Mathematical simulation of interdendritic solidification of low-Alloyed and stainless steels
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I.
INTRODUCTION
D U R I N G dendritic solidification, liquid flows transporting solutes may be formed due to several sources (electromagnetic stirring, bulging, shrinking, etc.), which makes the mathematical simulation of the process very difficult. In conventional microsegregation models, the treatment has been simplified, assuming a complete solute mixing (no solute transport) in liquid. This permits the mathematical treatments to be made in one volume element located in the mushy zone. In these models, the solute diffusion in solid is simulated using Fick's laws, and the interfacial (solid/liquid) compositions are related to the temperature using the information of binary phase diagrams. This information, however, may be insufficient when applied to multicomponent alloys (especially peritectic alloys). Due to this, the problem should be approached rather from a thermodynamic point of view. In the present model, the conventional treatment of solute diffusion (Fick's laws) was incorporated into a thermodynamic model, which relates the interfacial compositions to both the temperature and the phase stabilities. This kind of combined treatment permits the determination of stable phases and their fractions and compositions at any temperature during solidification. All calculations are made in the volume element shown in Figure 1. In order to test the model, some calculated results were compared with experimental measurements. The model also includes a routine for predicting the liquid undercooling caused by solute pileup formation ahead of the dendrite tips. However, the way these calculations and those of the base model could be combined still waits for a solution. In this article, only a very approximate treatment will be discussed.
ture is regular, having a hexagonal dendrite arm arrangement, and (b) the solute mixing is complete everywhere in liquid. Other typical simplifications made in the model are that (c) the thermodynamic equilibrium is reached at the phase interfaces a / L , y / a , and y/L, (d) the solute diffusion (one-dimensional in the volume element) is not influenced by other solutes, (e) no undercooling is needed before the nucleation of a new phase, and (f) the effects of surface tensions and differences in molar volumes are negligible. By proper mathematical treatments, some of these simplifications perhaps could be eliminated, but knowing the problems of solidification processes, one can never be sure that a certain mathematical improvement in a certain field will make the results more reliable. In the model, all calculations are made stepwisely. Due to this, the volume element was divided into steps, as shown in Appendix A. In this section, the main equations needed in calculations will be presented. A more precise description for these equations is given in Appendixes B, C, and D, and the way to apply the equations is given in Section III.
A. Chemical Potential Equalities One of the most common assumptions made in solidification models is that thermodynamic equilibrium is reached at the phas
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