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 (S _v), fraction of interdendritic liquid (φ), and primary dendrite spacing (λ 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 μm s⁻¹, it is observed that S _v=λ ₁ ⁻¹ S*^−0.33 (3.38−3.29 φ+8.85 φ ²), where S*=D _l G_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.