A three-dimensional mathematical model of electromagnetic casting and testing against a physical model: Part I. The math
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I.
INTRODUCTION AND PREVIOUS INVESTIGATIONS
ELECTROMAGNETIC casting (EMC) has been an accepted technique for "continuous" casting of aluminum for several years. This technology permits the metal to be solidified without contacting a mold; instead, the solidifying metal is partly supported by electromagnetic forces and partly by previously solidified metal beneath it. The electromagnetic forces originate with an inductor, carrying alternating current, that surrounds the melt. The inductor generates an alternating magnetic field which induces currents in the aluminum. The interaction of the magnetic field and the induced currents results in the forces that support the melt, as well as causing stirring in the melt. In most casters, the magnetic field is modified by a "screen" or "shield" which is a conducting band (usually of stainless steel) partly inserted between the inductor and melt. Several groups of investigators (e.g., References 1 through 13)have described mathematical models aimed at predicting the electromagnetic fields and electromagnetically driven flow in EMC; recent modeling includes simulation of heat transport and solidification in EMC B4,15]and investigation of the stability of the aluminum-free surfaceJ tr~ The majority of the models have entailed first a numerical solution of Maxwell's equations and Ohm's law to obtain the current density within the aluminum and the magnetic field in and around the metal. In some instances, t4~31these calculations are coupled to an iterative calculation of the electromagnetically supported melt surface (henceforth, "meniscus") so as to arrive at self-consistent results for the meniscus shape and electromagnetic fields. The current density and magnetic field having been computed, the Lorentz forces throughout the melt are immediately ob-
D.P. COOK, Mathematical Modeler, is with the Reynolds Metals Co., Richmond, VA 23261. J.W. EVANS, Professor of Metallurgy, is with the Department of Materials Science and Mineral Engineering, University of California, Berkeley, CA 94720. Manuscript submitted April 4, 1994. METALLURGICAL AND MATERIALS TRANSACTIONS B
tained and used as input to the fluid flow calculations. The time-averaged Navier-Stokes equations are then solved with turbulence incorporated using well-known models for turbulence such as the k-e model.t171 With few exceptions,t6,s] the models referred to previously have been two-dimensional (2-D), i.e., for a cylindrical (axisymmetric) geometry or for EMC of a strip with one horizontal dimension much greater than another.t9] Although there exist cylindrical casters (producing extrusion billet), there are as yet no strip casters in industrial use and the vast majority of EMC product is ingot of approximately rectangular cross section with rounded corners. Therefore, the fields, flow, and meniscus shape are inherently threedimensional (3-D) in the majority of casters employed in industry. The objective of the investigation described in this article was a 3-D model* for the fields, meniscus shape, *The term 3-D is us
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