United States EPA-600/7-82-0376
Environmental Protection
Agency May 1982
r/EPA Research and
Development
VERIFICATION AND TRANSFER OF
THERMAL POLLUTDN MODEL
Volume V. Verification of
One-dimensional Numerical Model
Prepared for
Office of Water and Waste Management
EPA Regions 1-10
Prepared by
Industrial Environmental Research
Laboratory
Research Triangle Park NC 27711
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RESEARCH REPORTING SERIES
Research reports of the Office of Research and Development, U.S. Environmental
Protection Agency, have been grouped into nine series. These nine broad cate-
gories were established to facilitate further development and application of en-
vironmental technology. Elimination of traditional grouping was consciously
planned to foster technology transfer and a maximum interface in related fields.
The nine series are:
1. Environmental Health Effects Research
2. Environmental Protection Technology
3. Ecological Research
4. Environmental Monitoring
5. Socioeconomic Environmental Studies
6. Scientific and Technical Assessment Reports (STAR)
7. Interagency Energy-Environment Research and Development
8. "Special" Reports
9. Miscellaneous Reports
This report has been assigned to the INTERAGENCY ENERGY-ENVIRONMENT
RESEARCH AND DEVELOPMENT series. Reports in this series result from the
effort funded under the 17-agency Federal Energy/Environment Research and
Development Program. These studies relate to EPA's mission to protect the public
health and welfare from adverse effects of pollutants associated with energy sys-
tems. The goal of the Program is to assure the rapid development of domestic
energy supplies in an environmentally-compatible manner by providing the nec-
essary environmental data and control technology. Investigations include analy-
ses of the transport of energy-related pollutants and their health and ecological
effects; assessments of, and development of, control technologies for energy
systems; and integrated assessments of a wide range of energy-related environ-
mental issues.
EPA REVIEW NOTICE
This report has been reviewed by the participating Federal Agencies, and approved
for publication. Approval does not signify that the contents necessarily reflect
the views and policies of the Government, nor does mention of trade names or
commercial products constitute endorsement or recommendation for use.
This document is available to the public through the National Technical Informa-
tion Service, Springfield. Virginia 22161.
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VERIFICATION AND TRANSFER
OF THERMAL POLLUTION MODEL
VOLUME V: VERIFICATION OF ONE-DIMENSIONAL
NUMERICAL MODEL AT LAKE KEOWEE
By
Samuel S. Lee, Subrata Sengupta
and Emmanuel V. Nwadike
Department of Mechanical Engineering
University of Miami
Coral Gables, Florida 33124
NASA Contract No. NAS 10-9410
NASA Project Manager: Roy A. Bland
National Aeronautics and Space Administration
Kennedy Space Center
Kennedy Space Center, Florida 32899
EPA Interagency Agreement No. 78-DX-0166
Program Element No. EHE 624A
EPA Project Officer: Theodore G. Brna
Industrial Environmental Research Laboratory
Office of Environmental Engineering and Technology
Research Triangle Park, North Carolina 27711
Prepared for:
U. S. Environmental Protection Agency
Office of Research and Development
Washington, D. C. 20460
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PREFACE
This final report is the summary of a two-year effort portraying
the development, calibration and verification of a one-dimensional variable
cross-sectional area numerical model. This is one of the publications
of the thermal pollution group at the University of Miami.
This effort covers the applications of this model to Cayuga Lake
in New York and Lake Keowee in South Carolina. The conclusions and
derivations are accompanied by an abundance of figures and tables.
These figures and tables are presented in such a way as to make it
possible for use in calibrating or verifying other one-dimensional models.
This work was conducted under funding from National Aeronautics
and Space Administration (NASA), Kenedy Space Center and the Environ-
mental Protection Agency (EPA).
u
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ABSTRACT
A one-dimensional model for studying the thermal dynamics of cool-
ing lakes has been developed and verified. The model is essentially a
set of partial differential equations which are solved by finite difference
methods. The model includes the effects of variation of cross-sectional
area with depth, surface heating due to solar radiation absorbed at the
upper layer and internal heating due to the transmission of solar radia-
tion to the sub-surface layers. The exchange of mechanical energy be-
tween the lake and the atmosphere is included through the coupling of
thermal diffusivity and wind speed. The effects of discharge and intake
by power plants are also included.
The numerical model was calibrated by applying it to Cayuga Lake.
The performance was very good. The model was then verified through
a long term simulation using Lake Keowee data base. The comparison
between measured and predicted vertical temperature profiles for the
nine years is good. The physical limnology of Lake Keowee is presented
through a set of graphical representations of the measured data base.
MI
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CONTENTS
Foreword
Preface ii
Abstract Hi
Figures iv
Tables xi
Symbols xii
Acknowledgments xiii
1. I ntroduction 1
2. Conclusions 3
3. Recommendations 4
4. The Mathematical Model 5
Description 5
Boundary conditions 8
Numerical method 9
5. Model Calibration 12
Cayuga Lake application 12
Input quantities 12
Results 15
6. Model Verification 24
Lake Keowee application 24
I nput quantities 24
Results 25
References 57
Appendices 59
A. Derivation of the Mathematical Model 60
B. Measured Temperature Profiles Plots (1971-1978),
Stations 500 through 506 68
C. Averaged Temperature Profiles, Stations 500 through 506 . 125
iv
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FIGURES
Number
1 Vertical temperature profiles, Cayuga Lake 17
2 Vertical temperature profiles, Cayuga Lake 18
3 Vertical temperature profiles, Cayuga Lake 19
4 Stratification cycle cylindrical domain, Cayuga Lake 20
5 Stratification cycle paraboloid domain, Cayuga Lake 21
6 Variation of eddy diffusivity with depth cylindrical
domain, Cayuga Lake 22
7 Variation of eddy diffusivity with depth paraboloid
domain, Cayuga Lake 23
8 Map of Lake Keowee 3l*
9 Temperature profiles, Lake Keowee, 1971 35
10 Temperature profiles, Lake Keowee, 1972 36
11 Temperature profiles. Lake Keowee, 1973 37
12 Temperature profiles, Lake Keowee, 1974 38
13 Temperature profiles, Lake Keowee, 1975 39
14 Temperature profiles, Lake Keowee, 1976 40
15 Temperature profiles, Lake Keowee, 1977 41
16 Temperature profiles. Lake Keowee, 1978 42
17 Temperature profiles, Lake Keowee, 1979 43
18 Lake Keowee discharge vs no discharge, 1971 44
19 Lake Keowee discharge vs no discharge, 1972 45
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FIGURES
Number
20 Lake Keowee discharge vs no discharge, 1973 ............ 16
21 Lake Keowee discharge vs no discharge, 1 971 ............ 17
22 Lake Keowee discharge vs no discharge, 1 975 ............ 18
23 Lake Keowee discharge vs no discharge, 1976 ............ 19
21 Lake Keowee discharge vs no discharge, 1 977 ............ 50
25 Lake Keowee discharge vs no discharge, 1978 ............ 51
26 Lake Keowee discharge vs no discharge, 1 979 ............ 52
27 Variation of eddy diffusivity with depth ................. 53
28 Monthly variation of the depth of the thermocline ........ 51
29 Temporal variation of the surface exchange coefficient .... 55
30 Stratification cycle, Lake Keowee . ....................... 56
t
B-1 Lake Keowee measured temperature profiles, 1971,
Station 500 .............................................. 69
B-2 Lake Keowee measured temperature profiles, 1972,
Station 500 .............................................. 70
B-3 Lake Keowee measured temperature profiles, 1973,
Station 500 .............................................. 71
B-1 Lake Keowee measured temperature profiles, 1971,
Station 500 .............................................. 72
B-5 Lake Keowee measured temperature profiles, 1975,
Station 500 .............................................. 73
B-6 Lake Keowee measured temperature profiles, 1976,
Station 500 .......... . ................................... 7I*
B-7 Lake Keowee measured temperature profiles, 1977,
Station 500 .............................................. 75
B-8 Lake Keowee measured temperature profiles, 1978,
Station 500 .............................................. 76
vi
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FIGURES
Number Pa9e
B-9 Lake Keowee measured temperature profiles, 1971,
Station 501 77
B-10 Lake Keowee measured temperature profiles, 1972,
Station 501 78
B-11 Lake Keowee measured temperature profiles, 1973,
Station 501 79
B-12 Lake Keowee measured temperature profiles, 1974,
Station 501 ............................................... 80
B-13 Lake Keowee measured temperature profiles, 1975,
Station 501 ............................................... 81
B-1U Lake Keowee measured temperature profiles, 1976,
Station 501 ............................................... 82
B-15 Lake Keowee measured temperature profiles, 1977,
Station 501 ............................................... 83
B-16 Lake Keowee measured temperature profiles, 1978,
Station 501 ............................................... 84
B-17 Lake Keowee measured temperature profiles, 1971,
Station 502 ............................................... 85
B-18 Lake Keowee measured temperature profiles, 1972,
Station 502 ............................................... 86
B-19 Lake Keowee measured temperature profiles, 1973,
Station 502 ............................................... 87
B-20 Lake Keowee measured temperature profiles, 1974,
Station 502 ............................................... 88
B-21 Lake Keowee measured temperature profiles, 1975,
Station 502 ............................................... 89
B-22 Lake Keowee measured temperature profiles, 1976,
Station 502 ............................................... 90
B-23 Lake Keowee measured temperature profiles, 1977,
Station 502 ............................................... 91
B-24 Lake Keowee measured temperature profiles, 1978,
Station 502 ............................................... 92
vii
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FIGURES
Number Page
B-25 Lake Keowee measured temperature profiles, 1971,
Station 503 93
B-26 Lake Keowee measured temperature profiles, 1972,
Station 503 94
B-27 Lake Keowee measured temperature profiles, 1973,
Station 503 95
B-28 Lake Keowee measured temperature profiles, 1974,
Station 503 96
8-29 Lake Keowee measured temperature profiles, 1975,
Station 503 97
B-30 Lake Keowee measured temperature profiles, 1976,
Station 503 98
B-31 Lake Keowee measured temperature profiles, 1977,
Station 503 99
B-32 Lake Keowee measured temperature profiles, 1978,
Station 503 to°
B-33 Lake Keowee measured temperature profiles, 1971,
Station 504 101
B-34 Lake Keowee measured temperature profiles, 1972,
Station 504 102
B-35 Lake Keowee measured temperature profiles, 1973,
Station 504 1Q3
B-36 Lake Keowee measured temperature profiles, 1974,
Station 504 104
B-37 Lake Keowee measured temperature profiles, 1975,
Station 504 105
B-38 Lake Keowee measured temperature profiles, 1976,
Station 504 106
B-39 Lake Keowee measured temperature profiles, 1977,
Station 504 1°7
B-40 Lake Keowee measured temperature profiles, 1978,
Station 504 108
viii
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FIGURES
Number
B-U1 Lake Keowee measured temperature profiles, 1971,
Station 505 109
B-42 Lake Keowee measured temperature profiles, 1972,
Station 505 .............................................. 11°
B-43 Lake Keowee measured temperature profiles, 1973,
Station 505 .............................................. in
B-4U Lake Keowee measured temperature profiles, 1974,
Station 505 .............................................. 112
B-45 Lake Keowee measured temperature profiles, 1975,
Station 505 .............................................. 113
B-46 Lake Keowee measured temperature profiles, 1976,
Station 505 .............................................. 11H
B-47 Lake Keowee measured temperature profiles, 1977,
Station 505 .............................................. 115
B-48 Lake Keowee measured temperature profiles, 1978,
Station 505 .............................................. 116
B-49 Lake Keowee measured temperature profiles, 1971,
Station 506 .............................................. 117
B-50 Lake Keowee measured temperature profiles, 1972,
Station 506 .............................................. 118
B-51 Lake Keowee measured temperature profiles, 1973,
Station 506 .............................................. 119
B-52 Lake Keowee measured temperature profiles, 1974,
Station 506 .............................................. 1 20
B-53 Lake Keowee measured temperature profiles, 1975,
Station 506 .............................................. 121
B-54 Lake Keowee measured temperature profiles, 1976,
Station 506 .............................................. 122
B-55 Lake Keowee measured temperature profiles, 1977,
Station 506 .............................................. 123
B-56 Lake Keowee measured temperature profiles, 1978,
Station 506 .............................................. m
rx
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FIGURES
Number
C-1 Lake Keowee averaged measured temperature profiles.
Stations 501-506, 1971 126
C-2 Lake Keowee averaged measured temperature profiles,
Stations 501-506, 1972 127
C-3 Lake Keowee averaged measured temperature profiles,
Stations 501-506, 1973 128
C-4 Lake Keowee averaged measured temperature profiles,
Stations 501-506, 1974 129
C-5 Lake Keowee averaged measured temperature profiles,
Stations 501-506, 1975 130
C-6 Lake Keowee averaged measured temperature profiles,
Stations 501-506, 1976 131
C-7 Lake Keowee averaged measured temperature profiles,
Stations 501-506, 1977 132
C-8 Lake Keowee averaged measured temperature profiles,
Stations 501-506, 1978 133
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TABLES
Number
1 Oconee Nuclear Station Condenser Cooling Water
Flowrate (m3 /min) 27
2 Oconee Nuclear Station Condenser Temperature Rise,
AT(°C) 28
3 Monthly Average Fiowrates (m3 /sec) - Lake Keowee
Hydro Station 29
H Lake Jocassee Hydro Flows (cfs) 30
5 Lake Keowee, Wind Speed (cm /sec) 31
6 Lake Keowee Gross Solar Radiation (Langleys) 32
7 Lake Keowee Dewpoint Temperature (°C) 33
XI
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SYMBOLS
h
A(z)
Kz)
Q(z)
P
V
KzZ
Kz
zo
W*=
a
Vertical coordinate measured
upward from the deepest
point of the lake. As a sub-
script it marks the vertical
component of a vector.
Depth of the lake
Horizontal cross-sectional
area at height Z
Bottom-surface source of
mass per unit area
Bottom-surface source of
heat per unit area
Temperature
Density of water
Vertical velocity
Eddy diffusivity
Eddy diffusivity under
neutral condition
(T , ) Friction velocity
empirical constant
Richardson number
Surface temperature
Bottom surface heat flux
ct
Hfr)
B
n
6
A
I3
IA
dz
Volumetric coefficient of
expansion of water
Surface shear stress
Heat capacity
Heat source/unit volume
Average value of W*
Half the annual variation of W*
C_, C3, Cr C5 Phase angles
Solar radiation incident on
the water surface
Average value of <}>
Half the annual variation of $
Extinction coefficient
Absorption coefficient
Volumetric discharge
Condenser temperature change
Discharge temperature
Surface heat flux
Surface heat exchange coefficient
Equilibrium temperature
Average value of TE
Half the annual variation of TE
Lake surface radius
Area variation with depth
xii
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ACKNOWLEDGMENTS
This work was supported by a contract from the National Aero-
nautics and Space Administration (NASA-KSC) and the Environmental
Protection Agency (EPA-RTP).
The authors express their sincere gratitude for the technical and
managerial support of Mr. Roy A. Bland, the NASA-KSC project manager
of the contract, and the NASA-KSC remote sensing group. Special
thanks are also due to Dr. Theodore G. Brna, the EPA-RTP project
manager, for his guidance and support of the experiments, and to Mr.
S. B. Hager, Chief Engineer, Civil-Environmental Division, and Mr.
William J. McCabe, Assistant Design Engineer, both from the Duke Power
Company, Charlotte, North Carolina, and their data collection group for
data acquisition. The support of Mr. Charles H. Kaplan of EPA was
extremely helpful in the planning and reviewing of this project.
xiii
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SECTION 1
INTRODUCTION
Deep bodies of water provide a convenient source of condenser
cooling water supply to electric generating power plants. One of the
problems associated with this, that has gained increasing importance in
recent years is the thermal pollution caused by the discharge of waste heat
from these power plants. The degradation of the quality of these waters
usually occurs through the direct influence of the increased temperature
on aquatic life or through the lowering of the amount of dissolved oxygen
and other biochemical effects.
In temperate regions most deep bodies of water develop a thermo-
cline during their annual heating cycle. A warmer epilimnion at the top
is isolated from a cooler hypolimnion below by severe stable gradients
(thermocUne). Convective transport and heat addition caused by power
plant discharge result in disturbances in the thermocline.
The formation time, phasing, depth and severity of the thermocline
are crucial factors affecting the bio-chemical processes in an aquatic
ecosystem. The nutrient levels, species spectra and physical character-
istics are quite different in the two distinct domains below and above the
thermocline. This stratification is believed to be caused by nonlinear
interaction between the wind-generated turbulence and stable buoyancy
gradients. While being heated from above, a basin forms stable strati-
fication thereby inhibiting wind-generated turbulence. During early
spring, most temperate lakes exhibit a nearly homothermal temperature
distribution with a temperature of about 4°C (which is the temperature
of maximum density for water) extending all the way to the bottom. As
the air above the lake begins to warm, the lake receives heat, at an
increasingly rapid rate. During the early part of the warming season,
the lake continues to remain nearly homothermal, since the heat that is
received at the surface layers is transported to the deeper layers by
wind-induced currents and turbulence. As the rate of heating of the
lake continues to increase, the rate at which heat is received at the
surface layers soon exceeds the rate of heat removal to the deeper lay-
ers, and the temperature of the surface layers begins to increase.
During this period the temperature decreases monotonically with increas-
ing depth. As heating continues, a point of inflexion develops in the
temperature profile separating an upper well-mixed layer from a lower
well-mixed layer. The region around the point of inflexion is a region
of intense stable temperature gradient and consequently, low turbulence
levels. Turbulent diffusion through this region is minimal. Further
-------
heating merely increases the temperature of the well-mixed upper layer.
As cooling begins, the wind mixing is augmented by convective
mixing caused by static instability and the thermocline (the region con-
taining the point of inflexion) recedes into the deeper layers of the lake.
The epilimnion cools until the lake reaches its minimum heat content and
near homothermal conditions result.
This report describes the application of a one-dimensional model in
predicting the above phenomena. This model was calibrated using data
from Cayuga Lake and the calibrated model was applied to Lake Keowee.
The purpose of the effort is to develop a predictive ability for long-term
thermal behavior of cooling lakes.
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SECTION 2
CONCLUSIONS
A one-dimensional model which includes area change with depth,
vertical convection, varying diffusivity, thermal discharges, and internal
absorption of radiation has been developed. The model was calibrated
using Cayuga Lake data base and verified with Lake Keowee data base.
The application to Cayuga Lake indicates excellent performance.
A comparison of cylindrical and paraboloid approximations for the lake
indicate significant differences in thermocline depth, eddy-diffusivity,
and temperatures at mid-depths. This shows that effects of area change
with depth are not negligible. The insignificant surface temperature
difference is attributable to the fact that equal surface areas were used
in both simulations. However, these effects will be more pronounced
in real basins where decrease in area with depth is more severe than
the linear variation for the paraboloid case.
The long term (nine-year) simulations of temperature profiles,
formation and decay of the thermocline for Lake Keowee compare fairly
well with measured data. For these simulations, the model did not need
recalibration. While the influence of discharge and ambient conditions
were satisfactorily modeled, the calculated and measured profiles had
much larger differences for Lake Keowee than for the one-year simulation
in Cayuga Lake. Two reasons are believed to cause this relatively great-
er inaccuracy. The first is the complicated discharge patterns in Lake
Keowee caused by power plant, hydro and pumped storage operation
compared to ambient simulation for Cayuga Lake. The second cause is
that for Cayuga Lake; the assumption of horizontal mixing is more mean-
ingful since it is a single basin domain, while Lake Keowee is a double
basin domain connected by a canal. The model, however, can be adapted
for application to two connected basins.
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SECTION 3
RECOMMENDATIONS
The main disadvantage of a one-dimensional thermal model lies in
the fact that resolution is sacrificed for computational speed. Three di-
mensional models are bulky and time consuming but have much better
resolution, however, when long term simulations are necessary, a one-
dimensional model is recommended.
The model described here can be modified to include the single
effects of the various quantities involved in the surface heat transfer
phenomenon rather than using the equilibrium temperature concept.
This is particularly recommended for the user who is interested in
modeling the long term effects of one (for example, evaporation) of
the quantities involved in the surface heat transfer processes.
Furthermore, the model can be easily adapted to handle connected
multiple domains. This recommendation is discussed in the text.
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SECTION 4
THE MATHEMATICAL MODEL
DESCRIPTION
The one-dimensional variable cross-sectional area model used for
this study is described in this section. The detailed derivation of the
governing equations is described in Appendix A, while the boundary
conditions and the important terms occurring in the governing equations
are discussed here.
This model which assumes lateral uniformity was developed to pre-
dict the vertical temperature profiles for complete annual cycles in a lake.
The model includes the effects of variation of the horizontal cross-section-
al area with depth. Surface heating due to solar radiation absorbed at
the surface layer and the internal heating due to the transmission of the
unabsorbed solar radiation to the deeper layers of the lake are taken
into account. The exchange of mechanical energy between the lake and
the atmosphere is accounted for through the friction velocity and the
thermal diffusivity, since these quantities can be related to wind speed.
Finally, the effects of power plant discharge and intake are considered.
The full three-dimensional equations of mass and heat balance are:
= -7"-p"v (4.1)
O L
and
IT (PC T) = V- PC K- 7T - V- PC TV + H (4. 2)
dip p p
Applying the divergence theorem, and then differentiating and integrat-
ing, a set of one-dimensional equations are obtained:
Aujf^f^zjpv^iA- <«-a
and
A(z) ^ (pCpT) = j[pCpA(z)Kz ] - -^pCfiA(z)TVz) +QA' + A(z)H (z)
(4.4)
where,
z is the vertical coordinate, measured upward from the deepest point of
-------
of the lake. As a subscript it marks the vertical component of a
vector.
A(z) is the horizontal cross-section of the lake at height z.
p is the density.
t is the time.
V is the vertical velocity of flow.
C is the heat capacity.
P
T is the temperature .
K is the vertical component of 'K', the heat diffusivity tensor (including
turbulent diffusivity) .
H(z) is the source of heat per unit volume.
A
A 1S 35'
I is the bottom- surf ace source of mass per unit area.
Q is the bottom-surface source of heat per unit area.
The following variables occur in the governing equations and are
defined thus:
Density, p, is assumed to vary with temperature in the following form:
p = 1.02943 + (-2.0 x 10~5)T + (-4.8 x 10~6)T2 gm Ice (1.5)
Equation (4.5) and the discussion of the constants were given by Sengupta
and Lick (1974).
Eddy diffusivity, K , is a function of both thermal and current structure
of a lake. The form used here was deduced by Rossby and Montgomery
(1935).
Kz = KZQ(1 + a,!*.}' (4.6)
where,
K is the eddy diffusivity under neutral conditions.
Sundaram et al. (1969) gave an empirical form for the yearly variation
of Kzo'
K = 0.21 + 0.052 sindjpt + 2.61) cm1 /sec2 (4.7)
ZO
-------
A sinusoidal variation of K with a mean 0.21 cm1 /sec and amplitude of
0.052 cm27sec is assumed in°Equation (4.7).
The semi-empirical constant a- = 0. 1 is estimated in this study by
comparing the values used by Sundaram et al. (1969) and the original
value used by Monin and Obukhov (1954).
The third variable appearing in Equation (1.6) is R., the Richardson
number. This is defined as:
R =J (4.8)
Ki W*2 8z V '
where,
a is the volumetric coefficient of expansion of water.
av
is defined as:
av = 0.0 + 1.538 x 10 5{T-4) + (-2.037 x 10 7)(T-4)2 (4.9)
The constants in the above equation were taken from Sundaram et al.
(1970). a can also be estimated from Equation (4.5); however, the
use of Equation (4.9) leads to a higher accuracy.
g is the acceleration due to gravity.
w* is the friction velocity calculated from known conditions above the
lake.
The heat source, H, is that part of the solar radiation transmitted
exponentially through the depth of the lake. The relation used in this
study has been used by Dake and Harleman (1969) and by Mitry and
Ozisik (1976) :
H(z) = n(1-{3)A(z) * EXP(-n(z-h)) (4.10)
o
where,
g = fraction of the solar radiation absorbed at the surface, (0.5).
n = extinction coefficient, (0.75 cm ).
h = depth of the lake, (m).
A = net solar radiation reaching the water surface, (cal/cm2/day).
Equations (4.3) and (4.4) are parabolic and mathematically repre-
sent a diffusion process. The solution of these equations requires one
initial condition and two boundary conditions.
-------
Initial Conditions
The temperature of the lake at spring homothermy is taken as the
initial temperature.
Boundary Conditions
Surface:
q = pC K |^ | . = K (T_ - T ) (4.11)
Ms p z 3z | z=h s E s
where,
TE = equilibrium temperature, (°C).
T = surface temperature, (°C).
K = surface heat exchange coefficient, (cal/cm2-s)
TE Evaluation
Brady et al. (1969) showed empirically that fluctuations in the
equilibrium temperature may be conveniently estimated using the approxi-
mate relationship:
= Td + Hs/Ks (4.12)
where,
T . is the dewpoint temperature, estimated from known conditions above
the lake.
H is the gross rate of short wave solar radiation.
Since the dewpoint temperature tends to remain relatively constant through
a single day, Equation (4.12) indicates that the main source of hourly
fluctuations in T_ is the solar radiation component. This generally
reaches a maximum at solar noon, unless variable cloudiness interferes.
At nighttime, T- approaches the dewpoint temperature, which acts like
a relatively invariant datum for periods of 24 hours or less. On an
annual basis, however, both T . and H are generally much greater in
summer than in winter. The dominant Contribution to the amplitude of
seasonal fluctuation in Tp is the dewpoint temperature.
K Evaluation
The form used by Edinger and Geyer (1967) was also used in this
study:
K = 4. 5 + 0.05 T + Bf(w) + 0. 47f (w) (4.13)
s s
-------
3 is found by applying standard curve-fitting techniques to pub-
lished data pertaining to saturated vapor pressures at various tempera-
tures; a convenient representation given by Edinger and Geyer (1967)
is:
3 = 0.35 + 0.0157 + 0.0012T 2 (mmHg/°C) (4.U)
m m
T + T
T = -S 9. (IMS)
m 2
The evaporative wind speed function f(w) used is also similar to that of
Edinger and Geyer (1967).
f(w) = 9. 2 + 0.46W2 (W ~2mmH -1) (4.16)
m g
where W is the wind speed (m/s)
Bottom:
The second boundary condition is at the bottom surface of the lake
which is assumed to be perfectly insulated.
Numerical Method
Numerical integration of the governing equations require that these
equations be replaced by finite difference equations.
In this study, the following notations are used in the finite differ-
encing of the basic equations:
Az The thickness of a slice (the basin is divided into equal number
of slices). This is used as depth increment.
At The time increment chosen at the n integration step.
n, n-1 The present and last time at which computations were performed.
j The space subscript. This is used as
T." = T(jAz, tn) = T(z, t)
where z is the vertical coordinate.
Equation (H.H) can now be written in difference form as:
T ^ T ^~^
_] 1 _ e ' > ^~ I ' r / A ix . -i- » n~ 1
-------
I
±
r AW \n~
" (pAVz}|-i
where,
j.
I P J
JP
H
"'1
(pAKAT)"",1 = pAK
z j z
t.HT
n I j
"'1
- T"~11
j i
K = K [1 + a,(-
z.+- zo 1 w
(a ).z. -
v j j 3z
(av). =
A -
A
Terms like AT.., and AT. , can be defined as
J+l J-i
10
-------
AT. . = TV - T.
AT. , = T. - T. ,
J-i J J-1
The sequence in which calculations are performed is as follows:
The dependent variables, T, K , w*, a , p, T_, and K , are
initialized. The area of each slice is calculated. The title of5the pre-
sent year is printed. The values of the variables, K , w*, a , p, T£,
and K , are then calculated. The temperatures of thl slices are finally
calculated. If the temperature profile is unstable, mixing of the unstable
portion of the profile is undertaken.
During the next time step, the temperatures are updated, and the
dependent variables are calculated again.
The values of the temperature, T, eddy diffusivity, K , number of
days and the surface heat transfer coefficient, K , are printed every
time step, every day or normally at the end of each month. At the end
of the present year, the new-year title is printed and computations con-
tinue as listed above. A detailed description of the programs and flow
chart has been published in a user's manual. Lee and Sengupta (1980).
11
-------
SECTION 5
MODEL CALIBRATION
The model described in the previous section was applied to Cayuga
Lake, New York. The main reason was to calibrate the model. Cayuga
Lake was chosen for this calibration because of the vast amount of nu-
merical work done on this lake. These include Sundaram and Rehm (1971),
Sundaram et al. (1971), Sundaram and Rehm (1972), and Mitry and Ozisik
(1976). The model was also verified using Lake Keowee data base; this
is described in the next chapter.
CAYUCA LAKE APPLICATION
Cayuga Lake is one of the finger lakes in west central New York.
It is the longest, about 40 miles (64 km), of the group, and the second
largest, about 66 square miles (171 sq. km), with steep banks cut by
inflowing streams. It averages 2 miles (3 km) wide about 500 ft (150
meters) deep, with a surface elevation of 381 ft (115 meters). Tang-
hannock Creek and its 210-foot (64-meter) cataracts flow into the lake.
The lake is drained to the north by the Seneca River, which joins the
Oswego River and flows into Lake Ontario. Cayuga Lake is connected
by canal with Seneca Lake on the west and Erie Canal on the north.
The city of Ithaca lies at the southern end of the lake.
INPUT QUANTITIES
In the numerical integration of the governing equations for Cayuga
Lake, the following ralations were used. The numerical values were
taken from Sundaram et al. (1971) and Mitry and Ozisik (1976).
Density, p, varied as in Equation (4.5), Section 4, of this report.
Eddy djffusivity varied as in Equations (4.6) and (4.7) with
a. =0.1 and the Richardson number as given in Equation (4.8).
The volumetric coefficient of expansion of water, a , used is shown
in Equation (4.9).
A sinusoidal relation was assumed for the variation of the friction
velocity, w*, with time; normally, w* is defined as
w* =-/(TS/P) (5.1)
12
-------
The form used is
w* = A1 + B1 sin(|g|+ C^ (5.2)
where,
A- = Average value of w*, (3.048 cm /sec).
B. = Half the annual variation of w*, (0.762 cm /sec).
C. = Phase angle, (2. 61) (chosen in such a way that at time t = 0,
w* = initial value of the friction velocity).
The heat source,,H, is taken to vary as in Equation (4.10), with
3 = 0. 5, n = 0.75 cm and h = 70.0 m ( = 200 ft).
The net solar radiation, 4> , reaching the water surface is taken
+C) (5.3)
from Mitry and Ozisik (1976): °
where,
A = Average value of A , (6.14 x 10 cal/cm2s).
B» = Half the annual variation of A , (3. 52 x 10 cal/cm2s).
C_ = Phase angle, (0.049) (chosen in the same way as C.).
The heat flux, Q, in Equation (4.4) is defined as
Q = (pC ATQ )/(A(z)) (5.4)
" P
where,
Q = Volumetric discharge from the power plant. In this study
P Q = 1.508 x 108 cm3/sec; this value is chosen to correspond
to the pumpina velocity used by Sundaram et al. (1971), (1/4
ft /day).
AT = 10°C = Assumed temperature change through the condensers
of the power plant.
The pumping velocity V is defined as
<5-5'
The pumping velocity term effects are felt between the intake and the
level at which the heated effluent becomes neutrally buoyant (effective
discharge level).
13
-------
_
K = Assumed constant at 5.65 x 10 cal/cm2-s.
_»
For a postulated 3500 MW plant for Cayuga Lake, a 8. 79 x 10
cal/cm2s of waste heat will have to be rejected.
Surface area of Cayuga Lake is 66 sq. miles (171 sq. km).
The intake to the power plant is fixed at 38.1 m (= 125 ft) from
the surface of the lake.
Initial temperature is taken as the temperature of the lake at spring
homothermy (usually coincides with the minimum surface temperature
attained by the lake). For Cayuga Lake, this occurs around March,
and the temperature assumed is T = 2. 9°C.
Finally, a sinusoidal variation is assumed for the variation of the
equilibrium temperature with time.
TEE =A3 + B3sin(t + C) (5.6)
where,
A = Average value of TE/ (11°C).
B = Half the annual variation of T£, (16°C).
C = Phase angle, chosen in the same way as C^ or C .
The justification for using Equation (5.6) is described in Appendix
D of this report.
Two topographies are studied in the application of this mode! to
Cayuga Lake; these are a right circular cylinder and a circular para-
boloid approximations for the lake.
Cylindrical Topography
The area (radius, B = 7. 38 x 10 cm).,pf the lake is assumed con-
stant throughout the depth. Term A1 or -5 = 0; see Equations (4.3)
and (4.4).
Circular Paraboloid Topography
The lakeJs assumed to be a circular paraboloid with surface radius,
B = 7. 38 x 10 cm. The area at any depth, z (measured from the deep-
est point of the lake) is given by:
A' =~ (5.7)
Thus, A1 is a constant:
14
-------
(5.8)
RESULTS
Computations for a yearly cycle for Cayuga Lake are presented.
The verification data base consists of vertical temperature profiles com-
piled by Henson et al. (1961). The comparison of simulated and observed
vertical temperature profiles are shown in Figures 1, 2 and 3. Each
figure shows five profiles representing observed (M), discharge (cylin-
drical domain - CD); discharge (parabolic domain - PD); no-discharge
(cylindrical domain - CN); and no-discharge (parabolic domain - PN).
The no-discharge simulations are in good agreement with the data
(the data was for no-discharge conditions).
The parabolic case has somewhat better agreement since it represents,
qualitatively, the decrease in area with depth. However, the closeness
of the simulated results for the two cases is surprising. Most lakes have
the rate of decrease in area with depth greater than a paraboloid, which
has a linear decrease. Thus, when realistic area changes are used, a
greater difference between cylindrical and paraboloid cases can be ex-
pected .
The effect of discharge is significant only in the top layers till July.
This is because the heated discharge rises to the surface during the first
half of the year. For the later months, the discharge temperature is
lower than the surface temperature causing the discharge to reach static
equilibrium somewhere below the surface. Thus, significant thermal
effects of discharge are seen at mid-depths till December. The tempera-
tures are higher at these depths for the paraboloid topography. In
general, a temperature difference of the order of 3°C over the no-dis-
charge case, can be seen. At the end of the annual cycle, a residual
temperature increase of 1.75°C is detectable.
In the above paragraph, the discharge from the power plant is
treated as a plane source which is injected into the domain at a level
where the discharge temperature equals the temperature of that level.
This level could be called the effective discharge level. The effects of
the pumping velocity term, therefore, can only be felt from the intake
level to the effective discharge level. A temperature rise of 10°C through
the condensers was assumed. This temperature difference justifies the
use of density as a function of temperature.
Figures 4 and 5 show the annual stratification cycle. From these
figures it is evident that the surface temperature difference between the
four cases is less than 2°C over the annual cycle. However, at mid-
depths the paraboloid discharge case shows a 5°C difference compared
to the no-discharge case; for the cylindrical discharge case this difference
is 3°C. Generally, the highest surface temperatures are reached after
150 days, while the highest equilibrium temperature occurs after 120 days.
15
-------
Thus, there is approximately a 30 day lag in the surface temperature
response. The maximum temperatures at mid-depth occur after 240 days
for the no-discharge case; for the discharge case, the corresponding time
is 210 days. No significant phase lag between cylindrical and paraboloid
cases are observed.
Figures 6 and 7 show the eddy diffusivity variation with depth
(time in days used as parameter) for the cylindrical and paraboloid
cases respectively. It is observed that thermal discharge causes in-
crease in eddy diffusivity in the epilimnion owing to increased mixing.
No significant changes are seen in the hypolimnion. There is a temporal
increase in the differences between the discharge and no-discharge cases.
These observations are true for both domains. However, the diffusivity
values are larger for the paraboloid cases. Also, at any given time,
the paraboloid case shows deeper thermoclines.
-------
M
61
OJ
Depth
(m)
61 r
Depth
(m)
PN PD
PD
(March)
J 12 16 20
Temp.°C
(May)
0 4 8 12 16 20
Temp. C
Depth
(m)
61
Depth
(m)
(April)
048 12Ib20
Temp.°C
CN
(June)
8 12o 16 20
Temp. C
(PN=Paraboloid, No-Discharge; CN=Cylindrical
No-Discharge; PD=Paraboloid"+ Discharge;
(^Cylindrical + Discharge; M=Measured)
Figure 1. Vertical temperature profiles, Cayuqa Lake
(from 0 to 90 days)
17
-------
61
Depth
(m)
61
Depth
(m)
CN
(July)
048
12 16 20
Temp..°C
PN
CN
61 r
Depth
(m)
Depth
(m)
PN
(August)
048
12 16 20
Temp. C
(October)
0 H 8 12 16 20 24 048 12 16 20
Temp.°C Temp.°C
(P₯=Paraboloid, No-Discharge; CN=Cylindrical
No-Discharge; PD=Paraboloid + Discharge;
CD=Cyl£ndrical + Discharge;. M=Measured)
Figure 2. Vertical temperature profiles, Cayuga Lake
(from 120 to 210 days)
18
-------
61.
Depth
(m).
61
Depth
(m)
CN
\
PN--
A
'PD (December)
o 4 8 12 16 20
Temp.i °C
0 1 8 12 16 20
PN ,PD
6l
Depth
(m)
c
>
k
. CN
X
HI 61
In M'
Depth
(m)
1 (January)
^
* ' ft
i-
k
V
1
'^TN
(February)
T 8 12 16 20 0 l) 8 12 16 20
Temp . °C Temp . °C
(PN=Paraboloid, No-Discharge; CN=Cylindrical
No-Discharge; PD=Paraboloid + Discharge;
CD=Cylindrical + Discharge; M=Measured)
Figure 3. Vertical temperature profiles, Cayuga Lake
(from 240 to 330 days)
19
-------
28
25
20
15
10-
,TE
TE
M
CN
CD
Equilibrium temperature
SURFACE TEMPERATURES
Measured
Cylindrical, no discharge
Cylindrical + discharge
MIDLAYER TEMPERATURES_
MM Measured
CNM No discharge
COM Discharge
-5 u
Figure 4. Stratification cycle cylindrical domain, Cayuga Lake
20
-------
,TE
28
25
20
15
10
-5
TE
M
PN
PD
Equilibrium temperature
SURFACE TEMPERATURES
Measured
No discharge
Discharge
MIDLAYER TEMPERATURES
MM Measured
PNM No discharge
PDM Discharge
30 60 90 120 150 180 210 240 S70 300 330/360
Days
Figure 5. Stratification cycle parabolic domain, Cayuga Lake
21
-------
No Discharge
With Discharge
cm2/sec
30
27
24
21
18
15
12
9
6
3
300 days
240 days
90 days
0 6 12 18 24 30 36 42 48 56 60
Depth(Meters)
Figure 6. Variation of eddy diffusivity with depth cylindrical domain,
Cayuga Lake
22
-------
No Discharge
With Discharge
cm2/sec
36
33
30
27
24
21
18
15
12
9
6
3
300 days
240 days
90 days
6 12 18 24 30 36 42 48 56 60
DepthCmeters)
Figure 7. Variation of eddy diffusivity with depth paraboloid domain
Cayuga Lake
23
-------
SECTION 6
MODEL VERIFICATION
The model calibrated in the last section was applied to Lake Keowee.
There were two objectives. The first was to simulate the stratification
behavior of Lake Keowee for the period 1971 to 1979, the entire existence
of this river-dammed lake, which started receiving power plant heated
discharges in 1973. The second objective was to test the accuracy of
this model for a situation where horizontal uniformity (see Appendix A)
of temperature was somewhat suspected since there are two basins con-
nected by a canal.
LAKE KEOWEE APPLICATION
Lake Keowee is located 40 km west of Greenville, South Carolina.
It is the source of cooling water for Oconee Nuclear Power Station (ONS).
It was formed from 1968 through 1971 by damming Little and Keowee
Rivers. A connecting canal (maximum depth 30.5 m) joins the two main
arms of the lake. Flow out of the lake, except for negligible leakage
through Little River dam, is through the Keowee Hydro Station. Lake
Keowee also exchanges water with Lake Jocassee-pumped storage station.
The three-unit steam electric ONS with a net capacity of 2580 Mwe
started operating in July 1973. ONS operated at annual gross thermal
capacity factors (CTCF) of 11, 28, 69 and 59% in the years 1973 through
1976, respectively. Duke Power Company (1976). From 1977 through
1979 the CTCF varied from 65 to 75 , Duke Power Company (1979).
INPUT QUANTITIES
Since long term effects are the important factors in this sutdy, the
input data to the one-dimensional model were mainly monthly averages
supplied by Duke Power Company (1976 and 1979). These are:
1. Maximum depth of Lake Keowee = 45 m ( = 150 ft).
2. Initial temperature (minimum surface temperature of Lake Keowee)
= 7.5°C.
3. ONS condenser cooling water flowrate, Q , Table 1.
P
4. Temperature rise in condenser cooling water. Table 2.
5. Outflow through Keowee Hydro Station, Table 3.
-------
6. Flow through Jocassee-pumpted storage station. Table 4.
7. Wind speed. Table 5 (0.2 meters from surface).
8. Solar radiation reaching the water surface, $ , Table 6.
9. Dewpoint temperature, Table 7.
RESULTS
Computations were carried out for nine years, 1971 through 1979,
using the above monthly-averaged data. The measured temperature
profiles referred to in this discussion are horizontal-averaged data col-
lected from seven different locations in Lake Keowee, as shown in Figure
8. This is in approximate agreement with the lateral uniformity assumed
in the formulation of this and other one-dimensional models.
Computed (with discharge) and measured temperature profiles for
1971 to 1979 are presented together whenever possible, Figures 9 through
17. A second set of figures, 18 to 26, for the nine simulated years show
the computed temperature profiles (with and without discharge). Other
figures presented include: a typical variation of eddy diffusivity with
depth (time in days as parameter), Figure 27; and a typical temporal
variation of the depth of the thermocline. Figure 28. Finally, the
variation of the surface heat exchange coefficient with time in years
(Figure 29), and the stratification cycle for Lake Keowee (1971-1979),
Figure 30, are presented. All the above figures are computer drawn.
For 1971, the year the lake came into existence, Figure 9 shows
the computed temperature profiles (solid lines) and the measured tem-
perature profiles (broken lines). The surface temperature comparisons
between the measured and computed are excellent except for the start
of the heating season when the computed temperatures seem to lag the
measured. The maximum lag occurs in April and reduces with the
heating season and completely disappears in October when cooling is in
full swing. The hypolimnetic temperatures show good agreement with
measured data. The mid-layer temperatures compare perfectly for the
first half of the year. From August to November, the trend changes
and the computed profiles lead the measured. However, the modeling
of the formation, maintenance and decay of the thermocline, is good.
This can be seen by comparing the shapes of the measured and computed
temperature profiles. The comparison improves with increasing years,
Figures 10 to 17, and generally follows the same trend as outlined above.
The comparison of measured and computed temperature profiles cannot
be expected to be more precise for the following reason. The measured
values were taken on some specific day (different for some months)
during the month while the computed results show the temperature pro-
files at the end of the month.
To estimate the impact of ONS operations on the thermal dynamics
25
-------
of Lake Keowee, a set of simulations (with and without discharge) are
presented. These are summarized in Figures 18 to 26. Two years are
selected for detailed discussion, 1975 and 1978. The earlier year.
Figure 22, was a year of very high ONS annual gross thermal capacity
factor and the later year. Figure 25, is representative of the situation
towards the end of computations. In Figure 22, although the discharge
surfaced only during the early and later months of the year, the effect
of discharge (broken lines) is clearly evident during most of the months
at the subsurface layers (especially below the thermcline). Figure 25
shows the same situation for 1978. This time, however, the discharge
surfaced from January through May and then from October through
December, at total of three months more than in 1975. The hypolimnetic
temperatures were also higher. Finally, by the end of computations,
1979, there was a residual temperature difference of almost 3°C due to
discharge. ONS started operations in July 1973, and accordingly, the
first three years. Figures 18 to 20, show no differences between dis-
charge and no-discharge simulations.
Figure 27 shows the variation of the eddy diffusivity with depth
for 1975, one of the nine years simulated. In this figure, the maximum
diffusivity always occurs at the surface for every month. The non-
stratified months show a constant diffustvity while the stratified months
show very high mixing in the epilimnion and very low mixing in the
hypolimnion.
Figure 28 shows the temporal variation of the depth of the thermo-
cline for 1975. This figure is in agreement with one of the generally
accepted theories of the formation, maintenance and decay of the thermo-
cline.
Finally, a summary of the stratification cycle for Lake Keowee for
the nine years simulated is presented in Figure 30, and the variation of
the surface exchange coefficient for the same period is shown in Figures
29 and 30. The maximum surface temperatures occur around 240 days,
and the minimum surface temperatures around 60 days for each year.
The equilibrium temperature of the lake is superimposed. The maximum
equilibrium temperatures occur around 210 days. The surface temperature
therefore lags the equilibrium temperature by 30 days. Two curves are
shown for the mid-layer temperatures to highlight the effect of ONS
operations. The temporal increase of this effect can be seen clearly.
The maximum difference occurs during the last quarter of each year.
The variation of the surface heat exchange coefficient, as expected,
follows the same trend as the variation of the surface temperatures.
26
-------
TABLE 1. Oconee Nuclear Station Condenser Cooling Water Flowrate (m3/min)
Month
January
February
March
April
May
June
July
August
September
October
November
December
1973
1890.2
1910.3
2099. 5
2232.5
2170.7
3284. 6
1974
3069. 3
3069. 4
2976. 9
2807.3
2164.6
4171.8
5334.6
4727. 1
5961.4
4953. 4
4202.1
5225.6
1975
4612.4
3694. 9
5456. 8
5570.8
6494. 3
6574. 2
7104.2
751Q. 1
7201.6
6993.4
7467. 1
6850. 9
1976
6069. 3
4440.2
4874.3
4272.1
3970.7
51 97. 6
5830.0
7248. 3
6785. 4
5637. 8
5809.2
491 4. 8
1977
5045. 8
4985.2
5113.5
601 3. 6
6302.4
4385.3
5038. 6
5708.9
6964.0
6754.7
4697. 6
5854.6
1978
6176.7
6444. 6
5195.7
4811.8
4984. 2
5659.9
7058.8
7914.9
6557. 3
7407. 4
6065.1
6503.5
1979
7207.7
7319.9
741 9. 5
7275.8
4189.1
5381.2
4753. 3
27
-------
TABLE 2. Oconee Nuclear Station Condenser Temperature Rise, AT(°C)
Month
January
February
March
April
May
June
July
August
September
October
November
December
1973
5.3
4.6
5.3
7.3
7.7
4.1
1974
4.2
7.4
8.4
8.0
2.7
6.0
5.0
4.8
5.8
3.5
7.9
5.9
1975
6.3
4.8
8.2
8.3
8.8
8.8
8.3
7.8
7.4
7.7
8.5
9.4
1976
10.6
7.3
7.1
5.1
5.8
9.3
7.4
8.5
8.0
7.8
6.7
8.4
1977
12.5
11.4
10.4
11.4
9.4
8.4
7.4
5.0
5.0
3.8
6.2
7.9
1978
9.0
11.0
13.2
9.7
10.1
8.1
7.9
7.5
7.6
6.2
8.4
7.2
1979
10.3
10.4
9.6
9.9
8.2
7.1
5.0
28
-------
TABLE 3. Monthly Average Flowrates (m3/sec) - Lake Keowee Hydro Station
Month
January
February
March
April
May
June
July
August
September
October
November
December
. Annual Average
1971
22.6
14.9
18.5
8.8
16.2*
1972
17.5
32.9
9.0
3.0
21.2
22.9
14.3
15.8
20.2
6.1
30.8
17.3
17.6
1973
22.2
17.2
33.6
30.1
31.6
55.7
9.7
16.1
18.4
3.3
5.6
19.3
21.9
1974
71.2
118.1
20.7
16.7
35.9
11.8
43.1
28.8
17.0
5.1
23.0
12.4
33.7
1975
27.0
50.8
82.7
42.5
63.8
16.1
20.3
39.9
35.1
49.8
47.2
44.1
43.3 .
1976
39.5
36.2
42.4
40.7
53.3
53.2
33.2
10.8
16.0
39.2
27.2
30.1
35.2
1977
35.5
33.6
25.4
50.5
21.6
27.5
20.1
55.2
16.5
7.5
22.5
20.5
27.2
1978
30.1
22.6
25.7
18.7
52.3
43.1
12.2
28.7
33.1
25.2
13.4
13.4
32.1
Average of September through December.
29
-------
TABLE 4. Lake Jocassee Hydro Flows (cfs)
Month
January
February
March
April
May
June
July
August
September
October
November
December
1973
1974
1975
-163
554
765
615
301
1976
505
509
562
-79
555
1032
112
-747
39
304
-251
-915
1977
-608
206
1181
1046
27
489
-831
-460
101
152
448
461
1978
187
54
516
370
1762.2
180.2
1979
30
-------
TABLE 5. Lake Keowee, Wind Speed (cm/sec)
Month
January
February
March
April
May
June
July
August
September
October
November
December
1971
6.69
9.3
9.2
8.72
7.5
5.65
6.48
5.75
5.77
7.02
7.53
8.3
1972
6.69
9.26
9.20
8.7
7.53
7.95
6.64
6.07
5.47
7.17
7.13
6.8
1973
7.22
7.3
7.1
8.44
6.83
3.04
5.32
5.1
6.8
7.1
8.14
5.6
1974
5.8
5.8
7.7
8.7
6.8
6.96
5.2
5.87
6.74
5.7
7.2
6.9
1975
6.3
7.6
9.6
7.6
4.8
5.82
5.1
5.4
7.3
7.7
6.9
7.2
1976
7.4
8.5
7.9
7.6
7.3
6.4
5.9
6.7
7.13
7.21
7.27
8.2
1977
8.04
8.4
7.7
7.6
6.2
6.7
5.8
5.4
5.3
7.2
7.5
7.2
1978
7.9
6.8
7.6
7.6
6.7
4.7
5.7
5.1
5.7
6.6
5.8
7.3
1979
8.6
7.2
7.9
7.6
6.7
4.8
31
-------
TABLE 6. Lake Keowee Gross Solar Radiation (Langleys)
Month
January
February
March
April
May
June
July
August
September
October
November
December
1971
167,0
264.4
264.4
457.0
480.5
478.0
409.0
428.2
329.0
261.3
247.7
147.7
1972
176.0
257.6
352.5
448.0
433.6
564.3
493.8
453.5
386.3
298.1
220. 9
148.0
1973
162.7
279.5
348.5
449.3
449.5
507.7
496.9
391.6
338.4
341.7
247.6
154.0
1974
191.4
226.9
326.1
397.7
436.0
559.3
459.5
480.0
339.2
302.5
231.1
181.9
1975
191.4
226.9
326.1
397.7
436.0
559.3
459.5
480.0
339.2
302.5
231.1
181.9
1976
209.8
310.9
338.6
496.9
448.4
480.2
288.3
480.4
345.1
287.5
237.5
195.0
1977
205.5
317.6
328.5
427.3
473.0
543.3
551. 8
423.9
350.7
286.6
196.2
178.2
1978
227.0
308.0
408.0
429.0
513.0
598.0
568.0
461.0
385.0
369.0
232.0
191.0
1979
208.0
251.0
373.0
479.0
513.0
32
-------
TABLE 7. Lake Keowee Dewpoint Temperature (°C)
Month
January
February
March
April
May
June
July
August
September
October
November
December
1971
3.0
0.9
6.3
7.5
17.2
18.8
20.0
19.44
18. 33
13.88
2.88
5.5
1972
1.67
-2.22
1.11
6.6
11.11
13.13
18.77
22.22
18.8
11.5
5.9
4.0
1973
1.0
-1.0
10.0
7.7
14.3
20.25
22.2
21.7
20.8
13.5
7.2
3.2
: 1974
8.2
0.0
6.3
10.7
17.2
17.8
21.0
21.0
17.5
10.2
6.0
3.8
1975
3.0
3.5
2.2
7.2
17.5
19.0
21.3
21.0
16.2
12.4
7.9
2.0
1976
-1.0
3.2
3.9
11.2
14.0
18.3
19.8
18.0
15.4
8.2
1.0
-1.5
1977
-6.6
-2.78
6.0
10.2
15.4
18.0
20.2
20.7
18.7
9.2
7.0
0. 4
1978
-2.8
-5.0
1.2
9.6
14.0
19.4
20.8
20.8
15.5
9.3
9.0
0.4
1979
-3.33
0.0
5.0
9.2
14.0
19.4
20.7
21.0
15.7
10.0
10.3
0.9
33
-------
to
N
Lake
Jocassee
OCONEE COUNTY
Little'
River Dam
OCONEE
NUCLEAR STN.
DISCHARGE
Figure 8. Map of Lake Keowee
-------
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a
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-
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a
DO 20.00
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DEC
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Fiqure 9. Measured (broken lines) and predicted temperature profiles,
Lake Keowee, 1971
-------
UJ
01
o
o
_
[Z
xg
a
0
a_
H-
Lu
a
a
o_
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u_
a
o
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FEB. £
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-
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]' So
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TEMP.fC) a TEMP.fC) a TEMP.fC) a TEMP.fC)
Figure 10. Measured (broken lines) and predicted temperature profiles,
Lake Keowee, 1972
-------
oo
o
JRN.
LU U.OO 20.00
a TEMP.(C)
a
MRY
O z.OO
a TEMP.fC
o
Q_
FEB.
a
o
OO 20.00
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JUNE
a
a
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SEPT.
o_
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0
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OO 20.00
TEMP.(C)
RPRIL
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o.
JULY
a
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RUG
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a TEMP.(C)
-
NOV.
DEC
_ i.OO 20.00 JlJTJ.OO Zb.OO S^-00 Z'0.00 £jt).00 ZO.OO
a TEMP.(C) a TEMP.(C) a TEMP.(C) a TEMP.(C)
Figure 11. Measured (broken lines) and predicted temperature orofiles.
Lake Keowee, 1973
-------
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t
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a
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0
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>' So
/ "o
1
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r~* ^
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DEC.
.00 20.00 Sj^'.oo 20.00 ^^'.oo 20.00 CJ°Q'.OO 20.00
TEMP.fC) a TEhP.(CJ a TEMP-ICJ a TEMP.(C)
Figure 12. Measured (broken lines) and predicted temperature profiles.
Lake Keowee, 1971
-------
o
'O
*
"
t
~T" ^"**
'
I
j
/
1
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o.
"
u_
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FEB. t
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.
/
f
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20.00 uj^'-OO 20.00 {^^.00 20-00 Sj^-OO 20-00
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y "?§
(f 2-
1
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a TEMP.
o
a.
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i
r^
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f
->
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/ f __.
1
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fCJ o TEMP.fCJ a TEMP.fCJ a TEMP.(CJ
1 / V§
1 ' '
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-SEPT. t
;5o
/ 2g
' ^~
^d'cT. tz
< >
Xo
/ >^o
1 ^^
^ijov. t
x§
/
(
DEC.
^°?PMP f°'?° uj'V.OO 20.00 £3^.00 20-00 Sj^-OO 20.00
a TEMP.(C) a TEMP.(C) a TEMP.(C) a TEMP.fCJ
Figure 13. Measured (broken lines) and predicted temperature profiles.
Lake Keowee, 1975
-------
o
0-
*-4
f *
h-
U
/
j
1
!
o
Q-
' »
JRN. £
xg
y Q '
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FEB. t
1
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i
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1 °a
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xg
/
£
\
s
RPRI
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o TEMP.fCJ a TEMP.fCJ a TEMP.fCJ a TEMP.fCJ
o
* «o
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0-
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v-
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j
/
/
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a TEMP.
o
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I/ 5s
i ^
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/ So
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1 -
1 l_^
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o.
/NOV. E
H^
i
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1
DEC.
m u.uu tu.uu uj u"J" 20.00 ujTl.OO 20.00 uj^J.OO 20-00
o TEMP.fC) a TEMP.fC) a TEMP.fC) a TEMP.fC)
Figure 14. Measured (broken lines) and predicted temperature profiles.
Lake Keowee, 1976
-------
o
T-«O
Q_
Lu
jE§
f 2o
Q_
JRN. tZ
,_
^0
/
/
1
1
1
t
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0_
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1
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1
1
1
1
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o.
^t
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^_
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/'
/
1
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uj^Q-OO 20.00
a TEMP.fCJ
_ 1.00 20.00
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1.00 20.00
TEMP.fCJ
_ .00 20.00
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a.
f
MRY
Uj^'-OO 20.00
a TEMP.fCJ
o
JUNE
a
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TEMP.fCJ
_ LOO 20.00
a TEMP.fCJ
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a TEMP.fCJ
o
a
XO
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a
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a
*
a.
o
r
1 ~g
o_
/'NOV. tz
w_/
1.
DEC.
1.00 20.00
a TEMP.fC)
1.00 20.00
TEMP.fCJ
Figure 15. Measured (broken lines) and predicted temperature profiles,
Lake Keowee, 1977
-------
4=
10
o
. O
*"""*
1
u_
"T ^D
*
fc'tf.oo
a TEMP.
"o
T-.0
.
C3_
_,
1
u_
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1-°
s
.
1
1
/
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a
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-*
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t
u_
xg
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1
o
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Q.
v^
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JRN. tZ
.
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t
j FEB. |Z
/ nr f-~n
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/
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1
1
1
1
1
{
1
/
f
20.00 Sj^'.OO 20.00 uj^.OO
fC
/,
) a TEMP.fCJ a TEMP.
o ^D
/ ^-iO
"1
/ -"*
'NOV. {z
x°
r - ..
i
t
j
f
1
1
DEC.
uj u.uu 20-00
a TEMP.fCJ
i.oo
20.00
.00
20.00
a TEMP.fCJ a TEMP.fCJ
ujTJ.OO 20.00
a TEMP.fCJ
Figure 16. Measured (broken lines) and predicted temperature profiles.
Lake Keowee, 1978
-------
OJ
o
< > o-
EH *-«
fe
ffi
H
P-i
M 0-
O
t"H
JAN.
)
20
o
o -
i-H
FEB.
r-i .
) 20 o
o -
t i
MARCH
r"i
1
/
APRIL
20 0 20
TEMP.(C) TEMP.(C) TEMP.
ess.
S o '*
rc
H
ft,
w
p °c
/
TEMP.
0
PO-
1
P-.
0
TEMP.
/
/
0 -
MAY
26
(C)
J
J ,
0
/
/
)
X
O "
t-l
JUNE
20 0
TEMP/
0
1 SEPT .
20
(C)
O '1
0
/
(G) TEMP
o
rH 1
OCT.
/
/
t
(C) TEMP(C>
\
S '
»H
JULY
/
/
[ AUG.
I
20 0 20
. (C) TEMP.(C)
20 0
TEMP.(G) TEMP
o-
NOV.
O-.
DEC.
20 0 20
. (C) TEMP.(C)
Figure 17. Predicted discharge temperature profiles. Lake Keowee, 1979
-------
o
o
o.
1
U_
I r
a
o
-O
o.
JRN. £
jEo
o
Q.
FEB. {Z
.00 20.00 Itj^.QO 20.00 [h1^
TEMP.(C) a TEMP.tCJ a
CZl
.0
MRRCH £
00 20.00 fh^
TEMP.(C) a
/
I RPRIL
i
00 20.00
TEMP.f CJ
o
u_
a
o
u_
MRY
OO 20-00
TEMP.(C)
SEPT
Uj B.OO 20-00
a TEMP.fCJ
o
^-.Q
*o
O_
X°
u_Q
ii^'.OO 20.00
a TEMP.(C)
OCT.
1-00 20.00
TEMP.(C)
20.00
TEMP.(CJ
O
NOV.
a
1-00 20.00
TEMP.tCJ
a
o
.o
*0
20.00
TEhP.(CJ
DEC.
a
1.00 20.00
TENP.CCJ
Figure 18. Predicted nb-discharqe temperature profiles, Lake Keowee, 1971
-------
o
* «o
o_
1
U_
X°
H-O
UjT)
a
o
'O
--"
JRN. t
^y~ CD
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TEMP.(C) a
o
"*°
"
FEB. t
area
t-Q.
.00 zb.oo uj*^
TEHP-(C) a
CD
O.
MRRCH £
:ra
H-°
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TEMP.CC) a
I
I
\
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.00 20-00
TEMP.(C)
Ul
o.
MRY
UJ
a
I.DO 20-00
TEMP.tC )
.0
JUNE
i^-oo
a
zb.OQ
TEMP.(C)
JULY
OO 20.00
TEMP.(C)
RUG.
m^.OO 20-00
a TEMP.(C)
o
o
CD
SEPT
£^).00 20.00
a TEMP.tCJ
OCT
OO 20.00
TEMP.(C)
a
o
O
a
o.
NOV
00 20-00
TEMP.fCJ
DEC
O 20-00
a TEMP.fCJ
Figure 19. Predicted no-discharge temperature profiles. Lake Keowee, 1972
-------
o
a
o_
J-
LL_
y«s "*
O
, 0
2-
JRN. t
^ °
o
~a
o.
FEB. t
f o
1 O
1 "ZTa
/ 2-
MRRCH tZ
^0
1
1
/
RPR1L
C]T).00 20-00
a TEMP.CC)
__ 1.00 20.00
° TEMP.CC)
a
1.00 20.00
TEMP.CC)
1.00 20-00
a TEMP.CC)
a>
o
a
o_
MRY
UJ
a
i.OO 20.00
TEMP.CCJ
a
«°
-
JUNE
1.00 20-00
TEMP.CC)
JULY
JtptJ.OO 20.00
a TEMP.CC)
_ 1.00 20.00
a TEMP.CC)
o
>o
0
Q
o
O.
SEPT
Uj^-OO 20-00
a TEMP.CC)
OCT
a
1.00 20.00
TEMP.CC)
o
TO
NOV
a
.00 20-00
TEMP.CC)
O.
DEC
_ 1.00 20.00
a TEMP.CCJ
Fiqure 20. Predicted discharge (borken lines) and no-discharae temoerature orofiles.
Lake Keowee, 1973
-------
Q
- O
a_
h-
U_
i °
UJ "
a
o
T-.O
O_
I
UJ U
a
Q
Q_
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u_
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a. 4:
UJ "
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; -~o
o.
JRW. t
^?
;
i
o
Q_
FEB. t
^°
1
:
.00 20.00 Sj^.OO 20.00 u^'.QO
TEMP.(C) o TEMP.(C) a TEMP
J -s
MOV "
nn f t,_
/
1
/ ~Q
S V°
ofc
f*
JUNE t
*
>
.00 20-00 Cj^'-OO 20-00 ui'V.OO
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a
/ ~~
/SEPT. t
1 ^ l~°
/
a
o.
OCT. t^
1
'.00 20-00 u^^'-OO 20.00 u^^'-OO
o
*
MRRCH £
;
RPRIL
i
20.00 {ti^.OQ 20.00
(C) a TEMP.(C)
/ a
J """o
/^ 2-
JULY t
//
r RUG.
20. 00 aJ^'-OO 20. 00
tCJ a TEMP.(CJ
i 0
I *~lQ
2-
i NOV. t
1
1 ,., .., ' ' *
i
i
DEC.
20-00 uj^-OO 20.00
Fiqure 21. Predicted discharge (broken lines) and no-discharge temperature profiles.
Lake Keowee, 197U
-------
4=
09
o
.. |Q
K o
Q-
U_
i «
Q
'O
X 0
O-
JRN. t
[rf.OO 20
a TEMP.CC
So '
* °
o_
" x
t
U.
x°
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*
i
O
« »o
no
o_
FEB. tZ
O
»-«o
o_
MRRCH t
RPRIL
.00 [I^b'.oo 20.00 ^^'.oo 20.00 SJ^'-OQ 20.00
) a TEMP.(C) a TEMP.(C) a TEMP . ( O
2a ' ° . / Q i
* °
o.
^"^
! MRY tl
i _^
i
i
i-
Sj'V.OO 20
a TEhP.(C
o
V-.Q
*°
5-
^ ^
L_
r
u_
X<=>
h- .
J.
f f
/ 1
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i Q
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/
f °
, , Q
X °
O-
/
/ '-
/ ,
/ OCT. t
/ ^0
H-°
» «o
* °
0.
*-
/NOV. t
/ xo
j_ . ^
j
i
i
i
i
i
, DEC.
\
i
.00 SJ^-QO 20.00 SJ^'.QO 20.00 SJ^'-OQ 20.00
J a TEMP.(C) a TEMP.CCJ a TEMP.tO
Fiqure 22. Predicted discharge (borken lines) and no-discharae temperature orofiles.
Lake Keowee, 1975
-------
o
'O
,0
o_
^_,
U_
Xg
LU "
a
o
. 'O
1
U_
jEo
a
o
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(
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a
o
' . 0
O-
,_
JRN. tZ
i Q
0
0.
^ .
FEB. [Z
£°
O
a_
^_,
MRRCH tZ
^°
/!
/'
i
RPRIL
.00 20-00 Sj^'.OO 20-00 Sj^'-QO 20.00 Sj^'.OO 20.00
TEMP.(C) a TEMP. CO a TEMP.(C) a TEMP.(C)
/o
7°
MRY |Z
^°
/I CD
| 0
/f *°-
/ JUNE |Z
1 1 '"""
i O
x^ 0.
/ JULY {Z
| Q
^f
f\ RUG.
.00 20-00 Sj^'.OO 20-00 uJ^'-OO 20.00 uj^'.OQ 20-00
TEMP-CCJ a TEMP.(CJ a TEMP-CCJ a TEMP-(CJ
o
-------
o
~o
M
Q_
£
JE§
UJ U
0
,~,O
*
o.
JRN. JZ
xg
.00 20.00 m^
I o
1 ~g
IK
o_
FEB. £
|ES
.00 20.00 m^
o
. 0
M
O.
MRRCH [Z
^§
.00 20.00 fh*^
ti
I
/'
/:
I
RPRIL
.00 20.00
a TEMP.tCJ a TEMP.fCJ a TEMP. CO a TEMP.(C)
ut
o
uj a.oo 20.00
a TEMP.CO
JUNE
_ i.ao 20.00
a TEMP.CO
o
"O
*9
JULY
1-00 20.00
TEMP.(C)
o
»o
O-
RUG
OjTl.OO 20.00
a TEtlP.(C)
o
a
Q-
U-
Uj U-00 20.00
a _ JLFMP . ( O
0
SEPT.
QCT
UTo'-oa 20.00
a TEMP.(C)
o
7§
o_
NOV.
_1_
a
1.00 20.00
TEMP.JQ1
Q_
DEC
iZJTJ.OO 20-00
^Q TEMP. (O
Figure 24. Predicted discharge (broken lines) and no-discharge temperature profiles.
Lake Keowee, 1977
-------
o
. . o
*o
o.
h-
u_
t °
o
0
*
2-
JRN. tZ
-L ^T '-
o
o.
FEB. tZ
i
1 ^~" ,
i O
; -g
o.
MRRCH tZ
J- t~°
M
/'
RPRIL
uj J.OO 20.00
a TEMP.(C)
.00 20.00
TEMP.(C)
a
00 20.00
TEMP.(CJ
a
.00 20.00
TEMP.(C)
o
MRY
Sj^-OO
a
0
20.00
TEMP.(C)
o.
Li-
a.
O
JULY
_ .00 20.00
a TEMP.(C)
uj U.OO 20.00
a TEMP.(C)
RUO
o
o.
SEPT
oo t-oo
<00
/OCT
20.00
i-oa
a
Q
.00
TEMP
Q_
NOV
20.00
.CO
DEC
%-ee-
Figure 25. Predicted discharge (broken lines) and no-discharqe temperature profiles.
Lake Keowee, 1978
-------
U1
o
- o
»
o.
f
^
UJ U
a
o
o_
H-
U_
a_a-
UJ U
a
o
'O
H-
U_
^
a_
JRN. t
Q
« .Q
FEB. t
, 0
^-.Q
0,
MRRCH tZ
ii
\ i
/ ' '
RPRIL
.00 20.00 S^-OO 20.00 ^j^-OO 20.00 m^.QO 20 00
TEMP.(C) a TEMP. CO a TEtlP.(C) a TEMP.(C1
/ °
J ~0
/ "=
i
MR Y u_
/ "^
j ^ ' SEPT. |Z
f=°
O
a.
/\
//OCT. £
/ ' _
1 ! ^°
Q
-.o
o_
/NOV. t
/
I
i
DEC.
.00 20.00 Sj^-OO 20.00 Cj^'-OO 20.00 m^-OO 20-00
TEMP (CJ P TEMP-CCJ a TFMP . ( H ) a TEMP.(C)
Figure 26. Predicted discharge (broken lines) and no-discharge temperature profiles.
Lake Keowee, 1979
-------
30
60 90
DEPTH (FT)
120
150
Figure 27. Variation of eddy diffusivity with depth. Lake Keowee, 1975
53
-------
E^
fc
o
o
o
m
H
in
*
CO
01
<*
*
r-
O
H CN
A r-
8
in
00
ffi <^
tl CN
fe ?!
O vO
W
E-i O
en co
w .
Q (*'
"*
CO
"t
*
1-1
(N
JAN FEE MAR APR MAY JXJN JUL AUG SEP OCT NOV DEC
1975.~ , .
Figure 28. Monthly variation of the depth of the thermocline.
Lake Keowee, 1975
-------
en
on
O
O
CO
O
00
Q
o
m
o
0 365 730 1095
1460 1825
TIME IN DAYS
2190 2550 2920 3285
Figure 29. Temporal variation of the surface exchange coefficient, Lake Keowee, 1971-1979
-------
9S
TEMPERATURES (C)
14 21 28
Tl
(Q
C.
a>
u>
C/i
r*
CU
s
r*
o'
o
a"
Z
1
tD
(D
VJ
I
Hd
iO
(O
03 cn
n o
o H
# H-
f
f H-
H- 3
3 fl>
fD 0)
CO
9?
H- I
(0 &
O H-
y ui
0) O
0)
I 1
3
QJ
(D
l-i
n>
I
(D
H
OJ
ft
C
n
(D
CO
35
-------
REFERENCES
Dake, J. M. K and D. R. F. Harelman. Thermal Stratification in Lakes:
Anaytical and Laboratory Studies. Water Resources Research, Vol.
5, No. 2. pp 484-495. April 1969.
Duke Power Company. Unpublished Materials. 1971-1979.
Dutton, J. A. and R. A. Bryson. 1962. Heat Flux in Lake Mendota
Limnol Ocenanog. 7, 80.
Edinger, J. E. and J. C. Geyer. Heat Exchange in the Environment.
Sanitary Eng. and Water Resources Report. 1967.
Henson, E. B., Bradshaw, A. S. and D. C. Chandler. The Physical
Limnology of Cayuga Lake, New York. Memoir 378, 1961. Agri-
cultural Research Station, Cornell University, Ithaca, New York.
Kraus, E. B. and J. S. Turner. A One-dimensional Model for the
Seasonal Thermocline II. The General Theory and Its Consequences.
Tellus, Vo. 19, No. 1. pp 98-105. 1967.
Lerman A. and M. Stiller. 1969. Vertical Eddy Diffusivity in Lake
Tiberias. Verh. International Verein. Limnol. 17, 323.
Monin, A. S. and A. M. Obukhov. Basic Regularity in Turbulent
Mixing in the Surface Layer of the Atmosphere. USSR Acad. Sci.
Works of Geophys. Met., No. 24, 163. 1954.
Mitry, A. M. and M. N. Ozisik. A One-dimensional Mode! for Seasonal
Variation of Temperature Distribution in Stratified Lakes. Inter-
national J. Heat Mass Transfer, Vol. 19. pp 201-205. 1976.
Oconee Nuclear Station Environmental Summary Report. Duke Power
Company, 1971-1976, Vol. 1. November 1977.
Rossby, C. C. and B. R. Montgomery. The Layer of Frictional Influ-
ence in Wind Ocean Currents. Papers in Physical Oceanography,
Vol. 3, No. 3. p 101. 1935.
Sengupta, S., Lee, S. S. and E. Nwadike. A One-dimensional Variable
Cross-section Model for the Seasonal Thermocline. Proceedings of
the 2nd Conference on Waste Heat Management and Utilization.
p IX-A-3. 1978.
57
-------
Sengupta, S. and W. Lick. A Numerical Model for Wind-driven Circula-
tion and Heat Transfer in Lakes and Ponds. FTAS/TR-74-98.
Sundaram, T. R., Easterbrook, C. C., Piech, K. R. and G. Rudinger.
An Investigation of the Physical Effects of Thermal Discharges into
Cayuga Lake. Report VT-2616-0-2. Cornell Aeronautical Labora-
tory, Buffalo, New York. November 1969.
Sundaram, T. R. and R. G. Rehm. Formulation and Maintenance of
Thermoclines in Stratified Lakes Including the Effects of Power Plant
Discharge. AIAA Paper No. 70-238. 1970.
Sundaram, T. R., Rehm, R. G., Rudinger, G. and G. E. Merritt. A
Study of Some Problems on the Physical Aspects of Thermal Pollu-
tion. VT-2790-1-1. Cornell Aeronautical Laboratory, Buffalo, New
York. 1971.
Sundaram, T. R. and R. G. Rehm. Formation and Maintenance of
Thermoclines in Temperate Lakes. AIAA Journal, Vol. 9, No. 7.
pp 1322-1329. 1971.
Sundaram, T. R. and R. G. Rehm. Effects of Thermal Discharges on
the Stratification Cycles of Lakes. AIAA Journal, Vol. 10, No. 2.
pp 201H210. 1972.
Sweers, H. E. 1969. Two Methods of Describing the "Average" Vertical
Temperature Distribution of a Lake. J. Fish. Res. Bd., Canada
25. 1911.
58
-------
APPENDICES
59
-------
APPENDIX A
DERIVATION OF THE MATHEMATICAL MODEL
DERIVATION OF THE VARIABLE CROSS-SECTIONAL AREA MODEL
Assuming lateral uniformity (horizontally uniform lake) , the isotherms
coincide with the isopycnals. The equations for such a model are obtained
from the basic balance equations of mass and heat:
pv (A.I)
^(pC T) = 7 pCpK 7 pCpTV + H (A. 2)
where,
p = density.
t = time.
V = velocity of flow.
C = heat capacity.
P
T = temperature.
K = heat diffusivity tensor (including turbulent diffusivity) .
H = source of heat per unit volume.
Most models have started with the above equations and neglected
all the terms involving the two horizontal components (or dimensions).
The neglect of the horizontal divergence in such simple models is not
always justified. There are at least two reasons for the existence of
horizontal divergence in real lakes.
1. The variation of horizontal cross-sectional area of the lake with depth.
2. The existence of sources of heat and matter efflux on the bottom of
the lake at depths above the deepest point.
60
-------
The need to include these in the diffusion equations of lakes was
already felt; Lerman and Stiller (1969), Dutton and Bryson (1962) and
Tzur (1973). Only Tzur (1973) formulated corrected diffusion equations.
Another complication in applying a 1-D model to a lake is caused by
seiches and internal waves. When these are present, the height of an
isopycnal loses its meaning. According to Sweers (1968), averaging
simultaneous temperature profiles at different points of a lake or taking
profiles during a short interval of time in one or more points can lead
to distortion of the shape of the thermocline. He recommends averaging
the depths of isotherms instead. In a lake of constant depth this is
probably the best available approximation but fails for a lake of varying
depth.
The effects of area change with depth are included by the following
treatments of Equations (A.1) and (A. 2). Integrating Equation (A.1)
over the volume of water below height h measured from the deepest point
in the lake yields
dv = / v pVdv
V V
Using Gauss theorem on the right hand side yields
= -p/fi VdS (A. 3)
Vou S
where S is a surface completely surrounding the volume V, hence
dS = dC + dA
where C is the surface area of the part of the bottom of the lake that
is bounded by the contour at height z. As a subscript, z marks the
vertical component of a vector.
Using dV = Adz in Equation (A. 3) yields
h
4| Adz =-p/n VdC - p/n VdA
o l C A
i.e.
h h
/ff Adz = -/PVndC - PVzA(h)
o o
Integrating Equation (A. 2) over the volume of water below height h
measured from the deepest point in the lake yeilds
61
-------
K-VT)dV - ./VpC TVdV + / HdV
V P V
Applying the divergence theorem to the first two terms on the right
yields
8
/A(z) pC Tdz = /fl. (PC K-7T)dS - /(fi-pC TV)dS
O K K P
h
+ / A(z)H(z}dz
o r
Using dS = dA + dC yields
h
o A(z)lrpCDTd2 = /(PCDK< ^T) dA + / (pC R. VT) dC
I- o H O P
h _ h
- / (pC TV) dA - / (PC TV) dC
o P z o p n
h
+ / A(z)H(z)dz
o
i.e.
h 3 h
/A(z)PCpTdz = pCpA(h)[K-vTJz + / ^C
h
- pC AfhJTV - / p C T V
D 7 f f* f* r\
H Q c c c n
h
-f- / A(z)H(z)dz (A 5)
o
where,
z = the vertical coordinate measured upward from the deepest point of
the lake. As a subscript, it marks the vertical component of a vector.
C = the surface area of the part of the bottom of the lake that is bounded
by the contour at height z.
62
-------
n = subscript that marks the component of a vector that is perpendicular
to the lake-bottom positive upwards.
A(z) = the horizontal cross-section of the lake at height z.
As all the properties of the lake are constant over the horizontal
isopycnals, the integration over the isopycnals becomes simple multipli-
cation by their areas.
Differentiating Equations (A. 4} and (A. 5) with respect to the height,
a set of 1-D equations are obtained from Equation (A. ft) :
A3 0 Q A \ / t/3V*
at = "TzApVz * pcVnTT
(A. 6)
The last term in the above equation is the mass addition term (source
term) from Equation (A. 5):
+ AH
(A. 7)
In the above equation,
p C (K VT) = Conduction
c c n
'cCcTcVn
= Convection
Heat addition terms
Because the horizontal gradients vanish, Equation (A. 7) can be simplified
further by noting that
(A. 8)
Following a similar treatment used by Y. Tzur (1973), the slope of the
bottom of lakes is usually small; it rarely exceeds a gradient of 0.1,
Assuming this and considering an elemental area shown below,
dZ
dCU
63
-------
tan 8 = 2. = gradient of the lake bottom
Cos 6 = Cosltan"1 S,-] (A. 9)
also
Cos 9 = i~ (A. 10)
From Equation (A. 9) and (A. 10)
= Costtan"1 i~] (A. 11)
= Cos [arctan (gradient)]
= TI s average of Cos arctan
gradient of the bottom surface
at that depth.
From Equation (A. 11)
dA = ndC
or
dA dC
dC _ 1 dA
dz n dz
Since Cos arctan (0.1) = 0.995, n = - = 1.
Alternatively, n can be incorporated into the surface sources (equation
(A. 7)).
The 1-D equations can now be written in more concise and conveni-
ent forms:
+ IA1 (A. 12)
a (. at. £.
and
AfpC T) =|-JpC AK ~) - |-(pC ATV ) + QA« + AH (A.13)
dt p d Z p ZdZ dZ p Z
where,
A^^
64
-------
1 = bottom-surface source of mass per unit area.
Q = bottom-surface source of heat per unit area.
Boundary Contitions
Surface:
Equilibrium Temperature Concept
The bulk temperature of a vertically-mixed body of water under
natural conditions, Tn, tends to increase or decrease with time, accord-
ing to whether the sum, 2H, of its heat inputs (net solar and atmospheric
radiation, thermal discharge and heat outputs; back-radiation, evaporation,
and condensation) is positive or negative:
h (A. 14)
Following a similar procedure used by Edinger and Geyer (1 967), Equa-
tion (A. 14) can be transformed (without linearizing the temperature
dependence of the components of IH) to yield
dT
where,
K = surface exchange coefficient that depends on the water temperature
s and wind speed
T_ = equilibrium temperature, defined as the hypothetical water surface
temperature at which the net rate of surface heat exchange would
be zero.
Equation (A. 15) expresses this definition of T£ for the particular case
of a vertically-mixed water body ^n = 0 wnen T = T j. Edinger and
Geyer (1967) show that the two parameters are intrinsically coupled
together via water temperature and meteorological conditions. For this
reason iterative techniques are often used when applying Equation (A. 15)
in a predictive role and because the rate of convergence is extremely
rapid. The first step often yields satisfactory results.
Brady et al. (1969) have shown empirically that fluctuations in the
equilibrium temperature may be conveniently estimated using the approxi-
mate relationship:
Tc = T . + H /K (A. 16)
65
-------
where,
T . = dewpoint temperature.
H = gross rate of shortwave solar radiation.
Since the dewpoint temperature tends to remain relatively constant
through a single day, Equation (A. 16} indicates that the main source of
hourly fluctuations in T£ is the solar radiation components. This gene-
rally reaches a maximum at solar noon, unless variable cloudiness inter-
feres. At nighttime T_ approaches the dewpoint temperature which acts
like a relatively invariant datum for periods of 24 hours or less. On
an annual basis, however, both T . H are generally much greater in
summer than in winter. The dominant contribution to the amplitude of
seasonal fluctuation in T_ is the dewpoint temperature.
Exchange Coefficient Evaluation
The same form used by Edinger et al. (1967) was used:
KS = 4. 5 + 0.05 TS + 8f(w) + 0.47 f(w) (A. 17)
where.
T = surface temperature.
8 is found by applying standard curve-fitting techniques to published
data pertaining to saturated vapor pressures at various temperatures;
a convenient representation given by Edinger and Geyer (1967) is
8 = 0.35 + 0.015 T +0.0012T 2(mmHg/C) (A.18)
m m
where,
T + T^
The evaporative windspeed function f(w) used is also similar to that
of Edinger and Geyer (1967):
f(w) =9.2 + 0.46 W2(Wnf 2mmHg~1) (A. 20)
where,
W = wind speed (m/S).
The first boundary condition is at the surface and can now be stated as:
". = >CpKr-i2=h KS
-------
Bottom:
The second boundary condition is at the bottom of the lake which is
assumed to be perfectly insulated
fT|z=0 = ° (A'22'
and the initial condition is the temperature of the lake at spring homo-
thermy.
T. ... . = T (A. 23)
initial o
67
-------
APPENDIX B
MEASURED TEMPERATURE PROFILES
(Individual Stations)
68
-------
en
10
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1 ^
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0.
j
u_
P
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JRN . £
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00 20-00 [i^-OO 20-00,
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i
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J
(
i
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\
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ItMP.tC) a TEMP.fC)
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Q_
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1
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a
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00 20.00
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0
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0
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u_
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t
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1
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1
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.00 20-00
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Figure B-1. Lake Keowee measured temperature profiles, 1971 - Station 500
-------
o
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a
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__ 20-00
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Figure B-2. Lake Keowee measured temperature profiles, 1972 - Station 500
-------
0
1 - 1
!* ^J
kg O
. *
\ 'T>_
! "
i JRN. iz
£=?
i
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0
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t DEC.
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Figure B-3. Lake Keowee measured temperature profiles, 1973- Station 500
-------
N)
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0.
._
1
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,
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h-°,
uj u
0
o
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-------
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Figure B-5. Lake Keowee measured temperature profiles, 1975 - Station 500
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Figure B-6. Lake Keowee measured temperature profiles, 1977 - Station 500
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Fiqure B-7. Lake Keowee measured temperature profiles, 1977 - Station 500
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Fiqure B-8. Lake Keowee measured temperature profiles, 1978 - Station 500
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Figure B-9. Lake Keowee measured temperature profiles, 1971 - Station 501
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Figure B-10. Lake Keowee measured temperature profiles, 1972 - Station 501
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Figure B-11. Lake Keowee measured temperature profiles, 1973 - Station 501
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Fiqure B-13. Lake Keowee measured temperature profiles, 1975 - Station 501
-------
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Figure B-H. Lake Keowee measured temperature profiles, 1976 - Station 501
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Figure B-15. Lake Keowee measured temperature profiles, 1977 - Station 501
-------
00
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Figure 17. Measured temperature profiles. Lake Keowee, Station 502, 1971
-------
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Fiqure B-18. Lake Keowee measured temperature profiles, 1972 - Station 502
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Figure B-19. Lake Keowee measured temperature profiles, 1973 - Station 502
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Figure B-25. Lake Keowee measured temperature profiles, 1971 - Station 503
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Figure B-26. Lake Keowee measured temperature profiles, 1972 - Station 503
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Figure B-27. Lake Keowee measured temperature profiles, 1973 - Station 503
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Figure B-28. Lake Keowee measured temperature profiles, 1974 - Station 503
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Figure B-29. Lake Keowee measured temperature profiles, 1975 - Station 503
-------
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Figure B-30. Lake Keowee measured temperature profiles, 1976 - Station 503
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Figure B-31. Lake Keov/ee measured temperature profiles, 1977 - Station 503
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Figure B-32. Lake Keowee measured temperature profiles, 1978 - Station 503
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Figure B-37. Lake Keowee measured temperature profiles, 1975 - Station 501
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Figure B-39. Lake Keowee measured temperature profiles, 1977 - Station 504
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Figure B-11. Lake Keowee measured temperature profiles, 1971 - Station 505
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Figure B-43. Lake Keowee measured temperature profiles, 1973 - Station 505
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Figure B-45. Lake Keowee measured temperature profiles, 1975 - Station 505
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Figure B-46. Lake Keowee measured temperature profiles, 1976 - Station 505
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Figure B-18. Lake Keowee measured temperature profiles, 1978 - Station 505
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Figure B-19. Lake Keowee measured temperature profiles, 1971 - Station 506
-------
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Figure B-50. Lake Keowee measured temperature profiles, 1972 - Station 506
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Figure B-51. Lake Koewee measured temperature profiles, 1973 - Station 506
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o
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Figure B-52. Lake Keowee measured temperature profiles, 1974 - Station 506
-------
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Figure B-53. Lake Keowee measured temperature profiles, 1975 - Station 506
-------
ro
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Figure B-51. Lake Koewee measured temperature profiles, 1976 - Station 506
-------
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Figure B-55. Lake Keowee measured temperature profiles, 1977 - Station 506
-------
NJ
4=
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Figure B-56. Lake Keowee measured temperature profiles, 1978 - Station 506
-------
APPENDIX C
MEASURED TEMPERATURE PROFILES
(Average of Stations 500 Through 506)
125
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Figure C-1. Lake Keowee averaged measured temperature profiles. Stations 501-506, 1971
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M
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Figure C-2. Lake Keowee averaged measured temperature profiles. Stations 501-506, 1972
-------
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Figure C-3. Lake Keowee averaged measured temperature profiles, Stations 501-506, 1973
-------
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Fiqure C-4. Lake Keowee measured temperature profiles, Stations 501-506, 197f
-------
Q
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Fiqure C-5. Lake Keowee measured temperature profiles. Stations 501-506, 1975
-------
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Figure C-8. Lake Keowee measured temperature profiles. Stations 501-506, 1978
-------
TECHNICAL REPORT DATA
(Please read Jauruciiuns on the reverse before completing)
REPORT NO. ,
EPA-600/7-82-037e
2.
3. RECIPIENT'S ACCESSION-NO,
4 TITLE AND SUBTITLE
Verification and Transfer of
Thermal Pollution Model; Volume V. Verifica-
tion of One-dimensional Numerical Model
6 REPORT DATE
May 1982
6. PERFORMING ORGANIZATION CODE
s.s.Lee, S.Sengupta, and E.V.Nwadike
8 PERFORMING ORGANIZATION REPORT NO.
PERFORMING ORGANIZATION NAME AND ADDRESS
The University of Miami
Department of Mechanical Engineering
P.O. Box 248294
Coral Gables, Florida 33124
10 PRC.GRAM ELEMENT NO.
11 CONTRACT/GRANT NO
EPA IAG-78-DX-0166*
12 SPONSORING AGENCY NAME AND ADDRESS
EPA, Office of Research and Development
Industrial Environmental Research Laboratory
Research Triangle Park, NC 27711
13. TYPE OF REPORT AND PERIOD COVERED
Final: 3/78-9/80
14 SPONSORING AGENCY CODE
EPA/600/13
s SUPPLEMENTARY NOTES IERL-RTP project officer is Theodore G.Brna, Mail Drop
61, 919/541-2633. (*) IAG with NASA, Kennedy Space Center, FL 32899,
subcontracted to U. of Miami under NASA Contract NAS 10-9410.
s ABSTRACT The six-volume report: describes the theory Of a three-dimen-
sional (3-D) mathematical thermal discharge model and a related one-
dimensional (1-D) model, includes model verification at two sites, and
provides a separate user's manual for each model. The 3-D model has two
forms: free surface and rigid lid. The former, verified at Anclote An-
chorage (FL), allows a free air/water interface and is suited for signi
ficant surface wave heights compared to mean water depth; e.g., estu-
aries and coastal regions. The latter, verified at Lake Keowee (SC), is
suited for small surface wave heights compared to depth (e.g., natural
or man-made inland lakes) because surface elevation has been removed as
a parameter. These models allow computation of time-dependent velocity
and temperature fields for given initial conditions and time-varying
boundary conditions. The free-surface model also provides surface
height variations with time. The 1-D model is considerably more econo-
mical to run but does not provide the detailed prediction of thermal
plume behavior of the 3-D models. The 1-D model assumes horizontal
homogeneity, but includes area-change and several surface-mechanism
effects. _
17.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b IDENTlFlERS'OPEN ENDED TERMS
COSATI 1 it-td Croup
Pollution
Thermal Diffusivity
Mathematical Models
Estuaries
Lakes
Plumes
Pollution Control
Stationary Sources
13B
20M
12A
08H,08J
21B
13 DISTRIBUTION STATEMENT
Release to Public
19 SECURITY CLASS tThtsRtponi
Unclassified
21 NO OF PAGES
147
2O SECURITY CLASS (Thtipagt/
Unclassified
12 PRICE
£PA Form 2230 ! (»-73)
134
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