Prediction of dendrite arm spacings in unsteady-and steady-state heat flow of unidirectionally solidified binary alloys
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l1 5 120
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!
16 X 1/2 0 G0 (εs) TM D (12k) m DH G R
1/2
ultimately proved its utility in the unsteady regime, and so it is recommended for purposes of predictions for general terminal alloys. For secondary spacings, a Mullins and Sekerka type formula proved from the start to be adequate in both unsteady- and steady-state heat flows, and so it recommends itself in calibrated form,
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~ !!
4s D l2 5 12p X0 (12k)2 DH R
2 1/3
for future predictions. I.
INTRODUCTION
FINE dendritic structures in castings are recognized to yield superior mechanical properties to coarser ones, notably with respect to tensile strength and ductility.[1–4] The beneficial effects have been mainly related to faster homogenization, where microsegregation (coring) is reduced to an acceptable level when fine structures are present.[5,6] This situation has led many foundries producing highstrength castings to implement dendrite spacing measurements as an integral part of their quality-control programs. Numerous unidirectional solidification studies have been carried out with a view to characterizing primary and secondary spacings under various circumstances. These studies can be grouped into two categories: those involving solidification in steady-state heat flow and those in the unsteadystate regime. In the former, the temperature gradient, G, and the growth rate, R, are independently controlled and held constant in time, with the spacings reported as a funcDOMINIQUE BOUCHARD, formerly Postdoctoral Fellow, Brockhouse Institute for Materials Research, McMaster University, is Research Scientist, Industrial Materials Institute, National Research Council of Canada, 75 de Mortagne, Boucherville, PQ, Canada J4B 6Y4. JOHN S. KIRKALDY, Professor, is with the Brockhouse Institute for Materials Research, McMaster University, Hamilton, ON, Canada L8S 4M1. Manuscript submitted June 3, 1996. METALLURGICAL AND MATERIALS TRANSACTIONS B
tion of these known quantities. In the latter, the temperature gradient and growth rate vary freely in time and the spacings are reported as a function of the time-dependent cooling rate, GR, as measured by thermocouples. Two and three-dimensional steady-state theoretical models relating the spacing parameters to temperature gradient, growth rate, and thermophysical properties have also been developed,[7–16] and comparisons with experimental data obtained in corresponding conditions have been made.[7,8,10,15,16] Definitive predictions for the unsteady-state model are more difficult to carry out, because the calculations involve time-dependent quantities which add to the complexity and inaccuracy of the calculations. Reliable spacing predictions in the unsteady-state regime are of prime importance, since this class of heat flows encompasses the majority of industrial solidification processes. While computer-aided modeling has been increasingly employed as a tool to optimize casting properties, prediction of dendrite parameters under various cooling conditions remains one of several facets of the solidification process t
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