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The article presents a nonlinear mathematical model of heating a two-layer body with allowance of the finite velocity of heat propagation and the temperature dependence of the properties of materials. A numerical solution of the nonlinear hyperbolic heat conduction problem is obtained for the case when the absorption of the radiation energy is modeled by a volumetric heat source. The implementation of the grid method using a three-layer implicit difference scheme in solving a nonlinear heat conduction problem in a two-layer body with allowance for the relaxation of the heat flux and the conjugation conditions in the case of ideal contact at the interface junction is considered. The described algorithm for calculating the temperature field for high-intensity heating of a coated body, taking into account the dependence of the thermophysical characteristics of materials from temperature, is based on the implementation of the sweep method with iterative correction of the coefficients. Programs are developed and the results of calculating the temperature fields are presented using nonlinear hyperbolic heat conduction equations and the corresponding linear ones taking into account the average integrated thermal and optical characteristics of the materials. Based on a comparison of the results obtained the necessity of taking into account the temperature dependence of the properties of materials during the study of processes of high-intensity heating of bodies.The developed mathematical model on the basis of a system of nonlinear hyperbolic equations can be used to create technological processes using methods for processing the surface of multilayer bodies by laser radiation.

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.

The use of new self-supporting insulated wires and high-temperature wires in the operation of power lines allows increasing the capacity of lines and, as a rule, reducing operational costs. An optimal utilization of the power line load capacity depends on the precise determination of the permissible current loads. The values of permissible currents and steady-state temperature are the main parameters of the line operating mode, affecting the strength and sag of the conductor. The temperature of the wire depends on weather conditions and current load. There are methods for determining the temperature and permissible currents for widely used traditional wires such as AC. They are partially outlined in the EIS (Electrical Installation Standard) and the standard of PJSC FGC UES (Federal Grid Company of Unified Energy System) of 2013. However, there is lack of studies in new types of wires. The paper considers the effect of weather conditions and load on the temperature and real-power losses in insulated and high-temperature wire, and solar radiation is under special consideration. For comparison, we present the results of calculations on traditional AC wires. The research shows that solar radiation, being taken into account, provides an increase of real-power losses of about 2 % with the given values of load and weather conditions. Calculations of permissible current values according to the developed technique for classical AC wires reveal a high coincidence with the values from PJSC FGC UES standard. The relative error is within two percent, and the proposed method is more generalized. It allows simultaneous analysis of both uninsulated and insulated wires. Due to the widespread use of self-supporting insulated wires, power industry experts can use the developed software in the design and operation of modern power lines to optimize capacity.