A unified microscale parameter approach to solidification-transport process-based macrosegregation modeling for dendriti
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SPECIES segregation formations in dendritic solidification processes of alloy castings and ingots are multiscale phenomena, which are generally classified as microsegregation and macrosegregation,[1,2] and both will lead to adverse effects on the properties and performances of the solidified materials and parts. However, these two differently scaled, solidification defects are closely correlated; the inherent microsegregation behaviors upon the dendrite alloy solidification rise to the possibility for the macrosegregation formation, whereas the macroscale segregation, in turn, results in variations in the local mixture composition (an open system in regards to the microscale solute redistribution[3]). Therefore, a proper segregation model for practical dendrite-alloy solidification processes inevitably needs to treat both the two-scale phenomena simultaneously. In the first part of the present work,[4] it was shown that with a continuum, one-phase volume averaging approach to the microscale modeling in macrosegregation predictions for dendrite solidification, the coupling for these two-scale species segregation behaviors is embodied through the timedifferential mixture averaged composition (TDMAC) term in the macroscopic species mass-transport equation. When the two extreme cases of Scheil type (zero solid back-diffusion) and lever-rule type (sufficiently rapid solid back-diffusion) solidification models are applicable, the TDMAC term can naturally take pure differential expression forms. However, for a real dendritic-solidification process with any DAMING XU, Professor, is with the School of Materials Science and Engineering, Harbin Institute of Technology, Harbin 150001, People’s Republic of China. Contact e-mail: d m [email protected] Manuscript submitted October 20, 2000. METALLURGICAL AND MATERIALS TRANSACTIONS B
extent of solid back-diffusion, a general TDMAC expression for the macroscopic species mass-transport equation contains an integral term; i.e., ⭸(C )m /⭸t ⫽ (SCS)*⭸fS /⭸t ⫹
兰
fS
0
[⭸(SCS)/⭸t]d
⫹ (LCL)⭸fL /⭸t ⫹ fL⭸(LCL)/⭸t
[1]
This term will highly complicate implementing the numerical computations with such a model.[4] For a general dendrite-solidification case, an exact solution to the integral term in Eq. [1] also requires detailed geometric information for the changing morphologies of growing dendrites under the given solidification conditions. However, several investigators[5,6,7] have shown that complicity and difficulties may be avoided by replacing this integral term with a simple but equivalent solid/liquid (S/L) interfacial-species flux; i.e., ⭸(C )m /⭸t ⫽ (SCS)*⭸fS /⭸t ⫹ ⌽ ⭈ fS⭸(SCS)*/⭸t ⫹ (LCL)⭸fL /⭸t ⫹ fL⭸(LCL)/⭸t
[2]
where the parameter ⌽ should take a value range of [0, 1] in order to unify the solidification cases with any extent of solid back-diffusion including the two extreme cases of Scheil-type and lever-rule-type solidification models. Through the microscale modeling carried out in the first part of the present work, the unified ⌽ parameter was found to be a
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