A Mushy-Zone Rayleigh Number to Describe Interdendritic Convection during Directional Solidification of Hypoeutectic Pb-Sb and Pb-Sn Alloys

摘要

Based on measurements of the specific dendrite surface area (S_v), fraction of interdendritic liquid (phi), and primary dendrite spacing (lambda_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~(-1) and growth speeds ranging from 2 to 157 (mu)m s~(-1), it is observed that S_v = lambda_1~(-1) S*~(-0.33) (3.38 - 3.29 phi +8.85 phi~2), where S* = D_lG_(eff)/V m_1 C_o (k - 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.