Thermal optimization of the continuous casting process using distributed parameter identification approach—controlling the curvature of solid-liquid interface

Abstract

Thermal optimization of the vertical continuous casting process is considered in the present study. The goal is to find the optimal distribution of the temperature and interfacial heat transfer coefficients corresponding to the primary and secondary cooling systems, in addition to the pulling speed, such that the solidification along the main axis of strand approaches to the unidirectional solidification mode. Unlike many thermal optimization of phase change problems in which the desirable (target) temperature, temperature gradient, or interface position are assumed to be a priori known, a desirable shape feature of the freezing interface (not its explicit position) is assumed to be known in the present study. In fact, the goal is equivalent to attain a nearly flat solid-liquid interface that is featured by its zero mean curvature. The objective functional is defined as a function of the temperature and mean curvature of the freezing interface. The solidus and liquidus iso-contours of the temperature field are used to implicitly determine the solid-liquid mushy region, i.e., the freezing interface. The temperature field is computed by solving a quasi steady-state nonlinear heat conduction equation. A smoothing method is employed to ensure the differentiability of heat equation. The explicit form of first-order necessary optimality conditions is derived. A gradient descent algorithm is introduced to find local solutions of the corresponding optimization problem. Numerical results support the feasibility and success of the presented method.