A Probability Model for Random Fiber

LEE, SUNG WON. A Probability Model for Random Fiber Breakage. (Under the direction of JOHN WILLIAM BISHlR). Two fundamental questions in connection with the random breakage of fibers in a breakage system are set forth: Given a fiber group of any particular length, (1) What is the probability that a fiber of the length groups breaks at all? (2) What is the probability that a fiber breaks into s segments (s :2: 2)? Since the existing models so far are based on the restrictions of linear probability of fiber breakage with respect to fiber length and two-s egment fiber breakage per breaking fiber, the models are not suitable for the elucidation of the above basic questions. To clarify the questions above and to provide a more realistic interpretation to fiber breakage phenomena, a new fiber breakage model is derived which allows multi-segment fiber breakage and a more flexible relation of the probability of fiber break to fiber length, linear or nonlinear, in the breakage system. First, a probability model for the uniformly random breakage of fibers of uniform length is derived with multi-segment fiber breakages taken into account. The model expresses the output fiber weight distribution in terms of the input fiber weight distribution and a parameter m, the average number of break points randomly occurring on an initial fiber due to the breakage system. The parameter m is a quantitative characterization of the severity of fiber breakage in the system, and simply estimated by the method of moments. Then, the first model above is extended to the cas e where the input fiber distribution is a nondegenerate arbitrary distribution. In this model it is assumed that the average number m. of break points J occurring on an initial fiber of length .t. is a function of the fiber J length .t. such that m. = a.t~, where a and 13 are parameters, which J J J permits flexibility of the probability of fiber breakage with respect to length. This model is further put into a compact matrix expression of the form .& = p . !..: where .& and1. are (rxl) column vectors for output and input fiber weight distributions in discrete densities, respectively, and P is an (rxr) stochastic matrix. The applicability of the matrix model above is demonstrated on two different lots of cotton fibers which had different pre-fiber proces s histories in cotton ginning process. The parameters a and 13 are estimated by nonlinear least squares iterations using the GaussNewton method. The estimated and observed output distributions are in good agreement, which verifies the validity of the model. Through '" '" the estimates a and 13, some new criteria of practical importance and usefulness are computed which provides a quantitative characterization of the different modes of fiber breakage. The set-out basic questions are completely elucidated for both cottons. Thus a probability model which provides a realistic interpretation of fiber breakage phenomena, in terms of probabilities of fiber breakage and average number of breakage points, is established. Through the applied examples it is found that the probability of fiber breakage with respect to fiber length is nonlinear and some three-segment fiber breakages occur, especially in longer fiber groups. Four or more segment breakages occurred relatively infrequently. Of course, this will depend on the severity of fiber breakage. Therefore, it can be concluded that the consideration of a flexible relation between the probability of fiber breakage and length and multi-segment fiber breakage in a fiber breakage model is pertinent for realistic interpretation of fiber breakage phenomena.