Keywords: the space-time conservation element and solution element method, heat conduction, melting, freezing, enthalpy method.

The range of scientific and industrial applications of heat flow with solid-liquid phase change is very broad. For example, migration and solidification of magma in geology, planet's core solidification in astrophysics, global weather modelling, formation of ice on the oceans as well as on aircraft surfaces, casting and welding of metals, water purification through freezing, usage of latent heat storage systems in temperature control, ablation of bodies during re-entry, destroying cancer cells is cryosurgery, melting of electrical fuses, and preserving food through deep freezing, represent a short list of phase change related phenomena and industries. Because of its importance, accurate and robust methods of modelling phase change problems are of great interest.

There exist two principal approaches for modelling phase change problems: (1) Phase front fitting in which explicit tracking of the phase change boundary is followed by using moving grid schemes. Phase front fitting methods, however, often require complicated starting solutions [1], and are inapplicable for materials that change phase over a temperature interval rather than at a single specified temperature. Moreover, these methods are difficult to extend to multidimensional problems; (2) Phase front capturing methods, which automatically determine the location of the phase front as part of the solution of the problem. These methods offer a more suitable approach for modelling a general phase change problem. The most prominent of the phase capture schemes is the enthalpy method which by incorporating the latent heat of fusion in the formulation provides the location of the liquid/solid interface as an integral part of the solution. Numerical application of these methods, however, produces better results when the phase change occurs within a temperature range. For situations where the phase change occurs at a single temperature, the phase front is a moving discontinuity. Consequently all of these methods need regularization and special adjustments in order to achieve convergence and stability of the numerical solution [2] and to avoid oscillations in the location of the interface.

The method of space-time conservation element and solution element (CE/SE) was developed by Chang [3] at the NASA Glenn Research Centre in order to solve the conservation laws. Between 1991 and today (2004), the CE/SE method has been applied to a range of PDE's, mainly in the area of fluid mechanics. However, the fact that the CE/SE method is spreading successfully to disciplines other than that where it had originated, presents a strong confirmation of the method's robustness and generality.

The space-time CE/SE method was applied by the authors to heat conduction problems with isothermal phase change, for two-dimensional and axisymmetric as well as three-dimensional geometries. The results for several cases were compared to available analytical and semi-analytical solutions. The method's convergence and error behaviour were also studied and it was found to be effective and accurate for these applications. No non-physical oscillations in the phase change interface were detected. Therefore, the CE/SE scheme was recognized as being able to resolve the weakness mentioned for the numerical simulations of the enthalpy method for isothermal phase change.

In the present paper, the CE/SE formulation is presented, briefly, for one-, two- and three-dimensional, as well as axisymmetric problems. A more detailed discussion is presented on the convergence, error behaviour, and the order of accuracy for each formulation. All cases studied hitherto involved isothermal phase change. The method is then extended to also simulate phase change over a temperature range. A one-dimensional test problem was used for validating the new modules. The program was found to be accurate and efficient in simulating phase change over a temperature range. It was confirmed that the space-time CE/SE method has the potential for being considered as an effective alternative numerical scheme for general phase change problems.

1

Boley, B.A., "A General Starting Solution for Melting and Solidifying Slabs", Int. J. Engng. Sci., 6(89), 1968. doi:10.1016/0020-7225(68)90022-0

2

Celentano, D. and Pérez, E., "A Phase-Change Temperature-Based Formulation Including General Latent Heat Effects", Int. J. Numerical Methods for Heat and Fluid Flow, 6(8), 71-79, 1996. doi:10.1108/eb017558

3

Chang, S.C. and To, W.M., "A New Numerical Framework for Solving Conservation Laws - The Method of Space-Time Conservation Element and Solution Element", NASA TM 104495, 1991. doi:10.1007/3-540-56394-6_255

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