A cloth stops having holes before it stops passing air

Everything this collection has said about what goes through a fabric has been about the holes between its threads. The hole is where the water rises, where the soil is stopped, where the resin runs, where the light comes through. Close the holes and the account runs out — which sounds like a limit of the subject and is in fact a mistake in the model, because a thread is not solid.

A ring-spun cotton yarn is sixty per cent fibre and forty per cent air. That is what a packing factor is, and this collection has been dividing by it since the very first diameter was computed from a count. The spaces between those fibres are a few micrometres across, they run the length of the thread and across it, and they are a path from one face of the cloth to the other that does not pass between any two threads at all.

So a cloth has two paths in parallel, and only one of them can be closed by weaving.

The claim #

A woven cloth’s air permeability is the sum of two independent paths: the channels between the threads, which the construction controls and which go to zero, and the pores inside the threads, which the construction does not control and which do not.

The consequence is a floor. However closely a cloth of a given yarn is woven, calendered, shrunk or beaten, it passes at least what its own threads pass — and that number is a property of the fibre’s diameter and the yarn’s packing factor, both of which were fixed in the spinning mill.

For a 20 tex cotton at a packing factor of 0.6, in a cloth a third of a millimetre thick, the floor is 8.1 millimetres per second at a hundred pascals. A specification asking for less than that has asked for a fabric that cannot be made from that yarn by weaving it more closely, and the reason has nothing to do with the loom.

The argument #

The two paths are added because they are two routes between the same two pressures, and nothing about either depends on the other. That is the whole of the coupling argument, and it is worth saying plainly because a great deal of effort is often spent on averaging things that should be summed.

The channels are ducts and are computed as ducts, one per hole in the repeat, at the waist each one actually has. Their contribution falls as the cloth closes because their width does — steeply, since flow down a duct goes as its area times the square of its hydraulic diameter.

The word independent is doing real work there and is worth one more sentence. Two paths in parallel share their end pressures and nothing else, so their flows add and neither one’s presence changes the other’s resistance. Two paths in series would multiply their resistances and the small one would vanish; two paths that exchanged fluid would need solving together. A cloth’s channels and its threads are genuinely parallel — a molecule entering one never enters the other — so the sum is exact rather than a first approximation, and the small term survives precisely because it is being added to rather than divided into.

The threads are a packed bed of fibres and are computed as one, by Kozeny–Carman at the yarn’s own packing, over the fraction of the surface the threads cover. That contribution rises slightly as the cloth closes, because more of the surface is thread. It rises from 1.7 mm/s at 6.9 threads per centimetre to 6.4 at 34.2, while the total falls from 8,285 to 1,337.

Darcy is the whole of the second term and that is a claim rather than a convenience. The pore Reynolds number inside a yarn comes out in the thousandths, so there is no inertial term to have left out — asserted as the sweep runs, at every sett, because a bed model quietly used outside its regime is the commonest way an arithmetic like this goes wrong.

Why the sett never reaches the floor #

The obvious way to test the claim is to keep closing the cloth until the two terms cross. It cannot be done: the yarn jams first.

At the closest sett a 20 tex cotton will weave — 34.2 threads per centimetre, where its ends meet — a muslin’s channels still carry 99.5 per cent of the flow, and there is no cloth beyond that. The crossing is past the jam. So for an ordinary cotton the floor is not a limit anybody arrives at by weaving; it is a limit that becomes visible only when something else closes the channel.

Three things do, and this collection computes all three.

The thread can be flattened. A calender nip closes on the crimp crowns and squashes them, which raises the cover without changing the sett at all — the compression is spent for good, and a cloth calendered to a three-to-one section has a warp cover of 0.59 where it had 0.40.

The fibre can swell. Cotton in water grows across itself by a fifth, so a wetted cloth’s threads are wider and its channels narrower — which is why a wet cloth is set closer than it was woven, and why a sail or a tent becomes noticeably less draughty in rain before it becomes wet through.

The cloth can be covered. A coating bridges the holes and the whole channel term vanishes, which is why coated is a state rather than a treatment.

What was counted, and how #

The floor is Kozeny–Carman through a bed of fibres at the yarn’s own packing, over the whole surface, with the cloth’s own thickness as the path length:

• porosity 0.40, being one less the packing factor
• fibre diameter 15 µm, which is a cotton’s
• Kozeny constant 5, fitted and quoted as fitted
• thickness 342 µm, from Peirce’s closure at the muslin’s construction

which gives a permeability of the order of 10⁻¹³ square metres and a face velocity of 8.1 mm/s at a hundred pascals.

The path length is the largest thing wrong with it, and it is wrong in a known direction. A path through a thread is longer than the cloth is thick, because a thread is crimped and a path through it winds, so the true floor is lower than this. The number is therefore an upper bound on the floor, which is the useful direction: a specification below 8.1 mm/s might still be reachable, and one above it certainly is.

The Kozeny constant is the one fitted quantity in the whole calculation, it varies between about one and five in the literature, and the permeability goes as one over it — so the floor could be five times higher than quoted. That does not touch the claim, which is that the floor exists and is not moved by the construction. It touches only where the floor is.

Two further inputs are the yarn’s and not the model’s, and both are known to a tighter tolerance than the constant. The packing factor of 0.6 is the number Peirce’s own diameter rule implies when it is set against conservation of volume, which is a coincidence this collection thinks worth trusting. The fibre diameter of 15 µm is a cotton’s and is the same figure the wicking arithmetic uses, so the two accounts of these pores are being fed the same yarn.

What the floor would take to move #

The floor belongs to the yarn, and it is worth being precise about which properties of the yarn, because two of the three are the ones a spinner is already choosing for other reasons.

The packing factor, through the bed’s porosity. Kozeny–Carman’s permeability goes as the cube of the porosity over the square of one minus it, which is a steep function: raising the packing from 0.6 to 0.7 cuts the bed’s permeability by roughly a factor of three. So a compact-spun yarn, or a mercerised one, or a hard-twisted one, has a lower floor than a soft ring-spun yarn of the same count — and every one of those is a change made for lustre, strength or handle rather than for permeability.

The fibre diameter, squared. A fine fibre makes fine spaces between fibres, so a microfibre yarn’s floor is far below a cotton’s — a factor of six for a fibre half as thick. That is the same square that decides the fine pore’s wicking and the same one that decides its capillary head, which is a reminder that the three properties are one geometry asked three questions.

And the cloth’s thickness, which is not the yarn’s at all and is the one the table’s spread is made of. It enters as a path length, so a thicker cloth has a lower floor — and thickness is a construction property, so this is the one route by which a weaver reaches the floor after all. It is a weak route: the eight cloths’ thicknesses span 2.5 to one, against the channel term’s 3.5, and it moves the floor the wrong way for anybody wanting an open fabric.

So the ordering of levers is fibre first, packing second, thickness third, and the first two belong to the spinning frame. That is the same conclusion the wicking ladder reaches about the ceiling, from the same two pores, and the agreement between them is not a coincidence: both quantities are properties of the fine system, and the fine system is a yarn.

There is one place the two ladders disagree, and it is worth setting down because it says which of the two pores decides which question. The wicking ceiling is the fine system’s because a rise goes as one over a radius and a fine pore lifts higher. The permeability floor is the fine system’s because the coarse system has been removed and only the fine one is left. So the fine system wins the first by being better at the job and the second by being the only survivor — two quite different reasons for the same pore to be the answer, and a reader who took the agreement as a single principle would be carrying one argument where there are two.

Eight floors, and what they depend on #

The floor is not the same for every cloth, and what moves it is worth naming because it is not what moves the cloth’s permeability.

All eight share a yarn packing of 0.6, so the bed’s own permeability is the same number in every row. What differs is the path length, which is the cloth’s thickness, and the floor is inversely proportional to it. So the floors run from 4.7 to 11.5 mm/s — a factor of 2.4 — while the as-woven column runs over a factor of 3.5, and the two columns do not order the cloths the same way at all. The duck is sixth by what it passes and first by how little it would pass if its channels were shut, because it is the thickest cloth in the table and a path through it is the longest.

That is the practical shape of the thing. A designer moving down the as-woven column is choosing a construction; a designer who has reached the floor is choosing a thickness and a yarn, and the ranking they were using no longer applies.

Where the model stops #

The two paths are treated as independent and they are not, quite. Air leaving a channel and air leaving a thread meet in the same boundary layer on the downstream face, and at high flows the two interact. At the pressures a permeameter uses the interaction is small, and nothing here computes it.

The bed is uniform along the thread and it is not. A yarn is compressed at its crossings and free between them, so its packing factor varies along its own length — which this collection knows, having priced the flattening — and the bed term should be computed against the tightest section rather than the average. That lowers the floor again.

The floor is a floor for a weave, not for a fabric. A nonwoven has no channels at all and is nothing but bed, which is why its permeability is quoted in different units and why what holds a nonwoven together is a different question from the start. A knit is closer to the nonwoven case than to the woven one, for reasons that belong to the loop.

And a coating is not a closed channel. A film over a cloth is a barrier in series with the fabric rather than a removal of one of its paths, and a real one has pinholes — which is exactly the finding that a coated cloth fails at its holes. Treating a coating as the limit of closing the channel is a convenience for putting it on the same axis, and the fourth bar of the figure above is labelled channel closed rather than coated for that reason.

The generalisation #

A quantity that is the sum of a controllable term and an uncontrollable one has a floor, and the floor is invisible while the controllable term dominates.

That is the whole shape and it is very common. Leakage current, dark noise, background counts, baseline load, the parasitic capacitance of a package: in each case a design parameter drives one term towards zero while another sits there unmoved, and the second becomes the answer only when the first has nearly gone. Until then it is a rounding error that nobody has any reason to measure, and every extrapolation of the first term alone runs off to zero and is wrong.

The diagnostic is to extrapolate rather than to measure. The floor here was never observed in any weavable cloth; it was found by taking the model to a state the loom cannot reach and reading what was left. That is a legitimate use of a model and it is the only use that finds this class of limit.

The second lesson is about what a specification is written against. Windproof, breathable and airtight are all specifications on this quantity, and all of them are written as if the fabric were the thing being specified. Below the floor, the yarn is the thing being specified, and the person who can meet the requirement is not the weaver.

Who found it, and when #

The two-pore description of a fabric is standard in filtration and composites, where it is called dual-scale porosity and where its consequence is that resin fills the channels first and the tows afterwards — which this collection computed for a reinforcement and found to have a crossing at a computable sett.

Kozeny–Carman is from 1927 and 1937, Darcy from 1856, and both are used here as published results.

What appears to belong to this collection is the observation that the crossing which is reachable in a composite reinforcement — where the tows are coarse and the setts are open — is unreachable in an ordinary woven cloth, so the same two-term model has a visible crossing in one case and a floor in the other. The difference is a factor of thirty in the ratio of the two length scales, and nothing else.

Where the ladder goes next #

If the floor is above the threshold, the threshold cannot be met, and for windproofing it is: a windproof cloth is at its yarn’s limit, and the arithmetic says so before any cloth is woven.

Sideways, the steeply falling term is the one everybody quotes, and it is quoted in a form that is only true at the closed end: the fourth power is a close cloth’s rule, carrying two per cent of the pressure drop in an open scrim and most of it in a shirting.