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

Based on measurements of the specific dendrite surface area (Sv), fraction of interdendritic liquid (φ), and primary dendrite spacing (λ1) 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−1 and growth speeds ranging from 2 to 157 µm s−1, it is observed that Sv=λ1−1S*−0.33 (3.38−3.29 φ+8.85 φ2), where S*=Dl Geff/V m1Co (k−1)/k, with Dl being the solutal diffusivity in the melt, Geff being the effective thermal gradient, V being the growth speed, ml being the liquidus slope, Co 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.