The pressure that switches at the hopper

The pressure that stops growing found why a silo is not a tank. Grain has shear strength, so the bin’s wall carries part of the weight by friction; the friction is generated by the wall pressure, the wall pressure by the vertical stress, and the three make an equation whose answer saturates. Below a characteristic depth the stress in the grain stops growing, and the wall carries the weight of everything above. It ended with the bin, where most silos do not: they narrow below the bin into a conical hopper that funnels the grain to an outlet. That essay named what happens there as the next question: the discharge case, where the ratio between the stresses changes, and the switch pressure at the hopper’s top.

The slice, tilted #

The bin is the earlier one: 8 m across, holding 30 m of grain weighing 9 kN/m³, wall friction 0.45, and a ratio of horizontal to vertical stress $K = 0.45$. At its foot, Janssen’s stress has nearly saturated at 84.6 kPa vertical, and the bin wall carries 38.1 kPa normal to it. Below, a conical hopper of half-angle 20° from the vertical narrows to an apex 11 m down, with a wall friction of 0.40 and a grain whose internal friction is 35°.

The free body in the hopper is the bin’s slice, tilted. A horizontal slice a height $x$ above the apex has a radius $x\tan\beta$; it carries its own weight down, the stress from above down and the stress from below up, and it is held by the wall, by friction that is whatever it needs to be up to its limit and, in a moving solid, exactly at it. The wall now leans: it pushes inwards with a normal stress $p_n$, which has an upward component because the wall slopes, and drags on the grain with friction $\mu_h p_n$ along its slope. Write the normal stress as a ratio of the vertical one, $p_n = F p_v$, and the slice’s vertical equilibrium is Walker’s equation:

$$\frac{dp_v}{dx} = n\frac{p_v}{x} - \gamma, \qquad n = 2\left[F\left(1 + \mu_h\cot\beta\right) - 1\right].$$

It is the bin’s equation with the wall tilted — the same balance of weight against friction that the area not in the equation reduces to a force proportional to the normal force — and its solution, finite at the apex and equal to the bin’s stress at the transition, is a power law plus a straight line. Everything below follows from it, and from one fact about $F$.

The ratio that changes when the grain moves #

In the bin, $K$ is the ratio of a horizontal stress to a vertical one, and while the grain is at rest it is the same filling or emptying, give or take the discharge factor codes apply to it. In the hopper the ratio is to the wall’s normal, and it depends on how the grain is moving.

Filling, the grain settles into the cone and its largest principal stress stays roughly vertical. The wall’s normal stress is a little less than the vertical stress: Eurocode 1’s empirical $F_f$ is 0.895 for this hopper. Emptying in mass flow, every particle moves, converging on the outlet. A converging flow is a passive state — the grain is being squeezed sideways as the cone narrows, as soil is when a wall is pushed into it rather than away, the distinction the load that depends on what carries it drew for retained earth — and its largest principal stress swings from vertical towards horizontal, normal to the wall. The circle nobody draws is the reason a rotation of the principal axes changes every component at once: Walker’s emptying ratio, from the Mohr circle of a grain at failure against a wall at its own friction, is $F_e = 1.131$.

The two ratios meet the same vertical stress at the transition, 84.6 kPa, and that is the switch. Just above the transition the bin wall carries $K$ times it, 38.1 kPa. Just below, the hopper wall carries $F$ times it: 75.8 kPa filling, 95.7 emptying. The wall pressure doubles across a weld line, and when the silo empties it is two and a half times. The bin’s discharge factor, which multiplies its wall pressure by 1.35 for emptying, takes the bin wall to 51.4 kPa and leaves the hopper’s edge carrying nearly twice that.

The reason is geometric before it is about grain. A vertical wall carries the horizontal stress, the smaller one; a sloping wall carries a stress much nearer the vertical one, because it faces nearly upward. A 20° hopper’s wall is closer to a floor than to a wall, and a floor under 84.6 kPa of grain carries 84.6. The switch is the bin’s horizontal pressure becoming the hopper’s nearly vertical one at a line, and emptying turns the screw further by rotating the stress towards the wall.

Near the outlet the silo is forgotten #

Below the transition the wall pressure falls, filling and emptying. The vertical stress in the hopper dies away from its transition value as $(x/h)^n$, and under it a second term grows: a straight line through the apex.

The straight line is the part of the solution that does not know about the bin. Near the apex the transition’s term has died away, because $(x/h)^n$ is small when $x$ is, and what is left is $\gamma x/(n - 1)$: a stress proportional to the distance from the apex, set by the grain’s weight and by $n$ — that is, by the hopper’s angle and the friction of its wall and its grain. Emptying, $n$ is 2.75 and the line rises at 5.15 kPa per metre; a metre above the apex the stress is 5.2 kPa, out of 84.6 at the transition.

This is the radial stress field Jenike found for converging flow: near the outlet the stress is proportional to the distance from the apex and independent of what is above. It is the same kind of field — stresses fanning out from a point and growing with distance from it — that the ground under a footing carries in Prandtl’s solution, met here in a cone of grain instead of a wedge of soil. The grain above the transition is carried by the bin’s wall friction and by the hopper’s upper wall, which is exactly why the hopper’s upper wall carries so much. By the time the stress reaches the outlet the hopper has handed nearly all of it to its walls.

Why a silo empties at a steady rate #

The consequence is one of the more surprising facts about granular materials, and it falls out of the straight line.

Half a metre above the apex the emptying stress is 2.57 kPa with 5 m of grain in the bin and 2.58 kPa with 60 m. The bin above changes from a squat bin to one seven diameters tall; the foot of the bin feels it, its stress rising two and a half times before it saturates; the outlet does not feel it at all. A tank of water would show 144 kPa at its outlet for the shallow fill and 639 for the deep one, and its outflow, which goes as the square root of the head, would slow as it emptied.

Grain does not slow. The flow through an orifice in a hopper is set by the stress and the geometry close to the orifice — Beverloo’s correlation, fitted to experiments in 1961, gives the mass flow rate as a function of the outlet’s size, the particle size and the bulk density, with no head in it at all — and the stress close to the orifice is the straight line. A silo empties at the same rate full or nearly empty, which is why an hourglass works: the sand runs at a constant rate because it is a hopper, and the head above the neck is forgotten exactly as it is here.

A steeper hopper switches harder #

The hopper’s angle is a designer’s choice, and it is made for flow before it is made for pressure.

A hopper steep enough for mass flow — here steeper than about 25° from the vertical, by Jenike’s chart in the approximation of Arnold and McLean — makes every particle move, first in first out, with no stagnant grain sitting against the walls. That is what a designer wants for most products, because grain that sits still in a corner spoils, cakes or segregates. But steepness raises the switch. Emptying, a 10° hopper’s edge carries 3.06 times the bin wall’s pressure, against 2.16 at 30°: a steep wall makes the converging grain squeeze harder against it, and Walker’s ratio grows. Filling, the trend runs gently the other way, so the gap between filling and emptying is widest exactly in the steep hoppers mass flow asks for.

The steeper hopper also concentrates its load. At 10° the emptying pressure falls from 116.4 kPa at the top edge to 24.0 half way down; at 20°, from 95.7 to 36.7. A steep hopper’s exponent $n$ is large — 6.99 at 10° — so the transition’s stress dies away within the top part of the cone and the rest of the hopper carries little. The ring at the transition and the top few metres of a steep hopper’s shell are where its pressure is, and the design of a hopper is mostly the design of that band.

What the shell feels at the weld #

The switch is a pressure, and a shell carries pressure as hoop tension, $p r$ in a cylinder and $p_n r/\cos\beta$ in a cone. Just above the transition the bin wall, radius 4 m, carries $38.1 \times 4 = 152$ kN per metre of height in hoop tension. Just below, the emptying hopper’s edge carries $95.7 \times 4/\cos 20° = 407$ kN per metre. The hoop tension jumps 2.7 times across the weld that joins them.

At the same line the hopper hangs from the bin. Its wall carries the stress arriving at the transition, $84.6 \times \pi \times 4^2 = 4{,}250$ kN, and the weight of the cone of grain inside it, 1,660 kN, so 5,910 kN in all goes up the cone’s wall into the junction. Along the cone’s slope that is a meridional force of $5{,}910/(2\pi \times 4 \times \cos 20°) = 250$ kN per metre of circumference, and its inward component, $250 \sin 20° = 86$ kN per metre, squeezes a ring at the junction into a compression of $86 \times 4 = 342$ kN. The transition is therefore where three things meet at once: the largest hoop tension in the hopper, the change from cylinder to cone that turns the wall’s slope into an inward pull, and a ring in compression holding them apart. Silos fail at the transition more than anywhere else, and the switch is a large part of why.

The switch saturates with the bin #

The switch pressure is proportional to the bin’s stress at the transition, so it inherits Janssen’s saturation: it grows with the bin’s height until about four diameters and then stops, at about 100 kPa here. A silo eight diameters tall puts no more pressure on its hopper’s edge than one four diameters tall. The ratio between the edge and the bin wall does not move with the bin at all; it is a property of the hopper, the grain and the walls.

The switch by hand #

At the foot of the bin Janssen gives $p_v = (\gamma R/\mu K)(1 - e^{-z/z_0})$ with the hydraulic radius $R = D/4 = 2$ m and $z_0 = R/\mu K = 9.88$ m: $p_v = (9 \times 2/0.2025)(1 - e^{-30/9.88}) = 88.9 \times 0.952 = 84.6$ kPa. The bin wall carries $0.45 \times 84.6 = 38.1$ kPa.

For the 20° hopper with wall friction 0.40 ($\varphi_w = 21.8°$) and grain at 35°, Walker’s emptying ratio has $\varepsilon = 21.8° + \arcsin(\sin 21.8°/\sin 35°) = 62.2°$, and $F_e = (1 + \sin 35°\cos 62.2°)/(1 - \sin 35°\cos 102.2°) = 1.267/1.121 = 1.131$. The hopper’s edge carries $1.131 \times 84.6 = 95.7$ kPa, 2.51 times the bin wall’s.

The exponent is $n = 2[1.131(1 + 0.40\cot 20°) - 1] = 2.75$, so near the apex the stress is $9x/1.75 = 5.15x$ kPa with $x$ in metres: 2.6 kPa at half a metre, whatever the bin holds.

What the slice cannot see #

The slice averages across the hopper. Walker’s equation treats the vertical stress as uniform across each slice and the wall stress as a fixed multiple of it. Near the transition, where the stress field reorganises from the bin’s to the hopper’s, it is neither; the slice gives the switch as a jump at a line where the real stress rises over a short distance and overshoots. Walters’ theory of the switch, and measurements in real silos, put the peak higher than the slice does and spread it over a band of the wall.

The ratio is a constant. Filling and emptying are given one ratio each, where in reality the hopper passes from one to the other progressively from the outlet upward when discharge starts, and the switch travels up the hopper as it does. A wall checked for both states separately is checked for each end of that passage, not for every stage of it.

The flow is mass flow. Everything emptying here assumes the whole contents move. A hopper shallower than Jenike’s limit discharges in funnel flow, through a channel of moving grain inside stagnant grain, and the stagnant grain becomes the channel’s wall — with its own switch where the channel meets the bin, at a level nobody chose, which may move from one emptying to the next.

The silo is symmetric. An off-centre outlet, or a channel that forms against one side, makes the pressure round the circumference unequal; a circular wall that carries uniform pressure in hoop tension is bent by unequal pressure, and eccentric discharge is the load case that has split more silos than any other.

The angle is chosen twice #

A designer choosing a hopper’s angle is answering two questions that pull in opposite directions. Flow asks for a steep hopper, steeper than Jenike’s limit, so that the whole contents move and nothing sits against the wall to cake or spoil. The shell asks for a shallow one, because the steep hopper switches harder and concentrates its pressure into a narrower band below the transition. The two meet at the mass-flow limit, and most hoppers are built a few degrees steeper than it, where the switch is about two and a half times the bin wall’s pressure and the band that carries it is the top third of the cone.

The angle also decides the grain’s history, which no pressure figure shows. Mass flow empties first in, first out: grain loaded in autumn leaves before grain loaded in spring. Funnel flow empties last in, first out, through the channel above the outlet, and leaves the grain that went in first sitting against the walls until the silo is emptied completely. For a feed mill the first matters more than the pressure; for a cement silo, where a cake against the wall can collapse into the channel as a single mass and load the hopper suddenly, both do.

The ring at the transition #

They cannot show the ring at the transition. The hopper’s shell hangs from it, pulling inwards and downwards along the cone’s slope with all the weight the hopper’s wall carries; the inward pull puts the ring into compression and the shell above into a band of high meridional stress, exactly where the switch pressure presses outward on the hopper’s edge. They cannot show the grain itself, which is not a continuum and which arches across an outlet too small for it — the arching that decides what a buried pipe carries, here deciding whether the silo empties at all.

Still open: the silo whose outlet is not in the middle #

Every hopper here is concentric with its bin and empties through its apex. Many silos have outlets off centre, or several outlets that are used one at a time, and a bin emptying through an off-centre outlet develops a flow channel against one side. Where the channel touches the wall, the wall sees the low pressure of moving grain; opposite it, the stagnant grain presses as if the silo were at rest. Whether that unequal pressure round the circumference bends the wall more than the switch at the hopper loads it, and at what eccentricity of the outlet the bending begins to govern the wall’s thickness, is the question a hopper that empties to one side puts to a cylinder designed for pressure that is the same all the way round.