Cotton cost has always been the single major contributor to the cost of gray yarns. Consequently, economic cotton selection for meeting yarn specifications continues to be one of the main concerns of a spinner. Left to himself, any spinner would very much like to spin a sample of every single lot of cotton to yarn before taking the decision to buy the lot or not. However, the exigency of the situation rules out this most reliable basis for cotton selection. Mostly, one has to take decisions regarding the purchase of cotton on the basis of test reports alone.

In reading meaning out of test reports of the cotton under consideration, one has often mentally to balance the shortfall in one characteristic with the premium in another characteristic. Let us consider, for example, the two cottons in Table II – 1 – i. Cotton ZZ is shorter and coarser, but stronger than cotton XX. Which of the two cottons would be appropriate to spin NE 40 (15 tex) hosiery yarn? Again which of the two would be appropriate for 30 NE (20 tex) warp yarn? In order to answer these questions we need to be able to estimate the tenacity of yarns of desired count and twist from each of the two cottons. Over the years research workers have, therefore, expended considerable effort on the derivation of equations for the estimation of yarn tenacity from cotton fibre characteristics.

Most researchers in this area have followed either one of two methods to derive yarn tenacity equations. The first is the setting up of a regression equation between yarn tenacity and individual fibre characteristics. The second method is the setting up of a regression equation between yarn tenacity and one or more fibre-quality indexes that are especially constructed for the purpose. Morton (5) has pointed out the shortcomings of the resulting equations. Firstly, Morton asked, ‘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?’ Secondly, Morton remarked, that the use of count as one of the independent variables ‘does nothing to advance our understanding—indeed, it tends to obscure the issue—of how fibre properties determine spinning behaviour’. Count is, no doubt, a variable whose contribution has to be accounted for in yarn tenacity equations. One has, however, to be wary of the contribution of count to yarn tenacity pushing up the value of the correlation coefficient of the resulting equation. This is what happens by the spinning of each of the cottons used in a study to a number of counts. One introduces a large variation in the yarn tenacity values most of which is then accounted for by the easily quantified contribution of count. However, on the basis of the high value of the correlation coefficient, one concludes, erroneously, that the equation has effectively accounted for the contribution of cotton characteristics to yarn tenacity.

Many years ago Peirce (1) suggested what one should do to obtain a model that is free from the flaws that Morton cautions against: one should aim at deriving algebraic expressions that will help trace the well-known twist-tenacity curves of yarns, examples of which are given in Figure II – 1 -i. In this paper we will first review earlier unsuccessful attempts in this direction. We will then present the derivation of a general equation for yarn tenacity, that by far fulfills Morton’s expectation of such an equation, namely, that ‘it advance our understanding of how fibre properties determine spinning behaviour.’ More specifically, we will show that the proposed equation delineates the role of fibre length and fibre fineness in the translation of fibre tenacity into yarn tenacity.

The derivation of the general equation requires test data on a number of cottons, and on yarns of varying count and twist spun from them. In the work that is being reported, the first attempt to formulate the general equation used published data (7). This has two merits: saving of the considerable time that would be spent in carrying out the spinning and testing; enhancing the readers’ credibility in the accuracy that has been achieved in the yarn tenacity estimates from the equation. In the published data, on the basis of which the general equation was at first formulated, yarn tenacity is expressed in terms of CS, the British lea (count x strength) product. Other publications (4, 6, 16, 18,19) that have been used to assess the merit of the general equation also reported yarn tenacity as CS. Even today the British lea count strength product is in use both in trade as well as in papers in journals (27) and presentations in international conferences (6). So much so, Zellweger Uster incorporate, in their HVI, software to estimate CS of yarn spun from a cotton, immediately after the cotton has been tested.

Numerically speaking, the CS value can be readily converted to centinewton, the international unit for yarn tenacity by multiplying the former by (0.981/208.35). But technical clarity would require that the resulting value be qualified by the clause, ‘from tests on a 120-yard lea of 80 loops’ to distinguish it from the tenacity value in centinewton that is generally determined by tests on single threads. This is because, even for any one given yarn, the two determinations would yield significantly different values. To make matters worse, the ratio of the two values is dependent upon the irregularity of the yarn and the twist inserted in spinning the yarn. Nothing is then gained from the numerical artifice of converting CS to centinewton. In this paper, therefore, yarn tenacity is expressed in terms of lea count strength product, CS, the actual test result.

Starting from some basic considerations, Bogdan [19] developed an equation for the count x strenth – count – twist (CS-C-M) relationship of cotton yarns.

Therefore, there remains only one unknown, P.

Bogdan’s approach is unique in that it encapsulates the three-dimensional CS – C – M relationship in one parameter P, the intrinsic strength parameter.  Given a cotton, with just this one parameter one can draw the CS-M graph for any C.

Bogdan did not, however, conceive of the intrinsic parameter as a fibre-quality index. He proposed this parameter solely as a conceptual abstraction forming the core of the algebraic expression embodying the CS-C-M relationship. This means that the numerical value of P of any cotton can be determined only after spinning the cotton to any one yarn of specified count, at any one specified twist. Once we know any one set of the three values, namely, C, M, and CS for the cotton, we can determine the value of P of that cotton, from the Bogdan equation, by a process of iteration. We can, then, determine for that cotton, the (CS)’ of any other yarn of specified count, C’, and twist, M’, by substituting this value of P in Bogdan’s equation.

Subramanian, Bandyopadhyay and Ganesh (20) realized that Bogdan’s P offers a means of deriving a generalized equation for estimating cotton yarn tenacity from measurable fibre characteristics. They reasoned that, all that is required for this purpose is the setting up of an empirical equation to relate Bogdan’s P to a suitably constructed fibre-quality index. Once this is achieved for a given fibre-test system and spinning set-up, then any new cotton can be evaluated on the basis of the fibre-tests, without having to spin yarn from it. This entails three steps: i from the fibre-test values we calculate the fibre-quality index; ii using the empirical equation we estimate the value of P of that cotton; iii using this value of P we estimate the CS of the yarn of desired C and M from Bogdan’s equation.

Subramanian, Bandyopadhyay and Ganesh used Lord’s data (2) on carded, carded double-rove feed and combed yarns to investigate the feasibilty of this approach.

They derived the values of R and D, the regression coefficients, separately for each one of the three different spinnings in Lord’s data (2). The error in the estimate of Bogdan’s intrinsic parameter from the fibre-quality index was less than 3% for each one of the 37 yarns. The average error of estimate of the yarn CS was 2.7%. Only for two out of the 37 yarns was the error more than 6% -- about 8%.

In his experiments, Lord spun each cotton to one specified count only. This could have contributed to the accuracy of CS estimates from P, which is itself estimated from the fibre quality index. Subramanian and Ganesh (21), therefore, analysed U.S.D.A. data (22) to assess the correctness of using Bogdan’s intrinsic parameter as the foundation for a generalized CS equation. They found that Bogdan’s intrinsic strength parameter is an abstraction that has some basis. They, however, identified some inconsistencies. We give one example. In the case of carded yarns from the 55 American Upland Long-staple Cottons analyzed in the study, the estimate of P from NE 22 yarn is in most cases in agreement with the estimate of P from NE 50 yarn. However for these same cottons spun to combed yarns of the same counts, the estimate of P from the NE 22 yarn is generally larger than that from the NE 50 yarn. Yet again, in the case of NE 50 and NE 80 combed yarns from 52 American Egyptian Extra-long Cottons, the estimate of P from the coarser and finer yarns were in agreement.

There is also one practical observation that is in conflict with Bogdan’s premise that the CS-C-M relationship can be expressed in terms of one single intrinsic strength parameter. This is the intersection of the CS – C lines of two cottons of which one is longer, but weaker, and the other shorter, but stronger – a fact to which Lord (3) draws attention.

The conclusion is inescapable: Bogdan’s concept of the intrinsic strength parameter is far too simplistic to cope with the complexity of the situation.

In1982 Dhawan and Subramanian (23) proposed an improvement to Bogdan’s CS-C-M equation

where G is the CS  of yarn of zero-count incurring no obliquity loss, F is the expression for yarn tenacity build up with twist, is Bogdan’s obliquity correction factor, and N is the average number of fibres in the yarn cross section.

Making use of Brown’s yarn data (7), Dhawan and Subramanian first established that the new equation based on three parameters, namely, G, F and k is superior to Bogdan’s equation based on the single intrinsic strength paramete.

They then derived algebraic expressions for G, F and k in terms of the fibre characteristics. The resulting equation gives very accurate estimates of CS (23) for Brown’s data [7] and for Lord’s data [2]. For Brown’s data only for six out of the 123 yarns considered, does the estimates incur errors of more than 6%; out of these six, in only one case is the error more than 10%.

The equation has also some fundamental merits. The maximum possible yarn tenacity depends only upon fibre-bundle tenacity, and no other fibre characteristic. The yarn tenacity build-up with twist depends upon effective length, fineness, and gauge-length effect of bundle-strength; the rate of drop of tenacity with count depends upon fibre-fineness and effective length and is independent of twist. Only the constants in the expression for the rate of drop of tenacity with count have to be determined afresh for each new spinning set-up.

Dhawan and Subramanian equation has, however, some serious drawbacks. The expressionpurports to quantify “the fraction of fibres which contribute to yarn strength as against those which slip”. Obviously this fraction of fibres which contribute to yarn strength should increase with increasing twist. The algebraic expression for this fraction, however, does not subsume this. Really speaking, the expression is appropriate to quantify the drop in CS with increasing count, which is the result of the decrease in the fraction, (number of fibres at the place of break/ average number of fibres in yarn cross-section), with increasing count. The tenacity build-up with increasing twist, quantified by F, is wholly attributed to “ the decrease of l , the gauge-length at which fibre bundles are loaded”, when the yarn is loaded in tenacity test. Really speaking, the tenacity build up comprises two parts: one this shortening of gauge length of fibres under stress; the other a decrease in the number of fibres that slip and do not contribute to yarn tenacity; and both should be decreasing functions with increasing twist. Dhawan and Subramanian did not consider yarns of T.M. of 2.75 which are available in Brown’s data.

In 1985 Subramanian (32) realized that in articulating Peirce’s suggestion (1) of a general equation for yarn tenacity, one must ensure that the resulting algebraic expressions satisfy some basic requirements. Firstly the expressions should make it explicit that it is fibre tenacity that manifests itself as yarn tenacity. Secondly the equations should, at the same time, elucidate how fibre-fineness and fibre-length govern the effects of yarn irregularity and yarn twist in the translation of fibre tenacity into yarn tenacity.

This paper reports the successful completion of these objectives by work that was carried out by Subramanian on his own, during the period 1985 to 1991 (32). The belated publication of the paper after a lapse of almost fifteen years is justifiable. There is a greater chance of the acceptance of the general equation for use in industry today than in the eighties. This is so because of two developments: the availability today of instruments for the rapid measurement of cotton fibre characteristics like fibre-fineness and fibre-length distribution; the ubiquitous presence of the computer that has literally put numerical evaluation of the most complex algebraic expressions at the finger-tips (on the computer key-board and mouse).

The following passage in Physical Organic Chemistry by Louis Hammet aptly specifies the choice of the method to use to construct a model for yarn tenacity. “For this, as for any other human attempt to master nature, two kinds of approach are possible. One is the construction of broad, far-reaching principles from which the detailed properties of matter may be deduced. The other involves the bit-by-bit development of empirical generalizations, aided by theories of approximate or limited validity whenever they seem either to rationalize a useful empirical conclusion or to suggest interesting lines of experimental investigation.” The derivation of the yarn-tenacity equation from a set of first principles is indeed out of reach. The feasible alternative is the bit-by-bit development of an empirical model. In this approach one first writes down the framework of the equation relating yarn tenacity to fibre characteristics, so that it subsumes the features that are characteristic of the phenomenon of yarn break in tenacity testing. One then elaborates upon it by a set of algebraic expressions that are first deduced from empirical data, and then vetted for their predictive accuracy by regression analysis.  

The two significant features of the phenomenon of yarn failure in tenacity testing are as follows:

There are really speaking four reasons for the fibre tenacity being only a fraction of fibre tenacity.

The linear density of a cotton yarn is highly variable along its length. At the place of break the number of fibres will most likely be only a fraction of the number of fibres corresponding to the average count of the yarn. The count at the place of break will, therefore, be finer than the average count. Yarn tenacity is, however, reckoned on the basis of the average yarn-count, and not on the count at the place of break. Yarn tenacity can, therefore, be only a fraction of cotton fibre tenacity.

Further, of the fibres present at the place of break, only a fraction may break and contribute to yarn tenacity, while the rest may slip.

The fibres at the place of yarn break that break themselves and contribute to yarn tenacity, may not break at the ‘zero-gauge’, but at some finite gauge. In this context Lord’s concept (8) of a yarn-element that is subject to break in yarn testing is worth recalling. “Where a length of yarn is stressed the tension is not distributed uniformly along the length of each fibre. The yarn may be conceived as being composed of successive elements in the form of sections of fibre bundles. Not only will these bundles differ in fibre arrangement and linear density, giving tight and loose and also thick and thin places, but they will also differ in their length. The length of any given element is poorly defined, because the elements are not separate entities but each gradually merges into its neighbours at either end. Nevertheless, an average element length may be considered to exist. Amongst many other factors, the strength of the yarn will depend on the bundle strength of the elements, and in turn on the length of the elements. From this approach it is hardly surprising to find that, for carded yarns, the bundle strength measured at zero test length provides information less useful than measurements at 1/8-in. The increased regularity and parallelization of fibre arrangements in combed yarns suggests a decrease in effective length of yarn element, a feature which may also occur in doubling when fibres are brought into more intimate contact with each other. For such material the best test length for fibre-bundle strength determination may be one closer to zero than that found suitable for the looser, carded yarns.” 

Thus according to Lord, the differences in regularity and parallelization of fibre arrangements between carded yarns on the one hand, and combed and doubled yarns on the other, make it preferential to use the 1/8-in.fibre-bundle tenacity for carded yarns, and the zero-gauge length tenacity for combed yarns in the respective predictive equations for yarn tenacity. There is an important corollary to this. The appropriate gauge-length of cotton tenacity to use in the prediction equations for yarn tenacity should decrease asymptotically with increasing levels of twist in the yarn. Now, as the test length is decreased, cotton fibre-bundle tenacity is known to increase asymptotically to the zero-gauge value. We have to incorporate this fact in the equation for the translation of fibre-bundle tenacity into yarn tenacity. We can achieve this by the use of fibre-bundle tenacity at zero-gauge in general, and at the same time by the inclusion in the equations of a fraction that would give the tenacity at the gauge-length appropriate for the level of twist in the yarn.

There is a subtle difference between the way fibres are broken in the determination of bundle-strength and the way fibres are broken in the determination of tensile strength of yarn: all the fibres in the bundle under test, without exception, bridge the entire distance between the clamps; as against this some fibres in the bundle that breaks in yarn testing may have one end lying within the break-zone; the number of such ends is conceivably dependent on fibre length, and its variability. This has to be taken into account in the construction of the equation.

There is one more reason for the yarn tenacity falling short of the fibre-bundle tenacity. This is the obliquity of the fibres in the twisted yarn to the yarn-axis, which is the direction in which tensile stress is applied in the determination of the breaking load.

The value of R can be shown to be 208.35. In the lea-strength test 160 threads together resist the imposed stress. The theoretically possible maximum contribution per thread can be the zero-gauge fibre bundle tenacity, Z g/tex. Therefore the theoretical maximum breaking stress that the lea can sustain is 

Replacing tex by NE, and grammes by pounds, we find that the theoretical maximum load that a lea can sustain is

We note in passing that lea-CS can numerically be converted into lea cN per tex:-

Even when the same cotton is spun on the same set of machinery, the irregularity of the yarn increases with increasing count. There is an important consequence of this. The English count-strength product, CS, which was originally proposed as a measure of yarn tenacity, that is independent of the count of yarn, is in reality not so. The CS of yarns spun from the same cotton, and on the same set of machines decreases with increasing English count. The expression that is in industrial use to quantify this fall in CS with increasing count is

where S₁  and S₂ are the breaking strengths of yarns of countrespectively. The commonly used value of K is 18, irrespective of the cotton used to spin the yarn. This generalization yields estimates that are accurate enough for the purpose for which it is used, namely, the correcting of the CS for small discrepancies of the actual count from the nominal count.

This equation accounts for the drop in CS with increasing yarn-count as a result of increasing irregularity. In practice, the increase in yarn irregularity with yarn count is less for the finer of two cottons. This implies that the expression for irregularity effect on CS should be written not in terms of C, but in terms of N, the average number of fibres in the cross-section of yarn, that is in terms of both yarn-count and fibre-fineness, and not in terms of only the count of the yarn. Furthermore, practical observation tells us that K is, generally, the smaller for the longer of two cottons (26). This is because the increase in irregularity with increasing count is less for the longer of two cottons. Examination of experimental data suggests that a plausible expression for F1 :-

If at any twist, the number of fibres that slip and therefore do not contribute to yarn tenacity is U, then,

In writing the algebra relating U and g to M we are guided by certain premises, that seem quite plausible on basic considerations. The premises are as follows.

With increasing twist, the number of fibres that slip, and do not contribute to yarn tenacity decreases. The longer the fibre, the steeper will be this fall of the number of fibres that slip at breaking point. This fall could also be steeper for a yarn that has more fibres in the cross-section than for a yarn that has less fibres in the cross-section. The rate of fall of U with M is, therefore, determined by fibre-length, fibre-fineness and yarn count. Examination of the two broken ends of a high-twisted yarn shows that most the fibre-ends appear broken. We can therefore, take the asymptotic value of U to be equal to zero.

The gauge length of fibre-break in yarn-break is very likely to be determined by the intimacy of intermingling of fibre assemblies inside the yarn (8). The intermingling is quite likely governed by the number of fibres in the cross-section of the yarn. The fall of g with M is, therefore, likely to be dependent not upon fibre-length, but upon the number of fibres in the cross-section, being steeper for a yarn with more fibres in the cross-section than for a yarn with less fibres in the cross-section. The fall of g to its asymptotic minimum is thus dependent upon fibre-fineness and yarn-count. A corollary is that the minimum value of g may not necessarily be zero, but may itself be dependent upon fibre-fineness. We can therefore write,

where lm, the asymptotic minimum value, is dependent upon fibre-fineness, and L_M, the increment over lm at any finite twist, is dependent upon fibre-fineness, as well as upon yarn count and twist.

where M is the twist-multiplier in the English system of units.

Table – II – 1 - i Fibre Test Data on Cottons XX and ZZ

FIGURE – II – 1 – i Typical inverted parabola relationship between CS and M