Modeling of microsegregation in macrosegregation computations

A general framework for the calculation of micro-macrosegregation during solidification of metallic alloys is presented. In particular, the problems of back diffusion in the primary solid phase, of eutectic precipitation at the end of solidification, and of remelting are being addressed for an open system, i.e., for a small-volume element whose overall solute content is not necessarily constant. Assuming that the variations of enthalpy and of solute content are known from the solution of the macroscopic continuity equations, a model is derived which allows for the calculation of the local solidification path (i.e., cooling curve, volume fraction of solid, and concentrations in the liquid and solid phases). This general framework encompasses four microsegregation models for the diffusion in the solid phase: (1) an approximate solution based upon an internal variable approach; (2) a modification of this based upon a power-law approximation of the solute profile; (3) an approach which approximates the solute profile in the primary phase by a cubic function; and (4) a numerical solution of the diffusion equation based upon a coordinate transformation. These methods are described and compared for several situations, including solidification/remelting of a closed/open volume element whose enthalpy and solute content histories are known functions of time. It is shown that the solidification path calculated with method 2 is more accurate than using method 1, and that 2 is a very good approximation in macrosegregation calculations. Furthermore, it is shown that method 3 is almost identical to that obtained with a numerical solution of the diffusion equation (method 4). Although the presented results pertain to a simple binary alloy, the framework is general and can be extended to multicomponent systems.