C. Beckermann

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

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.

Modelling of macrosegregation: applications and future needs

Abstract Modelling and simulation of macrosegregation has experienced explosive growth since the pioneering studies of Flemings and co-workers in the mid-1960s. This paper presents a review of recent macrosegregation models, with particular emphasis on their application to selected industrially relevant casting processes. The successes and shortfalls of the models in predicting measured macrosegregation patterns are noted. Although it is still early in their development, advanced macrosegregation models that include a detailed consideration of the solidification microstructure are also reviewed. Recent studies involving direct numerical simulation of macrosegregation related phenomena on a microscopic scale are highlighted. Important issues deserving future research attention are identified.

Equiaxed dendritic solidification with convection: Part I. Multiscale/multiphase modeling

Equiaxed dendritic solidification in the presence of melt convection and solid-phase transport is investigated in a series of three articles. In part I, a multiphase model is developed to predict com-position and structure evolution in an alloy solidifying with an equiaxed morphology. The model accounts for the transport phenomena occurring on the macroscopic (system) scale, as well as the grain nucleation and growth mechanisms taking place over various microscopic length scales. The present model generalizes a previous multiscale/multiphase model by including liquid melt convec-tion and solid-phase transport. The macroscopic transport equations for the solid and the interdendritic and extradendritic liquid phases are derived using the volume averaging technique and closed by supplementary relations to describe the interfacial transfer terms. In part II, a numerical application of the model to equiaxed dendritic solidification of an Al-Cu alloy in a rectangular cavity is dem-onstrated. Limited experimental validation of the model using a NH4C1-H2O transparent model alloy is provided in part III.

Natural convection solid/liquid phase change in porous media

media with natural convection in the melt region. The model is based on volume averaged transport equations, while phase change is assumed to occur over a small temperature range. Experiments are performed in a vertical, square enclosure using gallium and glass beads as the fluid and the porous matrix, respectively. For natural convection, melting and solidification (with initial supercooling), the numerical results show reasonable agreement with the temperature and interface position measurements. Natural convection in the melt as well as heat conduction in the solid is found to considerably influence the interface shape and movement during both the melting and solidification experiments.

Formation of macrosegregation by multicomponent thermosolutal convection during the solidification of steel

The formation of macrosegregation by multicomponent thermosolutal convection during the solidification of steel is simulated by simultaneously solving macroscopic mass, momentum, energy, and species conservation equations with full coupling of the temperature and concentrations through thermodynamic equilibrium at the solid/liquid interface. The flow field, solid fraction evolution, and macrosegregation patterns for four cases are presented. The results show both the formation of channel segregates and the formation of islands of mush surrounded by bulk melt. In examining the solidification of a ten-element steel, the global extent of macrosegregation of an element is found to be linearly dependent on its partition coefficient (more severe segregation for small partition coefficient), although such scaling is not possible locally. Results for the solidification of a binary Fe-C alloy (with the same carbon content as the ten-element alloy) are similar to those for the ten-element alloy due solely to the large contribution of carbon to buoyancy driven flow in the ten-element steel chosen for study. While including only those elements that make significant contributions to buoyancy driven flow reproduces the global extent of macrosegregation seen in the ten-element alloy, local differences in the predictions are visible. Finally, comparison of results for the solidification of the same ten-element steel using two different sets of data to describe the partition coefficients and change in liquidus temperature with concentration of the elements shows completely opposite behavior,i.e., upward flow through the mushy zone for one case and downward flow for the other. Thus, the need to have accurate phase-equilibrium data when modeling multicomponent macrosegregation is illustrated. Together, the results give an indication of what areas require more careful examination if accurate modeling of multicomponent solidification is to be accomplished.

EFFECT OF A CENTERED CONDUCTING BODY ON NATURAL CONVECTION HEAT TRANSFER IN AN ENCLOSURE

The effect of a centered, square, heat-conducting body on natural convection in a vertical square enclosure was examined numerically. The analysis reveals that the fluid flow and heat transfer processes are governed by the Rayleigh and Prandtl numbers, the dimensionless body size, and the ratio of the thermal conductivity of the body to that of the fluid. For Pr = 0.71 and relatively wide ranges of the other parameters, results are reported in terms of streamlines, isotherms, and the overall heat transfer across the enclosure as described by the Nusselt number. Heat transfer across the enclosure, in comparison to that in the absence of a body, may be enhanced (reduced) by a body with a thermal conductivity ratio less (greater) than unity. Furthermore, the heat transfer may attain a minimum as the body size is increased. These and other findings are justified through a careful examination of the local heat and fluid flow phenomena.

Mathematical modeling of transport phenomena during alloy solidification

Mathematical modeling of mass, momentum, heat, and species transport phenomena occurring during solidification of metal alloys is reviewed. Emphasis is placed on the incorporation of the effects of the solid structure and the interactions between the solid and liquid phases on a microscopic scale into a (macroscopic) model of the transport phenomena occurring at the system scale. Both columnar and equiaxed growth structures, as well as laminar convection of liquid and solid crystals are considered. The macroscopic conservation equations are introduced via a volume averaging approach and commonly made simplifications are examined. Basic constitutive relations for the phase interactions occurring in alloy solidification are presented. Recent progress in including nucleation, microsegregation, undercooling and other microscopic phenomena in the macroscopic equations is reviewed. The specific areas where future theoretical and experimental research is needed are identified.

A multiphase solute diffusion model for dendritic alloy solidification

A solute diffusion model, aimed at predicting microstructure formation in metal castings, is proposed for dendritic solidification of alloys. The model accounts for the different length scales existing in a dendritic structure. This is accomplished by utilizing a multiphase approach, in which not only the various physical phases but also phases associated with different length scales are considered separately. The macroscopic conservation equations are derived for each phase using the volume averaging technique, with constitutive relations developed for the interfacial transfer terms. It is shown that the multiphase model can rigorously incorporate the growth of dendrite tips and coarsening of dendrite arms. In addition, the distinction of different length scales enables the inclusion of realistic descriptions of the dendrite topology and relations to key metallurgical parameters. Another novel aspect of the model is that a single set of conservation equations for solute diffusion is developed for both equiaxed and columnar dendritic solidification. Finally, illustrative calculations for equiaxed, columnar, and mixed columnar-equiaxed solidification are carried out to provide quantitative comparisons with previous studies, and a variety of fundamental phenomena such as recalescence, dendrite tip undercooling, and columnar-to-equiaxed transition (CET) are predicted.

A two-phase mixture model of liquid-gas flow and heat transfer in capillary porous media—I. Formulation

Abstract A model for two-phase transport in capillary porous media is presented, in which the two phases are viewed as constituents of a binary mixture. The conservation equations are derived from the classical separate flow model without invoking additional assumptions. The present formulation, owing to its analogy to conventional multicomponent mixture flow theories and to a considerable reduction in the number of the differential equations required for the primary variables, provides an alternative for the theoretical analysis and numerical simulation of two-phase transport phenomena in porous media. Several complicated problems such as boundary layer two-phase flows, conjugate two- and single-phase flows in multiple regions and transient flows are shown to become more tractable within the framework of this new formulation.

Natural convection in vertical enclosures containing simultaneously fluid and porous layers

A numerical and experimental study is reported of natural convection in a vertical rectangular fluid enclosure that is partially filled with a fluid-saturated porous medium. Velocities, stresses, temperatures, and heat fluxes are assumed to be continuous across the fluid/porous-medium interface, and the conservation equations for the fluid and the porous regions are combined into a single set of equations for numerical solution. Thermocouples as well as a Mach-Zehnder interferometer are used to measure temperature distributions and infer fluid flow patterns within the fluid and the porous medium. For various test cells, porous-layer configurations and fluid-solid combinations, the model predictions show excellent agreement with the experimental measurements. It is found that the intensity of natural convection is always much stronger in the fluid regions, while the amount of fluid penetrating into the porous medium increases with increasing Darcy and Rayleigh numbers. The degree of penetration of fluid into the porous medium depends strongly on the porous-layer geometry and is less for a horizontal porous layer occupying the lower half of the test cell. If penetration takes place, the flow patterns in the fluid regions are significantly altered and the streamlines show cusps at the fluid/porous-medium interfaces. For a high effective-thermal-conductivity porous medium, natural convection in the medium is suppressed, while the isotherms bend sharply at the fluid/porous-medium interface.