Analysis of Two-Dimensional Unsteady Temperature Fields in a Copper Panel During its Induction Heating

An electroconductive panel of rectangular cross-section is considered, which is inductively heated by a quasi-steady electromagnetic field. As a result, unsteady volumetrically distributed Joule heat sources appear in the panel. These sources create a non-stationary two-dimensional temperature distribution across the panel cross-section. The conditions of convective heat exchange with the external environment are fulfilled at the bases and side faces of the panel. A nonstationary two-dimensional heat conduction problem is formulated to determine the temperature field in the panel. The solution to the problem is constructed using the approximation of the temperature distribution along the thickness coordinate of the panel by a cubic polynomial. The coefficients of the approximation polynomial are represented by the temperature characteristics integral along the thickness coordinate and the conditions for the temperature boundary values at the bases and side faces of the panel. As a result, the original two-dimensional initial-boundary value problems on temperature are reduced to one-dimensional initial-boundary value problems on the integral temperature characteristics. The solution to the problems on the integral temperature characteristics was found using the Laplace integral transform in time and the finite integral transform in the transverse coordinate of the panel. It is obtained in the form of convolutions of functions corresponding to homogeneous solutions of initial-boundary value problems for the integral temperature characteristics and functions describing the available unsteady Joule heat sources and surface temperature values. The temperature regimes of a copper panel during its induction heating by a homogeneous quasi-steady electromagnetic field are analyzed numerically. Two characteristic modes of near-surface and continuous induction heating of the panel are considered. The results of calculations of temperature distributions depending on the parameters of induction heating and heat transfer conditions are presented in the form of 2D and 3D graphs.