The focus of the present work was on a low density, polymeric, fibrous medium, comprising a mix of blown micro-fibers, having a relatively broad fiber size distribution, and a second, so-called staple fiber component having a very narrow fiber size distribution. The airflow resistivity of such a material is usually considered to be its most important macroscopic property when it comes to defining the material’s acoustical properties (since the tortuosity and the porosity are very close to unity, for example). Thus, in the first instance, it is of interest to be able to calculate the flow resistivity of such a material on the basis of the distributions of the various fiber sizes, the densities of the fibers and the bulk density of the material. A recent survey of methods for predicting the flow resistivity of fibrous media has revealed a wide variety of approaches, largely based on a knowledge of a material’s solidity (1 minus the porosity), and mean fiber spacing, but in all cases it is assumed that the fiber radius is uniform. An example of such an approach is the work of Tarnow who has developed a model based on the viscous drag experienced by uniform-sized fibers positioned within randomly-spaced Voronoi cells. Recent measurements have shown that Tarnow’s “perpendicular random” model allows accurate predictions of flow resistivity for fibrous media comprising a single fiber component having a very narrow fiber size distribution. It has also been shown that Tarnow’s theory can be modified to account for multiple fiber components having different fiber size distributions. It is then assumed that the flow resistivity calculated in that way can be used to predict the acoustic properties of the medium, although the latter approach takes no specific account of the range of fiber sizes existing within the fibrous medium. Thus, in the present work, two specific issues are addressed: first, how accurate is the new method of calculating the flow resistivity, and secondly, can the effect of fiber size distributions be neglected when predicting sound propagation through such media. The execution of this work has primarily involved the use of numerical tools, GeoDict and Comsol. In particular, Fiber Geo was used to generate a variety of fiber arrays having different orientations and fiber size distributions, and FlowDict was used to compute the pressure drop resulting from low speed viscous flow through a cell of fibers, from which the flow resistivity can be calculated, for example. In the first stage of the work, fibers were modeled as occupying a fluid volume. It is possible to specify fiber sizes and orientations in order to represent materials consisting of fibers having a specified distribution of fiber radii, based for example of micro CT scans of real materials. Flow resistivity results for sets of fibrous media having differing solidities were computed in this way. These results are for practical purposes “exact”, and so provide benchmarks against which parameterized predictions such as presented in reference can be compared. Close agreement with the latter results has been found. Further, the effect of fiber orientation has been studied, which has allowed the prediction of direction dependent flow resistivity in non-isotropic fiber arrays. In the second phase of the work, the fiber geometries considered above were imported into Comsol. By using that software, all of the JCA parameters can be calculated, hence making it possible to calculate the acoustical properties of the fibrous media: e.g., complex densities and sound speeds. At the same time, finite element models of the fiber arrays can be generated, and then linearized visco-thermal models may be solved to yield the “exact” wave propagation characteristics of the fiber arrays. Initially, rigid models were studied: i.e., no motion of the fibers was allowed. Subsequently, full fluid-structure interaction models were implemented to allow for fiber motion in response to oscillatory acoustic flows. In this way, the propagation properties of limp porous materials may be predicted, as can the properties of elastic fiber networks when the fibers are connected. By using the models described here, wave propagation in fiber arrays having realistic bi-modal fiber size distributions has been predicted and compared with corresponding predictions made using conventional Biot-based models. As a result, it has been possible to draw conclusions regarding the ability of those models to accurately represent wave propagation in fibrous media having relatively broad fiber size distributions.
Fibrous media, Numerical modeling, Flow resistivity, JCA parameters, Acoustics
Acoustics and Noise Control
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