Mushy-zone rayleigh number to describe macrosegregation and channel segregate formation during directional solidificatio
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Mushy-Zone Rayleigh Number to Describe Macrosegregation and Channel Segregate Formation during Directional Solidification of Metallic Alloys S.N. TEWARI, R. TIWARI, and G. MAGADI A recently defined mushy-zone Rayleigh number (RaM) that includes side-branching contributions to the mushy-zone permeability has been examined for its correlation with the longitudinal macrosegregation and channel segregate formation. The Rayleigh number shows (1) a strong correlation between the extent of longitudinal macrosegregation and increase in the mushy-zone convection and (2) a good ability to predict the formation of channel segregates during directional solidification.
I. INTRODUCTION
DIRECTIONAL solidification of alloys, such as hypoeutectic Pb-Sn or Pb-Pb, where solute enrichment causes a reduced melt density, in a positive thermal gradient, with melt on top and solid below, produces a density inversion in the melt, both in the mushy zone and in the overlying liquid immediately ahead of the dendritic array tips. This causes convection that produces macrosegregation along the sample length;[1,2] when severe, it also produces channel segregates (or freckles).[3–13] Based on the measurements of dendrite specific surface area (Sv) in the quenched mushy zone of directionally solidified PbSb and Pb-Sn alloys, we recently reported[14] that Sv 11 S*0.33 (3.38 3.29 8.852), where 1 is primary dendrite spacing, is fraction interdendritic liquid, and S* DlGeff / [VmlCo (k 1)/k], with Dl being solutal diffusivity in the melt, Geff, the effective thermal gradient, V, the growth speed, ml, the liquidus slope, Co, the solute content of the melt, and k the solute partition coefficient. Incorporation of the above Sv dependence into the Kozney–Carman permeability () of a porous bed, 3/4.2 S v 2 (1 )2,[15] yielded a relationship that describes the processing parameter dependence of mushyzone permeability during directional solidification. Using this permeability, a mushy-zone Rayleigh number, RaM, was defined following the procedure described by Beckermann et al.[16] Since convection is known to be localized in the immediate vicinity of dendrite tips,[17] it was assumed[14] that the Rayleigh number calculated in the mush at a distance equal to 30 times the dendrite tip radius can be used to represent convection in the mushy zone. The tip radius and spacing of primary dendrites used in these calculations were those predicted by the model attributed to Hunt–Lu.[18] In a recent publication,[14] it was demonstrated that RaM, which includes side-branching contribution to the mushy-zone permeability, shows a marked improvement in correlating the extent of primary spacing decrease with increasing convection as compared with RaB, the Rayleigh number used by Beckerman et al.,[16] which used a different permeability relationship. S.N. TEWARI, Professor, and R. TIWARI, Graduate Student, are with the Chemical and Biomedical Engineering Department, Cleveland State University, Cleveland, OH 44115. Con
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