A calender's best cloth is the one its nip fills

A covered cloth cannot be calendered found the cap on a calender’s work: a thread flattened at constant area widens until it meets its neighbours, and a cloth of cover k lets a thread widen only to its own pitch, d/k. The plateau the nip can make falls to nothing at full cover. And since lustre is a length times a width — crown line, which rises with the sett, times plateau, which falls with it — the essay expected the product to peak somewhere in between, at a cover that would move with the weave, and left the multiplication undone.

Done, the multiplication says two things the expectation did not. On the cap alone the product has no peak: it falls from the most open cloth to the most covered. And the peak a real calender does have is not made by the cloth at all. It is made by the nip.

The product, written out #

Take one thread system — the one on the face, whose crowns are the lustre. Per unit of the cloth’s plan it carries crown line equal to its crown line per crossing, L, times its threads per unit width. At cover k and diameter d, the threads per unit width are k/d.

Each length of crown line carries a plateau: the flat run on top of the flattened section, which is the width within the specular tolerance. So the share of the plan that is specular is

$$S = \frac{L \cdot k \cdot \text{plateau}}{d}.$$

Three things are in it and only one is the weave. L is the draft’s: the crown line per crossing, a quarter for a 2/2 twill, three quarters for an eight-end satin, nothing for a plain weave. k is the construction’s. And the plateau is the finish’s — how far the calender has flattened each thread.

On the cap alone, it only falls #

Suppose the calender flattens every thread as far as the cloth allows: to the cap, where the thread is exactly as wide as its pitch. Then the plateau is the cap’s plateau, and k times the plateau over d is exactly the plateau as a share of the thread’s own width at the cap — the plateau share, the last column of the cap essay’s table.

That column falls monotonically: 0.928 at a cover of 0.3, 0.795 at 0.5, 0.577 at 0.7, 0.240 at 0.9, nothing at 1. So on the cap alone the product has no interior maximum. The crown line rising with the sett is exactly cancelled, in the first factor, by the thread’s pitch shrinking with it, and what is left is the plateau share, which only falls.

The expectation had the right two curves and multiplied the wrong things. Crown line per unit area does rise with the cover; plateau per thread does fall; but the plateau per thread falls faster than the cover rises at every cover, so the product is largest for the most open cloth a calender can reach the cap of.

A real nip does not reach the cap of an open cloth #

That last clause is where the peak comes from. A calender does not flatten every thread to its cap. It flattens by whatever its nip does — a pressure, a temperature, a dwell — and a calender works where nothing has been measured found that even its mildest setting, a flattening of ×1.25, is a shape strain of 11 per cent, past cotton’s measured recovery.

So each nip has its own flattening f, and on an open cloth the nip runs out long before the cap does: the thread is flattened by f and no further, and it is nowhere near its neighbours. On that side of the tent the plateau is fixed by the nip, and the product rises in proportion to the cover — more threads, each with the same plateau.

On a close cloth the cap runs out first: the thread meets its neighbours at a flattening less than f, and the extra pressure has nowhere to go. On that side the product is the falling plateau share.

The peak is where the nip exactly fills the cloth #

The two sides meet where the nip’s flattened width equals the cloth’s pitch:

$$k^* = \frac{d}{a(f)},$$

the cover at which a thread flattened by the nip exactly fills the space the sett gives it. At that cover the plateau share is the nip’s own flat run over its own width, and for a racetrack at aspect f that is exactly

$$S^* = L\,\frac{f - 1}{f}.$$

Neither expression has the yarn in it. The diameter cancels from d/a(f), because the racetrack’s width at a given aspect is a fixed multiple of the diameter, and from (f − 1)/f entirely. A fine yarn and a coarse one have their best calender cloth at the same cover.

Why the peak is (f − 1)/f #

The height of the peak has a picture behind it that is worth having, because it says what a calender is selling.

A racetrack at aspect f is a rectangle with a semicircle on each end, f times as wide as it is thick. Its flat top runs the whole width less the two rounded ends, and the two ends together are one thickness wide. So the flat run is f − 1 thicknesses out of f, and the share of a flattened thread’s width that is flat is (f − 1)/f — a half at ×2, two thirds at ×3, a fifth at ×1.25.

At the cover the nip exactly fills, the threads’ flattened widths tile the cloth edge to edge, so the flat share of each thread is the flat share of the plan. That is all the peak is: a calender buys the width, and at the best cover the width it buys covers the whole cloth with no room wasted between threads and no thread held back by its neighbours. Either side of that cover, one of the two is happening.

A mild nip wants a covered cloth #

Read the numbers at the nips a calender actually uses.

At ×1.25, the mildest setting in this collection’s series, the best cloth has a cover of 0.919 and the lustre there is 0.20 per unit crown line. At ×1.5 the best cover is 0.853 and the lustre 0.33. Only at ×2, a shape strain of 35 per cent, does the best cover fall to 0.75, and only at ×3, 55 per cent, to 0.63.

So a mild calender wants a nearly covered cloth. That is the finishing trade’s rule — a cloth must be closely set to take a good chintz or schreiner finish — and the cap essay said the rule had its sign backwards. On the cap alone it does. Against a real nip it is right.

The best cloth, in ends per centimetre #

The cover is a ratio, and a finisher sets a loom in ends per centimetre, so it is worth converting. A cover of 0.853 means a thread diameter’s worth of yarn in every 1/0.853 of a diameter’s width, and the sett is 0.853 over the diameter.

For a 20-tex cotton, 0.167 millimetres across at an ordinary packing, the ×1.5 nip’s best cloth is set at 51 ends a centimetre; the ×1.25 nip’s at 55. For a 30-tex yarn, 42 and 45. For a fine 10-tex yarn, 72 and 78. These are close setts — a shirting’s, not a sheeting’s — and they are what the tent says a mild calender is asking for: a warp set close enough that its threads, flattened by a fifth or a half, just meet.

The yarn cancels from the cover and not from the sett, which is the difference between the two numbers. Every yarn has its best cloth at one cover for a given nip, and each yarn reaches that cover at a sett of its own.

It also says why the calender trade talks in setts and gets away with it. A mill finishing one quality of yarn has, in effect, fixed the diameter, and for it a sett is a cover in different units. The rule “set it closer” is the cover rule in the only units that mill needs, and it fails only when the same rule is carried to a different count without rescaling.

Every cloth in the table is on the rising side #

The table of eight cloths used throughout has warp covers from 0.21 on a cheesecloth to 0.64 on a 44-end poplin.

Every one of them caps at a flattening of ×2.9 or more, above every nip in the calender series up to ×2.5. So at every nip short of the series’ hardest, every cloth in the table sits on the rising side of the tent, where more cover means more lustre in proportion; only the 44-end poplin at ×3 is just past its peak. For all the rest the trade’s advice is simply correct: set it closer and it will calender brighter.

The falling side, where more cover means less lustre, is reached only by a cloth set closer than the nip can fill — past 0.92 at ×1.25, past 0.85 at ×1.5. The dense shirtings are approaching it; nothing in the table is there.

The weave sets the height, not the place #

The one thing the cap essay was most wary of — that the peak would move with the draft — does not happen, because the weave enters only through L.

At ×1.5 the best cover is 0.853 for a twill, a satin and anything else, and the best share is a third times the weave’s crown line per crossing: 0.083 for a 2/2 twill, 0.167 for a 3/1, 0.200 for a five-end satin, 0.250 for an eight-end. The satin calenders three times as brightly as the twill at the same nip and the same cover, and it does so at the same cover.

A plain weave is the exception and it is a principled one. It has no crown line — its crowns are points, not lines — so its lustre under a calender comes from each crossing’s flattened top, a patch rather than a strip, and it is outside this product; no cloth shines under a sky found the same distinction deciding how a plain weave’s highlight scales with the light.

What the nip is spending #

The tent’s two sides differ in what the pressure does, and a calender spends the compression for good is the account of the second.

On the rising side the nip’s whole flattening goes into reshaping threads that have room to widen. The yarn keeps its area, the cloth gets thinner and wider-threaded, and the work done is the reshaping. On the falling side the threads meet their neighbours before the nip has finished; the pressure beyond that point cannot widen them further, and what it does instead is compact them — squeeze area out of the yarn — which is a different mechanism with a different energy and a recovery nobody has measured either.

So the best cloth is also the cloth on which a nip does the least work it cannot use. Set a cloth past k* and part of every pass through the nip is spent compacting rather than shining; set it short of k* and the nip’s reshaping is spread over fewer threads than it could have served. Calendering arrives at the other model — from Peirce’s circle to Kemp’s racetrack — and the best cover is where it arrives with nothing left over.

Mercerising moves a cloth along the axis #

A finish applied before calendering can move a cloth along the tent without touching its sett. Mercerising is a packing factor: the caustic swells the fibres, the yarn packs tighter, and its diameter at a given count falls. The cover is diameter over pitch, so a mercerised cloth has a lower cover than the same cloth before treatment, and on the rising side of every tent that means less calender lustre, not more.

That is worth stating because mercerised cotton is sold for its lustre. Whatever lustre mercerising adds comes from the fibre — a rounder, smoother section reflecting better — and not from the geometry this essay computes, which it moves the wrong way. A cloth meant to be both mercerised and calendered has to be set closer to stay where the nip wants it.

The model named #

The section is Kemp’s racetrack at conserved area: at aspect f its width is a = f·b and its flat run a − b, with b the thickness. The cap is the aspect at which the width equals the pitch d/k, solved by bisection because the racetrack’s width has no closed inverse. The nip flattens every thread by f unless the cap binds first. The specular share is the crown line per crossing of the lustrous system, times its threads per unit width, times the flat run.

Required of the arithmetic, not shown: at the cover the nip exactly fills, the plateau share is (f − 1)/f to twelve figures; and no cover on the swept grid beats it, the best grid point lying next to k*. The cap’s own identity — width equal to pitch at the cap — is the cap essay’s and is checked there.

What was counted #

Four nips, ×1.25, ×1.5, ×2 and ×3, each swept across seventy-one covers from 0.3 to 1; the best cover and share over sixty nips from ×1.05 to ×4; the same tents for a yarn of 0.35 mm; and the crown line per crossing of five named weaves.

What the model cannot show #

Recovery. Every flattening here is taken as it is made. The calender essay found that a cotton’s recovery from a calender’s shape strain has not been measured at any setting, and a thread that springs back loses plateau in proportion. The tents here are the lustre at the nip; the lustre in the shop is some fraction of it, and the fraction may differ between the rising and the falling sides.

The weft. Only the lustrous system is counted. A warp-faced satin’s weft is hidden and contributes nothing; a twill’s weft carries crown line of its own, which adds a second tent at its own cover.

And the tolerance. The plateau is the whole flat run, as if every part of it were within the specular tolerance. The width a real light source finds specular is narrower than the flat run on a slightly curved crown, and the length-times-width account of lustre carries that width separately.

Who found it, and when #

Calendering is centuries old and its rules are empirical: close setts, hot bowls, a friction calender for chintz and an engraved one for schreiner. The racetrack section is Kemp’s, from 1958, and the cap is this collection’s.

What is added here is the tent: that the calender’s best cloth is set by the nip meeting the cap, at k* = d/a(f) with a share of (f − 1)/f, and that the trade’s rule and the cap’s reversal of it are the two sides of one curve. At the nips a cotton survives every real cloth is on the side where the trade is right.

Still open: whether the falling side is ever worth reaching #

The falling side of the tent is where a cloth is set closer than its nip can fill. The dense shirtings are approaching it at the mild nips, and a heavier nip moves it away — but a heavier nip is exactly the one whose shape strain no fibre has been measured to recover from.

So the practical question is a trade between two unknowns: whether a cloth set past k* for a mild nip loses lustre, as the arithmetic says, or holds it because a mild nip’s flattening recovers less. A calender run on one yarn at a series of setts either side of 0.85, with the lustre measured before and after a wash, would say which side of the tent real cloth lives on.