Ensemble averaged two-phase Eulerian model for columnar/equiaxed solidification of a binary alloy

A new two-phase eulerian model for the columnar and equiaxed dendritic solidification has been developed. The mean conservation equations are derived by means of a statistical phase averaging technique, and the mathematical formulation of the model can be used for both columnar and equiaxed solidification. We present a new set of equations to simulate the segregations during the columnar solidification of a binary alloy. The use of such an averaging procedure has several advantages compared with the volume averaging technique which has been widely used during the last decade. Firstly, the statistical approach does not need the definition of a representative elementary volume. Secondly, it is consistent with the random aspect of the equiaxed grain germination and motion as well as the possible turbulent behaviour of the fluid flow. The averaged equations are closed by means of the cell model approximation. This technique can be successfully used to model the various interactions between the liquid and the solid. It may also incorporate the effects of the inhomogeneities of the various scalar fields, e.g., the solute and temperature gradients. First tests of the model are presented in two asymptotic situations, namely the wholly equiaxed and the columnar solidification.