Modeling of multidimensional solidification of an alloy
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I.
INTRODUCTION
A L T H O U G H it is well recognized that the solidification of alloys occurs with a highly convoluted solid-liquid interface and over a range of temperatures, calculations and numerical schemes to simulate solidification have mostly considered the movement of a planar interface and a pure metal. It has normally been implicity assumed that the solidification time calculated this way is close to the actual experimentally determined number. Calculations based on pure metal freezing for large alloy ingots or sand castings have been experimentally verified and found correct (e.g., the Chvorinov rule Ill is a good indicator for solidification time in sand castings). Calculations which assume that the solidification times for a pure metal and dilute alloy are equivalent have not been tested for small castings (e.g., splats, atomized droplets, etc.) nor have they been tested for carefully designed experiments at moderate cooling rates normally encountered for net-shaped castings. Part of the difficulty has been the nonavailability of good measurements (e.g., those obtained where fluid flow has been prevented). The other aspect has been the nonavailability of numerical techniques which simply and adequately treat multidimensional alloy solidification. Some notable previous modeling work has been directed toward alloy solidification by Hunt and McCartney [21 and Worster, [3[ who have developed elegant models which treat the solid-liquid region as a continuum with properties varying with fraction liquid or solid. The results obtained from these studies correctly predict the basic morphology of solid evolution. A previous model by Huppert and Worster [4[ is physically impossible because of the unlikely distribution of solid in the mushy region. Available physically correct models [2,3J are difficult to extend beyond specially formulated oneB. BASU is with the Tara Research Development and Design Center, Pune 411001, India. J.A. SEKHAR, formerly with the Defence Metallurgical Research Laboratory, Hyderabad 500258, India, is Associate Professor at the Department of Materials Science and Engineering, University of Cincinnati, Cincinnati, OH 45221-0012. Manuscript submitted March 30, 1988. METALLURGICAL TRANSACTIONS A
dimensional studies. Sekhar et al. ,[sj Rolph and Bathe, [6j and Oreper and Szekely t7[ proposed various multidimensional alloy solidification models based on simplifying assumptions. Sekhar etal. [5] formulated the multidimensional enthalpy-temperature technique to treat alloy solidification but circumvented the nonlinearity in the mushy region by assuming a linear change in the fraction solid from the eutectic to the liquidus. The other feature of the Sekhar et al. [51 technique was that the dendrite tip was assumed to be at the liquidus temperature and that the fraction solid was independently determined by the simplified Scheil equation. [5] This second drawback is not expected to be the cause for any perceptible error when determining solidification times but obviously affects knowledge of t
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