Numerical Study of Micro-Macro Coupled Transport Phenomena in the Unidirectional Solidification of a Binary System

A mathematical model that considers both microheat and mass transfer and macrotransport phenomena is presented to investigate the unidirectional solidification of a binary NH₄Cl-H₂O solution. It is shown that the solidus and liquidus locations predicted by the numerical coupling calculation are close to the experimental data. The variation of solute concentration approaches the Scheil equation. The mushy zone has good permeability and makes way for double diffusive flow. The deviation of numerical results of new phase growth from experimental data is probably caused by the inadequate estimation of permeability and microfluid flow in the mushy layer.     

The structure of plumes generated in the unidirectional solidification process for a binary system

The present article describes the structure of plumes generated during solidification of a binary system. A transparent aqueous ammonium chloride solution is employed for super-eutectic growth in a Hele-Shaw cell. The velocities of plume convection in the melt layer and interstitial fluid flow within the mushy layer are measured by the particle tracking and dye tracing methods, respectively. Several important features are identified for each convective flow. In particular, the plume convection is found to consist of the upward flow enveloped in the downward flow, i.e., double flow structure. The downward flow enhances the solidification in the neighborhood of the exit of the channel emanating the plume, like a volcano. Interstitial fluid within the mushy layer is observed to move downward uniformly, which is induced by the plume convection.     

The steady-state solidification scenario of ternary systems: Exact analytical solution of nonlinear model

A mathematical model describing the steady-state solidification of ternary systems with mushy layers (primary and cotectic) is formulated: solidification along a liquidus surface is characterized by a primary mushy layer, and solidification along a cotectic line is characterized by a secondary (cotectic) mushy layer. Exact analytical solutions of the model under consideration are found in a parametric form (thicknesses of mushy layers, growth rate of their boundaries, temperature and composition fields, solid fractions are determined in an explicit form). The velocity of solidification is completely determined by temperature gradients in the solid and liquid phases. This velocity coincides with similar expressions describing binary melt solidification with a planar front or a mushy layer. It is shown that the liquid composition of the main component decreases in the cotectic and primary layers, whereas the second (cotectic) composition increases in the cotectic layer, attains a maximum point and decreases in the primary layer.     

Mathematical model for turbulent flow,heat transfer,and solidification

A steady‐state, two‐dimensional numerical model has been used to describe coupled liquid steel's turbulent flow and heat transfer with solidification for Fe‐C binary alloy in a crystallizer of inverse casting. The solid‐liquid phase change phenomena have been modeled by using continuum formulations and considering the mushy zone as porous media. The turbulence flow in the crystallizer has been accounted for using a modified version of the low‐Reynolds‐number κ?ε turbulence model. The flow pattern in the liquid zone and the temperature distribution in the solid, mushy, and liquid regions have been predicted. The numerical analysis indicates that the residence time of the mother sheet in the crystallizer is one of the key parameters. The effects of some other main parameters on the solidification behavior have also been studied, such as the thickness and the initial temperature of the mother sheet, and the superheat degree of liquid steel. © 2003 Wiley Periodicals, Inc. Heat Trans Asian Res, 32(7): 582–592, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.10112     

A numerical study of unidirectional solidification of a binary metal alloy under influence of a rotating magnetic field

In this paper we present a numerical study of the fluid flow during directional solidification of a binary alloy (Pb85wt%Sn) in presence of a forced convection. The latter is driven by a rotating magnetic field (RMF) the strength of which, expressed by the magnetic Taylor number, varies between 10⁴ < Ta < 2 × 10⁶. The focus of this paper is the problem when cooling starts simultaneously with the acceleration of the melt from a state of rest. Thus, we study the interference of the so-called spin-up problem with the solidification of the melt. The numerical simulations are based on a mixture model formulation. We show that three distinct fluid flow phases exist. During the first two phases (initial adjustment and inertial phase) the acceleration of the liquid takes place which occurs in close similarity to the isothermal spin-up [P.A. Nikrityuk, M. Ungarish, K. Eckert, R. Grundmann, Spin-up of a liquid metal flow driven by a rotating magnetic field in a finite cylinder. A numerical and analytical study, Phys. Fluids 17 (2005) 067101]. The third phase is characterized by a braking of the fluid flow due to the progressive solidification increasing the aspect ratio of the liquid (2R₀/H_l) and decreasing the forcing. We show that as soon as the velocity of the secondary flow exceeds the velocity of the solidification front, a convex shape of the mushy zone can be observed. In parallel, Taylor–Görtler vortices advected by the secondary flow towards the mushy zone might impose a wavy substructure on the latter. At the end, predictions with respect to heat flux and macrosegregations are given.     

MODELING OF DIELECTRIC FLUID SOLIDIFICATION WITH CHARGED PARTICLES IN ELECTRIC FIELDS AND REDUCED GRAVITY

A mathematical model and an explicit finite-difference iterative integration algorithm for two-dimensional laminar steady flow and solidification of an incompressible, viscous, electrically conducting but neutrally charged melt containing electrically charged panicles and exposed to an externally applied electrostatic field were developed. The system of governing electrohydrodynamic equations was derived from a combination of Maxwell's equations and the Navier-Stokes equations, including thermally induced buoyancy, latent heat release, and Joule heating, while accounting for the mushy region. Physical properties were treated as arbitrarily temperature-dependent. Numerical results demonstrate the existence of strong electrothermoconvective motion in the melt and quantify its influence on the amount of accrued solid, deposition pattern of the electrically charged particles inside the accrued solid, and the melt/solid interface shape.     

Melting in a vertical cylindrical tube: Numerical investigation and comparison with experiments

The present work numerically investigates melting of a phase-change material (PCM) in a vertical cylindrical tube. The analysis aims at an investigation of local flow and thermal phenomena, by means of a numerical simulation which is compared to the previous experimental results .The numerical analysis is realized using an enthalpy–porosity formulation. The effect of various parameters of the numerical solution on the results is examined: in particular, the term describing the mushy zone in the momentum equation and the influence of the pressure–velocity coupling and pressure discretization schemes. PISO vs. SIMPLE and PRESTO! vs. Body-Force-Weighted schemes are examined. No difference is detected between the first two. However, considerable differences appear with regard to the last two, due to the mushy zone role.Image processing of experimental results from the previous studies is performed, yielding quantitative information about the local melt fractions and heat transfer rates. Based on the good agreement between simulations and experiments, the work compares the heat transfer rates from the experiments with those from the numerical analysis, providing a deeper understanding of the heat transfer mechanisms. The results show quantitatively that at the beginning of the process, the heat transfer is by conduction from the tube wall to the solid phase through a relatively thin liquid layer. As the melting progresses, natural convection in the liquid becomes dominant, changing the solid shape to a conical one, which shrinks in size from the top to the bottom.     

Finite element modeling of the porosity formation in castings

The aim of the present paper was to contribute to understanding the origin and effects of porosity in aluminum die-castings by characterizing the distribution and geometry of the porosity. It also seeks to develop a predictive model for microporosity formation during solidification through an analysis of the alloy solidification path, the presence of gas dissolved in the molten metal and the flow of liquid metal through the mushy zone. The finite element method was used for solving porosity formation problem jointly with the problem for heat and mass transfer.     

Directional solidification of binary melts with a non-equilibrium mushy layer

When the melt or solution solidifies a constitutionally supercooled mushy layer is frequently formed ahead of the phase transition boundary. This leads to nucleation and growth mechanisms of newly born solid particles within a mush. The latter is responsible for the structures and properties appearing in the crystal. The process of solidification with a supercooled mushy layer is analytically described on the basis of two joint theories of directional and bulk crystallization. Such characteristics as the constitutional supercooling, the solid fraction and the radial density distribution function of solid particles in a mushy layer are found. The complex structure of the non-equilibrium mushy layer is completely recognized.     

Mushy zone equilibrium solidification of a semitransparent layer subject to radiative and convective cooling

Equilibrium solidification in a semitransparent planar layer is studied using an isothermal mushy zone model. The layer is made up of a pure material being emitting, absorbing and isotropically scattering and is subject to radiative and convective cooling. The model involves solving simultaneously the transient energy equation and the radiation transport equation. An implicit finite volume scheme is employed to solve the energy equation, with the discrete ordinate method being used to deal with the radiation transport. A systematical parametric study is performed and the effects of various materials optical properties and processing conditions are investigated. It is found that decreasing the optical thickness and increasing the scattering albedo both lead to a wider mushy zone and a slower rate of solidification.     

Thermosolutal flow in steel ingots and the formation of mesosegregates

The influence of the thermosolutal convection of the liquid steel in the solidifying core of a 3.3-ton ingot on the formation of banded mesosegregates is investigated by a multiscale solidification model. We first show how the thermosolutal flow structure in the solidifying core depends on the relation between the interacting thermal and solutal buoyancy forces and the coupling by the phase-change kinetics. We further show that banded mesosegregates are triggered by instabilities of the solidification front, that their location is determined by flow instabilities, and that their “A” or “V” orientation depends on the global direction of the flow circulation. Moreover, the results show that local remelting is not necessary to develop a channel mesosegregate. Destabilization of the mushy zone with local variations of the solidification velocity is sufficient.     

Modeling of Solidification Process in a Rotary Electromagnetic Stirrer

A macroscopic model of the solidification process in a rotary electromagnetic stirrer is presented. The fluid flow, heat, and mass transfer inside a rotary stirrer are modeled using, 3-D swirl flow equations in which turbulent flow is simulated using a k ? ? model. A hybrid model is used to represent the mushy zone, which is considered to be divided into two regions: a coherent region and a noncoherent region. Each region is represented by a separate set of governing equations. An explicit time-stepping scheme is used for solving the coupled temperature and concentration fields, while an implicit scheme is used for solving equations of motion. The coupling relations also include eutectic solidification, which is an important feature in modeling solidification with electromagnetic stirring, especially in the context of the formation of semi-solid slurry. The results from the present numerical solution agree well with those corresponding to experiments reported in literature.     

MICROSOLIDIFICATION PROCESS IN MULTICOMPONENT SYSTEM

In conventional solidification of multicomponent mixtures, a mushy zone appears between the pure solid and liquid regions and promotes stable solidification by accepting the rejected solute regionally. From the standpoint that the fineness of inhomogeneity influences the mechanical properties in material processing, the linking of macro heat transfer and microsolidification in the mushy zone was studied. First, the crystal growth and its accompanying concentration field near the advancing front of the mushy zone were observed precisely by using the light absorption method. It was clarified that the mushy zone consisted of the leading front in which the frame structure formed with an accompanying concentration boundary layer and a growing region where the solidification proceeds by fattening of the crystals. Second, the mechanism of side-branch evolution was studied in conjunction with interfacial instability due to constitutional supercooling and curvature supercooling around the primary arm surface. Summarizing these results, the microsolidification process is discussed quantitatively in relation to macro heat transfer.     

PIV study on the development of double-diffusive convection during the solidification effected by lateral cooling for a super-eutectic binary solution

The solidification of a binary solution occurs in a variety of industrial applications. The fundamental mechanism of the development of “double-diffusive convection” during solidification was one of the main subjects of study in the past. This study focused on the effect of the initial concentration of a super-eutectic aqueous ammonium chloride solution on the development of double-diffusive convection during the solidification process. Particle image velocimetry (PIV) technique was employed to measure the flow velocity and observe the morphological conditions associated with the solidification effected by cooling from the side wall. The transient temperature distribution within a test cell was also measured by using the designed experimental system. PIV measurement revealed that the flow structure was composed of a major circulation flow within the test cell for the binary solution with low-concentration and multiple layers of circulation flows developed in sequence in the melt of the test cell for the high-concentration binary solution. Test-cell images captured by CCD camera indicated the presence of more A-segregates within the mushy zone as the initial concentration of the binary solution increased. For the low-concentration binary solution, under circumstances of without or with very weak double-diffusive convection, transient temperature distribution presented thermal stratification within the test cell. However, strong double-diffusive convection resulted in intersection on temperature curves for the high-concentration binary solution.     

Modeling and Measurements of Heat Transfer Phenomena in Two-Phase PbSn Alloy Solidification in an External Magnetic Field

The main aim of this work is to study numerically the influence of an external magnetic field on the solidification processes of two-component materials. Based on the continuum model of two-phase flow a mathematical model for the directional solidification of a binary alloy in a magnetic field is presented. The model includes mass, momentum, energy and species mass conservation equations written in compressible form and additional relationships describing the temperature-solute coupling. The geometry under study is a cylindrical mold with adiabatic walls and cooled bottom. The macroscale transport in the solidification of alloys is governed by the progress of the two-phase mushy zone, which is treated by means of a porous medium approach. The volume fraction of liquid and solid phases, respectively, is calculated from a 2D approximation of the phase diagram. The results of calculation are compared with experimental data.     

Buoyancy and surface tension driven flows in float zone crystal growth with a strong axial magnetic field

The buoyancy driven flow due to the temperature gradient in the melt of a float zone and the surface tension driven flow due to the non-uniform temperature distribution along the free surface of the zone are studied in the presence of a strong axial magnetic field. The non-cylindrical shape of the zone is found to have a profound effect on the melt motion. The results indicate that the regions near the free surface are controlled mainly by the thermocapillarity, while the inner region is dominated by the buoyancy driven flow. Some implications for the mass transport of dopants in the molten float zone are discussed.     

A Semi-Exact Solution for Solidification of a Binary Solution on a Cold Isothermal Surface below Eutectic Temperature

The solidification of a binary solution on a cold horizontal surface below eutectic temperature is solved using a semi-exact method. The temperature distributions in the solid and liquid zones are obtained by exact solutions, while heat transfer in the mushy zone is obtained by an integral approximate method. The locations of the interface between solid and mushy zones and interface between mushy and liquid zones are obtained by coupling the temperature distributions in the three regions. The effects of initial temperatures, wall temperatures, and initial concentrations on the solidification of the binary solution are investigated.     

A numerical study of the combined effects of microsegregation, mushy zone permeability and fllow, caused by volume contraction and thermosolutal convection, on macrosegregation and eutectic formation in binary alloy solidification

Numerical simulations of the columnar dendritic solidification of a Pb-20 wt% Sn alloy in a square cavity cooled from one side and fed by a rectangular riser are reported. Overall macrosegregation patterns predicted using Scheil and lever-rule type microsegregation models are found to be similar, although the predicted eutectic fraction is significantly higher with the Scheil-type model. The choice of mushy zone permeability function significantly affects the predicted number, length and orientation of segregated channels. The inclusion of shrinkage-driven flow leads to the prediction of the well-known inverse macrosegregation pattern. However, macrosegregation caused by thermosolutal convection readily masks the inverse segregation. The microsegregation models predict different solid concentrations and eutectic fractions, leading to different solid density distributions which, in turn, cause differences in the extent of contraction-driven flow.     

The study on polymer melt front,gas front and solid layer in filling stage of gas-assisted injection molding

The algorithms are developed to predict the polymer melt front, gas front and solid layer in gas-assisted injection molding. The simulation of two-dimensional, transient, non-isothermal and high viscous flow between two parallel plates with the generalized Newtonian fluid is presented in detail. During solidification while an injection mold fills, a solid-liquid interface moves and a two-phase zone exists; an enthalpy model is used to predict this interface in the two-phase flow problem. The model takes into account the three-phase flow including the effects of the gas front, solid layer and polymer melt front.