Mathematical Model
Representation of mass transport in materials that possess three mutually perpendicular axes of symmetry utilizes a second-rank tensor to describe the direction-dependent flow rates. In paper and paperboard, the orthotropic diffusion tensor characterizes how moisture or gases migrate along the machine direction, cross direction, and thickness direction. The tensor is essential for modeling barrier performance under varying environmental conditions.
Since paper is highly anisotropic, the diffusion rates in these three directions differ by orders of magnitude.
Directional Transport
Prediction of moisture penetration through a package wall requires precise knowledge of these directional coefficients. When using the orthotropic diffusion tensor, the calculation accounts for the high resistance of the Z-direction compared to the planar directions. The fibres lie predominantly in the sheet plane, creating a tortuous path for molecules moving through the thickness.
This anisotropic behavior means that planar diffusion is much faster, which affects the edges of the board. The model allows engineers to optimize the seam design of liquid packaging cartons.
Structural Application
Design of moisture-resistant containers depends on accurate simulation of these transport phenomena under tropical conditions. Utilizing the orthotropic diffusion tensor prevents the over-engineering of barrier coatings. It assists in calculating the exact lifetime of the packaged dry food product.
