A wick reaches its ceiling in the time its cloth takes to dry

A strip of cloth hung with its foot in water lifts it. How high a cloth wicks found two answers, one for each of a cloth’s pore systems: the holes between the yarns stop the water at about fourteen centimetres in a few seconds, and the spaces between the fibres would carry it more than six metres if they were given five days to do it.

They are not given five days, because the cloth is drying while it lifts. A drying cloth cannot lift what a sealed tube can balanced the water climbing the strip against the water leaving its faces and found the fine system standing at half a metre in an ordinary room rather than six. It ended on the question it could not answer: how long does the front take to get there? A steady state that takes a week to arrive is a ceiling nobody meets.

The answer turns out to be a time a reader already knows without having thought of it as a wicking time.

The balance needs one more term #

The steady balance has two terms: Darcy’s flow up the wetted strip, driven by the capillary suction at the front and opposed by gravity, and the evaporation from the strip’s two faces along the wetted length. It asks where they are equal.

A front that is still climbing has a third job. Every millimetre it advances is a millimetre of pores that must be filled, and the water to fill them has to arrive at the front after the faces behind it have taken their share. So the rate the front climbs is what reaches it, divided by the water a millimetre of cloth holds:

$$\varepsilon t\,\frac{dh}{dt} = \frac{K t}{\mu}\,\frac{P - \rho g h}{h} - E h.$$

The first term is what Darcy delivers to a front at height h; the second is what the two faces of the wetted length lose on the way. ε is the share of the strip’s volume the pore system occupies, t the strip’s thickness, K its permeability, P the capillary suction and E the evaporation from each face.

Set the left side to nothing and the drying balance’s steady height comes back exactly, which is the first check that the storage term was added to the right equation.

Without gravity it has an exact solution #

For the fine system, gravity is a small correction: at the steady height in an ordinary room it takes 8 per cent of the capillary suction. Drop it and the equation can be solved in closed form.

Write h* for the steady height and u for the front’s share of it. The equation becomes one line about $u^2$, whose solution is

$$h = h^*\sqrt{1 - e^{-t/T}}, \qquad T = \frac{\varepsilon t}{2E}.$$

And T has a meaning that owes nothing to capillarity. εt is the water the pore system holds per unit area of cloth; 2E is the rate the cloth’s two faces lose water. Their ratio is the time the room would take to dry those pores if the source were taken away.

So the front climbing a drying cloth approaches its ceiling on the time scale of the cloth drying. Neither the permeability nor the surface tension is in T; they are in h*, and they decide how high the ceiling is, but not how long it takes to arrive.

Early on it is Washburn’s front, and late on it is the ceiling #

The closed form contains both of the earlier answers as its two limits.

When t is much smaller than T the exponential is nearly linear and the front goes as the square root of time — Washburn’s law, which the first essay used for the front’s early progress and which knows nothing about drying because at the start there has not been time to dry anything. When t is much larger than T the front is at h* and stays there, which is the drying balance.

The two limits are joined by an identity that is worth stating by itself, because it is exact:

$$h^{*2} = D\,T,$$

where D is Washburn’s coefficient for the same pore system. The ceiling is exactly the distance Washburn’s front would travel in one drying time. That is why the ceiling falls as the root of the evaporation rate: a faster room shortens the drying time, and a front moving as the square root of time travels the root of the time.

The drying time is half an hour #

For the model sheeting — a 0.38-millimetre strip whose fine pores are the voids between fibres in yarns covering 73 per cent of its plan — the fine system holds 111 grams of water per square metre. An ordinary room at 100 grams per square metre per hour from each face takes it away at 200, so the drying time is 33 minutes.

The closed form then says nine tenths of the ceiling arrives at 1.66 drying times, 55 minutes, and the full integration with gravity says 53 — gravity slows the front’s approach to a lower ceiling slightly less than it lowers the ceiling, so the real front arrives a little early.

That answers the question the steady balance left open. Washburn’s front in the same slab, with the drying ignored, would pass half a metre at thirty-one minutes; the drying slows the last part of the climb, and still the front is nine tenths of the way up in fifty-three. It arrives within the hour, because the same evaporation that lowers the ceiling also shortens the time to it — and the ceiling and the time fall together, one as the root of the other.

A sealed tube is the limit where the clock stops #

The closed form also says what the sealed tube of the first essay was. With no evaporation the drying time is infinite — nothing ever dries — and the front’s only brake is gravity, which bites only as the front nears Jurin’s height. That approach is slow for a reason of its own: the suction left to drive the front is the small difference between the capillary pull and the weight of a column several metres tall, and the flow it drives has to cross all of those metres.

Integrated with no loss at all, the fine system reaches nine tenths of its 6.4-metre ceiling after 9.8 days. Open the room to the stillest air in the table, 20 grams per square metre per hour, and the ceiling drops to 1.06 metres and arrives in four hours. The loss does not merely lower the ceiling; it replaces gravity’s clock, which runs in days, with the room’s, which runs in hours.

That is the sense in which the sealed tube is a limit rather than a case. Every real cloth in every real room is on the drying clock, and the sealed number is what that clock reads when it is stopped.

The coarse system does not care what room it is in #

The holes between the yarns behave completely differently, and the figure shows why at a glance.

The coarse system’s ceiling is Jurin’s height, set by gravity against a suction a forty-sixth of the fine system’s, and it gets there in seventeen seconds because its permeability is a thousand times larger. Its drying time is half an hour too — it holds 103 grams of water per square metre in its holes — but that is irrelevant: a front that stops in seventeen seconds has lost almost nothing to the air, and the evaporation lowers its ceiling by under one per cent in an ordinary room.

So the two systems are governed by two different clocks. The coarse system runs on gravity’s clock and the fine one on the room’s. The strip test that records both is recording two processes whose time scales differ by a factor of two hundred in an ordinary room and more in a still one.

What a thirty-minute strip test reads #

The standard wicking test hangs a strip for a stated time and reads the height. How high a cloth wicks concluded that a thirty-minute reading is a Washburn number rather than an equilibrium one, since the equilibrium in a sealed tube is five days away.

That conclusion holds in still air and in nothing else.

In still air at 20 grams per square metre per hour the drying time is nearly three hours, the test reads 43 per cent of the ceiling, and it is indeed reading a front on its way up. In an ordinary room it reads 78 per cent of the ceiling, and above about 400 grams per square metre per hour it reads the ceiling itself, which is no longer a property of the fibres’ capillarity alone but of the room.

So the same test on the same cloth is measuring a different physical quantity depending on the draught in the laboratory. A standard that specifies the time and not the evaporation has specified half of what its number depends on.

A shorter test moves the boundary, not the problem #

The obvious repair is to shorten the test until it always reads the Washburn front. It does not work, and the reason is that the drying time is the only clock.

At ten minutes the ordinary room reads 52 per cent of the ceiling; a room at 800 grams per square metre per hour still reads 95. Whatever the test’s length, some room turns it into a ceiling measurement, namely any room whose drying time is shorter than the test. A test that wanted to read Washburn’s front in every room would need a sealed chamber, which is exactly what a garment is not in.

The pore water is the cloth’s, and the room is the wearer’s #

The drying time has two factors and they belong to different people.

The pore water per unit area is the cloth’s. It is the voids in the yarn times the yarn’s share of the cloth times the thickness — a construction property, fixed at the loom and the spinning frame. A yarn’s voids are not enough found that water swells a thread by more than its own air can take up, so a wetted cloth’s pores are not its dry geometry’s; the pore water here is the dry geometry’s, and a swollen cloth’s drying time moves with its swelling in a way this model leaves out. A cotton’s own water is a twentieth of what a cloth holds found that the geometry of a cloth decides how much water it can hold far more than its fibre does, and the same geometry decides this.

The evaporation rate is the wearer’s. It is the air movement, the humidity and the skin’s temperature, and it changes by a factor of a hundred between a still room and a run.

So the front’s arrival time is a product of a construction property and a use condition, and a cloth designed to move water quickly is one with little pore water to fill — a thin cloth of open yarns — worn where the air moves. That is the opposite of the design rule the steady height gives, which rewards a thick cloth: the thickness is the lever that raises the ceiling, and it lengthens the time to reach it in exact proportion.

Thickness buys height at the price of time #

The trade-off is exact, and it is the essay’s most practical number.

The ceiling goes as the root of the thickness, because the flow up the strip is proportional to its thickness while the loss is not. The drying time goes as the thickness itself, because the pore water does. So doubling a cloth’s thickness raises its ceiling by 41 per cent and doubles the time to reach it — and at a fixed time, well before the ceiling, the thick cloth and the thin one are at exactly the same height, because both are still on Washburn’s front and Washburn’s front does not know the thickness.

A thick cloth is therefore better only for someone prepared to wait. For anyone reading the front in the first drying time, the thickness is invisible.

Why the fine system is the one that matters for comfort #

Of the two systems, the coarse one does most of a garment’s visible wetting: a splash spreads through the holes between the yarns in seconds. But the coarse system cannot lift sweat off skin into a cloth held above it, because a fourteen-centimetre ceiling in a hole a fifth of a millimetre across is set by gravity against a weak suction, and the holes empty as soon as the cloth is lifted.

The pore that wicks is the pore that leaks found that a rise is set by the finest pore and a leak by the coarsest, so it is the fine system that does the lifting, and this essay adds its clock. The fine system reaches its ceiling in 1.66 drying times, so a garment moves sweat up to its steady height in about the time it would take to dry — which is a statement a wearer can test with a stopwatch and a damp shirt, and which contains no fibre property at all.

The model named #

The strip is a porous slab of thickness 0.38 millimetres with the model sheeting’s two pore systems: the holes between yarns, as Poiseuille tubes of the hydraulic radius four threads bound, and the voids between fibres, as a Kozeny–Carman bed at a packing of 0.6 with fibres 14 micrometres across. The permeabilities are the ones the steady balance used, so the ceilings here are its ceilings. The storage is each system’s own pore volume by the same convention its permeability is weighted by: the yarn’s voids over the plan area the yarn covers, through the whole thickness, and the open area through the whole thickness. The front is sharp, the flow behind it is Darcy’s, and the loss is uniform over both faces of the wetted length.

The equation is integrated from Washburn’s early solution to 99.95 per cent of the ceiling with a fourth-order step proportional to the front’s own time scale, so seconds and days are resolved alike. Two things are required of it, not shown: with gravity switched off, the integrated front must arrive at a half, nine tenths and ninety-nine hundredths of its ceiling within two parts in a thousand of when the closed form says; and the ceiling it arrives at must be the steady balance’s height exactly.

What was counted #

Two pore systems in nine rooms, from 10 to 3,000 grams per square metre per hour from each face — eighteen fronts, each integrated twice, with and without gravity, and read at half, nine tenths and ninety-nine hundredths of its ceiling. The strip test was read at ten and thirty minutes in each room.

What the model cannot show #

The front is not sharp, and it is not straight. Wicking is slower along a crimped thread, so a front in a real cloth runs faster along the less crimped system and is an ellipse rather than a line; and water in a real yarn advances as a ragged front with a wet tail behind it, fuller near the source and emptier near the top, and the storage term here treats every wetted millimetre as full. A partly filled tail holds less, so the real drying time is shorter than this one and the real front arrives sooner; the direction is certain and the size is not.

The evaporation is uniform. A strip dries faster at its top edge, where air reaches it from above as well, and faster where the front has just arrived and the surface is wettest. The rate here is one number per room, taken from the same bracket the steady balance used.

The two systems are separate. In a real cloth the holes between the yarns feed the yarns and the yarns feed the holes, and a front in one system is not independent of the other. The two clocks are real; whether they couple strongly enough to shift either is not computed.

And the contact angle is zero. Nothing here knows whether a fibre is scoured or finished, and the hairs decide the sign of the wetting on a raised cloth. At any other angle the suction falls as its cosine, the ceiling falls with it, and the drying time — which has no suction in it — does not move at all.

Who found it, and when #

Washburn’s law is from 1921 and Darcy’s from 1856; wicking with evaporation has been studied in porous media and in paper, where a strip’s steady height against evaporation is a known result and a front approaching it has been solved numerically many times.

What is done here is to read the closed form’s time constant as a quantity that belongs to the cloth and the room separately, and to find that it is the drying time. That the ceiling is exactly Washburn’s distance in one drying time, and that a strip test’s reading changes meaning at the point where the test’s length passes the cloth’s drying time, both follow in a line once the storage term is in.

Still open: what happens when the water runs out #

Everything above assumes the foot of the strip stands in an unlimited reservoir. Sweat is not one: a patch of skin supplies a small volume at a rate, and a garment wicking it is drawing a finite supply up a drying cloth.

The equation needs only a different boundary condition at the bottom — a flux instead of a pressure — and it should give a different answer in kind. With a limited supply, the front rises, stalls where the supply equals the whole wetted strip’s loss, and then retreats as the supply fails, and the height the cloth reaches is set by the supply rate rather than the suction. Whether a garment’s fine system ever reaches its drying-limited ceiling on a sweating body, or is always supply-limited below it, is the question that would turn these ceilings into a statement about comfort.