In, J. “co, Mrru zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Trcm.sf<~r zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Vol. 36. NO 18, pp. 4493-4497. lYY3 Pnnted I” Great zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Bntain zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 0017-93lO/Y3$6.Otl+O.O0 C 1993 Pergamon Press Ltd TECHNICAL NOTES Effective modeling/analysis of isothermal phase change problems with emphasis on representative enthalpy architectures and finite elements RAJU R. NAMBURU and KUMAR K. TAMMA+ Department of Mechanical Engineering, I I I Church Street S.E., University of Minnesota. Minneapolis, MN 55455, U.S.A. INTRODUCTION THE SUBJECT matter related to phase change and typical of solidification or melting is of importance in many engineering applications. Over the years, a number of analytical and numerical approaches have been attempted for the simu- lation of phase change problems and a great deal of interest and research endeavors are in progress both in industry and the research community at large. The purpose of the present paper is to describe effective yet simplified representative enthalpy-based formulations following the initial developments due to the authors [I]. In particular, emphasis on the applicability to a class of isothermal phase change problems is demonstrated via analogous representations with only minor modifications. Although the original developments which we strongly advo- cate are robust and geared towards the general (isothermal and mushy) phase change problems [l], the present rep- resentations provide certain simplifications with some added advantages and restrict attention to a class of isothermal phase change problems. Attention is confined to fixed grid methods for the purposes of illustration. NUMERICAL MODELS For the numerical simulation of phase change problems, both finite difference methods and the finite element methods have been extensively used. Because of the inherent flexi- bility, effectiveness for modeling complex shapes, and the several other advantages, the paper concerns the finite element method. Employing finite elements for modeling: analysis of phase change problems, the class of methods, namely, apparent heat capacity methods, fictitious heat flow or source-based methods, and enthalpy-based methods seem to be the more prominent methods customarily advocated. The apparent heat capacity methods with temperature fields as the dependent variables have been traditionally used in conjunction with finite elements [2, 31 since the basic form of the equation for phase change is analogous to that of the classical heat conduction equation. However, to handle the Dirac-d-type behavior for the heat capacity in phase change situations. an enthalpy function is introduced. And various approximations appear in literature for evaluating the effec- tive heat capacity, pC, of which the more commonly advo- cated methods are summarized in ref. [l] and references thereof. In conjunction with all of these approximations, a fnite interval width At, is assumed even for handling iso- thermal phase change problems. As a consequence of these approximation techniques, a correct heat balance is pre- served by avoiding the possibility of missing the peak values. _F Author to whom correspondence should be addressed. and much of the past work involves interpolation of enthalpy. H = s PC(O) dU, H = N,H,, rather than the direct evaluation of the heat capacity. In the fictitious heat flow or source based methods, the effects due to latent heat are introduced directly as a non- linear source related term. Most of these methods involve some sort of monitoring of the heat Row to represent the release of latent heat. Numerous strategies have been attempted by various researchers (see Ralph and Bathe [4]. Roose and Storrer [5], and references thereof) to include effective updating procedures for computing the resulting liquid fraction field from known temperature fields and the like. Although applicable to isothermal and mushy phase change problems, it has been observed to yield accurate freezing front locations with fairly coarse (large) sizes for mesh and large time steps with the exception for computing temperature fields which require much refined values. Although enthalpy based methods have been emphasized for phase change problems [6], and literature regarding their use in conjunction with the finite element method is limited (see refs. [7--91) in comparison to the apparent heat capacity methods, more recently. increased attention has been directed to demonslrate the effectiveness of enthalpy based representations for general phase change problems [I]. The distinguishing differences in the various formulations rel- evant to enthalpy-based methods lie in the representation of the resulting governing equation either in terms of the total enthalpy and therein employing the discretization process. or, in employing the discretization process firstly in the form represented by fi-(k$,,),, = e (1) and later introducing the representative relations for the temperature 0 in terms of the enthalpy H. In this paper it is this later form of representation in which we introduce modifications to our previous efforts to achieve certain added advantages for applicability to a class of isothermal phase change situations. y-FAMILY OF REPRESENTATIONS For handling general phase change problems, we first rep- resent the governing equations in conservation form as H, = 43.) + P (x,. t) E .Qx(o, T) (2) subjected to appropriate boundary and initial conditions. Following our previous effort [I], introducing the approxi- mations 4493