J. C. Heinrich

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.

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.

Finite element approximation to two-dimensional sine-Gordon solitons

Abstract The paper presents a finite element algorithm for the numerical solution of the sine-Gordon equation in two spatial dimensions, as it arises, for example, in rectangular large-area Josephson junctions. The dispersive nonlinear partial differential equation of the system allows for soliton-type solutions, an ubiquitous phenomenon in a large variety of physical problems. A semidiscrete Galerkin approach based on simple four-noded bilinear finite elements in combination with a generalized Newmark integration scheme is used throughout the paper and is tested in a variety of cases. Comparisons with finite difference solutions show the superior performance of the proposed algorithm leading to very accurate, numerically stable and physically consistent solitary wave solutions. The results support the confidence in the present numerical model which should be capable to treat also more complex situations involving soliton-type interactions.

Numerical model for dendritic solidification of binary alloys

Abstract A finite element model capable of simulating solidification of binary alloys and the formation of freckles is presented. It uses a single system of equations to deal with the all-liquid region, the dendritic region, and the all-solid region. The dendritic region is treated as an anisotropic porous medium. The algorithm uses the bilinear isoparametric element, with a penalty function approximation and a Petrov-Galerldn formulation. Numerical simulations are shown in which an NH4Cl-H2O mixture and a Pb-Sn alloy melt are cooled. The solidification process is followed in time. Instabilities in the process can be clearly observed and the final compositions obtained.

Macrosegregation patterns in multicomponent Ni-base alloys

A mathematical model of the dendritic solidification of multicomponent alloys, that includes thermosolutal convection and macrosegregation, is presented. The model is an extension of one previously developed for binary alloys. Numerical simulations are given for ternary and quaternary Ni-base alloys, and the evolution of macrosegregation during solidification is studied. The results show that the segregation patterns vary greatly with cooling conditions, adopting several shapes and levels of intensity. Calculations of segregation in rectangular molds and in molds with smooth and abrupt variations of the cross sections exhibit significant differences in the distribution of macrosegregation due to the change in geometry. In addition, the segregation patterns are found to be particularly sensitive to the values of the equilibrium partition coefficients of the alloy components.

Modeling dendritic growth of a binary alloy

A two-dimensional model for simulation of the directional solidification of dendritic alloys is presented. It solves the transient energy and solute conservation equations using finite element discretizations. The energy equation is solved in a fixed mesh of bilinear elements in which the interface is tracked; the solute conservation equation is solved in an independent, variable mesh of quadratic triangular elements in the liquid phase only. The triangular mesh is regenerated at each time step to accommodate the changes in the interface position using a Delaunay triangulation. The model is tested in a variety of situations of differing degrees of difficulty, including the directional solidification of Pb-Sb alloys.

Three-dimensional simulations of freckles in binary alloys

Abstract A tridimensional finite-element model was developed to calculate the thermosolutal convection and macrosegregation during the solidification of dendritic alloys. A single set of conservation equations is solved in the mushy zone, all-liquid, and all-solid regions without internal boundary conditions. The model is applied to simulate the directional solidification of a Pb–Sn alloy in cylinders of square and circular cross section. The calculations are started from an all-liquid state and the evolution of convection, solute and energy transport, and the mushy zone growth are followed in time. The results show details of the channels, which result in freckles, that are not observable in existing two-dimensional simulations. Several qualitative features of channels and freckles previously observed in experiments with transparent systems, like chimney convection, preference of channels to be on surfaces, and enhanced solid growth at the channel mouth (“volcanoes”) are successfully reproduced.

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

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,GL,vs the horizontal wave number of the perturbed quantities. For a gravitational constant of 1g,0.5g, 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 10-4g, calculations show that the system is stable for the thermal gradients investigated (2.5 ≤GL≤ 100 K-cm-1).

Thermosolutal convection during dendritic solidification of alloys: Part II. Nonlinear convection

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 10−4g0 (g0=980 cm·s−1) or when the lateral dimensions of the container are small enough, for stable temperature gradients between 2.5≤Gl≤100 K·cm−1 at solidification velocities of 2 to 8 cm·h−1.

Finite element analysis of directional solidification of multicomponent alloys

A finite element model of dendritic solidification of multicomponent alloys is presented that includes solutal convection and is an extension of a previously developed model for solidification of binary alloys. The model is applied to simulation of the solidification of ternary and quaternary Ni‒based alloys. The role of solutal convection in the macrosegregation and the formation of freckles is analysed. Calculations show the effects of geometry and material properties on the convection patterns and the attendant segregation.