Multi-domain Two-Region Methods

The prediction of only the temperature distribution during continuous casting of a metal does not usually require detailed modeling of the flow and species transport processes in the melt. For certain purposes, energy-equation based models can be useful to understand and optimize the process. However, such models cannot be used to predict inhomogeneities in the chemical composition of a solidified material (i.e. macrosegregation) caused by uncontrolled species transport due to melt convection. 

There are generally two main approaches for handling the mushy region in multi-domain methods. The first approach considers the planar interface between the solid and liquid regions and solves only transport equations for the liquid and solid phases. The interface is tracked explicitly with appropriate boundary conditions [87,89-92], 

In this kind of problems involving solidification from below, the transport equations for solid and liquid phases are  

Two flux conditions are typically applied at the unknown position of the interface h(t). Conservation of heat requires that the latent heat of solidification must be diffused away from the interface thus  

Assuming thermodynamic equilibrium, the temperature and concentration at the interface are related as  

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