Diffusion of active ingredients in textiles: a three step multiscale model

Most practical textile models are based on a two scale approach: a one-dimensional fiber model and a fabric model. No meso-level is used in between, i.e. the yarn scale is neglected in this setup. For dense textile substrates this seems appropriate as the yarns connect everywhere, but for loose fabrics or scrims this approach cannot be kept. Specifically when one is interested in tracking an active component released by the fibers, the yarn level plays an important role. This is because the saturation vapor pressure will influence the release rate from the fibers, and its value will vary over the yarn cross-section. Therefore, in this work we present a three step multiscale model: the active component is tracked in the fiber, the yarn, and finally at the fabric level. At the fiber level a one-dimensional reduction to a non-linear diffusion equation is performed, and solved on a as needed basis. At the yarn level both a two-dimensional or a one-dimensional model can be applied, and finally the yarn result is upscaled to the fabric level