Permeability for flow of interdendritic liquid in columnar-dendritic alloys
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I.
INTRODUCTION
and
IT is well known that the convection of interdendritic liquid in solidifying alloys is responsible for many types of macrosegregation. To model the convection and hence macrosegregation, D'Arcy's law is used. When inertial effects are negligible, D'Arcy's law is written as pg)
v = --(K/I~gL) ( V P -
where v = /z= gL = p= p= g= and K=
[1]
velocity of the interdendritic liquid, m 9 s -t viscosity of the interdendritic liquid, N 9 s 9 m -2 volume fraction of the interdendritic liquid pressure, N 9 m -2 density of the interdendritic liquid, kg 9 m -3 gravitational acceleration, m 9 s -2 specific permeability of the solid-liquid zone, m 2"
Obviously the specific permeability is a key parameter in modeling of macrosegregation; yet there are relatively few data available. Permeabilities in equiaxial dendritic structures have been reported by Piwonka and Flemings, ~ Apelian et al. ,2 Streat and Weinberg, 3 Takahashi et al.,4 and Murakami and Okamoto. 5 Permeabilities in columnar dendritic structures have been reported by Streat and Weinberg, 3 Murakami et a1.,7'8 and Nasser-Rafi et al. 6 Some of these works have been discussed in a previous paper, 6 with particular emphasis given to experimental techniques. Here the purpose is to analyze and discuss the permeability for columnar structures, so permeability data for flow in equiaxial structures are not considered further. Because of the directional nature of a columnar-dendritic structure, its permeability is anisotropic. 6'7's Here the principal flow directions are selected to be parallel and normal to the primary dendrite arms, then the components of the permeability are K r and Kx, respectively. Accordingly, the components of Eq. [ 1] are v, = -- ( K y / I . t g L ) ( O P / O y
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pgy)
I2]
D. R. POIRIER is Professor, Department of Materials Science and Engineering, The University of Arizona, Tucson, AZ 85721. Manuscript submitted June 23, 1986. METALLURGICAL TRANSACTIONS B
v~ = - ( K x / ~ g L )
(OP/Ox
-- Ogx)
[31
where vx, Vy = velocity components in the x and y directions, respectively. and gx, gy = components of g. Data illustrating this anisotropy of permeability in columnar structures were presented by Murakami et al. 7"8 for borneol-paraffin organics which have a dendritic morphology the same as that of metallic alloys. More recently, Nasser-Rafi et al. 6 considered the anisotropic permeability in Pb-20 wt pct Sn columnar alloys. Here we reexamine the data from those three worKs, 6'7'8 along with those of Streat and Weinberg, 3 and include additional results which have been obtained for Pn-20 wt pct Sn alloys. The collective data are analyzed by emp!rical regression analyses and by physical models, some of which have been previously used to represent permeability in porous media.
II. P E R M E A B I L I T Y DATA FOR COLUMNAR-DENDRITIC STRUCTURES Summaries of the data for permeability when flow is parallel to and normal to the primary dendrite arms are given in Tables I and II, respectively. The data listed in Tables I and II tha
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