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  4. 2025
  5. VOL 47, NO 5 (2025)
  6. pp. 105-125

MATHEMATICAL MODELS AND ANALYSIS OF THE TEMPERATURE FIELD RESULTING FROM HEATING OF ELECTRONIC DEVICES

V I. Havrysh

Èlektron. model. 2025, 47(5):105-124

https://doi.org/10.15407/emodel.47.05.105

ABSTRACT

Linear and nonlinear mathematical models for determining the temperature field and, subsequently, for analyzing temperature regimes in electronic devices described by an isotropic plate have been developed. This plate is heated by a heat flux concentrated in a rectangle on the boundary surface and uniformly distributed internal heat sources in a parallelepiped. For the case of a thermosensitive plate (the thermophysical parameters of the material depend on temperature), the Kirchhoff transformation has been applied, which made it possible to linearize the original nonlinear boundary value problems. As a result, linear second-order differential equations with partial derivatives and a discontinuous right-hand side, linear boundary and partially linearized boundary conditions were obtained. The temperature was approximated by the spatial coordinate on the plate boundary surface by a segment-constant function and as a result, a linear boundary value problem was obtained. The integral Fourier transform was applied to the boundary value problems and analytical and analytical-numerical solutions were obtained. For a thermally sensitive plate, as an example, the linear dependence of the thermal conductivity coefficient of a structural material on temperature, which is often used to solve many practical problems, was chosen. As a result, an analytical relationship was obtained for determining the temperature distribution in a thermally sensitive plate. Software tools were developed, using which numerical analysis and geometric representation of the temperature distribution depending on spatial coordinates were performed. The developed linear and nonlinear mathematical models for determining the temperature field in spatial thermally active environments allow analyzing their thermal resistance. As a result, it becomes possible to increase it and protect it from temperature disturbances that can cause destruction of both individual nodes and the entire structure.

KEYWORDS

temperature field; isotropic thermally sensitive plate; thermal conductivity of the material; convective heat transfer; heat flux; thermal sensitivity of the material; ideal thermal; contact, thermal resistance of structures.

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