Modelling of the binary alloy solidification process
Doctoral Thesis
1992
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University of Cape Town
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Abstract
A thermodynamically consistent model capable of modelling a binary alloy undergoing solidification or melting is developed. The theory is continuum based, and the solid-liquid phase change system is described macroscopically by a single set of conservation equations. The model is an extension of that presented in the literature. The thermodynamic theories of this type in the current literature are based on the assumption of local equilibrium. This assumption is not representative of most alloy solidification processes where the solid-liquid phase region, termed the mushy region, is of dendritic nature with the rates of diffusion in the liquid being orders of magnitude faster than that in the solid. The propose model includes the assumption of local non-equilibrium where solute diffusion in the solid phase is assumed to be zero. The thermodynamic formulation is expressed in terms of three thermodynamic variables: pressure, temperature and average solute concentration for both the equilibrium and non-equilibrium case. A generalized set of conservation equations of mass, energy, momentum and solute with the necessary constitutive equations is presented. A Finite Element (FE) formulation of a simplified form of the governing equations is developed. The reduced set of equations implemented in the FE formulation consists of a fully coupled heat conduction and solute diffusion formulation, with solid-liquid phase change, where the effects of pressure and convection are neglected. The FE formulation is based on the fixed grid technique where the elements are two dimensional, four noded quadrilaterals with the primary variables being enthalpy and average solute concentration. Temperature and solid mass fraction are calculated on a local level at each integration point of an element. A fully consistent Newton-Raphson method is used to solve the global coupled equations and an Euler backward difference scheme is used for the temporal discretization. The solution of the enthalpy-temperature relationship is carried out at the integration points using a Newton-Raphson method. A secant method employing the regula falsi technique takes into account sudden jumps or sharp changes in the enthalpy-temperature behaviour which occur at the phase zone interfaces. The Euler backward difference integration rule is used to calculate the solid mass fraction and its derivatives for the non-equilibrium case. Two solidification examples, using both the local equilibrium and the local non-equilibrium cases, are analyzed. The finite element results obtained for the two cases are compared, and the accuracy of the finite element model is checked. Both dendritic and eutectic phase change are tackled. Even though the discrete eutectic phase change is approximated using the fixed grid approach, the results are considered to be reasonable approximations to what occurs in reality. Favorable comparisons of the results are obtained with that in the literature and convergence of the finite element results for different mesh sizes are shown. For dilute alloy solutions, the solidification results for the local equilibrium and the local non-equilibrium cases are shown to differ markedly, whereas for near eutectic solutions little difference is observed. The use of the local non-equilibrium assumption in the finite element solidification model is shown to effect the macro-segregation of solute.
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Bibliography: leaves 129-132.
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Reference:
McAdie, R. 1992. Modelling of the binary alloy solidification process. University of Cape Town.