Criterion function for predicting freckles in CMSX-4 during directional solidification

In the present work the impact of curvature effects on freckle formation in CMSX-4 during directional solidification was modelled and simulated. Modelling work is based on a four-step criterion function for predicting freckles. First, only surface elements are taken into account. Second, critical cross-sectional areas are identified where freckle formation is possible. As a new aspect, third, curvature effects are taken into account for predicting freckles in radiation shadow. Fourth, the Rayleigh number is calculated based on a permeability function of dendrite arm spacing and fraction of liquid. In the thus identified areas sufficient thermo-solutal convection takes place to initiate freckle formation. With the new model freckles can only appear in areas where the thermal temperature profile is concave, that is, at locations where the temperature gradient is positive against the surface normal. The present model was tested simulating step samples. For that, the individual parts of our industrial Bridgman furnace were input and joined in one FEM model. The thermal results clearly show concave solidus isothermal at the shadow faces of the step sample in the cluster. Freckles are predicted at the corresponding locations. When comparing simulated freckle formation with experimental findings, good correlation was found. Furthermore, we could predict both the transition length that occurs when the cross-sectional area increases stepwise and the run length when the cross-sectional area is reduced.