Distribution of Fiber Intersections in Two-Dimensional Random Fiber Webs – A Basic Geometrical Probability Model

It is of great mathematical significance and practical importance that the number of fiber intersections and uniformity within a unit area of a non-woven fiber web changes with the number and lengths of fibers as well as with aggregate length. The numerical relationships, while most important for imparting the desired physical and mechanical properties of the resulting fabric, are complex and not easily understood due to the geometrical and probabilistic nature of fiber arrangement within a web that provides the intersection distribution. The distribution not only determines the uniformity of the basis weight but also such physical properties as strength, elongation, air and water permeabilities, acoustics, and filtering efficiency of the resulting fabrics as well as the optimal control strategies with respect to the desired properties [1, 2]. As the number of fibers per unit fabric area increases, the physical counting of the intersections becomes an almost impossible task, making the theorybased computer simulation the only viable alternative.1 The major thrust of this study is to understand the basic nature of fiber arrangement within a two-dimensional random fiber web in terms of geometrical probability under certain simplified assumptions, and to extend the results to a more general case. First, a given number of fibers of equal lengths will be considered within a unit area as a starting point. Of primary importance then is to compute the probability of intersection between any two fibers within a given area when, say, the midpoints of the two fibers are assumed to be located within the area. This will be the founding block for deriving the mean and variance of the number of fiber intersections in the area when n fibers are placed within what we call the “seeding area.” As Abstract Fundamental theories governing the number of fiber intersections in random nonwoven fiber webs were developed based on the planar geometry of fiber midpoints distributed in a two-dimensional Poisson field. First, the statistical expectation and variance for the number of fiber intersections in unit web area were obtained as functions of a fixed number of fibers with equal lengths. The theories were extended to the case of a two-dimensional Poisson field by assuming that the number and locations of the fibers are random. The theories are validated by a newly developed computer simulation method employing the concept of “seeding region” and “counting region.” Unlike all previously published papers, it was shown for the first time that the expectations and variances obtained theoretically matched that from computer simulations almost perfectly, validating both the theories and simulation algorithms developed.