In the general Equation F1 quantifies the distinctive contribution to yarn tenacity of yarn irregularity that is inevitable in roller drafting. F2, F3, and F4 quantify the distinctive contribution of twist that has necessarily to be introduced into the drafted fleece to convert it into formed yarn.

We have already noted that for any one given system of fibre and yarn tests, the set of numerical values in the expressions for is the same for all ring-spun yarns; also is unique, and the same for all ringspun yarns. 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. Therefore, any mill that wishes to use the General Equation needs to determine the constants in the expressions for F2 and F3 only once; they can then keep on using these constants for any ringspun yarn; they need to estimate afresh only the numerical constants in F1 for each drafting system.

In the ATIRA investigations all 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 all the fibres in the tuft bridge the distance between the two clamps. The yarns were all tested for tenacity on the conventional lea tester. The same set of numerical constants in the expressions for F2 and F3 could, therefore, be used to estimate the CS of yarns from any one of the many spinnings that were carried out in the investigations. Only the numerical constants in F1 were determined afresh for each drafting system.

H fibre-fineness, millitex, not micronaire

where l and m are specific to a drafting system, and have, therefore, to be determined afresh for each drafting system,

Table I – 6 – i, gives the numerical values of each step in the calculation. In this exercise we take l=4.9613 and m=-1.2583, the values from one of the ATIRA spinnings.

We can take advantage of the fact that the expressions for F2 and F3 are unique for any system of fibre and yarn tests, and are applicable to ring yarns from any different drafting system. For any one system of cotton and yarn testing, we need to derive the numerical constants in the expressions for F2 and F3 only once; we can then use them for any ring-spun yarn; only the numerical constants in the expression for F1 need to be determined anew for every different drafting system. The determination of the constants in F1 requires only the spinning of each of five different cottons, each to three different counts, each count at any one convenient T.M.

The procedure for determining the constants in F2 and F3 is explained in Part II; that for determining the constants in F1 is explained in the next section, Table – 6 - ii.

We can use the general equation to generate a wealth of information for use in cotton selection. Immediately after carrying out the cotton fibre tests, the laboratory can trace out the CS – T.M. curves for a range counts of yarn of likely interest to the spinner. All that we require is the storage in a computer of the requisite programme for the numerical evaluation of the expressions for F1, F2, f3 and F4 and for curve tracing.

Note: In this exercise we take l =4.9613 and m=-1.2583, the values from one of the ATIRA spinnings.

More decimals than are really required are given to help you verify your computer output.

Given a system of fibre and yarn tests, for every new drafting system we have to determine the numerical constants in F1. In such a situation, to get accurate enough values of the constants, we have to spin each of five disparate cottons to three different counts; we need to spin each count at any one convenient twist multiplier only. We determine the CSP of all the yarns. We then proceed with the calculation upto the stage,

in Table – 6 – ii. We calculate the value of Y for each of the three counts of yarn spun from each of the five cottons. We then prepare a plot of the fifteen Y-values against the corresponding Q –values. We will have three points over each of the five Q’s. We draw the best fitting straight line through the points. The intercept of the line gives the numerical value of l; the slope of the line gives the value of m. Alternatively, we can use the ‘solver’ of the computer to get l and m.

We calculate the value of Y for each of the three counts of yarn spun from each of the five cottons. We then prepare a plot of the 15 Y-values against the corresponding Q-values. We will have three points over each Q. We draw the best fitting straight line through the points. The intercept of the line gives the numerical value of l; the slope of the line gives the value of m. Alternatively, we can use the ‘solver’ of the computer to get l and m.