Finite Element Analysis of the Effect of a Non-Planar Solid-Liquid Interface on the Lateral Solute Segregation During Un
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FINITE ELENENT ANALYSIS OF THE EFFECT OF A NON-PLANAR SOLID-LIQUID INTERFACE ON THE LATERAL SOLUTE SEGREGATION DURING UNIDIRECTIONAL SOLIDIFICATION. F.M.CARLSON KRAR Incorporated, Bagdad Road, Potsdam, New York 13676 L-Y CHIN Phillips Petroleum Company, Bartlesville, Oklahoma 74003 A.L.FRIPP AND R.K.CROUCH NASA Langley Research Center, Hampton, Virginia 23665 ABSTRACT
The effect of solid-liquid interface shape on lateral solute segregation during steady-state unidirectional solidification of abinary mixture is calculated under the assumption of no convection inthe liquid. A finite element technique isemployed to compute the concentration field inthe liquid and the lateral segregation inthe solid with a curved boundary between the liquid and solid phases. The computational model isconstructed assuming knowledge of the solid-liquid interface shape; no attempt is made to relate this shape to the thermal field. The influence of interface curvature on the lateral compositional variation isinvestigated over arange of system parameters including diffusivity, growth speed, distribution coefficient, and geometric factors of the system. Inthe limiting case of a slightly non-planar interface, numerical results from the finite element technique are in good agreement
with the analytical solutions of Coriell and Sekerka obtained by using linear theory. For the general case of highly non-planar interface shapes, the linear theory fails and the concentration field in the liquid as well as the lateral solute segregation inthe solid can be calculated by using the finite element method.
INTRODUCTION The macroscopic interface shape strongly influences the lateral solute segregation during steady state unidirectional solidification. Coriell and Sekerka [11 analytically studied the dependence of lateral solute segregation on the curvature of a nearly planar solid-liquid interface during solidification of abinary alloy. Inthis paper the same.,physical problem without the small curvature restriction on the interface shape issolved numerically using a finite element method. A parametric.study isincluded which sheds light on how the magnitude of interface curvature,-affects the lateral segregation for various values of disfribution coefficient, diffusivity, growth speed, and system geometry.
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ANALYSIS A schematic diagram for unidirectional solidification with a parabolic interface shape is shown in Figure 1. The coordinate system is stationary with respect to the interface which moves in the positive y*-direction at a constant growth velocity V*. The binary mixture is held in a container of width 2x*. The origin of the coordinate system is located such that the intersection of the interface and the walls is at x* = ± xo* and
Liquid
Liquid
(Liquid
(0,
x*,•) (0O -
(x*,0)
Solid
Solid
d0
Yo
Yo
< 0 h. Concave: 0 > 0 0 o Figure 1. Geometry and coordinate system for unidirectional solidification with a non-planar solid-liquid interface. a.
Convex:
y* = 0. The assumptions made are: (i) all material properties are constan
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