Warmth is mostly the hairs

Warmth is a thickness of air is one of the cleaner results in this collection and it is entirely negative about fibre. A fabric is a tenth fibre and the rest air; the bounds any two-phase mixture must lie between are narrower, at that solid fraction, than the difference between wool and nylon; so the model cannot tell one fibre from another in a cloth, and eight fibres’ whole bands overlap. What it can say exactly is that twice the thickness is twice the warmth.

The essay then measured thicknesses and computed resistances and got sensible answers. What it did not ask is what the thickness is a thickness of, and the answer changes the arithmetic completely for any fabric anybody actually wears in the cold.

The cloth #

A flannel and a poplin can be made of the same yarn at the same sett in nearly the same weave, and one of them is a winter shirt and the other is not. Nothing in the previous essay’s arithmetic distinguishes them: same fibre, same solid fraction, same cloth thickness, so same resistance. The trade’s explanation is that the flannel has been raised, which is true and is not an explanation.

What raising does is take fibre out of the floats and stand it up. It does not make the fabric thicker in any structural sense — it makes it weaker, because the fibre comes out of the yarns that were carrying the load — and the cloth left underneath is thinner than it was. Everything gained is in a layer of loose fibre standing in the air.

That layer is where the whole of the difference is, and until now this collection had no way to say how deep it is or how much of it there is.

The claim #

A canopy of protruding fibre is two parts in ten thousand solid, so its conductivity is air’s to four figures, and every micrometre of it is worth eight micrometres of cloth. On a raised fabric it is worth thirty times the cloth it grows on. On an unraised one it is worth nothing at all, because the hairs cannot reach one another and there is no layer to hold air still.

And the shape of the transition is the part that is not obvious: a canopy’s depth grows as the logarithm of its density, so raising has a diminishing return that is not a saturation but a logarithm, and every doubling of the hair population buys the same fixed depth of nap.

Why a canopy is a better insulator than a fabric #

The arithmetic is one line and it is worth doing slowly because the conclusion looks too strong.

A two-phase mixture of fibre and air has a conductivity somewhere between its two constituents’, and at a solid fraction c the optimistic bound is c·k_fibre + (1 − c)·k_air. Fibre conducts at about 0.20 watts per metre-kelvin and still air at 0.026, a ratio of eight.

A woven cloth is a quarter fibre. A sheeting at 135 grams per square metre and 0.38 millimetres thick, with cotton at 1.52 grams per cubic centimetre, is 23 per cent solid. Its conductivity is therefore about 0.066 — two and a half times air’s — and its resistance is 0.037 clo.

A canopy is two parts in ten thousand fibre. The total protruding fibre length per unit area times a fibre’s cross-section, spread over the canopy’s depth, comes out at 0.0003. Its conductivity is air’s to four figures.

So the fabric conducts two and a half times as fast as the layer above it, per micrometre. Every micrometre of nap is worth two and a half micrometres of cloth by conduction alone, and once the nap is millimetres deep and the cloth is a third of a millimetre, the ratio of the two resistances is thirty.

The threshold, which is why an ordinary cloth gets nothing #

A layer of nearly-still air needs to be a layer. A hair standing alone on a bare surface does not hold air still; it stands in a moving stream and is convected round.

The criterion is the pure number n_A λ² — the hairs per square millimetre times the square of their own length — which asks whether a hair can reach its neighbour. It is 0.53 for a sheeting, and the mean spacing between hairs is therefore longer than the hairs are. There is no canopy.

So the model gives an unraised cloth no still air at all, which is the conservative choice and is asserted as such. A real bare cloth certainly does have a slightly thickened boundary layer above it; giving it zero is a decision to under-claim rather than to guess, and it makes the comparison between the raised and unraised states a lower bound on the raising’s worth.

The narrowness of that spread is the answer to the obvious next question. A weaver cannot make a warm cloth by construction: doubling the sett and halving the count leaves the hair density where it was. Everything that crosses the line crosses it in the finishing department, and there are only two ways over — raising, which multiplies the population, and a pile, which is a third thread system and a different construction entirely.

The logarithm #

Above the threshold the canopy’s top is where the hairs stop reaching one another, which is where n_A(h)λ² falls to one. The population is exponential, so

h_c = λ · ln(n_A λ²)

and the depth is the logarithm of the density. Every doubling of the hair population adds λ·ln 2 — four hundred and thirty micrometres, for cotton — and no more.

That is the arithmetic behind an observation every finisher makes and nobody quantifies: raising has a diminishing return. A second pass through the raising machine adds much less than the first, a fourth pass adds less again, and the returns fall away in a pattern that is neither a saturation nor a proportional decline. It is a logarithm, and the constant increment per doubling is the signature.

The assertion in the file is exactly that: the steps between successive doublings must all be equal to within two per cent. A model with a saturating canopy would fail it, and so would one with a proportional one.

A doubling is worth a fixed number of clo, and that is the whole exchange #

The logarithm has a consequence in the unit a garment is specified in, and it is a single number.

A doubling adds λ·ln 2 of depth — 433 micrometres for cotton — on each face, so 866 micrometres of still air in total. Divide by air’s conductivity and by the 0.155 watts per metre-kelvin that defines the clo, and

every doubling of the hair population is worth 0.215 clo.

That reproduces the worked example above: sixty-four times the bare population is a factor of 33.9 above the threshold, which is 5.08 doublings, which is 1.09 clo. And it inverts cleanly — one clo takes 4.7 doublings above the threshold, a population about fifty times the unraised cloth’s, whatever the cloth was.

The interesting part is what it is bought with. Raising spends the cloth’s strength by taking fibre out of the floats, and a pass that doubles the surface population removes some fraction of what is left in them. If that fraction is roughly constant from pass to pass — which is a scaling argument rather than a solve, and is offered as one — then the remaining load-bearing fibre falls geometrically in the number of doublings while the warmth rises linearly in it.

So the exchange rate worsens by a constant factor per doubling, and it worsens from the first one. There is no region of increasing return and no plateau to aim at: the first pass is the best value raising will ever offer, and every pass after it buys the same 0.215 clo for a larger share of what strength remains.

Which is a quantitative account of why finishers raise two or three times and not ten. The thermal side would happily continue — the logarithm has no ceiling in it — and the limit that stops the process is entirely on the other side of the ledger.

What a nap is worth in the unit clothing is bought in #

At sixty-fourfold the population — which is a moderate flannel, not an extreme one — the canopy is 2.2 millimetres deep on each face and the fabric’s total resistance is 1.1 clo.

One clo is the insulation of a business suit, defined as what keeps a resting person comfortable at twenty-one degrees. A single raised shirt weighing under two hundred grams per square metre reaches it, and the cloth underneath contributes three per cent.

That is worth putting beside the previous essay’s own worked example, which computed what a duvet is in millimetres and found that warmth is bought by thickness at a fixed rate whatever the filling. The same rate applies here; what has changed is the recognition that a fabric can carry its own thickness of air on its surface rather than needing a filling to hold one.

Which cloths can be raised, which is a question about floats #

The canopy is bought by raising and raising is not available to every fabric, so the thermal argument inherits a constraint from the finishing ladder.

Only a float can be raised is the essay that settled it: a raising machine’s wire teeth catch a length of thread lying loose on the surface and pull fibre out of it, and a thread that turns at every crossing has no such length. A plain weave cannot be raised. The census in that essay found that the drafts with raisable float are exactly the drafts with crown line — the same two censuses picking out the same cloths, which is why raising and lustre are alternatives rather than independent choices.

So warmth and shine are bought from the same account. A cloth with long floats can be calendered into a lustrous surface or raised into a warm one, and it cannot be both, because raising destroys the crowns that reflect. That exchange was priced three ways in the raising essay; what this one adds is the other side of the price, which is a full clo.

It also says which fabrics are simply excluded. A plain-woven poplin has no float to raise and therefore no route to a canopy at all, whatever its yarn or sett. A poplin cannot be made warm, and the reason is a property of a binary matrix.

Why the same layer costs nothing in wind resistance #

There is a pairing here that is worth drawing out, because it explains a property of raised fabrics that is usually described as a defect.

Thermal resistance is a sum and flow resistance is a series with a bottleneck. Adding a layer of air adds its resistance to the cloth’s, and both terms count. Adding a layer to a flow path adds its resistance too — but the resistances are wildly unequal, and a two-parts-in-ten-thousand canopy has essentially none. So a nap adds a full clo of warmth and no measurable air permeability at all.

A flannel is warm and not windproof, and the two facts are the same fact seen through two different kinds of arithmetic. The wind takes the air and not the cloth is the essay that established the second half: a wind strips the still air away, and a fabric’s measured resistance in a wind collapses to something much closer to the cloth’s own.

So a canopy’s entire contribution is conditional on the air in it staying put, and nothing in this arithmetic makes it stay put. That is the sharpest limitation in the essay and it is stated in the model.

What was counted, and how #

Four assertions, and the first is a shape rather than a value.

That each doubling of the hair population adds the same depth, to within two per cent, across five doublings. That is the logarithm, asserted as an equality of increments rather than as a fit.

That past a modest raising the nap is worth more than the cloth. That an unraised cloth is given exactly zero, which is the conservative half and would be quietly lost by any later change that gave a bare cloth a fractional canopy.

And, separately, that the canopy’s solid fraction really is small enough for the conductivity claim: the mixture is evaluated at the computed fraction rather than at an assumed one, so a model that produced a denser canopy would produce a worse insulator and the essay’s headline would weaken by itself.

Where the model stops #

Whether the air is still is not asked. The canopy criterion is a geometric proxy for it — hairs that reach one another obstruct a flow more than hairs that do not — and it is not a flow calculation. A canopy in a wind is not an insulator, and the essay’s numbers are for a fabric in still air.

Only conduction is computed. Radiation across a two-millimetre gap between two surfaces at body temperature is not negligible, and a canopy full of fibre is a good radiation barrier for a reason this arithmetic does not contain — it intercepts. Including it would make a nap worth more rather than less, so the direction is again conservative.

The canopy’s solid fraction is uniform and it is not. It falls exponentially with height like everything else in this ladder, so the top of a canopy is more nearly pure air than the bottom. Using a mean is right for a series resistance and would be wrong for anything nonlinear.

And only the long population is in it. The short cloud lies within a fibre diameter or two of the yarn and adds no depth, so it contributes nothing here — which is one of the few places in this ladder where leaving it out costs nothing.

What a specification would have to carry #

A thermal specification for a fabric is an areal resistance and a thickness, and the argument above says both are under-determined without a third thing.

The thickness has to say what pressure it was measured at. A nap measured at the standard kilopascal is a nap crushed flat: a gauge presses through the hairs at any pressure above a fifth of a kilopascal on a bare cloth, and the raised cloth’s crossover is fifteen. So a fabric’s thermal thickness and its mechanical thickness are two different numbers and only one of them predicts warmth.

And the resistance has to say which face. Raising is usually done on one side; the canopy is then on one face and not the other, and the resistance is half what both-faces arithmetic gives. A double-raised cloth is a different product from a single-raised one and the difference is a factor of two in the thing being sold.

Neither point is subtle and both are routinely lost, because a thermal test reports one number and a thickness test reports another and nothing in between records which surface either of them was about.

The generalisation #

An insulator’s job is to hold a fluid still rather than to be a solid, so the best insulating material is the one that is barely there.

The transferable form is that a resistance in series adds, so the useful figure of merit for an added layer is resistance per unit thickness — and for a two-phase mixture that is worst for whichever mixture has the most solid in it. Anything that occupies space without filling it beats anything that fills it, which is why a nap beats a cloth, a cloth beats a film, and a still-air gap beats all three.

The corollary is the warning. A layer that works by holding a fluid still fails completely when the fluid moves, and it fails without warning because nothing about the layer has changed. Every claim in this essay is conditional on a boundary condition it does not compute.

Who found it, and when #

That warmth is thickness and not fibre is old and well established, and this site’s own version computed the mixture bounds and found them narrower than the spread between fibres. That a nap contributes to that thickness is likewise standard: every textbook says a raised fabric traps air.

What is added is the arithmetic. The canopy’s depth, its solid fraction and its threshold come from a hair population rather than from a measured pile height, and the logarithm — which is a statement about diminishing returns that a finisher can act on — appears not to have been written down.

Where the ladder goes next #

The same canopy that holds air also blocks light, and the arithmetic is nearly identical: a projected area rather than a volume fraction. A hair layer veils a highlight finds that the blocking changes no contrast and the hairs’ own return destroys one, which is why a shot fabric has to be made of a fibre with no ends.

And the same canopy is what a drop meets. The hairs decide the sign of the wetting shows that a canopy amplifies whatever the fibre’s own contact angle already was, in whichever direction it already lay — so raising a cloth makes it wetter or drier depending on a decision made in the dyehouse.