Probabilistic Assessment of Irregularity in Random Fiber Arrays—Effect of Fiber Length Distribution on “Variance-Length Curve”

The exact distribution and moments are derived for the length (mass) aggregate of fibers to be found within an interval L of a random fiber array as a direct function of the constituent fiber length distribution and the interval size L. The moments immediately facilitate computation of irregularity in terms of the coefficient of variation or the better-known “variance-length curve”, B( L), without using an autocorrelation function required by the traditional time-series approach. The B( L) curve thus obtained reflects the effects of fiber lengths and their orientation on irregularity of the fiber array in the absence of other contributing factors. For a population of uniform fiber lengths, the new results are shown to be compatible with that obtained by using the time-series method. Effects of fiber length and fineness on the resulting yarn irregularity are examined theoretically for a number of selected yarn counts.