What a flattened yarn does to its cover

A cover factor is a sett times a diameter. It says what fraction of a cloth’s plan area a thread system occupies, and everything about how much a cloth hides, passes and shows is computed from it.

The diameter it uses is the round one, from the count and the packing factor. A thread in a cloth is not round, and the axis a cover factor cares about is the wide one.

The arithmetic #

Hold the area, since a yarn squashed out of round rearranges rather than compacting. Write the flattening as f, the ratio of the narrow axis to the round diameter.

Then the wide axis is d/√f, and a cover factor computed with the round diameter under-states the cover by exactly that factor.

At a flattening of 0.78 — what a knitted fabric’s own geometry demands, and roughly what a woven cloth’s section shows — that is 1.13.

Thirteen per cent, in a quantity everything about a cloth’s appearance depends on.

What inherits it #

Five quantities, and the list is the reason it matters.

The open area. A cloth’s holes are what is left when both systems’ covers are subtracted, and the subtraction is not linear — the two systems overlap where they cross. A thirteen per cent rise in each cover produces a larger fall in the open area, because the open area is a product of two complements.

The air permeability, which this collection computes from the open area and finds goes as roughly its square.

The opacity, likewise.

The shade, since a dyed cloth’s apparent colour depends on how much of it is thread rather than shadow.

And the wetting and wicking behaviour, which depend on the size of the channels between threads.

Every one of those is computed here from a round diameter.

Why the error is not thirteen per cent everywhere #

The propagation is worth doing rather than assuming, because it amplifies.

Take a cloth at a warp cover of 0.55 and a weft cover of 0.45, computed round. Its open area is the product of the two complements: 0.45 times 0.55, which is 0.248.

Now raise both covers by thirteen per cent: 0.62 and 0.51. The open area is 0.38 times 0.49, which is 0.186.

That is a fall of a quarter in the open area for a thirteen per cent rise in the covers.

And an air permeability that goes as the square of the open area falls by forty-four per cent.

So a thirteen per cent error in one input becomes a factor of nearly two in an output, which is the ordinary behaviour of a quantity computed by subtraction from something near one.

Which is why the trade uses empirical cover factors #

The error is large and the trade has not noticed it, and the reason is instructive.

A cover factor in practice is not computed from a diameter. It is computed from a count and a sett by a formula whose constant was fitted to cloths — the familiar forms where a cover factor is a sett divided by the square root of a count, times a number.

That fitted constant has absorbed the flattening, along with the packing factor, the fibre density and everything else between a count and a width.

So the trade’s cover factors are right and their derivation is not, and a person computing one from first principles gets a different and worse answer than a person using the empirical form.

That is a familiar shape in this collection: an empirical constant that works because it has absorbed several unmodelled effects, and that stops working the moment somebody changes one of them.

Where it stops working #

Two places, and both are real.

A new fibre. The fitted constant absorbed a fibre density and a packing factor. Use it on a fibre of different density and the answer is wrong by the density ratio’s square root.

A different flattening. A cloth finished differently — calendered hard, or not at all — has a different flattening, so its cover at a given sett differs. A fitted constant from ordinary cloths does not carry that.

The second is the more interesting because it is measurable: a calendered cloth covers more than the same cloth before calendering, at exactly the same sett and count, and this collection has computed the width a calender buys without connecting it to the cover.

What a knitted fabric’s cover does #

The knitted case is the one where the flattening is predicted rather than assumed, so the correction can be made rather than estimated.

A knitted fabric’s cover is computed the same way — a length of yarn per unit area times a width — and its width is the wide axis.

Since this work predicts the flattening from the fabric’s own geometry, the cover follows: a knitted fabric’s cover is 1/√f times the round-yarn value, with f from the tightness factor.

At an ordinary construction that is thirteen per cent, and at a tight one nineteen.

So the correction is computable for a knit and not for a cloth, which is the same asymmetry this work has found repeatedly: a knitted fabric’s section is structural and a woven one’s is historical.

What a measured cover would settle #

The disagreement between a computed cover and an empirical one is testable and the test is easy, which is unusual for anything in this subject.

Photograph a cloth against a light and measure its open area directly. That is a single image and a threshold, and it gives the open area without any diameter, sett or constant at all.

Compare it with the open area computed from a round diameter and from a flattened one.

The round calculation should over-state the open area by about a quarter and the flattened one should be close, and the difference is far outside anything an image measurement gets wrong.

That measurement has been made many times, mostly by people studying air permeability, and the comparison against a first-principles cover is not usually made — because the practical interest is in the permeability rather than in whether the geometry that predicted it was right.

So the data very likely exists and the comparison does not, which is the commonest shape of gap this collection finds.

What was counted, and how #

The section arithmetic is elementary and holds the area: an ellipse of minor axis f times a circle’s diameter has major axis the diameter over √f.

The cover propagation is this collection’s own — an open area as a product of complements, an air permeability as roughly its square — and is used unchanged.

The flattening for a knitted fabric is this collection’s prediction from its own geometry; for a woven cloth it is an assumed value in the range sections report, and the rung says so wherever it uses one.

The worked propagation uses a sheeting’s covers, computed round, and applies the factor to both systems.

The three quantities a cover is used for #

It is worth separating them, because the correction matters differently to each.

As a design index. A cover factor is used to say whether a construction is sensible: a cotton shirting sits in a band, a canvas in another. Used that way it is a comparison between cloths computed the same way, and a systematic thirteen per cent affects nothing.

As an input to a physical calculation. Air permeability, opacity, light transmission, and every quantity computed from an open area. Used that way the thirteen per cent propagates and amplifies, and it matters a great deal.

And as a limit. A cover of one means the cloth is closed, and a construction is checked against that. Used that way the correction moves the limit: a cloth reaches full cover at a thirteen per cent lower sett than a round calculation says.

So one of the three uses is unaffected and two are not, and the unaffected one is the commonest — which is the other half of why nobody has noticed.

Why a subtraction is where errors go #

The propagation above deserves a general statement, because it is the reason a small input error became a large output one and the pattern recurs.

An open area is one minus a cover, and a cover in an ordinary cloth is between a half and three quarters. So the open area is a small difference between two numbers of order one, and a relative error in the cover becomes a larger relative error in the difference — by the ratio of the cover to the open area, which is two or three.

Then a permeability goes as the square of that, doubling the exponent again.

So a thirteen per cent error becomes a quarter and then a half, entirely through the arithmetic and without anything else going wrong.

That is worth watching for wherever this collection computes a quantity by subtraction from something near one, and it does so in several places: an open area, a fibre volume fraction’s complement, a cover’s complement, and the fraction of light a cloth transmits.

Every one of those amplifies its inputs’ errors, and none of them says so.

Where the model stops #

The area is held. A calendered cloth’s threads have lost air as well as changed shape, so its wide axis grows by less than the constant-area arithmetic gives.

The section is an ellipse. A racetrack of the same area and flattening is slightly wider, so the correction is slightly larger than quoted.

And the cover model treats a thread as a rectangle of its own width. A round or elliptical thread’s projected width is its widest point, and a cloth’s cover is not simply the sum of widths because the threads are at different heights and shade one another obliquely.

That last is a limitation of the collection’s cover model rather than of this rung, and it runs in the direction of over-stating the cover.

Two corrections that partly cancel #

Since the rung has just named a second correction running the other way, it is worth asking whether they cancel and how nearly.

The flattening raises the geometric cover by 1/√f — thirteen per cent at 0.78.

The edge scattering lowers the effective opacity below the geometric cover, and this collection has computed that a cloth’s measured opacity exceeds its cover rather than falling short of it, because a thread’s own thickness blocks obliquely incident light that its projection would let past.

So the second correction runs the same way for opacity and the opposite way for open area, because opacity is about what is blocked and open area about what is not, and the two are not complements once the threads have thickness.

That is a genuine subtlety and it means the two corrections cannot simply be added. Which of them dominates depends on the quantity being computed and on the angle the light or the air arrives at.

Sorting it out needs an optical model this collection has and a geometric one it now has, run together — which is an afternoon and has not been done.

The generalisation #

The rung is a case of something worth watching for whenever an empirical constant exists.

A fitted constant absorbs everything between its inputs and its output, and its scope is whatever was in the fitting set.

The trade’s cover-factor constants were fitted to cotton cloths of ordinary construction and ordinary finish. Inside that set they are excellent — better than a first-principles calculation, because they contain effects nobody has modelled.

Outside it they fail silently, and they fail by exactly the amount the unmodelled effects differ.

So the value of computing a cover from a diameter is not that it gives a better number for an ordinary cloth. It is that it says which effects the empirical constant contains, so that somebody working outside the fitting set knows what has to be put back.

That is the standing justification for most of what this collection does, and it is worth restating here because the cover factor is the case where the empirical form is most obviously winning.

What a cover really is, restated #

The rung is a good place to say what a cover factor is measuring, because the correction makes the definition’s looseness visible.

A cover is meant to be the fraction of the cloth’s plan area that a thread system occupies. That is a projection: seen from directly above, how much of the cloth is thread.

The usual formula treats each thread as a rectangle of its own width and computes the fraction accordingly. That is right for a system of parallel opaque bars and it is an approximation for threads that are round or elliptical in section, because a curved thread’s edges are oblique and let light past at an angle.

So a cover factor is a geometric projection of an idealised section, and the flattening changes which section is being projected.

Two corrections, then, and they run opposite ways: the flattening makes the thread wider, and the curvature of its edges makes its effective opacity less than its width suggests.

This collection has the second — a cloth is more opaque than its cover says, because a thread’s edges scatter — and has never had the first.

The knitted cover, which nobody computes #

A last observation, because the correction is computable for a knit and the quantity it corrects barely exists.

A woven cloth’s cover factor is a universal index. A knitted fabric’s is not: knitters specify a tightness factor, a stitch density and an areal weight, and almost never a cover.

That is a gap rather than a preference. A knitted fabric’s opacity, air permeability and light transmission all depend on how much of its plan is yarn, exactly as a cloth’s do, and a knitted fabric’s cover is computable from the loop’s own solved path — a length of yarn per unit area times a width.

This work supplies the width, corrected. So a knitted cover factor is now available and would be worth having, because it is the quantity a knitted fabric’s appearance and permeability actually depend on and the trade’s own index does not measure it.

Two fabrics, one quantity, and the one where it can be predicted from first principles is the one nobody quotes.

Who found it, and when #

Cover factors are old and the empirical forms date from the nineteenth century in something like their modern shape.

That a thread in a cloth is flattened, and that its width therefore exceeds its round diameter, is universal knowledge.

What is this collection’s own is the propagation: showing that a thirteen per cent error in a cover becomes a quarter in an open area and a factor of two in a permeability, and identifying where the trade’s fitted constants have absorbed it and where they have not.

What a finisher changes without meaning to #

The rung has a practical consequence that a finisher would recognise as a familiar problem with an unfamiliar explanation.

Calendering presses a cloth between rollers and flattens its threads further. That raises the wide axis, raises the cover, and lowers the open area — so a calendered cloth is less permeable and more opaque than the same cloth before calendering, at exactly the same sett and count.

Both are observed and both are usually attributed to the cloth being “closed up” by the pressure, which is right and is vague. The mechanism is that the threads got wider.

Napping and raising do the opposite to the appearance and nothing to the section: they pull fibre out of the yarn, which raises the apparent cover without changing the geometric one.

And a resin finish may hold a flattening that would otherwise partly recover, so a resin-finished cloth keeps the cover the calender gave it.

Three finishing operations, three different relationships to the same quantity, and only one of them changes the geometry the cover factor is computed from. A finisher who knows which is which can predict which operations move a permeability specification and which do not.

Where the ladder goes next #

The flattening depends on the tightness factor, and the tightness factor contains the count. So the count decides the flattening, in a way that is not obvious and that runs opposite to intuition.

The count that decides how flat is the last of the contact ladder’s consequences and the one a spinner would care about.