Mathematical modeling and analysis of heat transfer in structures with foreign elements

Purpose. The study aims to develop linear and nonlinear mathematical models for determining temperature fields in isotropic three-dimensional media containing foreign thermoactive semi-through elements. This approach enhances the accuracy of temperature regime analysis under thermal loads and contributes to the improvement of design methods for devices whose individual units include foreign thermoactive elements. Methodology. To obtain analytical-numerical solutions of linear and nonlinear boundary value problems of heat conduction, asymmetric unit functions were used. As a result, the thermal conductivity coefficient for a structure with a foreign semi-through cylindrical element is represented as a unified whole. This ensures the fulfillment of ideal thermal contact conditions at the interfaces of dissimilar materials in the structure, reducing the problem to solving a single heat conduction equation with discontinuous and singular coefficients. For the nonlinear boundary value problem, a linearizing function was introduced, allowing the transformation into a second-order linear partial differential equation with discontinuous and singular coefficients, and a quasi-linear boundary condition. By approximating the temperature as a function of spatial coordinates on the inclusion surface and the layer’s boundary surface using piecewise-constant functions, the nonlinear boundary value problem was fully linearized. Findings. Linear and nonlinear mathematical models were developed to determine the temperature field and analyze thermal regimes in devices containing a foreign thermoactive semi-through inclusion. A linearizing function was proposed to simplify the nonlinear boundary value problem. Analytical-numerical solutions for both the linear and nonlinear heat conduction problems were obtained, allowing the determination of the temperature distribution as a function of spatial coordinates. Comparative analysis revealed a 7 % difference between results for constant and linearly varying thermal conductivity coefficients, explained by the small values of the thermal conductivity temperature coefficient for the selected construction materials. Originality. A method for linearizing the nonlinear mathematical model of heat conduction was proposed. Analytical-numerical solutions to the corresponding linear and nonlinear boundary value problems were obtained in closed form. The use of asymmetric unit functions allowed for a correct mathematical description of heat transfer processes in media containing foreign thermoactive semi-through elements. Practical value. The developed heat transfer mathematical models enable the assessment of media in terms of their thermal resistance, contributing to the improved performance of devices containing foreign thermoactive semi-through elements. This prevents overheating and extends their operational life. The results can be applied to practical problems of heat exchange and thermal insulation in industrial structures, including predicting temperature fields in mining equipment mechanisms, ventilation systems, and compressor stations. Implementing the proposed models improves the efficiency of ore extraction and processing, as well as reduces heat loss in industrial systems.