Modelling Non-Fickian Behavior in the Cell Walls of Wood Using a Fractional-In-Space Diffusion Equation

Abstract

Recent studies of diffusion processes in highly heterogeneous, fractal-like media highlight that the traditional diffusion equation may not adequately describe the movement of water in the pore network because of the evidently anomalous transport phenomena. It is postulated here that the diffusion of moisture in the cell walls of wood is anomalous (non-Fickian), and this motivates the use of a moisture potential defined in terms of a fractional-in-space operator involving the Laplacian raised to a fractional index. A coupled anomalous transport model for temperature and moisture content is then derived to simulate the absorption of water in the cell walls of wood. This model is discretized in space using the finite volume method to produce a large system of ordinary differential equations, which is advanced in time using a second-order exponential Euler method. A novelty of our computational strategy is the way in which the flux boundary condition involving the moisture potential is implemented to ensure an accurate global mass balance is achieved at each time step. The model is validated against experimental data available for beech, where good agreement is observed.

Keywords:

• Anomalous diffusion
• Fractional-in-space operator
• Matrix transfer technique
• Wood microstructure

ACKNOWLEDGMENTS

This research work was supported by the Australian Research Council Discovery grant DP0986766. We would also like to thank the anonymous referee for thorough reading of the text, and for offering suggestions on how to improve the overall presentation of the material.