Математичне моделювання та комп’ютерний аналіз теплових режимів електропровідної панелі при індукційній термообробці

Abstract

An electrically conductive panel of rectangular cross-section is considered. During induction heat treatment with a quasi-steady electromagnetic field, unsteady volumetrically distributed Joule heat sources arise in it. With appropriate parameters of the external electromagnetic field, thermal processes in the panel can occur under conditions of near-surface or continuous heating. A two-dimensional physical and mathematical model is proposed to study the regularities of its thermal modes under the above-mentioned heating conditions. This model consists of two stages. At the first stage, the component of the magnetic field intensity vector tangent to the panel bases is determined from Maxwell's relations. At the second stage, the Joule heat is calculated from this component. To construct a solution to the electrodynamics problem, we use a cubic polynomial approximation of the distribution of the magnetic field intensity vector component tangent to the panel bases along the thickness coordinate. The coefficients of the approximation polynomial are represented as a linear combination of the thickness-integral characteristics of the tangential component of the magnetic field intensity vector and the specified boundary values of this component on the panel bases. As a result, the original two-dimensional initial-boundary value problem on the determining function (the component under consideration) is reduced to a one-dimensional initial-boundary value problem on the integral characteristics of the determining function. The solutions to this problem are found by applying a finite integral transformation along the transverse coordinate of the panel using the given boundary conditions on the determining function at the end sections of the panel and the Laplace transform in the time variable. Numerical studies have been performed for a copper panel during its induction heat treatment with a homogeneous quasi-steady-state electromagnetic field. The thermal modes of the panel are analyzed numerically depending on the parameter characterizing the relative depth of induction current penetration for two typical cases of near-surface and continuous heating.