An asymptotic model for the formation and evolution of air gaps in vertical continuous casting

The formation of an air gap at the mould-metal interface in vertical continuous casting has long been known to have a detrimental effect on the efficiency of the process, and has therefore attracted attempts at mathematical modelling. While almost all current efforts consist of complex three-dimensional numerical simulations of the phenomenon, this paper considers instead an asymptotic model that captures the essential characteristics. The model is thermomechanical and is derived for a geometry, where the generalized plane strain approximation is appropriate. Although two-way coupling between the thermal and mechanical problems is accounted for, it is found that the problems decouple at leading order anyway, and that the thickness of the air gap does not depend on the constitutive relation used for describing the inelastic strains. Furthermore, a criterion for the onset of air-gap formation is derived in terms of the process operating parameters. Mathematically, we obtain a moving boundary problem for a parabolic partial differential equation with a degenerate initial condition and a non-standard Neumann-type boundary condition. Sample computations are performed using parameters for the continuous casting of the copper, and the results, qualitative trends and possible extensions are discussed.