numerical solution of heat conduction in a rectangular rod subjected to an N-cascade wall temperature

0735-1933/90 $3.00 + .00 Printed in the United States

INT. COMM. H E A T MASS T R A N S F E R Vol. 17, pp. 675-687, 1990 ©Pergamon Press pie

ANALYTICAI./NUMERICAL SOLUTION OF HEAT CONDUCTION IN A RECTANGULAR ROD SUBJECTED TO AN N-CASCADE WAI J. TEMPERATURE M.A. Ebadian and T.C. Yih Department of Mechanical Engineering Florida International University Miami, FL 33199

(Communicated by J.P. Hartnett and W.J. Minkowyet)

ABSTRACT An analytical/numerical method is employed to obtain the solutions for the steady state heat conduction in a rectangular homogeneous rod of infinite length. The rod is composed of an isentropic heat conducting material and is subjected to a multi-cascade wail condition with no heat source or heat sink. By using the Fourier integral transform, the temperature field in the rectangular rod is obtained either analytically or numerically, using an eigenvalue subroutine or a Runge-Kutta subroutine, respectively. In the case of a single cascade, the temperature variations in the transverse direction are plotted at various longitudinal sections along with the temperature variations for various points in the transverse coordinate. Also, for the same wall temperature condition, the generating lines of axisymmetric isothermal surfaces are plotted. The wall heat flux variation along the longitudinal direction is obtained and plotted against the longitudinal coordinate. To represent the total heat flow, a dimensionless heat flow coefficient at the cross-section corresponding to the wall temperature discontinuity is defined and its value is calculated. The expressions of the temperature field and heat flow for a rectangular rod subjected to a multi-cascade wail temperature case are obtained, and the significance of each is also discussed in detail.

Introduction An inspection of literature surveys, including Eckert, et al. [1,2,3], Kakac, et al. [4], Shah and London [5], Soloukhin and Martynenko [6], and Martynenko [7], indicates that significant attention has been devoted in recent years to the investigation of heat conduction in solid pipes with various cross sectional areas. This is mainly due to the advent of the digital computer that has allowed numerous investigators to report on comprehensive and sophisticated finite difference and finite element cedes for the numerical solution of heat conduction problems. Although these solutions seem quite convenient, their accuracy is often difficult to assess, and the results are less enlightening and systematic than those obtained from an analytical solution. Exact closed-form solutions are, however, not available except in the simplest case of a 675

676

M,A. Ebadian and T.C. Yih

Vol. 17, No. 5

semi-infinite rod with a fixed temperature imposed at the boundary, [8, 9, 10]. The problem considered here is a rectangular rod subjected to a general cascade wall temperature as shown in Figs. 1 and 2. The wall temperature distribution has (n+l) temperature discontinuities at the indicated longitudinal positions.

7 A IT~ /,. . . .

i

" I

I

: , ....

\ ~ - : i.... ~IJ-~ ---.-'

r<, 1, <,-1

i I

- ~ ~ !

/ \

I / \~ ~_,/

/

I i

z

"~/

, !-..-,~,....-

H .. Half height of the Rod T t , T s .... T., T . = Wall Temperature Change

Ll ' LII.... Ii

8 Step lengths

FIG. 1 Multi-Cascade Wall Temperature Condition

Y

]

a, m, I,~-,

r--w-... "~-

I

mtl~!,

I/"'-.. I : / ~ /

0

#

FIG. 2 Rectangular Rod Under a Step Wall Temperature Change The normal difficulty with this type of problem is the existence of a singular point, due to the discontinuity at the wall. This singularity strongly affects the entire temperature field. In the present paper, this difficulty is completely eliminated by using the method of Fourier integral technique in a truly exact manner, and a simple closed form temperature distribution is obtained in the entire field.

Moreover, this analysis is completely accurate everywhere

including the entrance region. The solution presented in Carslaw and Jaeger [! l], for parallel

Vol. 17, No. 5

HEAT CONDUCTION

IN A R E C T A N G U L A R

ROD

677

rectangular pipes deals with conduction in the single-cascade rod subjected to a temperature jump.

Their results are presented in the infinite summation.

Wibulswas [12], is the forced convection in rectangular temperature.

The solution presented in

rod subjected to constant wall

He has numerically obtained the temperature distribution for a single-cascade

rectangular duct. The problem with these types of solutions is that they lose their accuracy in the entrance region. Thus, it is obvious that there is a definite need for a straightforward approach to solving heat conduction in a rod with various cross-sectional areas. This objective is achieved by using the Fourier integral technique [13, 14]. In addition, since the problem is internally linear, the superposition principle remains valid. Accordingly, a single-cascade wall temperature is first solved to create a fundamental solution. Then all other multi-cascade wall temperature cases are obtained from the solution of the single-cascade wall temperature by applying the superposition technique. presented in three different ways. transversal

dimensionless

The results for temperature distribution are then

First, the temperature variation is plotted against the

coordinate

for various

values of dimensionless

longitudinal

coordinates, then the temperature variation is plotted against the dimensionless longitudinal coordinate for various values of dimensionless transversal coordinates. Finally, the generating lines of isotherms surface, which are the surfaces of revolution plotted against dimensionless transversal and longitudinal coordinates.

The expressions for the overall heat flow and

dimensionless heat flow coefficient are obtained through the rectangular rod. Also, the heat flow distribution curve is plotted against the wall surface. In light of the foregoing, it is believed that the availability of such an analytical/numerical solution dealing with such diverse fields as metal casting, welding extrusion, thermal stress analysis of material, automotive industries, and other manufacturing

processes will be an

important contribution for engineers. Enertv Eauatlon for a S i n g l e - C a s c a d e Wall Temnerature The partial differential equation describing the geometry shown by Fig. 2 is given as Oat + Oat + Oat . 0 8~ =

an ~

(1)

8f z

where E, '7 and ~"are the dimensionless coordinates of the rectangular rod and are defined by (-~, X

,=~, Y

~=~ Z

(2)

Here, H represents the half height of the rod, X and Y lie in the cross section of the rod perpendicular to the long axis and at the center of the rectangular cross section, and Z is the dimensional longitudinal coordinate. Also, t in Eq. (1) represents the dimensional temperature distribution.

As was explained before, the temperature distribution for a single-cascade wall

temperature is considered first. Then this solution will be expanded using the superposition technique to obtain the temperature distribution for a general multi-cascade wall temperature.

678

M.A. Ebadian and T.C. Yih

Vol. 17, No. 5

A single-cascade wall temperature problem is depicted in Fig. 3 in which the origin of the longitudinal axis is taken at the section where the applied wall temperature undergoes a sudden increase (step change in wall-temperature) of T 1. In addition, the temperature is measured from a value of

T1/2.

By this choice, an antisymmetric temperature distribution

in the

coordinate is maintained and thus, temperature satisfies the following conditions:

l.-~ Symmetry Condition:

=

0 for

~

0 for

~-0

0

i ~0~

T 1

-~- for Boundary Condition:

(3)

0

t I.urf~

for

T1 - - ~ - - for

f>0

and

~=+1

f=0

and

~=+1

f
and

~=+-I

(4)

This type of boundary condition can be satisfied if the Fourier integral is considered as t(f, r?, f) = i

[ A(f, ~/; a) cos a f + B(f, r/; ca) sin

f] da

(5)

where - ® < ~" < ® and,

A(~, ,1;a) = ~-

t(~, tl, f) COS a f df

(6)

B(~, ,I; ") = _l

o.

ff

t(f,,~, f) sin a f df

are the Fourier transform of temperature.

-t.,~---~

i

-t,,.~- ~ - - ~

tl

I

H !

FIG. 3 Single Wall Temperature Change The discontinuous boundary conditions of Eq. (4) can be satisfied (noting t is an odd function in f) if the Fourier-Sine integral of t is considered as

Vol. 17, No. 5

HEAT CONDUCTION IN A RECTANGULAR ROD

679

(7)

t(¢, ~, f) = 2 ~ B(~, 17;a) sin a f da which is subjected to the boundary conditions T1

T

2 i B(~w"Tw;a) sin a f da =

0

for f > 0 for

Tx --~-- for

f-0

(s)

f<0

Here, it must be noted that the condition t - 0 for f = 0 is a mathematical requirement for the Fourier transform and does not modify the physical condition of the problem. Now, if the Fourier-Sine integral of the unit step function [15], 1 for

f>0

0

for

f =0

_ I

for

f,0

T

1 i sin a f

D t¢

dog i

~g

(9)

is considered and compared with Eq. (8), one obtains

e(f,,, ~,,; a) = TI

(I0)

Q~

Also, Eq. (7) can be written in terms of dimensionless parameters r and ~ as

r(~'~'f) " i ~(~,~,a) sin a f da

(II)

where the dimensionless temperature r in Eq. (l I) is defined as f =

t Tx/a~

(12)

and the dimensionless Fourier transform of temperature j9 in Eq. (11) is defined as =

B

(13)

Tl/a,r

on the periphery ~(~w,~;a) = l

and

/~(~,~/w;a) = l

(14)

The integrand of the integral of t, Eq. (II), will satisfy Eq. (I) since it is a Laplace equation. Therefore, the partial differential equation

~ + a.v = ~,2 #

(15)

follows, where ~¢¢ and Ps~ in Eq. (15) indicates the double differentiation with respect to the dimensionless coordinates ~ and ~, respectively.

Solution of this equation considering the

680

M.A. Ebadian and T.C. Yih

Vol. 17, No. 5

symmetry condition of Eq. (3) and the condition of Eq. (10) can be numerically obtained using a finite difference technique. The values of//1,//2,//s ..... //23a are determined for the prescribed values of a by the inversion of a matrix of order 232 x 232. The numerical values of / / a r e substituted

into Eq. (11) and the integral

in Eq. (11) is numerically

evaluated

using the

Runge-Kutta technique for various values of n and f. In Fig. 4, the temperature variation is plotted against the dimensionless coordinate for various values of f. In Fig. 5, the temperature variation is plotted against the dimensionless coordinate ~" for various values of n.

T f-OO

¢- l . s ¢ - s.a ¢ - l.z ¢ - o.g

¢-

0.6

¢ - o.s

¢- 0.1 ¢-0 + ¢-0• ¢ - -O.S ¢,- - O . S ~

¢'- --O.S

¢ - -0.9 ~'- -1.1

¢- -La ¢ - -1.5

FIG. 4 Temperature Variation in the ~ Direction

T i

r,/2 J

w~

~,. &!

q..u Lv ~,,.u

FIG. $ Temperature Variation in the f Direction

Vol. 17, No. 5

HEAT CONDUCTION IN A RECTANGULAR ROD

681

On the basis of the temperature distribution Eq. (11), the generating lines of isotherm surfaces, which are the surfaces of revolution, are plotted in Fig. 6 showing seven different isotherms. These are plotted in the plane of ~ and f (both dimensionless coordinates) with the selected r values of r

T

r

=

(0.15, 0.2, 0.3, 0.5, 0.7, 0.9, 1.0) TI/2.

0.16

Iv -

=, 0 . 3

t" - 0 . 6 'r ~ 0 . 7

~

~" - 0 . g I, = 1.0

-2.0

O.t5

' r i,' 0 . 3

~v , . 0"5 ' r =, 0 . 7

i I

-1.5

-1.0

-0.5

0.0

v = 0.g I" . . l . O

0"5

1.0

1.5

2.0

FIG. 6 Lines of Isotherms As expected, it must be noted that an inspection of the curves in Figs. 4, 5 and 6 shows a definite point of singularity at ~"- 0, ~ = 1.0. In addition, all lines in all of these figures form double symmetric families relative to the horizontal and vertical reference axes. Clt|¢ulatlon of Heat Flux In a R e e t a n n l a r Rod To define the wall heat flux distribution in the longitudinal direction, the temperature gradient in the direction perpendicular to the wall is obtained from

Eq. ( l l ) as

aq

t(f,q,f) -

x/w

D(fw, qw;a) sin a f da

(16)

Now, in order to define a dimensionless wall heat flux coefficient, a reference heat flux is considered as the heat flux through a circular rod of 2R made of the same material as that of the original rectangular rod, and whose surfaces are maintained under the same temperature difference of T x [13].

2 ~ 8 Nf = ~" Jo ~ # (~w, ~w; a) sin a ~" da

(17)

682

M.A. Ebadian and T.C. Yih

Vol. 17, No. 5

The integration in Eq. (19) is numerically performed using the Runge-Kutta integration technique and the resulting wall heat flux coefficient is plotted against the dimensionless coordinate f in Fig. 7.

N¢ .

~

I

'

i

'

I

'

I

'

I

'

I

'

I

'

I

U

¢ FIG. 7 Dimensionless Heat Flux Distribution on Lateral Surface Another physically significant heat flow is the heat flow through the section f = 0 (considered positive when heat flows in the positive f direction) where the applied surface temperature undergoes a sudden jump T r The required temperature gradient in the f direction at the section

~

f =

0 is

~=o

~H

~(~,,7;a) da

08)

Thus, the heat flow through the section at f = 0 in the positive direction of the Z axis is 1 1 F = - 2kH(l)

~

f=o

where k is the thermal conductivity of the rod material. Comparing this heat flow with the reference heat flow, F*, based on the reference circular rod, one obtains F °= -2kH(l)

TI = 2H - kTl

(20)

Then the dimensionless heat flow coefficient for the section f = 0 is obtained as

F---~ = ~.

~ ( ~ , ~ , a ) da

df dq

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H E A T C O N D U C T I O N IN A R E C T A N G U L A R R O D

683

The value of this factor after a R u n g e - K u t t a technique integration is performed is N - 9.0718

Determination of EnerEv Eauation and Heat Flux for Double-Cascade Wall Temnerature As it was discussed before, since the problem is linear, a double-cascade wall temperature case is obtained by superimposing two single-calcade, as shown in Fig. 8. According to this figure, the distance between each single-cascade is Ha. and the temperature discontinuities for the first and second cascades are T 1 and T 2, respectively.

,..-÷ ..................

i

~. A ~ d e of'r I

I

..................

r'- lia "-I

I

¢ e t t ~ t" e~ead*

t.a= _o~T,)

'

k. . . . . . . . . . . . . . . . .

~' Combined doubla.--ommade

FIG. g Double-Cascade Wall Temperature Condition

The temperature

distribution

for single and double-cascade based on Eq. (I I) can be

defined respectively as

TI i

/~(~,q;a) sin a ~ da ,

T, i

~ ( ~ , ~ ; a ) sin a(~" - a ) da ,

¢(~,t?,~)- ..~

r ( ~ , ~ , f ) = -~-

for the single--cascade T I

for the single-cascade T 2

(22)

(23)

Therefore, the double-cascade temperature distribution can be obtained by superimposing the temperature distributions,

r(~,t?,~) = T

Eqs. (22) and (23) as:

/~(~,~;a) sin ~ f da + T

/~(~,~;a) sin a(f-a) da

(24)

684

M.A. Ebadian and T.C. Yih

Thus, the longitudinal

Vol, 17, No. 5

temperature gradients at the section ~ = 0 and f ffi a, respectively, are

f=o

*H

B ( g , t / ; a ) da + lrH

/ / ( ( , t / ; a ) cos a a da

f=o

~H

~ ( ~ , ~ ; a ) cos a a da + ~ H

j g ( ~ , ~ ; a ) da

(25)

(26)

Then, the heat flows through the sections f = 0 and f = a from left to right, using Eqs. (25) and (26), respectively, are Fo = - 2kH(l)

~

f=o

(28)

On the basis of the same reference

[T,]

heat flow F*, Eq. (20), the dimensionless

T,]

heat flow

coefficient at the sections f -- 0 and f = a are obtained, respectively, as No = 2

p+

Q

,

Na=__

p+

(29)

Q

where

o ~;ummur¥ and Conclusions A new procedure for solving steady state heat conduction problems is given that involves Fourier integral transform.

Instead of having a classical solution in terms of single or double

infinite summation [11], or a numerical solution [12], to solve the heat conduction problem in the single-cascade rod subjected to wall temperature proposed.

The difficulty

with the classical infinite

jump, a straightforward

approach

is

solution is the effect of numerical

r o u n d - o f f associated with the sum of very large numbers of terms (or more for two or three dimensions problem)

which can severely affect the accuracy.

In addition, in both cases of

infinite summation and numerical technique, they lose their accuracy at the entrance region dramatically.

However, the analysis of this paper indicates that by using the Fourier integral

technique, the above difficulties associated with infinite summation and numerical technique are overcome. Furthermore, the temperature field is accurate in every region, including the entrance section (temperature discontinuities).

The method proposed is very effective and is

applicable to any rod with various cross-sectional area [13,14].

Vol. 17, No. 5

HEAT CONDUCTION

IN A R E C T A N G U L A R

ROD

685

The problem considered herein is that of the rectangular rod subjected to step wall temperature change. The temperature distribution for the single-cascade wall tempera-ture is first obtained by using the Fourier integral technique and then on the basis of this temperature distribution, the heat flow and heat flow coefficient is obtained. The temperature distribution is presented graphically in three different manners as is shown in Figs. 4, 5 and 6. These illustrations can be effectively and easily used by engineers and practitioners to obtain the temperature

field at any section and any coordinate, rather than just rely on infinite

summation equation as is given by Carslaw and Jaeger [13]. The preceding analysis for single-cascade wall temperature is used to obtain the temperature distribution and heat flow coefficient for double-cascade and triple-cascade wall temperatures.

In fact, the analysis in

double and triple-cascade wall temperatures shows that the extension of the formulation to a multi-cascade wall temperature case with more than two discontinuities is evident. Acknowledumeut

The results presented in this paper were obtained in the course of research sponsored by the Department of Energy under Subcontract No. 19X-SC746. Nomenclature A

The first coefficient of the Fourier transform of temperature distribution, Eq. (6)

B

The second coefficient of the Fourier transform of temperature distribution, Eq. (6)

F

The heat flow through the section ~ = 0 in the positive direction of the axis, Eq. (19)

FI

The heat flow through the section f = a in the positive direction of the axis for a double-cascade surface temperature case, Eq. (28)

Fo

The heat flow through the section f = 0 in the positive direction of the axis for a double-cascade wall temperature case, Eq. (27)

F*

A reference heat flow defined by Eq. (20)

H

Half height of the rod, Fig. 2

Ha

Longitudinal distance between surface temperature discontinuities of a double-cascade wall temperature

k

The thermal conductivity of the rod material

N

The dimensionless heat flow coefficient for the section f = 0, Eq. (21)

Na

The dimensionless heat flow coefficient at the section of the second wall temperature discontinuity for a double-cascade wall temperature case, Eq. (29)

NO

The dimensionless heat flow coefficient at the section of the first temperature discontinuity for a double-cascade surface temperature case, Eq. (29)

Ne

The dimensionless wall heat flux coefficient, ECl.(17)

P

A function defined by Eq. (30)

Q

A function defined by Eq. (30)

686

M.A. Ebadian and T.C. Yih

Vol. 17, No. 5

T1

The first wall temperature discontinuity

T2

The second wall temperature discontinuity

t

The temperature distribution in the rod

X

The dimensional longitudinal coordinate along the line of discontinuity of the surface temperature, Eq. (2)

Y

The dimensional coordinate in the transverse direction, Eq. (2)

Z

The dimensional longitudinal coordinate, Eq. (2)

fmaxk.S.m2~Jl Og

Fourier integral variable The dimensionless Fourier transform of temperature, Eq. (13)

17

The dimensionless transverse coordinate, Eq. (2) The dimensionless longitudinal coordinate, Eq. (2) The dimensionless longitudinal coordinate, Eq. (2) Dimensionless temperature distribution in the rod, Eq. (12)

f

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E.R.G. Eckert, R.J. Goldstein, E. Pfender, W.E. Ibele, S.V. Patanker, S.W. Ramsey, T.W. Simon, N.A. Deckor, T.H. Kuehn, H.O. Lee, and S.L. Girshick, "Heat Transfer - A Review of the 1986 Literature," Int. J. Heat and Mass Transfer, 29, 1767-1842 0986).

2.

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3.

E.R.G. Eckert, R.J. Goldstein, E. Pfender, W.E. Ibele, S.V. Patanker, S.W. Ramsey, T.W. Simon, N.A. Deckor, T.H. Kuehn, H.O. Lee, and S.L. Girshick, "Heat Transfer - A Review of the 1986 Literature," Int. J. Heat and Mass Transfer, 31, 2401-2488 (1988).

4.

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5.

R.K. Shah and A.L. London, Laminar Flow Forced Convection in Ducts, Academic Press, New York (1978).

6.

R.I. Soioukhin and O.G. Martynenko, "Heat and Mass Transfer Bibliography -Soviet Literature," Int. J. Heat and Mass Transfer, 26, 1771-1781 (1983).

7.

O.G. Martynenko, "Heat and Mass Transfer Bibliography -Soviet Literature," Int. J. Heat and Mass Transfer, 31, 2489-2503 (1988).

8.

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9.

M.N. sizik, Heat Conduction, John Wiley & Sons, New York (1980).

10.

C.J. Tranter, Integral Transforms in Mathematical Physics, Methuen and Co. Ltd., London; John Wiley and Sons, Inc., New York (1956).

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H E A T CONDUCTION IN A R E C T A N G U L A R ROD

687

11.

H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids, Oxford University Press, London (1959).

12.

P. Wibulswes, Laminar Flow Heat Transfer in Non-Circular Ducts, Ph.D. Thesis, London University, London (1966).

13.

M.A. Ebadian and O.A. Arnas, "Heat Conduction in a Slab Subjected to an N-Cescede Wall Temperature on Both its Surfaces," Int. Comm. J. of Heat and Mass Transfer, 16,2, 283-293 (1989).

14.

M.A. Ebadian and H.C. Topakoglu, "Heat Conduction in a Circular Rod Maintained Under a Multi-Cescade Wall Temperature," Int. Comm. J. of Heat and Mass Transfer, 6,1, 65-77 (1989).

15.

R.V. Churchill and J.W. Brown, Fourier Series and Boundary Value Problems, McGrawHill, New York (1959).