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.
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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.
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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.
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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
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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
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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
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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
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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-
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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.
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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.
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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.
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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
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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)
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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.
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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
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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-
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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)
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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-
-------
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-
-------
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-
-------
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-
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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-
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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
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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-
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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-
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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-
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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-
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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-
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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-
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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-
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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-
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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.
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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.
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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-
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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
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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-
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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-
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52. Villemonte, J. R., Rohlich, G. A. and Wallace, A. T., Hydraulic
and Removal Efficiencies in Sedimentation Basins, Third Interna-
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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-
-------
1.090
1 1 .451
? . 28s
7.706
3.87?
14.503
15.288
1 2-7^0
1 3 . H 1 0
18.2^?
19.289
19.1*9
19.7*?
19.087
P0.9M
20 . 705
?1 .8^4
20.201
19.991
21-359
23.398
18.3^3
19. RM
18.398
15. 840
13.559
10. 1 74
9.510
8.749
H.250
(i ,U?8
13.076
6.940
-0.20 7
6.543
0 .497
5.695
-3.691
PFLATIVE "
0 .687
0.7?^
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
0.6S1
0.901
<+ . 0 1 -i
1 (i . <4 78
1 . -ic1 1
9. llt>2
8.892
14. 329
1 s. 4M 1
1^.871
1 H. 75 i
1-1.990
1 '->. 8
2.808
7.009
1 •+ . '') 3 6
15. 785
1 7.2o3
lo. 12*
10.077
1 6 . 244
1 D. 808
2 0 . J 0 1
1 8 . 962
20.022
19.251
21.225
21. 199
?2. 264
20.838
20 .423
22.320
18. 172
20.279
18. o 19
16. 749
1 O. ^1*3
13.665
12.216
7. 19 3
et.602
10.623
10.962
4.233
1.251
13.233
1.092
-4. 045
(DtCI^ALS)
0.899
0.660
0 .06!
0.664
0 . 83D
0 ,6dl
0 . 765
0 .86 7
0.670
0. 7H9
0 . dhl
0 .671
0. 771
0 . 7^4
0. 741
O.^bl
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
2?. 816
20.266
21.372
21.810
17.669
20.783
16.330
18. 101
1 7.673
b.635
14.964
10. 727
6. 783
10 .444
6.450
4.929
0.795
16.696
1.279
-4.841
m
0 .800
0.688
0 .544
0.609
0.546
0.7b9
0.852
0.924
0 .647
0.739
0.740
0 .666
0 .806
0.826
0 . 765
0 . 8bO
0.811
0.827
9.306
10.673
2. 1 76
4.008
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
-0.781
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
-263-
-------
0.741
0.8]8
0 .876
0 .855
0.818
0.855
0.8?9
0.870
0.873
0.800
0.751
0.846
0.855
0.794
O.H73
0.790
0.917
0.81?
O.H54
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
7048.488
7015.250
5559.433
79?4.957
6625.54?
7?83.171
3086.595
5713.58?
46^6. 1 ?8
8765.703
73^2.76]
357^.029
71^7.3^3
7006. 195
5692.394
7688. 05«
2635.140
64^-3.4^0
4987.58?
6296.011
2783.813
1848.651
491 3.996
4285.332
3541 .468
0 . 70 7
0 . '-i 1 6
0.873
0.832
O.H76
0.854
0 . * i f
0 . 945
0.978
0. 86?
0.800
0.720
0.805
0. 731
0.932
0.891
1.000
0.991
0.892
0.979
0.9<3H
bOLAR WAI)
3036.242
4277.375
440 7. 632
30 17.081
5221. 113
b36b. 847
3084. 714-
3279.037
6d46.n23
3«r90.46l
MJ 38 . 246
6334.777
b 9 2 0 . 195
3^73. 046
7«74.636
3089.930
b29 "5.636
6914. ?85
7922. 9b7
8<+27- 1 0 1
^027.007
7423.613
3184. 9b7
67b5.55M
600b.98M
2620.025
2491.281
hh 79. 105
6lbj. 191
4bl3.261
5133.597
4644. 334
0. 703
0.83?
0.875
0 . 8 J4
0. 859
0. 337
0.960
0.915
0. 785
0.748
0 . 7n9
0.816
0.816
0 . 8bl
0.^78
0 .866
0.921
0 .860
0.822
1 AT I ON I
1<+05. 145
4075.889
4458. b60
4 « « 2 . 390
J3 12.2 71
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9.827
4.052
10.200
4.583
8.561
9.885
4.487
8.813
7.507
7.866
8.573
M.354
8.484
8.072
8.405
8.276
8.430
8.231
8.321
8. 180
9.249
9.481
10.075
8.277
9.117
W.828
9.653
9.257
10.713
INFLOW » (PPM) .
10.142
10.614
9.708
9.600
9.213
10.681
9.462
9.171
8.713
9.041
8.677
8 . 4b8
8.293
8.329
8.H13
8.894
8. 550
8.076
7.H91
8.608
8.341
B. 284
7. 793
8.965
9. J67
8.279
8.301
8.354
9.295
9.213
9. 178
9.904
9.678
9. 712
9.028
10.101
10.053
9.204
9.572
8.370
8.536
8.980
8.955
8.883
8.516
8.207
7.855
8.911
6.575
6.909
7.948
8.020
8.251
7.569
8.334
9.449
7.992
9.389
7-930
H.640
8.475
9.0 79
9.052
9.6H4
10.674
10.158
10.191
10.399
9.401
9.531
8.856
8.954
a. 322
8.599
8.679
8.675
7.617
8.377
7.547
9.099
7-173
7.436
7-95a
8.107
8.640
8.091
8.202
8.303
9.003
9.989
8.480
8.827
8.694
b. 778
9.829
9.883
11.496
10.282
9.553
10.281
10.108
9.435
9.074
8.510
8.869
9.268
8.836
8.752
8.829
8.309
7.966
7.938
7.483
7.971
8.092
7.994
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
9.894
9.590
11.619
9.407
6.832
8.785
8.928
9.879
9.020
8.795
8.692
8.439
8.559
8.956
7.505
7.621
8.185
7.570
8.430
8.047
8.372
9.293
9.270
9.869
9.453
7.928
8.643
8.525
8.226
9.555
8.768
11.649
9.956
9.780
10.234
10.141
8.950
8.682
8.584
9.698
8.908
9.050
8.686
8.457
8.557
9.078
8.817
7.575
8.176
8.590
7.920
7.928
7.691
9.596
8.775
9.900
9.782
8.466
8.494
8.641
8.009
9.851
8.758
-267-
-------
H.9R1
9.113
8.696
8.677
9.SOO
9.380
10 .993
9.020
8. 785
4.749
9.970
9.422
10.501
9.415
8.604
9.390
9.524
9. 782
9.496
9.817
8.612
9.780
8.162
9.692
10.089
10.524
8.929
tt.928
8.122
9.302
11.549
9.350
8.232
9.724
8.472
9.155
11.708
9.342
8.322
9.315
8.779
9.497
11.326
9.288
8.086
9.472
8.798
9.643
10.870
* HOD IN INF LOuJ. ASSUMED (PPM)
P.O «.0
1
rt . 0 0.0
0.05 0.00 10
o.O 8.0
9 9 J06 306 58
1
5.24 15.24
1.0
283279.9 164968(1. 42491
99.
1.0 396.
24 396.24 1.0
7243872. 10643231. 14487744. 21286463. 30027672.
4046R560 .
*
1
LFNGTHS. (M) .
770.278 10Mb3
.07 160 77
.3^
23480.33 2821
1.80 34552
.62 41038
.27 43259
.17
45737.56
*
WIND SPFEhb»
>-.193
1 .6] 8
2.651
^.603
9.85?
2.6HO
3.187
0.788
2. 2*1
0.533
2.899
1.586
0.979
0 . ^ 4 •*
1.058
1.041
?.3«6
1.359
1.464
1.257
0.623
1.345
?.250
0.6Q1
0.741
0.7?9
("I/DAY) .
2.405
0.992
3.775
1 . 949
3. 32o
3.288
2.649
1.355
1.862
0.152
0.721
1.550
0 . 887
0.251
0.802
0.639
1.435
1.506
O.M91
0.849
1.211
1.215
0.842
0 . 6<+6
0.592
(i .229
1.421
1.035
6.938
6.130
5.095
4.248
1.969
1.731
1.942
4.120
1. 734
3.277
1.967
1.081
0.534
0.410
0.929
0.909
1. Ibl
1.628
0. 824
1.028
1.151
0.819
0.328
0.121
3.429
2.853
4.893
5.487
6.488
5.726
0.756
0.430
1.666
2.054
2.097
4.091
1.293
1.867
0.586
1.129
1.309
1.157
1.291
0.885
1.176
0.717
0.870
1.134
0 .456
1.180
6.901
1.644
1.839
3.184
3.390
5.434
0.322
0.721
1.205
0.259
0.844
2.774
0.875
1.564
1.413
1.561
1.676
0.727
0.909
2.238
0.724
1.313
0.654
1.564
0.346
1.802
7.768
0.889
2.643
8.166
4.515
3.313
2.455
0.972
4.386
2.256
1.101
1.630
2.457
0.382
1.089
0.163
1.735
2.642
1.662
0.843
0.462
1.690
0.497
1.488
2.400
2.178
5.466
0.642
3.086
3.361
3.134 10.144
6.914
2.674
2.674
4.126
2.521
2.721
0.454
0.217
0.903
2.107
1.029
1.268
2.760
1.550
1.517
1.027
0.777
1.483
0.733
0.390
1.654
0.745
2.115
6.889
8.374
1.958
2.821
4.524
2.672
1.541
0.202
1.006
1.478
1.094
0.721
2.133
1.329
2.137
0.482
0.789
1.681
2.076
0.714
0.924
1.705
1.J02
-268-
-------
1
1
2
3
1
1
1
1
10
0
3
0
0
.6?6
.315
.737
.889
.394
.780
.655
.729
.203
.33?
.984
.590
.59?
* ATMOSPHFH1
5359
47^1
60^9
i+844
55*3
5066
6647
7<*76
6121
67«4
7??9
7*79
6979
7456
7371
8397
79a8
83H5
711 1
74P4
H<+5*
80^1
7172
761?
6904
749?
6446
60P9
56H3
6? 12
64?H
4690
6455
5 ?HQ
534?
61??
5075
5729
4974
.863
.516
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.61 3
.699
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.836
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. 160
.809
. 8fl3
.7*9
.OH?
.379
.031
. 141
.5?7
.?34
. ?«5
.746
.773
.590
.234
.547
.56?
.^59
.547
. 3?8
.039
.40?
.273
.746
.687
.590
.9??
.4??
.309
.340
.980
O.?41
1 . 37rf
1.000
2.956
1.261
2.100
1 .234
H.414
12.457
0.569
1. 179
0 .966
0.745
1C KA01AT
56HH. 148
5168.273
5946. 840
5268.578
b222.957
62<+* . 648
72U4. 191
7381.379
651 7.695
7338.680
7303.203
7024. 7b8
72t»3.109
7296. 3b7
7024. 102
W2 36. 1 29
7782.410
7^33.770
7430 .930
7312.430
791 1.74?
811 1.734
7778.11 3
7500. 57H
7334.477
7^26.094
f 2?3. 926
62b<+.922
5787.109
5745. 5bl
5739.641
560 1.930
6696.348
5732.281
5226.633
6620 .082
^»2b.215
6006.320
5551.066
0.
2.
1.
1.
1.
2.
1.
1.
6.
1.
0.
6.
I O'M, (ft
6561.
5329.
5860.
4951.
6089.
6990.
7<+69.
7621.
7202.
7170.
7D06.
6530.
7*31.
7184.
7124.
8109.
7603.
8112.
7484.
8139.
7910.
8672.
7246.
7621.
7337.
7384.
7535.
6<+92.
6 0 8 a .
5771.
5ti42.
6322.
6213.
5407.
5103.
7186.
4758.
4583.
439
961
324
661
455
028
435
010
320
388
346
726
0.
1.
2.
0.
2.
0.
0.
0.
0.
3.
0.
6.
635
881
234
962
289
763
875
303
787
839
423
212
4.
0.
1.
0.
1.
0.
0.
0.
4.
3.
0.
3.
610
439
934
555
284
500
703
415
825
702
320
251
1.279
0.650
2.138
0.956
2.372
0.471
0.960
0.474
2.617
4.136
1.007
0.388
3.110
0.328
1.951
1.208
5.438
1.273
0.631
0.381
1.822
0.841
3.952
1.767
1.121
4.608
0.515
0.381
7.725
0.713
1.459
2.876
0.319
1.305
1.731
7.885
CAL/M-M-UAY) .
992
066
254
293
039
770
715
145
363
375
473
402
371
180
500
238
152
770
805
949
773
945
906
980
852
020
242
477
355
69b
367
152
141
797
184
b74
594
145
6726.
5536.
5818.
4768.
5304.
7127.
7473.
7414.
7320.
6572.
7809.
5955.
8076.
7413.
7408.
8148.
8224.
8157.
7619.
7657.
8286.
8511.
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-
-------
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-
-------
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-
-------
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-
-------
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-
-------
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
-------
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-
-------
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)
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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)
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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
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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)
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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 )
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