The use of heat flow modeling to explore solidification phenomena
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I.
INTRODUCTION
MATHEMATICAL modeling is proving a powerful tool for process optimization and control in a number of areas, largely as a result of the rapid development of numerical techniques and the widened availability of computing facilities. Within the solidification field, numerical models have been developed to treat a wide variety of situations 1-5including recent developments such as rheocasting, directed energy, and rapid solidification processing, 6-~~and considerable interest has been generated by specialist conferences such as that of August 1980 held at Rindge.tt Descriptions of the progress of solidification (and/or melting) are normally obtained by modeling changes in the thermal field resulting from external injection/extraction of heat. The art of developing a numerical model lies in striking a workable compromise between retaining viability in terms of programming complexity, system processing time, data storage space, etc., and representing the physical phenomena being modeled by means of simplifications that are acceptably realistic. In solidification processes there are, broadly speaking, two sets of phenomena which must be incorporated into the model: firstly, the exchange of heat between the material to be frozen/melted and its environment, taking account of the symmetry of the system, etc., and secondly, the internal redistribution of heat (as it is influenced by the constitution of the material, its freezing/ melting characteristics, the effect of melt convection, etc. ). This internal redistribution is in principle completely described by the correct manipulation of the thermal field equation, which (for constant thermophysical properties) reduces in the one-dimensional case to dT
02T + "~q
Ot = a OX---~
[l]
pc
where a is the thermal diffusivity, p is the density, c is the specific heat, a n d / / i s the local rate of heat generation. In most modeling schemes, attention is biased toward those boundary conditions describing heat exchange with the environment, with factors such as the role of microsegregation in changing the form of the mushy (liquid plus solid) zone and the effect of melt convection ignored or T. W. CLYNE is Lecturer in the Department of Metallurgy and Materials Technology, University of Surrey, Guildford, Surrey GU2 5XH, England. Manuscript submitted February 3, 1982. METALLURGICALTRANSACTIONS B
grossly simplified. While such approaches may be acceptable for many applications, more exact representation of how the material freezes (particularly with respect to the coexistence of liquid and solid) can be of considerable importance, especially for cases where the mushy zone may be of appreciable extent. In a recent publication,~2 the present author has described an explicit finite difference model for directional solidification of alloys, detailing techniques by which the thermal effects of solute microsegregation and melt convection may be realistically simulated. In the present publication, results obtained with this model are presented and used to illustrate some
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