Numerical calculation of permeability in dendritic alloys.

Abstract

A new technique of analyzing flows through dendritic microstructures for permeability calculations is presented. The method of applying a Navier-Stokes solver for analyzing flows normal and parallel to columnar dendritic structures and past equiaxed grains is elucidated. The permeability for flow normal to arrays of circular cylinders is first calculated to verify the computational method. Calculated results are extended to volume fraction of liquid as high as 0.98 to achieve relevancy to solidification modeling. The study showed a weak dependence of nondimensional permeability on the Reynolds number(Re), up to Re = 60. Dendritic microstructures, obtained by quenching an alloy during solidification are captured with an image analysis system. Complex solid-liquid dendrite interfaces are then represented in a computer program by retrieving and digitizing the pixel data. A mesh generator is used to subdivide the quenched liquid into quadrilateral finite elements. Using a Navier-Stokes solver, the velocity and pressure at the nodes are calculated at the microstructural level and used to calculate nondimensional permeability. It is found that the calculated results for flows normal to the primary dendrite arms at high liquid volume fractions merge well with the empirical permeabilities obtained at lower volume fractions. The analyses are extended to calculate permeability for flow past equiaxed dendritic grains and past globular grains. A method has been devised to extrapolate results from two-dimensional flows to three-dimensional flows. The results, considering specific surface area as characteristic length, compare to the analytical results for flow through different arrays of spheres. Numerical calculations are also performed to obtain permeabilities for flow parallel to primary dendrites in columnar structures, with high liquid volume fraction (g(L)), where physical experiments fail. For g(L) > 0.6, the results of the present work agree with the permeabilities calculated for fully developed flow parallel to an array of circular cylinders. Also, when the roughness factor is introduced, permeabilities obtained by the present technique merge well with the experimental data. The range in g(L) where the roughness factor should be introduced, however, is yet to be resolved. This merits, therefore, further work involving three-dimensional flow analyses.