Ztschr. t. nnjiea.. 158 (; o 1 d s t e i 11 , Calciilation of the Siirfact Teinperatiire of Simple Bodies Math. ,and M~&.

On the Calculation of the Surface Temperature of Geometrically Simple Bodies. By Syrluiey Goldstein in Cambridge (England).

Beredmung der Oberflachentemperatur geometrisch einfacher Korper. Zusammenfassung. Der Verfasser zeigt, dafj die von G. P i i s c h l mit Hilfe von Reihen-

entwicklungen gewonnenen Zahlenwerte fur die Obeiflachentemperatur bei Beginn der Ab- kiihlung einer Platte, eines Zylinders oder einer Kugel (G. Po s cli 1 , Rerechnung der OberflBchentemperatur geoiiietriscli einfacher Korper usw., diese Zeitschrift, Bd. 12, 1932, S. 280) in einfaclier Weise mi t Hilfe der H e R v i s i t l c - Met.hode gewonnen werden kfinneii (sielie G o 1 (1s t e i 11 , The application of H e a v i s i d e s operational method etc., cliese Zeit- sclirift, Bd. 12, 1'338, S. 234). Fur den betracliteten Fall, dab die Kikper, tlie die Anfangs- teriiperatur 1 haben sollen. sich in einer Umgebung mit der konstanten 'I'enipcratur 0 befindcn, ergeben sich fur den Beginn des iibkiihlungsvorganges die einfachen Nii.lierungsfor~ncln, fiir die Oberflaclienteiriperatur it.,, fiir die Platte: B , = f , (I)), fiir den Zylinder: ('3, = f , (11)

+ ',;, f, ( I ? ) , fur die Kugel : B, = f , ( 1 1 ) + 7- f ; ( ' I ) . 1)arin ist 'I eine Variable, die init 11 t ( t = &it) proportional ist, j , eine Konstante (13edeutung siehe unten), und es ist

1 1 d

Zum Vergleich mit den Zdilenwerten von Piis c 11 1 sind drei Tabellen berechnet, die den 'I'empcraturverlauf fiir Ebene, Zylinder und Kugel wiedergeben.

1. Introduction. In the Z. f. angew. Math. u. Mech. for October. 1032 (vol. 12, pp. 2tU3-287I Herr Ilr. G. P i i s c h l calculates the surface temperature of (1) an infinite slab, (2) an infinitely long circular cylinder and (3) a sphere, where, in each case, the hody is at a given constant temperature at time t = O and is placecl in a medium a t another given, lower, constant temperature, into which radiation takes place from its surface. If 2 1 is the thickness of the slab, R the radius of the cylinder or sphere, and IG the thermometric conductivity, Herr P b s c h l ' s calculations relate to the case in wliich c c t / l z or nt /R2 is small. In such cases the infinite series given by the usual solution converges very slowly, and numerical computation beconies extraordinarily laborious, with a correspondingly large chance of error. Thus Herr P i i sc l i l states that lie sometimes liad to calculate more than 30 terrns of the series; and each term itself requires a fair aniount of calculation. The object of this paper is to show that in sucli cases calculation from the series is unnecessary, the results being very simply obtainable otherwise. The required formulae are most easily deduced by nieans of H e a v i - s i tl e's operiitional method. This method was surnrnarised and illustrated in a paper i n the %. f . angew. Math. u. Mecli. for August, 1932 (vol. 12, pi). 234-213). The problems of the infinite slab, infinite cylinder, and cylinder of linite length were considered, and it was shown Iiow, from the operational solution, formulae suitable for computation for small values of a t / P or at / R2 could be found. In that paper I was concerned with illustrating the method, rilthcr than with tlie siniplification and numerical evaluation of the results; this paper may be considered, tlicn, as a continuation of that one, suggested by Herr P o s c h 1's work.

In what follows we sliall suppose, as we tnay without loss of generality, that the initial teiuperature 'of the body is unity and that of the mediuni zero. We use -9 to clenote temperature, 6, tlie surface temperature, a tlie therinometric coritluctivity, a i i d h the relative emissivity. We also put

p I = c i t / Z 2 or c l t / R 2 )

j , = h I . . . . . . . .

11 = p' j , z h (d)' and

The symbols p, j , , 11 are not introduced by P i i s c l i l , who uses X instead of 1 , and 6)o/f9c instead of 8,. The other symbols have the same meaning here as i n Pbschl's paper.

We are concerned with computation when p is small.

159

2. The infinite slab. The formula suitable for the calculation of B when ,b is small was given in the previous paper (eq. (26), p. 238). To find the surface temperature, we must put z - Z or 2 - - 1 in that formula.

Band 14, Heft 3 Jiiiii 1934 ( t o 1 d s t e i 1 1 . Cnlculatiori of tlw 8irrf:wP Temperature of Simple Bodiefi

The result is

~ - - - -- ---

10- I - 20 ~ 50 1 100 200 - - - ---

0,l 1 O,B

0,998 0,990 I 0.981 i 0,926 0,831 i 0,704 1 0,467 I 0,2m 1 0,157

where

500-11000 1 2000

0,065 I 0,033 I 0,016

. . . . . (:3) ,

and is the ,,Felileriotegral" of Jn l i n k e and l311ide. For large valucs of s

1

and since we are concerned with cases in which P - 5 is large, we may take

The smallest value of ,!I-: with which we shall be concerned is 10 (for smaller values, calculation from the infinite series is fairly easy) and it is clear that to obtain results to three or four places of decimals, we may neglect the second term in { in (2) with an abundant margin of safety. We have then tlie simple formula

190=e72(1-erf17). . . . , . . .. . . . . . . . . (41.

For sinall values of 11, this gives

Tor large values of 11 it gives

and for other values of ? I , 8, is very easily found from the tables'). I have prepared the following table for comparison with Herr P d s c h 1's Table I (see p. 285 of Iris paper); but i t should be stressed that the correct way to prcsent numerical results would be to tabulate 19, against tlie single variable q . The calculation from tables of the exponential and error fuiictions is so simple that it hardly seems worth doing.

T a b l e 1. 8, for a n i i i f i n i t e s l a b .

0,0010 0,0025 0,0050 0,0100 -

1 ) Herr P o s c h l refers to certain Darmstadt dissertations i i i which the error function ccciirs. 1 h a w had no oplmrtuiiity of coiisiilliiig thrse Inriiioirs.

Ztschr. f .angew. Math. ,,,,d Mecl,. 160 (i o 1 tl s t e i n , Calculation of the Surfaw Temperature of Simple Bodies

3. The infinite circular cylinder. A formula for the cylinder, suitable for tlie calculatioii If we put of A when f i is small, will he found on p. 2-10 ol tlie previous paper (cq. (30)).

r = R in that formula, we find

but in deriving the formula, it was assutned that y was not large, so that when p is small, 31 is also sinall. (The assrimption occurs in equation (24) on p. 2.10, where not only 1 /(q R), but also h / q , is assumed small.)

I f we return to tlie operational snlution in eq. ($)) on 1). 23s of tlie previous paper, we can easily tind a formula applicable for all v:ilues of 7 , so long as 1; is small. I n equation (!I), 1). 238, we put r = R, and then use tlic asymptotic expatisions of l , (qR) and I, (qR) , iintl ~~roceed as below. We find

' . . ' (7).

We must now interpret. and therefore

If 1' is tlie usual H e i t v i s id e operator, q was defined by 1' = ( 1 q?,

(cp J e f f r e y s , Operational mctliods in matliematical l)liysics, (1931), p. W). Wlieri we divide by Ir, and differentiate with respect to 17, we find

(!)).

Substitute from (8) arid (9) into (7), and use (1). This gives

where

G o 1 d s t e i n , Calculation of the Surface Temperature of Simple Bodies 161 Band 14. Heft 3 Jui i i 1934

0,998 I 0,!)90 I 0,981 1 0.926 1 0,830 I 0,702 1 0,465 I 0,285 1 0,155 I 0,064 1 0,032 0,049 0,025 0,036 ~ 0,017

Equations (10) and (11) give tlie required formulae. The first term in (10) is the result found for an infinite slab in eq. (4). The second term in (10) is sufficiently small throughout the whole of the range for 7 and y to justify the omission of the following terms in (7).

0,016 0.012 0,009

For small values of 11,

f, (7) = -

and, for large values,

. (12)

(13).

l'lie following table is inserted for comparison wit11 Herr P6schl's Table 2 (see p. 2% of his paper); but it niust again be empliasised that the best way to present numerical results would be to tabulate f, ( y ~ ) and f 2 (11) against 1 1 .

0,0003 0,0005 0,0010

o , m o 0,0025

4. The sphere. The operational solution for a sphere was not given in the previous paper, but is easily obtained by the usual method. If 1' is distance froin the centre, the temperature B satisifes the equations

. . . . . . . . . (l.t),

B = 1, when t = 0 * (151, . . . . . . . . . . . . .

a s ~ + h 8=0, when r = R . . . . . . . . . . . (16). Br

If 1) = 1:. q= (171, . . . . . . . . . . . . . . . . .

the auxiliary equation is

Since -9. must remain finite when r vanishes, tlie required solution is

sinh q r B = l + A - - - , r

and, from (IG), A ( q R cosli q R - sinh q R + h H sinh q R) = - h R2,

so that sin11 q r

r . . h R' y R cosli q R -- sinh q R + h R sin11 q R

1 9 = 1 - . - ___ (19).

11

162 ~ ~ ~ . ' ~ & ~ ~ M " , " , " h :

The surface temperature is found by putting r = R, and a formula suitable for small values of d / R 2 is found by expanding on the assumption that q R is large, and then interpreting term by term.

G o 1 d s t e i n , Calculation of the Surface Teniperaturc of Simple Bodies

Hence we find

-- q R cosli q R + ( h Iz - I) sinli q R

h R(1 ~ c - - " r i H )

q R+ hR-1 +e-"R(q R - h R f 1) - - I - - -

h R = 1 - -- - (sufficiently accurately)

-- - ~-

031 0,5 1 4 10 1 20 ' 50 100 200 ROO I 1000

0,0003 0,998 0,990 0,981 0,925 0,829 ' 0,700 ' 0,462 0,283 0,153 0,063 0,032 0,0005 0,997 0,987 0,975 0,906 0,787 ' 0,638 ' 0,391 0,227 0,119 0,049 0,024 0,0010 0,996 0,982 0,964 0.869 0,717 0,54.5 ' 0,300 0,164 0,084 0,034 0,017 0,0025 0,994 0,971 0,944 0,801 0,603 0,413 0,199 0,103 9,0.52 0,081 0,011) 0,0050 0,9% ~ 0,959 0,920 0,733 0,503 , 0,316 1 0,140 , 0,071 0,035 0,014 0,007

Comparison with (7) and its interpretation in (lo), sh

2000

0,016 0,012 0,008 0,005 0,004

WY that for a sphere we may tak

(20).

!

where f , (q ) and f 2 ( ) i ) are given by (ll), and y and 11 are defined in (1). To this order of approximation, the results for the slab, cylinder and sphere are in arithmetical progression.

The following table is inserted for comparison with Herr P 6 s c h l ' s Table 3 (see p. 387 of his paper).

T a b l e 3. 6, f o r a s p h e r e .

Summary. If a is the thermometric conductivity and h the relative emissivity, simple formulae are given for the surface temperature, 6,, of (1) an infinite slab of thickness 2 Z(2) an infinite cylinder of radius R (3) a sphere of radius R, which, a t time t = O , is i t unit temperature and is placed in a medium a t temperature zero, into which radiation takes place from the surface. The formulae apply when atlP or at/Ra is small, when the usual solution as an infinite series converges very slo\vly. The formulae are

6, = f, (?I) for the slab,

1 = f, (q) + 2; f, ( ) I ) for the cylinder, /

1 Y

= f, ( ' I ) + - - f, ('1) for the sphere,

where f, (q) and f 2 (q) are given in equation (ll), erf being defined in equation (3), and y and q in equation (1). Some numerical values are given for comparison with values calculated froni the series by Herr Dr. G . P o s c h l . 336