Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST)

on October 9, 2015.

NATIONAL BUREAU OF STANDARDS REPORT

4795

SECOND EDITION AUGUST 8, 1972

SOME OBSERVATIONS ON THE USE OF

UNDERGROUND RESERVOIRS AS HEAT SINKS

Report to

Office of the Chief of Engineers

Department of the Army

Washington, D. C. 20305

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
NATIONAL BUREAU OF STANDARDS

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NAHOiMAL, BUREAU OF STANDARDS REPORT

4795

SECOND EDITION AUGUST 8, 1S72

SOME OBSERVATIONS ON THE USE OF

UNDERGROUND RESERVOIRS AS HEAT SINKS

Report to

Office of the Chief of Engineers

Department of the Army

Washington, D. C. 20305

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU Of STANDARDS

Preface

This report is being reissued at this time because of the current

national concern for energy conservation. The technical content is

pertinent to those processes where available thermal energy must be

stored for use at a later time. The best transfer analysis mathematic

and techniques in this report were developed in regard to underground

heat sinks but can be used for problems dealing with the storage of

thermal energy.
.
SOME OBSERVATIONS ON TEE USE OP UNDERGROUND

RESERVOIRS AS ELAT SINKS

by

B. A, Peavy and R. S. Dill

Abstract

Disposal of waste heat from such equipment as

Diesel engines and refrigerating or air conditioning machines is a practical necessity for

the occupation of large underground installations and the operation of their facilities.

Cooling water supply from outside may be

interrupted during emergencies so the use of

underground reservoirs for the disposal of

waste heat Is under consideration. Recirculation of the water and use of the reservoir as

a heat sink have the advantages that the

water is conserved for other purposes and that

the total heat absorbing capacity is augmented

by the capacity of the rock surrounding the

reservoir to absorb heat. The objectives of

the present work were (1) to find whether a reservoir provided for an existing underground

installation is large enough to meet design

criteria and (2) to determine for the general

case the advantage, in terms of Increased heat

capacity, of recirculating the water and

utilizing the heat absorbing capacity of the

surrounding rock. An approximate analytical

solution of the problem was sought and Its

precision was investigated by means of tests

with the large reservoir and with a smaller

reservoir provided for experimental purpose..

INTRODUCTION

This paper covers analytical and experimental work

undertaken to establish a means of predicting, for

erg insuring design purposes, the heat capacities of

v-ater -filled underground reservoirs. The capacity of

the contained water is, of course, the product of its

weight and temperature rise, but the total heat capacity

of the reservoir is augmented by heat transferred to the

surrounding rock and the rate of heat absorption by the

surrounding rock is dependent upon the rate of temperature rise of the water in the reservoir.

It is probable that underground reservoirs will be

useful as emergency heat sinks in connection with

underground protective structures assigned to resist

attack in wartime, since heat from engines or from airconditioning equipment can be absorbed by such a reservoir if the outside water service is cut off due to

enemy action or other causes.

An analytical method for predicting- water temperature as a function of time and heat input rate was

adapted from the literature* and experiments were conducted

with an existing reservoir to corroborate the findings.

MATHEMATICAL- ANALYSIS AND ASSUMPTIONS

To develop a theoretical approach to this problem,

considerations are based on a hypothetical, infinitely

long cylindrical cavity of radius, a, from which heat is

transferred radially from Its lateral surface to the

surrounding material (rock) that is assumed to be boundlees. The thermal properties of the surrounding material

o.re thermal conductivity, k, and thermal diffusi vi ty, ©o . The hollow cylinder contains a fluid (water) of specific

heat, c

1 , Btu per pound degree F and mass, M 1 , pounds

per foot of length of the cylinder. At any time, t, the

temperature 6 of the fluid is assumed to be the same as

the temperature at the surface of the cylinder. A constant heat flux Q., Btu per hour per foot length of cylinder,

is added to the fluid from ah external source and the heat

balance becomes;

at radius a

- J>~

M 1 c

1 dG

at

- k dQ

dr

= 4,

From Carslaw and Jaeger, “Conduction of Heat in Solids",

p. 2dii , the temperature of the fluid at any time, t, is;

0 = 2ft_

Trie

/ - e d y

y3

The above equation

,

*18 it f(-^~-i G)

for which numerical

Bureau of Standards

figures 1 and 1A.

2 IT a2 k

C~ M- C

!

written in dimensionless

( 1 )

solutions , evaluated by

Computation Laboratory,

On these figures*

groups is;

the National

appear in

Symbols appear at the end of this report.

Tunnels used for underground reservoirs have irregular

or non-circular cross sections, and finite lengths . The

equivalent radius of the tunnel is computed from che

average perimeter of the actual irregular cross section,

hxcept for cases in wnich the width to height ratio is

large, this gives approximately the same heat flow path

as the cylindrical case. For a reservoir or tunnel of

finite length some heat is transferred into the rock at

the closed end and into the rock surrounding the opening

at the open end. This heat is not accounted for in the

theoretical treatment and can often be neglecced because

(1) this heat provides a factor of safety in design

computations, (2) this heat is negligible \%rhen the ratios

of length to width or height are great, since in such a

ca-o the surface area of tuo cna is small in comparison

zo the lateral surface area, and (3) relative time for

use of the reservoir is small.

EXPiR I MENTAL UNDERGROUND- hEBERVOIR

The experimental underground reservoir is 66 foot

long with an irregular cross section on the order of

seven feet square. The equivalent cylindrical radius

was computed to be I4.2 feet from averaging perimeter

measurements taken every two feet- along the length.

The reservoir is surrounded by greenstone except for

one end which was closed during tests by a three-foot

concrete dam with a plywood diaphragm above it. The

properties of the greenstone are: thermal conductivity,

1 . 1-5 Btu/hr-f t-{ deg F); specific heat, 0.2 Btu/lb

(aeg F); ana density, I06 lbs/ft3 . Water at the dam

was pumped through a tank containing electric heaters

and discharged back to the reservoir. Pleat input to

the water by electric 'heater s and pump was measured by

a watthour meter. Temperature measurements were made

at various positions in the rock, at the rock surface

and in the water by means of thermocouples.

OBSERVATIONS WITH FOG-NOZZLE GRAYING OF HOCK

SURFACES ABOVE WATER LEVEL

For the duration of this test, approximately

11,390 gallons or 95*000 pounds of water was maintained

in the reservoir. The water was discharged back to the

reservoir through a two-inch header with fog-nozzles

spaced every seven feet along the length of the reservoir. Nozzles were directed at the exposed rock

surface above the water level.

Heat was added to the water at a rate of 55*^00

Btu per hour for hours. Figure 2 shows the imetemperature relationship for the water, rock surface

and rock at depths of 2 and 5 feet. The values for

the dimensionless groups of equation (1) are:

G =

2TT (L; .2) 2 X l.ltS X 66 . 95,000 x 1.0 x 0.0395 68

0k. = 1.15 x 86 x e C 55,000

0.002270

cct 0.0395t

12' = ' 747272 " 0 . 0022i|t

The result of substitution of experimental data in the

above equations is shown on figure 3 along with the

theoretical plot for G=3«5*

The experimental results agree closely with those

yielded by equation (1) closely up to ect/a^ = 0.2.

The discrepancy between the theoretical and experimental

results for values of ^ct/a2 greater than 0.2, can be

explained by the fact that the theoretical treatment

does not take into account the heat transferred into the

end of the reservoir. This can be demonstrated more

clearly by comparing the heat balances as follows:

Heat Balance Based on At End of 250 Hours

Temperature Increases

heat in w at er 2,.97 X

0

H 0 Btu

heat in rock cylinder 10.• 25 X 10 6 Btu

heat in the end 1..07 X 10 6 Btu

Total lia

.

.20 X 10° Btu

Total he at adaea

indie ated by Watthour

meter s 111.,27 X 10" Btu

Approx. Losses from

System 70,000 Btu

- 6 -

Theoretical Heat Balance Based on liquation (1) At

End ox 2$S Hours

heat in water 3*45 * 10^ Btu

L heat in rock cylinder 10.75 * 10 Btu

Total ll|.2 x 10 6 Btu

The heat balances show that for the same amount of

heat added to the system, the distribution assumed for

the theoretical case was different from that which

actually occurred during the experiment and that the

water temperature rise was less for the experimental

case. On the basis of the total amount of heat added,

the theoretical heat absorbed by the water was approximately 3 percent greater than the experimental. An

empirically determined relationship for values of

*ct/a^ greater than 0.2 can be written:

/0k} TM / Qk n 0 s Q3kt 0 > ^ 'actual " 'theoretical ” M * ^

which closely follows the experimental plot beyond

«s: t /a

2 — 0.2.

Unaccountable heat losses from the system include

water seepage into the reservoir from fissures in the

rock and out the overflow pipe, and water vapor transfer

and heat loss through openings above the dam. As a percentage of the total heat added these losses can be

considered negligible.

Q3SERVATI0NS WITHOUT 3 PRAYING OF SURFACES

ABOVE WATER LEVEL

For the duration of this test, approximately

15,200 gallons or 120,600 pounds of water was maintain©-

in the reservoir-. Circulating water was drawn from the

reservoir near' the bottom of the dam, passed through

the heater and pump and returned to the reservoir

through a long pipe which extended the length of the

tunnel and delivered the water at the far ena.

-7-

Heat was added to the circulating water at a T ate

of 61;

,

500 3tu/nr for 256 hours and figure 6 shows the

time- temperature relationship obtained for the water,

rock surfaces arid rock depths of 2 and 5 feet. The

values for the dimensionless groups of equation ( 1 ) ore:

2fr(6.2 ) 2 x 1.14.5 A 86 _ , 126,600 x 1.0 x 0.0395

~ *'

1.65 x 06Q

66,500

0.0395t

C6.2) 2

0.00193©

0. 00226

t

The result of substitution of experimental data in the

above equations is shown on figure 5 along with the

theoretical plots for G=2 . 5 and 3.0. The experimental

results are slightly above those yielded by equation

(1) for G=2,76 and for values of «ct/a^ less than 0.6.

For greater values of ect/a^ they fail below the results

of equation (1). The discrepancy between the theoretical

and experimental results when <ct/a^ exceeds 0.6 can be

reconciled as in the case of the spraying test by the

following empirical relationship:

/ ©k) = / ©k y __ 0. 02kt

q

; actual Q, theoretical M"7

DISCUSSION AND CONCLUSIONS

Two assumptions are essential for this analytical

study of the problem which cannot be met in a practical

sense. They are as follows:

•NOS'DC — 0 —

1 . That heat is transf erred radially from the

lateral surface of the hollow cylinder and that the

ends of the hollow cylinder are insulated from heat

flow. This was not true during the experiments

because there was heat flow into the ends and its

effect gave lower water temperatures than predicted

by theory.

2 . That water is so well stirred that the temperature of the water is the same as the temperature of the

surface of the reservoir. This impl 5.es that there is

no thermal resistance between the water and the rock

surface. In the two tests (figures 2 and I4 } there was

a temperature difference between the water and the

surface. This effect vias more evident when there was

no spraying of rock surfaces above the water level.

The above two effects appear to counteract each

other and the greatest effect after a period of time

was shown to .be heat stored in the end of the reservoir.

This was borne out by lower water temperatures than

predicted by the theoretical approach, as shown in the

empirical equations (2) and (3).

COMPARISON OF RESULTS WITH AND WITHOUT SPRAYING

OF ROCK SURFACES

Comparison between the two tests cannot be directly

made because of the difference in heat input rates, but

when equation (2) is subtracted from (3), the following

relation results:

0 , OlQt

M 1

where 9 . and 9^ are the temperature rises fer the nonspraying and spraying tests, respectively. By assuming

a steady heat flux, Q, of 700 Btu/hr-ft, mass, M 1 , of

1350 lbs /ft, and time, t, of ipQG hour 3 5 the temperature

difference between the two cases becomes 2F. This

represents an increase of approximately 16 hours

attributable to spraying water on rock surfaces above the

water level.

DLitJIGN APPLICATIONS

The results of the experiments indicate that

equation ( 1 ) and figure 1 can be used for estimating

heat capacities of underground reservoirs with reasonable accuracy in most practical cases. For values of

exceeding 0.2, the Pieat capacity of the rock

surrounding the ends of a reservoir becomes appreciable

and makes the total capacity greater than that predicted

by equation (1) and figure 1. Correction factors are

suggested in equations (2) ana (3). However, underground reservoirs used as heat sinks are likely to be

larger -ihsun the experimental reservoirs so that values

of exceeding 0.2 are not likely in practice and

the correction is expected to be unnecessary in most

cases. For the sake of example, assume an equivalent

radius of 10 feet and «?= O.Ob ft^/br, men the time

necessary to reach a value of &?t/a^ = 0.2 is 500 hours

or 21 days. This is probably a considerably longer

time than is anticipated for the use of a reservoir

a.a an emergency heat sink.

F.or a sample problem, consider an underground

reservoir with dimensions 230 ' long by 2k 1 wide and

29* high holding 1.5 million gallons of water. The

thermal properties of the rock are the same as that

for the experiments and the initial water temperature

is 5 OF. The value for the equivalent radius is;

a

2 ( ih * 29)

2 T, = 20 feet

Assuming ; and a hea- values fo:maximum of 50? temperature rise of the wate:

input rate to the water of 6 x 10° Btu/hr

the dimensionless groups are:

th

2 Tf (20 ) 2 x l 0 l\5 x 2 30

1,500,000 X 0.33 X 1.0 X 0.0395 x 7

4

*-3T>

to

-10

oC t 0 . 0 3 9 9

1

~T2oT^~ = x io*^t

Gk 90 x 1,45.x 230

§

=

6 i "lO 5 0.00278

kef erring to figure 1A, cct/a^ = 0.012, and the tirae

equals 121 hours or a savings of 16 hours due to heat transfer

to the surrounding rock.

The above example is taken from an actual underground reservoir where results of a test on this

reservoir closely substantiate the above figures.

The value of G with 'the reservoir completely

filled is:

Gu = S££-

p^c 5

with water in the reservoir

pc

G ~ 31*2

where p and c are the density and specific heat of the

urrounding rock, respectively. For these experiments

= 186 lbs/ft-3 and c = 0.2 Stu/lb-°F, then G = 1.2.

her efore, optimum conditions are reached when the

eservoir is completely filled with water.

- 11 -

SYMBOL6

Symbols and constants used for the underground

reservoir are as follows:

a = radius of equivalent cylinder, feet

= I4.2 feet

c» = specific heat of fluid, Btu/lb - (deg F)

= 1.0 (water)

G = mathematical quantity, used in equation (1)

k = thermal conductivity of rock, Btu/nr - ft

- (deg F) = 1 . 1*5 (greenstone)

M* = weight of fluid per unit length of cylinder

WnTt.

Q, = heat flux to fluid from external source,

Btu/hr - ft

t = time, hrs

p ** = thermal diffusivity of rock, ft“/hr

= 0.0395 for greenstone

pc = volume specific heat of rock, Btu/ft^ - (deg ?) = 36.7 (greenstone)

p

! c* = volume specific heat of fluid, Btu/f t

J - (deg F)

= 62. ^ (water)

temperature rise of water above the initial

temperature at time, t, deg F.
.

' ’
rJitS-.’i.

C.03C 1 -

^ ^ O ^ -u

0.02 0.04 o.os

0. 020 ; —

0.05 O.io

0.005

; O' Oi

0.0 i 5

:—

0.0 c

‘t

LMP£

MATURE

RISE

ABGV

3 HOURS

I

rj u I ^

O o

0.4

0.5

0.6

FI G.URE 4 .. 1--

w-0>\ - GPRAYsNG TEST ’

T!f.)£ - .TEMPERATURE; --PLOT-—

. KiLAT: IMPUT- RATH- :

.

; . £ 4, 300. s-TL/ciR.

;

HOURS
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NOT FOR PUBLICATION OR FOR REFERENCE