The modeling of pool profiles, temperature profiles and velocity fields in ESR systems
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IN recent years there has been a growing interest in the development of suitable mathematical models for the representation of transport phenomena in the ESR process. In the earlier work interest has been concentrated on the representation of the pool profiles and the temperature fields within the system 1-5 through the use of empirically fitted "effective thermal conductivities" and postulated boundary conditions, specifying temperatures or fluxes at the slag-metal interface. This earlier work was very useful for interpreting measurements, but could not provide the needed insight into the actual mechanism of heat transfer, nor could it lead to the development of predictive relationships between the key process parameters. A more fundamental view of the system was taken in more recently reported work, where allowance has been made for the thermally and electromagnetically driven flow in the slag and the metal phases of ESR systems. 6-9 Of this work Kreyenberg and Schwerdtfeger, 9 considered fluid flow and heat transfer in the slag phase only, while Dilawari and Szekely examined the behavior of both slag and metal phases, allowed for the heat transfer between the metal droplets and the slag and accounted for the movement of the electrode. It is noted, however, that even this latter, more comprehensive treatment involved quite drastic simplifying assumptions, because the shape and size of the molten metal pool had to be specified and heat transfer in the mushy zone and in the ingot was neglected. Thus, while the formulation presented by Dilawari and Szekely grouped together the salient features of the ESR operation, it could not be addressed to the metallurgically important question of how to relate the shape and size of the molten metal pool and the mushy zone to the operating parameters. Furthermore, the somewhat arbitrarily assumed pool shape in this previous work precluded a meaningful comparison between the experimentally measured pool profiles and M. CHOUDHARY is Research Associate, and J. SZEKELY is Professor, Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139. Manuscript submitted May 11, 1979.
temperature fields, with predictions that could be based on the model. The work to be described in the present paper is to be regarded as a substantial extension of the previous efforts, in the following areas: i) While retaining the comprehensive formulation given in Ref. 8 the pool profile is no longer postulated, but rather calculated from first principles. Furthermore an allowance is made for the existence of a two-phase (mushy) zone. ii) Allowance is made for heat transfer in the ingot. iii) A more up-to-date k-e model is used to represent the turbulent viscosity and the turbulent thermal conductivity. iv) In some of the calculations an approximate allowance has been made for the temperature dependence of the electrical conductivity of the slag. v) The previously presented computed results were given for a fixed electrode position. In the present work calcu
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