In our first attempt to derive algebraic expressions for  in terms of readily measurable cotton fibre-characteristics we made use of data reported by Brown and co-workers (7). Brown and co-workers spun each of six cottons to three counts, NE 18, NE 36, and NE 72; and each count to a number of twist multipliers ranging from 2.75 to 5.75. The fibre test data on the six cottons are reproduced in Table – II –2 –i, and the experimentally observed CS values in Table II – 2 – ii. The procedure that we followed to derive the algebraic expressions is given in Appendix II – 2. The final set of equations is follows.

We denote the fibre UHML in mm by L_G, the fibre fineness in millitex by H, the Stelometer zero-gauge tenacity in g/t by Z, and the gauge-length effect on fibre tenacity by G.

all the expressions in the R.H.S. are now only in terms fibre characteristics and yarn count and twist. In other words we have a regression equation to estimate tenacity of yarn of any specified count and twist spun from a cotton solely from the fibre characteristics of the cotton.

We can get an overall assessment of the accuracy of the estimates of CS from the equation by a graphical method. For this we consider any one count of yarn from any one of the six cottons. We prepare a graph of CS against M and plot on it the experimental values of CS of this yarn. Then, using the equation, we estimate the CS for the yarn under consideration for a number of values of M. Over the points on the graph, we draw the smooth curve generated by these estimates. We do this for the remaining two yarns from the cotton. We repeat this procedure for each of the remaining five cottons -- Figure I – 2 – i. We can now assess the merit of the equation by examining the proximity of the experimental points to the smooth curve generated by the equation in each of the eighteen available cases. Most of the experimental points are close to the smooth curve generated by the tenacity equation.

The proximity of the experimental points to the smooth curve establishes that the general equation is indeed a reliable approach to the delineation of how fibre-length and fibre-fineness govern the translation of fibre tenacity into yarn tenacity. The specific manner in which fibre length and fibre fineness on the one hand, and yarn count and twist on the other govern the translation of fibre tenacity into yarn tenacity is highlighted in Table – II –2 –iii.

Table – II – 4 – iv gives the percentage errors in the CS estimates for each of the 133 yarns for which experimental values are available. The algebraic expressions yield estimates of CS of error less than +/- 6 % in most cases. Only in 14 cases out the total number of 133 does the error exceed 6%; out of these 14 cases, in only one does the error exceed 10%. There are, however, instances of feeble systematic errors. The question arises as to which of the expressions needs to be improved upon. To identify this we proceed as follows.

We rewrite the equation,

where X= () is the hypothetical maximum possible value of CS that yarn of any one specified count can have in the absence of obliquity loss.

Let us enunciate the conceptual significance of the entity X. The twisting of yarn contributes to an increase in yarn CS in two ways: firstly, it reduces fibre slippage when yarn is stretched to breaking point; secondly, it reduces the gauge length at which fibres break when a yarn element breaks. Simultaneously, the twisting of yarn renders the fibres inside the yarn oblique to the yarn axis. This obliquity of fibres reduces the resistance offered by them to the applied stress which is in the direction of the yarn axis. This is why CS does not increase monotonically with M, but reaches a maximum, and then decreases with further increase in M. In short, this is why the CS – M curve takes the typical inverted parabola shape. If, now, we imagine there were no fibre obliquity ensuing from twist insertion, there will only be the two positive effects of twist, reduced fibre slippage and reduced gauge length of fibre breakage. In this hypothetical situation, the value of CS at any twist will be

Obviously, at any M, (CS x D) will be larger than CS.

What about the maximum value of (CS x D)? In articulating algebraic expressions for the general equation, we have postulated that at very high levels of twist there will be no fibre slippage, and that the gauge length of fibre breakage will reach its minimum value. Quantitatively, therefore,

the maximum value of=1, and

the maximum value of 

In the absence of fibre obliquity, therefore, with increasing twist, the CS of yarn of any count will reach its asymptotic maximum value:

We can picture this hypothetical situation. We first calculate (CS x D) for each of the yarns spun from a cotton - - Table II – 2 - v. We then plot the observed value of (CS x D) against M - - Figure II – 2 - ii. The experimentally estimated value of X is given by the asymptotic maximum of the plot. These values are given in the last column of Table II – 2 – v.

We have now a method of identifying the expression in the general equation that needs improvement. We prepare a graph and plot the experimental values of CS against M for the three yarns from any one cotton. Over these points we draw the smooth curves generated by

For this, we use the graphically determined value of X, but compute the numerical values of F₂, f₃ and F₄  from the corresponding set of algebraic expressions. We do this for each of the six cottons - - Figure II – 2 – iii.

In each of the resulting 18 cases, the experimental points are in close proximity to the smooth curve generated by the algebraic expressions. Numerically speaking, these algebraic expressions yield estimates of CS of error less than +/- 3.5 % in most cases  -- Table II – 4 – v. We can, therefore, conclude that the algebraic expressions for F2, F3, and F4 are very plausible ways of quantifying the three-fold contribution of twist in the phenomenon of the translation of fibre-bundle tenacity into yarn tenacity.

A comparison of Figures II – 4 - i and II – 4 - iii shows that the experimental points are much closer to the smooth curves in the second set of graphs than in the first set of graphs. Two factors could account for this improvement.

The magnitude of the contribution of the first factor will depend upon the errors in the determination of fibre tenacity. The magnitude of the contribution of the second factor will depend upon any inadequacy of the parameter of fibre-length in Brown’s data (7) namely, UHML, to quantify the contribution of fibre-length to yarn irregularity. There is no way of estimating the errors of determination of fibre tenacity. We can, however, improve upon the algebraic expression for F₁  We will undertake this in Chapter 3.

Table II- 2 - ii Fibre Characteristics And Yarn Count And Twist
as Contributing Factors To F1, F2, F3 and F

TABLE – II – 2 – iv % Error in Individual CS Estimates, Brown’s Data

TABLE II – 2 – v (CS x D) and (CS x D)max

TABLE II – 2 – vi % Error in Individual CS estimates by X, F2, F3, F4

FIGURE II – 2 – i Proximity of Experimental Points of CS versus M to Smooth Curve from General Equation: Brown’s Data

FIGURE II – 2 – ii Plot of (CS x D) against M to Determine (CS x D)max

Figure II – 2 – iii Proximity of Experimental CS-M points to the Smooth Curve

Generated by CS =