Half the air goes through a tenth of the holes

An air permeability is quoted as one number, and every geometric account of one computes it the same way: work out the size of a hole, work out how much goes through it, multiply by how many there are. This collection has done exactly that, twice.

The multiplication assumes the holes are alike. They are not, and the reason has nothing to do with careless weaving.

A hole between two ends is the spacing less the diameter. The spacing is set by the reed and by the take-up and does not vary; the diameter is a yarn’s and varies by ten or fifteen per cent. A constant minus a varying quantity has a larger coefficient of variation than the varying quantity did — larger by the ratio of the diameter to the gap — so a cloth whose threads vary by fifteen per cent has holes varying by ten per cent when it is openly set and by forty-two when it is close.

That is the first half. The second half is that the flow through a hole is not proportional to its size.

The claim #

A fabric’s air does not leave through its holes. It leaves through its widest holes, and the closer the cloth is set the fewer of them matter.

At an ordinary muslin’s construction — twenty-four ends per centimetre of a yarn varying by fifteen per cent — the widest tenth of the holes carries 14.6 per cent of the flow, which is a mild thing. Close the same yarn to forty-four ends per centimetre and the widest tenth carries 53 per cent, and half the air is through 8.8 per cent of the openings.

Nothing about the cloth has changed except its sett. The yarn is the same yarn, with the same spread, and no hole has been made badly.

The argument, which is two multiplications that arrive together #

The concentration has two causes and they are not independent, which is why it climbs so steeply.

The spread in the holes rises as the cloth closes. A gap is p − d: subtracting a varying quantity from a fixed one leaves the same absolute spread on a smaller mean, so the coefficient of variation is multiplied by d/w. At sixteen ends per centimetre that factor is a third; at forty-four it is nearly three, and the ten per cent yarn is leaving forty-two per cent holes.

The exponent rises as the cloth closes too. A wide hole’s pressure drop is nearly all inertial, and an inertial drop hardly depends on the width at all — the flow is the area times a velocity that barely moves — so the flow goes as the square of the width. A narrow hole’s drop is viscous, and that is where the fourth power comes from. The effective exponent runs from a little over two to four across this axis.

So the flow through a hole is w to a power that is rising, over a population whose spread is rising, and the concentration is the product of the two effects rather than the larger of them. It is why the curve above is flat for the first third of its range and then bends hard.

The amplification, in one line #

The first multiplication is worth writing out, because it is the part that is arithmetic rather than fluid mechanics and it is the part that gets left out.

A gap is w = p − d. Its standard deviation is the diameter’s, because p contributes none. So

$$\mathrm{CV}_w = \frac{\sigma_d}{p - d} = \mathrm{CV}_d \cdot \frac{d}{w}.$$

The factor d/w is the ratio of thread to hole, which is precisely what closing a cloth increases: at twenty-four ends per centimetre a muslin’s is 0.67, at thirty-six it is 1.51, and at forty-four it is 2.78. So a single yarn at fifteen per cent leaves holes at ten, twenty-three and forty-two per cent respectively — three completely different populations of hole, made from one population of thread.

This is the same amplification that makes a jammed cloth impossible to specify by mean. It is worth noticing that it has no upper bound: as the cloth approaches closure the gap goes to nothing and its coefficient of variation goes to infinity, which is the arithmetic saying, correctly, that near jamming some holes are shut and others are not, and the cloth is no longer describable by an average anything.

What was counted, and how #

The hole population is built from the yarn’s, not modelled separately: the spacing is the cloth’s own, the diameter distribution is a lognormal at the yarn’s stated coefficient of variation, and the gap is the difference. The flow through each hole is this collection’s own two-term duct law — a viscous term and an entrance loss, solved together rather than chosen between — with the channel length taken as the cloth’s thickness.

The concentration is then a quantile integral and nothing more: order the holes by width, work out what each contributes, and ask what fraction of the total the widest tenth carries. Twenty thousand quantiles, evenly in probability, so that the widest tenth means a tenth of the holes rather than a tenth of the range of widths.

Three things are asserted as it runs, and each would catch a different mistake.

• The widest tenth always carries more than a tenth. This has to be true for any distribution and any increasing flow law, so if it ever failed the ordering or the integration would be wrong rather than the physics.
• Half the air goes through fewer than half the holes. The same statement from the other end, and the two would not fail together.
• A closer cloth concentrates more, asserted step by step across the sweep rather than at its ends, because a curve that is monotone in the middle and not at the edges is what a sign error in the gap looks like.

Two cloths to one specification #

The clearest way to see what this costs is to build the same fabric twice.

Take a cloth set at forty ends per centimetre — close, but nothing exotic — and weave it once from a well-made filament yarn at a coefficient of variation of eight per cent and once from an ordinary staple yarn at twenty. Both have the same mean count, so both have the same mean diameter, the same cover factor, the same open area, and the same everything that a specification sheet carries.

The two fabrics differ by half again in what they pass, and by a factor of two in how concentrated that passage is. Nothing distinguishes them on paper. They have the same construction, and a specification that names a count, a sett and a weave has said nothing about the difference — because the quantity that separates them is printed on the yarn’s own delivery note and never travels as far as the cloth’s.

The same comparison at an open sett is nearly nothing. At twenty-four ends per centimetre the two yarns give 0.3 per cent and 2.2 per cent excess respectively, and the widest tenth carries 12.3 against 16.3. So evenness is not a general virtue with a general size: it is worth almost nothing in an open cloth and a great deal in a close one, and the crossover is inside the range of ordinary shirtings.

Why this matters more than a per cent #

A permeability discrepancy of ten per cent is a nuisance. A concentration of flow into a tenth of the openings is a different kind of fact, because it changes what several other numbers mean.

A windproof specification is about the tail, not the mean. A fabric asked to pass less than a stated flow is being asked about its total, and its total is dominated by its widest holes — so two fabrics with identical mean holes and identical open areas can differ substantially in what they pass, and the one with the more even yarn wins. That is an argument for evenness that has nothing to do with appearance, and it is not how yarn evenness is usually justified.

A filtration rating is about the same tail, from the other side. A filter is rated by the hole it does not show — by its largest opening rather than its average one — and the two arguments are the same arithmetic asked with different questions. The widest holes carry the air and pass the particle. A cloth optimised to breathe is being optimised in exactly the direction that makes it a worse filter, and the trade-off is quantified by this one curve.

And an average measured downstream cannot see any of it. A permeameter reads a face velocity over a test area, which is a sum. Two fabrics that pass the same total through very differently distributed holes are indistinguishable to it, and the distinction is precisely the one that matters for both of the uses above.

A measurement that would see it #

The awkwardness of everything above is that it is invisible to the instrument. A permeameter reads a total, and a total cannot distinguish a cloth whose holes are all alike from one whose air is coming out of a tenth of them.

Except that it can, if the pressure is swept.

The flow through a hole follows a power law in the pressure whose exponent says which term is deciding: a half where the drop is entirely inertial, one where it is entirely viscous. A cloth whose holes were identical would sit at a single point on that scale, wherever its own hole width puts it. A real cloth does not: a wide hole is inertial while the narrow one beside it is viscous, so the fabric is a mixture of regimes at once and its measured exponent is pulled towards the middle by however much its holes vary.

That is a tenth of an exponent, which is far above the resolution of any instrument that can hold two pressures. So the spread of a fabric’s holes — the thing that decides its permeability excess, its windproofing, and its filtration rating, and that no direct measurement reaches — is legible in the curvature of its own pressure–flow line.

This is a prediction and it is offered as one. Nothing here has measured a fabric; the curve is this collection’s own duct law averaged over a hole population, and the assertion that runs with it is about direction and size rather than about a number anybody has confirmed. What makes it worth stating is that it costs nothing to test: any laboratory with a permeameter and a fabric of known yarn CV has both halves already.

Where the model stops #

The holes are treated as independent and they are not. Two holes on either side of one end share that end’s diameter, so a thick place makes both of its neighbours narrow at once. That correlation does not change the distribution of hole widths, and therefore does not change any number above; it changes their arrangement, which matters for anything asking about a path through the cloth rather than a total across it.

Each hole is given one width. A hole is a channel rather than an opening, with a waist somewhere in the thickness, and the waist depends on where the four threads round it sit. Giving the channel the spread of its projection is a simplification in the conservative direction: the waist varies at least as much as the projection does.

The thread’s own path is not given a spread. The channel length here is the cloth’s thickness computed from mean threads, and a thick end makes its own channel longer as well as narrower. Including that would raise the concentration further, so the numbers quoted are a floor rather than an estimate.

And the yarn’s variation is treated as independent from end to end. It is not — a spinning frame leaves periodicities — and a periodic variation in the gaps does something completely different from a random one of the same size, which is a separate argument with a separate instrument.

One number decides all of it #

The concentration is computed here by ordering twenty thousand quantiles, and the whole of it collapses onto a single parameter — which is worth having because it says what to control.

For a lognormal population of hole widths and a flow going as the kth power of the width, the share carried by the widest fraction q is

1 − Φ(z₁₋q − kσ), with σ = √(ln(1 + CV_w²)),

and the fraction of holes carrying half the air is

1 − Φ(kσ).

Both depend on the two inputs only through the product kσ — the flow exponent times the log-spread of the holes — and nothing else about the cloth enters at all.

The expressions reproduce the essay’s own numbers. At twenty-four ends to the centimetre the holes vary by ten per cent and the exponent is about 2.5, so kσ is 0.25 and the widest tenth carries 15 per cent against the computed 14.6. At forty-four the holes vary by 42 per cent and the exponent is about 3.4, so kσ is 1.36, the widest tenth carries 53 per cent, and the fraction of holes carrying half the air is 1 − Φ(1.36) = 8.7 per cent against the computed 8.8.

Two closed forms, one parameter, and the essay’s whole table.

Which says exactly how even a yarn would have to be #

The parameter’s usefulness is that it inverts. To hold the widest tenth below a fifth of the flow — a mild requirement — needs

kσ ≤ 0.44.

At a close cloth’s exponent of 3.4 that is σ ≤ 0.13, or a hole CV of thirteen per cent. And the hole CV is the yarn’s multiplied by the thread-to-gap ratio, which at forty-four ends per centimetre is 2.78. So the yarn would have to be at

CV = 4.7 per cent.

No spun yarn reaches that. An excellent combed cotton is eight per cent and a compact-spun one seven; four and a half is a filament yarn’s territory and nothing else’s.

So the finding has a hard edge that the essay’s table implies and does not state: a closely set cloth of spun yarn cannot be made to pass its air evenly, at any spinning quality available, because the amplification factor d/w is nearly three and eats whatever evenness the spinner supplies.

A filament yarn can. At a CV under one per cent, kσ is a twentieth and the widest tenth carries eleven per cent — a cloth that is, for this purpose, uniform.

That is a much sharper statement of why windproof and filtration fabrics are made of filament than the usual appeal to smoothness or strength. They are made of filament because the flow concentration is a function of a product, one factor of which is forced up by the very closeness the fabric needs, and the only lever left is the other factor.

And it says where the effort should go at an open sett #

The same product read the other way says when evenness is not worth buying.

At twenty-four ends per centimetre the amplification is 0.67 and the exponent 2.5, so a fifteen per cent yarn gives kσ = 0.25 and a five per cent yarn gives 0.08. The widest tenth carries 15 per cent against 12 — a difference of three points in a quantity that is barely doing anything.

Below about kσ = 0.3 the cloth is effectively uniform and evenness buys nothing measurable. Above about one it is running on its tail and evenness is the only lever. The crossover sits at a hole CV near ten per cent, which for an ordinary yarn is a thread-to-gap ratio near two-thirds — an open shirting.

So the single number a designer needs is kσ, it is computable from a sett, a count and a delivery note, and it decides whether the whole of this essay applies to their cloth or none of it does.

The generalisation #

Whenever a total is a sum of a convex function over a spread population, the total belongs to the tail, and the mean says almost nothing about where it came from. That is a statement about arithmetic rather than about cloth, and it has an immediate diagnostic use: if a system’s output is concentrated in its extremes, then improving the average of its parts is the wrong intervention, and narrowing their spread is the right one.

The test is cheap and worth running on anything: compute the answer at the mean, compute it over the distribution, and then ask what fraction of the answer comes from the top tenth. If that fraction is near a tenth, the mean is a good summary and every conventional correction applies. If it is half, the system is being run by its outliers, and the summary statistic everyone quotes is measuring the wrong population.

And the second lesson is that two rising quantities multiply. The temptation with a system that has two causes is to identify the dominant one and work with it. Here neither dominates: the spread and the exponent each roughly double across the range, and the concentration quadruples. A model carrying only the more obvious of them would have been half right in the exponent and entirely wrong about the behaviour.

Who found it, and when #

That flow concentrates in the widest pores is not a new idea — it is the standard argument for why a packed bed’s permeability cannot be predicted from a mean pore size, and it is why filtration practice has always specified a maximum pore rather than an average one. What appears not to have been done for woven cloth is the arithmetic that connects it to the yarn’s evenness, which is a number every mill already has, measured for entirely different reasons.

The two ingredients have been separately available for a long time. The amplification of a spread by subtraction is elementary. The two-term drop through a fabric’s channels is this collection’s own, from the work on what a cloth lets through. Putting them together gives a prediction with no fitted constant in it: a cloth’s permeability excess over its geometric value is settled by the yarn’s CV, the cover, and nothing else.

Where the ladder goes next #

The same distribution asked for its extreme rather than its total gives the two measurements this ladder turns on next: what a thickness gauge reads, which is a maximum over the crossings under a foot, and the sett a warp will actually take, which is a maximum over the pairs across a width.

Sideways, the tail that carries the air is the tail that lets a particle through, so this argument is the filtration one turned over — and a cloth cannot be improved at both ends of it at once.