Effect of gravity on the solidification of binary alloys—A numerical simulation

Acta Astronautica Vol. 13, No. 11/12, pp. 735-738, 1986 Printed in Great Britain. All rights reserved

0094-5767/86 $3.00+0.00 ,~', 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 numerically 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.

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 computation 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. PROBLEM STATEMENT

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

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

2.1 Flow in the two-phase zone

T=Tx) X

The flow through the interdendritic area is described by D'Arcys law[2]: Kx ~p #gt c3x

Vr

v~

=

~gl \Oz

,

(1)

,I,

(2)

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

Z

Fig. 1. Geometry used for calculation. 735

736

ROGER WEST

tions are the Navier-Stokes equations in two dimensions:

&',

&,,

&',

-

1 -

8t + v ~

8p

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

- - - ~ - .. ~ ' G /'~,-t,~ "~'"l1 ~,cx~

•

V, ~ '

1 815

1~:, - ¢b

4---

,~b i'?'Z

tb

V ,.

(16)

'

,

+V~ s x

pcJ.,:

\cz-

~,'x-/ 3. NUMERICAl, SOLUTION

~t

+ t : . - ~ + v ~ -~

" oz

ox

1 8,o 8T

rgT

~T

["

8:v.

[82T

82T~

8t + : -

---

+ v~--- = ~1~-~- + w ~ - ] ~ 8z 3x \ez¢'¥V

(7) (8)

a n d the continuity e q u a t i o n &,.

&,, + ~ = O. 8z cx

(9)

These equations are p u t in a dimensionless form by the same t r a n s f o r m a t i o n s as in the previous case and a dimensionless time a n d t e m p e r a t u r e are defined as = vt/d 2 and T = T - T o ~ T , - ~). T r a n s f o r m i n g the Navier-Stokes equations to the streamfunction-vorticity system one now obtains: 8T V cqT 0T :~+v~y-& - + V:¢

~?oJ

&o

=

~co

a~- + v " ~ + V ~

1 ~rrV2T

=

-

~,~

(10) (I 1 )

G r ~ T + V2o~

~x

(12)

where Pr = v/a and G r = gfl(T - To)d3/v 2. 2.3 Effect o f solidification shrinkage The effect of solidification shrinkage can be demo n s t r a t e d by assuming a velocity of the solid in the x direction. The e q u a t i o n of continuity is then given by: 8

8

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

(13)

The s t r e a m f u n c t i o n is now to be defined by:

g,V,,: = -~ gtVL~ + -P.,g., V,., p~

-

(14) ~¢ ?z

+-

P., v , , . Pt

(15)

The numerical technique used for the solution ot" eqns (10) and (12) is the A D I m e t h o d described by P e a c e m a n and Rachford[3] and Roache[4]. Equations (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 pcrmeabilities 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, = 7 2 ( 1 - g,) 23 3 + ( Z g - 3 "

l--g,

3 ;

g / > 1/3 K = K t + K 2.

(17)

Finally, density is assumed to vary linearly with x in the two-phase zone and the fraction liquid to vary linearly between 0.05 a n d 0.95. To c o m p a r e different models for the permeability, the two-phase zone can be assumed to extend all the way over the m o u l d and the interdendritic fluid flow can be calculated. Two calculations based u p o n data from Table 1 are shown in Figs 2a a n d 2b, They show the influence of the coefficient Y2 on the flow field. The flow in the two-phase zone and in the liquid a h e a d of the solidification front can now be calculated simultaneously in terms of the streamfunction 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 convection is driven by a small t e m p e r a t u r e difference in the liquid. Figure 4 shows the effect o f a decrease in density in the two-phase zone during solidification and the d e v e l o p m e n t of a c o u n t e r r o t a t i o n in the box. This effect can be more clearly studied by setting the t e m p e r a t u r e difference to zero in the liquid and only taking the interdendritic 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=0.1m g = 9.81 m/s 2 v~.r = 10-3 m/s ?l = 6 . 4 . 1 0 - ~ 3 m 2 72=8.8.10 IIm2 Ap i = 1000 kg/m 3

Solidification of binary alloys

737 I

Fig. 2 b Streamfunction for

F i g 2a. Streamfunction for 7~ = 0, A0 = 0.001

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F i g 3 Velocity plot for the case of no density differences in the two-phase zone, Gr = 1 6 , 104.

F i g 5 Streamfunction for Gr = 0, At) = 0 1

:::::::::::::::::::::::::::::: .

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

-

. . . . . . . . . .

~1117/!~~ I I Z / I ~ - -

. . . . . . . . . .

~'lllX/l'plI/ZZI

~-

. . . . . . . . . .

~ / l l X Z l P/llZZt

s~-

...... ......

~/li/Yl ~III~

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~ I I / Z : f ' t/IZZ I~-

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~IP-

i ~i~#p-

I S" I#"

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

of

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 "Fcchnology, 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).