Transient temperatures in slabs heated or cooled on one side

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Transient temperatures in slabs heated or cooled on one side

BY

T .

2.

HARiTaATHY nncl

j .

A. C . Blanchurcl

liepritzted fi.07tl

T h c Canadian Journal of Chcrnical Engineering 41: 128: June, 1963

X Publication of T h c Chemical Institute of Canada

48 liidcau Strcet, O t t a w a 2, Ontario liescarch Papcr No. 183

of the

Slabs Heated

'

1

ransient

'

1

'emperatures

or Cooled on One Side'

T. 2. HARiMATHY2 and

J .

A. C. B L A N C H A R D 2

Th.

e telnperature history of an infinite slab, one face of which 1s suddenly csposed to a fluid of temperaturc T , while thc other continues to be in contact with a fluid of tcmpcrature

?:.

may be of intcrcst in scvcral fields of cnginecrlng.

T h e mathematical

presentation

of thc problem is as follows:

form of the second of tlic above cxpressions is given in Table I*. Besidcs tlie gcncral solution, those pertaining to special boundary conditions and small vnlucs of iV,,,, as wcll as equations describing thc steady-state tcmperaturc distribution and defining tlic y,, valucs arc also tabulated.

T h e Gurncy-Lurie typc plots of the solution a r c givcn in Figures 1 to 6* for all combinations of (iV.\,,)i = 0.4, 1, 2, 4, 10, m,

(N,).,,),,,

= 0, 0.4, 1, 2, 4, 10, m, a n d s / L = 0, 0.2, 0.4, 0.6, 0.8, 1. Interpolation between the curves can best be done on ,,emi-lbgarithm paper. If higher accuracy is rcquired, it is advisable to calculate a few points of the curves, making use of the solution for small values of

N,,,,,

and of tlie fact that

if N,:, 2 _{0.2, the error resulting from ncglecting all higher} terms of the infinite serics is

generally less than t w o per cent:

T o facilitate such calculations, a chart is given in Figure 7* frorn which y l can bc tlcterrnined with a two-figure accuracy for any cornbination of (N,\.,,,) i and (N,\ ,,)

..

a

T

Ji,(T,

-

T )

+

k -

_as

= 0 a t r = 0 . .

. . .

( 2 )

If the slab is heated, 7',

>

7:,; if it is cooled, T,

<

To.

N o m e n c l a t u r e Although the analytical solution of tlic problcrn has been

lcno\vn for sornc years('#?), the labor involved in oilraining numerical values is so significant, that the mere avai1:ability of the analytical solution presents little hclp to the design

engineer.

Fortunately, tabulated valnes of thc

1 = coclficicnt of I~catTtransfcr, B.t.u./(hr.)(sq. ft.)("F.) k = thermal conciuctiviiy, B.t.u./(hr.)(sq. ft.)("F./fi.) L = thickness of slab, ft.

iVp, = al/L2, Fourier number iV.\,,, = JlL/k, Nussclt number 1 = t i n s , hr.

1' = iemperaturc, O F .

.v = clistar~ce, ft.

a = thermal clilfusiviiy, sq. ft./hr.

y = positive root of Ecl~~ation

_-

( 7 ) in Table 1'

r m

erfcx = 1 - erfr = - e-E2 d l , in Tablc 1'

4s

'

J

%

function for many conibinations of (N,,,),, (N,,),), and s,lI, arc also availablc('). This form of the solution is, howevcr, not well suited for graphical rcprcsentation, nor are its tabulatcd valucs for interpolation.

Subscripts

i = a t tlic surlace .v = 0, or (in conncciion with T ) a t x<< 0

o = a t the surface r = L, or (in conricciion \\,iih T) a t x

>>

L

?L = 1 , 2 , 3 , . . . m m = when NI;, + m By rearranging the solution into thc form

R e f e r e n c e s

(1) Thorne, C. J. Temperature Tables, Part I. One-layer, One- space \Jariable, Linear. U.S. Naval Ordnance T c s t Station. Rcport No. 5.762, Part I, 1957.

( 2 ) Hill, P. R . A A4ethod of Computi~lg the Transient Temper- aturc of Thiclc Walls. N A C A Technical N o t e 4105, 'CVash- ington, 1957.

( 3 ) Gurney, H. P . and J. Lurie. Ind. Eng. Chern., 15, 1170 (1923).

a function is obtained which is equal to 1 w h c ~ l

N,,,

= 0 for any value of (N,\,,):, ( N , , , ) , and s / L , and thus can be represented ' on scrni-logaritlirn paper by familics of curves similar to those obtaincd by Gurney and I,~~ric(:~). In fact, thc cascs tliscussed by Gnrney and Lurie can bc regarded as such special cascs in which

7'-

=

Ti

thrcogliout the solid. In tlic present case thc steady-statc terupcraturc is not, as a rulc, constant, but a function of x/I,.

T h e solution of Equations (1) to (4), rearranged into the

"Table 1 and Figures 1-7 of this paper have been deposited .m Document

No. 7544 wit11 the AD1 Auxiliary Publication Project, Photoduplication Service, Library of Congress, Washington 25, D.C. A copy be obtnined by citing the Document No. and by remitting $8.75 for photoprints, or

$3.00 for 35 nnn. microfilm. Advance paylnent is rcquired. Make cheques

payable to: Chief, Photoduplication Service, Libmry of Congress.

* * *

lMnntrscril>t received June 11, 1962; occeptecl Noveniber 8, 1962. ?Division of Blrilding Resenrcl~, Nntionnl Reseorcl~ Cmlncil, Ottown