D. R. Poirier

Permeability for Flow of Interdendritic Liquid in Columnar-Dendritic Alloys

Permeability data for the flow of interdendritic liquid in partially solid Pb−Sn and borneol-paraffin columnar-alloys are summarized. The data are used in regression analyses and simple flow models to arrive at relationships between permeability and the morphology of the solid dendrites. When flow is parallel to the primary dendrite arms, the important morphological aspects are the volume fraction liquid (gL) and the primary dendrite arm spacing (d1). When flow is normal to the primary dendrite arms, the permeability depends upon the secondary dendrite arm spacing (d2) as well asd1 andgL. The parallel permeability is best described by a model based on the Hagen-Poiseuille law for laminar flow through a tube; for the normal permeability an empirical multilinear regression gives the best fit to the data. However, those models are not appropriate for extrapolations beyond the range of the available data (0.19≤gL≤0.66), particularly asgL approaches 1. For extrapolations, models based upon the Blake-Kozeny equation for flow through porous media are recommended.

Conservation of mass and momentum for the flow of interdendritic liquid during solidification

In this paper, mass and momentum conservation equations are derived for the flow of interdendritic liquid during solidification using the volume-averaging approach. In this approach, the mushy zone is conceived to be two interpenetrating phases; each phase is described with the usual field quantities, which are continuous in that phase but discontinuous over the entire space. On the microscopic scale, the usual conservation equations along with the appropriate interfacial boundary conditions describe the state of the system. However, the solution to these equations in the microscopic scale is not practical because of the complex interfacial geometry in the mushy zone. Instead, the scale at which the system is described is altered by averaging the microscopic equations over some representative elementary volume within the mushy zone, resulting in macroscopic equations that can be used to solve practical problems. For a fraction of liquid equal to unity, the equations reduce to the usual conservation equations for a single-phase liquid. It is also found that the resistance offered by the solid to the flow of interdendritic liquid in the mushy zone is best described by two coefficients, namely, the inverse of permeability and a second-order resistance coefficient. For the flow in columnar dendritic structures, the second-order coefficient along with the permeability should be evaluated experimentally. For the flow in equiaxial dendritic structures(i.e., isotropic media), the inverse of permeability alone is sufficient to quantify the resistance offered by the solid.

Simulation of freckles during vertical solidification of binary alloys

A mathematical model of solidification that simulates the formation of channel segregates or freckles is presented. The model simulates the entire solidification process starting with the initial melt to the solidified cast, and the resulting segregation is predicted. Emphasis is given to the initial transient, when the dendritic zone begins to develop and the conditions for the possible nucleation of channels are established. The mechanisms that lead to the creation and eventual growth or termination of channels are explained in detail and illustrated by several numerical examples. Predictions of the pattern and location of channels in different cooling situations are in good agreement with experimental observations.

Macrosegregation in a multicomponent low alloy steel

Macrosegregation theory is extended to predict the formation of channel-type segregation for multicomponent systems. Specifically, calculations are carried out for 0.7 pct C steel, by considering heat, mass and momentum transport in the mushy zone. In the model used for calculations the momentum transport equation and the energy equation were solved simultaneously. It is confirmed, by comparing calculated results with experimental results, that this model successfully predicts the occurrence of channel-type segregation. This analysis is also more rigorous than previous works on macrosegregation because previous analyses were done by solving for convection in the mushy zone with an “uncoupled” temperature field. Using the model, the effects of adjusting the compositions of silicon and molybdenum in steel were quantitatively evaluated in order to show how channel-type segregates can be avoided by adjusting alloy composition. A method of optimizing composition to minimize segregation is presented. It is recommended that this methodology be applied to alloy design so that ingots of alloys amenable to commercial practice can be obtained readily with a minimum amount of “trial-and-error” development work and expense.

A thermodynamic prediction for microporosity formation in aluminum-rich Al-Cu alloys

A computer model is used to predict the formation and the amount of microporosity in directionally solidified Al-4.5 wt pct Cu alloy. The model considers the interplay between so-called “solidification shrinkage” and “gas porosity” that are often thought to be two contributing and different causes of interdendritic porosity. There is an accounting of the alloy element, Cu, and of dissolved hydrogen in the solid- and liquid-phase during solidification. Consistent with thermodynamics, therefore, a prediction of forming the gas-phase in the interdendritic liquid is made. The local pressure within the interdendritic liquid is calculated by macrosegregation theory that considers the convection of the interdendritic liquid, which is driven by density variations within the mushy zone. Process variables that have been investigated include the effects of thermal gradients and solidification rate, and the effect of the concentration of hydrogen on the formation and the amount of interdendritic porosity. These calculations show that for an initial hydrogen content less than approximately 0.03 ppm, no interdendritic porosity results. For initial hydrogen contents in the range of 0.03 to 1 ppm, there is interdendritic porosity. The amount is sensitive to the thermal gradient and solidification rate; an increase in either or both of these variables decreases the amount of interdendritic porosity.

Estimation of the surface tensions of binary liquid alloys

A simple method to estimate the surface tensions of binary alloys has been developed by assuming that the partial molar excess free energies are proportional to the number of nearest neighbors in both the bulk solution and in the surface itself. In order to estimate the surface tension of the alloys, excess free energies of the alloys and the surface tensions of the pure components are required. This method has been applied to ten alloys exhibiting positive, positive as well as negative, and negative deviations from ideal solution behavior. The method depends upon the reliability of the thermodynamic data for the bulk solutions, and, further, it is important to use an interpolation scheme that is consistent with the Gibbs-Duhem requirement, when the thermodynamic data are presented in tabular form as a function of composition. To accomplish this interpolation, a special calculation technique is presented.

The energy and solute conservation equations for dendritic solidification

The energy equation for solidifying dendritic alloys that includes the effects of heat of mixing in both the dendritic solid and the interdendritic liquid is derived. Calculations for Pb-Sn alloys show that this form of the energy equation should be used when the solidification rate is relatively high and/or the thermal gradients in the solidifying alloy are relatively low. Accurate predictions of transport phenomena in solidifying dendritic alloys also depend on the form of the solute conservation equation. Therefore, this conservation equation is derived with particular consideration to an accounting of the diffusion of solute in the dendritic solid. Calculations for Pb-Sn alloy show that the distribution of the volume fraction of interdendritic liquid (gL) in the mushy zone is sensitive to the extent of the diffusion in the solid. Good predictions ofgL are necessary, especially when convection in the mushy zone is calculated.

Permeability for cross flow through columnar-dendritic alloys

Experiments for measuring permeability in columnar-dendritic microstructures have provided data only up to a volume fraction of liquid of 0.66. Hence, the permeability for flow perpendicular to the primary dendrite arms in columnar-dendritic microstructures was calculated, extending our data base for permeability to volume fractions of liquid as high as 0.98. Analyses of the dendritic microstructures were undertaken first by detecting the solid-liquid interfaces with a special computer program and then by generating a mesh for a finite-element fluid flow simulation. Using a Navier-Stokes solver, the velocity and pressure at the nodes were calculated at the microstructural level. In turn, the average pressure gradient was used to calculate the Darcy permeability. Permeabilities calculated by this versatile technique provided data at high volume fractions of liquid that merged with the empirical data at the lower volume fractions.

Permeability for flow parallel to primary dendrite arms

The permeability for the flow of interdendritic liquid parallel to primary dendrite arms in columnar structures was calculated using the boundary element method for fully developed flow. The permeability was calculated because when the volume fraction of liquid (gL)exceeds approximately 0.60 –0.65, experiments fail. The calculated results, based on the microstructures of directionally solidified alloys, agreed with analytical results for flow parallel to circular cylinders arranged in square and triangular arrays. It appears that there is a transition in the behavior of the permeability at gL ≈ 0.65.

Convection in the two-phase zone of solidifying alloys

The analysis is applicable to alloy solidification which proceeds horizontally to the center of a mold. The model follows the growth of the solid-liquid zone adjacent to the chill face (the initial transient), the movement of the zone across the mold, and the region of final solidification adjacent to the centerline (the final transient). During solidification the density of the liquid varies across the twophase zone. Consequently, there is natural convection which is treated as flow through a porous medium. The equations for convection are coupled with the equation of solute redistribution between the phases in order to calculate macrosegregation after solidification is complete. Results were computed for alloys which show: (1) “inverse segregation≓ at a cooled-surface; (2) macrosegregation resulting from solidification with the initial transient, a period with a complete two-phase zone, and a final transient; and (3) macrosegregation when the width of the two-phase zone exceeds the semi-width of the mold.