The fourth power is a close cloth's rule

There is a rule about fabric permeability that everybody quotes and it is a good rule. The air goes down the channels between the threads; a channel is a duct; the flow down a duct goes as the fourth power of its width; so a ten per cent change in the clear gap is a forty-six per cent change in the flow, and a sett is therefore a much bigger decision than it looks.

Every step of that is correct, and applying it to a muslin gives a face velocity of about eleven metres per second at a hundred pascals. A permeameter reads three.

It is worth being clear that the fourth power itself is not the suspect. The clear gap between two threads really is what a channel’s width is, the cover factor really does decide it, and a duct’s rate really does go as its area times the square of its hydraulic diameter, which is a fourth power of a length. Every one of those has been derived on this site and none of them is being withdrawn.

The error is a factor of four, it is not a calibration, and finding it needs nothing but reading the numbers the model itself returns. Eleven metres per second is not a small velocity. Air moving at eleven metres per second carries a dynamic pressure of eighty-one pascals — most of the hundred that was applied — and Poiseuille’s result assumes the whole hundred went on shearing the air against the walls.

The claim #

The pressure across a woven cloth has two parts and which of them dominates is a property of the cloth, not of the air.

$$\Delta P \;=\; \frac{2\,f\!Re\,\mu L v}{D_h^{2}} \;+\; K\,\tfrac{1}{2}\rho v^{2}$$

The first term is viscous — Poiseuille’s, generalised to a duct of any section — and it goes as one over the square of the hole’s size. The second is inertial: the cost of accelerating still air up to the speed it must have in the hole, plus what is lost when the jet spreads out again on the other side. It does not depend on the hole’s size at all.

So the ratio between them is a function of the geometry, it moves steeply as a cloth is closed, and the familiar rule becomes true rather than being true.

The argument #

A channel through a cloth is about as long as it is wide. A muslin’s hole is 250 µm across and its thickness is 342 µm; a poplin’s is 168 across and 332 long. That is not a duct in the sense Poiseuille assumed — a duct long enough for a parabolic velocity profile to develop and then persist. It is entrance region from end to end, and the entrance region is where a fluid is being accelerated rather than sheared.

The second thing the model returns is the Reynolds number, and it is the one that settles the matter. In a muslin’s hole at a hundred pascals the air is moving at about nine metres per second and the hole is a quarter of a millimetre across, which gives a Reynolds number of six hundred. Laminar, comfortably — but nowhere near the creeping regime where inertia can be dropped, and a factor of six hundred is a lot of inertia to leave out of a balance.

So the drop is written with both terms and the resulting quadratic is solved for the velocity, rather than one term being chosen and the other apologised for. The solution is written in the form that keeps its accuracy when the inertial coefficient is tiny, because that is the close-cloth limit and it is exactly the case the argument is about.

Where the two exchange places #

Held at one yarn and swept from a fifth of the jam to the jam itself, the viscous share runs from 1.0 per cent at 6.9 threads per centimetre to 61.6 per cent at 34.2, which is where a 20 tex cotton’s ends meet. It does not reach half until the jam.

Three things are asserted as that sweep runs and none of them is a property of the arguments it was given:

• A closer sett passes less air. Trivially true and worth asserting, because a sweep in which it failed would mean a solver had gone backwards.
• The threads carry more of what is left. Follows from the first, since the bed term moves only with the cover while the channel term is vanishing.
• The viscous share rises. This is the claim. It has no value in it — it says the rule gets closer to being true as the cloth closes, which is a statement about the two terms’ forms rather than about the numbers a particular cotton produces.

There is a second way to see the same thing, and it is the one a weaver would reach for. The viscous term carries a length in it — the channel’s own length, which is the cloth’s thickness — and the inertial term does not. So the ratio between them is a slenderness: how many hole-widths long the passage is. A cheesecloth’s hole is 795 µm across in a cloth 420 µm thick, which is a passage half as long as it is wide and barely a duct at all. A sheeting’s is 170 across and 382 long, which is more than two to one. Closing a cloth does two things at once — it narrows the hole and, because the crimp deepens, it lengthens the passage — and both of them push the same way.

The consequence for a quotation is severe at the open end and mild at the closed one. Quoting the fourth power alone at a cover of 0.115 overstates the flow ninety-eight-fold. At the jam it overstates it by 62 per cent. A rule that is wrong by three fifths is a rule worth using with a caution attached; a rule that is wrong by ninety-eight is a different function.

What was counted, and how #

The viscous coefficient needs the duct’s shape as well as its size, and the shape enters through a dimensionless number that depends on the section and on nothing else: 16 for a circle, 14.23 for a square, 24 for an infinitely thin slit. Those are Shah and London’s, from 1978, quoted rather than derived — deriving them needs the series solution of Poisson’s equation on a rectangle, which this collection has no business rebuilding and every reason to name.

The inertial coefficient is an entrance-plus-exit loss for a short square-edged orifice, about 1.5, with a bracket from 1 to 2. It is quoted the same way.

Nothing in the regime claim depends on either constant. The viscous term goes as L/Dₕ² and the inertial term does not depend on the hole’s size, so their ratio moves with the geometry whatever the two coefficients are. Doubling the loss coefficient moves the crossing cover by a few hundredths and moves nothing else. The constants decide where the crossing is; the forms decide that there is one.

The two terms are required to account for the whole applied pressure as the solution is returned — a check that would catch a sign error or an algebraic slip in the quadratic, and that costs nothing.

What the two terms do to a permeameter’s own reading #

The two-term form has a consequence for the instrument that is worth separating from the consequence for the model, because it says what a permeability figure means when it is quoted without a pressure beside it.

A viscous resistance gives a flow proportional to the pressure and an inertial one gives a flow proportional to its square root. So the ratio of the reading at two hundred pascals to the reading at a hundred is two for a close cloth and √2 for an open one, and everything in between is a mixture. The exponent is a measurement of where the cloth sits on the sweep, available from any permeameter that can be run at two pressures.

That is a cheaper experiment than anything else in this essay and it needs no calibration at all, because it is a ratio of two readings on one instrument. The prediction is sharp: a cheesecloth should read very nearly √2 and a sheeting at its jam should read nearer 1.8, and a scatter of ordinary shirtings should lie between them ordered by cover.

It also explains a practical nuisance the standards handle by fiat. A permeability quoted at one pressure cannot be converted to another without knowing the exponent, and the exponent is a property of the cloth. That is why the test methods specify the pressure — a hundred pascals in most of them, two hundred in some — and why figures from two standards are not interconvertible by a constant however carefully anybody tries.

The trade’s usual assumption in making that conversion is proportionality, which is the viscous limit, so converting an open cloth’s reading from one pressure to another overstates the higher-pressure figure by up to the ratio of the two exponents. For a doubling of pressure on a scrim that is a factor of √2 — forty per cent, from an arithmetic step nobody records as an assumption.

And the direction is the awkward one again: the conversion is safest for the close cloths, where the reading is small and nobody is worried, and worst for the open ones, where the number is large and is the reason the fabric was chosen.

Where the model stops #

The entrance loss is a straight orifice’s and a cloth’s hole is not straight. The channel bends: its waist is at the mid-plane and its mouths are bounded by different threads. A bent passage loses more than a straight one, so the true inertial term is larger than this and the crossing is at a closer sett than quoted.

The flow is treated as steady and a permeameter’s is. Wind is not, and a fabric flapping in a gust presents a different problem in which the cloth’s own motion matters. Nothing here is about that.

The cloth does not move and a real one does. Every hole here is held at the size Peirce’s closure gives it under no load. Air at a hundred pascals presses on a square metre of fabric with ten newtons, which a mounted permeameter specimen resists and a hanging curtain does not; a cloth that bows also stretches, and a stretched cloth’s holes are not the holes it was woven with. The instrument’s own clamp is part of the measurement.

Turbulence is not modelled and is not needed. At six hundred the flow in the hole is laminar, so the inertial term is an acceleration cost rather than a turbulent loss. At the pressures used for a spinnaker or a parachute the Reynolds number is several thousand and a different account is needed.

The floor is in every number here and is never the point. Every total in this essay includes what passes through the threads themselves, which does not go to zero when the channels do. At the setts a shirting is woven at it is a fraction of a per cent of the total and it does not affect the crossing; at the jam it is everything. The viscous share plotted here is the channels’ own, because the bed term is viscous by construction and including it would make the curve rise for a reason that has nothing to do with the argument.

And the holes are all the same in this sweep. The sweep is run on a plain weave, where they are. In any other weave they are not, and the flow is a sum over a distribution — which raises the total, because a fourth power is convex, and does not move the crossing.

The generalisation #

A power law quoted without its regime is a limit presented as a rule, and the failure is always at the end where the neglected term is largest.

The shape is everywhere. Stokes drag on a falling particle goes as the radius and is quoted freely until the particle is large enough that the wake costs more than the shear. Ohm’s law describes a resistor until the current is high enough to heat it. A beam’s deflection goes as the cube of its length until it is short enough for shear to matter. In every case the quoted law is the low-something limit of a two-term balance, the second term has a different power, and the rule fails in the direction that makes the answer too large.

The textile version has a particular sting in it. Air permeability is one of the few fabric properties with a cheap, standard, universally available instrument, so it is measured constantly and there is no shortage of data. What the data does not come with is a model, and the model people reach for when they need one is the fourth power — because it is the one in the books, and because it is right about the cloths the books are mostly about.

The diagnostic is to compute the neglected term rather than to bound it. The error here was not found by measuring anything: it was found by taking the velocity the model returned and asking what dynamic pressure a stream at that velocity carries. Eighty-one pascals out of a hundred, which is not a correction. Any model that returns a velocity can be asked that question, and it costs one line.

And there is a lesson about the shape of the disagreement. The fourth-power rule is not wrong by a constant factor; it is wrong by a factor that varies from ninety-eight to less than two across the range of ordinary cloths. So a fitted correction — a constant divided into the answer, chosen to fit a few fabrics — would work on those fabrics and fail badly on an open one, while looking like a calibration all the way.

Who found it, and when #

Poiseuille is 1846 and Hagen 1839. The two-term form with a velocity-squared correction is Forchheimer’s, from 1901, written for flow through porous beds at higher rates and used ever since in packed columns and in soil mechanics.

Applying it to fabric is not new either: the standard textile-physics treatments of air permeability note that the relation between pressure and flow is not linear except at low rates, which is the same statement.

What appears to belong to this collection is putting the crossing on a cover-factor axis, so that the question stops being at what pressure does the relation bend and becomes at what construction does the familiar rule start being true. The answer is a cover of 0.571, which is a cloth at its own jam, and it means the rule is not comfortably right about anything an ordinary cotton weaves — it is least wrong about a dense shirting and wrong by two orders about a curtain.

The two coefficients are both quoted from elsewhere, which is this collection’s standing practice when a number is published physics rather than cloth — as with Peirce’s constant, Munden’s constants and the Kozeny constant. What is computed here is the geometry they are applied to.

Where the ladder goes next #

The rule being a limit at the closed end is what makes a threshold specification decidable, and the decision comes out negative: a windproof cloth is at its yarn’s limit, because the floor set by the yarn is already above the threshold and no sett gets near it.

Sideways, the same duct arithmetic asked about the shape of a hole rather than its size gives a result at constant open area: a satin’s hole is a slot, and drawing a hole out at constant area cuts its flow by more than half.