To approximation of the temperature field of slabs during heating in continuous furnaces by a quadratic polynomial

Introduction. In the context of increasing requirements for the quality and efficiency of metal heating before rolling, the task of creating and improving the algorithmic support of automated pro-cess control systems (APCS) for heating furnaces is quite relevant. Purpose of the study. Development and specification of new options setup the parabolic approach to describing the temperature field of billets during heating in continuous furnaces. Materials and methods. The paper contains a physical, mathematical and numerical study of the extent to which the determination of the parabola coefficients from the so-called “most important temperatures in the body” provides a satisfactory representation of the temperature field of slabs during heating in furnaces by a quadratic trinomial. Results. Two options setup the quadratic polyno-mial are presented. A computational study has shown that these options provide a significantly more accu-rate reproduction of the temperature field of slabs during heating in furnaces than the solution given in the previously published article (Panferov V.I., Panferov S.V. On parabolic approximation of the tempera-ture field of slabs during heating in continuous furnaces. Bulletin of the South Ural State University. Ser. Metallurgy. 2025;25(1):44–52.). In terms of reproducing the actual temperature distribution across the slab cross-section, the algorithm variants are virtually equivalent, with the error in approximating the temperature field being negligible. It was also noted that the second setup procedure is significantly simpler and, crucially, can be implemented solely using current furnace measurements, which is clearly more accurate in practice. Conclusion. The results of this study can be used in the development and im-provement of algorithmic support for automated process control systems for continuous furnaces.