A spin leaves the water a fibre swelled by

A load of washing comes out of a 1,400 rpm spin feeling damp rather than wet, and weighing about half as much again as it did dry. The spin has taken out more than half the water the load held when the drum stopped filling, and then stopped taking any more: a faster machine does a little better, and no machine does much better.

A cotton’s own water is a twentieth of what a cloth holds counted where a wet cloth keeps its water — inside the fibre, in the channels between the fibres of a yarn, and in the holes four threads bound — and predicted that the three would leave in that order under a rising head, as a staircase. It ended by naming the experiment: saturate a cloth, apply a series of heads, weigh it after each.

A washing machine runs that experiment every time it spins, and the answer it gets is not the one that essay would have predicted. The reason is a number the earlier count took from the wrong table.

The earlier count put the fibre’s water at its regain #

The three-reservoir account gave the fibre 8.5 per cent of its dry weight as water, which is cotton’s regain: the water a fibre takes up from air at sixty-five per cent humidity. It said at the time that a soaked fibre takes more, and that doubling the figure would still leave it the smallest of the three.

Doubling is not enough, and the correct figure is already in hand. What water does to a thread established that a cotton fibre in liquid water is a fifth wider and a hundredth longer. A fibre that has grown by that much across and along has grown in volume by

$$\frac{\Delta V}{V} = (1 + s_d)^2 (1 + s_a) - 1 = 1.2^2 \times 1.012 - 1 = 0.457,$$

and the thing that filled that volume is water. At a fibre density of 1.52 grams a cubic centimetre, 0.457 of the fibre’s volume in water is 30 per cent of its dry weight. That is the water a soaked fibre holds inside its own substance, and it is three and a half times the regain.

The regain is not wrong; it answers a different question, and it is the same fibre whose swelling makes a wet fibre stiffer by its fourth power. It is the fibre in equilibrium with humid air, where the water is taken up molecule by molecule from vapour. A fibre in a drum is in equilibrium with liquid water, and a cellulose fibre in liquid water keeps swelling long after it has passed its humid-air content. The swelling is the measurement of how far it went.

The swelling moves the other two reservoirs as well #

A fibre that swells also changes the pores around it, and both changes run through the arithmetic.

The yarn’s channels grow with it. A yarn’s voids are not enough showed that a cotton yarn cannot take its fibres’ swelling into its own air, so the yarn itself grows; if it grows at constant packing, its voids grow by the same factor as its fibres, and every channel widens by the fibre’s fifth. A 2.33 micrometre channel becomes 2.80, and the head it holds against falls from 6.37 metres to 5.31.

The cloth’s holes shrink. The threads grow into the hole between four threads at the spacings they were woven at, which is what a wet cloth is set closer than it was woven is about. A sheeting’s open area falls from 24.5 per cent to 15.5, its holes from 92 micrometres in hydraulic radius to 73, and the head they hold against rises from 16 centimetres to 20.

So a soaked sheeting holds 140 per cent of its dry weight rather than 113: 30 inside the fibre, 64 in the swollen yarn’s channels and 46 in the narrowed holes. The fibre’s share is not a twentieth but a fifth, and it is the one share nothing but evaporation can reach.

A thin layer drains as a staircase onto a floor #

The simplest drainage is a layer of cloth so thin that every pore in it sees the same head. A pore empties when the suction on it exceeds its entry pressure, $2\gamma\cos\theta/r$ — the same pressure that, read with the other sign, decides which pore of a repellent cloth leaks — so a layer loses the whole of one reservoir at one head and nothing more until the next.

Two things in that picture matter for everything below. The steps are a factor of twenty-six apart in head, so there is a wide range in which the holes are empty and the channels are full — the state of a cloth hung to drip. And the floor under the second step is the fibre’s own water, which is where the old reading and the soaked one disagree by a factor of three and a half.

The shaded band is what makes a spin interesting. At 1,400 rpm the head a spin applies to a full load rises from nothing at the wall to 32 metres at the load’s inner face, passing 5.4 metres ten millimetres in, so it straddles the second step. A spin works exactly where the channels give up their water, and that is why its result depends on everything that decides where in the load that step is crossed.

A spinning drum is a suction that grows inward #

A load pinned to the drum by the spin is a porous layer draining outward through the perforated wall. At the wall the water is at the pressure of the air outside the drum. A layer a depth $z$ inside the load is pulled outward by the column of water between it and the wall, spinning with it, which is

$$P(z) = \rho\,\omega^2 \!\int_{R-z}^{R} r\,dr = \rho\,\omega^2 \left(Rz - \tfrac{1}{2}z^2\right).$$

That is the suction a pore at depth $z$ has to hold against. It is zero at the wall at every speed and it rises inward almost linearly, because $z$ is small beside the drum’s radius.

Setting the suction equal to the channels’ entry pressure gives the depth within which they keep their water,

$$z^{*} = R - \sqrt{R^2 - \frac{2P_e}{\rho\,\omega^2}} \;\approx\; \frac{P_e}{\rho\,\omega^2 R}.$$

At 1,400 rpm in a 250 millimetre drum that is 9.9 millimetres. The holes, whose entry pressure is twenty-six times lower, keep their water only within 0.4 millimetres of the wall — about one cloth’s thickness.

The cloth against the drum never drains #

The consequence is the reverse of what a picture of a centrifuge suggests. The cloth pressed hardest against the drum is the cloth least drained, because the suction is measured from the outlet, and at the outlet it is zero.

This is familiar in a different trade. Petroleum laboratories measure how tightly a rock holds its oil by spinning a core in a centrifuge and weighing what comes out, a method Hassler and Brunner set out in 1945. The first thing the method has to correct for is the end effect: the face of the core against the outlet stays saturated at every speed, because the suction there is zero, and a naive reading of the weight lost attributes the retained water to the rock rather than to the geometry of the test. A washing machine is the same instrument with a cloth in it, and the layer against the drum is its end effect.

It is the layer that feels wet when the load is taken out. The inside of a knot of sheets can be almost dry to the touch while the fold that was flattened against the drum is still cold and heavy, and on this arithmetic that is not uneven spinning; it is the spin working exactly as the suction says it must.

What the spin leaves, against the speed #

Put the three reservoirs and the suction together. The fibre keeps its swollen water at every speed. The yarn’s channels keep theirs in the share of the load within $z^*$ of the wall, weighted by radius because an outer layer of an annulus is a larger layer. The holes keep theirs within their own, much smaller, depth.

Below about 550 rpm the whole load lies inside the kept depth, the holes have emptied, and the load keeps 94 per cent of its weight in water and more: the curve is nearly flat. Past that speed the kept depth falls through the load, the channels drain from the inside outward, and the curve drops steeply to 63 per cent at 800 rpm. After that each extra hundred revolutions buys less, because $z^*$ falls as the square of the speed and there is less of it left to lose.

The two readings differ by 25 points at 1,400 rpm, and the difference is the fibre. The regain reading predicts a load that comes out at 16 per cent — dry to the touch, almost ready to fold — and no domestic spin has ever delivered that. The soaked reading predicts 41.

Where the energy label draws its lines #

A number a household can check is on the machine. The European energy label grades a washing machine’s spin by the moisture a standard cotton load keeps after it: class A below 45 per cent of the dry weight, B from 45 to 54, C from 54 to 63. A machine sold as spinning at 1,400 rpm typically sits in class B, near half the load’s weight.

Against that, the regain reading is wrong by a factor of three and cannot be rescued: it would put every machine faster than about 650 rpm in class A, and a label whose best class every machine reached would not have been drawn. The soaked reading puts a 1,400 rpm spin at 41 per cent and a 1,000 rpm spin at 51, low by something like ten points at each speed, which is the size and the direction of the one thing the model leaves out. It takes a drained channel as empty. A real one keeps a film on every fibre and a ring of water at every contact between two fibres, and that water is a share of the drained channels rather than of the full ones — so the gap should widen a little as the speed rises and more channels drain, and never close.

The label’s class A boundary sits fifteen points above cotton’s floor. No speed reaches the floor, because the layer against the wall never drains; but a machine that could push the kept layer to nothing would still hand over a load holding 30 per cent of its weight in water, and that 30 per cent is a property of cellulose, not of any drum.

A floor, not a law of diminishing returns #

It would be easy to read the flattening of the curve as the ordinary shape of any engineering improvement — each doubling of effort buying half as much. That reading is wrong about the mechanism and wrong about what would change it.

The curve flattens because it is approaching a level, not because the effort is getting less effective per revolution. Between 1,400 and 1,800 rpm the kept depth falls from 9.9 to 5.9 millimetres, which is a large fractional improvement in the thing the speed acts on; it moves the residual only four points because by then most of what the speed can reach has already gone. Almost all of what is left is inside the fibre.

So the lever that would move a cotton load’s residual much further is not a faster drum. It is a fibre that swells less. Polyester and cotton in the same construction come out of the same spin some thirty points apart, and all but one point of that difference is on the floor.

Five fibres in one construction #

The channel water kept by the wall layer is nearly the same across all five fibres, because the construction is the same and the channels differ only by how much the fibres widened them. What separates the five is almost entirely inside the fibre. Viscose, a regenerated cellulose whose fibres swell by half again in area and more, keeps twice what cotton keeps; polyester, which does not swell at all, keeps only the wall layer’s channels.

That ordering is the domestic experience: a viscose dress comes out of the machine heavy and slow to dry, a polyester shirt comes out almost ready to wear, and cotton sits between. What the arithmetic adds is that the ordering is not about absorbency in the sense a towel is sold on. The three-reservoir count found that a polyester and a cotton cloth of one construction hold almost the same water when saturated. They differ by thirty points after a spin because the spin removes the water that is between the fibres, and leaves the water that is inside them.

The laboratory measures the same floor #

Fibre scientists have a test that is this floor measured on purpose. The water retention value of a fibre is the water left in a small tuft of it after centrifuging at a few thousand g — far beyond any drum, and on a sample thin enough to have no layer worth speaking of against the outlet. Reported values put cotton near half its dry weight, viscose near its whole dry weight, and polyester at a few per cent.

The swelling arithmetic gives 30, 59 and 0.1 for the same three. It is low throughout, and low for a known reason: a cotton fibre has a lumen, the collapsed tube down its centre, which holds water the swelling of the wall does not count, and the tuft’s own fibre contacts keep their rings of water however hard it is spun. But the ratio is right: the model has viscose keeping 1.96 times cotton’s water, and the laboratory has it keeping about twice. That ratio comes from nothing but two diameter-swelling measurements taken under a microscope, one on each fibre, and it predicts a centrifuge result that was never part of its input.

A half load comes out wetter #

The kept layer is a depth, fixed by the speed and the drum, and not a share. So a load that lies thinner on the wall has a larger share of itself inside that depth.

A half load comes out of the same spin nine points wetter than a full one, on this arithmetic. That runs against the intuition that a lighter load spins drier because the drum has less to carry, and it is a prediction rather than a report: a half load in a real drum may not lie as an even layer at all, and a load bunched to one side of the drum leaves some of the wall bare and some of the load deep. What the argument does fix is the direction of the effect on an evenly spread load, and why a machine’s spin rating is always quoted at a stated load.

The speed is the wrong specification #

A spin is sold by its speed, and two machines at one speed do not apply one suction.

The kept share is $z^*$ over the load’s depth $D$, and to first order $z^*$ is the entry pressure over $\rho\omega^2 R$. So the share is the entry pressure over $\rho\omega^2 R D$ — which is the suction at the load’s inner face, the one number that combines the speed, the drum and the load. A larger drum at the same speed applies more; a deeper load at the same speed reaches more. Plotted against that suction the four machines are one machine.

The g figure sometimes quoted beside a speed, $\omega^2 R$ at the wall, is halfway there: it carries the drum’s radius and not the load’s depth. And the class the energy label assigns is measured at a stated load, which is the other half of the correction made by specifying the test rather than the quantity.

A hanging sheet keeps a wet band at its hem #

Gravity is the gentlest head a cloth meets, and the same arithmetic fixes what it leaves.

A cloth hung by its top edge drips from its hem, and while a drop hangs there the hem is the outlet: water at the air’s pressure. A point a height $h$ above it is held by the column below, at $\rho g h$. So the holes drain above the hole head and not below it, and the band that stays wet is a fixed depth — 20 centimetres on a sheeting — whether the sheet is half a metre long or three.

That band has a name in another field. Above a water table, soil is saturated to a height set by its pore size and dry above it; hydrologists call the saturated strip the capillary fringe, and its height is exactly this arithmetic with a grain of sand in place of four threads. The yarn channels are the same arithmetic again at 5.3 metres, which no domestic sheet reaches, so on a line they never drain at all: a sheet stops dripping at 94 per cent of its dry weight and does the rest by evaporation — which is where a drying cloth’s own lift takes over from gravity.

Two practical readings follow. A sheet folded over the line dries from the fold, because the fold is the top of two hanging legs and each leg’s band is at its own hem. And a closely woven cloth has a deeper band than an open one, because its holes are finer; a voile’s band is half a sheeting’s.

What the spin leaves for the dryer #

Everything the spin leaves has to be evaporated, and here the difference between the two readings stops being academic.

Every ten points of residual is a hundred grams of water per kilogram of dry load, and evaporating a hundred grams of water takes about 0.24 megajoules of latent heat, some 67 watt-hours, before a dryer’s own losses. Going from a 1,000 rpm spin to a 1,400 rpm spin removes ten points on this arithmetic, which is about half a kilowatt-hour of latent heat saved on a seven-kilogram load. Going from 1,400 to 1,800 removes four points.

The water no spin removes costs nearly three times what the channels the spin leaves cost. On a 1,400 rpm load of cotton, the fibre’s own 30 per cent is three-quarters of what remains; on viscose it is more than four-fifths. Nothing in the drum changes that. A dryer’s work on a cotton load is fixed mostly by what the cellulose swelled by, and the spin speed decides only the smaller part on top.

How the numbers were produced #

Each cloth’s yarn length, dry mass, fibre volume and yarn volume come from its setts, counts and crimps and a packing factor of 0.6, exactly as in the three-reservoir count. The fibre’s soaked water is its volume swelling, from the tabulated diameter and length swellings, taken as the volume of water absorbed. The yarn’s channels are the dry voids scaled by the same volume factor, which is exact for a yarn that swells at constant packing; their radius is the packed-bed hydraulic radius widened by the fibre’s diameter swelling. The holes are the wetted cover’s open area times a thickness grown by the same fraction, with a radius from the swollen threads at the woven spacings. Each pore’s entry pressure is Young and Laplace’s $2\gamma\cos\theta/r$ at a contact angle of nought.

The drum’s suction is the rotating column integrated from the depth to the wall; a load is an even annulus of the stated depth, with its mass weighted by radius. The drum is 250 millimetres in radius and a full load 68 millimetres deep, both illustrative.

Seven things are required to hold, and a change that broke any of them would stop the numbers above from being produced. A soaked cotton fibre holds more than three times its regain, and a soaked polyester less than one per cent. No spin leaves a load below its fibres’ swollen water, a faster spin never leaves more, and the fastest in the sweep is still above the floor. The layer at the wall keeps its channels full at every speed. A thinner load is left wetter at one speed. The suction agrees with a numerical sum of the rotating column to a part in a million, and the kept depth inverts it. A thin layer’s residual against a head has exactly two steps. A hanging cloth’s band is the same depth at three different lengths.

What the arithmetic leaves out #

A drained channel is taken as empty. It is not: a film on the fibres and a ring at each contact keep some of it, and that is the likeliest reason every soaked-reading residual sits below a real machine’s. The size of that remnant is a measurement, not a geometry, and it is the next thing owed.

Every pore of one kind has one size. Real channels and holes have a spread of radii, which rounds each step of the staircase and smears the kept depth into a gradient. The positions of the steps survive; their sharpness does not.

The spin is taken to equilibrium. Water leaves a two-micrometre channel slowly, through a long path of other channels, and a spin of a few minutes may stop before the layer just inside $z^*$ has finished draining. That too would leave a real load wetter than the model.

The load is an even layer. A real load is a crumpled annulus with folds, knots and trapped air, and its depth is not one number. The collapse onto one curve is a statement about even layers.

The lumen and the volume contraction on mixing are both missing from the fibre’s water, in opposite directions: the lumen adds water the swelling does not count, and the mixing makes the swelling slightly overstate it.

Who measured it first #

Jurin gave the height a capillary holds in 1718 and Young and Laplace the pressure across a curved meniscus in 1805; both are used here as published. The centrifuge method for a porous body’s capillary pressure, with its end effect at the outlet face, is Hassler and Brunner’s of 1945, for oil-bearing rock. The capillary fringe is soil physics of the same period. The swelling figures are the standard ones tabulated in Morton and Hearle’s Physical Properties of Textile Fibres. Reading a washing machine as a Hassler–Brunner centrifuge on a two-pore cloth, putting the fibre’s soaked water at its volume swelling rather than its regain, and the floor, the wall layer, the half-load and the inner-face collapse that follow, were worked out here.

Still open: how much water a drained channel keeps #

The one quantity standing between the model and a machine’s label is the water a drained yarn channel keeps as films and contact rings, and it has a clean measurement. Spin a single thin layer of a cotton cloth — thin enough to have no end effect worth the name — at a series of speeds past the channels’ step, and weigh it. On this arithmetic the result should sit flat at 30 per cent of the dry weight plus whatever the drained channels retain, independent of speed once the step is passed; the height of that flat line above 30 is the remnant.

The same experiment on a polyester cloth of the same construction separates the two things a cotton result mixes. Polyester has no swelling water, so its flat line is the remnant alone — or its contact angle’s refusal to wet the channels at all, which the hairs decide for a raised cloth and which how high a cloth wicks needs for any other. Subtracting one line from the other would give cellulose’s swelling water measured through a drum, to set against the thirty per cent a microscope gave.