MODEL OF THE EFFECT OF ALLOY CONTENT ON SHELL STRENGTH DURING SOLIDIFICATION OF BINARY ALLOYS

A thermomechanical model of unidirectional solidification of binary alloy systems is presented. The goal of the model is to begin to explore the effect of alloy content on the mechanical behavior of the solidifying shell by first examining the effect on lateral strength. The shell solidifies onto a semi-infinite mold proceeding behind a mushy zone that grows into an initially quiescent fluid. Deformation of the shell is modeled with a thermohypoelasioviscous constitutive law that allows for examination of the idealized case of elastic deformation of the casting as well as the case where strain rate relaxation due to viscous creep predominates. Any effects of alloy content on the coefficients in the constitutive model are ignored so that the calculated effects on strength arise entirely from the size of the mushy zone. Aluminum-magnesium alloys solidifying onto a copper mold are considered as specific examples using a linearized portion of the Al-Mg phase diagram. The material with the smallest alloy content exhibits the greatest shell strength for the same cooling histories. That material with the widest freezing range has the lowest strength. For the elastic model, the average strength always increases with time, whereas for the elastoviscous case it can decrease with time to the point where the alloy content has virtually no effect on strength.     

A THERMOMECHANICAL MODEL OF SOLIDIFICATION ON A MOLD MOVING WITH CONSTANT VELOCITY

A thermomechanical model of pure metal solidification on a moving mold plate is considered. The goal of the model is to obtain a formula for the contact pressure at the shell/mold interface as the mold moves into the molten liquid. From the contact pressure it is possible to infer the effects of the mold velocity and the mold microgeometry on the time and location of gap nucleation which results from irregular distortion of the shell as it grows from the melt. The mold, which moves at a constant velocity into the molten liquid, has a sinusoidal surface with a low aspect ratio: this means that its wavelength greatly exceeds its amplitude. The mold is of infinite area and is assumed to be perfectly conducting and thermomechanically rigid. We therefore neglect the complexities associated with the physics of edge constraints and/or free boundaries of the solidifying shell and the interacting distortions between deformable mold and shell materials along their interface. The ratio of the velocity of the solid/liquid interface to the mold velocity is identified as another dimensionless parameter in the analysis. In order to arrive at an analytical solution for the contact pressure along the shell/mold interface, we assume that this parameter is small. This makes the velocity ratio a convenient perturbation parameter for the analysis of thermomechanical distortion of the thin shell material as it grows from the melt. This necessarily limits the analysis to situations where the mold moves at faster rather than slower speeds. It is assumed that there is zero tangential shear stress between the fluid and the solidifying shell. As the molten liquid flows over the mold, it perfectly wets the surface. This precludes wetting effects due to surface tension. A hypoelastic constitutive law, which is a rate formulation of thermoelasticity, is assumed to govern deformation of the shell as it grows from the molten liquid. Latent heat liberated at the freezing front is extracted across a constant contact resistance at the shell/mold interface. Peculiar fluid motion at the tip is neglected. A solution for the contact pressure that is valid near the liquid surface (i.e., the meniscus) is derived from the main theoretical developments. Beyond the time of gap nucleation at the shell/mold interface, the model is no longer valid since it cannot account for gross distortion of the shell (i.e., distortions that greatly exceed the spatial perturbations considered in the model).     

Sharp interface numerical simulation of directional solidification of binary alloy in the presence of a ceramic particle

A sharp interface technique is employed to study the interaction of a solid–liquid interface in a solidifying binary alloy with a ceramic particle in the melt. The application targeted is solidification of a metal–matrix composite. A level-set based sharp interface numerical method is used to study the directional solidification process in the presence of the particle. The transport of solute and heat are computed. The directional solidification calculations are first validated against stability theory. The Mullins–Sekerka stability spectrum is reproduced with good agreement with the theory. The interaction of the cellular interface with a ceramic particle in the melt is then computed. It is shown that, in contrast to the case of a pure material, the ratio of thermal conductivity of the particle to the melt plays no role in determining the front morphology and the result of the particle–front interaction. The diffusion of species controls the evolution of the phase front around the particle. The implications of the results for particle–front interactions in a binary alloy are discussed.     

GROWTH INSTABILITY DURING PLANAR SOLIDIFICATION WITH A MOLD OF FINITE THERMAL CAPACITY

The temperature and the stress fields in the solidified layer and in the mold of finite thickness for a unidirectional casting process are investigated. Earlier solutions are extended to include the effect of the thermal capacity of the mold on the freezing front growth instability. A numerical solution is obtained for both the heat conduction and the residual stress problem. The results show that the perturbation in contact pressure tends asymptotically to a maximum value at larger times for the lower values of the thermal capacities of the mold materials. The magnitude of the contact pressure perturbation is decreased by the inclusion of the thermal capacity of the mold material, and this effect is enhanced for less distortive and thicker molds. The present article assumes that the thermal and mechanical problems are uncoupled along the casting mold interface. Despite this limitation, the results presented in this article indicate that a mold with a higher thermal capacity (or lower thermal diffusivity) might be less susceptible to thermoelastic instabilities associated with the contact pressure and its dependence on the thermal contact resistance at the casting mold interface.     

SOLIDIFICATION OF A PURE METAL WITH FINITE THERMAL CAPACITANCE ON A SINUSOIDAL MOLD SURFACE

Previous models of mold microgeometry-induced gap nucleation during pure metal solidification neglected the thermal capacitance of the solidifying shell: this is equivalent to the assumption that the shell has a small Stefan number. Although this assumption leads to steady heat conduction in the shell, and hence simplifies the solution for the thermal field, the corresponding assumption of a small Stefan number material is generally not appropriate for metals. In the present work, we remove the small Stefan number restriction used in a previous model for solidification of a pure metal on a rigid, perfectly conducting mold. The mold has a sinusoidal surface microgeometry for which the ratio of the amplitude to the wavelength is much less than one. This makes the aspect ratio a convenient perturbation parameter. Molten metal initially at its fusion temperature is assumed to wet the mold surface perfectly, which is held a constant temperature below the fusion temperature. The temperature field in the growing metal shell is numerically evaluated, and the instantaneous temperature field is incorporated into an analytical solution for the stress field in the shell. The evolving thermomechanical distortion of the shell is modeled assuming that the shell material follows a thermohypoelastic constitutive law that is a rate formulation of thermoelasticity. The contact pressure profile at the shell/asperity interface, which is indicative of shell distortion due to the asperity geometry, is obtained from the stress field. The effects of the mold wavelength and shell thermal capacitance on the contact pressure, temporal and spatial evolution of gap nucleation at the shell/mold interface, and mean shell thickness are examined for pure aluminum and iron shells.     

Solidification of a supercooled liquid in stagnation-point flow

The solidification of a thermally supercooled liquid in stagnation-point flow is investigated. Due to the advancing solidifying front, both the temperature and flow fields are time dependent. A numerical solution to the problem using an interface tracking method is compared to analytical solutions obtained for instantaneous similarity (short time solution) and quasi-steady state (long time solution). The results show that the velocity of the solid-liquid interface eventually reaches a constant value and that the magnitude of the interface velocity increases with greater thermal supercooling. The solution to this problem provides insight into more complicated solidification problems relating to crystal growth.     

Linear stability analysis of the solidification of a supercooled liquid in a half-space

The instability of the solidifying front of a supercooled liquid in a half-space is investigated by introducing a small disturbance at the solid-liquid interface. A relationship between the thermal properties of the material and the disturbance growth rate is obtained using the heat balance equation at the interface, including the effects of surface curvature on the equilibrium temperature. The heat balance equation is solved numerically and compared to the analytical solution obtained by neglecting the effects of surface curvature. The results show that the thermal gradients increase the growth rates of disturbances at the solid-liquid interface and that the effect of surface curvature results in a decrease in the disturbance growth rates. Further analysis shows that marginal stability occurs in both the longer wavelength and capillary regions.     

Thermal effect of surface tension on the inward solidification of spheres

Effect of surface tension (Gibbs-Thomson effect) on the inward solidification of a liquid in a spherical container is investigated analytically by solving the unsteady heat equation via a small-time series expansion technique. A nonlinear Shanks transformation is adopted to improve the convergence property of the series solution at large time. The results show that at fixed Stefan number, the effect of surface tension is to increase the growth rate of the freezing front. A local minimum in the freezing rate is found to develop for all surface tension parameter values considered in this study. Also, analytic expressions for the relations between the growth rate of the freezing front, Stefan number and surface tension parameter are derived under the asymptotic condition of small Stefan number.     

DESIGN OF TWO-DIMENSIONAL STEFAN PROCESSES WITH DESIRED FREEZING FRONT MOTIONS

This paper presents a methodology for the design of two-dimensional solidification processes, It is aimed particularly at the calculation of the boundary flux or temperature for a body that solidifies with a desired motion of the freezing front. Such problems are of particular technological significance, considering that the freeing front motion is directly related to the quality of casting. Solution of these problems can be used to achieve a desired east structure, to optimize the time of casting, to prevent choking of the liquid flow, or to control liquid feeding to the contracting front so that cavities and partial solidification in complex castings are avoided. Spatial smoothing and a modification of Beck's future information method are used to solve this ill-posed design problem. A moving and deforming finite-element formulation is employed. The accuracy of the method is demonstrated through several two-dimensional numerical examples.     

Spatial variation of heat flux at the metal–mold interface due to mold filling effects in gravity die-casting

Most of the research work pertaining to metal–mold heat transfer in casting solidification either assumes no spatial variation in the air gap formation or limits the study to only those experimental systems in which air gap formation is uniform. However, in gravity die-casting, filling effects induce variation in thermal field in the mold and casting regions. In this paper, we show that this thermal field variation greatly influences the time of air gap initiation along a vertical mold wall, which subsequently leads to the spatial variation of air gap and in turn, the heat flux at the metal–mold interface.In order to study the spatial variation of heat flux at the metal–mold interface, an experimental setup that involved mold filling was devised. A Serial-IHCP (inverse heat conduction problem) algorithm was used to estimate the multiple heat flux transients along the metal–mold interface. The analysis indicates that the fluxes at different mold segments (bottom, middle, and top) of the metal–mold interface reaches the peak value at different time steps, which shows that the initiation of air gap differs along the mold wall. The experimental and numerical results show that the heat transfer in the mold is two-dimensional during the entire period of phase change, which is initially caused by the filling effects and further enhanced by the spatial variation of the air gap at the metal–mold interface.     

Pressure corrections for the potential flow analysis of Kelvin–Helmholtz instability with heat and mass transfer

Pressure corrections for the viscous potential flow analysis of Kelvin–Helmholtz instability at the interface of two viscous fluids have been carried out when there is heat and mass transfer across the interface. Both fluids are taken as incompressible and viscous with different kinematic viscosities. In viscous potential flow theory, viscosity enters through normal stress balance and effect of shearing stresses is completely neglected. We include the viscous pressure in the normal stress balance along with irrotational pressure and it is assumed that this viscous pressure will resolve the discontinuity of the tangential stresses at the interface for two fluids. It has been observed that heat and mass transfer has destabilizing effect on the stability of the system. A comparison between viscous potential flow (VPF) solution and viscous contribution to the pressure for potential flow (VCVPF) solution has been made and it is found that the effect of irrotational shearing stresses stabilizes the system.     

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.     

INTEGRATED FINITE ELEMENT MODEL FOR TRANSIENT FLUID FLOW AND THERMAL STRESSES DURING CONTINUOUS CASTING

A finite element computational methodology is presented for predicting the temperature distribution, fluid flow, and thermal stresses evolving in a solidifying ingot, which itself is growing in length, during the start-up phase of a continuous casting process, with a particular reference to aluminum casting. The approach is based on the coupling of a thermal and flow model with a stress model, The thermal flow model is developed using a deforming finite element method with an Eulerian-Lagrangian transformation to account for the fact that the ingot itself is also growing at a prescribed casting speed. The stress model is developed also by the finite element method, with mechanical deformations in the solidifying materials described by a hypoelastic-viscoplastic constitutive relation. The integrated model has been applied to study the dynamic development of temperature, flow, and stresses in the solidifying ingot during the start-up phase for continuous casting of aluminum. The results show that the fluid flow and temperature distribution experience a rapid change at the initial stage but that the change slows down later in the process as it approaches to the steady state. Computed results compare reasonably well with experimental measurements for temperature distributions in the ingot. It is found that the thermal stresses in general evolve from small to big in magnitude and from compressive to tensile in the solidifying ingot. The hoop stress is larger than other stress components, in particular in the outer surface region. The air gap formed between the ingot and the bottom block increases initially and decreases afterward as a result of stress relaxation.     

A comparison of hyperbolic and parabolic models of phase change of a pure metal

The solidification history of individual thermal spray particles has been the subject of many experimental and theoretical studies. Yet it is customary to assume that solidification occurs at the equilibrium temperature, and that heat propagates according to Fourier’s Law. To account for a finite thermal diffusion speed, a hyperbolic heat conduction equation is usually adopted to analyze heat transfer. However, under certain circumstances, this equation can violate the second law of thermodynamics, and so others have modified the original hyperbolic equation via theories of extended irreversible thermodynamics. In this work, we study non-equilibrium effects of rapid solidification of a pure metal particle, and compare the so-called parabolic, hyperbolic and modified hyperbolic equations for heat transfer, to predict the interface undercooling due to thermal effects and velocity as a function of time, for different relaxation times. Results indicate that differences are limited to the early part of the solidification process, when undercooling is most significant, the interface velocity is highest, and non-equilibrium effects are most evident. As solidification progresses, the non-equilibrium effects wane and solidification can then be properly modeled as an equilibrium process.     

Inverse alloy solidification problem including the material shrinkage phenomenon solved by using the bee algorithm

In the paper we solve the one-phase inverse problem of alloy solidifying within the casting mould, including the shrinkage of metal which results from the difference between densities of the liquid and solid phases. The process is modeled by means of the solidification in the temperature interval basing on the heat conduction equation with the source element enclosed, whereas the shrinkage of metal is modeled by the proper application of the mass balance equation. The investigated inverse problem consists in reconstruction of the heat transfer coefficient on the boundary of the casting mould on the basis of measurements of temperature read from the sensor placed in the middle of the mould. Functional expressing the error of approximate solution is minimized with the aid of Artificial Bee Colony Optimization algorithm.     

Thermal Performance of a Phase Change Material‐Based Latent Heat Thermal Storage Unit

An experimental analysis is presented to establish the thermal performance of a latent heat thermal storage (LHTS) unit. Paraffin is used as the phase change material (PCM) on the shell side of the shell and tube‐type LHTS unit while water is used as the heat transfer fluid (HTF) flowing through the inner tube. The fluid inlet temperature and the mass flow rate of HTF are varied and the temperature distribution of paraffin in the shell side is measured along the radial and axial direction during melting and solidification process. The total melting time is established for different mass flow rates and fluid inlet temperature of HTF. The motion of the solid–liquid interface of the PCM with time along axial and radial direction of the test unit is critically evaluated. The experimental results indicate that the melting front moves from top to bottom along the axial direction while the solidification front moves only in the radial direction. The total melting time of PCM increases as the mass flow rate and inlet temperature of HTF decreases. A correlation is proposed for the dimensionless melting time in terms of Reynolds number and Stefan number of HTF. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21120     

Effect of Initial Concentrations on Solidification of Ammonium Chloride Water Solution

The effect of initial concentrations on solidification of ammonium chloride water solution is numerically investi-gated in detail.The solidifying process,with the cold wall temperature lower than the eutectic temperature,isassumed to be one-dimensional,and controlled by heat conduction only.The simulation reveals that:(1)Thesolid-mush interface grows in a linear manner,while the growth rate of the mush-liquid interface decreases ina parabolic manner,with increasing initial concentrations.(2)The temperature field in the whole region hasparabolic characteristics,but it shows a linear feature in the solid zone and mushy zone.(3)The concentrationalways has linear characteristics in the mush.(4)The solid fraction distribution is strongly affected by the ini-tial concentration.The solidification process shows quite different features,especially at small and high initialconcentrations.     

Analysis of Infiltration,Solidification, and Remelting of a Pure Metal in Subcooled Porous Preform

The parts fabricated by selective laser sintering of metal powders are usually not fully densified and have porous structure. Fully densified parts can be obtained by infiltrating liquid metal into the porous structure and solidifying the liquid metal. When the liquid metal is infiltrated into the subcooled porous structure, the liquid metal can be partially solidified. Remelting of the partially solidified metal can also take place and a second moving interface can be present. Infiltration, solidification, and remelting of metal in a subcooled porous preform obtained by laser sintering of metal powders are analytically investigated in this article. The governing equations are nondimensionalized and the problem is described using six dimensionless parameters. The temperature distributions in the remelting and uninfiltrated regions were obtained by an exact solution and an integral approximate solution, respectively. The effects of porosity, Stefan number, subcooling parameter, and dimensionless infiltration pressure are investigated.     

A predictive model for the evolution of the thermal conductance at the casting–die interfaces in high pressure die casting

An analytical model is proposed to predict the time varying thermal conductance at the casting–die interface during solidification of light alloys during High pressure Die Casting. Details of the topography of the interface between the casting and the die are included in the model through the inclusion of solid surface roughness parameters and the mean trapped air layer at the interface. The transitory phase of the interfacial thermal conductance has been related to the degradation of contact as solidification progresses through the casting thickness. The modelled time varying thermal conductance showed very good agreement with experimentally determined values for different alloy compositions and casting geometries. The analysis shows that the parameters that govern the thermal conductance are different for the first stage of contact compared to the second stage of contact when the alloy begins to solidify.     

Slow solidification of a cylinder with constant heat efflux

An asymptotic solution for the slow solidification of an infinite cylinder with constant surface heat flux is presented. The solidification of a 10 cm. diameter aluminum cylinder in 30 minutes corresponds to a value of 0.00901 for the small parameter. The predictions of the solidification front position based on the first three terms in the asymptotic expansion are presented. The extension to steady-state continuous casting is discussed.