Optimal control of ladle stirring

This thesis studies the mathematical modeling, simulation, and optimization of an industrial process called steel ladle stirring. In this process, gas is injected continuously from the bottom of the bath and rises by buoyancy through the liquid steel, thereby causing a turbulent stirring, i.e., a mixing of the bath. The process has been extensively studied in the literature both experimentally and numerically in order to understand the influence of control parameters on the stirring and to improve the mixing conditions in the industrial practice. Nevertheless, optimal control problems in mathematical sense have still to be explored in this area. The main contributions of this thesis can be divided into three parts. First, multiphase modeling of ladle stirring can become computationally expensive, especially when used within optimal flow control problems. This is why this thesis focuses on simplified models based on the single-phase incompressible Navier–Stokes equations. Three variants are formulated: a 2d Cartesian model, where the effect of the gas is modeled as a vertical boundary velocity, a 2d axial-symmetrical one with a central nozzle where it is modeled as a buoyancy force, and a 3d model of a laboratory-scale real ladle with two excentric nozzles, where the gas also appears as a volume force. The main differences with existing models from the literature are highlighted, and numerical simulations are compared with experimental measurements. Second, optimal control problems are investigated. The main difficulties come, on the one hand, from the formulation of the actual industrial problem, and, on the other hand, on the mathematical formulation of the control and cost functionals. In practice, the main control parameter is the volumetric flow rate of the injected gas. In addition, process constraints have to be taken into account. Due to the complexity of the industrial problem, several overlapping objectives are involved, such as maximize homogenization, minimize treatment time, minimize concentrations of inclusions, etc. A mathematical translation of the practical control and constraints is given, leading to so-called box constraints, and several cost functionals are proposed to describe the stirring efficiency. Numerical simulations are performed and conclusions are drawn for the industrial practice. Finally, as part of the cooperation project with the industrial partner, the main technological solutions for ladle stirring control are reviewed, leading to the choice of vibrations sensors. Thus, an experimental investigation of the vibrations of ladle stirring is conducted. This allows to formulate some practical recommendations for the industrial practice and improves our understanding of vibration phenomena for future modeling work