On the General Solution of the Fundamental Equation of Thermal Conduction in Bodies Whose Thermal Coefficients are Affected by Temperature

Publisher’s version / Version de l'éditeur:

Technical Translation (National Research Council of Canada), 1960

READ THESE TERMS AND CONDITIONS CAREFULLY BEFORE USING THIS WEBSITE. https://nrc-publications.canada.ca/eng/copyright

Vous avez des questions? Nous pouvons vous aider. Pour communiquer directement avec un auteur, consultez la première page de la revue dans laquelle son article a été publié afin de trouver ses coordonnées. Si vous n’arrivez pas à les repérer, communiquez avec nous à PublicationsArchive-ArchivesPublications@nrc-cnrc.gc.ca.

Questions? Contact the NRC Publications Archive team at

PublicationsArchive-ArchivesPublications@nrc-cnrc.gc.ca. If you wish to email the authors directly, please see the first page of the publication for their contact information.

For the publisher’s version, please access the DOI link below./ Pour consulter la version de l’éditeur, utilisez le lien DOI ci-dessous.

https://doi.org/10.4224/20386687

Access and use of this website and the material on it are subject to the Terms and Conditions set forth at

On the General Solution of the Fundamental Equation of Thermal Conduction in Bodies Whose Thermal Coefficients are Affected by Temperature

Sawada, M.

https://publications-cnrc.canada.ca/fra/droits

L’accès à ce site Web et l’utilisation de son contenu sont assujettis aux conditions présentées dans le site

LISEZ CES CONDITIONS ATTENTIVEMENT AVANT D’UTILISER CE SITE WEB.

NRC Publications Record / Notice d'Archives des publications de CNRC: https://nrc-publications.canada.ca/eng/view/object/?id=d6ea7a87-9f74-41fd-a8d5-5fe7986dac9c https://publications-cnrc.canada.ca/fra/voir/objet/?id=d6ea7a87-9f74-41fd-a8d5-5fe7986dac9c

One of the projects of the Fire Section of the Division of Building Research concerns the development of a method

for calculating the fire endurance of building elements. The main theoretical problem in this work is that the

classical solutions of the Fourier equation, based on con-stancy of properties, cannot be used because of the Wide variations in temperature involved in this experimental work.

Three papers by Masao Sawada that deal ｾｾＱｴｨ problems

of the heat conduction encountered when the heat capacity and thermal conductivity of the solid are variable, have been translated and issued as NBC Technical Translations

Nos. 895, 896 and 897. From the work of Sawada it is seen

that the more perfectly the mechanism of heat conduction within the solid is approached, the more limited the ap-plicability of the method becomes, as far as the shape of the solid or the boundary conditions are concerned.

Although the introduction of various numerical methods and

computer ｴ･｣ｐｾＱｱｵ･ｳ largely reduce the practical value of

such analytical methods, there are certain fields where they are indispensable.

Ottawa,

September 1960

B.F. Legget, Director

Title:

Author: Referenoe: Translator:

Technical Translation 897

On the general solution of the fundamental equation of thermal conduction in bodies whose thermal

coefficients are affected by temperature Masao Sawada

J. Soc. Mech. Engrs., Japan, 37 (201): 15-21, 1934 K. Shimizu

ARE AFFECTED BY TEMPERATURE

Abstract

This is a continuation of my article "On the general solution of the basic equation of thermal conduction", J. Soc. Mech. Engrs., Japan, 05 (183), 1932 (NRC TT-895). Here a solution to the fundamental equation is sought which will be applicable to finite as well as semi-infinite bodies, where thermal conductivity and thermal capacity can be regarded as a function of temperature. A few problems of thermal conduction under simple

bound-ary conditions are considered. Since the law of

super-position is not directly applicable with respect to temperature in this case, the author proposes that a "thermal function" be sought by solving a transformed fundamental equation.

1. Introduction

Thermal conductivity and thermal capacity vary somewhat as an

effect of thermal conduction. It is not easy to treat thermal

con-duction analytically in this case. Even if the solution to the

fundamental equation is possible, it is not easy to apply this to

problems concerning a finite body. The author has sought a

solu-tion· to the fundamental equation by assuming that thermal

conduc-tion Has a funcconduc-tion of temperature. The method follows that of

Van Dusen··. In the case of an ordinary homogeneous body, the

superposition of temperature is valid. It is not valid in the

present case •

•

••

See abstract.

Van Dusen puts V = _AO

f

f(v)dv =

"0

J

f(v)arvdr where AOf(v)

=

conduct1v1ty and 1n dea11ng w1th the fundamental equat10n and

re-lated problems he treats th1s as a "thermal funct10n". The author

be11eves th1s to be one of the more conven1ent methods. Only w1th

the estab11shment of th1s thermal funct10n 1s 1t poss1ble to apply

the pr1nc1ple of ｾｵｰ･ｲｰｯｳＱｴＱＰｮＬ and the problems relat1ng to f1n1te

as well as sem1-1nfin1te bod1es w1th var10us surface cond1t10ns, but not surface rad1at10n, are then access1ble to solut1on.

2. Outline of the Fundamental EQuat10ns Qf Thermal Conduct10n

[1°] The case where the body 1s not a heat source

For conven1ence, let the thermal capac1ty be ｰｾ

=

g1(X,y,Z) •

g2(V) • g3(t) and the thermal conduct1v1ty A

=

_{f 1 (x,y,z) • f 2(v) •}

f 3(t). Then, generally, the following cond1t10n holds:

pru,v=div(,logradv) (1)

Further, 1f we put

!fJ,V)dV=v, !fs(t)/Ks(t)odt= T,gz(v)l/J,v)=F{V)

the follow1ng equat10ns can be estab11shed.

KI(x,y, z)·F{V)UTV=div[};(x,y, z) grad VJ. .... (1_o)

KI(x,y, Z)!fI(X,y, z)0fitV)uTV

=div(grad V)+grad Vodiv[,l.d,(x,y.z)

.. F{V)uTV=[U.,(fI(X.Y.ｺＩｕＮＬｖＩＫｵｾＨｨｸＮｹＬ z)u" V)

+u.(fl(x,y, z)u. V)]/KI(x,y. z), [ru,UI(r, 0, z)'rUrV)+uo(fl(r, 0, z)UoV)

+ru.[JI(r,0, z ]u.V)]I,-...-·g-,I(.-r.---,;O..-,z--.--).

{sin2

0.ur[tl(r ,O,IP )·y2ur V]+sinOUe[sinOf,(r,O,IP)Ue V]

+ｵｾ｛ｊｩＨｲＬｏＬｱｊＩＸｾ V]}/rsin20KIlr ,O.IP ) (11)

If g2(V)

=

_{f 2(v), then F(V)}

=

1, and the fundamental equat10n

of the same type as 1n the case where ｰｾ and A are merely funct10ns

of d1mens10ns, 1s obta1ned. Thus, 1f g, (x,y,z)

=

g(l)(x) • g(2)(y)

• g(3)(z),

ｾ

(x,y,z)

=

f(l)(x) • f(2)(y) • f(3)(z), there are

cases wh1ch are capable of solut10n.· It w1ll be recalled that

Van Dusen'" discussed a solution of the problem of constant flow in

the case where A!P'Y

=

constant, and A= f_{2 ( v ) .} In short, if p'Y and

A are a function of the temperature v, the problem should be treated

by putting v equal to

J

r,

(v)dvV.

In the case of a long and slender cone or a cylinder, let F be the cross-sectional area, zf the perimeter of the cross-section, E the rate of heat diffusion and va the outside temperature, then

the following equation is obtained:

fJrFf),v=8.(FA8.v)-t· zf· (v"': Va) (12) ••

Further, for an annular or a radial fin, let z be the thickness

of the fin and 2nra the length of circular arc. Then

fJrzr8,v=8r(zrA8,.v)- 2£r(v-va) ••...(13) • •

fJrz(2n:r(J)8,v = 8_{r(2n:r(JzA8,.v)-}2[£lZ+ £2(2n:r(J)](v-va )

... (14),,/<

In (12), (13) and (14 ) F, Zf, z and 0 are either constants or

functions of the ｾＱｳｴ｡ｮ｣･Ｌ and A, p'Y, e, B_{1 , B 2} and v are

const-a

ants or functions of dimension, temperature or time. Of these

variables, the case in which A, P'Y, F, Zf, z, and a are functions of dimension has already been d1scussed.

[2°] The case where the body is a source of heat

If Q be the amount of heat emitted per unit time and volume,

then since Q

=

q1 (x,y,z) • q2(1)(v) • q3(1)(t), the requ1red

funda-mental equation will be obtained by merely adding Q to the right

side of (1). Similarly Q

=

q1 (x,y,z) • q2(V) • q3(T) should be

added to (1₀ ) , FQ.

=

Fq 1(x ) • q} 1) (v) • q 3(1) (t) to (1

2 ) , rzQ =

rzq1(r) • q2(1)(v) • q3(1)(t) to (1 3), and (2nro)zQ = (2"ra)zQ1(r) •

q (l)(v) • q (l)(t) to (1 )

2 3 4 •

•

••

J. Soc. Mech. Engrs., Japan, 33 (161), 329. Abstract 124 •

Dr. Saihara. J. Soc. Mech. Engrs., Japan, 35 (180),318;

Primarily, A, ｰｾ and Q are complex functions of dimension and

time. However, integration is possible only when the fundamental

equation CRn be expressed as a product of separate functions of di-mension, time and temperature.

The author will not consider the case in which

A

and ｰｾ are

functions of time. In this paper he will deal mainly with the case

in which the body.is not a source of heat, but will discuss the

case in which

A

and ｰｾ are functions of dimension and temperature.

3. The Solution of the Fundamental Equation for Finite

Body Problems

[1°]

The case in whiCh

A

= ａｾｊｖＩＬ ｰｾ

=

ａｾＰＲ

•

g2

_{1xl ,}

limited to rectangular coordinates

l/ao!·g.f..V)8IV=f),/,fj,v)f)",v)+ｦＩｾｕＲＨｖＩｦＩＬｰＩ +f).(J2(V)f).V)...(2)

(i) ｾ =±ik"alx±ik.'a2y±ik.'z-ik'a02{r-t.

W,'al)2+(i k.' a.,j2+_(ik.'a3)2=kk{r-.

Transform (2) by putting k_{1 I ,} k_{2 '} , k_{3 '} , and k ' as 3 or 4. Here the

first constant of ｩｮｴ･ｲＬｲｾｴｩｯｮ will be neglected.

ｧＲＨｖＩｦＩｾｖ =ｦＩｾｕＲＨｖＩｦＩｾＱＧ (20) .'. ex p,{fV2(V)/ f g2(v)dvｊ､ｶｽ］ａ･ｾ (21) integers, Developing Now let ... II fi v)=1+1,;;·...v..., g2(V)=1+ Lk"v". Am and kn ｾ 0,

and assume m and n to be positive integers (nesative positive or negative fractions are not impossible). further, we have

JViV)/ J g2(V)dVJdV

=Ag[v·.(v+ OI)"(V+ ( 2)••... (v+ 0,,)'.J

,., V"(V+01)"(V+02)" (v+O,,)·. =Aet _ＨＲｾＩ

Note that the left side, when expanded, becomes a multiple degree function of v and the right side becomes identical With the

three dimensional solution in the case of a homogeneous body. If

When

e

can be expressed as follows:

fM'=ao2_{[fJ,/ 8 + fJ,N' + fJ.}2_{8 ] (2}

5)

but cannot be applied directly to problems.

(11) v

=

w ) + u where U

_o

is a constant.

(x,y,z,t 0

...

..

fiv)=1+

l)...

v..., g2(V)= 1+

Lk..

v",

and m and n are pos1t1ve 1ntegers, we have, from (2):

where

...

ｾＱＫ

tk..

ＨｗＫＱｉｯＩｾ fJlw '=ao2diV{[I+ 1'A",(U'+uo)"'}radow} (26) 1+ t k..(W+Uo)"=klOl[I+i k l..I;"} 1+ iA",(W+110)'"= A(Ol[1 +

ｪ［ａＨＢＧｗＢＧｾ

•

In the above express10n

..

klOI=

r

k ..tto". o

..

r

11' kIll= -- '-k"u ..-I (n-l)1 0 , I

..

k(2) - '"" II! k .._2 - L.J 2!(1I-2)1 ..Uo , 2

..

k(3) _ ' " " 11! k ft-3 - .LJ 3'(n-3)! ..uo • , kC..+l)=k.._1ＫＱｉｫＢＭｬｴｾＰ［ k(") =k.., •

.o. (1 +

r

kl..IW")fJIW=a2diV[ (1 +

r

Al"'IW'")gradW

...(27 )

exp

.{! [

(1+ ) ;A("'I

w'")/w(

1+

t

kl..1w" j1+1I)

Jdw}

=ａ･ｾ (28)

Of the

ｾ

in (2

1) , the coeff1c1ent

ＱｫＧ｡ｯＲｾＲ

of t 1n th1s case 1s

k' 2 (O)/k(O)

1 a_{O A} • Thus, 1f w(x,y,z,t) and the constant u

_o

are super-posed, (2) reduces to (2_{e ) , a} s1m1lar type. If v

=

w( t) +

x,y,z, u(x,y,z)' th1s no longer holds.

We obtain from (2)

(iii) Let

!f2(V)dV= v, g2(V)!f2(V)=A.V).

F(V)01V"; a02[0.,z V+ 0,,2 V+0.2V] .' ＨＲｾＩ

Thus,

In this case, ｾ is identical with ｾ in [10

J,

(i). (25) holds

for 8.

(iv) If F(V)

=

1, we obtain immediately from (29 ) :

｡ｉｖ］｡ｯＲ｛ＰｾＲｶＫＰＬＬＲｶＫＰＮＲｖｊ .'. ｖ］ａ･ｾ ...(2u )

AI

pry

=

).'0/a

O:2.

This reduces to the case in which X = _XOf:2(v),

if the flow is constant, ｾＲｖ

=

0 :. V

=

ｂ･ｾＮ

,=

I1IX+112Y+ 11a,Z. 1112+I1l+I1l=o.

v=W<..", •.I)+U<e.".•), !ll.l(",...)=w(...)+G(...).

v=!ll.l(",...)+u(..)

However, Putting

and sUbstituting in the fundamental equation, ｾｬ･ obtain:

Ol!ll.l=ao2_{(O.,z!ll.l +01l25ffi +0.2!ll.l). ｡ｾＲｵＫＰＬＬＲｵＫＰＮＲｵ］ｯ ...}

... .(212) , (213)

Example i. The case in which

A=_Ao(l+AIV), or=Ao/ao2•(1+klv)

.', v(1+!k_lｖＩＲｾＧｫＬＭＱ =Ae:

kl=2AI...ｶ］ａ･ｾＬ kl=AI ..ﾷﾷﾷﾷｖＨｖＫＡａｬｶＩ］ａ･ｾ

Further if A=Aeo"l", or=AoItZo2

•t"'"

Example ii· where

"

. '. exp.[kl/(AI-kl)e<).'-",)tI]=Ael

A=Ao(l+'IV+A2V2), pr=_{Aoao2.(1 +kl}fl + k₂v2)

. " v(v+ＸＩｾ［ＨｶＫ ＸＲＩｾＧ =A expoW16)

81•2=3(kl+r/JII2)/4k2, r/J=k12_ (16/3)k2• t'l=(16/3)k2[A281 - ·Al+(3/2)kl/k 2]r/J1I2

t'l=(16/3)k2[AI-A282-(3/2)kl/k2]/r/J1I2 •

[2°] The case in wh1ch X

=

f,(x,y.z) • f 2 (v)

•

f

3 ( t ) . pry

=

8._t (x.y.z) •

r,

(v) •

s,

(t) reduces to the case in which we

put F(V)

=

1 1n the fundamental equation (1,1

If V

=

W(X,y,z,t) + ｕｾＬｹＬｺＩＧ then substituting this in (2 _{14 ) we} have:

K"b'-,y, r)8 rW=div[Jlr,y, z) gradW],

div [Jl(X,y, z) gradU]=0.

Thus, by giving sUitable forms to f 1(x,y,z) and g1(X,y,Z) we can obtain a three-dimensional solution with respect to all

co-ｯｲ､ｩｮｾｴ･ systems, and one which will be of practical use under

simple boundary conditions.

(i) The case where the flow is constant:

div[Jl(X,y, z) gradU]=0 (2₁&)

For instance, with respect to the rectangular coordinate by putting

we obtain:

J

l!ll(X,y, z)'dx=x,

f

l!ll(X,y, z).dy=Y,

J

l!ll(X,y, z)odz=Z

8Zl u + 8 l u + 8i1zu=o .'.U=ff2(v)dV=Be l ... (216 )

For two-dimensional ｰｲｯ｢ｬｾｭｳ analysis by means of complex

numbers is possible. It would be interesting to investigate other

coordinate systems.

where m' expresses curvature. In case

v=Jf2(V)dV T=j'fa(t)/K"3(t)'d/

.. V=exp.(-ao:!,62a2T){It...",,(2ayyP)/y,-,] ...(217)

=exp.(-ao2_{q2a2T){J±",(2aY;;;;)/y,-,].}

[ C? S_Sin(1I8)J, 111=(mI2+411)1/2tp (211)

ｾ = cos

a,

m = integer.

The above solutions are limited to problems of a

non-vibration-al type. Further, 1f mt _{= 0, r can be made equal to c}

±

_{x. If}

A = A

_o

_{em,x f 2 ( v ) • f3 ( t ) ,} A/P'Y = a02e-PXg3(t), then substitute eX

for x. If mt _{= 1 or 2, we can solve the fundamental equation by}

and A= Ar/..c±y)1+"',!y"" ·f2(V) ·f3(/),

Atpr］｡ｯＲｹＢＢＨ｣ﾱｹＩｾＢＡｙＢＢ •ga(l) , res pec t 1ve ly ,

i.e., one should seek suitable functions which eliminate the curva-ture. Example 111: Solution: where constant flow. A=Ao(l+A1v). PQrv]r.=t.(a-v.). [AQrV]r,= '.(Vi- h). 2 '=-l/AI +[1/A1 2+_2(Af(y)+ｂＩＯａｾＩｾ｝ｉｦＲ _{..:... .(2} 20) A =[Q+(Q2_1'R)1/Z]/p. B= AoA[Q/P- t.1'.""(1+A1a+ａｉＭｉｴＮｙＮＢＢｪｬｩＩ｝ＯＨｴｾｹＮＢＬＧＩＲ + Ao[a(l- Ala/2)/(tiy;",')2-b(1--A1h/ 2)/(t . l'. ""f] /P , P= 1/(6.1'0"'r-1/(ti";"")Z - - - _Q=Ao[<1+Ala)/t.y."" m' 0 1 2 ｾ｟｟｟｟｟｟｟｟｟｟｟｟ _{+(1+A1h)/ti y;",']+N}

f(y) r Agy -1/1' _{R=(a-b)(a+h+2/A I).}

M r; Agy. -1/1'.

N Y.-Yi Agyo/Y, 1/Yi-1/y•

.-

- _

.._ '

-For the valueR of f(r), M, and N see the table.

Example 1v: Constant flow.

-A=A o

(1+

l'A",V"'). m = positive integer

AO[V+

i

A",V1+"'/(1+tn)]=Ax+B.

To determine A and R under sUitable conditions, and to solve

for v.

Lemma: Boundary conditions: ｾ｡ v

=

+ g(v-c) where

r

a

u

r

A

=

ｾｯｦ 2 (v>.

Consider v

=

w(x,t) + u(r). Since ｾ

F

0, if g ｾ

=

0, if ｾｏＯｧ = 0, w

=

0, u

=

0, g

=

finite

Ao/2(w+ u)Qr(w+ u)=±t(w+u-c)

Next, by expandLng f _2(w + u), Ao/2(u)QrU=±t(u-c)

A., [J2(W+u)Qrw+wf'(u)'QrU+!llrf"(u)QrU+ ...]=±tw.

Thus, unless E is finite, solutions are not possible in

applied problems.

Ins1de the body

put

Cond1t1ons:

prutV=u"OU"V) where _{A=A of2(V), A!pr}=_ao2

v= !f2(V)d7'. V=W(".t)+U(..), v=[w(w)+G(IO,n)]+U(u)=!m(".l)+U(,,),

ul'1J3=ao2U,,2!m , ['1J31".=｛｜ｉｬＮｾ｝ＢＬ］ｯＬ

[llB]I=O=[V]t=o-U, U11=O=ID. U,,2U=O.

[U]".=[

v];::.

[U]",=｛ｕ｝ＺｾＺＮ

u=

[V]l='"

.1.".{[

v]t=o-U}sin IITrr;-x. dr; (222 )

"" X'i-X'a

Example: The case 1n wh1ch f₂ (v)

=

1 + A1v ,

Subst1tute the following in (2_{2 2} ) :

V=v(1 +!A1V), [V]t=o= !,6(x)[1+!A l!,6(X)], U= [V]t=", = Ax+B= u(1+!Alll),

u= -lIAl+[1/A12-2(Ax+_B)/A

oAl]l/2. A =(6-a)[1 +!Al(a+ 6)]/(Xi- x,,), B= [0(1 +!A la)xi-b(1+!A16)x.]/(Xi-X.)

V=(w+ u)[1 +!Al(W+ u)],

Problem 11: 'Nhere [V]".=a, [u"v] .... =O. [v]t=o=!,6(x).

A and A/P'Y as before.

Cond1 t10ns

u

t'1J3=4ul'1J3 , [!m]".=o. [u,,'1J3].... =o.

[!ID]I:O= [V]I:O- U. [V]I:O= lD(x).

Further _{[U"U] .... =o. u= [V]t=",}

.', v=l,) +_2_ ｾ･ｸｰＬ｛｟Ｈ｡ｊｬｉＫＡ｜ｔｲＩＧｴｊ｣ｯｳｻＨＲＱＱＫＱＩｔｲ X-Xi} x.-x,L.J

_..

X.-Xi 2 X.-Xi

1.".[

]

{(211+1)Tr r;-Xi } • 1D(r;)-U cos - 2 - - - dy (2_{l J )} "I . x.-xi

Example: The case 1n wh1ch f₂ (v) = 1 + A1 V ｖｾｶＨＱ +!A1V), [V]t:o= lD(x)=!,6(x)[1 +!A l!,6(X)] U= u(I+!A1U) ;' Ax+B. A=O, B=a(I+!A la). ｔｽｾ･ above expr-e sstons are substituted 1n (22 3 ) .

Problem 111: AO[U"V]".=t,,(a- [v]".), Ao[U..V] .... = t,{[V]"i- 6), ｛ｖ｝ｴｾｯ］ lD(x).

Inslde the body atv

=

a02ax2V where €1

=

_{€!/AO' €a}

=

€a/AO·

Condltlons are classlfled ln Table I. The present problem ls

not exact. Table I Boundary condltlons t U<ｾ COfts'ta ...t • floW' ｛ｯＢｕ｝ＬＬ［］ｾｻ｛ｖ｝ＢｉＭｬｉＩ [o ..ｱ｝ｾＮ =ｾＨ｡Ｍ [v]:r.) ｕ］｛ｖ｝ｉｾＧＢ [o"W]"I=!i[wlr l [o ..W] ...=Ｍｾ｛ｷ｝ｾＮ [W]I=O= dI(x)- U Example: ee .. V=U+2};r,I5011

..

Ｈ｡ｾＧＫ !..i'Xx.-x;)+(a,,'+ｾＡＮＮＡＮｘＡＮＮｩＫ !.!!)/a,,'+!/)

.f...

[dI(71)-uJ[｡ｾ｣ｯＹ｡ｾＨＷＱＭｸ［ＩＫＡｩｳｩｮ｡ｾＨＷＱＭｸ［Ｉ｝､ｊｬ ...(2₂₄₎

ttl:v)=1+A1v, V=W(",I)+U<.). V= W<e.I)+U<.)

V=v(1+!A1v)=W+U ... W=w(1+!A1w)+A1lCilt . ,

U=n(l+!A_tu), OIW=ao'O.IW. fJ..'u=o. U=Ax+B.

Dependlng on condltlon A and Bare determlned from Further,

Then AO(l + A1W)axW =

±

€W

error. Consequently, thls

However,

approx1-and lf ｾＱ ls small, let A1ax(WU) ｾ 0.

clearly holds. However, there ls some

case, strlct1y speaklng, ls not valld.

In problems 1 and 11, lf the external temperature ls lnfluenced by a functlon of tlme, they must be handled by Duhamel arlthmetlc,

although thls may be cumbersome. Further, a solutlon ls ｰｯｳｳｬ｢ｾ･

ln the case

The solutlon ls also posslb1e ln two or three dlmenslons. ln a general case, the calculation 1s 1mposs1b1e except by mat10n through expanslon.

If A

=

AOf2 (vIdv , p'Y

=

A

_o/ao

2 • g2(V), problems 1 and 11 can

However, this is theoretically somewhat unsound, and will not be

discussed here. This is due to the fact that the constant flow

value must be a V function, although there is nothing wrong in using

the

e

function. An exact treatment is possible in such problems as

｛ｶ｝ｾＮ］｡Ｌ ｛ｶ｝ｾ［］｢Ｌ [V]t=o=q,(,-)

r"'"PTatv=｡ｾＨＩＮｾＢＬＧ｡ｾｶＩ

A=_{Aorl+"'-"''l2(V), A/PT}=Ao/ao2,r--", ,,=m/f>,

one involving a semi-infinite body. Problem iv:

Inside the body where ...(225) Where m'

=

curvature number. In accordance \'1i th [20 ] , putting v= ff2(V)dV, v=ｗＨｾＮｴＩＫｕＨｾＩ then, ｛ｗ｝ｾＮ］ｯＬ ｛ｗ｝ｾ［］ｏＬ ｌｕ｝ｾＮ］｡ＧＬ ｛ｕ｝ｾ［］｢ＧＬｾＭﾭ

[W]t=o= [V]t=o- U. [V]t=o= (1)(r), u= [V]t=oo,

Inside the body ＬＮＬＮＫＢＧＭｬ｡ｴｗ］｡ｯＲ｡ｾＨｲＱＫＧＢ｡ｾｷＩＬ ｡ｾＬＭｴＫＢＧ｡ｾｕＩ］ｯ

co

.', v= [V]t=co+PL'il3,a/exp.(-ao2p2a/ t ) ,

-

ＭＭｾｕＬＬＨｾｩｾﾷ r1l+...: I[ (1)(r)-｛ｖ｝ｴ］ｯｯ｝ｾＬＬＨｾ､ｲ

. ｾ［

ｕｮＨｾ］ ｛ｊＬＬＨＡＺＩＺＡＭＬＬＨｾＩＭ ｊＭＬＬＨＮＦｊＬＬＨｾ｝ＯｶｲＭＮ !:.=2a,vr",

root of U,,(ra)=0 'il3,=(11"/sin"11")2(8,2-1).

8,=ｊＬＬＨｾＩＯｊＢＨＡＺｩＩ］ ｊＭＢＨｾＩＡｊＭｫｩＩＧ

Example: _{fJV)=l+A1V, V=v(l+tAlv),}

(1)(r)=q,(r)[l +!A1q,(X)], ｛ｖ｝ｾＮ =[u(l +!A1U)]..=a =a',

｛ｖ｝ｾ［］ [u(l +tAIU)]..=b=b'.

[V]t=oo=U= [u(l +tAlu)]t=oo='1'1+'I'Jr"',

｛ｖ｝ｾｾｾ］｡Ｇ］ＧｉＧＱ +'I'Jra.... [VJr:;;,=b'='I'I+t'2Ir ....

,', U=[V]t=oo=[a'/rr-b'/ra'"

-(a' -b')/r"']/(l/rr -l/ra"') ,

The above values are to be substituted in ＨＲＲｾＩＮ

｛ｖ｝ｾＮ］ 'Fa(t). ｛ｖ｝ｾ］ 'Nt) using Duhamel arithmetic.

ａｏ｛｡ｾｖ｝ｾＮ］Ａ｡Ｈ｡Ｍ [v] ...).

ｾｯ｛｡ｾ ｖ｝ｾ［］ !.{[vl.;-b), [V]t"o=(1)(1')

17"+--1at v=｡ｾｲＱＫＢＧ｡ｾ V).

Problem v: Inside the body

As in problem iii, consider V

=

W(r,t) + U(r). Then, the

following results easily follow, althoug1l they are inexact.

co

r:[V]t=oo+P.L'il3"a/exp,(-ao2p2a,2t)U,,(r)

,

Where

[No.1]= Ｍ｛ｰＨＡｪＩＯＲｲｩｨｩ｝ｊＭＨｦｴＫ､ｾＬＩＫ J-.(!j),

[No.2]=[P(!::i)/2r,h i] "ｉｮＫｴＨｾＩＫ In(rJ) ,

vg-{v)8Iv=kRoflA [v)-S(v-vaVs(v) ＧＨｾＹＩ

t=

J

ｖｧＨｾＩＯ｛ kRoN/i(v)-S(v-va)/s(v)}dv+AI (2eo)

as 1 s a pos1 t1 ve root of [U/(r)]r.+h..Un(r,,)= 00):iE1R. h"=E"I).r,, h,=Eil).o,

!H,={[(pa.fr"p+h"2r,,2- mh,,ra]ra'"[ｕｲＨｾＩ｝Ｒ

- [(pa.fr!'-2 -mh;/ri+hl](sinnTT:/TT:f}-1 •

Example: /2(v)=1+).lV, V=v(l+t).t v),

(1)(r)=¢(r)[1+t).l¢(X)], u,,=a', ui=h'

u= [V]I=", =u(l+!).IU)=%"l+ rJrm

%"1=[hJ(m/h"r"l+m_r,,-m)a'(m/h,r/+m+r,-;")]/q,

r2=(a J-h')/q, q=m[(l/h"r"I+,"+l/h,r/+m+(r,-m_ r,,-m)].

The above expressions are subst1tuted 1n (22 6 ) .

In the present problem, as 1n problem 111, 1t was assumed that

Newton's Law holds approx1mately. Thus, the solut1on 1s not exact.

[3°]

Thermal condupt1on of heat source 1n wh1ch pos1t1onal

changes 1n temperature are 19nored

In the case of objects such as an electr1c conductor, 1f we add

to the r1ght s1de of the fundamental equat10n (12) the quant1ty of

heat em1tted per un1t t1me and per un1t length Q

=

koI2 / F (where 0:

spec1f1c res1stance, I: strength of electr1c current, and k: constant), we cbta1n:

a can be cons1dered a funct1on'of x and v; and I, a funct10n

of t. If we let axv

=

0,

.PTFa1v =k aI2/F - t. "l:j" (v-va) ··· (228)

Next, cons1der that PT=g(v), a::fl(v), I2=I02, E=/sEv) , 1ntegrate (22 8 ) w1 th

respect to d1stance, and let V:

J

Fdx: volume, R: a

J

l/F • dx =

ROf1 { v ) : total res1stance, S

=

J

ｾｦ､ｸＺ

surface area.

Then we have

Example v: The case 1n wh1ch

PT=PoTo(l+hv), R=Ro(1+R1v),

Let

Then

4.

VPorol(tut1S)=a,

VPuroh=lJ, (kRoN HoVaS)/(totlS) =C,

[kRoRII02-Eo-S{l-ElVa) ]/(totlS)= Zd,

8₁= d + (c + d 2)IIZ, 8_{z= d - (c + d}2)1f2,

'r_{1= (a+}lJ0_{1)1(81- 8z),} 'r_{Z=(a+ lJOz)1(81-8z).}

[V], ...=V...=81

ＨｖＭＸＱＩｾｉＯＨｶＢＧＺＧＸｺＩｾＱ =A"r' A"］Ｈｖ｡ＭＸＱＩｾｉＯＨｖ｡ＭＸｺＩｴｬ

•'. ｛ＨｶＭｶＢＮＩＯＨｶ｡ＭｶＢＮＩ｝ｾｉ｛ＨｶＫｶｭＭＲ､ＩＯＨｶ｡ＫｶｭＭＲ､ｴｾＱ =rl

...(231)

The Solution of Fundamental Equation Applicable to Semi-infinite Body

[1°] One dimensional body: The case in which A

=

ａｾｩｬｬＬ

P'Y

=

A

rft!

₀2 • g2ill

Let v= !1J..v)dv, G(V)=gJ..v)lJz(V),Tile obtain from (1

1 ) :

G( V)8,V=ag2/y_'.8T'(r""8T'V) (3)

Now, letting!;

=

r 2 / 4 t _{and transforming (3), we have:}

fNV+[ｇＨｖＩＯ｡ｧｚＫＨＱＫｴｮＧＩＯＲｾ｝｡ｴｖ］ｏ (31)

., V=A!ｾＭＨｬＫｭＧＩＯＲＮ･ｸｰＮ｛ -ao-2

·f

ｇＨｖＩ､ｾ ｽｾＫｂ (3z)

If G(V)

=

I,

If G(V)

=

G(!;), (32 ) can be solved. However, in general,

special methods are reqUired, and (3_{2 )} will be of little value.

Example i: The case in lilhich

｛ｖ｝ｾ］ｯ］ Vo. ｛ｖ｝ｾ］ｃｘｬ］ｏＬ G(V)=l

Vo= [!JJ..V)dV

leo'

Thus, the case reduces to that of ordinary homogeneous infinite

body. However, mt _{must be equal to zero.}

!Jz(V)dV= VoI(arrl /2)ｾｃｘｬ ｲｾＯ｡ＮﾷｾｉＯＲ､ｾＬ ｾ =r/4t...(34)

If [V]T'=O= tP(t) , used.

[2°] Two-dimensional. homogeneous ｢ｯ､ｾｲＺ The case in which r..

and ｰｾ are the same as before, and

V

｡ｮｾ｟ｇＨｖＩ are also

the same as ｢ｾｦｯｲ･

By letting or

ｾ］ _{[(x-x,)2+(y-y'n/4t}

ｾ =[,-2+,..'2-2rr'cos(O-O')]l4t

we obtain the transformed equation and solution of (3 ₁) and (3

2)

easily from (1 _{1 ) .} However, m'

=

1.

[3°] Three-dimensional homogeneous body_ r.. and ｰｾ same

as before Transform (1 ₁) by letting or ｾ =[(X-X')2+ (y-y,)2+(z+z'n/4t ｾ］ [r+,..'2-2rr' cos(O-O')+(z-z'nl4t. or ｾ］ {r+,..'2-2,.,..'[cosO'cosO'+sinO.sinｯＧＮ｣ｯｳＨｾＭｾＧＩ｝ｽＯＴｴＮ

It ｷｩｾｬ reduce to the same form as (3 1 ) and (32 ) with m'

=

2.

Problem ii:

Intide the bOdy ｰｾ｡ｴｖ = 0t(r..0xv) where r..

=

r..

_of

2 ( v ) , ｲＮＮＯｐｾ

=

a

o

2 •

Apply (22 2 ) , the solution to the problem of finite body,

assuming initially [v]x=O

=

b, [v]xa

=

O.

. 2

I'"

v=[V]c=",+-Xa

..

exp.[ -(aonlr/xaft]sin[ nrr(x- xa)/xa]_.

As x_a ｾ

00,

let nn/x = a, or nix = da.

a a

Letting ｛ｖｊｾＺＺＧＧＧＧ］｢ＧＬ He have the required solut ion:

where V=<b'[l-ｾ fooosinax-e-4 ...cdaLOOSina7}d7}]

2L'"

L'"

+ - 0(7})sina7}d7} e-lJ . · " · 'sin axda Ir 0 0 =2b'/lr1f2.f. . ｟ｲｾＧ､ｹ ",{2lJ.'/c 1 roo +2rrvtJo ＰＨＷｽＩ｛･ＭＨｾＭＢＧＩＧｉＢＢＧｴＭ･ＭＨｾＢＢＢＧＯ｣ＢＢＧ｝､Ｗｽ ...(36) V=

J

fz(v)dv, ｛ｖ｝ＬｾＢＬ］ !fz(U)du=Ax+B, Q)(x)=

[VJ;;t

C ",\

If f 2( v ) = 1 + A,V, the constant flow value ｛ｶ｝ｴ］ｾＧ when

[v] -0 = b and [v] = 0, can be expressed by the following

expres-x- xa

slon. Thus, when x_a ｾ ｾＬ

｛ｖＩｬ］ｾ =-1/A1-+[(1/A1+W-(2/A1+b)bx/xa]I/Z

" [v):::'''''=b. !Jen,e,U=[V)l=,.,= b(l+!A1b)=b'.

When ｰｾ

=

A

_o/a

₀2 • g2(V), exact solution is possible by the use of

the

e

function.

[4°] ｏｮ･Ｍ､ｩｭ･ｮｳｩｯｮ｡ｾｬ bOdy ｷｬｴＮＮｨｾ､ｵ｡ｬ change in property:

The case in which A = Aof2{v)rl+m-m'

ｯｾ

=

ａｾｾ

f (v)r2- p -:2 Problem ill: where where Problem lv: Ins1de the body where

[V)r=O=V" [V)r=",,=O, Ｈｖ｝ｬ］ｾ］ＡｰＨｲＩＮ 1"'"pralv=al(Ar'" arv)

A=Ao!Z(V)rl -m- ..., pr=Ao/ao2ﾷＡＲＨｶＩｲｾＭｬＧＬ m'=0,

1 and 2, m/p

=

n, further,

1: A solution to the problem of finite body to be obtained by

assum1ng [V]r=r = O.

a

v=U+PilB.UkJexp.(

Ｍ｡ｯｺｰｾ｡ｬＩ

J:r0yP_-m-l [W(r)

.

Here as 1s a pos1 t1ve root of ｕＬＬＨｾＩ］ｵ lB.=[ra".f"-I(Za.Vra,,)]-1 == [ra"JJ"+I(Za.Vra,,)]-I •. CD(r)=

[f

J2(V) dvｊＺＺＺｾｲＩ ,'. _{v=v,[ l-(r/ra)"'- I(d.V?)"J,,(za.ViP)} •

exp.(-ao7J2a,2;"

J/[nll)(a.

v

r

,il: f"+I(Za.v'ra")]

+[X] (310)

Where

01

[X]=)IIB.U,,(dexp.(-iZo7J2a/t)

f:'

ｲＢＮＮＮＮ｟ｉ｛ＱｰＧＨｾ｝ｕＬＬＨｲＩ､ｲＮ

•

11: In the case 1n wh1ch

[V]x

=00

=

0, when r a ｾ 00,

a n ｉｮＨｸｾＩ _{• xa}

=

O. Therefore Next ".1here 111: ｊＢＫｉＨｾＩ］［ＨｚＯｲｲｾＩＱＯＲ sinＨｾＭＨｚｉｉＫ 1)rr/4) =｛ｚＯｲｲｾＭ ｊＬＬＲＨｾＩｊＱＧＲ］ＨｚＯｲｲｸｾＩＱＯＲＮ

a.=a, da=rr/Z,/xa" .'. Za.+1v'ra"-Za.Vr?=;rr

., ｖ］ｖＬｻｬＭｾｷ｜ f"'a"-IU,,(d·exp.(-ao7J2a2t)da}+[X] I ".f)0 ...(3₁₁₎ [.xl=2p

L'"

exp.ＨＭ｡ＯＩｾ｡ｾｴＩﾷ ｕＬＬｾＩｕＮＨＮＭＡＺＩ｡､｡

LOll

ｸｰＭｭＭＭｬｾＨｸＩ､ｸ =p

L'"

ｾＨｸＩｸＢＢＢｾＭＭｬ･ｸｰＮＨ ｟ｲＺｾｾｮﾷ ＱＶｾｸＩＢＢ I. (zv'(rx)")d .. iZo7J2t " ao7J2t x .

However, n >- ＭｾＮ and it is needless to add that [X] is not influenced by changes in surrounding temperature with respect to time.

Thus, when one extends the problem from that of a finite body

to that of a semi-infinite body, n

=

m!p >- ｾ［ however, when the

problem is treated directly as that of a semi-infinite body,

n' > 0 in ( 3e ) •

[5°]

Relation between time and distance in temperature in

a body with a gradual change in properties

m' express curvature.

Consider temperature, taking r as a parameter: 1 +n>O [V]I=O= [V]l=oo=0,

l+n<O [V]I=O=O. [V]I=oo=OO.

l+n=O [V]I=O=O' [V]l='" = finite and a function of r.

That is to say, if 1 + n > 0, the temperature is always equal

to zero at the two extremeties of time. Further, if we let

r P/a₀2_p2 ｾ _{0, further consideration gives:}

a1v= v«(J/tr-(l+n)/t

••• [81V] 1=0 = [81VJl=oo=[a1v] 1=0'/(1+..) = 0 atv= v[ (J4/t2- 2(2+n)(J2/t+(1 +n)(2+ n)]/t2

.", _[aI2V]1:0=[atV]I=OO = [812V]I: O'/[2+n!<2l-n)li'] =0

where

[a

I2V] t =O.(o ...) =-(l+nr a negati ve Quantity.

Thus, when t = 0

2/(1

+ n)

=

rp

/ &o2

p2(1 + n), the temperature

is maximum, and since 1 + n > 0 and p ｾ _{0, vmax• is inversely}

proportional to ｲｾｰＫｭＮ

The above results show that, if p > 0, [v]t=o ｾ 0 except at

r

=

0; and if p > 0, [V]t=O

=

0 except at r

=

00. Further the

10C4S of v_{rna x•} is the curve t

=

ｲｾｐＯ｛｡ＰＲｰＨＱ

+ n)]; and it is

Next consiner the ｾｵ｡ｮｴｩｴｹ of heat flow. The amount of heat

flowing through a unit volume per unit time is ｑ］ｰｲﾷｶ］ＱＯ｡ｲｬﾷｾｲｐＭＲｶ

i.e. ,

Examination by letting r P/a02p2

=

e

shows that, ｾｨ･ｮ I + n > 0,

｛ｑｊｬｾｏ］ [Q]l="" =0,

Qm..=[PPV]l=9'/Cl+,,)=ｾｯＯ｛ a02(An2/1l1Inc)t+"r 1+"" J

i.e., so long as

e

remains finite, Q is inversely proportional

"max,

to rl+m.

If at a certain distance (point, line or surface) a fixed

quantity Q_{O 1s added to a body, the thermal conduction becomes:}

v=Qo(Aa2t)-(I+") exp.( -rPlao2p2t).

Further, the quantity of heat flowing through an infinitesimal

volume dQ

=

ｰｾ • v dx dy dz. Let ｾ

=

_{r P/(a0}2 p t )

and I +n > 0, and

consider the total quantity of heat: K

=

A

_o[a

₀2(Aa 2 ) I + n ] ,

Q".,=o-= QoKf_""';lr. (rPlt)I+" exp.( -rPjao2p2t)dr = 2QoKf..ao2p2)l+"JPＰｊＺ｣ｯ･ＭＧｾＢＫｉＭｬ､ｾ

= 2QoKf..ab2)1+"/p°l\l+n) ' ' . ' ' ' ' ' .(3₁₆₎

That is to say, in an infinite body, p is specially defined as a positive even number or a fractional even number; however, in

a semi-infinite body, p is simply assigned a positive real value

and is one half of the value shown above. In short, since the

quantity of heat which is added to an infinite or a semi-infinite body is constant, A can be determined from the equation

QO=2QoKf..ao2p2)1+"/p. F(l+n),

Again, the values of A can be determined for a circular cylinder or a sphere from the following equations:

Q""=1=_{2rc'QoKfo""1'-1. (rPlt)I+" exp.( -rPjao2p}2t)dr

=2rcQo(ao2p2)i+,,/pol\l+n). K , , ,(317)

In general, if n > 0, by letting R

=

JrP in (2'7) we have:

v= Bfo""(za)l+n e x p.( - ｡ｯｾｾ｡ｾｴＩＺＡｮＨｚ｡ｒＩＨ R··d(Za) (319)

］ｂＨｚＯ｡ｯＧｊｰｾｴＩｉＫｮ exp.( - ｒＲＯ｡ｯＧｊｰｾｴＩ (3₂₀)

Further, it is possible to assume that

R2= (v'Y"P )2+ (v'r'P)2_Zv'(yyl)P cos(O-O').

5. Conclus,ions

1. The solution of the fundamental equation, in which thermal

conductivity and thermal capacity are assumed to be functions of

temperature, is possible only in the case of rectangular coordinates. It is possible to derive an equation analogous to the equation of

ordinary homogeneous heat flow by using

e

or V, certain functions

of temperature. V becomes a heat function only when thermal

con-ductivity is constant, and can be applied to problems of finite or

semi-infinite bodies. Exact oolution is not possible where there

is a natural surface cooling.

2. The solution of the fundamental equation in which thermal

conductivity is a function of temperature and dimension, and

thermometric cmlductivity is a function of dimension, is possible

with respect to all coordinate systems. By introducing V, a heat

function, it is possible to apply it to the solution of problems

concerning finite and semi-infinite bodies. The problems of eddy

current conduction can also be solved as an extension of the problems of finite bodies.