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A mathematical model of fiber orientation in sliversBohuslav Neckář a , Dipayan Das b & S.M. Ishtiaque b

a Department of Textile Technology , Technical University of Liberec , Liberec , CzechRepublicb Department of Textile Technology , Indian Institute of Technology Delhi , New Delhi , IndiaPublished online: 31 Aug 2011.

To cite this article: Bohuslav Neckář , Dipayan Das & S.M. Ishtiaque (2012) A mathematical model of fiber orientation inslivers, The Journal of The Textile Institute, 103:5, 463-476, DOI: 10.1080/00405000.2011.586153

To link to this article: http://dx.doi.org/10.1080/00405000.2011.586153

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A mathematical model of fiber orientation in slivers

Bohuslav Neckář a, Dipayan Dasb* and S.M. Ishtiaqueb

aDepartment of Textile Technology, Technical University of Liberec, Liberec, Czech Republic; bDepartment of TextileTechnology, Indian Institute of Technology Delhi, New Delhi, India

(Received 27 February 2011; final version received 28 April 2011)

The directional arrangement of fibers significantly determines the internal structure and the mechanical propertiesof the slivers and the ultimate yarns produced from such slivers. In order to evaluate the directional arrangementof fibers in slivers, the experimental method according to Lindsleys’s idea, based on the weighing of suitablecombed-out and cut-out fringes, is most frequently used in practice. But, the traditional Lindsley’s evaluation takeson some empirical ratios of fringe weights only. In this work, the experimental method according to Lindsley’sidea is analyzed and a set of relevant equations are derived for the theoretical determination of the measured fringeweights. The new characteristic of fiber orientation represents the harmonic mean of cosines of angles among theshort fiber segments and the longitudinal direction of the sliver. The probability density function of the directionalarrangement of fibers in a sliver is also estimated. The experimental data obtained on polyester-drawn sliver iscompared with the theoretical results and a satisfactory correspondence between them is observed. The fiber orien-tation function reported in this article can be used in practice as a new quality parameter for the slivers and alsoto judge the effectiveness of the fiber preparation processes.

Keywords: fiber; orientation; sliver; model; experiment

Introduction

The arrangement of fibers in the slivers is describedby the directional as well as the packing arrangementof the fibers in the slivers. The directional arrange-ment of the fibers in the slivers is generally known asfiber orientation. It has been observed that the orienta-tion of fibers plays a very important role in determin-ing the mechanical properties of the slivers and themechanical properties of the ultimate yarns producedfrom the slivers. Also, the orientation of fibers in theslivers is a very useful parameter for evaluating theeffectiveness of the fiber preparation processes,namely carding and drawing processes. Moreover, thefiber orientation is known to determine the fiber lengthutilization in slivers. But, the measurement of fiberorientation in the slivers is a very complex task as thecross-section of a sliver typically contains severalthousands of fibers. Lindsley (1951) developed anapparatus and described a methodology that has beenwidely followed in practice for the measurement andevaluation of fiber orientation in the slivers. Later on,Simpson and Patureau (1969) modified Lindslay’sapparatus (IIT Delhi, New Delhi, India) to measureand evaluate the orientation of fibers in the slivers

more accurately and comprehensively. A large numberof research studies have been reported to use Linds-ley’s apparatus and methodology in order to character-ize the orientation of fibers in the carded and drawnslivers and also to establish the effect of carding anddrawing process parameters on the fiber orientation inthe slivers (Garde, Wakandar, & Bhaduri, 1961;Ghosh & Bhaduri, 1968; Kumar, Ishtiaque, & Salho-tra, 2008; Perel, 1982; Rao & Garde, 1962; Simpson,Sands, & Flori, 1970). However, Lindsley’s methodol-ogy for evaluation of fiber orientation in the slivers isunderstood as empirical only and there is no scientificbasis reported on this till date in the literature. In thiswork, a mathematical model of fiber orientation inslivers is derived based on Lindsley’s methodologyand demonstrated in support of a practical example ofpolyester-drawn sliver.

Mathematical model

Background

The mathematical model presented in this article isbased on the methodology reported by Lindsley (1951)which is often used in practice for determination

*Corresponding author. Email: dipayan@textile.iitd.ac.in

The Journal of The Textile InstituteVol. 103, No. 5, May 2012, 463–476

ISSN 0040-5000 print/ISSN 1754-2340 onlineCopyright � 2012 The Textile Institutehttp://dx.doi.org/10.1080/00405000.2011.586153http://www.tandfonline.com

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of fiber orientation in slivers. The step-by-step proce-dure of Lindsley’s methodology is described below insupport of an apparatus which is schematically shownin Figure 1.

Step 1: Take a sufficiently long sliver. Twist oneend of the sliver slightly so as to mark the direc-tion by which it was delivered by the machine,i.e. forward or backward. Take out the top threeplates P, Q, R (Figure 1) and place the sliveronto the bottom plate D and clamp it by reposi-tioning the top three plates.Step 2: Comb gently the sliver in the forwarddirection in order to remove all the loose fibersthat are not clamped by the plates Q and D. Dis-card the combed-out fibers. Then cut the fibersusing a sharp razor blade at the right edge of theplate Q and weigh the cut fiber portion. Let thisweight be Wf.Step 3: Remove the top plate Q. Then, comb thefibers held below it. Retain the combed-out fiberportion and weigh it. Let this weight be Cf.Step 4: Put the top plate Q back to its originalposition. Then, cut all the fiber ends that areextending beyond the edge of the plate Q. Col-lect the cut fiber portion and weigh it. Let thisweight be Ef.Step 5: Remove the plate Q again. Then, cut thefibers at the right edge of the plate R. Collect thecut fiber portion and weigh it. Let this weight beNf.Step 6: Repeat the steps from 1 to 5 for thebackward direction. Let the correspondingweights of fibers portions be Wb, Cb, Eb, and Nb,respectively.Step 7: Repeat the steps from 1 to 6 for manymore samples of the sliver. Then, calculate the

average of the weights. Use these weights todetermine the index of fiber orientation by usingthe following formulas: index of fiber orientationin forward direction nf = 1 � Ef/Nf and index offiber orientation in backward direction nb = 1 �Eb/Nb.

Modeling scheme

The model of fiber orientation in slivers reported hereis based on Lindsley’s methodology described earlier.Initially, let us introduce the following assumptions:(1) all fibers have the same (straight) length l1 and thesame linear density (fineness) t, (2) all fibers have thesame waviness so that the shorter crimped length a ofeach fiber is the same (see Figure 5 preliminarily), (3)fibers are distributed randomly along the sliver, and(4) the number of fibers is very high in the cross-sec-tion of the sliver.

In the present model, the random organization ofthe crimped fibers in the sliver is represented by theparallelograms as shown in Figures 2(a0), 3(a0) and4(a0). Let us imagine that a part of such sliver isfirmly gripped by a bottom plate and top two plates Rand Q. The vertically shaded plate R permanentlygrips the sliver used, but the dotted plate Q isremoved twice during the process of measurement.

We must consider the following three cases:

Case 1: The crimped fiber length a is longer thanthe width d of the top plate Q,Case 2: The crimped fiber length a is shorterthan the width d of the top plate Q, but thestraight fiber length l is longer than d,Case 3: The crimped fiber length a as well as thestraight fiber length l is shorter than the width dof the top plate Q.

Figure 2 illustrates the first case (a > d) – see sliver inFigure 2(a0). Consider that the fibers protruding fromthe right-hand edge of the top (dotted) plate Q arecombed as shown in Figure 2(a). As a result, thefibers which were not gripped by the top and bottomplates are removed and the fibers which were grippedby the plates are straightened. The straightened fibersprotruding from the right-hand edge of the top plate Qare then cut and the first fringe of fibers of weight Wis thus obtained. In the next step, the top plate Q isremoved and the rectangle of crimped fibers is seen tobe lying under it – see Figure 2(b0). Let us imaginethat we straighten all these fibers so that we obtain awider rectangle of straight fibers as shown in Figure 2(b). Of course, in reality, only the fibers protrudingfrom the right-hand edge of the top plate R arestraightened by combing, whereas the other fibers arecombed out and contribute to the fringe weight C.

Figure 1. Schematic diagram of the apparatus used todetermine fiber orientation in slivers.

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Figure 2. Scheme of modeling in accordance with Case 1 when a > d.

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Figure 3. Scheme of modeling in accordance with Case 2 when a < d and l > d.

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Figure 4. Scheme of modeling in accordance with Case 3 when a < d and l < d.

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Now the plate Q is replaced back on the straight fibersgripped by the plate R. This is shown in Figure 2(c).The protruding fibers are then cut and a small fringeof fibers of weight E is thus obtained. In the finalstep, the top plate Q is once again removed and thefibers protruding from the right-hand edge of the topplate R are observed as shown in Figure 2(d). Theprotruding fibers are then cut and a fringe of fibers ofweight N is obtained.

Figure 3 illustrates the second case (a < d and l >d). The description of the procedure used can be quitethe same, only the shapes of the fringes are different.(Compare the corresponding schemes displayed inFigures 2 and 3.)

Figure 4 illustrates the third case (a < d and l <d). The commentary of the procedure used can also bequite the same. We only notice another shape of thefringes and remark that the “cut off fringe” shown inFigure 4(c) does not exist – its weight E is equal tozero.

It can be noted that the aforesaid step-by-step pro-cedure can also be followed by putting another topplate P at the left-hand edge of the left-hand side topplate for backward direction of the sliver.

Fiber orientation

Each fiber is interpreted as a chain of small straightsegments of constant lengths dl as shown in Figure 5.The projection of a segment in the direction of theaxis of the sliver is denoted by the “crimped length”da, and its angle of inclination to the axis of the sliveris indicated by #. They are related as follows:

da ¼ dl cos#: (1)

The quantities da and cos# are random variables withmean at da and cos#, respectively. It is then valid towrite

da ¼ dl cos#: (2)

It is valid to express cos# as follows

cos# ¼Zp=2

�p=2

cos# f (#) d# ¼ kn (3)

where f (#) denotes the probability density function ofangle #. The last integral is also called as kn accordingto Neckář and Ibrahim (2003). Using this, we canrewrite Equation (2) as follows:

kn ¼ da=dl: (4)

According to Neckář and Ibrahim (2003), it is oftenpossible to use the following expression for kn:

kn ¼ 2g arctan (ffiffiffiffiffiffiffiffiffiffiffiffiffig2 � 1

p)

p(ffiffiffiffiffiffiffiffiffiffiffiffiffig2 � 1

p)

: (5)

This is the result of integration of Equation (3), whenthe following probability density function f (#) ofangle # is used:

f (#) ¼ 1

pg

g2 � (g2 � 1) cos2 #; # 2 � p

2;p2

� �: (6)

Let us remark that g ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif (0)=f (p=2)

pP 1 is valid

for the parameter g; f(0) represents the maximum (inthe longitudinal direction of sliver) and f(p/2) repre-sents the minimum (in the cross direction of sliver) ofthe probability density function f (#) according toEquation (6). (g = 1 corresponds to the isotropicorientation and the higher value of g determineshigher level of anisotropy.)

Because of the assumptions made earlier, thefollowing expression is valid to write:2

a=l ¼ kn: (7)

Let us denote the linear density (fineness) of afiber by the symbol t which is defined by t = m/l,

Figure 5. Scheme of a fiber and its segments.

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where m indicates the mass of the fiber. Let us dividethe mass m of the fiber by the crimped length a ofthe fiber and a quantity ta is obtained such that ta =m/a, where ta is called as the linear density of thecrimped fiber. It is then valid to write that m = tl =taa. Using Equation (7) the following expression canbe written:

tat¼ l

a¼ 1

kn: (8)

If it is assumed that there are n number of fibers pres-ent in the cross section of the sliver then the followingexpression is valid to write:

T ¼ nta ¼ nt

kn; n ¼ T

tkn (9)

where T denotes the linear density of the sliver.

Geometry and weights of fringes

Fringes when a > d

The detailed geometry of these fringes, introducedgenerally in Figure 2, is characterized in the followingfour schemes displayed in Figure 6.

The first fringe, earlier shown in Figure 2(a), isnow shown in detail in Figure 6(a). Here, the numberof fibers gripped along the clamping line BD of lengthy is equal to the total number n of fibers present inthe cross-section of the sliver. This fringe of straightfibers has the triangular shape BC0D with the longestfiber of length l. The total length of fibers in such afringe is n l=2. By applying Equation (9), we obtainthe expression for the weight (mass) W of the fringeusing fiber fineness t as follows:

W ¼ nl

2t ¼ T

tkn

l

2t ¼ T

l

2kn: (10)

The second fringe, introduced in Figure 2(b), isnow displayed in Figure 6(b). The fibers after straight-

Figure 6. Schematic representation of fiber geometry in Case 1 when a > d.

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ening are shown by a set of horizontal lines; the con-tinuous lines show the fibers gripped by the clampingline FG and the dashed lines show the fibers to becombed out. The earlier position of edge of plate Q ismarked by the thin dotted line BD. The oblique dashline AOD represents the ends of the fibers in the par-allelogram of fibers before straightening of fibers, thedotted line OD0 represents it after straightening offibers. The distance AB is equal to the crimped lengtha of fibers. (Compare it with Figure 2(b0).)

The crimped length d (plate width) is elongated tothe straight length d

0and we obtain the following

equation valid similarly to Equation (7):

d0 ¼ d=kn: (11)

Let the number of combed-out fibers be n1 andthese fibers are lying at a distance y1, the remainingn2 fibers are lying at a distance y2. Let us take that thedistance y = y1 + y2 is proportional to the number offibers n = n1 + n2, y1 is proportional to n1 and y2 isproportional to n2, all with a common constant of pro-portionality.

It is shown that the triangle ABD is similar to thetriangle OHD. Therefore,

a

y1 þ y2¼ d

y1

a

d¼ y1 þ y2

y1¼ 1þ y2

y1(12)

y2y1

¼ a=d � 1:

According to the proportionality, the following equa-tions are valid to write:

y2=y1 ¼ n2=n1 (13)

n1 þ n2 ¼ n (14)

Combining Equations (12), (13), and (14) it is possi-ble to write that

n2 ¼ n1(a=d � 1) ¼ (n� n2)(a=d � 1)

n2 þ n2(a=d � 1) ¼ n(a=d � 1)

n2½1þ (a=d � 1)� ¼ n2a=d ¼ n(a=d � 1) (15)

n2 ¼ na=d � 1

a=d¼ n 1� d

a

� �

and

n1 ¼ n� n2 ¼ nd

a: (16)

By using of Equations (9) and (11) it is possible towrite the following expression for the total weight ofthe sliver (fibers in the rectangle FGD0B0):3

GTOTAL ¼ nd0t ¼ T

tkn

d

knt ¼ Td: (17)

Using Equations (16), (11), (7), and (9) the weightof the combed-out fibers (dashed lines in the triangleOH0D0) is expressed as follows:

C ¼ n1d0

2t ¼ n

d

a

� �1

2

d

kn

� �t ¼ n

d

lkn

d

2knt

¼ T

tkn

d

lkn

d

2knt ¼ T

d2

2lkn: (18)

Figure 6(c) displays the arrangement of fibers aftercombing-out the second fringe of fibers and returningthe top plate Q back to its original position. (Compareit with Figure 2(c).) The straight fibers, touching theedge BD of the plate Q, are lying in the rectangleBB0D0D in two parts – in the rectangle BHH0B0 and inthe triangle O0DD0. Let us consider that the number offibers held in the distance y3 is n3. It is then valid towrite that

d0

y1¼ d0 � d

y3

y3=y1 ¼ 1� d=d0: (19)

Nevertheless, according to the proportionality it mustbe also valid to write that

y3=y1 ¼ n3=n1: (20)

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Substituting Equations (20) and (11) into Equation(19) and then using Equation (16) the followingexpression for n3 is obtained:

n3 ¼ n1(1� kn) ¼ nd

a(1� kn): (21)

By using Equations (11), (15), (21), (9), and (7) theweight E of the third fringe of fibers (rectangleBB0H0H and triangle O0DD0) is expressed as follows:

E ¼ n2(d0 � d)t þ n3

d0 � d

2t

¼ t(d0 � d) n2 þ n32

� �

¼ td

kn� d

� �n 1� d

a

� �þ 1

2nd

a(1� kn)

� �

¼ tdn1

kn� 1

� �1� d

aþ 1

2

d

a� 1

2

d

akn

� �

¼ tdn1

kn� 1

� �1� d

2a� 1

2

d

akn

� �

¼ tdT

tkn

1

kn� 1

� �1� d

2lkn� 1

2

d

lknkn

� �

¼ Td(1� kn) 1� d

2l� d

2lkn

� �: (22)

Figure 6(d) displays the arrangement of fibers aftercutting the third fringe of fibers. The weight N of thelast fringe of fibers, lying in the rectangle FBO0K0,can be expressed by using of Equations 17, 18, and22 as follows:

N ¼ GTOTAL � C � E

¼ Td � Td2

2lkn� Td(1� kn) 1� d

2l� d

2lkn

� �

¼ Td 1� d

2lkn� 1þ d

2lþ d

2lknþ kn � dkn

2l� d

2l

� �

¼ Tdkn 1� d

2l

� �:

(23)

Naturally, an identical result can also be obtainedby using the geometrical dimensions directly fromFigure 6(d), i.e. after rearrangement of the followingequation:

N ¼ ½(n1 � n3)d=2þ (n2 þ n3)d�t:

Fringes when a < d and l > d

The detailed geometry of these fringes, introducedgenerally in Figure 3, is characterized in the followingfour schemes shown in Figure 7.

The first fringe, introduced in Figure 3(a), is nowshown in Figure 7(a). This scheme is fully analogousto the scheme shown in Figure 6(a) so that Equation(10) is valid for the weight W in this case too. Also,Equation (17) is valid for the total weight GTOTAL ofthe sliver, i.e. for all fibers in the rectangle FGD0B0 inFigure 7(b) – gripped fibers (continuous line) and to-be-combed-out fibers (dashed lines).

The fibers gripped by clamping line FG are lyingin the trapezoid OJA0F and in the triangle GE0O. Nev-ertheless, the triangle GE0O can be “replaced” ontothe position FA0O0. If the length y = GF = OO

0, then

the weight of gripped fibers can be expressed as fol-lows:

U ¼ nl

2t ¼ T

tkn

l

2t ¼ Tkn

l

2: (24)

The weight C of the combed-out fibers (dashed lines)is given by the following equation:

C ¼ GTOTAL � U ¼ Td � Tknl

2¼ T d � lkn

2

� �: (25)

Figure 7(c) displays the arrangement of fibers aftercombing-out the second fringe of fibers and returningthe top plate Q back to its original position. (Compareit with Figure 3(c).) The straight fibers, touching theedge BD of the plate Q, are lying in the triangle KHJ.The length of the abscissa HJ is l�d. Let us denotethe length KH by y4 and the corresponding number offibers protruding beyond this length by n4. Because ofthe proportionality, the following relation is valid towrite:

y4=y ¼ n4=n: (26)

The triangle KHJ is similar to the triangle O0OJ andthe line O0O has the same length as that of FG. Then,

y4y¼ n4

n¼ l � d

l; n4 ¼ n

l � d

l: (27)

The weight E of the triangular fringe KHJ can beexpressed using Equations 9 and 27 as follows:

E ¼ n4l � d

2t ¼ n

l � d

l

l � d

2t ¼ T

tkn(l � d)2

2lt

¼ Tkn(l � d)2

2l(28)

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The last fringe, after removing the plate Q, is shownin Figure 7(d). The weight N of this fringe is equal tothe weight U, derived earlier, minus the weight E ofthe triangular fringe KHJ. This is written below:

N ¼ U� C ¼ Tknl

2� Tkn

(l � d)2

2l

¼ Tkn2

l � l þ 2d � d2

l

� �¼ Tdkn 1� d2

2l

� �: (29)

Fringes when a < d and also l < d

The detailed geometry of these fringes, introducedgenerally in Figure 4, is characterized in the followingfour schemes displayed in Figure 8.

The first fringe, introduced in Figure 4(a), isnow displayed in Figure 8(a). This scheme is fullyanalogous to the scheme shown in Figure 6(a) sothat Equation (10) is valid for the weight W in thiscase too. Also, Equation (17) is valid for the totalweight GTOTAL of the sliver, i.e. for all fibers in therectangle FGD0B0 in Figure 8(b) – gripped fibers(continuous line) and to- be-combed-out fibers(dashed lines).

In analogy to the previous discussion for obtain-ing Equation (24), the fibers gripped by the clampingline FG are lying in the trapezium OJA0F and in thetriangle GE0O, and the triangle GE0O can be“replaced” onto the position FA0O0. So, Equations(24) and (25) are also valid in this case for theweight U of the gripped fibers and for the weight Cof the combed-out fibers (dashed lines).

Figure 7. Schematic representation of fiber geometry in Case 2 when a < d and also l > d.

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Nevertheless, in this case, no one fiber is touch-ing the edge BD after returning the top plate Qback to its original position. This is shown in Fig-ure 8(c). Therefore, the following equation is validto write:

E ¼ 0: (30)

The last fringe obtained after removing the plate Qis shown in Figure 8(d). The weight N of this fringeis equal to the weight U of the fibers gripped by the

plates along the edge FG. Applying Equation (24) weobtain

N ¼ U ¼ Tknl

2: (31)

Summary of results

The results of weight of the fiber fringes derived ear-lier are summarized in Table 1.

Figure 8. Schematic representation of fiber geometry in Case 3 when a < d and also l < d.

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Generalization for fiber length distribution

The model, derived for geometry and weights of fringes,assumes a constant length l of fibers and a constant valueof the parameter kn. However, both of them are usuallyrandom quantities, hence they are generally described bya conjugate probability density function. Usually, wecannot find the variability of the parameter kn, but it isvery easy to determine the distribution of fiber lengths.Therefore, let us keep the constant value of kn for allfibers, but think about the distribution of fiber lengths.

Let the probability density function of mass distri-bution of fibers in relation to their length be c(l). Thenthe relative frequency of mass of fibers in each ele-mentary class defined by the lower limit of length land the upper limit of length l + dl is c(l)dl. Then the“elementary sliver” has the fineness (linear density)dT = Tc(l)dl, where T denotes the fineness (lineardensity) of the sliver. Then it is valid to write the fol-lowing expressions:

W ¼Zlmax

0

W(10)c(l)dl (32)

C ¼Zlmax

0

Cc(l)dl

¼Zd=kn

0

C(25)c(l)dl þZlmax

d=kn

C(18)c(l)dl (33)

E ¼Zlmax

0

Ec(l)dl

¼Zd

0

E(30)c(l)dl þZd=kn

d

E(28)c(l)dl

þZlmax

d=kn

E(22)c(l)dl (34)

N ¼Zlmax

0

Nc(l)dl

¼Zd

0

N(31)c(l)dl þZd=knd

N(29)c(l)dl

þZlmax

d=kn

N(23)c(l)dl: (35)

The numeral subscripts in the brackets indicate thenumber of relevant equations, which correspond toTable 1.

However, in practice, the integrands mentioned inEquations (32)–(35) are often difficult to solve analyti-cally. But, they can be solved numerically. The easiestof the approaches to solve them is stated below. If Mdenotes the weight of the original sliver and L denotesthe length of the original sliver, then the linear density(fineness) T of the original sliver can be expressed byT = M/L.

Let us assume that this sliver consists of m numberof partial slivers (i ¼ 1; 2; . . . ;m) according toFigure 9. Each partial sliver has the common length Land is created from fibers of constant length li. (Weassume that the original sliver has only the fibers oflengths l1; l2; . . . ; lm.) The linear density (fineness) Tiof the ith partial sliver is Ti = Mi/L, where Mi denotesthe weight of such partial sliver. It is then possible towrite that Mi = Mci, where ci denotes the relative fre-quency of weight of ith partial sliver in the weight ofthe original sliver. Then each partial sliver will have

Table 2. Weight-based length distribution of polyesterfibers.

Length (mm) Relative frequency (–)

36 0.6562535 0.1979230 0.0781214 0.06771Total 1.00000

Table 1. Summary of results of weight of fiber fringes.

Fiber length⁄⁄Case 1 Case 2 Case 3l > d/kn l 2 ðd; d=knÞ l < d

Weight W Equation (10) Equation (10) Equation (10)Weight C Equation (18) Equation (25) Equation (25)Weight E Equation (22) Equation (28) Equation (30)Weight N Equation (23) Equation (29) Equation (31)

**Equation (7) is used for the crimped fiber length a.

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values for Wi, Ci, Ei, and Ni according to the fiberlength li – see the Table 1. The summation of each ofthese values for all partial slivers will give the corre-sponding values for the original sliver, that isW ¼ Pm

i¼1Wi, C ¼ Pmi¼1Ci, E ¼ Pm

i¼1Ei, andN ¼ Pm

i¼1Ni.

Practical example

A polyester drawn sliver of 5.37 ktex linear densitywas taken for this study. The weight-based fiber lengthdistribution of this sliver is shown in Table 2. Thefineness of the polyester fiber was found to be 1.67dtex.

The methodology as reported in mathematicalmodel was followed to obtain the values of W, C, E,and N in the forward and the backward directions ofthe sliver. The average of 100 such readings carriedout on the aforementioned sliver is reported in the col-umn named experimental in Table 3. The width of theplate d was kept at 12.7 mm. A computer programwas developed to find out the corresponding values byusing the equations derived earlier. They are reportedin the column named theoretical in Table 3. It wasobserved that the summation of the squares of thedeviations between the experimental and theoreticalreadings for the four variables corresponding to theforward and backward directions was found to beminimum at kn = 0.73 and kn = 0.71, respectively.The coefficient of determination (R2) was found to be0.9405 and 0.9123, respectively, at the forward andbackward directions. It can be therefore said that thetheoretical results are in good agreement with theexperimental results. By using Equation (5), the valuesof the fiber orientation parameter g in the forward andbackward directions were obtained as g = 1.6 and g =1.43, respectively.

Let us mention that a set of experimental results oforientation of fibers in the carded viscose websobtained by using the well known tracer fiber tech-nique was presented by Neckář and Ibrahim (2003).The values of g were reported around 1.8 which is nottoo far from the values obtained in the present work.It shows that the idea of transformation of fiber orien-tation from the web to the sliver without too signifi-cant change, presented by Neckář and Ibrahim (2003),can be principally right.

By putting the values of g in Equation (6), theprobability density function of fiber orientation in theforward and backward directions of the sliver was

Table 3. Comparison between experimental and theoreticalresults.

Weight (mg)

Variable Experimental Theoretical at kn = 0.73

Forward directionWf 0.0590 0.0663Cf 0.0157 0.0185Ef 0.0067 0.0098Nf 0.0461 0.0399

Weight (mg)

Experimental Theoretical at kn = 0.71Backward directionWb 0.0568 0.0645Cb 0.0163 0.0190Eb 0.0048 0.0104Nb 0.0466 0.0388

Figure 10. Behavior of probability density function oforientation of fiber in the polyester-drawn sliver.

Figure 9. Scheme of original and partial slivers.

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obtained as f (#) ¼ 0:5096=(2:56� 1:56cos2#) andf (#) ¼ 0:4554=(2:0449� 1:0449cos2#), respectively,where # 2 (�p=2; p=2). The behavior of these func-tions is displayed in Figure 10.

Conclusion

A novel mathematical model of fiber orientation inthe slivers was derived and demonstrated with thehelp of a practical example of polyester-drawn sliver.The theoretical results were found to be in goodagreement with the experimental results. The fiberorientation function kn thus found can be used inpractice to judge the effectiveness of the carding anddrawing processes and also to compare a set of sliv-ers produced by using different fiber materials andemploying different manufacturing processes alongwith different process parameters. In future, fiberhooks can be considered to be included in the modeland attempts can be made to compare the carded anddrawn slivers based on the fiber orientation functionreported in this article.

AcknowledgementsThis work was supported by the Grant Agency of theCzech Republic (GAČR) Project Number 106/09/1916. Thissupport is gratefully acknowledged. We also gratefullyacknowledge Ms Pragya Dixit of Department of TextileTechnology, Indian Institute of Technology, Delhi, forcarrying out a part of the experiment discussed in thisarticle.

Notes1. This assumption will be generalized later on.2. We use this equation also more generally; each crimped

fiber length divided by the straitened fiber length is equalto kn in our model.

3. This result is immediately visible with a view to the sli-ver shown in Figure 2(b)’.

ReferencesGarde, A.R., Wakandar, V.A., & Bhaduri, S.N. (1961). Fiber

configuration in sliver and roving and its effect on yarnquality. Textile Research Journal, 31, 1026–1036.

Ghosh, G.C., & Bhaduri, S.N. (1968). Studies on hook forma-tion and cylinder loading on the cotton card. TextileResearch Journal, 38, 535–543.

Kumar, A., Ishtiaque, S.M., & Salhotra, K.R. (2008). Measure-ments of fiber orientation parameters and effect of prepara-tory process on fiber orientation and properties. IndianJournal of Fiber and Textile Research, 33, 451–467.

Lindsley, C.H. (1951). Measurement of fiber orientation. TextileResearch Journal, 21, 39–46.

Neckář, B., & Ibrahim, S. (2003). Structural theory of fibrousassemblies and yarns: Structure of fibrous assemblies. Lib-erec: Technical University of Liberec.

Perel, J. (1982). Characterization of fiber length in slivers. Tex-tile Research Journal, 52, 376–379.

Rao, J.S., & Garde, A.R. (1962). Theoretical computation ofcombing ratios of ideal and quasi-ideal slivers. Journal ofTextile Institute, 53, T430–T445.

Simpson, J., & Patureau, M.A. (1969). A method and instru-ment for measuring fiber hooks and parallelization. TextileResearch Journal, 40, 956–957.

Simpson, J., Sands, J.E., & Flori, L.A. (1970). The effect ofdrawingframe variables on cotton fiber hooks and parallel-ization and processing performance. Textile Research Jour-nal, 40, 42–47.

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