Mathematical models for determining and analyzing thermal regimes in mining industry mechanism structures

Abstract

Purpose. To develop linear and nonlinear mathematical models of heat conduction for isotropic heterogeneous media with internal heating. This will allow for an increased accuracy in determining temperature fields, which will subsequently impact the effectiveness of designing mechanisms, devices, and individual components of structures that have a layered structure and are subjected to heat stress. Methodology. For the development of linear and nonlinear mathematical models of the temperature field and the analysis of temperature regimes in layered media with internal thermal heating, the coefficient of thermal conductivity is described as a whole using asymmetric unit functions. This makes it possible to solve a differential equation with singular coefficients in both linear and nonlinear boundary value problems of heat conduction with appropriate boundary conditions. Findings. Quadratic equations are obtained to determine the analytical solutions of linear and nonlinear boundary problems of heat conduction for a layered plate with internal heat load. Originality. The scientific novelty lies in the given method of linearization of the nonlinear mathematical model of heat conduction and obtaining analytical solutions, in a closed form, of the corresponding linear and nonlinear boundary value problems for isotropic layered media subjected to internal heating. Practical value. The developed linear and nonlinear mathematical models for determining the temperature distribution in layered structures with internal heating make it possible to analyze heat exchange processes and ensure the thermal stability of such structures. This also makes it possible to increase the heat resistance of structures and protect them from overheating, which can lead to damage to individual components and elements of mechanisms, as well as to the entire structure as a whole. The resulting analytical solutions can be used to predict temperature fields in mine shafts, underground environments and mechanisms of mining equipment, in particular, in drilling and underground compressor stations, ventilation systems and other equipment, which improves work efficiency and reduces useful energy consumption.