Yarn Diameter from Linear Density

Treat a yarn as a solid circular cylinder of uniform bulk density and the diameter follows from the count in two lines. Linear density divided by bulk density is an area — kilograms per metre over kilograms per cubic metre leaves square metres — and that area is the yarn's cross-section:

\[ A = \frac{\rho_L}{\rho}, \qquad d = \sqrt{\frac{4A}{\pi}} = \sqrt{\frac{4\rho_L}{\pi\rho}}. \]

The only judgement in the whole thing is $\rho$, the bulk density of the yarn, which is not the density of the fibre. A yarn is fibre plus air, and the fraction that is fibre is the packing factor. Cotton fibre is about 1.52 g/cm³; a real ring-spun cotton yarn packs to roughly 0.55 to 0.65 of its volume, giving a bulk density near 0.85 to 0.99 g/cm³. Textile texts usually state the same information as a specific volume $v = 1/\rho$ in cm³/g, and in that form the relation is written $d\,[\text{cm}] = \sqrt{\mathrm{tex}\cdot v}\,/\,280.25$, where the 280.25 is exactly $\sqrt{\pi \times 10^{5}}/2$ and carries nothing but the unit bookkeeping.

Where the old rule comes from. The trade has used $d = 1/(28\sqrt{N_e})$ inches for well over a century, and it is this equation with one particular density baked in. Run it backwards. Set $d = 0.0254/(28\sqrt{N_e})$ metres and $\rho_L = 590.5412/N_e$ tex, and solve for $\rho$: the count cancels — as it must, or the rule would not work at every count — and you are left with

\[ \rho = 0.9137\ \mathrm{g/cm^3}, \qquad v = 1.0944\ \mathrm{cm^3/g}, \qquad \text{packing factor} = \frac{0.9137}{1.52} = 0.601. \]

So the 28 is not a constant of nature. It is a cotton yarn at sixty per cent packing, written as a whole number, and that is a perfectly reasonable thing for a nineteenth-century rule to be. Knowing what it assumes tells you when it stops being right: a bulky rotor yarn packs looser and is fatter than the rule says, a compact-spun yarn packs tighter and is thinner, and a polyester yarn has a different fibre density altogether.

Hold on to the number 28, because it comes back. On the cover-factor page it appears again, and it is provably the same 28 — the cover factor is nothing but this diameter divided into the thread spacing, and 28 is the value it reaches when the cloth is completely covered.

Two honest limitations. A yarn in a woven cloth is not circular: it flattens against the threads it crosses, and Peirce's own geometry later had to be extended with elliptical and racetrack cross-sections to handle firmly set cloths. And a yarn has no sharp boundary at all, thanks to the hairs standing off its surface, so "the diameter" is a modelling convenience rather than a measurable length. Use it as one.