Simulation of Liquid Flow Permeability for Dendritic Structures during Solidification Process

Abstract

Alloy solidification is a two stage process, which starts by nucleation and ends by growth of solid phases. Subsequently, number, distribution and morphology (dendritic or nondendritic) of the grains are formed during the solidification. Some critical defects such as micro/macro segregation, micro/macro porosities and micro/macro shrinkage take place in the solidification stage. The micro-defects are located in the interdendritic space, which are micro-channels that fluid flow through them in the last stage of the solidification. Herein, the region in the grain growth stage is introduced as a mushy zone (or porous media), where the solid phase is constantly progressing; and the ability of fluid to flow into the mushy zone is known as permeability of interdendritic liquid. Therefore, formation of the micro-defects depends on controlling of the permeability factor. In a great number of studies micro/macro solidification models have been simulated based on the permeability factor using Darcy’s law (Ganesan & Poirier, 1990; Nandapurkar et al., 1991; Poirier, 1987; Worster, 1991). Interdendritic flow, in many CFD documents, is described using Darcy’s law, which relates the fluid flow rate to the pressure gradient, fluid viscosity, and permeability of the porous medium. To obtain an expression for the permeability as a function of the porosity of the porous medium, one generally considers flow through an idealized medium geometry, since it is impractical to solve the flow equations for the complex flow between the particles. Fig. 1 presents two viewpoints for investigation of the permeability in the porous media: metallurgical view and non-metallurgical (or common) view. As shown in Fig. 1a two of the most commonly used geometries for analytical models are capillaries (Carman, 1937; Chen et al. 1995; Williams et al., 1974) and an array of spaced particles. A more realistic approach will be introduced that assumes geometry of a periodic or random array of cylinders. Since it is not possible to solve analytically for this type of flow over the full range of porosities, two limiting closed form solutions are used for lubrication and point-particle (dilute) models in low and high porosities, respectively. Analysis of permeability for Stokes flow through periodic arrays of cylinders were done by Sangani & Acrivos (1982), Sparrow & Loefler (1959) and, Larson & Higdon (1986). The effect of fluid inertia on pressure drop required to drive the flow is a function of Reynolds number. Several authors computed the fluid flow through periodic arrays of