hf. J. Hcut Mau lian~fir. Vol 31, No. I, pp. 3546. 1988 0017-9310/88$3.00+0.00 Prmtcd in Great Br~tam :r‘ 1988 Pergamon Journals Ltd Natural convection solid/liquid phase change in porous media C. BECKERMANN and R. VISKANTA Heat Transfer Laboratory, School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, U.S.A. (Received 21 February 1987 and in final form 29 May 1987) Abstract-A combined numerical and experimental study is reported of solid/liquid phase change in porous media with natural convection in the melt region. The model is based on volume averaged transport equations, while phase change is assumed to occur over a small temperature range. Experiments are performed in a vertical, square enclosure using gallium and glass beads as the fluid and the porous matrix, respectively. For natural convection, melting and solidification (with initial supercooling), the numerical results show reasonable agreement with the temperature and interface position measurements. Natural convection in the melt as well as heat conduction in the solid is found to considerably influence the interface shape and movement during both the melting and solidification experiments. INTRODUCTION interaction between the solid/liquid interface motion and natural convection. With the exception of the Sot_m/liquid phase change in saturated porous media studies by Weaver and Viskanta [lO-121, no exper- occurs in a wide variety of systems in nature and imental investigations of solid/liquid phase change of engineering. Applications include the freezing and porous media contained in an enclosure with natural melting of soils [l], artificial freezing of ground for convection in the melt region have, apparently, been mining and construction purposes [2], thermal energy reported in the literature. storage [3], freezing of soil around the heat exchanger Due to the fundamental nature and practical rele- coils of a ground based heat pump [4, 51, food pro- vance of such problems, the objective of the present cessing, etc. The present study is, however, motivated study is to develop a general model for solid/liquid by solidification of multi-component mixtures. Due phase change in a porous medium with natural con- to the extended freezing temperature range of a vection in the melt region. In order to account for mixture, a mushy zone might exist, consisting of a fine the complicated interfacial structures of the various meshwork of dendrites growing into the melt region. constituents, the model equations are based on volu- In the past, this mushy zone has been modeled as metric averaging of the microscopic conservation a porous medium [6]. All of the above systems are equations. Although it may be possible to obtain simi- characterized by microscopically complicated struc- lar model equations directly from macroscopic bal- tures of the solid/liquid interface. In addition, natural ance considerations, it is believed that a rigorous aver- convection of the fluid occupying the void spaces of aging process will offer additional insight into the the solid matrix may strongly influence the heat trans- assumptions involved [14]. In addition, the volume fer and phase-change processes. averaged equations employed in the present study are While liquid/gas phase change (i.e. condensation of a more general nature than equations which are and evaporation) in porous media has received con- primarily based on empirical relationships (e.g. siderable research attention [7, 81, little work has been Darcy’s law). For example, the averaged equations done in the area of melting and solidification of a are valid for any volume fractions of the porous matrix liquid saturated porous medium [S]. Freezing and and the liquid and solid phases. Hence, the present melting of water saturated porous media contained in approach offers the advantage that the entire domain various enclosures has been studied by Weaver and can be treated as a single region governed by one set Viskanta [l&12]. Their freezing experiments in a ver- of conservation equations. In other words, the same tical rectangular cavity [12] clearly show the influence equations can be used for the melt as for the fully of natural convection on the solid/liquid interface solidified regions. Theoretically, the model can also shape and motion. O’Neill and Albert [13] have be utilized for solidification of multi-component mix- numerically modeled solidification of porous media tures, where the porous matrix is formed by the solid in the presence of natural convection using a finite phase of the fluid alone (i.e. dendritic growth). Again, element method. The energy equations for the frozen the model equations would be valid for the pure liquid and unfrozen zones have been coupled by an appro- and solid regions as well as for the porous mushy priate interface energy balance. Their computed zone. results for a vertical square cavity indicate a strong In the present study, the averaged conservation 35 5 C. BECKERMA~N and R. VISKA~TA NOMENCLATURE A aspect ratio, H/L 9 dimensionless vertical coordinate, y/L specific heat [J kg- ’ K - ‘1 0 dimensionless temperature, FY inertia coefficient (T- 7’c)I(Tn - Tc) Da Darcy number, K/L’ K permeability ratio, K&/K 9 gravitational acceleration (m s-‘) A thermal conductivity ratio, keff/kl h enthalpy [J kg _ ‘1 p dynamic viscosity [N s mm ‘1 Ah latent heat of fusion [J kg- ‘] V kinematic viscosity [m’s_ ‘1 H height of enclosure [m] 5 dimensionless horizontal coordinate, x/L k thermal conductivity [W rn- ’ K- ‘1 z dimensionless time, toll/L2 K ~rmeab~lity ]m”] cf, quantity, see equation (2) L length of enclosure [m] 0. thermal capacitance ratio, jZ/( p,c,). P pressure [N m- ‘1 P dimensionless pressure, pL,“/(p,v,cc,) Subscripts Pr Prandtl number, v& C cold Ra Rayleigh number, g/$(Tn - Tc)L3/(v& eff effective Ste Stefan number, c,( 7’n - Tc)/A.h f fluid t time [s] H hot T temperature [K] initial u velocity [m s- ‘1 : liquid u dimensionless velocity, uL/ct, m fusion V volume [m ‘1 p porous matrix x horizontal coordinate fm] ref reference Y vertical coordinate [ml. S solid c( constituent Greek symbols E porosity. thermal diffusivity, k,/(~c,) [m’s_ ‘1 ; coefficient of thermal expansion [K- ‘1 Y liquid fraction, see equation (1 b) Superscripts 6 liquid fraction, see equation (lc) * microscopic & porosity, see equation (la) average. -I equations are applied to phase change of a fluid inside each phase, are integrated over a small volume a vertical rectangular enclosure filled with a porous element, V. As shown in Fig. l(a), such a volume matrix. The governing equations are solved numeri- element is, in general, composed of the porous matrix cally. Some aspects of the model are verified by con- VPand the fluid V, which, in turn, may be in the solid, ducting a limited number of natural convection, melt- V,(t), or liquid, V,(t), state. Hence, we can define the ing and solidification experiments in a rectangular test following volume fractions cell heated and cooled from the sides. The porous matrix used in the experiments consists of spherical fraction fluid in volume element glass beads. Because of its low melting temperature and well established thermophysical properties, gal- lium is chosen as the fluid occupying the void spaces fraction liquid in fluid and undergoing phase change. ANALYSIS Mathematical formulation fraction liquid in volume element. The complicated interfacial geometry of the porous matrix and the solid and liquid phases prohibits a In the melt region, the fluid is completely in the liquid solution of the microscopic conservation equations state so that y = 1 and 6 = E. On the other hand, in for mass, momentum and energy. Therefore, some the pure solid region, both y = 6 = 0. Note that E is form of a macroscopic description of the transport independent of the state of the fluid. As sketched processes must be employed. The present model is in Fig. l(b), the solid/liquid interface may be highly based on the volumetric averaging technique [14] for irregular due to the presence of the porous matrix. obtaining the macroscopic conservation equations. In This is especially true for large differences in the ther- this technique, the microscopic equations, valid for mal conductivities of the porous matrix and the fluid Natural convection solid/liquid phase change in porous media 37 (1) The flow and heat transfer are two-dimensional and laminar. (2) The thermophysical properties of the porous matrix as well as of the solid and liquid phases are homogeneous and isotropic. (3) The porous matrix and the fluid (solid or liquid) are in local thermal equilibrium (i.e. T, = T, = T, = T). (4) The porous matrix and the solid are rigid (i.e. (4 UP = II, = 0). 2 (5) The porous matrix/fluid mixture is incom- t Solid pressible and the Boussinesq approximation can be invoked. (6) The thermophysical properties are constant, but may de different for the porous matrix (p), liquid (1) and solid (s). 0 0.5 IO (7) The dispersion fluxes due to velocity fluctuations are negligibly small. (b) (8) Velocities due to density change upon phase change are neglected (i.e. p, = pS = pr). ,,o!_____ - The averaging process of the microscopic con- servation equations for each constituent is described in detail in refs. [14, 16181 and does not need to be repeated here. It should be mentioned, however, that the mass and energy equations are obtained by com- bining the averaged conservation equations for each constituent, while momentum conservation needs to be considered for the liquid phase only. The averaged 0 + I T,-AT Tm T,+ AT mass and momentum conservation equations are T v*u=o (4) (4 PI au PI FIG.