Warmth is a thickness of air

Fibres are sold on warmth. Wool is warm, linen is cool, and a great deal of the price of a garment is the claim that one fibre keeps heat in better than another.

The claim has a computable content and the computation is short, because a cloth is a two-phase mixture of exactly two things and nothing else: fibre, and the air between it. A muslin is nineteen per cent fibre by volume. Everything else in it is air. That figure is not weighed: it falls out of the same geometry that gives a fabric its weight and a thread its diameter, which means it moves with the sett and the count rather than being an independent property.

A mixture of two materials in known proportions has a conductivity somewhere between two bounds that Wiener gave in 1912 and that no arrangement can escape. Heat flowing along a parallel arrangement sees the volume average; heat flowing across a series arrangement sees the harmonic one; and every real arrangement — every weave, every crimp, every fibre lying at every angle — lies between them.

Those bounds, at a cloth’s solid fraction, are narrower than the difference between wool and nylon. So the model cannot tell them apart, and saying so is worth more than a number computed at one bound and quoted as though it were the answer.

The claim #

A woven cloth’s thermal conductivity is air’s, to within the width of the bracket the model itself carries, and its thermal resistance is therefore its thickness divided by a number that is very nearly the same for every fabric.

Resistance is thickness over conductivity. If the conductivity is fixed at about 0.03 W/m·K whatever the cloth is made of, then twice the thickness is twice the warmth, exactly, and the fibre contributes a few per cent that the model’s own uncertainty swallows.

The argument #

Write φ for the fraction of the cloth’s volume that is fibre. The two bounds are

$$\lambda_{\parallel} = \varphi\lambda_f + (1-\varphi)\lambda_a, \qquad \lambda_{\perp} = \left(\frac{\varphi}{\lambda_f} + \frac{1-\varphi}{\lambda_a}\right)^{-1}$$

and the arithmetic is decided by φ being small and by λ_f being of the same order as λ_a.

Air is 0.0257 W/m·K. Wool is about 0.05 and nylon about 0.25 — a factor of five between the extremes of the fibre table. Put φ = 0.19 into the parallel bound and wool gives 0.031, nylon gives 0.069. That looks like a real difference until the other bound is computed: the series bound gives 0.028 for wool and 0.031 for nylon, because a series arrangement is dominated by the worst conductor and the worst conductor is the air in both cases.

So wool’s band is 0.028 to 0.031 and nylon’s is 0.031 to 0.069, and they touch. Cotton’s is 0.029 to 0.034 and sits inside nylon’s. Zero pairs out of fifty-six are separated — asserted rather than reported, over the whole table, because the claim is about the relation and not about any two fibres.

What was counted, and how #

The solid fraction is computed rather than weighed, from the same geometry every other argument here uses. In one cell of the repeat — p₁ by p₂ — there is one warp segment of length l₁ and one weft segment of length l₂, each a cylinder of its own diameter, and each of those cylinders is itself only packing fibre. So

$$\varphi = \frac{\text{packing}\cdot\left(l_1 \pi d_1^2/4 + l_2 \pi d_2^2/4\right)}{p_1 p_2 t}$$

with the thickness t from Peirce’s closure and the packing factor from the count. For a muslin that is 0.191: thirty-two per cent of the cloth’s volume is thread, and threads are sixty per cent fibre.

The conductivities are quoted as bands rather than as values, because different sources disagree about them by more than the differences anybody uses them to make. That is a decision worth defending: quoting a single figure for cotton and a single figure for wool would produce a clean separation that is an artefact of two authors’ choices.

The resistances come out in the units clothing is specified in. A muslin is 0.108 tog; a duck is 0.179; a batiste is 0.077. All eight cloths lie between 0.077 and 0.179, which is a factor of 2.3, and their thicknesses lie between 240 and 589 µm, which is a factor of 2.5. The two factors are nearly the same number because the conductivity is nearly the same number.

The eight cloths, and what separates them #

Read the table by thickness and it is very nearly the same order as by tog. The one row out of place is the cheesecloth, and it is out of place for the reason the whole essay is about: it is the openest cloth here, its solid fraction is half everything else’s, and being nearly all air makes it a better insulator per micrometre than the shirtings. It is also a scrim that anybody can see through, which is a reminder that this arithmetic is about conduction alone and that a cheesecloth stops no wind whatever.

Everything in that table is a cotton, so the fibre is held fixed and it is the geometry that moves. Holding the geometry fixed and moving the fibre instead — which is the comparison the trade makes — moves the tog by less than the width of the band each entry already carries.

Where the bracket would open again #

The bands close because the solid fraction is small and the contrast is modest, and it is worth asking what would have to change for a fibre choice to matter — because the answer says which fabrics the conclusion does not cover.

Raise the solid fraction and the bounds separate. The gap between the parallel and the series bound grows with φ, so a fabric compressed to half its thickness at the same mass is twice as solid and its bracket is correspondingly wider. At a solid fraction near a half, the parallel bound is dominated by the fibre and the series bound is still dominated by the air, and the two are far enough apart that no single number describes the mixture at all. A cloth crushed flat is the case where the model can say least, which is the opposite of the intuition that a denser thing is easier to describe.

Raise the contrast and they separate too. The fibres in the table span a factor of five against air; a metal fibre spans four orders of magnitude, and a conductive-yarn fabric at nineteen per cent solid has a parallel bound near a thousand times its series bound. That is a fabric whose conductivity is genuinely a property of its arrangement, and it is why a metallised textile’s performance is quoted per construction rather than per fibre.

And wetting does both at once, which is why the wet case is named as the model’s boundary rather than as a correction. Water fills the air spaces, so the minority phase becomes a majority one and its contrast against the fibre is small — an entirely different mixture, with the bounds close together again around a much higher value.

It is worth adding that all three conditions are satisfied by every fabric anybody wears in ordinary conditions, which is why the conclusion is worth stating flatly rather than hedged. A shirt, a sheet, a coat lining and a fleece are all mostly air, all made of fibres within a factor of five of one another, and all dry most of the time.

So the conclusion is not that a mixture rule can never distinguish two materials. It is that at a fabric’s own solid fraction, against air, over the fibres anybody wears, it cannot — three conditions, all of them checkable, and all three doing work.

What a duvet is, in millimetres #

The arithmetic runs the other way and is worth doing, because the answer is a length that anybody can check against an object.

A duvet sold at ten tog has a thermal resistance of 1.0 m²K/W. At a conductivity of 0.03 W/m·K that is a thickness of thirty millimetres, and there is no way round it: to hold a tenth of a square-metre-kelvin per watt at any conductivity near air’s takes three centimetres of something.

A down fill does it at a solid fraction near one per cent, which is why it can be compressed into a bag and why its warmth collapses when it is wet or crushed. A polyester wadding does it at three or four per cent and is bulkier for the same tog. Neither of those numbers is a conductivity — both are packing arrangements, and the fibre named on the label is the least informative thing on it.

The same calculation says what a single layer of shirting is worth. A muslin at 0.108 tog is a hundredth of a square-metre-kelvin per watt; the still-air layer that clings to any surface out of the wind is about 0.12, which is twelve times as much. A shirt is very nearly a way of carrying its own boundary layer around.

The one thing that does move it #

The bracket that eats the fibre does not eat the thickness, and it does not eat the solid fraction either.

A fabric at half the solid fraction is nearly all air and its conductivity approaches air’s from above; a fabric at twice it moves further from air. So the way to change a fabric’s conductivity is to change how much fibre is in a given thickness, which is what a raised or brushed surface does — and what raising a cloth costs is exactly what this collection has computed elsewhere.

That is the mechanism behind every warm fabric there is, and it is a mechanism about arrangement rather than about material. A napped wool, a fleece, a wadding, a down fill: each of them holds a large thickness at a very low solid fraction. It is also why a nonwoven can be a better insulator than a woven of the same mass — it has no crimp to spend and no interlacings to pull it flat. Down works because it holds several centimetres of air at a solid fraction of about one per cent, and at one per cent the two bounds are indistinguishable from air’s own conductivity to three figures.

The measurement, and why it agrees #

None of this would be worth much if it disagreed with the instrument, and it does not, for a reason that is itself instructive.

A fabric’s thermal resistance is measured on a guarded hot plate, and the standard procedure measures the plate with the specimen and the plate without it and takes the difference. What that subtracts is the boundary layer, so the result is the fabric’s own resistance — and the fabric’s own resistance is the small number computed above. Measured values for ordinary shirtings come out in the range this arithmetic gives, between about 0.07 and 0.2 tog, and they order themselves by thickness.

Where the measurement and the arithmetic part company is on compressible fabrics, and the parting is not a disagreement about physics. A hot plate presses on the specimen, a fleece under a plate is not the fleece on a body, and its thickness — which is the whole answer — depends on the pressure. So the standard specifies the pressure, and a fleece’s tog is quoted at it. This collection has the machinery for that too: what a compression does to a thickness is computed here for a thread, and the same question for a lofted structure is a packing problem it has not solved.

Where the model stops #

Wiener’s bounds are the widest honest ones and better bounds exist. Hashin and Shtrikman’s are tighter for an isotropic mixture, and a fabric is not isotropic. Using tighter bounds would narrow the bands, and narrowing them enough to separate two fibres would need a bound that knows the arrangement — at which point the arrangement is the answer and the fibre still is not.

Conduction is not the only transport. Air in a fabric can move, and once it does the problem is convection rather than conduction; radiation across a still gap matters at these temperatures and is not in the arithmetic at all. Both of those are about the arrangement too, which is the same conclusion arriving by another route.

Nothing here is wet. Water’s conductivity is 0.6 W/m·K, twenty-three times air’s, so a damp fabric is a different mixture entirely and the bracket that closes here opens wide. That is the real difference between fibres in use and it is a difference in how much water they hold rather than in how they conduct — which this collection has as what water does to a thread, and which is a matter of regain rather than of conductivity.

And a garment is not a fabric. The thermal resistance a person feels is the fabric’s plus the still-air layer clinging to it, and the second is much the larger. It is also the one that seams, cuffs and openings defeat, in the way that a garment cut dry and worn wet defeats a pattern.

The generalisation #

When a mixture is mostly one constituent, the other’s properties are bounded out of the answer, and no amount of care about the minority constituent recovers them.

The controlling quantity is the product of the volume fraction and the contrast ratio. Here φ is 0.19 and the contrast between nylon and air is ten, so the parallel bound moves by a factor of two and a half while the series bound barely moves at all — and the gap between the bounds is what decides whether a difference is real. Once the gap exceeds the effect, the model has stopped being able to see it.

The mistake this prevents is a common one and it always looks like precision: computing at one bound, or with one closed-form mixture rule, and reporting the difference between two materials as though it were a prediction. It is a prediction of the rule.

The diagnostic is to compute both bounds and compare their separation to the effect. It costs one extra line, it is available whenever a mixture rule is being used, and it converts a confident number into an honest interval — which here is more useful, because the interval is what says stop asking about the fibre.

Who found it, and when #

Wiener’s bounds are from 1912 and are the elementary ones for a two-phase mixture; Hashin and Shtrikman’s improvement is 1962. Neither is textile work.

That fabric insulation is essentially proportional to thickness is one of the oldest quantitative results in clothing physics, established in the 1940s during military clothing research, and the clo unit was defined in that period for exactly this reason: it is a resistance, and a garment’s clo value is very nearly its total thickness of trapped air.

What appears to belong to this collection is doing the arithmetic on the geometry rather than on a measured density — taking φ out of Peirce’s closure and the packing factor, so that the solid fraction is a consequence of the sett and the count rather than an input — and then observing that the bracket which results is wider than the fibre effect it was computed to evaluate. The conclusion is old and the demonstration that the model cannot support the opposite conclusion is what the bracket adds.

Where the ladder goes next #

If the fabric is the smaller half of the resistance, the larger half is worth looking at, and it turns out to be the half that moves: the wind takes the air and not the cloth, and it takes it from the boundary layer before it takes it from anywhere else.

Sideways, the same solid fraction that decides conduction decides what passes: a cloth stops having holes before it stops passing air, and the path that survives there is the same air the heat is going through here.