| A general multidimensional model of alloy solidification is presented in which a velocity-dependent freezing temperature is coupled with the macroscale energy equation. The velocity dependence of the freezing temperature ( Tf∼v ) results from the microscale species diffusion for microstructures with coupled eutectic growth. At solidification rates ( ∼ 1--10 mm/s) that are representative of gravity permanent mold and die casting processes, consideration of the nonequilibrium conditions at the interface affects the prediction of the macroscale thermal field. Near-eutectic alloys freeze with a macroscopically discrete solid-liquid interface at a temperature below the equilibrium eutectic temperature.;The model is illustrated with unidirectional solidification of a near-eutectic alloy in a finite domain and solved numerically with a fixed-grid Galerkin finite element method. The numerical algorithm includes inexpensive steps to compute the interface speed explicitly. By nondimensionalizing the governing equations the effect of coupled eutectic growth on heat transport is clearly identified so that the model's sensitivity to important parameters can be investigated. Additionally, the average eutectic spacing can be determined with the temperature field, rather than post-determination from a standard, uncoupled solution of the energy equation. The eutectic coupling results indicate that the predicted solid-liquid interface location lags behind the uncoupled solution; therefore, decreasing the amount of solid formed, increasing the total solidification time, and increasing the average eutectic spacing. A procedure is also illustrated for computing mechanical properties using experimental correlations and the computed interface velocity history.;The effect of the eutectic undercooling is then studied in a square domain and a realistic three-dimensional production casting geometry. In order to address the multidimensional cases, a phase-field formulation is developed. Although the interface is considered to have a small finite thickness, the utility of the method is good for complex evolving solid-liquid interfaces and the velocity-dependent freezing temperature is satisfied implicitly. After demonstrating sufficient numerical accuracy, numerical results are presented for the square domain and three-dimensional geometry. Limitations of the phase-field method are discussed, and the conjugate heat transfer problem is studied to address boundary condition issues. |
