Construction of mathematical models of heat exchange for digital devices with local near-surface and internal heating

This study investigates heat exchange processes for thermally sensitive media with local near-surface and internal heating. As a result of the thermal load, significant temperature gradients arise. To establish temperature regimes for effective operation of electronic devices, linear and nonlinear mathematical models for determining the temperature field have been constructed, which could allow further analysis of temperature regimes. Based on the stated linear and nonlinear boundary value problems of thermal conductivity, their analytical and numerical solutions have been derived. Using these solutions, numerical calculations of the temperature distribution in spatial coordinates for given geometric and thermophysical parameters have been performed. Reliability of the results has been confirmed by experimental findings and the determined numerical values of temperature distribution in the medium. For an effective description of local heating, the theory of generalized functions was used. A technique for linearizing nonlinear mathematical models has been introduced. As a result, linear second-order differential equations with partial derivatives and a singular right-hand side have been derived. The numerical results reflect the temperature distribution in the medium in spatial coordinates for the given geometric and thermophysical parameters. The number of divisions of the interval (0; x*) was chosen to be 9, which made it possible to derive numerical values of temperature with an accuracy of 10–6. The obtained numerical values of temperature for silicon under a linear temperature dependence of the thermal conductivity coefficient differ from the results obtained for its constant value by 2%.