Does the expression for A work for viscose fibres also? Can Viscose, for which short-fibre content = 0, and long-fibre content = 100, be construed to be an asymptotic case of cotton? Do the cotton expressions offer plausible estimates of CSP of viscose yarns? In short, can the general equation stand extrapolation to viscose?

The cotton CS equation is

In this equation the constant in Bogdan’s expression for the obliquity fraction, namely 0.014, may not be applicable to viscose. If we denote the value for viscose by d, we get,

Viscose does not incur the gauge-length effect in fibre-tenacity testing. Therefore G=0, for viscose. The equation thus simplifies itself to

To estimate the numerical value of d we proceed as follows. At any twist multiplier sufficiently larger than the optimum, U=0. Therefore, beyond the optimum twist, the equation gets further simplified to

This equation gives a method of estimating d for viscose. Table II –A -4 - i gives values of d obtained from each of a number of pairs of values of CS taken from published data (9). On the basis of these figures we take d = 0.0225 for viscose.

The CS equation for viscose is, therefore,

For commercial viscose fibre of any staple, the %-age fibre shorter than 12-mm is = 0, and the %-age fibre longer than 24-mm is = 100. Even so, for any count of yarn, a plot of CS against M yields an inverted parabola. There is, however, a very important difference between the viscose and cotton curves:  the viscose curve peaks up much faster than the cotton curve. Can the cotton expressions for  help trace accurately this rise and fall of the viscose curve too?

There is a simple way of checking this. For this we restate our problem: for any one viscose of known denier, is the rise and fall of experimental CSP with M in agreement with the curve traced by the equation

More specifically, does the experimental value of M at which CS is maximum agree well enough with that predicted by the equation? Now for a given viscose of known length and denier, the R.H.S. of the equation reaches its maximum when the numerical value of the expression

reaches its maximum. Once we know the M and C to which the yarn has been spun, and the length and denier of the viscose from which it has been spun, we can calculate the numerical value of the expression

with the help of the cotton expression for U.

Now, at any one count, with increasing twist, the value of the expression

will give increasing fractional values up to the optimum twist, equal the maximum fractional value at the optimum twist, and, with further increasing twist give decreasing fractional values. We, therefore, have a method of estimating the optimum twist for viscose yarns. Table II – A – 4 - ii compares the experimentally observed optimum value of M (9) that gives the maximum CS with the estimated optimum value of M at which the fraction given by R.H.S. of the equation reaches its maximum. The agreement is very close indeed.

There are two important points to note here. In the case of viscose, the %-age fibres shorter than 12-mm is = 0, and the %-age fibres longer than 24-mm is = 100. Because of this absolute uniformity of fibre-length, the effects on the optimum twist 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 facts - -Table II – A – 4 - ii. Secondly, for viscose which has no test-length effect on fibre-tenacity, F3 is = 1, at all M; and yet the original cotton expression for F2 accounts for the contributing fibre fraction in this case also.

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.

We have, now, to check the overall accuracy of the viscose CS equation:-

Now, data on the CSP of viscose yarns show that for viscose yarns the rate of drop of CSP from NE 20 to NE 60 is as follows:

There will be many expressions that will give accurate enough estimates of CSP. In making a choice we were guided by an important criterion: the expression which represents f, the fraction of fibres that do not have free fibre ends inside the breaking element in yarn-tenacity testing should yield numerical values that are compatible with those of comparable cottons, and they should have an asymptotic maximum with increasing Q.

Here Q is the same function of fibre-length, short fibre %, and long fibre % as for cotton. The only difference is that for cotton we use the comb-sorter effective length, and for viscose we use the staple length.

This gives us

For a yarn of known C and M spun from viscose of known staple, denier and tenacity, we can calculate the numerical value of the L.H.S. of the equation. Also we know the values of  and Q for this yarn. Therefore, if we have three or more yarns from a given viscose, we can determine, by trial and error . In this we can take the help of the solver in the computer to examine the errors of estimate, and the compatibility of the values of f. For published data (9) a, b, f”(Q), and p are as follows:

Table II – A - 4 – iii gives the errors in the estimates of CSP. If we scan the data 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 incapability of the f values between groups of viscose. This would mean letting the regression analysis take over, and surrendering a desideratum fundamental in our model for the phenomenon. We, therefore, retained these expressions, that give accurate enough estimates.

Table II – A – 4 - iii shows that the values of f 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.