The performance of non-woven fibrous media, generally used for air filtration or purification, can be evaluated via their efficiency as well as their pressure drop, which depends on their permeability at low Reynolds numbers, in accordance with Darcy's law. Real filters are generally composed of a distribution of fiber diameters, complicating the experimental determination of their characteristics.

Permeability corresponds to the square of the fiber diameter divided by a packing density function f(α). Several authors have studied equivalent diameter models to adjust this relation for fiber bimodal distributions; considering the resistance due to a distribution of fibers as the resistance of a structure composed of a single equivalent fiber diameter. Assuming all fibers have the same length, different expressions of equivalent diameter from the literature can be summarized in equation 1 ...

... These different models illustrate that no consensus has been reached in the literature. The aim of this study is therefore to establish the best approach for determining the permeability of media composed of bimodal to multimodal distributions of fibers.

To avoid various possible experimental biases, perfectly characterized structures were generated using a numerical tool, the GeoDict (2022) software. To cover a wide range of different structures, while avoiding slipping phenomena, fiber diameters ranging from 1.5 µm to 30 µm were selected, with packing density ranging from 1% to 20%. For a given fluid velocity, the pressure drop was simulated, from which the permeability of the microstructure was deduced. For each microstructure (given compacity and fiber size distribution), five simulations were carried out with a different initialization thanks to a random seed, in order to calculate an average permeability for each type of structure.

The literature models, combined with the modified Happel packing density function (Thomas et al. 2023), were compared with permeability simulations of microstructures with a bimodal fiber distribution (figure 1). The diameters d2,1 and d2,2 which exhibit significant discrepancies are not shown graphically. These results highlight that no equivalent diameter in the literature can accurately predict the permeability value of bimodal media. Thus, a new mixing law model was developed, from which a new equivalent diameter was derived (eq. 2), providing a more accurate agreement with our simulated values.