A basic model of the transport phenomena occurring during solidification of multicomponent mixtures is presented. The model is based on a two-phase approach, in which each phase is treated separately and interactions between the phases are considered explicitly. The macroscopic transport equations for each phase are derived using the technique of volumetric averaging. The basic forms of the constitutive relations are developed. These relations link the macroscopic transport phenomena to microscopic processes such as microstructure development, interfacial stresses, and interfacial heat and mass transfer. Thermodynamic relations are presented, and it is shown that nonequilibrium effects can be addressed within the framework of the present model. Various simplifications of the model are examined, and future modeling needs are discussed.

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A volume-averaged two-phase model for transport phenomena during solidification

The study of metallic melts, their physical properties, and their solidification behaviour is of relevance to understanding of fundamental principles and to metallurgical applications. Many, if not...

Containerless processing in the study of metallic melts and their solidification

As a molten alloy or any multi-component liquid is cooled and solidified the growing solid phase usually forms a porous matrix through which the residual liquid can flow. The reactive two-phase medium comprising the solid matrix and residual liquid is called a mushy layer. Buoyancy forces, owing primarily to compositional depletion as one or more of the components of the alloy are extracted to form the solid phase, can drive convection in the layer. In this review, I present an account of various studies of buoyancy-driven convection in mushy layers, paying particular attention to the complex interactions between solidification and flow that lead to novel styles of convective behavior, including focusing of the flow to produce chimneys: narrow, vertical channels devoid of solid. I define an ‘ideal’ mushy layer and argue that chimneys are an inevitable consequence of convection in ideal mushy layers. The absence of chimneys in certain laboratory experiments is explained in terms of nonideal effects.

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Convection in mushy layers

The solidification of melts can be profoundly influenced by convection. In alloys, compositional convection can be driven by solute gradients generated as one component of the alloy is preferentially incorporated within the solid, even when the thermal field is stabilizing. In this paper, two modes of compositional convection during solidification from below are uncovered using a linear-stability analysis : one, which we shall call the ‘mushy-layer mode’, is driven by buoyant residual fluid within a mushy layer, or porous medium, of dendritic crystals; the other, which we shall call the ‘ boundary-layer mode ’ is associated with a narrow compositional boundary layer in the melt just above the mush-liquid interface. Either mode can be the first to become unstable depending on the thermodynamical and physical properties of the alloy. The marginally stable eigenfunctions suggest that the boundary-layer mode results in fine-scale convection in the melt above the mushy layer and leaves the interstitial fluid of the mushy layer virtually stagnant. In contrast, the mushy-layer mode causes perturbations to the solid fraction of the mushy layer that are indicative of a tendency to form chimneys, which are vertical channels of reduced or zero solid fraction that have been observed experimentally. Particular attention is focused on the mushy-layer mode and its dependence upon the thermodynamical properties of the alloy. The results of this analysis are used to make a number of interpretations of earlier experimental studies such as the observations that some systems are less prone to form chimneys and that the regions of melt in these systems evolve to supersaturated conditions, while the melt evolves to unsaturated conditions once chimneys have formed. In addition, good quantitative agreement is found between the results of the linear-stability analysis and the experimental results of Tait & Jaupart (1992) for the onset of the mushy-layer mode of convection.

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Instabilities of the liquid and mushy regions during solidification of alloys

In the present work, a general implicit source-based enthalpy method is presented for the analysis of solidification systems. The proposed approach is both robust and efficient. The performance of the method is illustrated by application to a number of problems taken from recent metallurgical literature.

A general enthalpy method for modeling solidification processes

This paper describes the simulation of thermosolutal convection in directionally solidified (DS) alloys. A linear stability analysis is used to predict marginal stability curves for a system that comprises a mushy zone underlying an all-liquid zone. In the unperturbed and nonconvecting state .e.}, the basic state), isotherms and isoconcentrates are planar and horizontal. The mushy zone is realistically treated as a medium with a variable volume fraction of liquid that is con-sistent with the energy and solute conservation equations. The perturbed variables include tem-perature, concentration of solute, and both components of velocity in a two-dimensional system. As a model system, an alloy of Pb-20 wt pct Sn, solidifying at a velocity of 2 X 10-3 cm s-1 was selected. Dimensional numerical calculations were done to define the marginal stability curves in terms of the thermal gradient at the dendrite tips,G
 L
 ,vs the horizontal wave number of the perturbed quantities. For a gravitational constant of 1g,0.5
 g, 0.1g, and 0.01g, the marginal stability curves show no minima; thus, the system is never unconditionally stable. Nevertheless, such calculations quantify the effect of reducing the gravitational constant on reducing convection and suggest lateral dimensions of the mold for the purpose of suppressing convection. Finally, for a gravitational constant of 10-4
 g, calculations show that the system is stable for the thermal gradients investigated (2.5 ≤G
 L
 ≤ 100 K-cm-1).

Thermosolutal convection during dendritic solidification of alloys: Part i. Linear stability analysis

A mathematical model of thermosolutal convection in directionally solidified dendritic alloys has been developed that includes a mushy zone underlying an all-liquid region. The model assumes a nonconvective initial state with planar and horizontal isotherms and isoconcentrates that move upward at a constant solidification velocity. The initial state is perturbed, nonlinear calculations are performed to model convection of the liquid when the system is unstable, and the results are compared with the predictions of a linear stability analysis. The mushy zone is modeled as a porous medium of variable porosity consistent with the volume fraction of, interdendritic liquid that satisfies the conservation equations for energy and solute concentrations. Results are presented for systems involving lead-tin alloys (Pb-10 wt pct Sn and Pb-20 wt pct Sn) and show significant differences with results of plane-front solidification. The calculations show that convection in the mushy zone is mainly driven by convection in the all-liquid region, and convection of the interdendritic liquid is only significant in the upper 20 pct of the mushy zone if it is significant at all. The calculated results also show that the systems are stable at reduced gravity levels of the order of 10−4
g
0 (g
0=980 cm·s−1) or when the lateral dimensions of the container are small enough, for stable temperature gradients between 2.5≤G
l≤100 K·cm−1 at solidification velocities of 2 to 8 cm·h−1.

Thermosolutal Convection during Dendritic Solidification of Alloys: Part II. Nonlinear Convection

The purpose of the paper is to present a method of calculating the enthalpy of a dendritic alloy during solidification. The enthalpies of the dendritic solid and interdendritic liquid of alloys of the Pb-Sn system are evaluated, but the method could be applied to other binaries, as well. The enthalpies are consistent with a recent evaluation of the thermodynamics of Pb-Sn alloys and with the redistribution of solute in the same during dendritic solidification. Because of the heat of mixing in Pb-Sn alloys, the interdendritic liquid of hypoeutectic alloys (Pb-rich) of less than 50 wt pct Sn has enthalpies that increase as temperature decreases during solidification.

Enthalpies of a binary alloy during solidification

For analyzing dendritic solidification, convection in the mushy zone is based on a momentum equation for a porous medium with a spatially varying fraction of liquid. Results are compared with those obtained with a less complete momentum equation. Calculations are done for directional solidification, with isotherms and isoconcentrates that move with a constant velocity. Thermosolutal convection as a function of time is studied by perturbing the solutal field. The total kinetic energy, the nature of the convection cells, and the isoconcentrates are significantly different for the two models. Hence the complete form of the momentum equation should be used

Momentum equation for dendritic solidification

This paper presents a mathematical model for directional solidification of a binary alloy including a dendritic region underlying an all-liquid region. It is assumed initially that there exists a nonconvecting state with planar isotherms and isoconcentrates solidifying at a constant velocity. The stability of this system has been analyzed and nonlinear calculations are performed that show the effect of convection in the solidification process when the system is unstable. Results of calculations for various cases defined by the initial temperature gradient at the dendrite tips and varying strength of the gravitational field are presented for systems involving lead-tin alloys. The results show that the systems are stable for a gravitational constant of 0.0001 g(0) and that convection can be suppressed by appropriate choice of the container's size for higher values of the gravitational constant. It is also concluded that for the lead-tin systems considered, convection in the mushy zone is not significant below the upper 20 percent of the dendritic zone, if al all.

P. Nandapurkar

Papers

Thermosolutal convection during dendritic solidification of alloys: Part i. Linear stability analysis

Thermosolutal Convection during Dendritic Solidification of Alloys: Part II. Nonlinear Convection

Enthalpies of a binary alloy during solidification

Momentum equation for dendritic solidification

Thermosolutal convection during dendritic solidification