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

Abstract

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.

Air gaps in vertical continuous casting in round moulds

“…Furthermore, from the analysis in [12], it became clear that, for that geometry (where t = x, n = y), σ n = −p a would be inconsistent with a solution structure that assumes that the shear stresses are zero: in particular, the solution must be of the form σ y ≡ −p(z) ( = 0) if τ xy = 0. In order not to lose analytical tractability, Vynnycky [12] retained (2.46), even after the gap had formed. However, one benefit of the cylindrical geometry is that we can make analytical headway for both cases and can therefore explore the consequences of both assumptions.…”

“…Here, as in other continuous-casting models [12,19], we invoke the generalized plane strain approximation, by which we mean that the length scale in the z-direction is much greater than that in the r -direction; this enables us to reduce (2.19)-(2.21) to…”

“…An exception to all of the above is a recent model [12] that uses asymptotic methods; that model was, however, derived for an idealized two-dimensional configuration and the purpose of this paper is to extend that model to a geometry of actual industrial interest.…”

“…Furthermore, this configuration presents an ideal case for extending the asymptotic air-gap model in [12], since substantial analytical progress can be made and it proves possible to test alternative hypotheses regarding the appropriate boundary conditions for the onset of air-gap formation and gap evolution; in addition, the model solutions obtained can be later used as benchmarks for testing 3D codes.…”

The Influence of Peritectic Reaction/Transformation on Crack Susceptibility in the Continuous Casting of Steels

“…In order to see the complete model details where one can compute the required location of the boundaries simultaneously, the reader is referred to earlier work. [21,22] The first boundary condition is expressed as…”