Mechanical deformation of dendrites by fluid flow
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Mechanical Deformation of Dendrites by Fluid Flow
Pro -
v.p.d Re = - r/ 3.2 • 10 -3
= 0.22
[1]
so that it is reasonable to assume streamline flow and that there would be negligible turbulence on the downstream side of the dendrite arm. The shear stress acting along the tangential direction on the surface of the cylinder, rr0, due to drag and the hydrostatic pressure Pro would be Is]
J. PILLING, Associate Professor, and A. HELLAWELL, Professor, are with the Department of Metallurgy and Materials Engineering, Michigan Technological University, Houghton, MI 49931. Manuscript submitted March 13, 1995. METALLURGICAL AND MATERIALS TRANSACTIONS A
2r
sin (0)
[2] cos (0)
Fy = fro" sin (0) + PrO" COS (0)
It is generally accepted that liquid agitation during alloy solidification assists in crystal multiplication, as in dendrite fragmentation and the detachment of side arms in the mushy region of a casting. Even without deliberate stirring by electromagnetic or mechanical means, there is often vigorous interdendritic fluid flow promoted by natural thermosolutal convection. Interdendritic fluid flow rates in metals might be as high as 10 m m s-l.m It is the purpose of this article to examine whether such fluid flow can cause mechanical deformation of dendrites, sufficient to cause side arms to bend or break. Metals are so ductile at their melting points that applied forces could only be expected to cause bending, as opposed to fracture, although there are no reports of which we are aware of dendritic arms being mechanically bent in this way. The following estimates demonstrate why even bending is not to be expected. Figure 1 is an example of an ammonium chloride dendrite growing into an aqueous solution between two glass slides, under steady state conditions, with no fluid flow. In this "alloy," there is a very small solid fraction and wide primary spacing--this is a fairly extreme example and in a majority of " r e a l " cases the dendrite side arms would be shorter and thicker. As is usual, there is considerable ripening of side arms and most of them develop narrow necks at the roots where they attach to the primary dendrite stem. There is no significant side arm detachment unless the growth rate falls and/or the temperature is raised, t2,3] In this analysis, we shall estimate the stress at the root of a secondary dendrite arm of aluminum arising from the action of a flow of molten metal past the dendrite arm. The schematic geometry of the dendrite is shown in Figure 2. Both the root and main sections of the dendrite are assumed to be cylindrical with diameters dr and d and lengths Lr and L, respectively. In typical castings, the main diameter would vary from 10 to 25 /xm, while the root diameters might be between 5 and 10/zm. With flow velocities up to 10 -2 m s 1 and viscosity of 3.12 • 10 -3 kg m -1 S-l, [4] the upper bound for the Reynolds number would be
10 - 2 " 2 . 7 • 103"25 • 10 -6
2r 3~/v
The force acting in the y direction (parallel to the flow) would be
J. PILLING and A. HELLAW
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