Illustrated Physics
physics, drawn
Fluids

The outlet that jams by chance

A hopper whose outlet is a few grains wide does not flow slowly. It flows at the full rate and then stops dead, when a handful of grains wedge into an arch across the opening, and it flows again only when something knocks the arch down. The number of grains that pass between one jam and the next is random and exponentially distributed, as if every grain leaving had the same small chance of being the last. Its average rises so steeply with the opening's size that it looks like a threshold, and whether there is one is still argued: two laws agree wherever an outlet can be watched, and they part only after millions of grains.
14 min read 5 figures Order out of the randomWhere the model stops

Assumes: The hourglass that keeps time · The heap that becomes a solid

The hourglass that keeps time found a granular law with a strange exponent: grain leaves a hopper at a rate set by the five-halves power of the outlet’s width and not at all by how much grain is above it, because the hopper’s walls carry the weight. The law was right over a wide range and wrong at its narrow end, where it predicts a small, steady trickle and the hopper delivers nothing. That essay noted the cause, an arch of grains wedged across the opening, and left it there. The arch is a subject of its own, because the way an outlet jams turns out to be a matter of chance rather than of size — and that makes a design rule followed in every factory into a statement about probability.

A hopper that runs in bursts. Grains discharged against time, on a logarithmic scale, from a silo of 1 mm glass beads through outlets 3.5, 4.0, 4.4 grain diameters wide; each time the outlet jams it is cleared after 2 s (a tap on the silo) and the flow restarts. Avalanche sizes are drawn at random from an exponential distribution with each outlet's mean. At 3.5 diameters the mean avalanche is 53 grains and the outlet jammed 57 times in two minutes, passing 2,537 grains; at 4.0 diameters the mean avalanche is 1,000 grains and the outlet jammed 37 times in two minutes, passing 33,568 grains; at 4.4 diameters the mean avalanche is 45,821 grains and the outlet jammed 3 times in two minutes, passing 116,983 grains. While it flows, each outlet runs at its full Beverloo rate; a narrower outlet does not flow more slowly, it stops more often, and the flat stretches are the waits.
Fig. 1 Grains discharged against time, on a logarithmic scale, through outlets 3.5, 4.0 and 4.4 grain diameters wide in a silo of 1 mm glass beads; each jam is cleared after 2 s and the flow restarts. Mean avalanches of 53, 1,000 and 45,821 grains; 57, 37 and 3 jams in two minutes. While flowing, each runs at its full Beverloo rate.

Full rate, then nothing

Watch a narrow outlet and what it does is unlike a narrowed pipe. A pipe half as wide carries less water, steadily. An outlet four grains wide passes grain at the rate Beverloo’s law gives for its size — several hundred millimetre beads a second — and then, after a few hundred or a few thousand grains, stops completely. Nothing comes out. The grains above it are motionless, held up by the walls as before, and the grains at the opening are held up by each other: a few of them, each pressed against its neighbours, form a curved bridge from one side of the opening to the other, and the bridge carries the load of everything above it. A tap on the side of the silo or a puff of air through the outlet breaks the arch, and the flow restarts at its full rate, until the next one forms.

The first figure follows three outlets through two minutes of this, with each jam cleared after two seconds. The flat stretches are the waits. The outlet three and a half grains wide jams fifty-seven times and passes a few thousand grains; the one 4.4 grains wide jams three times and passes more than a hundred thousand. The difference between them is not the rate while flowing, which differs by about a factor of two and a half, but how long each burst lasts before an arch closes it off.

Every grain the same small chance

Avalanches of every size, in one shape. How often an avalanche of a given size occurs, for 4,000 avalanches at each of three outlets (3.5, 4.0, 4.4 grain diameters), with the size measured in units of each outlet's own mean (53, 1,000, 45,821 grains), on a logarithmic scale. All three fall on one straight line, e^(−s/⟨s⟩): the exponential distribution, which is what results if every grain that leaves has the same small chance of completing an arch, whatever has passed before. The outlet's size enters only through the mean. There is no typical arch-forming moment and no memory: a hopper that has just been running for a long time is no more and no less likely to jam in the next second.
Fig. 2 How often an avalanche of a given size occurs, for 4,000 avalanches at each of the three outlets, with the size measured in units of each outlet’s mean (53, 1,000 and 45,821 grains), on a logarithmic scale. All three fall on one straight line, e^(−s/⟨s⟩): the exponential distribution, with no memory.

How many grains pass between one jam and the next is random, and its distribution has been measured carefully. In 2005 Iker Zuriguel and colleagues discharged thousands of avalanches of glass beads through the outlet of a three-dimensional silo and counted each. They found the sizes exponentially distributed. On a logarithmic scale the count of avalanches of a given size falls in a straight line, and when each outlet’s sizes are measured in units of its own mean, the lines of different outlets lie on top of one another, as in the second figure.

An exponential distribution has a simple meaning, and it is the one a nucleus with no clock found in radioactive decay: there is a fixed probability per event and no memory. Each grain that reaches the outlet has the same small chance of completing an arch, whatever happened before it. A hopper that has just run for an unusually long time without jamming is not due a jam; the next grain is as likely to be the last as the first one was. The outlet’s size enters the whole description through a single number, the mean, which is the reciprocal of that chance per grain.

The absence of memory is a little surprising, because an arch is a structure and structures take time to build. It means the arch is built in the instant the last few grains arrive, from whatever configuration they happen to arrive in. If the configuration at the opening is one of the rare ones that can bear load in compression across the gap, it jams; if not, the grains fall through and the next configuration is drawn. Each configuration is a fresh throw of the dice.

Why a few grains can hold up a silo

An arch of grains is a structure with no cement, and it stands for the same reason a masonry arch does. Each grain in it is pressed against its two neighbours, and if the arch is curved the right way those contact forces lean towards the vertical at the ends and towards the horizontal at the top, so that their sum carries the load from above down and out to the edges of the opening. Every contact is in compression. No grain needs to pull on another, which is just as well, since dry grains cannot.

That one requirement explains most of what is known about the arches’ shapes. Photographs of jammed outlets in two-dimensional hoppers, where every grain can be seen, show arches of a few grains, typically one more than the number of grain diameters across the opening, and nearly always convex: a chain whose direction turns steadily downwards from one edge to the other. A configuration that is not convex has a grain somewhere pushed outwards by its neighbours rather than held between them, and it falls. Friction at the contacts widens the range of angles at which a pair of grains can press on each other without slipping, so rough grains jam more readily than smooth ones at the same outlet; the arch does not need friction, but friction makes more arrangements count.

The arch also explains why the opening’s size matters so steeply. A wider opening needs a longer arch, and a longer arch needs more grains, each arriving at an acceptable angle to the last. The chance that a randomly arriving cluster of grains happens to satisfy all those conditions at once is a product of one chance per grain, which is why the probability of jamming falls so quickly as the opening widens, and why it is natural to expect it to fall exponentially rather than to stop.

A rise so steep it looks like a wall

Two laws that agree wherever an outlet can be watched. The mean number of grains between jams against the outlet's size in grain diameters, on a logarithmic scale. Solid: a law with a critical size, ⟨s⟩ ∝ (R − R)^(−γ), R = 4.94 and γ = 6.9, as fitted to three-dimensional silo measurements; it runs off the top at 4.94 and says a larger outlet never jams. Dashed: a law with no threshold, ⟨s⟩ ∝ exp(bR²), the chance of an arch falling with the number of grains it must recruit across the outlet's area, matched to the first at 3.0 and 4.4 diameters. Between those sizes they never differ by more than a factor of 2.7. At 4.8 diameters the first gives 5.1 × 10⁸ grains between jams and the second 1.1 × 10⁶; at 6 the second gives 6.6 × 10¹⁰, and the first none at all. Distinguishing them needs avalanches longer than any experiment has the patience to wait for (shaded: beyond about 10⁶ grains).
Fig. 3 The mean number of grains between jams against the outlet’s size, on a logarithmic scale. Solid: a law with a critical size, (Rc−R)−6.9(R_c - R)^{-6.9} with RcR_c = 4.94, as fitted to silo measurements; dashed: a law with no threshold, growing exponentially with the square of the size, matched to it at 3.0 and 4.4 diameters. Between those sizes they never differ by more than a factor of 2.7; at 4.8 they give 5.1 × 10⁸ and 1.1 × 10⁶ grains. Shaded: avalanches too long to wait for.

The mean is what the outlet’s size controls, and it controls it violently. In the third figure, scaled so that an outlet four grains wide passes about a thousand grains between jams, an outlet three grains wide passes seven and one 4.4 grains wide passes forty-six thousand. Zuriguel and his colleagues fitted their measurements with a law that diverges, ⟨s⟩∝(Rc−R)−γ\langle s\rangle \propto (R_c - R)^{-\gamma}, with the critical outlet size RcR_c = 4.94 grain diameters and an exponent γ of 6.9. Read literally, the law says an outlet wider than about five grains can never jam: the mean avalanche is infinite, which means the probability per grain of completing an arch is zero. That is a jamming transition with a sharp threshold, of the kind the heap that becomes a solid found when a packing of grains first becomes rigid.

There is another way to read the same data. The arch has to span the opening, and the wider the opening, the more grains must be caught in exactly the right arrangement at once. If each extra grain the arch must recruit multiplies the chance of success by a fixed factor, the chance falls exponentially with the number of grains in the arch, and the mean avalanche grows exponentially with the outlet’s size — or with its area, if what matters is every grain across the opening in three dimensions. The dashed curve in the figure is such a law, with no threshold at all, matched to the critical one at three and at 4.4 diameters.

Between those sizes the two curves are never more than a factor of 2.7 apart, which is within the scatter of the measurements. They part only above a million grains between jams: at 4.8 diameters the critical law gives five hundred million and the exponential one a million. Telling them apart needs avalanches that last hours. The shaded band in the figure is where an experiment has to sit and watch an outlet flow without interruption for a very long time to see a single jam, with any vibration or impurity in the meantime counting against the measurement. Experiments since have argued both ways. Douglas Durian and his collaborators measured the fraction of grains that jam over a wide range of outlets and hopper angles and found no sign of a critical size, only a smooth, very steep decline; others have found data that a threshold fits better. The question is not settled, and it is the same question that runs through the physics of glasses: whether a relaxation time that grows explosively becomes truly infinite at some temperature or only too long to wait for.

A practical difficulty makes the two readings hard to separate even where both make predictions. The mean of an exponential distribution is estimated from a sample with an uncertainty equal to the mean divided by the square root of the number of avalanches counted. Measuring a mean of a million grains to ten per cent needs a hundred avalanches, a hundred million grains through the outlet, without a single disturbance that might trigger or prevent a jam by some other route. At the largest outlets anyone has measured, each data point is a long, fragile experiment.

What the outlet actually delivers

What a hopper actually delivers when it keeps jamming. Grains delivered per second, averaged over flowing and jammed time, against the outlet's size, when each jam is cleared after 0.5 s, 2.0 s, 10.0 s, beside Beverloo's rate for an outlet that never jams (dashed). Above about five diameters the jams are so rare that the hopper delivers its Beverloo rate whatever the clearing takes. Below, the delivered rate collapses: at 3.5 diameters the outlet would pass 421 grains a second while flowing, and delivers 84 if jams are cleared in half a second and 5 if they take ten. The size of an outlet below the jamming range sets how often it must be attended to, more than how fast it flows.
Fig. 4 Grains delivered per second, averaged over flowing and jammed time, against outlet size, with each jam cleared after 0.5, 2 or 10 s, beside Beverloo’s rate for an outlet that never jams (dashed). Above about five diameters jams are too rare to matter. At 3.5 diameters the outlet passes 421 grains a second while flowing and delivers 84 if jams are cleared in half a second, 5 if they take ten.

For anyone who has to empty a hopper, the mean avalanche matters through what it does to the delivered rate. The fourth figure combines the Beverloo rate while flowing with the waits between flows. Above five diameters the jams are so rare that the hopper delivers its full rate whatever the clearing method. Below, the delivered rate collapses, not because the flow is slow but because it is mostly stopped. At three and a half diameters the outlet runs at 421 grains a second while it runs, but each burst lasts about a tenth of a second, and a jam that takes two seconds to clear leaves the outlet idle most of the time. Clear it in half a second and it delivers 84 grains a second; leave each jam for ten and it delivers 5.

This is why vibrating feeders and air-assisted outlets work. They do not widen the outlet or speed the flow. They shorten the wait, by breaking each arch as soon as it forms, and in the arithmetic of the figure the clearing time is as important as the outlet’s size. The same reasoning is used for crowds, where the arch is a group of people wedged at a narrow exit, and for the flow of sheep through gates, both of which have been measured to produce exponential bursts between stoppages; the remedies proposed in each case, from an obstacle placed just in front of the exit to a change in how hard the crowd pushes, act on how often arches form and how long they last.

How long an outlet runs between jams

How long a hopper runs between jams. The mean time between jams, in hours, for 1 mm glass beads leaving through an outlet of the given size at Beverloo's rate, under the two laws for the mean avalanche, the one with a critical size and the one without. With a critical size the time runs off the top at 4.94 diameters and an outlet wider than that is said never to jam. Without one, it grows without limit but never becomes infinite: a jam an hour on average at 5.01 diameters, one a year at 6.04, and one a century at 6.49. The engineering rule of an outlet six to eight times the largest particle is, under either law, a statement about how rarely jams can be tolerated, not a line beyond which they cannot happen.
Fig. 5 Mean time between jams for 1 mm glass beads leaving at Beverloo’s rate, under the two laws for the mean avalanche. With a critical size the time runs off the top at 4.94 diameters. Without one it grows without limit but stays finite: a jam an hour on average at 5.01 diameters, one a year at 6.04, one a century at 6.49.

The last figure converts the mean avalanche into the quantity a designer cares about: hours between jams. Under the law with a critical size, the curve goes vertical at 4.94 diameters, and the design question has a clean answer: make the outlet wider than that and it will never jam. Under the law without a threshold, it gets steeper and steeper without becoming vertical: an outlet of five diameters jams about once an hour, one of six about once a year, one of six and a half about once a century.

The industrial rule of thumb — an outlet six to eight times the largest particle — sits exactly where the two laws stop mattering for practical purposes. Under either, a hopper built to it will not jam within the life of the plant. But the rule was not derived from either. It encodes experience, and the figure shows what kind of statement it is: not a line beyond which jamming is impossible, but a choice of how rarely it can be tolerated, made before anyone could say which law was right. Real hoppers jam above it anyway, for reasons outside both laws. A grain larger than the rest, a lump of grains stuck together by moisture, or a cohesive powder whose particles attract one another can all form arches far more readily than dry, uniform beads.

Where the description holds

Everything here is about dry, non-cohesive grains of roughly one size, moving slowly enough that the air between them does not matter. Fine powders behave differently, because the air must flow in to replace the grains leaving and can hold them back, and cohesive or moist grains form arches by sticking as well as by wedging. Mixtures of sizes jam at outlets set by their larger grains. The figures also assume a flat-bottomed silo; a hopper with sloping sides directs the grains towards the outlet at an angle, and steeper hoppers jam less, though not by enough to change the shape of the laws. The Beverloo rate used for flowing is itself a fit, and its constants depend on the grain.

The exponential distribution is the most robust thing in the essay; it has been found for beads, for irregular grains, in two and three dimensions, and for crowds. The steepness of the mean’s rise with size is robust too. What is fragile is any statement about what happens beyond the largest avalanche anyone has measured.

What the curves cannot show

The curves are statistics. They do not show an arch, and an arch is a geometrical object worth seeing: in a two-dimensional hopper, where it can be photographed, it is a chain of a few grains, usually convex, each pressing on its neighbours along their line of contact, so that the whole carries the load above it as compression, like a masonry arch. The figures do not show how an arch breaks, either. Vibration loosens it by momentarily relieving the contact forces, and the time an arch survives under steady vibration has been found to have a broad, heavy-tailed distribution — so broad, in some experiments, that the mean time to unjam seems not to exist, and an outlet under gentle vibration can stay blocked for very long times. That is a second statistical law, about breaking rather than forming, and it is not drawn here.

Still open: whether there is a size beyond which an outlet cannot jam

The question the third figure leaves open has a practical side and a theoretical one. Practically, the difference between the two laws is the difference between a hopper that will never jam and one that will, very rarely, without warning, after months of faithful service; for a hopper feeding a process that must not stop, that matters. Theoretically, it is whether the arrest of a flowing granular material at a narrow outlet is a true phase transition, with a critical size, or a smooth crossover in which the probability of arrest falls continuously towards zero. Recent experiments that run silos for days, and simulations that follow billions of grains, are pushing the measured avalanches into the range where the laws part. Some find the threshold-free description holding further than the critical law allows, and others report a critical size that depends on the hopper’s shape. Whether a single answer exists for all granular outlets, or whether it depends on the grains’ friction, their shape and the outlet’s geometry, is unresolved.

The habit worth carrying away is to ask whether a threshold is a measurement or an extrapolation. Grains leave a narrow outlet at the full Beverloo rate in bursts whose sizes are exponentially distributed — every grain the same small chance of completing an arch, with no memory — and the mean burst rises so steeply with the outlet’s size that a law with a critical size at 4.94 diameters and a law with none agree within a factor of 2.7 everywhere they can be tested, parting only beyond a million grains between jams, where an hour between jams at five diameters and a year at six are the difference between them. A curve steep enough to look like a wall is not evidence that the wall is there.

Part 11 of 12

This essay is one argument about Granular matter. The others:

How a layer of grains moves down a slope. The velocity through a steady layer of 0.5 mm glass beads 20 grains deep flowing down slopes of 22°, 25° and 28°, against height above the rough base in grain diameters; the dashed curve is a Newtonian liquid's parabola, scaled to the 25° surface speed. Because every depth must have friction tan θ, every depth has the same inertial number, and the shear rate grows with the square root of the pressure above it: Bagnold's profile, u ∝ h^(3/2) − (h − z)^(3/2). Its surface speeds are 8.0, 41.0, 113.7 cm/s; compared with a liquid's parabola the shear is spread more evenly through the depth, because the shear rate falls off only as the square root of the depth below the surface rather than in proportion to it. The sand whose friction depends on how fast it flows Part 10 — A heap of sand stands at its angle of repose and starts to slide a few degrees steeper; between those angles there are steady avalanches, and their speed is decided by one number. The inertial number compares how long a grain takes to fall into a gap opened under it with how long the flow takes to open the gap. Friction rises with it, from the static value to a ceiling, and that single dependence explains why grains flow steadily on only a twelve-degree window of slopes, why a flowing layer has the velocity profile Bagnold measured, and why a deeper avalanche is faster — but not as a liquid would be. The force that does not count the water. The force of a water bridge between two touching spheres of radius R, in units of 2πRγ, against the bridge's volume on a logarithmic scale running over six decades, for three contact angles. Each curve is nearly flat: from a millionth of R³ to a thousandth, the force on a wetting grain falls only from 0.99 to 0.92 of 2πRγ, and it stays close to cos θ for the others. A droplet between two grains of sand pulls almost as hard as one a thousand times larger. This is the reason sand wet with one per cent of water and sand wet with ten per cent hold together with about the same strength. The curves fall where a bridge is large enough for its neck to stop being small compared with the grain, and bridges of that size merge with their neighbours. The sandcastle a spoonful of water holds Part 12 — A dry heap of sand stands at thirty degrees and no steeper. Add a spoonful of water to a bucketful, and the same grains will stand in a vertical wall and carry a person's weight. Add ten times as much water and the wall is no stronger, because a bridge of water between two grains pulls with almost the same force whatever its size. Its strength belongs to the grain and the surface tension, and the amount of water hardly enters.

Where this goes next

The hourglass that keeps time. Discharge against how much is left above the opening, for grain and for liquid through the same 40 mm hole, each as a fraction of its own rate at a full hopper. The grain rate is a horizontal line: it does not appear in Beverloo's law at all, because the pressure at the outlet does not depend on the head — the walls carry the weight, which is what Janssen's argument establishes, so the grains at the opening are pushed by their immediate neighbours and by nothing else. The liquid falls as the square root of the head and is down to 32 per cent by the time a tenth is left. That is why an hourglass keeps time and a water clock does not, and why the water clocks that worked were built with a float and an overflow to hold the head constant. The hourglass that keeps time Grain leaves a hopper at a rate that does not depend on how much is above it, and that goes as the orifice to the five-halves power rather than the one half a liquid gives. Both facts follow from the same thing: the weight is carried by the walls, not by the grains at the opening. Where a heap stops flowing. The number of motions that cost nothing, against the mean number of contacts each grain has, for a patch of 37 grains. Every point is the rank of an actual rigidity matrix — one row per contact, one column per coordinate — subtracted from the number of coordinates, so it is a measurement of the network rather than a formula about it. The dashed line is Maxwell's count, two coordinates a grain minus half a contact each, which is what the counting alone predicts. The measured curve reaches its floor of three free motions — the rigid-body ones — at a coordination of 4.11, and Maxwell's line reaches zero at exactly four, which is twice the dimension of the plane. That is the isostatic point and the difference between the two numbers is the boundary of a finite patch, whose outer grains have fewer neighbours than they would in an infinite one. Above the threshold the two curves separate: the measured floor stays at three while the count keeps falling, and the gap is the redundancy — contacts whose forces no equilibrium equation can determine. Below it the pack has genuine mechanisms and will rearrange under any load at all. The transition is in the count and not in the material, which is why a fluid and a solid here are made of exactly the same grains. The heap that becomes a solid Sand poured into a jar flows; the same sand shaken down and pressed does not. Nothing about the grains has changed — not their size, their hardness, their friction or their density by more than a per cent — and the thing that changed is a count of contacts, which crosses a threshold set by the dimension of space and by nothing else. Decay, and the ensemble it is a property of. 400 nuclei followed for 4 half-lives. The smooth curve is the exponential; the stepped traces are 3 independent runs in which every nucleus was given its own decay time and told nothing about the others. The number surviving halves at each dashed line — 200, 100, 50, 25 — and it halves again over the next interval regardless of how long the sample has already been sitting there, which is the property no ordinary clock has. The traces wander further from the curve as the numbers get small: at the end only about 25 are left and the scatter is a visible fraction of that. A nucleus with no clock A half-life is a precise number and no individual nucleus has one. Each has the same chance of decaying in the next second as it had on the day it formed, and the exponential curve is a property of the population rather than of any member of it.

The objects named here

The third axis, after the field and the reading path: the things themselves, and every essay that touches each one.

CloggingCritical pointExponential distributionForce chainsGranular flowJammingMemorylessnessPoisson process