A mathematical model of slag and metal flow in the ESR Process
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1. INTRODUCTION
THE
widespread application of the ESR P r o c e s s has stimulated considerable interest in the mathematical modeling of these systems. A good review of recent work is available in a paper by Mitchell e t a l , 1 which may be summarized by stating that while very useful work has been done by Pridgeon, Mitchell, Elliott and others on modeling the pool profiles by fitting coefficients to F o u r i e r ' s equation, no predictive relationships have been developed for the fluid flow fields in the slag and the m e t a l pool. Complementing some very elegant physical modeling work Campbell2 cited some relationships between current and melt velocities, however, these could not be used even for predicting the o r d e r s of magnitude involved. Notwithstanding the fact that the prediction of the melt velocities in the ESR process is a difficult task because it requires the solution of Maxwell's equations together with the turbulent Navier-Stokes equations, such work is thought to be very worthwhile, for a number of reasons, enumerated in the following. The development of a predictive t h e r m a l model of the process ( e . g . the computation of the pool prof i l e s for various geometries and current inputs) is not possible without the knowledge of the spatially distributed heat generation in the slag and the convective heat transfer processes in the pool, which in turn depend on the fluid flow. It is stressed in this context that the models that a r e available at present, while very useful, are essentially interpretive, r a t h e r than predictive. Another, equally important, point is that both the refining process and the structure of the ingot produced a r e critically dependent on the flow fields A. H. DILAWARI and J. SZEKELY are VisitingEngineer and Professor of Materials Engineering, respectively, Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA02139. A. H. Dilawari is presently on leavefrom Institute of ChemicalEngineeringand Technology, Punjab University, Lahore-l, Pakistan. Manuscript submitted August 10, 1976. METALLURGICAL TRANSACTIONS B
and turbulence levels in the slag and in the molten m e t a l pool. The profound effect of fluid flow on the morphology of the solid phases formed is readily appreciated as a result of recent work by Flemings, 3 Tsavaras ~ and others on various solidification processes. The work t o be described in this paper is part of a continuous effort a i m e d at the development of a comprehensive, predictive mathematical r e p r e sentation of the ESR process, with the ultimate purp o s e of establishing predictive relationships between the system geometry, current input, pool profiles and other process variables. In the present paper we shall present a mathematical formulation, together with some computed results for the electromagnetic f o r c e field, turbulence parameters and melt velocities for both laboratory and industrial s c a l e systems, having a somewhat idealized pool geometry, and operated with a direc
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