On the Permeability of the Two-Phase Zone during Solidification of Alloys

ROGER WEST

This paper makes some comments on the calculation of convection in the two-phase zone of solidifying alloys. The conclusion is that the permeability has to be modeled differ- ently compared with previous attempts.

Calculation of flow through porous media is often per- formed with the assumption that the permeability, K, is described by the relation

KI = Y I P 2 [1]

where P equals the porosity or the fraction liquid and Yl is a constant.

This relation has been used to describe the complete two- phase zone 1'2'3 even though it gives permeabilities which differ considerably from those obtained experimentally, l The model used appears to be applicable to the later stages of solidification where the solid phase forms a coherent network, but fails in the early stage of solidification.

Brinkman 4 has given an equation describing the flow of a fluid through an array of spherical particles. This equa- tion can be used to obtain a different expression for the permeability.

K = 18 3 4 , / 8 )

+ 1 ~ 3 1 - P 3 [21

4 7 r N R 3 = 1 - P 3

where R equals the radius of the spheres and N is the number of spheres.

This can be rewritten as

( 4 3 ~ / 8 3) K2 = 112(1 - p)ZJ3 3 + 1 - ~ 1 - P

[3] This equation gives K2 = 0 for P = 1/3, and rapidly increasing permeability with increasing P. This gives a smoothly increasing permeability if one simply adds this contribution to Eq. [ 1] to get the permeability of the mushy zone. This model can be used in the interval 0 < P < 1 and gives very high values of the permeability when P ap- proaches 1.

ROGER WEST is Research Associate, Department of Casting of Metals, Royal Institute of Technology, Stockholm, Sweden.

Manuscript submitted July 5, 1984.

O l .k

:E r

>- i - ra ..,J em

u.I :E r e LU Q.

o ,-I

0.1 - 5 , , )

LL

[ ] [ ] "

- 6 [ ]

[ ] [ ] [ ] [ ]

-7

- 8

-9 ~}~ [][] I

-10 " []

-11 , - 1 . 0

0 . 2 0 . 5 1.0

I I I , I I I i

- 0 . 8 - 0 . 6 - 0 . 4 - 0 . 2 0

LOG (FRACTION LIQUID ) Fig. l - -Permeabi l i ty vs fraction liquid; squares represent measurements by Piwonka (Ref. I). Solid line calculated from Eq. [1] and Eq. [3].

Experimental results by Piwonka I can be used to get val- ues of Y~ and Y2. The permeabilities at P = 0.125 and P = 0.9 give after some calculations Y~ = 6.4 �9 1 0 - 9 c m 2

and 112 = 8.8 �9 10 -7 cm 2. The experimental data are com- pared with the model in Figure 1. The improvement in agreement between theory and experiment when perme- ability is calculated in this way is considerable.

The conclusion is that proper modeling of the perme- ability necessitates that one considers the mushy zone as a transition from capillarities in a solid body to particles in a liquid. This means one has to combine at least two different models to describe the resistance to flow in the two-phase region.

REFERENCES

1. T. S. Piwonka and M. C. Flemings: Trans. TMS-A1ME, 1966, vol. 236, pp. 1157-65.

2. R. Mehrabian, M. Keane, and M. C. Flemings: Metall. Trans., 1970, vol. 1, pp. 1209-20.

3. A.L. Maples and D.R. Poirier: Metall. Trans. B, 1984, vol. 15B, pp. 163-72.

4. H.C. Brinkman: Appl, Sci. Res. A] , 1947, pp. 27-34.

METALLURGICAL TRANSACTIONS A VOLUME 16A, APRIL 1985 693