Heat transport and solidification in the electromagnetic casting of aluminum alloys: Part II. Development of a mathemati
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I.
II.
INTRODUCTION
P A R T Its1 of this two-part series described two experimental campaigns carried out on a pilot-scale electromagnetic caster at Reynolds Metals Company (Richmond, VA). In these campaigns, extensive temperature measurements were made on the aluminum being cast. In most cases, the measurements were made using sacrificial thermocouples moving downward through the liquid aluminum pool and into the solidifying metal. In this way, the temperature distribution throughout the metal could be obtained, together with the location of the liquidus and solidus isotherms. Measurements were carried out on two ingot shapes at various casting speeds and various flow rates of the water sprayed on the outside of the ingot. The major inference from the results reported in Part I is that heat transfer into the water at the ingot surface is not rate determining for the solidification of the major part of the ingot, although the flow rate can affect heat transport in the near-surface region. This second part describes the development of a mathematical model aimed at predicting heat transport and solidification in electromagnetic casting. The predictions of the model are then compared with the measurements described in Part I.
D.C. PRASSO, Engineer, is with Intel Corporation, Aloha, OR 97007. J.W. EVANS, Professor, is with the Department of Materials Science and Mineral Engineering, University of California, Berkeley, CA 94720. I.J. WILSON, Project Director, is with Reynolds Metals Company, Muscle Shoals, AL 35661. Manuscript submitted July 13, 1994. METALLURGICAL AND MATERIALS TRANSACTIONS B
D E V E L O P M E N T OF THE M A T H E M A T I C A L MODEL
A. Governing Equations and Assumptions
The mathematical model starts with the general heattransport equation in rectangular coordinates for a fluid of constant density:" *A list of symbols appears at the end of this article.
pC.t. -Ot -+v~
=-
+v~ ~
+Vz 7 -
ox oy oz O ( OT'} O / OV'~ O[ OT~
oyt oy) ozt, oz)
=
(1)
The equation was simplified by assuming negligible generation of heat by viscous dissipation. In a first mathematical model, Joule heating was ignored; for a representative case, Joule heating was estimated to be 19 kW compared to the enthalpy inflow with the liquid aluminum of 623 kW with respect to room temperature. In a second model, allowance was made for Joule heating, as described subsequently. The velocities appearing in Eq. [1] will be functions of position and, because the flow in the liquid pool is turbulent, time. One semirigorous approach to computing the heat transport in the liquid pool would be to solve the time-averaged version of Eq. [1] with the timeaveraged velocities first obtained by a solution of Maxwell's equations, for the electromagnetic forces, and the time-averaged Navier-Stokes equations, t:,3,41 Fortunately, the distribution of the inflowing aluminum by the spoutsock, the electromagnetic stirring of the melt, and its high conductivity result in heat transport within the melt being facile, compared to that in the solid. T
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