ON T H E

SOLUTION

PROBLEMS

WITH

OF

NONSTATIONARY

VARIABLE

HEAT-CONDUCTION

HEAT-TRANSFER

COEFFICIENT UDC 536.21

V. N. Kozlov

T h e a r t i c l e d e s c r i b e s an exact method f o r calculating the t e m p e r a t u r e field in solids when they a r e heated in a m e d i u m with a v a r i a b l e h e a t - t r a n s f e r coefficient and a nonuniform initial t e m p e r a t u r e distribution. In [1] a method for the exact calculation of the t e m p e r a t u r e field of a solid object undergoing heat e x change in a m e d i u m with a v a r i a b l e t e m p e r a t u r e and a v a r i a b l e h e a t - t r a n s f e r coefficient was d i s c u s s e d for a l a r g e n u m b e r of Bi(Fo)functions o f p r a c t i c a l i n t e r e s t , as applied to an infinite plate. F o r 8(1, Fo), the t e m p e r a t u r e of the heated s u r f a c e , we found in [1] an o r d i n a r y differential equation with v a r i a b l e coefficients which is s o l v a b l e by o p e r a t i o n a l methods [2]. The initial t e m p e r a t u r e distribution was a s s u m e d to be z e r o . We shall now show, using the e x a m p l e of a plate, how to deal with a nonuniform initial distribution. We shall a s s u m e that the t e m p e r a t u r e of the m e d i u m is z e r o . Heat t r a n s f e r t a k e s p l a c e at the plate s u r f a c e X = 1, while the s u r f a c e X = 0 is t h e r m a l l y insulated. T o solve the p r o b l e m , we m u s t e s t a b l i s h how a|

Fo)/OX v a r i e s with 0(1, Fo).

It was shown in [3] that if F| > 0, the function 0|

Fo)/0X can be r e p r e s e n t e d as a convergent s e r i e s r

O@(1, Fo) aX

~ ' ~ Z, (Fo), z.a

(1)

i=[

in which Zi(Fo), i = 1, 2, . . . .

a r e d e t e r m i n e d f r o m the solution of the o r d i n a r y differential equations T~2i (Fo) + Z~ (Fo) = 2T~6 (1, Fo), i = 1, 2, . . ,

(2)

with initial conditions Zi(0 ) = Z~, uniquely d e t e r m i n e d by the initial t e m p e r a t u r e distribution function. F o r the equations in (2) we have 4 Ti : ( 2 i - 1)~n 2 The solutions Z i (Fo) of t h e s e equations with initial conditions Z ~ which a r e nonzero at t i m e F| = 0 - 0 {before the s t a r t of the p e r t u r b a t i o n ) w i l l be identical for F| __> 0 + 0 (after the s t a r t of the perturbation) with the solutions Yi (Fo) of the equations Tiyi(Fo) -j- y~ (eo) = 2T~O(1, eo) + TiZ~

i = 1, 2, . . ,

(3)

with initial conditions which a r e z e r o at t i m e F| = 0 - 0 [4]. H e r e 6(Fo) is the Dirac &function. Summation of the left and right sides of Eq. (3), taking account of (1) and the identities Zi(Fo ) - Yi(Fo), which a r e valid for F| -> 0 + 0, yields: ~o

r162

r

ox i~l

i=1

x_a i=l

Now we multiply each equation of (3) by T i and differentiate t e r m by t e r m : T 2,yt'"(Vo) + T,y, (Fo) = 2T~6(I, Fo) + T2Z~ $(Vo), i = 1, 2, . . .

(5)

F. E. D z e r z h i n s k i i Heat Engineering Institute, Moscow. T r a n s l a t e d f r o m I n z h e n e r n o - F i z i c h e s k i i Zhurnal, V o l . 2 0 , No. 5, pp. 921-924, May, 1971. Original a r t i c l e submitted M a r c h 25, 1970. 1973 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without permission of the publisher. A copy of this article is available [rom the publisher for $15.00.

657

Summing with r e s p e c t to i in (5), we obtain the sum ~ T i ] i ( F o ) and substitute the resulting e x p r e s s i o n into (4): ~=1

2

T~y'i(F0)--6(1 , Fo)

2

2 r ~ + o ( 1 , Fo)

2

2T, -- 0e(1, Fo)

6(Fo)

T+Z~ 8(Fo)

2

T, Z~.

(6)

P r o c e e d i n g with r e p e a t e d t r a n s f o r m a t i o n s of this kind we finally obtain an ordinary differential equation for | Fo)=

Z

Fo> +O,l, ox Vo,+ 2

a+ d-~o" o (t,

n~l

b,.

(Fo)

(7)

tn~0

with the coefficients co

a+ = (-- 1)n+z'~2TP, n = 1, 2, ...

(8)

i=1

bm = (-- I)'~§ "~ Tin'4-1, m = 0 , A~ Z o~-i

1. . . .

(9)

i=1

Taking account of the boundary condition of the third kind for the case of a medium at z e r o t e m p e r a ture,

09(1, Fo) = Bi(Fo)O(1, Fo),

(10)

OX

we finally a r r i v e at the equation Bi(Fo)O(1, F o ) +

a,~-~-~ne(1, Fo)= n~l

8 ( Fd-~o o).b,~

(11)

m~O

It follows f r o m the method used for obtaining Eq. (11) that h e r e the initial conditions for Fo = 0 - 0 will be z e r o . A s s u m e , as in [1], that Bi (Fo) = Bio -- [o (Fo),

(12)

where Bi0 = const and f0(Fo) is r e p r e s e n t a b l e by a rational combination of sines (or cosines), polynomials, and exponents. P r o c e e d i n g in a nmnner analogous to [1], for a solution of Eq. (11), in the image domain, we make use of the "bifrequency t r a n s f e r function" method of [2]. According to [2], 0(1, s ) = rots L 0(1, Fo)= Z

1 dVi-l [(q--qi)ViW(s' q)]' (~?j2.1)t dqVF1

(13)

q=qi

w h e r e the sum is taken over all the qj-poles of the second argument of the function W(s, q), and ~ is the multiplicity of these poles. If (12) is satisfied, we can obtain the bifrequeney t r a n s f e r function W(s, p) in the f o r m of the absolutely and uniformly convergent s e r i e s W(s, p) = ~ W~(s, p).

(14)

v=O

In the p r o b l e m under consideration the zeroth t e r m of this s e r i e s yields the formula oo

++, k~O

where

658

ii+,

co

(16)

(s) = Bio + X a:~' k~l

"

and the a k and b k a r e the coefficients (8) and (9) of Eq. (11). It should be noted that the sum in (16) is a s e r i e s expansion of the function ,Is th ( s , and in (15)

z?

bhsk=i=0

Therefore

S ~

- -

T,

, z0

=

W o(s, p)

(17)

~1

k=0

(18)

~

PtF(s) .= s +

TI 1

'

9 (s) = Bio + V's-th ]Z~ All the subsequent (v = 1, 2 . . . .

) t e r m s of the s e r i e s (14) a r e found by the r e c u r s i o n formula:

'~

(vj

W~(s, p)=

(19)

I

dv/-____~ i)! dqvi-I [(q--qyiW,(s, q)W~_l(s-- q, p--q)].

(20)

q=q]

The sum in (20) is taken over all the q -poles of multiplicity v. of the second argument of the b i f r e quency t r a n s f e r function Wu(s , q), which in ~ur p r o b l e m has the for)m

~ . (s, q) = F~ - ~ -(q) ,

F0(q) = ~..qL:o (Fo)

After determining the t e m p e r a t u r e | Fo), the t e m p e r a t u r e field of the plate | f r o m the solution of the p r o b l e m with a boundary condition of the f i r s t kind.

(21) Fo) can be found

NOTATION | L X a

t x = x/L

Fo = at/L z Bi (Fo) = o~ (Fo)L/k

is is is m is is is is is is

the the the the the the the the the the

temperature; thickness of plate; space coordinate; t h e r m a l dtffusivity; t h e r m a l conductivity; h e a t - t r a n s f e r coefficient; time; dimensionless coordinate; F o u r i e r number; Biot number. LITERATURE

1. 2. 3. 4.

CITED

V . N . Kozlov, Inzh. F i z . Zh., 1__88,1 (1970). I . N . B r i k k e r , Art| i Telemekhan., 8 (1966). V . N . Kozlov, Inzh. Fiz. Zh., 15, No. 5 (1968). A . V . Solodov, L i n e a r Automatic-Control Systems with Variable P a r a m e t e r s [in Russian], Fizmatgiz (1962).

659