Transactions on Modelling and Simulation vol 10, 1995 WIT Press, ISSN X

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1 Numerical analysis of melting process in a rectangular enclosure J. Goscik Department of Mechanical Engineering, Technical University of Bialystok, Bialystok, Wiejska 45C, Poland Abstract In the paper some results of numerical analysis concerned with the natural convection dominated melting inside a rectangular enclosure with the heating vertical and horizontal walls are presented. Finite volume computational technique, including the enthalpy method to model the latent heat absorption at the moving liquid-solid interface and Boussinesq approximation to description the convective flow in the melt region is utilized. The findings, most interesting for engineering practice, i.e. data referring to time variations of the melted phase change material (PCM) volume fraction and the Nusselt number, are compared with the published correlations. 1 Introduction Problems associated with quantitative descriptions of liquid-solid phase change processes are a permanent subject of interest. Recently, these interests have been pushed forward by needs of analysis of the heat storing systems that utilize thermal effects of the phase changes. In this case particular attention is paid to influence of natural convection in the liquid phase of a (PCM). There are some excellent reviews confirming this fact for a generally stated problem [1]. Also recently numerous efforts done in the above area and referring to the rectangular storing units have been summarized [2]. Taking results of these achievements into account we can conclude that for rectangular geometry the investigations concerned with two somewhat limited approaches to the problem prevail, as follows: i) thefirst,when driving forces represented by temperature gradients are orientated parallel to the gravity field force (the Rayleigh-Benard problem), ii) the second, when

2 128 Computational Methods and Experimental Measurements temperature gradients are perpendicular to the gravity field force. Actually, only a few trials can be found aimed to describe the problem when driving force being the combination of the above mentioned cases governs the melting process evolution. Despite common practical reasons, e.g. for the exchangers equipped with finned heat transfer surfaces where often frequently the storing substances have markedly lower the heat conduction coefficient, this state of achievements in the area still has been maintained. On the basis of the numerical solutions obtained on an assumption of the two dimensional (with respect to the space coordinates) model, the work presented is aimed mainly to estimate how much credibility can be paid to such descriptions of real processes occurring in the enclosure having complex heat transfer surfaces. Referring to the above, the obtained numerically data describing time variations of amount of the melted medium as well as the Nusselt number (averaged spatially and with respect to time) are compared with selected referenced correlations. 2 Formulation of the problem The subject of interest is a computational analysis of the melting process of a mediumfillinga rectangular enclosure that model representation is shown in Fig.l. The description of the process refers to a separated part of the enclosure cross-section marked as shaded area in Fig.l. Figure 1: The physical model of the enclosure It is assumed that the enclosure is partiallyfilledwith a PCM that enthalpy

3 Computational Methods and Experimental Measurements 129 h can be represented by the well known superposition [4], as follows: /o c<w and -o=0* where: c - the specific heat, 0* - the phase change temperature, g - the density, AA* - the latent heat of phase change, % - the liquid fraction defined as unique temperature function. The PCM has constant thermodynamic properties independent of phase form. The viscosity represents here the only exception to the above assumption, so that this property is treated as an effective in further considerations and is defined as a function of the liquid fraction by the following form: where: 77* - the dynamic viscosity at a reference temperature, cons^ - a constant value, taking order of 1(P. Consequently, the Boussinesq approximation is applied here to describe the buoyancy effects. For purposes of the analysis it has been assumed that energy storing by thermal effects of the liquid-solid phase changes occurs at isobaric conditions. Moreover, an assumption is introduced that liquid phase of the PCM behaves as an incompressible Newtonian fluid in the laminar flow and the flow is treated as spatially two dimensional. In further considerations the model applied has been reformulated to get dimensionless forms of the equations involved. These were derived by introduction the following scale ratios: geometry SQ = /i -> x = x/li ; time ^ = (/?/a)-a;-* -> r = Fo-u (w = Ste/(Ste+l)- time scaling factor); temperature & = A0(= 0*-%,) -» 0(0-0,)/A0; velocity & = y/fi _» = (i///i).i; ; pressure ^ = ^(i///i)2 _» p = ^.(/i/f/)2.p; viscosity Sr, = rjo -> T/ = T//^. Thus, the dimensionless, primitive variable^ continuum conservation equations for phase change and convection in two dimensions are as follows: 0 = divi +^2^2; (3) w a = -^i(pr. ui 6 - %a) - %(f r ^ 6 - %4 - (1 - w) 9,%. (4) Moreover, it is assumed that initially region of interest (a rectangle D of /i x /2 dimensions) isfilledwith a PCM (being in the solid phase) that has a uniform temperature 0, across the region which is equal to 0* (the initial subcooling of the solid phase is negligibly small). The melting process is

4 130 Computational Methods and Experimental Measurements commenced by a rapid increase in temperature on vertical and horizontal walls up to OH > 0". Therefore, the initial conditions are a(z) = 0, u(z) = 0, Vz G D, T = 0. (5) Energy transfer conditions between the enclosure and its surroundings across the corresponding walls are homogenous and permit to be represented by boundary conditions of thefirstorder. Thus s(:r)_l, VxeriVT2,,s(x] _ 0, Vx TS V T<, I ^ With the use of the scale ratios as those introduced above, description of the melting process involves a set offivedimensionless numbers: A = /2//i - ratio of the characteristic dimensions, Pr rj/ g a - the Prandtl number, #ai = 2 0 A0 (/i)3/(;/ a) -the Rayleigh number, gfe = c (0* - #*)/AA* - the Stefan number, Sc - the parameter of subcooling (defined similarly as the Stefan number). Equation system (l)-r(4) accompanied by initial condition (5) and boundary conditions (6) was solved by means of code denoted as LHS-FREE. This code has been tested and the corresponding results can be found in [5]. Moreover, after comprehensive and multicases sensitivity tests of the solutions obtained with respect to spatial mesh increments and stepping in time, the results presented were calculated using a uniform spatial mesh 64x96 (at constant value of the mesh parameter A# = 1/64) and constant value of the time step AT = 1 10"^. 3 The results of calculations The analysis presented was carried out taking into account results of a process simulation computed for the following parameter values assumed: A = 1.5, Pr = 50, Ra = l- 10\ Ste = 0.133, Sc = 0.0. Consequently, all the conclusions and remarks have been formulated on this basis. 3.1 The determining of the melted volume fraction Marshall [6] recorded by a camera variation of the melted phase for many different arrangements of the transfer surface of a rectangular heat storing unit. However, he did not recommend for engineering practice any useful formula describing variations he investigated. Therefore, it may be said that proposals of a simple description related to are given only by Ho and Viskanta [3], and Viskanta [1]. They [3] suggest a simple formula X(o) = 51.3 T -«" Ste * A-, (7)

5 Computational Methods and Experimental Measurements 131 where melted fraction X(o) is defined as a ratio of instantaneous melted PCM volume i? to the initial volumetf<>.moreover, a slightly different proposal done by Viskanta [1] consists in the determining of the melted volume fraction with respect to maximum volume 7?rna*, i.e., (8) 0.0 A= 1.5,Pr=50, Ste = 0.133, S = Time? Figure 2: Time variation of the melted volume fraction According to correlations (7) and (8) and the numerical solution a dependency on time of the liquid phase fraction is presented in Fig.2. The plot shows a reasonable good agreement in the results obtained numerically and by empirical correlation (7). Thus, it may be concluded that an assumption about invariable volume of the phase changing medium does not introduce essential discrepancies and the numerical approach presented appears to be almost alternative here (excellent agreement for time interval that the melting process takes on to be completed) with respect to experimental treatments. At the same time the course of the curve found numerically displays the evidence of various stages of the melting occurring in reality, as well. This fact is demonstrated by somewhat nonlinear character of changes at the initial stage (when conduction governs the melting) and at the final stage when departure from the linear behaviour can be explained as resulting from decreasing of the driving force in course of the process running. Marked differences between the computational curve and empirical correlation (7) can be reported as caused by neglecting of the density changes. Overflow of the solid phase due to the expanding the liquid from the melt and seeking additional volume, as observed [3], leads to decrease in the driving force affecting the vertical zone, and thus to reduction in course velocity of the melting process. Consequently, experimentally estimated quantities can reach lower values.

6 132 Computational Methods and Experimental Measurements 3.2 Heat transfer Another quantity of essential importance in engineering practice that has been involved into comparisons presented is the Nusselt number averaged spatially and in time, where: a - the averaged heat transfer coefficient, lh - the characteristic length, A - the heat conduction coefficient. Here one follows method of averaging as reported by Marshall [6]. Accordingly to his approach the averaged heat transfer coefficient can be defined with the use of relation where: Q - the total amount of heat stored in the system, F - the heat transfer surface, A0 - the driving force (here = OH 0*). The characteristic length seen in (9) has been chosen as equivalent for one dimensional formulation of the problem considered. Therefore, for the geometry assumed the characteristic length is defined as Then, taking into account relations (10) and (11), and combining these with method of determining Q recommended by Marshall [6] one finds, A However, the Rayleigh number formulated with the use of the l^ becomes A ^ (13) After substitution values X(num) of the melted fraction, as computed, courses of Nu(x) and Raeq(x) are obtained from eq.(12) and (13). Fig. 3 shows the corresponding plot of Nu(x) and Ra,eq(x)- At the same time after a simple rearrangement the most preferred relation i.e., Nu = f(ra^q) has been obtained. As it can be seen the relation Nu = f(rdeq) reveals numerous irregularities for the initial stage proper a complex nature of forming a well developed convective flow pattern. However, since Ra w 1-10* the curve appears as being a regular, adequate for melting when natural convection is fully developed. For this region application of the least square method of fit gives = 0.21 #a f*, I.IO* < Raeq < *. (14)

7 Computational Methods and Experimental Measurements 133 Then, expression (14) demonstrates an excellent accordance to those proposed by Ho and Viskanta (in the considerations presented the same definition of the Nusselt number is applied) who having performed a number of experiments recommended the following correlation determining the Nusselt number valid within region 2 10^ < Ra,eq < 6 10^ (15') or alternatively (15") However, Marshall [6] investigating the configuration as considered carried out numerous experiments and the statistical analysis has found finally that the best expression describing the Nusselt number takes the following form: = (16) , Pr= 50, Ste = 0.133, S = C Ra = 10 & a Rayleigh number Raeq Figure 3: The Nusselt number versus time A somewhat self-evident remark refers to difference in values of the exponents appearing at the Rayleigh number. Ho and Viskanta explained that first of all this difference is caused by the scattering of data that was taken by Marshall [6] to develop the correlation. However, it seems that these discrepancies can be due to a solid phase behaviour. Marshall has not described precisely how the experiments were running and was behaviour of the solid phase. Then, it is not completely clear whether in Marshall's experiments melting in so called close contact has been recorded and whether displacement of the solid phase caused by the gravity has been observed. When experiments were carried out by Ho and Viskanta setting the solid phase to be kept at rest has been established as an important factor. Therefore, the

8 134 Computational Methods and Experimental Measurements numerical solutions presented and the results of experiments published in [3] are markedly close by. Then, such agreement between the simulation results as presented and correlations (15') and/or (15") may refer to a melting process occurring at no displacement of the solid phase. 4 Concluding remarks The main feature of the solutions presented consists in that the process analyzed can be described by two dimensional models. This assumption has been introduced taking into account considerably limited performance of the commonly available computational resources. The qualitative results clearly disclosed here discrepancies not only caused by a reduced spatial description but as well by omitting changes in the medium density that can emerge for different phases. At this stage of investigations drawing conclusions concerned with answering a question which disturbance, among as mentioned the above, may has stronger influence on the results obtained is difficult. It seems that the form of the model proposed provides sufficiently accurate data useful in engineering practice. References 1. Viskanta,R. Natural convection in melting and solidification, Natural Convection: Fundamentals and Applications, eds. S.Kakac et al., pp , Hemisphere Publishing Corporation, Washington, DC, Goscik,J.& Lach,J. Analysis of transport phenomena occurring in rectangular latent heat storage units, Parts II and III - Description of storage cycle, to be published in Trans, of the Institute of Fluid Flow Machinery, Gdansk, 1995 (in Polish). 3. Ho,J.C.& Viskanta,R. Experimental study of melting in a rectangular cavity, in Heat Transfer 1982 (eds. U.Grigull et al.), pp , Proceedings of the 7th Int. Heat Transfer Conference, Munchen, Germany, Hemisphere Publishing Corporation, Washington, DC, Voller,V.R. Implicit finite-difference solutions of the enthalpy formulation of Stefan problems, IMA J.Numer.AnaL, Vol.5, No.2, 1985, Goscik, J. Modeling of unsteady transport phenomena in a heat exchanger with the presence of liquid-solid phase change processes, Ph.D. Thesis, Institute of Fluid Flow Machinery, Gdansk, 1995 (in Polish). 6. Marshall,R.H. Natural convection effects in rectangular enclosures containing a phase change material, Thermal Storage and Heat Tr. in Solar Energy Systems, eds. F.Kreith et al., pp 61-69, ASME, New York, 1978.

NUMERICAL STUDY OF MELTING OF TIN WITHIN A RECTANGULAR CAVITY INCLUDING CONVECTIVE EFFECTS

NUMERICAL STUDY OF MELTING OF TIN WITHIN A RECTANGULAR CAVITY INCLUDING CONVECTIVE EFFECTS Christiano Garcia da Silva Santim, chrisoff22@yahoo.com.br Luiz Fernando Milanez, milanez@fem.unicamp.br Universidade