A Probability Model for Random Fiber Breakages

Abstract

Applications of the probability model for random fiber breakage which was derived in Part I are demonstrated. The parameters of the model for an arbitrary initial fiberlength distribution are estimated by the nonlinear least-squares iterative method. Some practical uses of the estimates of parameters are shown. Different modes of fiber breakage for two different lots of cotton due to opening-carding process are assessed in terms of average number of break points and probability of fiber breakage, thus, providing complete clarification to the fundamental questions raised at the beginning of the study and a realistic interpretation to the random fiber breakage phenomena.Results of the application show that the probability of fiber breakage on fiber length is moderately nonlinear and sizable three segments fiber breakage was seen to be present, especially for long fiber groups, although in an over-all sense it was fairly small. Fiber breakages into more than four segments were negligible. These will, of course, depend on the particular breakage process under study. It is demonstrated that the model can be used to assess analytically whatever the mode of ordinary random fiber breakages, allowing more flexible relation between breakage probability and fiber length and taking multi-segments fiber breakage into account. Estimation and Practical Uses of ParametersProbability models for random fiber breakage for the cases of uniform-length input fibers and arbitrary-length input fibers were derived as shown in Equations 18 and 42, respectively [3]. In this paper, estimation of parameters and some informative interpretations, through the estimates of parameters, pertaining to different modes of fiber breakages are demonstrated on two different lots of cotton. The fiber breakage model (Eq. 18, [3]) for uniform-length input fibers may seldom .be used directly in textile practice. Equation 42 [3] for an arbitrary-length input fiber-weight distribution is more pertinent for conventional textile fibers.For this model, in which two parameters a and are involved nonlinearly in the model, a nonlinear iterative solution by the standard Gauss-Newton method is used for the estimation of parameters.Since the model for uniform input fibers can be regarded as a special case of the model for arbitrary input fibers as discussed for Equation 45 [3], any estimation procedure for the latter may apply to the , former' case. However, a separate estimation of the parameter m in the former model is presented here, since a simple and explicit expression for the estimate by the method of moments is readily available and the estimate bears a descriptive physical meaning. Uniform-Length Input FibersBy directly using the model (Eq. 18, [3]), it is found that the first reciprocal moment of the weight distribution function has a very simple linear expression in the parameter m. Let Exll1 X be the first reciprocal moment for Equation 18 [3~.Then, Ex'° can be written as = 1 For Part I, see reference [3].

The Mechanics of Stretch Breaking

“…In fact, the author himself expresses that his theory is applicable to any process in which a random size reduction of one dimensional substance takes place [6].…”