To the solution of the problem on the controllability of heating of metal in industrial furnaces

In the working space of the chamber or in the zone of through heating furnaces, ingots or billets with different thermal and other characteristics are simultaneously heated. As a result, the heating trajectories that are optimal for the selected criterion for each ingot and each billet will be different. At the same time, due to the design features of heating furnaces, all the ingots or billets have one controlling effect - the temperature of the working space (provided by the corresponding fuel consumption into the working space) of the chamber furnace or in the passage zone. In this connection, the question naturally arises whether it is fundamentally possible in such a situation to create individual heating trajectories for each ingot or billet, i.e. The paper considers the solution of the problem of the controllability of the process of heating a metal in industrial furnaces for rolling production. To solve this problem, a mathematical description of the heating process has been proposed. This description is a matrix system of first-order differential equations. Found conditions complete controllability of the object. In this case, the controllability matrix was calculated, it was shown that its determinant can be expressed in terms of the Vandermonde determinant. This circumstance greatly simplified the definition of the conditions of controllability. It has been established that for complete controllability of an object it is necessary that the heated ingots or billets have different heating time constants, which is caused by different thermal or geometric characteristics. In most practical cases, this difference is usually not, therefore, the process of heating the metal in industrial furnaces, as a rule, is not fully controlled. As a result, the maximum effect from the point of view of the selected criterion of the quality of heating can usually not be obtained.