The hole between four threads

A fabric used as a filter is bought for the size of its holes. Not for their shape, their arrangement or their number, but for one length — and the trade calls it an apparent opening size, measures it by sieving glass beads through the cloth, and prints it on the roll in micrometres. What that measurement actually finds is the largest hole rather than the typical one, which matters on every weave but a plain one.

The measurement is a genuine one and it is worth asking what it is measuring. For a woven fabric the answer is unusually clean, and the cleanness is a property of the repeat rather than of the fabric’s quality.

Four threads bound a square, and every square is the same #

The geometry is one line. Two neighbouring warp ends are a spacing p apart and each is d across, so the clear gap between them is p − d. Two neighbouring picks leave the same, and the hole they bound between them is that gap square.

And because a weave is a repeat, every hole in the cloth is that size. Not on average, not to within a tolerance: the repeat tiles the plane, so the pore-size distribution of an ideal woven fabric is a single value with no spread at all.

That is the sentence that separates the two halves of the geotextile trade, and it is worth stating carefully because it is a consequence of periodicity rather than of care in manufacture. A nonwoven’s pores are the gaps in a random fibre web; they have a distribution, and the specification has to name a percentile of it. A woven’s pores have no distribution to take a percentile of.

The open area is one minus the cover, which the site already knew #

The second number a filter cloth is specified by is its percentage open area — the fraction of the surface that is hole rather than thread. For a square-set cloth it is the gap over the spacing, squared:

open area = (1 − n·d)²

and 1 − n·d is one minus the cover factor, which is the quantity this site derived the trade’s constant 28 from. So the filter’s second specification number is the complement of the shirting’s marketing number, and the two trades have been computing the same thing for different reasons for a century.

The square is the interesting part and it is the standard pass’s own result, arrived at there for a completely different purpose. The gap a person holds a sheet up to the light to see moves as the square of one minus the cover, so small changes in the sett make large changes in the open area: at 20 threads per centimetre of a 150 µm monofilament the cloth is 49 per cent open, at 30 it is 30 per cent, at 40 it is 16, and at 50 it is 6.

Both routes to the open area are computed and asserted equal in the code, because they are the same statement in two notations and a disagreement would mean one of them had a factor wrong. That the check has never fired is not the point; the point is that the two quantities are one quantity, and the assertion is what says so.

What the opening size is measured against #

The hole is only half of a specification. The other half is what has to be held back, and the trade states it as a grain size: D85, the sieve size that 85 per cent of the soil passes.

The criterion is then an inequality between two lengths, and it comes from practice rather than from geometry. A common form for a woven geotextile under dynamic loading is

O95 ≤ 0.2 · D85

with less severe factors — 0.5, or 1 — for static loading, coarser soils and less critical applications. Nothing on this site derives that, and the code says so at the point of use: the factor is an argument, the criterion is labelled as a design rule fitted to filtration tests, and the verdict is quoted with the factor beside it.

What this site supplies is not the criterion but the geometry it is applied to. The opening size is computed exactly, from the sett and the diameter, with no fitted constant anywhere in it — and that is the half a specification usually has to be measured for.

The finest soil each yarn can hold, which is a property of the yarn #

Put the two halves together and something falls out that neither contains: there is a finest soil a given yarn can filter at all, however it is set, and it is proportional to the yarn’s diameter.

Retention wants the hole no bigger than f·D85, so it wants a spacing of at most d + f·D85. Open area wants 1 − d/p at least √A, so it wants a spacing of at least d/(1 − √A). Put those together and everything cancels but a ratio:

D85 ≥ (d / f) · √A / (1 − √A)

At a fifth of D85 and four per cent open, the constant is 1.25. A 150 µm monofilament cannot filter a soil finer than about 190 µm at any sett whatever, and no amount of weaving skill changes it, because the constraint is not about weaving.

The closed form is checked against a search over the setts, and the two agree to within a fifth of a per cent. That is this site’s standing habit for a threshold, and here it earned itself twice: the search’s tolerance had to be derived rather than chosen, because an error in the sett is amplified on its way to a soil size by p/(p − d), which at the threshold is exactly 1/√A. Five, at four per cent open. A quarter of a per cent in the sett is one and a quarter per cent in the answer, and knowing that was the difference between a tolerance that meant something and a tolerance that had been widened until it passed.

Why the sieve and the arithmetic do not agree, and by how much #

The measurement and the computation are of different things, and the gap between them is estimable rather than mysterious.

A glass bead arriving at the cloth does not have to fit through the square in plan. It can arrive over a corner, where the clear distance is the square’s diagonal — √2 times the side — so a bead up to 41 per cent larger than p − d can find a way through a cloth whose plan opening is smaller than it is. Against that, the passage is not a straight tube: it narrows where the threads cross above and below, and a bead of exactly the diagonal has to travel a curved path to use it.

So the sieve’s answer sits between the side and the diagonal of the computed square, and which end it sits nearer depends on the weave: a plain weave’s passage is short and nearly straight, while a twill’s is offset between its two faces and behaves more like a tube with a bend in it. The arithmetic gives a lower bound on the measurement, and a factor of at most 1.41 covers the difference.

That is a satisfying place for a geometric model to be. It cannot replace the sieve, it brackets the sieve, and the bracket is a number rather than a shrug — which is worth more than a fitted correction that would agree with one laboratory’s beads.

The nonwoven is specified by a percentile because it has a distribution to take one of #

The contrast with a fibre web is not rhetorical, and this site has the machinery to make it precise.

A nonwoven’s coherence is a percolation threshold: fibres dropped at random on an area hold together once there are enough of them to form a spanning network, and an earlier essay here measured the crossing at a stick density between 5.4 and 5.8 against a published constant of 5.637. Everything about that construction is a distribution — how many contacts a fibre has, how far it runs before one, and how large the gaps between them are.

The two constructions therefore need different specifications for a structural reason. A woven fabric’s holes are congruent because a repeat is a repeat, and the number to quote is the hole. A nonwoven’s are not, and the number to quote is a percentile of a distribution whose shape depends on the fibre length, the density and the needling. Quoting O95 for a woven cloth is quoting the 95th percentile of a delta function, which is why the woven’s specification looks tighter than it is entitled to and why the woven’s failures are different in kind: a woven filter blinds, and a nonwoven passes a fine tail.

The same hole, four trades apart #

The gap between two threads is the most reused length on this site, and this is the essay to say so plainly.

It is the resin channel of a woven reinforcement: the passage the flow runs down, whose width sets the permeability and whose closing makes a part unfillable. It is the room a sewing needle finds as it comes down through the cloth, and whether it finds enough decides whether a seam’s stitches weaken the fabric they join. It is the slack a tear pulls out of the weave before the threads at its tip take the load, which is what an earlier essay here found the geometry of tearing reduces to. And it is the filter’s pore.

Four trades, four purposes, one expression: p − d. Nothing in that is an analogy. The same two lengths are being subtracted, the same figure could be drawn for any of them, and the reason the applied field keeps arriving back at it is that a fabric is mostly hole and the size of the hole is the fabric’s only wholly geometric property.

The corollary matters for reading a specification. A trade that quotes a sett is quoting one of the two lengths and leaving the other unstated, which is exactly the failure thread count commits — and any specification of a hole that does not name a diameter is not a specification of a hole.

The diagonal is not available to a sphere #

The bracket between the side and the diagonal is offered as covering the gap between the arithmetic and the sieve, and it is worth being precise about which particles can use the diagonal, because the answer is not the ones the test uses.

A sphere cannot. The largest circle that fits inside a square of side g is the inscribed circle of diameter g, not the diagonal — a sphere larger than the side simply does not enter the opening, whatever angle it arrives at, because a sphere presents the same width in every direction. So for a glass bead

the passage is exactly p − d, and the bracket collapses to one.

An elongated grain can. A particle whose short axis is under g can turn and present that axis, and its sieve size — which is what a D85 reports, and which is nearer the intermediate axis — may be up to √2 times the opening before the diagonal runs out. The 1.41 is real and it belongs to angular soil rather than to beads.

So the laboratory and the field are not measuring the same passage. The apparent opening size is measured by sieving glass beads, which cannot use the diagonal; the fabric in service meets angular soil, which can. The measurement is therefore an optimistic number for the thing the specification is about, by a factor of up to 1.41 depending on how angular the soil is.

Which is part of what the design factor is covering #

That has a consequence for reading the criterion. O95 ≤ 0.2 · D85 is quoted as a design rule fitted to filtration tests, and a factor of five is a great deal of margin for a geometric comparison between two lengths.

Some of it is now identifiable. Up to 1.41 of the five is the diagonal, present in the field and absent from the bead test that produced the O95 on the roll. What remains — a factor of three and a half — covers the things the geometry genuinely cannot reach: the cake, the blinding, the dynamic loading, the grading of the soil either side of its D85.

That is worth separating because the two behave differently. The diagonal factor is a property of the soil’s shape and could be measured; a rounded fluvial sand needs less of it than an angular crushed rock, and the criterion applies one number to both. The rest is a genuine safety allowance on mechanisms nobody has a model for.

So a specifier facing a rounded soil is carrying a factor of 1.41 they do not need, and one facing a very angular soil may be carrying less margin than the number suggests. The criterion is calibrated on the average angularity of whatever soils the tests were run on, which is not a quantity anybody records.

And it says which measurement would tighten it #

The separation gives a cheap experiment. Sieve the same cloth with glass beads and with crushed grit of the same nominal grading, and the ratio of the two O95 figures is the diagonal factor for that soil, directly.

If it comes out at one, the diagonal is not being used and the whole of the design factor is covering mechanisms rather than geometry. If it comes out near 1.4, the geometry is eating a fifth of the margin and the criterion could be tightened for rounded soils and should be loosened for angular ones.

Either answer is worth having and neither needs a fabric that does not exist, which is unusual for a proposal in this collection. It needs two sieve runs on one cloth, and the second one uses a material the soil laboratory next door already has.

Where the model stops, and it stops in four places #

A real cloth’s holes are not all the same size. Setts vary along a piece, monofilaments vary in diameter, and a fabric that has been tensioned or heat-set has holes that differ between its middle and its selvedge. The distribution is narrow rather than absent, which is why the trade measures a percentile — O95 — of something the model says has no spread.

The hole is not square and it is not flat. Two threads crossing leave an opening whose walls are curved, and the narrowest section through it is not the square in plan: a bead a little larger than p − d may still pass by finding the diagonal, and a bead smaller than it may not pass if it arrives at an angle. Sieving measures a hydraulic-ish diameter of a three-dimensional passage, and the arithmetic gives a plan dimension.

Nothing here is a model of filtration. Soil arriving at a filter builds a cake, and the cake does the filtering after a few minutes: the fabric’s job is to hold the first grains and then not to blind. Blinding — the pores closing with trapped fines — is the commonest real failure and it has no representation in this geometry at all.

And the thickness has been ignored. A woven filter is one thread thick; a needled nonwoven is millimetres thick, and its tortuous path is the reason it can hold fines without a single small opening. Comparing the two on opening size alone gives the woven a flattering picture of a job it does differently.

What was counted, and how #

Nothing in this essay needed an enumeration, and it is worth saying why the one place a count could have gone was left alone.

The opening size is arithmetic on two lengths, so the checks are consistency checks rather than counts: the open area is computed from the gaps and again from the cover factors and the two are asserted equal; a closer sett is asserted never to leave a larger hole or more of the surface open, over every ladder of setts any figure uses; and a cloth set past its own jam is refused rather than reported, because a negative gap is not a small hole.

The finest-soil threshold is the one number here that could have been quoted and is instead computed twice — the closed form above, and a search over a sett ladder scaled to where the answer will be. What the pair of them catches is not an arithmetic slip but a modelling one: if the two criteria had been applied to different definitions of the opening, or the open area had been taken as the gap rather than its square, the closed form and the search would disagree by a factor that no single calculation would have shown.

The census this site could have run here is a different question and is deliberately not answered: how the opening size is distributed over the 22,874 four-by-four drafts. It is not a well-posed question in this geometry, because the plan opening of a non-plain weave depends on which two neighbouring threads are being asked about and on which face — a satin’s surface has runs of parallel floats with no crossing between them, and the hole under a float is not the hole between two crossings. That is a real gap in what this arithmetic covers and it is recorded rather than smoothed over.

Who found it, and when #

The cover factor is Ashenhurst’s arithmetic from the 1880s, and the opening-size specification of a geotextile is from the 1970s, when the material became civil-engineering practice and needed numbers a specifier could write into a contract. The retention criteria — Terzaghi’s for graded filters, and the O95-against-D85 rules that followed — came out of the same period’s testing.

The two halves of this essay therefore arrive from a century apart and from trades that never met: a cotton weaver’s rule for how much of a surface a thread hides, and a soil engineer’s rule for what a fabric must not let through. They are the same expression. One minus the cover, squared, is the open area, and the spacing less the diameter is the pore — and the reason they coincide is that both trades were measuring a hole between four threads.

Where the ladder goes next #

The two numbers this essay computes pull opposite ways, which makes a filter cloth a specification rather than a fabric: retention wants a close sett and flow wants an open one. The next rung draws the interval the two leave between them, finds that it is sometimes one sett wide and sometimes empty, and arrives at the nonwoven from the direction of a woven that has run out of room.