Analytical Solutions for Transient Heat Conduction Problems in Axis-Symmetric Domains

Abstract This study presents analytical solutions for transient heat conduction problems in axisymmetric domains. A stepped temperature change and a time-periodic temperature are specified as boundary conditions. The resulting temperature field is obtained as a linear superposition of two distinct components: a transient contribution arising from an initial thermal step and a periodic steady-state component. Each solution is obtained separately using appropriate nondimensional scaling. The spatial and temporal evolution of the temperature fields are examined with particular focus on the thermal penetration depth. Analytical solutions of such heat conduction problems are well known for plane domains and for stationary problems. However, transient and axisymmetric configurations are more complex due to the coupled dependence of temperature on both radial position and time, requiring more elaborate mathematical treatment. In doing so, the present work complements existing eigenfunction-based approaches by providing closed-form, fully nondimensional solutions for transient and time-periodic heat conduction in axisymmetric cylindrical domains.