Ensemble averaged multiphase Eulerian model for columnar/equiaxed solidification of a binary alloy: I. The mathematical model

A new multiphase Eulerian model for 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. The model uses three different phases, respectively, the columnar, the equiaxed solid and the liquid. The new set of equations enables us to simulate the columnar-to-equiaxed transition (CET) during the directional solidification of a binary alloy. Owing to the statistical nature of the model, we are able to treat rigorously the coexistence of equiaxed and columnar structures and consequently the CET phenomena. 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. An envelope model is used to parametrize the small scales, i.e. the dendrite scale. This leads us to distinguish two types of liquid, namely, the extra-dendritic and the inter-dendritic liquids. In part I the equations are rigorously derived in the purely diffusive case, whilst in part II we will present one-dimensional simulations of Sn?Pb and Al?Cu directional solidification experiments involving CET phenomena. Quasi-steady state CET maps are also computed.