A calender works where nothing has been measured

Twice this account has stopped at the same place. Calendering arrives at the other model found that a calender moves a cloth from one of this account’s thread sections to the other and had no number in it, because the amount of the move was free. A calender spends the compression for good supplied the number as a pressure and then said, in its own closing lines, that the missing piece is plasticity — “until this account has a law that says what fraction of a flattening survives, every finishing essay here computes what the machine does and not what the customer gets.”

The law is missing for a sharper reason than nobody having written it down. Nobody has measured anything at the strains a calender works at.

A flattening is a strain, and it is a large one #

Kemp’s racetrack at conserved area is this account’s model of a pressed section: the yarn’s cross-section keeps its area and takes an aspect ratio f, so its width goes as $\sqrt{f}$ and its thickness as $1/\sqrt{f}$. The strain that implies is the logarithmic one,

$$\varepsilon = \tfrac{1}{2}\ln f,$$

which is this account’s own logarithmic shape strain and is what a fibre at the extremity of the section actually experiences.

The mildest setting in this account’s own calender series is eleven per cent. For comparison, cotton breaks in tension at six to ten.

It does not break in a calender, because the strain is transverse and compressive and a fibre in compression across its axis simply flattens. But the number says where the machine is working, and it is a long way from anywhere a fibre’s elastic behaviour has been characterised.

Where recovery has been measured, and it is not there #

This account carries elastic-recovery data for seven fibres, and it carries it as measurements because nothing predicts it — the function that reads it refuses outside its measured span in both directions, on purpose, because extending cotton’s last segment to eight per cent gives a recovery of 0.16 and to ten per cent gives minus seven.

Six of the seven have no recovery data at any calender setting at all, and at the heaviest setting flax is twenty-seven times past its last measured point.

Wool is the exception and it is a narrow one. Its recovery is measured to twenty per cent, which covers a flattening of 1.25 and nothing above it — the next setting in the series, ×1.5, is 20.27 per cent and already outside. So the one fibre with data has it for one of five settings, and that setting is the one nobody calenders at because it buys almost nothing.

Which means the missing law is a missing experiment #

That changes what the account’s own shortfall is. It was recorded as a gap in the model — a law this account has not derived. It is a gap in the data, and the two want quite different work.

A missing law can sometimes be reasoned to. This one cannot, and the reason is in the shape of the recovery curves themselves. Cotton recovers 0.45 of a 5 per cent strain; the trend over its two measured points is steeply downward; and the whole content of this account’s refusal to extrapolate is that continuing that trend is not a prediction, it is a fabrication. A recovery fraction is bounded in [0, 1] and every plausible extrapolation of every fibre’s curve leaves that interval before it reaches a calender.

So the honest statement is the one this essay can make and the previous two could not: the finished cloth’s section is somewhere between the cloth as woven and the cloth in the nip, and nothing in this account narrows it.

The rank might survive even though the value does not #

There is one thing the table can be read for that does not need an extrapolation, and it is worth separating from what it cannot.

Each fibre’s recovery at the top of its own measured range orders them: nylon 0.89, polyester 0.70, silk and flax 0.65, cotton 0.45, wool 0.35, viscose 0.32. The strains those are measured at differ — flax’s is 2 per cent and wool’s is 20 — so the ordering is not a comparison at a common strain and cannot be read as one.

But a rank is a much weaker claim than a value, and if the curves do not cross between the measured range and the calender’s, the rank at 5 per cent is the rank at 35. Nothing here says they do not cross; what can be said is what would follow if they did not: a calendered nylon would hold its lustre best and a calendered viscose worst, with cotton in the lower half.

That is a prediction with a cheap test and it is the one the trade’s own practice bears on. A schreiner finish on a synthetic is famously durable and the same finish on cotton is famously not — cotton is resin-treated before calendering precisely to make the flattening stay — and the rank above puts cotton at 0.45 against polyester’s 0.70. The practice is consistent with the rank and is not evidence for it, because a hundred other things differ between the two fibres; it is worth one sentence and not more.

How wide the bracket is, in the quantities the account computes #

The bracket is not a wide error bar on a known answer. It is the whole interval, and the interval is large in exactly the places the lustre account cares about.

Lustre. A calender buys the width found that pressing a cloth multiplies its specular area by twenty-four, and that none of that comes from the crown line and all of it from the width of the plateau. If nothing recovers, the customer gets twenty-four; if everything recovers, the customer gets one. The bracket on the single quantity a calender is bought for is a factor of twenty-four.

Cover. What a flattened yarn does to its cover has a flattened yarn a third wider than a round one of the same area, so every opacity and permeability computed from a round diameter moves in one direction — and how far is the same unknown fraction.

Thickness. The section’s thickness goes as $1/\sqrt{f}$, so at a flattening of two the cloth in the nip is 71 per cent of its unpressed thickness, and the finished cloth is between 71 and 100 per cent.

And the evenness. A finish spends a spread before it spends a mean computed calendering’s log-slope at 1.4 and read off that the operation makes a cloth less even. That result is a slope and slopes survive a scale factor — so it is the one essay of this account the missing recovery does not touch, and it is worth saying which results are robust rather than only which are not.

The one end of the bracket that is not a guess #

There is a reason to prefer the pressed end as the working assumption, and it is not that it is convenient.

A calender’s nip is not a momentary squeeze. The cloth is in it for a dwell set by the roller diameter and the line speed — a few milliseconds on a fast machine — and it is usually hot, because a calender bowl is steam-heated to a hundred and fifty degrees or more. Both of those push the deformation towards the permanent end: a viscoelastic material held at strain for longer relaxes more of its stress, and a material above its glass transition sets rather than springs.

That is why calendering is described in the trade as a heat-setting operation on synthetics and as a temporary finish on cotton. Polyester’s transition is around eighty degrees and a calender bowl is well past it; cotton has no transition to cross dry, and the operation on cotton is mechanical flattening with nothing to hold it.

So the fibre table’s rank and the machine’s temperature say the same thing from two directions, and the honest reading is that the recovered fraction is small on the melt-spun fibres and large on the cellulosics. That is a statement about which half of the bracket, not about where in it, and it is the most this account is willing to say.

Why the trade gets away with it #

A bracket of twenty-four on the quantity a machine is sold for ought to make the machine unusable, and calenders have been in every finishing works for two centuries.

The reason is that a calender is not specified, it is set. A finisher runs a sample, looks at it, and turns the nip until the cloth looks right — which is a closed loop round the unmeasured quantity, and a closed loop does not need the open-loop law. What the missing law costs is not the ability to finish cloth; it is the ability to predict a finish, to transfer a setting between machines, and to say in advance what a nip pressure will produce on a cloth nobody has run.

That is exactly the list of things a specification is for. A specification names a cloth by its yarn, its setts and its weave and says nothing about its finishing state, and this is the arithmetic behind why: the state cannot be computed from the machine’s settings, so it has to be carried as a sample.

And the sample is a physical object, which is the one notation in this account that cannot be transmitted, searched or checked — so a trade that finishes by eye is a trade that has to keep swatches.

What the experiment would be, and how small it is #

The measurement the whole account is waiting on is narrow enough to state precisely, which is worth doing because a shortfall stated as “plasticity” invites a research programme and this is an afternoon.

Take a single yarn. Press it between flats at a stated line load. Release it. Measure its width. That is a transverse compression to a known aspect ratio and a recovery from it, and the quantity wanted is the ratio of the released aspect ratio to the pressed one, at shape strains of 0.11 to 0.55.

Three things make it easy. The strains are large, so the recovered and unrecovered sections differ by tens of per cent rather than by a few — nothing needs fine measurement. The section is visible: a yarn’s flattened cross-section can be photographed, and this account’s own arithmetic converts a width to an aspect ratio at conserved area. And the answer is one number per fibre per strain, which is a table of about twenty entries.

What would change if it existed. Every essay of this account would acquire a second column — the machine’s number and the customer’s — and the A calender buys the width would become a factor. Mercerising, which is permanent and therefore has no recovery, would become the reference case rather than a separate mechanism: it is the same arithmetic with the fraction set to one, and it is the only finishing operation in this account whose customer number and machine number are known to agree.

The same absence, in a place nobody was looking #

The gap does not stop at the lustre account, and naming where else it reaches is the honest accounting.

A thickness gauge presses. The relaxed cloth’s contact force reads a cloth’s own crossing force out of a thickness measurement, and the reading is taken under a foot at a stated pressure — which flattens the sections while it measures them. The strains are smaller than a calender’s, but the same recovery decides how much of the flattening is still there when the foot lifts, and every thickness in this account’s tables is a pressed thickness with an unknown amount of the pressing already recovered.

A seam presses. A crease cannot cross a seam computes a fold across a stack of layers, and a pressed seam is a calender applied to four thicknesses of cloth at once.

And a roller in any machine presses. A cloth passes through a padder, a stenter’s rollers and a batching drum before anybody measures it, and each of those is a nip.

So the finished number a specification quotes has been through several unmeasured recoveries rather than one. That does not widen the bracket — the bracket is already the whole interval — but it does say that the cloth in the warehouse is not at either end of it, and that the question “what does a calender leave” is really “what does a sequence of nips leave”, which is the same missing measurement asked of a history rather than of an operation.

What was counted, and how #

The shape strain is this account’s own logarithmic shape strain, ½·ln(f), which is the strain at the extremity of a racetrack section at conserved area. It is a logarithmic strain rather than an engineering one because the deformation is large and the two differ by a third at a flattening of two — quoting the engineering strain would have understated the gap.

The recovery data is this account’s own table and it is used through the function that refuses outside its measured span. That refusal is what produced this essay: the first attempt at the plasticity law called recovery reader at a calender’s strain and was refused, which is the arithmetic doing its job and is a better result than an extrapolated number would have been.

The comparison is required rather than described. The census required that exactly one fibre reaches any setting, that it is wool, and that it reaches only one of the five — so a table that grew a fibre with wider data, or a series that grew a lighter setting, would change the check rather than quietly change the claim.

And the bracket’s ends are both computed. The unpressed end is this account’s round section and the pressed end is the racetrack at the stated flattening; neither is an estimate. What is not computed is anything between them, and the figure draws that as a band rather than as a line for exactly that reason.

What this cannot say #

The recovery data is tensile and the deformation is transverse. Every number in the fibre table is an elastic recovery measured by stretching a fibre along its axis and letting it go; a calender squashes it across. Those are different deformations of different parts of the fibre’s structure, and there is no reason the tensile recovery at eleven per cent would be the transverse recovery at eleven per cent even if it had been measured.

So the gap is wider than it looks. The table says no fibre is measured at the strain; it is also true that no fibre is measured in the mode. A transverse-recovery measurement at small strains would not close the gap either, because the strains are large; the experiment has to be both.

And a yarn is not a fibre. A yarn’s stiffness is a bracket because its fibres may slide or may not, and a flattened yarn recovers partly by its fibres springing back and partly by their sliding back — the second of which has a friction in it and no fibre measurement will ever supply. The single yarn-level experiment above measures the combination, which is what the account needs, and it means the answer is a property of the yarn’s construction rather than of its fibre.

Who found it, and when #

Calendering is old — the word is mediaeval and the machine is eighteenth-century — and every finishing text describes the effect and none gives a recovery. Elastic recovery of textile fibres is a standard measurement, made by Meredith and others from the 1940s, and the standard test is tensile, at strains of a few per cent, because a few per cent is where a textile fibre is used.

That is the whole of the explanation, and it is a good one. The recovery data exists at the strains a fibre is worn at, and a calender works at the strains a fibre is processed at, and the two are an order of magnitude apart. Nobody left a gap; two different questions were asked and only one of them was about clothes.

What is this account’s is the comparison, and it needed the shape strain to be computed from the section model rather than estimated — which is why it could not have been made before the racetrack and the circle were shown to be one yarn at two moments.

Still open: whether the bracket is wide in practice or only in principle #

A factor of twenty-four is the bracket the absence leaves, and the trade’s behaviour suggests the real answer is near one end of it.

A calendered cloth that lost its lustre on leaving the nip would not be sold, and calendered cloths are sold, so the recovered fraction cannot be near one. A calendered cloth that kept all of it would not need re-calendering after washing, and it does, so the fraction is not zero either.

Those two observations bound it from outside the arithmetic, and they are worth writing down because they are the only bounds this account has. They say the answer is interior; they do not say where; and turning them into a number needs the same yarn-level experiment, which would also say how much of the recovery is immediate and how much arrives over the days a finished cloth spends in a warehouse before anybody looks at it.