Abstract
With readily available and ever-increasing computational resources, the modelling of continuous casting processes—mainly for steel, but also for copper and aluminium alloys—has predominantly focused on large-scale numerical simulation. Whilst there is certainly a need for this type of modelling, this paper highlights an alternative approach more grounded in applied mathematics, which lies between overly simplified analytical models and multi-dimensional simulations. In this approach, the governing equations are nondimensionalized and systematically simplified to obtain a formulation which is numerically much cheaper to compute, yet does not sacrifice any of the physics that was present in the original problem; in addition, the results should agree also quantitatively with those of the original model. This approach is well-suited to the modelling of continuous casting processes, which often involve the interaction of complex multiphysics. Recent examples involving mould taper, oscillation-mark formation, solidification shrinkage-induced macrosegregation and electromagnetic stirring are considered, as are the possibilities for the modelling of exudation, columnar-to-equiaxed transition, V-segregation, centreline porosity and mechanical soft reduction.
Highlights
Whilst there is certainly a need for this type of modelling, the focus in this paper is on an alternative approach more grounded in applied mathematics, asymptotic methods in particular, which lies between overly simplified analytical models and multi-dimensional simulations
This approach is well-suited to the modelling of continuous casting processes, which often involve the interaction of complex multiphysics
A novel “ moving-point” formulation was derived, which unlike the volume-of-fluid (VOF) formulation adopted in [56,58,59,60,61], attempts to track explicitly the location of the point at which solidification first begins. This approach suggests that if solidification has not occurred within a capillary length from the top of the meniscus, the meniscus profile predicted by Bikerman [62], and which emerges naturally in the asymptotic analysis in [32], will collapse, leading to overflow; further work is required to extend the isothermal model derived in [32] to determine where the temperature first becomes low enough for the steel to start to solidify
Summary
Whilst there is certainly a need for this type of modelling, the focus in this paper is on an alternative approach more grounded in applied mathematics, asymptotic methods in particular, which lies between overly simplified analytical models and multi-dimensional simulations. In this approach, the governing equations are nondimensionalized and systematically simplified to obtain a formulation which is numerically much cheaper to compute, yet does not sacrifice any of the physics that was present in the original problem; in addition, the results should agree quantitatively with those of the original model. Appendix A shows in more detail how the practical asymptotics approach is applied to one of the sub-problems considered in the paper
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