A stitch takes back what the parallel rule cannot see

Two layers are warmer than they are thick divided a cloth’s yarn into two layers of half the count and found the pair $\sqrt{2}$ as thick and more than $\sqrt{2}$ as warm — 18.8 per cent more on a muslin — because the same fibre now fills more volume and the mixture conducts less. It closed on the question every double-cloth weaver would ask next. Every stitch that binds the two layers is a thread crossing between them, and a thread is a conductor. How much of the bonus does a stitch take back?

The question could not be asked of that essay’s arithmetic, and the reason it could not is the first finding here. The rule it used for mixing fibre with air is the one rule that cannot see a stitch at all, and it is also the rule that makes the bonus as large as it can possibly be.

Three ways to mix fibre and air #

A fabric’s conductivity is somewhere between air’s, 0.026 watts per metre-kelvin, and fibre’s, about 0.20. Where exactly depends on how the fibre is arranged relative to the heat’s path, and there are three standard answers.

Parallel. Fibre and air as slabs side by side, each running the whole way through the cloth. The heat has a fibre highway and an air highway and takes both; the conductivity is the volume-weighted mean. It is the highest a mixture can conduct.

Series. Fibre and air as slabs stacked across the heat’s path, so every joule must cross both in turn. The conductivity is the harmonic mean, dominated by the air. It is the lowest.

Maxwell’s rule for cylinders. Long round fibres lying across the heat’s path in a matrix of air, not touching one another. That is a woven cloth: yarn lies in the cloth’s plane, heat crosses the plane, and the fibres are cylinders the heat must go round or through. The rule’s answer sits between the two bounds and much nearer the lower one at the fibre fractions cloths have.

Warmth is a thickness of air established that air does the insulating; which rule describes the fibre decides how much the fibre spoils it.

The parallel rule gives the biggest bonus #

The double cloth’s bonus is a dilution. Divide the yarn, the stack gets $\sqrt{2}$ thicker, the fibre fraction falls by $\sqrt{2}$, and the conductivity falls with it. How much it falls depends on how much of the conduction the fibre was carrying, and the parallel rule gives the fibre the most: every fibre a highway.

So the parallel rule gives the largest bonus of the three, and the numbers are not close. On the muslin:

• parallel: 18.8 per cent beyond the thickness;
• Maxwell’s cylinders: 8.4 per cent;
• series: 5.3 per cent.

It is natural to expect the reverse — that a rule which lets the fibre conduct less would leave the division more to gain — and it is backwards: a rule that gives the fibre less of the conduction gives the division less to dilute. The finding survives in its sign and its ordering: under every rule a double cloth is warmer than its thickness, and under every rule a close cloth gains more than an open one. What changes is the size. For yarn lying in its plane the bonus on a muslin is 8.4 per cent, and 18.8 is its ceiling.

Why the parallel rule cannot see a stitch #

A stitch in a double cloth is a yarn of one layer carried into the other and back: a back end lifted over a face pick, or a face end dropped under a back pick. For the length it spends crossing between the layers, that yarn runs through the cloth rather than along it.

Under the parallel rule that changes nothing. The rule already counts every fibre in the cloth as though it ran straight through, so turning a little of it to run that way adds no path the rule did not already have. The stitch is invisible, and the rule gives the same warmth for a cloth stitched at every intersection as for one not stitched at all. That is required of the arithmetic, not merely observed: a stitched double cloth computed by the parallel rule must come out identical to an unstitched one, to the last digit, and it does.

Under Maxwell’s rule the stitch is the one piece of fibre the rule was not describing. Everything else lies across the heat’s path; the stitch lies along it, and it conducts as a fibre highway exactly the parallel rule’s way.

A stack of three slabs #

The stitched double cloth is then a small calculation.

Each stitch crosses from the centre of one layer to the centre of the other and back, so it bridges the middle half of the stack. The outer quarters conduct by Maxwell’s rule alone; the middle half conducts by Maxwell’s rule over most of its area and as solid fibre over the share of the plane the stitches’ fibre occupies. The resistance is the three slabs in series.

That share is small and it is computable. A muslin layer has 528 intersections to the square centimetre. Stitched at one position in five — a satin’s scattered plan, one stitch on every pick of a five-end repeat — it carries 106 stitches a square centimetre, each two crossings of a ten-tex yarn, and their fibre fills 1.39 per cent of the plane. Fibre conducts eight times as well as air, so a sliver of it in the middle half matters.

One stitch a repeat, one a pick, every hidden position #

Where a stitch can hide found how many positions a face weave allows, and they give three stitching plans on a five-end satin face.

One stitch a repeat — the fewest that make two layers one cloth, which is one at any layer count — is 4 per cent of the intersections. It leaves the muslin 7.6 per cent of its 8.4.

One stitch a pick, scattered so that no two share a pick or a gap — the satin’s own arrangement — is 20 per cent. It leaves 4.7 per cent: the stitches have taken back 44 per cent of the bonus.

Every position the face can hide — the fifteen of twenty-five a five-end satin allows under the covering rule — is 60 per cent. It leaves −1.5 per cent: the stitched double cloth is now less warm than its thickness alone would make it, and everything the division bought has been spent on thermal bridges.

The loss is linear, and it breaks even at half #

The bonus falls in a straight line with the stitch density, because each stitch adds its own sliver of fibre highway and the slivers do not interact. On the muslin the line crosses zero at 49.5 per cent of the intersections: stitch half the cloth and the division has bought nothing but thickness.

The break-even is much the same on every cloth, between 43 and 59 per cent for everything but the cheesecloth, which is so open that it has too little bonus to lose and too little fibre to lose it with; it breaks even only at 81 per cent. The close cloths — the ones the warmth essay found gaining most from being divided — are also the ones that lose it fastest to stitching, because their stitches are the heaviest yarn.

The face that hides most stitches can spend the whole bonus #

Put the three face weaves beside one another and the hiding census turns against itself.

A 2/2 twill face can hide a stitch at a quarter of its intersections and no more. Stitched at all of them, a double muslin keeps 3.8 per cent. A twill face cannot erase the bonus, because it cannot hide enough stitches to.

A five-end satin can hide three fifths, and an eight-end satin three quarters. Both can take the cloth well past the break-even — to −1.5 and −3.4 per cent. The satins are the faces a double-cloth designer chooses precisely because they hide stitches so well, and hiding stitches well is exactly the property that lets a stitching plan spend the warmth.

So the census that a double cloth’s stitching was built on has a thermal reading nobody gave it. The count of hidden positions is also a count of places the warmth can leak, and the question a designer should ask of a stitching plan is not how many stitches the face can hide but how few the cloth needs.

Stitches and softness are the same cost #

The stitch has now been found working against the double cloth twice.

A double cloth is only softer if its yarn is set found that stitching the layers together pushes the cloth’s bending rigidity from its sliding bound toward its fused bound — a stitched double cloth bends more like one thick cloth than like two thin ones. This essay finds that stitching pushes its warmth from the divided cloth’s toward the single cloth’s. Both are the stitch doing the one thing it is for, which is making two layers act as one, and both of the things a double cloth is bought for — drape and warmth — are properties of its being two.

The minimum stitching is therefore the right stitching on both counts, and for the same reason. One stitch a repeat joins the layers; every stitch after that is paid for in drape and warmth and buys only a firmer join.

A stitch also pins the thickness #

The figures price a stitch as a bridge and nothing else, and a stitch does one more thing that runs the same way. It is a thread under tension carried from one layer to the other, so it pulls the two layers together along its line, and the stack is thinner there than between stitches.

The double cloth’s thickness is held by nothing but its yarn’s own springiness — a cloth compresses along its own bearing curve, and the light end of that curve is the soft end. A stitch drawn tight dimples the stack, and the warmth goes with the thickness. That is the familiar look of a quilted cloth: puffed between the stitch lines and flat along them, with the flat lines the cold ones.

So the figures here understate a stitch’s cost in two ways at once, by leaving out the thickness it pins and the contact it forces between the layers, and neither can make the minimum stitching look worse than the full one.

More layers, fewer stitches each #

The repeat allows four layers and the loom allows two, and a stack of separate fabrics allows any number. The stitching arithmetic scales in an unexpected direction as layers are added: one stitch between the outermost layers is enough to join any number of them, so the minimum stitching per layer falls as the count rises, while the dilution bonus it has to protect grows. The more layers a cloth has, the smaller the share of its warmth its joining has to cost, and the heavy stitching that spends a double cloth’s bonus is a choice rather than a necessity at every layer count.

What an interchange costs instead #

There is a way of joining two layers with no stitch at all. An interchange joins what a stitch would have had to: the layers change places at a block boundary, and the boundary is the join.

At the boundary every end and pick crosses from one face to the other, so for one thread’s width the whole of both layers runs through the cloth. That is a line of fibre highway rather than a scatter of points, and its share of the plane is one thread spacing per block width. A block eight ends wide spends one end in eight at the boundary — the same order as a scattered satin stitching — and a larger block spends less. An interchange is a stitching plan with its stitches gathered into lines, and the arithmetic here prices it the same way: by the share of the plane that fibre crosses the interface in.

The model named #

The cloths are the table’s eight, each divided by count into two layers at the same sett, so the pair carries exactly the single cloth’s yarn — the warmth essay’s own construction, and its thicknesses and fibre fractions unchanged. The in-plane yarn conducts by Maxwell’s rule for long cylinders across the flow, with air at 0.026 and fibre at 0.20 watts per metre-kelvin. A stitch is two crossings of one layer’s yarn, as solid fibre, bridging the middle half of the stack from centre to centre; the rest of the stack is three slabs in series. The stitch densities are the hiding census’s: one per repeat, the scattered maximum, and every covered position.

Required of the arithmetic, not shown: that the parallel rule reproduces the warmth essay’s double-cloth figures exactly; that the three rules order the bare bonus parallel, Maxwell, series on every cloth; and that the parallel rule gives an identical answer stitched and unstitched.

What was counted #

Eight cloths under three mixing rules unstitched, twenty-four bonuses; the same eight under three stitching plans on each of three face weaves by Maxwell’s rule, seventy-two more; and a sweep of the stitch share from nothing to every intersection in forty steps for each cloth, from which the break-even is found by bisection.

What the model cannot show #

Maxwell’s rule is for fibres that do not touch. At a cloth’s fibre fractions fibres touch one another at every crossing, and each contact is a small conduction path across the plane. That moves the true value from Maxwell’s toward the parallel rule’s, by an amount set by how the contacts are arranged, and nothing here computes it. The bracket is the essay’s honest result: the bonus is between 8.4 and 18.8 per cent on a muslin, and the stitches’ cost is a share of whatever it is.

The bridge is a guess at a length. A stitch is modelled as crossing the middle half of the stack; a stitch that ran the whole thickness would conduct half as well per unit of fibre and cost correspondingly less, and one pulled tight into a shallow dip would cost more. The linearity and the break-even’s insensitivity to the face survive any fixed choice; the break-even’s position moves with it.

And there is no interface resistance. Two layers resting on one another have a plane of imperfect contact between them, which is a resistance of its own and is part of why a real double cloth may be warmer than any of these rules says. A stitch pulls the layers into contact along its line, so it removes some of that resistance as well as adding a bridge — a second cost, of the same sign, that is not in the figures.

Who found it, and when #

Maxwell’s rule for the conductivity of a dilute suspension is from 1873, and its form for cylinders across the flow is Rayleigh’s, from 1892. That a fabric’s conductivity sits well below the parallel bound is a standard result of textile heat-transfer measurement, which is why effective-medium models are fitted to fabrics rather than volume averages.

That stitches through a batt or between layers carry heat is known to quilters and to anybody designing a sleeping bag, whose baffles exist to avoid sewing straight through the fill. What is added here is the price in the double cloth’s own terms — a share of intersections, read off the same census that decides where a stitch can hide, against a bonus that turns out to be less than half what the parallel rule promised.

Still open: what the contacts are worth #

The bracket between Maxwell’s rule and the parallel rule is the largest uncertainty left in the double cloth’s warmth, and it is set by one thing: how much heat crosses the plane through fibre-to-fibre contacts at the thread crossings.

The crossings are already counted, and the force at every one is already computed. A contact’s thermal conductance goes with the size of the contact patch, and the patch goes with the force, so most of what is needed to put a number between 8.4 and 18.8 is already in hand. Whether a double cloth’s real bonus is nearer the floor or the ceiling, and whether a firmly beaten cloth — more force, larger patches — gains less from division than a loosely beaten one, is the calculation that would close it.