A Rayleigh number based criterion is developed for predicting the formation of freckles in Ni-base superalloy castings. This criterion relies on finding the maximum local Rayleigh number in the mush, where the ratio of the driving buoyancy force to the retarding frictional force is the largest. A critical Rayleigh number for freckle formation of approximately 0.25 is found from available experimental data on directional solidification of a Ni-base superalloy. If the Rayleigh number in a superalloy casting is below this critical value, freckles are not expected to form. Full numerical simulations of freckling in directional solidification of superalloys are conducted for a large variety of casting conditions, alloy compositions, and inclinations of the system with respect to gravity. For the vertical cases, the Rayleigh numbers at the starting points of the predicted freckles are in good agreement with the critical value established from the experiments. The simulations confirm that the same critical Rayleigh number applies to different superalloys. The simulations for inclined domains show that even a small amount of inclination (less than 10 deg) significantly lowers the critical Rayleigh number and moves the freckles to the sidewall of the casting, where the mushy zone has advanced the most relative to gravity. In application of the Rayleigh number criterion to complex-shaped superalloy castings, the absence of freckles near upper and lower boundaries and in sections of insufficient crosssectional area or height needs to be taken into account as well. The criterion can be used to study the tradeoffs between different superalloy compositions, applied temperature gradients, and casting speeds. Additional experiments, in particular for other superalloys and for a range of inclinations, are desirable to confirm the critical Rayleigh numbers found in the present study.

The formation of macrosegregation and freckles by multicomponent thermosolutal convection during the directional solidification of single-crystal Ni-base superalloys is numerically simulated. The model links a previously developed thermodynamic phase equilibrium subroutine with an existing code for simultaneously solving the macroscopic mass, momentum, energy, and species conservation equations for solidification of a multicomponent alloy. Simulation results are presented for a variety of casting speeds and imposed thermal gradients and for two alloy compositions. It is found that for a given alloy composition, the onset of convection and freckle formation occurs at a critical primary dendrite arm spacing, which agrees well with previous experimental findings. The predicted number and shape of the freckle chains in the unstable cases also agree qualitatively with experimental observations. Finally, it is demonstrated how the onset and nature of convection and macrosegregation vary with alloy composition. It is concluded that the present model can provide a valuable tool in predicting freckle defects in directional solidification of Ni-base superalloys.

Melt convection and macrosegregation in casting of a large steel ingot are numerically simulated. The simulation is based on a previously developed model for multicomponent steel solidification with melt convection and involves the solution of fully coupled conservation equations for the transport phenomena in the liquid, mush, and solid. Heat transfer in the mold and insulation materials, as well as the formation of a shrinkage cavity at the top, is taken into account. The numerical results show the evolution of the temperature, melt velocity, and species concentration fields during solidification. The predicted variation of the macrosegregation of carbon and sulfur along the vertical centerline is compared with measurements from an industrial steel ingot that was sectioned and analyzed. Although generally good agreement is obtained, the neglect of sedimentation of free equiaxed grains prevents the prediction of the zone of negative macrosegregation observed in the lower part of the ingot. It is also shown that the inclusion of the shrinkage cavity at the top and the variation of the final solidification temperature due to macrosegregation is important in obtaining good agreement between the predictions and measurements.

The formation of spurious grains during the directional solidification of a Ni-base superalloy is studied by modeling the movement and remelting of dendrite fragments originating in channels inside the mush. Such channels exist because of thermosolutal convective instabilities during solidification and persist as freckle chains in the solidified material. The fragment model is linked to a phase equilibrium subroutine for multicomponent Ni-base superalloys, as well as to a previously developed solidification and convection code. A parametric study is performed to investigate the effects of initial fragment location and size on the fragment paths and survivability in the melt for one of the channels predicted in a typical directional solidification simulation. It is found that only a small window of initial conditions exists which leads to spurious grain formation. This window corresponds to medium-sized fragments originating near the mouth of the channel. Other fragments either remelt completely or sink into the channel. The need for an accurate fragment generation model is discussed.

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