Modeling and computer analysis of temperature profiles in an aluminum panel during induction heat treatment

Abstract

A physical-mathematical model for determining the temperature field in a non-ferromagnetic panel is considered. This model consists of two stages.In the first stage, Maxwell’s equations are used to determine the transient electromagnetic field and Joule heating caused by an external transient electromagnetic field. In the second stage, the transient temperature field is determined from the heat conduction equation, in which Joule heating is taken as the heat source. The panel is referenced to a Cartesian coordinate system. Its cross-section is a rectangle. Two-dimensional initial-boundary value problems in electrodynamics and heat conduction have been formulated for the panel under consideration. The determinant functions selected are the component of the magnetic field stress vector tangent to the panel’s bases and the temperature.The solution to these problems is obtained using a cubic approximation of the determinant functions with respect to the panel’s thickness coordinate. The coefficients of the cubic approximation polynomials are expressed as linear combinations of the integral characteristics of the sought functions—integrated over the thickness coordinate—and their boundary values at the panel edges. The corresponding one-dimensional boundary value problems were formulated to determine the integral characteristics. The solutions to these problems are obtained using a finite integral transform with respect to the transverse coordinate of the panel’s cross-section and a Laplace integral transform with respect to time.