A study of the heat parameters during bidirectional solidification

The temperature field and heat parameters are important in controlling metal liquid crystallinity in unidirectional and bidirectional solidification. The temperature field can be divided into three cases: a liquid temperature field; solid temperature field; and a temperature field on the solid-liquid (S–L) interface. Heat parameters can be divided into two cases: technical heat parameters; and solidification heat parameters. The temperature field on the S–L interface and solidification heat parameters are the most important for the structures and properties of materials. The temperature field on the S–L interface is determined by the alloy system, and solidification heat parameters are related to the temperature ield of the environment and technical heat parameters. The temperature ield on the S–L interface is closely related to the solidiication heat parameters.

A theoretical model describing precisely the temperature field on the S–L interface during bidirectional solidification was proposed. A series of heat parameters, including temperature gradients G, solidification rate R, cooling velocity V and characteristic temperature T_c have been derived from this model. A superalloy has been chosen as the experimental object in order to verify the theoretical model. The theoretical calculations are found to be in agreement with the experimental results.     

Computer model of unidirectional solidification of single crystals of high temperature alloys

Abstract

I t has been common practice to use mould withdrawal unidirectional solidification to produce single crystal castings. To grow single crystals successfully, it is important to control several solidification parameters, such as the morphology of the solidification front (solid/liquid interface), thermal gradient, and growth rate during solidification. It is the aim of this study to develop a solidification model that can predict such solidification parameters for various design and operating conditions. The solidification phenomena in the process modelled are basically controlled by two heat transfer mechanisms: conduction and radiation. A set of heat transfer equations and boundary conditions were employed to describe mathematically the heat transfer phenomena. Then the finite difference method was used numerically to solve these equations for specified boundary conditions to obtain the temperature distribution and temperature variation in the casting. The solidification parameters can subsequently be deduced from these temperature data. Several thin plate castings were tested using the model developed. The following design and operating conditions were evaluated: susceptor temperature (power input), withdrawal speed, changes of cross-sectional area in the casting, and geometrical arrangement of the casting tree.

MST/1422     

In-situ observation of solid-liquid interface transition during directional solidification of Al-Zn alloy via X-ray imaging

The morphological instability of solid/liquid(S/L) interface during solidification will result in different patterns of microstructure. In this study, two dimension(2 D) and three dimension(3 D) in-situ observation of solid/liquid interfacial morphology transition in Al-Zn alloy during directional solidification were performed via X-ray imaging. Under a condition of increasing temperature gradient(G), the interface transition from dendritic pattern to cellular pattern, and then to planar growth with perturbation was captured. The effect of solidification parameter(the ratio of temperature gradient and growth velocity(v), G/v) on morphological instabilities was investigated and the experimental results were compared to classical "constitutional supercooling" theory. The results indicate that 2 D and 3 D evolution process of S/L interface morphology under the same thermal condition are different. It seems that the S/L interface in 2 D observation is easier to achieve planar growth than that in 3 D, implying higher S/L interface stability in 2 D thin plate samples. This can be explained as the restricted liquid flow under 2 D solidification which is beneficial to S/L interface stability. The in-situ observation in present study can provide coherent dataset for microstructural formation investigation and related model validation during solidification.     

Study of non-equilibrium dendrite growth in an undercooled alloy

ABSTRACT

Nonequilibrium thermodynamics and transportation kinetics near the propagating solid–liquid interface dominates the rapid solidification process, which is far from a thermodynamically stable state. Rapid solidification process can be described more precisely using quantitative thermodynamic calculation of phase diagram with nonlinear liquidus and solidus and evaluating the nonequilibrium effect in diffusion kinetics. Based on these basic principles, we have used a current nonequilibrium dendrite growth model to describe the rapid solidification process and the recalescence temperature of deeply undercooled alloys. Evolution of the key fundamental solidification parameters was also evaluated. The experimental data agree well with the model prediction.     

Estimation of the Heat Transfer Coefficient in a Liquid–Solid Fluidized Bed Using an Artificial Neural Network

A non-iterative procedure has been developed using an artificial neural network (ANN) for estimating the fluid–particle heat transfer coefficient, h_fp, in a liquid–solid fluidized system. It is assumed that in a liquid–solid system, the liquid temperature is time dependent, and the input parameters and output parameters for the ANN are considered on a linear scale. The output configuration yields an optimal ANN model with 10 neurons in each of the three hidden layers. This configuration is capable of predicting the value of Bi in the range of 0.1–10 with an error of less than 3%. The heat transfer coefficient estimated using the ANN has been compared with the data reported in the literature and found to match satisfactorily.© Koninklijke Brill NV, Leiden and Society of Powder Technology, Japan, 2008     

Numerical approximation of a thermally driven interface using finite elements

A two‐dimensional finite element model for dendritic solidification has been developed that is based on the direct solution of the energy equation over a fixed mesh. The model tracks the position of the sharp solid–liquid interface using a set of marker points placed on the interface. The simulations require calculation of the temperature gradients on both sides of the interface in the direction normal to it; at the interface the heat flux is discontinuous due to the release of latent heat during the solidification (melting) process. Two ways to calculate the temperature gradients at the interface, evaluating their interpolants at Gauss points, were proposed. Using known one‐ and two‐dimensional solutions to stable solidification problems (the Stefan problem), it was shown that the method converges with second‐order accuracy. When applied to the unstable solidification of a crystal into an undercooled liquid, it was found that the numerical solution is extremely sensitive to the mesh size and the type of approximation used to calculate the temperature gradients at the interface, i.e. different approximations and different meshes can yield different solutions. The cause of these difficulties is examined, the effect of different types of interpolation on the simulations is investigated, and the necessary criteria to ensure converged solutions are established. Copyright © 2003 John Wiley & Sons, Ltd.     

单晶铸件凝固过程工艺优化的数值模拟

采用ProCAST软件系统研究了LMC(Liquid Metal Cooling)以及HRS(High Rate Solidification)工艺下,不同工艺参数对单晶铸件凝固过程中纵向温度梯度、温度梯度角、凝固界面位置的影响。结果表明:HRS工艺受型壳厚度影响很小,型壳表面的辐射散热是HRS工艺的主要影响因素,型壳的导热或者型壳和合金之间的换热是LMC工艺的主要影响因素;提高保温炉温度有利于提高纵向温度梯度;拉速是影响定向凝固最重要的参数,随拉速的增加,单晶铸件的纵向温度梯度先增大后减小,因此,制备不同合金铸件时应当采用不同的拉速;不同浇注温度时,经过10min的静置时间后,单晶铸件的初始温度分布趋于一致,对后续凝固过程影响很小。提出了以纵向温度梯度G∥、温度梯度角θ以及凝固界面位置Rp考察定向凝固工艺参数优劣的标准,纵向温度梯度、温度梯度角、凝固界面位置是评价定向凝固参数优劣的有效手段。     

Interface morphology evolvement and microstructure characteristics of hypoeutectic Cu–1.0 wt%Cr alloy during unidirectional solidification

Effect of unidirectional solidification rate on microstructure of hypoeutectic Cu–1.0%Cr alloy was investigated. The microstructure evolution of Cu–1.0%Cr alloy was noticed especially during the unidirectional solidification with the different solidification rates. It is shown that eutectic (α+β) and primary α(Cu) phase grew up equably in parallel to direction of solidification. A kind of fibriform microstructure will appear when unidirectional solidification rate is up to some enough high certain values. When temperature gradient was changeless, the interface morphology evolution of the primary α(Cu) phase underwent to a series of changes from plane to cell, coarse dendrite, and fine dendrite grains with increasing the solidification rates. Primary dendrite arm spacing λ1 of α(Cu) phase increases with increasing the solidification rate where the morphology of the solid/liquid (S/L) interface is cellular. However, λ1 decreases with further increasing the solidification rate where the S/L interface morphology is changed from cell to dendrite-type. Its rule might accord with Jackson–Hunt theory model. An experience equation obtained is as follows: . On the other hand, secondary dendrite spacing λ2 of primary α(Cu) phase will thin gradually with increasing the solidification rate. Moreover, secondary dendrite will become coarse in further solidification. Another experience equation about relationship among secondary dendrite arm spacing (λ₂), temperature gradient G_L and the velocity of the S/L interface (V) is that: λ₂=−0.0003+0.0027(G_LV)^−1/3. In addition, the volume fraction of eutectic will decrease with the increase of solidification rate.     

Numerical Simulation of Morphology and Microsegregation Evolution during Solidification of Al-Si Alloy

A stochastic model coupled with transient calculations for the distributions of temperature, solute and velocity during the solidification of binary alloy is presented. The model can directly describe the evolution of both morphology and segregation during dendritic crystal growth. The model takes into account the curvature and growth anisotropy of dendritic crystals. Finite difference method is used to explicitly track the sharp solid liquid (S/L) interface on a fixed Cartesian grid. Two-dimensional mesoscopic calculations are performed to simulate the evolution of columnar and equiaxed dendritic morphologies of an AI-7 wt pct Si alloy. The effects of heat transfer coefficient on the evolution of both the dendrite morphology and segregation patterns during the solidification of binary alloys are analyzed. This model is applied to the solidification of small casting. Columnar-to-equiaxed transition is analyzed in detail. The effects of heat transfer coefficient on final casting structures are also studi     

Simulation of temperature field,flow field and solidification structure for Ag–28Cu–2Ge–0.4Co alloy

ABSTRACT

The temperature field, flow field and solidification structure of Ag–28Cu–2Ge–0.4Co alloy during solidification under water cooling, air cooling and slow cooling conditions were investigated, respectively. The results indicate that the temperature distribution is the most uniform and the solid–liquid phase region is the widest under slow cooling condition. The temperature gradient at the solidification front is the largest under water cooling condition. The solidification rate increases with the distance away from the sidewall under three cooling conditions. Under water-cooled conditions, the liquid flow at the front of the liquidus is much smaller than that in the other two conditions. The ratio of equiaxed crystals and columnar crystals in the solidified structure is different under different cooling conditions.     

Effect of Melt Superheating Treatment on Directional Solidification Interface Morphology of Multi-component Alloy

The influence of melt superheating treatment on the solid/liquid (S/L) interface morphology of directionally solidified Ni-based superalloy DZ125 is investigated to elucidate the relationship between melt characteristic and S/L interface stability. The results indicate that the interface morphology is not only related to the withdrawal velocity (R) but also to the melt superheating temperature (Ts) when the thermal gradient of solidification interface remains constant for different Ts with appropriate superheating treatment regulation. The interface morphology changes from cell to plane at R of 1.1 μm/s when Ts increases from 1500°C to 1650°C, and maintains plane with further elevated Ts of 1750°C. However, the interface morphology changes from coarse dendrite to cell and then to cellular dendrite at R of 2.25 μm/s when Ts increases from 1500°C to 1650°C and then to 1750°C. It is proved that the solidification onset temperature and the solidification interval undergo the nonlinear variation when Ts increases from 1500°C to 1680°C, and the turning point is 1650°C at which the solidification onset temperature and the solidification interval are all minimum. This indicates that the melt superheating treatment enhances the solidification interface stability and has important effect on the solidification characteristics.     

Interfacial evolution in Sn–58Bi solder joints during liquid electromigration

For Sn–58Bi low temperature solder alloy, local molten induced from electromigration Joule heating might change the atomic diffusion and interfacial behavior. In this paper, the diffusion behavior and interfacial evolution of Cu/Sn–58Bi/Cu joints were studied under liquid–solid (L–S) electromigration in molten solder and were compared with the interfacial behaviors in solid–solid (S–S) electromigration in solid solder. L–S or S–S electromigration was realized by applying a current density of 1.0?×?10⁴ A/cm² to molten solder at 150 °C or solid solder at 25 °C, respectively. During S–S electromigration, Bi atoms were driven towards anode side under electromigration induced flux and then accumulated to form Bi-rich layer near anode interface with current stressing time increasing. During L–S electromigration, Bi atoms were reversely migrated from anode to cathode to produce Bi segregation at cathode interface, while Cu atoms were rapidly dissolved into molten solder from cathode and migrated to form large amounts of Cu₆Sn₅ rod-like phases near anode interface. The reversal in the direction of Bi atoms may be attributed to the reversal in the direction of electromigration induced flux and correspondingly the change on effective charge number of Bi atoms from negative to positive.     

Phase-field simulation of directional solidification of a binary alloy under different boundary heat flux conditions

Complicated morphologies of directional solidification structures attract a lot of theoretical studies and commercial uses. As known, the boundary heat flux has an important significance to the microstructures of directional solidification. In this article, the interface evolution of directional solidification with different boundary heat fluxes is discussed. In this study, only one interface has heat flow, and Neumann boundary conditions are imposed at the other three interfaces. From the calculated results, it is found that different heat fluxes cause different microstructures in the directional solidification. When the heat flux equal to 18 W/cm², the growth of lengthways side branches is accelerated and the growth of transverse side branches is restrained. At the same time, there is dendritic remelting in the calculating domain. When the heat flux equal to 36 W/cm², the growth of the transverse side branches and the growth of the lengthways side branches compete with each other. When the heat flux equal to 90 or 180 W/cm², the growth of transverse side branches absolutely dominates. The temperature field of dendritic growth is also analyzed and the relation between heat flux and temperature field is found.     

Effect of cooling rate on solidification of electron beam melted silicon ingots

Solidification rate has a significant influence on purification of silicon due to segregation of impurities at a liquid–solid interface of a solidifying silicon ingot. A mathematical model is developed to evaluate time-dependent position of the liquid–solid interface and solidification rate of electron beam melted ingots. A series of solidification experiments with different cooling rates are conducted to measure position of a line which separates directionally grown columnar crystals visible in cross-sections of the solidified ingots. Results show that not the whole ingot solidifies directionally when the reduction rate of the beam current is larger than 1.67 mA/s. The position of the dividing line depends on cooling rate and the experimental trend is consistent with that resulted from theoretical simulations. Modeling shows that the solidification rate changes fast when the beam current reduces linearly that is detrimental for segregation of impurities. It also predicts that an exponential reduction of the beam current leads to a uniform solidification rate which is beneficial to segregation of impurities, though not all exponential current reductions lead to this kind of solidification behavior.     

Kinetics of triple-junctions in eutectic solidification: a sharp interface model

The kinetics of triple-junctions (TJs) in eutectic solidification is modeled by the thermodynamic extremum principle (TEP). It consists of two parts. First, TJs as the interaction of interfaces follow the interface kinetics according to which the temperature and concentration at the TJs are determined. This interface part of TJ kinetics is closely related to the eutectic point in the kinetic phase diagram. Second, TJs have their specific kinetics according to which their morphology (e.g., the contact angles in two dimensions) is determined. Using a new solution of solute diffusion in liquid, the TJ kinetics is incorporated into the current lamellar eutectic growth model. The model is applicable to the concentrated alloy systems and can be extended to any kind of eutectics. Simulation results of the rapid solidification of a lamellar Ni₅Si₂–Ni₂Si (γ–δ) eutectic show that both parts of TJ kinetics can play important roles in eutectic solidification and need to be considered to improve the current eutectic theory.     

Three-dimensional axisymmetric model for convection in laser-melted pools

Abstract

A three-dimensional axisymmetric model of the fluid flow and heat transfer in a laser-melted pool is developed. The model corresponds to the limiting case when the scanning velocity is small compared with the recirculating velocity. This model is also valid for spot welding. Non-dimensional forms of the governing equations are derived, from which four dimensionless parameters are obtained: the Marangoni number, the Prandtl number, the dimensionless melting temperature, and the radiation factor. Their effects and significance are discussed, and numerical solutions are obtained. The position and shape of the solid/liquid interface are obtained by an iterative scheme. The quantitative effects of the dimensionless parameters on pool shape are presented. In the presence of the flow field, the heat transfer becomes convection dominated. The effect of convection on isotherms within the molten pool is discussed, and experimental results are presented.

MST/535     

定向凝固温度场的实验研究

本文研究了Bridgman—Stockbarger法定向凝固的温度场。着重研究了有限长试样从下向上凝固时,固液界面上的温度梯度和生长速度的变化,装置的环境温度分布,试样轴向温度分布以及提高温度梯度的方法。实验发现,有限长试样的中间部分凝固时的工艺条件是稳定且可控的。在炉底部安装一个内径很小的加热圈是提高温度梯度最有效的方法。     

Application of vibrational parameters to locate the solid/liquid interface during solidification and melting of metals

A method to locate the solid/liquid interface with vibrational parameters during solidification is proposed for the first time. The sufficient difference in resistance to shear stresses between liquid and solid phases of metals and alloys permits the application of vibrational parameters to locate the interface in real time and in a situation during solidification. Based on the solidification theory, continuum mechanics, vibrational modal analysis and sensitivity analysis, the mechanical model and the dynamic equations of a typical Bridgman solidifying system have been established, the sensitivity of eigenvalues of the Bridgman system to the location of the solid/liquid interface has been derived, and the formulae concerned calculated. The experimental results are in good agreement with the computed ones.     

Numerical model for microsegregation in ductile iron

Abstract

A numerical, non-steady state microsolute redistribution model is presented for ductile iron. The model takes into account solute diffusion in the solid and liquid phases, interface movement, a non-linear growth rate for the austenite phase and total solute conservation in the microvolume sphere. Preliminary calculations show that interface movement can be ignored and a linear austenite growth rate can be used for solidification conditions occurring during directional solidification experiments and keel block solidification. The numerical calculations of the solute distribution in the liquid and solid phases show reasonable agreement with the available experimental measurements.     

Analysis of zonal shrinkage in alloy steels

Abstract

Thermal behaviour of the solidifying steel structure is important for understanding of the defects during ingot solidification. During solidification and cooling, most metals shrink. As a consequence in upper part of solid ingot, pores and pipes of typical shapes and size are formed. Forming of pipes is closely related to the casting and solidification processing parameters. In the present paper the influence of liquid temperature, chemical composition and temperature gradient on the shrinkage intensity are investigated. The ratio of the pipe depth to total ingot height as a criterion of the pipe size is used. The values of the temperature gradients on the base of the numerical model solidification are obtained. The experimental measurements of temperature change have been carried out on laboratory steel ingot. The results by numerical model are compared with the experimental ones and showed a good agreement.