Evaluation of a Rayleigh-number-based freckle criterion for Pb-Sn alloys and Ni-base superalloys

A criterion for predicting the formation of freckles for Pb-Sn and Ni-base superalloys is evaluated using available experimental data. The criterion is based on a maximum value of the Rayleigh number, which indicates that the magnitude of buoyancy forces is largest with respect to the retarding frictional forces. The definition of the Rayleigh number involves a characteristic length scale. The two options explored in this study are the distance from the liquid mush interface and the scale represented by the ratio of thermal diffusivity to the casting speed. Additionally, two alternative ways of computing the mushy-zone permeability are explored. Only two of the possible combinations of length scales and permeability relations provide a maximum value within the mush that can be chosen as a representative value for the given conditions. It is found that the Rayleigh number that uses the ratio of thermal diffusivity to the casting speed as a length scale provides the best separation between freckled and nonfreckled data. Critical values for the Rayleigh number are in the range of 38 to 46 for vertically solidified Pb-Sn and within the range of 30 to 33 for Ni-base superalloys. Although not exactly the same, the proximity of these intervals indicates that the critical Rayleigh number is not very sensitive to alloy-specific system parameters. The influence of sample pulling inclination angle on this critical value is assessed for Ni-base superalloys. A preliminary relationship between the critical Rayleigh number and the inclination angle is developed.