Indian Journal of Fibre & Textile Research Vol. 25, September 2000, pp. 22 1 -224

Computational and simulation studies of jute fibre length distribution

S K Biswas'

Textile Physics Section, Institute of Jute Technology, 35 Ballygunge Circular Road, Calcutta 700 0 19, India

Received 21 June 1999; accepted 24 August 1999

The computer simulation of four frequency distributions of jute fibre length depending on the methods of sampling and testing is presented. These four distribution curves give six points of intersection. The intersection position along the fibre length axis for any pair of frequency distribution depends on the average length of fibre population, which, in tum, helps to find out the average length of fibre population from the relevant pair of experimental frequency distribution of fibre length.

Keywords : Computer simulation, Fibre length, Jute fibre, Length distribution

1 Introduction In fibre length measurements, two kinds of sample

are experimental ly obtainable through two kinds of sampling method, viz. end-biased and length-biased. An end-biased sample is defined as the one in which each fibre of popUlation has equal probability of being included since each fibre has two ends independent of its length or any other characteristics. The end -biased sample is a random sample. A length-biased sample is the one in which the probability of a fibre being included is directly proportional to its length. In fibre testing, a random selection is restricted due to the essential nature of the fibre that it is much longer than it is thick. Because of this, it is too easy to take a sample in such a way that it contains far more long fibres than it should. The knowledge of the nature of the bias in favour of the longer fibre would help avoid the bias. In some contexts in which length - biased sampling is used, it is reasonable to regard length -biased distribution as the object of a stud/ .

The frequency distributions of fibre length in jute slivers and yarns have been deduced analytically by Banerjee2 for different methods of sample preparation. The expected average length of fibres obtained on the basis of number and weight of fibres has been estimated for each of such distributions. It has been shown earlier3•4 that the gamma function provides a mathematical short-cut for the analysis of fibre length distribution and average length of jute fibres depending on sampling and testing methods to arrive at the results as obtained by Banerjee2. The two sampling methods, viz. end-biased and length -biased,

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have been considered. For each of such sampling, the expected distribution of fibre length obtained on the basis of number of fibres and weight of fibres has also been estimated. Thus, the following four types of length distributions are obtained : (i) the frequency of length I tested on the basis of number from end-biased sample [ fEN (l)), (ii) the frequency of fibre of length I tested on the basis of number from a length-biased sample [fLN ( I ) ], (iii) the frequency of fibre of length I tested on the basis of weight from an end - biased sample [ /Ew ( I )] , and (iv) the frequency of fibre of length I tested on the basis of weight from a length­biased sample [Aw ( I ) ] .

The end-biased and number-based length distribution of fibre /EN ( I ) which is expected to be similar to the basic type has been used by Banerjee2 to obtain AN ( I ) and /Ew ( I ) and in tum AN ( I ) has been used to obtain Aw ( l ). Biswas3.4 used the gamma function to obtain the same results very quickly and considered /Ew ( I ) to obtain Aw (I) . He also considered all the other possible routes of transformation of one type of length distribution to the other types and presented their rederivation through gamma function to achieve a considerable amount of simplifications. The summary of the results of the distribution is given in Table 1 .

However, the proliferation of so many distribution curves depending on the method of sampling and testing may give several points of intersection when these are superimposed. In fact, the reports on experimental observation of intersection are available in some cases and attempts have been made to attribute suitable technological significance to it6.7• In the present work, an attempt has been made to find

222 INDIAN J. FIBRE TEXT. RES. , SEPTEMBER 2000

Table I -Summary of results of derivation

Method Relationship for frequency Average distribution length

Basic population number-based

I -f( l ) = 7 exp (-I I I ) I

End- biased sampling I _ number-based !EN ( I ) = 7 exp (-1 / 1 )

Length-biased sampling number-based

End-biased sampling weight-based

Length-biased sampling weight-based

1

I -AN ( l ) = - exp (-I I 1 )

J2

12 _ !Ew ( l ) = - exp (-I I I )

21'

1 3 -Aw ( l ) = � exp (-I I I )

61

[

2 1

4 1

out analytically all the intersections and show that the intersections depend on the average length of fibre population . Computer simulations of frequency distribution of jute fibre length are also presented.

2 Intersections

2.1 Intersection OffEN ( 1 ) and AN ( 1 ) At this point of intersection of these distributions,

the frequency of fibre of length I tested on the basis of number from end-biased smple [ !EN ( I ) ] and the frequency of fibre of length I tested on the basis of number from a length-biased sample [fLN ( I ) ] would be equal. If this condition is satisfied at 1 = I ) , then

From Table I , Eq. ( l ) may be written as :

1l - 1 --=- exp (-I) / I ) = * exp(-/) / I ) I 1

which gives

2.2 Intersection off EN (I) andf EW (I)

. . . ( 1 )

. . . (2)

At this point of intersection of the distributions, the frequency of fibre of length 1 tested on the basis of number from end- biased sample [fEN (I)] and the frequency of fibre of length 1 tested on the basic of

weight from end - bisased sample [fE W (I)] would be equal. If this condition is satisfied at I = 12, then

From Table ! , Eq. (3) may be written as :

1 - Ii --= exp(-/2 / I ) = � exp(-/2 / l ) I 21 '

which gives

2.3 Intersection off E N (I) andf LW (I)

. . . (3)

. . . (4)

At this point of intersection of the distributions, the frequency of fibre of length 1 tested on the basis of number from end- biased sample [fEN (I)] and the frequency of fibre of length 1 tested on the basis of weight from length-biased sample [fLw (I)] would be equal. If this condition is satisfied at 1 = 13• then

From Table ! Eq. (5) may be written as :

1 - I; --= exp( -/1 / l ) = -=-4 exp( -1,,/ I ) I .

61 .

which gives

2.4 Intersection off LN (I) andf EW (1)

. . . (5)

. . . (6)

At this point of intersection of the distributions, the frequency of fibre of length I tested on the basis of number from length-biased sample [fLN (I)] and the frequency of fibre of length I tested on the basis of weight from end-biased sample [fEW (I)] would be equal. If this condition is satisfied at 1 = 14. then

From Table ! , Eq.(7) may be written as :

I - [2 _

_42 exp( -/4 / l ) = �1 exp( -14 / l ) 1 2[ '

which gives

. . . (7)

BISWAS: COMPUTATIONAL AND SIMULATION STUDIES OF JUTE FIBRE LENGTH DISTRIBUTION 223

. . . (8)

2.5 Intersection off LN (I) andf LW (I) At this point of intersection of the distributions, the

frequency of fibre of length I tested on the basis of number from length-biased sample [fLN (I)] and the frequency of fibre of length I tested on the basis of weight from length-biased sample [fLw (I)] would be equal. If this condition is satisfied at I = 15. then

From Table 1 , Eq.(9) may be written as :

Is - I� -� exp( -Is / I ) = -=;j' exp( -Is / I ) I 61

which gives

2.6 Intersection off EW (I) andf LW (l)

. . . (9)

. . . ( 1 0)

At this point of intersection of the distributions, the frequency of fibre of length I tested on the basis of weight from end-biased sample [fEW (I)] and the frequency of fibre of length I tested on the basis of weight from length-biased sample [fLW (I)] would be equal. If this condition is satisfied at I = 16• then

From Table 1 , Eq.( l I ) may be written as :

1 2 [ 3 6 - 6 --=-3 exp( -16 / l ) = --- exp( -16 / I )

21 ' 61 4

which gives

. . . ( I I )

. . . ( 1 2)

The summary of the results of intersection is given in Table 2.

3 Computer Simulation The computer simulation of four frequency

distribution functions, viz. fEN (I), AN (I) , fEw (I) and Aw (I) , were carried for I = 5 cm. The computer programme in F77 is given in Appendix-I .The resulting curves obatined with the help of a graphic software are presented in the Fig 1 . The intersection points in the figure corroborate with the results presented in Table 2

Table 2- Summary of results Of intersection

Intersection

!EN (I) and AN (I)

!EN (I) and!Ew (I) !EN (I) andAw (I) AN (I) and!Ew (I)

AN (I) andAw (I)

!Ew (l) andAw (l)

Intersection along fibre length axis

I

'-12 [ 61/ 3 [ 2 [

61/2 [ 3 I

I ntersection along fibre length axis for

1 = 5 cm

5.0

7. 1

9. 1

1 0.0

1 2.2

1 5 .0

20 r---------------�------------�

Fibre lengt h ,cm .

Fig. I-Theorical length distrubution of jute fibre in slivers and yams [A --end - biased number-based method; B - length­biased number -based method; C - end - biased weight - based method ; and D - length -biased weight- based method]

4 Conclusion

The intersection position along the fibre length axis for any pair of frequency distributions depends on the average length of fibre population, which, in tum, can be determined from that pair of experimental frequency distrubition of fibre length . Thus, while attributing any technological significance to the intersection of any pair of frequency distribution, it should be taken into account that the intersection is not independent of the average length of fibre population.

224 INDIAN J. FIBRE TEXT. RES. , SEPTEMBER 2000

Acknowledgement

The author wishes to thank Mr. G Bhattacharya of Jadavpur University for computational help.

References 1 Cox D R, Proceedings, Symposium on the Foundation of

Survey Sampling, edited by N L Johnson & H Smith (Jr) (John Wiley, New York), 1 969, 506.

2 Banerjee B L, lndian J Text Res, 5 ( 1980) 98.

3 Biswas S K, Indian J Text Res, 1 4 ( 1 989) 145.

4 Biswas S K, Jute Development J, 10 ( 1 990) 26.

5 Biswas S K, I A P Q R Transactions. 2 1 ( 1 996) 67.

6 Sinha N a, lndian J Phys, 49 ( 1975) 245.

7 Sinha N a, J Text Assoc, 36 April/June ( 1 975) 5 1 .

Appendix l-Computer programme for

l - -f(l) = -=2 exp (-I I I ) for I = a

l

F (I) = ( I * exp (- [ja)) I (a * a) Write ( *, *) , Input value of a , l = ' Read ( *, *) a , I Write ( *, *) , Input value of interval h =' Read ( *, *) h Write ( *, *) , Input value of required N =' Read ( *, *) N Open (Unit = I , File = 'ECT. DAT' , Status = 'New') Do 10 1 = 0, N -l Y = F (l) Write (1,*) I,Y Write (*,*) I ,Y I = I + h 1 0 Continue Close ( 1 ) End