A mathematical model for solute redistribution during dendritic solidification.

Abstract

SynopsisTo quantify the solute redistribution during solidification, an approximate solution of the Brody-Flemings model of dendritic solidification has been derived on the assumption of parabolic solidification and a constant partition ratio, and compared with an exact solution of the model. It has been shown that the approximate solution has good accuracy.Further, on the basis of the derived equation, an extended mathematical model is proposed, incorporating a thermal model of solidification into the analysis without assumption on the solidification rate. The extended model includes the treatment of multi-component alloy and of 1 transformation, the temperature dependence of diffusion coefficients and the estimation of the diffusion path. The extended model provides the results consistent with the available data.Key words: mathematical modeling; solidification; segregation; sion; carbon steel. dif f u- I. IntroductionIn the earliest description with the Scheil equation for solute redistribution during solidification, the diffusion of solute into solid phase (back diffusion) was neglected.1'2~ Brody and Flemings3~ have extended the theory to include the back diffusion effects based on their solidification model; the model assumed 1) complete diffusion in liquid phase and incomplete back diffusion, 2) a plate-like dendrite geometry, 3) a constant diffusion coefficient, and 4) parabolic or linear solidification rate. Both approximate equations derived by Brody and Flemings for linear and parabolic solidification rate are, however, invalid for rapid diffusion in the solid phase.Clyne and Kruz,4~ and Ohnaka51 have modified the Brody and Flemings equation for parabolic solidification rate; both the modifications are valid for infinite and infinitestimal diffusion coefficient. Further, the present author has derived an exact solution of the solidification model of Brody and Flemings for parabolic solidification rate, and has shown that the accuracy of all the approximate solutions mentioned above are often bad.6~ Although the solidification model of Brody and Flemings with parabolic growth has been rigorously solved, several problems still remain in applying the model to solidification processes : 1) calculation of the exact solution is uneasy because the solution has an infinite series of the confluent hypergeometric functions, 2) no realistic method has been found to estimate the solidification rate or the local solidification time, of which value is necessary for the calculation,

Restrictions of Physical Properties on Solidification Microstructures of Al-based Binary Alloys by Cellular Automaton

“…A seen, the latent heat of the solidification is regarded as the functions of the solute concentration in order to fit with Kobayashi's model. 8) The cooling rate in intermediate position Ṫ is set as 0.7, 2.3 and 5.0 K/s. The heat transfer coefficient h corresponding to each alloy and Ṫ is obtained beforehand through several attempts.…”

“…(8). Thus the evolution of the solidification microstructure is directly affected by the physical properties of the alloys.…”

“…An unsteady two dimensional heat transfer combined with the solute redistribution at solid/liquid (S/L) interface 8) during Al-Si and Al-Cu alloy solidification processes is considered. The temperature T nϩ1 , solid fraction f S,nϩ1 and concentration C nϩ1 can be determined by iterative solution August 20, 2010 ) The solidification microstructure evolution and the Columnar to Equiaxed Transition (CET) during Al-Si and Al-Cu binary alloy solidification processes are analyzed by the help of Cellular Automaton-Finite Difference (CA-FD) model.…”

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.…”