Numerical simulation of heat,mass and momentum transport behaviours in directionally solidifying alloy castings under electromagnetic fields using an extended Direct‐SIMPLE scheme

A Direct‐SIMPLE scheme is further extended for numerical simulations of heat and species mass transfer, and liquid flow behaviours in solidification processes of shaped alloy castings under various electromagnetic (EM) fields, based on a binary continuum model (a solidification transport phenomena/processes (STP)‐based dendrite solidification model). Unlike in a SIMPLE scheme, no iterative computations are needed to achieve the final pressure and velocity corrections in the extended Direct‐SIMPLE procedure, therefore extremely high computational efforts can be avoided. Using three different types of model alloys, pseudo‐binary In718 base‐4.85 wt% Nb, γ(TiAl)‐55 at% Al and Al‐4.5 wt% Cu systems, sample computations for solving strongly coupled solidification transport phenomena in directionally solidifying shaped castings under static and harmonic EM‐fields of different strengths are carried out, to demonstrate the feasibility and efficient calculation performance of the present model and numerical methods. Copyright © 2004 John Wiley & Sons, Ltd.     

On the interest of microgravity experimentation for studying convective effects during the directional solidification of metal alloys

Under terrestrial conditions, solidification processes are often affected by gravity effects, which can significantly influence the final characteristics of the grown solid. The low-gravity environment of space offers a unique and efficient way to eliminate these effects, providing valuable benchmark data for the validation of models and numerical simulations. Moreover, a comparative study of solidification experiments on earth and in low-gravity conditions can significantly enlighten gravity effects. The aim of this paper is to give a survey of solidification experiments conducted in low-gravity environment on metal alloys, with advanced post-mortem analysis and eventually by in situ and real-time characterization.     

Eutectic solidification patterns: Interest of microgravity environment

The solidification of binary eutectic alloys produces two-phase composite materials in which the microstructure, that is, the geometrical distribution of the two solid phases, results from complex pattern-formation processes at the moving solid–liquid interface. Since the volume fraction of the two solids depends on the local composition, solidification dynamics can be strongly influenced by thermosolutal convection in the liquid. In this contribution, we review our experimental and numerical work devoted to the understanding of eutectic solidification under purely diffusive conditions, which will soon be tested and extended during the microgravity experiment TRANSPARENT ALLOYS planned by the European Space Agency (ESA).     

Use of experiment and an inverse method to study interface heat transfer during solidification in the investment casting process

A technique to determine the thermal boundary conditions existing during the solidification of metallic alloys in the investment casting process is presented. Quantitative information about these conditions is needed so that numerical models of heat transfer in this process produce accurate results. In particular, the variation of the boundary conditions both spatially and temporally must be known. The method used involves the application of a new inverse heat conduction method to thermal data recorded during laboratory experiments of aluminium alloy solidification in investment casting shell moulds. The resultant heat transfer coefficient for the alloy/mould interface is calculated. An experimental programme to determine requisite mould thermal properties was also undertaken. It was observed that there is significant variation of the alloy/mould heat transfer coefficient during solidification. It is found to be highly dependent on the alloy type and on the vertical position below the initial free surface of the liquid metal. The aluminium casting alloys used in this study were 413, A356, 319 (Aluminum Association designations), and commercially pure aluminium. These alloys have significantly different freezing ranges. In particular, it was found that alloys with a high freezing range solidify with rates of heat transfer to the mould which are very sensitive to metallostatic head.     

The effect of solutal undercooling on double‐diffusive convection and macrosegregation during binary alloy solidification: a numerical investigation

In this paper, we present a macroscopic numerical model that is capable of capturing the interaction between the double‐diffusive convective field and a localized fluid flow on account of solutal undercooling during non‐equilibrium solidification of binary alloys. The model is essentially based on a fixed‐grid enthalpy based control volume approach. In the present model, microscopic features pertaining to non‐equilibrium effects on account of solutal undercooling are incorporated through the formulation of a modified partition‐coefficient. The effective partition‐coefficient is numerically modelled by means of a number of macroscopically observable parameters related to the solidifying domain. This feature has made the present treatment different from micro‐macro modelling of alloy solidification, which involves certain parameters that may not be macroscopically resolvable. Numerical simulations are performed for the case of two‐dimensional transient solidification of Pb–Sn alloys (both hypoeutectic and hypereutectic) in a rectangular cavity, employing the present model. The simulation results are also compared with the corresponding experimental results quoted in the literature, and the agreement is excellent. From the results, it can be concluded that non‐equilibrium effects on account of solutal undercooling result in a more enhanced macrosegregation. Copyright © 2002 John Wiley & Sons, Ltd.     

Comparison between Lever and Scheil Rules for Modeling of Microporosity Formation during Solidification

Segregation and microporosity formation are two important physicalphenomena that occur during solidification of binary alloys. The aim ofthis study is to investigate the effect of the model of solute diffusionat the local scale (which means at the local scale of the microscopicrepresentative elemental average volume (REV)) on solute transport andthe microporosity formation during this process. The Scheil rule and thelever rule are used to describe the solute diffusion at the local scale.Results indicate that solute diffusion at the local scale is animportant factor in microporosity formation. Also, microporosityformation slightly reduces inverse segregation because it partiallycompensates for shrinkage. The increase of the external pressure at thefree surface or the decrease of the initial hydrogen concentration inthe molten alloy can be effectively utilized to control microporosityformation.     

Effect of “Stefan” currents of a solidifying melt on the process of thermal convection

The calculated results, represented by graphs, show that in the initial period of solidification the motion of a melt is fully determined by the shrinkage at the front of crystallization. The effect develops more strongly at lower Grashof numbers and higher Stefan numbers. As the rate of solidification and the temperature gradient decrease the process of natural thermal convection develops in the liquid phase. The calculated results are compared with experiment.     

Numerical Approaches for microscopic modelling of solute redistribution during solidification of binary alloys

In the present study, two numerical approaches for single-domain modelling of microsegregation during solidification of binary alloys are presented. In the first approach, the concentration jump at the moving solid/liquid interface is formulated using a volumetric term and a Boolean function. The governing solute redistribution equation, valid for the whole domain comprising the solid and liquid regions, is derived in terms of the liquid phase composition. The effects of microstructure coarsening on microsegregation has been described and included in the model. In the second approach, the continuum mixture theory is utilized to derive a single domain solute redistribution equation in terms of the mixture composition. The solidification front motion and dendrite arm coarsening effects are accommodated by considering the representative elementary volume to consist of solid, interdendritic, and extradendritic liquid phases. Numerical solutions have been obtained using a control-volume based finite-difference method with a fixed grid. Good agreement has been observed between the predictions of the present fixed-domain models and the exact analytical and experimental results.     

Weak Non-Linear Analysis of Moderate Stefan Number Stationary Convection in Rotating Mushy Layers

The effects of rotation on a mushy layer, during the solidification of binary alloys, is considered. A near-eutectic approximation and large far-field temperature are employed in order to decouple the mushy layer from the overlying liquid melt. The current study employs a new moderate time scale for mushy layers exhibiting Stefan numbers of unit order of magnitude. The weak non-linear theory is used to evaluate the leading order amplitude. The results of the weak non-linear theory are then used to establish the nature of the bifurcation, that is whether the bifurcation is forward or inverse.     

Influence of chemical composition on the hot tearing formation of cast steel

Hot tears initiate under stress and strain, when the solid crystals are partially separated by the liquid film. At this stage, the overall strength of the hot spot of the casting is very low. The tendency of alloys to hot tearing depends on the temperature range in which the cracks can initiate. It has been assumed, that process of the cracks formation starts at the temperature of grains interlocking. The change of the length and chemical composition of liquid film separating the grains during solidification have been studied, using the simulation model for the growth of equiaxed grains. Mechanical properties of samples of the different cast steels in the solidification range, have been tested to define the lowest temperature of the hot tearing. Thus, for each chemical composition of the tested steel the range of brittleness could be calculated. The fractured surfaces were examined using Kevex X-ray analyzer coupled with scanning microscope. It was found that the examined regions were, at least partly, covered by the liquid phase rich in sulfur.     

不等径颗粒间液桥力学参数及形态的试验研究

作为一种自然界中广泛存在的力, 液桥力的研究对制药、重金属回收、颗粒分离等领域具有十分重要的意义. 利用纳米多功能拉伸试验机测量不等径颗粒间液桥拉伸过程中的液桥力?位移曲线, 同时配合CCD工业相机记录拉伸全过程液桥形态的变化. 首先分析了液桥力?位移曲线形态、最大液桥力、断裂距离随粒径比及液桥体积的变化规律, 其次基于圆环假设及Y-L方程对本文试验结果的合理性进行验算, 最后针对圆环假设在液桥力计算中存在的不足分析了其原因, 并结合重力对液桥形态的影响对液桥拉伸全过程的形态变化进行了具体分析. 结果表明: 最大液桥力受粒径比的影响较大而受液桥体积的影响较小, 与最大液桥力相反, 断裂距离受液桥体积的影响较大而受粒径比的影响较小; 圆环假设可以较好地预测最大液桥力大小但对拉伸过程中的液桥力预测不准, 这是由于当液桥力达到最大值后液桥的外轮廓已不能用圆环表示; 根据重力对液桥形态的影响, 将拉伸过程液桥外轮廓的变化简化为重力影响可以忽略时的圆环形?抛物线形, 重力影响处于过渡阶段或影响较小时的长轴与短轴之比不断增大的椭圆形, 以及重力影响不可忽略时的“冷却塔形”?双曲线形.      

Numerical solution of solidification in a square prism using an algebraic grid generation technique

The solidification of an infinitely long square prism was analyzed numerically. A front fixing technique along with an algebraic grid generation scheme was used, where the finite difference form of the energy equation is solved for the temperature distribution in the solid phase and the solid–liquid interface energy balance is integrated for the new position of the moving solidification front. Results are given for the moving solidification boundary with a circular phase change interface. An algebraic grid generation scheme was developed for two-dimensional domains, which generates grid points separated by equal distances in the physical domain. The current scheme also allows the implementation of a finer grid structure at desired locations in the domain. The method is based on fitting a constant arc length mesh in the two computational directions in the physical domain. The resulting simultaneous, nonlinear algebraic equations for the grid locations are solved using the Newton-Raphson method for a system of equations. The approach is used in a two-dimensional solidification problem, in which the liquid phase is initially at the melting temperature, solved by using a front-fixing approach. The difference of the current study lies in the fact that front fixing is applied to problems, where the solid–liquid interface is curved such that the position of the interface, when expressed in terms of one of the coordinates is a double valued function. This requires a coordinate transformation in both coordinate directions to transform the complex physical solidification domain to a Cartesian, square computational domain. Due to the motion of the solid–liquid interface in time, the computational grid structure is regenerated at every time step.     

Petrov-Galerkin methods for natural convection in directional solidification of binary alloys

A Petrov-Galerkin finite element method is presented for calculation of the steady, axisymmetric thermosolutal convection and interface morphology in a model for vertical Bridgman crystal growth of nondilute binary alloys. The Petrov-Galerkin method is based on the formulation for biquadratic elements developed by Heinrich and Zienkiewicz and is introduced into the calculation of the velocity, temperature and concentration fields. The algebraic system is solved simultaneously for the field variables and interface shape by Newton's method. The results of the Petrov-Galerkin method are compared critically with those of Galerkin's method using the same finite element grids. Significant improvements in accuracy are found with the Petrov-Galerkin method only when the mesh is refined and when the formulation of the residual equations is modified to account for the mixed boundary conditions that arise at the solidification interface. Calculations for alloys with stable and unstable solute gradients show the occurrence of classical flow transitions and morphological instabilities in the solidification system.     

Using nucleators to control freckles in unidirectional solidification

Buoyancy-induced fluid flow, which is responsible for most forms of macro-segregation and channel-type segregates in castings, is not directly controllable. If left uncontrolled, natural convection will contribute to non-uniform distribution of alloy constituents and grain structure during solidification of a casting. Non-uniform distribution of chemical composition and physical structure in an alloy casting can significantly affect the reliability of mechanical components. Therefore, materials with acceptable defects can be produced only by trial-and-error and their acceptability is determined by costly inspections. We present a novel technique to control the formation of chimneys and resulting freckles in the mushy zone during the solidification of ammonium chloride that is cooled from below. This is done by placing metallic nucleators in particular arrangements on the bottom cooling plate. With this technique, freckles in a casting might be avoided and/or be forced to form where stresses are expected to be lower during use of the part.The objective of this study is to investigate the effects of the arrangement, spacing, and size of the nucleators on finger formation, plume structure, and the solidification process. Results showed that it is possible to obtain a relatively large area free of channel-segregates in a metal analog directionally solidified upward by placing nucleators in certain arrangements at the bottom of the casting. The outcomes of this study will serve as a baseline for subsequent investigations that will examine the solidification of binary alloys, and could be used to test and develop mathematical and numerical models.     

Computation of Liquid Hydrazine Depressurisation with a Mach-Uniform Staggered Scheme

A numerical method (pressure-correction method using a staggered grid) is coupled to a thermodynamic model for compressible liquid hydrazine. The method is applied to the venting of liquid hydrazine into space, during which the fluid undergoes a large pressure drop. Below the saturation pressure vaporisation occurs. This takes place near the outlet and induces variations of temperature, which may cause solidification and pipe clogging. In order to assess the risk of phase changes, numerical simulations of the venting line have been performed using a quasi one-dimensional approach. The numerical method can handle compressible flows of fluids with nonconvex equation of state at the low Mach numbers that occur during hydrazine venting. A numerical study of the liquid behaviour during strong depressurisation is performed. The method is validated using experimental data, and allows prediction of pressure evolution and vaporisation location along the pipe. This revised version was published online in July 2006 with corrections to the Cover Date.     

On Perturbation and Marginal Stability Analysis of Magneto-Convection in Active Mushy Layer

This present study considers the problem of steady magneto-convection in a horizontal mushy layer with variable permeability and an impermeable mush–liquid interface during directional solidification of binary alloys. We model the flow by introducing a uniform magnetic field in the mushy layer which is considered as a porous medium where Darcy’s law holds and the permeability is a function of the local solid volume fraction. Basic-state solutions are obtained analytically using the no-flow condition. With the help of multiple shooting techniques, we obtain numerical solutions to the linear perturbation system for non-magnetic and magnetic cases. Numerical results are presented showing the effects of the magnetic field and the permeability of the layer. These results demonstrate that the application of an external magnetic field has stabilizing effects on the convection and can reduce the tendency for chimney formation in the mushy layer. In addition, variable permeability, which corresponds to an active mushy layer, indicates more stable and realizable flow system as compared to the case of constant permeability.     

Theoretical and experimental study of the isothermal mechanical behaviour of alloys in the semi-solid state

An isothermal constitutive model for semi-solid alloys based on the concepts of mechanics of continuous media and the theory of mixtures is presented. The model is applicable to semi-solid states obtained either by solidification from liquid state or partial remelting from solid state in which each of the solid and the liquid phases is contiguous. During deformation their behaviours are coupled: the densification of the solid matrix considered as a porous viscoplastic medium saturated with a liquid drives the fluid flow behaviour, and the resulting pressure distribution in the liquid affects in turn the stresses and the densification of the solid. The identification procedure of the model uses two types of mechanical tests: uniaxial compression and drained die pressing (filtration) carried out with A356 alloy. The identification results are then validated using drained triaxial compression.     

Multiphase transfer model for intermittent microwave-convective drying of food: Considering shrinkage and pore evolution

Intermittent microwave convective (IMCD) drying is an advanced drying technology that improves both energy efficiency and food quality during drying. Although many experimental studies on IMCD have been conducted, there is no complete multiphase porous media model describing the physics of the process available in the literature. A multiphase porous media model considering liquid water, gases and the solid matrix of food during drying can provide in-depth understanding of IMCD process. Currently there is no IMCD model that have taken shrinkage and pore evolution during drying into consideration. In this study, first a multiphase porous media model with shrinkage (IMCD2) has been developed for IMCD. Then the model has been compared with IMCD model without shrinkage (IMCD1). Simulated temperature, moisture content, density, porosity from IMCD2 are then validated against experimental data. The profile of vapour pressures and evaporation during IMCD are also presented and discussed.     

Rheology and stircasting of (organic) alloys

Partially solidified metal alloys behave thixotropic during stircasting. Some transparent, low melting organic alloys such as NPA behave similarly. We studied the rheological behaviour of NPA during and after stirred nucleation. The shear stress reaches a maximum _max and subsequently decreases. This decrease is explained by the coarsening of dendrites into spherical particles. The maximum shear stress increases with increasing cooling rate and also with decreasing final temperature. The latter increase is more pronounced if the shear rate during solidification is lower. Both effects can be explained in terms of coarsening and particle sizes.