J O U R NA L O F M AT E R I A L S S C I E N C E L E T T E R S 1 7 ( 1 9 9 8 ) 2 0 2 9 ± 2 0 3 1

Three-zone liquid metals solidi®cation model

S. I . BAKHTIYAROVSpace Power Institute, 231 Leach Center, Auburn University, Auburn, AL 36849, USA

R. A. OVERFELTMechanical Engineering Department, 201 Ross Hall, Auburn University, Auburn, AL 36879, USA

Numerous mathematical models have been devel-oped to predict a solidi®cation process of metals.Tiedje et al. [1] developed a numerical model forsolidi®cation of dendritic and cellular arrays. Themodel treats directional solidi®cation, and it makesno prior assumptions of the shape or scale of themicrostructure. The model shows that primarydendrite spacing is proportional to the square rootof the local solidi®cation time, and it is in goodagreement with experiments. A multiscale=-phaseapproach to the modeling of dendritic alloy solidi-®cation was proposed by Wang and Beckermann [2].The volume-averaging technique has been used todevelop macroscopic transport equations for solid,interdendritic and extradendritic liquid phases. Thismodel has been applied to investigate equiaxeddendritic solidi®cation of an Al±4 wt % Cu alloy ina rectangular cavity. The numerical solution of theFourier heat-transfer equation has been used by Fraset al. [3] for thermal analysis of solidi®cationkinetics in high-carbon Fe±C alloys. Predictionsare in close agreement with the experimental data.Applying the adjoint method and a functionaloptimization scheme, Yang and Zabaras [4] solvedan inverse solidi®cation problem with natural con-vection. The optimal heat ¯ux at the mold-wallboundary on both the solid and liquid mold sides hasbeen found for directional solidi®cation processes.Prescott and Kim [5] used a continuum model tosimulate thermosolutal convection during the solidi-®cation of metal alloys. It is concluded that ®rst-order integration methods are not suited for thisproblem and that higher-order schemes are needed.The continuum approach has also been used by Chenand Tsai [6] to calculate ¯uid ¯ow and solidi®cationpatterns in castings. Sundarraj and Voller [7] intro-duced a micro-scale model that would be suitable foruse in a micro-macro model of solidi®cation. Topredict microporosity evolution due to solidi®cationshrinkage and the rejection of dissolved gas duringcasting solidi®cation, Suri and Paul [8] developed athree-dimensional numerical model that comparedwell with experimental data for sand-cast aluminumalloy Al±7 wt % Si.

However, all these approaches did not consider thein¯uence of inter-phase stresses on the solidi®cationcharacteristics. Some researchers [5, 9] consideredthe stress and pressure ®elds at the mold=castinginterface, but not at the interface between differentphases of casting.

It is commonly recognized that liquid agitationduring metal solidi®cation assists in crystal agglom-eration, as in dendrite fragmentation. There arenumerous methods for moderate stirring duringmetal solidi®cation (electromagnetic, acoustic, mec-hanical, etc.). But even a natural thermosolutalconvection promotes vigorous interdendritic ¯uid¯ow, and ¯ow velocity might reach 1 cm=s [10]. It isthe aim of this letter to develop a mathematicalmodel that will predict the relationship between theinterphase stresses and the solidi®cation character-istics of liquid matrices.

Consider a differential control volume, of lengthdx, and width w, with three solidi®cation zones:liquid, mushy (interdendritic ¯uid ¯ow) and solid, asshown in Fig. 1. The mushy zone thickness is ä.There is a mass transfer across surfaces ab and cd ofsub-differential control volume abcd. Because con-trol surface ad is adjacent to a solid boundary, amass transfer through this surface is the same as thesolidi®cation rate _ms. To maintain the volume of themushy zone constant, there will be mass transferacross surface bc. We assume that:

· the process is isothermal and temperatures at allthree zones are the same;

· there is steady ¯ow at liquid and mushy zones;· there is incompressible ¯ow at liquid and mushy

zones; and· there are no pressure forces.

0261-8028 # 1998 Kluwer Academic Publishers

Liquid Zone

Ul

Mushy Zone δ

ms

Solid Zone

dx

y

x

c

da

b

Figure 1 Differential control volume of three-zone solidi®cation

model.

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We will apply continuity and linear momentumequations using the differential control volume:

= . r~V � @r@ t� 0, (1)

rD~V

Dt� r~g ÿ = p: (2)

The continuity equation for the control volumeabcd is:

_mab � _mcd � _mad � _mbc � 0, (3)

where

_mab � ÿ�ä

0

ru dy

( )w

is the mass ¯ux through surface ab;

_mcd ��ä

0

ru dy� @

@x[r udy]dx

( )w

is the mass ¯ux through surface cd;

_mad � rQ0 � rv0wdx

is the mass ¯ux due to solidi®cation (through surfacead).

Substituting the expressions for _mab, _mcd and _mad

into the continuity Equation 3 we obtain the mass¯ux between liquid and mushy zones (throughsurface bc):

_mbc � ÿrv0wdxÿ @

@x

�ä0

rudy

" #dx

( )w: (4)

Now we apply the x component of the momentumequation to control volume. From the basic equation:

@

@ t

�CV

urd8 ��

CS

ur~V . d~A � FSx � FBx (5)

where FSx and FBx are surface and body forces,respectively. We would assume that ¯ow is steadyand FBx � 0. The shear force acting between themushy and solid zones (ad) is:

Fad � ÿôwwdx

We then have the following equation:

ÿôwwdx � momab � momcd � mombc � momad

(6)

where

momab � ÿ�ä

0

uruwdy

is the momentum ¯ux through surface ab;

momcd ��ä

0

uruwdy� @

@x

�ä0

uruwdy

" #dx

is the momentum ¯ux through surface cd;

mombc � Ul _mbc

is the momentum ¯ux through surface bc.Substituting these expressions and Equation 4 into

the x momentum Equation 6, we obtain

ôw

rU 2l

� v0

Ul

� @

@x

�ä0

u

Ul

1ÿ u

Ul

� �dy (7)

If we consider the mushy zone as a porousmedium with the permeability k, then super®cialvelocity u can be expressed by the Darcy equation:

u � ÿ kì

@ p

@x� k

ì

ôw

L(8)

Combining Equations 7 and 8, we obtain thefollowing expression:

Prÿ2 � v0

Ul

� ä

L

krUl

ìLPrÿ2 1ÿ krUl

ìLPrÿ2

� �� �(9)

where

Pr � Ul

rôw

� �1=2

is the Prandtl number that represents the ratio ofinertia forces to wall shear forces.

Dimensionless parameter

ø � krUl

ìL

represents the ratio of inertia forces to ®ltrationforces.

In Fig. 2, we show computation results that displaya variation of the dimensionless solidi®cation rate(v0=Ul) as a function of the Prandtl number atø � 1 and ä=L � 0:1, 0.5, 1.0. As seen from the®gure, the Prandtl number plays a predominant rolein determining the solidi®cation rate. At Pr � 1(when inertia forces are equal to wall shear forces) asolidi®cation rate (v0=Ul) is not a function of themushy zone thickness ä at a given value ofdimensionless parameter ø. At Prandtl numbers lessthan 1, a solidi®cation rate increases with increasingthe mushy zone thickness. At Pr . 1, a solidi®cation

1.00E104

1.00E102

1.00E100

1.00E202

1.00E2041.00E201 1.00E100 1.00E101

Prandtl Number

υ 0/U

l

Figure 2 Variation of the solidi®cation rate with the Prandtl number at

ø � 1 ( ä=L � 0:1; ä=L � 0:5; ÿ ÿ ÿ ä=L � 1:0):

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rate decreases with increasing the mushy zonethickness.

Fig. 3 represents a variation of the dimensionlesssolidi®cation rate (v0=Ul) as a function of thePrandtl number at ä=L � 1 and ø � 0:1, 0.5, 1.0.Again, when inertia forces are equal to wall shearforces (Pr � 1), dimensionless parameter ø does notaffect a solidi®cation rate. At Pr , 1, a solidi®cationrate increases with parameter ø. However, at Pr . 1a solidi®cation rate decreases with increasing para-meter ø.

A three-zone mathematical model has been devel-oped to predict the relationship between the inter-phase stresses and the solidi®cation characteristics ofliquid matrices. An expression is obtained to relate asolidi®cation rate to Prandtl number, and a dimen-sionless number that represents the ratio of inertiaforces to ®ltration forces. Numerical simulationsshow that at Pr � 1, a solidi®cation rate does not

depend on the thickness of the mushy zone or on theratio of inertia forces to ®ltration forces.

AcknowledgmentThe authors gratefully acknowledge the ®nancialsupport received from NASA's Of®ce of Life andMicrogravity Sciences and Applications under Co-operative Agreement No. NCC8-128.

References1. N. T I E D J E , P. N. H A N S E N and A . S . P E D E R S E N,

Metall. and Mater. Trans A 27A (1996) 4085.

2. C . Y. WA N G and C . B E C K E R M A N N, in `̀ Proceedings of

the ASME Transport Phenomena in Solidi®cation Confer-

ence'', HTD-284=AMD-182 (1994) 75.

3. E . F R A S , W. K A P T U R C K I E W I C Z , A . B U R B I E L KO

and H . F. L OÂ P E Z , Metall. and Mater. Trans B 28B (1997)

115.

4. G . Z . YA N G and N. Z A BA R A S , in `̀ Proceedings of the

ASME National Heat Transfer Conference'', HTD-323

(1996) 1.

5. P. J. P R E S C OT T and B. K I M , ibid. HTD-323 (1996) 223.

6. J. H . C H E N and H . T S A I , in `̀ Proceedings of the AFS

100th Casting Congress,'' 96 (1996) 177.

7. S . S U N DA R R A J and V. R . VO L L E R , in `̀ Proceedings of

the ASME National Heat Transfer Conference,'' HTD-323

(1996) 35.

8. V. K . S U R I and A . J. PAU L, in `̀ Proceedings of the ASME

Micro=Macro Scale Phenomena in Solidi®cation Confer-

ence'', HTD-218=AMD-139 (1992) 101.

9. F. Y I G I T and J. R . BA R B E R , ibid. HTD-218=AMD-139

(1992) 141.

10. A . H E L L AW E L L , J. R . S A R A Z I N N and R . S . S T E U B E ,

Phil. Trans. R. Soc. London A345 (1993) 507.

Received 9 Julyand accepted 22 July 1998

1.00E104

1.00E102

1.00E100

1.00E202

1.00E2041.00E201 1.00E100 1.00E101

Prandtl Number

υ 0/U

l

Figure 3 Variation of the solidi®cation rate with the Prandtl number at

ä=L � 1:0 ( ø � 0:1; ø � 0:5; ÿ ÿ ÿ ø � 1):

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