Temperature Control of the Lower Zones of Methodological Furnaces

Abstract

Introduction. The problem of ensuring the identity of the heating trajectories of the upper and lower halves of the slab (slab heating symmetry) is considered for significantly different numerical values of the external heat transfer parameters for its upper and lower surfaces. This is due, in particular, to poor thermal insulation of the hearth pipes. Purpose of the study. To develop an algorithm for calculating the ratio of settings (tasks) required for symmetry for the temperature controllers of the working space of the upper and lower heating zones of continuous furnaces. Materials and methods. The analysis and generalization of literature data on the problem is carried out. A mathematical description of asymmetric heating is proposed, an optimization problem is posed and solved. Results. Algorithm variants provide for the possibility of describing external and internal heat transfer both in linearized and in the original non-linear form. For the case of representation of the heating process by a fully linearized heat conduction equation with a convective form of external heat transfer, the desired result is represented by an analytical formula. This, as is known, makes it possible to trace the influence of the initial process parameters on the optimal ratio of settings. Numerical research shows that observance of the calculated ratio of the settings ensures a satisfactory coincidence of the mass-average temperature trajectories of the upper and lower halves of the slab in the case of asymmetric heating itself (inequality of external heat transfer coefficients). A simplified solution of the problem under consideration is also proposed, when the dynamics of mass-average temperatures of the upper and lower halves of the slab is initially described by a first-order ordinary differential equation obtained from the heat conduction equation by integral temperature averaging over the thickness of the slab. The simplified version also provides, as a rule, practically acceptable proximity of the heating trajectories of the upper and lower halves of the slab and is even more “transparent” as to how the initial data affects the desired result. Conclusion. The use of the optimal ratio of settings allows, for example, when solving problems of optimal process control, to reduce the dimension of the desired control vector by half, which is very important in the development of algorithmic support for automated process control systems for heating furnaces.