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2nd European Thermal-Sciences and 14th UlT National Heat Transfer Conference 1996
G.P. Celata, P. Di Marco and A. Mariani (Editors) © 1996 Edizioni ETS, All rights reserved.
FINITE ELEMENTS BASED SOLUTION FOR HEA T TRANSFER WITH PHASE CHANGE IN A RECTANGULAR ENCLUSURE Ismail, K.A.R. Gonçah'es, M.M. and Scalon, V.L. Departamento de Engenharia Térmica e de Fluidos-FEM-UNICAMP Caixa Postal 6122, CEP 13083-970, Campinas (SP), Brazil FAX 055-0192-39-3722, E-mail:
[email protected]
ABSTRACT This paper present a numerical solution of the heat transfer problem with phase change inside a rectangular fmned cavity filled with pcm. The two dirnensional problem is formulated using the enthalpic method and the finite element representation. Three discretization schemes were tested to choose the most appropriate among the Lee's scheme, the modified Predictor-Corrector and the implicit scheme. It is found that the last schemeis the most accurate but consumes a lot of computing time. Numerical results 01' lhe temperature profiles, interface location and total energy stored as predicted from the implicit model are presented and discussed .
1. INTRODUCTlON The numerica1 modeling of heat transfer problems combined with phase change represents possible solutions to many practica1 problems in thermal storage systems, food freezing, steel industries and many others. Numerous methods have been proposed for the numerical solution of heat conduction processes with phase change. Some of these methods may dea1 only with situations of pure materiaIs and others are restricted to one dimensional cases. The great majorty of these studies uses in their solutions conduction based models combined with finite difference representation. When treating irregular geometries, convective effects, phase change over a temperature range and complicated boundary conditions, the numerical procedures are found to be complicated and time consuming. With the continuous advances in techniques based upon tinite elements representation, more progress is achieved in the specific area of heat transfer with phase change including the presence of convection inthe liquid phase. Bonnerot and Jamet [I] described a finite element method for the one dimensional Stefan problem. The elements are quadrilaterals of the space time plane which are determined at each time step in reference to the position offree boundary. The methodappears as a generalization of the classica1 Crank-Nicholson scheme and has the advantage of providing a simpIe and accurate determination of the free boundary. While this, Morgan, Lewis and Zienchiewicz [2] presented simplifications to the algorithm published by Comini, DeI Guidice Lewis and Zienchiewicz [3] for handling the phase change in non linear heat
conduction problems. They demonstrated that good accuracy may be achieved. Later Meyer [4] described the method of fractionaI steps and the method of lines, applied them to the solution of a general two dimensional Stefan type probIems and discussed the numericaI aspects and their relative performance. Morgan [5] presented an explicit tinite eIement algorithm for the soIution of the basic equations describing the combined conductive and convectíve heat transfer in materiaIs undergoing liquid-solid phase change. The results obtained seem to agree qualitatively with experimental observations. Later Rolph and Bathe [6] presented a simple and effective tinite eIement procedure for the solution of heat transfer with phase change. Y00 and Rubinsky [7] reported a solution of the multidimensional problerns of heat transfer with phase change using finite element approach. In their method the energy equation in the media and the energy equation on the interface are regarded as independent governing equations which, when solved through the use of a tinite element formulation, yielded the temperature distribution in the media as well as the continuous movement of the interface. Pham [8], described a method based upon using the lumped capacitance in the finite element solution of heat conduction problems with phase change. He adopted a weak solution method and consequently, a single partial differential equation is used to describe the heat transfer process throughout. The behaviour of the specific heat near the phase change point was overcome by Iumping the therrnal capacitance at the nodes. He obtained fast and accurate results. Albert and O'Neil [9] solved the two
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dimensional heat conduction phase change probicms using a moving boundary-moving mesh approach. A transfinite mapping technique successfully controls the interior mesh motion and the numerical results are found to compare well with analytical solutions. Yoo and Rubinsky [lO} reported a numerical technique for the analysis oftwo dimensional transient solidification in . the presence of time dependent natural convection in the melt. In their method they used a new Galerkin formulation for the energy balance on the interface. The finite elernent solution to the Galerkin formulation yielded the displacement of individual nodes on the interface. The displacement of the nodes is expressed by uncoupled components in the x and y directions. Dalhuijsen and Segal [11] presented a comparative study of finite element techniques for solving solidification problems. The accuracy of the computed temperatures of a liquid in a comer region under freezing conditions are compared for various fixed grid finite element techniques adopting the anaIytical solution as a reference. They also compared different time stepping techniques and different methods for the evaluation of latent heat, Storti, Criveli and Idelsohn [.12] described a method for straightcning curved interfaces arising in phase change problems. The method works on the isoparametric finite elements, performing a second transformation which maps the master element onto a new element in which the interface is a straight line. This allows using the current Gaussian integration techníque to evaluate the integrais over each phase. The method provides a better estimation of the latent heat comribution and produced good results. Comini, Dei Guidice and Saro [13] presented an algorithm for multidimensional conduction with phase change. The algoriOun was verified in the solution of one dimensional case and compared the results with available analytical and numerical solutions. They also presented results for the case of inward solidification in a cylinder as a two dimensional study case. McDaniel and Zabaras [14] studied a two dimensional solidification and melting problems for pure metaIs using an energy preserving deforming finite elements model. A least squares variational finite elements method formulation is implemented for both the heat and the fluid flow equations. The Boussinesq approximation is used to generate the bulk fluid motion in the melt. The method is shown to produce accurate numerical solutions for convection dominated phase change problems. The objective of this paper is to study a two dimensional phase change problem by using the enthalpy approach associated with the fmite elements representation. Investigations of the fin thickness, fin height and the boundary conditions and its effects on the solidified mass and .the interface movement are also included. This simple geometry is chosen in order to facilitate the comparisons, test and verification of some discretization schemes. 2. FORMULATION
AND NUMERICAL TREATMENT
A general sketch of the problem is shown in figure (1). The formulation of this problem is based upon lhe enthalpy method which includes the latent heat of the phase change
as a correction to the thermal. capacity in the phase change region. Knowing the temperature field, it is possible to calculate the enthalpy at each point and consequently the apparent thermal capacity of the PCM. The energy equation can be written as: I [pc
\
)
er -+-
Piat
õ
õx
lrk-aT) õx.
a(
aT) +- k=0 ày
(I)
ày
y
c..AViTY
J
Wrnt
Pc..
Figure 1: Layout ofthe problem The equations relating the enthalpy and the temperature are: H =
(pcp)
s
solid phase
(T - Tr)
H=(pcp)JTf-Tr)+P :Ar-(Tf - ~;)] H = (pCp)}T - 7;.) + PL+(pcp)/(T - Tf)
mushy range liquid phase
where ~T ís the phase change temperature range and L is the latent heat of fusion. In this way the phase of change problem is inserted into a simpler problem oftransient heat conduction. To solve this equation by a finite elements it is necessary to use lhe weighted residual method. Considering that the conductivity is coastant into each element and \jf a weight function:
er f (pc -dn+ o
p).
'a!
f o
a T dn+ f \jfk-a T dn=O 2
\jfk-2
õx.
2
o
2
(2)
ày
In this case we adopted a lumped formulation for the transient termo Converting ,lhe above expression into a weak form: aT f&\jf &T ( PCp ) i f at'dn+ k ax2 ax2 dn n n
(3)
+kf &\jf &T dO -kf\jf aT dT = O o ày2 ày2 T an-
Considering the application of finite elements to the above problem we have:
(4)
where N, is a interpolation function and Ti represents the temperatüre at the nodes.
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l
Also we adopt the Galerkin formulatiou:
O
O KN '(T
(:,J,]
(5)
The surface integral term is considered only at the frontiers and not at the internal points. Hence the problem ean be represented by the equation:
õÍ' (pc ) ( N. ) -+k Pe
(ÔN. ôNj ôN. ÔN )_ __ I __ + __ 1 -- j T =0 àyày 1
(6)
axax
lôt
where the operator
( ) represents the integration
over the
element volume. Adopting a second order discretization upon Taylor series for the transient term:
( pc )m( N. \(fm+l_fm-llj 2M 1
Pe
1
1
scheme
ôNJ. +__ ôN.1 __ ÔNJ.) +k (ÔN __ I __ ày ày
1
.
m+1 _
'I·'
T
m)
(11)
.,
Upon calculating the new thermal capacmes, it is possible to repeat the same procedure and obtain newer va!ues for the temperature field and so on. This is a quick apresentation ofthe procedure for solving the conduction based phase change probIem by the enthalpy method and the finite elements technique. In order to verify lhe numerical calculations, this method was applied to the same problem treated by VoIler and Cross [15] and under the same conditions.as shown in figure (2) . As can be seen the agreement for big time intervals are better than for the case of smaIl time intervals. --Volk:r and Cron • - - - - Present Mud~1
ax ax
(7)
(lTm-1 + Tm + Tm+1lj =0 1
based
I
1
3 which is valid for the third order temperature field and higher. To calcuIate the second order temperature field we use a fist order approximation:
{ \m( \PCp/e
')1 \1(-m+l T; -·T-m j
1 ôN iI e», j
Nj{----~~--tk\ ax ax
àNj + ày
a;J (8)
f.!Il + 1"."1+1, (
""Nj.\
1/_
-~--2-}--j = O Rearranging
equation (7), one obtains: Figure 2: Comparison
The soIution of equation (9) determines the temperature field and, subsequently, the enthalpy for alI the points can be obtained using the equation (2). Morgan, Lewis and Zienchiewicz [2] presented a long discussion on the approxirnation techniques for the heat capacity evaluation. In this study we adopted a first order approximation to the derivatives in the form: (10) The value of the element therrnal capacity can be caIculated as an average of the respective therrnaI capacities of the elements. By using an adequate formulation, one can also use the value of each corrected therrnal capacity in its node. To realize this, we include the values of thennal capacity in a diagonal matrix:
ofthe model prediction with VoIler & Cross results [15].
Although the Lee's algorithm predicted reasonably good results it showed very high oscilIatory behaviour special\'yvery near the interface. Because of this difficulty we had to test two more algorithms The first algorithm due to Comini, DeI Guidice and Saro [13] was tried and some modifications were introduced. In their original algorithm, they proposed a predictor corrector method with a second order aproximation in time,while in the present work, we used a first order approxirnation. The predictor uses the thermal capacity caIculated in terms of the temperatures aIready known. In the present modified version the temperature field calculated is used to reevaluate the thermal capacity and, by using the corrector, recaIculate the temperature field. The second method uses the implicit calculation scheme for evaluating the thermal capacity with the corrector equation recaIculated as many times as necessary until the temperature field converges. Pham [8] commented that these type of methods are inadequate for numerical simulations due to the high computational time in cpu but, with the rapid evolution of the computers, it seems that they are more preferred and getting widely used because of their precision and implementatíon simplicity.
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corrector method are away from lhe predictions from the other two methods.
3. RESITLTS A!\"'D DISCUSSION
The pcm used in the numerical simulatons is n-eicosane whose orooerties are shown below Properties n-Eicosane Solid Li~~ Saturation temoerature [K] 36,40 Latent heat IkJ kl!-'l 247,30 0,150 Thermal condutivitv rw m" KIl 0,150 Specific heat [J kg-I] 2,010 2,210 Densitv [kg m-31 778,0 856,0
_1.00
50 ~
100
150
Lee's Algorit m
~;:; ~1fi;:ep;edictor Corrector -1.2
~ -1.2
::3
-+-' o
•.. -1.4
~-1.4
E
,(l)
-1.6
-0-1.6
The proposed problem is solved using the three methods presented, keeping the same parameters as time step and gridsize (51 x 26) as constants. In the implicit scheme the convergence parameter used is O.Ü5 K as the maximum oscillation in the temperature field and 30 as the limiting number of iterations to achieve this limit. It is worth mentioning here that the relative performance of the three methods is evaluated in terms of a modified temperature defmed as Tm,i = Ti - Tr, where Tr is the phase change temperature and Ti as the temperature at some arbitrary point. In order to evaluate the relative performance of the three methods, we present the predictions of the temperature at the two points (0,100; 0,100) and (0,233; 0,500) of the domain, are shown in figures (3) and (4). These two points were chosen because the available results are in terms of these coordenares. As can be seen from figure (3) the . modified predictor-corrector rnethod seems to predict lower temperatures than the other two methods . Similar results and similar tendencies are presented in figure (4). Although the differences between predictions are big, one can notice that there is no oscilations in the temperature values.
Q)
~
U -: .8
~-;.8
100
Time
50 ~
Lee's
100
25g0
150
A1gorit m
28 L.!....!...!2 Impliclt Scheme 26 ~ Modified Predictor Correcto
28 26
~ 24 ::3 22
2+ 22
220 18
20 18
E-16
16
+-'
Q)
1-12
14 12
~10 _ 8
10 8
-o 5
6
Q)
o
2
14
4
4 2
2
O~~~~:==========O 50
100
Time
150
ZOO
200
2502.0
Figure 4: Temperature profile for position (0,233; 0,500)
~.
-:>
2:.- 100000 c-, o>
Q,
80000
C
W
-o
60000
60000
40000
40000
(j)
o>
o
'-
o .•.... VI
~ ~
-8 o f--
300
150
(hours)
50
Lee'a Algorithm ImpIiCl1:Scheme Moditied Predictor 100
150
20000 Correc tor 200
25~
Time (hours) Figure 5: Total energy stored Although the implicit scherne produces good and accurate results, the computation time involved is relatively high, as can be seen in table (1). Table 1: CPU Time for schemes using a Sun Sparc 20 machine Time (min) Scheme Lee's Algorithm 45,18 Modified Predictor-Corrector 89,26 Implicit Scheme 224,30
2502
(hours)
Figure 3: Temperature profile for position (0,100; 0,100). Figure (5) shows the total energy stored calculated by the three methods. Again one can notice that the predictions froro the modified predictor corrector method are higher than the predictions froro the other two methods. For the time interval of 10 hours, figure (6) shows the solid-liquid interfaces as predicted by the three methods. The results seem to be dose one to the other. For the time instant of 250 hours, the results predicted by the modified predictor-
Figure (7) shows the posmon of the solid liquid interfaces for the time intervals between 10 to 250 hours as predicted by the implicit scheme. As can be seen the profiles are smooth not showing any oscillatory behaviour. .It is interesting to mention that the above scheme linking the enthalpy approach to the tinite eleroents representation seems to predict precise and fast results specially when the geometry is very complexo The present roodel is being implemented in studies on heat storage using pcm of fixed fusion temperature as well as for materials with relativelly
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wide fusion temperature range. Also the mode! is bcing used in some research using the pcm as a thermal insulating material for wall panels. Pem storage configurations of the cylindrical type with a•.xial and radial fins are also being studied using the enthalpy finite difference representation scheme.
ACKNOWLEDGEMENTS The authors wish to thank the National Research Council, CNPq, for the financial support.
REFERENCES 1. Bonnerot, R. and Jamet, P., (1974),"A Second Order
Finitc Element Method for the One-Dimensional Stefan Problem" , Inl. J. for Numerical Methods in Engineering, vol.S, pp 811-820. 2. Comini, G., DeI Guidice, S., Lewis, R.w. and Zienkiewickz,C., (1974), "Finite Element Solution of Non-Linear Heat Conduction Problems with Special Reference to Phase Change" , Inl. J. for Numerical Methods in Engineering, vol 8 , pp 611~24.
NOMENCLATURE specific heat, kJ/kg K enthalpy, kJ/kg thermal conductivity, W/rnk Latent heat, kJ/kg time, s temperature, K phase change temperature, K modified temperature, K horizontal and vertical coordinates, m
.cp
H k L t T Tr
T-Tr x,y
0.0 0.1 0.2 0.3 0.4 0.5 1.1 I I I I I 1.11.0 0.9
Greek symbols D.T
- 0.8
.â ... ..- 0.7
- 0.7
..,
phase change range, K density, kg/m".
. p
0.8
til
~•..
PCM
0.6
- 0.6
o o
'
•...
e-,
í.O
I
0_9
I.LI
•.~,.*$-~-O! e-"'" ~.-~ •.......•• ;-..,
I
il] Ht
0.8
u{ u~tt
0.7
0.9
0.3
- 0.8
0.2
.
\
g,:.
- 0.6 '"
$I)
- 0.5 ct '§" - 0.4 "'--'"
t~ \. ""'::t:~_.-+
0.3 0.2
PCM
,t\
0.4
0.4
o
(lt ~\..
0.5
1.0
- 0.7~n
r:I
0.6
-
0.3 •....• t-+ -I- t-0.2 ~~
.• ~
I"\.
- 0.1
0.1 I
I
I
I
0.5
I
O·'ú.o 0.1 0.2 0.3 0.4 0.5°·0 : ~~coordinate (m) Ttme Time 250 hours 10 hours •...•....•. Implicit Scheme r-o _ :::Modified Predictor Corretor - ~ ++++ Lee's Algoritbm Figure 6: Comparation between the algorithm's solid liquid interface position.
- 0.5
- 0.3
~-~-~----------------0.1 0.0
... -.. ----.-----.--= O. 1 I
I
I
I
I
0.0
0.0 0.1 0,2 0.3 0.4 0,5 x coordinate (m)
________ T= 10h ------ T= 50 h - - - T= 100h
T= 150h T= 200 h T= 250 h
Figure 7: Solid liquid interface position for implicit scheme. 3. Morgan, K., Lewis, R.w. and Zienkiewicz, O'C; (1978), "An Improved Algorithm for Heat Conduction Problems with Phase Change" , Int. J. for Numerical Methods in Engineering, vo112, pp 1191-1195. 4. Meyer, G.H., (1978), "Direct and Iterative OneDimensional Front Tracking Methods for the Two Dimensional Stefan Problem", Numerical Heat Transfer, vol l, pp 351-364. 5. Morgan, K., (1981), "A Numerical Analysis of Freezing and Melting with Convection", Comp. Meth. Applied Mechanical Eng., vo128, pp 275-284. 6. Rolph III, w.n and Bathe, K.J., (1982), "An Efficient Algorithm for Analysis of Nonlinear Heat Transfer with
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Phase Change", Int. J. for Numerioal Methods in Engineering, vo118, pp 119-134. 7. Yoo, J. and Rubinsky, B., (1983), "Numerical Computation Using a Finite Elements For the Moving Interface in Heat Transfer Problems with Phase Transformation", Numerical Heat Transfer, vol 6, pp 209-222. 8. Pham, Q.T., (1986), "The use of lumped capacitance in the finite-element solution of heat conduction problems with phase change", Int. J. of Heat and Mass Transfer, vo129, pp 285-291. 9. Albert, M.R. and O'Neil, K., (1986), "Moving BoundaryMoving Mesh Analysis of Phase Change Using Finite Elements with Transfinite Mappings", lnt. J. for Numerical Methods in Engineering, vo123, pp 591-607. 10.Yoo, J. and Rubínsky, H.. (1986),"A Finite Element Method For the Study of the Solidification Processes in the Presence of Natural Convection", Int. J. for Numerical Methods in Engineering, vol 23, pp 17851805. 11.Dalhuijsen, AJ. and Segal, A, (1986), "Comparison of . Finite Element Techniques for Solidification Problems", Int. J. for Numerical Methods in Engineering, vol 23, pp 1807-1829. 12. Storti, M.; Crivelli, L.A. and Idelson, S.R., {1987), "Making Curved Interfaces Straight in Phase Change Problems", Int. J. for Numerical Methods in Engineering, vo124, pp 375-392. 13. Cornini, G. D. G. and Saro, 0., (]990), "A Conservative Algorithm for Multidimensional Conduction Phase Change",Int . .l. for Numerical Methods in Engineering, \'0130, pp 697-709. 14.McDaniel, D. and Zabaras, N., (1994), "A Least-Squares Front Tracking Finite Elemement Method Analysis of Phase Change with Natural Convection", Int. J. for Numerical Methods in Engineering, vol 37, pp 27552777. 15. VoIler, VR and Cross, M., (1981), "Accurate Solutions of Moving Boundary Problems Using Enthalpy Method", Int. J. ofHeat and Mass Transfer, vo124, pp 545-556.
