Abstract:Theoretical equations are developed which give the average length and the length distribution of the whole fiber population from the scan of a fibrogram type specimen. The average fiber length is shown to be equal to twice the area under the fibrogram curve plus an offset which is dictated by the amount of fiber loss in the clamp. The length distribution, in terms of coefficient of variation, is shown to be the square root of three times the length at which the centroid of the fibrogram is located divided by t… Show more
“…According to the definition of the Ib sample, the number ci can be written as where k is a constant factor. Now if N = E N, and c TABLE Il There are four definition equations: Equation 8 for the numerical mean length, Equation 9 for SFN, equivalent Equation 15 for SFN in terms of F'(~, and Equation 13 for SFW (see also Equations 3 and 7). Equation 10 for the SFW is obtained by substituting fw from Equation 21 into Equation 13.…”
Section: Substituting Equation A9 Into Equation a 5 And Simplifying Ymentioning
confidence: 99%
“…The equivalent Equations 16 and 17 can be obtained by integrating Equation 13 by parts, with substitutions from Equations 21 and 3; that is, -1 and Equation 16 follows. To obtain Equation 17, we use Equation 14 and split the integral into two terms:…”
Section: Substituting Equation A9 Into Equation a 5 And Simplifying Ymentioning
The mathematical fundamentals of fiber length distribution are reviewed in a unified way, with particular emphasis on the derivation of short fiber content (SFC) by number and weight and other related length parameters. The concept of "similar distribu-
“…According to the definition of the Ib sample, the number ci can be written as where k is a constant factor. Now if N = E N, and c TABLE Il There are four definition equations: Equation 8 for the numerical mean length, Equation 9 for SFN, equivalent Equation 15 for SFN in terms of F'(~, and Equation 13 for SFW (see also Equations 3 and 7). Equation 10 for the SFW is obtained by substituting fw from Equation 21 into Equation 13.…”
Section: Substituting Equation A9 Into Equation a 5 And Simplifying Ymentioning
confidence: 99%
“…The equivalent Equations 16 and 17 can be obtained by integrating Equation 13 by parts, with substitutions from Equations 21 and 3; that is, -1 and Equation 16 follows. To obtain Equation 17, we use Equation 14 and split the integral into two terms:…”
Section: Substituting Equation A9 Into Equation a 5 And Simplifying Ymentioning
The mathematical fundamentals of fiber length distribution are reviewed in a unified way, with particular emphasis on the derivation of short fiber content (SFC) by number and weight and other related length parameters. The concept of "similar distribu-
“…The reported fiber length characteristics are estimated from the Fibrogram, which is the second integral of the length distribution of the original sample. [11][12][13][14] The length parameters commonly used by the cotton industry are the upper half mean length (UHML, the average of the longest 50% of fibers by weight), the mean length and the uniformity index. The uniformity index, the ratio of the mean length to the UHML, is used as a primary indicator of the width of the distribution.…”
mentioning
confidence: 99%
“…The reported fiber length characteristics are estimated from the Fibrogram, which is the second integral of the length distribution of the original sample. 11–14 Figure 1.The top portion is a schematic diagram of the arrangement of the High Volume Instrument (HVI) beard and the jaws/clamps for the HVI strength measurement. This is aligned with its associated hypothetical Fibrogram shown in the bottom portion of the figure.…”
This study explores the influence of cotton fiber length characteristics on the High Volume Instrument (HVI) strength measurement. A set of cotton samples cut at different lengths from a common parent sliver was used. HVI strength data exhibited a consistent trend as a function of the fiber length properties. This data was analyzed using the working hypothesis that the HVI estimates the total mass of fiber at a position between the jaws, rather than the true mass, which contributes to the breaking force. A quantitative model was developed to correct for this overestimation based on the shape of the Fibrogram. It was demonstrated that the Fibrogram can be adequately modeled as a straight line in the region of the HVI strength measurement. Based on this, it was found that the required correction factor is a function of the mean fiber length and various geometrical parameters of the HVI instrument (i.e. (a) the distance from the base of the beard to the inside edge of the clamp nearest the base of the jaws, and (b) the actual position of the beard's fiber length determination relative to the clamping jaws). Importantly this correction factor is independent of the shape of the fiber length distribution. Application of this correction factor approach was able to remove the effect of fiber length on the corrected strength values. The same model was also applied to other published data and, again, a simple correction factor based on mean fiber length adequately 'removed' the observed bias.Cotton length and strength are two important fiber quality parameters that are commonly measured using the Uster High Volume Instrument (HVI) (e.g. ASTM Test method D 4605-86 1 ). For example, the Agricultural Marketing Service of the United States Department of Agriculture (USDA) specifies a range of cotton fiber quality parameters, including fiber strength and length characteristics based on HVI measurements.Average length and strength values of US cotton have both steadily improved, 2 and it is assumed that these values reflect real and separate improvements in both characteristics. Indeed, plant breeders develop strategies for new varieties with improved length and strength on the assumption that both are separate parameters that require attention.Cotton fiber length is generally quite variable within a sample and description of the length distribution has been the subject of a wide range of studies (recent work includes Krifa 3,4 and Lin et al. 5 ). Fiber breakage during ginning is a major contributor to the broadening of the length distribution, and as Krifa 3 notes, modeling of the length distribution is dominated by applications of 'breakage models'. 6-8 Krifa 4 demonstrated that a mixture of two Weibull probability density functions adequately fits the experimental length distribution data.
“…This multiple is equivalent to the total length of all the fibers. Since Hertel presented this theory, this axis has been considered as the weight axis [6,7], the number or relative number axis [1,2,3,4], or the probability axis [6,8]. In this work, we show that the ordinate axis can be considered as a proportional mass or mass probability axis, and we examine the fibrogram for additional information pertaining to the length of the sample.…”
The theory of the fibrogram has been derived based on the probability of randomly catching and holding a fiber. The amount axis has been shown to be proportional to the relative mass when related to the original specimen. Tangential equations are used to show that theoretically, for a length l or longer, the proportional mass, the mean length by number, the relative number, and the mean length by mass of fibers in the sample can be obtained from the fibrogram, as well as the number and mass arrays. Finally, the theoretical percentage of fiber by number or by mass extending a length l or longer from a clamped random sliver of the sample is obtainable from the fibrogram.The theory of the fibrogram was first published by . Hertel [2]. He considered the abscissa as the length axis and stated that the ordinate for I = 0 is NL where N is the total number of fibers in the sample and L is their mean length by number. This multiple is equivalent to the total length of all the fibers. Since Hertel presented this theory, this axis has been considered as the weight axis [6,7], the number or relative number axis [1, 2, 3, 4], or the probability axis [6,8]. In this work, we show that the ordinate axis can be considered as a proportional mass or mass probability axis, and we examine the fibrogram for additional information pertaining to the length of the sample. ' . Theory . Statistical texts generally write P[x) for the probability that the stochastic variable is less than or equal to the real number x. P[x] is called the cumulative distribution function of the variable. The derivative is called the distribution function of x.The general function that leads to the fibrogram must be defined differently. Consider the probability that the stochastic variable is greater than or equal to the real number l; q[l is then the complement of the cumulative distribution function. Since the fibrogram relates to fiber lengths, constraints must be applied to q[/]. Fibers shorter than 0 and longer than the longest fiber within the sample do not exist. The constraints applied to the probability q[I are that where 1m is the length of the longest fiber(s) in the sample. q[l is then the probability of the existence of a fiber in the sample having a length I or longer. The derivative is the distribution function of l.Since the total number of fibers in the sample is N, where N(l ) is the number of fibers in the sample having a length equal to I or longer. This gives Differentiating Q(1) gives the frequency distribution or
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