A model of binary alloy solidification

Abstract

SUMMARYA linear model for the solidification of a dilute binary alloy is presented. In this model the solidus and liquidus curves are linear. As a consequence internal energy depends linearly upon temperature and concentration. The formulation is a generalization of the well-known enthalpy method to treat a phase change problem involving coupled heat and mass transfer. Both analytic and numerical formulations are given. Results from the latter are presented and compared with an explicit solution of Rubinstein for a Stefan-like problem posed in a semi-infinite slab. Some remarks on the behaviour of the explicit solution are given.

A numerical method for the Rubinstein binary-alloy problem in the presence of an under-cooled liquid

“…The performance of this ''node jumping" scheme matched that reported by Crowley and Ockendon [14]. Alexiades and Solomon [13] and Wilson et al [16] have highlighted an important shortcoming with the Crowley and Ockendon enthalpy solution. When the Lewis number (the ratio of thermal to solute diffusivity in the liquid phase) is large, the solute rejected into the liquid as the solid forms is not easily removed from the region ahead of the solid-liquid interface.…”

“…These models require the tracking of a sharp solid-liquid interface-driven by coupled heat and mass transport-in the presence of an undercooled liquid. A well known related test problem in the existing literature [12][13][14][15][16] is a one-dimensional solidification controlled by coupled heat and mass transport-the binary-alloy problem. Prior numerical solutions of the binary-alloy problem, however, steer clear of cases of a sharp interface in the presence of liquid under-cooling.…”

“…Prior numerical solutions of the binary-alloy problem, however, steer clear of cases of a sharp interface in the presence of liquid under-cooling. The previous solutions either ignore cases where under-cooling will occur [14,15] or allow for the formation of a mushy region (a non-sharp front) when and under-cooled condition is reached [16]. In this paper, the binary-alloy problem as introduced by Rubinstein [12] is revisited and specific focus is placed on solutions of the problem in the presence of liquid under-cooling.…”

“…(2) In a physical setting, any small solid imperfection forming on the interface will find itself in a local environment favorable for further growth, resulting in a break down of the solid-liquid interface into a mushy region, where, at the scale of the problem, distinct solid and liquid regions cannot be identified. Wilson et al [16] take account of both of these consequences in developing a numerical solution that is able to handle the formation of a mushy region. In essence the use of this modified solution assumes that once a region of constitutional under-cooling is formed the sharp solid-liquid interface will break down into a mushy region.…”

Macroscopic global modeling of binary alloy solidification processes

“…Numerical approximations for essentially the same model have been discussed by Fix [10], Crowley-Ockendon [7], Meyer [14], Bermudez-Saguez [3], Wilson-Solomon-Alexiades [24]. The appearance of a "mushy" zone was noted for some values of the parameters which could not be entirely attributed to numerical errors.…”