A nonlinear axisymmetric transient heat conduction problem for a circular plate

Аuthors

Shlyakhin D. A., Guryanov S. A.^*, Nosikov I. S.

Samara State Technical University, SSTU, 244, Molodogvardeyskaya str., Samara, 443100, Russia

*e-mail: guryanovsa@mail.ru

Abstract

A new closed-form solution has been developed for the nonlinear axisymmetric transient heat conduction problem in a circular thick plate, subject to third-kind boundary conditions on its surfaces. Accurate modeling of temperature distributions in such thick plates is of considerable importance in various engineering applications, including thermal protection systems, industrial heating processes, and advanced material design. The nonlinear behavior of the system is introduced through a logarithmic dependence in the definition of volumetric entropy density, which is subsequently expanded into an infinite Taylor series. This representation allows the systematic consideration of temperature-dependent thermophysical properties and other nonlinear thermodynamic effects. 
To address the inherent nonlinearity, a stepwise solution approach is employed. In the initial stage, the problem is linearized by considering only the first term of the Taylor series. In subsequent stages, the solution is refined iteratively by solving an analogous problem in which the contributions of the nonlinear terms are incorporated through an auxiliary function derived from the results of the previous calculation. This iterative procedure provides a practical and efficient framework to account for nonlinear effects while maintaining an analytical formulation, thus bridging the gap between purely analytical and purely numerical approaches. 
The governing relations of the solution are obtained using the method of incomplete separation of variables. The temperature field is expressed through finite integral transforms with respect to the radial and axial coordinates, providing an explicit analytical representation of the transient temperature distribution throughout the plate. This formulation not only facilitates parametric analysis but also allows straightforward computation of temperature gradients and thermal fluxes at both the surfaces and interior of the plate. 
Analysis of the numerical results demonstrates the influence of nonlinear effects on the evolution of the temperature field during heating. The stepwise solution effectively captures the interaction between heat conduction and nonlinear thermodynamic behavior, providing a more accurate and detailed description of the transient thermal response. Overall, the proposed methodology offers a robust and versatile analytical tool for the study of nonlinear axisymmetric heat conduction in thick plates, combining theoretical insight with practical applicability. This work contributes to the advancement of thermal engineering analysis by providing a closed-form framework capable of addressing complex nonlinear heat transfer problems in engineering practice

Keywords:

circular plate, nonlinear heat conduction equation, finite integral transforms, numerical analysis, axisymmetric problem, temperature distribution, Taylor series

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