A windproof cloth is at its yarn's limit

Windproof is a threshold. A fabric sold as windproof is one whose air permeability is below a stated figure — commonly five millimetres per second at a hundred pascals, with a millimetre per second for a membrane — and those figures are trade practice fitted to what people report feeling, in the way that a cover factor is not, in the same way that a filter cloth’s retention criterion is practice fitted to filtration tests. Nothing here derives them and this does not pretend to.

What can be asked is whether a woven cloth can reach one, and for once the answer to a construction question is not a construction.

An ordinary shirting yarn at the closest sett it will weave passes 1,337 millimetres per second. The threshold is five. That is a factor of two hundred and sixty-seven, and there is no more sett to be had: the ends are as close as their own diameters allow and Peirce’s closure has no solution past it.

The claim #

Windproofing is the one fabric specification on this site whose answer is set by the yarn rather than by the construction, and the arithmetic cannot decide it, because the floor the yarn imposes lands on the same order as the threshold with a fivefold bracket around it.

That is an unusual conclusion for a collection whose habit is to compute the number and say what it is. It is worth stating plainly rather than dressing up: the fitted constant in the model varies by a factor of five across the literature, the correction for path tortuosity is unknown and goes the other way, and the answer sits inside the resulting interval.

What the arithmetic can say is sharper than a number, and there are three of them.

The three routes, and what each is worth #

Closing the sett is worth nothing. The setts a loom can reach are bounded above by the reed; what bounds a cloth here is neither the reed nor the loom. The closest weavable cloth is 267 times the threshold, and the reason it stops is that the yarn jams. Every further attempt is a request for a cloth whose ends overlap.

Closing the channel is worth a factor of three. A calender flattens the crimp crowns and raises the cover from 0.40 to 0.59 without touching the sett; a wetting swells the fibres by a fifth and does something similar. Each buys about a threefold reduction, and combining them buys less than the product because both are spending the same clearance.

Note that the two are not in a fixed order, and asserting one would be an assertion about how far each was taken rather than about cloth: a three-to-one calender closes more than a twenty per cent swelling, and a two-to-one calender closes less. That is why the figure above lists them in the order the arithmetic returned rather than in an order of merit.

Layering is worth almost nothing, and this is the surprise.

Why layers do not help #

Put n identical cloths in series and each of them takes a share of the pressure. At a hundred pascals across the stack, each layer sees a hundred over n.

If the resistance were viscous, velocity would go as pressure and the stack would pass one nth as much. But in an open cloth the resistance is almost entirely inertial: the pressure is being spent accelerating the air rather than shearing it, and there velocity goes as the square root of pressure. So the stack passes one over √n.

Five hundred and twelve layers of muslin — a wad four centimetres thick — still passes six times the windproof threshold. To reach five millimetres per second by layering alone would take something close to half a million layers, which is not a garment.

That is why a woollen blanket is not windproof and a thin nylon shell is. It is not a matter of how much cloth is in the way. An open structure’s resistance is inertial, inertial resistances combine as square roots, and no practical amount of them adds up.

The same arithmetic run on a close cloth behaves quite differently, because there the drop is mostly viscous and the layers do add in proportion. Layering works exactly where it is least needed, which is a shape this collection has met before and which is the natural consequence of a two-term law with a crossing in it.

And the layers have to line up #

There is a second thing wrong with layering, and it makes the arithmetic above optimistic.

Two identical cloths laid over each other do not present a fixed obstruction. In register, every hole sits over a hole and the pair is as open as one cloth; half a spacing out, each hole is half blocked. The average over registrations is exactly the product of the two open areas — which is what everybody assumes — and the value at any particular registration is not.

So a stack is a lottery, layer by layer. A folded curtain, a doubled sleeve, two plies quilted together: each interface takes whatever registration it happens to have, and the ones that land in register contribute almost nothing. The √n above is what a stack achieves on average; a stack that has settled into register achieves less.

Where the floor lands, and the bracket around it #

With the channels gone entirely, a cloth still passes what its threads pass. For a 20 tex cotton at a packing factor of 0.6 in a cloth a third of a millimetre thick that is 8.1 millimetres per second, which is above the threshold.

Two things move it and they move it in opposite directions.

The Kozeny constant is fitted and varies from about one to five, and the permeability goes as one over it, so the floor could be anywhere from 8 to 40 millimetres per second. That is the whole reason this essay does not end in a number.

The path length is understated. A route through a thread is longer than the cloth is thick, because the thread is crimped and the route winds, so the true floor is lower than the arithmetic gives. Nothing here computes by how much.

And the yarn itself moves it. The floor goes as the square of the fibre diameter, so a finer cotton has a lower one: at 12 µm rather than 15, in a sheeting’s geometry, it comes to 4.6 millimetres per second — under the threshold.

Two thresholds, and only one of them is about wind #

The threshold this essay is written against is not the only one on the same axis, and setting them side by side is what makes the difficulty visible.

Down-proof is a much older specification and a much easier one. A cambric that holds down feathers has to stop a quill, which is tens of micrometres across and stiff — a geometric obstruction rather than a flow limit — and a densely woven, calendered cotton achieves it comfortably. It is a statement about the largest hole, in the same family as a filter’s rating.

Breathable is the awkward one, because it is a floor rather than a ceiling on almost the same quantity. A garment shell is asked to be under five millimetres per second to the wind and over some figure to water vapour, and the two are not the same transport — vapour diffuses through a still film and does not need a channel — which is exactly why the membrane exists as a product category. A cloth cannot be specified at both ends of one number.

So of the three specifications a shell fabric carries, one is a maximum over the holes, one is a limit on the mean flow, and one is not about the holes at all. They are routinely quoted together as though they were three settings on one dial.

What the square root does to a garment’s layers #

The layering arithmetic reads as a curiosity because half a million layers is not a garment, and it has a consequence at ordinary counts that is worth stating, because it changes how a two-layer or three-layer construction should be thought about.

A shell over a lining over a shirt is three layers, and by the square-root law that is a factor of 1.73 rather than the factor of three an additive intuition gives. Adding a fourth buys a further thirteen per cent. The marginal return on a layer falls as one over the square root of how many are already there, so the second layer is worth 29 per cent and the tenth is worth 5.

That is a much sharper statement than “layering does not help much”, and it says where the effort should go. If a shell fabric is a factor of two hundred above the threshold, no arrangement of the layers a garment can carry will close the gap — but a single layer that is itself a factor of two hundred better closes it completely. The whole of the improvement has to come from one interface, and that is exactly what a coating or a membrane is.

It also explains an observation anybody has made about clothing. Putting on a second jumper is warm and does not stop the wind; putting on a thin nylon shell over one jumper stops the wind entirely. The two garments differ enormously in mass, thickness and thermal resistance, and the one that works is the one that is not an open structure at all — so the comparison is not between quantities of fabric but between two different regimes of the same law.

That difference in how the two quantities stack is worth carrying on its own, because it is the reason the two specifications sit so awkwardly together on a datasheet: one is additive in layers and the other is not, and a garment designer trading them against each other is trading quantities that respond to the same decision by different powers.

And the square root is why the wind is the specification a garment is designed around rather than a fabric. Heat, which is a thickness of air, does add across layers in proportion, so warmth genuinely is bought by piling fabric on. Windproofing is not, and the two are usually spoken of as though a warm garment were a windproof one with more of the same in it.

What the arithmetic can be held to #

An honest bracket is not an absence of a claim, and there are two here that a measurement could refute.

A cloth at its floor should stop responding to its construction. Every millimetre per second a fabric at the floor passes is going through its threads, so closing it further — another pass through a calender, a closer sett if one is available — should move its permeability very little. A cloth well above the floor should respond steeply. That is a difference in slope, it needs no absolute calibration at all, and it is measurable on any permeameter by giving the same fabric a second finish.

A cloth at its floor should be sensitive to what a cloth above it is not. The floor goes as the square of the fibre diameter and as the cube of the yarn’s porosity. So two windproof cottons of the same construction in yarns of different fineness should differ, and two open cottons of the same construction in the same two yarns should not. That is a paired comparison, and it is the kind that survives a fitted constant, because the constant cancels.

The prediction this collection is prepared to make from the bracket is therefore not woven cloths cannot be windproof. It is that the densely woven cottons that are sold as windproof are sitting on their yarn’s floor, and that everything about their behaviour should say so.

Where the model stops #

Every threshold quoted here is practice. Five millimetres per second at a hundred pascals is a figure from the trade, and different sources give different ones for different garments; the arithmetic is about the geometry a threshold is applied to.

The specimen is clamped and flat. A permeameter measures a disc held taut. A garment is not taut, is not flat, and has seams, closures and openings whose combined leakage is generally far larger than the fabric’s — which is why a windproof jacket is a construction problem as well as a fabric one, and why the fabric arithmetic is a necessary condition rather than a sufficient one.

The wind is steady here and is not in the world. Gusts, flapping and the pumping action of a moving body drive air through a fabric far more effectively than a steady pressure does, and none of that is modelled.

And a coating is not the limit of closing a channel. The last bar of the first figure is labelled channel closed rather than coated for that reason: a film over a cloth is a barrier in series with the fabric, and a real one has pinholes, which is exactly the finding that a coated cloth fails at its holes. The route that reliably reaches the threshold is a coating or a membrane, and the reason is that it removes the geometry from the problem entirely.

The generalisation #

A specification whose target lands inside the uncertainty of a model’s floor cannot be settled by that model, and the useful response is to find the claims that survive the uncertainty rather than to narrow it.

The two offered above have a common shape and it is a general one: fitted constants cancel in slopes and in paired comparisons, and do not cancel in absolute values. So a model with a fivefold constant in it is worthless for answering is this fabric windproof and perfectly good for answering will another calendering pass help or which of these two yarns will do better. Reaching for the second class of question when the first is unanswerable is not a retreat; it is usually where the information was anyway.

The second lesson is the square root. Whenever resistances are being stacked, the exponent relating flow to driving force decides what stacking is worth, and the intuition that more of the same must eventually be enough is an intuition about linear resistances. Under a square root, a factor of seven hundred costs half a million layers. Under linearity it costs seven hundred. The same physical arrangement, the same materials, and a difference between an engineering solution and an impossibility.

Who found it, and when #

The windproof thresholds are trade practice. The Forchheimer form and its consequences for stacking are standard in packed-bed hydraulics, where the same square root appears whenever a bed is deepened in the inertial regime.

Kozeny–Carman’s constant and its spread are as reported; the observation that it varies from one to five and that permeability goes as its reciprocal is not this collection’s.

What appears to belong here is the pairing: computing a yarn-imposed floor for a woven cotton, finding that it lands on the same order as a threshold the trade sets, and declining to resolve it. The alternative — quoting the floor at one value of the constant and concluding that woven windproof fabrics are impossible — would have been a confident, memorable, and unsupported result, and the fabrics that would have refuted it are on sale.

Where the ladder goes next #

The air a cloth passes matters mainly because of what it takes with it, and what it takes is heat: warmth is a thickness of air, almost none of which is in the cloth, and the wind takes the air and not the cloth.

Sideways, the same threshold arithmetic asked about water rather than air gives a much sharper answer, because water’s pressure scale is set by a pore radius rather than by a flow: the pore that wicks is the pore that leaks.