A mushy-zone rayleigh number to describe interdendritic convection during directional solidification of hypoeutectic Pb-Sb and Pb-Sn alloys

Abstract

Based on measurements of the specific dendrite surface area (S v ), fraction of interdendritic liquid (), and primary dendrite spacing ( 1 ) on transverse sections in a range of directionally solidified hypoeutectic Pb-Sb and Pb-Sn alloys that were grown at thermal gradients varying from 10 to 197 K cm Ϫ1 and growth speeds ranging from 2 to 157 m s Ϫ1 , it is observed that S v ϭ 1 Ϫ1 S* Ϫ0.33 (3.38 Ϫ 3.29 ϩ 8.85 2 ), where S* ϭ D l G eff /V m 1 C o (k Ϫ 1)/k, with D l being the solutal diffusivity in the melt, G eff being the effective thermal gradient, V being the growth speed, m l being the liquidus slope, C o being the solute content of the melt, and k being the solute partition coefficient. Use of this relationship in defining the mushy-zone permeability yields an analytical Rayleigh number that can be used to describe the extent of interdendritic convection during directional solidification. An increasing Rayleigh number shows a strong correlation with the experimentally observed reduction in the primary dendrite spacing as compared with those predicted theoretically in the absence of convection.

Mushy-zone rayleigh number to describe macrosegregation and channel segregate formation during directional solidification of metallic alloys

“…It not only shows a very good correlation between increased convection and decreased primary spacing, [14] it also shows an excellent correlation between experimentally observed longitudinal macrosegregation and convection. The parameter, R aM , can be used as a predictive tool to describe channel formation during directional solidification.…”

“…The Mushy-zone Rayleigh number (R aM ) defined recently, [14] which incorporates a permeability relationship that accounts for dendrite side branching, is a good parameter to represent the intensity of interdendritic convection during directional solidification of alloys where increasing solute content decreases the melt density. It not only shows a very good correlation between increased convection and decreased primary spacing, [14] it also shows an excellent correlation between experimentally observed longitudinal macrosegregation and convection.…”

“…The SNT sets also contain many samples that are not listed in Table I, especially no channel segregate samples, which were already included in a recent publication. [14] Data marked as Song-SNT are for Pb-32 and Pb-41.9 wt pct Sn alloy samples that were DS in 1.8-cm i.d. crucibles.…”

“…This causes convection that produces macrosegregation along the sample length; [1,2] when severe, it also produces channel segregates (or freckles). [3][4][5][6][7][8][9][10][11][12][13] Based on the measurements of dendrite specific surface area (S v ) in the quenched mushy zone of directionally solidified PbSb and Pb-Sn alloys, we recently reported [14] that S v ϭ 1 [15] yielded a relationship that describes the processing parameter dependence of mushyzone permeability during directional solidification. Using this permeability, a mushy-zone Rayleigh number, R aM , 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 R aM , 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 R aB , the Rayleigh number used by Beckerman et al, [16] which used a different permeability relationship.…”