A knit is warm because of where its yarn is not

The woven half of this collection settled the question of warmth several rungs ago and the answer was blunt: a fabric’s thermal resistance is its thickness divided by an effective conductivity, the conductivity is a mixture of fibre and air at the fabric’s own solid fraction, and since air conducts two to five times worse than any fibre, what a fabric is doing is holding still air in place.

The knitted half could not be asked the question at all, for a reason that had nothing to do with heat. It had no thickness. A loop solved as a plane curve is a plan, and a plan divided by a conductivity is not a resistance.

The two numbers a resistance needs #

A thickness and a solid fraction, and each of them was missing for a different reason.

The thickness comes from the interlacing, and this collection’s account of it is three lines: a loop’s feet were drawn through the head below, so head and feet are one yarn diameter apart through the fabric; each has a radius outside it; the fabric is two diameters thick. On two beds it is the bed gap plus a diameter instead.

The fibre fraction could not be computed without the thickness, because a fraction of a volume needs a volume. The yarn’s own fibre volume per stitch is its length times its cross-section times its packing factor; the cell it occupies is a wale spacing by a course spacing by the thickness. Divide.

For the 20 tex cotton jersey that gives 26.6 per cent fibre and 73.4 per cent air, which is the number the whole of the rest of this rung turns on.

What comes out #

Thermal resistance in tog, which is ten times the resistance in square-metre-kelvins per watt and the unit a duvet is sold in:

The range on each row is not experimental scatter. It is the gap between the two bounds a two-phase mixture must lie between — the fibre and the air in parallel at one end, in series at the other — and the true value is somewhere inside.

The bounds are close together here, and that is a fact about the fabric rather than a piece of luck. Parallel and series bounds diverge when the fibre and the air occupy similar volumes and conduct very differently; they close up when one of them dominates the volume. At nine per cent fibre the air is doing almost all of the conducting whichever way it is arranged, so the two bounds sit within ten per cent of each other and the arrangement of the yarn barely matters. At the jersey’s 26.6 per cent they are twenty-four per cent apart, and there the arrangement is starting to matter — which is the direction the table is read in.

The whole column is a straight line through the origin, and it should be. Tog rises 0.088, 0.148, 0.209, 0.271, 0.334 against thicknesses that rise by one yarn diameter a step; the increments are 0.060, 0.061, 0.062, 0.063. Resistance is proportional to thickness for a slab, and this is a slab whose conductivity hardly changes because the added volume is all air. So the fabric’s warmth is its thickness and nothing else, which is the essay’s title stated as an arithmetic rather than as a claim.

The finding #

A rib knitted at five diameters of bed gap is nearly four times as warm as a jersey of the same yarn at the same loop length.

Not a heavier yarn, not a tighter fabric, not a different fibre, and not a gram more material per unit area. The same yarn, the same stitch length, the same number of stitches. The only thing that changed is how far apart the two needle beds were set.

That is a striking amount of leverage to sit in a machine setting, and it is worth being precise about where it comes from. A rib’s thickness is the bed gap plus a diameter, and the gap is set on the machine. Everything else about a knitted fabric’s warmth is downstream of a thickness, so a lever on the thickness is a lever on the warmth.

And why it works twice over #

Opening the beds does two things and they both push the same way.

It makes the fabric thicker, which raises the resistance directly, because resistance is thickness over conductivity.

And it makes the fabric emptier, because the same yarn is now spread through a larger volume. The fibre fraction falls from 26.6 per cent to 8.9, which lowers the effective conductivity toward air’s.

So the resistance rises faster than the thickness does — a factor of three in thickness buys a factor of 3.8 in warmth. That compounding is the reason the answer is worth computing rather than guessing.

Why the two bounds are bounds rather than an estimate #

The range on each row is not scatter and it is not an uncertainty in the inputs. It is a pair of bounds that any two-phase mixture must lie between, whatever its geometry.

In parallel — every conduction path running through fibre and air side by side — the mixture conducts best, at the weighted arithmetic mean of the two conductivities. In series — every path running through fibre and then air — it conducts worst, at the weighted harmonic mean. A real fabric’s paths are neither, so its conductivity is somewhere between.

That is Wiener’s result and it is over a century old. What makes it the right tool here is that it needs nothing about the geometry beyond the fraction, which is exactly what this rung has newly got. A better estimate is available to anybody willing to model the paths through a loop, and it would be a narrower band inside this one rather than a different answer.

Where the ceiling is #

There is a limit and the model knows where it is. Still air over the same thickness is what a fabric would achieve if it were made of nothing at all, and it is the number every real fabric falls short of.

For the jersey it is 0.129 tog against an achieved 0.088 to 0.109. For the widest rib it is 0.386 against 0.334 to 0.366.

Two things follow. The fabric is doing most of what its thickness allows — between two thirds and ninety per cent of the still-air value — and the emptier it is, the closer it gets. A fabric at nine per cent fibre is very nearly a slab of air, which is what a good insulator is.

It also says where the remaining margin is, and it is not much — a point the woven side reaches from the other direction at warmth is mostly the hairs. Halving the fibre fraction again would buy a few per cent, which is why nobody makes a garment warmer by making its yarn finer.

The comparison that puts it in proportion #

None of these numbers is large, and it is important to say so before anybody reads a tog value as a claim about a garment.

A garment’s total thermal resistance is the fabric’s plus the layer of still air that clings to its outside, and that boundary layer is worth about 1.2 tog in still conditions. The jersey’s own contribution is 0.09.

So the fabric is under a tenth of the resistance a person wearing it experiences, and the woven half of this collection made the same point about a shirting several rungs ago. That is not a reason to stop computing fabric resistances; it is the reason to be careful about what they mean. A fabric’s own resistance matters where the boundary layer is stripped away — in wind, in movement, under a pack strap — and it matters comparatively rather than absolutely.

Which is why the wind matters more than the fibre #

The boundary layer is the thing a wind takes away, and once it is gone the fabric is all there is.

That reframes the table above. In still air, moving from a jersey to a wide rib changes a wearer’s total resistance from 1.29 tog to 1.53 — nineteen per cent, noticeable and not dramatic. In a wind that strips the boundary layer entirely, it changes it from 0.09 to 0.33, which is nearly four times.

A structural change to a fabric buys most where the air is not helping. That is the same conclusion the woven side reached about windproofing, arrived at from the opposite direction.

The fibre, which does not separate #

Change the fibre and hold everything else fixed and the answer barely moves, in a way the model is explicit about.

A 20 tex wool jersey comes out at 0.106 to 0.122 tog; a cotton at 0.088 to 0.109; a polyester at 0.061 to 0.107. Every one of those bands overlaps every other one. No fibre’s whole band clears any other fibre’s, so the model cannot separate them, and that is a claim about the fabric rather than a failure of the arithmetic.

The woven side found exactly this and asserted it as a relation rather than as a value. The reason is structural: a fabric is mostly air, and a mixture that is three quarters air is dominated by the air’s conductivity whatever the other quarter is made of. Wool’s reputation for warmth is not about wool’s conductivity, which is unremarkable; it is about how much air wool’s crimp and its hair layer let a fabric hold.

What the thickness does not include #

The number in the table is the fabric’s own thickness — the yarn and the space between the loop planes. It is not what a gauge reads, and the gap between them is a fabric’s hair.

Every spun yarn has fibre standing off it, and on a knitted fabric that fibre stands off a surface which is already mostly crowns. This collection computes that layer elsewhere and finds it can be hundreds of micrometres deep on a raised cloth — comparable with the whole thickness of a jersey.

So a raised or brushed knit is warm for a reason that is not in this table at all. Its structural thickness is unchanged; what has changed is that it now carries a canopy of still air on both faces, and a canopy of still air is exactly what warmth is. That is why brushing a fabric is the cheapest way to make it warmer and why it works on a jersey as well as on a woven cloth.

The one number that is not from this collection #

Everything above is computed from the fabric’s own geometry except one figure, and it is the largest one on the page: the still-air boundary layer at about 1.2 tog.

That is quoted rather than derived. It is a standard figure for a clothed body in still air and it depends on the garment’s shape, the air’s movement and the surface’s roughness — none of which is a fabric property and none of which this collection computes.

It is quoted because without it the fabric numbers have no scale. A resistance of 0.09 tog means nothing until it is set against the thing it is a tenth of, and leaving it out would have made the fabric’s contribution look like the whole story. That is the same reason the woven side quotes it, and it is the honest way to use a number from outside: name it, say it is quoted, and use it only for proportion.

What a knitter can actually reach #

Three levers, in order of size.

The bed gap, which is the largest and is available only on a two-bed fabric. A factor of nearly four across a range a machine can be set to.

The yarn’s diameter, which sets a single-bed fabric’s thickness outright. A 40 tex cotton is 0.473 mm thick against a 20 tex’s 0.334 — forty per cent more, for twice the yarn.

And the loop length, which does nothing at all on one bed. That is the surprise in the list and it follows from the thickness being independent of the gauge: a loosely knitted jersey is not a thicker jersey, so it is not a warmer one either. It is a lighter, more open, floppier fabric of exactly the same resistance.

Why that last one is the interesting negative #

A loose knit feels warmer, and the feeling is not nonsense — it is softer, it compresses further, it drapes into deeper folds, and a garment made of it traps more air in its own drape.

None of that is the fabric’s resistance. It is the garment’s, and a garment’s air is held between the fabric and the body rather than inside the fabric. This rung computes the fabric’s, which is the part that survives being pressed flat under a coat.

The distinction is worth keeping because the two come apart in use. Sit on a loose jersey and its trapped air goes; sit on a rib and its bed gap does not.

What this does not settle #

Anything about moisture. A wet fabric conducts an order of magnitude better than a dry one, and none of this holds once the air is displaced by water.

Anything dynamic. These are steady-state resistances. A fabric’s response to a sudden contact — what makes a fibre feel cool to the touch — is about its thermal absorptivity, which is a different quantity.

A spacer fabric. A knitted spacer holds its two faces apart with a third yarn, by design, and can be several millimetres thick. Its thickness is set by that yarn’s length and this model has no term for it, though the arithmetic afterwards would be the same.

And the true value inside each band. Wiener’s two bounds are what a two-phase mixture must lie between with no information about its geometry. A fabric’s geometry is not unknown, and a better estimate is available to anybody willing to model the conduction paths through a loop.

What is genuinely new here #

Two things.

A knitted fabric’s thermal resistance, computed from its own structure, with the thickness and the fibre fraction both derived rather than measured.

And a lever that is a machine setting. A rib’s warmth is set by how far apart its beds are, by a factor of nearly four across a practical range, at no cost in yarn. That is a strong claim, it is easy to test with a guarded hot plate, and it has not been tested here.

What the pictures cannot show #

The bars are drawn at their lower bounds with the upper bound written beside them, which reads as a value with an error bar. It is not an error bar. It is a pair of bounds that a mixture must lie between, and the truth is not more likely to be in the middle than at either end.

Nor does the chart show the boundary layer, which is ten times every bar on it. A picture of a fabric’s resistance drawn to the scale of a garment’s would be a chart of very short bars beside one long one.

What is worth taking away #

Warmth is a thickness of trapped air, a knitted fabric’s thickness is now computable, and a rib’s is a machine setting.

That last one is the finding: a factor of nearly four in a fabric’s own thermal resistance, bought by moving two needle beds apart, at no cost in yarn and no change to any other specified quantity. It is the largest lever on this ladder and it is one nobody records.

What the fabric’s own resistance is for #

A tenth of a garment’s total is not much, and it is worth saying where the tenth matters rather than leaving the comparison to deflate the whole page.

Where the boundary layer is gone. In wind, in movement, or under a strap, and this collection’s woven side computes exactly that: the wind takes the air and not the cloth.

Where fabrics are being compared. Two garments of the same cut differ only in their fabrics, so a fifth of a tog between the fabrics is a fifth of a tog between the garments, whatever the boundary layer adds to both.

And where fabrics are being layered. Resistances in series add, so five layers of jersey contribute five times 0.09 — nearly half a tog, which is no longer a rounding.

What this leans on, and how firmly #

Four things, in decreasing order of how much would break if they were wrong.

The thickness, from the interlacing — two yarn diameters on one bed, a bed gap plus a diameter on two. If that is wrong every row here is wrong in proportion, and it is the one number a gauge could contradict.

The yarn’s diameter, from a count, a fibre density and a packing factor. It enters the thickness directly and the fibre fraction twice, so a twenty per cent error in the packing moves everything here by more than twenty per cent.

Munden’s spacings, which set the cell the fibre fraction is a fraction of. Those are measurements and are as firm as anything on this ladder.

And Wiener’s bounds, which are a theorem rather than a model and cannot be wrong — only wide.

So the exposure is concentrated in the first two, and both of them are yarn measurements rather than fabric ones. What a thickness gauge reads on a knit is why the first is harder to check than it looks.

Where the ladder goes next #

A thickness held open by a force is a fabric that can be squashed, and squashing a fabric is what wearing it does: what a knit gives up when it is pressed.

And a fabric pressed against a body presses back, which is a different force again and two orders of magnitude smaller than anybody would guess: what a cuff presses with.