Publisher’s version / Version de l'éditeur:
Quarterly Journal of Mechanics and Applied Mathematics, 15, 4, pp. 393-399,
1962-11-01
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Steady temperature under the corner of a plate and under polygonal
areas on the surface of a semi-infinite solid
Brown, W. G.
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TH1
N21r 2
no.
174
c. 2
NATIONAL
RESEARCH
COUNCIL
CANADA
DIVISION O F BUILDING RESEARCH
STEADY TEMPERATURE UNDER T H E C O R N E R OF A PLATE
A N D UNDER POLYGONAL A R E A S O N THE S U R F A C E
O F A S E M I
-
INFINITE S O L I D
BY
W. G. BROWN
R E P R I N T E D FROM
QUARTERLY JOURNAL OF MECHANICS AND APPLIED MATHEMATICS VOL. XV. PART 4. NOVEMBER 1962. P. 3 9 3
-
3 9 9R E S E A R C H P A P E R NO. 174
OF T H E
DIVISION O F BUILDING RESEARCH
OTTAWA
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STEADY TEMPERATURE UNDER THE CORNER
OF
A PLATE AND UNDER POLYGONAL AREAS ON
TI-IE SURFACE OF A SEMI-INFINITE SOLID
B y W. G. BROWN ( D i v i s i o ? ~ of Building Research, National Research Council, Ottazucc, Ca?zccdcc)
[Received 28 J u n e 1961. Rcvise received 3 January 19G2]
The steady temperature regime under a wedge-shapecl plate on a semi-infinite solid is obtained b y t h e substitution of coordinates which reduce the Laplace equation to the two-dimensional form. The tenlperature regime under polygonal areas then follo~vs clirectly fronl superposition, which can be carried out graphi- cally.
1. Introduction
,4 S I M P L E and iilgeilious method of solution for a large class of steady and transient heat conclnction problems in a semi-infinite solid has recently been developecl by Lachenbrnch ( 1 ) malring use in part of grapl-Lical means used previously by Birch (2). The class coverecl is essentially that of a surface area of arbitrary shape on which either a step change in temperature occurs or where the whole surface temperature varies sinusoidally with time but the mean temperature ancl ainplitucle of the variation are different within ancl outsicle the given area. Lachenbruch's method consists of subclividing the surface area into sections of a circular annulus, for xvhich an exact solution for the temperature under the apex exists, then summing the effects of all area elements. Among the great variety of problems that can be approached in this way are: the location of perma-frost under lakes and rivers in the far north; freezing and thawing of roads and aircraft runways; temperature conditions under cities and city street intersections; and heating and cooling of basement- less buildings. The method is icleally suitecl to determiniilg the tempera- ture a t specific points but cloes not leilcl itself readily to the lnappiilg of isotherms. For the special case of steady temperature and fairly simple polygonal areas the isotherms can be mapped readily b y the method outlined below. The solution for the temperature distributioil uncler the corner of a wedge-shaped plate mill be clerivecl first: the temperature regime under polygonal areas then follo~vs from super- position.
[Quart. Journ. Mech. and Applied Math., Vol. XV, Pt. 4,19621
5092.60 Dd
2. Temperature regime under a corner
With the apex of a wedge-shaped plate of corner angle 0 on the surface of a semi-infinite solid as origin the Laplace equation for temperature v
in rectangular coordiilates is
~vhere z is depth, x is distance along t h e plate centre-line and y is perpendicular thereto. If r2 = x2 +y2 +z2, the substitutions
transform (1) into
With the plate surface temperature constant a t v = v, and t h e remaining surface a t v = 0, the teinperature a t ally point in t h e solid can be expressed in functional form as
v/vo = y(x/z,
YIZ)
(4)from which it is seen that the temperature remains constant in the radial direction. It may be noted t h a t
5
and r ] are themselves functions of x/z and y/z; hence av/ar = 0, and equation ( 3 ) becomes:The boundary temperatures in the two-dimensional case of equation (5)
corresponding t o the surface temperature conditioils for the plate are, (from (2))
151
<
tan(O/4), r ] = 0, v = V,It1
>
tan(0/4), r ] = 0, v = 0The solution to this two-dimensional problem is elementary, isotlierins being co-axial circles passing through t h e points (-tan(0/4), 0) a n d (tan(0/4), O), (Fig. 1). The equation relating temperature t o
5
and q isAfter expanding the first tern1 in (7) the temperature is obtained in explicit form, i.e.
Tv/vo = cot-I where cc. = tan(0/4).
STEADY TE$IPERATUl%li: UNDER THE CORKER O F il PLA'I'E 395
FIG. 1. Steady temperature in two dimensions (infinite strip on t h e surfacc
of a semi-infinite solid).
FIG. 2. Steady temperature tlistribution undcr t h e corner of a flat plate ( t e m ~ ~ e r a t u r e v,,), on the s ~ ~ r f a c e of the semi-infinite solid (temperature 0).
Purthermore by substituting for and 17 equation ( 7 ) becoines
By expanding terms, separating out { ( x / ~ ) ~ +(y/z)2
-t
l}* and squaring t h e resultant equation i t will be found that equation (9) talies the hyperbolic form i.e.It is thus a simple matter to deternliile the loci of points x/z and y/z lying on any isotherm. As an example, t h e temperature distribution for the important case of a corner of angle n/2 is given in Pig. 2.
3. Polygonal areas
The illost elemeiltary s~zperposition of t h e above results is t h a t of two plates of angle n/2 arranged as in Pig. 3a with their apices in contact. From physical symmetry the temperature vertically under the apex will be 0 . 5 ~ ~ . This substantiates the derivations given above for, by super- position, the temperature under the apex is 2 x 0 . 2 5 ~ ~ = 0 . 5 ~ ~ . Sinlilarly the temperature under the apex of two plates of angle n-12 lying side by side (Fig. 3b) is also 0 . 5 ~ ~ as for the case of a single llalf plane. In addition, the temperature far reinoved froin the apices along t h e line of contact tends to v = vo as required.
The necessary superposition for obtaining the temperature under a rectangle is shown in Big. 3c, where three distinct operations are to be carried out: first, the rectangle is formed from two plates of angle n/2; then the two external corners are 'filled out' by two inore plates of angle ~ / 2 . Tor a triangular area two superpositions would be necessary. Big. 3d shows the sinlple situation that would occur a t a n intersection such as a city street. Big. 3e is an irregular polygon fornled by t h e superposition of two rectangles and a triangle.
It will be realized, that except where only a few specific temperatures are to be evaluated, the method of superposition call be carried out most readily by graphical means. The author has done this for a rectangle, from which it was found t h a t the method proceeds rapidly, although considerable care is required in plotting the basic isothernls of Big. 2. The temperatures were found to agree with those calculated from t h e
STEADY TEMPERATURE UNDER T H E CORNER O F A PLATE 397
Ce)
FIG. 3. Exainples of polygonal areas on the semi-infinite solicl ~vhose
telnperature clistributions can be obtained by superposition of solutions for the corner of a plate.
following explicit equatioi~ given by Lachenbruch for the steady tempera- tare under the rectangle ( -nz
<
x
<
+ m ) , ( -n G y<
+n) (centre as origin),z(z2+ (x+ nz)" ( y + 1 ~ ) ~ ) ~
I-
- tan-' (x--m)(y+?z)
z { 2 + ( ~ - r n ) ~ + ( ~ + n ) ~ } ~
I-
Here 2112 and 2n are the lengths of sides, and the coordinates z and y are different from those used in this paper. The simplest comparison of
398 STr. G. BlZOWN
equation (11) with the result of superpositioil of eclnatioil (8) is obtained b y evaluating the temperature a t the center of a square for a clel~th of
z = 9.12 = n. For this condition equation (11) becoilles
'CVhen y = 0 , (8) reduces t o
Wit11 z = 9 n d 2 , z = nt, 91 = d3-d2 a n d a -. tan(r/S) = 2/2-1, this becomes
vo
=
ancl represents the contributioii to temperature due t o one internal corner. Similarly, with n: = - 9 4 2 t h e contributioli of a n external corner (see Pig. 3c) is
For all four corners the result is
which is the same result as obtainecl with equation (11).
4. Conclusion
It has been shown t h a t t b e three-dimensional steacly temperature regime under a n y polygonal areas on tlle semi-infinite solicl can be deternliiled directly by superposition of t h e elementary solutions for t h e wedge-shapecl plate. The method used here applies also t o a n y conical solid for which a11 boundary temperatures reinnin constant i11 the raclial direction, for exainple, a rectangular corner having t h e three faces a t temperature v,, vl, v2. For this class of problem we require t o have only the solution for t h e two-dimensional temperature distribution in t h e plane.
STEADY TEMPERATURE UNDER T H E CORNER O F A PLATE 399 Acknowledgements
The author is indebted t o Professor J. C. Jaeger of the Australian National University, ancl to Dr. A. H. Lachenbruch of t h e U.S. Geological Survey for their comments. The useful suggestioils of the referees are also gratefully acknowledgecl. This paper is a contribution from the Division of Building Research of the Natioilal lZesearcli Council of Callads a n d is published with t h e approval of t h e Director of the Division.
R E F E R E N C E S
1. A. H. L A C ~ ~ E X B R U ~ H , 'Three-cli~llensional heat conduction in pornlafrost beneath heated buildings', Geological S t o v e y Bulletin 1052-B. (U.S. Cov't printing office. TTTashington, D.C., 1957).
2 . F. BIRCH, 'Flow of heat in t h e Front Rangc', Colorccclo. Geol. Soc. America
B t ~ l l e t i n 61 (1950) 567-630.
3. I<. I ~ O P P , Elements of the t h e o ~ y of functio?zs, Chapter 111, 1st cdn. (Dovcr, h-c.rv Yorlr, 1952).
