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Masters Theses Student Theses and Dissertations

1961

Temperature distribution in an infinite plate as a function of the Temperature distribution in an infinite plate as a function of the

input heat flux input heat flux

George Franklin Wright

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TEMPERATURE DISTRIBUTION IN AN INFINITE PLATE

AS .A FUNCTION OF THE INPUT HEAT FLUX

BY

GEORGE FRANKLIN WRIGHI'

A

THESIS

submitted. to the faculty of the

�OOL OF. MINES AND METALLURGY OF THE UNIVERSITY OF MIS90URI

in partial fulfillment of the work required for the

Degree of

MASTER OF S::IENCE IN MECHAIICAL ERGIIEERIIG

Rolla, Missouri

1961

.- "··,

���· ·�� �cu�

ACKNOWLEDGEMENT

The author wishes to thank Professor Aaron J. Miles and Profess or

Ralph E. Lee and the Computer Center Staff for their assistance while

making this investigation.

ii

iii

TABLE OF CONTENTS

Page

Acknowledgement • • • • • • • • • • • • • • • • • ii

Tables • • • • • • • • • • • • • • • • • • • • • • • • • • • iY

Plates • • • • • • • • • • • • • • • • • • • • v

Figures • • • • • • • • • • • • • • • • Ti

Introduction • • • • • • • • • • • • • • • 1

Review of Literature • • • • • • • • • • • • • • • 3

Discussion • • • • • • • • • • • • • • • • • • • • • s Example Problem • • • • • • • • • • • • • 15

Conclusions • • • • • • • • • • • • • • 20

Summary • • • • • • • • • • • • • • • • 24

Vita • • • • • • • • • • • • • • • • • • • • • 25

Bibliography • • • • • • • • • • • • • • • • • • • 26

Table I

TABLES

Center Temperature in an Infinite Plate of .Stainless Steel Subjected to an Exponentially Decreasing Heat Flux

iv

Page

20

Plate 1

PLATES

Center Temperature in an Infinite Plate of stainless Steel Subjected to an Exponentially Decreasing Heat Flux

v

Page

21

Figure 1

FIGURES

Computer Program for Royal-McBee LGP-30 Digital Computer for Evaluation of Equation 21.

Yi

Page

19

INTRODUCTION

It is the purpose of this thesis to investigate the effects of a

variable input heat flux impinging equally on both sides, on the tem­

perature distribution within an infinite plate. An infinite plate

may be defined as one in which the dimensions are infinite in two

directions and finite in thickne••· For practical purposes such a

plate could oe thought of as one in which the thickness was small

enough, compared to the other two dimensions, so that a temperature

gradient exists only normal to the surfaces of the plate. This

assumption will be made to establish one-dimensional heat flow within

the plate. Variable input heat flux means that the heat input may be

expressed. as.some function of tillle; in this particular problem the

function in question will be an expenential function of time.

The subject in question could be of importance in the field of

space technology as a body in space is receiving heat and emitting

heat in an enYironment where temperatures haYe little aeaniag, due to

the absenc� of aatter. Therefore it would be valuable to ha.Ye a

method of predicting temperature distributions in a body as a function

of heat input alone.

The study of temperature distributions in plates is important due

to the stresses set up by changes in temperature and rates of chanqe

in temperature. The solutions to this type of problem at present are,

for the most part, ba•ed on variable surface temperatures rather than

oa the premise of the Yariable external heat flux itself.

2

In-conclus ion, then, the problem to be investigated in this thesis

is to determine the temperature distribution with1n an infinite plate

as a function of time and the variable external heat flux. The mode of

heat transfer to the plate, and the variation in surface temperature asa

boundary condition are not required as arguments in this development.

REVIEW OF LITERATURE

The study of temperature variations in an infinite plate as a

function of time has been undertaken by many investigators during the

past years. Much of this work is recorded in the standard texts on

heat transfer. The method of approach differs widely from one investi-

gation to another, however the assumptions made and the resu,lts obtained

are similar.

The strictly analytical approach to this problem may be found in

the more advanced texts; notable among these are the works of H. S.

Carslaw (l)* and also H. S. Carslaw and J. C. Jaeger (2). These

investigators have undertaken their analysis in much the same way • . .

A somewhat simplified, but completely valid, approach to the

problem may be found in the works of Schneider (3), Kern (4), and Jakob

(5). The material presented by these authors also represents the ana-

lytical approach but in'simpler form with many of their results pre·

sented in the form of charts and graphs.

McAdams (6} presents considerable material dealing with the

numerical solutions to problems involving transient heat conduction

in an infinite plate. For the most part this material represents a

collection of the experimental and analytical results of other inYes-

tigators.

Some work regarding this study has been presented by Brown and

Marco (7) in an elementary text on heat transfer. The method of

* All typed numbers refer to references in the bibliography.

solution presented by them eaploys.dimensional analysis, supplemented

by experimental and other analytical processes.

The effects of temperature variations as applied to the problem

of thermal stresses have been investigated and are presented in several

works. Notable among these investigators is B. E. Gatewood (8).

It is interesting to note that in all of the material presented in

this review of literature the approach to the problem in question has

been to develop an expression for the temperature distribution in an

infinite plate fro• the standpoint of a given or assumed surface tem­

perature argument. The autaor cf this thesis was unable to locate any

work using the approach of this investigation, namely, that of deter­

aining an expression for the temperature distribution in an infinite

plate as a function of tiae and a variable external heat flux.

DISCUSSION

Determining the temperature at any point within an infinite plate

at any time as a function of the applied heat flux required a

knowledge of the temperature behaviour of a solid as it undergoes

transient heat conduction as a result of this applied heat flux. It

was ass\lllled that the solid under consideration in this paper is

homogeneous and that the applied heat flux is distributed uniformly

over the surface of the infinite plate. Under the conditions stated,

the heat flow will be one-dimensional and normal to the surfaces.

The flow of heat in a solid takes place by conduction and is a

function of time, temperature gradient, and the physical properties

of the solid. A general conduction equation based on these parameters

was then necessary to determine·the effect of the applied heat flux.

The generally accepted differential equation describing the

phenomenon of conduction heat transfer in solids are the various forms

of the Fourier Field Equations (9):

(I)

where�� is the temperature qradient or the rate of change in tem­

perature with respect to the distance X .) 0( is a physical property of

the material called the thermal diffusivity, and 9will be used as

the symbol for tiae, and the Fourier Conduction Equation (10):

at i=-KAdX (Z)

6

where q is the amount of heat flowi'nq per unit time, A is the area

normal to the direction of the flow of heat and will be considered

unity for the purposes of this iaYestiqation, and� is a proportion-

ality factor called the ther11a.l conductivity.

In order to determine tae temperature of a body undergoing

transient, one-dimensional heat conduction, as a function of time,

positiQn, initial conditions, and boundary conditions, the solution

of eq11atioa I must be ebtained.

K?lny particular solutions are aYailable in the standard texts

dealiaq with heat transfer and from other sources; however the boUBdary

conditions goYerning these solutions are all expressed in teras of a

function of surrounding temperature or surface temperature.

Ia this· inYestiga.tion the solu.tion of equation I was undertaken

using a Tariable external heat flux applied to.the sll.rface of the solid

as a. boundary condi tioa. The function which was used in this iaTeati-

gation expressed the heat flllX at the surface as an exponential f11aetion

of tiae:

(3)

It JDAY be noted that in this particular function the heat flux, q,

deer••••• exponentially froa its initial Talue, q0, at time zero, ••

the time, i!3 , increases. The constant, c, in tae exponent determine•

the rate at which the heat flux deereasea.

In order to solYe the differential equation in qllestioa,

7

with the equation

as one of the boundary conditions it was necessary to determine other

initial and boundary conditions.

A sketch of the solid under consideration ·in this discussion was

in.order at this j11ncture, as an aid in presenting the required initial

and:hound•ry conditions.

-L �L

-c• �.: fJotZ

The first boundary condition ca.me as a result of the combination

of fquations Z and 3 to yield

-cs -3a�

In order to simplify the calculations involved, an initial uniform

temperature of 0°F i• ns\llDed although any u.nifora t•perature 1110uld be

acceptable. The temperature level resulting from this investigation

will represent a temperature difference between the actual temperature

of the point in questioa and the initial uniform temperature level of

the body-. · This condition of zero initial temperature le'Yel throuqhout

the body sets the following initial coaditiont

x::: x e = o . J

The second boundary condition comes from an examination of the

sketch of the body in question. It .. Y be aoted that th.e same heat

a

flux is being applied to the body from either side. This, coupled with

the fa·ct that the • terial is considered to be hoaogeneous with constaat

and unifora physical propertie•, would indicate that the center plane of

the body coald be considered a.n adiabatic wall. This would give rise to

the second boundary conditioa:

at ('x.,e) ax - 0 ,,..,hen x= o..., e= e .

Taking eqltatioa Z and writing it to. fit the problem, at the

boundary, in this discussion results in:

o('L e) =. -K rat(�e)l o J LJx Jx==L

Combining the resultiag eq11•tion with equation� gi'Yes

(�)

At this juncture it was decided to use the Laplace transform

aethod of solution to aolTe equation I under the stated iaitial and

boundary conditioaa.

Rewriting I and takiaq the Laplace Tranafo I1ll vi th respect to e :

Working with the left hand side of equation O (11):

J

where

Working with the right hand side of equation S (12):

= -t (x.1 0)+ P£. [tf . ,

where ·Pis defined by the equation:

�[ffu!] =_f �-P/'cW du

Combining eq\lAtions 8 and 6 gives

a�i oc -. = - t /x o) + Pt ax-

V;.,

Making uae of the fir at lni tial condition, 't (X-' O):: 0,

results in the following:

9

(S.)

(�)

(7)

(8)

(�)

or

(lo)

Applying th.e operator method of solution to equation /0 (13):

ThllS

_ , 11,x , -/Ix c=C,€ +se (11)

or (14):

i= C, cosh/lx+ � s1nhdx

Taking the Laplace Transform of the second boundary condition .

results in (15):

(a.)

Differentiating equation 12 with respect to X yields:

S! = c;/G smh/:x + �/fcosh�x (13)

Evaluating /3 at X= 0 ; a.ad equating to equation a.:

10

c)t,(b., e) � ax = o = o + c:a ,�- . In this cue either Cz or /G m11st equal zero. It -Y be

noted that � cannot equal zero due to the fact that such a con­

dition lft>uld lead to a trivial solution in equation /3. Therefore

c� must equal zero.

Hence:

t = C, cosh /: X (/4)

or (16)

(IS)

Taking the Laplace Transform of the first boundary condition

gi Ye& (17):

Differentiatin9 't in equation /!!, with respect to X :

(17)

Substituting eqation /� in equation I 7 ll1lder the condi tiona

of the first boundary condition, i.e. A= L :

11

12

or

(IB)

Replacing C, in equation /S-with th.e value developed in equation

AB·, the resulting equation is

or (18):

Hence:

or:

a.ad:

_, J. = __ �a/ii[£ r. I '-' R s { ,,,,Ot--c

. . l

Co.sh If PJ{

J . ;,o:t S/nh ;_; pJt (20)

13

In order to evaluate the inYerae Laplace Transfona it is necessary

to 111.ke use of the Convolution Theorem of Laplace Transform Calculus (19) .

Symbolically stated, the Convolution Theorem is:

e = f F,(,,\) ,c;. ('e-?l) dn 0

In this case:

f; (P}= P:c

-CB

· ;:; r-e) = IZ , (20)

and cash t'jf)P .11 F, (p) = pis. .Stnh(Jit-J p)i

;=;(e) =(¥J[�(lf / o'}/r •)l

. , • E. (·.x -'- J�/«in-e7

F, (s) = (.lf)(t: ";.7e)-� e _,.,,. £L - � ,n, /. L.�

• . /'111-•

- . (TTe)-I;f e-b-'L �-.z,,.,Jl/1'or6

h:s-..

(21)

(22)

14

Therefore from the statement' of the Convolution Theorem:

c =- ("21)

Equation�/ cannot be integrated analytically, therefore a

nwaerical aethod of approximation must be undertaken. The method used

in the solution ef this integral will be described in the following

example problem.

EXAMPLE PROBLEM

F or the _ purpose of illustrating the usage and a method of solution

of the results of the preceding discussion, the following example

problem is presented.

Consider a plate Bll!lde of Chrome-Niekel steel, (18Cr, 8Ni, V2A ) ,

otherwise known as stainless steel, with · dimensions large enough with

reapect to the thickness so that one-dimensional flow of heat through

the plate, perpendicular to the faces, is realized.. The physical pro­

perties of the material in question ( 23), and the characteristics of

the heat flux are :

K - 9. 4 Btu/Hr Ft2 �F/Ft L -= 1 /8 in

/'= 488 lb/Ft3 x - 0

Cp= . 11 Btu/ lboF qo• 500 Btu/Hr Ft2

OC = . 1 72 Ft2 /Hr c - +2 Hr-l

e• . 001, . 01, . OS, . 1, . 5, 1 , 2, 3. 5, 5, Hrs.

For purposes of this example problem the Talues of the Tario•s

physical properties of the material will be considered constant over

the raage of temperatures encountered, and uniform throughout the plate.

While equation �/ derived in the discussion portion of this thesis

describes a temperature distribution through the plate, this example

problea will be concerned with the temperature at the center of the

plate at any time after the heat flux is firat applied.

Equati on 21 may be written as

_ . . �" -.zt's-,}J °S e-f.o�04 Un -;J/}!'8S� �- 1�. 4478.s- e �-------a��--------------

. 0

when the constants listed above are introduced.

The equation may be further simplified by inspect ion of the

infinite series ia the integral. It may be noted that the series

may be written as

1/!:- K - £.0.10 � f2n - 1)1 /. (.�t!J �

2L e, h=o J

due to the fact that the terms on either side of n = 0 are equiva-

lent, term. for term, until n = k where the last term n s -k does

not equal the term for n = k. This fact however ia of little conse -

quenee in the light of the fact that :n must be made large enough at

n • k so that the value of the kth term is negligible, and approaches

The above equation is the form of the equation which vaa ued

to calculate C . For purposes of evaluating the integral it was decided to use

Simpson' s Rule (24) as the inethod, due to the fact that it is & very

accurate method and is adaptable to the digital computer. Simpson' s

Rule may be atated as follows :

16

usinq four interYals.

In order to eYaluate the inteqral by Simpson ' s Rule it was

necessary to determine the value f;, •• the coap11ter cannot handle

the case where the function is zero of infinity.

The function .f'o , which is the Yalue of th.e function in the

integral at )l = o , may be evaluated as follows:

where

The aeries DlY be eva.luted as a aingle . tera as long as n is finite.

Therefore :

Hence :

YA [i + ';. ... fl� .,.. �� + . - . . . - . ]

r,r;r .,- a +- a.• a' J L , l'J 'jx ;."-is +

A"

J.j + - - - . . . . .

-2 re - A ) .

IX + � ... ,;,C:.'u ... ;;_, + . . . . . . • . . .

-.ze __ tZ __________ - 0 �

O +- oo + co + oo -,. · - - - · · - ·

17

If n remains finite then the value of the function for ,l; , is equal

to zero.

1 8

The computer was then programmed to evaluate the integral by

Simpson' s Rule as shown i n Fi g. l. The program sholfl\ in Fig. 1 i s for

the Royal McBee LGP 30 Digital Computer using the floating point system.

The results of this evaluation are gi•en ·in Table l and Plate 1.

FIGURE l

OOMFUIER PIDGRAM FOR · RO�L MCBEE LGP-30 DIGITAL OOMEUTER FOR EVAllJATION OF EQUATION 21.

;°�000 ' /OOd.+OOO '

xr6300'xudi;OO ' i0200'xliOOOO'xle0000 1xlc0000 1b02J.4 'h0200'b0212 'h0202 ' xeOOOO 'r04o4 'ue4QO'xr6300. 'xuC400'm0208'a0200'h0200'b0202 'a02o6 1h0202 ' xeOOOO' rpl.1.04. 'uC400'xr6300 1xu.Oll.oo•ho100 1m02o4 'a0200 'h0200'b0202 •a02o6 1

h0202 'lz0010 1b0200' ?0lOO'h0200 'm02o6 'd0210 'xmOOO.O 'xmOOOO 'xpiOOOO ' xpOOOO'xzOOOO'

···

O '+OOf- '

O'+oo+' 2 '+oo+• O'+oo+ ' 4 ' +0o+' 3 '+0o+ ' O '+Oo+ ' O' +oo+ ' ull09 '

; Oodi.400'/0000000'

r6300,'u0li.OO'u4405 ',eOOOO'uOOOO'b5oo6'm5002 ' s50C4 'h4800'm4800 'm5000 ' d4202.'hOOOO' h4802•a5016 'h5016 •eoooo 1b6000' , 0000001 1w()()g3800' r6;oo' udt.OO'b!i002 1 a5·oe4•h5.002 •b5012 1 s4802't4405 'b4202 'rOOOO'h48Ql+.,'b50o8 ' s4202 'm5oo6 'yOOOQ 'hOOOO 'm5 010 'm5 016 'd48C4 'm5oo6 1 mOOOO 'pOOOO' dOOOO' h5030 'P5002 'b5014 ' h5002 'h5016 'b5030 'u4403 ' r6300 'uC400' iOOOO'u4003 '

. Ooot&449 '

5000 1

15721 • -oa .. • O '+oo+ ' 1 •+oo+.• 2 1+oo+ • 1 '+0}• I

1244785 '+05-- ' 5 '+07� I

O'+QO+ •. O '+oo+• g ' 42o6' 1 '+04.- ' g ' 4212' l '+c:4- '

:f '

TABLE l

CEIITER TEMPERATURE II AN INFINITE PLATE OF S?AINLF.SS srEEL

SUBJECTED TO AN EXPONENTIALLY DECREASING HEAT FLUX.

S a Time (Hrs )

t = Temperature (°F)

Q0• 500 Btu/Hr Ft2

t 1 . 50937

12 . 10758 48 . 837-i3 88 . 50325

289. 0171 390 . 9820 441 . 1898 447. 2708 447 . 9973

e . 0010 . 0100 . osoo

. 1000

. sooo

1 . 000 2 . 000 3 . 500 s . ooo

i L _ .. ! i ' 1-� i � I : · o

i ......_,

5 0 0

· - -4- 0-0

�-- ... p. -· · •• • • • •

: (1) : . . . L i ::::, i ---.p

! . . . i ; , -[

l --·· ···-l · · · -· · · 1· -s. - ----

+-·t=-·· .. �­i · l·· - -�-.. i ·

l

1 . . 1:. _ __ _ _ , .. ' ! · _: ... ...:.: L - ! ·r --

• • 4 __ .J

� l - -·--··--t- -·· - - - -·-- -i- · -··· · -- ·- .

. ! - :t· · . .. . . . ... !. : ----- ·•· ......

I . . - -�-i I

i i \ .

l.-... . . . .... ... . .. .... .

l . .. .

. . . .. . J ... ... _ J ........ L. . · !

i � I . f · . I ·· : . ' . . ; . �- - _(I) - -io: o - ·- · ·-·---- r - -- - - - : --- ---1----- --A -R-P.lj.tei c;l- --: .. H+e_at) . . EJ.u ?< - _ : ... . .. ..

i

; ! , . . \

L I

- 1

f-l !

. ·- : ·· · · j : -J _··}� -. i qj � to'f ·tJ : r : · .· : i !_ --- . . - . . . . • __ _ _ , _____ _ . - - -- --- - ·- ! ------ -- __ .. 4-- ---- . �--- . - i-� - --�- _ _:__ _ __ J _qb;; .. $.QQ_ . B.tuJ __ Hr __ _ . F, - .. t: .. __ .. i . r - . i . : - . I . - - i i : l 1- -� .?.0

1 · J :: ' . . j .. ; - . Lci�J- tr.��I:. ; ·: i :. , + - . . : J

1 -- � · · ·· · - - --··· -- - · 1 . .. . .. · ·· r ···_; 1 : ·· 1 ··· ·· r� � - ·· ; _ _ __ - 1 . --- - r- - : - - 1 · -- - -1-- - - -, - T . ! T ·· 1

1 -� - - . . ,{) ------�---+-----+--�---..-....------.---...__�-..---------I . ' · . 9 1 l ·· · • - i • - J. .- - f- i 1 • - i · f : 1

1

: _ __ _ _ ·__ _ _ _ _ __ _ _ _ . .!,; -- - i _ _ _ , _ __ ________ T

_ __ __ __ . . L . .. .. . �T-l.m.eJH�L_sJ - '. - --- - - � - - -- _ !-- - -- -- t- .. . . : · - ---·· f---- -- - -� - - ----L .. . ... · . . . . - -- ·- · - - _ _ _ ! ! I I i J · . : 1 • I

1 ' ,

I : ! - -- ' I

! i . . '. . -- j - : . l - : j .. . � - . I .. f . .. . j . . . I_ - . ..

L ___ ___ _ ___ , _______ .,.. _______ : ------ · _____ ___ . _____ _____ _I _ __ _ _ __ __ _ _ _ . .\ __ ___ _ __ . · -- ' - --· __ , __ I __ ·__ ! _________ __ J ____ , __ ___ i. ____ � ___ l. ____ ___ ' _____ : __ _ _ _. ___ i ____ , __ _j

CONCLUSIONS

The temperature at any Jl)Oint within an infinite plate as a

function of time and a variable external heat flux impinging equally

on both sides -Y be determined by the use of equation 2 I . The form

of the expression for the Tariable external heat f lux .. Y be fouad in

equation 3

The aethod described in this thesis employs a function of the

external aeat flux and is not dependent on surface temperatures as

is most of the published material on the subj ect. Thia fa.ct will make

this approach Yalltable in aituations where surface temperat11res are

difficult to obtain.

It is, of course, possible to use any expression for the external

heat flux and it might be desirable to develop several solutions, of

the type of equatioa �/, involving 111&ny of the standard functions such

as periodic f\lllctions.

Equation 21 is aa exact solution to the problem in questi••· How­

ever it is necessary to eTaluate the iDteqral by numerical aethoda.

The proce•• outlined ia the example problem deJBOnatratea this process.

It va.s found that Siapaon' s Rue was a good method of sol utioa. The

accuracy of the results will depend upon the size of the increment takea

for use vi th Simpson's Rule. However this aethod is noted for its high

degree of accuracy.

The solution of eqll4tion �/ requires the use of some type of com­

puting aachine if accuracy, combined with reasonable computing ti•• ia

desired. The tiae required to obtain reasonable accuracy without the

use of a c omputer would be prohibiti�e.

· The program outlined in the example problem is designed for cal­

culating the te11pera t•re a t the center plane of the pla te. HoveYer a

program could be easily written which would compute the temperature

at any point wi thin the plate at any ti.lie.

23

In the forqoing work the author haa made use of Fourier ' s general

expression for one-dimensional, transieat heat conduction and applied

initial and boundary conditions, to arrive at the •olution to the

problem of teaperature distributioa ia an infinite plate as a function

of a prescribed external heat. flux.

The resulting eqaation is ueful when used in conjunction with some

type o f computing deTice, and demonstrates a method which could be used

in the s olution of ,1a11ar problems inTolving Tariable external heat

fluxes.

VITA

The authGr was born December 1, 1935 to Mr. George F. Wright

(Deceased ) and Mrs. Agnes V. Wright at Springfield, Missouri.

He attended the public schools of Springfield, Missouri and

was graduated. from Central Hiqh School with· the class of 1953. He

entered Drury College in September 1953 oa the Three-Two Plan for

engineers which was set up with the Missouri School of Mines, and

became the first student to complete the program from Drury College.

In September 1956 the author entered the Missouri School of Mines and

was graduated with the clase of 1958.

The author hold• tllO baccalaureate degree s, a B.A. in Hathema.ties

and Physics fr•• Drury College, and a B.S. in Mechanical Enqiaeering

from Mi seouri S chool of Mines.

The author ia Mrried to BeYerly Jo Hooks Wriqht and aae oae

dauqhter, Kristin Lynn.

In February, 1958, he was appointed an In structor in Mech.a.aieal

Engineeriag at M1a1ouri Sc:h.ool of Mines aad has s erYed ia that capacity

to the present ti••

BIBLIOGRAPHY

1. H. S. Carslaw, Introduction to the Mathematical Theory of the Conduction of Heat in Solids, 2nd Ed. , H. Y. , Dover Publications , Am. Ed. , 1945, pp. 47-50.

2. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd Ed. , London, Oxford University Pre�s, 1959, p. 7 and pp. 50-77.

3. P. J. Schneider, Conduction Heat Transfer, lat Ed. , Mass. , Addison-Wesley, 1955, p. 234 .

4. D. Q. Kern, Process Heat Transfer, lst Ed. , N. Y. McGraw-Hill, 1950, p. 645.

S. M. Jakob, Heat Transfer, Vol. l, 1st Ed. , N. Y. , Wiley, 1949, · PP· 258 " 260.

6. W. H. McAdams, Heat Transmi ssion, 3rd Ed. , N. Y. , McGraw Hill, 19�4, pp. 35-38.

7. A. I. Brown and S. M. Marco, Introduction to Heat Transfer, 3rd Ed. , N. Y. , McGraw-Hill, 1958, pp. 265 - 269.

8. B. E. GateWGod, Thermal Stresses , I. Y. , McGraw-Hill, 1957 , pp.

43-64.

9. P. J. Schneider, Coaduction Heat Transfer, 1st Ed. , Maas. , Addiso.n-Wesley, 1955, p. 4.

19. W. H. McAdams, Heat Transmission, 3rd Ed. , I. Y. , McGraw-Hill, 1954, p. 3.

11. R. V. Churchill, Operational Mathematica, 2nd Ed. , Ji. Y. , McGrav­Hi ll, 1958, p. 112.

12. E. J. Scott, Traaafora Calculus With An Introduction to Complex Variables, 1st Ed. , I. Y. , Harper and Brothers, 1955, p. 67.

13. L. M. Cells, Elementary Differential Equations, 4th Ed. , N. Y. , McGraw-Hill, 1954 , pp. 128 -130 .

l� . Matae11atical Tables from Handbook of Cheaiatry and Physics, 7th Ed. , Ohio, Chemical Rubber Publishing Co. , 1941, p. 268.

15. R. V. Churchill, Operational Mathematics, 2ad Ed. , I. Y. , McGrav­Hill, 1958, p. 112 .

16. · Mathematical Tables From HABd.book of Chemistry and Physics, 7th Ed. , Ohio, Chemical Rubber Publiahinq Co. , 1941 , p. 268.

17. E. J. Scott, Transform calculus With An Introduction to Coaplex Variables, 1 st Ed. , I. Y. , Harper and Brothers, 1955, p. 31 8.

1 8. Mathematical Tables From Handbook of Chemistry and Physics, 7th Ed. , Ohfo, Chemical Rubber Publishinq Co. , 1941 , p. 268.

19. E. J. Seott, Traasfora Calcuhts With An Introduction to Compl ex Variables, 1st Ed. , N. Y. , Harper and Brothers, 1955, pp. 75-77.

20 . Ibid, , E. J. Scott, p. 31 8.

27

21. A. Erdelyi, Tables of Integral Transforms, 1st Ed. , Vol. 1, N. Y. , McGraw-Hill, 1954 , p. 258.

22. Ibid. , A. Erdelyi, p. 388.

23 • . E. R. G. Eckert and R. M. Drake Jr. , Heat and Mass Transfer, 2nd Ed. , .N. Y. , McGraw-Hill, 1959, p. 497.

24. W. A. GranTille, P. F. Smith, W. R. Longley, Elements of Calculus, 6t� Ed. , Maas. , Ginn and Compaay, 1946, p. 1 21 .

25. F. S. Woods, Advanced Calculus, 3rd Ed. , Ms.as. , Ginn and Company, 1934, p. 13.