Mathematical modeling of heating processes of bodies under influence of concentrated energy flows based on nonlinear hyperbolic heat conductivity equation

The article is devoted to numerical methods for solving nonlinear heat conduction problems with considering for the relaxation of heat flow. A mathematical model is developed on the basis of a non-linear heat equation of the hyperbolic type for calculating the temperature field in an infinitely extended (unlimited) plate. The implementation of the grid method using a three-layer implicit difference scheme for solving the nonlinear hyperbolic heat conduction problem is presented for the case when the absorption of radiation energy is modeled by a volumetric heat source. A numerical solution of the nonlinear heat conduction problem in an unbounded plate is obtained taking into account the relaxation of the heat flow on the basis of the finite difference technique using the sweep method and iterative refinement of the coefficients. A calculation algorithm with a graphical representation of the results of calculating the temperature field in an unbounded plate under the influence of concentrated energy flows is described. A comparison of the results of calculations of temperature fields in mathematical modeling on the basis of the nonlinear hyperbolic heat equation and the corresponding linear model using the mean integral values of thermophysical and optical characteristics is presented. The significant differences obtained between the temperature fields corresponding to the nonlinear and linear problems justify the need to take into account the temperature dependence of the thermophysical characteristics and the absorptivity in the study of high-intensity processes of heating the bodies. The developed nonlinear mathematical model of body heating with allowance for the finite speed of heat distribution and the temperature dependence of the material properties, can be used to select the modes for processing mode bodies with high-intensity energy flows.