We have seen that the appropriate fibre length parameter to use in F1, the irregularity fraction, is Q, a function of effective length and the two critical percentages of the fibre-length distribution. Just a reflection shows that , the fraction of fibres at the place of break in yarn tenacity testing that does not slip, but actually contributes to yarn tenacity, should also be dependent upon this length parameter. We carried out two spinnings to gather data for the derivation of the improved algebraic expressions for F1, and F2.

In the first, we spun five cottons, two to more than one count, and the remaining four to a single count; each count to a number of twist multipliers. We chose the five cottons so as to represent the astonishing variety of frequency distribution of cotton fibre length that is met with in practice- - Figure – I – 4 -i. We included in this spinning three cottons, one of which, S-4, though coarser and of less effective length than the other two, VL and DCH, was much more uniform in length than the other two. We also included cottons that were exceptionally strong, but short. In the second experiment we spun each of the twelve cottons of Figure – I – 4 -i. to three distinctly different counts, and all three counts at the same twist multiplier. We carried out the two spinnings on two different drafting systems.

From the data collected in the first spinning we formulated the algebraic expressions for F1, and F3 by making use of the methods that were used while dealing with Brown’s data (7), and are explained in Part II. We also took the opportunity to modify the expression for the differences between the two sets of data in fibre-test methods. The final set of equations is as follows:-

An appeal! Please do not be put out by the formidable expressions that appear in the calculation. After all, the computer will carry out the calculations in a few blinks of the eye. You have already seen how very misleading easy-to-use expressions can be.

H fibre-fineness, millitex, not micronaire

There is one point that needs clarification. From basic considerations, one would expect that for any one given system of fibre and yarn tests, the expressions for should be the same for all ring-spun yarns. In the present investigations the cotton fibre length tests were carried out by Baer sorter; the fibre bundle tenacity tests were carried out on tufts held by a pair of clamps, in such a way that and all the fibres in the tuft bridge the distance between the two clamps. The yarns were tested for tenacity on the conventional lea tester. The numerical constants in the expressions for F2 and F3 can, therefore, be used to estimate the CS of any ring spun yarn so long as the same methods are used to test cotton and yarn. The values of l and m, the numerical constants in F1, the expression for the irregularity fraction are, understandably, specific to the drafting system used to spin the yarn. These have, therefore, to be determined for every drafting system afresh. By way of example, l=3.166 and m=0.858 for one of the ATIRA spinnings.

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 difference. In the Brown’s-data expressions we use only UHML in F1, and F2. In the ATIRA-data equation, we derive a set of parameters from three fibre-length test data for use in F1, and F2.

Figure - I - 5 - i shows that the experimental points of the first ATIRA spinning are in close proximity to the smooth curves generated by the expressions for F1, F2, F3 and F4. Quantitatively, 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. We have already noted that the expressions for from the first spinning can be used for the second spinning also, and that the constants in the expression for could be different for different drafting systems. Accordingly, to assess the accuracy of estimates of CS in the second spinning, only the numerical constants in were estimated from the experimental data from this spinning. In this case l =4.961, and m =1.258. Table – I – 5 – i shows that in the case of the second ATIRA spinning also the errors of estimation are satisfactorily low.