Let us recall a pet sum of our middle-school arithmetic teachers. Chidambaram invested Rs.1000 at simple interest of 10% payable with the principal at the end of six years. Rashiklal invested Rs.1000 at interest of 8% accruable half-yearly for a period of six years. Who will get more money at end of six years?

Chidambaram will get 1000x(1+(10/100)x6)=1000x1.6=1600.

Rashiklal will get 1000x(1+(8/2x100))^12=1000x1.601=1601.

Would you want the second formula to be free of the power function just for the sake of simplicity? There is an important message for us here.

The over-simplified SITRA equation cannot tell us anything about how fibre length and fibre fineness govern the contributions of yarn irregularity and twist to the translation of fibre-tenacity into yarn-tenacity. On the contrary, it leads to an erroneous conclusion. According to the equation, the yarn tenacity of a NE 45.4 (13 tex) yarn at optimum twist is given by

Therefore a 25% increase in either fibre tenacity or fibre length of the cotton used to spin a NE 45.4 yarn will result in  a 5% increase in the CS of the yarn. This is contrary to what one would expect. The inference is obvious: over-simplification leads the SITRA equation to give a distorted picture.

Now let us consider the requirements of a general equation for yarn tenacity. Evidently, yarn tenacity is the manifestation of fibre tenacity. However we come across instances wherein length plays a decisive role: of two cottons of equal fibre tenacity, the one that is weaker but longer spins a yarn of equal – even more – strength than the stronger but shorter cotton. We explain this fact by reasoning that the longer cotton spins a more even yarn.

Were we asked to amplify our statement, we would say some thing along these lines. “The longer fibre spins a more even yarn than the shorter fibre. In other words, for any one given yarn count, along the yarn axis the longer-fibre yarn has less variation in the number of fibres in the cross-section than the shorter-fibre yarn. This means that, at the place of break in yarn testing, the yarn from the longer fibre will contain more number of fibres than the yarn from the shorter fibres.

This can possibly explain why the yarn form from the longer, but weaker fibre is stronger than yarn from the stronger but shorter fibre. Q.E.D.” An interesting surmise emerges from this analysis. A 25% increase in fibre tenacity would result in a proportionate increase, or a near 25% increase, in yarn CS. A 25% increase in fibre length would, however, result in a rather smaller increase in yarn CS. Also one would expect that the extent of increase in CS with increasing length would be more in the short to medium staple range than in the case of long to extra long staple range.

To sum up: other fibre characteristics remaining the same, an increase in fibre tenacity will result in a proportionate increase in yarn tenacity; the effect on yarn tenacity of an increase in fibre length will, however, be less than proportionate, and will be subject to a diminishing return. How can, then, one grudge the use of power functions, logarithms, growth curves and the like when one is dealing with cotton that is such an expensive raw material? After all, we have computers to take over the evaluation of the most complex functions! There is an imperative need for a CSP equation that is capable of  more accurate estimates and of meaningful interpretation.