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A THREE-DIMENSIONAL NUMERICAL METHOD BASED ONTHE SUPERPOSITION PRINCIPLEJules Thibault aa Department of Chemistry and Chemical Engineering , Royal Military College of Canada ,Kingston, Ontario, CanadaPublished online: 30 May 2007.

To cite this article: Jules Thibault (1984) A THREE-DIMENSIONAL NUMERICAL METHOD BASED ON THE SUPERPOSITIONPRINCIPLE, Numerical Heat Transfer: An International Journal of Computation and Methodology, 7:2, 127-145

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Numerical Heat Transfer, vol. 7, pp. 127-145,1984

A THREE-DIMENSIONAL NUMERICAL METHOD BASED ON THE SUPERPOSITION PRINCIPLE

Jules Thibault Department o f Chemistry and Chemical Engineering, Royal Military College of Canada, Kingston, Ontario, Canada

A threedimensio~l numerical method based on the superposition principle for the sdution of the hmt dffusion equation is derived for Cortesion cowdhtes and tested for three di f fmnt boundnry conditionr: a constant hmt flux density, a convective- type swface heat fhtx, and a sudden cooling of the surface to a constant temperature. In oddition. this three-dimensionul numerical method ir compred with the pop&? three-dimensional B ~ n ' s al ter~ting direction implicit {AD!) method. The method lmsed on the superposition principle has the same degree of acnrmcy in most cares as the method normally used for these types of mlculations. In addition, its olgo- rirhm is considerably simpler to formulate and easier t ~ ' ~ r o g m m , and it requbrs about half the computing time needed to solve the problem when using Brian's AD1 method. On the other hand, M n ' s AD! method is unconditio~lly stable, but the method bused on the superposition principle is not.

INTRODUCTION

With the advent of computers, researchers have been able to solve complex prob- lems that would otherwisk be impossible. A problem that has attracted considerable attention is the solution of the unsteady-state heat diffusion equation in solids of various shapes and materials. One can rarely rely on analytical solutions and must therefore solve this problem by a numerical method.

For three-dimensional problems, several versions of the alternating direction im- plicit (ADI) method extended to three space dimensions by using two intermediate values have been proposed. Among the more popular ones are those proposed by Brian [ I ] and Douglas [2]. The AD1 methods have been successfully used for many years. Their main advantage is that the system of equations is reduced to a tridiagonal coefficient matrix, for which a simple algorithm affords a straightforward solution 131. Another numerical method, which appears to have been used for the first time by Archambault and Chevrier [4] to solve for the two-dimensional temperature distribution is a quenched vertical cylinder, is based on the superposition principle. Thibault and Hoffman [5] corrected some irregularities introduced by Archambault and Chevrier when they defined their boundary condition equations. Tests on a two-dimensional heat conduction problem showed that the numerical method based on the superposition principle had the same degree of accuracy as an AD1 method [5, 61. This numerical method has not been used by other researchers and it is not mentioned in a recent book on the subject [7].

The objective of this work was to determine the effectiveness of the superposition principle in a finite-difference form in solving three-dimensional heat conduction prob- lems. In this paper the method will be derived mathematically and then compared with

Copyright 0 1984 by Hemisphere Publishing Corporation 127

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128 J. THIBAULT

NOMENCLATURE

A surfacearea,ml z zax i . coordinate, m Bi Biot number [Eq. (B 1 I ) ] ZL half-length of parallelepiped along Cp thermal capacity, J1kg.K z axis, m h heat transfer coefficient, W/m2 . K a thermal diffusivity, m2 /s (= X/pCp) NI number of mesh points in x direction stability criterion'[Eq. (35)] NI number of mesh points in y direction e average error, K NK number of mesh points in z direction h thermal conductivity, W/m. K q / ~ heat flux density, W/ml p roots of transcendental equations t time,s p density, kglm' T temperature, K U xaxis contribution to T, K V yaxis contribution to T, K

Subscripts

W z-axis contribution to T, K a analytical solution x xaxis coordinate, m avg average XL half-length of panuelepiped along i mesh point in x direction

j mesh point in y direction x axis, m y y-axis coordinate, m k mesh point in z direction YL half-length of parallelepiped along t a t t i m e t

y axis, m 0 a t time zero

the AD1 numerical method proposed by Brian [ I ] for cases for which analytical solu- tions are known.

MATHEMATICAL FORMULATION

The problem can be formulated in the following terms. Consider a finite plate (-XL G x < +XL, - YL 9 y G +YL, and -ZL G z < +ZL) (Fig. 1) having a constant thermal diffusivity ol and initially at a uniform temperature To. At time t > 0, the parallelepiped is allowed to lose heat at the surface. The heat flu density at the surface can be constant or time-varying and the temperature distribution within the parallelepiped

Fig. 1 Coordinate system: parallelepiped.

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A METHOD BASED ON THE SUPERPOSlTlON PRINCIPLE 129

can be calculated as a function of time by solving the unsteady-state heat diffusion equation:

The initial and boundary conditions, assuming that symmetry prevails, are:

t = O T = T o for O G y G Y L (2)

These boundary conditions indicate that it is only necessary to consider one-eighth of the parallelepiped and the heat flux density is constant over each face of the parallele- piped.

Based on the hypothesis that, under certain conditions, individual solutions of partial differential equations for simple problems can be combined either as a product or a sum to yield the solution of more complex problems, the solution for the temperature distribution within the plate can be expressed in the form

f i s form implies that the total temperature change over the time period t can be repre- sented as the sum of the independent one-dimensional temperature changes in the three orthogonal directions x, y, and z . Generally, the characteristic that allows this combina- tion is the linearity property of the governing differential equation and a constant surface heat flux. This superposition equation must satisfy both the unsteady heat diffusion equation and all boundary condition equations. Substitution of the right-hand side of Eq. (9) into the diffusion equation yields

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which can be reorganized as

Apart from the trivial solution U = V = W = 0, this equation is satisfied if and only if each term inside the bracket is equal to zero.

and the initial and the boundary conditions become

t = O U = T d for O Q x G X L

V = T o for 0 9 y G Y L

W = T o for O < z < Z L

The superposition principle in its additive form has been used successfully to obtain analytical solutions to simple' problems where the faces of the parallelepiped are sub- mitted t o a constant heat flux. By extension in the numerical method based on the super- position principle, Eqs. (1 2H14) are solved independently by an implicit finite-difference approximation for each mesh point within the parallelepiped, and then, at each time step, the temperature distribution is calculated by summing the independent solutions in the three different directions. By these successive solutions and summations over relatively short time steps, it is possible to evaluate the temperature distribution within the parallelepiped as a function of time. This procedure is similar to Brian's AD1 scheme,

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A METHOD BASED ON THE SUPERPOSITION PRINCIPLE 131

except that in the latter, extra terms in the other two directions are carried and the solu- tion is obtained for half the time step in each direction. In both methods, the system of equations is reduced t o a tridiagonal matrix system, which is solved with a rapid and simple algorithm.

Figure 2 shows the notation for the mesh grid used t o solve t h s problem by finite differences. The x axis is divided into (NI - I ) slices of thickness Ax = XL/(NI- I ) , the y axis into (NJ- 1) slices of thickness Ay = YL/(NJ- l), and the z axis into (NK - 1) slices of thickness Az = ZL/(NK - 1 ) . In this way ( M N J N K ) mesh points are defined and correspond t o ( i , j , k) , where i varies from 1 to NI, j from 1 t o NJ, and k from 1 t o NK.

If represents the temperature a t mesh point ( i , j , k ) a t a given instant t , Eqs. ( 1 2 H 1 4 ) in finite-difference form become

where Ui , j , k , V iJSk , and WiJ, represent the one-dimensional temperature distribution at . time t + At. Reorganization of Eq. (24) gives

with AI = a ~ t l h r ~ . This equation is solved for

The boundary conditions are formulated by the following equations:

1. At x = 0 (i = l ) , aU/ax = 0, the fmite-difference equation becomes

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J. THIBAULT

2. At the outer surface of the parallelepiped (i = NT), there is a heat flow out the surface. This heat flux density is expressed by

In finite-difference form, Eq. (29) is best represented by the central-difference form [5]

where (NI + 1) is a fictitious point located at a distance Ax beyond the boundary. This formulation allows the fictitious point to be replaced in E q . (27) t o obtain:

Equations (27) , (28), and ( 3 1 ) form a tridiagonal matrix system of Ni' equations and NI unknowns. The resulting matrix can be solved easily and rapidly with an algorithm presented by Carnahan et al. [3] . To obtain the complete one-dimensional distribution it is necessary to solve this tridiagonal system (NJNK) times.

Similar equations can easily be derived to solve for Vi,i.k and W i , j , k . Therefore, to obtain the three independent one-dimensional solutions, the tridiagonal matrix of coeffi- cients must be solved [(NJNK) + (NINK) + (NINJ)] times at each time step. This is also the number of solutions required when an AD1 method is used.

The three-dimensional temperature distribution at time t -t At is obtained by adding at each mesh point the three one-dimensional temperature variations:

which gives

In this section, the finite-difference scheme was derived for the particular case where the solid body is allowed to lose heat at its outer surface. The heat flux density can be constant or can change as a function of time and/or surface temperature. This numerical technique can also be used in cases where the surface temperature is suddenly lowered and maintained constant. In that case the problem is solved exactly the same way, except that whenever i = NI, j = N J , or k = NK, the value of U,,,,, , ViVj, k , and Wi, j ,k is set to this constant wall temperature.

Although for the derivation of the superposition equation it was necessary to as- sume linearity of the heat diffusion equation, this condition can be relaxed provided the time step and the grid spacing are chosen small enough that the material properbes and the surface heat flux do not vary appreciably over the time step considered.

Brian's AD1 algorithm is briefly described in Appendix A.

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A METHOD BASED ON THE SUPERPOSITION PRINCIPLE 133

R ESU LTS

To study the effect of the type of boundary conditions on the relative accuracy of the two numerical schemes used in this investigation, three different boundary conditions will be applied. These are described in detail in Appendix B. For all cases, the results are presented in terms of an average error E , defined as the square root of the average of the squares of the temperature differences between the exact solution and that predicted by the numerical method for all mesh points within the parallelepiped, namely

Each case is discussed in turn. All calculations were performed assuming a constant ther- mal diffusivity a, which was set at 1.072 cm2/s, that of pure copper.

Case I: Constant Heat Flux

Consider a parallelepiped, initially at a uniform temperature T o , which is submitted to a constant heat flux q /A on all its faces. It is allowed to cool for a sufficient time to permit the initial transient to dissipate and obtain the conditions where each point within the parallelepiped cools at the same rate. Figure 3 shows theLcooling curves for the comer and the center temperatures for a cube 5 cm* on a side. It took less than 3 0 s to achieve the pseudo-steady-state condition whereby a temperature difference between any two points within the cube was less than 0.001" at a subsequent time. A cooling period of 30 s was chosen when the longer side of the parallelepiped was 5 cm and 45 s when it was 7.5 cm.

Figures 4 and 5 show the average error e when a cube 5 cm on a side is submitted to heat flux densities of 20 and 50 W/cm2 with a grid of 10 mesh points in each direc- tion. For small time steps the accuracy of both numerical methods is very goad; the method based on the superposition principle shows slightly better accuracy. However, Brian's method, which is unconditionally stable, shows extremely good accuracy at larger time steps, while the superposition principle scheme become unstable at approxi- mately 0.2 s. The method based on the superposition principle shows a kind of resonance over an order of magnitude of time steps and appears to come back, again showing better accuracy than Brian's algorithm.

In addition, both methods were run in double precision (72 bits, versus 36 bits for single precision on an Honeywell CP6 system), and the results are shown on the same figures. The double-precision AD1 scheme is extremely accurate, showing an average error E of less than 0.002" up to a time step of 1 s and then overlapping the single- precision method. The double-precision version of the method based on the superposition principle shows excellent accuracy, but goes through a maximum at a time step of about 0.3 s and then regains its high accuracy, which persists for larger time steps than Brian's scheme. Even at a time step of 4 s 6 is only 0.007'.

*All dimensions now refer to half-lengths XL, YL, and ZL.

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0 1 0 20 30 40

TIME (SEC)

Fig. 3 Cooling cuwe.

-08 , I 1 I

- 0 7 - CASE I

W I - Y J . I K . I0

XL.YL. t L . 5 am '06 - ( * / A ) e0 w,ome -

0

t .OS - P R E C I S I O N

a 0

8I IBl .L DOUBLL - : .04 sup'CR O YI BRIAN Q

W

.03 - W > a

.02 -

.OI -

O 2 5 2 5

10" 10-a to" I 10

TlME STEP ( 8 )

Fig. 4 Average error versus time step at 30 s.

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A METHOD BASED ON THE SUPERPOSITION PRINCIPLE

TIME STEP (5) Fig. 5 Average error versus time step at 30 s.

Figure 6 shows the results obtained when a parallelepiped measuring 7.5 X 5.OX 2.5 cm is submitted to a constant heat flux density of 20 W/cm2 with grids of 15,10, and 5 mesh points, At small time steps, the method based on the superposition principle is far better than Brian's scheme. At a time step of 0.1 s, the two methods show the same high degree of accuracy. The method based on the superposition principle becomes unstable at the same time step as in the case of the cube. It also shows a similar resonance peak.

Table 1 shows the results of 15 different solutions for various sizes of the parallel- epiped in the case where the heat flux density is constant at 20 W/cmZ and the time step is 0.05 s. Both methods show extremely good accuracy.

These results show that very accurate temperature distributions can be obtained with either numerical method. Brian's method is unconditionally stable and yields ac- curate results up to a time step in the vicinity of 2 s for the geometry considered here. On the other hand, the method based on the superposition principle has a region of time steps where the solution is unstable. To the right of this resonance peak, the accuracy is very good in most cases, considering the large time steps used. This relatively low mini- mum average error beyond the resonance peak was obtained at a time step of approxi- mately 4 s and seemed to be independent of the geometry and heat flux density. At the same time step, Brian's scheme shows huge average errors. It is not known why the numerical method based on the superposition principle shows the peculiar behavior

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TIME STEP ( a ) Fig. 6 Average error versus time step at 45 s.

Table I Average Error for Case I with q/d = 20 W/cm and At = 0.05 sa

Average error e

(ZL, NK) XL Y L (cm) NI (cm) NJ (2.5.5) (4:0,8) (5.0, 10) (6.0, 12) (75,151

OThe upper numbn represents the average error for the method based on the superposition principle nnd the lower number for Brian's method.

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A METHOD BASED ON THE SUPERPOSITION PRINCIPLE 137

which is referred to here as a resonance peak because of its similarity to the phenomenon of,resonance. It is not suggested that time steps beyond the resonance peak should be used.

To characterize the onset of instability, the heat diffusion equation was solved for various values of At, &r, Ay, Az, and q/A. The method based on the superposition principle was stable provided

In a few cases, stable solutions were obtained even when this limit was exceeded. This is three times the value for an explicit scheme to be stable and the same as the value for the pure AD1 method extended to three space dimensions [ 3 ] . Calculations performed by Barakat and Clark [8] showed that a value of l = 2 could be used for an uncondi- tionally stable numerical method while still retaining adequate accuracy. The present results show that this limit could be safely exceeded for Brian's scheme while maintain- ing adequate accuracy. However, to represent adequately the temperature distribution at earlier cooling times or larger heat fluxes, the limit proposed by Barakat and Clark should be respected.

The CPU computation time on the Honeywell CP-6 system was shorter for the method based on the superposition principle. It took approximately 85% more computa- tion time to obtain the same solution with Brian's algorithm. In all situations described in this section the heat balance defined by Eq. (B2) was respected to within 0.1% for the coarser mesh and was smaller for a finer mesh.

Case II: Constant Wall Temperature

In this case, the surface temperature of a parallelepiped, maintained at a uniform initial temperature of 200°C, is suddenly lowered and maintained at 10o°C. The heat diffusion equation is therefore solved with this new boundary condition for the first 5 s of the cooling period. Since the boundary conditions for this problem lead to a solution that approaches a steady-state condition at late time and the derivatives of the solution decay exponentially with time, the time step for the solution will be progres- sively increased according to

Figure 7 shows the variation of the average error E as a function of the initial time increment for a cube of dimension 5 cm, using a grid of 10 mesh points in each direc- tion. Both numerical methods behave similarly under these extreme conditions. At small time increments the round-off errors are predominant, and at large time increments the accuracy of the methods deteriorates. For approximately two orders of magnitude of time steps, the average error for both methods is extremely small.

Figure 8 shows the results for a parallelepiped measuring 7.5 X 5.0 X 2.5 crn, with mesh refinements of 15, 10, and 5 points. The curves of average error as a function of time step for both numerical methods are similar in shape, but Brian's algorithm shows

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MI m YJ . WK m I0

XL. YL ZL I om

t I

TIME STEP ( 8 )

Fig. 7 Average error versus time step a t 5 s.

: . . : .El 5 . ,+J..

1 I I 1 1 h r n I I I 2 I I 8 I 8 L

1u4 10'' I O - ~ IQ'

TIME STEP (11 Fig. 8 Average error versus time step a t 5 s.

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A METHOD BASED ON THE SUPERPOSITION PFUNCIPLE 139

slightly better accuracy throughout the range of time steps than the method based on the superposition principle.

Figure 9 shows the average temperature difference error as a function of cooling time. The average errors for both numerical methods are very close, so that separate curves were not drawn. For short cooling periods, both numerical methods fail to repre- sent the analytical temperature distribution within the parallelepiped. It takes a certain time before the predicted temperature distribution becomes representative of the actual distribution. The average error at small cooling times is due mainly to the poor prediction of the temperature of the first interior mesh points. It is therefore not surprising that the average error decreases as a finer mesh size is used. To obtain good accuracy at short cooling periods, it would be necessary to use a finer mesh size and relatively small time steps.

Case I I I : Constant Heat Transfer Coefficient

In this case, a parallelepiped, maintained at an initial uniform temperature of 10O0C, is suddenly brought in contact with a fluid having a uniform temperature of 200°C and a uniform heat transfer coefficient h of 3.668 W/cm2.K on aU faces of the parallelepiped. The value of h is chosen to be the same as the thermal conductivity of copper, so that the Biot number becomes equal to the half-length of the parallelepiped.

O s e s e s 2 s e lo* 10" I 10

COOLING TIME (a ) Fig. 9 Average error as a function of cooling time,

I I l l I l l I I CASE I1

- XL. YL = 21 = S c r - To 2 0 0 -c T", = LOO .C - At4= 0 .004 @.a -

LOCUS OF 0 I( I .NJ wWK.8

MAXIMUM O N l = N J . M K . 8 ,

O nl = nJ 8 IIU 8 1 t

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Since the boundary conditions of this problem, as in case 11, Iead to solutions that approach a steady-state condition at late time, the time step can be progressively in- creased. Let the time step grow according to

This time progression was chosen arbitrarily. No attempt was made to optimize the progression.

Figures 10 and 11 show the solutions obtained for a 5 c m cube with grids of 8 and 12 mesh points in each direction after a cooling period of 5 s. The two methods yield concave curves with approximately the same accuracy for the range of time steps investi- gated. At small time steps the average errors are caused by round-off, and at large time steps the numerical methods, even though they are stable, fail to describe the actual problem.

Under these severe conditions (Bi = 5), to obtain good accuracy the diffusion equation has to be solved with very small time increments, and either method is per- fectly acceptable since this region of higher accuracy is located well before the unstable region of the method based on the superposition principle.

Figure 12 shows the average error E as a function of cooling time for three different

111 . w4 . MU . 8

X L - YL ZL = g a r

To . IOOeC

OlOT # - 0

SUPER 0 enlrn o

TIME STEP ( 8

Fig. 10 Average error as a function of time step at 5 s.

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A METHOD BASED ON THE SUPERPOSITION PRMQPLE

CASE Ill - M I - Y J M I 8 19

X L - Y L - Z L - B o r n

To 1 0 0 *C • ' T ~ - 9 0 0 9 c - alor 4 - s -

3 : - - t a . SUPER @ . . .. . a. . B R I A N a .. . .. # - -

. a . .. . .. . . - . . . . - -

- -

- - - "

I I 0 8 P 8 P 8 P 5

lo-* lo-= 10-a lo-'

TIME STEP (s 1 Fig. 11 Average error as a function of time step at 5 s.

grid refinements for a 5 c m cube. The average errors for both numerical methods are very close, so that separate curves were not drawn. The numerical methods are highly inaccurate for short cooling times. These large average errors at short cooling times are due entirely to the inabiiity of the numerical methods to predict the temperature of the surface mesh points, and, as expected, the average error decreases with an increase in the number of mesh points in the grid. Temperature differences as high as 28' between the analytical solution and the numerical solution of the surface mesh points were ob- served for the coarser mesh refinement. Both numerical methods failed at short cooling times to represent adequately the cooling at the outer surface, but the temperature distribution for all inside points was satisfactory.

CONCLUSION

In this paper the three-dimensional numerical method based on the superposition principle was derived mathematically and tested for three types of boundary conditions for which analytical solutions are known. Futhermore, it was compared with a popular three-dimensional numerical method, Brian's AD1 scheme.

It was shown that the numerical method based on the superposition principle behaves in a similar fashion t o Brian's scheme. Both numerical methods gave good results over a large range of time steps. During the initial transient, both numerical methods

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I l l I r I I I

CASE Ill - XL YL ZL = 8 cm - LOCU8 OF YAXlMUY To - loo-c

ERROR \. T,= noo"c - elor # = e - 0 Y I . Y J * M K = 8

Q M I = Y J * W K . I O - A n l - w - n ~ = t s

-

-

- -

t o P I P e P 10-~ 10" I 10

COOLING f IME (s) Fig. 12 Average error as a function of cooling time.

failed to represent adequately the temperature at the surface for high heat fluxes (case Ill) or the temperature next to the surface for the constant wall temperature (case 11). To obtain an accurate temperature distribution at short cooling times, a finer mesh and snaller time steps would be required.

Brian's scheme is unconditionally stable, but the method based on the superposi- tion principle becomes unstable at a value of = 1.5. However, under conditions leading to large temperature gradients (cases 11 and III), values of [ much smaller than 1 were required for sufficient accuracy t o be maintained.

Under most conditions the numerical method based on the superposition principle can be safely used for the solution of three-dimensional heat conduction problems, provided the time step is chosen with care. Its algorithm is considerably simpler to for- mulate and easier to program, and it requires nearly half the computation time of Brian's algorithm.

APPENDIX A: DESCRIPTION OF THE AD1 NUMERICAL METHOD

To verify the validity of the three-dimensional numerical method based on the superposition principle, it was compared to an AD1 numerical method. The AJJI method used was the one proposed by Brian [ l ] , which essentially solves the unsteady diffusion equation for two intermediate values T* and T** at the half-time step.

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A METHOD BASED ON THE SUPERPOSITION PRINCIPLE 143

Each equation is solved in turn for each mesh point in the grid, where one dimen- sion at a time is made implicit while leaving the other dimensions explicit. Each equation generates a tridiagonal system of equations solved by the same algorithm as in the method based on the superposition principle. Therefore, to obtain a complete temperature distri- bution at each time step, it is also necessary to solve the same number of tridiagonal systems of equations. However, generating the tridiagonal matrix of coefficients is much more laborious than in the method based on the superposition principle.

APPENDIX B: DESCRIPTION OF THE DIFFERENT CASES USED TO TEST THE NUMERICAL METHODS

Three different cases were used to test the two numerical methods investigated in this study. Each is discussed in turn below and an analytical equation expressing the temperature distribution within the parallelepiped is provided.

Case I: Constant Heat Flux

Consider the parallelepiped of Fig. 1 maintained at a uniform initial temperature T o . At time zero, the outer surface is exposed to a constant surface heat flux density q/A. The three-dimensional temperature distribution within the parallelepiped as a func- tion of time is not easily obtainable analytically. However, it is relatively easy to derive an equation expressing the relative temperature distribution within the plate when the initial transient period has elapsed. Under this pseudo-steady-state condition, each point inside the plate cools at the same rate. In the notation of Fig. I , the following equation gives the temperature at any point (x , y , z ) within the parallelepiped:

To determine whether the prediction of the temperature distribution by the nu- merical method is exact, it suffices to substitute into the equation the temperature at the center, as predicted by the numerical method, to determine the theoretical temperature distribution. The only uncertainty in this procedure is that the center temperature could be incorrect. However, if the predicted temperature distribution is exactly the same as the theoretical distribution and at the same time the overall heat balance is respected, then the validity of the numerical method is assured. The overall heat balance is given by

4 Q = P Cp (XL YL ZLXT, - Tavg,r) = - 2(XL YL + XL ZL + YL ZL) t A

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Case I I : Constant Wall Temperature

Consider the parallelepiped shown in Fig. 1, maintained at some initial uniform temperature To. The surface temperature is suddenly lowered and maintained at a con- stant temperature T,. The temperature distribution within the plate as a Function of time has been given by Luikov [9] :

where

Case Ill: Constant Heat Transfer Coefficient

consider the parallelepiped shown in Fig. 1 , maintained at some initial uniform temperature T o . At time zero, the parallelepiped is brought in contact with a fluid having a uniform temperature T, and a uniform heat transfer coefficient h . The heat transfer coefficient is the same on all faces of the plate. Luikov 191 showed that the solution may be represented in the form of a product of solutions for three infinite plates, the inter- section of which gives the parallelepiped. He obtained the following equation:

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A METHOD BASED ON THE SUPERPOSlTlON PRlNaPLE

where

2 sin p A =

p + sin p cos 1.1

p is the solution of the transcendental relation

1 cot p = - p Bi

and Bi is the Biot number defined by

REFERENCES

1.. P. L. T. Brian, A Finite-Difference Method of Higher-Order Accuracy for the Solution of Three-dimensional Transient Heat Conduction Problems, AIChE I . , vol. 7, p. 367, 1961.

2. J. Douglas, Alternating Direction Method for Three Space Variables, Numer. Math., vol. 4 , p. 41, 1962.

3. B. Carnahan, H. A. Luther, and J. D. Wilkes, Applied Numerical Methods, Wiley, New York, 1969.

4. P. Archambault and J. C. Chevrier, Distribution de la tempirature au sein d'uncylindre t rempt dans un liquide vaporisable, Int. J. Hear Mass Transfer, vol. 20, p. 1, 1977.

5. J . Thibault and T. W. Hoffman, Specification of the Correct Boundary Conditions, Int. J . Heat Mass Transfer, vol. 21, p. 368, 1978.

6. J . Thibault and T. W. Hoffman, A Heat Flux Meter t o Determine the Local Boiling Heat Flux Density during a Quenching Experiment, Inr. J. Hear Mass Transfer, vol. 22, p. 177, 1979.

7. L. Lapidus and G. F. Pinder, Numerical Solution o f Partial Differential Equarions in Science and Engineering, Wiley, New York, 1982.

8. H. 2. Barakat and J . A. Clark, On the Solution of the Diffusion Equation by Numeri- cal Methods, J. Heat Transfer, vol. 88, p. 421, 1966.

9. A. V. Luikov, Analytical Heat Diffusion Theory, Academic, New York, 1968.

Received July 13,1983 Accepted August 30,1983

Requests for reprints should be sent t o Jules Thibault.

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