Analytical solution of the Langmuir model for moisture diffusion in cylindrical coordinates

Moisture diffusion in polymer composites and bio-based materials frequently exhibits non-Fickian behaviors, such as delayed saturation or two-stage sorption kinetics. While the Langmuir model accurately captures these phenomena by accounting for the dynamic trapping of mobile water molecules, exact analytical solutions have historically been restricted to planar geometries.
In this study, the coupled partial differential equations of the Langmuir model are solved analytically in cylindrical coordinates. Exact closed-form solutions for both the local spatio-temporal moisture distribution and the macroscopic mass uptake are derived via the Laplace transform and Cauchy’s residue theorem. The thermodynamic stability of the temporal eigenvalues is mathematically proven, and the exactness of the series is validated against a finite difference numerical scheme, yielding a maximum relative deviation of 0.201%.
This exact analytical framework eliminates the computational costs and meshing dependencies of numerical solvers, providing a direct and robust tool for identifying transport parameters in cylindrical engineering systems such as natural fibers.