How high a cloth wicks

A towel is hung with its bottom edge in a tray of water and left. Some hours later there is a damp line across it, and the height of that line is what the trade means by wicking. The test is standard, the tray is standard, and the number is printed on the specification.

The question worth asking is which part of the cloth the water went up. A woven fabric is threads with gaps between them and fibres with gaps between those, and the two sets of gaps are not the same size — they are not within an order of magnitude of the same size. Whichever of them decides the damp line, the other one is doing something else entirely.

Two systems, and the drawn one is the wrong one #

The hole between four threads is the one this site has already computed, and it is one minus the cover read as a length. Spacing less diameter, the same size everywhere, because a weave is a repeat and a repeat tiles the plane — which is why a woven filter cloth is bought against a single opening size where a nonwoven needs a whole distribution. At 24 threads per centimetre of a yarn 167 µm across, that hole is 250 µm square.

The second system has been in every calculation on this site since the foundation and has never been looked at directly. A spun yarn is not solid. Its diameter comes from its count only after a packing factor has been assumed — the fraction of the cross-section that is actually fibre, about 0.6 for a ring-spun cotton, and the number mercerising moves — and the count-to-diameter arithmetic divides by it every time. That missing 40 per cent is air, and it is arranged as long thin channels running the length of the thread.

Those channels are what the water goes up. Not the holes. The holes are wide open, they fill in three seconds, and they stop at 119 millimetres.

What a pore radius means when the pore is not round #

Both published laws in this essay are written for a circular tube, and neither of these pores is one. The hole between four threads is a rectangle bounded by cylinders; the space inside a yarn is the interstice of a fibre bed. Giving each of them a radius is a convention, and stating which convention is the difference between a number and a decoration.

The convention used throughout here is the hydraulic radius, twice the flow area over the wetted perimeter. It has one property that recommends it over every alternative: for a circular tube it returns the true radius, so it does not quietly change the answer in the one case everybody’s intuition is built on. For the rectangle four threads bound, it is ab/(a + b), which for the square a balanced cloth leaves is exactly half the side. That is where the 125 µm above comes from — half of 250.

For a bed of parallel fibres of diameter d_f packed at φ, the same definition can be evaluated without any fitting at all. The void volume per unit length is (1 − φ); the wetted surface per unit length is 4φ/d_f; the quotient is

r = d_f (1 − φ) / (4 φ)

and there is no constant in it that anybody measured. A 14 µm cotton fibre at a packing of 0.6 gives 2.33 µm. Change the packing to 0.7 and it falls to 1.5; change the fibre to a 6 µm polyester microfilament at 0.7 and it falls to 0.64.

The wetted perimeter of the coarse hole is taken as the rectangle’s rather than as the four circular arcs the threads actually present. That is an approximation and it runs one way: arcs are longer than chords, so the true hydraulic radius is smaller and the true rise a little higher than quoted. It moves nothing here, because everything claimed is a ratio between two systems that are a factor of twenty apart at the closest.

Three published laws, used as published #

None of the physics in this essay is textile physics and none of it is derived here.

Young–Laplace, 1805: the pressure across a meniscus of radius r wetted at contact angle θ is 2γ cos θ / r. In the coarse pore of the cloth above that is 1.17 kilopascals; in the fine one, 62.4.

Jurin’s law, from James Jurin in 1718: the equilibrium height is that pressure divided by ρg. It gives 119 millimetres and 6.37 metres respectively, for water at 20 °C.

Washburn’s law, from Edward Washburn in 1921: with gravity left out, the wetted length grows as the square root of time, L² = γ r cos θ · t / 2µ. The bracket is a wicking coefficient with the units of an area over a time, and it is what a horizontal strip test measures.

What this site supplies is not any of those. It is r — the geometry a law is applied to, computed from a sett, a count, a fibre diameter and a packing factor that a weaver and a spinner between them chose. Every argument on this page needs a cloth somebody is designing; none of it survives as a statement about liquids.

What was counted, and how #

Eight constructions, each at the count, sett, fibre and packing it is actually made at. For each one the yarn diameter comes from the count by conservation of volume, the coarse pore from that diameter and the sett, the fine pore from the fibre and the packing, and both heights from Jurin’s law at perfect wetting. Nothing in the table is quoted from anywhere.

The closest the two systems come is 23, in a fine cotton shirting at 42 threads per centimetre — the most tightly set cloth in the table, which is exactly where the coarse hole is smallest and the comparison hardest. The widest is 102, in a cotton duck at 12 threads per centimetre of a 60 tex yarn, where the hole is 272 µm and the fibre spaces are still 2.67. Every row in between falls where the arithmetic puts it, and the ratio is never the fibre’s doing: it is the sett and the count, which are decisions.

The eighth row is worth its place because the model refuses it. A polyester monofilament is one filament, so φ is 1, the packed-bed formula gives zero, and asking for a rise from a pore of no size is asking for an infinite one. The machinery declines rather than returning a number, and the refusal is the right answer: a monofilament cloth has exactly one pore system, and whatever it lifts, it lifts by the hole between its threads. That is a fact about a whole class of industrial fabrics, and it arrived as an error message.

The one quantity that is assumed, and why it does not matter #

Nothing on this site knows any surface chemistry. Whether a cotton has been scoured, sized, waxed or given a fluorocarbon finish decides the contact angle entirely, and nothing here can compute it. So θ is an argument with a stated default of zero — perfect wetting, the best case — and every height above is a height at that assumption.

That is the sort of admission which usually ends an argument. Here it does not, and the reason is arithmetic rather than luck. Both systems are the same fibre with the same finish, so both take the same cos θ, so the ratio between them does not depend on the contact angle at all. The claim of this essay is a ratio. It survives every value of the one quantity that had to be assumed.

The refusal at 90° earns its place. cos θ goes negative past a right angle and the arithmetic will happily return a negative height, which is a real physical statement — the liquid is pushed out rather than drawn in — and is not a rise. Returning it as one is how a figure ends up with a cloth wicking downwards.

The faster system is not the higher one #

Here is the part that is genuinely surprising, and it is exact.

The equilibrium height goes as 1/r. Washburn’s coefficient goes as r. Every other quantity in both — surface tension, density, viscosity, the contact angle — belongs to the liquid and is shared, so it cancels. Therefore the coarse system is faster than the fine one by exactly the factor it is lower by, and that factor is the ratio of the two radii. In the cloth above it is 53 both times, and the assertion in the code checks the two numbers against each other rather than against a target.

The consequence is that a horizontal wicking strip and a vertical one rank the same cloth’s two systems in opposite orders, which is not a subtlety a test method would notice. Over half an hour, ignoring gravity, the coarse system’s front travels 2.86 metres and the fine one’s 391 millimetres. Under gravity the coarse system stops after 3.1 seconds at 119 mm and never moves again, while the fine one passes that same height after 2.8 minutes and is still climbing five days later.

That also settles what a thirty-minute strip test is measuring. It is not the equilibrium — the equilibrium in the fine system is five days away and several metres up — so the number on the specification is a Washburn number and not a Jurin one, and the two are different quantities that happen to share a unit. A cloth quoted at 120 mm in half an hour has not been shown to stop at 120 mm.

Where the two systems would meet, and why no cloth is there #

The bars in the ratio figure run from 23 to 102 and the reader is entitled to ask what would have to be true for them to close. The answer is a cover factor, it is computable, and it is a long way outside anything anybody weaves.

The two radii are made of different things. The fine one is the spinner’s: fibre diameter times (1 − φ) over 4φ, with no sett and no count in it, so it is 2.33 µm for a ring-spun cotton whatever the cloth is. The coarse one is the weaver’s: half the clear gap between threads, which is the spacing less the diameter, and it falls to nothing as the cloth closes.

Set them equal. The clear gap would have to be 4.66 µm — twice the fine radius — against a yarn 167 µm across. That is a gap of 2.8 per cent of a thread’s width, which is a geometric cover of 0.972.

No cloth is woven within three per cent of full cover, and this site has the reason two ladders away: threads have to bend past one another, so the sett at which they jam is half the covering sett for a plain weave and four fifths of it for an eight-end satin. The closest a real construction gets is 0.8, where the clear gap is still 42 µm and the ratio is still 18.

So the dual-scale description is not merely a convenience that happens to work for the eight cloths in the table. It is safe for every woven cloth of spun yarn there is, and the margin is the same margin that makes weaving possible at all. A cloth in which the two capillary systems were comparable would be a cloth whose threads could not cross.

Two smaller consequences fall out of the same comparison and both are checkable.

The tighter the cloth, the higher its coarse system lifts. The coarse height goes as one over the clear gap, so it rises steeply as the sett approaches the jam — from 119 mm in the shirting to nearly half a metre in a cloth set to four fifths of full cover. The fine system does not move at all. So a very closely set cloth is one in which the two mechanisms are closest to agreeing, and it is also the one whose wicking is least explained by pointing at its holes.

And a coarse open cloth is the case where the drawn system is most irrelevant. The duck’s ratio of 102 is not a fact about duck; it is a fact about a wide gap, and the wider the gap the less anything is holding water in it. A scrim, a net or a leno gauze has a coarse system that lifts almost nothing and a fine system unchanged, which is why an open cloth of spun yarn still wicks and a monofilament one of the same openness does not.

Where the model stops #

Neither pore is a tube. Both laws are written for a uniform circular capillary with fully developed flow in it. The coarse pore is a short rectangle between four cylinders, open at both ends and connected to its neighbours; the fine one is a tortuous, connected, non-uniform bed whose channels swell as they fill and whose front is ragged rather than plane. What transfers is the scaling — a height that goes as 1/r, a front that goes as √t, a coefficient linear in r — and every claim here is built on scalings and ratios rather than on absolute heights.

The two systems are treated as independent and they are not. A real cloth’s fine channels open into its coarse ones at every crossing, so liquid can and does transfer between them. Nothing here computes that exchange, and the honest statement of what has been shown is about the two systems’ own properties rather than about the cloth as one medium.

The equilibrium heights are large and are not observations. Six metres is what Jurin’s law says about a 2.33 µm tube and nothing has ever been seen to do it in a fabric, because evaporation, drainage, the finite height of a real sample and the five days it would take all intervene long before. The number is a ceiling on a mechanism, not a prediction of a measurement.

The contact angle is assumed, and the packing factor nearly is. φ = 0.6 for a ring-spun cotton is the accepted figure that Peirce’s 1937 constant turns out to imply, and it is not a measurement of any particular yarn, nor of a tow, which is not round and does not pack like one. The fine pore goes as (1 − φ)/φ, which is steep: at φ = 0.5 the radius is 3.5 µm and at 0.7 it is 1.5.

And every argument here needs a cloth. The two systems, the ratio between them and the trade that follows are statements about a fabric whose sett and count somebody is choosing. Take the cloth away and there is nothing left but the published laws, which belong to physics and are used here exactly as published.

Who found it, and when #

The physics is old and settled. Jurin’s height is 1718, Young’s and Laplace’s pressure is 1805, Poiseuille’s fourth power is 1846, and Washburn’s square root of time is 1921. None of it was found by anybody thinking about cloth.

The textile half arrived much later and in a different trade. The idea of giving a fabric two length scales rather than one belongs to the composites literature, where a woven reinforcement’s channels between tows and spaces between filaments differ by a factor of thirty and a resin fills them at wildly different rates. This site already computes that pair, in fabricPermeability, and refuses to average them. The claim of this essay is that a shirt is the same object: every woven fabric of spun yarn is a dual-scale porous medium, and the reason it is not usually described as one is that a person infusing a boat hull has to care and a person hanging a towel does not.

The word “wicking” itself is older than any of it and comes from the lamp. A candle wick is a loosely twisted bundle of cotton, which is a yarn with its twist and its packing factor deliberately made low, and the mechanism that draws oil up a lamp is the mechanism drawing water up the towel. Whoever first plaited a wick had found the fine system by hand about three thousand years before anybody wrote the law it obeys.

Where the ladder goes next #

The next rung takes the fine system seriously and asks about the path it runs along. The channels inside a warp end follow the warp end, and a warp end in cloth is not straight — it is longer than the fabric by its crimp, which this site has been computing since the foundation. A front travelling a thread’s own path arrives late in cloth coordinates, and by exactly how much turns out to be an identity with no free parameter in it at all.