WATER POLLUTION CONTROL RESEARCH SERIES • 16130DFX05/70
AN ENGINEERING - ECONOMIC
STUDY OF COOLING POND
PERFORMANCE
ENVIRONMENTAL PROTECTION AGENCY • RESEARCH AND MONITORING
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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 pollution in our Nation's waters. They provide a
central source of information on the research, develop-
ment, and demonstration activities in the Environmental
Protection Agency, through inhouse research and grants
and contracts with Federal, State, and local agencies,
research institutions and industrial organizations„
Inquiries pertaining to Water Pollution Control Research
Reports should be directed to the Head, Publications
Branch, Research Information Division, Research and
Monitoring, Environmental Protection Agency, Washington,
D. C. 20460,
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AN ENGINEERING-ECONOMIC STUDY OF
COOLING POND PERFORMANCE
by
Littleton Research and Engineering Corporation
95 Russell Street, Littleton, Massachusetts 01460
for
Environmental Protection Agency
Project 16130DFX05/70
Contract No. 14-12-521
May 1970
For sale by the Superintendent of Documents, U.S. Government Printing Qfflce, Washington, D.C. 20402 - Price $1.5
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EPA REVIEW NOTICE
This report has been reviewed by the Environmental
Protection Agency, and approved for publication.
Approval does not signify that the contents neces-
sarily reflect the views and policies of the
Environmental Protection Agency, nor does mention
of trade names or commercial products constitute
endorsement or recommendation for use»
11
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ABSTRACT
A procedure for predicting the temperature of a thermally loaded
captive pond is presented. Using this information, the cooling pond
is shown in a special case to have an economic advantage over a
cooling tower and to be not much more expensive than a natural body
(stream or ocean) of water. This, with the ecological and recrea-
tional assets of a captive cooling pond, would seem to encourage
their expanded use with large thermo-electric power plants.
This report was submitted in fulfillment of Contract No. 14-12-521
under the sponsorship of the Federal Water Quality Administration.
m
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FOREWORD
There is at present a growing need for industrial cooling in the manu-
facturing industries, in the electric power industries and in the pro-
cess industries. The primary method of effecting this heat transfer
from the plant to the environment at present is to pass river, lake or
sea water through heat exchangers or condensers in the plant and dis-
charge this water directly back to its source where it will subsequently
cool off by exchanging heat with the atmosphere. Our expanding indus-
trial production combined with our relatively fixed water resources are
producing substantial pressure on the ecology of our natural water re-
sources. These waters cannot continue to accept the ever-increasing
thermal waste energy without undergoing ecological change. We know
that heating of natural water in excess of certain limits results in a
degradation of water quality to the point where some species of aquatic
life will no longer be supported. The question of accepting the death of
some forms of aquatic life in favor of the added capacity of industrial
cooling will be resolved, of course, by a decision-making process in
which society is free to pick its choice, or degree of choice. The re-
cently established State-Federal water quality standards have set the
present tone of this decision.
The use of man-made cooling ponds has been suggested as a means of
relieving the thermal pollution of our natural waters. Such captive
cooling ponds were built to dissipate the waste energy from electric
power plants in the early nineteen hundreds in parts of the United States
where lakes and rivers were not readily available. The physical fac-
tors which control the cooling capacity of these pon/ls have long been of
fundamental interest to workers in the field of oceanography, limnology
and meteorology. As a result, considerable information relative to
these factors is available in the open literature. However, this inform-
ation is diffused over a large area. It is the purpose of this report to
investigate the economic feasibility of using man-made cooling ponds
for dissipating thermal loads. To this end the report presents a con-
cise method, with substantiating data, for determining the thermal cap-
acity of cooling ponds and considers the economic factors relating cool-
ing ponds to other thermal sinks - such as cooling towers and natural
water supplies. Attention has been focused on the cooling requirements
of large central electric power stations in view of the fact that they must
dissipate very large quantities of energy in a limited area.
IV
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SUMMARY
An analytical procedure for predicting the steady state and transient
temperatures of condenser cooling water obtained from a cooling pond
is presented. The predictions require knowledge of the monthly aver-
age climatic and power plant operating parameters. Measured water
temperatures for several operating cooling ponds distributed over a
wide region of the United States are compared to values predicted on
the assumption of fully mixed ponds and slug flow ponds.
An economic analysis of the use of captive cooling ponds is presented.
The factors considered in the appraisal include land costs, the influ-
ence of water temperature upon the efficiency and capital costs of the
power plant, and power required to provide water pumping capacity.
Results are presented in curve form so that it is not continually nec-
essary to return to the basic calculations.
It appears from the analyses that it is possible to obtain cooling water
temperatures in a captive cooling pond which are within 5 F of the
equilibrium temperature of a natural water supply with a pond area of
approximately four acres per megawatt. The water requirements for
a properly designed pond are not greatly different from those of a cool-
ing tower. Where areas of adequate size are available and not too ex-
pensive, the use of a cooling pond can result in lov/er overall electri-
cal costs than cooling towers and be reasonably competitive with those
of a natural water supply. Cooling ponds can pro/ide a positive con-
tribution to the recreational, aesthetic and ecological values of a com-
munity.
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TABLE OF CONTENTS
ABSTRACT iii
FOREWORD iv
SUMMARY v
TABLE OF CONTENTS vii
LIST OF FIGURES viii
LIST OF TABLES xi
CONCLUSIONS 1
RECOMMENDATIONS 2
INTRODUCTION 3
FACTORS REGULATING HEAT TRANSFER 5
POND OPERATING CHARACTERISTICS 19
CURVES FOR PREDICTING WATER TEMPERATURE 23
COMPARISON OF PREDICTED AND MEASURED 53
WATER TEMPERATURE
CURVES FOR PREDICTING WATER LOSS BY EVAPORATION 63
APPLICATION OF DESIGN CURVES TO PARTICULAR 69
POWER PLANTS
ECONOMIC ANALYSIS OF POWER PLANTS WITH COOLING 79
PONDS
MULTIPURPOSE OF COOLING PONDS 99
ACKNOWLEDGEMENTS 101
REFERENCES 103
NOMENCLATURE 107
APPENDIX A - Energy Balance Equations 111
APPENDIX B - Heat Transfer by Evaporation and Convection 121
APPENDIX C - Data Collected on Operating Cooling Ponds 141
vii
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LIST OF FIGURES
Fig.
No. Title
1 Energy Balance Terms for a Cooling Pond
12
2 Solar Radiation Reflectivity 10
3 Brunt Coefficient
4 Clear Sky Solar Radiation 12
5A Equilibrium or Mixed Steady State Pond Temperature
vs /, (or /, + Q ) For T > 32°F 25
1 1 pp
5B Equilibrium or Mixed Steady State Pond Temperature
vs /, (or /. + 6. ) For T < 32°F 26
1 1 pp
6 Linear Temperature-Depth Profile 27
7A-G Mixed Pond and Slug Flow Pond Transient 30-36
Temperature, or Slug Flow Pond Steady State
Temperature vs /
Lj
8 Comparison of Steady State and Transient Temperatures
for a Mixed and a Slug Flow Pond 47
9 Pond Discharge Temperature vs Pond Surface Area
for Mixed and Slug Flow Operation 50
10 Ratio of Mixed Pond Area to Slug Flow Pond
vs Approach Temperature 51
11 Measured and Predicted Pond Temperatures for the
Wilkes Plant 55
12 Measured and Predicted Pond Temperatures for the
Kincaid Plant 57
13 Measured and Predicted Pond Temperatures for the
Cholla Plant 58
14 Measured and Predicted Pond Temperatures for the
Mt. Storm Plant 60
15 Measured and Predicted Pond Temperatures for the
Four Corners Plant 61
16 Evaporation Loss Parameter for Mixed Pond in Steady
State as a Function of Temperature 64
vm
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17 Evaporation Loss Parameter vs Temperature
for a Slug Flow Pond Operating in Steady State 67
18 Temperature of Mixed Ponds, 2000 MWg Plant,
Design Climatic Conditions 72
19 Evaporated Water from Mixed Ponds near Phila-
delphia, Pa. and Winslow, Ariz., Design
Climatic Conditions 73
20 Temperature of Slug Flow Ponds for a 2000 MW
Plant located near Philadelphia, for Design
Conditions 74
21 Transient Temperature of Mixed Pond for a
2000 MW Plant near Philadelphia 76
e
22 Equipment Cost for Captive Cooling Systems 83
23 Exhaust Pressure Correction Curve 85
24 Condenser Back Pressure for a 2000 MW Plant
Q
near Philadelphia, Mixed Pond Cooling,
Design Climatic Conditions 86
25 Condenser Back Pressure for a 2000 MW Plant
£
near Philadelphia, Slug Flow Pond,
Design Climatic Conditions 87
26 Lost Capacity due to Condenser Back Pressure,
2000 Megawatt Electric Plant near Philadelphia,
Mixed Pond, Design Climatic Conditions 88
27 Lost Capacity due to Condenser Back Pressure,
2000 Megawatt Electric Plant near Philadelphia,
Slug Flow Pond, Design Climatic Conditions 89
28 Annual Cooling Cost, Mixed Pond - 2000 MW Plant,
Land and Development Cost - $500/Acre of Pond,
Design Climatic Conditions near Philadelphia 91
29 Annual Cooling Cost, Mixed Pond, 2000 MW Plant,
Land and Development Cost - $2000/Acre of Pond,
Design Climatic Conditions for Philadelphia 92
30 Annual Cooling Cost, Mixed Pond, 2000 MW Plant,
Land and Development Cost - $5000/Acre of Pond,
Design Climatic Conditions for Philadelphia 93
31 Annual Cooling Cost, Slug Flow Pond, 2000 MW Plant,-
Land and Development Cost - $500/Acre of Pond,
Design Climatic Conditions for Philadelphia 94
IX
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32 Annual Cooling Cost, Slug Flow Pond, 2000 MW Plant,
Land and Development Cost - $2000/Acre of Pond,
Design Climatic Conditions for Philadelphia 95
33 Annual Cooling Cost, Slug Flow Pond, 2000 MW Plant,
Land and Development Cost - $5000/Acre of Pond,
Design Climatic Conditions for Philadelphia 96
T *? 0
B-l Evaporation Rate vs Wind Velocity for Lake Colorado
City, Lake Hefner and the Meyer Equation
B-2 Comparison of Measured and Observed Pan Evaporatio 123
over the Atlantic Ocean
B-3 Elevation over Open Grass Land vs Ci 129
B-4 Evaporation Rate Given by Various Empirical Equations
and Eq. B-16, Summer Conditions T = T 131
\V cL
B-5 Evaporation Rate Given by Various Empirical Equations
and Eq. B-16, T > T 132
^ w a
B-6 Evaporation Rates given by Various Empirical Equations
and Eq. B-16, Winter Conditions, T = T 133
w a
B-7 Evaporation Rates Given by Various Empirical Equations,
and Eq. B-16, Winter Conditions, T > T 134
w a
C-l Sketch of the Wilkes Plant Pond 143
C-2 Wilkes Plant: Measured Temperature-Depth Profile -
January and June 1968 147
C-3 Kincaid Plant and Cooling Pond 153
C-4 Cholla Plant: Cooling Pond 155
C-5 Mt. Storm Plant and Cooling Pond 157
C-6 Mt. Storm Plant: Measured Temperature-Depth
Profile, August 158
C-7 Four Corners Plant: Measured Temperature-Depth
Profile, July 1969 159
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LIST OF TABLES
Title Page
Summary of Plant Characteristics r*
Cost of Equipment in Millions of Dollars 32
145
C-l Meteorological Data fpr Shrevesport, La. - 1968
C-2 Wilkes Data - 1968 148
C-3 Meteorological Data for Springfield, 111. - 1968 150
C-4 Kincaid Data - 1968 153
C-5 Meteorological Data for Winslow, Ariz. - 1967 -,rj
C-6 Cholla Data - 1967
(Effective Area =1/3 Actual Area) ^ 58
C-7 Cholla Data, 1967
(Effective Area = Actual Area) 159
C-8 Meteorological Data for Elkins, W. Va. - 1968 161
C-9 Mt. Storm Data - 1968 155
C-10 Four Corners Data - 1967
170
xi
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CONCLUSIONS
1. The simplified technique presented in the form of design curves
in the report can be used in conjunction with climatic data from nearby
Weather Stations to predict the monthly average value of condenser
inlet water temperature within about + 5 F. If this accuracy is suf-
ficient, it is not necessary to proceed to the more particular analysis
in this report.
2. Condenser inlet temperatures for a cooling pond can be made to
approach temperatures associated with once-through river water cool-
ing if the ratio of pond surface area to electric energy generation is
made equal to or greater than approximately 4 acres per megawatt.
3. For regions of the United States where the humidity and rainfall
are moderate to high, a cooling pond size can be selected which will
result in condenser inlet water at the same temperature or lower than
water from natural draft cooling towers. Likewise, for such ponds,
the amount of water evaporated in the natural draft towers will exceed
the difference between the water evaporated from the pond and the
water gained by the pond in the form of direct precipitation.
In semi-arid regions of the United States the loss of water by evapora-
tion from the pond surface will be substantially greater than in other
locations which have the same equilibrium temperature. This addi-
tional water loss may result in an economic advantage for cooling
towers in semi-arid regions.
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RECOMMENDATIONS
1. Of the two analyses presented in this report, namely, a general
analysis and a simplified analysis, comparisons between measured
and predicted water temperatures were made only on the basis of the
simplified analysis because the general analysis requires an iterative
solution which is prohibitively tedious if done by hand. It is recom-
mended that the general analysis be computerized so that it would be
necessary to supply only the monthly weather conditions as obtained
directly from Weather Bureau data and the monthly power plant load-
ings as input data in order to obtain condenser inlet water temperature
and total water loss by evaporation as the output of the computer pro-
gram. Furthermore, it is recommended that operating data for a
longer period of time than used in the present report be sought for the
two most sensitive (high ratio of thermal waste energy to pond surface
area) ponds representative of high and low humidity regions respect-
ively. Two such ponds couldbe the Mt. Storm pond in West Vriginia
and the Four Corners pond in New Mexico.
It is anticipated that the predictive technique could be refined by use
of the computerized general analysis so as to make possible the pre-
diction of cooling water temperatures to within less than + 5°F.
2. It is recommended that a program be initiated to collect data on
operating cooling towers located in various regions of the United States
and compare the engineering-economic parameters of towers and cool-
ing ponds at the various sites. The cooling tower data to be collected
in this program could be used to evaluate recently developed predict-
ive techniques for cooling towers such as that reported by Winiarski,
Tichenor and Byram [31] and by Llung and Moore [32j.
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INTRODUCTION
At the present time the major portion of electric power is generated in
thermal power plants. That remaining is generated in hydroelectric
facilities. This distribution between thermal and hydroelectric plants
cannot be substantially shifted to remove the load from the thermal
plants and in fact will move in the other direction. As a result of this
rather static distribution and with the tacit assumption that no signifi-
cant breakthrough will be made in the near future on a commercial
scale (electric utility level) in direct energy converting methods, con-
siderable engineering effort is now being expended to reduce the over-
all cost of electric power generation in thermal plants. To date this
pressure has lead to the adoption of three concepts, namely: nuclear
plants, mouth-of-mine coal driven plants and very large fossil fuel
plants. Unfortunately, all three concepts sharply increase the thermal
pollution burden.
In the case of nuclear plants, the increased thermal pollution danger
arises as a result of their lower energy conversion efficiency. Such
plants are "heat-engines" in the thermodynamic sense, and are thus
subject to the Carnot efficiency limit which dictates that the efficiency
decreases as the temperature difference between the steam generation
temperature and the surrounding (usually a river or ocean) tempera-
ture decreases. At present the temperature at which steam can be
generated in a nuclear plant is considerably below the corresponding
temperature in a fossil plant. As a result, the thermal energy rejected
to the surroundings in the form of heat transfer to a river or ocean per
unit of electric energy generated is greater in the nuclear plant. How-
ever, because nuclear thermal energy is less expeasive than fossil
fuel in some parts of the United States, the overall cost of generation
per unit electric power may be lower in the new nuclear plants than
organic plants. Nuclear plants to be economically efficient must be
large units so the amount of rejected heat is large and concentrated.
In the case of mouth-of-mine coal plants, the problem of thermal pol-
lution can be sharpened because the location of the plant is dictated by
the source of coal, leaving little room for optimizing location with
respect to thermal energy disposal. In this case also, economic con-
siderations favor very large plants with large localized heat rejections.
In the case of substantially larger coal, oil and gas driven plants, the
danger simply arises from the high concentration of thermal energy re-
jection at one location.
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There is, of course, the possibility that the aquatic thermal pollution
problem will be essentially eliminated if a major breakthrough were
to be made (at the commercial level) with one of the direct energy con-
verting methods which are not "heat-engines" and therefore not sub-
ject to the Carnot efficiency limitation. Anticipation of such a break-
through in the forseeable future does not seem to be realistic.
Faced with the economically dictated need to build large plants (on the
order of 1000 to 2000 megawatts of electric power) and relatively few
sources of fresh water that can accept the thermal waste from such
plants without violating the State-Federal water quality standards as
applied to aquatic thermal pollution, the electric power generating in-
dustry can no longer anticipate the unrestricted use of natural waters
for thermal energy sinks and must now look for more acceptable heat
sinks. The possibilities are the ground, the ocean, the atmosphere,
and ultimately space.
This report is concerned with the use of the atmosphere as the ultimate
heat sink and an isolated cooling pond as the intermediate thermal sink.
In particular, predictive models are developed and their validity is
assessed by comparing predicted and measured pond temperatures
under various climatic conditions and power plant loads. The economic
influence of the pond on the capital cost and on the operating cost of the
plant is also developed.
In order that the results of this work can be readily used, they have
been presented in a set of "design curves" so that it is not continually
necessary to return to the basic calculations.
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FACTORS REGULATING HEAT TRANSFER
A Brief Historical Review
A cooling pond serves its function as an intermediate heat sink by re-
ceiving the thermal energy rejected in the plant condensers and subse-
quently rejecting that energy to the atmosphere. Energy is added to
the pond water by direct heat transfer in the condenser of the power
plant, by absorption of short-wave solar radiation, by absorption of
long wave atmospheric radiation and by make-up water that is received
by the pond. Energy is removed from the pond by thermal radiation,
by conduction to the atmosphere, by evaporation and by water which
flows from the pond. Heat transfer between the pond water and the
ground can be safely neglected when compared to the other quantities
listed above.
The energy of the pond, and hence its temperature, at anytime is de-
termined by the time history of the various mechanisms that put energy
into or remove energy from the pond.
The concept of calculating the temperature of a natural body of water
by taking such an energy balance appears frequently in the literature.
One of the earliest discussions was presented by Cummings and Rich-
ardson in 1927 [l],
Lima [2] in 1936 was one of the first workers to compare measured
power plant cooling pond temperatures with predicted values. Lima
developed a set of en pirical curves from data collected by various power
power companies tha* had operated cooling ponds. These curves could
be used to determine an overall pond-to-atmosphere heat transfer coef-
ficient (K = heat dissipated to the atmosphere per unit area per unit
time per unit difference between the water vapor pressure in the air
and the water vapor pressure corresponding to the pond surface temp-
erature) if one knew the wind speed, air temperature and power plant
loading on the pond. Although Lima's empirical curves could be used
to predict the temperature of the ponds used in the study to within ap-
proximately +_ 5 F, they could not be confidently extrapolated to condi-
tions beyond those studied. Likewise this empirical approach led only
to the prediction of the mean pond temperature without giving insight to
the influence of vertical or longitudinal temperature gradients, or the
influence of the thermal capacity of the individual ponds involved in the
study.
Later; in 1951, Throne [3] devised an analytical procedure for pre-
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dieting cooling pond temperatures based on the energy balance con-
cept. Throne presented his technique in a set of curves from which
pond temperature could be determined if one knew the air tempera-
ture, wind speed and power plant loading. Throne compared his pre-
dicted results with data from each of 298 months of operation of a
plant in Colorado. The predicted and measured values agreed within
+_ 5°F for the vast majority of cases. However, in order to construct
the required curves, it is necessary to know the measured equilibrium
temperature of the lake in question when no waste energy load is being
imposed by the power plant. In addition, the curves assume a uniform
lake surface temperature and a 4°F vertical temperature variation
from surface to bottom. As in the case of the empirical curves de-
vised by Lima, Throne's technique does not account for non-steady
state behavior of the pond nor is the validity of extrapolation obvious.
In 1953 Langhaar [4] presented an analytical technique, based on the
energy balance concept, which was general enough to predict cooling
characteristics of ponds without the necessity of measuring the equili-
brium temperature of the particular pond when no waste energy load
is being imposed by the power plant. Likewise, his approach was cap-
able of being used to take into account the influence of longitudinal
temperature gradients and non-steady state operation. Langhaar's work
had been proceeded by proposed analytical techniques for determining
the cooling capacity of flowing streams by Le Bosque [5j in 1946 and
by the extensive experimental energy balance study on Lake Hefner as
reported, for example, by Anderson [6] in 1952. Langhaar presented
his work in the form of nomographs and compared his results against a
single pond in which a noticeable longitudinal (that is, in the direction
of flow) temperature gradient existed in contrast to the pond investi
gated by Throne in which the entire pond surface was found to be at a
uniform temperature.
In 1959 Velz and Gannon [?J modified the work of Langhaar and com-
pared predicted values of temperature for a cooling pond with a pro-
nounced longitudinal temperature gradient located in Shreveport, La. ,
and for a river in Michigan that received waste thermal energy. In
both cases the agreement between predicted and measured tempera-
tures was within approximately _+ 5°F.
Messinger [8] reported on an experimental study on a thermally loaded
stream in 1963. In this study the energy budget technique was used to
predict the temperature profile along a section of the West Branch of
the Susquehanna River in Pennsylvania below a point at which waste
heat is added to the river. The predicted temperatures were as much
as 5°F higher than measured values (based on four hour study periods).
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Messinger attributed the discrepancies to inadequacies in the measure-
ment of solar and atmospheric radiation of partially shaded water sur-
faces.
A comprehensive review of the energy balance technique was reported
by Edinger and Geyer in 1965 [9]. Edinger and Geyer present the
cooling capacity of a body of water in terms of a heat exchange coeffi-
cient by linearizing the energy balance equation. The results of this
study are presented in equation, chart and table form for ponds with
and without longitudinal temperature gradients. They applied the lin-
earized energy equation to the steady state steady operation of a "mixed"
pond (that is, one without longitudinal or vertical temperature grad-
ients) and to a "flow through" pond (that is, one with a longitudinal
temperature gradient but without a vertical temperature gradient) in
order to compare the two modes of pond operation. Although no sub-
stantial comparison is made between predicted and measured tempera-
tures for these two modes of operation, the authors and others [10]
later; in 1968, reported on a number of field sites that have been sel-
ected for the purpose of gathering rather extensive data on the cooling
characteristics of river, lake and tidal plants.
A Review of the Energy Balance Terms
The significant energy fluxes for a cooling pond are shown in Figure 1.
Each flux term is discussed in the following pages; however, before
considering the radiation terms, it is helpful to recall that Stefan's
Law states that all bodies radiate energy by electromagnetic waves and
do so at a rate proportional to the fourth power of their absolute terrv-
perature. These electromagnetic waves are not monochromatic, but
rather cover a range of wavelengths. The energy transported by these
waves is not the same at all wavelengths, but rather it is highly con-
centrated around a wavelength A given by Wien's Law as ^rn = Ci/T
[11 j, where Cj is a constant and T is the absolute temperature of the
body radiating energy. Thus the hotter the radiating body, the shorter
the wavelengths at which most of the energy is concentrated and vice
versa. Thus the sun, which is at a surface temperature of approxi-
mately 11,000 R will give off most of its energy at wavelengths which
are short compared to wavelengths given off by the pond which has a
surface temperature of approximately 500° to 570 R or, for example,
compared to radiation given off by water vapor and by carbon dioxide
that might be present in the atmosphere at temperatures of the order
of 400° to 600°R.
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1) Q = solar radiation
incident on pond
2) Q = solar radiation reflected
s
r from pond
3) Q = atmospheric radia-
tion incident on
pond
4) Q = atmospheric radiation
ar
reflected from pond
5)Q = back radiation from pond
surface v
8)m c0 = enthalpy
PPi PPi flux —
into pond
due to
condenser
water
10) rn c0 = enthalpy flux
P P «—.
into pond due
to direct
precipitation
12) m c0 = enthalpy
mu mu -
6)m h = enthalpy flux out of
pond due to evapora-
tion
7) Q = energy flux out of pond due
to heat conduction v
9)m c8 = enthalpy flux out
PP PP
o o of pond due to
condenser water
11) m c6 = enthalpy flux ouf ot
pond due to seeping
water and outflow
flux into
pond due
to inflow
and make-
up water
Note: All terms are in units of energy per unit time per unit of pond
surface area.
Figure 1 - Energy Balance Terms for a Cooling Pond
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Solar Radiation (Q }
s
According to Wein's Law the value of Am for the sun at a surface tem-
perature of about 11, 000°R is approximately 0. 5 microns. In particu-
lar,. Wein's Law indicates that 99% of the sun's radiated energy is
associated with wavelengths shorter than 4 microns. The amount of
radiation from the sun which reaches the outer atmosphere of the earth
is a function of the time of day, latitude, and the season and can be cal-
culated without undue difficulty. However, only some of this radiation
strikes the surface of the earth as short wave (4 microns or less) rad-
iation. The amount that does strike the earth surface is called short-
wave solar radiation or simply solar radiation.
Of the total radiation from the sun which strikes the outer atmosphere
of the earth some is reflected back into space, some is transmitted
through the atmosphere and strikes the earth surface (the solar radia-
tion) and some is absorbed by the gases in the atmosphere, primarily
by ozone in the upper atmosphere and water vapor and cloud cover. As
a result of the complexity of these three possibilities, the solar rad-
iation term (Q ) is more reliably measured than calculated. The in
S
strument used to make this measurement is a pyranometer. This de-
vice consists of a flat horizontal circular disc which is housed inside
a lime-glass bulb. The disc is separated into a white surface center
circle and a white surface outer ring by a blackened intermediate ring.
Thermopiles are used to measure the temperature difference between
the black and white surface. This temperature difference is a function
of the radiation flux penetrating the lime-glass bulb. The bulb mater-
ial is selected so that the device will be sensitive to radiation of •wave-
lengths equal to 4 microns or less. Selected local weather bureau sta-
tions routinely measure the solar radiation and annual summaries of
these data are available from the National Weather Records Center,
Asheville, North Carolina. Values of the solar radiation, averaged
over a period of years, are given in map form for the United States in
Ref. [12].
The solar radiation (Qs) averaged over a twenty-four hour period var-
ies with the geographical location and the time of year in the range of
400 to 2800 btu/ft2 day.
•
Solar Radiation Reflected from the Pond Surface (Q )
^ sr
A convenient way to characterise this quantity is to state the ratio be-
tween the solar radiation reflected from the pond surface and the solar
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radiation incident upon the pond surface, thus, Rsr = Qsr/Qs- This
ratio is the solar reflectivity which has been measured and reported
in the literature. In. particular, empirical reflectivity curves which
show the solar reflectivity as a function of the sun altitude for various
cloud cover were developed in the Lake Hefner studies as reported by
Anderson [6]. These empirical curves are reproduced here for con-
venient reference in Figure 2.
CO
4)
0)
bo to
CLEAR
HIGH
CLOUD
,K> .IS .10 IS
O .OS .10 .13 .10 ,t5 0 .OB .10 .13 .10 .J3 0
Solar Radiation Reflectivity, R
S = clouds scattered (I/ 10 to 5/10)
B = broken cloud cover (6/ 10 to 9/10)
O = overcast (10/10)
Figure 2 - Solar Radiation Reflectivity
(after Anderson [6j)
The cloud cover is characterized by the amount of cover as measured
in tenths of the total visible sky covered by clouds and by the height of
the cloud cover with "high" clouds designating those above 20, 000 feet
and "low" clouds designating those below 6500 feet. Although the solar
reflectivity will vary during the day as the sun altitude changes and thus
in principle the reflectivity and solar radiation should be known as a
function of the time of day, an average value of each has been used in
this work. The Weather Bureau reports the solar radiation, for exam-
ple, as energy per day per unit area averaged over a month. The aver-
age solar reflectivity was determined from the curves by using the
average sun altitude during sunlight hours for each month. Sun alti-
tude as a function of latitude and time of day is readily available in the
literature _ for example, see Ref. [l3j.
The average value of Rgr is in the range of 0. 04 to 0. 12 for the United
States.
10
-------
Atmospheric radiation incident on the Pond Surface (Q )
a
In contrast to the short wavelength (< 4 microns) associated with the
solar radiation, the wavelengths associated with the electromagnetic
radiation given off by the gases that constitute the earth's atmosphere
are predominately long, from 4 to 1ZO microns. The amount of this
long wave radiation which strikes the surface of the earth is known as
long-wave atmospheric radiation or simply atmospheric radiation.
Also in contrast to the solar radiation, the atmospheric radiation is
present in the night-time as well as daytime and on completely cloudy
days as well as sunny days. The intensity of this radiation is a com-
plex function of several parameters, including the ozone, water vapor,
and carbon dioxide content and distribution in the atmosphere and the
atmospheric temperature. Observations indicate that the atmospheric
radiation depends primarily on the air temperature and water vapor
content and increases with an increase in either of these two quantities.
It is this last characteristic of atmospheric radiation that results in a
substantial lowering of the temperature of a pan of water left exposed
to the night sky in an arid or semi-arid climate. Under these condi-
tions there is little water vapor in the air and as a result the pan of
water receives little sky radiation at night but does continue to rad-
iate energy to space essentially as a black body and hence experiences
a substantial energy or temperature depression.
Atmospheric radiation can be measured directly during the night using
a Gier-Dunkle flat plate radiometer or a Thornthwaite net radiometer
which measure all the radiant energy received on a blackened surface.
At least in principle i:he atmospheric radiation can be measured in the
daytime by taking the difference of two measurements, namely, the
total of atmospheric and solar radiation as measured on a radiometer
and the solar radiation as measured by a pyranometer. In practice
these measurements are not taken routinely by the Weather Bureau sta-
tions, and the atmospheric radiation for a particular site and time must
be estimated from one of the several empirical equations that have
been developed [14, 15]. One such equation that has been extensively
evaluated is that due to Brunt, as described by Koberg [9, 15], namely,
Q =CT€(T + 460)4 (C +. 223-/P") btu/ft2 day (Eq. 1)
a a ±5 a
Q ^ *
where CT = Stefan-Boltzmann Constant = 4. 12 x 10 btu/ft day (°R)
C = surface emissivity assumed constant of . 97
T = air temperature, °F
C = Brunt's coefficient. For convenience it is given in Fig. 3
11
-------
u
R = Ratio of measured to clear
sky radiation (Fig. 4)
28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92
Air Temperature °F
Fig. 3 - Brunt Coefficient (after Koberg - 1962)
3000 -
Feb Mar Apr May June July A.u^ Sept Get Mov Dec
Fig. 4 - Clear Sky Solar Radiation (After Koberg - 1962)
12
-------
as a function of the air temperature and the ratio of the
actual measured solar radiation (as obtained from Wea-
ther Bureau data) to the solar radiation that would be re-
ceived if the sky were clear (as obtained from Fig. 4).
P = water vapor pressure in the air, psia
ct
The atmospheric radiation intensity will vary with climatic conditions
and latitude but is approximately in the range of 1200 to 3000 btu/
ft^ day for most parts of the United States.
Atmospheric Radiation Reflected from the Pond Surface (Q )
_ ar
The reflectivity ( - Qar/Qa) of a water surface for atmospheric radia-
tion was shown to be approximately constant and equal to about 0. 03 by
Gier and Dunkle and reported in Ref. [14], pages 96-98. Thus the
atmospheric radiation reflected from a water surface may be conven-
iently taken as . 97 Q .
1 a
If a particular site that has been selected is under experimental study,
it is not necessary to measure the four radiation terms given above
separately, since they may be combined to give the net absorbed radia-
tion Qivr = Qc - QOT. + Q - CL , which in turn can be measured dir-
•LN & s J. cL 3- A
ectly by means of a Cummings Radiation Integrator (CRI) or with a
Gier-Dunkle and Thornthwaite device. The CRI consists of a shielded
shallow pan of water. The water volume is maintained at a constant
level and its temperature is taken in order to measure the net radia-
tion absorbed between specified time periods.
Back Radiation from the Pond Surface (Q, )
br
Water radiates almost like a perfect black body and since the water
temperature is in the vicinity of 50°F, the wave lengths will be long
(> 4 microns) compared to the incoming solar radiation and compar-
able to the incoming atmospheric radiation. Thus the back radiation
emitted by the water may be expressed as
2)
hS-l. W &
where C = emissivity of water surface assumed constant at .97
Wr
T = water surface temperature, F
8
It should be noted that the energy balance terms selected for this anal-
ysis allow for the independent evaluation of each radiation flux term.
13
-------
In order to emphasize this point it is helpful to consider the net
amount of energy lost by radiation from a body of water (which is al-
ways maintained at some constant temperature) on a clear night and
on a cloudy night. The amount of back radiation (as given by Eq. 2)
will be the same for both nights; however, the incoming atmospheric
radiation will be appreciably less on the clear night than on the cloudy
night. As a result the net energy lost by radiation is greater during
the clear night than during the cloudy night.
Since the water surface temperature may vary from 32 F to 120 F,
the back radiation can vary in the range of 2400 to 4500 btu/ft2 day.
Enthalpy Flux out of the Pond Due to Evaporation Water (rr^h^)
The enthalpy flux that leaves the pond as a result of evaporating water
is determined by the product of two terms, namely, the specific en-
thalpy (he) per pound of water vapor leaving the air-water boundary
and the rate at which water vapor leaves the air-water boundary (me).
The first term is a well documented thermodynamic property of water
and can readily be found for a given water vapor condition. The magni-
tude of the second term, however, depends on many factors, most not-
able of which are the average wind speed, vertical profile of wind speed,
the water surface temperature, and the water vapor pressure in the air.
The rate of evaporation of water from a natural body of water into the
atmosphere has been the subject of both analytical and experimental
study for a considerable length of time.
Of the analytical work the most prominent is that of Sverdrup [16] in
which he compares his predicted values with values based on observed
evaporation from an open pan on the deck of a ship at numerous loca-
tions.
1) The change in enthalpy between the liquid phase and the vapor phase
at the same pressure and temperature is known as the latent heat of
vaporization (L). It should be noted that in evaluating the energy bal-
ance on the cooling pond in the present study, the energy crossing into
or out of the pond has been identified separately with the result that
the latent heat vaporization, which is a difference in energy between
two phases at the same temperature and pressure, does not enter the
argument explicitly. This approach leads to flexibility in the analysis
in the sense that the make-up water temperature can be selected or
specified independent of the temperature at which water leaves the pond
by evaporation.
14
-------
Considerable experimental work has been done in order to formulate
empirical equations for the rate of evaporation under a various clim-
atic conditions. Although there is no way to directly measure the rate
of water evaporating from the surface of a natural body of water, it is
possible to obtain an indirect measure by making a water mass balance
study on the body of water. This is not an easy task for it requires an
accurate estimate of all inflows and outflows. The most comprehen-
sive experimental water budget study was that undertaken at Lake Hef-
ner, Oklahoma by a combined task force including the Geological Sur-
vey, the Weather Bureau, the U. S. Navy and the Bureau of Reclama-
tion. Marciano and Harbeck [19J showed that for Lake Hefner the rate
of evaporation could be found by use of the quasi-empirical equation
m = N W(P - P ) (Eq. 3)
e n w a
where N = an empirical coefficient for a particular lake when
evaporation is averaged over n days
W - wind speed
P = saturation water vapor pressure corresponding to
the temperature of the lake surface
P = water vapor pressure in the air above the lake
cL
Equations similar to the one given above have been developed by sev-
eral other workers. Prominent among these additional equations are
the ones reported by Koberg et al [20] for Lake Colorado City, Texas,
and the one reported by Meyer [21].
In addition to the analytical work and the water budget experimental
work referred to in the above discussion, the rate at which water is
evaporated from open pans placed on or somewhat above the ground
has been measured and reported by the Weather Bureau. Equations
are available in the literature for use in estimating the rate of evapor-
ation from natural bodies of water based on nearby pan evaporation
data.
The rate of evaporation appears to be highly dependent on the local top-
ography because of the resulting wind structure with the result that the
experimentally determined constants in the various reported semi-
empirical equations vary 19 the order of +_ 25% or more with respect to
computed versus actual evaporation rates. In addition to this variation
it is noted that all of the previous work has been limited to water at the
natural temperature or only a few degrees higher (as a result of other
than natural heat addition).
In view of the fact that evaporation accounts for the major portion of
15
-------
the energy transfer from the pond (approximately 40 to 70% depending
on the time of year), it is important to be able to make reliable estim-
ates of the rate of evaporation for ponds where specific experimental
evaporation data are not available, in particular for ponds that may be
subjected to sufficient heating to increase the water temperature to as
much as 30°F above the natural lake temperature.
In order to provide guidelines for making such estimates, an evapora-
tion equation was developed in Appendix B. However, it is not m
convenient form to use in the simplified analysis to be developed in
this report. In Appendix B it is demonstrated that the evaporation
equation proposed by Meyer with a particular value of the empirical
constant gives results reasonably close to those predicted by the dev-
eloped equation and is much easier to use. As a result, the following
form of the Meyer equation has been used to estimate the evaporation
rate in the simplified analyses:
m = (a, +a., W )(P -P) #m/ft2 day (Eq. 4)
e 1Z 13 m w a
where a , a - constants
J- L* J- .5
W = monthly average wind speed as obtained from
measurements taken at the nearest weather station
about 25 feet above the surface
Energy Flux out of Pond due to Convection (Q )
Since the process of convection of energy, that is, heat transfer, from
the water surface into the air above is similar to the avaporation of
water from the surface into the air above, it is possible to develop an
expression for the ratio of energy flux due to convection to the enthal-
py flux due to evaporation. Bowen [ZZJ, using diffusion theory, was
the first one to develop such a ratio. He considered three special
cases for which it was possible to obtain analytical solutions to his gen'
eral equation. Again, in order to evaluate the influence that heated
water may have on this ratio, the ratio was derived in Appendix B by
making use of the known relation between the mass transfer coefficient
and the heat transfer coefficient for smooth surfaces. The ratio was
found to be given by the expression: [See Eq. B-31)
V °< 0047, 'Tw - V PBAR
y - P • °°476
(P -P )14.67
e e w a
16
-------
where T = temperature of the water surface, F
T = air temperature, F
cL
P D = barometric pressure, psia
13 .A. xx
The above expression for y is the same as that given by Bowen with
the exception of the numerical coefficient which, however, is within
the range of the three cases calculated by Bowen.
Enthalpy Flux into (rh c9 ) and out of Pond (m c0 ) due to
- - PPi PPi -- — PPo PPQ -
Water Circulated through the Condenser
The rate at which cooling water is pumped through the condenser
(M.™. = MT-.T, = }vi ) and the temperature rise across the condenser
PPi PPo c
(0Dr> - 8 = AT ) are related to the waste thermal energy from the
c
plant by the equation:
M c(AT ) = WTE (Eq. 5)
c c
where c = specific heat of water, 1 btu/ F #m
Mpp. =
where A = pond surface area, ft
m = flow rate out of condenser per unit
of pond surface area, #m/ft day
where mpp = flow rate into condenser per unit
of pond surface area, #m/ft day
WTE = waste thermal energy from the plant to the pond,
btu/ ft2 day
Enthalpy Flux into Pond due to Direct Precipitation (m c9 )
_______ _ P P
The mass flow rate of direct precipitation into the pond will, of course,
vary with the time of year and location. The annual average values
fall between 10 inches/year in the semi-arid regions of the West and
100 inches/year in some areas of the Northwest. The direct precipita-
tion may be in the form of water, snow or hail. Thus the specific en-
thalpy can be expressed as:
17
-------
h = c9p. btu/#m for rain (0p. > 0) (Eq. 6)
i
h = -{-.4920p + 143.3} for ice and snow (9p. < 0) (Eq. 7)
Pi '
Enthalpy Flux out of Pond due to Seepage and Outflow (m c9g)
Again the specific enthalpy may be expressed in terms of the heat
capacity and the temperature of the water seeping and flowing out of
the pond, namely, h = c0 . The rate of seepage out of the pond or
lake is highly site dependent and no useful generalization can be made.
Enthalpy Flux into Pond due to Addition of Inflow and Make-up Water
(m c6 )
mu mu
The rate at which make-up water is added to the pond will depend on
the source of the make-up water. For example, if the make-up is run-
off water that is drained into the pond from surrounding land or is
provided by the inflow of a small stream, the make-up rate will vary
with the local precipitation and possible melting of snow cover. If, on
the other hand, a river, a lake or a reservoir are used to provide a
source of make-up, the rate may be made constant.
The specific enthalpy of the make-up water may be expressed as:
8)
18
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POND OPERATING CHARACTERISTICS
In addition to the climatic conditions imposed on the pond, its own
hydrodynamic and energy storage behavior will influence its capacity
to dissipate the waste heat received from the power plant. The com-
plete problem of pond hydrodynamics behavior would involve specifi-
cations of the inlet and outlet geometry, water condition, the shape and
depth of the pond, and the wind speed and direction. In addition, con-
servation of mass and energy equations and the equations of motion
for the pond would have to be solved in order to determine precisely
how the hot -water travels through the pond, how it mixes, and finally
the extent to which it is cooled in the process. Such a task is formid-
able if at all possible under the prescribed conditions. A simplified
approach is to be taken.
Since the problem involves three dimensions and time with associated
flow and turbulent mixing, it would seem helpful to break the problem
into 1) longitudinal flow and mixing (that is, in the direciicn of water
flow), 2) lateral flow and mixing (that is, perpendicular to the flow,
3) vertical flow and mixing and 4) the effect of thermal storage cap-
acity. Each of the four above considerations is discussed below with
the objective of developing a simplified approach to the complete pro-
blem.
1 & 2 The flow in the pond may, in the extreme cases, have a very
pronounced flow direction with little turbulent mixing or it may have
considerable turbulent mixing and no pronounced flow pattern. The
first extreme is conveniently called "slug flow" since the water dis-
charged from the condenser at a given time will tend to move through
the pond as a "slug" without mixing substantially with the water ahead
or behind it. The second extreme is referred to as a horizontally
mixed pond in the sense that the temperature of the water will be con-
stant in any horizontal plane as a result of horizontal (longitudinal
and lateral) flowing and mixing.
Toward which of these two extremes and to what extent a given pond
will operate depends on the geometry of the pond, the inlet and outlet
structures and the wind.
Slug flow operation is favored by a long narrow pond with inlet and
outlet at the two ends. If a pond is sufficiently long compared with its
width, the temperature in a horizontal plane will, for steady state con-
ditions, be a function of longitudinal distance only; lateral mixing due
to turbulence and density currents will exert sufficient influence to
19
-------
eliminate any lateral temperature gradients. The rate at which the
water flows through a given pond for a given plant is proportional to
the cooling water pumping rate.
Horizontally mixed pond operation is favored by a near unity ratio of
length to width with the outlet structure designed to float the hot water
on the surface aided by a wind that blows the discharge away from the
outlet structure.
In view of the fact that the cooling capacity is a surface area phenom-
enon, and since the heat release from the pond by evaporation, con-
duction and back radiation all inc.rease as the water temperature in-
creases, the cooling capacity of a slug flow pond is greater than that
of a mixed pond, all other conditions being held constant. The added
cooling capacity of the slug flow pond is a result of the continued higher
temperature at the pond inlet rather than the immediate achievement
of a uniformly mixed but lower temperature. In a mathematical sense
a mixed pond can be considered as a slug flow pond in which the tem-
perature rise through the condenser is allowed to approach zero while
the pumping rate is allowed to approach infinity since the temperature
of the water in the slug flow pond would then be everywhere equal at a
given instant.
In a given pond the surface area which is actively engaged in exchang-
ing heat with the atmosphere may be equal to the actual water surface
but in some cases will be less or more than the actual -water surface.
The decrease in effective area may result from channeling of the flow
or from the creation of "dead-water" zones. The increase in effective
area can result from wind generated waves.
_3 Vertical flow is induced in water when the top water cools, becomes
more dense, and subsequently sinks. In addition vertical mixing may
be present as a result of turbulence. Vertical flow is inhibited and in
the limit prevented when the upper layers are heated so as to become
buoyant as witnessed by the development of a. thermocline during the
summer in natural lakes. A vigorous wind will assist in mixing the
upper portion of a body of water.
The prediction of vertical temperature gradients in natural bodies of
water is a complex matter which has recently come under theoretical
and experimental study [23, 24j. The predictive techniques have not,
however, reached the state of development where they can be conven-
iently incorporated into a study of the present nature. In order to esti-
mate the influence of a vertical temperature gradient on the cooling cap
acity of a pond we have collected data from several operating ponds and
20
-------
this experimental information will be used to characterize pond oper-
ation.
Since the cooling capacity of a pond is a surface phenomenon and in-
creases with increasing surface temperature, it is desirable to spread
the heated discharge on the surface of the pond in order to enhance the
performance of the pond.
_4 Before dis cussing the question of thermal storage capacity of a
cooling pond, it is helpful to review the question of a natural pond with-
out added heat from a power plant. When a shallow natural pond (that
is, one which does not display a vertical temperature gradient) is sub-
jected to constant climatic conditions, the water temperature will ap-
proach a steady state value known as the "equilibrium" temperature.
The "equilibrium" temperature is the value to which the water will ad-
just itself in order to make the energy transfer into the pond exactly
equal the energy transfer from the pond. Thus, "when the equilibrium
temperature has been achieved and the weather conditions are assumed
to remain constant, there will be no additional change in the thermal
energy stored in the pond water; and hence no additional change in the
pond temperature.
If the natural pond is sufficiently deep, it will tend to divide into two
parts as natural heating progresses from the spring into the summer.
The upper region, or epilimnion, contains circulated rather turbulent
water of nearly uniform temperature which approaches the "equili-
brium" value. The lower region, or hypolimnion, contains relatively
undisturbed water-at a temperature considerably below that of the
epilimnion water.
Like the natural pond, a cooling pond of the "mixed" type, if shallow
and subjected to constant climatic conditions and power plant loading,
will approach a constant temperature. This "steady-state" mixed
pond temperature will be higher than the equilibrium temperature.
The "steady-state" temperature is the temperature that the pond water
will assume in order to balance the energy coming into and leaving
the pond. The time required to bring the mixed pond from some given
temperature to its "steady-state" value depends on the pond depth; the
deeper the pond, the longer the time required. If the pond has not
reached "steady-state" operation, it will be said to be in transient
operation.
A cooling pond of the slug flow type may also operate in either the
steady-state or transient condition. If the pond is initially at some
uniform temperature when the waste thermal energy load from the
21
-------
power plant is imposed and the climatic conditions are held constant,
the pond outlet temperature will rise during the transient phase of
operation and then reach its steady-state value. Again the steady-
state outlet temperature represents the coldest temperature along the
longitudinal temperature profile that the pond will assume in. order to
balance the energy coming into and leaving the pond. The duration of
the transient flow phase will become longer as the pond is made deep-
er, all other conditions being held constant.
Based on the above discussion the general problem may be simplified
by neglecting all lateral temperature gradients and developing a tech-
nique which will yield a solution for any combination of the remaining
six conditions, namely.
Mixed pond or Slug flow pond
Vertical Temperature gradients or no vertical temperature gradients
Steady-state operation or transient operation
The next section presents such a technique in the form of design curves,
22
-------
CURVES FOR PREDICTING WATER TEMPERATURE
In this section curves are presented which can be used to determine
the cooling capacity of a pond operating in any one of the eight possi-
ble configurations resulting from the groups of alternates described
in the previous section: name, 1) mixed or slug flow, 2) vertical
or no vertical temperature gradient, and 3) steady-state or trans-
ient operation.
Case 1 - Mixed pond, steady- state, no vertical temperature gradient
For this case the conservation of energy equation is shown to reduce
to the following in Appendix A (See Eq. A- 15) when certain assump-
tions are made.
= {6PP + [6N - a5 - 71
a , a |_ = coefficients to be used in the empirical equation
for rate of evaporation from the pond surface
(See Appendix B)
It is convenient to label the term in the square bracket of Eq. 9 in a
simple way, namely:
fl = [6N - "5 - (a!2 + a!3W)(al - 9aai4 ~ Pa)J
so that Eq. 9 can be written as:
23
-------
+ [ag + (a
(Eq. 11)
If the value of / and Q are determined, the pond temperature 6 is
easily found by solving lifq. 11. In order to avoid the need for contin-
ually finding solutions to this third order algebraic equation, the equa-
tion is presented in graph form in Figs. 5A and 5B for various values
of (&1? + a!3^)- If the pond temperature is found to be less than 32°F
(6<0), ice would form and the use of Eq. 11 (and hence Fig. 5B) would
no longer be valid. However, Fig. 5B does have an application at a
later point in the analysis.
In order to demonstrate the use of Fig. 5A, consider the example data
given below for the month of July in north -central United States.
WTE = 35. 0 x 109 btu/day
A = 50 x 106 ft2
Q,T = [Q - Q J + [Q - Q J
N s sr a ar
= C2050 - (. 07)(2050)j + [2860 - (. 03)(2860)J
= 4691 btu/ft2 day
P = . 294 psia
3.
6 = 75.8° - 32. 0° = 43. 8°F
a
W = 10. 0 mph
If the value of a and a suggested in Appendix B are used; namely,
a12 = 3730 and a13 = 37J, then the value of the function a^o + a!3W is
7460. The functions / and / + 6 have the following values:
1 1 pp &
f1 = [4691 - 2359 - 6714 (. 089 - (43. 8){. 00473) - . 294)J
fl = 5102 btu/ft2, day
- 51°2 + = 5
Using the above values, Fig. 5A gives a steady state pond temperature
of 6 = 79. OOF.
ss
It should be noted that the natural equilibrium temperature (9eq) for
this location and time of year can be found from Eq. 1 1 and hence also
24
-------
01
o o o o
o o o o
o o o o
m \o t^ co
10,000 20,000
PARAMETER f , or f -f Q , T > 32" F
1 1 pp
Figure 5A
-------
to
s
v
H
c
O
w
rtJ
0)
O
s
3
g.
W
-100
-300
-1000
-3000
PARAMETER f. or f, + Q , T < 32° F
1 1 PP
Figure 5B
-------
Fig. 5A by letting the value of Qpp go to zero. Thus in general the
natural equilibrium temperature is given by the expression:
fl =
> J8
eq
3
eq
2
eq
(Eq. 12)
For the example given above Eq. 12 or Fig. 5A gives an equilibrium
temperature of 75. 5°F.
Case II - Mixed pond, steady-state, with a specified linear vertical
temperature gradient
If the pond is assumed to have a linear vertical temperature gradient
as shown below in Fig. 6,
Fig. 6 - Linear Temperature - Depth Profile
then the surface temperature (6 ) and the bulk average temperature (8)
are related by the expression:
(Eq. 13)
s j
where )3 is the temperature decrease per unit foot of depth, or
'3'
8 = 6 + f(t,/Z) (Eq. 14)
Since back radiation, evaporation and heat conduction are all surface
phenomena, Eq. 12 can be expressed as:
27
-------
[6pp + fl3 = U6 + (a!2 + a!3W)(a2 + a!4)} [6 + ^(V2)}
+ la + a12 + a1
+ [ag + a12 + a13W)(a4)J {9 + /3(^/2)}3 (Eq. 15)
Comparison of Eqs. 12 and 15 shows that in steady-state operation the
temperature of the pond without a vertical gradient will be the same as
the surface temperature of a pond with a vertical gradient. This must
be the case since all heat exchange between the water and the atmos-
phere has been assumed to be a surface phenomenon.
If the condenser intake water structure is so designed that it draws
evenly from all levels so that the inlet water temperature is at the
bulk average temperature, then for the pond with a gradient, the con-
denser inlet water will be lower than in the case of no gradient by the
amount /3(A_/2). The data presented in Appendix C indicate that j3
does not exceed about 1. 0°F/ft even in operating ponds that have inlet
and outlet structures of such a design as to minimize mixing and en-
hance the flow of the hot water on the pond surface.
If a vertical gradient is anticipated, it is of course beneficial to ar-
range the intake structure to preferentially draw water from the lower
layers. If water is drawn from the lower layers, the gradient will
soon be diminished.
Case III - Mixed pond, transient operation, no vertical temperature
gradients
If, as in the case of steady-state operation, certain simplifying as-
sumptions are made, then the energy equation can be reduced to the
following form (See App. A, Eq. A- 13 and Eq. A- 14):
f - [6PP + V - {a6 + (a!2 + a!3W)(a2 + a14)}9
(Eq. 16)
where c = specific heat capacity of water, 1 btu/#m F
p = mass density of water, 62. 4 #m/ft^
V = volume of water in the pond
A = active surface area of pond
28
-------
Eq. 16 can not be solved in closed analytical form with the result that
numerical techniques must be used. Iri order to present the results in
a generally useful way and yet avoid repetitive use of tedious methods,
it is first helpful to note that the first term on the right hand side of
Eq. 16 can be related to the remaining terms on the right hand side by
use of Eq. 12, namely:
where 6 = the steady state temperature at which the given mixed
pond will operate referenced to 32°F (i. e. , 0 =
T - 32°F)
ss
Combining Eq. 16 and Eq. 17 yields:
f = l/cp(V/A) [a6 + (a13 + a^W) (a., + a14)}{ess - 0}
+ l/cp(V/A) [a? + (a12 + aJ3W) (a^j (0^ - 02}
+ l/cp(V/A) [ag + (a12 + a^W) (a4)J [0^ - 03} (Eq. 18}
An inspection of the above equation shows that the mixed pond approach-
es its steady-state temperature in an asymptotic fashion, taking pro-
gressively more time to proceed through the same temperature incre-
ment as the pond comes closer to its steady-state temperature. Eq. 18
is presented in graph form in Fig. 7A through 7G for various values of
the function (3-12 + a-j^W). In principle it is necessary to have a set of
curves for each value of 8 because of the two non -linear terms, name
Z "? ^ Q
!y> (®ss " & > and (0ss " )' However- tiie first term on the right
hand side of Eq. 18 is sufficiently greater than the remaining two right
hand terms so that reasonable accuracy can be achieved by using a lim-
ited family of curves, each member of the family being restricted to a
+ 5°F range of 0SS.
For the present case of a mixed pond in transient operation with no
vertical temperature gradients •, Fig. 7A through 7G was developed by
multiplying Eq. 18 by dt and integrating to obtain the temperature (0)
29
-------
CO
o
CD 20—:--.-•
COOLING POND
TRANSIENT TEMPERATURE
MIXED FLOW
SLUG FLOW
STEADY STATE TEMPERATURE ,1'.;1
SLUG FLOW
: j , .T •sV ,> v ,' r , *•:
AS FUNCTION OF /
.'::•. '. ' .' '. .i. :'
T or T = 80F
ss eq
\- \-\ r i rr-n-f. a,.,-].;-; > > \ i • '< : \ \ • , "fir '<."• > rrr^Ti" ' i" • ;
; '!'" !-:!::- , ;'!',.! '' ' ' I ! |:' ;' ! ' i" I' I f!:: '.
.;,,'; .', '' •,' ;.' i ; ! 1 ' .'. i :
• , ; ! : , . i, i i '•• i' I '' i ' ' i ' , ' • . : .'•>•'
0 -LlJ : ilLili liiUullJiLij
10
-4
5 7
10
jJ,LL.,LJJ _
-3 357
2
Figure 7-A
10
-2
-------
00
20
CD 10
i M!;!
i I • i
\ \ ;;.::5000. J U-!.' i "'!
* I V li i < ' I ! i
!,\\\\
411; STEADY STATE TEMPERATURE r .''
SLUG FLOW
AS FUNCTION OF /
T or T = 70°F
ss eq
0-2 -i-ilj-l' r • »- -,, •, ••- -rrrrr .'':•'•!
i i ! ! i-;!; il!i '•• J • i :! i !'•! i: !••! J I i ' i ! ;;i| '•{ '••
iilij it' , • i ' i !' i :!'!•! ' : ;i J!:;
i i"! 'I'f't i"' n It". • ' ! 'ii i i < i ' • ' ;
1 jrp^; :•.;; iiiTi.li.-ip.ii;:;,]-. -ri
! '.i i:.!.. 'r i ! . ' , ' i .'i ! i ' , : i i:
• I • i ; I r • t - I • . I ' , I • 1 • • I ' I i ' I I : . i • • I , •
'I :
..1-,,-j
.!-_:•:
10
-4
/.
Figure 7B
-------
50, ,
00
CO
40
30
o«
V
CD
CD 20
M
O
T"
i ,
"' '
i;
1 ,
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: .
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n
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i ;
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1 ,!i.':: ;.;.!.! i 1
!: •;:; ' . !•.' ' "' '•
•' i > •-
.|-,j... • , - .., , . ... .,
•\.\r- • • i • •! -;.;
;.' |ii ' '• i;i!.;i; ::i '•
i | -1, • | j ;i ,
i|i ;i >i |'j K; ',<]••_ •••
* 1 " ' 1 '
• i ' • • i , | i
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'• •' ''!• I!1; ' '''•!•
CD
I
CD
CD
<1
ca
CO
10
i i:
r
0-2
1 .,
i
- r
10
-4
i i i i.:
COOLING POND
TRANSIENT TEMPERATURE
MIXED FLOW
SLUG FLOW
;.!:STEADY STATE TEMPERATURE
SLUG FLOW
AS FUNCTION OF /
T or T = 60°F
S3 eq
-
Figure 7-C
-------
09
CO
CD
CD
II
CD
co
(0
p _
r i ' l j i •
t' I f '
ri hn
rt
rrrr
10
COOLING POND
TRANSIENT TEMPERATURE
MIXED FLOW
SLUG FLOW
liLi^U:. l..i'T. U^\]XX ^-LLl
i •!: !' \K.\\ ':, ;\-ii
! i M
0-2l4Jlf
UN |
i i I'i
iTTT!
0.-'-' i
10-4
STEADY STATE TEMPERATURE
SLUG FLOW
AS FUNCTION OF /
T or T = 50°F
ss eq
10
-3
Figure 7-D
-------
(J3
50
•"•,-Tp T|. 1 , . • -|
. : i < j
• t—t-*^»l<.»
' i
40
30
< -c
I •)••
:!', :
-U-U.U
20
'TT'T'I "] -,-Ty.rr-np
! ' I i i '..,'.•
.!,;••''•; ; ! !
.9000
;'..;8000
..-!' i j '! i
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!'iv:i;tir
TT
CD
I
CD
CD
.;. |
1J-
4 ^^ i. U-.!_u.-i. -L-L
7 1C'1
Figure 7-E
-------
so
CO
tn
Tff
rn-r
1.40'
o
CD
I
V
30
(0
(0
20
II
<
I I i ||l|l|i'Mllll!!'' |i|;:|' !IIM'|||!i''T'"l"yi I i '
' :;|lH|l|:!!yi!!gB!l-jt!;i i i i
^^'•;::i,i.\,ii:;:iy^v^iJi! ^-^^ -
j-i|i|j.
m
:il if :il'|.., ':Ui.| „
i'-i'l! •' ',•••••, ' •••[ -r •'!!:•
l • 1 j • I • • i
.Jui-uL.!.,,,!..!.. .;^L!ii..;-.1: :!J_I.,.: ii.'"i|:;,L
COOLING POND
TRANSIENT TEMPER/
MIXED FLOW
SLUG FLOW
STEADY STATE TEMPERATURE
SLUG FLOW
AS FUNCTION OF /.
10
T or T
ss eq
_ !;"!""l'n J ! ,
•i":i i i::." li-i i 1
:T;;;! i .! !''.!''•! !'
•."... i.: !.::.!,..:.. ::i t J L
!W i Hi
:j,jjjj-:.i:::l,i:.ll
= 30°F
., |... |, . .... ! ....
: ',;|::n : j:-|': : • f • -[-'-I i ! ^j !
' ial2''+' ai'3W =l"' '
• : ' ; '!' 'ii.l . ! i '!'!">! . •.• i '• j !"|T • i
^>rM;r?00! Ijjiiyil
•VN'-, '•{ '•'.. i ..-•''' •;' ; i •; ; , ij '•[ "i"
!p'f . C'\ix»!-;'i:'v .: i • ^'7000 i"
j 6000 ; ;
<•, :i i'''i-,'!"';-:iH" ; ,5oob': '• - 1 i
T ,,,... i— ;r.- v -i ;"<, *,;-;•->, . i • "T~7iT " ~i' "~7
: !-;.••/-, v;\l i ; ; :4000
ruRE :.j i ;< v-;..\:V;" i : ;-,-jj , ; i .
i !i j!- $••'•':' ''{\ V ' : ,; '
j-ilt/l !1> I'xK-Jvikii : ; L i:>il
CURE j-'i i .!,• ;.vT^, v i . i
ij1-;.! j i | '..••. .'!, \v :'\ '.'I ;': !;' i '
i v^ l\ \\\i '
: !• ! i •[ !.: ! • 'N^VV-' '
!.-:i:J • 1' :,::! ,ie L^2°F hX^"^::
! ;,--,!, 'i'ii-1 : ' ' : " ;
> t i
! '"
] ' '(
i i
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•.I
i
t
. i '-.
i '.
. . i
.1
' 1 • ;
i . i
• i
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| i
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— j*.-;—
....
.
. : .
. J
i ;
-„:_
•v.
10
10
Figure 7-F
-------
00
O5
COOLING POND
TRANSIENT TEMPERATURE
MIXED FLOW
SLUG FLOW
STEADY STATE TEMPERATURE
SLUG FLOW
AS FUNCTION OF /
T or T
ss eq
Figure 7-G
-------
after any elapsed time (t) as given by Eq. 18 -A.
0 t
Jd9 = / [{a + (a + a W) (a + a )} [6 - 6}
e=8att = 0 t=0 6 2 3 2 4 SS
Eq. 18 -A cannot be integrated analytically and as a result numerical
methods must be used. However ; once Eq. 18-A has been integrated
for a given value of (a,_ + aj_oW) and 0SS, a curve of pond tempera-
ture (0) vs time (t) can be plotted over as wide a range of 0 (or
8__ - 0) as desired. In addition since cn(V/A) can be assumed to be a
constant for any one pond during the period of analyses, the plot of 0
vs t can be made to apply to ponds of various ratios of volume to area
by plotting 0 vs t/cp(V/A) rather than 0 vs t. Fig. 7A through
7G is such a plot. The abscissa has been expressed as f? instead of
t/cp(V/A) because Fig. 7A through 7G will be used for slug flow ponds
also and in this latter application / will have a different definition.
L*
To demonstrate the use of the curves presented in Fig. 7, again con-
sider the example used in Case I. In addition to the previously given
information, the pond volume is 1000 x 10 ft^ and the pond is assumed
to be at a uniform temperature of 70°F on the last day of June. The
temperature of the pond during July is to be determined under the as-
sumption that the weather and waste thermal energy remain constant.
o
Since the steady-state temperature was determined to be 79. 0 F, the
appropriate curve to use is 7A. The following functions must be deter-
mined:
Initial value of (0cc - 0) = (79° - 70° = + 9°F
s s
Initial value of / - This is found from Fig. 7A by using (0SS - 0)- = 9 ,
and (a12 + a13W) = 7460.
/ = 7. 30 x 10" 3
i
The temperature at any day during July can now be readily found by add-
ing to f->. = 7. 30 x 10" ^ the appropriate value of t/cp(V/A) and mov-
ing to the corresponding value of /£ , along the curve for constant
(a!2 "*" al^W) = 7460. For example, on the 15th of July the pond tem-
perature will be found at a value of /2 , given by:
37
-------
f2f = /2. + cp(V/A)
_3 15 days
/2 =7.30x10 + (1)(62> 4)(1000 x 106)/(50 x 106)
= 7. 30 x 10"3 + . 01205 = . 01930
(9 - 9). = +0. 5°F
ss /
T = 79 - 0. 5 = 78. 5°F
Likewise on the 30th of July the temperature is found to be:
/ = 7. 30 x 10"3 -t- . 01930 = . 02660
(9 - 9), = 0. 0°
ss /
T = 79. 0° - 0 = 79. 0°F
o
Since the steady state temperatxire was previously found to be 79. 0 F,
steady state conditions have been achieved by the end of July. If the
analysis were to be continued beyond this point, the new climatic and
waste thermal energy loading for August would have to be used and the
process continued as long as need be. The above example clearly
demonstrates the asymptotic approach to steady state temperature
when it is noted that the pond temperature increased by 8. 5°F in the
first 15 days of July but increased only 0. 5°F in the last 15 days of
July.
Case IV - Mixed pond, transient operation, with a specified linear
vertical temperature gradient
Using the relationship between the bulk average temperature for the
water below the surface and the surface temperature given by Eq, 14,
the energy equation (Eq. 18) can be expressed as:
"dt * + ^T" = dT
6 12 13 2 14 ss 2 2.
+ l/cp(V/A){a + (a +a W)(a,)}{[9 + 0(— )J2 - 19 + j8(~)J2}
f ic. 15 5 ss 2 2
l/cp(V/A){a8 + (a1 + 2W)(a)}C[e + ^()J} (Eq.
38
-------
Fig. 7A through 7B can be used without alteration to yield solutions to
the above equation if 8 in Fig. 7 is replaced by [0 + ^(1
A comparison of Eq. 18 and Eq. 19 shows that if two similar ponds,
one without and one with a vertical temperature gradient, initiallially
have the same surface temperatures, they will continue to have the
same surface temperature as time goes on. However, as previously
noted, if the intake structure is so arranged as to preferentially draw
water from the bottom, the temperature of the incoming condenser
water will be colder for the pond with a vertical gradient.
Case V -Slug flow pond, steady- state, no vertical temperature
gx adi ent
If the pond is assumed to operate in steady-state, the energy equation
can be expressed as: (See Appendix A, Eq. A-24)
AT
AT
AT
AT
(Eq. 20)
where AT = temperature rise experienced by the cooling water
as it passes through the condenser. In steady-state
this must also equal the temperature drop through
the pond.
WTE = waste thermal energy from plant to pond, btu/day
= mc(ATc)c
^ _L
WTE m c
c
where m = the mass flow rate of cooling water through the
condenser, #m/day
Eq. 20 can not be solved in closed form and numerical techniques must
be used. However, it will be shown below that Fig. 7A - 7G can be
used to solve this equation if the relation for /j in terms of the equili-
brium temperature, as given by Eq. 12, is substituted into Eq. 20 to
yield:
39
-------
AT
If Eq. 21 is multiplied by dA and integrated to obtain the temperature
(9) after the slug has tranversed a surface area (A), the resulting ex-
pression becomes:
/d8 =JA -[{a 6 + (a13 + a W)(a2 + a14)}{e-e }
9=9 at A=0 A=0
ATc
(Eq. 21-A)
When Eq. 21 -A is compared to Eq. 18- A, it is noted that the two are
identical if (ATC dA)/WTE in Eq. 21-A is made equal to dt/cp(V/A) in
Eq. 18- A. As a result of this observation, Fig. 7A through 7G can be
used as the solution for Eq. 21 if the abscissa (/ ) is taken as
(AT )A/(WTE).
In the form of Eq. 21 the energy equation demonstrates the fact that a
"slug" of water approaches the equilibrium temperature as it flows
along the length of the pond. Like a mixed pond's approach to steady-
state temperature, the slug approaches the equilibrium temperature in
an asymptotic fashion with the downstream surface area being less ef-
fective than the upstream area.
To demonstrate the use of Fig. 7A - 7G for solving the steady- state slug
flow energy equation given above, consider the example given Case I for
a ATc = 10°F. (This is referred to as the "range". ) Since the natural
equilibrium temperature for this case is 75. 5°F, Fig. 7A is the appro-
priate set of curves to be used. In contrast with the mixed pond, two
possible situations must be considered.- The first situation arises when
a given value of the condenser inlet water temperature (which is often
specified as the equilibrium temperature plus a certain temperature dif-
ference called "the approach") is selected as a design criterion and the
40
-------
pond area is subsequently selected to force this to be the case. The
second situation arises when the following question is posed, "For a
given pond area, waste thermal energy load, and climatic conditions,
what will the approach temperature be?" Both situations can be dealt
with by using Fig. 7A - 7G; however, the second situation requires a
trial and error solution.
In order to demonstrate the procedure, both situations will be answered
using the data given in the example of Case I.
First consider the case where, as a design criterion, the approach and
hence the condenser inlet temperature has been selected and it is nec-
essary to determine the surface area required to achieve this. If the
approach is selected at 3. 5°F, the condenser inlet water temperature
will be 75. 5-+ 3. 5 = 79. 0 F. Since the equilibrium temperature is
75. 5 F, Fig. 7A will be the appropriate set of curves to use. The fol-
lowing functions must be determined:
Value of (6 - 8 ). at hot end of pond = [(79. 0 + 10) - 75. 5j = 13. 5°F
eq i
Value of /2- : This is found from Fig. 7A by using
(6-0 ). = 13. 5°F and a._ + a._W = 7460
eq i 12 13
.'. /-. = . 00540
^i
Value of (9 - 6) Qf pond = [79. o . 75> 5J = 3. 5°F
Value of /7f: This is found from Fig. 7A by using
(6 - 6 ), = 3. 5°F and a., + a10W = 7460
eq J 12 13
•'• f2f = • 0113°
A(AT )
Value of WTE = (/2 - /2.) = . 00590
The pond area can now be solved by use of the last calculated value,
that is:
A= .00590
A = . 00590 [(35xl09btu/day)
10°F
A = 20.6 x 106 ft2
41
-------
Thus, the slug flow pond requires less surface area than the mixed
pond in order to achieve the same condenser inlet temperature.
In order to demonstrate the second situation, again consider the same
example used in Case I, including the preselected surface area of
50 x 10^ ft . The condenser inlet water temperature is to be deter-
mined. As noted previously, this requires a trial and error solution.
This solution requires that two points along the appropriate curve in
Fig. 7A be fixed such that the difference between /2/ and f 2^ be the
required value given by the expression below and the difference between
(0 - 0<,«)f and (6 - 00J. be equal to the range, 10°F in this example.
ccj j ^4. 3-
(WTE)
f
'°°
f
(35 x lobWday)
By trial and error, using Fig. 7A, the two required points are:
(8 - 6 K = 0. 4° , . *. T = 75. 5° + 0. 4° = 75. 9°F
eq 7 /
(6 - 6 ) = 10. 4° , . '. T. = 75. 5° + 10. 4° = 85. 9°F
eq i i
Thus for the slug flow pond with the same surface area as the mixed
pond, the condenser inlet temperature is lower.
j3a.se VI - Slug flowpond, steady state, with a specified linear
vertical temperature gradient
As in the case of the mixed pond the energy equation is the same as for
Case V except for a change of 9 to the surface temperature [8 + fl(S,^/2) ].
Thus the surface temperature is given by the following equation:
dA
dA
(Eq. 22)
42
-------
Case VII, j- Slug flow pond, transient operation with no vertical
temperature gradient
This condition arises when the residence time in the pond (the time re-
quired for a slug of water to travel from pond inlet to outlet) is of the
same, magnitude or longer than the time over which the weather data are
averaged. There are two extreme or limited cases that should be noted.
First, when the residence time is very long and the ratio of V/A is
small, the condenser inlet temperature will be very close to the natural
equilibrium temperature associated with the time at which the water
enters the condenser. This case corresponds to the use of once-through
river water which has not been artificially heated upstream. The second
limiting case arises when the residence time is short (compared to the
time for averaging climatic conditions) and V/A is small. When this is
the situation, the condenser inlet temperature will be very close to the
steady state slug flow inlet temperature.
In studying the transient operation of a slug flow pond, the temperature
of a slug of water will change as it flows through the pond; before it
completes its pass through the pond, the average climatic conditions
may change. To describe the transient operation of the pond, it is nec-
essary to find the temperature-time relationship along the length of the
pond. Although this condition is best studied with the help of a computer
program, hand calculations with the aid of Fig. 7A - 7G are not excess-
ively tedious and they help to demonstrate the technique.
The procedure is to find a limited number of temperature-time-position
coordinates for a representative "slice" or "slug" of water as it travels
around the pond. When the slug reaches the cold end of the pond, the
range (AT ) is added to its temperature (to represent passage through
the condenser) and the slug is again allowed to pass through the cooling
pond. If intermediate temperature-time-position coordinates are needed,
additional representive slugs, appropriately spaced, can be followed
through the pond. The procedure is detailed into several steps and illus-
trated below.
Step 1. The condenser discharge temperature must be established at
some time, let this temperature be T. and the time be t = o or t . Thus
the slug will leave the condenser at T. and time t .
Step 2. Plot the equilibrium temperature, pond residence time
tr = (Vp)/mc = [Vpc(ATc)J/(WTE) and the wind parameter (a 12 + al3W)
as a function of time.
Step 3. With the use of Fig 7A - 7G determine the temperature of the
representative slug some preselected number of days (trO after time zero.
43
-------
To determine the temperature at tQ , first find the value of f 2^ from
Fig 7A - 7G and add to it the value of (tD - tQ)/ [pc (V/A)]. The position
of the slug along the pond can be found by equating the ratio of pond sur-
face area traversed or swept by the slug to total surface area with the
ratio of elapsed time to residence time. Thus the slug position at the
end of tD days is:
AA =
. 23)
T
where t = average residence time in period
t to t - days obtained from the plot
o D
Step 4. Repeat Step 3 until the slug just reaches the cold end of the
pond. The last time interval in this process will have to be selected
such that the final position (the cold end) is given by [L(AA/A)] = 1. 0.
Step 5. Let the slug pass through the condenser where its temperature
will increase by ATc and then repeat the previous steps.
To illustrate the use of this procedure, again consider the example
used in Case I. The cold end of the pond is assumed to be at 70 F on the
last day of June. It is desired to find the temperature of the water ar-
riving at the condenser intake during July. Only a few representative
values will be found to illustrate the technique. In order to avoid mak-
ing a plot of the equilibrium temperature, wind function and residence
time so that average values of these parameters will be available in this
example, it is assumed that these values are constant for July and equal
to:
T = 75.5°
eq
a + a W = 7460
JL £ i j
(ATc} (1000 x 106)(62.4)(1)(10) ._.,
*r = (WTE) = - (35x10^) - = 17. 8 days
The slug which leaves the "hot" end at time, t=o, with a temperature of
70 + 10° = 80°F will have a lower temperature at t = 1 0 days, namely
f? , = .0103
0 i n j
rt = -0103+- -
44
-------
•'• <9 -e >l-7 fil ^ = °- l°F
eq 17. 81 days
•'• Ti7 «! A = 75. 6°F
1 7. 81 days
After arriving at the cold end of the pond, the slug passes through the
condenser where its temperature is increased by AT (= 10°F) and sub-
sequently flows through the pond. The temperature during this second
pass is given below at the end of the next 10 days.
/ = . 0065
17. 8 days
/2 = .0145
27. 8 days
T__ = 75. 5° + 1.5° = 77. 0°F
27. 8 days
AT
27. 8 days _ 10
A = ITs = -561
When the slug arrives at the cold end of the pond for the second time,
its temperature is:
7 ft 1
/, = - 0145 + ^— :-r5- = . 0208
2__ , , .._ „..,. .lOOOx 10"
35. 6 days (62. 4)( 1) ( 5Q x lp6 )
T« LA = 0.4° + 75. 5° = 75.9°F
35. 6 days
Since the steady state slug flow temperature was previously found to be
75. 9°F at the cold end of the pond, steady conditions have been reached;
however, the climatic conditions and possibly the waste thermal energy
loading must now be changed to the appropriate values for August and
the process continued.
Case VIII - Slug flow pond, transient operation, with a specified
linear vertical temperature gradient
Again, as for the mixed pond, the assumption of a linear temperature
profile does not change the nature of the energy equation except to re-
place the bulk average temperature with the surface temperature. The
remarks made about Case IV are also pertinent to this case.
Concluding Remarks
For ease of comparison, the steady state and transient temperature
characteristics of the example pond, for both mixed and slug flow
45
-------
operation, are shown in Fig. 8.
Although it is more convenient to use the steady state analysis rather
than the transient analysis, such use has to be justified. The question
can always be answered precisely by applying both analyses to a given
problem but this is tedious. Fig. 7A - 7G can be used as a guideline in
making the decision; in particular, if it is noted that a value of the func-
tion /2 = t/[pc(V/A)] of about 0. 05 or greater, depending on 0sg/ for
a mixed pond will produce condenser intake temperatures within a de-
gree of the steady state pond temperature, then a rule of thumb for
using steady state analysis is the requirement that:
t > 0. 05pc(V/A) (Eq. 24)
where t = the time duration over which conditions are
averaged, in days
p - density of water = 62. 4 #m/ft
c = specific heat of water = 1.0 btu/#, F
V/A = pond volume to surface ratio
In view of the fact that weather and plant conditions are usually averaged
for a period of one month, the above guideline may be rewritten as:
V/A > 10 ft (Eq. 25)
Thus for mixed ponds of average depth of 10 ft or less, steady state
analysis can be used. Although this approximate rule was devised for a
mixed pond, it is also a valid guideline for slug flow ponds. As men-
tioned previously, a slug flow pond with sufficiently long residence time
and small depth (V/A) will not only be in steady state operation, but
the condenser intake temperature will be close to the equilibrium tem-
perature. Again Figure 7 can be used to generate the guideline if it is
noted that a value of the function f2= [A(ATc)]/(WTE) = A/(m c) of
about 0. 05, or greater, depending on 0eq for a slug flow pond will re-
sult in condenser intake temperatures within about 1°F of the equili-
brium temperature. This observation may be expressed as
A > 0. 05 mcc (Eq> 26)
Since the residence time and pumping rate are related by the expression
"c'r = PV (Eq. 27)
Equation 26 can be expressed as
tr > 3. 11 (V/A) (Eq. 28)
Thus for a 10-foot deep pond the minimum residence time that will
46
-------
86 -T
84-
82-
80 --
Mixed Pond Steady State
^-Mixed Pond Transient
July Equilibrium
Water Temperature
"-Slug Flow Transient
Slug Flow Steady State
68
66 ~
64
62
-I-
10
— July
20
30 Time, Days
Comparison of Steady State and Transient Temperatures
for a Mixed and a Slug Flow Pond
Figure 8
47
-------
cause the water to be cooled to the lowest possible value (equilibrium
temperature) is 31. 1 days or one month.
An inspection of Fig. 7A - 7G shows that for the same pond surface area,
the same difference in water temperature and natural equilibrium tem-
perature at the hot end of the pond, more cooling takes place (that is to
say, a lower exit temperature will result) when the equilibrium temper-
ature is high than when the equilibrium temperature is low. Therefore,
for a given pond surface area, cooling is greater in the summer than in
the winter for a given geographic location; and cooling is greater in re-
gions of high equilibrium temperature than in regions of low equilibrium
temperature at a given time during the year. Since Fig. 7A - 7G also
applies to the transient operation of a mixed pond when the reference
temperature is not the natural equilibrium temperature but the higher
"forced" steady state temperature (0 ), the above remarks also apply
to the rate at which a mixed pond approaches its steady state tempera-
ture. In addition, since a mixed pond may, mathematically, be consid-
ered to be a slug flow pond in which ATC approaches zero, a mixed
pond of given surface area o'perating in steady state experiences greater
cooling when the steady state temperature (6SS) is high rather than low.
To illustrate the point discussed in the above paragraph, consider the
example used in Case V, namely, a waste thermal energy load of
35 x 10 btu/day imposed on a slug flow pond operating in the steady
state condition. If the temperature rise through the condenser (^Tc)
and the approach temperature are again taken as 10 F and 3. 5°F res-
pectively, the pond surface area required to achieve this approach tem-
perature will be 20. 6 x 106 ft2, 31. 2 x 106 ft2 , and 42. 6 x 106 ft2 for
equilibrium temperatures of 80°F, 60°F , and 40°F respectively.
The physical reason for this behavior becomes apparent when it is re-
called that the three means by which thermal energy is removed from
the pond (other than by direct transport in escaping water), namely,
1) back radiation, 2) evaporation, 3) convection all depend on the
pond surface temperature whereas the means by which thermal energy
is added to the pond (other than by an inflow of water or the power plant),
namely, net solar and atmospheric radiation do not depend on the water
temperature.
In contrast to the greater cooling capacity at high equilibrium tempera-
tures vs low equilibrium temperatures for a given pond surface area
and the same difference between water temperature and equilibrium
temperature (at the hot end of a slug flow pond), it should also be recalled
that the plant efficiency decreases as the condenser temperature in-
creases with the result that additional cooling will be required if opera-
tion at the high equilibrium temperatures increases the condenser
48
-------
temperature. Thus, in general, when the equilibrium temperature is
increased, two opposing trends come into play, namely, increased cool-
ing capacity for a given pond surface and an increase in the waste
thermal energy load imposed by the plant on the pond.
The sensitivity of the water temperature to pond surface area is an im-
portant consideration when considering the economics of the pond as
well as when comparing predicted to measured water temperature. This
sensitivity depends on the difference between the water temperature and
the equilibrium temperature (for a slug flow pond) or steady state tem-
perature (for a mixed pond). In order to demonstrate this point, the
previous example is again considered, namely,
T = 75. 5°F
eq
W = 10. 0 mph (a + a W = 7460)
1Z 13
WTE =35 x 107 btu/day
Range = 10°F
/ = 5102 btu/ft2, day
The pond discharge (or condenser inlet) temperature for a mixed and
for a slug flow pond operating under these conditions is shown in Fig. 9
as a function of pond surface area. Again it is noted that for equal pond
surface areas, the slug flow pond will deliver colder water to the con-
denser. It is seen from Fig. 9 that the condenser inlet temperature is
not sensitive to pond surface area beyond approximately 50 x 10 ft. An
area of 50 x 10 ft^ and a waste thermal energy load of 35 x 10° btu/day
corresponds (at 40% thermal efficiency) to pond loading of about 4 acres
per megawatt of electricity produced by the plant. Therefore, in order
to test the accuracy of the predicting technique, it would be helpful to
have data available from ponds that have less thar. 4 acres per mega-
watts of electricity produced by the plant.
It is easier to see the relative cooling capacity of the two types of ponds
if the data of Fig. 9 are replotted as the ratio of mixed pond surface
area to slug flow pond surface area for the same condenser inlet water
temperature as shown in Fig. 10. An inspection of Fig. 10 shows that
the two ponds will have about equal cooling capacity as the difference be-
tween the condenser inlet temperature and the equilibrium temperature
(that is, the approach) is increased, whereas the slug flow pond is con-
siderably more effective at cooling the water if the condenser inlet tem-
perature is made to approach the equilibrium temperature closely. Al-
though Fig. 10 is for a specific equilibrium temperature (75. 5 F) and
for a specific WTE and AT (and hence the pumping rate), the indicated
behavior is true in general.
49
-------
CJ1
o
100
Ou
s
0)
H
90
M
0)
en
d
-------
12-
11-
10-
0
RATIO OF MIXED POND AREA TO
SLUG FLOW POND AREA VS APPROACH
TEMPERATURE
10 12
condenser inlet
28 30
) °F = Approach
equil
Figure 10
-------
COMPARISON OF
PREDICTED AND MEASURED WATER TEMPERATURE
Operating data of five cooling ponds were obtained from various elec-
tric power firms. These data consisted of the monthly average power
generation for the plant, the monthly average condenser inlet tempera-
ture rise across the condenser, the cooling water pumping rate of plant
heat rate, the nominal pond surface area, nominal pond volume or
depth, pond geometry and in some cases the vertical temperature grad-
ients.
The data obtained from the power companies together with monthly ave-
rage weather conditions at nearby weather stations (obtained from the
National Weather Record Center at Asheville, North Carolina) were
used to predict condenser inlet temperature for each month of the year.
Predicted and measured values are shown inFigs. 11 through 15. The
predicted values shown are for the condition of no vertical temperature
gradient in view of the fact that gradients comparable to the measured
magnitude do not appear to influence the condenser temperature strongly.
The detail calculations associated with the predictions are shown in
Appendix C. A brief summary of the pond characteristics is given in
Table I. Each of the five plants is discussed in the following sections.
Wilkes Plant
The Wilkes Plant (179. 5 MW.) is located in Jefferson, Texas. The
G
nearest weather station is at Shreveport, Louisiana, about 40 miles
away. The summer months are warm and rather hamid, the relative
humidity has an average value of about 90% during the early morning and
about 50% at mid-afternoon. Winter months are mild with any cold
spells being limited to a few days.
The "pond" at the Wilkes .Plant is in the shape of a river bend (See
Appendix C). The plant removes cooling water from one end of the
bend and discharges the heated water to the other end of the bend through
a discharge canal. Because of the large ratio of pond surface to gener-
ating capacity (~ 4 acres/MW ), the pond exit temperature should not be
sensitive to changes in the pond surface area. In addition, since the
residence time is large (compared to 3. 11 V/A), the exit pond tempera-
ture will approach the natural equilibrium temperature as shown in
Fig. 11.
From Fig. 11 it is seen that the measured condenser inlet temperatures
are within + 5° of the average value predicted by use of the two pond
53
-------
en
TABLE I
SUMMARY OF PLANT CHARACTERISTICS
Power
Plant
Wilkes
Kincaid
Cholla
Mt.
Storm
Four
Corners
Company
South-
western
Electric
Power Co
Common-
wealth
Edison
Co.
Arizona
Public
Service
Co.
Virginia
Electric
& Power
Co.
Arizona
Public
Serv. Co
V/A
ft
15
10
4. 5
21
(. 5V /A due
to flow
pattern)
40
t
r
days
46
9
8
16
(. 5 tr due
to flow
pattern)
35
3. 11 V/A
days
46
31
14
66
124
A/MW
e
acres/MW(e)
w 4
2*
» 3
w 1
w 2**
Ratio of Width
to Length in
Flow direction
~l/7
-1/ZO
~ 1/1
Intake loca-
tion below
hot water
out let
~ 1/1
Comments
Will operate close
to steady state.
Condenser inlet
temp, will be close
to equil. temp.
Can assume steady
state operation.
Condenser inlet
temp, should be
•well above equil.
temperature
Same as above
Pond will be in
transient
operation
Pond will be in
transient
operation
*Based on plant MWe installed capacity in 1968.
eventually doubled.
**Based on plant MWe rating in 1967. However,
raised the plant MWe rating considerably.
Horwever, it is anticipated that this capacity may be
Unit No. 4 was put into service in July 1969 and has
-------
or
loor
90.
80 —
60—
50
40
Calculated Mixed Pond (Steady State)
, Calculated Slug Flow (Steady State)
X = Measured
Equilibrium Temperature
30
I I
i I i i
FMAMJJASOND Month, of Year
Measured and Predicted Pond Temperatures for the Wilkes Plant
Figure 11
-------
models.
Kincaid Plant
The Kincaid Plant (~ 1000 MWe) is a mouth-of-mine plant located in the
coal fields of Southern Illinois. The location of this plant near the cen-
ter of North America results in a continental climate characterized by
warm summers and fairly cold winters. Summer weather tends to be
quite warm and humid. The winter does not have extended periods of
severe cold; however, sharp seasonal changes do take place during the
winter and summer.
The "pond" at the Kincaid Plant is in the form of three "arms", each
with a very irregular shore line (See Appendix C). The plant is located
between two of the three "arms" of the pond. The heated water dis-
charged from the condensers must flow down one arm and up the other.
The third arm serves as a storage reservoir.
The predicted and measured condenser inlet temperatures are shown in
Fig. 12. With the exception of the measured temperature for November,
the measured values agree within +_ 5°F of the average temperature pre-
dicted by the two models.
Cholla Plant
The Cholla Plant is located in Joseph City, Arizona, a semi-arid cli-
mate. The nearest weather bureau is located at Winslow, Arizona,
about 25 miles away. The terrain varies rapidly in the vicinity of the
Cholla Plant, with the Painted Desert just to the north and the White
Mountain area less than 75 miles to the southeast. As a result of rapid
change in terrain, the use of weather data from the Winslow Station in-
troduces some uncertainty. The Cholla pond is shallow (varying from
inches to a maximum of 12 feet) and is divided into two parts by a dike
and an associated inverted weir (See Appendix C). As a result of the
pond geometry, substantial channeling of the flow may be present be-
tween the inlet and outlet with the result that not all the pond area will
be effective.
The predicted and measured condenser inlet temperatures are shown in
Fig. 13. The predicted values are based on the assumption that only
one-third of the pond surface is effectively engaged in the cooling pro-
cess as a. result of channeling and subsequent dead water regions. The
value of one-third was selected because it results in the best fit with
the experimental data. Predicted temperatures based on the assump-
tion that the entire pond surface is effective are also given in Table C-4
56
-------
100
90
80
du70
a
fl
1 1
50
Highest Water
Level in Pond
30
Equilibrium
Temperature
Lowest pond
Level
Calculated Mixed Pond (Steady State)
-Calculated Slug Flow (Steady State)
X = Measured
JFMAMJ JASOND Month of the Year
Measured and Predicted Pond Temperatures for the Kincaid Plant
Figure 12
-------
ioor
en
co
Calculated Mixed Pond (Steady State)
Calculated Slug Flow (Steady State)
X = Measured
NOTE:
Equilibrium ^r+
Temperature ,
Abnormal
High Rain
Fall
(1967)
Calculated curves
based on assumption
that effective pond
area = 1/3 actual due
to suspected channel-
ing as a result of pond
geometry and small
depth
JFMAMJJASOND Month of the Year
Measured and Predicted Pond Temperatures for the Cholla Plant
Figure 13
-------
in Appendix C. From Fig. 13 it is seen that the measured and (aver-
age) predicted temperatures agree within +_ 5°F with one exception,
namely, December.
Mt. Storm Plant
The Mt. Storm Plant is located in West Virginia near the principal
storm tracks. As a result of its location it is subjected to frequent
weather changes throughout the year. The summer months are warm,
humid and showery. Severe cold spells'do occur but do not usually
last more than a few days.
The pond at the Mt. Storm Plant is quite deep (~ 100 feet) so that the
pond will operate in the transient mode rather than steady state. Be-
cause of the low ratio of surface area to generating capacity (~ 1 acre
per MWg , the highest thermal loading of any pond for which we have
data), the Mt. Storm reservoir provides a sensitive test for the valid-
ity of our temperature predicting techniques. The power plant is re-
quired to discharge a minimum flow of 2 cfs plus passing through the
facility and any flows discharged from the West Virginia Pulp and Paper
reservoir as low flow augmentation for the Potomac River.
The measured and predicted condenser inlet temperatures are shown in
Fig. 14. From Fig. 14 it is noted that the measured values are within
- 5°F of the average of the values predicted by the two pond models,
with the exception of April and May, which are - 9°F and - 6°F respect-
ively from the average predicted value. In this case the predicted
values overestimate the temperature which is the anticipated situation
due to the use of the Meyer Equation for a heavily loaded pond as dis-
cussed in Appendix B.
Four Corners Plant
The Four Corners Plant is located in Farmington, New Mexico. The
nearest weather station is at Winslow, Arizona, about 180 miles away
(the same station used in evaluating the Cholla Plant). Like the nearby
Cholla Plant, the Four Corners Plant is in a semi-arid environment and
experiences a rainfall of about 10 inches per year. Unlike the Cholla
pond, however, the Four Corners pond is relatively deep and channel-
ing should not occur. As a result it can be assumed that the entire pond
surface will be effective.
The predicted and measured condenser inlet temperatures are shown in
Fig. 15. With the exception of August, the measured values agree with-
in + 4°F of the average of the values predicted by the two pond models.
59
-------
100
90
80
a 70
0)
H
4J
-------
100
90
80
fc
D
d,
a 70
-------
CURVES FOR PREDICTING WATER LOSS BY EVAPORATION
In this section the equations for predicting the water loss from the pond
under both natural conditions and when the pond is receiving waste ther-
mal energy from the power plant are presented. As a matter or conven-
ience to the user, the results of these equations are presented in the
form of curves.
For either the mixed or slug flow pond it is assumed that the rate of
evaporation per unit surface area can be expressed in the simplified
form suggested in Appendix B by Eq. B-33, namely:
29)
where a = 3730
a^ = 373
P = water saturation pressure corresponding to the
temperature of the pond surface
Since the pond surface temperature is everywhere the same for the mixed
pond, the amount of water lost per day in a mixed pond is:
a. +a. W)
>
In order to present one set of curves to readily show this relation,
Eq. 30 can be rewritten as:
(Eq. 31)
where P = a, + a_9 + a 92 + a 63 (See Eq. B-37)
w 1 2 3 4
The left hand side of Eq. 31 is plotted in Fig. 16 as a function of pond
surface temperature (0 + 32°) and air vapor pressure. To demonstrate
the use of Fig. 16, consider again the example used in Case I, namely,
a steady state pond temperature of 79. 0°F, a pond surface of 50x10 ft ,
water vapor pressure in the atmosphere of . 294 psia, and a value of the
wind parameter (a + a.^ W) of 7460. For this case Fig. 16 gives the
following value:
!2
Thus the rate of evaporation from this pond is:
63
-------
01
rt
i
o
o
o
a
n)
CO
CO
o
C!
O
M
a
ni
1. 8
1. 6
1. 4
1. 2
1. 0
0. 8
0.6
0. 4
30
Atmospheric'1 i > o "^a
Vapor Pressure /, Q. 55
• - ,— ///
50 70 90 110
Pond Surface Temperature, °F
130
Evaporation Loss Parameter for Mixed Pond in Steady State
as Function of TemDcrature
Figure 16
-------
rhe - (,205)(50x 106)(-I~)(7460)
- 71. 5 x 106 #m/day
If the waste thermal energy received by the pond is reduced to zero,
then the pond temperature becomes the natural equilibrium temperature
which was previously found to be 75. 5°F. Again using Fig. 16, the rate
of evaporation under the previous conditions but for a surface tempera-
ture of 75. 5 F is found to be:
me = (. 152) (50 x 106)(y^)(7460) - 53. 1 x 106 #, /day
In the case of a slug flow pond the rate of evaporation per unit surface
area changes with distance along the direction of flow and this variation
must be integrated in order to determine the total water evaporated per
day from the surface of the slug flow pond. Thus, integrating Eq. 30
over the pond surface yields the total rate of evaporation, namely:
A a +a W
' 1070 "Pw - Pa>
Since P^ is constant along the pond surface, this equation can be re-
ct
written as:
A a +a W a H-a W
% = / ' 1070 'Pw " - ' !070 >PaA
-------
/ = / for temperature at the final or hot end of the pond
/ where the temperature is T ,
In order to facilitate the use of this procedure, values of / 3 are plotted
in Fig. 17 for a selected number of equilibrium temperatures. In order
to demonstrate the use of Fig. 17, again consider the previous example
where ATC = 10° F; A = 50xlObft2; Tj = 85. 9°F; Tf = 75. 0°F.
For this case Fig. 17 gives the following values:
/ = . 0465
i
/ = .0970
/
Thus the water lost by evaporation for this steady state slug flow pond
is given by the expression:
= 74. 5x 106 #m/day
As the ATC for the slug flow pond is allowed to approach zero by in-
creasing the pumping rate, the pond approaches a uniform temperature
along its length which is equal to the mixed pond temperature of 79. 0°F
and the evaporation approaches 71. 5x10 #m/day.
It should be noted that the terms selected for the water mass balance on
the pond allow for the estimation of each incoming and each outgoing term
independently. Thus, when the water mass balance is being considered,
attention must be paid to water loss from the pond by outflow and seepage
in addition to the evaporation considered above as well as to water gain-
ed by the pond as a result of inflow and direct precipitation falling on the
pond surface. For most cooling ponds the last item will be equal to a
substantial fraction of the water lost by evaporation. To demonstrate
this point it is helpful to consider the mixed pond example that was dis-
cussed above. In that example the rate of evaporation from the pond sur-
face was shown to be 53. 1x10 #m/day and 71. 5 x 10& #m/day for waste
thermal energy "loads of zero and 700 btu/day ft2 respectively. These
two rates correspond to a decrease in the pond depth of 0. 21 inches per
day and 0. 28 inches per day respectively for the 50 x 10^ ft2 pond used
in the example. An annual rainfall of 36 inches corresponds to an aver-
age increase in the pond depth of 0. 10 inches per day in the form of dir-
ect precipitation.
66
-------
05
•
:' i = sorr:
:;| EVAPORATION
1 0 j-T-i-.r;-| ;-M PARAMETER ^
! TEMPERATURE
i L_I . |. i: A SLUG FLOW POND
!"•'• "''T! OPERATING IN
,|i STEADY STATE
jTrrn~T
Figure 17
-------
APPLICATION OF DESIGN CURVES
TO PARTICULAR POWER PLANTS
The design curves developed previously are used to predict the per-
formance of cooling ponds for two "typical" power plants. One of these
is assumed to be located near Philadelphia, Pa. This location is near-
ly at sea level, has moderate temperatures, high humidity, and an ade-
quate water supply. The other is assumed to be located near Winslow,
Arizona. It is nearly a mile above sea level surrounded by even higher
terrain, and has a very dry but relatively mild climate. Thus these.
locations represent conditions under which normal and high evaporation
rates, respectively, are expected.
The power plants are assumed to generate 2000 MW and reject heat at
the rate of 38. 8 x 10" btu/day at each of these locations.
At each location the performance of two types of ponds are calculated:
a mixed pond, and a slug flow pond. These types of flow are extreme
models of a practical pond, the actual flow being somewhere between
the extremes.
The calculation of cooling pond performance requires the knowledge of
local weather conditions. These are obtained from Local Climatological
Data sheets [17, 18j, The Climatic Atlas [12], and Refs. [6] and [l3J-
Cooling pond performance is calculated for the summer months of June,
July and August, considered as design conditions. At the Philadelphia
plant the weather data for the summer months are summarized below,
according to source.
From Local Climatological Data sheets [l]
June July August
Wind speed, W, mph 8. 7 8. 1 7. 8
Mean temperature, Ta, °F 71.0 75.6 73.6
Normal daily maximum temperature,
T , °F 81.6 85.9 83.7
max
Normal daily minimum temperature,
T , °F 60.4 65.2 63.5
min
Relative humidity, 1 AM, H , % 79 82 81
Relative humidity, 7 AM, H , % 76 79 80
iL
Relative humidity, 1 PM, H , % 53 53 54
Relative humidity, 7 PM, H , % 60 62 65
69
-------
From Climatic Atlas
Mean daily solar radiation, Qg, langleys
Mean sky cover
From Reference [6] (See Figs. 2 and 4)
Shortwave solar reflectivity, R
2
Clear sky solar radiation, Qcg, btu/ft day
June July August
523 510 450
0.61 0.61 0.60
0. 06 0. 06 0. 06
2950 2850 2580
From Reference [13], page 472, after averaging:
Sun altitude, degrees 41. 7
39. 7
39. 0
Using the above weather data, the following quantities are calculated:
Pa = T
, :W+rT )]+p[T i(H+H)]} 0.251 0.300 0.286
mm 2 1 c. max ^ -> t
= 3730(1 + W/10)
o.
6 = T - 32 F
a a
C (from Fig. 3)
Q =4. OxlO-8(T + 460)4(C -0. 223v/PJ
a a .D a
Q = 0. 97Q +Q (1 - R )
N a s sr
a - a 6 - P )
1 14 a a
6980
39. 0
0. 74
2720
4460
-2420
4521
6750
43. 6
0. 74
2850
4530
-2820
4991
6640
41. 6
0. 74
2820
4290
-2620
4551
= 4688
' 6?9°
Then with /, + Q , the mixed pond temperature obtained from Fig. 5A,
the evaporation raretor from Fig. 16, and the evaporated water equal to
evaporation constant times A(a12 + a W)/1070, we obtain the following
performances for mixed ponds of five sizes. The pond sizes range from
0. 5 to 5 acres per MW , representing highly and lightly loaded ponds,
respectively.
Areas, A
in acres
1000
2000
4000
6000
10000
A/MW
e
0.5
1
2
3
5
•
Q
PP
8920
4460
2230
1487
892
/,+6
1 PP
13608
9148
6918
6175
5580
Evaporation
Constant
115
99
88
84
80
1.20
0.64
0. 38
0.31
0.23
Water Evap.
per day in
109 Ibs
0. 330
0. 351
0. 481
0. 507
0.637
70
-------
The average annual precipitation for the Philadelphia area is 40 inches.
Based on this value the difference between water lost by evaporation and
water gained by direct precipitation is shown below.
Water Gained Net Water Lost
Area, A by Precipitation (evaporation-precipitation)
in acres
1000
2000
4000
6000
10000
q
per day in 1 O7 Ibs
0. 025
0. 050
0. 099
0. 149
0. 248
per day in 10 Ibs
0. 305
0. 301
0. 382
0. 358
0. 389
The temperature and evaporated water variation with pond area for
mixed ponds at Philadelphia are shown in Figs. 18 and 19, respectively.
These curves show the expected trends: the temperature of the pond
increases rapidly as the heat loading increases (or equivalently as the
acres per MWe decrease) and the amount of total evaporated water in-
creases with increasing pond size. Fig. 19 also shows an evaporation
curve designated Average Over a Year. This curve is obtained by com-
puting the evaporation rate for each month and then averaging them.
The calculation of the temperatures of the steady state and slug flow
ponds requires the knowledge of the difference between inlet and outlet
temperatures, or equivalently the temperature rise through the conden-
ser. This temperature rise is a parameter in the power plant equip-
ment design and it is determined from an economic study of the power
plant operation. Such a study for the Philadelphia Plant is presented in
the next section. For the present, slug flow pond temperatures are cal-
culated for several condenser temperature rises, ranging from 10° to
30°F. The calculation of slug flow pond temperature involves the use of
charts (Figs. 7A - 7G) in an iterative fashion as explained previously.
The results for five pond sizes and five condenser temperature rises
are presented in Fig. 20. Comparing Figs. 18 and 20, we see that the
slug flow pond temperatures for corresponding pond sizes are lower than
those for the mixed pond.
The sensitivity of mixed pond temperature to the wind parameter-
al2 + a!1W' was evaluated- The results for a^2 + ai3^ equal to 4000
and 9000, representing the extremes of the wind parameter for which
curves are given, and corresponding to wind velocities of approximately
0. 7 mph and 14 mph are shown in Fig. 18. We see that the sensitivity
of pond temperature to the wind parameter is higher for highly loaded
ponds.
71
-------
130
120
110
100
90
-
nJ
M
-------
350
m Design
m Design
Philadelphia
me
Philadelphia
Average
over a Year
Cooling
Towers
Philadelphia
m - m
e p
50
Philadelphia
jAverage over
;a year
) 1 .2 3 4 5
Acres /Megawatt
Evaporated Water from Mixed Ponds Near Philadelphia, Pa. ,
and Winslow, Ariz. - Design Climatic Conditions
Figure 19
73
-------
1O n.^ r
i. U r
Design AT = 28 F
1 2 3
Pond Area, Acres per Megawatt
Temperatures for Slug Flow Ponds
2000 MWe Plant near Philadelphia
Design Conditions
Figure 20
74
-------
Transient temperatures in the mixed pond at the Philadelphia location
were calculated with the aid of the charts for predicting water tempera-
tures. The method for calculating transient temperatures consists es-
sentially of considering the pond to operate under steady climatic and
thermal waste energy for each month, and calculating the pond temper-
ature change during the month. The final pond temperature for one
month is taken as the initial temperature for the next month and a new
final temperature is calculated. The whole process is carried on for a
year. The temperatures for a 1-foot and 100-feet deep ponds are shown
in Fig. 21. The one-foot deep pond represents steady state operating
conditions, where no lag in pond temperature is expected. The 100-foot
deep pond shows a temperature lag in the early months, reaches the
same maximum at a slightly later time than the 1-foot deep pond, and
runs at a slightly higher temperature later in the year.
The performances of the mixed and flow-through ponds at the Philadel-
phia location are compared with a cooling tower design for that loca-
tion [28] in Figs. 18, 19 and 20. The cooling system of Ref. [28] con-
sists of two natural draft cooling towers, each 380 feet in diameter,
380 feet high with a design range of 28°F, a design approach temperature
of 16°F, and a design wet bulb temperature of 72°F. From Fig. 18 we
see that the mixed pond with 2 acres per MW provides the same cooling
G
as the cooling towers. Mixed ponds larger than 2 acres per MWe per-
form better than the cooling tower in terms of producing lower temper-
ature condenser cooling water. From Fig. 20 we see that slug flow
ponds larger than about 1. 3 acres/MW perform at lower temperatures
than the cooling towers. From Fig. 19 it is seen that the amount of
water evaporated by the towers is greater than the net difference be-
tween water evaporated from the pond and precipitation falling directly
on the pond surface for all pond sizes considered. The amount of water
evaporated by the cooling towers , as shown in Fig. 19, corresponds to
average conditions.
At the Winslow, Arizona, Plant the weather data taken from Ref. [18]
are summarized below:
June July August
Wind speed, W, mph 10.7 8.2 7.8
Mean temperature, T2> °F 69.7 78.3 75.6
Normal daily maximum temperature
Tmax , °F 92. 0 95. 7 92. 4
Normal daily minimum temperature
Tmin > °F 56- 5 64« 6 63- 1
Relative humidity, 5 AM, % 38 56 67
Relative humidity, 11AM, % 17 30 37
Relative humidity, 5 PM, % 14 27 32
Relative humidity, 11 PM, % 28 47 53
75
-------
100
90
80
70
V
(X
a 60
•O
c!
O
50
40
30
1 foot Depth / /
100 foot Depth
Pond Area, 1 acre/MW
Plant starts in January
with Initial Water Temp-
erature of 32°F
M A M J J A
Month of the Year
O N D
Transient Temperature of Mixed Pond for a 2000 MW
Plant near Philadelphia
Figure 21
76
-------
Then proceeding in the same way as for the mixed pond at the Philadel-
phia location, we obtain the performance of the mixed pond at Winslow,
Arizona, summarized below and plotted in Figs. 18 and 19.
Area, A
in acres
1000
2000
4000
6000
10000
A/MW
e
0. 5
1
2
3
5
T
113. 0
97. 5
86. 5
82.5
79. 0
Water evaporated
per day, 10° Ibs
0. 352
0. 412
0. 526
0. 670
0. 944
The equilibrium temperatures of mixed ponds at Winslow, Arizona, dif-
fer from those at Philadelphia by no more than two degrees. However,
ponds at Winslow evaporate much more water than at Philadelphia. In
addition, the average annual rainfall in the Winslow is only 7. 4 inches
with the result that only about 1/6 as much water will be added to the
Winslow pond by direct precipitation as at the Philadelphia pond.
77
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ECONOMIC ANALYSIS OF
POWER PLANTS WITH COOLING PONDS
The economic analysis of power plant construction and operation is a
well developed subject (See, for example, Ref. 29). The costs of pro-
ducing electric power are broken down into two categories: fixed
charges and operating costs. The fixed charges consist of interest,
taxes, insurance, and depreciation. The operating costs cover expend-
itures for fuel, labor, maintenance, supplies, supervision and operat-
ing taxes. The economic analysis of a power plant with a cooling pond,
and the economics of the cooling pond as a subsystem of a power plant,
can be subjected to such an analysis.
In order to apply the economic analysis summarized above, a power
plant design, with some detail, must be available. It should be clear
that the power plant design depends to some degree on the particular
site selected. The power plant •with a cooling pond may be significantly
different from power plants with other cooling systems in that more land
is required and the cooling pond may be more sensitive to the weather.
Therefore, the design and economics of a power plant with a cooling
pond are very much site dependent, and it is not possible, at this time,
to provide an all-inclusive procedure for the economic analysis of
power plants with a cooling pond without knowing the site and the details
of the design.
We suggest that some insight about the economics of cooling ponds may
be gained from an example. The power plant to be considered gener-
ates 2000 MWe, is located near Philadelphia and the performance
curves for the ponds for this plant are given in the previous section
designated "Application of the Design Curves to Particular Plants. "
This particular site is selected because the construction of a power
plant on the site is being considered and a plant with natural draft cool-
ing towers has been designed [28]. Hence this example affords a real-
istic economic comparison between cooling ponds and cooling towers.
The design with cooling towers for this location is known with a great
deal of detail [28]. The turbine is a General Electric design with a
known performance curve. The condenser is of selected design with
known tube size, length, gage, and material, and cooling water flow
velocity. The cooling water pumps are volute pumps with known head
and capacity. Cooling of water is Dy two very large natural draft cook-
ing towers.
The design wicn a cooling pond is presented in parametric form. Essent-
ially it is an extrapolation of the cooling tower design.
79
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Turbine It is assumed that the turbine of the cooling tower design is
used in the cooling pond design. There is a disadvantage to the cooling
pond in doing so, since the tiirbine is selected for a nigh temperature
rise through the condenser. For some ponds where the range of the
condenser is lower, a different turbine would be more economical.
Condenser The condenser design is extrapolated from that for the
cooling tower design. The basis of the extrapolation is the formulae of
Ref. |_3uj. The following are assumed to be known:
Heat transferred, Q = 1. 568 x 1010 btu/hr
Tube diameter, D = 1 1/4"
Tuoe thickness, t = 0.049"
Tube length, L = 96'
Tube material - Admiralty brass
Material factor, Cm = 1.0
Tube cleanliness factor; GC = 0. 85
Cooling water flow velocity, V = 7 ft/sec
Thus with the formulae of Ref. 30 we obtain
AC = 4. 33 x 10?/R (Eq. 35)
G = 3. 136 x 10?/R (Eq. 36)
T = T + R(l - e"L61Ct) (Eq, 37)
where AC = condenser area, sq. ft
G = cooling water flow, gpm
R = temperature range, °F
T^ = condenser inlet water temperature, F
TV = saturation temperature in condenser, °F
Ct = temperature correction factor, given in Ref. 30
The condenser area and the pumping rate for five temperature rises are
given below and compared with the cooling tower design:
R(°F) Ac(106 sq. ft) G(106gpm)
10 4.33 TTJ
15 2.89 2.09
Cooling Pond Design 20 2. 16 1 i>7
25 1.73 1.25
30 1.44 1.05
Cooling tower design 28 1.53 1 12
80
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Cooling water pumps The head loss in feet of water through 1 1/4"
tubes is 0. 25" per foot of travel [30J. The length of tubes is 96'. The
head loss in water boxes is 1. 4' L30], For the head loss in the pond and
conduits, take 10% of the sum of losses in tubes and water boxes. Thus
the total pump head required is:
H = [0. 25 (96) + 1. 4j 1. 1 = 28'
The power required is:
or
kw = 0. 7457 bhp = 0. 0062G (Eq. 38)
where bhp = brake horsepower
G = pumping rate, gpm
y = fluid density
T} = pump efficiency, assumed to be 85%
kw = power required, kilowatts
Cooling Pond
Both mixed and flow through ponds are considered. The design of these
is characterized by the performance curves for the Philadelphia site
presented in Figs. 18 and 20.
The remaining equipment, buildings, and facilities of the cooling tower
and cooling pond design are assumed to be the same. They are omitted
from the economic study. Thus the economic study is based on the cap-
ital cost and operating expenditures for that equipment which is different
in the cooling tower and cooling pond designs, namely, condensers, cool-
ing water pumps, cooling towers/ponds. The measure of economy is
the annual cost and operation of such equipment expressed in dollars. It
should be remembered then, that percentage differences between the
costs of the two designs are meaningless.
Cost of Equipment From Ref. [28] we find that for the cooling tower
design, the cost of condenser fabrication and installation is $4. 90 per
square foot of condenser area, and the cost of pumps is $1. 50 per gpm.
For the cooling pond design, assume that the condensers also cost $40 90
per square foot, but that the pumps cost $0. 50 per gpm, because the
required head is about one-third of that required for the towers. The
cooling towers cost $15,920, 000 [28]. The cost of the cooling pond will
include all expenditures for its construction such as the cost of land, site
preparation, construction of dams and dikes. Since this cost is highly
81
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site dependent, three values of pond cost will be considered: $500,
$2000. and $5000 per acre of effective pond area. These pond costs
are considered to cover the extremes and the average.
The cost of equipment for plants with cooling towers and cooling ponds
is summarized below. For cooling ponds five temperature rises through
the condenser, five pond areas, and three pond costs are considered.
TABLE II
COST OF EQUIPMENT IN MILLIONS OF DOLLARS
Cooling
Tower
Cooling Pond
R°F 10
15
20
25
30
Condensers
Pumps
Cooling Tower
Cooling Pond
$500/acre
$2000/acre
$5000/acre
7.
1.
15.
51
68
92
21.
1.
200
570
14.
1.
151
045
10.
0.
400
785
8.
0.
500
625
7.
0.
080
525
(acres)lOOO 2000 4000 6000
10000
0.5
2. 0
5. 0
1. 0
4. 0
10. 0
2.0
3. 0
8. 0 12. 0
20. 0 30. 0
5. 0
20. 0
50. 2
The above equipment costs are presented in Fig. 22.
As the condenser cooling water temperature increases, the turbine
back pressure will increase correspondingly. Subsequently the turbine
efficiency will decrease and less electric energy will be produced per
given unit of fuel energy expended. This loss of capacity due to in-
creased turbine back pressure can be a substantial factor in the operat-
ing expense of a cooling pond facility. In order to assess the magnitude
of this loss of capacity, the turbine "exhaust pressure correction curve"
(that is, the curve showing decrease in output electric energy per unit
fuel energy input as a function of back pressure) must be known. It
should be pointed out that the "exhaust pressure correction curve" will
vary from one particular turbine to another. Since the cost of lost capa-
city due to increased back pressure is one of the more significant terms
contributing to operating cost of a cooling pond facility, a particular
"exhaust pressure correction curve" (and hence a particular turbine)
must be selected in order to execute the economic analysis and thus the
economic analysis becomes a "cut-and-try" process. In principle a
number of possible turbines should be selected for the task of conducting
an economic analysis to yield the optimum pond size. In selecting the
group of turbines, the assumption should be made that the average oper-
ating back pressure may turn out to be quite low (as, for example, if the
cooling pond were to be very lightly loaded and located in the extreme
82
-------
1 2 3
Acres/Megawatte
Equipment Cost for Captive Cooling Systems
Figure 22
83
-------
north-central United States, say, northern Minnesota) or quite high (as
for example, if the the cooling pond were to be very heavily loaded and
located in a zone of high equilibrium temperature, say, in Louisiana).
Although in principle a number of possible turbines should be selected
for the study, only one will be used in the present example, namely,
General Electric Co. designation TC6F-38, thermodynamic rating 1,
112, 215 kw (See Fig. 23).
With the use of Eq. 37 and the pond temperatures given in Fig. 18 for
the mixed ponds and Fig. 20 for the slug flow ponds, the condenser back
pressure can be determined as a function of pond area. These values
are shown in Figs. 24 and 25.
With the back pressure established as a function of pond area, the
"exhaust pressure correction curve" shown in Fig. 23 can be used to
determine the lost electrical capacity due to increased back pressure as
a function of pond area. These values are shown in Figs. 26 and 27.
With the data in Figs. 22, 26 and 27, equations 36 and 38, and a main-
tenance cost for ponds of $2/acre, the economics of a power plant with
a cooling pond, as compared with a cooling tower, can be computed for
various temperature rises of cooling water, pond sizes, and pond costs.
An example is given below for a mixed pond.
1. Temperature range, AT = 10°
2. Pond size, A = 1000 acres = 0. 5 acres/MW0
c
3. Pond cost = $2000/acre of effective pond area..
4. Pumping rate = 3. 14 x 10" gpm
5. The cost of equipment = $24. 77 x 10 (Fig. 22)
6. Interest rate = 11.5%
7. Capital cost = 24.77x10(1.115) = $2. 848 x 106/yr
8. Lost capacity due to back pressure = 80. 6x10 kw (Fig. 26)
9. Cost of lost capacity due to back pressure* = 80. 6 x 10(10. 40)
= $0. 839 x 106/yr
10. Lost capacity due to pumping** = 0. 0062(3. 14 x 10^} = 19. 5 x 10\w
11. Cost of lost capacity due to pumping*** = 19. 5 x 10^(10 40)
= $0.202 x 106/yr
* Based on a loss rate of $10. 40/kw-yr due to increased turbine back
pressure above the turbine design pressure, [28]
** See Eq. 38.
*** Based on a loss rate of $10. 40/kw-yr due to loss in capability be-
cause of power required for pumps and fans. [28]
84
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EflfllflfflS :!
- I-:::.-): - -TC6F 38 T!^ llSO-B^OEiF?
fr —4 bu_
M:g^
F' —
. . —
,_' « rf yi: r'^4 w yt ^'L_,j_ _y _* ___J_W.L;- •' ^jj^ ^j.
•— —- ~"^r -\ •*
-^ r —Tfyriv
i^rrE3 EXHAUST F.I'j^'JSS CC.1.1ECT::.< FACTC3S r=rr;
(G.i. »2?SSSMCSCU^
Fig. 23 - Exhaust Pressure Correction Curve
85
-------
1234
Pond Area, Acres per Megawatte
Condenser Back Pressure for a 2000 MW Plant near
^^ *s
Philadelphia, Mixed Pond Cooling, Design Climatic Conditions
Figure 24
86
-------
Is
CO
,0
rt
0)
a
<«H
O
CD
o
ti
v
t-i
2
CO
CO
4)
h
OH
X
O
«S
n
CD
c
ID
O
O
1-7 O. D. Tubes
4
I
c Cooling Towers
I
01234
Pond Area, Acres per Megawatt
Condenser Back Pressure for a 2000 MW Electrical Plant
near Philadelphia, Slug Flow Pond, Design Climatic Con-
ditions
Figure 25
87
-------
I Cooling Towers-i
1234
Pond Area, Acres per Megawatt
0 e
Lost Capacity due to Condenser Back Pressure, ZOOO Megawatt
Electric Plant near Philadelphia, Mixed Pond, Design Climatic
Condition?
Figure 26
88
-------
CO
4)
Q
CO
n
o
(0
ID
u
nS
ffl
Cooling Towers
Rj
u
•M
in
O
-2
12345
Pond Area, Acres per Megawatt
Lost Capacity due to Condenser Back Pressure,
2000 Megawatt Electric Plant near Philadelphia. Slug Flow
Pond, Design Climatic Conditions
Figure 27
89
-------
12. Replacement cost of lost power* = (80.6 + 14.5)10 (21.00)
= $2. 10 x 106/yr 3 6
13. Cost of power for pumps* = 19. 5 x 10 (21.00) = $0. 409 x 10 /yr
14. Maintenance = 2(1000) = $0. 002 x 106/yr
Total cost = sum of items 7, 9, 11, 13, 14 = $6.40x10
The cost of a power plant with cooling towers can be evaluated similarly
except that equipment cost should be taken from Table II, and the pump-
ing head is approximately three times higher.
The economics of cooling ponds of various sizes with five temperature
rises through the condenser, and different unit pond costs, together with
their comparison with the economics of cooling towers is presented in
Figs. 28, 29 and 30 for mixed ponds, and Figs. 31, 32 and 33 for slug
flow ponds. For the mixed ponds the optimum temperature rise through
the condenser is 20°F and the pond size is approximately 3, 2 and 1. 5
acres/MWe for pond costs of $500, $2000 and $5000 per acre of effect-
ive pond area, respectively. For the Slug flow ponds, the optimum
temperature rise through the condenser is 25°F and the pond size is ap-
proximately 2. 5, 1. 5 and 1 acre/MWe for the above pond costs, res-
pectively. The optimum for a practical cooling pond would be some-
where between those for the mixed and the flow through ponds. Figs.
28 through 33 show that the cooling pond, for reasonable land costs, is
competitive with cooling towers at locations where an adequate supply
of cooling water is available. At site locations where the climate is dry an;
and cooling water is scarce, the cost of replacing evaporated cooling
water should be considered.
It should be noted that the values in Figs. 28 through 33 are based on
annual cost for constant design conditions (that is, average summer
conditions). Since the summer time imposes the most adverse condi-
tions, the back pressure on which these figures are based is higher than
what the actual back pressure will be for a considerable part of the
year. As a result, the pond operating cost tends to be overestimated.
However, the technique presented herein can be readily extended by
considering the average climatic conditions for each of the four seasons
or, if desired, by considering the average climatic conditions for each
month. When the economic evaluation is extended in this manner, a fam-
ily of turbines and condensers must be included in order that the optimum
hardware can be selected on the basis of seasonal or monthly average
climatic conditions (as well as power plant loading). Since the economics
* Based on 7000 hours of operation per year at an energy cost of $0. 003/
kw-hr, or $2l. 00/kw-yr.
90
-------
3. 5
rCooling Towers
1 2 3
Pond Area, Acres per Megawatt
Annual Cooling Cost, Mixed Pond, 2000 MWe Plant
Land and Development Cost - $500 per Acre of Pond
Design Climatic Conditions for Philadelphia
Figure 28
91
-------
3.5h
nJ
t>0
'H
0)
w
h
nt
,— i
,— I
o
Q
«M
O
TO
Tt
c
tti
CO
EJ
O
^
H
2. 5
(0
O
U
1. 5
T 234
Pond Area, Acres per Megawatt
G
Annual Cooling Cost, Mixed Pond, 2000 MWQ Plant
Land and Development Cost - $2000 per Acre of Pond
Design Climatic Conditions for Philadelphia
Figure 29
92
-------
4. 5
%
*
«J -
to -
-------
3.5
1234
Pond Area, Acres per Megawatt
Annual Cooling Cost, Slug Flow Pond, 2000 MWe Plant
Land and Development Cost - $500 per acre of Pond
Design Climatic Conditions for Philadelphia
Figure 31
94
-------
CO
ctf
r—I
O
Q
CQ
TJ
C
n!
o
A
H
CO
O
U
3. 0
2. 5
)ling Towers
A <-r
- D 1 r> T7>
_-^
^s*^
2 0
1. 5
1.0
0. 5
1234
Pond Area, Acres per Mega\vatt
Annual Cooling Cost, Slug Flow Pond, 2000 MW Plant
Land and Development Cost - $2000 per Acre
Design Climatic Conditions for Philadelphia
Figure 32
95
-------
4. 5
1234
Pond Area, Acres per Megawatt
Annual Cooling Cost, Slug Flow Pond, 2000 MW Plant
Land and Development Cost - $5000 per Acre G
Design Climatic Conditions for Philadelphia
Figure 33
96
-------
of the pond are dependent on the turbine back pressure characteristics,
the variation of climatic condition during the year and the variation
on power plant loading during the year, the task of identifying opti-
mum plant equipment and pond size are strongly coupled problems and
are substantially site dependent.
97
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MULTI-PURPOSE USE OF COOLING PONDS
The economic production of electric power is complicated by the fact
that it cannot be stored economically in large quantities and must
therefore be produced at a rate corresponding to the demand. The
demand varies considerably with the time of day. However, from the
viewpoint of the total cost, capital plus operating cost, it is desirable
to structure an electrical power system such that the fossil or nuclear
driven power plants operate at capacity in as steady a fashion as pos-
sible. When operating in this steady mode, the plants are referred to
as "base-loaded" plants. One technique that has been used to allow
base-loading of major size fossil (or nuclear) plants in a particular
power system and still match the system's electrical.power output with
the peak in the customer's demand curve is to include hydroelectric
plants in the system. Since enei gy used to drive the hydroelectric
plants, namely, water at the top of an abrupt elevation change, can be
"stored" to some extent and since such plants can be started and
stopped quickly compared to steam plants, the hydroelectric plants can
be used to provide the power to accomodate the demand peaks. Such
peaking power plants are restricted to power systems that are fortun-
ate enough to be located near feasible sites for the construction of hy-
droelectric plants.
A more generally applicable method for making it possible to base-
load the major plants in a system is to use the surplus electric power
generated by the base-loaded plants during periods of low consumer
demand to pump water from a lower level to a high level. Subsequently
this energy is used to generate electric power during periods of high
consumer demand by allowing the pumped water to return to the low
level by passing through the same hydraulic and electrical machine
functioning in this case as a turbine and generator. The overall effic-
iency of this storage system can be of the order of 70%.
Within recent years large reversible pump-turbines are being develop-
ed for peaking power plants. These are generally of the variable pitch
Kaplan type so that the same electrical equipment is used both as a
motor or a generator and the same hydraulic equipment as a. pump or
turbine. The direction of rotation is changed for pumping or generat-
ing and a wide range of flows and corresponding power levels for the
same synchronous rpm (except sign) can be handled by changing the
pitch of the pump-turbine.
It is desirable to use the combination of storage ponds that are required
for the pumped storage system for cooling of the condenser water. If
one of the water reservoirs is a river or lake in which thermal pollu-
tion is unacceptable, the power plant can be located on the upper
99
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reservoir which will function as a cooling pond. In this case the
channel to the pump-turbine should be on the inlet side of the conden-
ser so that cool water is interchanged with the lower reservoir, and
only the upper reservoir is significantly above the ambient tempera-
ture. The location of a fossil fuel power plant on the upper level is
desirable for air pollution reasons. It will be necessary to locate the
hydraulic peaking power plant at the lower level because the pump can-
not lift water more that a few feet without cavitating.
It will generally be desirable to have the power capacity of the hyd-
raulic peaking power plant about the same size as that of the power
plant. It will be found that at this rating the flow in the peaking power
plant will be much larger than that required for the condenser cooling
water and if the upper reservoir is not deep, the required storage cap-
acity will provide adequate surface area for cooling. In this case
there is no additional cost for the cooling pond.
At the end of a long power demand it might be anticipated that the level
of the upper reservoir will be down so that there will be a reduced sur-
face area for cooling the condenser water. However, since the flow
rates of the cooling water is only a small percentage of that of the
storage water*, the water for the turbines will be primarily the cooler
subsurface water and the warmed condenser water will not reach the
turbine intake. Even when operating at a small percentage of rated
capacity as a purnp, the hydraulic unit will deliver cool water to the
condenser.
The water lost by evaporation, as shown earlier, is only slightly
greater than that lose by natural evaporation, and so a significant sav-
ing in water is attained when the cooling pond is combined with the
pumped storage project.
* The water required by the condensers = 0. 01 52[KW x (l-TfcJ/fy AT] cfs
where TJ,. = plant thermal efficiency (neglecting stack losses) The water
flowing in the pumped storage system - 11. 8[KWr)p/AhJ cfs, as a pump
where TJp = efficiency of pump plus motor plus duct; Ah = ft difference
in head between reservoirs and = 11. 8 KW/(AhT) ) cfs, as a generator.
O
100
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ACKNOWLEDGEMENTS
The authors wish to acknowledge the advice and helpful information
provided by Mr. Walter Shade and the technical staff of Gilbert Assoc
iates, Reading, Pennsylvania.
The cooperation of the following electric power companies which
utilized captive ponds for condenser water cooling is appreciated.
These companies provided month by month information regarding
thermal loading and water temperature in their cooling ponds:
Southwestern Electric Power Company
Commonwealth Edison Company
Arizona Public Service Company
Virginia Electric and Power Company
Mississippi Power and Light Company
Texas Power and Light Company
South Carolina Public Service Authority
101
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Physical Review, Vol. 30, October 1927
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March, 1936
3. Throne, R. F. , How to Predict Lake Cooling Action, Power,
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4. Longhaar, J. W., Cooling Pond May Answer Your Water Cooling
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6. Anderson, E. R. , Energy Budget Studies in Water-Loss Investi-
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Environmental Science Services Admin. , Washington, D. C. ,
June 1968
103
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13. ASHRAE - Handbook of Fundamentals, American Society of Heat-
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N. Y. , 1967, pp. 470-474
14. Hutchinson, G. D. , A Treatise on Limnology, John Wiley and
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15. Koberg, G. E. , Methods to Compute Long Wave Radiation from
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face, U. S. Geological Survey Professional Paper, Washington,
D. C., 1962
16. Sverdrup, H. U. , On the Evaporation from the Oceans, Journal
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U. S. Dept. of Commerce, Environmental Science Service Admin. ,
Asheville, North Carolina
18. Local Ciimatological Data for Winslow, Arizona - 1967, U. S.
Dept. of Commerce, Environmental Science Service Admin. ,
Asheville, North Carolina
19. Marciano, J. J. and Harbeck, G. E. , Mass Transfer Studies in
Water-Loss Investigation: Lake Hefner Studies, U. S. Geological
Survey Professional Paper 269, Washington, D. C. , 1954
(This material previously published as U. S. Navy Electronics
Lab. Report No. 327 and U. S. Geological Survey Circ. No. 229,
1952)
20. Koberg, G. E. , Harbeck, G. E. and Hughes, The Effect of the
Addition of Heat from a Power Plant on the Thermal Structure and
Evaporation of Lake Colorado City, Texas, U. S. Geological Sur-
vey, Professional Paper No. 272-B, 1959
21 Linsley, R. K. , Kohler, M. A. and Paulhus, J. L. , Hydrology for
Engineers, McGraw Hill Book Co. , New York, N. Y. , 1958, pg 98,
Also see Ref. 7.
22. Bowen, I. S. , The Ratio of Heat Losses by Conduction and by
Evaporation from any Water Surface. , Physical Review, Vol. 27,
June 1926
104
-------
23. Sunderam, T. R. , Easterbrook, C. C. , Prech, K. R. and Rud-
inger, G. , An Investigation of the Physical Effects of Thermal
Discharges into Cayuga Lake, Cornell Aeronatuical Lab. ,
No. VT-2616-0-2, Buffalo, N. Y. , November 1969
24. Beer, L. P. and Piper, W. O. , Environmental Effects of Con-
denser Water Discharge in Southwest Lake Michigan, Waukegan
Study on Lake Michigan, Commonwealth Edison Co. , Chicago, 111.
25. Steam-Electric Plant Construction Cost and Annual Production
Expenses - Twentieth Annual Supplement, Federal Power Com-
mission, Washington, D. C. 1968
26. Chemical Engineers' Handbook, Edited by John H. Perry, McGraw
Hill Book Co. , New York, N. Y. , 1963, pp. 14-20, 21
27. Holman, J. P. , Heat Transfer, McGraw Hill Book Co. , New York,
N- Y. , 1963, pg. 383
28. Private Communication from Walter Shade, Gilbert Associates,
Inc. , Reading, Pennsylvania.
29. Skrotzki, B. G. A. and Vopat, W. A. , Power Station Engineering
and Economy, McGraw Hill Book Co. , New York, N. Y. , I960
30. Standard Handbook for Mechanical Engineers, Seventh Edition,
Edited by T. Baumeister, McGraw Hill Book Co. , New York,
N. Y. , 1967
31. Winiarski, L. D. , Tichenor, B. A. and Bryan, K. V. , A Method
for Predicting the Performance of Natural Draft Cooling Towers,
U. S. Dept. of Interior, FWPCA - Pacific Northwest Water Lab. ,
Corvallis, Oregon, January 1970
32. Leung, P. and Moore, R. E. , Water Consumption Determination
for Steam Power Plant Cooling Towers: A Heat-and-Mass Balance
Method, Presented at the Winter Annual Meeting of ASME, Los
Angeles, Calif. , November 1969
105
-------
NOMENCLATURE
a a constant coefficient
A pond surface area
c specific heat of water
c Brunt's Coefficient
13
c, friction coefficient
c specific heat of air at constant pressure
c . constant in Wien's Law
C concentration of water vapor per unit volume
D diffusion coefficient
/ a term in the energy balance equation defined as
/_ a parameter used to plot the steady state behavior of a slug
flow pond (in which case / - A(ATC)/WTE , or to plot the
transient behavior of a mixed or slug flow pond (in which
case / = t/pc(V/A) .
/ a parameter used to plot the water lost from a slug flow
pond by evaporation
g acceleration of gravity
h specific enthalpy of water as it leaves the pond by evaporation
(with respect to the reference point h =0 at 32°F)
h mass transfer coefficient
h specific enthalpy of precipitation (with respect to the
P reference point h = 0 at 32°F)
P
Ah change in the specific enthalpy of water between the liquid
state at the make-up temperature and the vapor state at
the pond surface temperature
Ah change in the specific enthalpy of water between the liquid
state and vapor state at constant temperature
H. R. annual average power plant heat rate
/ pond width at x=0
i pond length at y=0
107
-------
& pond depth at a given location
m mass flow rate per unit surface area
m mass flow rate through the condenser
m mass flow rate per unit surface area out of pond due to
evaporation
m mass flow rate per unit surface area into pond in the form
mu , , .
of make-up water
m mass flow rate per unit surface area into pond in the form
of precipitation
m mass flow rate per unit surface area out of pond due to
seepage
M molecular weight of air
cl
M molecular weight of water
w °
M mass flow rate
M mass flow rate into pond in the form of make-up water
M mass flow rate into pond from condenser
M mass flow rate out of pond to condenser
PPo
MW net electrical power produced by the plant in megawatts
N empirical coefficient used to relate evaporation rate to the
wind speed and relative humidity
P water vapor pressure in the atmosphere
P barometric pressure
' JBAK.
P saturated water vapor pressure corresponding to the
temperature of the water at the pond surface
Q atmospheric radiation incident on pond
3-
Q atmospheric radiation reflected from pond
3-1*
Q back radiation emitted from pond surface
Q energy flow rate per unit surface area
U net absorbed radiation
Q solar radiation incident on pond
s
Q solar radiation reflected from pond
sr
108
-------
R Reynolds Number
ey '
R ratio of reflected to incident solar radiation
sr
R universal gas constant
u B
S Schmidt Number
c
t time
t residence time of water in the pond
T temperature
T absolute air temperature
aa r
T absolute water temperature
wa c
AT temperature rise through the condenser
u fluid velocity
V pond volume
w humidity ratio
W wind speed
W wind speed in Colorado City Equation
Ci
W wind speed in Lake Hefner Equation
H
W wind speed in Meyer Equation or mean speed for one month
M
WTE waste thermal energy from a power plant
x a coordinate distance
y a coordinate distance
z pond depth
a thermal diffusivity
$ linear temperature profile gradient
C emissivity
C emissivity of water
w
6 temperature measured with respect to 32° F (6 = T-32°F)
8 equilibrium pond temperature measured from
Cq 32°F (8eq = Teq - 32°F)
0 temperature of direct precipitation measured from
Pi 32°F (8 = T - 32°F
P.- P.-
109
-------
} temperatxire of water leaving the condenser measured from
PPi 32°F (9 - T - 32°F)
PPi PPi
9 temperature of water entering the condenser measured from
o
PP~ 32°F (9 = T - 32°F)
PP
*" o o
9 water surface temperature measured from
S 32°F (9 = T - 32°F)
s s
9 steady state mixed pond water temperature measured from
SS 32°F (9 - T - 32°F)
ss ss
X pond width
X wavelength
V kinematic viscosity
p density
p density of boundary layer material
Cr Stefan-Boltzmann constant
110
-------
APPENDIX A
ENERGY BALANCE EQUATIONS
Completely mixed pond, no vertical temperature gradient
In the case of the completely mixed pond, it is assumed that the inlet-
outlet structure, the wind, turbulence, and vertical currents resulting
from cooling at the surface and subsequent sinking of the denser fluid
result in complete vertical mixing so that no temperature difference
exists between the water at the top and at the bottom of the pond.
Water is lost from the pond by evaporation to the atmosphere and by
seepage into the surrounding ground. Water is added to the pond by
precipitation on the pond surface, by precipitation run-off that flows
directly into the pond and by the addition of make-up water which may
be obtained from a stream, a storage reservoir or wells. For the pur-
pose of analysis, it is assumed that the flow of make-up water into the
pond is continuously adjusted so that the volume of water in the pond
remains constant. To check the reasonableness of this assumption, it
is helpful to note that based on two acres of pond surface per megawatt
of electricity; a 35% thermal efficiency and 2/3 of the cooling to re-
sult from evaporation, the pond volume decreases by only 0. 2 inches
per day due to the extra thermal load in the absence of seepage. On
the average, for an annual rainfall of 36", direct precipitation would
add 0. 1 inches per day of water to the pond.
The mass balance for the entire pond is given by the expression:
[& +M + Am ] - [M + Am + Am ] = ~ (pV) = 0 (Eq. A 1)
mu pp. p PP s e ot
where
M = flow of makeup and inflow water into the pond, #m/day
mu
M = flow of water from the power plant condenser into the
^i pond, #m/day
rn = precipitation flux falling directly on the pond surface,
P #m/day ft2
M = flow of water from the pond to the power plant conden-
* o ser, #m/day
m = flux of water from the pond through outflow and to the
ground by seepage, #m/day ft
m = flux of water from the pond to the atmosphere by
evaporation, #m/day ft
111
-------
A = pond surface, ft
p - density of water, #m/ft
V = volume of pond water, ft
The energy balance for the entire pond can be expressed as: (See
Fig. 1)
{M c9 + M CQ + Am h +A[(Q -Q ) + (Q -Q )]}
mu mu pp. pp. p p s sr a ar
- {M c9 + Am c9 + A[Q, + h m + Q ]}
c -r-v-r-* o KT* £* f* r*
PP
br e
o
= rJ7J'pcedv (Eq. A- 2)
where c = specific heat of water at constant pressure,
-1. Obtu/#m°F
6 = pond temperature in excess of 32 F = T-32 F, F
Q - solar radiation incident on pond surface, btu/day ft
s
Q = solar radiation reflected from pond surface, btu/day ft
sr
•
Q = atmospheric radiation incident on pond surface,
,a btu/day ft2
Q = atmospheric radiation reflected from pond surface,
3.T" *>
btu/day ftZ
Q = energy radiated from pond surface, btu/day ft
br
m = mass flux from the pond to the atmosphere by evapora-
tion, #m/day ft2
h = enthalpy of evaporated water, btu/#m (Ref. 32 F. fluid)
• ^
Q = energy transfer from the pond to the atmosphere by
convection, btu/day ft
h = enthalpy of precipitation (Ref. 32 F, fluid)
P = c6 , btu/#m°F, ifT > 32°F
Pi P!
= +0. 492(9) - 143. 3, btu/#m°F, if T < 32°F
Pi
Since the temperature is everywhere equal in the pond at a given in-
stant, the right hand side of Eq. A-2 may be expressed as:
c9dv = ~ (pc9V) = ce~(pV)+ c(pV)~ (Eq. A- 3)
However, the first term on the right hand side of the last equation is
112
-------
zero as a result of the assumption applied in Eq. A-l, thus the last
equation becomes:
ce dv = c(pV)|| (Eq. 2-4)
The first term in Eq. A-2 represents the energy introduced into the
pond by the addition of make-up water and can be expressed with the
aid of Eq. A-l, as:
M c 6 = A(m + m - m ) c 0
mu mu s e p mu
where m - m =0, since there is no loss of cooling water in
PP PP
i o the condenser.
Thus,
M c9 A(m - m ) c6 + A(m h ) (Eq. A-5)
mu mu s p mu e mu
where h = c6 = enthalpy of the water added as make-up
mu mu
Substituting Eq. A-4 and Eq. A-5 into Eq. A-2, the energy equation
becomes:
i ii in 9 iv t
{A[m -m ]c9 +M c6 + Am h + A[(Q -Q ) + (Q - Q )]}
s p mu pp. pp. p p s sr a ar
v vi vii vm.
- {M c0 + Am c9 + A(Q, ) + A[m (h - h ) + Q ]} = c(pV)—
pp s br e e mu c dt
(Eq. A-6)
It is now helpful to express the terms in Eq, A-6 as polynomials of 6
and then collect all terms of equal order.
Term I represents the energy introduced into the pond by the addition
of water to compensate for seepage and precipitation and is not a func-
tion of the pond temperature.
Terms III and IV do not depend on the pond temperature, 8, and may
be left as they are.
Terms EL and V may be combined and expressed as:
• •
M c0 - M cQ = waste thermal energy = WTE, btu/day
PP. i PP
1 ° (Eq. A-7)
Term VI is already expressed as a function of the pond temperature to
113
-------
the first power.
Term VII represents the energy radiated from the pond surface, and
since water radiates as almost a black body, this term may be ex-
pressed as:
Q = AC or(9 + 32 + 460)4 (Eq. A- 8)
br w
where C = emissivity of water = .97
w -3 2 o 4
a = Stefan- Bolt zman constant 4. 15 x 10 btu/day ft ( R )
Eq. A-8 can be expanded in binomial form to give the expression:
] (E,. A-9)
At the cost of introducing an error of less than 7/10 of one percent for
pond temperature up to 100°F, the last term on the right hand side of
Eq. A-9 may be neglected. Thus Term VII may be expressed as:
Q = A(a_ + a,9 + a _92 + a063} btu/day (Eq. A-10)
br 5 b I o
where ar = 2358. 74 btu/day ft
D
a, = 19. 176 btu/day ft2 (°F)
?
a = 5.850 x 10-2 btu/day ft2 (°F)
a0 = 7. 923 x 10"5 btu/day &2 (°F)3
o
Term VIII represents the heat transfer to the atmosphere as a result
of evaporation and convection, and may be expressed as:
A{Cm (h - h ) + Q ]} = A{m (Ah) + Q } (Eq. A-ll)
where m Ah = net energy transfer that results from water evapor-
ating from the pond and being replenished. The
value of Ah depends on the temperature of the evap-
orating water and make-up water. Since the make-
up water temperature can vary from 32°F to about
90°F, and the maximum pond temperature will be
about 25 F above the make-up water,- Ah could vary
from 1085. 7 to 1053. 6. Thus within an error of
Hh 1. 5%, the value of Ah may be taken as a constant
equal to 1070 btu/#m.
114
-------
In Appendix B it is shown (Eq. B-39) that Eq. A- 11 can be expressed
in the form of a polynomial in 8, namely,
4)9} (Eq. A-12)
(See Appendix B for values of coefficients)
where P = water vapor pressure in atmospheric air, psia
ct
6 = air temperature referenced to 32 F (9 =T - 32 F)
a a a
Substituting Eqs. A- 7, A- 10 and A-12 into the energy balance equa-
tion, A-6, yields:
(Eq. A-13)
, 2
where Q = WTE/A, btu/ft day
PP
If in addition to the previous assumptions it is assumed that the en-
thalpy of the precipitation and seepage are equal to the enthalpy of the
water in the pond, then the first terms on the right hand side of the
above equation will goto zero, namely:
{rh (h - c8 ) - m (c9 - c9 )} = 0 (Eq. A-14)
p p mu s mu
Completely mixed pond> simplified case, steady state operation
The general energy balance equation can be simplified considerably if
Eq. A-14 is assumed to be valid and if the ratio of volume to area of
the pond is assumed to be very small (or if the time over which weather
115
-------
conditions are averaged is long compared to the residence time in the
pond). These assumptions lead to the steady state energy equation:
(Eq. A-15)
Slug flow pond, no vertical temperature gradients
For this situation it is helpful to look not at the entire pond as was
done for the mixed pond, but at a differential control volume which
moves at the constant velocity, u, imposed by the plant pumping rate
at any given time. Water is lost from the control volume by evapora-
tion to the atmosphere and by seepage into the surrounding ground.
Water is added to the control volume by precipitation directly on the
exposed surface and by precipitation run-off that flows into the control
volume and by additional make-up water which may be obtained from a
storage reservoir, a stream or wells. Again as in the case of the
mixed pond, it is convenient for the purpose of analysis to assume that
the flow of make-up water into the control volume takes place contin-
uously at such a rate that the volume of water in the control volume re-
mains constant.
The mass and energy balance for the moving control volume can be
developed in a way similar to that used for the completely mixed pond.
However, in this case we must make allowances for variable tempera-
ture rise across the condenser, variable pond depth and variable pond
width. The pond geometry is sketched below along with the control vol-
ume.
Flow
Direction
x
116
-------
z - depth at any x dimension
SL - length at y = 0
£4
I = length at x=0
dX
-*! h-
Surface area = y(dX), a function of x
Volume = V = y(dX), a constant
-K Velocity, u, a function of x
The mass balance for the moving control volume is:
y(dX)[m + m J - y(dX)[m +m ] = ~Lpy )(dX)] = 0 (Eq. A16)
mu p s e o t z
where m = flux of make-up water into control volume, #m/ft day
mu
Eq. A- 16 can be solved for the rate of flow of make-up water required
to maintain a constant water volume in the pond and yields:
m = m 4- m - m (Eq. A- 17)
mu s e p
The energy balance for the control volume is:
y(dX){m c9 + m c8 + [(Q - Q ) + (Q - Q )}
J mu mu P P s sr a ar
-y(dX){mc9 + Q +mh + Q } = ^-[ c6pzy d6] (Eq. A-18)
However, the right hand side of Eq. A-18 can be expressed as:
5 3ft
~ [cSpzy dXj = pc (yz dX)|^ (Eq. A- 19)
since T— (yz dX) = 0 from Eq. A- 16.
Thus, be substituting Eq. A-19 and A- 17 into A-18, the energy equa-
tion can be expressed as:
117
-------
I III , IV
{(m - m )c9 + m c9 + [(Q - Q ) + (Q - Q )]}
cv s p mu PP s =^ a aT*
sr
ar
vi yii vm t 90
- {m c6 + Qu + Cm (h - h ) + Q ]} = cp z —
s br e e mu c o t
(Eq. A-20)
When the differential equation is solved for this case, it yields the
temperature of the fluid in the moving control volume. Subsequently
the temperature at any location along the pond flow length at any de-
sired time can be found from the relationship between time and dis-
tance traveled, namely:
(Eq. A-20A)
x = J u dt
t=0
where u = water velocity in longitudinal direction, ft/sec
Since the terms in Eq. 19 correspond to the like labeled terms in
Eq. A-6, Eq. 19 can also be expressed as:
-= {m (h -c9 )-m(ce-ce )
dt p P rnu s mu
(a
!2
(Eq. Q-21)
It is usually more convenient to find the temperature not after a cer-
tain longitudinal distance, x, has been traveled by the control volume
of "slug", but to find the temperature when the slug has swept out a
certain amount, A, of pond area. Thus for the variable width and
depth pond under consideration.
/-dA
->-Flow
118
-------
the continuity equation yields:
(dt) (m ) = pzdA (Eq. A-22)
c
or
dt = §r dA
-------
AT
where the temperature decrease experienced by the water as it
flows through the pond must be equal to the temperature rise
(ATC) experienced by the water as it passes through the con-
denser for steady state operation. This demand of equality
determines the magnitude of the inlet and outlet pond temper-
ature for given climatic and plant conditions.
120
-------
APPENDIX B
HEAT TRANSFER BY EVAPORATION AND CONVECTION
Evaporation
Many empirical relationships have been developed in order to predict
the rate of evaporation from a natural water surface exposed to the
atmosphere [19, 20, 2lJ. However, all the experimental data for
large bodies of water exposed to the atmosphere are limited to un-
heated or very slightly heated water, with the result that these data
have been collected under conditions where the equilibrium tempera-
ture and water temperature differ by only a few degrees. Since it is
a matter of experience that the equilibrium temperature and air temp-
erature generally differ by only a few degrees, it may then be said
that these data were collected for conditions of only a few degrees dif-
ference between air and water temperature. In view of the above con-
siderations, the empirical equations developed on the bases of these
data can not, a priori, be extrapolated to the anticipated situation
where the water temperature may be as much as 30°F in excess of
the air temperature as a result of heavy thermal energy loading on
the pond. In addition, evaporation rates predicted by the various em-
pirical formulae differ by more than a factor of two among each other.
(See Fig. B-l). Likewise, equations based on fundamental principles
have been devised, of which the work by Sverdrup is the most noteable
[16]. Nevertheless, when he used his method to compute evaporation
rates in various regions of the Atlantic Ocean and compared the re-
sults to measurements made by Wust, the computed values differed
from the measured values by as much as + 20% and - 35% (See Fig.
B-2).
In view of the fact that evaporation accounts for the major portion of
the heat transfer from the pond (approximately 40 to 70%), it is import-
ant to be able to make accurate estimates of the evaporation rate for
ponds where specific experimental evaporation data are not available, -
in particular, for heated ponds with temperature substantially above
natural pond temperatures. It is therefore desirable to develop an
analytical expression which displays reasonable agreement with some
of the known measurements in the range of their validity, but which
can be extrapolated to regions of present interest.
In order to develop the desired relationship, the analogy between mass
and momentum transfer will be used. The relationship will be devel-
oped on the basis that the water surface can be assumed to be a smooth
surface.
121
-------
to
Relative Humidity = 80%
6 8
Wind Velocity, mph
10
12
14
Figure B-l
-------
mm IN
24 HOURS
40° N 30'
20'
LATITUDE
10' 0*
10'
20'
30*
NOTE: Predicted curves are based on the assumption
that the sea surface is characterized by a con-
stant roughness parameter (E ) or by a smooth
surface (E ). [After Sverdrup, Ref. 16]
Fig. B-2 - Comparison of Measured Pan Evaporation and
Observed Values over the Atlantic Ocean
123
-------
If it is assumed that the water surface can be represented by a
smooth flat surface, the analogy between mass and momentum trans -
rer can be expressed as: [27, p. 337]
v q 2/3
no c/- •. 12
For Laminar Flow: ~ = — = 0. 332Reyx (Eq. B-l)
CD
h s 2/3
For Turbulent Flow. -—7 - -r^ = 0. 0288Rey " 5 (Eq>
U u A-
where c = local friction coefficient
h = mass transfer coefficient, ft/sec
S = Schmidt Number
C 2
= f/D (" = kinematic viscosity, ft /sec,
D = diffusion coefficient, ft /sec)
U = free stream air velocity, ft/sec
CO J
Rey = Reynolds Number at a distance, x, from the leading
X edge or Rey = (U^
If it is assumed that the boundary layer starts at the edge of the water
surface, the local value of the average mass transfer coefficient up to
any distance, x = L, from that leading edge is found by combining
Eq. B-l and Eq. B-2 to yield:
x= 300, 000—MAX U Ux ,,
*D - L LJ 0 2/3 v< J^/v V '
ave x=0 S
c
x=Rey (~-)MAX U^ Umx .
00 _J * ~ " (« Uo o o) ( " / cLx J (£j<^. B — 3)
x=300, 000 (V/U^) S V
The first term on the right hand side represents the contribution to the
diffusion coefficient over the zone where the boundary layer is laminar
and the second term represents the contribution where the boundary
layer is turbulent. However, before the second term can be integrated,
the upper limit of integration must be established. If the water and the
atmosphere were not exchanging heat, and if in addition, the wind was
steady and moving parallel to the water surface, the turbulent boundary
124
-------
layer would continue to thicken and the upper limit of x would be the
length of the lake. However, even in the absence of heat exchange
between the water and the atmosphere, the wind will not be steady but
will gust and may not be moving parallel to the water surface. These
conditions are likely to vary rapidly in time, but most likely result
in substantially smaller influences than the combination of averaging
the wind speed over as much as 30 days and the effect of heat ex-
change from water to the atmosphere. Thus, it seems reasonable as
a first approximation to take as the upper limit for x that distance
at which the heated and therefore buoyant boundary layer tends to rise
vertically from the pond surface, with the result that free stream air
fills in to form the beginning of new boundary layer material. The
distance at which this happens, xo , can be estimated by setting the
ratio of buoyant force to inertia force equal to unity. Thus:
B.F. . P~gXo3-pB.L.gXo3
where p = free stream air density
p = density of material in the heated boundary layer
-D. .Li.
g = acceleration due to gravity
U = air free stream velocity
Solving Eq. B-4 for the boundary layer thickness at x = x yields:
x = - - (Eq. B-5)
o P_
g(:
<144>
(53. 34)(T )
(53. 34, ,Twa,
(E,. B-7)
125
-------
o
where T = absolute air temperature, R
aa Q
T = absolute water temperature, R
wa
w = humidity ratio
P = barometric pressure, psia
BAR
P = water vapor pressure corresponding to saturation
at the temperature of the water surface, psia
Using Eq. B-5 as the upper limit for integrating Eq. B- 3 results in
the expression:
ave Xo S
c
+ .0288(~ )1/5(|)[xQ - (3 x 105)^-]4/5} ft/sec (Eq. B-8)
o>
Eq. B-8 may be solved for specific climatic conditions and substituted
into the expression for the net diffusion of water vapor into the atmos-
phere given below.
™ = hr^
-------
Eq. B-ll could be solved for a given water temperature and given
climatic conditions; however^ before doing so, it is helpful to note
that th,e laminar boundary layer extends for only a distance of
3x10 f/Um , which at 70°F and 5 mph is equal to
=
7. 34 ft/sec
Since this is small compared to the anticipated pond dimensions and
since the air will be initially turbulent, we can approximate Eq. B-ll
as if the boundary layer were turbulent from the edge of the pond.
Thus the expression for h from Eq. B-8 becomes:
ave
c
When Eq. B-5 and B-7 are substituted into Eq. B-12 and the results
are substituted into Eq. B-ll, the expression for evaporation becomes:
me = <1.047xlo4)^->2/V/5U,2/5
c
T (P ) P - P
r wa BAR ,-ii/5 r w a n
T )PRAp -P )(l+w)
aa BAR w wa
where S = r- and since D varies approximately at T [Ref. 26]
c D a.bs
,_J _ w 537 3/2
= V (— - ) ( ^ - )
D77°F Taa
D = 2. 75 x 10"4 ft2/sec at 77°F [Ref. 27]
Thus Eq. B-13 becomes:
m. .
e->-- — 7/15LT JLT (PQAT3-P )
I/ wa aa BAR w
(Eq. B-14)
Since the wind velocity is measured at some height, b, (often ~ 26 ft),
Eq. B-14 must be corrected for the velocity profile effect by substit-
ing into Eq. B-14 the expression:
127
-------
U = ( — )U (Eq. B-15)
00 a b
where U - velocity of the wind at some specified elevation
above the water surface, b feat, ft/sec
U = free stream wind velocity, ft/ sec
00 J
Ci - ratio of wind speed at elevation b to free stream wind
speed. This ratio will depend on U^ and location. The
value of tti as measured above open grassland is shown
in Fig. B-3. Although tt^ will vary with specific loca-
tion, a variation of + 50% results in a variation of the
evaporation rate of only _+ 18%.
Thus, the final expression for the rate of evaporation is:
TT T T CP }
1 ,3/5 b r aa wa' BAR1 ,1/5
x [P - P ] (Eq. B-16)
Of the formulae available in the literature, two have been selected
which have no zero wind velocity term, namely, an equation from the
Lake Hefner study and one from the Lake Colorado City study, and
one formula with a zero wind velocity term, namely, the Meyer Equa-
tion. In each of these three cases, the rate of evaporation is given as
a function of local wind speed and the difference between the satura-
tion vapor pressure corresponding to the water surface temperature
and the water vapor pressure some distance above the surface. The
three equations are:
Lake Hefner Eq. [19] : m =(• 6l4)WfP -P)#m/dayft3 (Eq.B-17)
e ri w a
c w
The Meyer Equation [Zl]:
Lake Colorado City [20j: m^ = (, 897) WJP_ - PJ #m/day ft (Eq. B-18)
W
+ ~p)(P -P ) #m/dayft2 (Eq.B-19)
where C^ is a constant for a given location and ranges from 10 to
15 depending on depth and exposure of the water under
study as well as the frequency of the available meteor-
ologic measurements. For surface accumulation, C is
taken near the higher value whereas for large deep
128
-------
1.0,
0. 8
0.6
•
-------
bodies of water, C is taken near the lower limit.
W = wind speed, mph
W = W in Lake Hefner Equation measured about 26 feet
above the ground and taken as the average over a
three-hour period
W = W in Lake Colorado City Equation measured about 26
feet above the ground and taken as the average over a
24-hour period
W = W in the Meyer Equation and taken as the monthly
average wind speed value from measurements made
at the nearest Weather Station about 25 feet above
the surface
P = equilibrium, or saturated, vapor pressure corres-
ponding to the temperature of the water at some
specified point near the surface
P = water vapor pressure of the atmospheric air, mea-
sured at the same height and averaged in the same
way as the wind speed, psia
Figs. B-4, B-5, B-6 and B-7 show a comparison of the three empir-
ical equations and the analytical solution, Eq. B-16, based on the an-
alogy between mass and momentum transfer for heated water under
summer and winter conditions respectively with T = T and with
T >T . W "
w a
Since T = T in both Fig. 4 and Fig. 6, the condition under which
the original experimental data were collected (that is, water near the
equilibrium temperature) is satisfied. From Fig. 4 and Fig. 6 it is
noted that the evaporation rates predicted by Eq. B-16 for the cases
where TW = Ta follow the trend of the Meyer Equation with Cj = 10
closely over the range of wind speeds pertinent to monthly average
values (~ 4 to 15 mph), but are somewhat lower (about 10 to 20%).
Whereas in Fig. 5 and Fig. 7 (with TW greater than T by 30°F) the
evaporation ratio predicted by Eq. B-16 also follows the trend of the
Meyer Equation with C^ = 10 over the wind speed range of present
interest, but are somewhat higher (about 20%). On the basis of these
observations it appears reasonable to use the Meyer Equation with
Cj = 10 to determine the evaporation ratio for pond temperatures that
are only slightly above air or equilibrium temperature. However, for
heavily loaded ponds, the water temperature will be well in excess of
the equilibrium temperature and the rates should be estimated by use
of Eq. B-16. If the simpler Meyer Equation is used for heavily loaded
130
-------
2. 0
CO
1.6
1. 2
fi
=tfc
- 0. 8
fl
o
• r-4
•s
o
cu
ni
W
0. 4
r:;-, a;:
= 75°'F
'Meyer Equation
, |,l :Meyer Equation
1 i ' .:•!..•
!/ ;: ':;; •
i ' ' ' ' '
! | ' , ] i •
1 • , . , i
'! = 70% (F
t , *
: ! ' ; I i ;
1 i , | j
n .^*
r^^*"^ , t ;
; , . . | _
t
n _[^ — ~"
' ^^
l^\
;' ! .
• ' i - • • I ; ' ; < • j ' ; ' ' 1 ! ' • ' i '
1 ' ' ' j ' *'•',{',.' '
1 , . ' '. ' ' i ' 1 . , '_';-.'„;'
, :j. K.:i ;;,; , , :,.• | • ; ' • :i ;
;.i J'; ;' ;i:::> ! ! ;::!'';.j' /! . . ,.! :.": .. - : .1.'-.'. ... '
; , > i ' • i ' . ' ' 1 ' • ! '• • '. i , j ! ' • ' ! ' ' •
'!• . ' •' • I1 '• ' '-I ! - ! . ; ! ^
••, \ :>'.\. •!..,:!.,. '• "T ;r i "I . . '• ^^
>. . = .0.!300 psia) '.. :;: 1 '.<• L- . '. l^^/^
a: ;;: ! !,• i •= , :•' !•• !' J^^/^ ' '.
:'"! >l'''>^^^^>'''-\---
! ;. ' i ^^^ , "^ ; ' ' ' •
"•' ' l*^^^ '' ' '/^ ' ' ' ' ' -
^^^ ! i ' j. '!'"., 'x^Ijaike . Colorado City ' " "'
.; .''i . !'.' i. ;. Jx^ ::.:_!•--' ^^-—---^^^^^
X^T-f PI' ' "i. \\^^<^^& Hefner !___._
Eq.i B-16|^-^^"^; 1 : I !
i-^T "!•• !V •:'. •: T" "' i'":' ;'-"". ""
^^^ • i • i ! , ' . i I , • , ; ; ! • i
rX^' !' i .' !': • ' ••!.' '!.. : ' ' i' i; ' <' '••'
1 , r . •; .,';:" " .;'~;i [.'""• f"' ";•"'" '
., .i:i'::L;^!:::J\:^.:L.u^-.:^l:.:L:; .^.L...
. i ; ! • i i . ' i . i ' I ' i : •-'•': . • ' ; !!,,'-!'..!:.;.;• • : .
1 : . 1 ' • ' i ' ' . • • . i ' ' I • : ' ' . ' ! : - ' " i , . '' . • ' ' i :
1 i ' ; i
j . ..!..!
; . • i
^^\
\\
'• i !
|
, !
i
' j
' i
i
i
i
, : i
\
6 8
Wind Velocity, mph
10
12
14
Evaporation Rate Given by Various Empirical Equations and Eq. B-16
Summer Conditions T = T
w a
Figure B-4
-------
CO
CO
Relative Humidity = 70%
; (P = 0,,300 psia)
.3. i
2 46 8 10 12
Wind Speed, mph
Evaporation Rate Given by Various Empirical Equations and Eq. B- 16
Figure B-5
T > T
w a
-------
1.0
co
oo
T = 35 F
TW=35°F
a
Meyer Equation!
Relative Humidity, = 25%
(C = 15)
(P = 0. 025 psia)
-"\Lake Colorado City
Meyer Equation
Lake Hefner
Eq. B-16
6 8 10
Wind Speed, mph
Evaporation Rates given by Various Empirical Equations and Eq. B-16
Winter Conditions, T = T
Figure B-6
14
-------
GO
4. 0
T = 50 F
w
T ' = 35°F
a
2. 0
0)
3
Relative Humidity = 50%
(P = 0. 0499)
cl •
O
Cu
rt
W
1.0
Meyer Equation
= 10)
6 8
Wind Speed, mph
10
12
Evaporation Rates Given by Various Empirical Equations and Eq. B-16
Winter Conditions, T > T
w a
14
Figure B-7
-------
ponds, the predicted temperatures will be too high.
When comparing the curves in Figs. 34 throxigh 37, it is helpful to
note that coefficients were also calculated for the Lake Hefner and
Lake Colorado City equations by using data from the nearest weather
station, as in the case of the Meyer Equation, and these coefficients
were found to agree within 5% with the previous ones. This agree-
ment is not surprising if the only substantive difference in the two
measurements is the height at which the observations are made in
view of the fact that the 26-foot height is well above the laminar and
buffer zone of the boundary layer and is thus in an area of active tur-
bulent mixing and relatively slow velocity change with elevation as
can be seen from Fig. B-3. The apparent difference between the var-
ious empirical curves, however, is far more significant. The dif-
ference amounts to a factor of almost 1. 5 between the two curves that
do not have a zero wind velocity term, and to a considerably higher
factor •when equations without the zero wind term are compared to the
equation with the zero term at speeds below 5 mph. A major portion
of these apparent discrepancies most likely result from the differ-
ences in local topography and the inherent difficulties of estimating
the mass of evaporated water by the indirect process of making a
mass balance.
Convection
Bowen [22J developed an expression for the ratio of heat transfer by
convection to heat transfer as a result of evaporation on the basis of
diffusion theory. He was able to find analytical solutions to his equa-
tions for three special cases and subsequently selected one of these
cases as the most probable for application to bodies of water,- namely:
Q T - T
y =
-------
U c
)2/3
oo
where c = local friction coefficient
x
h = local mass transfer coefficient, ft/sec
x
and the latter is given by the expression (See page 136, Ref. 27):
h Cf
p<5 U
^ °
(Eq. B-22)
where h = local coefficient of heat transfer, btu/sec ft F
x
P = Prandtl number = U/tt
r 2
Oi = thermal diffusivity, ft /sec
When both mass and heat transfer take place simultaneously as they
do at the air-water interface, the ratio of the heat transfer coeffic-
ient to the mass transfer coefficient can be found by dividing Eq. B-22
by Eq. B-21 to yield:
4 = >V5>2/3
-------
Ah = change in enthalpy between the vapor and liquid
state at the same temperature. This varies from
about 1075 btu/#m at 32°F to 1037 btu/#m at 100°F
If the expression given in Eq. B-10 for m is substituted into the
above equation, the value of Q is given as:
C
M (144)
Qe = (^T)hD£-Br7f - (pw - PJ] (3600 x24) btu/day ft2 (Eq.B-26)
u wa
The desired ratio, y , can now be found by dividing Eq. B-24 by
Eq. B-26 and substituting Eq. B-23 into the resulting expression:
c (?:)2/3 (T - T )
>. B-27)
wa
P M (144)
where p = - — — — - , here it is assumed that P = P
R T a
and M = molecular weight of air,
28. 8 j^n/#m-mole
Since both the thermal diffusivity, Ci, and the diffusion coefficient, D,
vary as the absolute temperature raised to the 3/2 power (to within
5% over the temperature range of interest), the value of Ci/D can be
expressed as its value at any one temperature, say, 77°F.
a /ax , 2. 34 x 10"4V ,
" <'° = '' = ^ ' ^
Substituting this value into Eq. B-27 together with the values of M
and M lead to:
w
T (T - T ) P
-, w a r -, .„ .
-I TS fTT L , , .-J (Eq. B-29)
5. 05 r wa -, w ~ a r BAR n
T (P - P ) 14. 67
aa w a
In order to compare Eq. B-29 with the expression proposed by Bowen
as the most probable when T was not excessively high, it is conven-
ient to note that for modest values of TW compared to Ta (say within
10°F), the ratio of (Twa/Taa) can be assumed to equal unity within an
error of about 2%, with the result that Eq. B-29 becomes:
137
-------
IT - T ) P
v - 5.05 l w a' BAR B_30)
y - AhT (Pw-Pa) 14.67
If in addition to the above assumption the value of Ah,,, is taken to be
the value of saturation condition corresponding to 69 F, namely,
AhT = 1060btu/#m, Eq. B-30 becomes:
y * 0. 00476 g' ; /,' ^ (E,. B-31)
w a
The value of . 00476 for the coefficient in Eq. B-31 is in reasonable
agreement with the value of . 00494 in Bowen's most probable equation
as given in Eq. B-20. When Twa is appreciably greater than Taa,
the use of Eq. B-29 for y is recommended over the simpler form
given by Eq. B-30.
Combined Convection and Evaporation
General Case
The combined heat transfer by evaporation and convection can be ex-
pressed by combining the appropriate expression for evaporation heat
transfer and Eq. B-29. In the most general form, Eq. B-16 would be
used for the evaporation heat transfer rate to yield the following ex-
pression for combined energy transfer by evaporation and conduction:
l U
T
. aa . ._ _ . . , .. . . .
- ~ 32)
wa Tw
Simplified Case
The mathematical difficulties associated with the use of Eq. B-32 can
be avoided at the expense of making approximations for m by using
Eq. B-19, with a value of Cj = 10 selected as a result of an examina-
tion of Fig. B-4 through Fig. B-7, in place of the analytical expression
given by Eq. B-16. Thus:
me = (3. 49 + 0. 349W) (PW - PJ (Eq< B_33)
138
-------
and
Q = Ah (3. 49 + 0. 349W) (P -P) (Eq. B- 34)
G JL "W cL
If in addition to the above assumption Eq. 30 is used in place of
Eq. 29 (for water temperature 30 F in excess of air temperature
this represents an error of 5% in the ratio Qc/Qe), the expression
for combined energy transfer by evaporation and convection becomes:
Qc = AhT(3. 4, + 0. 349W)(Pw- Pa)tl
Q
(Eq. B-35)
However, in the generalized analysis presented in the text of the re-
port, it is more convenient to have the combined net transfer of
energy that results from convection (Q ) and from water evaporating
from the pond at the pond surface as discussed in Appendix A,
Eq. A- 11. This net transfer of energy can readily be found by using
Eq. B-35. Thus:
/g _ a \ p
A (Ah) + Qc = AM3.49+0. 349W)(Pw-Pa)[l^ ^_^ _M|
w a
where Ah = 1070 btu/#m (See Eq. A- 11) (Eq. B-36)
In order to obtain the final form of the energy equation as a poly-
nomial in 6, the water vapor pressure corresponding to saturation
conditions at the temperature of the pond surface, PW , in Eq. B-35,
can be expressed by the following equation with an error of less than
2%.
P = a, + a6 + a fl2 + a B3 (Eq. B-37)
w 1 2 3 4
where a = . 089 for 0 < 6 < 70; a = . 089 -32<0<0
-3 -3
= 3. 50 x 10 for 0< 0 < 7.0; a = 4. 00 x 10 -32<9<0
/ 2 ^
=5.68x10" forO<6<70; a = 7. 30 x 10" -32<0<0
a
a
.
3 / 3
"
/ _
a = 1. 13 x 10" for 0 < 6 < 70; a . = 5. 20 x 10" -32<0<0
4 - 4 -
where Q = pond temperature in excess of 32 F
6 = T - 32°F
In order to reduce the number of variables in the final energy balance
equation without introducing errors greater^than 5% in the Qc term for
elevations up to 1000 feet above sea level (Q is itself only about 10%
139
-------
as much as meAh with the result that the total error introduced by
this approximation is small), the value of p;gAR/14' 67 in Eq. 36 can
each be taken as a constant, namely;
P
- - ; — = 1, that is, sea level conditions are assumed.
14. 67
Using this assumption, the coefficient a is defined as:
5. 05 PBAR 5. 05 , 14. 67 v ™^ /IP r, ,o\
ai4 = -Ah~ TT67 = To7o(l4T67) = -00473 (Eq- B'38)
When Eq. B-38 and Eq. B-37 are substituted into Eq. B-36, the final
simplified expression for the net transfer of energy that results from
convenction (Qc) and from water evaporating from the pond at the pond
surface temperature and being replaced by water at the make-up
temperature (m Ah) becomes:
f C(a12+al3W)a4Je
where a = Ah (3. 49) = 3730
a = Ah (.349) = 373
0 = air temperature in excess of 32 F or
a 8 = T - 32°F
140
-------
APPENDIX C
DATA COLLECTED ON OPERATING COOLING PONDS
Several electric power companies furnished data on cooling ponds
presently in operation. Sufficient data to allow a comparison of pre-
dicted and measured temperatures were made available on five
plants, namely,
1. Wilkes Plant - Jefferson, Texas
2. Kincaid Plant - Kincaid, Illinois
3. Cholla Plant - Joseph City, Arizona
4. Mt. Storm Plant - Mt. Storm, West Virginia
5. Four Corners Plant - Farmington, New Mexico
The data for each plant are discussed in the following section.
Wilkes Plant
Southwestern Electric Power Company supplied data for their Wilkes
Plant in Jefferson, Texas, together with an aerial photograph of the
pond. These data are shown below together with a sketch of the pond
site (Fig. C-l).
CIRCULATING WATER REPORT 1968
Month
January
February
March
April
May
June
July
August
September
October
November
December
Average
IN
50
53
57
69
77
84
87
89
82
74
64
55
Temperature, °F
OUT AT
c
87
91
93
96
98
105
108
110
103
94
85
92
37
38
36
27
21
21
21
21
21
20
21
37
Average Monthly Load
(MW )
174. 2
173. 9
174. 1
173. 0
173. 3
176.9
173. 5
173. 8
171. 4
164. 6
175. 3
173. 7
141
-------
JOHNSON CR3EK DAM AMD RESERVOIR
Wilkes Power Plant
Water Temperatures
Drainage Area: 11.0 sq. miles
Capacity: 10,100 Acre feet
Area of Lake: 6$1 Acres
Average Depth: 15.5 feet
Max. Depth: 43.0 feet
Min. Depth: 0.0 feet
Station Jan. Feb. Mar. Apr. May June July Aug., Sept. Oct. Nov. Dec.
Surface
2'
5'
10'
15'
19 '4"
Surface
2'
5'
10'
15'
18' 6"
Surface
2'
5'
10'
15'
22 1911
Surface
2'
5'
10'
15'
32' 6"
Surface
2'
5'
10'
15'
23 ' 8"
58
61
49
48
47
46
52
52
47
46
45
45
50
50
48
47
46
46
48
48
47
47
46
45
46
46
45
45
44
44
67
64
55
54
54
54
58
58
56
55
55
55
57
57
56
54
54
54
53
53
53
53
52
51
52
52
52
52
52
52
65
64
63
57
55
54
66
61
57
56
56
56
54
54
54
54
54
54
54
54
54
54
54
52
54
54
54
54
54
54
78
76
67
66
65
63
71
71
70
66
65
65
74
74
73
66
65
63
70
70
69
66
65
57
65
65
65
65
65
60
v
88
88
87
78
75
73
85
85
83
77
75
72
82
82
81
78
75
66
80
80
80
78
74
57
80
80
80
80
79
62
95
95
95
89
82
78
93
93
92
87
80
75
91
91
91
88
81
67
89
89
89
87
79
59
89
89
89
89
81
65
93
93
93
87
84
82
89
89
89
88
84
81
98
88
88
86
84
71
86
86
86
65
84
58
86
86
86
85
85
65
96
95
91
89
87
84
92
92
90
88
86
80
92
92
91
88
86
71
89
89
83
83
87
58
89
89
89
89
88
69
89
88
82
80
80
79
86
85
81
79
79
78
85
85
82
80
80
75
82
82
81
79
79
59
74
79
79
79
79
71
79
79
74
72
72
72
75
75
73
72
72
72
75
75
74
72
72
72
73
73
73
72
72
60
73
73
73
73
72
72
68
66
64
62
62
62
63
63
63
60
60
60
64
64
62
61
61
61
63
63
63
63
62
61
63
63
63
63
62
61
70
65
58
57
56
56
63
62
57
56
56
56
61
60
58
56
56
56
62
60
57
56
56
55
55
55
55
55
54
54
142
-------
Power Plant
SKETCH OF THE WILKES PLANT POND
Fig. C-l
143
-------
In addition to these data, the energy conversion efficiency of the plant
must be known in order to determine the waste thermal energy load
imposed on the pond. For the Wilkes Plant the average annual heat
rate (btu of chemical energy into the plant per kw-hr of net generation)
is given as 9, 854 in Ref. 25. Of this 9, 854 btu, 3413 btu are con-
verted to electric energy; a small fraction (say G^) is rejected dir-
ectly to the atmosphere primarily in the exhaust gases and the re-
mainder is rejected to the condenser cooling water.
The waste thermal energy (WTE) rejected to the pond can then be
expressed:
WTE = [(H. R. )(1 - tt ) - 3413J (MW ) (24 x 1Q3) btu/day (Eq. C-l)
c* 6
where H.R. = annual average heat rate in btu per kw-hr
(X is assumed to be constant at 10% for the plants
under consideration.
However, the heat rate will vary somewhat with the time of year.
This variation is usually small (~ _+ 2%). Here it has been approxi-
mated as eight hundredths of one percent for each °F deviation of the
condenser cooling water from its yearly average value. Thus the
heat rate to be used to calculate the waste thermal energy as given by
Eq. C-l for any given time of the year becomes:
(heat rate) = (heat rate) , +.0008(T - T )
annual ave condenser
average condenser outlet
outlet
(Eq. C-2)
where T = 96. 9°F for the Wilkes Plant
ave
condenser
out let
The climatic data were obtained from the nearest Weather Station at
Shr eve port, Louisiana, about 40 miles away, and are given in Table
C-l.
Since the measured solar radiation was not available in the "Annual
Summary" given in Table C-l, values had to be taken as the average
value over a period of years as given in Ref. 7 and repeated below.
144
-------
TABLE C - 1
SHREVEPORT, LOUISIANA
METEOROLOGICAL DATA FOR 1968
LATITUDE
LONGITUDE
ELEVATION (ground)
32° 28' K
93° 49' »
254 Feet
Month
JAiN
FE6
MAR
APR
KAY
JU N
JUL
AUb
StP
OCI
KOV
DEC
TEAR
Temperature
Averages
1
^ E
'B
I 1
53.2
54.4
67.3
76.5
92.2
89.6
90.4
91.5
83.8
78.0
64.4
57. 0
74.0
1
^ E
>• .a
" C
Q i
37.5
32.8
44.4
56.4
63.1
70.5
71.5
72.4
63.4
j>-
.C
C
2
45.4
43.6
55.9
66.5
72.7
80.1
81.0
82.0
73.6
55.2 66.6
42.9
36.2
53.9
53.7
46.6
64.0
Extremes
_
.c
5
76
69
82
85
90
97
96
95
92
88
83
7U
97
®
Q
31
27
10
22
24
12
18
5
$
18
21
23
39
51
59
62
25+ , 64
Z
2*
1
27
JUN.
12
56
38
28
24
o
a
3
22
1
6
>.
a
•a
«t>
Dl
&
604
614
308
49
1 i 0
27
5
29
27+
29
12
24
LAN.
18
8
0
0
0
0
57
346
560
2538
Precipitation
~n
£
8.33
2.22
1.89
9.38
6.05
2.78'
4.68
1.89
9.59
1.90
5.85
3.27
57.63
c
•5
£ -c
O 5
2.85
0.97
5
&
8-9
14-15
0.78 J21-22
2.50 ^7-28
1.38
1.63
2.12
1.09
5.14
0.94
1.65
1.39
5.14
9-10
26
18
10-11
14-15
9
30
12
SEP.
14-15
Snow, Sleet
~n
f.
1.0
0.4
1.5
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
I
2.9
s
w
V
1 £
O S
,
>• -a
— 3
a! "u
3
3
o
D
24
15
i 21
6 ! 17
10
11
12
12
8
12
6
4
92
14
13
12
8
9
7
14
16
170
o
0 ^
'^ o
t: -c
u £
15
6
11
12
12
10
8
11
6
5
9
9
114
a
JS o
M j=
* .1
s o
(rt —
1
0
1
0
0
0
0
0
0
0
0
0
2
o
E
a
c
3
H
1
0
4
7
2
2
58
s°
Temperatures
Maximum 1 Minimum
•^ TJ
X
3
0
0
1
1
0
0
1
2
J
1
1
13
c o
s!
0
0
0
0
1
20
22
24
7
0
0
0
74
|
-------
Solar Radiation, for Shreveport, Louisiana
.. , Solar Radiation, Qa
Month /T , /j \
(Langleya/day)
January 232
February 292
March 384
April 446
May 558
June 557
July 578
August 528
September 414
October 354
November 254
December 205
The parameters necessary to use the curves presented in the text of
the report can be calculated from the above data and the desired tem-
peratures determined. The results are shown in Table C-2 for steady
state pond operation.
Fig. C-2 shows the measured temperature-depth profile for the var-
ious stations along the flow direction for January and June. From
this figure it can be concluded that considerable lateral and longitud-
inal mixing occurs near the hot end of the pond where the heated water
enters the pond from the discharge canal. It is also apparent that the
upper few feet (up to 5 feet) of water is well mixed vertically so that
the temperature in this upper region at any one station is essentially
uniform.
If the measured temperature-depth profile is approximated with a
straight line, the slope of the line (0 ) would be approximately 0. 3°F/ft
for January and 1. 0°F/ft for June.
146
-------
100
90 !£.
Y
F
00
O
Average '
Condenser
Range :,.
, jAverage Condenser i
' i Range j
June 1968
-January 1968
10
Station No.
Near Inlet 1
2
3
4
Near Outlet 5
15 20
Pond Depth, ft
25
30
35
See Sketch of Plant Site for Station Locations
(Fig. C-l)
Wilkes Plant: Measured Temperature - Depth Profile - January and June 1968
Figure C-2
-------
TABLE C-2 - Wilkes Data 1968
Jan
Feb
Mar Apr May June July Aug Sep Oct
Nov
Dec
CO
(3 btu/ft2 day
N
Ta, "F
P , psia
3.
W, mph
a12+a!3W =
3730+3730W/10
-3730(1+W/10)
x (. 089)
-9 (. 0042)-P )
Si fa
Q =WTE/A
PP
'i
'i+app
A(ATc)/WTE
AT , °F
c
T .. , °F
equil
T ,°F
mixed'
(steady state)
T °F
slug flow '
(steady state)
2880
45.4
. 105
8. 5
6900
515
776
1036
1812
. 0475
37
45. 2
53, 2
45.4
3202
43. 6
. 114
8. 5
6900
552
787
1395
2182
. 0481
38
48. 8
56. 3
49. 0
3670
55.9
. 156
9. 3
7190
1293
792
2604
3396
. 0454
36
58. 8
65- 0
59. 9
4108
66. 5
. 245
9. 0
7080
2260
792
4009
3801
. 0340
27
69. 4
75. 0
69. 5
4820
72. 7
. 280
8. 0
6720
2570
801
5031
5832
. 0263
21
77. 2
82. 5
77. 3
5240
80. 1
. 384
6.8
6260
3130
79
6011
6802
. 0265
21
85. 5
90.. 0
85. 5
5260
81. 0
. 417
6. 3
6080
3470
821
6371
7192
. 0255
21
88. 0
92. 7
88. 0
5050
82. 0
. 386
6. 1
6000
3200
834
5891
6725
. 0251
21
86. 0
90. 6
86. 0
4458
73. 6
. 322
6. 1
6000
2610
805
4709
5514
. 0261
21
78. 5
83. 5
78. 6
3818
66. 6
. 228
6.4
6110
1850
750
3309
4059
. 0266
20
67. 6
73. 3
67. 9
3132
53. 7
. 150
8.6
6920
1140
778
1913
2691
. 0269
21
54. 1
60. 3
54. 9
281
46. 0
. 120
9. 0
7080
841
788
1299
2087
. 0472
37
47. 3
54. 7
47. 7
-------
Kincaid Plant
Commonwealth Edison Company supplied data for their Kincaid Plant
in southern Illinois together with a map of the site. These data to-
gether with a sketch of the site follow.
Monthly Average Power Generation - 1968 - Kincaid
Month
January
February
March
April
May
June
July
August
September
October
November
December
Gross Condenser Inlet Actual Heat Rate
Generation Water Temp° F btu/net kw-hr
239, 530 MWHR
259, 072 "
232, 382 "
65, 256 "
329, 160 "
556,984 "
535,687 "
608, 715
395, 030
293,811 "
364, 708 "
635, 320 "
35
38
41
55
62
80
86
88
79
68
67
51
10,200
10, 470
10,570
10, 800
10, 340
10, 340
10, 390
10, 200
10, 150
10, 300
10, 140
Total 124,515,655 "
1968 Avg. 376,304.6 "
Total Surface Area of Cooling Lake - 2700 acres
Average depth of cooling lake - 10 feet
Total volume of cooling lake - 35, 000 acre-ft
AT = about 18°F at full load
c
Eq. C-l was used to determine the waste thermal energy rejected to
the pond from the given heat rate and electric energy generating rate.
The climatic data (except for solar radiation which was taken as the
average monthly values from Ref. 12) were obtained from the nearest
Weather Station at Springfield, Illinois. These data are given in
Table C-3, followed by the data for solar radiation.
149
-------
TABLE C - 3
SPRINGFIELD, ILLINOIS
METEOROLOGICAL DATA FOR 1968
LATITUDE
LONGITUDE
ILEVATION (ground)
39" 50' H
89° 40' »
588 Feet
SPRINGFIELD, ILLINOIS
CAPITAL AIRPORT
1959
Month
JAN
Fco
MAR
APR
HAY
JUN
JOL
AuG
££p
ocr
/«uv
UtL
TLAfi
Temperature
Averages
£
*• w
1 I
31.1
34.6.
54.1
64.1
69.3
85.5
86.4
85.4
78.5
66.7
48.7
36.5
61.7
|
>.£
& 1
L&-4
16. a
31.4-
41. J
46.7
63.*
65.2
65.0
54.8
42.9
33.9
21.6
41.8
£
•2
2
23.8
25.6
4.2.8.
5'2.9
5»,0
7S..8
T5...2
a&.7
54.3
41,3
2-9'. I
51.8
Extremes
S
A
JZ
E
57
59
7B
79
87
9&
95
96
8-a
86.
74
5*
96
0
&
_
I
.3
it -14
I
28
30*
15*
30*
13
23*
22*
15
1
12
AUG.
23*
- i
3
29
37
50.
51
49
44
26
19
- 5
-14
>.
a
o
3. a
Q
Q
j
7 ' 1271
22 , 1134
13 \ 684
6 358
6
203
28* 8
3 1
28 2
27*
29
13
31
JAN.
7
31
353
703-
1107
sess
Precipitation
"5
£
1.79
1.15
1.25
2.44
5.69
3.25
4.67
0.99
3.29
1.43
3.08
2.64
3-1.67
=
S .-
! J 1 «
0 S
0.52
Snow. Sleet
•;
c
I *
Sit s
Relative
humidity
•a
:•
3
•<
e
o
o
z
:a ndard
time
Q ,2 a s 5 ' CEJim
1 .
I1
13-14* h 13.0
1.09 31-1 1.7
O.55 * IS ' 4.5
5.4 12-13 ' 76 76.
a.s
3*4
6—7 ' 69 7iJ
12 J 65 70'
1.S7 3 T I j, » .0
O.O 01^?
: o.o.
31.0:
!' 74
0.7 G/.3 V
4.7 l.S
|'
14-lS 24.6
""*
70
7T
8O.
SI
7"*
73
77
74
(0
86.
86-
8T
80 83
30 7S
JAM.
1Z-LJ
74
so-
78
ise J3
0
a
a
o-
o
10
IZ
13
0
a
j»
17
12
2
0
a
0
' 0
0
0
0
0
LI
i *Z
Minimum
•a
= i
8 1
27
11
o £
7
28 ; 2
16 ' 0
5
0
0
0
0
0
4
15
25
120
0
0
0
0
0
0
0
0
1
10
150
-------
Sangamon River
State_Rt. #104
Pawnee
i\_ r-'f-" ;i , -,
-W-1--t-v^-1 i!
Purchase
Area
C. & I. M. R. R.
#10
i— i
0 1
, Kincaid
"I
1—J Station
Miles
234
"'I
Kincaid Plant and Cooling Pond
Figure C- 3
-------
Solar Radiation for Springfield, Illinois
Solar Radiation
Month btu/ft2 day
January 608
February 885
March 1200
April 1513
May 1883
June 2070
July 2050
August 1870
September 1582
October 1132
November 738
December 534
Although the pond surface is 2700 acres, only two arms of the three-
arm "pond" is involved in the cooling circuit; the third arm provides
storage. In addition to the question of effective area due to the three-
arm shape of the "pond", the highly irregular shape also increases
the uncertainty of the effective area because of the possibility of
zones of dead water. From the map provided with the data it was
estimated that 60%- of the 2700 acres (or 1620 acres) was effective.
The predicted pond temperatures, based on an active surface area of
1620 acres-r are shown in Table C-4 for steady state operation.
152
-------
TABLE C-4 - Kincaid Data 1968
Jan
Ol
co
Feb Mar Apr May June July Aug Sept Oct Nbv Dec
Q , btu/ft^da?
T , °F
a
P , psia
3,
W, mph
a12+a!3W
= (3730 +
3730W/10)
-3730(1+W/10)
x[. 089 - 9a
x(. 00473)-Pa]
Q = WTE/A
PP
'i
'1*°*,
A(AT )/WTE
c
AT , °F
c
T °F
j. . -i i -*-
equil
T . , °F
mixed
(steady state)
T , °F
slug flow'
(steady state)
2060
23. 8
. 024
11.4
7980
-830
988
-1061
-49
. 0182
18
19. 0
will
ice
32. 0
23. 0
2345
25.6
. 041
12. 4
8350
-650
1202
-585
617
. 0149
18
25. 5
will
ice
38. 8
32.0
3008
42. 8
. 082
14. 7
9200
406
1013
1152
2165
. 0177
18
43. 2
51. 6
44. 9
3560
52.9
. 119
13. 5
8760
1125
304
2429
2733
. 0593
18
54. 6
57. 1
54.6
4033
59. 0
. 166
11.4
7960
1630
1485
3434
5064
. 0121
18
63. 5
74. 2
66. 8
4620
74. 5
. 256
9.6
7300
2690
2520
5071
7761
. 0071
18
75. 8
90. 2
80. 8
4690
75. 8
. 294
8. 0
6710
2770
2260
5241
8011
. 0080
18
77. 0
93.5
83. 4
4440
75. 2
. 302
8. 7
6960
2910
2580
5121
7701
. 0070
18
77. 0
91. 2
82. 2
3940
66. 7
. 229
9. 0
7090
2790
1745
4493
6238
. 0103
18
73. 1
83. 0
76.6
3270
54. 8
. 1385
11.4
7980
1340
1208
2359
3567
. 0149
18
55. 5
64.5
57. 2
2565
41. 3
. 102
10. 8
7750
465
1635
744
2379
. 0110
18
44. 8
56. 1
47. 8
2138
29. 1
. 059
14. 6
4160
0394
2610
-549
2061
. 0069
18
26. 2
will
ice
50. 5
40. 4"
-------
Cholla Plant
Arizona Public Service Co. supplied data for their Cholla plant in
Joseph City, Arizona, together with a sketch of the site. These
data are given below and in Fig. C-4.
1967 Station Net Output (MW-HR)
Jan 53,341.8 July 70,303.9
Feb 54,562.8 Aug 88,501.9
Mar 64,635.1 Sept 81,903.9
April 63,488.2 Oct 72,289.6
May 1,377.5 Nov 77,748.9
June 40, 978. 0 Dae 65, 024. 1
Average 61, 1 76. 3 MW-HR
Pond surface area: 380 acres
Total plant rating in megawatts: 115 MW
Average Temperature into Condenser by Month - 1967
Month ^F Month ^F
Jan 44 July 82
Feb 52 Aug 84
Mar 58 Sept 80
Apr 60 Oct 72
May 60 Nov 65
June 70 Dec 53
Average depth of pond: 4. 5 ft
Maximum depth of pond: 12 ft
Minimum depth of pond: 1 in.
Typical temperature depth profiles in pond:
(Pond too shallow for enough variation in temperature to
matter)
Temperature rise across the condenser: 20°
For the Cholla Plant the average heat rate is given as 9,838 btu/
kw-hr in Ref. 25. The waste thermal energy rejected to the pond
was calculated by use of Eq. C- 1 and C-2 where for this plant
rr 0
T , = 84 9 F
average condenser outlet
154
-------
Discharge
Canal ,
en
Intake
Canal
Dike
Condenser Temperature Rise - 20 F
Cholla Plant: Cooling Pond
Figure C-4
Dike
Inverted
Weir
-------
The climatic dataware obtained from the summary of the Winslow,
Arizona, Weather Station for 1967, about 25 miles away, except for
the solar radiation (not available at Winslow) which was obtained
from the records at Albuquerque. These data are given in Table
C-5 (Solar radiation data given below).
Because of the shallowness of this pond together with the pond shape,
it is most likely that substantial channeling of the flow occurs with
the result that only some of the pond surface area will effectively
take part in the cooling circuit. It is assumed that only 1/3 of the
total pond area of 380 acres (or 127 acres) will be effective. The
tabulated parameters and predicted temperatures are shown in
Table C-6 for steady state pond operation.
If it is assumed that the entire pond surface is effective in trans-
mitting energy to the atmosphere, the predicted pond temperatures
will be lower than the values given in Table C-6 and are given in
Table C-7.
Solar Radiation for Albuquerque
Solar Radiation
Month (Langleys/day)
January 303
February 386
March 505
April b!8
May 695
June 729
July 677
August 524
September 541
October 449
November 325
December 274
156
-------
TABLE C - 5
WINSLOW, ARIZONA
METEOROLOGICAL DATA FOR 1967
LATITUDE
LONGITUDE
35° 01' H
110°
ELIVATION (ground) 4895
Month
44' »
Feet
HINSLOff, ARIZONA
KUNICIPAL AIRPORT
1967
Temperature
Averages
E
3
>. J
^ X
Q I
FE6
£
a
* E
= c
& E
57.1 21.7
MAR 67.0 31.2
APR 68.3 32. L
MAY
77.9 42.2
JUN 85.9 53.5
JUL 92.2
AUG
90.6
SEP 84.3
OCT 76.0
64.4
60.6
54.2
38.0
NOV 63.4 ' 30. S
DEC 33.0
TEAR 70.3
9.8
37.8
>•
.c
c
o
S
„
«»
s:
to
X
39.4 71
49.1
50.2
60.1
69.7
78.3
80
79
92
96
101
75.6 96
69.3
92
57.0 | 87
47.1
75
21. » 59
54.1
101
Extremes
—
&
IT~
13
23
3
23+
30-*-
3*
26+
4
1
15
7
IUL.
3+
5
i
3
10
13
22
Si
O
>-
-a
0)
0)
J=
o S
0.10
0
Snow. Sleet
~n
3 f.
1 a
T
0.01 20 0.1
c
M
a
3 _c
6 s
0.1
0.24 ; 0.19 ' 29 0.1 ! 0.1
*40 , 0.10 0.05 9
24 1 182
43 6
59
6
8* 0
0.27 0.25 30
1.06 , 0.69 !l8-19
2.67
0.78 i 12
0.4 0.4
0.0 0.0
0.0
0.0
0.0 0.0
S.
Relative
humidity
5
AM
S
t
11
AM
:and
Lme
5
ra
ird
jsed
11
PM
Q ' MOUNTAIN
J
20 1 46
5
45
12 | 46
55 , 21 0 1.09 0.48 28 0.0 0,0
43 14 i 11
25 30 252
23 14 528
-12 22
-12
DEC.
22
134*
*997
0.49 ' 0.21 24
0.11 ! 0.11
3
0.36 I 0.29 22
3.73 1 1.51 13-14
10.23
1.51
DEC.
13-14
31
40
54
59
0.0 0.0 ' 61
0.0 0.0 1
44
0.2 0.2 j 29 54
39.6 17.0 '13-14 ! 76
»0.»
1
DEC.
17.0
13-H i 53
29
22
21
17
23
36
39
38
24
37
65
34
18
17
17
14
18
34
30
29
15
27
63
27
33
33
34
25
32
9
9
2
1
6
73
44
1
11 Resultant
Month
1
I
5
JAN ' 20
FE8 i 26
MAR ! 20
I
Q.
01
2.7
2.6
-a
a
o
<
7.0
Fastest mile
«
1
43
o
"I
3
20
8.2 44 22
6.7 10.8 1 46
APR i 20 llO.l 13.3 | 52
MAT 23 F5.1
JUN i 21
i
JUL ' 23
AUG
14
SEP i 19
OCT 24
NOV 19
OEC ' 23
TEAR
21
6.4
3.6
1.4
1.7
2.0
2.3
3.1
J.7
10.0
10.7
8.2
7.8
8.3
7.6
7.0
7.8
8.9
3u
37
35
35
32
26
29
39
52
19
18
23
19
26
30
21
34
19
22
18
0
a
5
14
29
12
5
5
4
23
24
29
28
16
APR.
12
1
Jr
O Q
11
s; S
£ 3
•
•3 2
O
O) «
< 3
5.1
Number of days
Sunrise to sunset
IB
41
U
14
3.5 . 16
5.7
3.2
4.1
3.0
6.5
4.9
4.2
2.5
3.7
4.5
4.2
12
20
16
19
5
12
14
21
16
16
181
>•£
~ =
S. -3
6
6
6
4
8
a
12
10
11
7
9
4
91
o
0
11
6
13
6
7
3
14
9
5
3
5
11
«
B 1
.2 !
!a O
•s. -S
3 S
rx ^
1
1
3
3
3
3
11
9
7
1
3
6
93 51
1
£
o
c
« :
2 o
™ "g
5 °
0
0
0
0
0
0
0
0
0
0
0
5
5
1
2 a,
S X
0)
]
5
P X
0 0
0 0
0
0
3
5
16
14
7
0
0
0
0
0
0
0
o
0
0 0
o; 4
45
Temperatures
Maximum
Minimum
T3
C 0)
s 4
0
0
0
0
2
10
20
21
4
0
0
0
57
13
c £
•a _o
S J>
5
0
0
0
0
0
0
0
0
0
0
15
20
-0
c *•"? *
S Ijo 1
28
5
25 0
17 0
22 i 0
4
0
0
0
J
7
22
31
156
0
0
0
0
0
0
0
9
14
157
-------
TABLE C-6 - Cholla Plant - 1967
en
CO
Jan Feb Mar Apr May June July Aug Sept Oct Nov Dec _
6 ,btu/ft2 day
Ta' °F
P , psia
cl
W, mph
a!2+a!3W +
(3730 + 3730W/10)
-3730(1+W/10)
xC.089 - 6a
x(. 00473)- Pa]
6 = WTE/A*
PP
'i
fi + QPP
A(AT )/WTE
AT , °F
c
T ... °F
equil
T . , (steady
mixed ...
state)
T
slug flow
(steady state)
2212
31. 2
. 024
7. 0
6350
-438
1599
-585
1014
. 0125
20
23. 5
will
ice
45. 0
39. 0
2615
39. 4
. 025
8. 2
6780
-203
1680
53
1733
. 0119
20
32. 2
52. 6
41. 0
3300
49. 1
, ObO
10. 8
7750
364
2034
1305
3339
. 0098
20
46. 5
63. 7
53. 5
3825
50. 2
. 052
13, 3
8690
425
2004
1891
3895
. 0099
20
50. 8
64. 7
56.8
4450
60. 1
. Ob8
10. 0
7450
758
45
2849
2894
. 444
20
60. 0
60.2"
60. 0
5270
69. 7
. 097
10. 7
7710
1432
1338
4343
5681
. 0148
20
/O. 0
/8. 2
71. 2
5030
78. 3
. 206
8. 2
6780
2260
2385
4931
7316
, 0084
20
76. 6
90. 3
80. 6
4750
75. 6
. 193
7. 8
6640
2050
3030
4991
7471
. 0066
20
74. 3
91.5
84. 3
4160
69. 3
. 154
a. 3
6820
1640
2766
3441
6207
. 00b8
20
66.8
84. 0
76. 8
3450
57. 0
. 084
7.6
6550
740
2355
1831
4186
. 0085
20
53. 5
/3. 1
64. b
2700
47. 1
. 062
7.0
6350
280
2505
021
3126
, 0080
20
40. 6
66. 0
54. 6
2085
21. 4
. 041
7. 8
6640
-664
2010
-938
1072
. 0099
20
18. 6
will
ice
45. 8
41. 0
*The effective area, A, is assumed to be 1/3 of actual surface area.
-------
Jan
TABLE C-7 - Cholla Plant, 1967 (Cont. )
Feb. Mar Apr May June July Aug Sept Qct Nov. Dec.
6 =WTE/A*
PP
/, +Q
1 pp
T • ^
mixed
(Steady state)
532
-53
31. 5
will
ice
559
612
40. 2
675
1980
52. 7
668
2559
56. 0
15
2864
60. 1
432
4775
76. 8
786
5717
81.5
1010
5451
80. 2
915
4356
72. 9
778
2609
60. 7
828
1449
50. 0
665
-273
28. 1
will
ice
* Effective area is assumed to equal actual surface area of pond.
en
UD
-------
Mt. Storm Plant
Virginia Electric and Power Co. provided data on their Mt. Storm plant
at Mt. Storm, West Virginia, together with a sketch of the cooling pond
(Fig. C-5). Some of these data are presented below.
Circulating Water Temperature, F
Month
Intake
Discharge
Net Generation
for Month, kw-hr
January
February
March
April
May
June
July
August
September
October
November
December
45. 0
43. 5
43. 5
50. 0
61. 0
69. 0
77. 0
81. 5
78. 0
64. 5
53. 0
44. 0
64. 3
62.8
62. 8
69.3
80. 3
88. 3
96.3
100.8
97. 3
83. 8
72. 3
63. 3
718,415, 900
576, 140, 500
705, 117, 000
582, 202, 600
577,455, 100
603,031, 300
533,818, 300
594, 162, 000
519,600, 000
204,414, 500
311, 167, 000
462,693, 000
Pond surface area = 1125 acres (at full pool)
Pond volume = 48, 000 acre-ft (at full pool)
Temperature rise across the condenser - 19. 3 F at full load
Using the same assumptions as for the Wilkes and Cholla plants and the
yearly average heat rate of 9,405 btu from Ref. 25, the waste thermal
energy can again be expressed by Eq. C-l and Eq. C-2 where:
average condenser outlet
= 78. 4°F
The climatic data were obtained from the data collected at the Weather
Station at Elkins, West Virginia, about 70 miles away. Solar radiation
data had to be taken from the maps in Ref. 12. The weather station data
is given in Table C-8.
Solar Radiation - Elkins
Month btu/ft day Month btu/ft day Month btu/ft day
Jan
Feb
Mar
Apr
725
1133
1392
May
June
July
1732 Aug
1620
2170
2010
1850
Sept
Oct
Nov
Dec
1520
892
555
545
160
-------
TABLE C - 8
ELKINS, WEST VIRGINIA
METEOROLOGICAL DATA FOR 1968
LATITUDE
LONGITUDE
ELEVATION (ground)
38° 53' N
79° 51' 1
1970 Feet
ELKINS, WEST VIRGINIA
ELKINS-RANDOLPH CO AP
1 963
Month
JAW
FEB
MAR
APR
HAY
JUN
JOL
AUG
4EP
OCI
MOV
DEC
TEAR
Temperature
Averages
g
. B
= a
0 S
38.6
33.0
55.5
64. u
66.9
78.7
82.1
81.0
74.4
63.5
51.7
39.0
60.7
G
>. M
~ d
S !
11.4
11.0
28.1
35.8
44.6
51.9
56.2
58.8
48.1
39.0
32.7
18.1
36.3
>,
2
"c
I
25.0
22.0
41.8
49.9
55.8
65.3
69.2
69.9
61.3
51.3
42.2
28.6
48.5
Extremes
V
.c
£
54
59
79
78
80
89
98
87
83
78
«
2
22
1
22
14+
a
I
3
-17
- 5
7
«
'ia
a
2
23+
14
21 2
15+ 25 I 7
30
39
21
18+ 40 ! 5
23 +
24
39 29
34
14 20
73 1
51 28
LUN.
30
>•
•u
O
D)
&
1233
1239
7U
444
279
67
16
46
122
31 425
15 14 673
_ 5
11
JAN.
il \ 30 -17 2
1121
6376
Precipitation
5
£
i.ae
1.80
4.28
1.54
7.30
2.U5
3-46
4.07
3.01
3.21
3.20
3.04
38.84
G
•s
« BJ
1 -=
6 s
0.76
0.69
1.28
0.45
1.86
U.42
1.01
1.24
1.29
«
a
13-14
29
12-13
4-5
23-24
Snow. Sleet
"*
1
14.1
24.1
10.3
0.1
0.0
11 O.u
25 ! 0.0
7
10
1.U5 118-19
0.97 ! 6-7
0.69 4
MAY
1.86 ^3-24
0.0
0.0
0.5
c
—
« ni
S .£
6 s
4.6
o
1
13-14
8.6 20-21
S.3 £9-1
0.1
0.0
0.0
0.0
0.0
0.0
11
0.5 29
11.9 8.9 12
19.7
80.7
4.0
18
NOV.
Relative
humidity
1
AM
7
AM
1
PM
7
PM
Standard
t
line
jsed :
EASTERN
83
70
77
81
84
96
97
8.9 12
86
74
33
86
87
95
97
97
98
95
39
82
89
65
49
53
52
60
57
55
62
59
56
67
75
56
57
48
65
72
69
77
82
76
79
70 , 77
59 | 69
•OTE: Station operated less than full tl»e after July. Suaaary based on available data.
Month
JAN
FEB
MAR
APR
MAY
JON
JOL
AUG
StH t
OCT
MOV
otc
rtAR
Wind
Resultant
a
S
B
28
28
28
27
27
28
1
1
2.2
5.7
5.0
2.8
3.4
2.5
•a
8
a.
1
o
6.9
9.6
9.5
7.8
8.4
5.5
Fastest mile
-a
Z
a
31
29
37
40
37
25
21
JS
16
23
30
35
4V
I
S
30
31
28
30
31
12
30
25
16
14
30
28
30
3
1
•3,
«
&
7
17
23
14
3
a
22
7
18 +
18
19
5
APR.
14
11
§ c
s §
0, »
r" cove!
jnset
-M a
: 2
a> o
2 -S
1 1
7.1
6.3
7.2
7.7
8.4
7.7
7.0
7.0
7.0
6.5
9.1
8.7
7.5
Number of days
Sunrise to sunset
S
0
,
7
6
4
2
3
5
3
4
8
1
2
53
if
S .2
a. u
3
9
"O
a
20
13
5 20
5 21
4
9
9
13
12
7
4
6
86
25
18
17
15
14
16
25
23
227
c a
.2 S
"3 o
||
1 3
15
£
1 S
$ c
c o
6
11 7
17 1 3
14
18
13
10
13
7
11
17
22
168
0
0
0
0
0
0
0
2
8
26
1
1
T)
G
3
iS
0
0
3
3
4
9
8
1
>
11
S £
0
0
0
0
u
0
0
0
0
0
0
0
u
1 *
Minimum
!
1 sil §
[VJ O CM V 01
co .2,0 j3,o ja
8 : 28 5
14
2
0
0
0
0
0
0
0
2
10
36
29 5
18 ' 0
13 0
4 0
0
0
0
0
8
15
30
145
0
0
0
0
0
0
3
13
161
-------
Power Station
Stony
River
\
Run
(
\
( jSample Points
Laurel -•
Run /
Mt. Storm Plant and Cooling Pond
Figure C-5
-------
RESERVOIR TEMPERATURES
MT. STORM RESERVOIR
ALL TEMPERATURES IN DEGREES FARENHEIT
Date
Buoy
Time
Depth
1
2
3
4
5
10
20
30
40
50
60
70
80
90
100
110
120
130
140
Bottom
Ft &°F
8-28-68
#1
9:25 A
85. 0
85. 0
85. 0
85. 0
85. 0
84. 8
84.4
82. 6
82. 0
81. 8
81.8
81. 5
81. 0
80. 9
92'
80.0°
8-28-68
#2
11:20 A
84.4
84.4
84. 4
84. 6
84. 6
84. 0
83. 1
82. 5
82. 2
81. 8
81. 4
81. 0
80.6
88'
79.9°
8-28-68
#3
9:52 A
83. 4
83. 4
83. 4
83. 5
83. 5
83. 2
83. 0
82. 8
82. 3
81. 8
81. 6
81. 4
81. 2
81. 0
80. 6
79. 1
117'
75.5°
8-28-68
#4
1 0: 2 0 A
83. 0
83. 0
83. 0
83. 4
83. 4
83. 4
83. 0
82. 8
82. 5
82. 0
81. 8
65'
81.5°
8-28-68
#5
10:30 A
83. 4
83. 4
83. 4
83. 5
83. 5
83. 2
82. 7
82. 4
82. 2
47'
81.6°
8-28-68
#6
11:40 A
84. 4
84. 4
84.4
84. 2
84. 2
84. 1
83. 2
82. 8
82. 2
81. 8
81. 2
80. 6
80. 1
87'
79. 8°
8-28-68
#1
1:30 P
83. 8
83. 8
83. 8
83. 5
83. 5
83. 0
82. 3
81. 6
81. 3
45'
81.0°
8-28-68
#8
1:45 P
82. 6
82. 6
82. 6
82. 5
82. 5
81. 5
79. 0
77. 5
30'
77.5°
8-28-68
#9
1:55 P
82. 0
82. 0
82. 0
82. 0
82. 0
81. 3
79.6
22'
78.0°
8-28-68
#10
2:35 P
83.6
83. 6
83. 6
83. 6
83. 6
83. 3
82. 1
81. 8
81. 5
80. 5
80.5
59'
82.3°
8-28-68
#11
2:50 P
83. 3
83. 3
83. 3
82. 8
82. 8
82. 7
15'
82. 3°
CO
-------
The "pond" at Mt. Storm is very deep v/ith the intake and outlet located
in the same region of the pond. The intake structure is 93 feet below
the surface and the two discharge structures are near the surface. As
a result of this geometry, when considering the slug flow operation, it
is necessary to consider the pond to be divided into two ponds each
with a depth of 1/2 the actual value. In the upper "pond" the slug flows
away from the intake, and in the lower "pond" the slug flows toward
the intake structure. As a result of this assumption, the slug spends
only one half of its time exchanging heat with the atmosphere and
spends the other half returning undercover to the intake structure.
The tabulated parameters and predicted temperatures are shown in
Table C-9 for steady state and transient operation.
Fig. C-6 shows the measured temperature-depth profile for the Mt.
Storm pond in August. From Fig. C-6 it is noted that although the con-
denser range is 19. 3 F (the average inlet and outlet condenser tempera-
tures for August were 81. 5°Fand 100. 8°F respectively), the measured
points show relatively small temperature variation (small compared to
19. 3°F) with either depth or horizontal displacement. As a result it
can be concluded that considerable mixing in all three directions takes
place in this pond. It should be pointed out that the intake and dis-
charge structure for the Mt. Storm Plant consists of large diameter
pipes rather than canals as in the Wilkes and Four Corners Plants. As
a result, stronger mixing would be anticipated in the Mt. Storm pond.
The upper few feet (up to about 5 ft) of water at any given station is
essentially uniform in temperature. If the measured temperature-
depth profile was to be approximated as a straight line, the slope of the
line would be about 0. 05°F/ft.
164
-------
I j
CJ1
100
: I I
90
M 80
at
ex,
s
ill
H
M
0}
13 70
Average
Temperature
Range, AT
of condenser
20
40
80
60
Pond Depth, ft
Station 1 (Near outlet, outlet near surface)
Station 9
Station 3 (Near inlet, inlet near bottom)
100
See Fig. C-5
for Station
Locations
120
Mt. Storm Plant: Measured Temperature - Depth Profile, August
Figure C-6
-------
TABLE C-9 - Mt. Storm Data, 1968
Jan. Feb. Mar. Apr. May
0 , btu/ft2day
T , °F
a
P , psia
a
W, mph
a12 + a!3W =
(3730+3730W/10
-3730(1+W/10
T. 089 - 6a
x(.00473)-Pa]
Q = WTE/A
PP
'i
'i + 6PP
A (AT )/WTE
c
AT , °F
c
(t30. 5)/CP
T °F
-L . i t -^
equil
T . , , °F
mixed
(steady state)
T . , °F
mixed
(transient*)
2180
25. 0
. 044
6.9
6300
-490
2310
-6/1
1629
. 0083
19. 3
0115
23. 3
will
ice
51. 5
45.0
47. 4
48.9
2335
22. 0
. 035
9.6
7300
-736
1842
-70b
1136
. 0105
19. 3
. 0115
23. 0
will
ice
45.4
50.9
48. 4
47. 2
3075
41. 8
. 090
9. 5
7260
342
2260
1053
3313
. 0086
19. 3
. 0115
44. 5
64. 3
46. 5
54. 2
58. 7
3620
49. 9
. 116
7.8
6640
744
1893
2005
3898
. 0107
19. 3
. 0115
55. 1
70.8
61. 3
66.0
68.4
3790
55. 8
. 164
8. 4
6850
1315
1948
2746
4694
. 0099
19. 3
. 0115
61. 3
75. 5
69. 4
72. 6
74. 0
June July Aug. Sept. Oct. Nov. Dec.
4500
65. 3
. 246
5. 5
5790
1816
2090
395 /
6047
. 0092
19. 3
. 0115
74. 0
88. 0
74. 6
81. 2
84. 8
4410
69.2
. 277
4, 3
5340
1940
1900
3991
5891
. 0105
19. 3
. 0115
76.4
89. 1
86. 3
87. 8
88. 4
4320
69. 9
.291
4. 1
5250
1475
2150
3961
6111
. 0090
19. 3
. 0115
75.6
90. 5
88. 7
89. 3
90. 0
3763
61. 3
. 231
4. 4
5370
1500
1890
2904
4794
. 0102
19. 3
. 0115
67.2
82. 0
90. 3
86. 3
84. 1
2950
51. 3
. 152
5. 0
5590
860
701
1451
2152
. 0275
19.3
. 0115
51,9
59. 5
83. 0
75.5
69. 8
2495
42. 2
. 108
6.9
6300
416
1030
552
1582
. 0187
19. 3
. 0115
39. 7
51. 7
66. 0
60. 7
57. 9
2190
28. 6
. 062
6.9
5300
-69
1523
-238
1285
. 0126
19.3
.0115
31. 0
will
ice
48. 2
55. 5
53. 0
51.2
en
CPi
-------
TABLE C-9 - Mt. Storm Data, 1968 (Cont. )
Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec.
T
Slug flow °F
(transient
operation)
43. 5
41.5
44. 5
53. 5
62. 5
70. 5
78. 5
81. 5
78. 5
65. 0
52. 0
45.5
*The three temperatures correspond to the 1-day, the 10. 2-day and the 20. 4-day of each month.
NOTE: For the transient case, it is assumed that the pond on January 1 has a water temperature
equal to the measured intake value of 45. 0°F.
cr>
-------
Four Corners Plant
Together with data for the Cholla Plant, Arizona Public Service Co.
also supplied data for their Four Corners Plant in Farmington, New
Mexico, together with a sketch of this site. These data are given below.
1967 Station Net Output (MW-HR)
Jan 371,666,1 July 406,318.8
Feb 356,133.5 Aug 373,721.1
Mar 355,936.1 Sept 363,212.1
April 300,292.0 Oct. 35/, 595. 8
May 393,629.6 Nov 244,555.4
June 377,698.1 Dec. 319,519.1
Average 351, 607. 3 MH-HR
Pond surface area: 1200 acres
Total Plant rating .in megawatts: 575 MW *
Average Temperature into Condenser by Month, 1967
Month
January
February
March
April
May
June
o
F
41
45
50
56
58
67
Month
July
August
September
October
November
December
o
F
75
76
74
66
55
43
Average depth of pond: 40 feet
Maximum depth of pond: 110 feet
Minimum depth of pond: 3 feet
Typical Temperature-Depth Profiles in Pond, 1969
Depth
1'
10'
20'
30'
40'
*1»T T • , g .
F
82.5
82. 0
81. 5
81. 0
79. 0
Depth
50'
60'
70'
80'
F
76. 5
71. 5
64. 5
58. 0
1969- This will raise the plant
rating considerably. However, not enough time has elapsed to give the
data required, relative to cooling pond involving No. 4.
168
-------
Pond Geometry
outlet canal
Temperature rise across the condensers: 18 F
Using the same assumptions as for the Cholla and Wilkes plants and the
yearly average heat rate of 10, 278 btu from Ref. 25, the waste thermal
energy can again be computed by Eqs. C-l and C-2 where
average condenser outlet - 76. 8 F
The climatic data were the same data used for the Cholla Plant since
these two plants are within a distance of about 175 miles of each other
and the Winslow station is the nearest Weather Bureau.
Since this pond is relatively deep and of a regular shape, it is assumed
that all 1200 acres of pond surface effectively enter into the cooling por-
cess. The tabulated parameters and predicted temperatures are shown
for transient and steady state pond operation in Table C-10.
Fig. C- 7 shows a typical measured temperature-depth profile for the
Four Corners Pond in July. The closer the water is to the surface,
the slower the temperature changes with depth. If the measured profile
is approximated with a straight line, the slope (/:?) would be about
0. 3°F/ft.
169
-------
Jan
TABLE C-10 - Four Corners Data - 1967
Feb. Mar Apr May June July Aug Sept Oct.
Nov.
Dec.
QN, btu/ft^ day
T , °F
a
P , psia
Si
W, mph
a, +a. 0W =
12 13
! 1
I 1
SAME CLIMATIC CONDITIONS AS USED FOR CHOLLA PLANT
(3730 + (3730W/10)
-3730(1 +W/10)
x[.089 - 0a
x( .- 00473) -Pa]
Q = WTE/A
PP
fi
f> + %
A(AT )/WTE
c
AT , °F
c
(t30.5>/Cp(V/A)
T ... °F
equil
T „, °F*
mixed
^transient)
1282
-585
697
. 0140
18
. 0122
23.5
will
ice
41. 5
41. 4
41.4
1245
53
1298
. 0144
18
. 0122
32. 2
41. 4
43. 4
44.8
1268
1305
2573
. 0142
18
. 0122
46. 5
46. 1
51. 6
54. 3
1090
1891
2981
.'0164
18
. 0122
50.8
56. 1
57. 5
58. 1
1440
2849
4289
. 0124
18
. 0122
59. 9
58. 4
64. 5
67. 9
1392
4343
573b
. 0126
18
. 0122
70. 0
69.4
74.7
77. 0
1570
4931
6501
. 0114
18
. 0122
76. 6
77. 7
82. 6
84. 8
1498
4441
5936
. 0119
18
. 0122
74. 3
85. 6
84. 5
83. 8
1400
3441
4841
. 0128
18
.0122
66.8
83. 4
79.2
77.6
1347
1831
3178
.0133
18
.0122
53. 5
76.9
71. 9
68. 2
885
611
1496
. 0203
18
. 0122
40. 6
66. 8
60. 8
57. 8
1115
-938
177
. 0161
18
. 0122
18. 6
will
ice
54.9
49. 0
45. 7
-------
TABLE C-10 - Four Corners Data (Cont, )
Jan. Feb. Mar Apr. May June July Aug. Sept. Oct. Nov. Dec.
T , °F
slug Flow
(Transient)
T . °F
mixed
(steady state)
37. 0
41.4
37. 7
47. 9
43. 0
57. 8
51.2
58. 8
55. 7
70. 9
65, 2
78. 3
75. 0
86. 4
78. 0
83. 2
74.2
76.2
65. 0
65. 1
54. 0
50. 6
46. 0
34. 0
* The three temperatures correspond to the 1-day, 10, 2-day and 20. 4-day of each month.
NOTE: Predicted transient operation temperatures are based on the assumption that the pond on
1 January has a water temperature at the intake equal to the measured value of 41. 5°F.
-------
90 |—
40 60
Pond Depth, ft
80
100
Four Corners Plant: Measured Temperature Depth Profile
July 1969
Figure C-7
172
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Accession Number
Subject Field &, Group
05D
SELECTED WATER RESOURCES ABSTRACTS
INPUT TRANSACTION FORM
Organization
Littleton Research and Engineering Corporation
Littleton, Massachusetts
Title
AN ENGINEERING-ECONOMIC STUDY OF COOLING POND PERFORMANCE
10 Authors)
Hogan, W. T. ,
Liepins, A. A., and
Reed, F. E.
16
21
Project Designation
FWQA Contract 14-12-521;
DFX
Note
22
Citation
FWQA, R & D Report No. 16130DFX05/70
23
Descriptors (Starred First)
"Thermal pollution, *Cooling water, *Ponds, *Design, *Economic Evaluation,
*Energy budget, Heated water, Heat transfer, Thermal Power Plants, Economics,
Capital costs, Operating costs, Evaporation, Convection, Design data
25
Identifiers (Starred First)
*Cooling ponds
27
Abstract
A procedure for predicting the temperature of a thermally loaded captive pond is
presented. Using this information, the cooling pond is shown in a special case to
have an economic advantage over a cooling tower and to be not much more expensive than
a natural body (stream or ocean) or water. This, with the ecological and recreational
assets of a captive cooling pond, would seem to encourage their expanded use with
large thermo-electric power plants.
This report was submitted in fulfillment of Contract No. 14-12-521 under the sponsorship
of the Federal Water Quality Administration.(Hogan-Littleton)
w4
bs tractor
T. Hoean
Institution
Littleton
WR:102 (REV. JULY 18691
WRSI C
Researrri
SEND
ft 1
TO:
^nnincpr-ino
Corpnrati
fm-
wSTE'R'R'ES'tftfrlCES SCIENTIFIC INFORMATION C EN 1
U S DEPARTMENT OF THE INTERIOR
WASHINGTON. D. C. 20240
rER
* CPO: 1969-359-339
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