Two layers are warmer than they are thick

Six essays of this account have taken a two-layer cloth apart from the draft’s side: the same draft finished four ways, where a stitch may go, what the beams cost, what the loom allows and whether the thing is any softer.

None of them has asked what the reader gets. A double cloth is twice the material and it is not twice the cloth, and how its thickness, its weight and its warmth compare with a single cloth made of the same total yarn is a question about the stack rather than about the draft.

Two of the three answers are already in this account and the third is not, and the third is the one people buy the cloth for.

The comparison has to hold the yarn fixed, and there is only one way to do it #

“The same total yarn” is not one construction, it is two, and the account’s own division names them.

Divide by count. Each layer carries a yarn of half the count at the same sett. There are twice as many threads, each half the mass, so the mass per unit area is unchanged — and because a yarn’s diameter goes as the root of its count, each layer is $1/\sqrt{2}$ as thick and the pair is $\sqrt{2}$ times the single cloth.

Divide by sett. Each layer carries the same count at half the sett. The mass per area is again unchanged, each layer is exactly as thick as the single cloth was, and the pair is twice as thick.

The second is the cloth nobody weaves, because a layer at half the sett is a layer with half the cover and a double cloth of two gauzes shows its own inside. The first is what a double cloth actually is and it is the one measured here; every ratio below is the count division.

The weight question therefore answers itself and it is worth saying plainly. A double cloth of the same total yarn weighs exactly what the single cloth weighs — not approximately, by construction — and any claim that a double cloth is heavier is a claim about a cloth using more yarn, which is a different comparison.

The thickness is the square root of two, which is the account’s own law #

The thickness ratio comes out at 1.409 to 1.414 across all eight cloths in the table of cloths here, against $\sqrt{2} = 1.4142$.

That is not a new result; it is the division law the essay before it computed for bending, applied to thickness instead. It is quoted here as a check rather than a finding: eight cloths of quite different constructions all land within four parts in a thousand of the same number, which is what a geometric law looks like when it is right.

And a thickness ratio is what a double cloth is sold on. Twice the material, half again as thick, the same weight — a bulkier fabric for nothing. If that were the whole of it, this essay would be arithmetic.

Warmth is not thickness, because the air fraction moves too #

Warmth is a thickness of air is this account’s own result: almost none of a fabric’s insulation is in the fibre, and what a cloth does is hold a depth of still air in place. The thermal resistance is the thickness over the conductivity of the mixture, and the mixture is decided by one number — the share of the volume that is fibre.

Dividing the yarn moves both. The thickness rises by $\sqrt{2}$ and the same fibre now occupies $\sqrt{2}$ times the volume, so the fibre fraction falls by $\sqrt{2}$ and the mixture is more air than it was. A muslin’s fibre fraction goes from 0.177 to 0.125; a sheeting’s from 0.233 to 0.165.

Fibre conducts about eight times as well as air — 0.20 against 0.026 watts a metre-kelvin — so diluting it matters. The resistance ratio is the thickness ratio times the conductivity ratio, and the second factor is between 1.12 and 1.22:

So a double cloth of the same yarn is 41 per cent thicker and 59 to 72 per cent warmer, and the extra is not a bonus anybody has costed because the thickness is the number the trade quotes.

And the cover goes up as well, for a reason nobody would have guessed #

The division makes each layer more open — a muslin’s layer covers 0.470 where the single cloth covered 0.621 — so the obvious expectation is that a double cloth hides less. It hides more.

Two layers are the product on average is this account’s own rule: two independent openness fractions multiply, so the pair’s cover is 1 − (1 − c)², and at a layer cover of 0.470 that is 0.719 against the single cloth’s 0.621.

Every cloth in the table covers between 10 and 27 per cent more as a double cloth of the same yarn, and the gain is largest on the openest cloth — the reverse of the warmth ordering, which is largest on the closest. The two mechanisms are different: warmth is bought by diluting a conductor and cover is bought by giving the holes two independent chances to be blocked.

So the same division improves two things a cloth is bought for and the improvement is worst on one exactly where it is best on the other. There is no cloth on which a double construction is a bad trade and none on which it is the best possible trade, which is an unusually flat recommendation for this subject and is worth having as one.

The ordering is the mechanism #

The gain is not the same on every cloth and the way it varies says what is happening.

A cloth that is already mostly air gains almost nothing extra. A cheesecloth at a fibre fraction of 0.089 is 91 per cent air already, its conductivity is nearly air’s, and halving its fibre fraction cannot take much more out: it gains 12.3 per cent over its thickness.

A close cloth gains most. A sheeting at 0.233 has a fifth of its conductivity coming from fibre, and diluting that is worth 21.6 per cent.

So the rule is the opposite of the intuition that bulky cloths benefit from bulking. The division is worth most to the cloth that is least bulky to start with — and “close and balanced” is exactly the description of the cloths double-cloth constructions are usually made from.

The bonus does not saturate, and the loom does #

Nothing in the argument stops at two layers. Each further division divides the fibre fraction again, so the warmth keeps outrunning the thickness, and the excess grows without any sign of a limit.

At eight layers the cloth is 2.83 times as thick and 4.35 times as warm, on the same yarn, and the excess is still climbing. In the limit the fibre fraction goes to nothing, the conductivity goes to air’s, and the resistance goes as the thickness alone — which is $\sqrt{k}$ without bound.

And a shaft loom weaves two. The repeat allows four layers and the loom allows two is this account’s fifth essay: the harness’s strain budget buys thirteen shafts, a layer costs its own weave’s shaft count, and two differing layers already want a beam each.

So a woven double cloth collects the first step of a bonus that has no top, and the ceiling is the harness’s rather than the physics’. That is the third time this account has found the loom binding where the repeat was not, and it is the first time the quantity being capped is one a wearer would notice.

The number a specification would have to carry #

Nothing in a fabric specification distinguishes the two cloths compared here. A specification names a cloth by its yarn, its setts and its weave, and a double cloth’s specification names two yarns and two setts and a two-layer weave — from which every number above follows, and none of which is the number anybody quotes.

What would have to be added is one quantity, and it is not a new measurement: the fibre volume fraction, which is the mass per unit area divided by the fibre’s density and by the thickness. All three are on the specification already or trivially derived from it, and their combination is what decides the warmth.

It is worth naming because it is the quantity that makes the whole comparison legible. A cloth’s thickness says how much room it occupies; its fibre fraction says how much of that room is fibre; and warmth is the first over the second, with the fibre’s own conductivity as the constant. Every result on this page is one division away from numbers a mill already has, and the reason nobody does the division is that the two halves of it belong to different documents.

Which is the argument for the constructions that are not woven #

The obvious response is that a stack of separate fabrics reaches every layer count the loom cannot, and it does — a quilt is exactly this arithmetic with k of ten or twenty. What a double cloth has that a stack does not is that the layers are held in register by the weave itself, so the air gap between them cannot be squeezed out at one place and doubled at another.

That is not a small distinction. The whole calculation above assumes the thickness is held; a cloth compresses along its own bearing curve, and thickness under pressure is the first thing a fabric loses. A stack of four gauzes has nothing at all holding its layers apart; a four-layer woven cloth cannot be woven.

So the practical range of this argument is exactly two, and the reason is the loom at one end and the absence of any structure at the other. The bonus at two layers is 18.8 per cent on a muslin, which is real and is worth having, and it is the most any woven cloth can take.

What the four quantities do together #

Set the four side by side and the double cloth stops being a bulking trick and becomes a construction with a shape.

Weight: unchanged, exactly. Thickness: ×1.41. Cover: ×1.10 to ×1.27. Warmth: ×1.59 to ×1.72. Bending rigidity: unchanged at the free bound, ×0.5 at the set bound, ×3.3 fused — which is A double cloth is only softer if its yarn is set and the one thing in the list that can go the wrong way.

Four of the five improve and the fifth depends entirely on the yarn and the stitching. That is why a double cloth is a good idea and why it is not a universal one: the whole construction turns on whether its yarn has been set, because an unset yarn’s bending rigidity is the number of fibres across the width times the stiffness of one, and dividing the yarn does not change either.

And the stitching decides the same thing twice. Stitch the layers hard enough to act as one section and the bending rigidity goes up by more than three; stitch them at all and the interface starts conducting. The construction’s two failure modes are the same operation, which is the cleanest reason to keep a stitching plan as sparse as the hiding census permits.

What was counted, and how #

The thickness is this account’s own Peirce solve for each construction, run at the divided count, which is why the $\sqrt{2}$ is a result rather than an assumption — nothing in cloth-state solve knows about layers.

The mass is the construction’s own: counts times setts, and it is identical between the single and the divided cloth by construction. The census required it rather than assuming it, because a division that quietly changed the mass would make every ratio here meaningless.

The mixture is taken in parallel — conductivity is the volume-weighted mean of air’s and fibre’s. That is the crude bound, it is the right one for a fibre network at these volume fractions, and it is the bound that understates the effect, because a parallel average lets the fibre’s conductivity count in full. A series or a Maxwell mixture would put the single cloth’s conductivity lower and the gain higher.

The conductivities are this account’s own constants, 0.026 for still air and 0.20 for fibre, the same pair the raising and warmth work uses, so this essay and that one cannot disagree.

And every claim is required over the whole table rather than shown on one cloth: that the mass is held, that the thickness is $\sqrt{k}$ to a part in a hundred, that the warmth exceeds the thickness on every cloth, and that the excess is monotone in the single cloth’s fibre fraction. The last of those is the one that could have failed and did not.

What the model cannot show #

There is no interface in it. Two layers touching are treated as one medium of the combined thickness, and they are not: where the layers meet there is a plane of contacts that conducts better than air and worse than fibre, and a gap where they do not touch that conducts like air. Which of those dominates depends on the stitching, and stitching is exactly the variable this account has spent two essays on. A heavily stitched double cloth should be measurably colder than a lightly stitched one of the same yarn, and nothing here computes by how much.

Nothing here is convection. Still air at 0.026 is a fiction in any layer thick enough for air to move in, and a four-millimetre gap is thick enough. At the thicknesses in this table — under a millimetre — the assumption is safe; at the eight-layer end of the sweep it is starting not to be, which is a second reason the tall end of that chart is a limit rather than a prediction.

And warmth is not the only thing a thickness buys. The wind takes the air and not the cloth — a double cloth’s extra air is also extra air to be blown out of, and the same division that raises the resistance raises the permeability by opening every layer up. The two move in opposite directions and this essay computes only one of them.

Who found it, and when #

That a fabric’s insulation is its trapped air is nineteenth-century and is the basis of every clothing-comfort text; the clo unit and the still-air model are Gagge’s, from 1941. That dividing a yarn at constant mass raises the thickness as the root of the division is A double cloth is only softer if its yarn is set and is elementary.

Putting the two together is what has not been done, and the reason is that the two live in different trades. A double cloth is a weaver’s object and its arithmetic is drafts, beams and shafts; a clo value is a physiologist’s, measured on a hot plate and reported as a number with no construction attached. The quantity that connects them — the fibre volume fraction — is computed by neither, because a weaver has no reason to divide a mass by a density and a physiologist has no reason to care which yarn produced the thickness.

The result is a straightforward consequence once the connection is made, and the part worth carrying is the sign: the extra warmth is larger than the extra thickness, always, and by more the closer the cloth. Anybody quoting a double cloth’s bulk is understating what it does by about a fifth.

Still open: how much of the bonus a stitch takes back #

Every stitch that binds the two layers is a thread crossing the interface, and a thread crossing the interface is a conduction path of fibre through what was air.

The count is available. Where a stitch can hide measures how many positions in a repeat may carry a stitch and how few of them a given face weave permits; a stitching plan’s density is therefore a number this account already computes. What is missing is what one stitch conducts, which is a fibre cross-section times a path length and is the same arithmetic as everything above.

The prediction has a sign and a shape. A stitched double cloth is colder than an unstitched one; the loss is linear in the stitch density; and the two ends of the range are the unstitched cloth, at the full 18.8 per cent bonus, and a cloth stitched at every permitted position, which is as near a single cloth as the construction gets. Where a real stitching plan falls between them is a measurement with a hot plate and two cloths, and it would say whether the thing a double cloth is bought for survives being held together.