The General Equation for CSP is

In this equation

What do these terms stand for?

Let us now imagine the notional zero-count yarn. Let us furtherimagine that this yarn has been so twisted that the fibres inside it have acquired perfect inter-locking, and have, therefore, no slippage, and yet at the same time have not incurred any obliquity to the yarn axis. For this yarn, obviously, N=infinity. Therefore for this yarn ; similarly are both =1; also F₄=1.

The rewritten General Equation tells us that this notionally perfect zero-count yarn that does not incur fibre obliquity inside it will have a strength of

and not

Why is this so?

We now recall the three important considerations that we noted must be taken care of in the construction of the General Equation.

The term

takes  care of these three aspects of the situation. Table I – 7 - igives values  of the entity, ‘f’, for each of the twelve cottons used in ATIRA spinning 2. S – 4 cotton has less effective length than the two cottons VL and DCH; yet, of all the twelve cottons, S-4 has the lowest percentage fibres shorter than 12 mm., and the highest percentage fibres longer than 24mm. As a result, S–4 cotton incurs less drafting system related irregularity, and has fewer fibres with free ends in the breaking zone in yarn tenacity testing, than all the other ten cottons. At any yarn count and twist, therefore, S–4 yarns lose a smaller fraction of the fibre tenacity on account of these two factors than yarns from any one of the other eleven cottons. Rightly, therefore, S-4 has the minimum value of Q, and the maximum value of ‘f’.

The General Equation and the Rule-Of-Thumb Equation for Count Correction

Really speaking, the rule-of-thumb equation for correcting count for small deviations of the actual count from the nominal, where K=18, is just that, and nothing more. The imposing of this straight-line relationship between CS and C over a large range of C leads to anomalies.

Grover and Hamby (15) have pointed out that even within the practical range of counts the slope of the CS – C line is larger for combed yarns than for carded yarns, and that this anomaly can be resolved only by a slightly curvilinear relationship between CS and C.

The General Equation accepts this curviliear relationship between CS and C. Furthermore, the General Equation uses N, the average number of fibres in the yarn cross-section that is dependent both upon yarn count and fibre fineness to account for the effect of count related yarn irregularity. Evidently, this is more appropriate than only C, the yarn count, which is used for this purpose in the rule-of-thumb equation. According to the General Equation for any one cotton, the CSP at any count is given by

Therefore, for yarns from a given cotton, and spun on the same drafting system, the  drop in CS between NE 20 and NE 60 is given by

We note that, for any one cotton, the drop in CS between any two counts is not unique over all counts, but depends upon the count range over which the drop is calculated, and upon the twist inserted as well. Further the drop is cotton dependent: it depends upon the zero-gauge-tenacity, the gauge-length parameter, fibre-length, and its uniformity, and fibre-fineness. This is the outcome of the quite tenable premises that we used to set up algebraic expressions for

Table II- -4- -ii gives the estimated drop in CS from NE 10 to NE 30 for yarns from three cottons, in each case for two levels of twist in the yarn. The very good agreement between the observed and estimated values of CS in the two ATIRA spinnings vouches for the correctness of these values.

There is yet another shortcoming of the rule of thumb CS – C linear equation: it does not envisage the intersection of the CS – C plots of yarns from two cottons, a practical observation to which Lord (2) draws attention. The General Equation subsumes this phenomenon.

Thus, The General Equation not only agrees with practical observations where these are valid generalizations, but also brings out nuances that practical observation misses. One such nuance that The General Equation tells us is that fibre-length uniformity, as much as  effective length and fineness of fibre, determines the rate of drop of CSP with increasing count.

To understand the interaction of count and twist in their contribution to CSP we have prepared Table II – 4 - iii. We note that yarn of NE 36 from Jyoti cotton contains on an average 104 fibres in the cross-section. In the element of beak in yarn testing, however, the number of fibres in the cross section is far fewer; furthermore, of the available fibres some have a free end in the break zone and do not share the load. In effect, in the break zone there are only 37 fibres that have both their ends outside the breaking element. Now, out of these 37 fibres, only 17 really break at a TM of 2.75, and the remaining 20 fibres slip. The number of fibres that contribute to yarn tenacity increases with increasing twist, and reaches its asymptotic maximum possible value of 37.

To isolate the contribution of the count-associated irregularity on yarn tenacity, we first calculate the theoretical maximum CSP at NE “zero”, 5, 20, 30, 40, and 50. These CSP values would be realized in a yarn that has been so twisted as to endow the fibres inside it with the maximum fibre-interlocking and zero fibre slippage, without at the same time introducing any obliquity of the fibres to the yarn axis. In calculating these values, however, we have allowed for the important fact that we just discussed: the inevitable presence in the yarn of some fibres that have one end within the yarn element that breaks when the yarn breaks in tenacity test, and, therefore, do not share the tensile load.

Table II –4 -iv, which gives us data on Jyoti cotton, tells us that the “ideally twisted zero-count yarn” has a CSP of  4316; and as the count goes on increasing, that is as the yarn gets finer, the CSP goes on reducing. Please recall that these CSP values are only notional: they can only be achieved by so twisting the yarn that maximum interlocking of fibres in the yarn is achieved and there is no fibre-slippage when the yarn fails, but at the same time there is no obliquity of fibres to yarn axis. This is merely a conceptual exercise to isolate the count-related irregularity effect. In practice these CSP values will be reduced by the obliquity fraction F₄. The practical CSP values achievable at a T.M. of 4.75, which is near about the optimum for Jyoti cotton, are also given in the Table II – 4 -iv.

The fact that the term is by itself adequate to account for the drop in CS with C leads to an important inference: we are justified in the partitioning of the irregularity term into two parts. We use part one to account for the three factors that we listed earlier: 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. We use the second to account for the decreasing trend in CS with count as a result of the increasing yarn irregularity.

Yarn tenacity is the manifestation of fibre tenacity. One would, therefore, expect that a 25% increase in fibre tenacity would result in a proportionate increase, that is a near 25% increase, in yarn tenacity. The contribution of fibre length to yarn tenacity is, however, indirect: through the contribution that fibre length has to yarn irregularity and fibre slippage. One would, therefore, expect that a 25% increase in fibre length would not result in an identical increase in yarn tenacity. Also one would expect that this increase in yarn tenacity would be more when fibre length increases from the short to medium staple, than from long to extra long staple. In other words, the increase in yarn tenacity with increasing fibre length will understandably be in the nature of a growth curve. The general equation yields estimates that are in accordance with these expectations, as can be seen from Table II – 4 - v.

These CS estimates bring out a very pertinent point. A low percentage of fibres shorter than 12 mm and a high percentage of fibres longer than 24 mm is of advantage in the translation of fibre tenacity into yarn tenacity. This is as it should be: much of the improvement in yarn tenacity that results from combing a cotton results from this improvement in fibre length uniformity. Most practical methods of testing cotton for fibre length, however, miss this important point.

It is in this regard that the Uster AFIS and similar instruments offer very valuable information.

** notional yarn that has been so twisted that the fibres achieve maximum interlocking, and incur no slippage in tensile testing, and yet the fibres do not incur obliquity of fibres to yarn axis.

@ TM=4.65
**No increase in EL, but a reduction in % fibres shorter than 12 mm, and an increase in % fibres longer than 24mm over C-1 that would result from combing.