A covered cloth cannot be calendered

A calender buys the width found where a calendered cloth’s lustre comes from: not from the length of its crowns, which moves by six per cent, but from their width, because a flattened section has a plane on top of it and a plane has one normal rather than a fan of them. The arithmetic refused to put any of the gain in the other factor, and the figure it quotes is a multiplication by twenty-four.

Nothing in that essay asks how far the flattening can go.

It cannot go far on a close cloth, and the reason is not the machine or the yarn. A calender widens a section at conserved area, and a thread that widens eventually meets the threads on either side of it. Where that happens is set by the cover the cloth already has, and the consequence is a frontier the lustre account has been working above without noticing.

The cap is a width meeting a pitch #

A cloth of cover k has its threads occupying k of every unit of width, so consecutive threads sit at a pitch of d/k with d the round diameter. Kemp’s racetrack is a rectangle with semicircular ends at conserved area: at an aspect ratio f it is a = f·b across and b = κ(f)·d through, with κ(1) = 1 exactly.

The thread may widen until a equals the pitch, and no further:

$$a(f_\text{max}) = d / k.$$

It is worth being explicit that this is not $f = 1/k^2$, which is what an ellipse at conserved area would give. A racetrack’s width grows faster than the root of its aspect ratio, because the flat run contributes area in proportion to the width while the two semicircular ends do not — so the cap is solved rather than written down, and the closed form that looks right is wrong by a third at a cover of a half.

At full cover the cloth admits no flattening at all. That is not a limit approached; it is the exact value, because a thread already as wide as its own pitch has nowhere to widen into, and the racetrack at f = 1 is the circle it started as.

Which makes the lustre and the opacity one constraint #

The last column is the one worth holding still for. The plateau is the whole of what a calender sells, and at a cover of nine tenths a nip can put 53 micrometres of it on a 200-micrometre yarn — a quarter of the width, against 93 per cent on a scrim.

So the two quantities a finisher would most like to have together are in direct opposition.

One minus the cover is a cloth’s openness, which is this account’s own covering rule. A cloth at cover 0.95 is nearly opaque and admits a flattening of 1.15. A cloth at cover 0.5 has a quarter of its area as hole and admits a flattening of 4.87.

A cloth that cannot be seen through is a cloth a calender cannot help, and the exchange rate between the two is the table above. That is a sharper statement than the usual finishing-trade advice, which is that a cloth must be dense to take a good calender finish — a claim that has the sign exactly backwards and survives because dense cloths are finished well for a different reason, which is the next section.

The warp caps first, on every cloth in the table #

The two thread systems have their own covers and therefore their own caps, and they are not the same.

The warp caps first on all eleven, which is required rather than observed: every construction in this account’s table is warp-dense, because a warp is set by a reed and a weft by a gear and the loom can choose one of them ninety-nine times more finely than the other.

That has a consequence a nip cannot avoid. Past the warp’s cap the warp cannot widen and the weft still can, so further pressure is put into one system only — and a cloth pressed past its warp’s cap is being weft-flattened while its warp is being compacted. Those are two different mechanisms happening at once in two different systems, and every number on the lustre account assumes one mechanism happening in both.

On this account’s own table the effect is narrow. Nine of the eleven cap above ×4, comfortably outside the ×1.5 to ×3 a calender uses; the 40-end poplin caps at ×3.57 and the 44-end poplin at ×2.91, which is inside. So the cloth this account once called impossible is also the one whose warp a calender can close.

The cap has no yarn in it #

The frontier is drawn for a 200-micrometre yarn and it is the same frontier for every yarn, which is worth establishing because it is what makes the cap a construction limit rather than a material one.

Both the section’s width and the thread’s pitch are proportional to the diameter — the first by the racetrack’s own scaling and the second because the cover is a sett times a diameter. So the condition a(f) = d/k has the diameter on both sides and it cancels: the flattening a cloth admits depends on its cover and on nothing else.

The plateau does not cancel, because it is a length: at a cover of a half a 200-micrometre yarn takes 318 micrometres of plateau and a 350-micrometre yarn takes 556, in exact proportion. So a coarse yarn buys more plateau at the same cover, and that is simply because there is more of it.

Which puts the two halves of a calender’s arithmetic in their places. The cap is dimensionless and belongs to the construction; the plateau is a length and belongs to the yarn; and the specular area is their product with the crown line, which belongs to the weave. Three factors, three owners, and the account has been quoting all three as though they belonged to the machine.

Why the trade’s advice has the sign backwards #

A finishing manual will say a cloth must be closely set to take a chintz or a schreiner finish well, and the table above says a closely set cloth admits less flattening. Both are true and they are about different things.

The cap is on the flattening of one thread. A close cloth reaches its cap sooner and gets less plateau per thread.

The lustre is the plateau times the number of threads. Lustre is a length times a width: the specular area factors into a length of crown line, which the draft and the sett supply, and a width of section, which the yarn and the finish supply. A close cloth has more crown line per unit area in exact proportion to its sett.

So the two effects pull opposite ways and the product is what a finisher sees. At a fixed yarn, doubling the sett doubles the cover, halves-and-more the plateau available, and doubles the crown line — and the plateau falls faster than linearly over most of the range, so the product has a maximum at an interior cover rather than at either end.

Locating that maximum exactly needs the crown-line arithmetic run against the cap, which is one multiplication of two curves this account already has, and it is the thing this essay most obviously leaves undone. What can be said now is that it is not at full cover, because there the plateau is zero and the product is zero with it.

Where one model stops and another has to start #

The cap is not only a number; it is a boundary between two mechanisms, and saying which side of it a cloth is on decides which arithmetic applies.

Below the cap, pressure reshapes. The racetrack conserves area: the yarn gets wider and thinner, its fibres slide past one another laterally, and the energy is a shape strain computed from the section’s aspect ratio.

At the cap, reshaping stops. The width is pinned by the neighbours and area can no longer be conserved by trading thickness for width, so further pressure has to take area out of the yarn — which is compaction, a rise in the packing factor, fibres moving closer together rather than sideways.

Those have different energies, different pressures and, critically, different recoveries. Nothing in this account knows what fraction of a flattening survives and there is no reason to expect the two mechanisms to recover alike — a reshaped section springs back by its fibres’ lateral elasticity and a compacted one by whatever holds a yarn’s packing factor, which is friction rather than elasticity.

So a cloth pressed past its cap is in a regime this account has no model for at all, and the essay before it’s honest bracket becomes two brackets with a boundary between them that nobody has measured either.

What a finisher could do with the cap #

Two practical readings follow, and both are the kind that can be checked at the machine.

A cloth that will not take a shine is not necessarily under-pressed. If its warp cover is above about 0.8 its cap is under ×1.75, and a heavier nip is putting its work into compaction rather than into plateau. The remedy is a different cloth, not a harder press — and the arithmetic says which: the same weight in a finer yarn at a lower sett, which holds the cover down while holding the mass.

And a finish that will not hold is not necessarily a recovery problem. Compaction is what a nip does past the cap, and compaction is undone by wetting, because water swells the fibres and pushes the packing factor back down — which is mercerising’s mechanism running in reverse. A cloth finished past its cap should lose its finish in the first wash more completely than one finished below it, and the two cloths would differ by nothing a specification records.

The same cap, arriving on three other accounts #

A limit stated as “a thread meets its neighbours” is the jamming argument, and once it is recognised as that, three results elsewhere in this account turn out to be the same statement.

A jamming sett falls as a yarn flattens. What a flattened yarn does to its cover has a flattened yarn a third wider than a round one of the same area, so a cloth of flattened yarn covers more at the same sett — which is this cap read from the sett’s side rather than from the flattening’s. The two are one inequality with a different unknown.

A specification’s setts are bounded by its weave’s crossings. A specification can name a cloth that is not there measured how much of a grid of specifications names nothing, and the bound there is the same width-per-repeat argument with the crossings included. A calender moves a cloth within that grid — it raises the cover without changing the sett or the count — so a cloth finished past its cap has been moved into the empty region by the finisher rather than by the designer.

And the beat-up has a ceiling for the same reason. A pick density is a force budget found the loom’s force ceiling cutting the top off the range of reachable pick densities, and noted that a finish moves the geometric ceiling — a calender flattens the yarn and lets the same threads sit closer. This essay is the other half of that sentence: the finish moves the ceiling, and the ceiling moves back and stops the finish.

That is three accounts meeting at one inequality, and it is the reason the cap is worth stating as a quantity rather than as a caution. A caution applies once; an inequality applies wherever its terms appear.

What was counted, and how #

The section is this account’s own racetrack at conserved area, and the cap is found by bisection on its width rather than by a closed form. The closed form was tried first and is wrong: an elliptical section at conserved area gives f = 1/k², which at a cover of 0.5 predicts ×4 against the racetrack’s ×4.87 — a fifth out, and out in the direction that makes the cap look tighter than it is.

The identity behind the cap is required at every row: the section’s width at the cap is exactly the thread pitch, to nine figures. That is what the bisection was solving, so it is a check on the search rather than a result, and it is the check that would have caught the closed form.

The covers are this account’s own cloth-state solve, from each construction’s counts and setts through Peirce’s diameter, so the caps here and the covers in Where the cover factor comes from are one calculation.

And the warp-first claim is required over the whole table rather than observed on a few rows, so a construction added to the table with a denser weft than warp would fail the check rather than quietly weaken the sentence.

What the cap cannot say #

It is a geometric limit and a real cloth passes it. Threads are not incompressible and a nip at enough pressure will push one end over its neighbour rather than stopping at contact. The cap is where the model stops conserving area, and what a real cloth does past it is the compaction regime this account does not have — so the cap should be read as “beyond here the arithmetic changes” rather than as “beyond here nothing happens”.

The cover used is the single-system cover and a cloth has two. The covering rule combines them as $k_1 + k_2 - k_1 k_2$, and that combined figure is what an opacity is computed from; the cap is set by one system’s own cover, because a warp end’s neighbours are warp ends. The two are different numbers and the essay above is careful to use each where it belongs, but a reader carrying “cover” as one word will conflate them.

And nothing here is about the crimp. A flattened thread in a cloth is also a thread on a curved path, and the plateau that a calender leaves is only specular where the crown is level — a float’s own crown line is what decides how much of the plateau is flat enough to reflect. The cap bounds the width factor and says nothing about the length factor, which is the other half of the product.

Who found it, and when #

That a cloth cannot be flattened indefinitely is obvious and is nowhere written down as a quantity. The reason is the same one that keeps appearing on this account: finishing is described operationally, by pressure, temperature and speed, and the cloth’s own geometry enters only as “a close cloth finishes differently”.

The cap needs three things the trade does not put together: a section model that says how width depends on flattening, a cover that says where the neighbours are, and the observation that the first must stop at the second. This account has carried the first two since its foundation and has been computing calender results without the third.

The specific number worth carrying is the one at the top of the range. At a cover of nine tenths a calender can flatten a thread by 32 per cent and no more, which is a sixth of what the machine is capable of and a quarter of the plateau the same yarn would take in an open cloth. Anybody who has pressed a fine dense cambric and been disappointed has met that number without having it.

Still open: where the product’s maximum is #

The cap falls with the cover and the crown line rises with it, so the specular area has an interior maximum in the sett — and finding it is one multiplication of two curves this account already holds.

The reason it is not done here is that the crown-line factor depends on the weave as well as the sett: a satin’s crown line spans a factor of sixty across the four-by-four catalogue, so the maximum is a maximum over two variables and its location moves with the draft. A one-variable answer for plain weave would be a few lines and would be quoted as though it were general.

What the answer would settle is the oldest question in the lustre trade: what construction takes the best calender finish. The ingredients are all here, the arithmetic is a product, and the reason to be careful is that the result would be a recommendation rather than a measurement — which is the kind this account is slowest to make.