A sweating cloth wicks as high as the room can dry it

Every strip of cloth in the essays on wicking has stood with its foot in a dish. How high a cloth wicks found two pore systems and two heights; a drying cloth cannot lift what a sealed tube can brought the finer one down from six metres to half a metre by letting the cloth dry while it lifted; and a wick reaches its ceiling in the time its cloth takes to dry found how long that half metre takes to arrive.

The dish is the assumption all three share, and it is the one a garment never meets. The water a shirt wicks comes out of skin, a few grams a square metre an hour at rest and a few hundred on a run, and the skin does not hold a pool of it at the cloth’s foot waiting to be drawn. It delivers a rate. A strip fed at a rate cannot take more than arrives, however hard its pores pull.

That one change at the bottom boundary changes the answer in kind, and most of what the three earlier essays found about the cloth stops mattering.

A supply instead of a reservoir #

The strip is the one the earlier essays used: a sheeting-weight cloth 0.38 millimetres thick, its fine pores the spaces between fibres inside its yarns, water moving through them by Darcy’s law behind a sharp front and leaving through both faces at a rate $E$ set by the room. With its foot in a dish, the foot sits at atmospheric pressure and the strip takes whatever its suction can drive up the wetted length.

Fed instead at a fixed rate $Q$ per metre of width, the foot’s pressure is no longer fixed; it is whatever it has to be to pass $Q$. The balance on the front is the same as before, with the supply written in:

$$\varepsilon t\,\frac{dh}{dt} = Q - 2Eh,$$

where $\varepsilon t$ is the water the fine pores hold per square metre of cloth — 111 grams for this sheeting, a construction property rather than a fibre one, as a cotton’s own water is a twentieth of what a cloth holds found for the whole of a cloth’s water — and $2Eh$ is what the two faces of the wetted length lose. The permeability, the pore size and the surface tension have all gone. They decided how much water a front could be sent; with the supply fixed, they no longer decide anything, provided only that the pores can pass what arrives.

The height has no cloth in it #

Set the left side to nothing and the steady height is

$$h_Q = \frac{Q}{2E}.$$

The front stops where the wetted strip loses exactly what the skin supplies. A run at 500 grams of sweat per square metre per hour, fed into the strip from a contact patch fifty millimetres tall, is a supply of 25 grams per metre of width per hour; in an ordinary room losing 100 grams per square metre per hour from each face, the front stands at 125 millimetres. Light work stands at 37.5, rest at under four, hard work at 300.

None of those numbers knows what the cloth is made of. A cotton sheeting, a polyester jersey and a paper towel of the same supply in the same room reach the same height, because the height is a statement about water in and water out and the cloth is only the place the accounting happens. What distinguishes one cloth from another is how much it can carry before the accounting breaks down.

The capacity is where the dish comes back #

The balance holds only while the foot is drawing — while the pressure there is below atmospheric, so the strip is pulling sweat off the skin rather than needing it pushed in. At a steady front that pressure is

$$p_{\text{foot}} = -P + \rho g h + \frac{\mu E}{K t}\,h^2,$$

and it rises steadily with the height the supply holds the front at. It reaches atmospheric at exactly one height, which is the drying ceiling of the dish-fed strip — 501 millimetres in an ordinary room — because at that height the strip is doing precisely what it did with its foot in water.

So the dish case is the top of the supply case rather than a different problem. The strip’s capacity is $2E$ times its drying ceiling, 100 grams per metre of width per hour in an ordinary room: every millimetre of the ceiling is being fed at the rate its two faces lose. Below that supply the height is $Q/2E$ and the cloth is irrelevant; above it the height is the drying ceiling, the cloth is everything, and the surplus has nowhere to go but out of the fine system.

A fed front climbs in a straight line #

The transient has a closed form below capacity, and it is not Washburn’s:

$$h = h_Q\left(1 - e^{-t/T}\right), \qquad T = \frac{\varepsilon t}{2E}.$$

At the start the front climbs at a constant speed, the supply divided by the pore water per unit area — for a run’s sweat, 225 millimetres an hour. A front fed from a dish starts at the opposite extreme, racing up as the square root of time because a short wet column offers almost no resistance. A fed front is never in that regime, because the supply rather than the resistance is what limits it: the pores could take the water faster and are not offered it.

And the time constant is the one the dish-fed front had. $T$ is the pore water over the rate the faces lose it, the time the room would take to dry the cloth — 33 minutes here. The approach is slower in its shape, though: an exponential takes 2.30 drying times to reach nine tenths of its height, where the dish-fed front took 1.66. A run’s front is at 90 per cent of its 125 millimetres after 77 minutes.

When the sweating stops #

Take the supply away and the equation has nothing on its right but the loss. The wetted length falls as $e^{-t/T}$, on the same clock, halving every 23 minutes.

So one time governs all three stages — the climb to a supply-limited height, the approach to a dish-fed ceiling, and the retreat after the supply fails — and it is the one that owes nothing to capillarity. A garment’s wet region follows a sweating body with a lag of about half an hour in an ordinary room, a few minutes in a wind, and a couple of hours in still air, and the lag is set by how much water the cloth holds and how fast the air takes it, not by how well its fibres wick.

That is a direct reading of the question the essay on arrival times left: a garment on a sweating body is always supply-limited below its drying ceiling unless the sweating is hard, the contact broad or the air still, and in the supply-limited regime the ceiling is simply not the number that matters.

The two pore systems do not share the sweat #

A cloth has two pore systems, and in a dish both drink. The holes between the yarns rise to 137 millimetres in seventeen seconds while the spaces between the fibres climb slowly towards their half metre.

Fed at a rate, they do not both drink. The water enters wherever the suction is strongest, and the fine system’s suction is 62 kilopascals against the coarse holes’ 1.3 — forty-six times stronger. Everything that arrives goes into the fibres. The coarse holes can hold water only where the pressure in the wet strip is above minus their own suction, and a supply-limited strip is under suction everywhere: at a run’s supply the foot sits at 58 kilopascals below atmospheric, far past anything a hole a fifth of a millimetre across can hold against.

The holes flood in the last per cent #

The suction at the foot eases as the supply rises, but slowly, because it has the whole fine-pore pressure to spend. It crosses the coarse holes’ threshold at 98.9 per cent of capacity in an ordinary room, and at 98.8 to 98.9 per cent in every room from still air to a gale. The number is nearly a constant of the cloth, because it is fixed mostly by the ratio of the two suctions.

So the holes between the yarns go from bone dry to filling over the last one per cent of the supply range. Below it the cloth carries every gram of sweat inside its yarns and its holes stay open — its air still passes, and half of it goes through a tenth of the holes, all of which are clear. Past it the surplus has nowhere to go but into those holes, which fill to their own ceiling and then shed onto the skin or down the cloth — and a hole full of water passes no air, so the cloth’s breathing goes with them, from the largest holes down, the order in which a cloth stops having holes before it stops passing air found them carrying it.

The spreading wet patch a splash makes is therefore a threshold on a sweating body, not a gradient. It appears when the supply passes what the yarns can carry to a drying front, and not before. That is the one surprising connection this calculation makes, and it turns a familiar sight into a measurement: a shirt that shows a spreading dark patch is a shirt being fed faster than its fine pores’ capacity, and the size of the patch is the surplus rather than the sweat.

What floods a shirt #

The capacity is a supply per metre of width, and turning it into a sweat rate needs a contact patch — the height of skin feeding the cloth above it. At fifty millimetres, an ordinary room floods the holes at 1,980 grams per square metre per hour; at a hundred millimetres, 990; at two hundred, 495, just under a run.

The room moves it the other way from what intuition expects. The capacity is $2E$ times a ceiling that falls as the root of $E$, so it rises as roughly the root of the evaporation. Still air floods a cloth first: at twenty grams per square metre per hour a hundred-millimetre contact floods at 421, below a run, while a breeze at four hundred needs 2,021. The same run in the same shirt stays inside the yarns on a windy day and soaks through them on a still one — not because the wind dries the patch after it has formed, but because a drying cloth can carry more water to its own faces.

Thickness buys capacity and costs nothing in height #

In the dish case, thickness bought height at the price of time: a thicker cloth reached a higher ceiling more slowly, and at any early moment the two were level.

Fed at a rate, the two statements come apart. The height is $Q/2E$ and has no thickness in it, so a thin cloth and a thick one of the same fibres end at the same place. The drying time is proportional to the thickness, so the thick one gets there more slowly. And the initial speed is the supply divided by the pore water, so the thin cloth is ahead at every moment, not merely level early on.

What the thickness does buy is capacity. The strip’s delivery grows with its cross-section, and its capacity nearly as the root of its thickness: 71 grams per metre of width per hour at 0.19 millimetres, 100 at 0.38, 139 at 0.76. A thick cloth floods later and wets sooner at nothing; a thin one spreads sweat faster and gives up sooner. Neither is better at wicking, because below the flood neither is doing anything the room did not decide.

The finer fibre floods first #

The fibre enters only through the capacity, and there it runs against every other wicking result on this account.

Finer fibres make finer pores between them. The suction rises as one over the pore’s size, which is why a sealed tube of 5-micrometre fibres would lift water 17.8 metres and one of 35-micrometre fibres only 2.5: the fine-fibre cloth wicks higher by every sealed-tube measure. But its permeability falls as the square of the pore’s size, and the capacity goes as the root of suction times permeability. The fine-fibre cloth carries less.

At 5 micrometres the capacity is 62 grams per metre of width per hour; at 35, 140. The ordering is exactly the reverse of the sealed-tube ranking. It is the same as the dish-fed drying ceiling’s, and necessarily so: the capacity is $2E$ times that ceiling, and the essay that brought the ceiling down found that it is the system that pulls hard and delivers little which a drying room starves. The fine fibre is that system in its extreme form.

On a sweating body below capacity, every one of these cloths reads the same height; at capacity, the fine-fibre cloth is the one that floods. The packing works the same way through the permeability: loosen the yarn from 0.6 to 0.45 and the capacity rises from 100 to 147; tighten it to 0.75 and it falls to 58.

Which wicking number ranks for a body #

Two numbers are quoted for how well a cloth wicks, and they rank fibres in opposite orders.

The first is the rise, the height a sealed column of the pores would reach, which is suction against gravity and nothing else; the pore that wicks is the pore that leaks found the finest pores setting it. It is the number behind every claim that a finer fibre wicks better, and on a sweating body it ranks cloths backwards: the fine fibre with the great rise is the one that floods first.

The second is a timed strip test, a strip stood in a dish and read after ten or thirty minutes. Early on its front is Washburn’s, whose distance grows as the root of suction times permeability over the pore water — and the capacity is the root of suction times permeability times thickness. For a change of fibre, which moves neither the pore water nor the thickness, the two are the same combination. The routine test ranks fibres in exactly the order a body floods them, for a reason nobody designed it around.

It does not rank thicknesses that way. A Washburn front has no thickness in it and a capacity grows with the thickness, so a strip test calls a thin cloth and a thick one equal where a body finds the thick one carries four-tenths more before its holes fill. And below the flood no test ranks anything, because a body’s cloths all stand at $Q/2E$. The measurement that would rank for a body directly is the fed strip — a known supply at the foot, a known room, and the supply at which the first hole fills.

The model named #

The strip is the drying strip of the earlier essays: a porous slab 0.38 millimetres thick with the model sheeting’s two pore systems, the fine one a Kozeny–Carman bed of 14-micrometre fibres at a packing of 0.6, weighted by the plan area the yarns cover, and the coarse one the holes between four threads as Poiseuille tubes. The front is sharp, the flow behind it is Darcy’s, and the loss is uniform over both faces of the wetted length. The supply enters at the foot at a fixed rate per metre of width, and the strip takes the lesser of that rate and what its suction delivers with the foot at atmospheric pressure; the surplus, when there is one, is counted and not followed.

The sweat rates are a bracket rather than data: 15, 150, 500 and 1,200 grams per square metre per hour for rest, light work, a run and hard work, round figures for whole-body sweating spread over an adult’s surface. A patch of skin can sweat several times its body’s average, and the patch height is a stand-in for how much of that skin a garment touches; both are stated rather than measured.

What was counted #

Four rooms and five supplies, each integrated for two hours of sweating and three of drying; the supply-limited fronts were required to match the closed form within two parts in a thousand at every step and to have a foot below atmospheric pressure at every step. Capacities and flood shares in nine rooms from 10 to 3,000 grams per square metre per hour; the flood rate for three patch heights in each; three thicknesses and seven fibre diameters in an ordinary room; three packings. The flood share, 98.8 to 98.9 per cent, is the one number that barely moved across all of it.

What the picture cannot show #

The supply is not at the foot of a hanging strip. A garment touches skin over an area and is fed across it, not at one edge, and the cloth above the contact is not a strip hanging in air: it is draped, folded and pressed, and its inner face is against a layer of humid air rather than the room. The contact-patch height is a single number standing in for all of that. What survives is the form of the answer — a height set by supply and loss, and a capacity above which the holes fill — not the millimetres.

A yarn full of water is not invisible. Wetting the fibres changes how a cloth scatters light even with its holes empty, so a cloth carrying sweat inside its yarns darkens somewhat. How much, against the much larger change of water standing in the holes, is an optical question this account does not compute; the claim is about where the water is, and what shows is inferred from it.

The front is sharp and the two systems are separate, as in every essay before this one. A real front has a partly filled tail, which would shorten the drying time, and it runs faster along whichever thread system is less crimped, because wicking is slower along a crimped thread — which moves the capacity, since that goes with the permeability, and leaves the supply-limited height exactly where it was, since that has no permeability in it; and the coarse holes near the flood do not fill as a switch but as a gradient along the strip, beginning at the foot, where the suction is weakest. The share at which the first hole fills is computed; how fast the flooded length grows past it is not.

The pores are the dry cloth’s. Water swells a thread, and a yarn’s voids are not enough to take the swelling up, so a wet yarn’s fine pores are narrower than the dry geometry says. That raises the suction and lowers the permeability together, which is the fine-fibre direction: the capacity falls, and a swollen cotton floods sooner than its dry numbers suggest. The contact angle is zero, and on a raised or finished cloth the hairs decide the sign of the wetting; a worse-wetting fibre lowers the suction and the capacity with it, and leaves $Q/2E$ untouched for as long as the fibre wets at all.

Who found it, and when #

A strip fed at its root and losing along its length is the book-keeping of a cooling fin, and a wetted length balancing a supply against evaporation is a result that paper and soil physics have used in many forms. Washburn’s law is from 1921 and Darcy’s from 1856; the sweat rates are exercise physiology’s round figures.

What is new here is the reading. The steady height’s independence of every cloth property is one line once the boundary is written down; the finding that the foot’s suction falls to the coarse holes’ threshold only within the last per cent of capacity, in every room, is what makes a sweat patch a threshold rather than a gradient; and the reversal of the fibre ranking between a sealed tube and a fed front is a consequence nobody reading a wicking height would expect.

Still open: whether a garment is ever fed from its foot #

Everything here feeds the cloth from one edge. A shirt on a back is fed from its whole inner face, and a cloth fed across its area and losing from its outer face is a different geometry: the supply and the loss are in the same place, the strip’s length plays no part, and the question becomes whether the fine system can carry sweat through the thickness faster than the skin supplies it, which is a thickness-wise capacity rather than a length-wise one.

That capacity should be much larger, because the path is a fraction of a millimetre rather than tens of millimetres, and if it is, a cloth pressed flat against skin never floods — its holes fill only where it lifts away and the sweat must travel along it to find a drying face. Whether a flood starts at the places a garment touches or the places it does not is the question that would say where on a body the first wet patch forms.