The hourglass that keeps time

The silo that does not weigh what it holds establishes that the pressure at the bottom of a column of grain stops growing with depth: the walls take the weight through friction, and below a couple of diameters the floor feels nothing more however much is added.

That result has a consequence that turns an ancient instrument into a good one.

Why an hourglass works and a water clock does not #

A water clock has an obvious defect that its makers spent two thousand years working around: the flow slows as it empties. Torricelli’s law gives the outlet speed as $\sqrt{2gh}$, so the rate falls as the square root of the head, and a vessel that takes an hour to empty spends its last quarter-hour delivering a trickle.

The workarounds were ingenious — a float and an overflow to hold the head constant, a shaped vessel whose cross-section compensates, a bank of vessels feeding one another. All of them are engineering around a physical fact.

An hourglass needs none of it. The rate is constant from full to empty, so equal intervals of time correspond to equal amounts of sand, and the instrument can be calibrated once. That is not a property of sand’s fluidity; it is a property of sand’s rigidity, and it follows directly from the wall friction that carries the weight.

The grains at the opening do not know how much is above them. They are pushed by their immediate neighbours, at a pressure set by the local geometry, and the weight of the column has been transferred to the walls long before it reaches them.

The exponent that is not two #

If the grains left at a fixed speed the rate would go as the area, and the exponent would be two. It is five halves, and the extra half is a statement about what sets the speed.

The argument is due to Hagen and is a dimensional one. Nothing above the opening supplies a speed — the head is screened out — so the only length available is the orifice diameter $D$, and the only acceleration is $g$. A speed built from those is $\sqrt{gD}$, and the rate is that speed times the area:

$$W \sim \rho \sqrt{gD}\, D^2 = \rho\sqrt{g}\, D^{5/2}.$$

The picture behind it is a free-fall arch: a dome of about the orifice size spans the opening, the grains inside it are in free fall, and they emerge having fallen a distance of order $D$. It is a cartoon rather than a derivation, and it is the right cartoon — the exponent it predicts is what is measured, over three decades of orifice size, in every material anybody has tried.

The same reasoning gives the right answer for a liquid and shows where the two part company. A liquid’s outlet speed is $\sqrt{2gh}$ because the head is available to it; a grain’s is $\sqrt{gD}$ because the head is not. Both are free-fall speeds through the only length the problem offers, and the two problems offer different lengths.

A power that is not an integer is usually a sign that two things are being multiplied, and here the two are an area and a speed that both depend on the same length.

The grain size, and where it enters #

Beverloo’s contribution was the correction, and it is a single subtracted length:

$$W = C\rho\sqrt{g}\,(D - kd)^{5/2},$$

with $k$ between one and three for ordinary materials. The interpretation is an empty annulus: a grain whose centre passes within about a radius of the rim cannot get through, so the opening behaves as though it were a grain diameter or so smaller than it is.

What makes that more than a fitting parameter is the collapse. Scaling the rate by $\rho\sqrt{g}\,d^{5/2}$ and the orifice by $d$ puts every grain size on one curve exactly. That is a strong statement about the mechanism: it says the grain size matters solely by shrinking the effective hole, and not by changing how fast the grains leave, how they pack on the way out, or how they interact.

If the grains arranged themselves differently at different sizes, or if the exit speed depended on the grain as well as on the orifice, the collapse would fail. It does not, over the whole range anybody has measured, which is why two fitted constants are considered a satisfactory theory here rather than an embarrassment.

How the screening actually reaches the outlet #

The head-independence deserves a closer look than “the walls take the weight”, because the outlet is at the bottom of the silo and the walls are at the sides.

Janssen’s argument gives a vertical stress that saturates at a value of order $\rho g R/(2\mu K)$ — set by the silo’s radius rather than by its height. That is the stress on the floor, and the floor is mostly solid; the opening is a small part of it. So the question is what the stress is just above the hole.

The answer is that it is much smaller still, and for a second reason. Near the outlet the material is flowing and dilating, the force chains that were carrying the load terminate on the intact floor around the hole, and the region directly above the opening is partly unloaded — the same arching that jams a narrow opening is, at a wide one, merely relieving it. Measurements of the stress at the rim of a discharging orifice find it lower than the Janssen value by a large factor and roughly proportional to the opening rather than to the silo.

So there are two screenings in series: the walls remove the height, and the arch above the outlet removes most of what is left. What reaches the grains that are about to leave is a stress set by the local geometry, which is why the only length in the discharge law is the orifice.

A quantity that has been screened twice is a quantity the answer cannot contain, and this is why the law is simple: three of the four obvious variables have been removed before the physics of the exit is reached.

The exponent nobody agreed on #

The literature before Beverloo contains exponents from two to five, reported by careful people, and the last figure explains all of them at once.

$(D - kd)^{5/2}$ is not a power law in $D$. Its local logarithmic slope is $\tfrac52 \cdot D/(D-kd)$, which tends to five halves for wide openings and diverges as the opening approaches the jamming size. A researcher working with a four-millimetre grain and openings between fifteen and thirty millimetres is on the steep part of that curve and will report an exponent near three; one working with fine powder and large openings will report five halves.

Fitting a power law to a curve that is not one returns a number that depends on the range, and this is one of the cleaner examples of a general hazard. The fix is not a better fit but a better functional form, and the diagnostic that a form is wrong is exactly that the fitted exponent moves with the window.

The same law over a wider range #

Drawing the extremes together makes the structure of the law visible in a way the middle range does not.

At large openings the two curves are nearly parallel and offset by a small amount: the annulus is a millimetre or a centimetre, the opening is twenty, and subtracting a fixed length from a large one barely changes it. Both curves are then effectively $\rho\sqrt{g}D^{5/2}$ and the grain size has stopped mattering.

At small openings they separate completely. The eight-millimetre gravel needs an opening of eleven millimetres before it flows at all according to the law, and in practice needs forty before it flows reliably; the half-millimetre sand is already flowing at one.

The grain size sets where the law turns on and the orifice sets how fast it runs, and the two roles are cleanly separated because they enter through a difference rather than a ratio. That separation is what the collapse figure verifies, and it is the reason the law extrapolates: a measurement on one material at one size predicts another at another size, which is a great deal to ask of two fitted constants.

Where it stops: the arch #

The law’s most conspicuous failure is that it predicts a small positive flow for openings just above $kd$, and what actually happens there is nothing at all.

Below about five grain diameters, an arch forms across the opening — a ring of grains wedged against each other, each held by its neighbours, carrying the load above. It is the same mechanism as the heap that becomes a solid, operating across a hole rather than through a bulk, and it is stable indefinitely until something disturbs it.

The transition is not sharp in the way a phase transition is. Between about five and ten grain diameters the flow is intermittent: it runs, jams, is freed by a vibration, and runs again, and the distribution of avalanche sizes between jams broadens as the opening narrows. The mean avalanche size appears to diverge somewhere near five diameters, though whether it is a genuine divergence or a very rapid rise is still argued about.

That is why industrial hopper design uses a rule of thumb — make the opening at least six to eight times the largest particle — rather than a calculation. The law says how fast it will flow; nothing says reliably whether it will.

What is being measured when a hopper is timed #

The law is used in two directions and it is worth separating them, because the second is the less obvious and the more common.

Forward, it predicts a rate from a geometry, which is what a silo designer needs: how long to empty a bin, what size opening a feeder needs to deliver a required throughput, whether a rotary valve will be the limiting element.

Backward, it is a measurement of the material. The two constants are properties of the grain — $C$ reflects how efficiently the free-fall region converts into throughput and $k$ how much of the rim is unusable — and both change when the material does. A powder that has taken up moisture flows more slowly through the same opening, a grain that has been damaged in handling has a different $k$, and the discharge rate is a cheap continuous readout of both.

That is why a flow-rate measurement is a standard quality check in bulk handling, and why a rate that drifts is treated as a signal about the material rather than about the equipment. An instrument whose reading is stable to first order is an instrument whose drift is informative, and the head-independence is exactly what makes the drift attributable.

The same argument is why an hourglass was worth making. Its rate depends on the sand and on the opening and on nothing that changes during the run, so all its error is in the calibration and none of it accumulates. A water clock’s error accumulates by construction, since the rate depends on a quantity that the running of the clock changes.

Where the model stops #

The hopper is assumed wide compared with the opening. A hopper whose walls converge steeply onto the outlet has a different flow pattern — mass flow rather than funnel flow — and the rate can differ by tens of per cent. The law describes a flat-bottomed bin with a hole in it.

The material is dry, cohesionless and monodisperse. Damp sand arches at much larger openings, a mixture segregates on the way out — the big one comes to the top is that effect during handling — and cohesive powders can refuse to flow through openings of any size.

Air is ignored. For grains below about a hundred micrometres the interstitial air matters: it has to flow in to replace the discharged solid, and the resulting pressure gradient can halve the rate or, in a fluidised state, multiply it. Fine powders do not obey Beverloo’s law and are not usually described as obeying anything simple.

And the two constants are fitted. $C$ and $k$ are measured for each material, they vary between about 0.55 and 0.65 and between one and three, and no theory predicts either from the grain shape or the friction coefficient. The law’s content is the functional form and the collapse, not the numbers.

The rate is a mass rate and the material’s density is assumed fixed. A flowing granular material dilates — it has to, to shear at all, which is the heap that becomes a solid read backwards — so the bulk density at the outlet is a few per cent below the packed density in the bin. That difference is absorbed into the fitted constant $C$, which is one of the reasons $C$ is not predicted from first principles.

And the law says nothing about the velocity profile. It gives a total throughput and no distribution: whether the material leaves as a plug across the opening or as a jet from the centre with stagnant regions at the rim is a separate question with a different answer in every hopper geometry. For an hourglass that does not matter. For a blender fed by a hopper it matters a great deal, because the residence-time distribution decides whether the product is uniform.

What the pictures cannot show #

Every figure draws a smooth curve, and the flow at a real orifice is not smooth. Grains leave in bursts, the instantaneous rate fluctuates by tens of per cent, and the smooth curve is an average over many seconds. The fluctuation spectrum is itself informative — it carries the signature of the intermittent arching — and the drawing has thrown it away.

The collapse figure claims the curves coincide to a part in $10^9$, and that is a statement about the formula rather than about any material. Real data collapse to a few per cent, which is impressive and is not what the figure shows. The check performed is that the algebra is consistent with the claim being made about it.

A third omission is the sand itself. Everything drawn treats the grains as identical spheres of one diameter, and an hourglass is filled with rounded, graded, hard grains chosen over centuries by trial: too fine and it clumps with humidity, too coarse and the rate fluctuates, too angular and it wears the neck and changes its own calibration. The material is as much of the instrument as the geometry, and none of the figures has a place to say so. The best hourglasses were filled with ground marble or with crushed eggshell, dried and sifted, and the choice was made empirically by people who had no law at all.

Where the ladder goes next #

The granular-matter ladder began with the silo that does not weigh what it holds, passed through the angle that does not know the size of the heap and the heap that becomes a solid, and reached the big one comes to the top. This rung asks how fast the material leaves and finds a law with the wrong exponent for a fluid and no dependence on the head at all. The rungs after it: the jamming transition itself, where the intermittency becomes the subject; the flow rules that relate stress to strain rate in a dense granular flow, which is the constitutive question the law sidesteps; and the effect of the interstitial gas, which is where powder handling parts company from grain handling.

The habit worth carrying away is that a screening effect changes which variables a problem has. Once the walls have taken the weight, the head is not a variable, and the law that results is simpler than the fluid one and better behaved — an unusual direction for a complication to run in.