On the permeability of the two-phase zone during solidification of alloys
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ROGER WEST
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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 differently compared with previous attempts. Calculation of flow through porous media is often performed with the assumption that the permeability, K, is described by the relation [1]
KI = Y I P 2
where P equals the porosity or the fraction liquid and Yl is a constant. This relation has been used to describe the complete twophase 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 equation can be used to obtain a different expression for the permeability. K=
18
3 +
4 1~
47rNR3 3
,/8 3 1 -
= 1 -
P
) 3
P
3 ~ / 1 -8 P
3)
[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 approaches 1.
ROGER WEST is Research Associate, Department of Casting of Metals, Royal Institute of Technology, Stockholm, Sweden. Manuscript submitted July 5, 1984.
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Fig. l--Permeability vs fraction liquid; squares represent measurements by Piwonka (Ref. I). Solid line calculated from Eq. [1] and Eq. [3].
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where R equals the radius of the spheres and N is the number of spheres. This can be rewritten as K2 = 112(1 - p)ZJ3( 3 + -1 ~4
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Experimental results by Piwonka I can be used to get values 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 compared with the model in Figure 1. The improvement in agreement between theory and experiment when permeability is calculated in this way is considerable. The conclusion is that proper modeling of the permeability 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.
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