Acta Astronautica Vol. 13, No. 11/12, pp. 735-738, 1986 0094-5767/86 $3.00+0.00 Printed in Great Britain. All rights reserved ,~', 1986 Pergamon Journals Ltd

EFFECT OF GRAVITY ON THE SOLIDIFICATION OF BINARY ALLOYS--A NUMERICAL SIMULATIONt

ROGER WEST Department of Casting of Metals, Royal Institute of Technology, S-100 44 Stockholm, Sweden

(Received 27 May 1986)

Abstract--The paper illustrates the influence of gravity on convection in the mushy zone of solidifying alloys by numerical solution of the governing equations for the fluid flow.

Different models for the permeability are compared and the effect of convection in the liquid ahead of the solidification front is taken into account.

1. INTRODUCTION

Convection in the liquid ahead of a solidification front has a definite influence on the formation of structure and segregation in binary alloys. Compared with solidification with a planar front, where the boundary conditions are obvious, the presence of a mushy zone makes calculation of the convection difficult as that the flow in the mushy zone has to be taken into account as well. In a recently published paper a new model for the permeability of the mushy zone is presented and shown to fit experimental results in a more reasonable way than models used earlier. In this paper the influence of gravity on convection in the mushy zone is simulated by numer- ically solving the equations governing the flow, and the effect of convection in the liquid ahead of the solidification front is taken into account.

The major purpose of this paper is to compare different models for the permeability and to calculate the fluid flow for high values of the fraction liquid.

2. PROBLEM STATEMENT

Solidification is assumed to proceed horizontally in a square mould, shown in Fig. 1, where the top and bot tom are thermally insulated and the right wall has a temperature slightly above that of the solidification front.

2.1 Flow in the two-phase zone

The flow through the interdendritic area is de- scribed by D'Arcys law[2]:

Kx ~p Vr

#gt c3x

=

v~ ~gl \Oz

(1)

(2)

tPaper IAF85-998 presented at the 36th Congress of the International Astronautical Federation, Stockholm, Sweden, 7 12 October 1985.

735

In order to simplify as much as possible a steady state is assumed. The equation of continuity is in this case given by:

~x (glv~) + ~z (glv:) = 0 (13)

where g~ = volume fraction liquid. The equations are put in a dimensionless form by

the transformations V = dr~v, X = x/d, Z =- /d , p = pd2/pv 2. To facilitate the computat ion of the flow field a stream function is introduced[2]:

~z - gl I(~, ~.~ = g/V.. (4)

Now eqns (1-3) can be rewritten by eliminating the pressure in terms of the steam function to obtain an elliptic differential equation. The differential equation describing the flow in the two-phase region is now given by:

2.2 Calculation o f the flow field

It is now time to consider the thermal convection ahead of the solidification front. The governing equa-

T=To

Z

T=Tx) X

, ,I,

Fig. 1. Geometry used for calculation.

736 ROGER WEST

tions are the Navier-Stokes equat ions in two dimen- sions:

- - .. - - ~ ' 1 ~ G "l ~ ,cx~ &', &,, &', 1 8p /'~,-t,~ "~' - - ,

8t + v ~ +V~ sx pcJ.,: \ c z - ~,'x-/

+ t : . - ~ + v ~ -~ ~t " oz o x

1 8,o [" 8:v . (7)

8T rgT ~T [82T 82T~ 8t + : - + v~--- = ~1~-~- + w~- ] (8) --- ~ 8z 3x \ e z - ¢ '¥V

and the cont inui ty equa t ion

&,. &,, + ~ = O. (9)

8z cx

These equat ions are pu t in a dimensionless form by the same t rans format ions as in the previous case and a dimensionless t ime and tempera ture are defined as

= vt/d 2 and T = T - To~T , - ~). Transforming the Navier-Stokes equat ions to the

s t reamfunct ion-vor t ic i ty system one now obtains:

8T V cqT 0T 1 - - ~rrV2T (10) & + : ~ + v ~ y -

V : ¢ = - ~,~ ( I 1 )

~?oJ &o ~co G r ~ T + V2o~ (12) a~- + v " ~ + V ~ = ~x

where Pr = v/a and G r = gfl(T - To)d3/v 2.

2.3 Effect o f solidification shrinkage

The effect of solidification shr inkage can be dem- ons t ra ted by assuming a velocity of the solid in the x direction. The equa t ion of cont inui ty is then given

by:

8 8 &-~ (p,gtvL, + p~g.,v,,.~) + ~ (ptg, v,,:) = 0. (13)

The s t reamfunct ion is now to be defined by:

g,V,,: = -~ (14)

gtVL~ + P.,g., V,., ~¢ P., - - + - v , , . ( 1 5 ) p~ ?z Pt

The flow in the two-phase zone is now described by

1 815 1~:, - ¢b V / , - V, ~ - 4 - - - V ,. (16)

• ' ,~b i ' ? ' Z t b '

3. NUMERICAl, SOLUTION

The numerical technique used for the solution ot" eqns (10) and (12) is the ADI method described by Peaceman and Rachford[3] and Roache[4]. Equat ions (5) and (11) are solved by a fast multi-grid relaxation procedure[5,6]. The solidification takes place in a box, which means that one can assume that the velocities perpendicular to the walls are zero, thus ¢ = 0 at the sides of the box. The pcrmeabili t ies are assumed to be equal in the x and - directions and are calculated in the same way as ira a previous paper[I]:

KI = 71g~

K, = 72 (1 - g,) 23 3 + ( Z g - 3 - 3 " l - - g , ;

g / > 1/3

K = K t + K 2. ( 1 7 )

Finally, density is assumed to vary linearly with x in the two-phase zone and the fraction liquid to vary linearly between 0.05 and 0.95. To compare different models for the permeabili ty, the two-phase zone can be assumed to extend all the way over the mould and the interdendri t ic fluid flow can be calculated. Two calculat ions based upon data f rom Table 1 are shown in Figs 2a and 2b, They show the influence of the coefficient Y2 on the flow field.

The flow in the two-phase zone and in the liquid ahead of the solidification front can now be calcu- lated s imultaneously in terms of the s t reamfunct ion and using a 32 x 32 mesh. Figure 3 shows an example where it is assumed that no density differences are present in the two-phase region and convect ion is driven by a small t empera ture difference in the liquid. Figure 4 shows the effect of a decrease in density in the two-phase zone dur ing solidification and the development of a counte r ro ta t ion in the box. This effect can be more clearly studied by setting the tempera ture difference to zero in the liquid and only taking the interdendri t ic density differences into

Table 1. Selected data for calculations; alloy: Sn 5%Pb

P~,h = 10.66044).00139 (T - 327.4) (Mathiak[7]) P~n = 6.98014).00075 (T - 231.9) (Mathiak[7])

p ~ (220 K) = 7180 kg/m 3 VSn = 2.6* 10 -7 m2/s (Mathiak[7]; Thresh[8]) fl = 1.1.10 -4 1/K

Pr = 0.014 (p~ _ pl)/pt = 0.04

d = 0 . 1 m g = 9.81 m/s 2

v~.r = 10 -3 m/s ?l = 6 . 4 . 1 0 -~3m 2 72=8.8 .10 IIm2

Ap i = 1000 kg/m 3

Solidification of binary alloys 737

F ig 2a. Streamfunction for 7~ = 0, A 0 = 0.001

I

Fig. 2b Streamfunction for 72=88"10 ~tm 2 ,A¢ = 0 1

. . . . . . . . . . . . . . o .

. . . . . . . . . . . o , , # .

. . . . . . . . . . . . o ~

. . . . . . . . . . . o o . o

. . . . . . . . . . . . o *

. . . . . . . . . . ° o o | °

. . . . . . . . . . . . * ~ °

. . . . . . . . o ° . o o o

. . . . . . . . . . . o . i

. . . . . . . . . . . ° , .

. . . . . . . . . . . . , o l o

. . . . . . . . . . . . . o ~ o

. . . . . . . . . . . . . . ~ °

# I s ~

I / . , , % %

..~ % 1

. % % %

. . . . . . . . . . . . . . . . J i l l . . . . . . . . . . .

Fig 3 Velocity plot for the case of no density differences in the two-phase zone, Gr = 1 6 , 10 4.

Fig 5 Streamfunction for Gr = 0, At) = 0 1

: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : . . . . . . . . / /

/ \ \ . . . . . . . f

. . . . . . . |

. . . . . . 0 i

. . . . . . . i

. . . . . . . I

. . . . . . . i

. . . . . . . i

. . . . . . , I

. . . . . . , i

. . . . . . 0 i

. . . . . . ° i

. . . . . . , i

. . . . . . t I

. . . . . . . i

. . . . . . 0 i

. . . . . . , I

. . . . . . , I

. . . . . . . i

. . . . . . . i

. . . . . . i i

. . . . . . . . \ \

. . . . . . . . . , . , ,N\ / / . . . 2 . . 2 % - _ / I /

j o . . . . . . . .

i o . . I

° . , i

. . 0 1

° . , i

, . t l

i , . t l

l . i

I t • ' ° l ; , o , i

I h ' ° ' J

L ° o

, . o ° O r l .

~ . . - . s t l °

b . . . . . . . . .

Fig. 4. Velocity plot showing the combined effect of thermal convection and a density difference in the two-phase zone

. . . . . . ~ 1 1 7 7 7 7 -

. . . . . . ~ 1 t 7 7 L J -

. . . . . ~ 1 1 1 7 / ! ~ -

. . . . . ~ I I Z / I ~ - -

. . . . . ~'lllX/l'-

. . . . . plI/ZZI ~-

. . . . . ~ / l l X Z l s -

. . . . . P / l l Z Z t ~ -

...... ~/li/Yl ~-

...... ~III~ IP-

. . . . . . ~ I I / Z : f ' -

. . . . . . t / I Z Z I ~ -

. . . . . . ~ / l l Z , i ~ -

. . . . . ~ / l l l , i ~ -

. . . . . ~ / t 1 1 , 1 ~ -

. . . . . . p / l l l [ l ~ - - s i l l I # p -

. . . . . . . ~/II I S"

. . . . . ~ s e # I I # "

Fig. 6. Velocity plot showing the combined e ~ c t of solidification shrinkage and convection.

738 ROGER WEST

account. Figure 5 shows a case where the liquid flows out of the two-phase zone and forms a convection cell in the liquid. Recalculation of Fig. 5 with the solidification shrinkage added gives the flow field in Fig. 6, where the interaction between the circulating

flow and the flow due to solidification shrinkage is clearly seen.

4. CONCLUDING REMARKS

The interaction between convection in the liquid and fluid flow in the two-phase zone of solidifying alloys has been studied by numerical simulation. The rapidly increasing permeability for high values of the fraction liquid gives a strong interaction with thermal convection in the bulk liquid and the flow velocities at the solidification front could be calculated.

Acknowledgements--This work was sponsored by the Swed- ish Board for Space Activities. The author expresses his

gratitude to Prof. Hasse Fredriksson and Dr Laszlo Fuchs for stimulating discussions.

REFERENCES

1. R. West, On the permeability of the two-phase zone during solidification of alloys. Met. Trans. 16A, 693 000 (1984).

2. R. A. Greenkorn, Flow Phenomena in Porou~ Media. Marcel Dekker, New York (1983).

3. D. W. Peaceman and H. H. Rachford, J. Soc. hul. Appl. Math. 3, 28-41 (1955).

4. P. J. Roache, Computational Fluid Dynamic.~. Hermosa Albuquerque, New Mexico (1972).

5. A. Brandt, Mathematics of Computation 31, 333 390 (1977).

6. L. J. Fuchs, TRITA-GAD-4, Royal Institute of "Fcch- nology, Stockholm (1980).

7. E. Mathiak, W. Nistler and W. Waschkowski, Pr~izisionsmessungen der Dichte von geschmolzenem Gallium, Zinn, Kadmium, Thallium, Blei, Wismut. Z. Metall. 74, 793--796 (1983).

8. H. R. Thresh and A. F. Crawley, The viscosities of lead, tin, and Pb-Sn alloys. Met. Trans. I, 531 535 (1970).