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Numerical Heat Transfer, Part B: Fundamentals: AnInternational Journal of Computation and MethodologyPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/unhb20

NONITERATIVE PROCEDURE FOR THE FINITE-ELEMENTSOLUTION OF THE ENTHALPY MODEL FOR PHASE-CHANGE HEAT CONDUCTION PROBLEMSZhen-Xiang Gong a & Arun S. Mujumdar aa Department of Chemical Engineering, McGill University, Montreal, Quebec, H3A 2A7,CanadaPublished online: 15 Mar 2007.

To cite this article: Zhen-Xiang Gong & Arun S. Mujumdar (1995) NONITERATIVE PROCEDURE FOR THE FINITE-ELEMENTSOLUTION OF THE ENTHALPY MODEL FOR PHASE-CHANGE HEAT CONDUCTION PROBLEMS, Numerical Heat Transfer, Part B:Fundamentals: An International Journal of Computation and Methodology, 27:4, 437-446, DOI: 10.1080/10407799508914966

To link to this article: http://dx.doi.org/10.1080/10407799508914966

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NONITERATIVE PROCEDURE FOR THE FINITE-ELEMENT SOLUTION OF THE ENTHALPY MODEL FOR PHASE-CHANGE HEAT CONDUCTION PROBLEMS

Zhen-Xiang Gong and Arun S. Mujumdar 1 Department of Chemical En~ineering, McGi l l Uniuersity, Montreal,

Based on a lumped-mass model, a compulahulahoMUy eficient nonitemtive procedure is developed for fhefinite-element solution of the enthalpy model proposed by Shamsundar and Sparrow 111. Thisprocedure uses Gauss elimination as the equation sober and incorporates a three-time-level time integration scheme. Sample results are presented and discussed.

INTRODUCTION

The enthalpy model proposed by Shamsundar and Sparrow [I] has been one of the most popular approaches to the numerical solution of melting and freezing problems. In this model, both enthalpy and temperature are included in the governing partial differential equation as the variables to be solved. Therefore, n equations containing 2n dependent variables appear in the final discrete algebraic equation system. This feature of the enthalpy model prevents the use of direct solvers. Gauss-Siedel iteration, or an equivalent nonsimultaneous iteration proce- dure, must be applied to solve the algebraic equation system along with the equations of state. If the enthalpy model is solved by the finite-difference tech- nique, there is no additional difficulty to the solution procedure of the algebraic equation system, since the finite-difference approach is always associated with indirect equation solvers. However, if the enthalpy model is solved by the finite- element technique, indirect solvers are inefficient since they provide slow conver- gence, especially due to the nonlinearity inherent in the phase-change model. This results in high computing time. Also, existing finite-element programs are always equipped with direct equation solvers. Therefore, the finite element has seldom been implemented for solving the enthalpy model.

The enthalpy concept was applied to phase-change problems in 1974 by Comini et al. [2], and improved versions of this method have been reported consistently during the last two decades (e.g., Morgan et al. [3], Pham [4], and

Received 24 May 1994; accepted 20 December 1994. Z.-X. Gong gratefully acknowledges the financial support of the Canadian International Devel-

opment Agency (CIDA) and McGill University in the form of a McGill/CIDA Fellowship. Research support of Natural Science and Engineering Council of Canada as well as Exergex Corporation is also acknowledged.

Address correspondence to Dr. Arun S. Mujumdar, Department of Chemical Engineering, McGill University, 3480 University Street, Montreal, Quebec H3A 2A7, Canada.

Numerical Heat Transfer, Part B, 27:437-446,1995 Copyright O 1995 Taylor & Francis

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438 Z.-X. GONG AND A. S. MUJUMDAR

NOMENCLATURE

specific heat specific heat matrix elements of specific heat matrix heat load vector elements of heat load vector convection heat transfer coefficient enthalpy per unit mass enthalpy of liquid at melting point enthalpy of solid at melting point thermal conductivity conductance matrix elements of conductance matrix mass matrix elements of mass matrix outward normal of boundary shape function rate of internal heat generation time temperature ambient temperature

initial temperature melting temperature space coordinates a time integration parameter boundary time step latent heat density solution domain

Superscript

n nth time step

Subscripts

e element I, J node number of an element I liquid phase s solid phase

Comini et al. [51). The procedures developed in [2-51 have been well established and are effective in solving conduction phase-change problems. However, those procedures are, in fact, combinations of the enthalpy concept with the equivalent heat capacity method and use the enthalpy concept not in the way of Shamsundar and Sparrow. Gong and Mujumdar [6] developed a simultaneous iteration proce- dure to solve the enthalpy model of Shamsundar and Sparrow. This procedure successfully implemented direct solvers for the resulting system of equations.

In this article we present a noniterative procedure for the finite-element solution of the general multidimensional enthalpy model. This procedure utilizes Gauss elimination as the equation solver and employs a three-time-level scheme for time integration. It does not need iteration and therefore results in a great reduction in execution time.

MATHEMATICAL FORMULATION

According to Shamsundar and Sparrow [I], the governing equation for phase-change heat conduction problems based on enthalpy can be described by

in which

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SOLUTION OF PHASE-CHANGE CONDUCTION PROBLEMS

Equation (1) is coupled with the following equations of state:

FINITE-ELEMENT FORMULATION

After spacewise discretization of Eq. (11, subject to the boundary condition of the third kind (convection boundary condition), viz.,

is accomplished using the standard Galerkin method [Ill, we obtain the following semidiscrete matrix system:

in which the superposed dot denotes differentiation with respect to time. Typical elements of the matrices in Eq. (5) are

JN, JN, JN, JN, dN, dN, - -+--+-- - d n + ~ / h ~ , ~ , d r (7) dx ax ~ y d y J Z J Z re

It is reiterated that the set of equations (5) is highly nonlinear due to the phase-change nonlinearity.

The discretization of the time derivative in Eq. (5) is most often achieved with a finite-difference technique. Although many time-stepping schemes are available, the most popular are two-time-level methods in which iterations within each time step are required. This is very time consuming for a highly nonlinear problem. To circumvent this disadvantage, a three-time-level scheme (see Dalhuijsen and Segal [7]) is adopted. According to this scheme, Eq. (5) can be approximated as

in which

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440 Z.-X. GONG AND A. S. MUJUMDAR

in correspondence with {HI,

Here the superscript * designates that the matrices and vector are evaluated at time level I* = t,,,, or t* = I,.

Corresponding to t*,

If the enthalpy rate is selected as the variable to be solved and Eq. (11) is substituted into Eq. (9), we obtain

[M*]{H) + (0.5 + a ) AI[K*](T) = {F*) - [K*](( l - a ) T n + a T n - ' ) (14)

Since the following relationship between enthalpy and temperature exists in the absence of phase change,

we obtain the following equation if a lumped heat capacity model [4] is employed:

in which [C] is the specific heat matrix, which is diagonal. Since [C] is a diagonal matrix, it is easy to obtain

{TI = [CI- '(H)

Substitution of Eq. (17) into Eq. (14) yields

[[M*] + (0.5 + a) A~[K*:I[C*]- ' ] (H)

= (F*) - [K*](( l - a ) T n + aT"-I) (18)

If phase change takes place in the heat conduction process, Eq. (17) is not valid due to the fact that the temperature rates of those nodes in which phase change is taking place are equal to zero but the enthalpy rates are not, and therefore, Eq. (18) is also not valid for phase-change problems. However, Eq. (18) can still hold true for conduction problems with phase change if the conductance matrix [K*] on the left-hand side consisting of the global matrix of the final discrete equation system is modified such that all the elements in column i are set equal to zero if phase change is taking place in node i.

For further explanation, take an element (bilinear quadrilateral element employed in this study), for example, if there is no phase change at all nodes of the

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SOLUTION OF PHASE-CHANGE CONDUCTION PROBLEMS 441

element, the following equation holds true:

(19)

while if phase change is taking place, for example, at node 1, we have T~ = 0. Then

in which [ K * @ ] , is the modified conductance matrix.

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442 Z.-X. GONG AND A. S. MUJUMDAR

[ K * ] is originally a symmetric matrix. The modification described above, as well as the multiplication of [C*]- ' on the right-hand side of [ K * @ ] , makes [K*@:I[C8]- ' asymmetrical. This can be explained at the element level by referring to the following equation:

[ K * @ ] ~ [ c * ] , =

This is a disadvantage of this procedure. However, it is a good trade-off that such treatment of the conductance matrix as described above makes it possible to use direct solvers and thus to implement a noniterative time scheme.

After the treatment described above, Eq. (18) becomes

[ [ M * ] + (0.5 + a ) A ~ [ K * @ : I [ c * ] - ' ] ( H )

very and step

It is convenient to solve Eq. (22) by Gauss elimination. After ( H ) is obtained, + I ) is updated by Eq. (10).

The three-time-level scheme used in Eq. (9) was found, with a = f, to work satisfactorily for a nonlinear model equation [13] and to be stable, accurate, free from oscillations for solidification problems for a moderately large time

. size [7,8]. Due to the strong nonlinearity of the problem, the element conductance

matrices have to be calculated at each time step and the calculation occupies a large portion of the total computational work. Generally, the calculation of the element conductance matrices is performed by numerical quadrature. For the isoparametric bilinear/quadrilateral elements adopted in this article, four-point quadrature is often required. This is very time consuming. In order to speed up the computation, a stabilized one-point quadrature algorithm developed by Liu and Belytschko [9] is used for the calculation of the conductance matrix in this study.

TEST EXAMPLES

1. Accuracy Test

To demonstrate the accuracy of the procedure described above, two illustra- tive problems are presented.

Problem 1: Solidification of a n infinite s lab of liquid. A liquid at a uniform temperature, 1O0C, which is above its freezing point (O°C), is confined to a half-space x > 0. At time t = 0, the boundary surface at x = 0 is lowered to a

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SOLUTION OF PHASE-CHANGE CONDUCTION PROBLEMS

Figure 1. Finite-element mesh.

temperature, -2O0C, and maintained at this temperature for t > 0. The thermo- physical properties are as follows:

Two-dimensional elements are utilized to solve this problem although it is physically one-dimensional. The finite-element mesh is displayed in Figure 1 in which BC = 1.0 m. Computations were carried out using 40 elements with a time step of 200 s, as well as 80 elements with a time step of 200 s, respectively.

Figure 2 displays a comparison of the progress of the freezing front between the computed results and the analytical solution [12]. As the slab computed is finite, comparisons with the theoretical infinite-slab solution are terminated when temperature begins to change appreciably at boundaly DC. It can be seen from this figure that the computed results are in very good agreement with the analytical solution.

Problem 2: Solidification of a corner region. The comer region of a liquid body extending in the positive x and y directions is frozen by bringing the surface temperature to - l.O°C at time t = 0. The thermophysical properties are k, = k , = 1 W/mK, c, = c, = 1.0 J/kgK, p = 1 kg/m3, T,, = 0°C. Two cases were computed; one is T, = 0°C and A = 1.5613 J/kg, the other is T, = 0.3"C and A = 0.25 J/kg.

The finite-element mesh is shown in Figure 3. Figure 4 is the comparison between the present finite-element results and the analytical solution [lo] of the freezing front. It is found that the results are also in very good agreement with the analytical solution.

Figure 2. Comparison of the computed interface position with analytical solu- tion.

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Z.-X. GONG AND A. S. MUJUMDAR

Figure 3. Finite-element mesh.

Figure 4. Comparison of the computed 0 0

o s , , interface position with analytical solu- XI= tion.

Figure 5. Temperature histories for the case of 40 elements (curve 1 for x = 0.05 m; curve 2 for x = 0.15 m).

Figure 6. Temperature histories for the case of 80 elements (curve 1 for x = 0.05 m; curve 2 for x = 0.15 m).

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SOLUTION OF PHASE-CHANGE CONDUCTION PROBLEMS 445

2. Time-Step Size Test

Problem 1 was also computed to study the effect of the time-step size. Computations were carried out using 40 elements with time-step sizes of 50, 800, and 1,000 s and 80 elements with time-step sizes of 50, 400, and 800 s, respectively. Figure 5 shows the temperature histories at x = 0.05 m and x = 0.15 m for the case of 40 elements. Curves 1 and 2 are for x = 0.05 m and x = 0.15 m, respectively. The solid line, dashed line, and dotted line represent time-step sizes of 50, 800, and 1,000 s, respectively. From this figure, we can see that there is little difference among temperature histories using a time-step size in the range of 50-800 s. This indicates that the accuracy is independent of time-step size over a wide range of time-step sizes. For a time step of 50-800 s, a general oscillation of temperature is observed after the freezing point. This is caused by the enthalpy model itself, not by the solution procedure. In the authors' experience, such forms of oscillation can be observed no matter what time integration scheme is used when large meshes are used. It can be alleviated by mesh refinement. For a time-step size of 1,000 s, a moderately strong oscillation is observed after the freezing point, but it disappears immediately and stable solution is obtained.

Figure 6 displays the temperature histories at x = 0.05 m and x = 0.15 m for the case of 80 elements. Curves 1 and 2 are for x = 0.05 m and x = 0.15 m, respectively. The solid line, dashed line, and dotted line represent time-step sizes of 50, 400, and 800 s, respectively. It is found that the larger the time step, the stronger is the oscillation after the freezing point. Another observation is that oscillations appear earlier as the time-step size increases using 80 elements than using 40 elements. This is due to the refinement of the space step. When the time step reaches 400 s, a moderately strong oscillation occurs after the freezing point. For the time step of 800 s, the oscillation is stronger and disappears more slowly than that for 400 s.

It was also found that the solution of interface position is much less influenced by the time-step size in spite of the oscillation of the temperature.

CONCLUDING REMARKS

A computationally efficient noniterative procedure has been developed for the finite-element solution of multidimensional enthalpy model. This procedure does not need iterations and allows relatively large time steps. Numerical examples have demonstrated the accuracy obtainable with the new procedure.

REFERENCES

1. N. Shamsundar and E. M. Sparrow, Analysis of Multidimensional Conduction Phase Change via the Enthalpy Model, J. Heat Transfer, Trans. ASME, vol. 97, pp. 333-340, 1975.

2. G. Comini, S. D. Guidice, R. W. Lewis, and 0. C. Zienkiewicz, Finite Element Solution of Non-linear Heat Conduction Problems with Reference to Phase Change, Int. J. Numer. Meth. Eng., vol. 8, pp. 613-624, 1974.

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446 Z.-X. GONG AND A. S. MUJUMDAR

3. K. Morgan, R. W. Lewis, and 0. C. Zienkiewicz, An Improved Algorithm for Heat Conduction Problems with Phase Change, Int. J. Numer. Meth. Eng., vol. 12, pp. 1191-1195, 1978.

4. Q. T. Pham, The Use of Lumped Capacitance in the Finite-Element Solution of Heat Conduction Problems with Phase Change, Int. J. Heat Mass Transfer, vol. 29, pp. 285-291, 1986.

5. G. Comini, S. D. Guidice, and 0. Saro, A Conservative Algorithm for Multidimensional Conduction Phase Change, Int. J. Numer. Meth. Eng., vol. 30, pp. 697-709, 1990.

6. Z. X. Gong and A. S. Mujumdar, A Simultaneous Iteration Procedure for the Finite Element Solution of the Enthalpy Model for Phase Change Heat Conduction Problems, Int. J. Numer. Meth. Heat Fluid Flow, in press.

7. A. J. Dalhuijsen and A. Segal, Comparison of Finite Element Techniques for Solidifica- tion Problems, Int. J. Numer. Meth. Eng., vol. 23, pp. 1807-1829, 1986.

8. B. G. Thomas, I. V. Samarasekera, and J. K. Brimacombe, Comparison of Numerical Modelling Techniques for Complex Two-Dimensional Transient Heat-Conduction Prob- lems, Met. Trans. B, vol. 15, pp. 307-318, 1984.

9. W. K. Liu and T. Belytschko, Efficient Linear and Non-linear Heat Conduction with a Quadrilateral Element, Int. J. Numer. Meth. Eng., vol. 20, pp. 931-948, 1984.

10. H. Budhia and F. Kreith, Heat Transfer with Melting or Freezing in a Wedge, Int. J. Heat Mass Transfer, vol. 16, pp. 195-211, 1983.

11. 0. C. Zienkiewicz, The Finite Element Method, McGraw-Hill, New York, 1977. 12. A. V. Luikov, Analytical Heat Diffurion Theory, Academic Press, New York, 1968. 13. M. A. Hogge, A Comparison of Two- and Three-Level Integration Schemes for Non-

linear Heat Conduction, in R. W. Lewis, K. Morgan, and 0. C. Zienkiewicz (eds.), Numerical Methods in Heat Transfer, pp. 75-90, John Wiley, Chichester, 1981.

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