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 field of the environment and technical heat parameters. The temperature field on the S–L interface is closely related to the solidification 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.     

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.     

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.     

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。     

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.     

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     

Efficient thermo‐mechanical model for solidification processes

A new, computationally efficient algorithm has been implemented to solve for thermal stresses, strains, and displacements in realistic solidification processes which involve highly nonlinear constitutive relations. A general form of the transient heat equation including latent‐heat from phase transformations such as solidification and other temperature‐dependent properties is solved numerically for the temperature field history. The resulting thermal stresses are solved by integrating the highly nonlinear thermo‐elastic‐viscoplastic constitutive equations using a two‐level method. First, an estimate of the stress and inelastic strain is obtained at each local integration point by implicit integration followed by a bounded Newton–Raphson (NR) iteration of the constitutive law. Then, the global finite element equations describing the boundary value problem are solved using full NR iteration. The procedure has been implemented into the commercial package Abaqus (Abaqus Standard Users Manuals, v6.4, Abaqus Inc., 2004) using a user‐defined subroutine (UMAT) to integrate the constitutive equations at the local level. Two special treatments for treating the liquid/mushy zone with a fixed grid approach are presented and compared. The model is validated both with a semi‐analytical solution from Weiner and Boley (J. Mech. Phys. Solids 1963; 11 :145–154) as well as with an in‐house finite element code CON2D (Metal. Mater. Trans. B 2004; 35B (6):1151–1172; Continuous Casting Consortium Website. http://ccc.me.uiuc.edu [30 October 2005]; Ph.D. Thesis, University of Illinois, 1993; Proceedings of the 76th Steelmaking Conference, ISS, vol. 76, 1993) specialized in thermo‐mechanical modelling of continuous casting. Both finite element codes are then applied to simulate temperature and stress development of a slice through the solidifying steel shell in a continuous casting mold under realistic operating conditions including a stress state of generalized plane strain and with actual temperature‐dependent properties. Other local integration methods as well as the explicit initial strain method used in CON2D for solving this problem are also briefly reviewed and compared. Copyright © 2006 John Wiley & Sons, Ltd.     

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。     

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.     

Control of the freezing interface morphology in solidification processes in the presence of natural convection

This paper presents a methodology for the solution of an inverse solidification design problem in the presence of natural convection. In particular, the boundary heat flux q₀ in the fixed mold wall, δΩ₀, is calculated such that a desired freezing front velocity and shape are obtained. As the front velocity together with the flux history q_ms on the solid side of the freezing front play a determinant role in the obtained cast structure, the potential applications of the proposed methods to the control of casting processes are enormous. The proposed technique consists of first solving a direct natural convection problem of the liquid phase in an a priori known shrinking cavity, Ω_L(t), before solving an ill-posed inverse design conduction problem in the solid phase in an a priori known growing region, Ω_S(t). The direct convection problem is used to evaluate the flux q_ml in the liquid side of the freezing front. A front tracking deforming finite element technique is employed. The flux q_ml can be used together with the Stefan condition to provide the freezing interface flux q_ms in the solid side of the front. As such, two boundary conditions (flux q_ms and freezing temperature θ_m) are especified along the (known) freezing interface δΩ_I(t). The developed design technique uses the adjoint method to calculate in L₂ the derivative of the cost functional, ∥θ_m – θ( x , t; q₀)∥, that expresses the square error between the calculated temperature θ( x , t; q₀) in the solid phase along δΩ_I(t) and the given melting temperature. The minimization of this cost functional is performed by the conjugate gradient method via the solutions of the direct, sensitivity and adjoint problems. A front tracking finite element technique is employed in this inverse analysis. Finally, an example is presented for the solidification of a superheated incompressible liquid aluminium, where the effects of natural convection in the moving interface shape are controlled with a proper adjustment of the cooling boundary conditions.     

The Extinction Problem for Three-dimensional Inward Solidification

The one-phase Stefan problem for the inward solidification of a three-dimensional body of liquid that is initially at its fusion temperature is considered. In particular, the shape and speed of the solid-melt interface is described at times just before complete freezing takes place, as is the temperature field in the vicinity of the extinction point. This is accomplished for general Stefan numbers by employing the Baiocchi transform. Other previous results for this problem are confirmed, for example the asymptotic analysis reveals the interface ultimately approaches an ellipsoid in shape, and furthermore, the accuracy of these results is improved. The results are arbitrary up to constants of integration that depend physically on both the Stefan number and the shape of the fixed boundary of the liquid region. In general it is not possible to determine this dependence analytically; however, the limiting case of large Stefan number provides an exception. For this limit a rather complete asymptotic picture is presented, and a recipe for the time it takes for complete freezing to occur is derived. The results presented here for fully three-dimensional domains complement and extend those given by McCue et al.[Proc. R. Soc. London A 459 (2003) 977], which are for two dimensions only, and for which a significantly different time dependence occurs.     

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.     

The melting point,latent heat of solidification,and enthalpy for both solid and liquid α-Al2O3 in the range 550–2400 K

In this paper the authors describe the use of a high-temperature drop calorimeter with autoadiabatic control for the measurment of the enthalpy of -Al₂O₃ in the temperature range 550 to 2400 K for both solid and liquid phases. Equations representing the enthalpy of both solid and liquid states are obtained from the data with the use of a computer. In addition to the melting point, T _m=2328±7 K, the latent heat of solidification, L=1137.90 J · g^–1, has also been determined. The results of the present work are compared with those reported in the literature.Paper presented at the Ninth Symposium on Thermophysical Properties, June 24–27, 1985, Boulder, Colorado, U.S.A.     

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.     

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。     

Non-equilibrium solidification of undercooled droplets during atomization process

Thermal history of droplets associated with gas atomization of melt has been investigated. A mathematical model, based on classical theory of heterogeneous nucleation and volume separation of nucleants among droplets size distribution, is described to predict undercooling of droplets. Newtonian heat flow condition coupled with velocity dependent heat transfer coefficient is used to obtain cooling rate before and after nucleation of droplets. The results indicate that temperature profile of droplets in the spray during recalescence, segregated and eutectic solidification regimes is dependent on their size and related undercooling. The interface temperature during solidification of undercooled droplets rapidly approaches the liquidus temperature of the alloy with a subsequent decrease in solid-liquid interface velocity. A comparison in cooling rates of atomized powder particles estimated from secondary dendrite arm spacing measurements are observed to be closer to those predicted from the model during segregated solidification regime of large size droplets.     

Computer Modeling of the Evolution of Dendrite Microstructure in Binary Alloys During Non-isothermal Solidification

Two-dimensional simulations of the evolution of dendrite microstructure during isothermal and non-isothermal solidifications of a Ni-0.41Cu binary alloy are carried out using the phase-field method. The governing evolution equation for the phase field variable, the solute mole fraction and the temperature are formulated and numerically solved using an explicit finite difference scheme. To make the computations tractable, parallel computing is employed. The results obtained show that under lower cooling rates, the solidification process is controlled by partitioning of the solute between the solid and the liquid at the solid/liquid interface. At high cooling rates, on the other hand, solute trapping takes place and solidification is controlled by the heat extraction rate. An increase in the cooling rate is also found to have a pronounced effect on the dendrite microstructure causing it to change from poorly developed dendrites consisting of only primary stalks, via fully developed dendrites containing secondary and tertiary arms to the diamond-shaped grains with cellular surfaces. These findings are in excellent agreement with experimental observations.     

Modeling constrained dendrite growth in rapidly directional solidification

Incorporating the effect of nonlinear liquidus and solidus, a constrained dendrite growth model was established for rapidly directional solidification, where both the interface and the bulk liquid are under local non-equilibrium conditions. The effect of nonlinear liquidus and solidus was first introduced into the interface kinetics, based on which the marginal stability criterion was then derived. By this way, not only the temperature dependent equilibrium partition coefficient k _e, equilibrium liquidus slope m _l and equilibrium solidus slope m _s but also the derivation of k _e with temperature are considered. The model is more physically realistic than the works in which the effect of nonlinear liquidus and solidus is not incorporated into the interface kinetics and hence the temperature dependent m _s is not embodied. Applying the model to rapidly directional solidification of Ag–15 wt% Cu alloy, the effects of nonlinear liquidus and solidus and local non-equilibrium on the constrained dendrite growth were studied.     

The propagation and basal solidification of two-dimensional and axisymmetric viscous gravity currents

The effect of basal solidification on viscous gravity currents is analysed using continuum models. A Stefan condition for basal solidification is incorporated into the Navier-Stokes equations. A simplified version of this model is determined in the lubrication and large-Bond-number limit. Asymptotic solutions are obtained in three parameter régimes. (i) A similarity solution is possible in the following cases: the two-dimensional problem when volume per unit length (V) is proportional to time (t) raised to the power 7/4(V = qt ^7/4) and the Julian number (v ³ g ² /q ⁴ ) is large, where v is kinematic viscosity, q is a constant of proportionality and g is the acceleration due to gravity; the axisymmetric problem when volume is proportional to time raised to the power 3 (V = Qt ³) and the dimensionless group vg/Q is large, where Q is a constant of proportionality. In both cases, the front is found to depend on time raised to the power 5/4, as it does in the absence of solidification, but the constant of proportionality satisfies a modified system of equations. (ii) In the case of large Stefan number and small modified Peclet number (Pe ² 1, where Pe is the Peclet number and is the aspect ratio), asymptotic and numerical methods are combined to produce the most revealing results. The temperature of the fluid approaches the melting point over a short time-scale. Over the long time-scale, the solid/liquid interface is determined from the conduction of latent heat into the solid. Strong coupling is observed with the basal solidification modifying the flow at leading order. The solidification may retard and eventually arrest the front motion long before complete phase change has taken place. (iii) In the case of constant volume and large modified Peclet number (Pe ² 1), similarity solutions are found for the solidification at the base of the gravity current on the short time-scale. The coupling is weak on this time-scale with the solidification being dependent on the front position but not influencing the fluid motion at leading order. Over the long time-scale, the drop completely solidifies. Analytical solutions are not obtained on this time-scale, but scalings are deduced.