Analytical Modelling of Orthotropic Transient Heat Conduction in the Thermal Therapy Mask Within the Symplectic Framework

The thermal therapy mask, as a wearable device, requires precise thermal management to ensure therapeutic efficacy and safety, which necessitates a detailed investigation of its heat conduction behavior under complex conditions. However, the heat convective behavior of an orthotropic thermal therapy mask with an embedded line heat source under practical operational conditions has not yet been rigorously investigated. Therefore, this study addresses this specific problem by abstracting it into a 2D orthotropic transient heat conduction problem with a line heat source under Robin BCs, and derives its analytical solution using the SSM without any assumption of solution form. The SSM first transforms the governing equation into the frequency domain via the Laplace transform technique and reformulates it within the Hamiltonian framework. The original problem is then decomposed into two subproblems, which are solved by the method of separation of variables and the symplectic eigen expansion. The final analytical solution is obtained through superposing the solutions of the subproblems, and its accuracy is validated through comparison with the finite element method. The influence of the heat convection coefficient on the thermal behavior is systematically analyzed, revealing that increasing the heat convection coefficient accelerates the procedure from transient to steady state and results in reduced steady-state temperature. Furthermore, the analysis of orthotropic thermal conductivity reveals a "short-plank effect", where the temperature evolution is limited by the smaller thermal conductivity. This study provides benchmark results for accurate and efficient thermal prediction and may enable an extension to broader applications in flexible electronics such as wearable sensors and displays.