Parameters of cotton fibre-length like comb-sorter effective-length, upper half mean length, upper-quartile length, 2.5 % span length have been in use for deciding upon the optimum distance between roller-nips in drafting systems. They are eminently suited for this purpose. These parameters are, however, not foolproof measures of the propensity of cotton to incur irregularity, during roller drafting. This is because the above-listed parameters are not parameters that characterize the frequency distribution of fibre length. Indeed, the situation is very complex. The frequency distribution of the length of cottons does not follow any known theoretical type. Although the shape of length distribution seems to be typical of a variety, it varies considerably from variety to variety. Figure II – 3 - i shows the various shapes encountered in commercial cottons in India: left-skewed unimodal, bi-modal, nearly rectangular, and right-skewed unimodal. The parameters of no known statistical distribution can, therefore, characterize all cottons for length and its variability, let alone characterize cottons for draftability. We have, therefore, to identify parameters that account for the draftability of cotton specifically for use in the irregularity fraction.

Yarn irregularity is the outcome of thin and thick places in the yarn. Almost all thin places and most thick places in the yarn are the result of out-of-turn movement of fibres in the drafting zones. Why do fibres move out of turn in the drafting zone?

The length of most of the fibres is less than the distance between the nips of the front and second pairs of rollers in drafting systems. There is, therefore a duration of time between the trailing end of any fibre leaving the grip of the rear roller-nip and the leading end of the fibre getting gripped by the forward nip. During this period of time in the transit of a fibre from the rear to the front nip, the fibre is prone to move, out of turn: It is likely to move at the speed of the forward nip, if a length of the fibre is sufficiently intermingled with other fibres that are already in the grip of the front rollers. In the front zone of roller drafting systems, aprons are provided to counter this. The pressure between the aprons restrains fibres from being pulled forward out of turn by their merely being entangled with other fibres already in the grip of the forward rollers; yet the fibres are free to move forward when once they are themselves gripped by the front rollers. Thus with the provision of aprons, not all the fibres are likely to get pulled forward out of turn. Fibres that are longer than a critical length will have their trailing part under the restraining influence of the aprons for most part of their transit from the rear nip to the forward nip. They will change over to the speed of the front-roller nip only on their leading ends being gripped by this front nip. The aprons, however, are not effective in restraining the out-of-turn movement of all fibres because their influence is effective only upto a certain distance forward of the middle roller nip. Fibres that are shorter than a critical length may not, therefore, have their rear portion under the restraining influence of the aprons for the later part of their transit from the rear to the forward nip. Any such fibre is quite likely to get pulled forward out-of-turn, if its forward portion is sufficiently intermingled with fibres that are already under the grip of the front-roller nip. Some such fibres may, for a duration of time, move alternately at the speed of either the middle rollers or the front rollers. This out-of-turn and irregular movement of fibres in the drafting zone is what results in thick and thin places in the drafted fleece flowing out of the front nip -- in other words, in yarn irregularity. Thus the propensity of cotton fibres to incur thin and thick places can be quantified by two critical lengths. Fibres longer than the first critical length are least prone to incur out of turn movement in drafting, and therefore, are least likely to incur thin and thick places in the drafted fleece; fibres shorter than the second critical length are most prone to incur out of turn and irregular movement in drafting, and therefore, are most likely to incur thin and thick places in the drafted fleece. We conducted a special investigation to identify these two critical lengths.

In this investigation we used all the twelve cottons that represent the widely different types of length distribution encountered in Indian commercial cottons – Figure II – 3 - i. The fibre characteristics of these cottons are in Table II – 3 –i. We spun each of the 12 cottons to Ne 22, 30, and 40. Then by correlating Uster counts of thin and thick places in yarns with the fibre characteristics of the cotton used to spin the yarns, we identified the critical fibre lengths to be 12-mm and 24-mm.

The set of regression equations that we obtained in this exercise are given in Appendix II - 2. The equations contain two independent variables: i) the function, f(Q;H) of Q, a derivative of effective length, and the percentages of fibres shorter than 12-mm and longer than 24-mm, and H, the fibre fineness; and ii) yarn count. Table II – 3 – ii compares a number of cottons for their propensity for irregular drafting through their respective values of the function, f (Q;H).

The equations give very accurate estimates of the USTER counts of thin and thick places in the yarn. The power of these two critical lengths to discriminate between cottons for their susceptibility to drafting irregularity is noteworthy. The effective length of S-4 cotton is much less than that of VL and DCH-32 cottons. Naturally, therefore, VL and DCH contain some fibres of very long length, that are not present in S-4. S-4 is also much coarser than the other two cottons. However, S-4 is superior to VL and DCH in a unique respect: it has much less percentage of fibres shorter than 12-mm, and also much more percentage of fibres longer than 24-mm, than the other two cottons. The regression equation based on these two percentages correctly reflects the observed fact that, in the propensity to incur thin and thick places during drafting, S-4 is comparable to the longer and finer VL and DCH. This justifies the use of Q as the length parameter in F1, the irregularity fraction of the yarn tenacity equation.

The Need to Revise the Fibre Slippage Fraction, F₁

Fibre length is an obvious contributing factor to F₂, the fibre slippage fraction. A little reflection suggests that, as in the case of the irregularity fraction, here again the use of a single length parameter is not adequate.

Improved Expressions for F2 and F2

To derive the improved algebraic expressions for F2 we spun five cottons, two to more than one count, and the remaining three to a single count; each count to a set of twist multipliers.

We chose the cottons that were used in these spinning so as to represent the variety of frequency distribution of cotton fibre length that is met with in practice- - VL, DCH, S-4, MCU-7, F 414, and JYOTI in Figure II – 3 - i. Of the three cottons in the spinnings, namely, VL, DCH and S-4, S-4, though coarser and of less effective length than the other two, was much more uniform in length than the other two, and gave the more even yarn. We also included F 414 that is shorter, but stronger, than S –4. Data on the yarns spun in this investigation are in Table II – 3 - iii

From these data we simultaneously built up expressions for F₂ and for. The method that we followed to derive the expressions is similar to the one we used in dealing with Bogdan’s data. We, however, took the opportunity to adjust the expressions for the difference between Brown’s and ATIRA data in the cotton test methods.

Figures II – 3 – ii, iii, iv which compare the observed value of = with the estimated value of  show that the two are in very close agreement over the range of values from 0.34 to 1.00.

Improved Expressions for the Irregularity Fraction in the General Equation

To arrive at the algebraic expression for the irregularity fraction, we used the data from this spinning, as well as from another. In the second we spun each of the twelve cottons,  Table – II – 3 – i, to the same set of three counts, and each yarn to the same twist multiplier. The experimentally observed CS values are in Table II – 3 – iv.

The final set of equations could be written in terms of the following fibre characteristics: L, the Baer sorter effective length in mm; S, the percentage fibres by number shorter than 12 mm; B the percentage fibres longer than 24 mm; H, the fibre fineness in millitex; Z, the zero-gauge bundle tenacity in g/t; F, the 3.2 mm (1/8-in) bundle tenacity in g/t.

We, therefore, have algebraic expressions in terms of fibre-characteristics and yarn parameters for all the four fractions in

There is a striking similarity between the two sets of expressions, the one derived from Brown’s data and the other from ATIRA data. There is, however one significant difference. In the Brown’s-data equation, we use only UHML in the expressions for F1, and F2. In the ATIRA-data equation, we use parameters that are derived from three fibre-length test data in F1, and F2.

Figure II – 3 - v shows that the experimental points of the first ATIRA spinning are in close proximity to the smooth curves generated by the regression equation. The errors in the estimates of yarn tenacity from fibre characteristics are mostly within the acceptable limits.

The ringframe used for the second spinning was different from the one used for the first. From basic considerations, for any one given fibre and yarn test system, the expressions for F₂ and F₃ should be the same for all ring-spun yarns. The expressions for F₂ and F₃ from the first ATIRA spinning can, therefore, be used for the second ATIRA spinning also. The constants in the expression for F₁  could, however, be different for different drafting systems. Accordingly, to assess the accuracy of estimates of CS in the second spinning, only the constants in F₁ were estimated from the experimental data from this spinning.  In this case l =4.961, and m=1.258. Table II – 3 – v shows that in the case of the second spinning also the errors of estimation are satisfactorily low.

The estimates of CS from the general equation are in very good agreement with the corresponding actual values for yarns spun from very disparate cottons, and over a wide range of count and twist. We can, therefore, use the estimates from the general equation to grasp how fibre length and fibre fineness govern the contributions of yarn irregularity and twist to yarn CS. 

Figure II – 3 – v Proximity of Experimental Points on CS-M Plot to Smooth Curve Generated by General Equation