Modeling of inverse segregation and porosity formation in directionally solidified aluminum alloys
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I.
INTRODUCTION
MACROSEGREGATION, a casting defect which results in heterogeneous solute concentrations and mechanical properties, can be induced by several mechanisms. The most common source of macrosegregation is associated with the motion of the liquid, but the settling of equiaxed grains or the deformation of the solid skeleton of the mushy region can also modify solute distribution on the ingot scale, t~,2] Most numerical studies dealing with this subject have focused on the macrosegregation induced by convection,t3-n] and among these, two types of problems have been solved. The movement of the free liquid, which partially penetrates into the mushy zone and carries some of the segregated interdendritic liquid along, and the solutal convection initiated in the mushy zone, possibly leading to the formation of segregated channels (freckles), have been simulated under the well-known Boussinesq approximation.t 3-9] A continuum formulation of the system based upon volume-averaged equations is usually considered for the conservation of mass, solute, energy, and momentum. These coupled equations are solved using finite volume or finite element formulations in two dimensions. The other type of segregation, called inverse segregation, is associated with the feeding of the mushy zone, which is necessary to compensate for solidification shrinkage.[lo,~,~2] In directional solidification, the flow of the interdendritic fluid is opposite to the movement of the isotherms, since the density of the solid is usually larger than that of the
P. ROUSSET, Graduate Student, and M. RAPPAZ, Professor, are with the Department of Materials, Ecole Polytechnique Frdrrale de Lausanne, CH-1015 Lausanne, Switzerland. B. HANNART, Process Engineer, is with the Prchiney Center of Research of Voreppe, F-38340 Voreppe, France. Manuscript submitted August 23, 1994. METALLURGICALAND MATERIALS TRANSACTIONS A
liquid. For a partition coefficient smaller than unity, the liquid entering a small volume element of the mushy zone therefore has a concentration higher than the nominal concentration, thus increasing the final average local concentration. This effect is particularly noticeable at the surface of the casting in contact with the chill, and the concentration profile in this region is "inverse" to the initial solute transient associated with the growth of a planar front. Therefore, the simulation of inverse segregation must account for the suction of the liquid in the mushy zone, and accordingly, the two-phase medium can no longer be considered as incompressible. Such a situation has been modeled in one dimension, and the average conservation equations of heat, mass, and solute* have been solved *The momentum equation is not required in one-dimensional problems when there is no microporosity formation.
using analytical and finite difference methods.t~o.n.121 The inverse segregation models developed so far neglect the formation of microporosity in the alloy. As is well known, microporosity forms in the mushy zone as a result of the pressure drop
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