Textile Structure
how cloth is put together
Cloth doing a job

A filter is rated by the hole it does not show

A woven filter cloth is sold on two numbers pulling opposite ways: an opening small enough to hold the soil and an open area large enough to pass the water. Both are computed from the same holes, and they are not computed from the same statistic of them. One reads the maximum and the other reads the mean, so a change that improves either can worsen the other without moving a single measurable property of the cloth.
15 min read 6 figures Structure before fibreA weave is a matrix

Worth reading first: A filter cloth has two jobs · A hole is a channel, not an opening · Only a plain weave has one size of hole.

A woven filter cloth is specified by exactly two numbers, and this collection has computed both of them. The apparent opening size must be small enough to hold the soil back, which wants a close sett. The percent open area must be large enough to pass the water, which wants an open one. Both are inequalities in the sett, they pull opposite ways, and for a fine enough soil the interval between them is empty.

That arithmetic was done on a cloth with one hole in it — the same hole everywhere, because a weave is a repeat. A cloth has a distribution of holes instead, and the moment there is a distribution the two specifications stop reading the same thing.

A rating is a maximum. A flow is a mean. The soil grain that gets through is the one that finds the largest hole; the water that gets through is the sum over all of them. Those are different statistics of the same list, they respond differently to everything, and no cloth has both of them printed on it.

Seven weaves at one construction, and what each of them really passes. Every weave here is drawn at the same filter — the same yarn, the same sett, the same cover and therefore the same open area, equal to twelve decimal places. Each one shows an opening of 263.7 µm to anybody looking straight through it, and that is the number a specification quotes. The bar is what each will actually let past, which is the narrowest section anywhere along the channel rather than the narrowest view down it. The plain weave passes exactly what it shows and it is the only weave here that does: its ends transit at every gap, so the four threads round every hole are level in pairs and the waist is at the middle. A float leaves two ends together at the top of the cloth, their neighbours at the bottom, and a passage between them that is wider than its own mouth — up to 12.8 per cent over the rating, for the basket.
Fig. 1 Seven weaves at one filter cloth’s construction — the same yarn, the same sett, the same cover, and therefore the same open area to twelve decimal places. Every one of them shows the same opening to anything looking straight through. What each will actually pass is the bar, and only the plain weave’s bar lands on the line. A specification written on the shown opening is therefore a specification about a plain weave, quietly, and is wrong by up to fifteen per cent for anything else.

The claim

A filter cloth’s rating is a property of the largest hole in its repeat and its permeability is a property of the mean, so a weave chosen to improve one of them worsens the other, and neither change is visible in the cloth’s open area.

The first half is geometry and was settled by the hole census: in every weave but the plain weave, the largest passage exceeds the shown opening, by nine per cent for the twills and satins and fifteen for a basket.

The second half is the one with a consequence. Holding open area fixed and moving holes between the ends of the distribution — which is exactly what changing the weave does — raises the flow and raises the rating together. The cloth passes more water and holds back less soil, and the two numbers that were meant to be traded against each other have moved the same way.

The argument

The rating first. A retention criterion is a rule from practice, fitted to filtration tests, and this collection has always named it as one: the opening must not exceed a stated fraction of the soil’s own grain size, commonly a fifth of the eighty-fifth percentile for a fine soil under dynamic loading. Nothing here derives that rule and this does not pretend to.

What the rule is applied to is the geometry, and the geometry is where the substitution happens. The rule is written about the largest opening the cloth has. The number substituted into it is the opening the cloth shows. In a plain weave those are the same number; in a twill the second is nine per cent smaller than the first, and the substitution is in the unsafe direction — the cloth is credited with holding back a grain it will pass.

Now the flow. What a channel passes goes as its area times the square of its hydraulic diameter, so it is very nearly a fourth power of a length. A fourth power is convex, so spreading a fixed total of open area across holes of unequal size passes more than concentrating it in equal ones. That is Jensen’s inequality and it needs no textiles in it at all. It is the same convexity that makes a long float pass more than its share of everything that scales steeply with a length.

So both quantities move up together as a weave’s holes become unequal. The trade that a filter specification is built around — closer for retention, opener for flow — is a trade in the sett, and it is a real one. Changing the weave is not that trade. It is a move that helps one specification and hurts the other, and it is available to a designer who has been told the two numbers describe the cloth.

Seven weaves at one construction, and what each of them really passes. Every weave here is drawn at the same muslin — the same yarn, the same sett, the same cover and therefore the same open area, equal to twelve decimal places. Each one shows an opening of 249.6 µm to anybody looking straight through it, and that is the number a specification quotes. The bar is what each will actually let past, which is the narrowest section anywhere along the channel rather than the narrowest view down it. The plain weave passes exactly what it shows and it is the only weave here that does: its ends transit at every gap, so the four threads round every hole are level in pairs and the waist is at the middle. A float leaves two ends together at the top of the cloth, their neighbours at the bottom, and a passage between them that is wider than its own mouth — up to 15.2 per cent over the rating, for the basket.
Fig. 2 The same excess on an ordinary cloth, for scale. Every weave has holes larger than its nominal one and the rating is set by the largest — so the number a filter is sold on is a statement about a tail, and the tail is wider on the weaves that pass most.

What was counted, and how

The eight cloths of this collection’s standing table, asked both questions at once:

cloth shows passes open area air at 100 Pa
cheesecloth 795 µm 795 µm 64.9% 6,723 mm/s
voile 287 287 49.3 4,808
batiste 194 194 40.1 3,622
muslin 250 250 37.9 3,505
poplin 168 168 34.0 2,901
duck 336 336 30.4 2,820
filter 264 264 29.1 2,584
sheeting 170 170 24.5 1,907

Every row is a plain weave, so every row’s two opening columns agree — which is the control. Woven as a 2/2 twill instead, at exactly the same construction, each cloth’s passes column rises by 9.89 per cent and every other column in the table stays where it is.

Every row is also a cloth this collection has met elsewhere — the duck under what a fabric weighs, the batiste under how close threads can be set — so the table is the same eight cloths asked a new question rather than eight new cloths.

The ordering is where the two statistics part company most usefully. Read down the table by air and it runs cheesecloth, voile, batiste, muslin, poplin, duck, filter, sheeting. Read it by the head of water each cloth can hold back — which is set by the coarsest pore, and is therefore the same ranking as by hole size — and the duck moves from sixth to second. It passes less air than the batiste and leaks at three fifths of the batiste’s pressure, because it has few large holes where the batiste has many small ones.

That is the whole content of the disagreement, in one pair of rows, and it is not an artefact of any model: the duck’s hole is 336 µm and the batiste’s is 194.

The eight cloths on air and on water, which do not agree. Each of this site's eight cloths, with what it passes in air at 100 Pa beside the head of water it holds back at a contact angle of 120°. Both come from the same holes and the two orderings are not the same. Air permeability is decided by how much of the surface is pore — an average over all of them — and the head is decided by the coarsest single pore, because water needs one path and takes the cheapest. So the duck passes less air than the batiste and leaks at 57 per cent of the batiste's pressure: few large holes against many small ones. A maximum and a mean do not order a set of cloths the same way, which is the whole content of the disagreement and is why one fabric cannot be specified by one number.
Fig. 3 The eight cloths on air and on water side by side. The bars are computed from the same holes and the two orderings are not the same; the rows that swap are named underneath. A cloth’s flow reads the mean of its holes and its water resistance reads the maximum, and a set of cloths ordered by a mean is not the set ordered by a maximum.

The measurement that would catch it, and why nobody takes it

There is a straightforward test that separates the two statistics and it is already standard: a bubble point. Wet the cloth, raise the air pressure beneath it, and the pressure at which the first bubble appears is set by the largest through-pore, because that is the one that gives way first. Raise it further and the flow that develops is set by the whole distribution.

So the instrument exists, and the two numbers it gives are exactly the maximum and the mean.

What is usually measured instead is an opening size — optically, which is the same quantity a person holding a cloth to the light is judging, or by sieving glass beads through the cloth and finding the size at which ninety-five per cent are retained. Both of those are measurements of the projection. The optical one is exactly the projection by construction. The bead one is closer to the passage, since a bead can move sideways as it descends, and that is precisely why the two disagree on anything but a plain weave: two methods that were calibrated against each other on plain-woven wire will not agree on a twilled cloth, and the disagreement is not an error in either.

What it does to the number the field actually uses

This collection has already computed the sharpest form of the two-sided specification: the finest soil a woven cloth of a given yarn can filter at all, found by pushing the sett until the retention criterion and the open-area floor meet. The answer came out with no sett in it — the finest soil is a fixed multiple of the yarn’s own diameter, 1.25 times it, so a cloth of 150 µm yarn cannot hold back a soil finer than 187.5 µm however it is set, and the only route to a finer filter is a finer yarn.

That constant is a plain weave’s. Substitute the passage rather than the projection and it moves by exactly the excess: 1.374 for a twill, so the same yarn’s floor rises from 187.5 µm to 206.0.

Eighteen and a half micrometres does not sound like much, and the form it takes matters more than the size. It is a constant with no free parameter in it, of the kind this collection goes out of its way to find, and it turns out to have carried an assumption about the weave the whole time. A number that survives every sett and every count is exactly the sort of number that gets quoted without its conditions, because there appear not to be any.

How open a filter is, against where the light is. One minus the cover is a cloth's openness to a light directly behind it, and for this filter it is 29.1 per cent. Move off the normal and a line of sight has to clear the hole at the top of the cloth and the same hole one thickness below, so what it can use falls as t·tan α is taken off each side — reaching nothing at 28.8° across the ends. Averaged over the whole sky with Lambert's cosine weighting, the cloth is 2.54 per cent open — 11.4 times less than the covering rule says, and the whole of the difference is the thickness. One minus the cover is the openness of a cloth with no thickness, and no cloth has none.
Fig. 4 How open the filter is to the whole sky, which is the quantity a maker would rather quote. It is a mean over every hole and it says nothing about the largest — so the two numbers a filter carries are a mean and a maximum, and only one of them decides what gets through.

Why the two statistics respond oppositely to unevenness

The claim that improving one specification worsens the other is worth putting as a single sweep rather than as a pair of observations, because then it says which direction every decision points.

Take a cloth’s holes and hold their total area fixed while making them more unequal. That is not an exotic operation — it is what changing the weave does, and what a drifting thread does, and what any variation in the reed’s spacing does.

The rating gets worse, monotonically. The rating reads the largest hole, and moving area from small holes to large ones is the definition of making the largest larger. There is no arrangement of a fixed area into unequal holes whose maximum is smaller than the equal arrangement’s.

The flow gets better, monotonically. What a channel passes rises faster than its area, so redistributing area towards the large holes gains more than it loses. Jensen’s inequality says the mean of a convex function exceeds the function of the mean, and both terms of a duct’s resistance are convex in the size.

So the equal-hole cloth is simultaneously the best rating and the worst flow that a given open area admits, and every departure from it moves both numbers in the unhelpful direction at once. Only a plain weave sits at that point, which is why the seven-weave figure has one bar on the line and six above it.

That is the sharpest statement of the essay’s finding. The two specifications are not merely reading different statistics; they are reading statistics that are ordered oppositely by the same one-parameter family of changes. A designer with an open-area target and a free choice of weave has a dial that trades the rating against the flow, and no specification records which way it has been turned.

What the drift arithmetic does to the flow

The same sweep prices a drifting thread twice, and the second reading is the one that makes the fault invisible rather than merely undetected.

Pushing an end sideways by δ leaves the open area exactly unchanged, which the essay establishes. Read against the sweep above it does more than that: it is a pure increase in unevenness at constant area, so it raises the flow as well as the rating. A cloth whose ends have drifted passes more water than it should and holds back less soil than it should, and the first of those is the number a mill is likely to be watching.

That is an uncomfortable pairing. A filter cloth whose threads have moved will test as freer-flowing than its specification, which reads as a good cloth or as a generously made one, at the same time as it has stopped meeting the retention criterion it was bought for. The one measurement that would look like a warning looks like a bonus.

The size of the flow effect is small — a few per cent for a drift that costs a fifth of the rating, because the flow is a sum over all the holes while the rating is one of them — and its smallness is part of the problem. A change too small to investigate accompanies a change large enough to fail the cloth, and the two have the same cause.

Where a drifting thread comes in

There is a second route to a large hole, and it is worse than the weave because it has no draft to be read off.

The ends are held where the reed put them by friction at their crossings, and nothing else. Push one end sideways by δ and the gap on one side becomes g − δ while the gap on the other becomes g + δ. The sum is unchanged. So the open area is unchanged, to twelve decimal places, and a percent-open-area measurement returns exactly what it returned before. The largest hole has grown by δ — one micrometre of rating for every micrometre of drift, with no factor between them.

One end out of place in a filter, and what still measures the same. A filter's warp seen from above, with one end pushed 60 µm out of place. The gap on one side of it falls to 204 µm and the gap on the other rises to 324 µm. The sum is unchanged, so the cloth's open area is unchanged to twelve decimal places, and a percent-open-area measurement returns exactly what it returned before. The largest hole has grown by 60 µm — one micrometre of rating for every micrometre of drift, with no factor between them — and the largest hole is the whole of what a filtration rating means. What holds the end in place is friction at its crossings, which this site computed from the capstan and found falls smoothly to nothing as the cloth opens, with no threshold to warn anybody.
Fig. 5 One end of a filter cloth pushed sixty micrometres out of place. The gap on one side falls and the gap on the other rises by the same amount, so the cloth’s open area is exactly what it was. The largest hole is now twenty-three per cent over the specification, and the measurement that would notice is not the one anybody takes.

What resists the drift is the grip of the crossings, which this collection computed from the capstan and found falls smoothly to nothing as a cloth opens, with no threshold to warn anybody. So the cloths most exposed to this are the open ones — which are the ones used for filtering.

Where the model stops

The retention criterion is practice and is not derived here. The factor of a fifth is an argument with a stated default, quoted with every verdict it produces. Nothing in this essay improves it; the argument is entirely about which length is substituted into it.

The finest-soil constant inherits everything the criterion does. It is the retention factor and the open-area floor combined, both of them practice, and moving it from 1.25 to 1.374 changes only which geometry was substituted. If the factor of a fifth is wrong the constant is wrong by more than the weave moves it.

A soil grain is not a sphere and a fibre is not a grain. The channel’s narrowest section is a good description of what stops a rigid compact particle and a poor one for a flake, a fibre or an aggregate. Real filtration is also not a single-pass sieving problem: a cake builds on the upstream face within minutes and thereafter the cake is the filter, which is why the criteria are fitted rather than computed.

The threads do not move in the geometry and they do in the cloth. The drift arithmetic above prices the consequence of a displacement and says nothing about how large a displacement to expect, which needs the lateral force a crossing can resist — a quantity this collection has for the pick and not for a sideways push on an end.

And the flow model is a duct model. Each hole is treated as a short channel with a viscous and an inertial term, which is discussed where it belongs; the convexity argument that unequal holes pass more needs only that the exponent be above one, and both terms satisfy that.

The generalisation

Whenever a specification has a pass criterion and a throughput criterion, the first is almost always a maximum and the second almost always a mean, and a designer told only the mean can improve the throughput by worsening the pass.

The pattern recurs wherever a population is being screened by a barrier. A mesh, a membrane, a sieve, a firewall rule set, a queueing discipline with a deadline: the failure is always the extreme member and the capacity is always the average one, and any single number that claims to describe the barrier has quietly chosen one of them.

The diagnostic is cheap. Ask what would happen if the population’s variance rose at constant mean. If one of the two specifications improves and the other worsens, they are reading different statistics, and no amount of care with the single quoted number will keep them consistent.

The second lesson is about substitution. A criterion fitted on one class of specimens carries that class’s coincidences inside it. Retention criteria were fitted on woven geotextiles, and woven geotextiles are overwhelmingly plain weaves and twills — so the criterion has never had to distinguish the shown opening from the real one, and the constant it was fitted with silently contains whichever it was measured on.

Where Poiseuille's rule starts being true of a cloth. Every account of a fabric's air permeability quotes the fourth power: flow down a channel goes as the fourth power of its width, so a small change of sett is a large change of flow. That is Poiseuille's result and it is about the viscous drop. A channel through a cloth is about as long as it is wide and the air in it is moving fast, so most of the pressure goes on giving that air its kinetic energy instead — and the inertial term does not depend on the hole's size at all. Plotted against cover, the viscous share of the drop runs from 0.8 per cent in an open filter to 51.8 in a close one, passing half at a cover of 0.572. Quoting the fourth power alone below that is wrong by a factor rather than by a correction: at the open end it overstates the flow 133-fold.
Fig. 6 And what the same holes do to the flow. A filter’s channels are long enough for the fourth-power rule to hold, so the largest hole passes disproportionately more than its share — the hole that decides the rating is also the hole that carries the flow.

Who found it, and when

The retention and open-area criteria are geotextile engineering practice from the 1970s onward, fitted to filtration tests, and are quoted here as such.

The bubble-point method is much older — it is the standard way of finding a membrane’s largest pore — and the fact that it measures the maximum while a flow measures the mean is textbook filtration.

What appears to belong to this collection is joining the two to the weave matrix: that the largest pore of a woven cloth is a computable function of the draft, that it exceeds the optical opening in every weave but one, and that the excess has a ceiling of 15.2 per cent at a four-by-four repeat, reached by the basket.

Where the ladder goes next

If the rating is a maximum and the maximum drifts with the threads, the question is what holds the threads — and there is one construction whose answer is not friction at all. A leno’s hole cannot drift, because its crossing locks the spacing geometrically, which is why every bolting cloth and every mesh worth the name is a leno.

Sideways, the same distribution asked about a fluid rather than a particle gives a different surprise: what a cloth passes does not go to zero when its holes do, because a cloth stops having holes before it stops passing air.

Where this goes next

A filter cloth has two jobs. Opening size against sett for a woven filter cloth, with the two limits that specify one: the soil is retained below the retention line, and the water passes above the open-area floor. They pull opposite ways, so the answer is an interval in the sett — and for a fine enough soil there is no interval at all. A filter cloth has two jobs Hold the soil back and pass the water. The first wants a close sett and the second wants an open one, so a filter cloth is not a fabric but an interval — one that is sometimes a single sett wide, and for a fine enough soil is empty. The empty case is the useful one, because it says the answer is not a woven cloth at all. The channel through a muslin in plain. A cut across a muslin woven plain, at 373 µm per hundred pixels, showing one hole between two warp ends. The ends sit at the same level here, so the clear width between them is 250 µm at the waist against 250 µm straight through. The profile on the right is that width at every height in the 334 µm the threads occupy: the passage is an hourglass, its narrowest section is 250 µm across, and the band that every level contains — what a straight line of sight can use — is 250 µm. The two differ by 0.0 per cent, and the difference is the weave and nothing else. A hole is a channel, not an opening Every hole drawn so far has been drawn from above, and the sentence underneath every one of them says the same thing: the hole between four threads is the spacing less the diameter, and every hole is the same size. That is a picture of a cloth's shadow. A cloth has a thickness, so its hole has a length, and the narrowest place along it is not the place a person looking through can see. The 14 drafts whose holes are all one size. Every four-by-four draft in which each end and each pick interlaces — 22874 of them — built and asked whether all sixteen of its holes pass the same thing. At a muslin's construction 14 of them do. Set the same yarn square and 170 do; turn the cloth over and it is 14 again. Only 2 drafts are in all three lists, and they are the plain weave and its complement, marked. The other 12 are uniform because this cloth's warp is set closer than its weft, so the gap across the ends is the smaller of the two and binds every hole whatever the picks are doing — a fact about the sett wearing a fact about the weave's clothes. Two of them carry floats of three, which is as long as this repeat allows. Only a plain weave has one size of hole A weave's holes come in kinds, and the kinds are read off the matrix. Asking which weaves have only one kind looks like a question with an obvious answer and a one-line proof. Every draft at four by four was built and asked instead, and the count came back fourteen — of which twelve turn out to be telling the truth about the sett rather than about the weave.

What links here

Computed from the collection rather than written here: the essays that point at this one.

Reads more easily once this is understood

Essays that name this one as worth reading first.

Shares its objects with

Essays naming at least two of the same things, that neither author linked.

Named objects

A flat tag is an object no other essay names yet.

Apparent opening sizeChannel waistClear openingHydraulic radiusOpen areaRetention criterionSettWeave matrix