JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

44, 2, pp. 367-379, Warsaw 2006

SIMPLIFIED MODELS OF MACROSEGREGATION

Bohdan Mochnacki

Institute of Mathematics and Computer Science, Technical University of Częstochowa

e-mail: moch@imi.pcz.pl

Józef S. Suchy

Department of Foundry Processes Modelling, AGH, Cracow

e-mail: jsuchy@agh.edu.pl

Themacrosegregationprocess takes place during typical solidification of
alloys. Fractions of alloy components in a liquid and solid sub-domains
are time-dependent and determined by the course of border lines on the
equilibrium diagram. From the mathematical point of view, the process
is described by a system of partial differential equations (diffusion equ-
ations) and boundary-initial conditions. The process is coupledwith the
solidification one. In this paper, simplified models of macrosegregation
are discussed. The volumetric solidification and the ’sharp’ solid-liquid
interface are considered. Examples of computations are also shown. It
seems that forpractical applications, themethodsproposedare sufficien-
tly exact. Additionally, they are very simple for numerical realization.

Key words: macrosegregation, solidification, numerical simulation

1. Introduction

The proceeding of a macrosegregation process in the casting domain is
described by a system of equations in the form (Crank, 1984)

x∈Ωm :
∂zm(x,t)

∂t
=∇[Dm∇zm(x,t)] (1.1)

where m = 1,2 correspond to the liquid and solid sub-domains, zm(x,t) is
the alloy component concentration, Dm – diffusion coefficient, x, t – spatial
co-ordinate and time.



368 B. Mochnacki, J.S. Suchy

On the moving boundary Γ12 limiting the liquid and solid sub-domains,
the following boundary condition is given (Fraś, 1992; Majchrzak et al., 1998)

D2
∂z2(x,t)

∂n

∣

∣

∣

x=ξ
−D1

∂z1(x,t)

∂n

∣

∣

∣

x=ξ
=(1−k)

dξ

dt
z1(ξ,t) (1.2)

where ∂/∂n denotes the normal derivative, x = ξ is the solid-liquid interfa-
ce, k = z2/z1 is the partition coefficient. It should be pointed out that the
solidification rate dξ/dt results from the solution of the solidification model.
The position of ξ corresponds to the liquid border temperature TL or to the
equivalent solidification point defined as follows

T∗ =

TL
∫

TS

C(T)T dT

TL
∫

TS

C(T) dT

(1.3)

where TS is the temperature corresponding to the endof solidification, C(T) is
the substitute thermal capacity of the alloy (Mochnacki andMajchrzak, 1995;
Mochnacki and Suchy, 1997).

On the outer surface of the system, the no-flux condition is accepted. This
means

x∈Γ0 :
∂z2(x,t)

∂n
=0 (1.4)

Additionally, for time t=0: z1(x,0)= z0.

The model presented can be useful if we describe the alloy solidification
using the classical Stefan approach (Mochnacki and Suchy, 1995) because the
position of solid-liquid interface and solidification rate must be known (see:
Crank, 1984; Mochnacki and Suchy, 1995). On the other hand, however, the
solidification proceeds in an interval of temperature and the Stefan model
concerning puremetals is not entirely acceptable.

The obtainment of numerical solution to the problempresented is possible,
of course, but taking into account mutual connections between the solidifica-
tion and macrosegregation models, the task is rather complex. Additionally,
as was mentioned, the temporary position of the solid-liquid interface must
be known, in other words we can consider the process in which the ’sharp’
solidification front is generated. In the case of volumetric solidification, such
an approach to the macrosegregation modelling is useless.



Simplified models of macrosegregation 369

2. The macrosegregation during volumetric solidification

The simplest and the well known criterion determining the type of the
solidification is based on the ratio K = ∆T/(TL − TS), where ∆T is the
maximum change of temperature in the casting domain. If K ¬ 1 then the
’sharp’ front appears, if K ≈ 1 then the volumetric solidification takes place.
Below the approach based on the lever arm rule and the Scheil models

for such a situation will be presented. At first, we assume a constant value of
the mass density and then, in the place of mass balances, we can analyze the
volume ones.
Let t and t+∆t denote two successive levels of time. Then

V2(t)z2(t)+V1(t)z1(t)=V2(t+∆t)z2(t+∆t)+V1(t+∆t)z1(t+∆t) (2.1)

Using the Taylor formula, one obtains (m=1,2)

Vm(t+∆t)=Vm(t)+
dVm(t)

dt
∆t

(2.2)

zm(t+∆t)= zm(t)+
dzm(t)

dt
∆t

Introducing theabove formulas tobalance (2.1) andneglecting the components
containing ∆t2, one arrives at

f2
dz2
dt
+
df2
dt
z2+f1

dz1
dt
+
df1
dt
z1 =0 (2.3)

where

f2(t)=
V2(t)

V
f1(t)=

V1(t)

V
f2(t)= 1−f1(t) (2.4)

Next, introducing the partition coefficient and using the dependence
f2 = 1− f1, we obtain the final form of the balance equation. It should be
solved for the initial condition in the form: z = z0: f1 = 1. Assuming the
constant value of the partition coefficient k, we find

f1 =
z0−kz1
(1−k)z1

(2.5)

The above solution corresponds to the solution resulting from the well known
lever-arm principle, in other words in equations (2.1) D1 →∞,D2 →∞. The
same equation can be used in order to find the solution to the so-called Scheil



370 B. Mochnacki, J.S. Suchy

model (diffusion in the solid state is neglected, D2 = 0, D1 → ∞). Let us
assume that dz2/dt=0 and then

f1 =
(z0
z1

)

1

1−k

(2.6)

The knowledge of temporary f1(t) in the casting domain (this value results
from the solidification model) allows one to determine z1(t) and next to cor-
rect the values of border temperatures TL and TS. Solutions (2.5) and (2.6)
have been obtained an the assumption that the functions f1(t) and f2(t) are
uniform in the whole casting domain. In reality, the local values of solid or li-
quid volumetric fractions can change from 0 to 1. So, a better approach to the
mass balances results from the introduction of the control volume approach.
The casting domain is divided into n control volumes and then one obtains
the following formulas determining temporary values of z1(t)

z1(t)=
Vρ1z0

k
∑n
i=1∆Viρ2f2i(t)+

∑n
i=1∆Viρ1[1−f2i(t)]

(2.7)

or

z1(t
p)=

VρLz0−
∑p−1
s=1

∑n
i=1∆Viρ2z2(t

s)(fs2i−f
s−1
2i )

k
∑n
i=1∆Viρ2(f

p
2i−f

p−1
2i )+

∑n
i=1∆Viρ1(1−f

p
2i)

(2.8)

where V is the casting domain, ∆Vi are the control volumes. Additionally,
it is assumed that the mass densities of solid and liquid phases are different.
Formula (2.7) concerns the lever-armmodel, while formula (2.8) concerns the
Scheil one. In the case of Scheil approach, we must remember the ’history’ of
the solidificationprocess and t0 =0, t1, t2, . . . , tp, . . .denote thepoints forming
the time grid (Majchrzak and Szopa, 1998; Mochnacki et al., 1999).
In the quoted papers, the examples of numerical computations are also

presented. As an example, the solidification of spherical casting (R=0.05m)
made of Cu-Zn alloy (10%Zn) was considered (Mochnacki et al., 1999). The
following thermophysical parameters were introduced there: λ1 = λ2 = λ =
120W/(mk), c1 = c2 = c = 390J/(kgK), ρ1 = ρ2 = ρ = 8600kg/m

3,
LV = 1.63 · 106kJ/m3 (latent heat), k = 0.855, the function TL = f(z1)
is of the form TL =1083−473.68 · z1, T0(r) = 1080◦C (initial temperature),
z0 = 0.1 (initial concentration of Zn). On the outer surface of the casting,
the Robin condition was assumed (heat transfer coefficient α=35W/(m2K),
ambient temperature T∞ =0◦C).
InFigure 1, the kinetics of solidification (the course of f2(t)) is shown.The

next figure illustrates the cooling curves for r = R (casting surface), at the
same time the numbers 1, 2, 3 correspond to the model without segregation,



Simplified models of macrosegregation 371

lever-arm model and the Scheil one. In Fig.3, changes of the solidification
point are marked.

Fig. 1. Kinetics of solidification

Fig. 2. Cooling curves

3. The models of macrosegregation in the case of ’sharp’ interface

In the papers by Mochnacki et al. (2003, 2004), Suchy and Mochnacki
(2003), the approximation of the alloy concentration in themoltenmetal sub-
domain by the broken line aas discussed. The first part of this function corre-
spondsto theboundary layer δ (Suchy, 1983),while the secondonecorresponds



372 B. Mochnacki, J.S. Suchy

Fig. 3. Changes of the solidification point

Fig. 4. The broken line model

to the sub-domain in which the convectional mass flow causes equalization of
the function z1 (Fig.4).

The amount of information concerning the physical aspects of the process
(solidification rate, thickness of the boundary layer, etc.) assures the univocal
determination of the parameters of the assumed function. The starting point
of the algorithm consists in computations of the direction of a sector corre-
sponding to the boundary layer. Next, the mass balance of alloy components
allows one to determine other parameters of the broken line model.

We consider the solidification problem for which the temporary position
of interface and the function determining its dislocation are known. Themass
balance for the neighborhood of themoving boundary leads to condition (2.2).



Simplified models of macrosegregation 373

If themass transfer in the solid body is neglected (D2 =0) and a 1Dproblem
(plate of thickness L/2) is considered, then

−D1
∂z1(x,t)

∂x

∣

∣

∣

x=ξ
=(1−k)

dξ

dt
z1(ξ,t) (3.1)

On the basis of the last formula, we determine the slope of the first section of
the broken line for x= ξ (ξ is amultiple of the assumed step ∆x=h). Next,
on the basis of the balance for time t corresponding to x= ξ, namely

ξ
∫

0

z2(x) dx+

ξ+δ
∫

ξ

z11(x) dx+z12
(L

2
− ξ− δ

)

=
L

2
z0 (3.2)

where z11 is a linear function approximating the concentration field in the
domain of boundary layer, z12 is a constant value (see: horizontal sector in
Fig.4), we can calculate the alloy component concentration for x = ξ. In
this way, the set of parameters determining the course of the broken line is
known.
As an example, a plate (2L = 0.018m) made of Al-Si alloy (z0 = 0.05)

has been considered. The constant solidification rate dξ/dt = 2 · 10−6m/s,
partition coefficient: k=0.2, diffusion coefficient: D1 =3.5·10−8m2/s, thick-
ness of boundary layer δ = 0.5, 1, 1.5mm have been assumed, respectively
(Suchy, 1983). In Figure 5, changes of the concentration in the liquid sta-
te for x = ξ(t) are shown. The solution presented in Fig.5 shows that the
assumption concerning the thickness of the boundary layer does not cau-
se essential differences in the calculated courses of boundary and internal
concentrations.
In Figure 6, the concentration profiles for different times are shown. The

thickness of the boundary layer equals 1.5mm.
The solutionpresented inFigure 7hasbeen found for variable solidification

rate.Thewell knownequation ξ=β
√
thasbeentaken intoaccount (β=6.32·

10−5) and the solidification rate resulted from differentiation of the formula
discussed. From the numerical point of view, such a problem is not more
complicated than the problems discussed previously.
Similar considerations can be done in the case of cylindrical or spherical

geometry (Mochnacki et al., 2005).
The collocation method presented inMochnacki and Suchy (1995) can be

a base for other segregationmodel (1D task is also considered). Themathema-
tical description of the process bases on the Fick equation. Themass transfer
process in the solidified part of the casting has been neglected, while the do-
main of liquid metal in which the Fick equation is obligatory corresponds to



374 B. Mochnacki, J.S. Suchy

Fig. 5. Concentration for x= ξ(t)

Fig. 6. Concentration (δ=1.5mm)

a certain layer δ close to the solidification front. For the remaining part of
the liquid sub-domain, we assume a constant value of the alloy component
concentration. Thus, we have

x∈ (ξ,ξ+ δ) :
∂z1(x,t)

∂t
=D1

∂2z1(x,t)

∂x2
(3.3)

For x= ξ the condition (3.1) is given.



Simplified models of macrosegregation 375

Fig. 7. Concentration field for v(t)

In order to assure the constant mass of alloy component, the following
condition should be formulated, see (3.2)

z0
L

2
=

ξ
∫

0

z2(x) dx+

ξ+δ
∫

ξ

z1(x,t) dx+z1(ξ+ δ,t)
(L

2
− ξ− δ

)

(3.4)

In order to estimate the co-ordinate ξ and the parameter v, one should find
a numerical solution to the solidification problem or assume the knowledge of
solidification rate.
In this place, the model of solidification based on the control volume me-

thod (Mochnacki andCiesielski, 2002)] can be used. Here, some remarks con-
cerning the model of segregation will be discussed. In the layer δ, we distin-
guish the set of points x0,x1, . . . ,xn. The concentration field for time t+∆t
is assumed in the form of the algebraic polynomial

z1(x,t+∆t)=
n
∑

j=0

ajx
j (3.5)

The first and the second derivatives of (3.5) are equal to

dz1(x,t+∆t)

dx
=
n
∑

j=1

jajx
j−1

(3.6)

d2z1(x,t+∆t)

dx2
=
n
∑

j=2

j(j−1)ajxj−2



376 B. Mochnacki, J.S. Suchy

The numerical approximation of the Fick equation for x=x2, . . . ,xn−1 takes
a form

n
∑

j=0

ajx
j
i = zi(xi, t)+D1∆t

n
∑

j=2

j(j−1)ajx
j−2
i (3.7)

It should be pointed out that the course of z1(x,t) is known from the initial
or pseudo-initial conditions. For x=x0, we have

a0ν(1−k)+D1a1 =0 (3.8)

Themass balance leads to the equation

z0
L

2
=

ξ
∫

0

z2(x) dx+

ξ+δ
∫

ξ

n
∑

j=0

ajx
j dx+z1(xn, t+∆t)

(L

2
−ξ− δ

)

(3.9)

Equations (3.7)-(3.9) create a linear system fromwhich the coefficients ak can
be found. Next, we can define the continuous function z1(x,t+∆t).

As itwasmentioned, thebroken linemodel gives the solution in the formof
C0 type. So, the thus obtained function z1(t) is not differentiable and it is, to
a certain extent, the fault of themethod proposed. For the same assumptions,
it is possible to construct the distribution of z1(t) in the form (a 1D task is
considered)

x∈ [ξ,ξ+ δ] : z1(t)= a0+a1x+a2x2

x∈ [ξ+ δ,L] : z1(t)=A0 = const
(3.10)

The parameters of the above distribution result from the mass balance, bo-
undary condition given on the liquid-solid interface and the assumption con-
cerning continuity of the first derivative for x = ξ+ δ (∂z1/∂x = 0). The
number of unknownparameters corresponds to the number of conditions, and
the temporary values of a0,a1,a2 can be easily found (on the assumption that
the solidification rate is known).

As an example, distributions of z1(t) in the domain of a plate (L=2cm)
made ofAl-Si alloy (z0 =0.05) have been determined. In the first version, the
constant solidification rate ν = 2 · 10−6m/s has been assumed. In Figure 8,
the solutions for 0.5 and 1.5mm boundary layers are shown.

In the second version of the solution, the model of solidification basing
on the CVM algorithm has been introduced, while the macrosegregation one
resulted from the parabolic approximation of z1. In Figure 9, the results obta-
ined for δ=1 and 1.5mm and the physical parameters quoted inMochnacki
and Suchy (1995) are presented.



Simplified models of macrosegregation 377

Fig. 8. Distribution of z1, (a) – 0.5mm, (b) – 1,5mm

Fig. 9. Distribution of z1 (solidification)

The testing computations show that the model incorporating the parabo-
lic approximation gives good results for rather small solidification rates (e.g.
system casting-sand mixmould).

Summing up, the numerical solutions discussed in this paper concern 1D
problems. It is, of course, the self–evident limitation of their applications. On
the other hand however, in the initial stages of solidification (at that time
the heat andmass transfer proceed very intensively and determine the further
courseof theprocess analyzed) the real geometry of thedomain isnot essential,
and the 1D solution is quite acceptable. So, the applications of presented
models go beyond the 1D limit.



378 B. Mochnacki, J.S. Suchy

Acknowledgement

The paper has been sponsored by StateCommittee for ScientificResearch (KBN)

under Grant No. 3T08B00428.

References

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Simplified models of macrosegregation 379

Uproszczone modele makrosegregacji

Streszczenie

W pracy przedstawiono opismatematyczny procesu segregacji składników stopo-
wych w objętości krzepnącego odlewu. Wskazano na trudności związane z rozwiąza-
niem odpowiedniego problemu brzegowo-początkowego, a w dalszej części artykułu
przedstawiono propozycje rozwiązań przybliżonych. Rozpatrywano zarówno problem
krzepnięcia objętościowego, jak i klasyczne zadanie Stefana. Rozważania teoretyczne
zilustrowano przykładami obliczeń numerycznych.

Manuscript received December 15, 2005; accepted for print January 23, 2006