Illustrated Physics
physics, drawn
Thermodynamics

The heap that sets itself alight

A haystack, a coal stockpile, a bin of oily rags or a silo of damp grain can catch fire with no spark at all. A slow reaction inside makes a little heat; the heap holds it in; the warmer middle reacts faster and makes more. Whether that ends in a warm heap or a burning one is decided by a single number built from the heap's size, the air's temperature and the reaction's speed — and past one value of it there is no steady temperature at all. The critical size shrinks steeply as the weather warms, and the heap gives almost no warning before it goes.
16 min read 5 figures Where the model stopsThe shape decides

Assumes: The equation that only runs forwards, and the walk underneath it · The exponential that decides everything

Farmers have known for centuries that hay stacked damp can set itself on fire weeks after it is made. Coal miners and shippers knew it of coal: piles of freshly mined coal in stockyards and the holds of ships burn from the inside with no flame having come near them. Oil-soaked rags thrown into a bin, linseed oil especially, have burned down workshops overnight. In each case the fire starts in the middle of the heap, not at its surface, and the material was at the air’s temperature when it was piled.

The mechanism is a feedback between a slow chemical reaction and the heap’s own insulation, and its decisive feature is a threshold. Below it, the heap warms by a few degrees and stays warm indefinitely. Above it, there is no temperature at which it can settle, and it heats until it burns. The threshold depends on the heap’s size and shape, on the reaction, and — exponentially — on the weather. Nikolai Semenov worked out its first version in 1928, while studying explosions in gases, and David Frank-Kamenetskii the version that includes the heap’s internal temperature in 1939.

Heat made against heat lost

The reaction in a heap is slow — the oxidation of the oils in hay or rags, the reaction of coal with air, the respiration of microbes in damp grain — and it makes heat at a rate that depends on temperature through the Arrhenius factor e−E/RTe^{-E/RT}, where EE is an activation energy. The exponential that decides everything found that factor deciding which molecules can climb a barrier; here it decides how fast a heap makes heat, and for typical activation energies it doubles every ten degrees or so.

The heap loses heat through its surface, to the air, at a rate roughly proportional to how much warmer it is. Semenov treated the heap as having one temperature and compared the two rates.

Heat made against heat lost. Semenov's picture of a self-heating heap, its temperature lumped into one value: the rate at which the reaction makes heat, rising exponentially with temperature (black), against the rate at which the heap loses heat to the air, rising in proportion to its excess temperature, for three heaps — the loss slope falls as the heap grows, since heat is made in its volume and lost through its surface. Temperature in units of RTₐ²/E, the air's at −1. A small heap's loss line crosses the gain curve twice: at the lower crossing it settles a little above the air's temperature, stably, and the upper crossing is a tipping point it never reaches. The critical heap's line just touches the curve, one unit of RTₐ²/E up — about ten kelvin for a typical organic material. A larger heap's line never meets the curve: heat is made faster than it is lost at every temperature, and the heap runs away.
Fig. 1 Semenov’s picture of a self-heating heap with its temperature lumped into one value, in units of RTa2/ERT_a^2/E, the air’s at −1: heat made by the reaction, rising exponentially (black), against heat lost to the air, rising in proportion to the excess temperature, for three heaps. A small heap’s loss line crosses the gain curve twice; the critical heap’s just touches it, one unit above the air; a large heap’s never meets it, and the heap runs away.

The heat gain is a steepening curve, the heat loss a straight line starting at the air’s temperature, and steady states are where they cross. A small heap loses heat easily — its surface is large compared with its volume — and its loss line is steep: it crosses the gain curve twice. At the lower crossing the heap settles, a little above the air’s temperature; if it warms slightly, it loses heat faster than it makes it and cools back. The upper crossing is a tipping point: a heap pushed above it would run away, but nothing pushes it there. As the heap is made larger, its volume grows faster than its surface, its loss line flattens, and the two crossings move together. At one size they merge, the line just touching the curve: the critical heap. A larger heap’s line misses the curve entirely. At every temperature it makes heat faster than it loses it, and it heats until the reaction changes character — until it burns.

The temperature at which the critical heap’s line touches is only about RTa2/ERT_a^2/E above the air: for an activation energy of 80 kilojoules per mole at 40 °C, about ten kelvin. A heap at the edge of ignition is not hot. It is a few degrees warm.

Heat that has to be conducted out

A real heap is not at one temperature. Heat made in its middle has to be conducted to its surface through the heap itself, which is a poor conductor — that is what makes it dangerous — so the middle is warmer than the edges. Frank-Kamenetskii wrote the steady heat equation with the reaction as a source, the equation that only runs forwards with its time derivative set to zero and the Arrhenius heating added. In a dimensionless temperature θ=E(T−Ta)/RTa2\theta = E(T - T_a)/RT_a^2, it is

∇2θ+δeθ=0,\nabla^2\theta + \delta e^\theta = 0,

with θ=0\theta = 0 at the surface, held at the air’s temperature, and

δ=QAEρ r2kRTa2 e−E/RTa,\delta = \frac{Q A E \rho\, r^2}{k R T_a^2}\,e^{-E/RT_a},

a single number that weighs the rate of heating at the air’s temperature against the rate of conduction across a half-width rr.

The unit of temperature, RTa2/ERT_a^2/E, is the natural one for a reason worth seeing. Near the air’s temperature, the Arrhenius exponent −E/RT-E/RT changes with temperature at the rate E/RTa2E/RT_a^2 per kelvin, so a rise of RTa2/ERT_a^2/E multiplies the reaction’s rate by ee. For a typical activation energy this is about ten kelvin, which is why the whole drama of a heap’s ignition is played out in a few tens of kelvin above the air before anything visibly happens. Frank-Kamenetskii’s approximation — replacing the exact Arrhenius factor by eθe^{\theta} — is good while the temperature rise is small compared with TaT_a, and it is exactly in that range that the threshold is decided. Everything about the heap — its size, its conductivity, the reaction’s heat and speed, the air’s temperature — enters only through δ\delta.

The steady temperature inside a self-heating slab. The steady temperature across a slab of self-heating material whose faces are held at the air's temperature, in units of RTₐ²/E, for Frank-Kamenetskii parameters δ = 0.3, 0.6, 0.8 and the critical 0.8785: θ = θ₀ − 2 ln cosh(√(δe^θ₀/2) x). The centre is hottest, where heat has furthest to go. As δ rises — a thicker slab, a warmer day, a more reactive material — the centre climbs: 0.17, 0.43, 0.75, 1.19. At δ = 0.8785 it reaches 1.19, and no steady state exists beyond: a slab with any larger δ has nowhere to settle and its centre runs away. A stack whose middle has warmed by much more than RTₐ²/E above the air is past the last steady state it had.
Fig. 2 The steady temperature across a self-heating slab whose faces are held at the air’s temperature, in units of RTa2/ERT_a^2/E, for δ = 0.3, 0.6, 0.8 and the critical 0.8785: θ=θ0−2ln⁡cosh⁡(δeθ0/2 x)\theta = \theta_0 - 2\ln\cosh(\sqrt{\delta e^{\theta_0}/2}\,x). The centre is hottest and climbs as δ rises — 0.17, 0.43, 0.75, then 1.19 at the critical value — and beyond it no steady state exists.

For a slab the equation can be solved exactly, and the steady profile is a logarithm of a hyperbolic cosine, hottest at the centre. As δ\delta rises the centre climbs, slowly at first and then faster, and at δ=0.8785\delta = 0.8785 the steady solution ceases to exist. The centre’s temperature at that point is 1.19 units of RTa2/ERT_a^2/E above the air — again only about a dozen kelvin for a typical material. A slab with a larger δ\delta cannot settle, however long it waits.

The fold

The same equation for other shapes must be integrated numerically, and the clearest way to see the answer is to draw every steady state at once: the centre’s temperature against δ\delta.

The fold beyond which no heap is steady. The centre's steady temperature rise, in units of RTₐ²/E, against the Frank-Kamenetskii parameter δ, for a slab, a long cylinder and a sphere, found by integrating the steady heat equation outward from the centre: each shape has two steady states for every δ below a critical value — a cool, stable one and a hot, unstable one — and none above it. The fold is at δ = 0.878, 2.000, 3.322 for the slab, cylinder and sphere: a sphere tolerates nearly four times a slab's δ at the same half-width, because its surface is larger in proportion to its volume and heat from its centre has more ways out. A real heap, which is neither, has a critical δ between, and fixing it for a given shape is the first step in calculating the size at which a store becomes unsafe.
Fig. 3 The centre’s steady temperature rise against δ for a slab, a long cylinder and a sphere, found by integrating the steady heat equation outward from the centre. Each shape has two steady states for every δ below a critical value — a cool, stable one and a hot, unstable one — and none above. The folds are at δ = 0.878, 2.000 and 3.322.

Each curve folds back on itself. Below the fold there are two steady states for every δ\delta — the cool, stable one the heap settles into and a hot, unstable one, the distributed version of Semenov’s upper crossing. At the fold the two merge, and beyond it there are none. The critical values, 0.878 for a slab, 2.000 for a long cylinder and 3.32 for a sphere, are numbers of the same standing as π\pi: they depend on the shape alone. A sphere tolerates nearly four times a slab’s δ\delta at the same half-width, because its surface is larger in proportion to its volume and heat from its centre has more ways out. A real heap — a cone of hay, a long ridge of coal, a cube of stacked bales — has a critical value between, which can be computed for its shape.

The fold is the same kind of event that the calm that is the ghost of a cycle followed in an oscillator: two steady states, one stable and one not, approaching each other as a parameter changes and annihilating. What happens just beyond the fold is the most practically important part of the whole problem.

Weeks of warming, then hours

A heap just past the critical value has no steady state, but it has the ghost of one. Its temperature rises towards where the steady state used to be, slows down there, and lingers — and only after that does it run away.

Weeks of warming, then hours. The temperature of a lumped self-heating heap against time, in units of RTₐ²/E and of the heap's cooling time, for heating parameters ψ = 0.3, 0.36, 0.37, 0.4; the critical value is 1/e = 0.3679. Below it the heap warms and levels off, at about one unit for ψ = 0.36, just under critical. Just above, at 0.37, it warms to the same level and lingers there — the curve is passing the place where a steady state almost existed — for 55 cooling times before running away in a few; at 0.4 the run-away comes after 12. That long plateau is why heaps of hay, coal, or oily rags seem safe for weeks before they ignite, and why a temperature probe pushed into a stack can read a modest, steady-looking rise in a heap that is already past the point of no return.
Fig. 4 The temperature of a lumped self-heating heap against time, in units of RTa2/ERT_a^2/E and of its cooling time, for heating parameters ψ = 0.3, 0.36, 0.37 and 0.4; the critical value is 1/e = 0.3679. Below it the heap levels off — at about one unit for 0.36. Just above, at 0.37, it warms to the same level and lingers for 55 cooling times before running away in a few; at 0.4 the run-away comes after 12.

The closer to critical, the longer the linger: a heap 0.6 per cent past the threshold spends fifty-five cooling times — for a large haystack, whose cooling time is days, that is weeks — at a modest, almost steady temperature, then heats through hundreds of degrees in a fraction of that time. From the outside, and even from a thermometer pushed into its middle, it looks like a heap that has warmed a little and stopped. That is exactly the history farmers report: hay stacked in early summer, warm but apparently stable for weeks, then smoke. It is why regulations for storing materials that heat themselves require the temperature to be monitored and set an alarm at a modest rise rather than at a high one, and why a slowly rising temperature that has not yet levelled off is treated as an emergency.

How long is a cooling time? It is the time heat takes to diffuse from the heap’s centre to its surface, which grows as the square of the half-width divided by the material’s thermal diffusivity — the scaling the summer that reaches the cellar in December found setting how long a season takes to soak into the ground, and the ice that grows more slowly the thicker it gets found slowing a lake’s freezing. For a haystack a few metres across, with the diffusivity of loose dry plant matter, it is of order days to weeks; for a binful of rags, hours. So the same dimensionless linger of fifty cooling times is a few days for the rags and months for the stack — and the stack, which gives the longest warning, is also the one least likely to be watched for that long.

How large a heap can be

Since δ\delta grows as the square of the heap’s half-width, the critical half-width is the size at which δ\delta reaches its critical value. And since δ\delta contains e−E/RTae^{-E/RT_a}, that size depends exponentially on the air’s temperature.

How large a pile can be on a warm day. The largest half-width a slab-shaped pile of a self-heating material can have and still settle to a steady temperature, against the temperature of the air around it, on a logarithmic axis, for an illustrative organic material with an activation energy of 80 kJ/mol whose 1-metre-thick slab is just critical at 40 °C. At 20 °C the same material is safe up to 2.7 m thick; at 60 °C only to 42 cm. Each 20 °C warmer roughly halves the safe size, because the reaction speeds up exponentially while the conduction does not. The dots are cubic sample baskets of 5, 10 and 20 cm, which become critical in an oven at 123, 100, 80 °C: testing small baskets hot and extrapolating along this line is how the safe size of a store is set without ever setting one on fire.
Fig. 5 The largest half-width of a slab-shaped pile that can settle to a steady temperature, against air temperature, on a logarithmic axis, for an illustrative organic material with an activation energy of 80 kJ/mol whose 1 m slab is just critical at 40 °C. At 20 °C it is safe up to 2.7 m thick; at 60 °C only to 42 cm. The dots are cubic sample baskets of 5, 10 and 20 cm, critical in an oven at 123, 100 and 80 °C.

For the illustrative material drawn, a stack that is safe at 2.7 metres thick on a 20 °C day is safe only to 1 metre at 40 °C and 42 centimetres at 60 °C. Each 20 °C warmer roughly halves the safe size. A warehouse that has stored a material safely through a cool year can ignite in a hot summer with nothing else changed, and a stack that is safe in the open can ignite when the same material is stacked against a warm wall or under a sunlit roof.

The exponential dependence is also how safe sizes are measured. Nobody determines the critical size of a coal stockpile by building larger and larger piles. Instead, cubes of the material in wire-mesh baskets a few centimetres across are placed in ovens at a series of temperatures, and for each basket size the oven temperature is found at which it just ignites. Small baskets need high temperatures — 123 °C for a 5-centimetre cube of the material drawn — and larger ones lower. Plotting the results in Frank-Kamenetskii’s variables gives a straight line, and extrapolating it down to ordinary temperatures gives the critical size of a full-scale store. The method, standardised in Europe for bulk materials, rests entirely on the theory: on δ\delta containing the size and the temperature in exactly the combination drawn.

Volume against surface

The physics underneath is the old competition between volume and surface that the size at which a body becomes round found deciding which bodies gravity can shape. Heat is made throughout a heap’s volume and escapes through its surface; doubling the heap doubles the distance heat must travel and quadruples, relative to the rate of escape, the heat it has to carry. Small things cool; large things accumulate heat. A handful of oily rags is safe; a binful is not. The insulation that makes a wire lose more heat found the same competition in the other direction, a thin wire shedding heat more easily when its surface was enlarged by insulation; here enlarging the heap reduces its surface in proportion to its heat, and nothing rescues it.

The exponential makes the competition sharp. If the reaction’s rate grew only in proportion to temperature, a larger heap would simply run warmer, in proportion to its size, without any threshold. It is the Arrhenius factor, growing faster than any power of the temperature, that makes the heap’s own warming feed back on itself, and that turns a smooth dependence on size into a fold.

The same fold in a flask of gas and in a battery

Semenov did not start from haystacks. He was studying why mixtures of gases explode when a vessel containing them is heated past a definite temperature, and the theory he wrote for them is the one drawn above: the gas reacts slowly, heats itself, loses heat to the vessel’s walls, and above a critical combination of temperature, pressure and vessel size has no steady state. His work on the other route to explosion — chain reactions, in which the number of reactive fragments multiplies rather than the temperature — won him a Nobel Prize in 1956, and the two mechanisms, thermal and chain, between them account for when a fuel and air ignite of their own accord. The thermal one is also why an engine knocks: the unburned mixture ahead of the flame, compressed and heated, reaches its critical condition and ignites before the flame arrives.

The newest instance is in batteries. A lithium-ion cell has several reactions between its electrodes and electrolyte that are negligible at room temperature and grow exponentially with temperature, and it loses heat through its surface. Above a critical temperature, a fault or an internal short raises it past, those reactions make heat faster than the cell can shed it, and the cell runs away — venting, burning, and heating its neighbours past their own critical points, which is how one failed cell can take a whole battery pack with it. Pack designers space cells, add barriers between them and size cooling systems by the same comparison of heat made against heat lost, and test cells by heating them in calorimeters until they run away, the battery’s version of the basket test. The reaction that cannot go all the way followed reactions that stop at an equilibrium; the reactions in a runaway are the opposite case, kept far from equilibrium by the heat they themselves release.

What the figures leave out

The figures treat a reaction with a single activation energy, unlimited fuel, and a heap of uniform conductivity whose surface is held at the air’s temperature. Real heaps consume their reactant, which caps the runaway and is how a heap can heat and then cool again without igniting; they contain moisture, whose evaporation and condensation carry heat far more effectively than conduction, which is why damp hay is the dangerous kind and why the heat often moves upward in a heap; and they let air flow through them by convection, which supplies the oxygen the reaction needs and also cools. Biological heating, by bacteria and fungi in damp organic material, raises a heap to about 70 °C, where the microbes die, and chemical oxidation takes over from there; the single-reaction model covers only the second stage. The size figure is for an illustrative material, not a measured one. The domain is a conductive heap with a single dominant reaction, before it has consumed much of its fuel.

Still open: what large heaps actually do

The theory is clean and the basket tests are standard practice, but predictions for very large stores — silos of wood pellets, coal stockpiles of hundreds of thousands of tonnes, landfills — remain uncertain, because in such large masses the moisture transport, the airflow and the changing composition of the material as it reacts can matter more than conduction, and the extrapolation from small baskets over two or three orders of magnitude in size compounds every error in the activation energy. Wood-pellet storage, which has grown rapidly as pellets replaced coal in power stations, has had a series of self-heating fires and explosions, and the models that include moisture and gas flow are still being tested against them. How to monitor a heap of a hundred thousand tonnes for the modest internal rise that precedes ignition is a practical problem in its own right.

The threshold is exact. A heap whose reaction makes heat as e−E/RTe^{-E/RT} and loses it by conduction has a steady temperature only while δ=(QAEρr2/kRTa2)e−E/RTa\delta = (QAE\rho r^2/kRT_a^2)e^{-E/RT_a} is below its shape’s critical value — 0.878 for a slab, 2.000 for a cylinder, 3.32 for a sphere — where its centre is barely a dozen kelvin above the air; past it, after lingering near the ghost of the vanished steady state, it runs away; and since δ\delta grows as r2e−E/RTar^2 e^{-E/RT_a}, each 20 °C of warmer weather roughly halves the safe size. A heap burns not because it is hot but because it is large, on a day warm enough for its size.

Part 14 of 14

This essay is one argument about Diffusion. The others:

Insulating a thin wire makes it lose more heat. The heat lost per metre from a wire 2 mm across held 40 K above the air, against the outer radius of the PVC insulation round it (k = 0.19 W/m K), with the surface losing 10 W/m²K to still air by convection and radiation, on a logarithmic radius axis; dashed, the bare wire. The first layers of insulation increase the loss, from 2.51 to a maximum of 12.11 W/m at an outer radius of 19 mm — the critical radius k/h — because they add surface faster than they add resistance. Beyond the critical radius the loss falls again, but so slowly — the resistance of the insulation grows only as the logarithm of its radius — that it does not get back down to the bare wire's until the PVC is 178 km thick. Any sheath a cable could carry makes it lose more heat, not less. The insulation that makes a wire lose more heat Part 13 — Wrap a hot water pipe in glass wool and it loses less heat. Wrap a thin electrical wire in PVC and it loses more — four times more, at the best thickness — and runs cooler for it. The insulation adds a resistance to heat flowing through it, but it also adds surface for the heat to leave from, and around a thin enough cylinder the surface wins. The crossover is a radius, k/h, set by the insulation and the air; below it, more insulation means more heat lost, and a small enough ball cannot be insulated by anything wrapped round it.

Where this goes next

Insulating a thin wire makes it lose more heat. The heat lost per metre from a wire 2 mm across held 40 K above the air, against the outer radius of the PVC insulation round it (k = 0.19 W/m K), with the surface losing 10 W/m²K to still air by convection and radiation, on a logarithmic radius axis; dashed, the bare wire. The first layers of insulation increase the loss, from 2.51 to a maximum of 12.11 W/m at an outer radius of 19 mm — the critical radius k/h — because they add surface faster than they add resistance. Beyond the critical radius the loss falls again, but so slowly — the resistance of the insulation grows only as the logarithm of its radius — that it does not get back down to the bare wire's until the PVC is 178 km thick. Any sheath a cable could carry makes it lose more heat, not less. The insulation that makes a wire lose more heat Wrap a hot water pipe in glass wool and it loses less heat. Wrap a thin electrical wire in PVC and it loses more — four times more, at the best thickness — and runs cooler for it. The insulation adds a resistance to heat flowing through it, but it also adds surface for the heat to leave from, and around a thin enough cylinder the surface wins. The crossover is a radius, k/h, set by the insulation and the air; below it, more insulation means more heat lost, and a small enough ball cannot be insulated by anything wrapped round it. Long stretches of order, broken without warning. 1800 successive values of the logistic map at r = 3.828427 − 0.00002, a distance of 2.0 × 10⁻⁵ below the setting at which its stable three-cycle is born. The shaded stretches are calm: the orbit repeats itself to within 0.004 every third step, cycling through three values as though the three-cycle already existed. Between them the orbit bursts through the whole interval with no discernible pattern, and then, at an unpredictable moment, is captured into another calm. In a run of 400,000 iterates at this setting the calms last 173 iterates on average, and the channel the orbit creeps through — measured on the map's own third iterate — has a gap of 4.1 × 10⁻⁵ and a longest passage of 253 iterates. Nothing random is added: the sequence is the same every time it is computed from the same start. The calm that is the ghost of a cycle Just before a chaotic system settles into a stable cycle it does something stranger than either: it behaves perfectly periodically for long stretches, and then, at moments nothing in the record predicts, bursts into disorder and back. The calm is a cycle that does not exist yet, creeping through the narrow gap where it is about to be born, and how long each calm lasts is set by the square root of the distance to that birth. A ratio in the exponent. The Boltzmann factor against the energy of a state measured in units of kT, with the logarithm up the axis so that the straight line is the whole content. Every kT of energy costs a factor of e, so 10, 20, 30 times kT are factors of 10^-4.3, 10^-8.7, 10^-13.0. At 300 K, kT is 25.9 meV, so a barrier of 0.35 eV is 13.5 kT and a factor of 1.3e-6. That is the sense in which a third of an electronvolt is not a small energy: it is small compared with a chemical bond and enormous compared with kT, and it is the second comparison that decides whether anything happens. The exponential that decides everything Maximising the number of ways a reservoir can arrange what is left after taking E out of it gives one factor, e to the minus E over kT. Its exponent is a ratio, which is why a barrier of a third of an electronvolt — nothing at all by chemical standards — is the difference between instantly and never.

The objects named here

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

Activation energyArrhenius lawBifurcationHeat conductionSaddle nodeSelf heatingSteady stateThermal explosion