The Nonlinear Heat Transfer With Phase Change in Homogeneous Materials

B. B. Budkowska Civil and Environmental Engineering Department

University of Windsor Windsor, Ontario, Canada

I. Kreja Civil Engineering Department Gdansk Technical University

Gdansk, Poland ABSTRACT

The paper presents the rormulation or the heat conduction problem including phase change. The analysis uses the strong and the weak rorm or the heat propagation equations with discontinuous thermal properties on the boundary of moving interface associated with change of state. The weak formulation constitutes basis for the application of the finite element method with respect to spatial variables. Temporal integration is obtained by implementation of the Euler backward procedure. Nonlinear time dependent thermal properties are taking into account by employing the incremental decomposition concept. A new treatment is used for the analysis of the discontinuity surface connected with the release or absorption or latent heat term. High nonlinearity of the problem necessitated the application of Newton-Raphson equilibrium iterations at each time step.

NOMENCLATURE

c - heat capacity per unit mass [JIkgK],E1j - the permutation symbol, J - the Jacobian matrix,

- thermal conductivity tensor [WImK],kij

L - latent heat per unit volume [J1m3], N - shape function, n - unit vector normal to the moving interface [m], ~ - the heat flux [W1m2], u - the specific internal energy (J1m3] T - phase change interface velOCIty vector [ml s], Xl' ~, ~ - global coordinate system.

~ - the specific internal enthalpy [J/m3], \1' {2' {at - local coordinate system,

P - the mass density [kg/m3],

9 - the variable temperature fc], 9 f - the phase change temperature fc],

t+Atn9 11 - the temperature gradient for the time instant t+At t~],

INTRODUCTION

The differential equations governing heat transfer with phase change (constituting strong formulation) can be solved by means of different numerical methods, which in fact transform the differential equation into a the system of algebraic equations.

The solution of the variational equation which is called the weak formulation to the differential problem of heat transfer is very useful when the finite element method is to be applied. It is also worth mentioning, that weak solutions are important, since certain discrete numerical schemes converge to the weak solution of the original boundary value problem, despite discontinuities of the temperature gradient at the phase change front. The importance of the phase change problems has resulted in the development of a large. number of different methods fl,2,3,4,5,6,7,8,lO,ll,l2,l3,l4,l5,l6,l71, which will not be discussed here because of umited space. In the paper the detailed formulation of the non-linear heat transfer with phase change is presented. Nonlinearity connected with material properties is considered by introducing the incremental decomposition with respect to time. The analysis uses the finite element method with respect to spatial variables, while integration with respect to time is obtained by means of the Euler backward procedure. A new treatment of the latent heat connected with the phase change is solved by employing the properties existing between unit normal vector defining moving surface and velocity vector describing speed of moving phase-ehange surface. To speed up the convergence of the solution, the Newton-Raphson method was used. In order to demonstrate the effectiveness of the presented formulation and algorithm, a numerical example is presented.

THE STRONG FORMULATION OF THE HEAT TRANSFER PROBLEM WITH PHASE CHANGE

Let us consider a control volume [} with a fixed boundary 00. The control volume [} is divided by the interface of moving boundary ret) into two regions, 01

and ° 2, of different phases shown in Fig.l.

a.!t.

Fig.l The analyzed control volume and moving interface ret)

534

The heat flow equilibrium equation, is given in the following form:

(1)

The heat flux ~ conducted per unit area is related to the temperature gradient

by means of the Fourier law, which in generalized form is written as:

oe q. = - k .. ~ (2)1 1J uXj

For most general case can be the function of space, time and temperature.kij Confining the considerations to isotropic medium, the conductivity tensor kij reduces

to one scalar value equal to k. Employing this property and combining the relation (2) with equation (1), the following equation valid for conduction is obtained:

8 oe e e OX; (k.. ox;) = pc9 or ! e (k!9) = pe9 (3)

i IJ Xj

The eq. (3) must satisfy the prescribed temperature boundary conditions at specific points and surfaces of the body, i.e.

9 = 9* on BOT (4)

The fact of the phase change of the medium, should be emphasized in boundary conditions. Thus on the moving phase change interface r(t) the following condition, according to Eringen [18] must be satisfied:

for 9 = [k~a]e n + p L 'Yen = 0 on r(t) (5)a f where [] defines. (after Eringen [18]) the jump on the moving phase change surface.

The eq. (3) with boundary conditions (4) and (5) completely defines the heat transfer problem including phase change in a homogeneous continuous medium.

INTEGRAL FORMULATION OF THE ENERGY CONSERVATION LAW

The mathematical form of the energy conservation law in the absence of internal heat sources [181 has the following form:

Dm6pu dV = 10 k~aen dA (6)

In thermal problems, the internal energy is combined indirectly with temperature ,through the specific internal enthalpy E, which enables one to write the eq. (6) in the absence of flow as follows:

535

(7)

For the material derivative of a medium possessing discontinuous material properties, according to Eringen [18], the following relation is valid:

(8)

The right-hand side of 'the relation (7) according to the Green-Gauss theorem for the medium with discontinuity [18] should satisfy the following relationship:

(9)

'Combining (8) with (9), the law of conservation of energy for the two-phase medium is given as:

(10)

On the phase change boundary, the specific intemal enthalpy is related to the latent heat in the following form:

(11)

In the temperature range excluding phase change, the rate of change of specific internal enthalpy is proportional to the rate of change of temperature. The coefficient of proportionality is equal to the heat capacity, i.e.

8 Be· (12)1K (pE) = pc or = pea

Taking into account (12) &; (U), eq.(10) has the following form:

(13)6(pea •

- 2e(k28»dV - f ([k2a] +pLv)en dA = 0

Introducing a new function w from the variational space, such that the 111 satisfies boundary conditions for a, we get the weak form of the equation ,far transient heat flow with phase change.

e (l.()6wpea dV - 6~ e (k28) dV - f wfk28]en dA - f wpLven dA = 0

Integration by parts of the second term in eq.(14) lea.ds to the relationship.

536

e6wpcS dV + 6fwe(kfS)dV = fwpLven dA (15)

which constitutes the weak form of the transient heat propagation in homogeneous medium including phase change. The equation (15) can be interpreted in the following way: the first term on the left-hand side represents the generalized variation of the transient heat flow, the second term defines the variation of the influx conducted into volume n and the right-hand term stands for the variation of the latent heat on a moving surface.

NUMERICAL APPROXIMATION OF THE WEAK FORM OF BEAT TRANSFER PROBLEM WITH PHASE CHANGE

The unknown temperature variable e as well as variation w will be

approximated by their finite dimensional approximation Sh and J. respectively, according to following formula:

(16)

. where e *h satisfies the temperature boundary conditions (5) on anT and

hT = 0 on anT'

hAssuming that e can be described by means of polynomials in the following form: n . n n *

Sh = E N.e. = E N.T. + E N.S, (17)11'111 '111 '11= 1= 1=

where Ni are well known shape functions, we introduce the finite set of arbitrary

functions wj, j=1,2,3 ..... n instead of J..

With these approximations, eq.(15) has the following form:

_h e*h *heh (18)6wfCT dV+hfWt(kf'l-)dV=fwILven dA-h wre dV-hfWt(k!e )dV

IMPLICIT INTEGRATION WITH RESPECT TO TIME FOR BEAT TRANSFER PROBLEM WITH PHASE CHANGE

. The time dependent variables appearing in eq. (18) imply the necessity of Integration with respect to time. The incremental decomposition with respect to temporal variable is introduced [19], which enables one to define time dependent quantities appearing in eq. (18) in the following form:

t+AtT = tT + T and V(t+lltT ) = V (tT) + VT (19) "Where T stands for the increment of temperature, llt denotes the increment of time,

537

The integration with respect to time is obtained by employing Euler backward method according to following scheme:

t+.6t· t+.6tT - tT T (20)T = ilt = At

Introducing the Bubnov-Galerkin concept [20] for the choice of the weighting function Wj in eq.(lS), we have:

W. = N. (21)J J

Taking into account (19) + (21) and employing matrix notation, the incremental equilibrium equation (IS) IS written In the following form:

(22)

where T is the vector of nodal temperature increment for time step .6t,

tc = 6NT tp tc N dV - denotes the incremental capacity matrix, (23)

tK = 6BT tk B dV - stands for incremental conductivity matrix, (24)

t+.6tQL = f NT t+.6tp L t+.6tv.t+.6tn dA - is vector of heat flow due to

t+.6t phase . change effects, (25)

t+.6tQ* = 6NT tp tc t+.6t9* dV + 6BT tkV (t+.6t9*) dV - is vector of heat

flow due to temperature boundary conditions, (26)

tQ = 6B T tkV(tT) dV - means vector of the "balanced heat flow" at time t (27)

Consequently, the introduced linearization assumptions imply, that the results obtained by means of eq. (22) constitute the approximation to the exact solution. Moreover, the nonlinearity of the problem necessitates the application of Newton-Raphson equilibrium iterations in order to improve the accuracy of the solution.

THE TREATMENT OF LATENT HEAT AT THE INTERFACE

Now, the analysis is concentrated on the latent heat term defined by the formula (25). For two-dimensional analysis (plane x1~) we introduce the local coordinate system e,e ,e3, shown in Fig. 2, with following properties:1 2

538

1) axes el'and e2, belong to the interface surface r (t+at) at the instant (t+at) and axis e3, is perpendicular to r(t+at).

2) axes e and e lay in plane xl' ~ and consequently axis e is perpendicular to 2 3 1 the plane xl' ~.

3) the natural coordinates ei are confined to the range of absolute value of unit

i.e. -1 ~ ei ~ 1 for i = 1,2,3.

~-------~~~---------+X1

Fig. 2 Natural coordinate system on the moving boundary.

The velocity vector components can be described in following way: dt+atx at+at de a~+atx de

t+at i Xi ~3 i ~2 Vi = a t - ae3 at + ae2 at (28)

and consequently at+atx3 tf+atx. tf+atx. de

t+atv.t+atn. dA = _ J 1 3 dt de (29)1 1 ae1 eij ae2 ae3 (If"" ~1 ~2

for i,j = 1,2

Since the motion of the phase change surface is in the coordinate system e2

, eit implies that after inte~ation the vector of heat flow due to phase change effects can be replaced by single (linear) integral, Le.

539

3

(30)

u = ;}+8.t~ v = ;}+8.t~ (32)a(2 a(3

It is easy to notice that det (J) can be expressed by means of vectors u and T. Since the vectors u and T are perpendicular, their dot product is equal to zero.

Consequently, this implies, that vector u also has the following form: ;}+8.t~

a(a ;}+8.txu= 1 (33)a(a

Mter some manipulations, the following relation can be shown to be valid: de3 1/2 as 2 ae 2 -1/2 de dt det (J) = [det(J)] [(at+At ) + (t+At )] (- at) (34)

Xl a ~

It is worth to indicate that when the phase change front moves toward liquid, which is the consequence of the decrease of temperature, then the velocity vector v

of moving interface r(t) has opposite sign to the rate of temperature ~~. Inserting (34) in (30), vector of heat flow due to phase change effects is now expressed as:

at+8.t t+8.tQ = -211 , NT t+8.tpL9 (t+8.tllvell)-l J det(l) de (35)

L -1 a(1 2

where the temperature gradient is defined as shown below:

540

It will be useful to introduce the following geometrical interpretation. Namely, in the coordinate system Xl ~ we can introduce vectors u and T coinciding with , coordinate system el' e2 and defined in the following way:

;}+8.tx ;}+8. tx 1 1

--~8~(2~ a(a

1/2t+~tllvell -[( 8e )2 + ( 8e )2] (36)

- at+AtXl at+at~

•Regarding (16), the rate of change of temperature e in (35) can be approximated as:

• • .* 9=T+9 (37)

Application of Newton-Raphson procedure with respect to, relation (35) gives the result:

t+~tT·(i)_ t+~tT·(i-l)+ ~T(i) (38)- at

Substituting (37) and (38) into (35) enables one to define vector of heat now due to phase change eflects in more convenient form to numerical analysis. Thus,

t+~tQ _ t+~tC'_ (i-I) 1 ~T(i) + t+~tQ *(i-l) + t+~tQ (i-I) (39)L - - -L At L L

where t+~tC1,(i-I) - is the additional capacity matrix defined as

, at+~t (i-I) t+~tC'_ (i-1)=2 II Xa NT t+~t/i-1) L N Xdet (t' d{ (40)

-L -1 Del t+ 2t nv911 1-1)

t+~tQL* (i-I) _ denotes the heat now vector due to phase change in the region with

prescribed temperature given as:

and last term t+~tQL(i-I) i.s in fact another form of vector of heat now due to

phase change effects defined by (25).

t+~tQ (i-I) _L -

r ) ,

t+~t -2 II 8 "3 .-1 NT Hdtp(i-l)LHdti(i-l) i. det (~l de (42)

-1 ae 1 t+ tnvsl! 1-1) 2

541

NUMElUCAL EXAMPLE .

For the presented theoretical formulation and then developed numerical algorithm, some computer examples were anallzed and one of them will be presented here. The similar problem was analyzed in l21], however the formulation was based. on different numerical algorithm. The presented example is the discrete representation of Neumann problem which can be treated as one dimensional case. The geometry of the strip, finite element mesh, the temperature boundary conditions are shown in Fig. 3.

90~-3~~ 4+ I !: 1 i lo.lm1111111111 1 : I I I I

o C l ,," f •L

O.02m O.08m "

Fig. 3 The finite element mesh for freezing of a semi-infinite body of water with step change in thermal conductivity and capacity.

The thermal properties of the analyzed medium were as follows:

- 4.186 • 106 J /kg KCL C - 2.06 • 106 J /kg Ks kL - 0.56 W/mK

ks - 2.3 W/mK

L - 3.33 • 108J/m3

The obtained results for different time increment are shown in Fig.4.

It is worth mentioning, that running program has the possibility of taking into account different types of the finite elements, application of regular Gaussian quadrature procedure as well as reduced integration technique, and variable time step.

Generally, in comparison with the results published in scientific literature [2,3,21,22], the obtained numerical results indicate good accuracy and convergence to the exact solution with application relatively rough finite element mesh. Typically for nonlinear problems, all tested examples are sensitive to the applied time increment, which is intrinsic1y connected with the required number of equilibrium iterations in Nemon-Raphson method.

542

SLAB FREEZING

.018

2000 4000 6000 8000 10000 S

LEGEND - SLAB 1 - 100 STEPS --- SLAB 2 - 20 STEPS .-.-. EXACT SOLUTION

Fig. 4 Frozen thickness for different time steps.

CONCLUSION AND FINAL REMARKS

The paper presents detailed formulation of the nonlinear heat transfer problem including phase change in homogeneous medium. The analysis employs strong and weak form of heat propagation with phase change. With respect to spatial variables the finite element method is used, while integration with respect to time variables is obtained through application of the Euler backward procedure. Nonlinear mechanical (density), and thermal properties as well as variable temperature are analyzed. by introducing the concept of incremental decomposition with respect to time. New approach is applied in the analysis of the latent heat connected with phase change. The concept is based on the properties existing between unit normal vector defining moving surface and vector of velocity describing speed of moving phase change surface. The transformation mapping of the coordinates was linked with the rate of temperature changes. In the scope of the finite element method, vector describing phase change was divided into three parts. In order to speed up the convergence of the solution, the Newton-Raphson method is used. The accuracy of the presented formulation and numerical algorithm is exemplified by one of the tests and compared

543

with the exact solution. On the basis of various tests, it is concluded that the proposed procedure gives the results of higher accuracy with application of rough finite element mesh.

ACKNOWLEDGEMENT

The authors wish to express their gratitude to the Natural Science and Engineering Research Council of Canada. for providing financial support under Grant No. OGP 0004686.

REFERENCES

1. Salcudea.n, M. and Abdullah, Z., "On the Numerical Modeling of Heat Transfer During Solidification Process", Int. J. Numer. Meth. Engng., Vo1.25, 1988, pp. 445-473.

2. Lunardini, V.J., Heat Transfer in Cold Climates, Van Nostrand Reinhold Co., New York, NY, 1981.

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

4. Comini, G'l DelGiudice, S., Lewis, R.W. and Zienkiewicz, O.C., "Finite Element Solution 01 Nonlinear Heat Conduction Problems with Special Reference to Phase Change", Int. J. Numer. Meth. Engng., Vol. 8,' 1974, pp. 613-624.

5. Bell, G.E. and Wood, A.S., "On the Performance of the Enthalpy Method in the Region of a Singularity", Int. J. Numer. Meth. Engng., Vol. 19, 1983, pp. 1583-1592.

6. Shamsundar, N. and Sparrow, "Analysis of Multidimensional Conduction Phase Change via the Enthaply Model", J. Heat Transfer, 1975, pp. 333-340.

7. Voller, V., "Interpretation of the Enthalpy in a Discretized Multidmensional Region Undergoing a Phase Change", Int. J. Heat Mass Transfer, Vol. 10, 1983, pp. 323-328.

8. Voller, V. and Cross, M., "An Explicit Numerical Method to Track a Moving Phase Change Front", Int. J. Heat Mass Transf., Vol. 26, 1983, pp. 147-150.

9. Lewis, R.W., Mor§an, K. and Habashi , W.G. (eds.), "Numerical Methods in Thermal Problems, Proc. Fifth International Conference, Montreal, Canada, June 29 - July 3, 1987, Pineridge Press, Swansea, U.K. 1987.

10. Voller, V., Cross, M. and Markatos, N.C., "An Enthalpy Method for Convective Diffusive Phase Change", Int. J. Numer. Meth. Engng., Vol. 24, 1987, pp. 271-284.

11. Roose, J. and Storrer, 0., "Modelization of Phase Changes by Fictitious Heat Flow", Int. J. Numer. Meth. Engng. Vol. 20, 1984, pp. 217-225.

12. Goodman, T.R., "The Heat Balance Integral and its Applications to Problems Involving a Change of Phase", ASME Trans., Vol. 80, 1958, pp. 335-345.

13. Bell, G.E., "A Refinement of the Heat Balance Integral Method Applied to a Melting Problem, Int. J. Heat Mass Transfer, Vol. 21, 1978, pp. 1357-1362.

14. Kececioglu, I. and Rubinsky, B., "A Mixed-Variable Continuously Deforming Finite Element Method for Parabolic Evolution Problems. Part I: ' The Variational Formulation for a Single Evolution Equation", Int. J. Numer. Meth. Engng., Vol. 28, 1989, pp. 2583-2607.

15. Kececioglu, I. and Rubinsky, B., "A Mixed-Variable Continuously Deforming Finite Element Method for Parabolic Evolution Problems. Part II: The Coupled Problem of Phase-Change in Porous Media", Int. J. Numer. Meth. Engng., Vol. 28, 1989, pp. 2609-2634.

544

16. Kececioglu, I. and Rubinsky, B., "A Mixed-Variable Continuously Deforming Finite Element Method for Parabolic Evolution Problems. Part III: Numerical Implementation and Computational Results", Int. J. Numer. Meth. Engng., Vol. 28, 1989, pp. 2715-2760.

17. Kececioglu, I. and Rubinsky, B., IIA Continuum Model for the Propagation of Discrete Phase-Change Fronts in Porous Media in the Presence of Coupled Heat Flow, Fluid Flow and Species Transport Processes", Int. J. Heat Mass Transfer, Vol. 32, No.6, 1989, pp. 1111-1130.

18. Eringen, A.C., Mechanics of Continua, John Wiley &; Sons, Inc., New York, 1967.

19. Bathe, K.J., Finite Element Procedures in Engineering Analysis, Prentice-Hall, Inc., Englewood Cliffs, New York, 1982.

20. Hughes, T.J.R., The Finite Element Method-Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1987.

21. Pham, Q.T., "The Use of Lumped Capacitance in the Finite-Element Solution of Heat Conduction Problems with Phase Change", Int.J.Heat Mass Transfer, Vo1.29, No.2, 1986, pp.285-291.

22. Jumikis, A.R., Thermal Geotechnics, Rutgers University Press, New Jersey, 1977.

545