GEOLOGICAL SURVEY CIRCULAR 282 

THE USE OF RESERVOIRS AND LAKES 

FOR THE DISSIPATION OF HEAT 

UNITED STATES DEPARTMENT OF THE INTERIOR 

Douglas McKay, Secretary 

GEOLOGICAL SURVEY 

W. E. Wrather, Director 

GEOLOGICAL SURVEY CffiCULAR 282 

' THE USE OF RESERVOIRS AND LAKES FOR THE DISSIPATION OF HEAT 

By G. Earl Harbeck, Jr. 

~ashington, D. C., 1953 

Free on application to the Geological Survey, W&shington 25, D. C. 

PREFACE 

This report on the use of reservoirs for the dissipation of excess heat was prepared in the Water Re• 

sources Division of the U. S. Geological Survey, 

C. G. Paulsen, Chief Hydraulic Engineer, under the 

administrative supervision of R. W. Davenport, Chief, 

Technical Coordination Branch, and under the technical direction of W. B. Langbein, Hydraulic Engineer. 

The study was originally suggested by J. D. Doherty, 

Synthetic Liquid Fuels Branch, u. s. Bureau .of Mines, 

II 

who also furnished useful information concerning appro:,cimate quantities of heat to be disposed of as a 

result of. certain industrial processes. 

The advice and helpful suggestions of E. R. Anderson 

and J. F. T. Saur of the U. S. Navy Electronics 

Laboratory; San Diego, Calif., are gratefully 

acknowledged. 

THE USE OF RESERVOIRS AND LAKES FOR THE DISSIPATION OF HEAT. 

By-G. Earl Harbeck, Jr. 

CONTENTS 

Abstract ................................................. . 

Introduction ............................................. . 

Theory ................................................... . 

Basic data ............................................... . 

Page 

1 

1 

1 

2 

Computation methods ................................... . 

Results ..................................................... . 

Conclusions ............ _ .................................. . 

References.-.............................................. . 

Page 

3 

5 

5 

6 

ILLUSTRATIONS 

Figure 1. Reservoir temperature rise resulting from the addition of heat_ ............................................. .. 

Page 

4 

2. Comparison between cooling-tower water use and increase in evaporation from Lake Hefner ........ . 4 

TABLES 

Table 1. Selected hydrologic and meteorological data for Lake Hefner, Okla ......................................... .. 

Page .3 

2. Assumed hydrologic and meteorological data for hypothetical lake ........................................... .. 3 

3. Percent of total heat added that is dissipated by each of various processes and percent of water 

saved ...................................................................... _ ................................................. . 5 

ABSTRACT 

A method is presented for estimating the possib~e 

saving in water resulting from the use of lakes or ex .. 

isting reservoirs instead of cooling towers for the dissipation of excess heat. For the two reservoirs studied, annual savings were found to "be 45 to 50 percent. 

INTRODUCTION 

Cooling by evaporation has long been used for the 

dissipation of unwanted heat in certain industrial processes. Where water supplies are plentiful, the consumptive use of water in cooling towers presents no 

problem, but where supplies are scarce and relatively 

expensive, other methods of cooling warrant 

consideration. 

An alternative method, which has long been used, is 

to withdraw water. from a reservoir or lake, let it 

absorb heat, and then return it to the reservoir or 

lake in such a manner as to prev~nt immediate reuse. 

It is obvious that evaporation from the lake will be increased thereby. If consumption of water is the only 

criterion, the practicability of the method depends on 

whether the amount of water lost by increased evaporation is less than that which would have been used by 

a cooling tower in dissipating the same amount of heat. 

Some theoretical aspects of the problem have been 

studied by other investigators, including Lima (1936) 

and Throne (1951). Although the general principles 

were understood, this is the first opportunity to explore the problem using field data and to apply the results of the Lake Hefner experience and refinements 

in theory to this problem. 

THEORY 

The energy budget for a lake or reservoir may be 

expressed as follows: · 

Oa· O,+Oa· Oar·Obs· Oe·Oh+OvrOw·Ow • O,. (1) 

In the equation above: 

Oa - solar radiation incident to the water surface •. 

0, • reflected solar radiation •. 

Oa • incoming long-wave radiation from the 

atmosphere. 

Oar ... reflected long-wave radiation. 

Ou = long-wave radiation emitted by the body of 

water. 

o. - energy utilized by evaporation. 

011 - energy conducted from the body of water as 

sensible heat. 

Ow. • energy advected into the body of water. 

Ow - energy advected out of the body of water. 

1 
2 DISSIPATION OF HEAT 

Ow energy advected by the evaporated water. 

0-6- "' increase in energy stored in the body of 

water. 

In order to illustrate the approximate magnitude of 

the various items in the energy budget, data obtained 

at Lake Hefner, Okla.-, during the interagency waterloss investigations (1952) have been used. For the 

12-month period Sept. l, 1950, to Aug. 31, 1951, average daily values of certain of the items in equation 

(1) in calories per square centimeter per day were 

approximately as follows: 

Qbs = 781 Q = 6 ro 

Oe = 222 Ow 6 

Q;,,;, 8 Q-a- = -2 

0·= •Vl 8 

Inserting the figures given for these items in equation 

(1), we find that Q8 ·Qr+08 ·0ar = 1,018 calor_!__~s 

per square centimeter per day. The data presented in 

table- 7 and figures 65 and 66 of the interagency report 

(1952) and the relation Qar = 0. 03 08 (as found by 

Gier and Dunkle and given in the same report) were 

used as a guide in apportioning the total of 1, 013 calories per square centimeter per day among the four 

items approximately as follows: 

Os = 432 

() 

•r 26 

Oa 622 

0 = ~ar 19 

Among the various energy-budget items, Qbs (the 

long-wave radiation emitted by the body of water), 08 

(the incoming long-wave radiation from the atmosphere), and Q8 (the incoming solar radiation) are the 

largest. Measurements of water-surface temperatures 

are all that is needed to determine Obs at any reservoir; few measurements of surface temperatures have 

been made, however. Eppley pyrheliometers have long 

been used to measure Q8 at certain Weather Bureau 

stations, and many such records are available. Except 

for the data obtained at Lake Hefner, there have been 

extre~ely few measurements of Q8 , and many additional data are needed to determine the areal variation in 

this important item. 

In studies of annual evaporation, it is customarily 

assumed that the change in energy storage in the body 

of water is negligible-for a period of a year. The 

figure of -2 calories per square centimeter per day 

for o,_ given above substantiates this assumption; of 

course, this figure is invalid for shorter periods of 

time. 

Consider a natural lake or a reservoir that is to be 

utilized for the purpose of dissipating excess heat. 

The addition of heat to the lake will not affect the 

following items in the energy budget: (!8 , Or• 08 , 

Oar• Ovi• and Q-6-, where Q,. is considered to be the 

increase in energy stored in the body of water resulting from natural processes only. Of these, Q8 , Or• Oa• 

and Ovi obviously are not affected by the watersurface temperature of the lake. Oar, the reflected 

long-wave radiation; is also independent of watersurface temperature, depending only on Q8 • Thus from 

equation (1) it 'follows that when heat is added to a lake 

(Ql,s • Qbs) + (9'e • Q;) + (Qh • Qh) + (Q'ro ·Oro)+ (Q'w • Qw) = Qc (2) 

(in which the unp:dmed symbols refer to the lake in its 

natural condition, the primed symbols to t;he lake after 

heat has been added, and Oc to the amount of heat added). Or, rewriting in a more simple form, 

t!Qbs + hQe + t!Qh + !l.Qro + !l.Ow .. Oc· (3) 

A direct solution for each of the terms on the left 

side of equation (3) for a given value of Oc is not possible. Each term, however, varies directly with the 

temperature rise, and for various temperature rises 

it is possible to solve equation (3) for corresponding 

values of Qc. 

It should be emphasized that all the terms on the 

left-hand side of equation ( 3), with the possible exception of t!Qro , are surface phenomena. They depend 

only on water-surface temperatures and characteristics of the ambient air but are independent of the thermal structure of the reservoir. Consider the case 

where water is withdrawn from a reservoir at some 

depth, used for cooling, and then the heated water is returned to the surface of the reservoir. Assuming complete lateral mixing over the surface, some of the heat 

added inevitably must be utilized in increasing heat 

storage in the lake. The rate at which this occurs will 

depend to a large extent on the amount of wind-induced 

mechanical turbulence. Thefollowing computations 

are based on the premise that equilibrium has been 

reached and that all heat added to a lake is dissipated 

to the atmosphere and is not used to increase energy 

storage in the lake, or in other words, over a long 

period of time t-.O,_= 0 • Until equilibrium is reached, 

the amount of heat dissipated from the surface, including that utilized to increase reservoir evaporation, 

will be less than the computed value. The addition of 

heat at the surface of the lake may tend to cause stratification, and equilibrium may be attained quickly if 

little mechanical mixing occurs. 

If all reservoir outflow occurs at the surface level, 

as in the case of an overflow dam, there is little or no 

error introduced by assuming the outflow temperature 

to be the same as the water-surface temperature. If 

outflow occurs at considerable depth, some error may 

be introduced by this assumption, but in most storage 

reservoirs the amount of heat removed in this manner 

is quite small compared with other items in the energy 

budget. For a run-of-the-river reservoir, inflow and 

outflow volumes are large relative to its capacity, and 

the resultant mixing reduces or eliminates vertical 

temperature gradients in the reservoir. 

BASIC DATA 

Selected data from the Lake Hefner investigations 

were used to compute Qc for arbitrarily chosen values 

of fiT (temperature rise), as given in table 1. 

In order to obtain some idea of the possible effect 

of reservoir size, inflow, outflow; and other hydrologic and climatologic factors. a hypothetical reservoir was selected for study, and the assumed cond1tions 

are given in table 2. 

Although no specific location was selected, the assumed climatic and hydrologic conditions are reasonably representative of southeastern Colorado. In comparison, the surface area of the hypothetical lake is 

22 percent, and the capacity is 8 percent, of that of 

Lake H~fner. 
COMPUTATIDN METHODS 3 

Table 1.-Belected hydrologic and meteorological data for Lake Hefner,· Okla. 

October January April July September 1, 1950, 

195{) 1951 1951 1951 to 

August 31, 1951 

Water-surface temperature ....................... °C •• 19.28 3.48 11.29 26.73 14.96-

Air temperature, 8-meter level .................. °C .. 19.02 2.52 12.85 26.45 14.57 

Wet-bulb temperature, 8-meter level ......... °C .. 13.72 -0.27 8.43 21.89 10.68 

Evaporation .................................. acre-feet •• 1271 459 496 1385 10,. 293 

Outflow ........................................ acre-feet .• 832 860 890 1346 11,724 

Reservoir surface area ....................... acres .• 2342 2232 2182 2316 2275 

Table 2. -Assumed hydrologic and meteorological data for hypothetical lake 

Drainage area above reservoir ................................................................................. square miles .• 100 

Reservoir capacity ...................................................................................................... acre-feet .. 5,000 

Surface area .................................................................................................................... acres .. 500 

3,300 

2~200 

1.100 

Annual inflow including rainfall on lake surface ................................................................. acre-feet .. 

Annual reservoir evaporation under natural conditions ....................................................... acre-feet •• 

Annual outflow (by difference) ........................................................................................ acre-feet •. 

Average annual air temperature ..................................... , ..••••.• ~ ................................................ oc •• 10 

10 

50 

Average annual water-surface temperature under natural conditions .............................................. oc .. Average relative humidity ................................................................................................ percent •• 

COMPUTATION METHODS 

The various terms in equation (3) were computed as 

follows (units: calories per organic c.entimeter per 

dav): · i\Qbs..; 0.970o-((~ '+ 273)4 • (T0 + 273)4] (4) 

in which o- = Stefan-Boltzman constant for black-body 

radiation (= 1. 71 x 1o-7 calories per square centimeter per (degree)4 _per day). 

llQe • pE'L' ·pEL (5) 

in which P • average density of evaporated water (• 1 

gram per cubic centimeter) 

E• average daily evaporation in grams per 

square centimeter per day("' centimeter per day) 

L• latent heat of vaporization in calories per 

gram at T0 (• 595.9 - 0. 545 T0 , very 

closely). 

But from mass-transfer theory, assuming no change 

in wind speed and that the possible effect of changes in 

atmospheric stability resulting from an increase iil 

water temperature is negligible, 

(5a) 

in which •o • saturation vapor pressure at,T0 in mb 

e8 = vapor pressure of the air at the 8-meter 

level in mb, determined from 7' a. and 1'8• 

I!.Qh •R'Q~ ·\~O~~R'pB'L' • RpEL (6) · 

in '#hich R • the Bowen ratio ·(\. 0.6iP(7'0 .ra> 

tt itm teO. &81 P = atmospheric pressure in Jllb, 

ll.Qvo•cp(T0 '·T0 )V0 (7) 

in which c • specific heat of water (• 1 calorie per 

..gram per degree) 

V0 = outnow volume in grams per square centimeter per day(" centimeter per day). 

AQ.,•cp(E'T0 ' -E7'0 ). (8) 

Using the Lake Hefner data, values of Oc for vari:-

ous assumed temperature rises (AT) were computed 

using equation (3) with the individual terms being computed using equations (4) to (8). Simiiar computations 

were made for the hypothetical reservoir under the 

conditions listed in table 2 and also for assumed relative humidities of 30 and 70 percent. 

After the values of Oc had been computed, the next 

step was to compare, for each value of Oc, the increase in evaporation ( E' - E) with the amount ot 

water that would be evaporated in a conventional cool.;. 

ing tower during disposition of the same. quantity of 

heat. For the purposes of this study, the amount of 

water evaporated from a cooling tower, E1, W!'-S computed by diyiding the _amount ~f heat to be dissipated, 

Qc, by The latent heat of vaporization of water at the 

original temperature, T0 • Admittedly, this is only an 

approximation, because both the heat removed by convection as sensible heat and the heat carried away by 

t_he evaporated water have been ignored. The heat removed by convection is small as compared with the 

heat removed by evaporation. At least partly counterbalancing this omission is the fact that no allowance 

has been made for drift or spray losses. 

It should be pointed out that the above method of 

computation probably results in smaller values of E1 

than would be observed in an actual cooling tower, because of the decrease in latent heat of vaporization, L, 

with temperature. If the evaporating water is at 10°C, 

as was assumed for the hypothetical reservoir, the 

disposal of 1 x 1013 calories per day (3. 97 x 1010 Btu 

per day) by evaporation only would use 1. 69 x 1010 

grams per day (4. 47 mgd). If the evaporating water is 

at 70° C, however, disposal of the same quantity of 

heat would use approximately 6 percent more water • 

Thus the values of Et, as computed, are probably conservative, despite the fact that certain small items in 

the energy budget for the cooling tower have been 

ignored. 
4 

-0 

0 3 

-

-w 

I 

w 

z 

0 

~ 3 

et::o 

oo 

n.-

~== w~ 

a:: 0::: 2 

o:;e 

> a::(/) 

w:E 

CJ)<( 

wo:: a::(!) 

z 

w 

CJ) 

<( 

w 

a:: 

(.) 

z 

DISSIPATION OF HEAT 

Lake Hefner 

2 3 4 5 6 7 8 9 

DEGREES CENTIGRADE. 

TEMPERATURE RISE (To-To) 

10 

Hypothetical 

reservoir 

Figure 1. -Reservoir temperature rise resulting from the addition of heat, 

MILLIONS OF GALLONS PER DAY 

2 4 6 8 ·to 14 

\'Z. 

2 3 4 . 5 6 

GRAMS PER DAY x 101

0 

WATER USED BY COOLING TOWER (Et) 

14 

0 

12() 

tIOz 

:::) 

..J 

<l 

82 

a:: 

LIJ 

J: 

t6:1: 

!!! 

t40: 

m 

2 

10 

>- <l 

0 

8ffi 

a.. 

(/) 

z 

0 

:::; 

6:;#, 

(!) 

ll.. 

0 

(/) 

4~ 

2 

..J 

:::! 

:E 

Figure 2. -comparison between cooling-tower water use and increase in evaporation from Lake Hefner. 
CONCLUS~ONS 5 

Table 3. -Perc;ent of total heat added that fs dissipated by each of the various processes and percent of water saved 

[For Qc = 1 x 1013 calories per day= 4 x 10l0 Btu per day] 

Relative 

Period humifii.ty l'l.Qvo l'l.Qbs l'l.Qe l'l.Qh l'l.Qw Percent saving 

(percent) i~ water 

Lake Hefner 

October ••..••••• ··5s. 3 1 15 59 24 1 40 

January •...•...• 58.1 1 19 40 39 1 60 

April. •...•..••••. 55.8 1 20 47 31 1 52 

July •.••••••..•••• 6?. 3 1 14 64 18 3 35 

Year ............. 62.9 1 15 54 28 2 46 

Hypothetical Lake 

Year ............. 30 1 

Do •••.••..•.••• 50 1 

Do •..•..•.....• 70 1 

RESULTS 

Figure 1 indicates that Qc, the heat added to the 

reservoir, is almost directly proportional to l'l.T, 

20 

15 

10 

the temperature rise, over the range investigated. 

For a value of Qc of 1 x 1013 calories per day (3. 97 x 

1010 Btu per day), the temperature rise in Lake 

Hefner for the 4 selected months would range from 

1.1°C in July to 2. 0°C in January, with an average of 

1. 6° C for the year. Temperature rises are approximately inversely proportional to water-surface temperatures. For the hypothetical lake, the temperature 

rise is much greater of course .. because its surface 

area is much smaller. On a unit-area basis, the Lake 

Hefner results agree reasonably well with the hypothetical lake results. However, the computed values 

of Qc as shown in figure 1, purposely were not presented on a unit-area basis in order to avoid any inference that generalization is possible. For a run-ofthe-river reservoir, for example, annual outflow may 

be many times greater than the capacity of the reservoir, and the amount of heat used to increase reservoir evaporation may be negligible as compared to the 

amount of heat carried away by the outflow. 

Table 3 shows the relative magnitudes of amounts of 

heat disposed of by the various processes, and it shows 

a comparison between the amount of water used by increased reservoir evaporation and that used in a conventional cooling tower. 

Figure 2 indicates that for Lake Hefner the present 

water saving in percent does not vary appreciably with 

Oc within the limits shown. All of the curves are 

slightly concave upward, as might be expected, because if Qc were allowed to increase without bound, 

the increase in evaporation, E'· -li; woald approach the 

cooling-tower use, Et • . 

The Lake Hefner results illustrate the seasonal 

change in amounts of heat disposed of by th~ various 

processes. Seasonal changes in l'l.Qvo~ !!.Qbs• and M)w 

are small compared with the changes in l!.Q . and .f'l.Qh. 

In January, for example, only 40 percent o' the total • 

heat is disposed af by increased evaporation, and 39 

percent is carried away as sensible heat, but in July 

the percentages are 64 and_ 18, respectively. The 

49 27 3 50 

52 30 2 48 

54 34 1 45 

greater amount of heat dissipated by convection in January results from the fact that the Bowen ratio is 

larger; dissipation of heat by convection is directly 

proportional to the temperature gradient and inversely 

proportional to vapor-pressure gradient. Temperature 

gradients are usually large in winter and vaporpressure gradients are small. The percentages for the 

hypothetical lake are of the same order of magnitude 

as for Lake Hefner. 

It will be noted that each figure in the column headed 

l'l.Q8 is al~~st exactly equal to the figure in the last 

column subtracted from 100. n· L, the latent heat of 

vaporizatiol\did not vary with temperature, they would 

agree exactzy, because 

!!.Q E'L'-EL 

.::.:J!. ---x 100 in percent 

Qc Qc 

and percent saving in water • 

100 [1. E' .. E] ·100 [1-~]. Et Oc 

The saving in water that would result from the use 

of Lake Hefner for cooling, as shown in the last column ·of table 3, is 46 percent for the year and ranges 

from 40 to 60 percent for the selected months. For the 

hypothetical reservoir, the saving is approximately the 

same. It should not be expected that these figures apply to all lakes or reservoirs (particularly to reservoirs in an extremely arid or extremely humid region 

or to run-of-the-river reservoirs). 

As has been previously discussed, the computed water savings are based on the assumption that the reservoir has reached thermal equilibrium. Until this condition has been attained (the time required may range 

from hours for a small pond to years for a large, deep 

reservoir), the computed increase in reservoir evaporation will be greater than that which actually occurs, 

and the saving in water will be correspondingly greater. 

CONCLUSIONS 

The results of the study indicate that, for the dissipation of excess heat resulting from certain industria.! 

processes, substantial savings in water often can be 
6 DISSIPATION OF HEA 't 

realized, as compared with conventional cooling-tower 

methods, if it is practicable to withdraw water from a 

reservoir or natural lake, to let it absorb heat, and 

then to return it to the reservoir •. 

It should not be assumed that a water saving will result if a reservoir is constructed solely for the purpose 

of dissipating excess heat. In that case. the increase 

in evaporation resulting from the construction of the 

reservoir must be added to the increase in evaporation 

resulting from the addition of heat, and their sum will 

in all probability be greater than the amount of water 

used by a cooling tower. 

There are many interrelated factors involved, and 

it would be unwise to assume that the figures of possible water savings shown for the two reservoirs studied 

will apply to all lakes or reservoirs. The theory presented is completely general, however, and may be applied to any body of water, if the necessary hydrologic 

and climatologic data a:re available. 

REFERENCES 

Lima, D. 0 •• 1936, Pond cooling by surface evaporation; Power, v. 80, no. 3, p. 142-4. 

Throne, R. F., 1951, How to predict lake cooling 

action; Power, v. 95, no. 9, p. 86-89. 

U. S. Geol. Survey, 1952, Water-loss investigations; 

volume 1-Lake Hefner studies, technical report: 

U. S. Geol. Suriey Circ. 229. Also published as 

U. S. Navy Electronics Laboratory report 327. 

INT.-DUP. SBC .• WASH., D.jl5J01'r