Hervé Combeau

Modelling of microsegregation in ternary alloys: Application to the solidification of Al–Mg–Si

Abstract A fast numerical model has been developed for the quantitative prediction of microsegregation during solidification of ternary alloys. Considering a small volume of uniform temperature, the back-diffusion equations in the primary solid phase are solved in a 1-dimensional configuration using an implicit finite difference formulation with a Landau transformation onto a fixed [0,1] interval. The other phases which may precipitate during solidification are supposed to be stoichiometric and at equilibrium while the liquid is in a state of complete mixing. These calculations are coupled with phase diagram data through the use of mapping files: the liquidus surface, the monovariant lines and all the pertaining information are mapped through calls to Thermo-Calc [B. Sundman, B. Jansson and J. O. Andersson, CALPHAD, 9, 153 (1985)], prior to starting the microsegregation calculation itself. This very efficient microsegregation model can thus be coupled directly to macrosegregation computations performed at the scale of a whole casting: from the average enthalpy and concentrations variations computed at each mesh point of a casting during one time step, this microsegregation model is capable of predicting the variations of temperature, of the volume fractions of the various phases, of the liquid concentrations and of the average density. The efficiency of this coupling between microsegregation calculation and thermodynamic mapping files is demonstrated in the particular case of the Al–Mg–Si system.

Modeling of microsegregation in macrosegregation computations

A general framework for the calculation of micro-macrosegregation during solidification of metallic alloys is presented. In particular, the problems of back diffusion in the primary solid phase, of eutectic precipitation at the end of solidification, and of remelting are being addressed for an open system,i.e., for a small-volume element whose overall solute content is not necessarily constant. Assuming that the variations of enthalpy and of solute content are known from the solution of the macroscopic continuity equations, a model is derived which allows for the calculation of the local solidification path (i.e., cooling curve, volume fraction of solid, and concentrations in the liquid and solid phases). This general framework encompasses four microsegregation models for the diffusion in the solid phase: (1) an approximate solution based upon an internal variable approach; (2) a modification of this based upon a power-law approximation of the solute profile; (3) an approach which approximates the solute profile in the primary phase by a cubic function; and (4) a numerical solution of the diffusion equation based upon a coordinate transformation. These methods are described and compared for several situations, including solidification/remelting of a closed/open volume element whose enthalpy and solute content histories are known functions of time. It is shown that the solidification path calculated with method 2 is more accurate than using method 1, and that 2 is a very good approximation in macrosegregation calculations. Furthermore, it is shown that method 3 is almost identical to that obtained with a numerical solution of the diffusion equation (method 4). Although the presented results pertain to a simple binary alloy, the framework is general and can be extended to multicomponent systems.

Free growth of equiaxed crystals settling in undercooled NH4Cl–H2O melts

Abstract Recently theoretical works concerning the effect of convection on the growth of isolated dendrites have been compared with experiments on NH 4 Cl settling equiaxed crystals. It was inferred that more accurate theories were still needed to describe properly the equiaxed crystal growth in the presence of convection. Some new results have been obtained using the following experimental set-up: in a tube containing an undercooled solution of NH 4 Cl–H 2 O, settling NH 4 Cl equiaxed crystals have been filmed with a video camera so as to determine the evolution with time of their size and of their settling velocity. After a careful comparison of the experimental results with some calculations involving the choice of a stability constant, no major discrepancy has been found to prevent the application of the theories in question to moving equiaxed crystals.

A new cellular automaton—finite element coupling scheme for alloy solidification

A cellular automaton (CA) method was developed in the 1990s for the prediction of solidification grain structures. It was coupled with a finite element (FE) method to solve the energy conservation equation at the scale of the casting. Such coupling is revisited in this contribution thanks to a recently proposed front tracking method that gives the correct solution to a heat flow problem in which the undercooling of the growing fronts that define the limits of the mushy zone is taken into account. Validation is provided by considering unidirectional solidification. Comparison with a formerly proposed coupling scheme of the CAFE model clearly demonstrates the improvement achieved. The new coupling scheme developed should thus be preferred for further developments of solidification models that include undercooling of the growth fronts.

Equi-axed growth and related segregations in cast metallic alloys

Abstract Full-scale trials of DC ingots and laboratory scale directional solidification experiments have been performed to study the effect of grain structure on macro-segregation in industrial cast products. An Al alloy sheet ingot was cast with constant casting conditions (speed, superheat, cooling rate) except for the grain refiner: the first half of the ingot was non-inoculated, while the second half was inoculated. The results indicate that the extent and intensity of the centreline segregation is modified via the grain-refinement treatment: the finer the grains are, the more intense is the macro-segregation. Numerical simulations of directional solidification of binary Al-Cu alloys have been carried out with the help of a 2D finite volume software which takes account of the movement of the liquid with respect to the solid in the mushy zone. It is possible to account for the segregation pattern of the directionally solidified ingots that exhibit columnar or coarse equi-axed grain structures. Contrarily, the intense segregation of the fine-grained ingots is not yet understood.

Effect of discretization of permeability term and mesh size on macro- and meso-segregation predictions

The effect of interpolation schemes used for discretization of permeability in numerical simulations on macrosegregation and channel segregation (mesosegregation) during solidification has been studied. The different ways to discretize the permeability term and its effect on the interdendritic velocity are illustrated by a simplified 1D model which solves the Darcy equation for a porous medium. The Darcy equation is solved numerically using the SIMPLE algorithm for the coupled velocity–pressure fields. For this simplified case, an analytical reference solution can also be obtained. In the numerical solution four different interpolation schemes for permeability discretization have been employed, and the results obtained are compared. For coarse mesh, different interpolation schemes produce large differences from the analytical reference solution. We thereafter, present simulation results for solidification of Sn–Pb alloy in a two-dimensional rectangular cavity using different discretization schemes. It is observed that solute-rich liquid flowing towards the bottom of the rectangular cavity in the mushy zone due to thermosolutal convection results in patches of thin structure known as channels. These channels are formed by perturbation by the interdendritic fluid flow in the mushy zone and in some cases by the localized remelting in some portions of the solid/mush interface. The role of discretization schemes and mesh size on the formation of channel segregates and macrosegregation is discussed.

Numerical model for prediction of the final segregation pattern of bearing steel ingots

Abstract Chemical heterogeneities arise during solidification of steel ingots due to long range solute transport. A model taking into account heat, mass and momentum coupled transports during solidification of multicomponent alloys in a mold, is briefly presented. A comparison between experimental and numerical results illustrates its abilities and limits in the prediction of segregation pattern in bearing steel ingots.

Modelling of heat transfer coupled with columnar dendritic growth in continuous casting of steel

The general aim of this work is to calculate the extent of the equiaxed zone in continuously cast steel products. Free equiaxed grains can grow only in undercooled liquid regions. Undercooling of the bulk liquid occurs because the columnar dendrite tips growing from the mould reject solutes in the liquid. The specific aim of this contribution is to calculate the thermal and physical state of continuously cast steel long products assuming a columnar solidification mode, taking into account the tip undercooling at the solidification front. A 2‐D heat transfer model has been developed where the columnar solidification mode is assumed. The calculation of the undercooling at the advancing solidification front is coupled with the heat transfer equation. The comparison between the results of the present model and the classical heat transfer model indicates the importance of modelling the undercooling phenomenon. The influence of the secondary cooling has also been studied.

Influence of Discretization of Permeability Term and Mesh Size on the Prediction of Channel Segregations

Macro- and meso-segregations correspond to compositional heterogeneities at the scale of a casting. They develop during the solidification process. One of the parameters that have an essential effect on these segregations is the mush permeability, which is highly nonlinear, and varies over a wide range of magnitudes. We present simulation results for solidification of a Sn-Pb alloy in a two-dimensional cavity, highlighting the role of (i) the numerical interpolation schemes used for the finite-volume discretization of the highly-nonlinear permeability term and (ii) of the mesh size on the prediction of mesosegregations and macrosegregation. We observe that solute-rich liquid flowing through the mushy zone due to thermo-solutal convection results in patches of thin channel structures, which develop into mesosegregations. We notice little sensitivity of the predicted macrosegregation to different discretization schemes for the permeability term. However, we found their influence on the prediction of channel segregates to be significant when using coarse computational grids, customary in the simulation of industrial castings. Mesh refinement is crucial for capturing the complex phenomena in the formation of channel segregates. With a very fine mesh channels have been captured with more than one grid point along their width, allowing the determination of their width.

Effect of Grain Refinement on Macrosegregation in Direct Chill Semi-Continuous Casting of Aluminium Sheet Ingot

An experimental investigation of the macrosegregation occurring in a commercial size direct chill cast of an 5182 alloy ingot is described. The extent of the chemical homogeneity in a sheet ingot depending of the grain refining practice is discussed. A complete analysis of 2D macrosegregation maps and of the associated grain structure in a non grain refined part and a grain refined part of the ingot is compared and discussed. The intensity of the centerline segregation is modified via the grain refining practice. The change of macrosegregation measured in the slices with and without grain refiner may be attributed to the effects of the grain motion but also the liquid movement.