A drying cloth cannot lift what a sealed tube can

How high a cloth wicks is this account’s foundation and it ends in an admission. The fine system inside the yarns lifts 6.37 metres against the coarse system’s 119 millimetres, a factor of forty-six — and the essay says in its own limitations that “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.”

Four candidates, none of them computed. This essay computes the first and finds it is the binding one, and the result is not that six metres becomes five.

The balance is one integration, and the width falls out of it #

A sealed tube reaches equilibrium because nothing leaves. A strip of cloth hanging with its foot in water loses liquid from the whole of the area already wet, so it is not at equilibrium — it is at a steady state, in which the flow arriving at any height equals everything evaporating above it.

Take a strip of width W and thickness t, losing E metres of liquid a second from each of its two faces. The flow crossing a section at height z must feed everything above:

$$Q(z) = 2EW(h-z).$$

Darcy’s law supplies it, against gravity and the pressure gradient, with the permeability the account computes for the hole between four threads and for the bed inside the yarn:

$$Q = -\frac{K}{\mu}\left(\frac{dp}{dz} + \rho g\right)Wt.$$

Integrating from the water surface, where p = 0, to the front, where the meniscus is fully curved at p = −Pcap:

$$P_\text{cap} = \rho g h + \frac{\mu E}{K t}\,h^2.$$

The width cancels exactly, appearing once in the supply and once in the loss. So the standing height is a property of the cloth and not of the sample, which is worth knowing before anybody argues about strip widths in a test method.

And at E = 0 the quadratic term vanishes and the height is Pcap/ρg, which is Jurin’s — the check the whole function is built to satisfy, and the one it is required against.

What that does to the two systems, which is not the same thing #

The quadratic’s coefficient is μE/Kt, and K is where the two pore systems part company. The coarse system’s permeability is 980 times the fine one’s — a hole a hundred micrometres across against a gap of two — while its capillary pressure is 46 times smaller.

So the fine system has the high suction and the low supply, and those are the same property. It pulls hard because its pores are small and it delivers nothing for the same reason.

In an ordinary room the fine system stands at half a metre instead of six and a third, which is 7.9 per cent of what a tube would give it. The coarse system loses a third of one per cent.

And above a drying wind the ranking reverses #

The crossing is not an artefact of the range chosen. At a large enough loss the gravity term is negligible and both heights go as $\sqrt{P_\text{cap} K t / \mu E}$, so the ratio tends to

$$\sqrt{\frac{P_f K_f}{P_c K_c}} = 0.218,$$

which is below one. The fine system wins at rest by 46 and loses in the limit by a factor of nearly five, so a crossover must exist — and the census required that it must before going to look for it.

It is at 1,553 grams a square metre an hour, where both systems stand at 131 millimetres. That is a real rate: a wet cloth in a brisk wind, or on a warm dry day, loses at about that. Below it the fine system lifts higher; above it the coarse one does.

Which inverts the account’s foundational result under conditions a garment actually meets. A shirt drying on a line in wind is a cloth whose coarse pores are doing the lifting; the same shirt hanging indoors is one whose fine pores are. Which system is carrying the water decides what a finish can do to it, since a finish is the only thing that chooses between wicking and resisting.

Why it is the fine system that suffers #

The mechanism is worth stating plainly, because it is the same mechanism in both directions and it is not obvious which way it should run.

A pore’s capillary pressure goes as $1/r$ — small pores pull hard, which is How high a cloth wicks. A pore’s permeability goes as $r^2$ — small pores deliver little. So the ratio of what a system can lift to what it can supply goes as $1/r^3$, and the fine system is 46 times better at one and 980 times worse at the other.

The balance above weights those two against each other with the evaporation rate, and the weighting is not symmetric. The gravity term is linear in h and the loss term quadratic, so the loss term dominates once the front is high — and the fine system’s whole advantage is that its front is high. It is starved exactly where it was winning.

Put the other way round: the coarse system never has to supply much, because it never stands very high. Its sealed height is 137 millimetres and the liquid it must lift to hold that is nothing against a permeability of 3.1 × 10⁻¹⁰ square metres. It is comfortable everywhere in the range.

The thickness is the lever, and it is the wrong way round #

The balance has exactly one construction variable in it that the account has not already swept, and it is the cloth’s thickness.

$$P_\text{cap} = \rho g h + \frac{\mu E}{K t}\,h^2$$

A thicker cloth stands higher, because t is in the denominator of the loss term: thickness is cross-section for the supply and adds nothing at all to the evaporating area, which is two faces however thick the cloth is.

That is the opposite of every other wicking result on this account, where thickness appears nowhere. A sealed tube’s height has no thickness in it — Jurin’s law is about one pore — so the whole account has been able to treat a cloth’s rise as a property of its pores. Once the cloth is losing water it is a property of its pores and its thickness, and the two are chosen separately.

It also explains a piece of practice with no explanation attached. A towelling or a fleece lifts water further than a shirting of the same fibre, and the usual reason given is that its yarn is bulkier and its pores finer. The arithmetic says the thickness alone does most of it: a cloth four times as thick stands twice as high at a fixed loss, before anything about its pores is considered — and a thickness is not a mean but a maximum over the crossings under the foot, which is a second reason it is badly specified.

What this changes about every rise on the account #

Three results are affected and they are affected differently, which is the useful part of the accounting.

The two-system finding survives and its number does not. The factor of twenty to a hundred between the two systems is a sealed-tube factor. In a room it is 3.7. The structure of the result — two systems, one high and slow and one low and fast — is unchanged, and the ratio anybody would quote is out by an order of magnitude.

The crimp identity is untouched. Wicking is slower along a crimped thread is a statement about path length: the front travels the thread’s path and arrives late by exactly (1 + c)². That is a geometric identity and nothing above it depends on where the front stops.

And the sett result gets sharper. The sett decides how much, not how high found the maximum rise to be a property of the yarn, identical at every sett, because the sett moves the coarse pore and not the fine one. That is true in a sealed tube. In a drying room the fine system’s height depends on the cloth’s thickness and on its permeability, both of which the sett moves — so the sett does decide how high, once the cloth is losing water, and the negative result is a sealed-tube result rather than a general one.

The number a test method is actually measuring #

The vertical wicking strip test is the standard measurement, and it is run in a laboratory at a controlled temperature and humidity, in still air, for a stated time. Everything above says what it is reading.

At still-air rates the fine system stands at about a metre, which is taller than any test strip anybody uses. So a strip test of ordinary length never reaches the steady state at all — it is measuring the Washburn front’s progress, which is the coefficient, and not a height — and that coefficient is the yarn’s own divided by the square of its crimp.

But the height is what the result is quoted as. A wicking test reports millimetres at a stated time, and a reader takes that as how far the cloth lifts. It is how far the cloth had got.

The two numbers separate exactly where the strip is long enough for the steady state to matter, and this arithmetic says where that is: at an ordinary room’s evaporation the fine system’s ceiling is 500 millimetres, so a strip longer than half a metre is measuring a ceiling and a strip shorter than that is measuring a rate. Standard strips are 200 millimetres, which is comfortably in the second regime and is the reason the distinction has never had to be made.

Raise the temperature and it does. At 400 grams a square metre an hour — a warm room with a fan, which is what a drying cabinet is — the ceiling is 255 millimetres, and a 200-millimetre strip is within a fifth of it.

The same balance says what a wick has to be #

A candle wick, a lamp wick and the wicking layer of a two-layer sports fabric are all doing this arithmetic, and the balance says what they must be made of.

A wick has to deliver, not to lift. Its job is to carry liquid to a place where it is consumed — burnt, or evaporated off the outer face of a garment — and the rate is the whole of what it is for. The balance’s supply term is K t, so a wick wants a coarse pore system and a thick section.

And a lifter has to pull. Its job is to get liquid up against gravity, and the capillary term is 1/r, so it wants a fine pore system.

Those are opposite requirements and the account has met the pair before. The pore that wicks is the pore that leaks found a cloth’s water resistance and its wicking to be one expression read at two ends of a contact angle, so the geometry cannot be chosen to give both. This is the same shape with the geometry rather than the chemistry doing the choosing: a fine pore lifts and does not deliver, a coarse pore delivers and does not lift, and no single pore size does both.

Which is the argument for a two-layer construction, and it is sharper than the usual one. A next-to-skin layer with fine pores lifts liquid off the skin; an outer layer with coarse pores carries it out to evaporate. The two layers are not doing the same job better and worse; they are at the two ends of an exchange whose rate this balance computes, and a fabric that used one pore size throughout would be at whatever point of it the yarn happened to give.

What was counted, and how #

The permeabilities are the account’s own pore-flux calculation: Poiseuille through tubes of the hydraulic radius for the coarse system, and a Kozeny–Carman bed weighted by the plan area the yarn occupies for the fine one. Neither is new here and both carry their own checks.

The capillary pressures are Young–Laplace at the same two radii the account has used since its foundation, so the sealed heights recovered here are the account’s own to the last figure — 137 and 6,374 millimetres, against the 119 and 6,370 the first essay quotes for a slightly different sett.

The evaporation rates are a stated bracket rather than a value, from still air at 20 grams a square metre an hour to a gale at 5,000. That is the one quantity here that belongs to the room rather than to the cloth, and the whole sweep is run across it for the same reason the contact angle is swept rather than assumed.

The crossover is found rather than assumed, by bisection, after the census has required that one must exist — which it does by comparing the sealed ratio against the large-loss limit and requiring the first to be above one and the second below.

And the monotonicity is required at every step: each increase in evaporation must cost the fine system more than the coarse one, and the coarse system must lose a smaller share of its own height. Either failing would mean the balance had been mis-integrated rather than that the prose needed softening.

What it means for a cloth that has to dry #

There is a reading of the balance that runs the other way and it is the one a wearer cares about.

A cloth that lifts water further is a cloth that spreads it over more area, and a cloth that spreads water over more area dries faster, because drying is an area process. So the same evaporation that limits the rise is accelerated by it, and the two are in a loop.

The loop’s fixed point is what the balance computes. At the steady height the cloth is wet over 2Wh of area and is losing 2EWh of water a second, all of it supplied from the foot — so the total loss is proportional to the standing height, and a cloth standing at 500 millimetres dries a puddle 3.7 times as fast as one standing at 137.

That is the practical value of the fine system and it is not the one it is usually sold for. The fine system is described as the one that lifts, and what lifting buys is evaporating area. A cotton’s own water is a twentieth of what a cloth holds is the collection’s account of where the water actually sits; this is the account of how fast it leaves.

It is also why what the wind takes is the air and not the cloth has a wet counterpart with a quite different sign: dry, a wind removes the still air a cloth holds; wet, it removes the water, and the cloth’s own geometry decides how much of it can be replaced.

And it means a cloth in a wind is in a different regime in two ways at once. The wind raises E, which lowers the standing height, which lowers the wetted area, which lowers the total loss — so the drying rate rises less than the evaporation rate does. Working the balance through, the standing height goes as $1/\sqrt{E}$ at the wet end, so the total loss goes as $\sqrt{E}$ rather than as $E$. Doubling the wind buys about forty per cent more drying, not twice as much, and the missing sixty per cent is cloth that has gone dry above the front.

What the model cannot say #

The two systems are solved independently and they are not independent. A real cloth’s fine and coarse pores are in contact everywhere, so liquid moves between them, and the coarse system can feed the fine one at any height. That exchange would raise the fine system’s ceiling and lower the coarse one’s, which means the crossover computed here is a bound rather than a location — the true curves are closer together than these.

The evaporation is uniform over the wetted area, and it is not. Loss is driven by the vapour gradient, which is steepest at the top of the wetted region where dry cloth is adjacent, so a real strip loses more near its front than at its foot. That concentrates the loss where it costs the most supply and would lower the ceiling further.

Nothing here is transient. The steady state is reached after the front has had time to arrive, and the fine system takes days to cover its own height. So the number computed is where the front would stand, and a cloth that dries out before it gets there never sees it — which is the fourth of the first essay’s four candidates, and it is the next one to compute.

And the meniscus is taken as fully curved at the front. That is the standard closure and it is right for a static front; for one that is advancing, the pressure at the front is less than the full suction by whatever is driving the advance, so the steady height is approached from below and is exactly right only in the limit.

Who found it, and when #

Evaporation-limited capillary rise is old physics and is standard in soil science, where it decides how much water a water table delivers to a surface and is computed exactly this way. The soil literature calls the ceiling the limiting rate of capillary rise and has tabulated it against texture since the 1950s.

It is absent from textile wicking entirely, and the reason is visible in the test method: a wicking strip is 200 millimetres and a laboratory is still, so the effect has never been large enough in a measurement to demand a model. The theory quoted in the textile literature is Washburn’s, which has no loss term in it, and the heights quoted are Jurin’s, which is a sealed tube.

What this account supplies is the connection, and it needed two things it already had: a permeability for each of the two pore systems, and the observation that they differ by a factor of a thousand while their suctions differ by a factor of forty-six. The whole result is those two numbers put into one balance, and the reversal is what happens when a ratio of 980 meets a ratio of 46.

Still open: how long the steady state takes to arrive #

The four candidates the first essay listed are evaporation, drainage, the sample’s height and the time — and this essay has done the first and made the fourth worse.

A front reaching 500 millimetres by Washburn’s law, at the fine system’s coefficient, takes about twenty-one hours. A front reaching 6.37 metres takes fifteen weeks. So the steady state computed here is nearer than the sealed-tube one by a large factor and is still not reached in an afternoon.

What is needed is the transient, which is the same balance with a storage term: the front’s position against time, with the cloth behind it losing water while the front advances. That is one differential equation rather than one integration, and its answer would say whether a cloth ever reaches its own ceiling before the source dries up — which is the practical form of every question on this account, and the one none of its four essays has asked.