A satin's hole is a slot

The float decides a great deal on this site: lustre, snagging, abrasion, tear, drape, how close a cloth can be set. It has never decided anything about what a cloth passes, because the open area is one minus the cover and the cover has no float in it.

That is still true, and it is where the cover factor comes from: two cloths at the same sett in the same yarn have the same open area whatever their weaves, asserted to twelve decimal places whenever the comparison is drawn. And they do not pass the same air.

What a duct passes is its area times the square of its hydraulic diameter, divided by a dimensionless number that depends on the shape of its section and on nothing else. Hold the area and draw the rectangle out, and both of those move against it.

The claim #

At equal open area, a hole drawn out into a slot passes less than a square one, and the loss reaches sixty per cent by an aspect ratio of twenty.

Two things cause it and neither is a correction to the other:

• The hydraulic diameter falls. For a rectangle w × h it is 2wh/(w + h), and at constant area the sum w + h is smallest when the two are equal — so drawing the rectangle out raises the perimeter and cuts the hydraulic diameter. Since the flow goes as its square, this alone costs a factor.
• The shape factor rises. The dimensionless number is 14.23 for a square and climbs monotonically to 24 for an infinitely thin slit. That is Shah and London’s result from 1978, quoted rather than derived, and it is the only place on this site where it does any work.

Where a weave puts a cloth on that curve #

A float is a run of parallel threads lying at one level with nothing between them, so the region beside a float is not divided into square cells by crossings — it is divided the long way and not the short way, and the hole is longer than it is wide.

How much longer depends on the float, and the honest statement is a bound rather than a number. A float of length n can present a hole up to n times as long as it is wide, if the threads within the float lie tight against each other; it presents a square hole if they stay at their reeded spacing. Where a real cloth sits between those depends on the lateral grip its crossings supply, which is the same quantity that decides whether an end drifts at all.

So the arithmetic gives the range a weave makes available:

• a plain weave has no float, so its holes are as square as its setts make them, and an unbalanced sett is the only thing that draws them out
• a 2/2 twill can reach 2:1, which is a three per cent loss
• an eight-end satin can reach 8:1, which is about seventy per cent of the square’s flow
• and a jacquard figure, whose floats are limited only by what a designer will allow, can reach further still

The upper end of that range is not reached by a well-set cloth and is reached by a badly set one, which turns a shape factor into a symptom: an unexpectedly low permeability at the right cover is evidence that the threads have grouped.

The other thing that draws a hole out #

A weave is not the only route to an elongated hole, and the commoner one is the sett.

An unbalanced cloth has different spacings in its two directions, so its holes are rectangles before any weave is applied. A poplin at 32 × 22 has a warp gap of 168 µm and a weft gap of 287 — an aspect ratio of 1.71, which is a two and a half per cent loss. That is small, and it is worth having because it is not zero: an unbalanced cloth passes slightly less air than a balanced one of the same open area, and nothing in the cover factor says so.

The two effects combine badly in one common case. A warp-faced sateen at a warp-dense sett has its long floats running along the direction that is already closer set, so the weave and the sett draw the hole out the same way rather than against each other.

What it is worth in a real cloth #

Two and a half per cent for a poplin’s unbalance and three per cent for a 2/2 twill are not large numbers, and it is worth saying where they sit against the other things that move a permeability.

A ten per cent change in sett moves the flow by tens of per cent. A calendering pass moves it by a factor of three. The floor set by the yarn is four hundred times below what an ordinary cloth passes. Against those, a few per cent from the shape of a hole is a detail.

It stops being a detail at two points. The first is a long-float cloth whose threads have grouped, where the aspect ratio is not two but eight or twelve and the loss is a third to a half. The second is a comparison — two fabrics measured against each other at nominally the same construction, where a few per cent is exactly the size of the discrepancy that gets attributed to the yarn, the finish or the instrument.

And it is worth having for a reason that has nothing to do with its size: it is the only route by which the float enters what a cloth passes at all. Every other quantity in this collection’s transport arithmetic — the open area, the cover, the pore radius, the floor — is blind to the weave. This one is not, and it is small precisely because the weave has so little else to work with.

What was counted, and how #

The shape factor is Shah and London’s polynomial in the aspect ratio, interpolating between the known ends: 14.23 at unity and 24 in the limit. It is quoted because deriving it needs the series solution of Poisson’s equation on a rectangle.

The sweep holds the area exactly and asserts it, at every point, because the whole argument is a comparison at constant area and a sweep that had drifted would be comparing nothing. It also asserts monotonicity: drawing a hole out never passes more.

Both terms are viscous. The inertial term does not know the hole’s shape at all — it is the cost of accelerating air to the velocity the area demands, and the area is held — so the slot penalty is a penalty on the viscous half of the drop. That means it is largest in a close cloth, where the viscous half is most of the drop, and it very nearly vanishes in an open one where inertia is paying the bill.

That interaction is worth stating plainly because it reverses the intuition. A satin’s slot costs it almost nothing in an open curtain fabric and costs it most of the forty per cent in a close shirting — and a close shirting is where somebody is measuring permeability and expecting the cover factor to explain it.

Why the penalty is a lower bound on what grouping costs #

The slot arithmetic prices the shape of a hole and holds its area, which is the right comparison to make and is not the whole of what grouping does to a cloth.

Threads that group do not merely stretch one hole out. They leave a larger gap on one side and a smaller one on the other, so a repeat that had four equal holes acquires two wide ones and two narrow ones — and that is a change in the distribution rather than in any one hole’s shape.

Both effects act at once and they act in opposite directions. The unequal sizes raise the flow, because what a channel passes is convex in its size and spreading a fixed area across unequal holes passes more. The elongation lowers it, by the arithmetic above. So a grouped cloth’s permeability is the difference between two effects of comparable size, and which wins depends on how the grouping divided itself.

That has an awkward consequence for the diagnostic offered above. A low permeability at the right cover is evidence of grouping; a normal permeability at the right cover is not evidence of its absence, because the two effects can have cancelled. The measurement is a one-sided test.

The pore-size measurement is the two-sided one. Grouping raises the largest hole under both effects — a wider gap is a wider gap and an elongated hole has a larger inscribed circle along its length — so a rating that reads the maximum moves the same way whichever effect dominates. That is why a bubble point catches a drifted or grouped cloth and a flow measurement may not, and it is a second and independent argument for the instrument the filtration rung recommends.

The two effects also have different reaches. Elongation is bounded by the float — a hole cannot be drawn out further than the run of parallel threads that bounds it — while the inequality of sizes is bounded only by the threads meeting, which is the whole spacing. So at a large enough grouping the convexity wins outright and a grouped cloth passes more than its cover says, which is the state a badly set satin actually reaches and is the opposite of what the slot arithmetic alone would predict.

The limit at the end of the curve #

The curve does not go to zero, and where it goes is worth naming because it is the shape rather than the value that is the result.

At constant area, as the aspect ratio grows, the hydraulic diameter approaches twice the short side and the shape factor approaches 24. So the flow approaches w h³ · ΔP / (12 μ L) — the exact result for flow between parallel plates, which is where the 24 came from in the first place. The whole curve is an interpolation between two exact answers, a square duct and a slit, and the interpolation is Shah and London’s arithmetic rather than this collection’s.

The important feature is the cube. At a fixed area the flow between plates goes as the cube of the short side, and the short side goes as one over the aspect ratio, so the flow falls as the square of the aspect ratio in the far limit. That is why the curve keeps falling rather than levelling: doubling the aspect ratio of an already-long slot costs a factor of four, indefinitely.

A weave cannot get there. An eight-end satin’s float is eight threads and the aspect ratio it makes available is eight, which is well inside the interpolation. The far limit belongs to a crack rather than to a cloth — and it is the reason a seal fails at a scratch rather than at a dent of the same area.

Where the model stops #

The hole is idealised as a rectangle and it is not one. A real hole beside a float is bounded by two cylinders the long way and two the short way, with rounded corners and a waist that is not at the mid-plane. The rectangle is the section the hydraulic-radius convention is defined for, and using it here is a convention rather than a measurement.

The threads’ lateral positions are not computed. Whether a float’s parallel ends group is a force problem — the crossings’ lateral grip against whatever pushes them together — and this collection has the grip for a pick along its own length but not for an end pushed sideways. Everything above about how far a weave draws a hole out is therefore a range with a mechanism named and not a prediction.

Fully developed flow is assumed for the shape factor and is not present. The channel is entrance region from end to end, and the shape factor for a developing flow is different from Shah and London’s; using theirs overstates the difference between shapes.

A balanced plain weave is not perfectly square either. A muslin at 24 × 22 has an aspect ratio of 1.15, so even the reference case in the table above is a little drawn out — by half a per cent of flow, which is below the precision of anything else here and is quoted only so that square is not read as a description of any real cloth.

And the census’s other effect points the other way. The same four thread positions that make a hole long also make its waist wider than its mouth, which passes more. Both are computed here and neither is a correction to the other; a satin’s hole is simultaneously wider than it looks and worse shaped than it looks.

The generalisation #

A quantity that is claimed to depend on an area usually depends on a shape as well, and holding the area fixed is the experiment that separates them.

The check costs nothing and is almost never made, because the area is the thing that is easy to measure and the shape is the thing that varies without anybody choosing it. A duct, a crack, a channel in a chip, a gap in a seal, an aperture in a screen: in each case the flow, the leakage or the throughput is quoted per unit area, and in each case the same area drawn out into a slot behaves differently.

The diagnostic is the perimeter. Anything whose resistance is a wall effect scales with the perimeter while its capacity scales with the area, so the ratio of the two — the hydraulic radius — is the governing length, and it is minimised at fixed area by the most compact shape. Any quantity that depends on that ratio is a shape-dependent quantity wearing an area’s units.

And the second lesson is about which half of a two-term law a correction lands on. The slot penalty affects only the viscous term, so its size depends on the regime: it is invisible where inertia dominates and full-strength where viscosity does. A correction quoted as a percentage, without its regime, is a percentage of whichever term the person quoting it happened to be in.

Who found it, and when #

The hydraulic-radius treatment of non-circular ducts is nineteenth-century engineering practice. Shah and London’s compilation of exact friction factors for rectangular and other sections is 1978 and is the standard reference.

That floats let threads group is a weaver’s observation of long standing — it is why a sateen is said to be sleazy and why a satin needs a close sett to be firm — and this collection has already used it for tear strength, where grouping is what lets yarns share a load.

What appears to belong here is the joining of the two: that grouping does not merely change how the threads share a force, it changes the shape of the holes between them, and that the shape has a computable consequence at constant open area. The consequence is not large — a factor of two and a half at the extreme — but it is a factor that a cover factor cannot see, in a quantity that is measured constantly.

Where the ladder goes next #

Both this and the drift argument are about threads not staying where the reed put them, and there is exactly one construction whose spacing is not held by friction at all: a leno’s hole cannot drift, which is why every gauze, mesh and bolting cloth is one.

Sideways, the whole planar apparatus — holes, shapes, waists — refuses a knit outright, and the refusal has a threshold in it: a knit has no hole to lose, because at any ordinary tightness its loops are already over-full.