FLX IN (N) -FLXOUT (N) ) /RHO/HCAP/DY*2 . 0 )
  1 16S T ( JM,?) =T ( JM, 1 ) +UtLT JM
       IF (V (2« 1) )  1166. 1167, lib?
       neLTl=or*( < 1 .ii-HtfA)-FL
-------
     <•"(>  TO 1 lh-3
     OKLT1 -[>T- ( ( 1.0-tfE ( A)*FLxLUC.o*ui_ ( J) +i( j) ) *PHI
      FLUXOT=(- LUXOI +IJ'-IIM«^.O-* (XL
 111^  T ( J.^) =T ( J.2) -  >-LT
 1117  no  111^  J=i » J^
 lll»<  T(J. 1) =T ( J.P)
C CH^CK  »F:ASON^riLEMt.SS uF  KtSULTS.
      IF ( Ays ( F ( JM,2) ) -100. u)  bO.nNb?
   S7 TSTOP =  T T
      GO TO hu
C SIP AVF~> ^1X£S  SU-x'KACt  LAYERS IN  THE  EVENT OF  A  SURFACE INSTABILITY.
   80 IF (T(.iM.p) +u.oi-| ( jM-1 ,^) )  b3. 779
C  S'JH AVEH> ut^Fo^MS CO.xlVtCTlVt MIAIN&  OF  TEMPERATURE  IN MIXING  LAYERS.
      CAl L AVt -> (r-J)
t-  s-iH S»FC'W  ^t^FUKV-, CO.MVECTIVE MlxI.Nb OF SPECIFIED  MATERIAL  IN  MIXING LAYEP
      CALL Srj^cjv (u)
      CALL SP^C'JT (-J)
C  SU« S^f-'C'tT  C^LCULAIEi  P^JPORTIDM  OF  SPECIFIED  MATERIAL IN OUTFLOW.
   7Q f;u-MTp!l)t
      f;T=')TT
C  SUH TOUT C^LCULATF-5 OUTFLOW FEMPEKATURE.
      CALL TOUT ( YNT« YS4TI )
      TOUTC =  YNT/YMTI
      TOUTF -  1. -i*T-j!)IC + 32.0
      IF (iM-MPn- )  100 « 1 00 « 80
   HO MPf-1 = MPR + ^PKl-MF
      WRITE  (^.»400) (-«H( I ), 1 = 1,20)
                      vl«EL( JM)
      F = F|..xiM(".j)
      i^RTTE  (6f^l^)  .11 M , E V A P , t-
      001 1 =  'JU'.Jl C!)
      WRITF  (h«^^b) ^it. I" A, flr< , w
      00 = 01"J (M
      W^ITt-  (H,-*)^)  I I F , P A;"j.U''J
      WRITE  (i»-^17) iih^lv.F
      GO TO  ( M-T , M^) , «ij
      (,;^TTr  (o,y[4) ilu^i^X ( 1 ) ,UIMAX ( 1) .TOUfC, TOUTF
      GO TO  ( *-*• ^^ >
      1.1 K j j I-  ( n • -* / M
      1,,- w T T i-  ( i- . ^ / > i )
      po 4,i  1 = 1,10
                                     -235-

-------
   go wPTTi-  lh.-y'2])  ( J , tL ( J > « T ( J , 1 ) , J= 1 » JP » 1 0 )
      IF  "4. 10)
100   IP (MUOCA- J001 )  1 fG4, 1 /10» 1709
1710  w^TTF ((?
      'JUH* T Ti- ( ^ . 3  ( J.tL ( J) .CC( 1 » J.3) »   J=J,JP»10)
      GO  TO  lOO
300?? I L^^O
      ir« i TK ( ^. .-i
      r^O  100^^  1=S1.LI.
      i"kTTfc  ((-.^1)  ( J.tL(JJ .CC( 1» J»2) »   J=I»JM.10)
322?? wPITir:  (n.-y.)?l)  COOTll.X1)
4021  FORMAT  (//' ut)  l'\l  OUTFLOW I'M PREVIOUS TIMESTEP  =«»F10.5  )
1704   IF  (f-'T-TSTOP) 20.1,1
    1  COMTIMUt
  900  FOPMftf  (2(iA4)
  401  F 0 ^ '-' a f  ( 1 (S I S )
  40?  FO^'^iaT  (hf in .S)
  406.  FQo^aT  (•  MU-^if-  OF  b^-IO PO i^T S= ' I 3» 1 7X« • SURFACE ELtVAT ION=» F7.2»
      1 Inx, nit '\iS  LT Y='t 12.b)
              ('  oUTLEf  LcVtL='I3,   2t.X, 'OUTLET ELE V AT ION= • F8 . 2» 18X »
              (' 1JY='F>.2.33\, 'dOTTOM  EuEV AT I ON= • F8. 2, 1 8X» «ETA= • F6. 3)
  907  FOP.^ar (' OT^'F^.^, 33x, ' INITIAL  TEMPEKATUPE='F6.2f 17X» «BETA=»F5.2)
  qn*  FOP^AT (' MIRIAM  -itTrt=»Fb.2.26X, • INFLOW STD.  DEV .= ' F6. 2» 20* » • COEF.
      IQKRifl  I i\j M^FA H'UK'-ii)LA= ' r. 12.b)
  90Q  FO^^^T (• ^TijP AT  TlMt=«F7.2«2^X» •OUTFLOW SPREAD CONST. = 'F5.2» 16X»
      I'^IOTn AT Y = o I'>i  AH'LA FOP-MULA=«E12.5)
  910  FO^waT (• F.KATURE=|Fb.2)
  Ql>  FOt-^AT (• i-oo. ,)h  OKiu polNTS=»13»20X» 'ELEVATION  OF INFLOW= 'F7.2»
      11 M ,««•£•. L M 1 vt Hu-^ 1 0 1 T Y = » F ^ . 2 )
  9]S  FORMAT (' LF.N/EL  OF 1.-JFLOW=' 13. 23X, 'EVAPORATION FLUX= • El 2.5 1 l1..).\  Cor.FMCIt^l='E12.lD»^X, 'RADIATION FLUX='E12.5»
              F L J-i ^ATr = 'F 1 1 . 1 )
  417  FOWMAT (' 'jOTFl.n.M  T t ^P viK AD I LNiT = • Frt . b» 1 2X . • HEAT  LOSS FLUX='E12.5»
                                   -236-

-------
   l i.v FE.'-iPl-.RATURE=»F6. 2, ' C  AND  'Fb.2,1  F.')
     u^MAT  ('   J    LLEV     Tt'V(C)'))
92] FORMAT  i  F AC FOH='Fb. 2)
    FuwMAT  ('  1--MH OF HiAl'.ib  L A YEK= ' Fb. ,^» 1 5X » ' MIXED INFLOW TEMP= •
   1 F h . •> )
    FOP'^AF  ('  H^IA'J H^TA=«F5.2*2bX, • ATMOSPHEKIC  H AUI AT ION= ' E 12 . 5»
     4X.«^[\0  SPt".tr)=»FS..^)
    CALL txlT
C M..')SS
C
c
                                 OUE  TO EVAPOKATION» CONDUCTION*  AND RADIATION,
              FLXOUT<\)
              f SURFACt LOOSES
            F 1 M( no) »*iNU( J10) »DO(310) »QI(J10)fUO(310)»P(50) »NPR
            uO^X (-•) ,IJI-1AX (/>) »UTTl»DTTA,DTSI(jH«DTFIN»DTwlNU»OTDD»DTQI
            uT 10, JM, JOUT» JIM,K01F,i? t-"
T F
              T\/ARO.EVAP,^AD.TAlR,PSI»DERIV»nAFDEL»E-PSlL»GJ
              Y^or.^N^Htf A,OAJM»DELCON»V (  60»i)»ui(  6o»i)»DTT
              R->0»MCAP,KMU»RMIX* JMiX«»f-ilXED,UMlX»KAREA»DATRAO»ATRAU(310)
              £>-y,.v.CO.CI »H (bO) «S(^0) «EX (60) «LXO(60) » A^F »UO ( 60 » 1 )
              ulM(310) . Ti.M(3lO) »CC(20»60«if) »CCC( 20*310) »COUT (20» 310)
              CCT (>>0. HO) .JQMIX (bij) »XINF (bO) «OUTF (60) »MIXH»MM
              SJK(- ( Jld) •oRAV»SLOiJt« V ISCOS . LAbT IM ( 310 )
              H-^ASOT (?0) »PMASIM(20) »ET»NTHAC(20) » ITR» I5TO» I SOI » 1 502
              ISTO'N, 1ST  Jl, TH1CK1,TH1CK2.L>OALE(60»20) , DO (306) ,800(306)
              NLtVE ( BOb) » VOL »Ni« « NOE f « Z » Z 1 » ODOC » NGDET » D80D
              (j^ LA«!)RATiJRlT USING  ROHiAiEw FORMULA.
                FIELO USING KOHLfcR  FORMULA.
                FIELD USING ROHW/EX  FORMULA.
            t>~
            OR
           = l
                  (M)
    L  =  R
    RP = R-l- LJA [ (L)
    i- =  KT/"TSl-:'lr!wTS+13.006d
                                             + 1 3.006y)
    GO  TO  ( 1
                 /r'O
                    ) )
                                   -237-

-------
c CALCULAT Tor--> r'j>-  LAr-nMAiu^r  USF P.OI-MER  FORMULA.
                                       >.}»( TS-IAIR) )
c UNITS  OF PAuiATlUM
       AP  = i>. /ni^nr-in   *( TAiP*^ 73. 16) **<+

       k.  = n. o
       P ( 1 0 )  = r^AP + PAII
       pi XOIJ r  = P(30)

C FOP  FIF.LD DATA.  MJ.'-JD 7>PEFJ  Ib I'M ""/SEC.
   ^n  P  - FT/OT "J 1 "ID
       I  = w

C FOS-  FIFI.O.  ATMosPr-iE^lc KAJIATION IS  INCLUDED  AS AN INRUT  TO PROGRAM.

       L  = P
       AP  = A[tvAM(L'fl)*(P — FLJAT(L))'U'(ATKAD(L*2)""ATRAD(L+1))
       PAf) =  1.13iDd^E~^i'*'(TS* • ^5»  i'U . --LO^S
c CALCULATION OF  FIELD CVAPOKM IOM USING KOHLER- FORMULA.
c VAPOR  PPF^SU^ES  r\i'-••!»-),

       FVA.3 =  H*!jc. +HCA^^LJc-;}r S + 372.0* (TS-TAIR)
       F V A P =  *: V J C 0' J '* v 10 •* iv «• E \/ A P
       p ( ?0 )  = tVAP + PA')
       Ft XOUT  = ^ (lii) /wPF
       Rf- TUP'')
c CALCULATION OF  FU.LU C.VAPOP/\I"ION USING ROHV/ER FORMULA.
   10  CHI =  P~!0#(H*.jF. + T:j*HC'»P*Dt+*'v
       FVAP =  C-i I-F W-1
       R ( T0 )  = cvyA^ + p
       F LXOUT  = P ( 'VI ) /
               I JF  TOUT ( Y >*T,Y Jfl)
c  cnxpuTF •>,£ii.i-iTti'j AVT.-fMbt.  OF OUTFLOW  TEMPE.KATURE.
C  USF  CHMPuTFU  lMsrt>0 JF  GIVrlN OUTFLOW RATt  f-OR YNT1.
C  Y'-JT1  j(/II.L  Ht.  '^t-'AfK-v  [riAix,  MOT F(IK  NARROW WITHDRAWAL  LAYERS.
C  HF^1CF. USF  bA-i(-  ^t:. THOU  TO  CALCULATE  J IN bOTH YNT AND YNT1.
C  CALCULATED  TOUT  =rV I/r Jf 1 .
       COMMON  1(^n,^)«FL (ru).XL(60),A(60),TI(310)»TA(310).SIGH(310)
               (• I-J (  Ufi) .^I'NiH no) .DDUIO) »(Jl ( 31 0 ) » QO (.31 0 ) »K (50) »NPR
               ijfj/.Ax (^) ,uJ 'AX (x) ,DTTi.DTTA,DTSIGM,DTFlN»UTFOUT»DTOD»DTQI
               ))1 :JO.JM,JUUT» J1K'. K u IF , K SUP , K On, KQ»K LOSS»YSUR»Y OUT » D T * DY
       CO 4^04  TST jP.t V^CJ.'M*u-1tuA»H/,SPREAO»SIbwAI. SIGMAO » ET ADY » T V ARI
       COMMON  1 VA'-'(uFv/iiP,wAlJ*TAIR.PbI»OERIV«HAl-OEL»tPbIL»bJ
   U'N-JITS OF ^flli[ATI():J  A-vb  ^CAL/M-M-uAY.
       COMMON  Y-T.UT . •J(-j.H!r,r«»UaJ^,DELCO!M» V ( 60»1)»UI( 60tl)»DTT
       COMMON  "-CJ.HC-'-'.S-M x,R./iU, UV!lXd,:«lIxE1J,QMlX,KAREA,DATRAD»ATRAD(310)
                 , .. rjur,Cu,C{ '-!(  60).S(  bO)»EX(  60)»EXO(  60)»ARF.UO(  60»1)
                l ., ( 11, i) . T i;\j( jio) ,CC (<^0*bu»^) »CCC(=?0»310) «COUT (20*310)
                                      -238-

-------
    COMMON)
    COMMON
    COMMON
    COMMON
    YNT  =
    YK'Tl  =
    JI|M  =
    M  =  JM
10
               (20»31o) »C.OS»LA(:>T IMI310)
            ^ -i A SOT (2u) .P-IAbl.N (20) ,ET , NTKAC (20) , I TR , 1STO, IS01 , IS02
            IS TON, IST01 . rH!CKl,THICK2,UOXLE(t>0,20) , D0(30t>) ,800(306)
            NLtvE ( 306) , VOL,NW,NDET,Z,Z1 , UOOC,NGOET »DBOO
            .O
                -^/ 2)
YN'T
YNTJ
JI|M
YMT
YM1
00 ^
M =
          = 1 . 5* (H ( J'
         = Ji-1-1

         = f'Ml *T ( 1. 1
                      J ) TH  is ri(Y)  =  i-t/*txp (OMEGA^Y) .
   vOxT^ATJUN MlK^.ULfl  (t-U'-jCT i->^0w)  USED Tu tVALUATt  PROBABILITY  INTEGRAL.
    COMMON I (r.n«2> «i-.L (bO> »XL (bO) « A (60) » TI (310) »TA(J10)»SIGH(310)
    COMMON i- IN ( UO) » "iNuM JlO) «0o(310) »QI ( 310 ) » QO ( 310) ,P (50 ) »NP«
    COMMON ijJ-^lAX" ( ^) , (JiMAX ( r-) »UTTI » UT T A » UTS I &H , DTP IN » DTFOUT » DTOD» DTQ I
    COMMON UTu'l, J-1. J'JUT* JlN»Ki.)IF,KbUK»^On,KU«KLOSS»YSUKf YOUT»OT»OY
    COMMON TSTOP, t\/p>CoN(«.r'itoA»rtZ» SPREAD, SI(3MAI»SIGMAO»ETADY»TVARI
    COMMON
    COMMON
    COMMON
    COMMON
    COMMON
    COMMON
            YdOT » i\N . ht T A , D A JM, DELCON , V (  60,1)._,UI(  60,1)»DTT
            wHO.HC4iJ.KMl A,r-Mlx* J ^1 1 X d , M I xtD , QM I X , K ARt A , OATkAD , ATR AO ( 3 1 0 )
            A-v, AlMuY«C'J,Ul ,rt ( bO),'S( 60>»EX(  bO),EXO( 60)»ARF,UO(  60,1)
            01 N ( U')) « TIN ( 310) , CC (20, t>0,2) »CCC (20,310) ,COUT (20,310)
            CCT ( ?0« UO) , 'J'JilX (60) .X INF (60) ,OUTF (bO) ,MIXH,MM
            SJ--F ( 31'.)) .'.>KAV,5LOPE« V I SCOS , LACjT H (310)
            PMAsOT (20) ,P.^ASli\(20) ,ET,NTHAC(20) ,ITR» ISTO, IS01»IS02
            IS TON, Is I 01 , TMlCx 1 ,THICK2'OOXL.t (60,20) , DO (30 6) , BOD (30 6)
            '\ii_EVE ( BO--.) , VOL »N'« , NOtT »Z t Zl , ODOC, NGQET , DrfOU
            MU^AXIAL  1 MlCKNtSn.
           ONLY M^LF  TnK wIT^OrVAwAL  TnlCKNtSS IS COMPUTED.
           =  (T ( JOuT+i , 1)-T ( JOUT-1 , 1) )/2.0/DY
 CRITERION F J-v FXfSIAiJCd  OF  A  WITHDRAWAL LAYER.
    IF (OE^l »/-!' -00 1 )  2*2«S
  2 JOUTl=J"jr+2
    f>o  ?no J = J'')')T 1 • Jvi
    IF ( ( r ( j+i • i > -r (J, i) ) /of-. ooi)  200,202,202
200 COMTI'-int
                                  -239-
COMPUTE  '•/ 1
     THAT

-------
       00  Til -,
           EL = t-L')t r (J-JOUT
      (if.)  TO <-.
C APPROXIMATING FO-^M'Ji.A USED  FO* DENSITY IS KnO=l . 0-0 . 00000663* ( T-<». 0 )**2.
    t, (-PSIL- ^.D-.:- (T Uoui , l)-4.0)/( 1-510 UO.O-(T ( JOUT » 1 ) -4 . 0 ) »*2) *QERI V
      GO  TO (3.1),  «. >i
C CAL.CUl.flT 10. •' Or  •/ 1 MiJ^ Ai'JAL  TnlCKgLSS  USINb KAO  FORMULA.
C CAl.CULATIO'; Of  "(ITHOrVAwAL  ThlCKNtSS  USING KOH  FORMULA.
     ?  HAFOFL = i)ELCOM/t:HSIL**0. 1666667
     ^  SIGMAO = riAFDtL/SH-vt AO
       IF (SI'j-.AO)  300».3(JU»30l
  301
    f

C FIP^T  COMPOTE  XAXI^.JM \/tiL'JCITIFS» THEN OTnEHS._
    7  XXI  = Pwjri(r^(jr,hL (JlAl) +P(31) .blbMAi)
       XXO  = HRJn ( YriUT. YiJUT + i)p*''SOSO»Sl-'OT-<;>.S3i  lb»10»10
    10  UO^AX(l)  =  OOUT (N) /SIo'^AO«CO
       GO TO ?->
    11  CO = P( .^)*FxP(-H(3J)-^SIoMAO*SIbMAO
    \c<  XQ = Co/Sl'b''/iAO/(P-'<0^(LL(JM) , YOUT + OMSOSQ«SIGMAO)-XXO)
       LKt^AX (1)  =  .)Uur (:\() "XU
       00 TO (s'r>» 3S) ,KAMEA
    ?<->  IF (FL( JM) -h'L( Jl \)-^.S^*SlG^Al)  35*35,30
    30  UIWAX(l)  =  Ol-^(i^) /Sio>IAl^CI
       GO TO ^u
 35     IF(jM-JiM)  40,40,41
L   THIS  1^ Tut 'UNlFO-^vi VELOCITY DIbTKlHUTION  IF  THE FLOW  IS SURFACE FLOW
C   ISTO  IS Tnt  -NU'-irsf-v OF bwlU POI'MTS  BELOW SUHFACE INTO WHICH FLOW  WILL ENTER
40     IST^Tr-iJC* l/Uf-U.5
       00 700  1 = 1. IS TOM
 /OO    UI(I.1)=0.
       00 701  I=IST01.JM
       1)1 (I«l)=QI\i(M)/(IbTO+.S)/OY/d(I)  *(1.0+HMIX)
       GO Tu S^
    4]  CI = P(3b)*EAP(-EL(JIiM)*OMEbA)
       P(??) = SIbMAl/CI
    4^  XI = Cl/SI(iMAI/(P^Ob(EL(JM) .EL(JIN) *P(31) . SIGMAI ) -XXI )
       UIMAX (!)  =  QIN(i\l) ^Al
    SO  GO TO tr ^/p(
       77 = U-I X/P (^S)
    S?  SAO = Slb^AO/CO
C  CO^iPUTF \/F.^TtCAL AOVECTIVE VELOCITY ANO  SOUKCE VELOCITY.

                                    -240-

-------
      L>i) 70  j=l . jv
      XU =  SAO<* (PWOrj(tL ( J) »YOUT+OMSOSQ»SIGMAO)-XXO)
      1 1- ( ,,M- J 1 M )  70 3 , 70 J » 56
      XI =  P <2^)*(PHOH(EL< J) ,EL( JIN) +P<31> ,SIGMAI)-XXI)
      UI ( J. 1)  = UIMAX(1)*£X (I)
703   CONTINUE
70    CONTINUt
C CO'VPUTF  EXPONENTIAL  PAkT OF SINK  VELOCITY  FOR USE IN bUb  TOUT AND FUNCT  UO,
      IF  < JM-£>»JOUT+1>  75,75,80
   7^ LUP  =  JOJT-1
      LOM  =  JM-JOUT
      IS  =  -1
      GO  TO  *s
   HO LUP  =  JM-JOUT
      LDM  =  JOJT-1
      IS  =  1
   HC E»n(JOUT) =  1.0
      00  100  1=1. LUP
      J =  JO'JT+1S*I
      AwG  =  S< 1+ I) /
      IF  (a-vi-,-^0.0)
      GO  TO  ^D
^9    FXO(J)=0.0
VO    IF  (I-LiJN) ^ !»-»!, 100
   ^1 JJ  =  JciUT-IS*I
      FXO( JJ)  = t xij( J)
  100 CONJTI'M'Jf-.
      IF ( JM-J1 \i) 706, 70o, 706
70S
      l-n  710 J=l,ISiil
 710   not J.I)  =I)OMAX ( 1 ) ->tAO( J)
      OO  7 1 ? J=IsTU 1 « JM
      nn (i,l) =jo^iax ( ] ) ^txo ( J)
 71?   COMTlN'Jt
      1-.0  TO ( Jl .32) ,K'^IX   '
J?    IF(J-JVI*-0  31«JJ,3J
   33 OQMIX ( J) =OlN(N) **'4lX./ (Ml
      HO (j.D =;jQ*I x ( J) /d ( J) /OY
      IF( J.E'j.J-")
      GO  TO 3b
   3] UO(.J. 1) =0.0
   3fi U0( J. 1 ) =UO( J, 1 ) +UtJ.v'AX ( 1 ) *EXO( J)
711   CONTTfMUtr.
      V ( i , 1 ) =0.0
      V(?. ] ) = ( JI
      JN:X=JI^+ 1
      HO  500  J=.i,JMX
      V(J. 1 ) = (V (J-l. 1 )-(A( j-d) +A( J-l) )/^.0+ (UI
     1 *DY ) / ( A ( j ) + ft ( J- 1 ) ) * <£ . U
  SOO CUMT
                                     -241-

-------
               i -jr.  avt^CM)
C PF^FOP^S CONVrCriVE MlxJNG OF  SUKf ACE  LAYERS.
               I C--0.2) «tL (60) tXL (60) , A(bO) , T I (310) , 1 A (310) , SIGH (310)
               i- I'M CM:))  < <,1J')Y,CO.CI,^ (  hO),S( 60)»Ex( 60)»EXO(  60)»AKF»UO(  60»1)
            'xi  -)\.v ( ilii) -T IN Cjl'j) »CC(d()*60.^) »CCC(?0»31U) »COUT (20»310)
            N  CL f (^i). 110) , 'JO-iIX (ofi) »X1NK C:>0) »OUTF (bO) ,MIXh»MM
            N  sj.yj- (j] <)) , ,,-VA\/» SLOPE, VlbCOS»LAGTIM (310)
            M  i->'i^oT (xu) ,P[V|/\^IN (?0) ,ET,NTKAC(^0) » I T R , I STO » 1 SOI , I S02
            N  IS I ;J1'.|.lSIONTHICKl,THiCK2,UOXLE(bO,20) »UO(3U6) , BOD (306)
            M  iMLEVE  ( J'l^) »VOI_»MW»NLJt:T,/:»Zl«L)OUC»NGOET»DBOO
       Dl^FNSf 'iNJ  \/V  (60) » V (DO) ^vH (20) »TT (60) f AA(60) , XXL (60) »C3(60)
       AVI =0.0
       JMM=JM_ i
       00 S I=l,jr.)r.1
       IF (T(J. 1 ) -1 ( JJ. 1 ) )  b, 7. /
     *>  rOMTINiJr.
       IF (  J-2)  '•^^


       (-.()  TO  7
     Q  |)fi  \0  i< = l.JJ
       KJ-J+1-K
       KJJ=KJ-1
       IF(JM-KJ)  ?,/,:^
     2  F6C=O.S
       GO  TO  6.
     3  FAC=1.'>
     a  A V 1 = A V ] + I ( K. J , 1 ) * n ( K J ) * t i C

       T A V = A \/1. / A \i f>
       IF (TAV-T (KJJ. 1 ) )  iO.^.,,x:)
    10  COMT INI it
    20  IFCJ.f uJM)  -!l«"i-^
       00   10  | =
-------
   XX, =  ( y-Y-.V ) /SI
   X =  ii^S(JA)
   IF < x-i n . o j   1,1.
 ? X = 10.()
 l XT =  i . <:/ (  i . (j + n
   xa =  r<^(_x-^x/^
   X A =  X A* (0.4 <•  Th-i->t.K.\ ru^r_- F
C Ll^f;Al-'  I!'-iT- .v->^i_ A T l  n  jr. ['JdcKi
           f>.\ |  (M'..^) , f-.Ll^u) .XL
           O'.J F  I g ( il ',),.*,!. jut il'i
                                JI HEAD IIM VALUES.
                                uttU  IN VALOtS.
                                .'D ,A(bO) »TI (310) »TA(310) » SIGH (310)
                                i>0( 310) »'jl (JlO) »QO< 310) »P (50) »NPR
           uO -'•'*•> ( ') .')! -'-^ (/) .OTT 1 iUTTA,ijrbIbH,UTFlN»DTWlND»OTOD»DTQI
                                                       YSU^»YOUT«OTiOY
                                       \/ (  60»1)»UI( 60»1)»DTT
           ^^'i,HC^J,-\ /i A»^"iIX» JM[Xd,MlxtL),QMIX»KAKtA,DATRAD»ATRAD(310)
           u-<. . ] M JY »CU»C[ »n (   »CCC(?0»310) »COUT ( 20 » 310)
           CC I (^u«  SI ) ) . .j.j.-{ x (^0) ,XlNr (60) «OUTF (60) »MIXH»MM
           -^ J-H- (.UO),'jKAv/.SLOi-1if:.vlbCJS»LAbTlM(310)
           H^«^,)( {  JD) . ^ -lAbL'M («?0) »t: T .MT*AC(^u) « 1TR» 1STO» IS01* IS02
           l^F )'-i. ISTO L. MiCKl ,TnlCrs^,UOXLt ( t.0 » 20 ) » DO ( 306) , BOD (306)
           NLt\/F ( ioi)
      w  = ir r/^T
      L  = -1
      MP  = ;v-h L
      T T i ' -j= T i ( >_
      kFTi)->  j
      F MO
                          n
                            2 > - r i ( L + 1 ) )
C CnupUTF
C RfAO  IV
            I !)••.)  t-
           INC') •'] '-
           y/ALUi ->
               f (T'
               F
                 If
      COM
               -J'l  S1L4K  K^JlATlDN  FROM KEAU  1 H VALUES.
                r^c^lEJ  AS  A STEP  FUNCTION.
               ',,2 ) »1L <^U) »AI_ (^0) ,A(^0) »T I (310) » FA(JlO) f SIGH(JIO)
               ( Hi)) • ".'I'MU ( 311')  »'">:J( 310) »Ul (310) « (JO (310) »P(bO) »NPR
           UJ^-x (^) ,ui "iAX (/) ,UT Tl ,L)TTA,OTSlbH,DTFlN»UTwlND»OTDD»DTUI
           Ml v'i, j^. j:jul ,.JiN,M)IF,KSUr<,KOH»KQ»KLOSS» YSUR»YOUT»OT»OY
           |'sTJP»E>/HCJM,0-i!-oA,rt/,SPWEAU»SU7MAI»SlGMAO»tTADY»TVARI
           T VA-'II . t-.V'i^,^ Au. 1 a IK , PSl . UbKl V»HAt- OEL»EPSlL» bJ
           r T1-!.) T • 'M'-I » '
-------
      f 7=nTTAFLOAT C-J)
      ^ =  FF/ijU" I'-'
      L. =  *
      F L J T N  = F 1 N (L. + 1 )
      fi- TiJW•.;
      FUA'CT ION O'J I N ( M)
C COMPUTE"  INF'LJ^ H>ATF  F*Ji«i REAiJ  IN  VALUES.
C RF^D  IN  VALUES T^F.aTEu AS A STFiJ  FUNCTION.
      COMMON T(^0.>?).EL(^0).xL(M))»A<60)«Tl(31Q)»TA(310)»5IGH(3lO)
              F {N ( 11 0 ) . >.v I -iiJ M 1 0 ) «J0< 31 0 ) , 0.1 < J i 0 > . 00 ( 310) »P (50) » NPR
              I.I;JVA< (^) .Ui-iAX(^) ,uTTl»UTTA.L)rSIGH,DrFIN»OTwlND»DTOD»DTUI
              UP 10, J-'« JOUT. JlM»^JlF,KSUK»KOH»K(j»KLOSS» YSU«»YOUT»DT»DY
              FSTO^ ' t VHCJN»0 ^F(jA«4Z»SPHEAD«SI&MAI tSIGMAO»ETADY » TVARI
              ) v " rv o . H; v a p , >H t\ j, T A IR , u S I »0 E H I V % H A I- 0 F L » E P SIL » ti> J
              VdOT .AtN.Htit 4, .)Ajvi,UELCO:\I.V ( 60» 1) »UI ( 60»1)»OTT
                       . K. vi I \ . *M £ X « JM i Xd » MI X £0 » QM I X
      L  =  ^
      OOIM='JI (L+ 1 )
      ^FTIJW'^
      t ^.| H
      FlJ'JCT ION OOUI ("•!)
  CO'MPIJTF  OOTr-uOi-i lVATt  F^uM KEAU  IN VALUES.
  PrAO  1M  VA! Oi-, Ti^t-.ATEO AS A STEP FUNCTION.
      COMMON I (nc, ,2) .tLlhO) , XL(60) ,A(60) .TI (310) » TA(310) » SIGH (310)
      COMMON r l\)(31u) »>MlNU(.UO) .00(310) ,ul (310) «UO< 310) »P (SO) »NPR
      COMMON oDMAX. ( ?) .U1.MAX (^) .UTTI»L)TTA,OTSIGH.OTFIiM»OTFOUT»OTL)D»DTOI
      COMMON ofwo,JM. JOUT « JIN.KOlF .KSU^.KOH,K(j»KLOSS» YSUR» YOUT»DT»DY
      COMM(JN TSTOP«FVt->CUN,0-'lEbA,^^,SPKtALJ»Si&MAI»SIGMAO»tTADY»TVARI
      COMMON I V^O*EVAP,KAO.TAI*.PSI»IJEKIV» HAFt)EL»EPSIL»CiJ
                                        »V ( 60»1> »UI( 60»1)»OTT
       hv  = FT/!)T'.HI
       L.  = ^
       001 )T = UJ(L
       ^tT'Jw"-!
       fc'K'O
C C'l-'^PUTF  DTFi-" JSI\/IT V  FKO^ -^EAO  IN VALUES.
C AN>  assii- F:> ^^vjiiTiosi ()p ]-H£  L)IFFUSIV1TY  MAY ^E PROGRAMMED  IN THIS FUNCTION
c HF-^F,  A  co••isTjr>iT  VALUE OF u  is ASSUMED.
       Covr^H T (.,,.,,-) ,I-L (6u) »XL(60) ,A (6U) . TI (310) .TA(310) » SIGH (310)
                                   -244-

-------
      (.ovMf.rvj (- [;g ( iio) , ,u\'U( 310) .DD(310) »UI (310) «QO(310) »P(bO) »NPR
      CUMMOM L) t ,1Cl< ('^) ,0i MAX (,>_) ,orri»l)TTA»UTSIGH,DTFIiM,OTFOUT»OTDD»OTQl
      f (iMMOxj DF'JO. JKU J'JiJT* Jl ^K01F,KSUK.KOH,KQ,KLOSS»YSUR»YOUT»DT»DY
                         : J'J»0/L'3A»^?»SH«t.AU«SiGMAl »SIGMAO»tTAOY*TVAPI
              YiOT . M J.Mt I A, !)A JM,(JtLCOi\l» V ( bO » 1 ) »U!
              H-10,HCa-',KMi^,KMlX.JMlXr-l»^lXtO»QMIX
      D =  00(1)
    CALCULflT [ON OF  COMHOSiTIdN OF  INFLOW
      CO^'^OiM f (-,j.) »DTTl,UTrA.i)TSIGH.DTFlN»OTWlND,OTDD»DTQI
            N uTiJO, JM, JOUT, JI.M»KOIF,KSUK,KOH,KU»KL(JSS» YSUk» YOUT » DT * OY
            h T^r')P,eV^CU\»OMtbAfHZ«SPwLAD»SHjMAI » SI GMAO » tT ADY » T V AR I
            'M 1 VA.-vO,tI\/flP,kAU» T AlW.PSI^DLKitf.HAF OFL» EPSIL* GJ
      COMMON Y-^OT • ^vl.-it-I f M,UAj^»i)tLCON»V ( 60»l)»'ul( 60»1)»OTT
                        Cu.Ci«8( bO),S(  bO)»EA(  6n).£XO(  6U),ARF,UO(  60
              ;JlNJ( 3 10) , fl'sl ( 310) »CC(\() * ( 1 . O + H'.-I l n )
       IF ( xo.^'v.0.0) i^o  TO  b
      GO  TO  ( _V)U«?n 1 ) ..-.i'jiJtT
C  ^ilLSF  If-Ul-LT IO--1 CALCULAF lOixl
300   IF(^i^)  1 lolO« 1 in 10, 1 101 1
11011 CONTI'-Jtit
      I in  1  ••*.= 1 . ,viry|
      Y'..: = 0.0
YO/X.'J
^1    CO^TTNUh

-------
.40    COMTlNHJt

    0  17000 J = JMI K'-iiJ^M

1700n YUO = YQ'J + CC ( ?« J. 1 ) *CO'J -HX ( J)
      CCC( 1 ,•'••) =f:j/XQ

      If- (M-^i.i)  1 /332. 17J32, 1 7333
17333 NX = 60
      C-0  TO   1 c'+'S
1731? MX=M
       IF (M- (iML^ + LAGTI'-H.vJL^) ))  18000»lHOQl»lbOOO
ISO01  CCC ( 1 »M =CCC( 1 »i\i) +'JQINJ(NL^) J>UDU(:NJLM) /XQ
       CCC(?.M) =CCC(^.'M) +0^l-J(NL^) »ri^OU(MLM) /X(J
18000  C
              I '-ir.  SPtC^L dM)
     CALCULATIO-J OH  i)I^T ^IriUTlON  OF SPEClFlEu INPUTS
              T(-,u,2).tL(6i)).XL(60)»A(&0),TK310)»TA(310)»SIGH(310)
              MM (.U r.) . vI,MU( HO) .00(310) ,01 (310) ,00(310) ,P( 50) »NPR
              UUMA\ (2) .iJlMAX (2) »DT Tlf UT 1 A.DTblbH,DTFIN»DTWlNO»DrOD»OTQI
       COMMQ.xi uTOU, J-^. JOUT, Jli\j.KOIF.KSUK,KOH,KQ»KLUSS»YSUK»YOUT»DT»DY
       COM Jin.xj f sTO^, tV^CON^OvitoA. HZ, SPREAD, SIGMA I, blGMAO»ETADY»TVARI
              TV/j^O«nVAP,km),TAlK,i-'Sl»L)EKlV.hAFDEL»EPbIL»GJ
              YiUT ,N 'vl« 4t T A , 0^ JEA»DATRAD»ATRA
       COMMON a>v,-.-fj;\,._)Y.CO»Cl«r. (  60),S(  bG)»EX( 60)»EXU( 60)»AKF»UO( 60
              >JlM(310) -> fi'\i( U(i) ^CC( 20*60,2) .CCC(?Oi310) * COUT (20 , 310)
              LC I ( 20 « Hi) ) *u'J"llX C->(J ) , X INK (hO ) »UUTF (60 ) ,MIXH,MM
              SJ^H (3 I'D ,OKUV,SLOPF,vlSCOS*LAbTiM(310)
              ^vijsOF (2-J) »Pr'^^l^(?0) ,t.I ,NTKAC (20) »ITH,ISTO,IS01,I502
       COM^ONi ISfOM, ISTOl, TnlLAl,THlCK2.L)uXLE(hO,20) »DO (30 6) , BOD (306)
                     30 6) , VUL , >\U »NOf. T * /.» Z 1, UOOC, NGOET , DBOO» JEUP
       DO 1 I=^,J:-1M
       I F ( 'j ( I . [ ) )  2,2,3
     ? OIJTF( l ) = {;jo( I, 1 )->H(l) *OY-V(I,1)MA(I)+A(I-1) )/2.0)*DT
       XIMF ( I) =-v (I + 1.1)-(A(1)+A(I + 1) )/2.0*L)T
       COMTTMUh
       GO TO'  1
     ? ni)TF( i) = ( j.;( i . i ) v.i(l)-[»Y*V ( I*l,l)*(A(I)+A(I + i) )/2.0)*DT
       x I MF (I ) ^ v ( ] , i) --- ( a (i) + A (i - i > ) /2. 0 *UT

                                    -246-

-------
     1 COMT
      Jf (MM)  1 I,, jo, 110 10, 1 lull
11011 COMTFNUt
      no  S3 'V=i.MM
      HO  SI ]=e'.JMM
      GO  TO  ( ^213.71 12) ,Mb.)tT
M12  IF (I  -Jr.U^) 7211. /21 1.72 12
/212  CO\'ST 1 = 0.0
      CO\'ST? = o.v)
      GO  TO  721 3
7211  IF (M-) )   / ih, M6.7J7
M6   CONsT !=/
      GO  TO 7J
      COMST1=X
/3S   COMTIratJb
^213  COMTlMUt
      IF (V( 1, 1) ) h*h,7
    * IF (V( 1+1 » 1) )  20.20.21
   ?o ccc-'.i .2) = (cc( u I .i)*4< i )*I;Y-OUTF (I) *cc(M,i,n -t-ccc(M,N)*ui (i.D*
     ]DT»-» ( I) *OY + X1NF (I) <>CC (v, 1 + i, i) )/A ( I )/UY-COi\lSTl^CC(2»I, 1)*DT
      G(J  TO  s
   ?] CC(M, I ,^) = (CC( y. 1 , 1 ) *4 (I ) *OY-OJTF (I)*CC(M»I»1) +CCC (M,N) *UI (I»D*
     IOT»M( I)*'JY+».1'1JF( I)^CC("'.l , 1) )/M D/UY  -CONST 1«CC ( 2 » I » 1 ) *OT
      GO  TO  s
    7 IF (V ( 1 + 1 , 1 ) )  22.22.23
   23 CC ( ••% I .2) = (CCCU i, 1) *A(1)*DY-OUTF (I) ttCC ( M, I » 1 ) t-CCC (M,N) *UI (I»l>*
     ]OT»^ ( I) *jY + xl'xjr ( I )-^CC(v'.l-l . 1) ) /A(1)/L)Y  -COfMSTl*CC(2,I» 1)*DT
      GO  TO  H
   22 CC ( vt , I . d) = ( CC ( "•< . I , 1 ) *A ( I ) *U Y-UO ( I » 1 ) *d ( I ) *0 Y»OT *CC (M» I » 1 )
     1-V ( T + l * 1 ) '- (A ( I) +A (1*1) )/2.Q*UT*CC(M,I + l,l) *CCC(M,iM)*UI (I»l)»
     107*^(1 >-JY + XIMF ( I)*CC (M.I-1 ,1 ) )/A(I)/UY  -CONST 1 *CC ( 2 » I » 1 ) *DT
->     IF  (CC(^«I.2) -n.lt-3D)  bf),t>0»bl
SO    CC (•"« 1.2) =0.0
•^1    COMTTMiJt
C  CALCUL4TIOM JN SU^t-ACc.
      GO  TO  (423.^-2^).
424   J=JM
      no«; A= 1^.^776-0. JS/y^T ( J. 1) +0.0043*(T ( J»
      CONST 1=0.0
      GO  TO
      COMST?-0
      COMTIMUK
      IF (y/ (J^. 1 ) )  S>. 9» lu
      CC (M.JM.2) = (CC(M, J.VL 1)*A ( JM)*DY/2.0-(00( JM»l)»d( JM)*DY/2.0*CC(M» JM
     1, ] )-CCC(-"»\)*Jf (JM,i)^
-------
     ice (>*• j^-i « i ) ) ---or ) /A < jiA) /L>Y»2.o -coNSTi*cc(2» JM» D*UT
     ? +COMST2* c;)asA-cc( i, j"U n
11    GO  rO  ( ?b.^S5»7bbSn) ,
7655^ JF(M-l)  76557, 7h5o7,
76557 CO'STl^
      CONST2=/1
      GO  TO  7b555
76558 COMST1=/
      COMST 2=0.0
C  CALCULATION  OM  BOTTOM
      IF (V ( -CONiTl*CC(2» 1»1)*OT
      GO TO £.
      CC ( '-^ 1 . 2) = ( CC ( ••% 1 . 1 ) » A ( 1 ) »OY/2 . 0- ( UO ( 1 » 1 ) *B ( 1 ) »OY/2. 0*CC (M , 1,1)-
                    «l)*d(l)*UY/2.0 + V<2»l)*(A(l)«-A<2) ) /2. 0*CC (Mf 1 , 1 ) )*DT)
                    -CO'->iSTl*CC(2»l» 1)*DT
      CONTINUE
      I F ( CC ( w , j v, , ? ) - o . 1 1- JO )  5^ » 5<+ . 5b
      CC (M« JM« 2) =0 • 0
      COMTTMUc
      IF  (CC(M«1«2)  -0.1E-3'J)  ^2»52»b3
      CC(M, 1,2) =0.0
11010 COMTlNUti
      CiU  TO  ( ^b'lh»7S.W) »NOUt£T
/S37  DO  3S47  J=1,JM
      DOSA=l'4.f77ft-0.35/9*T (J»2) * 0.0 043* (T
      If- (CC(
      CC( 1 . J
      GO  TO
      IF (CC( 1 • J«*) -DObA) 3S4 ? , 3547 »
      CC( 1 «J«2) =Oi)-3«
3547  COMTINUfc
C  Sj^FACt" ASSJ'''PT1«J'\I FOn1 00.
      GO  TO
I?SH? HO
      DOS A= 1^.4^7^-0. 3b/v*T (J,2) + 0.0 043* (T ( J»2)**2)
      CC( 1 .
      GO  TO
      CO  1 ^SrtM j=KCALC«JM
175BH  CC( 1 . J^) =t)OSA
1P5HQ  COM T I -Jut
            j
       SU".POU7 I ME SPEC A V (N>
       WAGlNG OF SPLCIHEO  VUHIHIAL IN MlxE'J  LAYERS
       COMMON T(-.!),^).t.L(6n)»XL(60)«A(60)»Tl(310)fTA(310)»SI6H(310)
       CO^^Oixj F IN ( 31 0 ) , A/ 1 iJU ( 310 ) »0 j ( 310 ) »QI ( J10 ) , QO ( 310) »P ( 50) »NPrt
                                    -248-

-------
11011
      COMMON
      COMMON
      COMMON
      COMMON
      JM J XH
      IF (MM)
      CO-NT !•»»-.
      Q ( i 1 M = 1
      >cc=o.o
                      ,ul *l AX (?) . DTTI.UTTA.DTSIGH.DTFlN.DTw/lND.OTDD.DTOI
             ur JL>. JM, JOUT » J IN.KU I F , KbUK , KOh.KQ.KLOSS. YSUR. YOUT»DT»DY
             1 sTu^fLVHCUNtOMr-GA.BZ. SPREAD* SIGMA] »SIGMAO»ETADY»TVARI
             T Vah»0. LtfA'-'.KAli, T A IK, PS I . DERI V* hAFOFL.EPS IL*GJ
                       t I A.u-'AjM.QELLOiM. V (  60*1) *UI( 60*1) »DTT
                       K«1lX.-.MlXED,QMIX.KAREA,DATRAD»ATRAD(310)
             A-v.wINDV.Cu.CI ,rt( 60)»S(  60).E*(  60)»EXU( 60)»ARF»UO(  60»1)
             ulN( 310) ,T1N< 310) *CC(20*60»2) *CCC(20*310) * GOUT (20, 310)
             CCT (20.310) . JQMIX ( bO ) »X JNF ( 60) »OUTF (60) »MIXH,MM
                  (J10) .oWAV.SLOPFf V1SCOS»LAGTIM(310)
                    (?0) .M^A^1N(20) f ET . iNTRAC ( 20 ) . 1 TR» ISTOt I SOI » IS02
             IbTtiN. IS T01. T H ICi^l* THICK 2 »OOXLt (60.20) ,UO(306) , BOD (30 6)
             NLtv£ ( iOfo) , VOL.N^/.NDLT.Z.Zl . UUOC. NGOET * OHOD
             M- -i i XH+ 1
             11010.11010,11011
      00
             -i. J.?) *A( J)*L)Y+XCC
      >CC=XCC*CC(M»JM,2)*A(JM)»OY/2.0
      XA = X A*A
      00 1 I=
    1 CC(M, 1,
    1 CONTINUb
      GO TO  ( ^b3b. 7-3 37)
C  ()0
i"537  DO
      DOSA = la.i+776-0. 1-3 /y* T ( J . 2) + 0.
      IF (CC( 1 . J.2) )
                                          (T ( J» 2)
      C-.U TO
      IF (CC( ] . J.?) -D
      CC ( 1 « J«2) ^Oi'rjA
      CONTINUE
11010 C
      P
      FNO
      SUH-?OUTIME  SPECOT(N)
L PROPORTION OF  SPECIEIED
      COMMON
      COMMON
      COMMON
      COMMON
      COMMON
      COMMON
      COMMON
                           INFLOWS  IN  OUTFLOWS
              r (fto,^) .CL((oO> ,XL(60) .A (60) .TI (310) ,TA(310) , SIGH (310)
              F I N ( 3 1 0 ) . v I N J ( 3 1 0 ) . i)0 ( 3 1 0 ) . U I < 3 1 0 ) . 00 ( 3 1 0 ) . P ( 50 ) » NPR
              DOM AX (2) .UI^AX (?) ,i)T n»L)TTA»L>TbIGH,DTFIN.DTrtlNL),DTGD,OTQI
              L)T(JO» JM. JOUT. J1N.KOIF , KSUR . KOH » KU . KLOSS* YSuR » YOUT » OT » DY
              1 STOP. EVHCUNiOMtGA.rtZ. SPREAD* S16MAI.S1GMAO»ETADY»TVARI
              T\/ARO.E\/AP.RAO»TAIR.»bI , i)ER I V * HAKDEL » EPS IL » bJ
              Y-iOf .NN.Ht I/-.OAJM.DELCON. V (  60,1).DI( 60.1) *DTT
WINDY
      COMMON
                       CJ.C1 «
                       TIN (31 i
                               (  oi)).b(  oO).r.X( 60)»EXO( 60)»ARF»UO( 60»1)
                               ) . CC(20.60.2) . CCC( ^0.310) »COUT (20.310)
                                       -249-

-------
          MON  CCn?(U 310) fJQMlX (60) ,X INF (60) »OUTF(60) ,i"1lXH»MM
           O'M P^ASOT (20) ,pvi4SIU(?0> *t-T»,MTHAC<20> » 1TR» 1STO» IS01 » IS02
      COMMON IS TON* IS TO If 1 Hi CM . THI CK?, UOXLt (60, 20 )» DO ( 306) , BOD (306)
      COMMON NLtVF ( 30 M « VOL»'Mw»NfjET»Z»Zl«DDOC»NriOET»DBOO
      JMM=JM- l
      IF (MM)  1K,)0, 1 1010, 1 1011
11011 CON'TINUtr
      no  i  -1=1. vH
21    XC = CC(v,, JM, i)-*(M (JM)*UY/2.0*UO(JM, 1)-QG1MIX (JM) ) +CC IJY / ? . 0 + CC ( M » 1 , 2 ) * A ( 1 ) *DY/2 . 0
      DO ?  J = f>»JMM
      IF ( J-JMIX.H) 1U, 11 » ] 1
   10 XC = XC + CC(M, J, ]. )*IJU*6( J)»DY -QQMIX(J))
    ? COMTlNUt
      IF (GOUT ( si) )  b0.bO»5
      COUT(M.N) =0.
      CCT(w.M) =0,
      GO  TO (H00« 1 ) « JbUdT
      GO  TO ^<
      XF = QOUT
      COUT (M,N)
      GO  TO ( ^Oi?» l ) ,-x|bi)tT
MO    CCT (M,,\J) =AC/'J J IN (/JM) /
^2    COMTINUK
              M) =XCC/0'Jl^(NM) /')TT
              M)  =CCT (:^»:M) *x,r ' +PMASOT (M)
     1 COMTTMUr.
      GO  TO (Hi 1,300) . -,'oULT
H 1 1   I F ( Nw-.xii/r. T }  30 0 « 3 J 1 ^ 300
301   WPITF(h,b) f T
              ('  ELAPStO  TIMt =•,  F7.2)
           F (h, y?7St)
           ATM T-'ACr.  COUT/^ASSIN    COUT      TKACOT       * REMAINING')
      WPTTF (f.'-f ) (X.CCT ( '1»,M) »COUT (M»N) »PMASOT (M) ,PMAS1N(M) »M=1»MM)
     4 FORMAT ( ] <+« tF 12.^3)
      h ' I// = 0
300   00  IS ./.= !,MM
      no  is i = i.jvi
    ]^ ccr-i. i . i) =cc (^.1.2)
    If- COMTIMUh
110 in co\iTi"jut-.
       i-UNCTIO\' OUO(N)
C  COMPUTE  INPJF 00  FKOM K£AiJ  IN VALUES

                                   -250-

-------
   COMMON l(M).2)»tL(b()),XL(60),A(60),TI<310)»TA(310)»SlGH(310)
   COMMON FI\|( 310) , wLNU( 310) ,1)0(310) »QI ( J10) * 00(310) ,P(bO> »NPR
   COMMON uuwflX (2> ,UIMAX (?) *i>TTi,urrA,L>TSIoH,OTFlN»DTwlNOtDTDD»DTai
   COMMON OT'JO. JM, JG'.JT, JIN»KOIF .KbUK «KOH, KG. ,KLOSS » YSUR, YOUT »OT »DY
   COMMON TbH^.tx/^COiNUO^tGAiB/SSPREAUiSlGMAI » S IGMAO »ETAOY » TVARI
   COMMON TVA^O,E\/AP,r,TAln,Pbl,OERl\/»HAFOELfEPSIL»bJ
   COMMON YLKJT , NN.HE F A , 0 4 JM , OELCON , V (  bO»l),Ul( 60»1)»L)TT
   COMMON *HJ.HCAP.KM.IA»-?MIX, JM I X b , M 1 XED , QM I X , K ARLA , D A TR AU , ATRAD < 3 1 0 )
   COMMON A-?,,*lNOr.CiJ,Cl,M 60), b<  60)»tX( bO)»EXO( 60)»AHF»UO(  60»1)
   COMMON ul^(31u) »TlN( 310) »CC(20»6U»2) »CCC »COUT (20 , 310)
   COMMON CCT (20. MO) ,'OCHIX ( bO ) » XiNF (60) »OUTF (60) »MlXri»MM
   COMMON bJ*F ( j] 0) «'j^Av/,SLUPE» V 1 SCOS » L AbT I M < 3 1 0 )
   COMMON PMASOT (2U) fPM«bIN(20) *ET,'MTkAC(20) « I TR» ISTOt I SOI » IS02
   COMMON IbTOM, ISfOl* THICK l»THICK2»UOXLt (60 « 20) » UO ( 306) ,800(306)
   COMMON NLtve (306) t \/uL , NW , NOET , i: , Z 1 , OUOC» NGOET , 0600
   NGOT=NLtVt(N)
   000=00(L+l)
   11- < L fl G T I -I (,M ) )  1,1,
   GO TO  ( J , 2) « '\H30 T
D00=00(L+l)-HHOUd
*UTT*(-/)))
CQMTINUt
                       J>( 1.-EXP(LAGTIM(N)*UTT»(-Z) ) ) / (EXP (LAGTIM (N)
   FIJMCTIOM
CALCULATED  1 -JPUT ^00 F*OM KEAO  IN  VALUES
           f(^o,2),EL(hO)»XL(60),A(60)»ri(310)»TA(310)»SIGH(310)
           r IN (31 'I) ,/;lNU( 310) ^ 00 (310) ,QI (310) ,QO(310) «P(bO) «NPR
           iJJ-'AX (P) »|JIMAX<2) ,OTTI»OTTA,OTSIOH,DTFIN»OTWINO»DTOD»DTQI
   COMMON  oT..^), jw, JOUT » JINi»KOIF ,KbUH ,KOH,KU,KLOSS» YSUH, YOUT ,OT ,OY
COMMON
COMMON
COMMON
COMMON
COMMON
COMMON
COMMON
                                        V»HAFOEL»EPSIL»GJ
           rnOT,1\fJ»HtTA«OAJM»L)ELCOi'J, V (  bO»l)»Ul( 60,1) »OTT
           ^HO.HCAP,KMIX»^MIX, JM 1 Xb , M I XEO , QM I X » K AREA , OA TRAD » ATRAO ( 31 0 )
           u.-v,'A'IN(jY»CO,CI,S(  60)»b( 60)»EX( 60)»EXO( 60>*ARF,UO( 60,1)
           iJl.N ( 310 ) »TIN< 3KU ,CC( 20,60*2) *CCC(20,310) »CUUT (20 , 310 )
           CCT (20. UO) ,'JO^IX (60) ,X1NF (60) ,OUTF (60) ,MIXH,MM
           hJ^F ( 3 ID) , u^ A V , bLOPE » V I bCOS » L AGT IM(310)
           PMASOT (20) ,PMAaliM(20) ,tr,NTRAC(20) , I TR , 1 STO , ISO 1 , 1 502
           iSTO'xl, IST01 , THlCM,THICK2»UOXLt ( 60, 20 ) ,DO ( 30 b) , BOO (30 6)
           NLEvE (3r^) , y/OLf Nta,iviOET,Z,Zl,Ou)OC»NGOET,L)riOL)
   fv)GOT=iM!.h V£ (N)
   GO TO  (2,1) , N>?0 F
      T"
         (L+ 1 > * (
                      EXP(|_AOTlM(iM)-*UTT*(-Z) ) )
                                -251-

-------
C (JMT iMUt
Ph T'.N'Xi
E'NH
                                   -252-

-------
                     APPENDIX II




       INPUT VARIABLES TO THE COMPUTER PROGRAM







Card 1, FORMAT 20A4





     WH = Alphanumeric variable used to print a title




          at beginning of output.  Anything printed on




          this card will appear as the first line of output




Card 2, FORMAT 20A4





     WH = Alphanumeric variable used to list units used




          in computation prior to output at each time




          step.




Card 3, FORMAT 1615




     JM = Initial number of grid points = number of the




          surface grid point.




   JOUT = IN umber of the grid point corresponding to outlet




          elevation.




   KulF = 1 for a constant diffusion coefficient.




        = 2 for a variable diffusion coefficient.




   KSUR = 1 for a constant surface elevation.




        = 2 for a variable surface elevation.




    KOH = 1 for use of Koh's Equation 2-49 for computing




          the withdrawal thickness.




        = 2 for use of Kao's Equation 4-26 of Huber





          and Harleman.




     KQ = 1 for computations with inflow and outflow.




        = 2 for computation with no inflow or outflow.






                           -253-

-------
      KLOSS = 1 for laboratory evaporation formula (Eq.  2-40).




            = 2 for Kohler field evaporation formula (Eq.  2-23





              of Huber and Harleman)•




            = 3 for Rohwer field evaporation formula (Eq.  2-43)




      NPRINT = Number of time steps between print outs of




               calculations.




      KAREA = 1 for laboratory reservoir calculations.




            = 2 for calculations for any other reservoir.




      KMIX = 1 for no entrance mixing.




           = 2 to include entrance mixing.




      MIXED = Number of grid  spaces in  surface layer for




              entrance mixing (defines  d  in Eq.  2-58).




Card 4. FORMAT 8F10.5




      YSUR = Surface elevation at beginning of calculations.




      YOUT = Elevation of outlet.




      i)T = Time step, At.




      TSTOP = Time at which progress ceases calculations.




      TZERO = Initial isothermal reservoir temperature.




      EVPCON = Constant, a, in evaporation formulas of




               Chapter 2 for  KLOSS = 1  or 2.  For KLOSS  =  3,




               EVPCON = 0.01.




      OMEGA = Constant u of Equation 5-2.




      BZ = Constant B  of Equation 5-2.




Card 5, FORMAT 8F10.5





      SPREAD = Number of outflow standard deviations, a  ,








                              -254-

-------
               equal to half the withdrawal thickness (see


               discussion of Equation 2-50).


      SIGMAI = Inflow standard deviation, a., Equation 2-51.


      ETA = Radiation absorption coefficient, r\, Equation 2-31


      BETA = Fraction of solar radiation absorbed at the


             water surface, |3, Equation 2-31.


      RKO = Water density, p.


      I1CAP = Water specific heat, c
                                    P*

      DELCON = Half the value of the constant of Equation 4-4b


               used to predict the withdrawal thickness,  6.


      RMIX = Mixing ratio, r , Equation 2-55.


Card 6, FORMAT 1615


      NTI = Number of inflow temperatures to  be read in.


      NTA = Number of air temperatures to be  read in.


      NSIGH = Number of relative humidities to be read in.


      NFIN = Number of insolation values to be read in.


      NSURF = Number of surface elevations to be read in.


      NDD = Number of values of the diffusion coefficient to


            be read in.


      NQI = Number of inflow rates  to be read in.


      NQO = Number of outflow rates to be read in.


Card 7, FORMAT 8F10.5


      DTTI = Time interval between  input values of TI.


      DTTA = Time interval between  input values of TA.
                              -255-

-------
      UTSIGH = Time interval between input values of SIGH.



      DTFIN = Time interval between input values of FIN.



      DSURF = Time interval between input values of SURF.



      DTDD = Time interval between input values of DD.



      UTQI = Time inverval between input values of QI.



      DTQO = Time inverval between input values of QO.



Card Group 8, FORMAT 8F10.5



      TI = Values of inflow temperatures, T. .



Card Group 9, FORMAT 8F10.5



      TA = Values of air temperature,  T .
                                       a


Card Group 10, FORMAT 8F10.5



      SIGH = Values of relative humidities, fy,  in decimal form.



Card Group 11, FORMAT 8F10.5



      FIN = Values of insolation,  d> .
                                   o


Card Group 12, FORMAT 8F10.5



      SURF = Values of surface elevations, y .
                                           J s


Card Group 13, FORMAT 8F10.5



      DD = Values of diffusion coefficients, D.



Card Group 14. FORMAT 8F10.5



      QI = Values of inflow rates, Q..



Card Group 15, FORMAT 8F10.5



      QO = Values of outflow rates, Q .
                                     o


Card 16, FORMAT 3F12.2



      SLOPE = Average slope at the inlet end of the reservoir.

-------
      GRAV = Acceleration of gravity = 3528000 cm/min2


             (KAREA = 1) and 73156608000 m/day2  (KAREA = 2).
      VISCOUS = Viscosity of water
Card 17, FORMAT 215


      NGDET = 1 for pulse injection solution.

            = 2 for D.O. calculation.


      NBOUND = 1 for entire euphotic zone saturated.

             = 2 for specified number of grid points for

               saturated region.


             = 3 for no saturation assumption, reaeration

               only mechanism.

Card 18, FORMAT 215

      ITR = Number of pulse injections to be traced (if NGDET = 1)

      or

      NDISSO = Number of input D.O.'s to be read in.
                                                    (if NGDET = 2),
      NBOD = Number of input B.O.D.'s to be read in.

      The following sequence holds if NGDET = 1.


Card Group 19, FORMAT 1615

      NTRAC(I) = Time steps at which pulse injections were

                 input.(This will depend on DT for example if

                 DT = 2 minutes and the first trace was input


                 at 10 min., NTRAC(I) = 5).


Card 20, FORMAT 15

      NDET = Number of time steps to be passed between printout


             of TRACOT  (Equation 3-38).


      Go to card 25.
                         -257-

-------
      The following sequence holds if NGDET = 2.




Card 19, 2F10.5




      DDOC = Time interval between input values of D.O.




      DBOD = Time interval between input values of B.O.D.




Card Group 20. FORMAT 8F10.5




      DO = Values of inflow D.O.




Card Group 21, FORMAT 8F10.5




      B.O.D. = Values of inflow B.O.D.




Card 22, FORMAT 15




      NPROF = 1 for a constant initial B.O.D.  and D.O.  profile.




            = 2 for a linear initial B.O.D. and D.O.  profile.




Card 23. FORMAT 4F10.5




      If NPROF = 1




      DOI = Initial D.O. value.




      BODI = Initial B.O.D. value.




      or if NPROF = 2




      DOB = Initial D.O. value at the reservoir bottom.




      DOT = Initial D.O. value at the surface of the reservoir.




      BODB = Initial B.O.D. value at the reservoir bottom.




      BOOT = Initial B.O.D. value at the reservoir surface.




Card 24, FORMAT 2F10.5, 15




      Z = First order decay constant for B.O.D. (Eq.  3-14).




      Zl = First order reaeration constant at surface.




      NDOCA = Time interval between printout of D.O.  profiles.

-------
Card 25, FORMAT 2F10.5




      THICK1 = Thickness of surface layer for lagtime




               calculation (Equation 2-92).




      THICK2 = Thickness of subsurface layer for lagtime




               calculation (Equation 2-92).




      If lagtime is not to be considered set THICK1 and




      THICK2 = 0.00001 meters.




      The following parameters are read in when KAREA = 2.




Card 26, FORMAT 1615




      NAA = Number of areas to be read in.




      NXXL = Number of lengths to be read in.




      NWIND = Number of wind values to be read in.




      NATRAD = Number of atmospheric radiation values to be




               read in.




      JMP = Number of grid points for which program variables




            should be initialized.  (This should be the




            maximum value of JM expected to occur in the




            calculations.)




Card 27, FORMAT 8F10.5




      DAA = Vertical distance interval between input values of AA,




      DXXL = Vertical distance interval between input values





             of XXL.




      UTWINU = Time interval between input values of WIND.




      DATRAD = Time interval between input values of ATRAD.
                            -259-

-------
      AAB = Elevation of first (lowest) value of AA.



      XXLB = Elevation of first (lowest) value of XXL.



      ARF = Area reduction factor, a  = 1.



Card Group 28, FORMAT 8F10.5



      AA = Values of horizontal cross-sectional areas, A.



Card Group 29.. FORMAT 8F10.5



      XXL = Values of reservoir lengths, L.



Card Group 30, FORMAT 8F10.5



      WIND = Values of wind speeds, w.



Card Group 31



      ATRAD = Values of atmospheric radiation, $ .
                                                3.
                             -260-

-------
                         APPENDIX III




         SAMPLE INPUT DATA FOR FONTAi\A P.O. PREDICTIONS










      This appendix contains typical input for the prediction of




temperature and D.O. profiles and outlet values. This particular




input set is for the case of initial B.O.D. = 0, initial D.O. =




8 ppm and K = 0.05 day   and saturation in the entire euphotic zone.




      Cards or card groups are separated by blanks in the computer




listing. This is only for illustrative purposes and would not be




present in the actual data deck. Data contained on the card or card




groups are titled with a card prefaced by an asterisk (*) that would




not appear in the actual computer input.
                                 -261-

-------
1FIF.LD DATA Fu-v FOMTANA RESERVOI^ FOK MARCH I TO DECtMBER 31» 1966.

OALL UNITS I "i METERS.  D4YS» K1LOCALORIES,  KILOGRAMS, AND DEGREES CENTIGRADE.
47 2?
493.0
1 . 9»S
306 306
1.0
» INFLOW
7.68
6.06
10.84
9.40
9.91
9.95
13.87
15. 6«
13.?6
12.59
15.80
16.36
17-98
19.39
19.73
19.47
20.43
23. ??
20.54
18.04
18.49
19.55
19.30
18.79
18.77
17.21
15.75
13.69
11. ?3
14. IP
12.1 1
8 . 43
12.2?
9. 7?
4.55
*i. ?l
8. 8?
6.4S
4.^9
1 2
443.0 1.
4.0 0 .
306 306
1.0 1.
TEMPERATURES
7.24
h.67
10.83
H.52
10.23
10.41
13.56
15.00
13.41
12.28
16.46
17.13
18.00
14.28
14.25
14.29
21.06
2] .44
P1.25
l8.<+5
18.71
2<> . db
14.66
18. 4h
19.00
14.43
16. 14
14.24
lo. *2
J3. 74
11.41
8.41
12.04
-.06
h . 64
7. 78
f-'. 47
6.63
4. -i 2
1 1
3 10
2 2
0 300.0 6.7
7b 0.
306 2
0 1.
. (DEGREES
6.93
7. hi
11.41
8.25
10.43
11.63
12.81
14.56
13.43
13.28
16.97
1 7 . 0 h
18.37
19. 34
18.53
18.76
20.75
21 .hi*
21.19
16.44
1 8.48
19.46
18.46
19.04
1 8 . 7 h
16.36
15.^9
14.44
10.84
1 J. 84
11.49
10.01
11 .60
4. t> 1
h.29
8.44
4.24
5. 42

50 997.
306 306
0 1.0
CENTIGRADE) .
8.27
8.29
11.39
8.27
9.75
12.29
13.31
14.79
14.38
14.30
1 7.25
16.20
16.65
19.87
la. 91
16.»1
22.16
22.30
21.03
16.27
17.83
19.89
18.62
ltt.84
18. 18
16.03
15.95
13.44
11.11
14.25
10.93
10.92
10.71
8.b3
b.33
9.88
6.4-6
4. OB

0



8.35
9.13
10.94
8.50
9.47
12.32
13.97
14.93
15.14
13.81
16.26
15.68
18.46
19.67
19.42
20.31
23.63
22.84
21.33
18.31
17.62
19.70
19.12
19.17
1 7.16
16.28
16.36
13.97
11.58
14.28
11.38
11.14
9.91
8.04
6.46
9.19
6.49
3.76

                                                  306.0
                                                       1.0
               (DhorvEES
6.034
                             CEMTIbRADE).
                                  H.7H6
                                        0.281
                                                        6.34
                                                        9.96
                                                       11.36
                                                        8.73
                                                        9.55
                                                       12.66
                                                       14.25
                                                       14.30
                                                       15.11
                                                       13.86
                                                       15.73
                                                       15.31
                                                       1.8.00
                                                       ^0.57
                                                       18.07
                                                       20.81
                                                       24.24
                                                       21.37
                                                       21.08
                                                       18.48
                                                       17.74
                                                       19.54
                                                       19.58
                                                       19.08
                                                       17.37
                                                       15.85
                                                       16.32
                                                       13.80
                                                       12.46
                                                       1J.04
                                                       12.03
                                                       11.87
                                                        9.64
                                                        8.55
                                                        6.27
                                                        8.24
                                                        6.59
                                                        4.12
-3.432
                1.0
                                                             5.62
                                                            10.70
                                                            12.11
                                                             9.83
                                                             9.45
                                                            12.45
                                                            15.29
                                                            13.59
                                                            13.81
                                                            14.16
                                                            15.81
                                                            15.78
                                                            18.58
                                                            19.39
                                                            18.42
                                                            20.55
                                                            23.73
                                                            20.65
                                                            20.63
                                                            18.48
                                                            18.14
                                                            19.18
                                                            18.84
                                                            18.26
                                                            17.30
                                                            16.00
                                                            15.20
                                                            13.98
                                                            12.79
                                                            13.02
                                                            10.81
                                                            11.14
                                                            10.14
                                                             9.47
                                                             5.68
                                                             7.22
                                                             6.46
                                                             5.05
                                                                -2.560
                      5.51
                     10.97
                     11.68
                      9.93
                      9.48
                     12.92
                     15.42
                     12.86
                     12.83
                     14.51
                     15.51
                     16.91
                     20.45
                     19.75
                     18.88
                     20.59
                     23.74
                     20.48
                     19.01
                     18.20
                     18.91
                     19.86
                     18.44
                     18.88
                     17.28
                     16.10
                     13.84
                     14.28
                     12.56
                     11.74
                      9.20
                     12.52
                     10.27
                      9.08
                      5.26
                      7.02
                      6.31
                      5.38
-1.432
                                        -262-

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1.090
1 1 .451
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7.706
3.87?
14.503
15.288
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15. 840
13.559
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9.510
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0 .687
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0.612
0.655
0 .4HO
0.59?
0.629
0.899
0.6?0
0 .849
0.693
0.811
0.704
0.751
0.740

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0.901
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8.892
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10.077
1 6 . 244
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19.251
21.225
21. 199
?2. 264
20.838
20 .423
22.320
18. 172
20.279
18. o 19
16. 749
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13.665
12.216
7. 19 3
et.602
10.623
10.962
4.233
1.251
13.233
1.092
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(DtCI^ALS)
0.899
0.660
0 .06!
0.664
0 . 83D
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0 . 765
0 .86 7
0.670
0. 7H9
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0 .671
0. 771
0 . 7^4
0. 741
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0 . 799
0 . * -1 8
7.473
10.118
1 .612
4.490
14.996
16.072
15.828
1 7.406
14.543
1 9 . 7 o 0
11. 156
20 . 1 JO
18. 722
20.839
21. 147
22.387
23.723
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20.266
21.372
21.810
17.669
20.783
16.330
18. 101
1 7.673
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14.964
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6. 783
10 .444
6.450
4.929
0.795
16.696
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0 .800
0.688
0 .544
0.609
0.546
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0.852
0.924
0 .647
0.739
0.740
0 .666
0 .806
0.826
0 . 765
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0.811
0.827
9.306
10.673
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10.948
16.669
16.199
16.866
12. 711
15.585
1 0.64b
20.709
ltt.485
21.466
22.369
24.086
21.759
23.756
20.037
20.924
22.092
19.054
20.950
16.854
16.357
16.044
9.879
17.684
14.354
9.233
10.746
5.769
5.464
2.252
10.667
2.979
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0.845
0.767
0.696
0.658
0.583
0.793
0.922
0.960
0. 708
0.945
0.882
0 .668
0. 858
0.798
0.776
0.819
0.772
0.900
12.868
15.065
6.157
6.440
9.377
15.946
16.398
13.309
19.122
17.954
12.878
20.640
16.408
20.298
20.963
24.779
22. 718
23.990
20.121
22.145
22.846
20.546
18.379
18.640
14.567
15.379
11.651
14.084
14.866
10.010
12.747
6.126
7.803
0.443
3.067
3.808
0.409


0.989
0.730
0.669
0.569
0.597
0.795
0.796
0.929
0. 7b7
0.781
0.871
0.683
0.806
0.906
0.825
0.848
0. 778
0.761
11.472
13.138
6.616
5.253
7.864
16.830
14.063
9.056
17.264
17.391
15.439
19.662
18.199
21.017
22.457
23.719
20.908
22.134
20.964
22.389
21.757
18.785
18.892
15.742
15.396
10.724
13.383
6.452
13.859
2.928
14.179
9.373
11.906
-0.513
2.058
3.705
4.788


0.992
0.943
0.797
0.609
0.785
0.715
0.699
0.888
0.562
0.915
0.948
0.739
0.751
0.774
0.848
0.833
0.828
0.801
11.003
4.107
7.283
6.247
9.398
16.717
12.950
12.204
18.814
17.497
17.855
20.206
18.628
21.295
21.289
24.490
21.449
20.767
20.303
22.619
20.887
19.387
18.478
17.059
13.470
8.225
16.243
10.002
11.152
-1.490
12.162
10.375
10.372
1.178
1.567
3.997
-1.212


0.807
0.841
0.637
0.652
0.567
0.653
0.750
0.684
0.565
0.806
0.950
0.730
0.697
0.753
0.846
0.874
0.847
0.743
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0.741
0.8]8
0 .876
0 .855
0.818
0.855
0.8?9
0.870
0.873
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0.751
0.846
0.855
0.794
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0.790
0.917
0.81?
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0.848
0.965
PREDICTED
3641.306
4?30.890
3880.664
50^8.949
4949.656
58?6.90?
5734. 1?8
?403.948
7044. ] 64
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6296.011
2783.813
1848.651
491 3.996
4285.332
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0.873
0.832
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0.854
0 . * i f
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0.978
0. 86?
0.800
0.720
0.805
0. 731
0.932
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3036.242
4277.375
440 7. 632
30 17.081
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NC^EASED BY
2366.663
4121.945
4787.433
5104.730
4769.910
4125.269
3135.408
4542. 37b
6851.671
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0.764
0.825
0.931
0.882
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0.853
0.951
0.858
0.789
0.836
0.880
0.887
0.814
0.866
0.812
0.858
0.816
0.937
0.867
0.776

0.775
0.859
0.886
0.837
0.818
0.824
0.846
0.873
0.059
0.867
0.832
0.925
0.861
0.937
0.840
0.843
0.874
0.831
0.853
0.825

0.920
0.858
0.875
0.917
0.901
0.786
0.899
0.793
0.904
0.971
0.920
0.895
0.951
0.967
0.894
0.981
0.749
0.883
0.745
0.904

0.912
0.878
0.853
0.807
0.878
0.804
0.823
0.851
0.846
0.809
0.969
0.937
0.869
0.949
0.873
0.940
0.639
0.917
0.739
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lb%» KCAL/M-M-DAY) .
3389.045
2184.633
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5294.582
4474.335
2184.1 12
3160.601
5b41 . 69a
6590. 136
3038.519
2816.871
8283.511
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7389. 199
5660.726
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6 725.464
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5967.421
2139.097
1979.240
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1662.454
3^80.351
3431.094
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3178.336
3463.941
4600.007
5027.503
5474.960
3667.411
5932.941
3717.163
6674.550
4536.125
5467.761
8105.980
5014.085
3080.518
6626.492
3504.713
7483.988
7893.617
8731.785
4487.039
3224.782
7236.601
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3936.839
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1959.093
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1644.110
2300.179
1375.295

3706.230
2257.337
4120.648
5256.492
3506.479
4911.644
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2505.267
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3737.138
3833.682
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5969.429
3079.269
3103.319
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3647.045
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3130.831
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3538.553
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4183.367
3540.808
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9.204
9.572
8.370
8.536
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8.883
8.516
8.207
7.855
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7.569
8.334
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8.475
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8.599
8.679
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7.617
8.377
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9.099
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8.202
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9.003
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8.510
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9.268
8.836
8.752
8.829
8.309
7.966
7.938
7.483
7.971
8.092
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8.595
7.385
7.884
9.341
9.498
9.738
8.315
8.662
8.696
8.201
9. 198
10.128
11.271
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9.590
11.619
9.407
6.832
8.785
8.928
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8.795
8.692
8.439
8.559
8.956
7.505
7.621
8.185
7.570
8.430
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8.372
9.293
9.270
9.869
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8.525
8.226
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8.584
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8.686
8.457
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9.078
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7.575
8.176
8.590
7.920
7.928
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9.596
8.775
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3.086
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7823.
7872.
7252.
7509.
7093.
5860.
6770.
6197.
5840.
6285.
5854.
5575.
4706.
7554.
5431.
4200.

355
418
348
641
258
250
594
383
734
730
570
070
301
969
723
437
410
992
887
551
695
437
852
359
727
039
6b6
102
586
914
879
121
039
625
9b3
Ob9
008
000

4716.
6b47.
6395.
4901.
5186.
6657.
7352.
7334.
7075.
6812.
7b39.
5804.
8137.
7594.
7457;
8150.
8191.
7899.
8073.
7533.
8054.
7937.
8033.
7780.
7731.
7437.
6696.
6032.
7«05.
7323.
5997.
6800.
5677.
5514.
5340.
6732.
5605.
4903.

891
926
285
270
629
32tt
336
086
352
238
875
230
871
602
172
547
645
594
980
656
516
074
828
195
406
426
422
348
184
012
723
262
031
562
715
836
648
750

4418.652
6811.750
6877.480
5650.805
5403.684
6409.637
7115.891
7499.074
6282.387
7898.969
7616.016
0149.875
7906.078
7555.770
7907.836
8283.824
8113.930
7615.902
7767.598
8023.031
8446.711
8026.960
8120.164
7038.840
7522.094
6908.430
7397.156
6270.711
6690.336
7402.449
6459.695
7113.973
5947.832
6404.789
4976. 117
5731.562
5335.344
5178.578

4470.258
6803.648
6804.520
5523.246
5479.039
5656.371
6959.629
7236.090
5659.367
7567.121
7670.324
6748.168
7472.082
7352.023
8425.734
8607.852
8767.758
7695.609
8149.699
7916.973
7904.047
8064.906
7847.305
7214.680
7469.223
6482.023
6460.887
7118.867
5657.578
7231.988
5706.117
7264.023
6686.859
6588.035
4839.344
5346.113
5526.586
5878.484

4479.535
6515.793
5295.703
5532.844
5328.777
5769.660
7336.344
6135.754
6227.539
7732.457
7732.340
6984.199
7157.809
7167.137
8442.891
7919.309
8590.437
7329.574
7942.430
8248.227
8006.719
7314.211
7812.555
7055.676
7047.305
6458.586
5851.910
6726.227
6612.137
6738.133
4758.484
7015.832
6690.953
6673.324
5500.930
5550.945
5338.875
4487.641

-269-

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                              APPENDIX iv




                       LIST OF TABLES ANH FIGURES




                                FIGURES





NUMBER                       TITLE                          PAGE




2.1        The Changing Inflow Level and Withdrawal Level




           Distribution of a Stratified Reservoir            17




2.2        Work Input to Displace a Partical of Fluid in




           a Stably Stratified Fluid                         22




2.3        Flow in Chemical Engineering Process Equipment    30




2.4        Constant Longitudinal Dispersion Coefficient




           Model                                             33




2.5        Control Volumes Illustrating Concervation of




           Mass and Energy in a Stratified Reservoir         39




2.6        Penetration of Radiation into a Reservoir         42




2.7        Control Volume and Schematization For Mathe-




           matical Model of an Idealized Reservoir           49




2.8        Laminar Flow Towards a Line Sink (23)             50




2.9        Determination of the Outflow Standard Deviation   53




2.10       Dye Concentration Profiles in Fontana Reservoir   55




2.11       Schematic Representation of Entrance Mixing       58




2.12       Two Layered Flow Schematization for Sinking




           Flow                                              60




2.13       Points of Evaluation of Equation 2-32             70




2.14       Numerical Dispersion                              75
                                 -270-

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NUMBER.                           TITLE                         PAGE




  3.1         The Graphical Temperature Prediction Model




              of Wunderlich                                     83




  3.2         Evaluation of the Bulk Depletion Factor           86




  3.3         The Graphical D.O. Prediction Method of




              Wunderlich                                        87




  3.4         Dissolved Oxygen Saturation vs. Temperature       97




  3.5         Control Volume for the Water Quality Model        99




  3.6         Boundary Conditions for D.O. and B.O.D. in




              The Numerical Scheme                             101




  3.7         The Distribution of an Input Under Stratified




              Conditions                                       103




  3.8        Schematic Curves Predicted for the Pulse




              Injection Solution                               112




  4.1         The Laboratory Flume                             115




  4.2         The Entrance Section                             117




  4.3         The Outlet Section                               118




  4.4         Fluorometer Calibration-Concentration vs.





              Dial Reading                                     120




  4.5         Monitoring of Fluorometer Reading With a




              Sanborn Recorder                                 121




  4.6         Fluorometer Calibration-Dial Reading vs.





              Sanborn Deflection                               122




  4.7         Fluorometer Calibration-Temperature Dependence   123
                                -271-

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NUMBER                       TITLE                           PAGE




  4.8         Movable Probe and Thermistor for Temperature




              Measurements                                   124




  4.9         Laboratory Insolation Calibration              126




  4.10        Dye Trace in a Laboratory  Flume (3 traces)     128




  4.11        Water Temperature vs.  Density                  133




  4.12        Input to Variable Inflow-Outflow,Variable




              Insolation,  Constant Surface Elevation




              Experiments                                     136




  4.12a       Temperature  Profiles                           137




  4.13        Concentration Predictions                       139




  4.14        Concentration Predictions                       140




  4.15        Cumulative Mass Out  Predictions                 141




  4.16        Cut Off Criteria For The Withdrawal Layer       144




  4.17        Temperature  Profile  Predictions-Sensitivity




              Analysis                                       148




  4.18        Temperature  Profile  Predictions-Sensitivity




              Analysis                                       149




  4.19        Outlet Temperature Predictions-Sensitivity




              Analysis                                       150




  4.20        Cumulative Mass Out  Predictions-Sensitivity




              Analysis                                       151




  4.21        Cumulative Mass Out  Predictions- Sensitivity




              Analysis                                       152




  4-22        Input to Constant Inflow-Outflow,  No  Insolation







                                  -272-

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NUMBER                          TITLE                        PAGE




              Experiments




  4.23        Temperature Profiles




  4.24        Concentration Predictions




  4.25        Concentration Predictions




  4.26        Concentration Predictions




  4.27        Cumulative Mass Out Predictions                154




  4-28        Inputs to the Variable Inflow-Outflow,Variable 166




              Insolation,Variable Surface Elevation Experiments




  4.29        Temperature Profiles                           j_6y




  4.30        Concentration Predictions                      169




  4.31        Cumulative Mass Out Predictions                179




  5.1         Map of Fontana Reservoir and Watershed         173




  5.2         Determination of Absorption Coefficient and    177




              Surface Absorbed Fraction for Fontana Reservoir




  5.3         Exponential Width-Elevation Relationship for




              Fontana Reservoir                              181




  5.4         Outlet Temperature For Fontana Reservoir       182




  5.5         Temperature Profiles For Fontana Reservoir     183




  5.6         Temperature Profiles For Fontana Reservoir     184




  5.7         Temperature Profiles For Fontana Reservoir     185




  5.8         Temperature Profiles For Fontana Reservoir     186




  5.9         Temperature Profiles For Fontana Reservoir     187




  5.10        Temperature Profiles For Fontana Reservoir     188




  5.11        Temperature Profiles For Fontana Reservoir     189






                                -273-

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NUMBER                        TITLE                           PAGE




  5.12        Temperature Profiles For Fontana Reservoir      190




  5.13        Fontana Reservoir Simulation of Various




              Pulse Injections                                195




  5.14        Outlet D.O. Concentrations  For Fontana




              Reservoir                                       203




  5.15        Outlet D.O. Concentrations  For Fontana




              Reservoir                                       204




  5.16        Dissolved Oxygen Profiles For Fontana




              Reservoir                                       205




  5.17        Dissolved Oxygen Profiles For Fontana




              Reservoir                                       206




  5.18        Dissolved Oxygen Profiles For Fontana




              Reservoir                                       207




  5.19        Dissolved Oxygen Profiles For Fontana




              Reservoir                                       208




  5.20        Dissolved Oxygen Profiles For Fontana




              Reservoir                                       209




  5.21        Dissolved Oxygen Profiles For Fontana




              Reservoir                                       210




  5.22        Dissolved Oxygen Profiles For Fontana




              Reservoir                                       211




  5.23        Dissolved Oxygen Profiles For Fontana




              Reservoir                                       212

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                           TABLES









Number                      Title                             Page






2.1              Reservoir Stratification Criteria             25




4-1              Peak Concentration  and Arrival Times-        142




                 Variable Inflow-Outflow and Insolation,




                 Constant Surface  Elevation




4.2              Cut Off Criterion                           146




4.3              Peak Concentration  Characteristics           160




5.1              Fontana Reservoir Areas, Lengths and         180




                 Widths




5.2              Comparison of  Predicted Cumulative           197




                 Mass Out Values with  the Detention




                 Times  of Wunderlich




5.3              B.O.D. Measurements in Fontana Reservoir     200




                 Inflows




5.4              The  Various  Initial Conditions in  the        202




                 D.O. Analysis
                                -275-

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                          APPENDIX V

                    DEFINITION OF NOTATION


Representative units of variables are given in cm}gm,min,cal, and  C.
a        Constant in evaporation formula (cm/min-millibar)•


a        Constant in Dougal—Bowmann equation (min  )


a        Atmospheric transmission coefficient

                             3
c        Concentration (gm/cm )


c'       Turbulent concentration fluctuations (gm/cm )

                                         3
c        Concentration of tracer E (gm/cm )
 £j
                                                          3
c .       Concentration in convectively mixed region (gm/cm )

                                       3
cout     Concentration in outlet (gm/cm )


c        Specific heat (cal/gm°C)

                                             3
c        D.O.  saturation concentration (gm/cm )
 S3 C

d        Depth of fluid (cm)


d        Depth of euphotic zone (cm)


d        Depth of surface layer for entrance mixing (cm)


d        Depth of entering stream (cm)
 s

         Depth for saturation in the  water quality model (cm)


e        Base of Kaperian  logarithms


e        Saturated water vapor pressure at temperature of air (milli-
 ci

         bars)

                                           2
g        Gravitational acceleration (cm/min )


h        Thickness of horizontal layer for lag time (cm)


i        Direction
                                   -276-

-------
j        Direction

                                    _ -I
k        Bulk depletion  factor (min  )


^       Reoxygenation rate  constant  (min  )'


k2       Reoxygenation rate  constant  (min  )


m        Optical air mass


n        number of  time  steps


n        Direction  parallel  to reservoir  bottom  (cm)


nmax     Location of maximum velocity for sinking flow  (cm)


p        Pressure (millibar)

                                                   2
q.       Inflow rate per unit  vertical distance  (cm /min)

                                                    2
q        Outflow rate per unit vertical distance (cm /min)


r        Normalized distance between the  sun and the earth


r        Stratification  criterion ratio
 tl

r        Entrance mixing ratio
 m
                             3
sinks    Sinks of mass (gin/cm  -min)
     m
                               3
sinks    Sinks of heat (cal/cm -min)


t        Time (min)


t.       Detention  time  (min)
 d

t.       Start of water  quality calculations (min)


t.1      Start of pulse  injection calculation  (min)


t        Time to drain volume  of water above center line of intake
 it

         (min)


t        Total lag  time  (min)
 Li

t        Horizontal lag  time component (min)
 LH

t        Time for incoming water to reach its  density level (min)
                                -277-

-------
u        Horizontal advective velocity  (cm/min)



u'       Turbulent advective velocity fluctuations  (cm/min)



u.       Interfacial velocity (cm/min)



u        Maximum velocity in lower layer of surface entrance  (cm/min)
 max


v        Vertical convective velocity (cm/min)




v        Voltage



v        Maximum vertical velocity in numerical scheme  (cm/min)
 max


w        Wind velocity (cm/min)



x        Horizontal distance (cm)



y        Vertical distance, elevation (cm)



y.        Reservoir bottom elevation (cm)



y.       Elevation of inflow (cm)



y .       Elevation of bottom of mixed convective layer  (cm)
'mix                                               J


y        Elevation of outflow (cm)
Jout                          v  '


y        Surface elevation (cm)
 s


z        Transverse direction (cm)
                                            2
A        Horizontal cross-sectional area (cm )



B        Reservoir width (cm)



B        Average width of surface layers subject to entrance



         mixing (cm)



B        Width at elevation zero (cm)



B.O.D.   Initial condition for B.O.D. (ppm)




C        Cloudiness




D        D.O. deficit
                               -278-

-------
                                 2
D,D      Diffusivity of heat  (cm /min)

                                                o
D^       Longitudinal dispersion coefficient  (cm /min)

                                 2
DM       Diffusivity of mass  (cm /min)

                                 2
D        Numerical dispersion  (cm /min)

                                    2
D        Dispersion coefficient  (cm min)

                                      2
Dr       Vertical eddy diffusivity  (cm /min)


D.O.     D.O. in outlet  (ppm)
    out                   ^^
                                          2
E        Turbulent diffusivity of h'eat (cm /min)


IF       Reservoir Froude number


G        Dummy variable


J        Number of spatial grid  points in finite difference equations


K        B.O.D. decay constant (min )


K        Decay constant  (min   )


L        Reservoir length (cm)


L1       Reservoir length for  lag time (cm)


L        Latent heat of vaporization (cal/gm)


M        Mass (gm)


P        Reservoir perimeter


P        Rate of photosynthetic  oxygen production  (min  )


P        Prandtl number
 r

Q        Volume rate of flow  (cm /min)

                               .   . 3,  . ,
0        Inflow rate to reservoir (cm /min;
xi
                                                ,  3,  . ,
0 '       Total inflow rate with  entrance mixing (cm  /mm)
xi

0        Portion of mixed inflow withdrawn from surface layers

          .   3,  . ,
         (cm /mm)
                              -279-

-------
                                        3

0        Outflow rate from reservoir (cm /min)
xo


Q        Vertical flow rate in reservoir (cm /min)
 v


R        Reynolds number



R        Rate of oxygen demand by algae (min  )




S        Dummy variable




S        Schmidt number
 c


T        Temperature (°C)



T1       Turbulent temperature fluctuation (°C)




T        Air temperature (°C)
 3.


T        Air temperature, measured two meters above surface  (°C)

 a2

T.       Inflow temperature (°C)




T. '      Inflow temperature with entrance mixing (°C)




T        Temperature of mixed surface layer




T        Average temperature of surface layers for use with




         entrance mixing (°C)




T .      Temperature of convective mixed layers
 mix                                       }



T        Initial uniform temperature (°C)



T        Outflow temperature (°C)




T        Reservoir temperature (°C)




T        Surface temperature (°C)
 S



T        Water temperature (°C)




U        Average advective velocity (cm/min)




U.       Inflow velocity (cm/min)




U        Maximum inflow velocity (cm/min)
  max
U
 m
Uniform outflow velocity from surface layer  subject



to entrance mixing (cm/min)


                        -280-

-------
U         Outflow  velocity  (cm/min)
 o

U         Maximum  outflow velocity (cm/min)
  max
                                              o
V         Volume of  reservoir above  intake (cm )


Vo        Vertical equivalent outflow advective velocity  (cm/min)


V         Volume  (cm3)


V         Volume of  inflow  (cm/min)

                              •3
¥         Reservoir  volume  (cm )

                         o
W         Work  (gm-cm/min )

                                  2
W,         Load  of  tracer  (gm-cm/min  )
a        Solar altitude  (degrees)


a,a      Parameters in Koh's prediction formula for the withdrawal

                      -2/3
         thickness  (cm    )


g        Fraction of solar radiation absorbed at water surface

                                         4
g        Vertical density gradient  (gm/cm )


Y        Specific weight (gm/cm-min)


6        Thickness of withdrawal layer (cm)


<5(x)     Dirac delta function


A        Increment  (tnin)


e        Radiative emissivity


e        Normalized density gradient (cm  )


e        Saturated vapor pressure at temperature of air (millibars)
 a

e        Saturated vapor pressure at temperature of water (millibars)
 s

D        Radiation absorption or extinction coefficient (cm  )


6        Dummy variable



                              -281-

-------
6        Angle between reservoir entrance slope on the reservoir


         surface


X        Dummy variable


p        Dynamic viscosity (gm/cm-min)

                                2
v        Kinematic viscosity (cm /min)


p        Density (gm/cm )

                                 3
p        Reference density (gm/cm )
 o

!L        Biochemical oxygen demand (ppm)


I .       B.O.D. in convectively mixed layers (ppm)
 mix

£        B.O.D. in incoming streams (ppm)

                                         2
o        Stefan-Boltzman constant (cal/cm -min-°K)


o.       Standard deviation of inflow velocity distribution  (cm)


a        Standard deviation of outflow velocity distribution  (cm)
 o

T        Dummy variable

                                2
T        Shear stress (gm/cm-min )
 o
                          2
        Heat flux (cal/cm -min)

                                           2
(J>        Atmospheric radiation flux (cal/cm -min)
 a
                                                     2
<}>,        Solar radiation absorbed internally (cal/cm -min)

                                     2
        Conductive heat flux  (cal/cm -min)

                                            2
<(>,-,       Total evaporation heat flux  (cal/cm -min)
 £j

4>        Evaporation heat flux from vaporization  of  surface  water


         (cal/cm -min)

                                                   2
4>T       Heat flux from surface heat losses (cal/cm  -min)
 jj

        Heat flux from heat transfer through  reservoir  sides

                2
         (cal/cm -min)
                                 -282-

-------
                                                      2
>         Solar radiation (insolation) heat flux (cal/cm -min)

                                                                2
         Longwave radiation heat flux from water surface (cal/cm -min)

                               2
         Solar constant (cal/cm -min)
 sc

>         Evaporation heat flux from heat advected from water surface

                2
         (cal/cm -min)


         Dummy variable


fy        Relative humidity


10        Parameter in reservoir width-elevation relationship (cm  )
                                   -283-

-------