A mushy-zone rayleigh number to describe interdendritic convection during directional solidification of hypoeutectic Pb-
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A Mushy-Zone Rayleigh Number to Describe Interdendritic Convection during Directional Solidification of Hypoeutectic Pb-Sb and Pb-Sn Alloys S.N. TEWARI and R. TIWARI Based on measurements of the specific dendrite surface area (Sv), 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 cm1 and growth speeds ranging from 2 to 157 m s1, it is observed that Sv 11 S*0.33 (3.38 3.29 8.85 2), where S* Dl Geff/V m1 Co (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. I. INTRODUCTION
NATURAL convection in the dendritic mushy zone is responsible for the nucleation of spurious grains[1] and the formation of channel segregates[2–9] in directionally solidified alloys. It produces radial[10,11,12] and longitudinal macrosegregations[13,14] and alters the cellular-dendritic and planar-cellular transitions.[15] It also appears to reduce the primary dendrite spacing.[13–18] Numerical simulations of the transport phenomena during alloy solidification have shown some success in predicting some of the solidification defects in metal alloys,[19–22] but they have two serious limitations. First, the two-order-of-magnitude difference between the thermal and solutal length scales makes the three-dimensional computation very time consuming, and, second, the interdependence of the mushy-zone morphology, its permeability, and its convection does not allow an exact analysis of the problem. For example, the extent of convection depends upon the permeability of the mushy zone, but the convection alters the primary dendrite spacing and, hence, affects the permeability itself. Therefore, several attempts have been made to describe convection in terms of a nondimensional mushyzone Rayleigh number.[4,7,23–26] The mushy-zone Rayleigh number proposed by Beckermann et al.[26] (RaB) has been successful in describing the freckle initiation in nickel-based superalloy castings. However, the critical Rayleigh number for the onset of freckling has not been tested against the extensive freckling data available in the literature on other alloy systems, such as Pb-Sn,[3–7,9] Pb-Sb,[4] and Al-Mg.[8] Describing convection through the mushy zone requires knowledge of the relationship between mushy-zone permeability (K) and its morphology, such
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