Modeling of Macrosegregation and Solidification Grain Structures with ...

ISIJ International, Vol. 46 (2006), No. 6, pp. 880–895

Review

Modeling of Macrosegregation and Solidification Grain Structures with a Coupled Cellular Automaton—Finite Element Model Gildas GUILLEMOT,1,2,3) Charles-André GANDIN1,2) and Hervé COMBEAU2) 1) CEMEF UMR CNRS 7635, Ecole des Mines, BP 207, 06904 Sophia Antipolis, France. 2) LSG2M UMR CNRS 7584, Ecole des Mines, 54042 Nancy, France. 3) LMPGM UMR CNRS 8517, Ecole d’Arts et Métiers, 59046 Lille, France. (Received on January 20, 2006; accepted March 10, 2006 )

A coupled Cellular Automaton (CA)–Finite Element (FE) model is presented for the prediction of solidification grain structures coupled with the calculation of solid and liquid flow induced macrosegregation. The model is applied to simulate the solidification of a Pb–48wt%Sn alloy in a rectangular cavity cooled down from only one of its vertical boundaries. The algorithm and the numerical implementation of the coupling between the CA and FE methods are first validated by considering a single grain developing with no undercooling. Such a CAFE simulation is shown to retrieve the solution of a purely FE method simulation for which the grain structure is not accounted for. Several applications of the model are then presented to quantify the effects of the grain structure on the final macrosegregation map. In particular, the effect of the undercooling of the columnar front, the presence of equiaxed grains nucleated in the undercooled liquid, as well as the transport and sedimentation of equiaxed grains are investigated. Although good validation is reached when comparing computed and measured segregation profiles available in the literature for the chosen configuration, it is concluded that refined experimental data are required to further validate the predictions of a coupled CAFE model. KEY WORDS: modeling; solidification; grain structure; macrosegregation.

1.

shape of the envelope is thus a direct consequence of the local environment seen by the grains.4) The transition from elongated columnar to round equiaxed grain shapes or viceversa is a direct output of the simulations. The detailed physics entering nucleation of the grains being little known, stochastic algorithms are used in order to mimic the formation of new grains on heterogeneous particles and impurities. Several drawbacks yet remain, that were not suppressed by the extensions proposed in the literature.5) It includes the need to model coupling with fluid-flow induced segregation as well as transport of the equiaxed grains. The goal of the present contribution is to present an extention of the stochastic Cellular Automaton (CA) model and its coupling with a Finite Element (FE) model for the prediction of grain structure and macrosegregation in casting. While this new coupling between the CA and FE method was first developed as an extension of the FE model published by Ahmad et al.6) and described in details in Ref. 7), it is now coupled with the FE model developed by Bellet et al.8) and Liu.9) As a consequence, the CAFE model and all calculations presented hereafter are an extension of the FE model presented in Refs. 8) and 9) for the study of macrosegregation. It is used to study the interplay between the developments of the grain structure and macrosegregation in casting. A detailed description of the coupled micro-macroscopic model is first given. Applications are then proposed through

Introduction

The present article is a contribution to the problem of solid and liquid flow induced macrosegregation1,2) in the presence of i-undercooled convecting liquid, ii-movement of free equiaxed grains and iii-columnar and equiaxed grain structures. The objective is to model the influence of the interactions between the development of the grain structure and the macrosegregation. These objectives are consistent with some of the basic phenomena identified to require increased research attention by the review on modeling of macrosegregation proposed by Beckermann.2) Evidences of the effect of the grain structure on macrosegregation in casting of metallic alloys have been found. Consequently, models were developed to predict both grain structure and segregation. The most advanced models are based on deterministic approaches.2,3) They account for the transport of the equiaxed grains. However, each grain is assumed to develop following a given shape. The transition from columnar-to-equiaxed grain structures observed in casting and the complicated morphologies of the grains observed in metallographic cross-sections of ingots are thus hardly reproduced. As a consequence, direct modeling has been proposed based on stochastic modeling. The formation of each grain is independently modeled by tracking the development of the boundary defined by the mushy zone and the liquid melt defining its envelope. The © 2006 ISIJ

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simulations based on the configuration of the experiments performed by Hebditch and Hunt.10,11) Implementation of the microscopic model is validated by considering a single grain that develops with almost no undercooling. As expected, the calculation retrieves the solution of a macroscopic calculation in which the grain structure is not accounted for.6) The effects of the undercooling of the columnar front, the presence of equiaxed grains nucleated in the undercooled liquid, as well as the transport and sedimentation of equiaxed grains are then investigated. 2.

flow induced macrosegregation. The liquid density is constant and equal to r 0 in all terms of Eq. (3) except for the gravity term where it is replaced by a function of the local composition of solute in the liquid, wl, and the local temperature, T:

r
r 0[1 b th(TT0) b c(wlw0l)]................(4) where b th and b c are the thermal and solutal expansion coefficients. References used to define the variations of the liquid density with the local composition of solute in the liquid and the local temperature are T0 and w0. In case of a fixed solid phase, the last term of Eq. (3) representing the volumetric friction force, M, is calculated by the relationship:

Modeling

The finite element (FE) model developed by Bellet et al.8) and Liu9) solves the conservation equations averaged over a representative elementary volume containing a mushy zone, i.e., a mixture of the solid and liquid phases. This model is hereafter named ‘macroscopic FE model’. The ‘microscopic cellular automaton (CA) model’ aims to simulate the development of the dendritic grain structure.4) The size of the cells should typically be chosen of the order of the secondary dendrite arm spacing in order to mimic the development of the growth front delimited by the boundary between the dendritic mushy zone and the liquid melt.4) For each of the procedure defined in the CA model for nucleation, growth and transport of the grains, a kinetics model is used. All these models are presented in this section, together with an algorithm explaining the coupling between the CA and FE methods. The latter section is of great importance since it explains the way the FE macroscopic model developed with assumptions of equal densities in the liquid and solid phases and a fixed solid can be coupled with the CA microscopic model extended to account for the transport of the grains.

M


µ l g v ................................(5) K

This force is a function of the permeability, K. For a mushy domain, the permeability is a function of the volume fraction of liquid and the dendrite arm spacing, l 2. In a purely liquid region, this parameter tends toward infinity and the last term of Eq. (3) vanishes to retrieve the NavierStokes equation. When decreasing the fraction of liquid, the permeability tends toward zero and the last term becomes dominant, while inertia and rheological terms vanish, yielding the Darcy’s relation. We will detail later the calculation of the permeability in accordance with the microscopic model as well as the modification of the volumetric friction force to account for the movement of the solid phase due to grain movement. 2.1.3. Energy Conservation The average energy conservation is written as follows8,9):  ∂H  ρ0   v ⋅ ∇H l   ∇ ⋅ (κ ∇T )
0 ..............(6)  ∂t 

2.1. Macroscopic FE Model 2.1.1. Total Mass Conservation For constant densities of the solid and liquid phases, the average total mass conservation writes8,9):

where v is a macroscopic flow velocity. The solid being assumed fixed in the present macroscopic model, its average velocity is equal to zero, vs
0, leading to:

where H is the average enthalpy per unit mass and H l is the average enthalpy of the liquid phase per unit mass. Approximation of constant and equal densities of the solid and liquid phases is used in Eq. (6). Further assuming constant and equal values of the heat capacity per unit mass for the liquid and solid phases, Cp, one can write:

v
glvl .....................................(2)

H
Cp[TTref.]glD ls Hf ........................(7)

where gl is the volumetric liquid fraction and vl is the average velocity of the liquid phase.

H l
Cp[TTref.]D ls Hf ........................(8)

∇ · v
0.....................................(1)

where D ls Hf represents the latent heat of fusion per unit mass and Tref. is a reference temperature, the later not being used explicitly in the integration of the equations. The average thermal conductivity, k , is taken as a constant in the following.

2.1.2. Momentum Conservation The average momentum conservation writes8,9):

ρ0

ρ ∂v  0l ∇ ⋅ (v  v )
∇ ⋅ ( µ∇v )  g l ∇p  g l ρg  M ∂t g

2.1.4. Solute Conservation The average conservation of the mass of a solute element in a binary alloy writes8,9):

...........................................(3) where m is the dynamic viscosity, p is the intrinsic pressure in the liquid, g is the gravity vector and t is the time. While this equation is written with the assumption of equal and constant densities of the liquid and solid phases, the Boussinesq approximation is introduced to compute fluid

∂w  v ⋅ ∇w l  ∇ ⋅ ( D l g l ∇w l )
0 ...............(9) ∂t 881

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study, NnF equals 3 for the triangular elements used. For the purpose of exchanging information from the FE nodes to F the CA cells, linear interpolation coefficients, cnn i, are defined between each node ni and a cell n . A variable available at the nodes, x ni, can thus be used to calculate an interpolated value at cell n , x n :

where w is the average composition of solute and wl is the average composition of solute in the liquid. Dl is the diffusion coefficient for the liquid phase. It has been chosen not to neglect this parameter for numerical stability reasons in macroscopic resolution. 2.1.5. Resolution An extended description of the strategy used to solve the above listed conservation equations as well as several validations of the implementations are given by Liu.9) A Cartesian two dimensional representation of the casting domain is considered. The computational domain is decomposed into fixed triangle elements of a FE mesh. The average enthalpy, H, is taken as the primary unknown in the energy conservation, unknown H l being replaced by inserting Eq. (8) into Eq. (6). The temperature in this equation is eliminated by a first order of Taylor’s expansion as a function of the average enthalpy as proposed by M’Hamdi et al.12) The primary unknown considered in the solute conservation Eq. (9) is the average composition of solute, w. The average liquid composition is eliminated following the work of Prakash and Voller13) who introduced a split operator technique with an Euler backward scheme, i.e. the value of is estimated using the value deduced from the previous time step. The microsegregation model used to link the average composition of solute and the temperature to the fraction of solid assumes local equilibrium with uniform compositions of both the solid and the liquid phases. Finally, a combination of the momentum conservation and the total mass conservation offers a set of equations to solve simultaneously the pressure and the average velocity fields.9)

ξν


niF ν ξni

.............................(10)

Similarly, when information computed at the level of the cells of the CA grid, x n , is required at the level of a FE node, x n, the following summation is performed: 1 ξn
Λn

N νn

∑c ξ

n νi νi

i
1

..........................(11)

where Nvn is the number of cells n seen by node n. For node n1F, it corresponds to all the cells drawn in Fig. 1. This averaging procedure is normalized by the summation over all interpolation coefficients:

Λn


N νn

∑c i
1

n νi

...............................(12)

2.2.1. Indexes The state index, InS, is used to characterize the phase and the neighborhood of a cell n . It is defined as follows: – Cell n is liquid: InS
0. – Cell n is no longer liquid and at least one of its nearest neighboring cell, m i is still liquid (∃i∈[1, N v]/I Sm i
0): InS
1. – Cell n is no longer liquid, neither are its nearest neighboring cell, m i (∀i∈[1, N n ]/I Sm i 0): InS
1. where N n is the number of cells defined in the neighborhood of cell n . The Moore configuration considering the first and second nearest neighboring cells is used, yielding to N n
8. The grain index, InG, is used to track the development of the mushy zone within cell n . It is defined as follows: – Cell n is liquid (InS
0): InG
0. – Cell n is no longer liquid (InS0) and the mushy zone is still developing: InG
1 with gnm1, gnm being the volume fraction of the mushy zone in cell n . – Cell n is no longer liquid (InS0) and its mushy zone is fully developed: InG
1 for gnm
1. The CA model described hereafter is based on rules to modify the state index, InS, and the grain index, InG, associated to each cell n of a square lattice with the goal to simulate the development of the mushy zone upon solidification. 2.2.2. Nucleation In order to initiate the solidification process at the microscopic scale, nucleation sites are randomly distributed among the cells of the CA square lattice. A nucleation site contained in cell n is characterized by a critical nucleation undercooling, D Tnnucl, and a random crystallographic orientation, q n . There are three tests performed to nucleate a new grain

Topological links between a triangular finite element mesh, F, and a square cell of the cellular automaton grid, F n . Interpolation coefficients, enni , are defined between F each cell and the Nn nodes of the finite element mesh, NiF, with i
[1, NnF] and NnF
3 in this figure.

© 2006 ISIJ

∑c i
1

2.2. Microscopic CA Model The square lattice of the CA method is superimposed on the same domain used for the macroscopic FE model as schematized in Fig. 1. Each cell n is uniquely defined by the coordinates of its center, Cn
(xn , yn ), located in a finite element mesh, F. Element F has NnF nodes, labeled niF in Fig. 1, which are locally indexed from i
1 to NnF. In this

Fig. 1.

N nF

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in a cell n containing a nucleation site: n1: The cell is still liquid: InS
0. n2: The cell temperature, Tn , falls below the local liquidus temperature: Tn TL(wn ) with TL(wn )
TM mwn , where TM is the melting temperature of the solvent in the binary phase diagram, m is a linear approximation of the liquidus slope of the binary phase diagram defined with the solute element and wn is the local composition of cell n . Temperature Tn is calculated from a conversion of the average enthalpy of cell n , Hn , considering a purely liquid cell (Hn
CpTn D lsHf ). n3: The temperature of cell n , Tn , deduced by interpolation from the nodes niF defining element F in which cell n is located (Eq. (10)), must fall below the critical temperature defined for nucleation, Tn  TL(wn )DTnnucl. This is of course only possible providing a nucleation site was previously ascribed to cell n . These tests are applied to all the cells containing a nucleation site. If the three tests are verified, the following actions are taken to initialize the growth of the grain: ni1: The state index of cell n , InS is updated following the definitions given above. Consequently, the state index of all cells defined in the neighborhood of cell n is also updated. ni2: The growth index of cell n , InG, is updated following the definitions given above. ni3: The center of the growing shape of the newly nucle-

Fig. 2.

ated grain, Gn , is located at the center of cell n , Cn . A square shape with main diagonals corresponding to the preferential
10 directions of the dendrites stems and arms are initialized with a very small size. Angle q n defines the orientation of the
10 direction of the grain, Gn Sn[10], with respect to the Ox axis. 2.2.3. Growth Figure 2 schematizes the growth propagation of the mushy zone from a cell n to a cell m . Within one time step, the four half-diagonals associated to the shape of cell n have grown from center Gn . As a result, the mushy zone has extended from the polygon delimited by the white continuous thin lines to the polygon delimited by the black continuous thin lines and identified by the Sn[ij] tips, [ij]∈[01, 10, 1¯0, 01¯]. Cell m was initially liquid (ImS
0, ImG
0) while the mushy zone has developed in cell n (InS
1, InG
1). After the propagation of the mushy zone in cell n , Cm is now engulfed by the growing shape Sn[ij]. Capture of cell m is thus achieved by its neighboring cell n . The initial growing shape associated to a cell m , Sm[ij] (hashed area delimited by a continuous bold lines), as well as the position of the growth center of cell m , Gm , are then calculated. Two other growing shapes are schematized in Fig. 2 for cell m . Tips Sm[ij] min (grey area delimited by dashed bold lines) represent the initial shape defined at a former time when edge Sn[01]Sn[10] reached position Cm , thus starting to engulf cell m . Tips Sm[ij] min thus define the minimum size associated with

The algorithm to model the growth of a cell n identified as part of a grain is schematized, as well as the propagation of the grain to a neighboring cell m . The growing shape associated to cell n has propagated from the white to the black continuous thin lines to reach positions Sn[ij], [ij] ∈ [01, 10, 1¯ 0, 01¯ ]. At that time, the growing shape has engulfed the center of cell m , Cm . The actual size of the growing shape associated to cell m , Sm [ij] (hashed area delimited by a continuous bold lines), is calculated. The initial growing center and shape associated to cell m , respectively and Sm [ij] min (grey area delimited by the dashed bold lines), as well as its maximal growing shape, Sm [ij] max (delimited by the dotted bold line), used to defined the volume fraction of the mushy zone associated to cell m , gmm, are also shown.

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the growth of cell m . When no movement of the grain is accounted for, the center of this first growing shape, Tm , is kept unchanged during the entire time of the simulation. If growth is maintained homothetic with respect to the velocity in all
10 directions, the maximal growing shape of cell m can be evaluated when the mushy zone has propagated to all the neighboring cells. In Fig. 2, this maximal growing shape corresponds to tips Sm[ij] max defined when the center of the south-east neighbor of cell m is reached by edge ¯ Sm[01]Sm[10]. All the neighboring cells of m are then captured. The current, minimal and maximal growing shapes, respectively Sm[ij], Sm[ij] min and Sm[ij] max, are used to define the volume fraction of the mushy zone, gmm, associated to cell m as:   Aµ  Aµmin g µm
Min  max ,1 ..................(13) min   Aµ  Aµ where Am, Ammin and Ammax are the areas associated with the current, initial and maximal growing shapes, respectively. A volume fraction of solid located in the envelope Sm[ij], gms m, will later be defined for cell m . The volume fraction of solid associated to cell m , gms , is then given by:

Fig. 3.

gms
gmm gms m ...............................(14) It is to be noticed that the growing shape associated to a given cell n is not a regular square as was the case in previous procedures.14,15) The reason is that the effect of the fluid flow on the growth kinetics of the four
10 directions is accounted independently for. Since each of the four directions Gn Sn[ij], [ij] ∈ [01, 10, 1¯0, 01¯], makes a different angle with the fluid flow direction, four different values of the growth velocities, vn[ij], are calculated. A similar extension was proposed by Takatani et al. for the study of grain texture formation in strip casting of thin steel sheets due to the effect of the relative fluid flow velocity on grain structure formation.16) The growth kinetics model developed to account for the effect of the fluid flow will be briefly summarized later.17)

d t1 the growing center moves to position Gntd t1 by GntGntd t1
vCt d t1
d ln1. The length of the projection of d lnlon axes Ox and Oy being smaller than the cell size, lCA, no translation is yet realized. At the next micro time step, position Gnt is reached with t
td t1d t2 and GntGnt
vCt d t1vCtd t1 d t2
d ln1d ln2. The projection of d ln1d ln2 on Oy is now larger than the cell size, |(d ln1d ln2) · y|lCA. All information contained in cell n is thus moved to cell m . This includes the value of the state indices, as well the transfer of several variables, e.g. Gmt
Gnt, Sm[ij]t
Sn[ij]t, [ij] ∈ [01, 10, 1¯0, 01¯], as shown in Fig. 3. Thus, since the transport procedure is applied to all the cells of the grains defining a cluster with the same velocity vector, vC, the entire cluster is translated. In order not to bias the transport of the information by the discrete nature of the jump from one cell to another, a residual component of the displacement vector, d ln1d ln2lCA y, is associated to the growing shape of cell m . It should be noted that, at each micro time step, because the center of the growing shape moves, a new maximal growing shape Anmax is calculated for each cell n of the cluster. It is deduced from the position of the new growing center as explained above and illustrated in Fig. 2. The volume fraction of the mushy zone, gnm, is thus updated at each micro time step according to Eq. (13) applied to cell n . Since grains move with their own velocity, impingement can take place. In this eventuality, they are kept into contact together for the rest of the simulation, using a sticking con-

2.2.4. Transport The transport of grains due to liquid flow and sedimentation is accounted for in the CA model. A velocity vector, vC, is calculated for each cluster of grains. A cluster of grains is defined by an assembly of several grains in contact to each other by at least one neighboring cell. This information is multiplied by the micro time step d t to obtain the cluster displacement d lC:

d lC
vCd t .................................(15) Displacement d lC is applied to the growing center of the cells defining the grains of the cluster, d ln
d lC. After several micro time steps, the total displacement becomes large enough to permit a horizontal and/or vertical translation of the information contained in each cell, i.e. a displacement of the cluster. Figure 3 gives an illustration of the transport procedure. The position of a growing shape associated to cell n is displayed at three consecutive times. At time t the growing shape is defined by its center, Gnt, and the four
10 tips, Sn [ij] t [ij] ∈ [01, 10, 1¯0, 01¯]. During the first micro time step © 2006 ISIJ

Schematics of the translation of the growing shape associated to cell n and its transfer to the underneath first neighboring cell m . Cell n is part of grain, itself part of a cluster. After two successive translations, d ln1 and d ln2, due to the integration of the velocity vector during micro time steps d t1 and d t2, the vertical component of the translation vector, d ln, is larger than the cell size, lCA y.  The growth center is transferred from Gnt to Gnt with t
td t1d t2, becoming the new growth center associat  ed to cell m , Gmt
Gnt . The same operation applies to the   growth shape transferred from Sn[ij] t to Sn[ij] t
Sn[ij] t with ¯ ¯ [ij] ∈ [01, 10, 1 0, 01 ]. The intermediate translation drawn with thin plain lines corresponds to the intermediate time td t1. This schematics is simplified assuming no growth takes place during the micro time steps d t1 and d t2.

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tact condition. This is indeed the way two single grains form a cluster, explaining the above presentation of the transport algorithms. The velocity of the new cluster C formed due to the impingement of NC clusters, vC, is then calculated using a balance of momentum:  N C  v C
 N νCi v Ci   i
1 

∑

2.3.2. Growth Each of the four half diagonals of the polygon developing in a mushy cell of the CA model, Gn Sn[ij], [ij] ∈ [01, 10, 1¯0, 01¯], grows with a velocity computed with a dendrite tip growth kinetics. The details of the computation of the growth velocity are given in Ref. 17). It is briefly summarized hereafter. Neglecting the kinetics and thermal contributions of the total dendrite tip undercooling, D T, one can write20):

 N C  N νCi  ..............(16)   i
1 

∑

where NnCi is the number of cells defining cluster Ci moving with velocity vCi. Procedures are also applied to check the validity of the translation of the state indexes and to account for impingement of several clusters. The limits of the simulation domain are considered as sticking wall on which the clusters stop. Thus, if a cluster moves to a position where at least one of its cells defines a limit of the simulation domain (at least one missing neighboring cell), or if a cell of a cluster has a neighboring cell belonging to a fixed cluster, the cluster velocity is set to zero and no movement is later possible. A fixed cluster is thus attached to an interconnected network of grains attached to the mold walls.

 1 ∆T
mw l ∞ 1 1(1 k )Ω c 

where G is the Gibbs–Thomson coefficient and r is the radius of curvature at the dendrite tip. Assuming equilibrium at the solid/liquid interface defined by the dendrite tip, the supersaturation, W c, is defined by the compositions of the liquid at the dendrite tip solid/liquid interface, wl s/l, the liquid composition far from the interface, wl ∞, and the segregation coefficient, k:

2.3. Kinetics Models 2.3.1. Nucleation It is assumed that nucleation of grains takes place on heterogeneous particles present into the melt as soon as a critical undercooling is reached. Because such nucleation behavior is only dependent on undercooling and not on cooling rate, it is considered as an instantaneous nucleation law. A Gaussian distribution is used to characterize the density of heterogeneous particles, n, as a function of the critical undercooling, D T, at which they become active: n( ∆T )


∫

∆T

∫

∆T

0




0

dn d∆T

Ωc


w l s/l  w l ∞ w l s/l (1 k )

σ *


Γ

Dl vp r

2

m( w

l s/l

(1 k ))

..................(20)

where vp is the velocity of the dendrite tip, Dl is the diffusion coefficient of the solute element in the liquid, and s * the marginal stability constant considered as21,22):

2  1  ∆T #  ∆TN   #  exp     d∆T  2  ∆Tσ 2π    .........................................(17)

nmax ∆Tσ

..........................(19)

Following the model proposed by Kurz et al.,21) the marginal stability criterion22) is used to relate the radius of curvature at the dendrite tip, r, to the minimum stable wavelength destabilizing a planar front,23) leading to:

d∆T #

#

 2Γ   r .........(18) 

σ *

1

...............................(21)

4π 2

The supersaturation, W c can be expressed thanks to a boundary layer correlation as a function of the dimensionless numbers17):

The three standard parameters introduced are the average undercooling, D TN, the standard deviation, D Ts and the maximum density of nucleation sites, nmax. This procedure follows the model introduced by Rappaz.18) The number of nucleation laws is proportional to the number of boundary conditions defined at the limits of the simulation domain with the macroscopic FE model. Thus nucleation boundary conditions are set on each mold wall. Bulk nucleation requires an additional set of the three parameters of the Gaussian distribution law. A detailed description of other possible nucleation boundary conditions built in a CA model is available in Ref. 19). Since the present model is two-dimensional, stereological relationships are used to link the maximum surface and volume densities entering Eq. (17) to maximum linear and surface densities, respectively.7)

 Ω c
Pe vp exp(Pe vp )  El (Pe vp )      4  El  Pe vp 1    .....(22a) D2 D3 D1 Re Sc sin(φ / 2)      Pe vp
Pe v1
Sc


885

r vp

.............................(22b)

2 Dl rvl ∞ 2 Dl

ν Dl

............................(22c)

...............................(22d)

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Re


1¯0, 01¯]. These values are used to integrate over successive micro time step the extension of the length of the
10 halfdiagonals of cell n , Gn Sn[ij]. Finally, it should be noted that since the growth velocity of all Gn Sn[ij] vectors evolves independently, [ij] ∈ [01, 10, 1¯0, 01¯], the maximum growing shape Anmax needs to be calculated for each growing cell. It is approximated considering a homothetic evolution from the current growing shape. The volume grain fraction, gmm, can thus be updated using Eq. (13).

4Pe vl .............................(22e) Sc

where Pevp is growth Peclet number, Pevl is the flow Peclet number, Sc is the Schmidt number and Re is the Reynolds number, E1 being the integral exponential function.24) The angle f is defined by the
10 growth direction of the dendrite tip and the relative fluid flow velocity far from the interface, vl ∞, and n is the kinematics viscosity. In the following variation domains of the dimensionless number, Pev ∈ [104, 10], Peu ∈ [102, 10] and Sc ∈ [50, 500], the parameters D1, D2 and D3 entering Eq. (22) are equal to (17):

2.3.3. Transport The velocity vector of a cluster, vC, is calculated considering a balance of the forces acting on a mushy zone surrounded by its liquid melt. Assumption is made that the cluster acceleration is negligible. Since the cluster is made of one or several grains connected and moving together at the same speed, an equivalent envelope is considered in the following. The main forces applying to this equivalent envelope are the weight force, P, the Archimedes force, PA and the drag force, FD:

D1
0.5773..............................(23a) D2
0.6596..............................(23b) D3
0.5249..............................(23c) Equations (22) and (23) were obtained by analogy with the Ranz–Marshall correlation25) modified to retrieve the analytical solution provided by Ananth and Gill for the flow around a paraboloidal dendrite tip held in an infinite reservoir of undercooled melt flowing opposite to the growth direction.26) In a purely diffusive regime (Pevl
0), it retrieves the Ivantsov function.20,27) For given values of the physical and thermodynamic properties Dl, n , k and m, using values D1, D2 and D3 given by Eq. (23) and the approximation of s * given by Eq. (21), the variables vp, r and W c can be calculated with Eq. (18), (20) to (22). This requires to provide the growth kinetics model with values for wl s/l, DT, vl ∞, wl ∞ and f . These values are deduced from the information at each cell n . The total undercooling, DT, is the local cell undercooling, DTn . It is calculated as the difference between the local liquidus temperature of the cell deduced from the average cell composition wn , and the cell temperature interpolated from the nodes of the finite element F containing cell n , Tn , using Eq. (10):

P
Me g ..................................(27) PA
M el g ...............................(28) FD


where Me is the mass of the equivalent envelope, Mel is the mass of the liquid in the volume of the equivalent envelope, r l is the density of the liquid, Ae is the area projected in the direction of the relative velocity of the cluster, vC r, and CD is the drag coefficient. Several other forces could be considered, which have been neglected in the following.28) After developing the equilibrium of the forces listed above, the relative grain envelope velocity vC r representing the difference between the cluster velocity, vC, and the fluid flow velocity surrounding the cluster, vl, is given29,30):

DT
DTn
TL(wn )Tn
TMmwn Tn ..........(24)

 2 g s m ( ρ s ρ l )Ve g  v C r
v C  v
  C l  ρ AeCD  

The compositions of the liquid at the dendrite tip solid/liquid interface, wl s/l, is calculated with the phase diagram of the binary alloy accounting for the Gibbs– Thomson effect via the relationship: Tν
TM  mw l s/l 

l

1/2

y .....(30)

where r s is the density of the solid phase of the alloy, Ve is the volume of the equivalent envelope, and g is the norm of the gravity vector acting in direction y. The drag coefficient, CD, is calculated using the correlation originally proposed by Haider and Levenspiel31) for large variation ranges of the Reynolds number, Re (25 000), and then extended by Ahuja29):

2Γ .....................(25) r

We assume here that the composition far from the interface, wl ∞, is given by: wl ∞
wn ..................................(26) This approximation is valid since the solute diffusion length ahead of the dendrite tip remains much smaller than the cell size. Finally, the liquid velocity vector far from the interface, vl ∞, is approximated at cell n by its interpolation from the values at FE nodes using Eq. (10), vl ∞
vnl, and angle f is simply deduced from the direction of the interpolated liquid velocity vector at cell n , vnl ∞, and the
10 directions of the grains growing in cell n . Since there are four
10 directions, four values are computed for the velocity of the growing shape associated to cell n , vpn[ij], [ij] ∈ [01, 10, © 2006 ISIJ

1 l ρ v C r v C r AeCD ....................(29) 2

CD


24Cf C3 .....(31) (1CgC1 ReC2 )  Cg Re 1(C4 / Re)

The two correction coefficients Cg and Cf are introduced by Ahuja29) to enable consideration of the grain envelope shape and porosity. Expressions for coefficients C1, C2, C3 and C4 entering Eq. (31) are computed using the formula given by Haider and Levenspiel.31) These formulas are direct functions of the shape factor, y , defined by the ratio of the equivalent sphere area divided by the equivalent area of 886

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the envelope of the cluster. Coefficient Cg is given with the correlation proposed by Ahuja et al.30) and corrected by de Groh III et al.32):  ψ  Cg
1.26 log10   .....................(32)  0.163 

The equivalent sphere being defined by a volume equal to the volume of the equivalent octahedron, one can write:

 2  tanh(β )    2 β  3 1 β     .........................................(33)

de

2

3

4 d  Ve
π  e  ...........................(40) 3  2  The shape factor y entering Eqs. (31) and (32) is defined as the ratio of the surface of the equivalent sphere divided by the surface of the octahedron:

...............................(34)

2 K

The permeability K is calculated thanks to the Carman–Kozeny relation34) considering an isotropic domain, a tortuosity equal to unity and a Kozeny coefficient equal to 5: K


(1 g Csm )3 5( As/l / V s )2 g Csm

2

ψ


π 1/ 3

................................(41)

3

The internal fraction of solid entering Eqs. (30) and (35) is calculated by averaging the internal volume fraction of solid of the cells n defining the cluster:

......................(35)

where As/l is the interfacial area between the solid and the liquid phases in the equivalent envelope. The internal volume fraction of solid, gCs m, is defined by the ratio (V s/V e), i.e. the volume of the solid phase in the equivalent envelope divided by the volume of the equivalent envelope. The specific area, Sv, defined by the ratio (As/l/V s), varies according to the authors. It has been chosen here to use the expression proposed by Jalanti35) for a hemispherical geometry: Sv


..............................(38)

d  Ae
4π  e  ............................(39)  2 

where coefficient b is defined as the normalized radius of the grain envelope with diameter de and of permeability K:

β


π 1/ 3

where de is the diameter of the equivalent sphere entering Eq. (34). The envelope area Ae and envelope volume Ve of the equivalent sphere entering Eq. (30) are then simply given by:

Neale et al.33) have proposed an extension of the Darcy law valid for highly porosity:  tanh(β )  C f
2 β 2 1  β  

2 dC

de


g Csm


3.

1 N νC

N νC

∑g

sm ν

........................(42)

i
1

Coupling of the CA and FE Models

While the coupling of the CA microscopic models with the kinetics microscopic model has been explained, the coupling with the macroscopic FE model is not straightforward and needs more explanations. In fact, the macroscopic FE model is based on the assumption of equal and constant densities of the solid and liquid phases, further assuming a fixed solid phase. The sedimentation of the grains is accounted for thanks to the CA algorithms, using a solid velocity computed for each cluster (Eq. (30)). This movement of the solid phase generates a movement of the liquid phase and the transport of all quantities that needs to be accounted for at the scales of the CA grid and the FE mesh. The coupling scheme presented hereafter allows for the transport while keeping almost unchanged the set of equations written and solved by the FE method. It is an extension of the coupling algorithm previously developed for energy.37)

6 .................................(36) λ2

Several variables presented above needs to be calculated using an equivalent envelope defined in the concept of the CA model. For that purpose, a regular octahedral shape with its six main directions corresponding to the
100 crystallographic directions in which the dendrite trunks and arms preferentially develop is used. It is to be noticed that, since most of the cluster are made of single equiaxed grains that develop in relatively uniform fields, this shape is in good agreement with previously proposed equivalent envelopes.15,36) The present CA model is two dimensional and four of the six
100 directions lie in the simulation domain (directions noted [ij], [ij] ∈ [01, 10, 1¯0, 01¯] above). The two
100 directions perpendicular to the simulation domain define two pyramids with a common equivalent square base delimited by the two dimensional cluster shape whose growth is simulated. The length of the edge of this square base, dC, is thus easily computed from the number of cells n located in the two dimensional cluster, NnC:

3.1. Transport Figure 4 shows the principle of the transport from an initial position (a) to a final position (b). Upon displacement of the cluster, a translation of the indexes of all its cells n , InS and InG, is operated as indicated by the arrows. The average enthalpy Hn , the temperature Tn , the volume fraction of solid, gns, the volume fraction of the mushy zone gnm, the internal volume fraction of solid, gns m, the average composition, wn , and the size of the mushy zone measured in all
10 directions from the growing center Gn , Gn Sn[ij], [ij] ∈

dC
øNnC lCA ...............................(37)

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ISIJ International, Vol. 46 (2006), No. 6

(12). For the cells n which are not concerned by a movement of the solid phase, vCn
0.

Fig. 4.

3.3. Conversion of Enthalpy and Composition Using Eq. (10), the variations of the average enthalpy, d Hn , and composition, d wn , at cell n are interpolated from the variations over the micro time step of the average enthalpy, d Hn, and composition, d wn, coming from the FE nodes. The temperature and internal solid fraction at time tmd t, Tn
Tntmd t and gns m
gns m tmd t, respectively, are then deduced according to the new updated values of the average enthalpy, Hn
Hntmd t
Hntmd Hn , and composition, wn
wntmd t
wntmd wn . Several cases are encountered for each cell n : c1: The cell is liquid (InS
0):

Schematics of the translation of the information of all the cells defining a cluster of three grains identified with three different grey levels. (a) The initial position of the three grains is translated downward to the underneath cells, as schematized by the short arrows. The liquid cells marked with crosses in (a) are captured due to the translation whereas the liquid cells marked with crosses in (b) are released in the final position of the grains. The information of the liquid cells marked with crosses in (a) is transferred to the cells marked with crosses in (b) located in the same column. This is schematized by the long arrows in (a).

Tn
(HnD slHf)/Cp ........................(44a) gns m
0.................................(44b) c2: The cell is mushy (InS0) with Tntm TE and gns m tm1: Tn
(HnD slHf(1gnmgns m))/Cp ...............(45a)

[01, 10, 1¯0, 01¯], are also moved down to the lower corresponding new cell (small vertical arrows in Fig. 4(a)). It has been chosen to transfer the properties of the liquid cells which are captured at the lower part of the cluster (marked with crosses in Fig. 4(a)) to the liquid cells located in the upper part of the cluster (marked with crosses in Fig. 4(b)), thus ensuring conservation of quantities. The latter transfer is schematized by the arrows going from Fig. 4(a) to Fig. 4(b). The transport of these quantities on the CA grid needs to be sent on the corresponding FE nodes. This is done by computing for each cell the variations associated to the movement of the clusters for the average enthalpy, d Hn, the temperature, d Tn, the volume fraction of solid, d gns, and the average composition, d wn. The corresponding variations at nodes for the average enthalpy, d Hn, the temperature, d Tn, the volume fraction of solid, d gns, and the average composition, d wn, are then calculated using Eq. (11).

gνs m


c3: The cell is mushy (InS0) with Tntm
TE and gns m tm1: Tn
TE ..................................(46a) gns m
1(HnCp TE)/D slHf ..................(46b) c4: The cell is solid (InS0) with TntmTE and gns m tm
1: Tn
Hn /Cp ...............................(47a) gns m
1.................................(47b) where Eq. (45b) gives the internal fraction of solid with the lever rule written for an alloy of average composition wn ,20) i.e., based on a mass balance of solute species in the solid and liquid phases assuming uniform compositions for each phase. The solid fraction of cell n , gns , is then deduced applying Eqs. (13) and (14). The new temperature and volume fraction of solid enable calculations of the variations at the microscopic scale during the micro time step, d Tn
Tn Tntm and d wn
wn wntm, respectively. These variations are reassigned to the FE nodes using Eq. (11) to give the average macroscopic variations, d Tn and d gns. Finally, at the end of the micro time step, the summation of the variations calculated at node n due to the transport of the cluster, dx n, and the conversion of enthalpy and composition, dx n:

3.2. Total Mass Conservation and Liquid Velocity In order to compute the movement of the liquid due to the transport of the solid phase, it is considered that the total mass conservation, Eq. (1), can be applied to each cell. The approximation of equal densities in the solid and liquid phases thus remains and one can write: [glvl]n
[gsvs]n
gnsvns
gnsvCn , where vns
vCn denotes the velocity at which cell n , located in cluster Cn , moves. It is calculated using Eq. (30). The summation of the velocity of the cluster, vCn , over the Nn n cells n seen by node n thus defines d [glvl]n, the variation of the liquid velocity at node n of the FE mesh moving opposite to the solid: 1 δ [ g v ]n
Λn l l

N νn

∑ i
1

cνn δ [ g l v l ]νi
 i

1 Λn

x ntmd t
x ntmdx ndx n ......................(48) with x the average enthalpy and composition, as well as the temperature and the volume fraction of solid.

N νn

∑c i
1

3.4. Friction Force Similarly to the Boussinesq approximation modifying Eq. (3) to compute fluid flow induced macrosegregation, a new approximation is introduced. It consists of a modification of Eq. (5) in order to account for the presence of a moving solid phase at velocity vs. The volumetric friction

n s g v ν i ν i Cν i

.........................................(43) The above equation is nothing but the application of Eq. (11), the normalization coefficient L n being defined by Eq. © 2006 ISIJ

1 TL ( wν )Tν .................(45b) 1 k TMTν

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ISIJ International, Vol. 46 (2006), No. 6

force, M, computed at the FE nodes, writes as follows38): M


g ns m


µ l l l g [ g v  g l v s ] .....................(49) K

K
([(1gf)K d]n[K f]n)1/n ..................(50a) n
0.176 log10(b d)0.275 ..................(50b) where Kd and K f are expressions of the permeability of the dendritic envelope and the extradendritic liquid, respectively. Their expressions are linked to the normalized radius b d and b f via the relationships38):

βf


de

.............................(51)

where vp[ij]n max is the maximum over all mushy (InS
1) growing (InG
1) cells n of the dendrite tip growth velocity of the [ij] directions of the growing shape, with [ij] ∈ [01, 10, 1¯0, 01¯]. Similarly, velocity vCmax is the maximum v over the cells of the displacement velocity of the cluster to which it belongs. Parameter a , with 0a 1 was previously introduced for a better control of the micro time step.19) Since the variations at nodes are calculated with the FE method over a large time step, Dt, time interpolation is also required. A simple linear interpolation is thus performed prior to using Eq. (10):

..............................(52)

2 Kf

The Carman–Kozeny relation34) for K d makes use of Eq. (35) together with the chosen definition of the specific area, Eq. (36). The normalized radius, b d, then writes:

βd


3 5 g s m de

λ2 (1 g s m )3 / 2

.......................(53)

The second normalized radius, b f, is calculated thanks to the model proposed by Happel40): 9 2(4 / 3)η 5 Cf  β
 (1 g f )  5 6 23η3η 2η Cg  2 f

δξn


1/2

if

0gf0.7:

δt ∆ξn .............................(58) ∆t

where D x n and dx n refer to the variations over the macro and micro time steps, respectively. Boxes are identifiable in Fig. 5 that relate to the initializations and solvers of the FE and CA methods. The update of the variables at nodes and cells at the end of each micro time step is also shown, ensuring a large part of the coupling between the two methods.

.....(54)

where h equal (1gf)1/3. The shape factor, Cg, is a function of the volume fraction of extradendritic liquid, gf: if

.............................(56)

d t
Min(a lCA/(vp[ij]n maxvCmax ), Dt)..............(57) n

2 Kd de

1 g nf

3.5. Algorithm Figure 5 presents a flow chart of the coupling scheme developed between the CA and FE methods. Two time stepping loops are shown. The outer and inner loops are performed by the FE and the CA methods, respectively. They correspond to FE macro time steps, Dt, and to a CA micro time step, d t. Adjustment of the micro time step is carried out automatically considering that development of the mushy zone due to the grain growth or the cluster displacement should not exceed one cell size, lCA. This consideration leads to the following limitation of the time step used in the integration scheme of the CA algorithms:

Churchill and Usagi39) and Wang et al.38) have proposed a correlation for the permeability that writes:

βd


g ns

Cg
y 2 ...................(55a)

 ψ  0.7 g f 1: Cg
1.26 log10   .....(55b)  0.163 

4.

Results and Discussion

The experimental configuration proposed by Hebditch and Hunt10,11) is considered for the applications of the CAFE model. It consists of a parallelipedic cavity of 100 mm long, 60 mm high and 13 mm thick. All faces are carefully insulated with the exception of the vertical lefthand-side thinnest mold wall with surface 60 mm13 mm. The Pb–48wt%Sn binary alloy is cooled down and solidified by imposing a circulation of water in a copper chill that is maintained in contact with this face. The measurements of macrosegregation in the final as-cast ingot are carried out at the locations shown in Fig. 6(M). As a consequence of this configuration, a two-dimensional Cartesian approximation of the transport phenomena can be made if one neglects the interactions of the fluid flow with the two largest faces of the mold.41) Table 1 gives the value of both the thermophysical data and the numerical parameters used for all FE simulations. Table 2 lists the complementary values of the thermophysical data and parameters required for a CA calculation.

where the shape factor y has been defined in Eq. (41). Cf is the porosity factor as presented in Eq. (33) in which b has to be replaced by the normalized radius used to describe the dendritic envelope permeability, b d, as defined in Eq. (53). Since the unknown in Eq. (3) is the average liquid velocity, the term proportional to the solid velocity entering Eq. (49) is introduce as a source term computed at each FE node n, (m /Kn) (gnl)2 vns. The average solid velocity at node, vns, is deduced from the velocity at cells, vns
vCn, using the summation over the Nnn cells seen by node n through Eqs. (11) and (30). Similar summations are used to compute the volume fraction of extradendritic liquid and the average diameter of the equivalent sphere, respectively gnf and de n. The internal volume fraction of solid at node n, gns m, is calculated by the ratio of the volume fraction of solid, gns, divided by the volume fraction of extradendritic liquid, (1gnf): 889

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ISIJ International, Vol. 46 (2006), No. 6

Fig. 5.

Flow chart of the coupling algorithm between the finite element and the cellular automaton methods. Table 1.

© 2006 ISIJ

Value of the properties and parameters used for a finite element simulation of the solidification of a Pb–48wt%Sn alloy.35)

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ISIJ International, Vol. 46 (2006), No. 6 Table 2.

Value of additional parameters required for a cellular automaton simulation of the solidification of a Pb–48wt%Sn alloy.

4.1. Numerical Validation Figure 6 presents the results of the simulations. Two calculations are superimposed. The first calculation, displayed as thick grey lines, is carried out using only the FE model. The second calculation is shown by thin black lines in the same figure. It makes use of the CAFE model with a single grain nucleated with no undercooling and growing with almost no undercooling. This is achieved by artificially decreasing the growth undercooling of the solidification front using the dummy growth kinetics parameters given in Table 2. As a consequence, all the cells are captured at a temperature very close to the local liquidus temperature and the volume fraction of solid starts to increase when the enthalpy falls below the enthalpy that corresponds to the local liquidus temperature. The segregation maps and the concentration profiles deduced from the CAFE model is found Table 3. Value of the Sn compositions [wt%] measured by Hebditch.11)

Fig. 6.

Purely macroscopic finite element (FE) simulation (thick grey lines) compared with a coupled cellular automaton–finite element (CAFE) simulation (thin black lines). Maps of the (a) temperature, T [°C], (b) solid fraction, gs [], and (c) relative average Sn composition, (w ¯w0)/w0 [%], are drawn 100 s after the beginning of the simulations. The thick black line labelled GF in (a–c) corresponds to the position of the growth front deduced from the CA model at the same time. When the ingot is fully solidified, (d) the relative composition maps are drawn, together with (e) the average Sn composition profiles, w ¯ [wt%], along the four horizontal lines superimposed on (M) the FE mesh. The FE and CA parameters of the simulations are listed in Tables 1 and 2, respectively. The symbols in (e) correspond to the measured data given in Table 3, with locations and line styles shown in (M). The growth kinetics parameters of the CA model are arbitrary chosen to reach a vanishing undercooling of the growing structure considering a single grain with
10 orientations aligned with the horizontal and vertical directions (i.e., q
0°).

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ISIJ International, Vol. 46 (2006), No. 6

to retrieve the results of a purely FE calculation8,9): plain black curves are superimposed on the grey curves. In fact, such a test validates the numerical implementation of the coupling algorithm developed between the CA and FE method for the growth of a columnar front with no solid transport. Other similar specific numerical simulations have been performed to test the implementation of the coupling algorithms defined for the transport of the solid due to grain movement.7)

7 d. This means that realistic values of the undercooling of the growth front has no large effect on the development of macrosegregation when a single columnar front develops. However, it is interesting to observe the large deviation with the propagation of the growth front. Indeed, 100 s after the beginning of the simulation, the mushy zone is found to be present in about two-third of the simulation domain in Fig. 6(b), to be compared with about one-third in Fig. 7(b). This is due to the small temperature gradient in the experimental configuration of the casting. The few degrees of the undercooling of the tip indeed correspond to a large zone of the liquid melt.

4.2. Growth Undercooling In Fig. 7 a single columnar grain was also nucleated on the left-hand-side wall of the simulation domain with no undercooling and its
10 directions aligned with the Ox and Oy axes (same nucleation procedure as for Fig. 6). The main difference with the previous simulation is that the dendrite tip growth kinetics is computed using Eqs. (18), (20) and (22) and realistic values of the parameters listed in Table 2. The first observation is that, compared to the purely macroscopic FE simulation presented in Fig. 6, little variation of the final macrosegregation map is reached in

© 2006 ISIJ

4.3. Crystallographic Orientation In Fig. 8, a single columnar grain was similarly nucleated on the left wall of the simulation domain with no undercooling but its
10 directions are misaligned with respect to the Ox and Oy axes as shown in Fig. 8(b). The main difference compared to Fig. 7 is observed in the final segregation maps shown in Fig. 8(d), showing perturbations more or less aligned with the
10 directions. This is to be com-

Fig. 7.

Same as Figs. 6(a)–6(d) for a cellular automaton–finite element simulation using realistic parameters for the CA model for the growth kinetics of the structure (cf. Table 2).

Fig. 8.

Same as Figs. 6(a)–6(d) for a cellular automaton–finite element simulation using realistic parameters for the CA model for the growth kinetics of the structure (cf. Table 2) and a single grain whose
10 orientations are misaligned by 24° with respect to the horizontal and vertical directions (i.e., q
24°).

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ISIJ International, Vol. 46 (2006), No. 6

pared with similar results presented in Ref. 42) where the authors could observe highly segregated bands of composition. In fact these bands were due to the quadrangles FE mesh used, aligned with Ox and Oy axes, which generated anisotropic flow velocities. These mesosegregation are still

Fig. 9.

found but are highly reduced by the use of triangular meshes shown in Fig. 6(M). 4.4. Transport of Equiaxed Grains The only difference between Figs. 9 and 10 is the possi-

Same as Figs. 6(a)–6(d) for a cellular automaton–finite element simulation using realistic parameters for the CA model for the growth kinetics of the structure (cf. Table 2) and nucleation of equiaxed grains which remain fixed with time. The final simulated grain structure is shown in (S).

Fig. 10. Same as Figs. 6(a)–6(d) for a cellular automaton–finite element simulation using realistic parameters for the CA model for the growth kinetics of the structure (cf. Table 2) and nucleation of equiaxed grains which are free to move in the liquid. The final simulated grain structure is shown in (S).

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ISIJ International, Vol. 46 (2006), No. 6

bility for the grains to move freely due to the transport of the liquid and sedimentation. As can bee seen, the final grain density in Fig. 10(S) is higher at the bottom of the ingot compared to Fig. 9(S). Explanations involve the following: (i) equiaxed grains accumulates in the bottom region of the ingot when grain movement is permitted and (ii) when the equiaxed grains are fixed, their growth is accompanied by a reheating of the local area that prevents nucleation of additional equiaxed grains to occur. This is well illustrated by comparing Figs. 9(a) and 10(a). Although the general trends of the segregation maps are the same in Figs. 9(d) and 10(d), it is interesting to observe that variations are also predicted at a smaller length scale in Fig. 10(d). These variations are not as well linked to the scale of the grain structure but their origin is the same: the development of equiaxed grains in preferred zones leads to an instability of the segregation field that further develops with the propagation of the mushy zone. To that respect, it is interesting to observe the formation of channels in the top-left side of Fig. 10(d). The mechanism that has led to the formation of this channel is an instability of the growth front. Equiaxed grains have moved and accumulated at a position close to the actual root of the channels, forming arms of fixed equiaxed grains. These arms served as a barrier to the fluid flow that then developed preferentially on its sides. As a consequence, a channel with enriched solute developed between the arms, similar to the mechanism leading to the formation of a freckle.

the segregation maps computed for single grains. It develops in conditions very close to those known by several authors44–46) to promote the formation of “superdendrites”, i.e. limited temperature gradient in the melt and a decreasing growth front velocity. Such a situation is believed to favor the formation of equiaxed grains due to fragmentation taking place in the mushy zone. However, although these conditions were reached, such a fragmentation is not modeled. Instead, because the columnar zone develops in a highly segregated liquid due to layering of Sn in the liquid, the supersaturation of the growth front decreases and lead to the formation of a columnar grain structures in the last solidified top right part of the casting as shown in Figs. 9(S) and 10(S). It is due to the difficulty for equiaxed grains to nucleate, the supersaturation of the melt being reduced by solute enrichment. This equiaxed to columnar transition is little predicted with grains structure models. The reason is that most of the models do not track the development of individual grain formed in casting. Only an average density of grain is considered and, even if it is found to decrease, no information is available on the elongation of the grains due to their growth in solute or temperature gradients. No detailed measurement of the average composition is conducted with the purpose to characterize a segregation map with a definition smaller than the grain size. It is thus difficult to validate the present predictions of mesosegregation. Furthermore, the measurements are often carried out with samples of several cubic millimeters. Such measurements do not permit to validate the simulations, unless the predicted segregation maps are also drawn by averaging the average composition over the same volume as the one used for the measurements. This procedure, however, has not been used here since it strongly rubs out the effect of the grain structure.

4.5. Comparison with Measurements The calculations show that, in the configuration of the experiments performed by Hebditch and Hunt,10) the macrosegregation is not directly influenced by the structural features. Indeed, the predicted composition range is not significantly modified from Figs. 6(d) to 10(d). The effect of the transport of equiaxed grains on the macrosegregation is yet well demonstrated in the literature,2,43) especially for larger ingot sizes. The limited size of the present Pb–48wt%Sn ingot does not favor such effect. It should also be said that the grain structure model only considers the formation of equiaxed dendritic grains: no globulitic grain with a high inner volume fraction of solid is taken into account. Such limitation could be detrimental for the application of the present model to the prediction of segregation induced by grain movement. Not only the morphology of the equiaxed grain structure plays a role on segregation, but also it is expected to considerably modify the fluid flow. Despite the fact that the latter effect is accounted for in the present model, the consequence on the overall fluid flow remains limited and macrosegregation is still mainly explained by the same thermosolutal considerations that lead to the segregation map computed with a purely macroscopic FE calculation.1) The effect of the grain structure on mesosegregation is less investigated in the literature. The fact that instability of the growth front leads to further instability of the segregation map is well established for the formation of channels that could give rise to freckle formation. The present model yet suggests that accumulation of equiaxed grains could also be used to explain the origin of the channels. Instability is predicted by the CAFE model and observed in © 2006 ISIJ

5.

Summary

A fully detailed description of a model based on coupled cellular automaton—finite element methods has been presented for the first time. The model is validated by a comparison with a purely FE calculation. Applied with realistic nucleation and growth conditions, the model does not predict large variations of the macrosegregation maps. Only mesosegregation is predicted, which differs from the purely FE calculation. Instabilities of the segregated maps are produced due to growth front instabilities and accumulation of equiaxed grains. Since the grain structure is difficult to extract from the configuration presented in Refs. 10) and 11) chosen for the simulations, and since the measurements were carried out on a very coarse grid and for relatively large samples (4 mm diameter rods were extracted from the as-cast structure), it is not possible to directly use these experimental results in order to validate the predictions of the CAFE model. Further experimental results should be collected with a detailed characterization of the link between the segregation and the grain structure. Further developments of the model are also required to predict the link between the grain structure and macrosegregation. In particular the possibility to treat solidification of a mushy zone by considering internal fraction of solid 894

ISIJ International, Vol. 46 (2006), No. 6

that is not directly linked to a simple microsegregation law would be useful.

22) 23) 24)

Acknowledgements

This work has been carried out within the framework of the Microgravity Applications Promotion project number AO-99-117 “CETSOL” of the European Space Agency. The cellular automaton model was initially developed within the finite element software calcosoft® (Calcom ESI, PS-EPFL, 1015 Lausanne, Switzerland). All results of simulations presented above were calculated with the cellular automaton model now implemented into the finite element software R2Sol® (SCC, Parc Giron, 42100 Saint-Etienne, France).

25) 26) 27) 28) 29) 30)

31) 32)

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7) 8)

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