Modelling convection in solidification processes using stabilized finite element techniques

Solidification of dendritic alloys is modelled using stabilized finite element techniques to study convection and macrosegregation driven by buoyancy and shrinkage. The adopted governing macroscopic conservation equations of momentum, energy and species transport are derived from their microscopic counterparts using the volume‐averaging method. A single domain model is considered with a fixed numerical grid and without boundary conditions applied explicitly on the freezing front. The mushy zone is modelled here as a porous medium with either an isotropic or an anisotropic permeability. The stabilized finite‐element scheme, previously developed by authors for modelling flows with phase change, is extended here to include effects of shrinkage, density changes and anisotropic permeability during solidification. The fluid flow scheme developed includes streamline‐upwind/Petrov–Galerkin (SUPG), pressure stabilizing/Petrov–Galerkin, Darcy stabilizing/Petrov–Galerkin and other stabilizing terms arising from changes in density in the mushy zone. For the energy and species equations a classical SUPG‐based finite element method is employed with minor modifications. The developed algorithms are first tested for a reference problem involving solidification of lead–tin alloy where the mushy zone is characterized by an isotropic permeability. Convergence studies are performed to validate the simulation results. Solidification of the same alloy in the absence of shrinkage is studied to observe differences in macrosegregation. Vertical solidification of a lead–tin alloy, where the mushy zone is characterized by an anisotropic permeability, is then simulated. The main aim here is to study convection and demonstrate formation of freckles and channels due to macrosegregation. The ability of stabilized finite element methods to model a wide variety of solidification problems with varying underlying phenomena in two and three dimensions is demonstrated through these examples. Copyright © 2005 John Wiley & Sons, Ltd.     

Analysis on fluid permeability of dendritic mushy zone during peritectic solidification in a temperature gradient

Compared with the growing applications of peritectic alloys,none research on the fluid permeability K of dendritic network during peritectic solidification has been reported before.The fluid permeability K of dendritic network in the mushy zone during directional solidification of Sn-Ni peritectic alloy was investigated in this study.Examination on the experimental results demonstrates that both the temperature gradient zone melting (TGZM) and Gibbs-Thomson (G-T) effects have obvious influences on the morphology of dendritic network during directional solidification.This is realized through different stages of liquid diffusion within dendritic mushy zone by these effects during directional solidification.The TGZM effect is demonstrated to play a more important role as compared with the G-T effect during directional solidification.Besides,it is shown that the evolution of dendrite network is more complex during peritectic solidification due to the involvement of the peritectic phase.Through the specific surface Sv,analytical expression based on the Carman-Kozeny model was proposed to analyze the fluid permeability of dendritic mushy zone in directionally solidified peritectic alloys.In addition,it is interesting to find a rise in permeability K after peritectic reaction in both theoretical predication and experimental results,which is different from that in other alloys.The theoretical predictions show that this rise in fluid permeability K after pedtectic reaction is caused by the remelting/resolidification process on dendritic structure by the TGZM and G-T effects during peritectic solidification.     

Pb-Sn合金侧向凝固过程A偏析的数值模拟

基于二元系凝固过程热溶质的传输行为,建立了描述A偏析形成及演化的数学模型,给出了固相分数与温度场及浓度场的耦合关系.先用已有的实验结果验证了模型的正确性,然后模拟计算了Pb-Sn合金侧向凝固过程A偏析的形成及演化过程,并研究了浮力数对A偏析形成位置及偏析程度的影响.结果表明,在糊状区中双扩散对流引起的密度变化,导致局部流动,形成偏析通道;为了维持偏析通道中局部液体的流动,枝晶间的液体通过糊状区从液相区得到补充.在相同的凝固条件下,浮力数越小,A偏析的形成时间愈迟,偏析的程度也越小.     

Macrosegregation and thermosolutal convection-induced freckle formation in dendritic mushy zone of directionally solidified Sn-Ni peritectic alloy

Compared with the growing applications of peritectic alloy,none research on the freckle formation during peritectic solidification has been reported before.Observation on the dendritic mushy zone of Sn-36 at.％Ni peritectic alloy during directional solidification at different growth velocities shows that the freckles are formed in two different regions:region Ⅰ before peritectic reaction and region Ⅱ after peritectic reaction.In addition,more freckles can be observed at lower growth velocities.Examination on the experimental results demonstrates that both the temperature gradient zone melting(TGZM)and Gibbs-Thomson(G-T)effects have obvious influences on the morphology of dendritic network during directional solidification.The current theories onKI Rayleigh number Ra characterizing the thermoso-lutal convection of dendritic mushy zone to predict freckle formation through the maximum of Ra can only explain the existence of region Ⅰ while the appearance of region Ⅱ after peritectic reaction cannot be predicted.Thus,a new Rayleigh number RaP is proposed in consideration of evolution of dendritic mushy zone by both effects and peritectic reaction.Theoretical prediction of RaP also shows a maximum after peritectic reaction in addition to that before peritectic reaction,thus,agreeing well with the freckle formation in region Ⅱ.In addition,more severe thermosolutal convection can be predicted by the new Rayleigh number RaP at lower growth velocities,which further demonstrates the reliability of RaP in describing the dependence of freckle formation on growth velocity.     

强磁场下Cu-50%Ag合金定向凝固过程中的组织及固液界面形貌演变

目的 研究强磁场下Cu-50%(质量分数)Ag合金定向凝固过程中的组织演变、固液界面形貌变化及溶质迁移行为,分析强磁场对金属凝固过程的作用机制,为强磁场下的金属材料制备提供理论借鉴和指导。方法 在不同的凝固速率与磁场条件下进行定向凝固和淬火实验,对合金的定向凝固组织、糊状区与固液界面形貌以及溶质分布行为进行考察。结果 强磁场破坏了凝固组织的定向生长,使凝固组织转变为枝晶与等轴晶共存的形貌;强磁场诱发了熔体对流,减少了糊状区中溶质的含量;强磁场改变了固液界面处的溶质分布和固液界面形貌,破坏了固液界面的稳定性。结论 强磁场通过洛伦兹力和热电磁力的共同作用,诱发了糊状区内液相的纵向环流,改变了固液界面及糊状区中的组织形貌与元素分布。     

An examination of effects of solidification parameters on permeability of a mushy zone in castings

A model describing the development of dendritic structure and the resulting gradient of flow resistance to interdendritic liquid is presented. The Hagen–Pousielle version of D’Arcy’s equation for flow through a porous structure is developed as a function of cooling rate and liquid volume fraction. Applied to finite elements in a unidirectionally cooled casting model, permeability gradient, feeding flow-rate required to prevent porosity, and mushy-zone liquid pressure drop at this flow rate are evaluated for the simple Fe–2Cr–0.5C and Al–5Cu castings exhibiting asymptotic and linear temperature profiles, respectively. The model shows permeability of the dendritic structure in the mushy zone dropping sharply, approaching the root of solidification front (solidus). Also shown is the effect of relative magnitude of primary and secondary arm spacing. If secondary dendrite arm spacing approaches primary arm spacing, the permeability for flow normal to primary dendrite arms approaches or even surpasses the permeability for flow parallel to primary dendrite arms.     

Effects of dendritic arm coarsening on macroscopic modelling of solidification of binary alloys

Abstract

A pure macroscopic two-dimensional numerical model has been developed, capable of capturing the effects of dendritic arm coarsening on the transport phenomena occurring during a binary alloy solidification process. The general continuum conservation equations are aptly modified to take into account shrinkage induced fluid flow. Simultaneously, the effective permeability of the mushy zone is numerically modelled according to the microscopic coarsening kinetics. Moreover, a new nodal latent heat updating algorithm is proposed that takes into account dendritic arm coarsening considerations. The numerical results are first tested against experimental results reported in the literature, corresponding to the solidification of an Al-Cu alloy in a bottom cooled cavity. It is concluded that dendritic arm coarsening leads to an increased effective permeability of the mushy region as well as an enhanced eutectic fraction of the solidified ingot. Consequently, an enhanced macrosegregation is predicted, compared with that dictated by shrinkage induced fluid flow alone. Physical insights are also developed regarding the effects of various parameters on the overall macrosegregation.     

Flow instabilities of liquid and mushy regions during alloy solidification and under high gravity environment induced by rotation

Linear flow instabilities of the liquid and mushy regions during directional solidification of a binary alloy are studied under a high gravity environment where the rotation axis is inclined with respect to the high gravity vector. Stability analysis and numerical computation are carried out to determine the results for the stationary disturbances at several values of the rotation rates and for given values of the other parameters. The results provide information about the effects of Coriolis force on various flow features in the liquid and mushy layers. The preferred structure of the mush-liquid interface is found to be that of longitudinal rolls. The main mode of convection is found to be able to generate double-cell structure in the vertical direction and is strengthened in the mushy layer by the Coriolis force. The Coriolis force appears to be generally stabilizing in the sense that the motion in the liquid zone is significantly weakened, the tendency for the chimney formation in the mushy zone is reduced and the critical values of the liquid and mush Rayleigh numbers and the wave number increase with increasing the rotation rate.     

Effect of Solid/Liquid Interface Morphology on Microstructure and Segregation of DSX40M Superalloy

The effect of solid/liquid intedece morphologies on the microstructure and segregation of a new type superalloy, DSX40M, was studied. It has been found that the primary arm spacing presents maximum value as the solid/liquid interface shape transforms from cellular to cellular-dendritic.As the alloy solidifies with a coarse dendritic interface, the solute segregation degree and the average size of the carbide reach the maximum values because of the widest mushy zone. A Zr-rich phase forms at this range. Within the solidificatin rate range of dendritic interface. the primary dendritic arm spacing and solute segregation decrease with the increasing of solidification rate and the Zr-rich phase disappears. It should be indicated that the change of the solid/liquid interface does nt vary the carbide type, but greatly affects the average size of the carbides.The quantitative results of the carbide size change in this alloy system with different solid/liquid interfaces is presented     

Transport Properties and Transport Phenomena in Casting Nickel Superalloys

Nickel superalloys that are used in the high-temperature regions of gas-turbine engines are cast by directional solidification (DS). In the DS processes, the castings are cooled from below, and three zones exist during solidification: (1) an all-solid zone at the bottom, (2) a “mushy zone” that is comprised of solid and liquid material, and (3) an overlying all-liquid zone. Computer simulations can be useful in predicting the complex transport phenomena that occur during solidification, but realistic simulations require accurate values of the transport properties. In addition to transport properties, the thermodynamic equilibria between the solid and liquid during solidification must also be known with reasonable accuracy. The importance of using reasonably accurate estimations of the transport properties is illustrated by two-dimensional simulations of the convection during solidification and the coincidental macrosegregation in the DS castings of multicomponent Ni-base alloys. In these simulations, we examine the sensitivity of the calculated results to measured partition ratios, thermal expansion coefficients, and viscosities that are estimated by regression analyses and correlations of existing property data.     

Effects of Re on surface eutectic formation for Ni-base single crystal superalloys during directional solidification

Effects of Re on the formation of surface eutectics have been investigated by using Ni-base single crystal superalloys with different Re additions. It was found that Re promotes the segregation of Al and Ta to the eutectic melt, leading to an increase of the surface and internal eutectics. In addition, the addition of Re also increased the freezing range, the local solidification time, and the permeability of the dendritic network within the mushy zone. These factors ultimately promoted the outflow of the interdendritic residual liquid with the action of solidification shrinkage, and led to the formation of more surface eutectics. In contrast, the addition of Re had no obvious influence on the surface eutectic microstructures.     

On nonlinear evolution of convective flow in an active mushy layer

We consider the nonlinear effect of convective flow in a horizontal mushy layer during solidification. The mushy layer that we consider has a permeable mush–liquid interface and is treated as an active porous medium with variable permeability. The nonlinear partial differential equations involved in this system are conservation equations for flow momentum, mass, temperature, and concentration. Numerical solutions to the resulting weakly nonlinear equations are obtained using a fourth-order Runge–Kutta integration scheme via a shooting technique. An evolution equation of Landau type is derived in terms of linear and first-order solutions by introducing an adjoint operator. The Landau constant is calculated for both cases: constant permeability and variable permeability. The analysis reveals that there is a slow transition of the flow to a steady state with smaller amplitude for an active mushy layer.     

Modified enthalpy method for the simulation of melting and solidification

Enthalpy method is commonly used in the simulation of melting and solidification owing to its ease of implementation. It however has a few shortcomings. When it is used to simulate melting/solidification on a coarse grid, the temperature time history of a point close to the interface shows waviness. While simulating melting with natural convection, in order to impose no-slip and impermeability boundary conditions, momentum sink terms are used with some arbitrary constants called mushy zone constants. The values of these are very large and have no physical basis. Further, the chosen values affect the predictions and hence have to be tuned for satisfactory comparison with experimental data. To overcome these deficiencies, a new cell splitting method under the framework of the enthalpy method has been proposed. This method does not produce waviness nor requires mushy zone constants for simulating melting with natural convection. The method is then demonstrated for a simple one-dimensional melting problem and the results are compared with analytical solutions. The method is then demonstrated to work in two-dimensions and comparisons are shown with analytical solutions for problems with planar and curvilinear interfaces. To further benchmark the present method, simulations are performed for melting in a rectangular cavity with natural convection in the liquid melt. The solid–liquid interface obtained is compared satisfactorily with the experimental results available in literature.     

糊状区渗氮对Cr10Mn9Ni0.7合金氮含量及凝固相变过程的影响

研究了糊状区保温对Cr10Mn9Ni0.7合金凝固过程和氮含量及相变过程的影响。结果表明,随着糊状区保温时间的延长,铸锭中的氮含量逐渐升高,同时铸锭中的气孔率逐渐降低。当氮气压力为0.1 MPa时,氮含量由0.17%升高到0.23%,而气孔率则从1.86%降至1.37%;当氮气压力为0.4 MPa时,氮含量由0.29%升高到0.37%,而气孔率从1.41%降至1.06%。糊状区保温的增氮机制可归结为:在糊状区保温会促进包晶反应进程,使更多的铁素体转变为奥氏体;同时糊状区保温能够提高残留液相中的氮含量,进而提高"通道状"奥氏体中的氮含量。糊状区保温能够消除铁素体阱,从而降低气孔率。     

Feeding distance of tin bronze castings: intrinsic and extrinsic estimates

Casting alloys with long freezing ranges are extremely difficult to cast because of the length of the mushy zone. During solidification, the liquid metal needs to find its way through the dendritic structure. Tin bronzes have a long freezing range and therefore, are known to be hard to cast because of their propensity for interdendritic porosity. In this study, the intrinsic feeding ability of the tin bronze is calculated based on the intrinsic fracture pressure of liquid metals. Calculations have shown that feeding distances increase with longer solidification times and that the metal is capable of feeding itself over long distances. The differences between calculated and observed feeding distances can be attributed to entrainment defects in the solidifying metal.     

Effect of rotation on the structure of a convecting mushy layer

Summary Nonlinear natural convection in a mushy layer during solidification of a binary alloy is investigated under a high gravity environment where the rotation axis is inclined to the high gravity vector. Asymptotic and scaling analyses are applied to convective flow within the mushy layer and in vertical chimneys. The main result is that, for some particular moderate rotation range, vertical velocity in the chimneys decreases rapidly with increasing rotation rate and appears to have opposite signs across some rotation dependent vertical level.     

Positive segregation as a function of buoyancy force during steel ingot solidification

Abstract

We analyze theoretically and experimentally solute redistribution in the dendritic solidification process and positive segregation during solidification of steel ingots. Positive segregation is mainly caused by liquid flow in the mushy zone. Changes in the liquid steel velocity are caused by the temperature gradient and by the increase in the solid fraction during solidification. The effects of buoyancy and of the change in the solid fraction on segregation intensity are analyzed. The relationships between the density change, liquid fraction and the steel composition are considered. Such elements as W, Ni, Mo and Cr decrease the effect of the density variations, i.e. they show smaller tendency to segregate. Based on the modeling and experimental results, coefficients are provided controlling the effects of chemical composition, secondary dendrite arm spacing and the solid fraction.     

Dendritic Microstructure Affecting Mechanical Properties and Corrosion Resistance of an Al-9 wt% Si Alloy

It has been reported that the mechanical properties and corrosion resistance of metallic alloys depend strongly on the solidification microstructural arrangement. The correlation of corrosion behavior and mechanical properties with microstructure parameters can be very useful for planning solidification conditions in order to achieve desired final properties. The aim of the present work is to investigate the influence of dendritic microstructural parameters of an Al-9 wt.% Si alloy on mechanical properties and corrosion resistance. The experimental results establish correlations between secondary dendrite arm spacings (λ₂) and ultimate tensile strength (σ_u), yield strength (σ_y), corrosion potential (E_Corr), and corrosion rate (i_Corr).     

Simultaneous control of solidus and liquidus lines in alloy solidification

A methodology based on inverse analysis techniques is presented for simultaneous control of solidus and liquidus lines, and hence the mushy zone thickness, in the directional solidification of alloy materials. In order to minimize the risk of freckle formation in the casting process, specific distributions of heat fluxes on the boundaries of the model are predicted that result in a narrow mushy zone. To this end, the heat fluxes should be so evaluated that the positions of the solidus and liquidus lines can be controlled simultaneously. The numerical formulation of the solidification process is carried out by a previously established truly meshless technique. The accuracy and effectiveness of the proposed method are verified through some 2D numerical examples. The numerical examples assess the great potential of the proposed method in confining the mushy zone to a narrow part of the domain.     

Metallography of a pulsed Nd: YAG laser weld in a RS/PM Al---8Fe---2Mo alloy

The microstructure of a pulsed Nd:YAG laser weld in a rapid solidification/powder metallurgy (RS/PM) Al-8.0 wt.% Fe-2.3 wt.% Mo (Al---8Fe---2Mo) alloy was investigated using light microscopy, scanning electron microscopy (SEM) and transmission electron microscopy (TEM) techniques. This analysis revealed significant microstructure variations across the weld fusion zone (FZ). Near the fusion boundary, a light-etching FZ microstructure was observed to contain submicron dispersoids “entrapped” in a matrix of fine-sized dendritic alpha aluminum. At the center of the FZ, the presence of relatively coarse-sized intermetallic particles that served as growth centers for coarser dendritic alpha aluminum promoted a dark-etching microstructure. In the boundary between successive melt zones, both a heat-affected zone (HAZ) containing acicular dispersoids and a fusion boundary region (FBR) containing irregular-shaped particles in a coarse-grained dendritic alpha aluminum matrix were observed. A HAZ comprised of slightly coarsened dispersoids was also observed in the base metal bounding the FZ. Knoop hardness traverses across the FZ indicated a maximum hardness of KHN 260 in the light-etching FZ and a minimum hardness of KHN 135 in the HAZ between successive melt zones.