The sett decides how much, not how high

The setting ladder on this site has, rung by rung, found the sett deciding almost everything a cloth is. It sets the cover and therefore how much light gets through. It sets the crimp and therefore how much the fabric gives. It sets the hole a filter is bought for. It sets the weight, and the weave decides how close it can go at all.

So the natural question for the wicking anchor is what the sett buys, and the natural expectation is that a close sett wicks higher. It does not. The sett does not move a cloth’s maximum rise by one millimetre, and this essay is about why, about what it moves instead, and about the interval that leaves.

The sett reaches one pore system and not the other #

The first rung split a woven cloth into two capillary systems: the hole four threads bound, and the spaces between the fibres inside a yarn. The first is spacing less diameter and the second is d_f(1 − φ)/(4φ), by the hydraulic-radius convention in both cases.

Look at what the sett appears in. It appears in the first — the spacing is ten over the sett, so a closer cloth has a smaller hole, exactly and by construction. It appears in the second nowhere at all. The fine pore is a function of the fibre diameter and the packing factor and of nothing else, and neither of those is something a weaver chooses. The spinner chooses them, at a machine the weaver never sees, some weeks earlier.

An equilibrium rise goes as one over the radius, so the fine system always lifts higher, so the cloth’s maximum rise is the fine system’s, so it does not depend on the sett. That is the whole argument and it fits in a paragraph. What takes an essay is that it is the opposite of what four previous rungs would lead a reader to expect, and that the quantity the sett does control turns out to be worth more than the one it does not.

What the sett does decide, and it is not small #

Poiseuille’s law says the volume rate through a tube goes as the fourth power of its radius. There are more holes in a closely set cloth than an open one — the count per unit area is the two setts multiplied — but the fourth power beats the count comprehensively.

Write the coarse system as a bundle of tubes of its own hydraulic radius and the Darcy permeability is N π r⁴ / 8. Over the weavable range of the cloth above it falls from 2.2 × 10⁻⁸ m² at eight threads per centimetre to 7.4 × 10⁻¹¹ at thirty-four: a factor of 292, against a change of 8.7 in the height it can reach with it. And it is monotone the whole way. There is no sett at which the flux is best, because the best flux is at no sett at all, which is a bedsheet with no threads in it.

So the trade is real and it has a shape. An open cloth carries a great deal of liquid and lifts it barely anywhere; a close one lifts it nearly nine times further and carries a three-hundredth as much. A towel is designed at the open end of that and a wick tape at the close end, and neither of them is designed at an optimum because there is not one.

It is worth being explicit about which power does which, because the asymmetry is the whole design lesson. Closing this cloth from eight threads per centimetre to thirty-four shrinks the pore radius by a factor of 8.5. The rise carries one inverse power of that, so it goes up 8.7 times. The flux carries four positive powers of it, which is a factor of 5,200 lost, of which the extra holes — the sett squared — hand back only 18. Net, 292 lost against 8.7 gained.

So the sett is a blunt instrument for raising a fabric’s reach and a very sharp one for cutting its throughput, and a designer who closes a cloth in order to make it wick further has spent a great deal of flux for a little height and has not moved the ceiling at all.

What was counted, and how #

The sweep runs the whole weavable range for a stated yarn, refusing any sett past the jam, and asserts four things at every step rather than reporting them.

A closer sett leaves a finer hole. So it lifts higher. So it carries less. And the rise inside the yarn does not move at all — that last one to twelve decimal places, row against row, because it is the claim of the essay and a claim that is not asserted is a sentence.

There is a fifth assertion and it is the one that makes the essay’s title true rather than merely likely. At every row the cloth’s maximum rise is taken as the larger of the two systems’ heights, and the sweep requires that larger to be the fine system’s in every single row. If a sett existed at which the hole between the threads out-lifted the spaces between the fibres, the sweep would stop rather than average them — and for a spun yarn no such sett exists below the jam, because the coarse pore at the jam is still an order of magnitude the coarser.

One exact check falls out of the arithmetic and is worth having. The coarse permeability can be written two ways: as N round tubes of the hydraulic radius, or as the open-area fraction times r²/8, which is the tube-bundle result for a medium of that porosity. They differ by exactly π/4 for every cloth, always, because one counts a circle inscribed in each square hole and the other counts the whole square. That is what the round-tube convention costs, it is a constant rather than a drift, and the assertion is there so that the two routes are never quietly averaged into a third number that is neither.

The fine system’s own permeability comes from Kozeny–Carman, which this site already runs for a fibre bed inside a preform, and which carries a fitted constant — so it is quoted at a stated k and never as a property of cloth. The ratio between the two systems runs from about two hundred thousand at eight threads per centimetre to eight hundred at twenty-eight. That is the dual-scale statement the applied field arrived at for resin, met again in a shirt.

The interval, and where it ends #

An applied requirement on this site is a pair of inequalities, and the useful answer is whether the interval between them is empty. Wicking gives a clean instance.

A specification asks for a rise of at least so much and a throughput of at least so much. The first is a floor on the sett, because the rise climbs as the cloth closes; the second is a ceiling, because the flux falls. Both are monotone, so the answer is always an interval and never a point, and quoting an optimum sett for a wicking fabric is quoting something that does not exist.

What closes the interval at the near end is not a specification but the cloth. Threads jam when they touch, and for a 167 µm yarn in a plain weave the Peirce circular model puts that at 34.6 threads per centimetre. Past it there is no fabric to have a pore radius, and the machinery refuses rather than extrapolating. So the flux argument does not run to zero: it runs to a computable place and stops, and how close that place is depends on the weave, because a weave that interlaces less can be set denser.

The one quantity that does move both #

Everything in this essay separates a sett decision from a spinner’s decision, and there is one input that reaches both systems and is worth naming because it is the only one.

The yarn count moves the coarse pore and leaves the fine one alone, so it behaves like a sett: a finer yarn at the same sett has a wider gap between threads and the same spaces between its fibres. That is the same negative result again, arriving from the other input a weaver controls.

The fibre’s own fineness moves the fine pore and, through the diameter, the coarse one too. A finer fibre at the same count packs to very nearly the same yarn diameter, so the hole between the threads barely moves — and the space between the fibres goes as the fibre’s diameter directly, so the ceiling moves in proportion. A fibre half as fine doubles the ceiling.

So the ladder of control is: the weaver moves the flux and nothing else; the spinner’s count moves the flux and nothing else; and the fibre’s fineness is the only lever on the ceiling at all. That is a three-level ordering with the fabric at the bottom of it, and it is why a moisture claim is made about a fibre rather than about a construction.

The packing factor is the second lever and it is smaller and awkward. The fine pore goes as (1 − φ)/φ, so a looser yarn has larger fine pores and a lower ceiling — which means the two things a spinner does to make a yarn better, spinning it finer and packing it harder, move the ceiling in opposite directions. A compact-spun yarn is packed harder than a ring-spun one at the same count, so its fine pores are smaller and its ceiling is higher, and the effect is a few tens of per cent rather than a factor.

Which puts the whole of the wicking specification at the spinning frame, and puts none of it at the loom. That is an unusual conclusion for this site’s setting field to reach and it is the one the arithmetic supports.

It also explains a division of labour the trade already has and does not justify. A moisture-management fabric is bought as a yarn specification with a construction attached, rather than as a construction with a yarn attached — which is the reverse of how a shirting or a filter cloth is bought, and which nobody defends because nobody has had to. The arithmetic above says why: for every other property the setting ladder computes, the construction is the lever and the yarn is a constraint; for this one the yarn is the lever and the construction cannot reach the quantity at all.

The one thing the construction still owns is worth restating so the conclusion is not read too far. The flux is entirely the weaver’s, it moves by a factor of nearly three hundred across the weavable range, and it is what decides how a garment behaves in the minutes anybody is wearing it. A specification that named only the yarn would have fixed the ceiling and said nothing about the rate — which is the mirror image of the mistake this essay is about, and is the likelier one to be made once the ceiling is understood.

What this does not say #

It does not say the sett is irrelevant to wicking. It says the sett decides the flux and not the ceiling. A garment is not usually operating anywhere near an equilibrium, and over the first minutes the coarse system moves far more liquid far faster — so a cloth’s behaviour depends heavily on its sett even though its maximum does not.

And it does not say the weave is irrelevant. The weave changes the crimp, and the crimp changes the apparent coefficient by (1 + c)² — a firmer weave with more interlacings has more crimp and therefore a slower fabric, from the identity on the previous rung. The weave also moves the jam and so moves the end of the interval. What neither the weave nor the sett can do is reach inside a thread.

Where the model stops #

The negative result has exactly one exception and it is instructive. A monofilament cloth has no fibre bed, so it has one pore system, so its maximum rise is the hole between its threads and is a function of the sett. The census on the first rung contains one such cloth, the model refuses to give it a fine pore, and the refusal is the boundary of this essay’s claim rather than a gap in it.

The two systems are treated as independent. They are not: the fine channels open into the coarse holes at every crossing, so a real cloth is one connected medium and liquid can pass between the scales. Nothing here computes that transfer, and what has been shown is about each system’s own properties.

The flux is quoted at a fitted constant on one side and not the other. Poiseuille through the coarse holes has no fitted constant in it; Kozeny–Carman through the fibre bed has k, reported values for which span a factor of five, and the permeability goes as 1/k. The two are run side by side and never averaged, which is this site’s standing rule for a pair of models that are not refinements of each other.

The packing factor is assumed and it is the sensitive input. The fine pore goes as (1 − φ)/φ, which is steep: the flat line moves from 3.5 µm at φ = 0.5 to 1.5 µm at φ = 0.7, so the yarn’s ceiling roughly halves across a range of packing that ordinary spinning covers. The claim that the sett does not move it is exact; the value it does not move is a number at a stated φ.

And every argument here needs a cloth. The flat line, the falling flux and the interval between them are statements about a fabric whose sett somebody is choosing and whose count somebody has chosen. None of it is a statement about liquids, and none of it survives the cloth being taken away.

Who found it, and when #

The trade has known the practical half for a very long time and states it the other way up. Wicking tapes, lamp wicks, candle wicks and surgical dressings are all made from loosely constructed fabric of finely spun yarn, and every manual that describes them says the yarn matters more than the cloth. That is this essay’s result, arrived at by making things that work.

The reason the arithmetic is worth doing anyway is that “the yarn matters more” is a comparative and this is an identity: the sett does not move the ceiling at all, and the whole of the ceiling was fixed before the warp reached the loom. A comparative can be traded against; an identity cannot.

The dual-scale idea itself belongs to composites processing, where the difference between the channels around a tow and the spaces inside it is the reason a large infused part has voids inside its tows and not between them. That literature is about filling a fabric under pressure rather than about a fabric lifting liquid on its own, and it reaches the same split for the same geometric reason. The transfer here is that a shirt is the same object as a glass-fibre preform in this one respect, which is not a sentence either trade would volunteer.

Nothing in the physics is new or textile. Poiseuille’s fourth power is 1846, Kozeny’s bed is 1927, Jurin’s height is 1718.

Where the ladder goes next #

Two things are outstanding and both are measurements this site cannot yet make.

The first is the exchange between the systems. Everything above treats the coarse and fine networks as separate media, and they meet at every intersection under pressure — so the next rung needs a model of contact between a warp and a weft, which is the same missing piece the criterion cannot see friction recorded at the compound-cloths field.

The second is the packing factor, which enters every number here and is assumed everywhere on the site. It is the one input a spinner actually controls, mercerising moves it deliberately, and a rung that computed the yarn’s ceiling against the twist that sets its packing would close the loop between the spinning frame and the damp line on the towel.