An approximate solution of heat conduction equation with mixed boundary conditions of a rectangular plate

Mechanics Research Communications, Vol. 25, No. 6, pp. 631-636, 1998 Copyright © 1998 Elsevier Science Ltd Printed in the USA. All rights reserved 0093-6413/98 $19.00 + .00

Pergamon

PII S0093-(~113(98)00081-0

AN APPROXIMATE SOLUTION OF HEAT CONDUCTION EQUATION WITH MIXED BOUNDARY CONDITIONS OF A RECTANGULAR PLATE J. Stefaniak, J. Jankowski Institute of Applied Mechanics, Poznafi University of Technology Poznafi, Poland

(Received22 April 1998; acceptedfor print 23 July 1998) Introduction In papers [1], [2] and [3] an approximate method of solution of linear problems of heat conduction in solids has been presented. This method is applicable to direct and inverse problems as well. In these papers some simple inverse problems and some one-dimensional direct problems have been presented. By direct problem we understand the solution of heat conduction equation with given initial-boundary condition and possibly the intensities of heat sources. One of the inverse problems consists in finding the intensities of heat sources, when the temperature of the considered solid is known. The other one consists in finding the boundary conditions, when their kind is known, i.e. when the temperature at the boundary is a known function of time, or the flow of heat at the boundary is known, or the linear connection between temperature and flow at the boundary is given. Also the mixed boundary conditions may be taken into account. The method in application to direct problems consists in introducing "fictitious" heat sources located at some points outside the region under consideration. There are to find such intensities of these sources, which assure the prescribed temperature at selected points of the boundary. By suitable choice of location of heat sources and sufficient amount of the sources and points at the boundary it may be assumed that the temperature field induced by the sources within the region is a good approximation of the temperature induced by the given boundary conditions. In this paper the application of this method to an example of mixed boundary-value problem in D-2 is considered. Method of solution Let us consider the following boundary value problem LT=0,

LIT = g(x°,t),

(1)

T(x,0) = 0,

where 10 L_V 2 ____ z¢ Ot

x~V, x ° ~SV,

631

632

~: = -

J. STEFANIAK and J. JANKOWSKI ,¢

- thermal conductivity, 2 - heat conductivity (heat conduction coefficient), c - specific

cp

heat, p - mass density, 7"- temperature difference, ct,fl - constant coefficients. The boundary condition reduces to the condition of the first kind, if fl= 0 and to the second kind, if a = 0 . Let us first consider an unbounded region. Let a source of intensity W " ( ~ , t ) act on an arbitrary surface S (i.e. ~ S ) ,

where S ~ ( V + 3 V ) = O

(Fig.l). Then the temperature T " ( x , t )

is

represented by the relation t

T'" (x,t) = ~"~F*" (x - ¢,t - r ) W " (~, r ) d r d s ,

(2)

'¢ 0

where F'" (x,t) is the unique solution of the equation

tF" :- 1--8(x) ~t)

A and 6(.) is the Dirac distribution. S

Fig. 1. The region under consideration V, its boundary cV, the surface S and the discrete distribution of heat sources ~, To solve the initial-boundary problem (l) the intensity W"(~, t) must be chosen in such a way that the temperature T**(x,t) satisfies equation (1)l and conditions (1)2 and (1)3. That means that the problem has been reduced to solution of an integral equation with respect to W" (~,t) obtained by inserting the temperature T " ( x , t ) given by (2), into the boundary conditions (1)2. The exact solution of that integral equation (i.e. the intensity W**(~, t)) introduced into relation (2) gives the temperature, which satisfies the equation (1)l in the region V and the conditions (1)2 and (1)3. The main difficulty lies in the exact calculation of the source intensity W " ( ~ , t ) . An approximate solution of the integral equation may be obtained by discretisation of the source in time and space. The method of discretisation is described in detail in paper [3]. The essential point of the method consists in replacement the intensity continuously depending on time by an intensity changing stepwise with -

m-I

time W*(~,t)=~-~W[(~) r l ( t - t ~ ) , k=0

HEAT C O N D U C T I O N IN RECTANGULAR PLATES

-

633

continuously distributed sources on S by n concentrated sources located at the points ~j eS, (Fig.l).

Then the approximate temperature can be written in the form n

m-|

T(x, t) = E E wk' F ( x - ~,, t - t k )r/(t - t k ),

(3)

/=l k=O

where F(x, t) is the unique solution of equation LF

= -15(x)q(t)

and r/(.) is the Heaviside function. It is not possible to find, in general, such a set of W,k that the temperature given by (3) satisfies the boundary condition (1)2 at each point and at each moment. But we are able to require the temperature (3) to satisfy this condition at n selected points xjo ~ d V , j = 1,2 ..... n and at the moments t t, l = 0,1..... m - 1. Replacing x ° by x~ and t by tj in the formula (3) and inserting them into boundary conditions (1) 2 one obtains the following set o f n x m algebraic equations for W,k: , ,,-i

E Z w: ,=l

(-7-

k=o

dF"

(x~ - ~j ,t / - t k ).1 0#l

]=

Substituting the obtained values W,k into relation (3) we obtain the approximate temperature T(x,t), which satisfies the equation (1)j in V and the boundary conditions at n points xj0 e d V and at m moments it. Formulation of the problem The above procedure has been applied to solve the following initial-boundary value problem. Let us consider a rectangular plate of sides a and b. Two adjoining sides are thermally insulated and at the remaining two the temperature is a prescribed function of time. In this example it has been assumed in the form: T = ToO- e -~' ). The initial conditions are assumed to be homogeneous. Then the problem is described by relations (Fig.2):

v 2 - ~l 4 T=0, r(x,0)=0,

~-t-7,y, )-0,

~t

' - 7 ' ) = o,

634

J. STEFANIAK and J. JANKOWSKI ,y 0.056 ylm]

r = To(1 -e~)

0.04

olo,,

~j

t xj

002

~x =

x [m]

~.025

0025

-0'01

"

o.~5 T = To(1 -e~t

0.05 0.07 x

0.02

-0.~

Fig. 2. Boundary conditions f o r

Fig.3.

a=O. 1 m and

b=O.08 m.

Location o f n points xj at sides of the rectangle and the "fictitious" sources at the outward rectangle

For the numerical calculations we assume: 1. The dimensions o f the plate: a=O. l m, b : O . 0 8 m. 2. Material coefficients: )7, = 165 W m -~, to= 0.6454 x 10- 4 m2s -1. This coefficients resemble those of duraluminium. 3. The coefficient a = 0 . 0 5 s -1. T0 = 100 °C. 4. Location o f "fictitious" heat sources at an outward rectangle o f the sides a',b' ,where a' b' .... 1.4 a b 5. 58 points x located at the sides of the rectangle, where the temperature and the gradient of temperature are prescribed (Fig.3). 6. 58 sources located at the outward rectangle in the points ~ (Fig.3). 7. Constant time interval Atk = At = 4,5 s. There are no rules concerning the amount of points at the boundary and the amount o f locations of heat sources. It depends on the required accuracy and the experience o f the researcher. Solution The general formula for the approximate temperature field in the boundary-value problem is given by the formula (3): n

m-I

T(X,/) = ~ , ~ _ W , '

F ( x - ~,,t - t k ) r l ( t - t ,

)

i = I k=o

where W,k is the intensity of/-heat source in the time interval A t k = tk+ ~ - t k . In D-2 the function F(x,t) takes the form F(x,t)=

1 Ei 4~r2

4Kt '

where [4]: E i ( - z ) = - ~ u e - " d u

HEAT CONDUCTION IN RECTANGULAR PLATES

635

The above solution shows that the calculated temperature has singularities only at the points, where the "fictitious" sources are located. That means that this temperature is a continuous function o f spatial variables and time within and at the boundary of the considered rectangle. Thanks to the continuity of the calculated temperature and its gradient at the appropriate sides of the rectangle it is possible to compare them with the prescribed ones at the boundary. This comparison at selected moments are shown in Figures 4-5, where the following notations have been used: -Temperature

prescribed temperature side thermally insulated side o prescribed value, • calculated value

Gradient of temperature:

o •

prescribed value, calculated value r[*c]

T [ *c] [*C/ml

~=

,vr [*C/ml

.~0

}On

~ocE~ ! I

i

1

i

~: ~ : :~ : : '

;;;;

.y[m]

i

.......

.~

a

:earn

•

.:'.

,~oo O

~/

z~,~

•

•

-y[m]

!

~

x [ml

-~::---

~

:

,~

, x [m]

r

Fig.5. Comparison at t: = 2o s

Fig.4. Comparison at t~ = lOs r [*c]

~®

[ *C/m]

35°C

rs00 I

t #'"

i

15°C i

i •

on

•..

"

!7,®

•

5°C

rz~

"

y [m] :

Fig. 6. Comparison at t, = 3 0 s

i,

"

10°C

\\

.x Ira]

Fig. 7. The temperature field at t, : 10s

636

J. STEFANIAK and J. JANKOWSKI

7~°C

60°C

....

..........

.....

\

hi

20°C ~ - - ~ 15°C

.... ,,

" ....

',,

30°C 25°C

IO°Ci-~_, _ i ~ :i i Fig. 8. The temperature field at t~= 2os

Fig. 9. The temperature field at t~ = 30s

The temperature fields at the same moments t~=10 s, 20 s, 30 s. are shown in Figures 7 - 9. Calculations have been performed with use of the program elaborated at Poznafi University of Technology, Institute of Applied Mechanics. Conclusions 1. The temperature in the considered rectangle and at its boundary is a continuous function. Therefore it is possible to compare at any moment t and at any point x ° ~ c7~ the approximate temperature and approximate gradient at appropriate sides of the rectangle with the prescribed ones. 2. Such a comparison may be assumed as a form of estimation of the error, at least at the boundary. Figures 4 - 6 shows that the approximate and strict solutions differ at the boundary slightly. 3. It is to underline that the number of assumed points of heat sources (n) and the number o f time intervals (m) are independent. 4. The time intervals have been assumed constant, but in this method this assumption is not necessary.

Acknowledgement This paper is related to the project No. 248/T07/97/12. Support of KBN (Komitet Badafl Naukowych - Committee of Scientific Research) is gratefully acknowledged. References l. J.Stefaniak, Controlling the Concentrated Sources in Some Problems o f Heat Conduction, J. Tech Phys., 26 ,pp.349-358, 1985 2. J. Stefaniak, J. Jankowski, Numerical Methods of Concentrated Heat Sources Control, J. Techn.Phys., Vol. 28, l, pp.87-96, 1987. 3. J. Stefaniak, The Method of concentrated Sources in Heat Conduction, Journal of Thermal Stresses, Vol. 19, No. 5, pp.481-493, 1996 4 A. Erdelyi, W. Magnus, F. Oberhettinger, F. Tricomi, Tables o f integral transforms, New

York, Toronto, London, Mc Graw-Hill, 1954.