Asymptotic approximation and numerical simulation of electromagnetic casting
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I. INTRODUCTION
THE technology of electromagnetic casting (EMC) of aluminum alloys is relatively new, but several million tons per year are cast in this way. During this process, the metal is continuously withdrawn from an upper pool of liquid aluminum into a descending bottom block. As casting proceeds, the aluminum solidifies due to water jet cooling and the bottom block moves downward at the casting speed. In direct chill casting, the upper part of the ingot is in contact with a mold, whereas in EMC, the liquid metal is supported by electromagnetic forces, which are produced by an alternating current of a few kA at several kHz, passing through an inductor surrounding the metal. There are two commercially established technologies in EMC, namely, the Russian type and the Kaiser type casters. In the former, a screen is interposed between the inductor and the molten metal to shape the melt surface (henceforth, meniscus), whereas in the latter, this screen is removed and the electromagnetic field is modified by appropriately choosing the shape of the inductor. The main advantage of EMC technology is that the faces of the resulting ingot are smooth enough to be rolled without the scalping process. For a more detailed description of this technology, the reader can consult, for instance, the articles of Evans,[9] Sakane et al.,[19] and references therein. Even though several physical phenomena are taking place in EMC—thermal, mechanical, electromagnetic, and hydrodynamic phenomena—it is with the last two ones that the present article is concerned. The liquid-solid interface is an input to our model, as in References 5 and 19 for instance. Articles devoted to the study of heat transport and solidification in EMC have been carried out by Prasso et al.[15,16] and by Drezet and Rappaz,[8] and nowadays the coupling between the thermal, hydrodynamic, and electromagnetic processes is a challenge for future research. The aim of this article is to present numerical models describing the electromagnetic and hydrodynamic behavior of EMC. Numerical modeling of such problems has already been considered by several authors (for instance, References 2,5,10, and 18, and references therein). But, to our knowledge, in this work, we present the new contribution of representing the first two terms of an asymptotic expansion for ´ DEZ, Professor, M.C. MUN ˜ IZ, Permanent Professor, and A. BERMU P. SALGADO, Doctoral Student, are with the Department of Applied Mathematics, Santiago de Compostela University, 15706 Santiago de Compostela, Spain. Contact e-mail: [email protected] Manuscript submitted December 12, 2001. METALLURGICAL AND MATERIALS TRANSACTIONS B
the magnetic induction field outside the metal through the solutions of two systems of partial differential equations, which are set up in a domain that does not include the metal. The present article completes the more theoretical one[3] in that it includes numerical simulation of the liquid metal flow driven by the electromagnetic field. However, to be more self-contained, we recall the mai
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