A calender spends the compression for good

Calendering arrives at the other model made the best observation available at the time: a calender flattens the threads of a cloth, this site carries two thread-section models that differ by exactly that, and so calendering is the process that takes a fabric from the first to the second.

It could not say how far. The racetrack’s aspect ratio was a free parameter, nothing on this site computed it, and the rung ended by drawing a cloth at a flattening somebody had chosen.

The conversion, which contains the surprise #

Two divisions and a multiplication, and the last of the three is where the interest is.

$$\text{pressure} = \frac{\text{line load}}{\text{nip width}}, \qquad N = \text{pressure} \times p_1 p_2$$

A machine setting of 30 newtons per millimetre over a five-millimetre nip is 6 N/mm², which is a substantial pressure and is what calenders run at — the same order as the pressure a wetting generates against a cloth’s own geometry. The area a crossing owns is the product of the two thread spacings — 0.137 mm² for a sheeting.

That area varies across this site’s table by a factor of eleven, from 0.104 mm² for a batiste to 1.111 for a cheesecloth. So the same nip gives a crossing of the batiste 0.63 newtons and a crossing of the cheesecloth 6.7, and neither the machine nor the setting says anything about it.

A finishing works sets a calender once and puts different fabrics through it, and the model says they are not being given the same treatment at all. That is not a small correction to a description; it is a statement about the operation of a machine, and it goes the awkward way round: the fine, closely set shirtings and sateens that lustre finishes are usually for are exactly the cloths whose crossings own the least area.

What the flattening buys #

The finish is sold for lustre and the physical basis of lustre is a flat.

A round thread reflects a highlight along a line and scatters everything else. A flattened one presents a genuine plane to the light — the flat of the racetrack, exactly (f − 1)·b wide — and a plane gives a specular reflection with an area rather than a line. So the quantity a calender is bought for is the fraction of the cloth’s surface that is flat, and the model computes it directly.

For a sheeting at a modest nip, the flat runs to 42 per cent of the thread spacing. That is nearly half the visible surface turned from curved to planar, and it is what the shine is.

The cover rises with it, from 0.52 to 0.72, because a flattened thread is wider. A cloth that was two thirds covered is now nearly fully covered, and its air permeability, its opacity and its handle all move — none of them by intention, all of them because the same flattening does all four.

A light nip and a heavy one are different processes #

Setting the line load down from thirty newtons per millimetre to eight makes the point that the arithmetic is not merely a scaling.

At eight, a sheeting’s crossing carries 0.22 newtons. Its sections flatten to about 1.4, the cloth thins from 0.39 mm to 0.30, and the cover rises from 0.52 to 0.59. Those are numbers inside the range the compression model was fitted in, at aspect ratios where a small-strain energy is defensible, and they describe a cloth that has been given a light finish and is still recognisably itself.

At thirty, the same crossing carries 0.82 newtons, the sections go to two and a half, the cloth is at 0.22 mm and the cover at 0.72. That is nearly a fully covered surface and a cloth of about half its original thickness.

The two settings differ by a factor of under four and produce cloths that would be described differently by anybody handling them. The nonlinearity is in the aspect ratio’s response rather than in the pressure, and it is why finishing is a craft of small adjustments: the useful range of a calender is narrow and its far end is not a stronger version of its near end.

Why a calender is hot, which this model does not explain #

Everything above is elastic and reversible. A cloth pressed and released springs back to a round section, and the model has no mechanism by which it would not.

Real calendering is hot — the bowls are heated, and for the brightest finishes one bowl is heated and runs faster than the other so the cloth is polished as well as pressed — and a hot-calendered cloth stays flat. It stays flat because the fibre is taken above the temperature at which its own structure can rearrange, and the flattened shape is then the shape it holds when it cools.

The heat is not a way of pressing harder. It is a way of not getting the energy back, and that is the whole difference between calendering and putting a cloth through a mangle. This model has no plasticity in it and therefore has nothing to say about which flattening survives — only about which flattening the nip produces while the cloth is in it.

That is a real limitation and it is worth being exact about what it costs. The pressure-to-flattening arithmetic is a statement about the cloth in the nip. Everything about the finished cloth — how much of the flattening survives, how it recovers in wear, how it comes back after washing — needs a constitutive law nobody here has.

There is a second reason the fine cloths lose out and it compounds the first. A crossing’s share of the pressure is the area it owns, and the area a crossing owns is the product of the two spacings — so it falls as the square of the sett, near enough, while the thread’s resistance to being squashed falls only as the square of its diameter, which is the count. A cloth made finer at constant cover therefore has both terms moving and they do not cancel: the force per crossing drops faster than the yarn’s resistance does.

Which is the arithmetic behind a thing finishers manage by feel. A fine cloth needs a harder nip than a coarse one to reach the same finish, and the setting does not scale with anything a specification records.

And it is spent, which is why the finish washes out #

The trade knows a calendered finish is not permanent on cotton. It softens in wear and it comes back substantially in the first wash, which is why a calendered cotton is sold as a finish rather than as a construction and why the permanent versions use resin.

The energy account says why in a sentence. The flattening stores compression energy; heat lets the fibres rearrange so the energy is dissipated rather than held; and what the cloth then has is a shape with no restoring force behind it. Wet the cotton and the fibres swell, the rearrangement partially undoes, and the section rounds up again — recovering a shape it has no memory of but that its own swelling drives it towards.

The same sweep on a much heavier cloth is what shows that the spending is a property of the cloth rather than of the machine.

The three things one setting changes at once #

A calender is bought for lustre and sold as a lustre finish, and the model says the nip changes four properties together with no way to separate them.

Lustre rises because the flat grows: at a modest nip 42 per cent of the thread spacing becomes a plane rather than a curve.

Cover rises with it, because a flattened thread is wider — 0.52 to 0.72 at a heavy nip for a sheeting. That is the same movement, seen from above rather than in section, and it closes the channels a cloth passes air down without touching the floor its yarn sets.

Thickness falls, from 0.39 mm to 0.22, and with it the handle: a thinner cloth of the same weight is a limper one, because bending stiffness goes as the cube of the thickness for a plate and something between the first and third power for a fabric.

Air permeability falls too, and steeply, because the pore between four threads is what a flattened thread is closing.

A finisher wanting one of those gets all four. There is no setting that raises the cover without thinning the cloth, and none that adds lustre without closing the fabric, because all four are consequences of a single geometric change with a single parameter. The nip has one knob and the cloth has four properties on it, which is why a lustre finish is specified by what it does to the handle as much as by what it does to the shine.

That is the honest limit of what a machine setting can be asked for, and it is a stronger statement than a list of side effects: the four are not independent quantities that happen to move together, they are one quantity read four ways.

The force is not the treatment, and the right variable is a product of covers #

An elevenfold spread in the force at a crossing is a striking number and it overstates the case, because a crossing that owns more area also presents more area to be pressed. What decides how far a thread flattens is the pressure across its contact, not the force on it, and the two differ by exactly the same geometry the conversion introduced.

Take the contact patch at a crossing as roughly a diameter square — which is what two cylinders crossing at right angles present once they have flattened at all. Then

pressure at the contact = nip pressure × (p₁p₂ ÷ d²) = nip pressure ÷ (K₁ K₂),

where K₁ and K₂ are the two geometric covers. The area term that produced the elevenfold spread appears once on top and once underneath, and what survives is the reciprocal of the product of the two cover factors.

That is a much better variable than the force, for three reasons.

It is smaller and it is still large. Across the table the product of covers runs from about 0.04 in the cheesecloth to a third in the batiste, so the local pressure spread is about eightfold rather than elevenfold — a correction, not a reversal.

It runs the same way, so the essay’s conclusion survives. A closely covered cloth sees a local pressure near the nip’s own, and an open one sees many times it. The fine, well-covered shirtings that lustre finishes are wanted on are still the cloths that get least, and now for a reason that does not depend on their crossings being small.

And it is on the specification already. A finisher setting a calender has the two cover factors in front of them, in the sense that the counts and the setts are on the order and the conversion is one constant. The area a crossing owns is not on any document.

The practical rule the arithmetic gives is therefore short. A calender setting scales with the product of the cloth’s two cover factors, so a cloth half as covered needs about half the line load for the same finish — and a finisher moving a setting from a sheeting to a scrim is delivering roughly six times the local pressure without touching the machine.

One caution, because the contact patch is the weakest step. Taking it as a diameter square is a statement about round threads, and the whole point of a calender is that the threads do not stay round: as the flattening proceeds the patch grows, the local pressure falls, and the process is self-limiting in a way the constant-patch reading does not show. That is a stabilising nonlinearity and it is the reason a calender has a usable range at all — a contact whose area did not grow with the load would flatten without limit once it started.

What was counted, and how #

The line load and the nip width are machine settings and are inputs. Nothing about them is derived and nothing about them is fitted; what the rung does is convert them, and the conversion is two divisions.

The area a crossing owns is the product of the cloth’s two spacings, which come from its setts, which are the construction. The flattening comes from minimising the same potential the compression ladder uses — bending energy plus compression energy plus the load’s work — over the aspect ratios and the division of the thickness together.

The one assertion in the calender table is written as a relation and not a value: the cloth that flattens most is the one whose crossings carry the most force. That would catch the commonest kind of error here, which is a conversion that lost the area term and made the flattening a function of the pressure alone.

The transverse modulus is the fitted constant the whole compression ladder rests on, read out of eight fabric thicknesses, and it carries every modelling approximation above it. So the aspect ratios here are quoted to two figures at most and the ordering is what they are for.

Where the model stops #

No plasticity, so no permanence. Stated above and it is the largest gap: this rung computes the flattening in the nip and cannot say what survives.

No heat, in a process that is defined by heat. Fibre modulus, transverse stiffness and friction all move substantially with temperature and none of them moves here.

No polishing. A friction calender runs its bowls at different speeds so the cloth is dragged as well as squeezed, which lays the surface fibres down along the cloth — a different mechanism entirely, closer to what raising does in reverse, and not modelled.

The aspect ratios are past the model’s warrant. At a heavy nip the sections come out between two and a half and five, and the compression energy is a quadratic in the log aspect derived for small strains. At an aspect of three the log strain is 0.55 and the energy is being asked to work outside where it was derived. The figures are drawn to show ordering.

And the residual thickness is a thickness under load. A gauge measures a calendered cloth after it has left the nip, and everything here is the cloth while it is in it.

The generalisation #

A machine setting and a material’s experience are separated by a geometry, and the geometry is usually the part nobody records.

A calender is set in newtons per millimetre and a crossing feels newtons, and between them is an area that belongs to the cloth. The same shape appears everywhere a distributed load meets a discrete structure: a press setting and the force on one rivet, a rolling load and the stress at one contact, a clamping torque and the pressure at one gasket rib. In each case the machine is quoted in the units the machine is built in, the material responds in the units the material is built in, and the conversion carries a term that varies by an order of magnitude across the things the machine is used on.

The practical form is a warning about process settings. A setting that has been optimised on one substrate is not a setting; it is a setting plus a geometry, and moving it to a different substrate moves the treatment by whatever the geometries differ by. Here that is a factor of eleven, entirely invisible on the machine.

Who found it, and when #

Calendering is very old and its mechanics are well understood in the trade: line loads, nip widths, bowl temperatures and speed differentials are all standard settings and are quoted as such.

Kemp’s racetrack of 1958 is the section a flattened yarn is modelled with, and using it to describe a calendered cloth is standard practice rather than an innovation.

What is this site’s is the conversion in both directions — the line load to a force at a crossing, and the resulting flattening from an energy rather than from a measurement — and the observation that a crossing’s share of a nip pressure varies elevenfold across ordinary constructions, which follows in one line from the areas and appears to be nobody’s stated result.

Where the ladder goes next #

Sideways, the same flattening is what makes a cloth’s cover a property of its history rather than of its construction alone, and what a thickness gauge is reading when it recovers the force a relaxed cloth carries.

Along the lustre ladder, the missing piece is plasticity. Until this site has a law that says what fraction of a flattening survives, every finishing rung here computes what the machine does and not what the customer gets — and mercerising, which is permanent, is a different mechanism with the same gap.