Mathematical analysis of solute redistribution during solidification based on a columnar dendrite model.

Abstract

An exact solution and an approximate solution have been derived for the columnar dendrite model proposed by Ohnaka,8~ in which a solidification rate, equilibrium distribution coefficient and diffusion coefficient in solid are assumed to be constant throughout the solidification. By comparing with the exact solution, it has been shown that the approximate solution has good accuracy. Further the model is extended on the basis of the derived approximate solution by incorporating a thermal model of solidification. The extended model includes multi-component alloy accompanying phase transformation in solid and temperature dependence of diffusion coefficients, and excludes the assumption of a given solidification rate. The calculated results of the extended columnar dendrite model agreed well both with the available experimental data and the calculated results of the extended planar dendrite model proposed previously. The assumption of the solidification geometry is unimportant for quantifying the microsegregation effects.

Methodological Progress for Computer Simulation of Solidification and Casting

“…Meanwhile, the combination of a microsegregation model with the macroscopic heat and solute transport equations is an efficient way for prediction of both the solidification path and the solidification grain structure. From another point of view, for simulations of actual castings, 54) Kobayashi's model [70][71][72][73] and Natsume's model, 84) based on a one-dimensional case, should presently be recommended.…”

“…Here, the solidification behavior under local equilibrium conditions is assumed to be controlled by the solute diffusion in both the liquid and the solid. In addition, the exact analytical solution of the Brody-Flemings equation by Kobayashi [70][71][72][73] should be introduced here, since it is not included in any commercial software packages today.…”

Computation of Partial Equilibrium Solidification with Complete Interstitial and Negligible Substitutional Solute Back Diffusion

“…13) Asymptotical analyses of all these modified Brody-Flemings equations 10,11) or completely new ones, 12,13) lead successfully to the equilibrium or lever-rule equation if…”

Solidification Path and Solute Redistribution of an Iron-Based Multi-Component Alloy with Solute Diffusion in the Solid

“…It is also meaningful to compare the results of this method in a simple binary system with the results of the GulliverScheil model, 11) the lever-rule, Kobayashi's solution 6) and the Brody-Flemings solution, 3) because the Gulliver-Scheil model and the lever-rule (equilibrium) are the limit cases of the diffusivity in the solid.…”

“…Several models have been proposed for the case in which the liquid is completely mixed and finite diffusion works in the solid. [3][4][5][6][7][8] In Wołczyński's method 9,10) for back-diffusion phenomenon during the crystal growth the mass balance problem at the solid-liquid interface is separately treated from the solute redistribution in the solid. Therefore, by using this method, we can easily calculate the solidification path of a multicomponent alloy such as Fe-C-X alloy.…”