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This lecture honoring Keith Brimacombe looks over the history, current abilities, and future potential of mathematical models to improve understanding and to help solve practical problems in the continuous casting of steel. Early finite-difference models of solidification, which were pioneered by Keith Brimacombe and his students, form the basis for the online dynamic models used to control spray water flow in a modern slab caster. Computational thermal-stress models, also pioneered by Brimacombe, have led to improved understanding of mold distortion, crack formation, and other phenomena. This has enabled process improvements, such as optimized mold geometry and spray-cooling design. Today, sophisticated models such as transient and multiphase fluid-flow simulations rival water modeling in providing insights into flow-related defects. Heat-flow and stress models have also advanced to yield new …