The temperature under heated or cooled areas on the ground surface

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Ser THl N21r2 no. 208 c . 2

BLDG

P R I C E 25 CENTS

NATIONAL RESEARCH COUNC I L CANADA

D I V I S I O N OF B U I L D I N G RESEARCH

T H E TEMPERATURE UNDER HEATED OR COOLED AREAS ON THE GROUND SURFACE

BY W .G. BROWN

R E P R I N T E D FROM

TRANSACTIONS OF THE E N G I N E E R I N G I N S T I T U T E OF CANADA VOL.

6,

NO. B - 1 4 , JULY 1963

P A P E R NO. EIC-63-MECH 3 RESEARCH P A P E R NO. 208 O F THE D I V I S I O N O F B U I L D I N G RESEARCH

2'71-779

OTTAWA DECEMBER 1963 NRC 7 4 1 4

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THE TEMPERATURE UNDER HEATED OR COOLED AREAS ON THE GROUND SURFACE

by

W.G. Brown'':

SUMMARY

Equations and graphical methods are presented which allow calculation of the steady or unsteady temperature in the ground under flat areas of any shape o n ' the ground surface. The equations, some of which are derived by superposition, are applicable to structures such as basementless buildings, shallow lakes and rivers, street intersections and ice-rinks.

The problem of determining ground temperatures under a surface area which is at a temperature different from that of the surroundings occurs when dealing with structures as varied as basementless building, ice-rinks, roads, street

intersections, and shallow lakes and rivers. For all of these, the temperature in the ground depends on the shape and size of the area covered, the temperature at which it is maintained, the temperature of the external ground surface and the geothermal gradient in the ground. It is not always appreciated that although the temperature regime in this case may be three dimensional and time dependent, it can nevertheless be determined with the help of the superposition principle and exact basic equations. In addition, it is not generally recognized that superposition itself, which is nothing more than direct addition of partial temperature effects, can be carried out graph'c lly. A significant contribution to this approach has been inade by Lachenbrucht2? who, however, restricted his considerations to basementless buildings in permafrost regions.

It will be the two-fold purpose of the present paper to bring together the basic equations in a manner suitable to direct application in the field, and to

indicate their use in graphical form. Development will proceed beginning with the steady-state solution under the edge of a large area an3 continue to the long strip, corner, rectangular and polygonal area, and the irregular area. Time dependence will then be considered in the same sequence.

All the equations are solutions of the general Laplace equation,

$: _{Building ~Lfvices Section, Division of Building Research, National Research}

For simplicity, the mean annual external ground surface temperature will generally be considered to be zero. To obtain actual temperatures, the true mean annual external ground surface temperature would then be added to the result obtained from the temperature equations.

THE TEMPERATURE REGIME IN THE STEADY-STATE

(1) Temperature near the straight side of a large area

In the absence of a geothermal gradient, the steady temperature regime near the edge of a large basementless building or shallow body of water would occur as depicted in Fig. la. Here the surface temperature under the structure is

3,

and the external surface temperature is zero. By inspection the elementary temperature relationship is

? / = 9

tan -1 z

Tr

With a geothermal gradient present, the resulting temperature, obtained directly by superposition, becomes

where GG is the rate of change of temperature with depth in the undisturbed ground.

In graphical terms, the procedure involved in superposing the geothermal grad- ient on the temperature regine of Fig. la is shown in Fig. lb. Here the horizontal lines are equally spaced isotherms due to the geothermal gradient, and the radial lines are those of Fig. la. By joining points of intersection of the two sets of lines having equal sums a new set of isotherms is obtained which is the solution for the combined problem. The graphical method of plott-

ing an isotherm proceeds rapidly when it is noticed that adjacent points on it lie on the opposite corners of quadrilaterals.

(2) Temperature in the ground near a strip area on the surface

The temperature distribution due to a river, road, or long basementless building of width 2a (Fig. lc) is obtained by superimposing the result of equation (2) on itself with a change in the coordinate y. Replacing y by (y-a) and (-~-a), summing the two temperatures and subtractingPo gives a temperature distribu- tion resulting from the strip at

y o

and the remaining surface at zero, i.e.,

Equation

(4)

can be rearranged into the following simplified form:

-

1 2a z

V =

90

_{tan )22.(}

'lT _Y

+ Z

-

a

Isotherms in this case (Fig. lc) are simply coaxial circles passing through the edges of the strip. The temperature influence of the geothermal gradient GGZ can now be added to equation (5) to obtain the complete solution.

Figure lc also illustrates the graphical procedure of super-imposing the two temperature regimes with the edges a distance 2a apart. In this case one of the two basic temperature regimes is reversed in direction. Again it will be

recognized that the circular isotherms under the strip result from joining points of intersection of the two basic regimes which have equal sums. It will be

noted that the summation procedure results in a temperature of 2-17, on the strip surface and $lo on the remaining surface. In order to have the external surface at zero it is only necessary to subtractv, from the temperature at all points. This results in the strip being at?, as in Fig. Id, where, by way of additional example, the influence of the geothermal gradient is again added to complete the solution.

( 3 ) Temperature near the corner of a large surface area It has been shown") that by replacing y by

y/ (x + x + y + z ), z b y z/ ( x + x + y + z ) ,

and a by tan 814 in equation (5) the resulting temperature is that for the three-dimensional case of a large wedge-shaped surface area of corner angle

8, i .e.,

( 6 )

TF

2 2 - 1 2

y2

+

z

-

(x + x + y + z ) tan (814)

The coordinate y is now the horizontal distance perpendicular to the centre- line of the corner, x is the distance along the centre-linemd z is the depth as before. The relationship between temperature and the coordinates can also be put into a form more suitable to determining isotherms:

cot

( " " 1 ~

0) 2 2 2

(r

+

sin (@,2)

-

(el2

cot (812)

=

cot (812) cosec

(fi919~). ( 7 )

The temperature regime for the important case of corner angle fl/2 is given in terms of the dimensionless coordinates x/z and y/z in Fig. 2a. The influence of the geothermal gradient can again be added directly.

(4)

Rectangular and polygonal areas

Having obtained the temperature regime under a single corner, the solutions for several corners can be superimposed graphically to yield the temperature distribution under polygons. By way of rxample, the formation of the

temperature regime under a rectangle is shown in Fig. 2b. Here, two corners (top and bottom) are superimposed, resulting in a temperature of 272, on the rectangle but with the remaining surface divided into two regions with tempera- tures o f P o and zero. Intermediate isotherms are thereby obtained, as in the previous sections, by joining points of intersection (in this case in the horizontal plane) where the sums of the temperatures due to the two corners

are equal. By superimposing two more corners externally (Fig. 2b, right and

left) the end result is a temperature of 2P0 on the rectangle andPo over all of the remaining surface. The temperature regime for a rectangle at tempera- turep,, with the remaining surface at zero, is now obtained by subtracting

vo

from the sum of the temperatures resulting from all four corners. The

temperatures in this case might be representative of a cold storage building in tropical regions or of the steady portion of the temperature due to a heated or cooled basementless building at any world location.

In Fig 2c the temperature solutions for four external corners of different corner angles could be superimposed graphically to give the steady temperature that might result at a street intersection, the ground externalto the street being assumed covered by buildings at t e m p e r a t ~ r e 9 ~

.

The steady temperature under a basementless building of the shape shown in Fig. 2d is obtained by

direct addition of the temperatures resulting from two rectangles at temperature P o * Again the procedure can be carried out graphically and the effect of the geothermal gradient would be added as a last step in the calculations.

(5)

Irregular areas

The above procedure for polygonal areas could also be used in principle for irregular areas such as those in Figs. 2 e a d 2f which might represent shallow lakes and rivers in permafrost regions. The details of such a calculation, however, would soon become prohibitive. A simpler and more direct approach

is that due to ~achenbruch(~) which consists of subdividing the surface area

into circular sectors of angle 0 _{and radius R. In the absence of a geothermal}

gradient, the temperature at a depth z under the apex of the sector is

It is now a simple matter to sum the temperature effects of all sectors, there-

by obtaining the resultant temperature under the common apex. The various radii

sector is obtained by entering a graph in which equation (8) is plotted. In the case of multiple areas and locations outside of them ( ~ i ~ . 2f) it will be realized that the sum of temperatures due to sectors with radii R1 will be subtracted from the sum for sectors with radii R2.

THE TIME-DEPENDENT TEMPERATURE REGIME

(6) Temperature near the straight edge of a large area

(a) The entire ground surface initially at zero temperature, on2 half-plane suddenly raised t o y o and maintained at this temperature.

The solution for this case, which might represent sudden heating or cooling of a large building in the absence of a geothermal gradient and with the out- side surface temperature constant at zero, is given on p. 419-420 in reference

(3). It is

-

1

where:

P

=

tan (z/~), and r

-

4

y2

+

z 2

,

(see Fig. la); is the ground thermal diffusivity and t is the time. Jn is a Bessel function of order n.

(b) One half-plane maintained at zero temperature, the other having a sinusoidal temperature variation of amplitude ,,go and period P, the mean temperature being zero.

This problem can be solved by the Laplace transfonm method used in (a) above. The solution is

-8-

(ur ) du ,

-.

where w

=

2 f i / ~ , a n d P is the period.

(c) By superimposing the solutions of 6(a) and 6(b) in combination with equation (12), the situation is obtained which might represent a large area

- . - -- - -- - - -

*

These equations are now being evaluated at NRC with the help of an electronic computer.

being suddenly heated (or cooled) to a temperaturefi above the mean annual temperature of the external ground surface, while this surface temperature varies sinusoidally, approximately as in reality under the influence of the annual climate cycle.

( 7 ) Temperature under the strip area on the surface

By replacing y by ( ~ - a ) and (-y-a) in equation (9) and summing the two temperatures, the solution is obtained for a sudden change in temperature of 2-30 on the strip surface and$o on the remaining surface. To obtain the effect of changing only the strip temperature toQo, the external surface remaining at zero, subtract the temperature that would result if the entire surface were suddenly raised from zero The equation for this

(reference (3), p. 60) is

The same procedure can be used with equation (10) but, in this case, the temperature to be subtracted is that due to the entire surface temperature varying sinusoidally, i.e. (see reference (3) p. 651,

-z TT

P I =

Po

e

'K

sin

(~zL.

-

z

.

P

K T

When both the periodic solution and sudden change solution are added together, a temperature regime is obtained which might occur under a long basementless building which is suddenly heated or cooled while the remaining external ground surface temperature varies sinusoidally, following the annual climate cycle.

(8) Irregular areas

The method for determining temperatures under areas other than those already discussed is due to ~achenbruch(~), and is similar to that used iri section 5 in that areas are subdivided into circular sectors of angle 8 and radius

R.

(a) Sudden step-change in temperature from zero t o P o on the sector. The temperature under the apex at depth z and time t is

( b ) S i n u s o i d a l t e m p e r a t u r e v a r i a t i o n o f p e r i o d P and a m p l i t u d e P o on t h e s e c t o r , r e m a i n i n g s u r f a c e a t z e r o . For t h i s c a s e t h e t e m p e r a t u r e under t h e a p e x i s ( I t s h o u l d b e n o t e d h e r e t h a t t h e t r a n s i e n t t e r m i n e q u a t i o n ( 1 4 ) r e q u i r e s n u m e r i c a l i n t e g r a t i o n , which h a s n o t y e t been c a r r i e d o u t . F o r d e t a i l s t h e r e a d e r i s r e f e r r e d t o r e f e r e n c e ( 2 ) . ) By summing t h e t e m p e r a t u r e s due t o a l l s e c t o r s , t h e t e m p e r a t u r e u n d e r a n a r e a whose s u r f a c e v a r i e s s i n u s o i d a l l y i s o b t a i n e d . Normally, t h e problem of i n t e r e s t would be t h a t w h e r e t h e e x t e r n a l s u r f a c e t e m p e r t u r e v a r i e s s i n u s o i d a l l y , t h e

i n t e r n a l a r e a b e i n g a t c o n s t a n t t e m p e r a t u r e . For t h i s c o n d i t i o n i t i s o n l y n e c e s s a r y t o s u b t r a c t from t h e p r e v i o u s sum t h e t e m p e r a t u r e t h a t would r e s u l t

i f t h e e n t i r e s u r f a c e o f t h e ground had a s t e a d y s i n u s o i d a l t e m p e r a t u r e v a r i a t i o n o f a m p l i t u d e

po

and P. T h i s i s t h e s o l u t i o n f o r o n e - d i m e n s i o n a l h e a t f l o w i n t h e ground a l r e a d y g i v e n by e q u a t i o n ( 1 2 ) . ( c ) S t e a d y t e m p e r a t u r e o f

P o

on a s u r f a c e a r e a , t h e r e m a i n i n g s u r f a c e tempera- t u r e v a r y i n g s i n u s o i d a l l y w i t h p e r i o d P and a m p l i t u d e

_P

0'

The t e m p e r a t u r e , which i n t h i s c a s e m i g h t be due t o a b a s e m e n t l e s s b u i l d i n g a f t e r s e v e r a l y e a r s o p e r a t i o n a t c o n s t a n t t e m p e r a t u r e , i s o b t a i n e d by a d d i n g t h e s t e a d y t e m p e r a t u r e e f f e c t a s o b t a i n e d u s i n g s e c t i o n s ( 1 ) t o ( 5 ) t o t h a t o b t a i n e d from ( b ) above. When t h e e f f e c t o f t h e g e o t h e r m a l g r a d i e n t i s t h e n added t h e c o m p l e t e s o l u t i o n i s o b t a i n e d .

DISCUSSION AND CONCLUSION

The equa$;ions p r e s e n t e d h e r e c o v e r e s s e n t i a l l y a l l c a s e s of t e m p e r a t u r e i n t h e ground u n d e r a n d a b o u t f l a t s u r f a c e a r e a s whose t e m p e r a t u r e s d i f f e r from t h e t e m p e r a t u r e o f t h e r e m a i n i n g e n v i r o n m e n t . The g r a p h i c a l p r o c e d u r e h a s b e e n e n l a r g e d upon i n a r e p o r t ( 4 ) which g i v e s c h a r t s a n d g r a p h s f o r a p p l i e d u s e .

In using the equations and graphical methods it would be simplest, of course, if data for the annual ground surface temperature variations and the geothermal gradient at different locations in the orld were available without the necessity of direct measurement. Jen- hu-Chang (5y has recently compiled ground surface temperature data from about 750 different world locations; Birch (6) gives values for the geothermal gradient. As an average value GG can be expressed as

where k is the ground thermal conductivity in ~ t u / (hr) (OF) (ft).

If it is desired, the variation of ground thermal conductivity with temperature can be accounted for exactly in any of the foregoing equations by making the substitution.

where kl is the conductivity at the base temperature TI and u T

-

TI.

In the transient case o( must be assumed constant. The proof for the procedure

is given in reference ( 3 ) , page

6

.

A factor that should be considered in dealing with small surface areas is the influence of the heat transfer wfilmw coefficient h. It would not generally be feasible to account for this effect in detail but, for practical purposes, the film can be treated as an extra thickness of soil over the ground surface, i.e.,

Here h is the film coefficient over the internal area and k is again the conductivity of the ground. Since the external ground surface must also be raised by a depth! in order that the whole surface remain a plane, the amplitude of the external temperature cycle at this elevation should be altered in accord with the simple relation derived from equation (12), i.e.,

where is the true amplitude of temperature variation at the remote ground '

a

surface. With the help of equations (17) and (18) the ground surface tempera- ture in the neighbourhood of the area can be estimated by setting _z

₌

1

.

REFERENCES

(1) Brown, W.G. Steady temperature under the corner of a plate and under

polygonal areas on the surface of a semi-infinite solid. Quarterly Journal of Mechanics and Applied Mathematics (to appear).

(2) Lachenbruch, A.S. Three-deimensional heat conduction in permafrost beneath

heated buildings. Geological Survey Bull. No. 1052-B, U.S. Govlt. Printing Office, Washington, 1957.

(3)

Carslaw, H.S. and J.C. Jaeger. Conduction of heat in solids. 2nd edition,

Oxford, Clarendon Press 1959.

(4)

Brown, W.G. Graphical determination.of temperature under heated or

cooled areas on the ground surface. (To be submitted for publication). (5) Jen-hu-Chang. Ground temperature, Vols. I and 11. Blue Hill Meteoro- logical Observatory, Harvard University, 1958.

(6) Birch, F. The present state of geothermal investigations. Geophysics,

Vol. 19, No.

4,

October 1954, p. 645

-

659.

LIST OF SYMBOLS

G~

=

geothermal gradient.

h

=

air llf ilmtl heat transfer coefficient.

k

=

thermal conductivity of ground.

1

k/h, equivalent depth of soil.

P

=

Period of sine wave.

R

=

Radius of circular sector.

t

=

time.

T

=

temperature; T1 = mean annual temperature of external ground surface;

T2

=

surface temperature of area.

x,y,z

=

coordinates; x, y, in the horizontal direction, z the depth below

W

=

thermal diffusivity of the ground.

13

n tan

-

1 (z/~) in equations (9) and (10).

Q

_{= corner angle of a wedge-shaped surface area or circular sector.}

A0

=

amplitude of a sinusoidal temperature variation at the surface.

/cc = temperature caused by sinusoidal temperature variation of amplitude

ko

over a circular sector, external surface at zero temperature.

A' = temperature caused by sinusoidal temperature variation over entire ground surface, mean temperature of zero.

73

=

T

-

T in steady-state. 1

6

=

T 2

-

T in steady-state. 1

THE TEMPERATURE UNDER HEATED OR COOLED AREAS ON THE. GROUND SURFACE

by W.G. Brown

LIST OF FIGURE CAPTIONS

Figure 1 (a) The steady temperature distribution near the straight edge of a large area on the ground surface in the absence of a geothermal

gradient.

(b) Graphical superposition of a geothermal gradient on the tempera- ture regime near the straight edge of a large area.

(c) Superposition of two temperature-regimes for the region near the

edge of a large area t o _{obtain the temperature regime under a}

strip area of width 2a on the ground surface.

(dl Superposition of a geothermal gradient on the temperature regime under a strip area.

Figure 2 (a) Isotherms in steady heat flow at constant depth under the corner

of a large area on the ground surface. The plate at temperature

y o

and the external surface at zero.

(b) Method of superposition of the temperatures due to four corners to obtain the steady temperature under a rectangle.

(c) Superposition of four corners forming an intersection.

(dl Superposition of two rectangles to form a more complicated area. (e) Subdivision of an area into circular sectors for determination

of the steady temperature under the common apex.

(f) Subdivision of multiple areas into circular sectors for determin- ation of the steady temperature under the common apex.

DISCUSSION BY W .A. WOLFE;? T h i s i n t e r e s t i n g p a p e r shows how t h e i n g e n i o u s c o m b i n a t i o n o f e l e m e n t a r y s o l u t i o n s o f t h e h e a t c o n d u c t i o n e q u a t i o n c a n be combined t o p r o d u c e s o l u t i o n s f o r more c o m p l i c a t e d g e o m e t r i c a l s h a p e s . T h e r e i s l i t t l e o n e c a n s a y a b o u t t h e c o n t e n t o f t h e p a p e r e x c e p t t o n o t e t h a t t h e c o m p u t a t i o n s become i n c r e a s i n g l y t e d i o u s a n d l o n g a s t h e c o m p l e x i t y o f t h e p h y s i c a l s i t u a t i o n i n c r e a s e s . P e r h a p s t h e a u t h o r c o u l d b e p e r s u a d e d t o e x t e n d h i s e f f o r t s t o w a r d s t h e p r e p a r a t i o n o f c h a r t s o r t a b l e s , b a s e d on t h e i d e a of s u p e r p o s i t i o n of s o l u t i o n s f o r t h e more i n v o l v e d g e o m e t r i e s . AUTHOR

'

S REPLY M r . W o l f e ' s comments on t h e p a p e r a r e much a p p r e c i a t e d . As h e p o i n t s o u t , c o m p u t a t i o n becomes t e d i o u s f o r complex s i t u a t i o n s , and c h a r t s o r t a b l e s would b e u s e f u l . I n t h i s c o n n e c t i o n c o p i e s o f a n e n l a r g e d v e r s i o n o f F i g u r e 2a a r e a v a i l a b l e a t t h e D i v i s i o n of B u i l d i n g R e s e a r c h , N.R.C., w h i l e Lachenbruch

i n c l u d e s g r a p h s w i t h h i s p a p e r which a r e u s e f u l f o r b o t h s t e a d y - s t a t e a n d time- dependence. For t h o s e r e a d e r s who w i s h t o make d e t a i l e d c a l c u l a t i o n s , t h e a u t h o r h a s p r e p a r e d a Bendix G-15 computer program f o r s t e a d y t e m p e r a t u r e s u n d e r a r e a s o f a n y s h a p e ( B e n d i x U s e r s ' Program No. 698) and t h e same p r o c e d u r e h a s been e n l a r g e d upon t o i n c l u d e time-dependence i n a n I.B.M. 1620 program p r e p a r e d by L.M. Marks o f t h e Computation C e n t r e , N.R.C.

R e a c t o r R e s e a r c h a n d Development D i v i s i o n , Atomic Energy o f Canada L i m i t e d , Chalk R i v e r .