Prediction of Shrinkage Pore Volume Fraction Using a Dimensionless Niyama Criterion
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TRODUCTION
THE Niyama criterion is presently the most widely used criterion function in metal casting. It is used to predict feeding-related shrinkage porosity caused by shallow temperature gradients. All casting simulation software packages calculate the Niyama criterion as a standard output; foundries worldwide use this criterion to predict the presence of shrinkage in castings. Foundry-simulation users view Niyama contour plots predicted by a casting simulation and expect that shrinkage porosity will form in regions that contain Niyama values below some threshold value. The Niyama criterion is a local thermal parameter defined as .pffiffiffiffi Ny ¼ G T_ ½1 [1]
where G is the temperature gradient and T_ is the cooling rate, both of which are evaluated at a specified temperature near the end of solidification. Niyama et al.[1] empirically determined the functional form of this criterion, but they also offered a physical model as justification. The model begins with Darcy’s law, which relates the interdendritic feeding-flow velocity to the pressure drop across the mushy zone. By making simplifying assumptions about the functional form of
KENT D. CARLSON, Research Engineer, and CHRISTOPH BECKERMANN, Professor, are with the Department of Mechanical and Industrial Engineering, The University of Iowa, Iowa City, IA 52242. Contact e-mail: [email protected] Manuscript submitted July 28, 2008. Article published online November 26, 2008 METALLURGICAL AND MATERIALS TRANSACTIONS A
the solid fraction-temperature curve and the permeability, they were able to analytically integrate the onedimensional (1-D) form of Darcy’s law to demonstrate that DP / T_ G2 (or DP / Ny2 ), where DP is the pressure drop across the mushy zone. By comparing experimentally observed shrinkage porosity patterns in cast-steel cylinders to corresponding casting simulations, Niyama et al. discovered that there was a threshold value of Ny below which shrinkage was typically present. While the presence of the temperature gradient in Eq. [1] is not surprising, and implies that shrinkage porosity will form in regions of low G, the presence of the cooling rate (albeit inside a square root) in the denominator may at first seem counterintuitive. The Niyama criterion predicts that shrinkage porosity increases with increasing cooling rate, because the feeding-flow velocities and the resulting pressure drop across the mushy zone are higher for higher solidification rates. It follows that the beneficial effect of a chill, for example, is due to the increase in the temperature gradient the chill provides; the concomitant increase in the cooling rate near a chill is actually counterproductive with regard to avoiding shrinkage porosity. To illustrate the correlation between low Niyama values and shrinkage porosity, a sample Niyama result, based on experiments conducted in a previous study,[2] is given for the end-risered plate 1 9 5.5 9 19 in. (25.4 9 140 9 483 mm) in size shown schematically in Figure 1(a). Fifteen such plates were cast from WCB steel (Table I) in a s
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