| The emphasis placed on the study is to develop a basic geometrical probabilistic model for random nonwoven fiber webs produced from fibers. This is a rational and essential starting point for both describing the non-uniformities of random fiber webs and relating their mechanical properties to the structures. Among others, the statistical expectation and variance of the number of fiber intersections in unit area are of great significance for determining various properties of random fiber webs. Especially, the distribution of the number of fiber intersections describes the nonuniformity of basis weight and can be used in designing optimal control strategies with respect to such physical properties as strength, elongation, air/water permeabilities, acoustics, filtering efficiencies, etc.; The basic geometric probabilistic models related to number of fiber intersection starts from the Buffon's needle problem. In classical areas, the probability of intersection between two straight fibers was obtained in the area in which there is no difference between the seeding region and the counting region. However this is not only an idealized assumption but also is not appropriate in defining geometrical probability in a specific area. This probability was obtained by assuming that all fibers lie entirely within that area, and did not take the "edge effects" into consideration. Furthermore, their equation for computing the variance of the number of fiber intersections substantially deviated from the simulation results.; In this study, we define the intersection probability between two fibers considering "edge effect" and obtain the statistical expectation of the number of fiber intersections in terms of them. For the first time, the variance of the number of intersections is obtained based on the covariance structure of two non-independent events that a given fiber intersects two other fibers simultaneously. These new results have established that all previous results from the independence assumption were in error.; The results on intersection geometry have been extended to the so-called "pore-size" distributions that axe crucial for determining the filtering efficiency and air and liquid transport properties of paper and nonwoven products. As a new measure of filtering efficiency, the probability that a random disk of radius r passes through the random fiber web has been developed along with a computer algorithm and program to enumerate the number of polygons. The computer simulation method facilitates comparison and validation of the existing theoretical work. |
