A filter cloth has two jobs

Behind a quay wall, under a road, along a drainage trench, there is a fabric between the soil and the water. Its job is to let the water out and keep the soil in, and those are two jobs rather than one — because the same hole does both, and they want it to be different sizes.

The specification is therefore not a fabric. It is a pair of inequalities on one decision, and the answer to a specification is an interval or nothing.

The two inequalities #

Both criteria come from practice, and this site names them as practice at every use rather than deriving what cannot be derived here.

Retention. The opening must be no larger than a fraction of the soil’s own grain size: O95 ≤ f · D85, with f a fifth for a fine soil under dynamic loading and up to one for a coarse soil in a static application. It is a rule fitted to filtration tests. What this site computes is the opening — exactly, from the sett and the diameter — and what the trade supplies is the factor.

Flow. The fabric must stay open enough to pass water at the rate the drainage needs, and the number specified is a percentage open area: a few per cent for a fine filter, ten or more for a drainage fabric. It is a proxy for permittivity, it is measured optically or computed from the geometry, and its floor is an engineering judgement about the flow the site has to carry.

One wants a small hole. The other wants a large one. Both are functions of the sett. That is the entire structure of the problem, and everything below is reading it.

The interval, which is sometimes one sett wide #

For a 150 µm monofilament against a fine sand at D85 = 500 µm, the retention line sits at 100 µm and the open-area floor bites above fifty threads per centimetre. The surviving interval is forty, forty-five and fifty threads per centimetre — narrow, but a designer can work in it.

Make the soil finer and the interval closes from both ends at once, because the retention line comes down while the open-area floor stays where it is.

That is worth pausing on, because it is a way for an honest computation to give a wrong answer. The band is found by testing a list of setts; a list that steps in fives when the answer is one wide will miss it. The routine that computes the threshold below therefore builds its own ladder, scaled to where the answer will be — and the tolerance on its agreement with the closed form is derived from the ladder’s step rather than chosen to pass.

The finest soil, and what lies below it #

Push the soil finer still and the interval empties. Where it does is a closed form, and everything in it cancels but a ratio:

D85 ≥ (d / f) · √A / (1 − √A)

At a fifth of D85 and four per cent open, that is 1.25 times the yarn diameter, whatever the diameter and whatever the fibre. A 150 µm monofilament stops at about 190 µm of soil; a 400 µm monofilament stops at 500.

Below that line the answer is not a sett. No woven cloth of that yarn satisfies the specification however it is set, and the options are a finer yarn — which costs strength and stiffness, and is why filter cloths are woven from monofilaments as fine as they can be handled — or a different construction.

The different construction is a nonwoven, and arriving at it this way is worth more than describing it. A needled fibre web has no single opening size; it is millimetres thick where a woven cloth is one thread thick; and its filtration comes from a tortuous path through a depth rather than from a hole in a plane. It can therefore hold fines that no woven cloth of the same permeability can hold, and the price is a pore-size distribution with a tail — some pores much larger than the mean — which is exactly what the percolation picture predicts of a random web.

This is the second time the applied field has arrived at a different construction by exhausting a woven one. A part too large to infuse needs a different process, not a better cloth; a soil too fine to retain needs a different fabric. In both cases the useful output of the arithmetic was that the interval was empty.

Three ways the interval can be empty, and only one of them is the soil’s fault #

The threshold above holds the two specification numbers fixed and moves the soil. Two other things can close the interval, and distinguishing them is what turns a computation into advice.

The soil is too fine for the yarn. That is the case above, and the answer is a finer yarn or a nonwoven.

The flow requirement is too high. A drainage fabric behind a wall may be specified at ten per cent open rather than four, and the closed form says immediately what that costs: √A/(1 − √A) rises from 0.25 to 0.46, so the finest soil the same yarn can hold nearly doubles. The specification has been made harder by a decision that had nothing to do with the soil.

The retention factor is severe. A dynamic application — wave action, traffic loading, repeated flow reversal — takes f from one down to a fifth, which is a factor of five in the finest soil the same cloth can hold. The specifier chooses that factor, and choosing it well is the difference between a fabric that exists and one that does not.

So an empty interval is a question rather than a verdict: which of the three numbers is doing the excluding, and which of them is genuinely fixed? The arithmetic answers it in one line, because each of the three enters the closed form separately.

What a permittivity actually is, and why the open area stands in for it #

The flow requirement above is written as a percentage open area, and the quantity an engineer actually needs is a permittivity — a flow rate per unit area per unit head, with the fabric’s thickness divided out of it.

The relationship between the two is not geometry alone. Flow through a hole a few tens of micrometres across at the heads a drainage trench works at is viscous, so it goes roughly as the fourth power of the hole’s size and the number of holes per unit area — which combines into something close to the open area times the square of the opening. Entry and exit losses, the shape of the passage between two crossing threads and the fabric’s thickness all sit in the constant.

None of that constant is claimed here. What is used is the comparison between setts, and the constant divides out of a comparison — so the figures above are honest about which sett is more open and say nothing whatever about litres per second. The open area is a proxy, it is the proxy the trade specifies, and treating it as the physical quantity would be exactly the over-claim this site’s figure rules exist to prevent.

A sensitivity that is exactly 1/√A #

One number falls out of the algebra that is not about filters at all, and it is the reason the threshold is delicate.

Near the threshold, the clear gap is a small fraction of the spacing — the cloth is nearly closed. So a small change in the sett is a large change in the gap: the sensitivity is p/(p − d), which at the threshold is exactly 1/√A. At four per cent open that is five.

A quarter of a per cent error in the sett is therefore one and a quarter per cent in the soil size the cloth can hold. Read as a manufacturing statement it says something a specifier should know: a filter cloth working near its own limit has almost no tolerance, and the ordinary variation of a sett along a piece is amplified fivefold in the property being specified. Read as a modelling statement it is what let the search’s tolerance be derived rather than tuned.

What was counted, and how #

The band is built by evaluating both criteria at every sett in a ladder and keeping the rows that satisfy both. Two monotonicities are asserted while the rows are built — a closer sett never leaves a larger hole, and never leaves more of the surface open — because the interval’s being an interval depends on them, and a non-monotone row would mean the band was a union of pieces and the answer was being read wrongly.

The threshold is computed twice. The closed form is four lines of algebra; the search bisects on the soil size, asking at each step whether any sett in a scaled ladder satisfies both. They agree to within two per cent, and the two per cent is the ladder’s step times the amplification derived above rather than a tolerance widened until the check passed. The far side is asserted too: at ten per cent below the threshold no sett satisfies the specification, which is the half of a boundary that is easy to leave untested.

The retention criterion, the open-area floor and the sieve-to-plan relationship are all stated inputs, and the figures print them on their own faces. That is not modesty: the whole value of separating them is that a reader with a different factor can move it and get their own interval, and a reader who cannot see which numbers came from practice cannot do that.

What the picture cannot show #

The trade-off figures draw a curve, two limits and an interval, and none of the three is a fabric.

They cannot show the cake that does the filtering after the first minutes, nor the blinding that ends the fabric’s life, nor the depth a nonwoven filters through — which is why the web figures beside them are drawn in plan and are honest only about the distribution rather than about the mechanism.

And an interval is a strange thing to draw. It is marked on an axis rather than depicted, because the object being reported is a set of satisfiable setts and there is nothing to see. The figures that draw actual cloth in this field — the pore in plan, the gap in section — are the ones that carry the geometry, and the trade-off figures carry the specification. Keeping the two kinds separate is the field’s own drawing rule.

Where the model stops #

Blinding is not modelled and it is the commonest failure. Fines arriving at a filter lodge in its pores; the cloth’s effective opening falls, its permittivity falls with it, and the drainage stops working. Nothing in a geometric model of a clean fabric can see that, and the trade’s answer to it is a larger opening than retention alone would ask for — which is a third inequality, pulling the same way as flow, and it is why the factors f above go up to one.

Nor is the cake. After a few minutes of service the soil itself is doing the filtering, and the fabric’s job has changed to holding the cake in place. A specification is therefore about the first minutes of a fabric’s life, and this arithmetic is about that period only.

The open area is a proxy for the permittivity, not the permittivity. The flow through a fabric depends on the thickness, the shape of the passage, the entry losses and the viscosity, none of which is here. What is here is a comparison between setts, which is what the interval needs, and the constant that would turn it into a flow rate divides out of the comparison.

And a real cloth’s openings vary. The threshold above is computed for a fabric whose holes are all the same size, and the trade measures O95 precisely because they are not quite. Every number in this essay is therefore a number about an ideal repeat, which is the same caveat the previous rung ended on and the reason the sieve is the arbiter.

The third inequality, and what it demands of the soil #

Blinding is named as the commonest failure and as a third inequality pulling the same way as flow. Written down, it does something the essay does not expect: it turns a condition on the fabric into a condition on the soil.

A pore much larger than a fine grain passes it; a pore comparable to it traps it. So the anti-clogging rule the trade uses puts a floor under the opening — in its usual form O95 ≥ 3·D15, with D15 the fine end of the soil’s grading and the three a practice factor like the others here.

Set that against retention and the opening is bracketed from both sides:

3·D15 ≤ O95 ≤ f·D85.

The bracket is non-empty only if D85 ÷ D15 ≥ 3 ÷ f — which is a statement with no fabric in it at all.

That ratio is the soil’s uniformity coefficient, and the condition says:

A uniform soil cannot be filtered at all under a severe retention factor. Not by a finer yarn, not by a nonwoven, not by any construction — because the two criteria have excluded each other before any fabric has been considered.

Which names the soil everybody has trouble with #

That is a strong claim and it is checkable against practice, because the trade already knows which soils are difficult.

A uniform fine sand — well sorted, a uniformity coefficient of three or four, the material of a beach or a dune — is the standing nightmare of geotextile filtration, and the literature is full of special provisions for it. On the bracket above it is not a difficult case; it is an impossible one at f = 0.2, and possible only if the loading is gentle enough to relax the retention factor to one.

A well-graded silty sand, with a uniformity coefficient of fifteen or twenty, is routine — and the bracket says why: it clears the condition at every retention factor in use.

So the empty interval has a fourth cause the essay’s list does not contain, and it is the one that cannot be fixed by changing the fabric. Three of the four causes are answered by a finer yarn, a nonwoven or a renegotiated specification; the fourth is answered by grading the soil, which means placing a graded granular layer between the soil and the fabric and is exactly what civil-engineering practice does when a fabric alone will not serve.

And it reorders the diagnosis #

The closed form for the finest soil a yarn can hold is a lower bound on D85. The uniformity condition is a bound on the ratio. They are independent, so an empty interval now has two separate tests, and running them in the right order saves work.

Test the soil’s uniformity first, because it needs no fabric and no yarn: compute D85/D15 and compare with 3/f. If it fails, no cloth exists and the answer is a graded transition layer.

Then test the yarn, with the 1.25·d threshold. If that fails, the answer is a finer monofilament or a nonwoven.

Then read the interval, which is where a sett gets chosen.

Three tests, in order of how little each needs to know, and the first two are one division apiece. That is the shape this field keeps producing — an applied requirement is a set of inequalities, the interval they leave can be empty, and knowing which inequality emptied it is the whole of the advice.

Why the answer is a sett rather than a fabric #

There is a habit of mind this field keeps rewarding, and this is the clearest case of it.

Asked to choose a filter cloth, the natural move is to compare fabrics: this roll against that one, this weave against that weave, an opening size and an open area printed on each. The arithmetic above says the comparison is the wrong shape. The fabrics on offer are samples from a one-parameter family, the parameter is the sett, the two criteria are monotone in it in opposite directions, and the answer is therefore an interval — which either contains one of the rolls on offer or does not.

Reading it that way changes what a specifier asks for. Not “is this fabric suitable” but “what is the interval, and does anything in the catalogue sit inside it” — and, when nothing does, which of the three inputs to move. The weave has hardly appeared in this essay, and that is not an omission: for the two properties being specified, a plain weave and a twill of the same sett and yarn differ by the geometry of a passage rather than by the size of a hole, which is a second-order effect on a first-order decision.

That is also why the essay has quoted no fabric by name. A style number identifies a fabric somebody has used successfully, which is the same information a reinforcement’s style number carries and the same limitation: it does not transfer to a different soil, a different flow requirement or a different loading, because none of those is in the number.

Who found it, and when #

Woven monofilament filter cloths and the specifications that go with them are a 1970s development, arriving with geotextiles as a civil-engineering material rather than out of weaving. The retention criteria descend from Terzaghi’s graded-filter rules for granular filters — where the “filter” was gravel — restated for a fabric whose opening could be measured directly.

The closed form above is not in the literature in that shape, as far as this site can tell, and it is not a deep result: it is the two standard criteria written as inequalities in the spacing and solved. What makes it worth a figure is that it turns two practice rules into a statement about a yarn — that a monofilament of diameter d cannot filter a soil finer than 1.25d — which is the form a purchasing decision actually takes.

Where the ladder goes next #

Filtration asks a fabric for a hole. The next two rungs ask it for a load path: a pressurised vessel puts twice the stress around its circumference as along its axis, so the cloth that holds it must be unbalanced in a computable ratio, and a braided one must be laid at one particular angle. The next rung starts from the pressure and works back to the sett.