Dual-beard sampling method for fibre length measurements

Transcription

1 Indian Journa of Fibre & Textie Research Vo. 39, March 14, pp Dua-beard saping ethod for fibre ength easureents H Y Wu & F M Wang a Coege of Texties, Donghua University, Shanghai, China Received 5 February 13; revised received and accepted 6 May 13 A ne ethod for fibre ength easureent using dua-beard saping ethod has been deveoped. The dua-beard saping ethod is used to produce a dua-beard sape on hich fibres are hed in the idde and cobed fro both sides. The dua-beard can be scanned fro the hoding ine synchronousy to generate fibrogras. The atheatica reationship beteen the fibrogra and the eight-based distribution of the origina sape is derived and the equations are estabished to cacuate ean ength by eight, coefficient of variation, oda ength and quaity ength of the origina sape. Finay, experients are perfored and the resuts sho that the dua-beard saping ethod can be used to easure fibre ength distributions. Keyords: Beard saping, Dua-beard saping, Fibrogra, Fibre ength, Length easureent 1 Introduction Length is one of the fundaenta properties of textie fibres. Usuay, fibres need to be aigned and cobed to produce beard for fibre ength anaysis. Therefore, beard saping is the critica process that affects the trueness and efficiency of ength easureent. At present, there are to different ays to produce fibre beards for fibre ength easureent. In the first ethod as used in coercia instruents, such as roer anayser 1, cob sorter and aeter 3, fibres are aigned to for a baseine in one end. This saping ethod is generay so, tedious and often yieds baseine fuctuations. Therefore, the first ay is not good enough for a rapid test ethod. In the second ethod as used in the high voue instruent HVI), fibres are caped randoy, aigned in the hoding ine, and cobed to for the beard. Soe researches 4, 5 pointed out that fibres are ikey to sip, be ost and be entanged in the saping process of HVI, and thus fibre distributions of the inner and outer of the cob are asyetric. Therefore, the atheatica reationship beteen the fibrogra and the fibre ength distribution of the origina sape is abiguous. Furtherore, soe fibres ay be entanged near the root of the cap and a singe fibre ay be hed at the hoding-ine utipe ties. Hence, the fibrogra scanning has to begin at a certain a Corresponding author. E-ai: fuei@dhu.edu.cn distance fro the hoding ine, and the beard cannot be scanned fuy. The fibrogra theory associated ith beard saping as first introduced by Herte 6. Based on the probabiity theory and a cob-saping ethod, Herte deonstrated that the fibrogra as an integra of the cuuative distribution function of fibre ength by nuber, or a doube integra of the fibre ength distribution function by nuber. This fibrogra theory as ater questioned by Weidong Pan 7. He asserted that for fibres ith different engths, the probabiities of fibres extending distance as east as great as L fro the baseine ere not equivaent. Afterards, other authors 8-1 considered the ordinate axis of the fibrogra as a proportiona ass or ass probabiity axis, rather than the nuber axis as defined by Herte. Prier 1 discussed the atheatica basis of the fibrogra fored by a pneuatic fibre ength anayer. According to the distribution of caped fibres in the cap, three types of fibrogras ere defined based on the conditiona probabiity and argina distribution theory, and the equations of the ean ength and the coefficient of variation ere deveoped. Taan 11 ater revised that the ean ength cacuation shoud be based on eight rather than on nuber as appeared in Prier s deveopent purport. Hoever, the fibrogra fundaentas in the previous orks have not fuy addressed the reationship in fibre ength distributions beteen the cobed sape and the origina sape, and that beteen the nuberbased distribution and the eight-based distribution.

2 WU & WANG: DUAL-BEARD SAMPLING METHOD FOR FIBRE LENGTH MEASUREMENTS 73 To continue investigating efficient and unbiased ethods for fibre ength easureent, there is sti a need for exporing ne fibre saping ethods and associated fibrogra theories. An effective saping ethod shoud possess four iportant features 1, 13, naey i) fibres ere caped randoy aong ength direction; ii) the quaity of the beard shoud be enough to get steady resuts; iii) the hed fibres cannot sip and be ost, and a hed fibre cannot be hed utipe ties; and iv) there is certain atheatica reationship beteen the fibrogra and the fibre ength distribution of the origina sape. With an ai to deveop an effective saping ethod, in this investigation a ne saping ethod, naey duabeard saping ethod, has been deveoped and associated fibrogra theories are studied. Aso, the experients ere perfored to verify the effectiveness of ne ethod. Methodoogy.1 Dua-beard Saping Method A ne saping ethod, naey dua-beard saping ethod, has been deveoped, hich can eet four iportant features of an effective saping ethod. Dua-beard saping ethod can produce a dua-beard sape hich has to syetrica ends and can be scanned fro the hoding ine to get to fu fibrogras for each sape. This ethod not ony eiinates possibe fibre entangeents in the cap and utipe hodings of fibres, but aso perits expicit atheatica equations of various ength paraeters to be derived directy fro the fibrogra. This ethod takes four ajor steps to produce dua-beard sape. Firsty, randoy seect a.-.3g sape fro tested fibres, open and ix fibres by hands, and reove trash partices ith a pair of teeers. Secondy, dra the sape three ties on a fibre dra-off device [Fig. 1a)] to prepare a siver in hich fibres are neary parae, straight and unifor [Fig. 1b)]. Thirdy, cap the siver randoy aong the fibre ongitudina axis. Reove oose fibres on the open end of the cap using a cob [Fig. 1c)]. Lasty, cap the siver ith another caper at the hoding ine of the first caper. Reease the first caper and reove a oose fibres on the open end of the second one [Fig. 1d)]. This dua-beard sape has a eight ranging fro g to 4g and a idth of 5c. The to beards are approxiatey syetrica, hich ao to fibrogras scans apart fro the hoding ine siutaneousy. Because fibres in the beards are e straightened and parae, the hoe fibrogra can be obtained.. Associated Fibrogra Theories It is paraount to study the atheatica reationship beteen the fibrogra and the fibre ength distribution of the origina sape. Ony hen this reationship is knon, the fibre ength distribution of the parent sape can be derived fro the fibrogra. To faciitate the study, soe paraeters and functions are defined such as is the fibre ength, is the axiu fibre ength, is the nuber ean ength of the origina sape, L is the span of a fibre protruding fro the hoding ine, p ) is the frequency density function by nuber of the origina sape, p ) is the frequency density function by eight of the origina sape, f ) is the frequency density function by nuber of the beard, and F L ) is the fibrogra...1 Length Distributions beteen Beard and Origina Sape In the dua-beard sape, the cap catches fibres randoy aong the ongitudina axis and the probabiity of fibres to be on the hoding ine is proportiona to their engths as e as the aount of fibres in the origina popuation. So, the caping probabiity of a fibre ith ength + d < ) is k p )d, here k is the ratio of the nuber of fibres in the beard to that in the origina sape. Fig.1 Dua-beard saping ethod a) fibre dra-off device, b) siver, c) cap dua-beard, and d) dua-beard

3 74 INDIAN J. FIBRE TEXT. RES., MARCH 14 Denote the tota nubers of fibres in the origina sape as N, and the nubers of fibres ith ength + d in the beard as N. Thus, N = N k p )d 1) The frequency density function by nuber of the beard, [ f ) ], can be expressed: N k p )d p ) f ) = d = N k p p ) d )d By the definition, e have ) = )d 3) p Substituting Eq. 3) into Eq. ) gives fooing reationship: f ) = p ) 4) It is knon fro the previous research 1, 14, 15 that the reationship beteen the ength distributions by nuber and by eight is ) p ) = p 5) Fro Eqs 4) and 5), e have f ) = p ) 6) This proves that the fibre ength distribution by nuber in the beard is equa to that by eight in the origina sape... Theoretica Equation of Fibrogra The fibrogra [ F L ) ] is the beard curve that shos the quantity of fibres at each protruding distance or span L ) fro the hoding ine. L L ) is a rando variabe hose frequency density function is F L ). The origin of the coordinates is set at the ocation of the hoding ine, and the abscissa and ordinate vaues are L and F L ) respectivey. F L) is contributed by fibres hose tota ength and protruding ength are not ess than L, as shon in Fig.. Fig. Diagra of fibres protruding fro the hoding ine Assue that the fibres in Fig. are idea perutation. Hence, for a hed fibre of ength L ), the probabiity, that the ength of the portion projecting fro the hoding ine by one side is at east as great as L [ L ]. The quantity of fibres ith ength + d is f )d, so the quantity L contributed for F L ) is f ) d. Therefore, the tota quantity of fibres protruding fro the hoding ine by one side to a distance L ) is L F L) = f )d 7) L Eq. 7) is the theoretica equation of the fibrogra...3 Matheatica Reationship beteen Fibrogra and Fibre Length Distribution In order to concude atheatica reationship beteen the fibrogra and the fibre ength distribution of the origina sape, substituting Eq. 6) into Eq. 7), e have L F L) = p )d 8) L By differentiation of Eq. 8) ith respect to L, e get L F L) = p )d L p ) = p )d L d L L p ) p L) = p L) d + L ) L L p ) = d L 9)

4 WU & WANG: DUAL-BEARD SAMPLING METHOD FOR FIBRE LENGTH MEASUREMENTS 75 And another differentiation yieds p ) [ )] L F L = F L = ) 1) L Sipifying Eq. 1) gives p ) ) = F 11) Lasty, Eq. 11) shos the atheatica reationship beteen the fibrogra and the fibre ength distribution by eight of the origina sape. Fro Eq. 11), it is cear that dua-beard saping ethod can satisfy the forth basic deands of the second saping ethod. So, if the fibrogra is obtained, the fibre ength distribution by eight of the origina sape can be obtained by Eq. 11)...4 Deterination of Paraeters of Fibre Length Distribution Since, in practice fibres are not fuy rando, parae and straight in sivers, the resuts contain any rando errors. Moreover, there is a agnification of rando errors inherent in taking derivatives of observed data. Thus, it is difficut to derive the fibre ength distribution by Eq. 11) 1. So, ength paraeters of the origina sape are extracted directy fro the fibrogra. We i give the equations to cacuate four ength paraeters, naey ean ength by eight, coefficient of variation of ength, oda ength and quaity ength that aays are used to represent fibre ength, as discussed hereunder Mean Length by Weight As others, 8, 1 e use atheatica expectation of to cacuate ean ength by eight, hich ay be expressed as foos: E = p 1) ) )d Substituting Eq. 11) into Eq. 1), e have E = F 13) ) )d Then, integrating by parts of Eq. 13) gives ) ) )d E = F F 14) Fro Eq. 9), e kno F ) =, so ) )d E = F 15) Then, integrating by parts of Eq. 15) gives ) )d E = F 16) Fro Eq. 16), it is seen that the ean ength by eight is equa to tice the area under the fibrogra curve. So the ean ength by eight of the origina sape can be obtained by cacuating the integra of the fibrogra...4. Coefficient of Variation of Length According to definition of variance, the variance of can be expressed as ) = ) )d D p 17) No returning to Eq. 11), e have ) = ) )d D F 18) Fro Eq. 18), after to successive integrations by parts, e have 3 ) = + )d ) D F 3 4 )d F ) = + F )6 4 )d 3 4 ) F ) = + = F F )6 4 )d ) = F )6 4 )d + Substituting Eq. 3) into the above Eq., the resut is ) = )6 4 )d + )d D F F = F )6 )d 19) So, e can obtain the coefficient of variation as foos D ) CV %) = 1 6 ) F )d = 1 ) After the fibrogra as extracted, Eq. ) can be used to cacuate coefficient of variation of fibre ength.

5 76 INDIAN J. FIBRE TEXT. RES., MARCH Moda Length The oda ength is the ength of fibres hose eight percentage is axiu in the ength distribution 1. We i derive oda ength fro the fibrogra Fig.3). Suppose F ) and F + d ) are to adjacent dates in the fibrogra curve ith the arc ength S. The tangents to F L ) at and + d intersect the vertica axis at E and D respectivey. F is the point of intersection and α is the cross ange of the to tangents. As the definition of curvature, the curvature at is expressed as foos k α = i S S When S is a constant, e can obtain k 1) α. Fro Fig. 3, it is readiy seen that α = BFE BFD and α DE. Therefore, it can be obtained. k DE ) In addition, the eight percentage of fibres ith ength ay be expressed as foos )d = )d )d + d 3) p p p No returning to Eqs 8) and 9), e have p )d = F L ) LF L ) 4) L When L = and L = + d respectivey, substituting Eq. 4) into Eq. 3) gives p )d = F ) ) d ) d ) d ) F F F + 5) Fig.3 Diagra of extracting oda ength Aso, Fig. 3 shos the reations F ) = OC F + d ) = OA F ) = CE and + d ) F + d ) = AD. Substituting the reations into Eq. 5) gives p )d = OC + CE OA AD = DE 6) So, e can obtain p )d DE 7) Fro Eqs ) and 7), the resut is k )d p 8) Fro Eq. 8), it is readiy seen that ony hen k is axiu, p )d is axiu. That is to say that in the fibrogra the abscissa of a point, hose curvature is axia, is the oda ength of the origina sape. This feature coud be used to cacuate oda ength of the origina sape Quaity Length Since the quaity ength is defined as eighteighted average ength of fibres hose ength exceed the oda ength ) 1, the quaity ength p ) ay be expressed as foos P = p )d p )d )d = ) ) + )d 9) Substituting Eq. 11) into the nuerator of Eq. 9) and integrating by parts, e have p F F F 3) Substituting L = into Eq. 4), it foos that p )d = F ) F ) 31) Then, substituting Eqs 3) and 31) into Eq. 9), the quaity ength is p = ) ) + )d F F F F ) F ) 3) Eq. 3) provides a ethod to obtained quaity ength of the origina sape fro the fibrogra.

6 WU & WANG: DUAL-BEARD SAMPLING METHOD FOR FIBRE LENGTH MEASUREMENTS 77 3 Materias and Methods 3.1 Coparison ith Singe Fibre Measureent Method Sape Six kapok sapes, to cotton sapes and one poyester sape ere seected, hich contained a broad range of ength-distribution properties Test Methods Dua-beard saping ethod for easuring fibre ength distributions as appied. First, the beard as produced by dua-beard saping ethod. Then, the beard as scanned to get transission ight. Afterards, coputer softare as deveoped to obtain the fibrogras Fig. 4) based on the inforation of beard s transission ight, and to cacuate ean ength by eight, coefficient of variation, oda ength and quaity ength using Eqs 16), ), 8) and 3) respectivey. The resuts of one easureent ere the average vaue of the to ends of the dua-beard. Finay, the coputer softare exported resuts. Singe fibre easureent ethod 16 as perfored on these sapes. Then the fibre ength distribution by nuber as obtained, hich coud be converted to the fibre ength distribution by eight by assuing constant eight per unit ength. Thus, fibre ength characteristics ere coputed on the basis of eight. The consistency of the to ethods is shon in Fig Coparison ith HVI and AFIS Five cotton sapes ere tested by Dua-beard saping ethod, HVI and AFIS. The three ethods can obtain one sae ength paraeter such as ean ength by eight Fig. 6). Fig.4 To fibrogras of the dua-beard 4 Resuts and Discussion Figure 5 shos the reationship beteen dua-beard saping ethod for fibre ength easureent and singe fibre easureent ethod. The square of correation coefficient very good agreeent beteen the to ethods. So these resuts sho that the equations derived offer a ethod of cacuating, fro fibrogra, R shos Fig.5 Consistency of the to ethods a) ean ength by eight, b) oda ength, c) quaity ength, and d) coefficient of variation

7 78 INDIAN J. FIBRE TEXT. RES., MARCH 14 are extracted. The experients perforedverify that the equations derived offer a ethod for cacuating fibre ength fro the fibrogra and dua-beard saping ethod can be used to easure fibre ength. Acknoedgeent The authors acknoedge the support given by the Centra Universities in China for founding the project Grant No.1D1138). The authors are aso gratefu to B G Xu and J J Yan for providing riting assistance. Fig.6 Coparison ith dua-beard saping ethod, AFIS and HVI ean ength by eight, oda ength, quaity ength and coefficient of variation. That is to say that the dua-beard saping ethod can be used for fibre ength easureent. Figure 6 shos that the ean ength by eight of Dua-beard saping ethod is very cose to that of AFIS s ethod, hich are sighty higher than HVI s. The square of correation coefficient is R =.991beteen Dua-beard ehtod and AFIS, and R =.9765 beteen Dua-beard ethod and HVI. So, Dua-beard saping ethod has ore agreeent ith AFIS than HVI. 5 Concusion A ne ethod for easuring fibre ength distributions using dua-beard saping ethod has been studied. Based on the ne saping ethod, the atheatica reationship beteen the fibrogra and a eight-based distribution of the origina sape is derived. Theoretica equations of ength paraeters of the origina sape References 1 Test Method for Length of Cotton Fibers-Part 1:Roer Anayser Nationa Standard of the Peope s Repubic of China, Beijing), 6. Cotton fibres Test ethod for ength Cob sorter ethod Nationa Standard of the Peope s Repubic of China,Beijing), 8. 3 Yu W D, Text Materia China Textie & Appare Press, Beijing), 6, Zhang Y X, J Text Res China, 51984) He X F, Liu W Y & Xu S D, J Donghua University Eng Ed.) China, 36) Herte K L, Text Res J, 1194) Pan W D, J Donghua University Natura Sci) China, 31959) Kroicki R S & Duckett K E, Text Res J, ). 9 Landstreet C B, Text Bu, ) 54, Prier H W & Sasser P E, Text Res J, 4197) Taant J D & Pittan R A, Text Res J, ) Liu R H, A study on the testing of cotton short fiber content, PhD thesis, Dong Hua University, Shanghai, China,. 13 Jiang S C, Lai K, Wang X D & Yao M, Northest Inst Text Sci Techno China, 151) Woo J L, J Text Inst, ) Zeidan M I & Batra S K, Text Res J, ) Li R Q & Song J C, Measuring Techniques for Textie Materias China Textie University Press, Shanghai), 1995, 5.