A spun cloth dries by its weight

A plain cotton cloth of 150 grams a square metre can be woven from a 20-tex yarn set close or a 200-tex yarn set open, and a weight fixes the fibre and not the drape followed that family from one end to the other. Every member contains exactly the same mass of fibre, which is what a fabric’s weight is a sum of. The count only decides how it is arranged, and the arrangement is not a small matter: the coarse cloth is 1.06 millimetres thick and the fine one 0.33, and the cover falls from 0.54 to 0.19 as the thickness triples.

That essay closed on a question it said was worth computing before anybody believed an answer. A thicker cloth has more room in it for water and a longer way for the water to come out, so the thick end of the line ought to dry more slowly. But the volume a weight fixes is also the volume a finish has to wet and the room has to dry, and if drying belongs to the fibre rather than to the room between the fibres, then drying time should belong to the weight and not to the count — the opposite of what the thickness says.

Both are right, and which one is right depends on one thing done to the cloth after washing.

Laid flat, the thickness decides #

A cotton’s own water is a twentieth of what a cloth holds counted the three places a wet cloth keeps its water — inside the fibre, in the channels between the fibres of each yarn, and in the holes four yarns bound — and the spin essay corrected the first of them: a soaked cotton fibre holds the volume it swelled by, 30 per cent of its dry weight, not its 8.5 per cent regain.

Run those three places along the weight line, with every pore full and every fibre a fifth wider than it was dry, and two of them do not move at all. The fibre’s water is 30.1 per cent of the dry weight at every count, because it is a property of the fibre and every cloth on the line has the same fibre. The channels hold 63.9 per cent at every count, because a yarn packed at 0.6 is three parts of fibre to two of channel whatever its count — and a yarn’s voids are not enough to absorb the swelling, so the yarn grows and its channels with it — and the line’s yarns all carry the same total fibre.

The holes are different. They are the coarser of the two pore systems every woven cloth has, and a coarse cloth of one weight is set open and lies thick, so its holes are both wider and deeper. At 20 tex the holes of the swollen cloth hold 34 per cent of the dry weight; at 200 tex they hold 497 per cent. Laid out flat after a hand wash, the coarsest 150-gram cloth carries 5.9 times its own weight in water and the finest 1.3 times, and the whole of the difference is in the holes.

At an evaporation of a hundred grams a square metre an hour from each face — an ordinary room in the bracket a wick reaches its ceiling in the time its cloth takes to dry works across, five times still air and a quarter of a breeze — the fine cloth laid flat dries in just under an hour and the coarse one in four and a half. That is the thickness line’s prediction, and for a cloth laid flat it holds: the thick cloth is the slow one, by its count.

Hung, the holes drain and the channels do not #

Hang the same cloths by an edge and gravity is a head. The spin essay found what it does to a sheeting: the holes drain everywhere above a band at the hem as tall as the head their radius holds, and the channels, whose head is over five metres, drain nowhere on any cloth a person hangs up.

Along the weight line the band’s height follows the holes. A fine 150-gram cloth, whose holes are small, keeps a wet hem 267 millimetres tall; the coarsest keeps 14. The coarse cloth’s holes hold much more water each but give it up much more easily, and the two nearly cancel: hung for a metre, every count on the line holds between 101 and 104 per cent of its dry weight. The thickness that decided the flat cloth’s drying has gone, and what is left is the fibre and the channels — 94 per cent — and a little hem.

So a line does most of what the count was doing. It does not do the rest, because the channels stay full, and the channels are most of what a hung cloth holds; from there a drying cloth’s own lift moves the water to the faces the air is taking it from.

Spun, every count keeps the same share #

A spin at 1,400 rpm empties the holes everywhere but a sliver against the drum’s wall and empties the channels everywhere beyond a kept depth. What it leaves in a full load of each cloth on the line is 40.9 per cent of the dry weight, the finest and the coarsest within a tenth of a point of each other: 30.1 in the fibre, 10.5 in the channels kept full near the wall, and between 0.2 and 0.3 in holes the drum could not empty.

That is the result the weight line’s surprise predicted. A spun cloth of one fibre and one weight carries the same water into the drying whatever its count and whatever its thickness, so it dries in the same time: a third of an hour at a hundred grams an hour from each face, an hour and a half in still air at twenty, for every member of the family.

And the time goes with the weight itself. The share is the same at every weight as well as every count — 41.0 per cent at 100 grams, 40.8 at 250 — so a spun 250-gram cloth carries two and a half times the water of a spun 100-gram one and takes 2.49 times as long to dry. Laid flat, the same two weights overlap so thoroughly, count by count, that a light coarse cloth outlasts a heavy fine one.

That the share holds across counts and across weights is no surprise once the reservoirs are counted: the fibre’s water and the channels’ are both proportional to the mass of fibre, and the holes, which are not, are gone. What is a surprise is the channels. The channels a spin leaves full ought to depend on how the yarn is built, because they are the yarn’s own pores and a spin only empties some of them — and they turn out not to.

A spin cancels the yarn’s packing #

A yarn packed loosely has more channel and a yarn packed tightly less: at a packing of a half the channels hold one volume of water for every volume of fibre, at seven tenths under half of one. That is a large difference between an open-end yarn and a compact one, and a hung cloth shows all of it. A 150-gram, 40-tex cloth of the loose yarn hangs at 136 per cent of its weight and one of the tight yarn at 81.

Spun, the two keep 40.94 per cent each, to the fourth figure.

The reason is a cancellation inside the drum and it is exact rather than approximate. A loose yarn has more channel water, but its channels are coarser, so they hold their water with less suction and the spin empties them further into the load. Both factors go as the share of the yarn that is channel over the share that is fibre — one up, one down — and they meet.

What a spin leaves in a yarn is its fibres’ surface #

The cancellation has a cleaner statement, and it changes what the question is about.

A drum’s suction at a depth $z$ from its wall is $\rho\omega^2(Rz - z^2/2)$, the spinning column of water between that depth and the wall. A load lying evenly against the wall has its mass spread in proportion to the radius, so the share of the load within a depth $z$ is $(Rz - z^2/2)/(RD - D^2/2)$ for a load $D$ deep — the same expression. At the kept depth, the suction is the channels’ entry pressure. So the share of the load whose channels stay full is the entry pressure divided by the suction at the load’s inner face, with nothing approximated.

The entry pressure is $2\gamma/r$ for a channel whose hydraulic radius is $r$, and a hydraulic radius is a pore’s volume over the surface that walls it. Multiply the channels’ volume by their entry pressure and the volume leaves:

$$\text{kept channel water} = \frac{2\gamma\,S}{\rho\,\omega^2\,(RD - D^2/2)},$$

where $S$ is the swollen fibres’ surface in a square metre of cloth. What a spin leaves in a yarn is the fibres’ wetted surface times the surface tension, over the drum’s suction. Neither the count nor the packing nor the thickness is in it. The weight is, through the fibre it fixes: a 150-gram cotton of 14-micrometre fibres has 34 square metres of swollen fibre surface in every square metre of cloth, and that is what holds its 10.5 per cent.

So the fibre’s fineness, which no other part of the water arithmetic cared about, is now the second thing after its swelling. A fibre half as wide has twice the surface in the same mass, and keeps twice the channel water: a cotton of 7-micrometre fibres would keep 21.1 per cent in its channels against 10.5, and 51 per cent all told against 41. A polyester swells by almost nothing, so its channels are nearly all it keeps: 10.0 per cent at 14 micrometres, which is why it comes out of the drum feeling nearly dry, and 27.2 for a polyester microfibre of 5 micrometres, close to three times as much. A microfibre cloth’s reputation for holding water is this arithmetic’s prediction rather than a property of the polymer. Both polyester numbers are taken at a contact angle of nought, which the hairs decide the sign of the wetting shows a polyester does not have; the angle enters the identity as its cosine, so a real polyester keeps less, by the same factor at every fineness.

The thickness line was the wrong guide #

A weight fixes the fibre and not the drape found that a cloth’s bending length, at the free bound of yarn stiffness, has neither the count nor the weight in it: only the fibre’s modulus, diameter and density, because a weight fixes how many fibres cross a unit width and each fibre bends for itself. This is the same fact arriving from the water. A weight fixes how much fibre surface a square metre has, and the surface is what a spin leaves water on.

The thickness, the cover and the yarn’s packing are all properties of the arrangement. Each of them decides how much water a cloth can hold, and none of them decides how much it keeps, once something has drained it. A thickness gauge, which reads the arrangement, is therefore a good guide to a cloth laid out to dry from soaking and a poor one to a cloth taken from a machine.

A towel is a head made of cloth #

There is a practical step between the two, used on the very garments that are laid flat because a spin or a line would stretch them: roll the wet piece in a dry towel and press.

A dry towel’s channels are fine, empty and pull with the channels’ entry suction, some fifty kilopascals; a wet cloth’s holes hold with under three kilopascals on the finest cloth of the line and under two hundred pascals on the coarsest. So every hole the towel is pressed against empties into it, until the towel’s own channels and fibres are full. A dry 500-gram cotton towel can take 470 grams a square metre that way.

On the weight line, one towel layer takes all the hole water of every count up to 100 tex, and leaves those cloths at 94 per cent of their weight — the fibre and the channels, the state a line reaches, with no hem. The coarsest cloth’s holes hold 745 grams and need a second towel. So the roll is the line without the hanging, and it removes the count from the drying exactly as far as hanging does. What it cannot do, any more than a line can, is reach the channels: for that the head has to be a drum’s.

How the numbers were produced #

Each cloth on the line is a balanced plain weave whose sett and crimp are solved together so that it weighs exactly 150 grams a square metre, with yarn diameters from the counts at a packing of 0.6; the same cloths the weight line draws. Its water is the soaked reading of the spin essay: the fibre’s swelling volume as water, the channels at constant packing scaled by the same volume factor, the holes at the wetted cover’s open area times a swollen thickness. Each pore’s entry pressure is $2\gamma\cos\theta/r$ at a contact angle of nought.

Laid flat, nothing drains. Hung for a metre, the holes keep their water below the hole head and the channels, with a head over five metres, keep all of theirs. Spun, the drum is 250 millimetres in radius with a 68-millimetre full load, and each pore keeps its water within its kept depth of the wall. The drying time is the constant-rate period: a wet surface loses water at the rate the room sets, from both faces, so $W$ grams a square metre leave in $W/2E$ hours.

The calculation is required to put every cloth on the line at the weight the line draws to a part in a billion; to give the fibre and the channels one share at every count; to keep every spun share within half a per cent of every other and every hung share within four, while the flat shares spread more than fourfold and rise with the thickness; to reproduce the closed form above to a part in a billion at three fibre widths and three counts; to leave twice the channel water for a fibre half as wide; to cancel the packing when spun and not when hung; and to refuse a drying state it does not have, a room that does not dry, a count that cannot make the weight, and a towel with no weight or no layer.

What the arithmetic leaves out #

The last stage of drying. The constant-rate period ends when the remaining water is inside the fibre, and from there the rate is set by how fast the cellulose gives up its water rather than by the room. That stage holds the same water in every cloth of one fibre and weight — the fibre’s water never varied — so to the extent its rate is the fibre’s it adds the same time to every member of the family. A thick cloth also gives its vapour a longer path out, which would lengthen its last stage a little; neither is computed.

A drained channel is taken as empty. Films on the fibres and rings at their contacts keep some water after a spin, and on the identity above that remnant is also a surface quantity. It would raise every spun share alike, and it has not been measured for a spun cotton.

One channel size and one hole size. Real yarns have a spread of channel radii, which rounds the kept depth into a gradient; the cancellation of the packing then holds on average rather than exactly.

An even load and a uniform yarn. A crumpled load has folds against the wall and air inside it. And a real spinning system does not pack every count alike: a fine yarn is often packed a little tighter, which a line would show and, on the identity, a spin would not.

The evaporation rate is the room’s. A cloth hung in still air dries from both faces at a rate the air sets, and a cloth laid flat on a towel dries mostly from one. The flat cloths’ times are therefore if anything too short, and the gap between flat and spun too small.

Who measured the parts #

The centrifuge as a measure of a porous body’s capillary pressure, and the way a spinning sample keeps its water at the outlet face, are Hassler and Brunner’s of 1945. The constant-rate and falling-rate periods of drying belong to the chemical engineering of drying from the 1920s on. The hydraulic radius as a pore’s volume over its wetted surface is Kozeny’s, from the same decade. Reading what a spin leaves in a yarn as its fibres’ surface, and the count, thickness and packing dropping out of it along a weight line, were worked out here.

Still open: what a drained channel keeps, read as a surface #

The spin essay ended on one measurement, and the identity here sharpens it. If the water a drained channel keeps as films and rings is a surface quantity — so many grams on every square metre of fibre — then it scales with the fibre’s fineness exactly as the kept channels do, and the two cannot be told apart by changing the count, the packing or the weight. They can be told apart by changing the drum: the kept channels go as one over the square of the spin speed, and a film does not depend on the speed at all.

So the experiment is the thin-layer spin at a series of speeds, run on two cloths of one weight and construction spun from fibres of two widths. On this arithmetic the water above the swelling floor falls as the inverse square of the speed towards a constant, and that constant, divided by the fibres’ surface, is the film’s thickness — the same for both fibre widths if the account is right.