MATHEMATICAL MODEL AND METHOD OF SOLVING THE GENERALIZED NEYMAN PROBLEM OF HEAT EXCHANGE OF PARABOLOID OF ROTATION

Abstract

Currently insufficiently studied the distribution of temperature fields in the workpieces with a new method of heating, carried out by rotating the workpieces in a magnetic field of direct current, which is created in exciters with superconducting windings without knowledge of which it is impossible to implement its technical implementation with high technical and economic performance. A small number of scientific papers are devoted to the study of electromagnetic and thermal phenomena inside the workpiece during its rotation in a magnetic field of direct current. Therefore, the task of developing mathematical methods for modeling temperature fields in workpieces during induction heating of metal using innovative technology, the solution of which is devoted to this work, is very important. The article constructs a new generalized spatial mathematical model for calculating temperature fields in workpieces, in the form of a paraboloid of rotation rotating at a constant angular velocity, taking into account the finite velocity of heat propagation as a boundary value problem of mathematical physics, and finding solutions to the boundary value problem. For the first time, a mathematical model for calculating temperature fields in a paraboloid of rotation, taking into account the finite velocity of rotating heat, is constructed as a boundary value problem of mathematical physics for hyperbolic equations of thermal conductivity with Neumann boundary conditions. An integral transformation for a two-dimensional finite space is constructed, using which the temperature field is found in the form of convergent series by Fourier functions. The solution of the generalized boundary value problem of heat exchange of a rotating paraboloid, taking into account the finiteness of the value of heat propagation, can be used to modulate the temperature fields arising from induction heating by rotating the workpieces in a magnetic field of direct current generated in the excitation. with superconducting windings.