B R I E F C O M M U N I C A T I O N S

T R A N S I E N T T E M P E R A T U R E IN A P O R O U S T U B E

M. D . M i k h a i l o v UDC 536.245:532.546

An analyt ica l solution is given for the t r ans ien t t e m p e r a t u r e dis tr ibut ion in a porous tube with genera l i zed boundary conditions at the inner and outer s u r f a c e s .

Porous cooling is an efficient method of t h e r m a l protec t ion under conditions of high t h e r m a l loads; it is used in cooling the nozzles of rocke t m o t o r s , the walls of ae rodynamic tubes, expe r imen ta l ins t ruments for sho r t - l i ved phenomena, e tc . In connection with th is , g rea t in te res t a t taches to the t e m p e r a t u r e d i s t r i - bution a c r o s s the th ickness of a porous wail . The s t eady- s t a t e p rob l ems are studied in [1-8].

It can happen in many ca se s that the expenditure of coolant will be exces s ive ly g rea t if it is d e t e r - mined f r o m the solution for a s t a t ionary t e m p e r a t u r e field, and will be found to be within reasonab le l imi t s if it is de te rmined on the bas i s of a solution for a nonsta t ionary s y s t e m . T h e r e f o r e the t rans ien t t e m p e r a - tt tre of a f lat porous wall is studied in [9-13].

A solution which can be used to ca lcula te the tmnsta t ionary t e m p e r a t u r e dis tr ibut ion in a porous tube was obtained in [14] by the method of finite in tegra l t r ans fo rma t ions , although its applicat ion to porous cooling was not shown. La te r in [15] a solution was obtained by the method of separa t ion of var iab les for the t e m p e r a t u r e of a porous tube for two different boundary conditions at the outer sur face of the tube :con- vect ive heat exchange with the surrounding medium; and a constant heat flux at the su r face . After uncom- pl ica ted t r a n s f o r m a t i o n s the solutions obtained [15] can be p resen ted in a m o r e compact f o r m . Because the solutions in [15] a r e r a t h e r c u m b e r s o m e and requ i re a l a rge amount of calculat ing work, in [16] it was p roposed to use the method of finite in tegra l t r a n s f o r m a t i o n s . It must be said that the solution obtained in [16] is l e ss c u m b e r s o m e than that in [15] since the authors have solved the p rob lem with s impl i f ied bound-

a ry condit ions.

An analyt ical solution for the nonsta t ionary t e m p e r a t u r e dis t r ibut ion in a porous tube is given in the p re sen t note on the bas i s of a solution of the genera l ized t r anspo r t equation given by the authors in [17].

The t e m p e r a t u r e f ield of a porous tube is desc r ibed by the equation [15]

00(~, Fo) 1 O (~ 00(~, Fo) ~ 2v O0(~, Fo) ~-Po(~), (1)

~o'< ~ ~ ~1, Fo~O,

where ~, Fo, and Po(~) a re the d imens ion less coordinate , the F o u r i e r number , and the in ternal heat source , r e spec t ive ly , while u is a p a r a m e t e r of the coolant flow r a t e .

Equation (1) will be solved for the boundary conditions

0 (~, O) = fo (~), (2)

Ao 00 (~o, F o) + Bo O (~, Fo) = b o, (3) og

A1 O0 (.~1, Fo) + B10 (~1, Fo) = bl, (4) og

where A0, B0, A1, BI, b0, and b 1 a r e fixed cons tan ts . By an appropr ia t e choice of these constants , it is pos - s ible , without difficulty, to obtain as pa r t i cu l a r e a se s the solutions given in [14-16], where f0(~) = const and

Po(~) = o.

Lenin Higher Mechanical E lec t ro techn ica l Inst i tute , Sofia. T r a n s l a t e d f rom Inzhenerno-F iz ichesk i i Zhurnal , Vol .21, No. 6, pp. 1101-1104, D e c e m b e r , 1971. Original a r t i c le submit ted March 22~ 1971.

, �9 1974 Consultants Bureau, a division of Plenum Publishing Corporation, 227 ~('est 17th Street, New York, N. Y. 10011. No part of this publica~tion may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission of the publisher. A

copy of this article is available from the publisher for $I5.00.

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We note in passing that an approximate calculation of the three-dimensional heat-conduction problem is reduced to an analogous one-dimensional problem in [18]o

Equation (1) can be presen ted in the form

~I-2v O0(~, Fo) __ 0 (~l-2v 00(~, Vo) )@~l_2Vpo(~) ' (5) ]

from which it is seen that it is a par t icular ease of a more general t ranspor t equation the solution of which is given in [17] in the form of s e r i e s of eigenfunctions.

In the case under examination the eigenvalues /~i and the eigenfunctions r (~) a re determined by the following S t u r m - Liuville problem :

t--2v (6) r (~) + - - t , ' (~) + ,,~, (~) = o,

Ao~' (~o)" Bor (~o) = O, (7) &,K (~,) + G ~ (h) = 0. (8)

Comparing (6) with the general ized Besse l equation obtained by Douglas and presented in [19], we find

, (~) = ~, {c,J,, (vg) + GY,, (~)}.

The boundary condition (7) is sat isf ied if we set

cl = ~ 0 ~ (.~o) + , ~ A J . _ , (,%),

G = - - BoJ , ( ~ o ) - - ~AoJ~_~ (~o) ,

while (8) gives the following charac te r i s t ic equation for determining Pi:

B0g , (~;) + ~AoY,~ 1 (~0) B y , (gh) + ~AY,L ( ~ ) BoJ,~ (~t~o) + ~tAoJ,~-i (P~o) Bj,, (~t~) + ~A~J,~_ 1 (p~)

(9)

(10) (11)

(12)

Then the solution of [17] is obtained in the following form~

0(~, Fo)= {boAl~I-~v--blAo~l-2v+ (~o~1)I-2v [ boB1 i ' - -

.G

+(blBo__boB1) .f ~ 1 + [Bo~_2v j" ~d~l-2v Ao]

G G ~~

d~ ~l--2v

1

%o

t g= l

~o i=1

g~ 2 . BoJ,, (~igo) + t~iAoJ, . (.aigo) __ b0 ~ + gl-2, ~, (g) Po (~) d~ ,

go

(13)

where

= - A2,11 For boundary conditions of the second type at both surfaces of the tube, i.e., when b0 = bl = 0, the

solution of [17] has the form

(14)

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0(~, Fo)-- ~, ~'-2~f~ ~ A~ ~-2~ d~ ~0

b h ~ g

~0 go b h ~ g

h ~ g

go go go ~.

g1-=, (g)

~-~+ fo(~) d ~ - - . . . bl i=1 ~o

AoJ,-1 (~g0) b0 - -2~ ~-2~ ~ (~) Po (~) dg . (15) X All~-t (~h~l) ab

b From Eqs. (13) and (15) for ~ = 1/2, 0, a n d - l / 2 solutions may also be obtained for bodies of simple

form without taking into account convective heat transport.

LITERATURE CITED

1. S. Weinbaum and H. L. Wheller, J r . , J . Appl. Phys., 20, No. 1, 113 (1949). 2. I. Friedman, J. Am. Rocket Soc., No. 79 (1949). 3. L. Green, J r . , J . Appl. Mech., Trans. ASME, 1__.9, No. 2, 173 (1952). 4. E. Mayer and J. G. Bartas, Jet Propulsion, 2__44, 366 (1954). 5. N.N. Gvozdkov, Vestn. Mosk. Univ., Ser. Mat., Makh., Astr . , Fiz~ Khim., No. 1 (1958). 6. P. Grootenhuis, J . Royal Aeronaut. Soc~ No. 578 (1959). 7. G . T . Sergeev, in: Heat and Mass Exchange in Capillary-Porous Bodies [in Russian], Minsk (1965). 8. M.D. Mikhailov, Inzh.-Fiz. Zh., 1__1, No.2,264 (1966). 9. P . J . Schneider and J. J. Brogan, ARSJ, 3._22, 233-236 (1962).

10. A . R . Mendelsohn, AIAA J., 1, No. 6 (1963). 11. W . P . Manos and D. E. Taylor, International Developments in Heat Transfer, 1961 International Heat-

Transfer Conference of ASME, AIChE, IME, ICE, Pt. 4 (1961), pp. 731-736. 12. M.D . Mikhailov, Nonstationary Temperature Fields in Shells [inRussian],Energiya, Moscow (1967). 13. M.D. Mikhailov, Nonstationary Heat and Mass Trasfer in One-Dimensional Bodies [in Russian], ITMO

AN BSSR, Minsk (1969). 14. E . M . Gol'dfarb, Izv. VUZ. Chernaya Metallurgiya, No. 1, 167 (1963). 15. L . M . Dzhidzhi, Raketnaya Tekhnika i Kosmonavtika, No. 11, 120 (1965) 16. G.A. Surkov and L. A. Konyukh, Vestsi Akad. Nauk BSSR, Ser. Fizika-Energetychnykh Navuk, No. 1,

120 (1970). 17. M.D. Mikhailov, in: Problems of Heat and Mass Transfer [in Russian], l~nergiya, Moscow (1970). 18. N.A. Yaryshev, in: Heat and Mass Transfer [in Russian], Minsk (1968), p~ 19. P. Schneider, Engineering Problems of Heat Conduction [Russian translation], IL (1960).

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