Modeling solidification and mushy zone deformation of alloys

Modeling solidification and mushy zone deformation of alloys

E. M. da S. Rizzo^I; R. Gomes dos Santos^II; C. Beckermann^III

^ICentro Federal de Educação Tecnológica do Espírito Santo, Coordenadoria de Metalurgia, C.P. 5139, 29040-333 Vitória, ES. Brazil

^IIUniversidade Estadual de Campinas, Departamento de Engenharia de Materiais, Cx. P. 6122, 13083-970 Campinas, SP. Brazil. [e-mail]

^IIIDepartment of Mechanical Engineering, The University of Iowa, Iowa City, IA 52242, USA

ABSTRACT

The deformation of the partially solidified strand during near net shape continuous casting of thin slabs is only one example of a technologically important process where a mushy zone is mechanically altered in order to achieve improved grain structures and homogeneity. However, the phenomena occurring during plastic deformation of a solidifying mush are poorly understood, have not been observed directly, and have not been modeled in detail. A mathematical model that predicts the temperature and solid fraction profiles, liquid concentration evolution, and solid phase displacement during the deformation of a solidifying mush is presented. This model is derived considering a two-dimensional case, the volume averaging method, the enthalpy technique, and a coordinate transformation to avoid a changing mesh in a numerical solution. The numerical results are found to match the experimental behavior of metallic alloys during solidification with or without deformation.

Keywords: Near net shape, solidification, deformation, volume averaging, and modeling

Introduction

The manufacture of almost every man-made material involves the solidification process at some stage of the manufacturing process. Some of the important processes that involve solidification are foundry, welding, casting of ingots and continuous casting. Continuous casting is of such importance because it is a very economic method of forming a metallic component.

The composition variation within a solidified product is known as segregation. Segregation in an alloy is a result of solute rejection at the solidification front followed by its redistribution by diffusion and mass flow. Depending on the extension of composition variation, this defect is classified as microsegregation, relative to grain scale, or macrosegregation, relative to the product scale. Macrosegregation affects the mechanical (ductility, strength, etc) and chemical properties (corrosion resistance) of materials in the product extension. It also affects precipitation of weak second phases, that was not expected if the initial composition was achieved, and porosity may be generated.

The formation of macrosegregation during metallic alloys' solidification is mainly caused by macroscopic movement of liquid metal and/or solid phases. Considering the case of continuous casting process, the following phenomena have been established as driving forces for this movement (Beckermann & Viskanta, 1993; Prescott & Incropera, 1994; Miyazawa & Schwerdtfeger, 1981):

- density changes due to phases transformations during the solidification and cooling;

- density changes due to temperature and compositional changes within a phase (natural convection);

- the action of gravitational field on a density gradient (buoyancy driven flows);

- drag forces from melt convection or from solid motion;

- external forces such as magnetic fields and forces applied on the surface (forced convection);

- solid stress due to solidification contraction or expansion (as a result of temperature variation or solid state phase transformation), metalostatic pressure, bulging between rolls and roll bending.

A review of the models previously employed to study the macrosegregation formation induced by deformation, particularly the centerline macrosegregation, demonstrates the range of approaches used by researchers (Moore, 1980; Lait & Brimacombe, 1982; Irvwing, 1993, Miyazawa & Schwerdtfeger, 1981; Lesoult & Sella, 1990; El-Bealy & Fredriksson, 1991). These approaches can be evaluated by considering the physical principles and the simplifying assumptions used to derive the correspondent equations that the authors assumed to explain the phenomenon of centerline macrosegregation.

Recently it has been proposed that the deformation of the partially solidified strand during near net shape continuous casting of thin slabs could reduce the defect of centerline macrosegregation. However, to the author’s best knowledge, no mathematical model or even sets of complete experimental results that could be used to relate the process parameters and the macrosegreation reduction have been presented, i.e., when and how to apply the deformation. Examples of process parameters are the amount of deformation, the deformation rate, solid fraction and temperature at the moment of deformation and the effects of alloy composition.

Model Description

The model development starts with a microscopic description. Conservation laws are introduced at the microscopic scale (Fig. 1). Assumptions are made about the microscopic behavior of phases and interfaces, and accordingly, constitutive equations are also introduced at the microscopic scale (e.g. Newtonian fluid, Fickian diffusion, etc.). Resulting field equations are then averaged to obtain macroscopic field equations. Often in order to make the averaging procedure tractable and to obtain desirable results, many assumptions are made in the averaging process. These assumptions are typically related to the spatial and/or temporal distribution of properties, expected or magnitude of various terms, and existence of certain relations among various properties.

The main advantages of the averaging approach are that the available information at the microscale is passed on to the macroscale; the interfaces and their thermodynamic properties are explicitly taken into account; and the microstructure is more appropriately modeled. The disadvantage of this approach is that the types of restrictions imposed, especially those justifying neglect of various microscopic and macroscopic terms, may seriously limit the generality of the results and the opportunity to systematically explore fundamental behavior of complex phenomena. Also, construction of a full thermodynamic theory at the macroscale becomes intractable.

Nomenclature

A = total frontal area of the air flow, area, m²

b = pin height, m

A = interphase area, m²

c = specific heat, J/(kg K)

C = species concentration, dimensionless

Cmix = mixture concentration, dimensionless

f_s = solid fraction, dimensionless

J = Jacobian

k = thermal conductivity, W/(m K)

S_k = volumetric source term which includes production or annihilation of Y_k.

t = time (s),

T = temperature, (^oC)

U = contravariant component of velocity vector,

v = velocity

V = volume, m³

V = contravariant component of velocity vector,

X,Y = fixed Cartesian coordinate system,

x,y = moveable Cartesian coordinate system

Greek Symbols

DP = pressure drop, Pa

a_ij= metric tensor components

f_k= solid fraction of a given phase k, dimensionless

r_k= density of a given phase k, kg/m³

L_k= diffusion coefficient of a given phase k,

y_k = general scalar specific quantity, dimensionless

Subscripts

k = a given phase,

s = solid phase

l = liquid phase

Figure 2 presents the sketch for representing, at the grain scale (macroscopic level), the liquid and solid phases in the mushy region. The mushy zone comprises two interpenetrating phases (Wang & Beckermann, 1993). A porous medium is defined as a portion of space that is consisted of a solid phase and a void space. The interconnected voids are occupied by one or more fluid phases, which can flow through by the material.

On the void scale (microscopic scale) the flow quantities (velocity, pressure, etc.) will clearly be irregular (Nield & Bejan, 1992). However, in typical experiments, the quantities of interest are measured over areas that cross many voids, and such space-averaged (macroscopic scale) quantities change in a regular manner with respect to space and time, and hence are amenable to theoretical treatment.

Consider the transport of a general scalar specific quantity Y_k associate with a phase k in a multiphase mixture. For an arbitrary fixed control volume V of surface A, which is larger than a phase element but smaller than the characteristic domain dimension, conservation of Y_k can be expressed, in differential form for each phase (Ni, J.; Feller, J.; Beckermann, C., 1991; Bennon & Incropera, 1987; Voller & Brent, 1989), as a balance of accumulation, net outflow by advection and diffusion, and volumetric production in a general form as follows:

or

where t = time, f_k= solid fraction of a given phase k, r_k = density of a given phase k; L_k = diffusion coefficient of a given phase k, S_k= volumetric source term which includes production or annihilation of Y_k.

In order to consider the grid changing due to deformation application (Fig. 2), it is necessary to apply a coordinate transformation to the Eq.(1). If we consider a transformed, structured and orthogonal form (Maliska, 1995; Shyy et al., 1996), the general equation in coordinates can be written as follows:

The metric tensor components a_ij (Shyy et al, 1996; Maliska, 1995 ) and the Jacobian of the transformation, are given by

where:

The metrics of transformation are defined as

The contravariant components of velocity vector, and , are defined as follows (Shyy et al, 1996; Maliska, 1995, Zeng & Faghri, 1994):

Equation 3 can be rewritten as

The grid velocity is also a very important variable when we consider a coordinate transformation. In our case, the coordinate system of the solid phase relative to the moveable wall (x,y) is fixed at position (0,y). Hence, the grid velocity is equal to zero at this position and it increases in the direction of x = Sc. The Cartesian components of the grid velocity vector (x_t and y_t) are calculated, for a given position i at a given time l, by

The grid velocity also varies according to the grid deformation pattern assumed for a given situation.

Numerical Implementation

For the results presented in this paper, an implicit, control-volume-based finite difference scheme has been used to discretize the conservation equations, and a power-law scheme has been used to evaluate the finite-difference coefficients. The velocity-pressure coupling in the momentum equations was handled using the SIMPLER algorithm (Patankar, 1980). A uniform grid deformation for the mesh is assumed during the deformation stage.

Results And Discussion

The model was applied to a case of deformation of an organic alloy (SCN-acetone). Succinonitrile (SCN) is a transparent organic substance in the liquid state that is used as an analogue material to study some phenomena related to the solidification processes. This material solidifies in a dendritic morphology. Experimental works with this alloy are been performing to compare the numerical results. Acetone was added to obtain an alloy behavior, i.e., to get a segregated profile during solidification. The initial composition for the experiments presented in this study is 11.3 wt% of acetone. The shape of the domain is rectangular and the initial and final sizes are 50 x 170 mm and 40 x 210 mm respectively. The boundary condition at the interface heat exchanger/SCN-acetone is the temperatures measured by thermocouples placed on this interface. The other boundaries are considered as adiabatic (Fig. 2).

The thermophysical properties of SCN-Acetone alloy used for numerical simulations are summarized in Table 1. The initial conditions used supposed a melt alloy with a uniform temperature of 42.35^oC and uniform composition, i.e., initial concentration for acetone = 11.3 wt%. The cold wall was set at 5 ^oC during the simulations. The thermophysical properties for the alloy were taken using the mixture rule based on the properties for the pure components, provided by Gebhart, 1993 and Glicksmann et al., 1976.

Space limitations prohibit the presentation of variables evolution plots for many times or during the deformation stage. Only two times were select to present the variables evolution. The first plot set (Fig. 3) shows the variables at the time relative to the deformation beginning, i.e., at 26246 s from cooling start. The second plot set (Fig. 4) is relative to the end of deformation end (26301 s). Figure 3 shows the (a) mesh, (b) temperature field, (c) solid fraction contours, (d) mixture concentration, (e) velocity field and liquid concentration contours and (f) a detail the velocity field for the time just before deformation was applied. Figure 4 shows the same results group for a time just after deformation.

Figure 4e

Since the liquid is cooled as it falls along the left wall, the mushy zone is thicker near the bottom of the casting (Fig. 3c) as well as the temperature drop is higher (Fig. 3b). Figure 3e shows the convection pattern (buoyancy driven flow) in the bulk melt and in the mush zone typical of metallic alloys solidification. The liquid concentration contours in Fig. 3e and solid fraction in Fig. 3c show that the counterclockwise flow in the melt and the downward flow trough the mush zone carry some of this enriched liquid into the bulk melt.

Conclusions

The mathematical model and the numerical code presented can be useful to determine the suitable process parameters in order to reduce the macrosegregation defect in the continuous casting process. The numerical result patterns agree with experimental and other numerical codes results for the stage without deformation and with experimental observations when a domain deformation is applied. These preliminary simulations are now being compared to laboratory experiments with organic and metallic alloys.

Acknowledgments

This work was supported by FAPESP (grant number 1997/2036-9) and Finep (Recope). One of the authors (EMSR) thanks The University of Iowa College of Engineering

Paper accepted: July, 2002

Technical Editor: José Roberto de França Arruda

Presented at COBEM 99 – 15th Brazilian Congress of Mechanical Engineering, 22-26 November 1999, São Paulo. SP. Brazil.

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Publication Dates

• Publication in this collection
  18 Mar 2004
• Date of issue
  Apr 2003

History

• Accepted
  July 2002