Mathematical analysis of solute redistribution during solidification with diffusion in solid phase.

Abstract

SynopsisSolute redistribution during solidification has been analyzed by a profile method which has approximately considered the two-dimensional diffusion in solid phase. Derived equations have been compared with conventional equations, numerical analyses and experimental data. It was found that a newly derived simple equation can estimate the liquid concentration better than Brody-Flemings' equation but numerical calculation or further improvement of the analytical treatment is required for estimating the segregation ratio correctly. 7n addition, calculations showed that the dendrite shape, growth mode (parabolic and constant growth) and diffusion in the liquid phase did not affect so much the solute redistribution at least for f<0.9.The diffusion path, which is thought to be a more important factor, has also been discussed and an equation is proposed for the case where diffusion path or the partition coefficient changes stepwise. I. IntroductionIn the solidification of alloys, especially steels, the diffusion of solute into solid phase (back diffusion) affects greatly the microsegregation.Although Brody and Flemingsl~ have firstly analyzed a model which assumed complete diffusion in liquid and incomplete back diffusion, they have not solved the diffusion equation and hence the application of their result is limited to a slow diffusion. Recently Clyne and Kurz2~ derived a better mathematical expression which asymptotically approaches the Scheil's equation and the equilibrium equation for infinitestimal and infinite diffusion coefficient, respectively. Matsumiya et a1.,3~ however, showed that in some cases this equation did not agree so well with numerical results. Further Kobayashi4) has solved Brody-Flemings' model more rigorously.In order to make clear the two-dimensional effect of dendrites and to select a suitable equation for estimating the microsegregation, this paper presents an analysis based on a profile method which does not solve directly the diffusion equation but solves a overall solute mass balance equation and compares derived equations with conventional equations, numerical analyses and experimental data.

Modeling of manganese sulfide formation during the solidification of steel

“…Microsegregation was predicted using the stepwise Ohnaka's model [20] based on thermodynamic databases. With segregated concentrations in the residual liquid, the PSD method was applied to describe the size and number evolution of manganese sulfides.…”

“…In the coupled model, the solute enrichment in the residual liquid steel was estimated using Ohnaka's model [20]. The thermodynamic equilibrium at the solidification interface was calculated using ChemApp [21] and the ChemSage datafile.…”

“…Microsegregation was calculated using Ohnaka's model [20] with local partition coefficients. In the calculations, the analytical solution of Ohnaka's model [20] was integrated into Eq.…”

“…Microsegregation is estimated using Ohnaka's model [20]. The thermodynamic equilibrium is calculated with ChemApp [21] to determine the liquidus temperature and solute partition coefficients at the solidification interface based on commercial databases.…”

Mathematical Model for Prediction of Composition of Inclusions Formed during Solidification of Liquid Steel

“…Many analytical equations of microsegregation with different assumptions and simplifications have been developed to predict solute redistribution and related phenomena. [16][17][18][19][20][21][22] Numerous studies on microsegregation have been carried out for binary alloys. 22) The heart of most of the simple microsegregation models is the assumed relationship between solute concentrations in interdendritic liquid as function of solid fraction.…”