In 1992 Neelakantan (6) constructed an equation to estimate yarn strength at any count and twist. This equation has not merely a high correlation coefficient, but yields CS estimates of impressive accuracy. “Only 17 out of 132 cases showed errors of prediction beyond 6% and only in two cases errors were beyond 10%.”  How did Neelakantan achieve this?

Neelakantan started with a meaningful equation

where,P is the theoretical maximum possible value of CS

 is the obliquity correction, M being the twist multiplier,and, k and t are constants for a given cotton implying thereby that they are functions of fibre properties.

Neelakantan predicated his equation on the premise that P=208.35 * S₀, where S₀ is the fbre-bundle tenacity at zero gauge. To cross check this assumption Neelakantan derived, by regression analysis, the numerical values of k, t and P, individually, for each of the six cottons for which data are available in Brown’s spinnings (7). He, then, found that even with a separate regression equation for each individual cotton, the regressed values of P did not equal (208.35 x zero-gauge bundle strength) in the case of three out of six cottons. He reasoned that, the “assumptions used in deriving equation namely that there is no interdependence between count and twist multiplier may not be correct.” He, therefore, rewrote it:

Then, by regression analysis, he related the three coefficients k, t, and d to fibre characteristics:

In rewriting the equation, Neelakantan sacrificed the distinctiveness of the contribution of twist and irregularity that is explicitly stated in his original equation. His sole concern from then on was to minimize the errors of prediction. Naturally therefore, he did not invoke any plausible hypothesis in the construction of the fibre quality indexes for use in the expressions for k, t, and d. The resulting equation, therefore, is devoid of any meaningful interpretation regarding the role of fibre-length and fibre-fineness in the translation of fibre-tenacity into yarn tenacity. Consequently Neelakantan’s model also incurs Morton’s criticism (5), that it ‘does nothing to advance our understanding—indeed, it tends to obscure the issue—of how fibre properties determine spinning behaviour’

The distribution of errors of estimates from the general equation for Brown’s data is given in Table II – 5 - i. Only in 14 cases out of 133 cases is the error more than 6%; and only in one case is the error more than 10%. The general equation thus offers estimates that are as accurate as Neelakantan’s equation. The general equation has the additional merit that it achieves this by the use of expressions that are obtained from a tenable set of postulates concerning the phenomenon of yarn break in tenacity testing. Thus the general equation is also capable of delineating the way in which fire length and fibre fineness determine the contribution of yarn irregularity and yarn twist to the translation of fibre tenacity into yarn tenacity. Neelakantan’s equation does not have this fundamental merit.

How General Is the General Equation?

We assessed the general equation by applying it to two extreme cases of fibre-length distribution. We will now discuss these two exercises.

Applicability of Model to Yarns from Hand- ginned Cottons

In the first exercise we applied the general equation to yarns from lints that were obtained by factory-ginning (25) and hand-ginning each of three varieties of cotton. As would be expected, the hand-ginned lint is very much more uniform in length than the factory-ginned lint –Table II – 5 - ii.

We first considered the data from the factory-ginned lint only. We retained the earlier ATIRA expressions for F2, and F3, and Bogdan’s expression for F4, and evaluated only the constants in F1. Then we estimated the CSP of the yarns from the factory-ginned as well as the hand-ginned lint from this equation. The accuracy of the estimates was good for both the sets of yarns.

Thus the equation derived from factory ginned cottons stands extrapolation to hand-ginned cottons which contain very much less percentage of short fibres, and very much more percentage of long fibres than the corresponding factory-ginned lint.

We, therefore, derived the values of l and m by taking together the data for the two sets of yarns: l = 3.937, and m = -1.049. Table II – 9 - ii shows that one common equation gives accurate enough estimates of the CSP of yarns from factory-ginned as well as hand-ginned cottons. 

Applicability of model to viscose yarns

As a fibre, viscose differs from cotton in two important aspects. The first difference is that viscose does not incur the gauge-length effect in fibre-tenacity, or in other words for viscose G=0. As a result of this, for viscose,

The second difference between viscose and cotton is that in the case of commercial viscose, the %-age fibres shorter than 12-mm is = 0, and the %-age fibres longer than 24-mm is = 100.

The question is whether, in spite of these two striking differences, the cotton expression for F2 is applicable to viscose yarns.

To answer this question we analyzed published data (9). The analysis brought out two differences between viscose and cotton yarns. i) The K-value of yarns from viscose is larger than that of carded yarns from cotton of comparable length and fineness.ii) The constant in Bogdan’s expression for the obliquity fraction, namely 0.014, is not applicable to viscose yarns.

We had, therefore, to rewrite the CS equation for viscose. The procedure we followed is given in Appendix II - iv. The final equation is

For the published data (9) that we made use of, the values of a, b, f”(Q), and p are as follows:

The algebraic expression for U is the same as for cotton.

Earlier we noted two distinguishing features of viscose fibres in comparison to cotton: for viscose F3 is = 1, at all M; for viscose fibre length is absolutely uniform. These features of viscose render the CS – M relationship of viscose yarns strikingly different from that of cotton. The effects on the optimum twist for maximum yarn tenacity of fibre length and fineness are very feeble; the increasing trend in the optimum twist with increasing count, however, is discernible. The cotton-data based expression for F2 convincingly accounts for these experimental findings concerning viscose yarns - - Table II – 5 – iii.

Therefore, two important inferences emerge. Firstly, the partitioning of the increase in CS with increasing twist into the two components, F2 and F3, is not a computational artifice, but is based on reality. Secondly, the numerical values we get from the algebraic expressions for F2 and F3 are not altogether arbitrary, but reflect the actual.

Table II – 5 –iii brings out two more conclusions. Firstly the errors of CS estimates are generally acceptable; secondly, the numerical values of

for viscoseare compatible with the values of f of comparable cottons.

We, therefore, have parameters derived from fibre-length test data that help us in characterizing cotton as well as viscose for susceptibility to irregularity in drafting, and in delineating the role of fibre length and fibre fineness in the translation of fibre tenacity into yarn tenacity.

If we scan the errors of the CS estimates of minutely, we can discern some bias: positive errors for all three counts for the 39.7-mm ~ 167-millitex normal fibre, and the 47.6-mm ~ 139-miilitex extra-strong fibre; negative errors for the 50.8-mm ~ 167mllitex normal fibre and the 39.7-mm ~ 111-millitex extra-strong fibre. We could make these errors much less by alternate values of the power of N, but such improvement is at the cost of incompatibility of the f values between groups of viscose. This would mean letting the regression analysis take over, and surrendering the fundamental desiderata in our model for the phenomenon. We, therefore, retained these expressions, that give accurate enough estimates.

Why do we need such complex expressions in the general equation?

One criticism of the general equation could be that most of the algebraic expressions, particularly those for A, I,and Q, the derivatives from fibre-length test data, contain functions that look formidable, even contrived. But a closer look will show that these functions are just examples of growth curves. Why do we need to use growth curves in these expressions? Yarn irregularity is known to be a decreasing function of fibre length; further this decreasing trend is known to be not monotonic, but subject to the law of diminishing return. Therefore, the way in which fibre characteristics govern the effect on yarn tenacity of processes like drafting and twisting is best quantified by growth curves. Let us consider two specific cases.

The Use of Growth Curves in the Expression for The Irregularity Fraction

We recall our analysis of how thin and thick places materialize in the drafted fleece. On the basis of this analysis, we reached an important conclusion: in order to discriminate between cottons for draftability, we should take into account, apart from the effective length, the percentage of fibres that are most prone to irregular movement in drafting, and the percentage of fibres that are least prone to irregular movement. We then constructed a notional effective length, A, compounded out of L, the effective length, S, the % fibres shorter than 12-mm, and B, the % fibres longer than 24-mm. We then constructed Q, an inverse function of A. We used Q and fibre fineness to arrive at equations that could estimate accurately the counts of thin and thick places that materialize in the yarn during drafting. Figure II – 5 – i depicts the relationship between the length parameters on one hand, and A and Q on the other, for the set of illustrative values of L, S, and B in Table II –8 – iv. Both the curves are essentially growth curves.

The usefulness of the postulated notional effective length is manifest in its resolving an apparent anomaly in the experimental data. The frequencies of thin and thick places in yarns from S-4 cotton are comparable to those in yarns from VL cotton, and less than those in yarns from DCH cotton. This is so, in spite of the fact that S-4 is inferior to the other cottons in terms of effective length and fibre fineness. The values of A, the notional effective length, very correctly reflect the superior draftability of S-4:

This is why we decided upon the use of Q in the expression for the irregularity fraction,

The accuracy of the estimates of drafting-induced thin and thick places in the yarn, and the accuracy of estimates of the CSP of yarns corroborate the appropriateness of the use of growth curves in the expressions for estimating these.

Let us consider another example: the number of fibres that slip, and therefore do not contribute to yarn tenacity. Obviously this number decreases with increasing twist. Conversely, the number of fibres that are well interlocked, and therefore, contribute to yarn tenacity increases with increasing twist. We can make a logical premise regarding the nature of the increase of F₂, the fraction of fibres that contribute to yarn tenacity, with increasing twist: in the initial phase the increase will be slow; there will then be a phase of rapid increase; in the final third phase the effect will taper off, and a plateau,F₂=1, will be reached. In other words, the increase of F₂ with M willfollow a growth curve. We can make one more premise: a longer and finer fibre will reach F₂=1 at a lower twist than a relatively shorter and coarser fibre. The algebraic expression for F₂, the contributing-fibre fraction, has to reflect these expectations.

To meet these requirements we again made use of the notional effective length, A, to construct expressions for the parameter, I. We then used this I and fibre fineness to construct an appropriate growth curve to depict the increase in F₂with M.

The graphs in Figure II – 5 – iii show, by way of example, the plot of F₂ against M for three cottons, JYOTI, S-4, and VL. All the three plots yield growth curves. We have already seen that the algebraic expressions for these curves represent reality: on plots of CS against M, the experimental values are very close to the smooth curve generated by the general equation – Figure II – 5 - iv. In these plots any discrepancy between the experimental value and the curve is the cumulation of errors from each of the four expressions, F₁,F_2,F₃, and F₄Even so, the curves are accurate enough representation of the experimental data, in terms of the CS-M relationship.

A closer look at the plots in Figure II – 5 – iii brings out some interesting points. For the coarse and short Jyoti cotton the curve reaches unity at M=5.25; for the relatively long and fine S-4 cotton the curve reaches unity at M=4.25. This is as one would have expected. There is, however, an observation that one may not have anticipated. The curve for VL cotton, that is longer and finer than S-4, reaches unity at M=4.50. Why is this so? The value of A, the notional effective length, is larger for S-4 than for VL. Being a function of A, the contributing-fibre fraction increases more rapidly with M for S-4 than for VL.

Possibly, with further effort, some of the expressions can be algebraically simplified while they retain the trend they are meant to portray.  But this is none too essential        when one has the use of a computer to deal with the most complex of functions.     

The general equation delineates the contribution of fibre-length and fibre-fineness in the translation of fibre-tenacity into yarn tenacity. It yields accurate estimates of tenacity of yarns spun from cottons of astonishing differences in the distribution of length, and of large differences in fineness and strength. With some minimal modification the Equation is applicable to yarns from viscose which has practically no variation in length, and which, unlike cotton, does not incur the gauge-length effect in tenacity test.

The general equation takes cognizance of the three important aspects of the phenomenon of tenacity test of yarn: fibres with free fibre ends inside the yarn breaking zone; the contribution of the drafting system to irregularity; the basic difference in the calibration of the two instruments used to test fibre and yarn tenacity. Indeed, the equation quantifies the contribution of these three aspects by the entity f. From basic considerations, the value of f should be dependent upon the fibre-length distribution only. This is correctly reflected in the general equation: the algebraic expression for f is in terms of Q, a function of Baer effective length and the percentages of short and long fibres.

One likely criticism of the model is of a statistical nature: the number of regression coefficients in the yarn tenacity equations is so large that the accuracy of estimates is not surprising. This criticism misses an important point. The numerical constants in the functions for V, W and Q the derivatives from Baer sorter effective length, percentage fibres shorter than 12-mm and percentage fibres longer than 24-mm were originally arrived at in an altogether different context. These constants were fixed in regression equations for the prediction of Uster counts of thin and thick places in yarns from fibre characteristics. These numerical constants do not, therefore, take away any degrees of freedom from the residual sum of squares of the tenacity equation. The constants in the functions for I are common for all the ATIRA cotton spinnings, and they are applicable to viscose as well. The number of constants that were determined by regression to use in the tenacity equation for any one set of data is, therefore, really much less than the number of pairs of dependent- independent variables in almost all the instances.

Figure II – 5 - iv gives plots of f against Q. The smooth curve through the set points of any one spinning is compatible with the curve of any other spinning; and as one would expect each curve tends to an asymptotic maximum value. However, the curves do not all merge into one unique one – even in the case of cottons. This is understandable because the entity f quantifies the contribution of three factors to yarn tenacity: drafting system, free fibre ends in yarn breaking element in tenacity test, and incompatibility of calibration of instruments for fibre and yarn tenacity tests. The drafting systems used in the four ATIRA spinnings are different, and to this extent the smooth curves will be separate. However, one would expect the asymptotic maximum value of all the cotton yarn curves to be the same: because this is the contribution of solely the incompatibility of calibration of instruments for fibre and yarn tenacity tests, and in these investigations the instruments used are the same.

In this respect The General Equation attracts Morton’s reproach (5): ‘why is it that, when the same investigator, using a consistent technique, divides his entire population of samples into one or more groups, he arrives at different—and sometimes markedly different—prediction formulas for the different groups?’

Where did we go wrong? In the General Equation, as it stands, we have let f quantify three aspects of the phenomenon: the fraction of fibres potentially capable of sharing the applied tensile stress; any contribution of the specific drafting system in use; the incompatibility in the calibration of the instruments used to test fibre tenacity and yarn tenacity. Really speaking, we should provide two expressions: one growth-curve type of function of Q to account for the fraction of fibres with free fibre ends in the breaking element, which expression should be applicable to all ringspun yarns; another growth type of function of Q to account for differences between drafting systems. For any one spinning set-up, then, we need determine only these drafting-dependent parameters. The incompatibility in the calibration of the instruments used to test fibre tenacity and yarn tenacity can, then, be taken care of by a numerical constant that is unique for any one test set up.

experimentally observed; ERR:  error in estimate
fg factory-ginned; hg hand-ginned.
L, effective length; S, % fibres shorter than 12 mm.; B, % fibres longer than 24 mm
H, fibre fineness millitex;
Z, Stelo zero-gauge fibre tenacity; F, Stelo 1/8-in gauge fibre tenacity;

OBS: experimentally observed;
EST: estimated from general equation.
NA: not available

L Effective length; S % fibres < 12mm.; B % fibres > 24mm.;
A Notional effective length;
Q=f(L,S,B) Floating fibre index used in irregularity fraction:
Larger the value of Q, the more the proportion of floating fibres,
and therefore, the more the propensity for yarn irregularity

Figure II – 5 –i
Growth curve for A With increasing serial number, S decreases, and B increases,
L remaining the same
Table II – 5 - iv

Figure II – 5 – i Decay Curve Type of Relationship between Q
and fibre length distribution: With increasing serial number, S decreases, and B increases, L remaining the same~ ~ Table II – 5 - iv