Illustrated Physics
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Heat engines — the series

12 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A Carnot cycle on pressure–volume axes. Two isothermal steps joined by two adiabatic ones, forming a closed loop. The gas expands 9.6-fold in reaching the cold reservoir at 0.50 of the hot one, and the area enclosed is the net work done over one cycle.

    The ceiling on every engine, set before it was designed

    There is a maximum efficiency no heat engine can exceed, and it depends on nothing but two temperatures. Not the fuel, not the working substance, not the cleverness of the engineer.

    part 1 · thermodynamics
  2. The ceiling, inverted. How many joules of heat a perfect machine can move per joule of work, against the outside temperature, with the inside held at 21 °C. The upper curve is heating — T_h/(T_h − T_c), which is what the Carnot argument becomes when the cycle is run backwards — and the lower one is cooling the outside, T_c/(T_h − T_c). They differ by exactly one everywhere, to 1.8e-15 across the whole range as drawn, because the work put in is delivered as heat along with whatever was moved. The dashed line at one is a resistive heater, which is 100% efficient and is the worst option on the figure. At 7 °C and −7 °C the ideal coefficients are 21.0 and 10.5; a real machine reaching 25% of the ideal gets 5.3 and 2.6, which is still several times what burning the same energy would give.

    The engine that pays back more than it takes

    Carnot's argument puts a ceiling on how much work a flow of heat can be made to do. Run the same cycle backwards and the ceiling inverts into a floor that is greater than one — so a machine can deliver three or four joules of heat for every joule it consumes, and a perfectly efficient electric heater is the worst way to warm a room.

    part 2 · thermodynamics
  3. The ceiling, the estimate, and three power stations. Two efficiencies against the ratio of the cold reservoir's temperature to the hot one. The upper curve is Carnot's 1 − Tc/Th, which is a ceiling on the work per unit of heat and is reached only by an engine that runs infinitely slowly, because a reversible heat flow needs a vanishing temperature difference to drive it and therefore infinite time. The lower curve is 1 − √(Tc/Th), the efficiency of an engine with finite thermal contact run for the most power rather than the most work. Three measured plants are marked: West Thurrock, coal runs at 36 per cent against a ceiling of 64 and a finite-time estimate of 40; CANDU, nuclear runs at 30 per cent against a ceiling of 48 and a finite-time estimate of 28; Larderello, geothermal runs at 16 per cent against a ceiling of 33 and a finite-time estimate of 18. Every one of them is closer to the lower curve — within 4.4 points at worst, against 16.5 at best from the ceiling. The second law is not what limits a working power station. What limits it is that somebody wants the electricity this year.

    The engine that has to finish

    Carnot's ceiling is exact and it is reached only by an engine that takes for ever, because a reversible heat flow needs a vanishing temperature difference to drive it. Ask instead for the most power rather than the most work per joule of heat, and the answer is a different function of the same two temperatures — and three measured power stations sit on it rather than on the ceiling.

    part 3 · thermodynamics
  4. One temperature, four pawls, and nothing gained. The net rate of a ratchet whose gas and whose pawl are at the same temperature, against the load, for notches 2, 5, 10, 20 times the thermal energy deep. Every curve passes through zero at zero load and is negative everywhere else. The device is not merely unable to lift a weight; under any load at all it turns the wrong way and lets the weight down, converting its potential energy into heat in the gas. Making the notch deeper slows everything down — an exponential in the depth — and does not change the sign anywhere. That is the second law arriving as a mechanism rather than as a prohibition. Nothing was assumed about entropy; the pawl was simply allowed to be as warm as everything else, and its own fluctuations undo exactly the rectification it was there to provide. Any rectifier small enough for thermal noise to matter has this problem, and the rectifier being clever does not help, because the same noise reaches the cleverness.

    The engine a fluctuation cannot run

    A ratchet lets a shaft turn one way and not the other. Put a paddle in a gas on the same shaft and molecular collisions appear to become a lifted weight — an engine running on one reservoir. It does not work, and following exactly why turns the second law from a prohibition into a mechanism: the pawl is as warm as the gas, and it lifts whenever it is asked to.

    part 4 · thermodynamics
  5. The rectangle every cycle is equal to. An ideal Otto cycle for air on a temperature–entropy diagram: compression ratio 9, intake at 300 K, 1400 kJ/kg added at constant volume. Compression takes the charge to 722 K, combustion to 2672 K, expansion back to 1110 K, and the exhaust cools at constant volume. The shaded loop is the work; the region under the lower curve is the heat rejected. Heat enters over a range of temperatures, and its entropy-weighted mean — heat divided by the entropy it brings — is 1491 K; the heat leaves at a mean of 619 K. The dashed rectangle between those two temperatures has exactly the loop's area, and 1 − 619/1491 = 58.5%, which is the Otto efficiency, checked to rounding. A Carnot engine between the coldest and hottest points of the same cycle would reach 88.8%.

    The temperature an engine really takes its heat at

    Carnot's ceiling is set by two temperatures, and no engine that burns fuel takes its heat in at one temperature or gives it out at another. It takes heat over a range, from the moment combustion starts to the moment it ends. For any reversible cycle there is an exact replacement for Carnot's two numbers: the average temperature at which heat arrives and the average at which it leaves, each weighted by the entropy the heat carries. The gap between a real cycle and Carnot is a gap between those averages and the extremes.

    part 5 · thermodynamics
  6. Two bodies an engine draws together. Two equal bodies of 4.186 kJ/K — a kilogram of water each — one at 90.0 °C and one at 10.0 °C, against the heat drawn from the hot one. The solid curves are the best possible engine running between them, a reversible one, which leaves the product of the two temperatures unchanged and brings both to the geometric mean, 320.7 K (47.5 °C). It draws 177.8 kJ from the hot body and delivers 20.8 kJ of work, C(√T₁ − √T₂)², checked against the heat balance. The dashed lines are the same bodies simply touching: they meet at the arithmetic mean, 323.1 K, having exchanged 167.4 kJ and delivered nothing. The 2.5 K between the two endpoints is the work, left behind as heat.

    The work left in two buckets of water

    Carnot's ceiling assumes reservoirs so large that taking heat from one and giving it to the other changes neither temperature. Two buckets of water are not reservoirs. Run the best possible engine between a hot one and a cold one and both temperatures move, the efficiency available shrinks as they do, and the engine stops when they meet — at the geometric mean of the starting temperatures, not the ordinary one. The work it delivered is exactly the difference between those two meeting points, and it is far less than the starting temperatures promise.

    part 6 · thermodynamics
  7. A fridge with no work going into it, and its ceiling. How much heat a three-reservoir machine can lift out of a cold space per unit of heat supplied to drive it, against the temperature of the driving heat, for 3 cold temperatures and an ambient of 300 kelvin. No work enters or leaves: the machine takes heat in at the top, takes heat in at the bottom, and rejects the sum at ambient. That it can do anything at all is the surprise — the second law allows heat to be moved up a gradient provided a larger flow is moved down one, and the accounting is a single inequality in the three entropy flows. The ceiling is the product of two familiar expressions, and at 450 kelvin driving a 253-kelvin space it is 1.79. The marks are what real machines achieve, which is a fifth to a third of it — absorption refrigeration is not efficient and is chosen when the heat is free and the silence and the absence of moving parts are worth something.

    A fridge with no work going into it

    Every engine here so far turns heat into work or work into a heat flow. A machine exchanging heat with three reservoirs and doing no work at all can still move heat from cold to hot, and the ceiling on how much is the product of two Carnot expressions — an engine's efficiency times a fridge's coefficient of performance. A gas flame makes ice, and the accounting is one inequality in three entropy flows.

    part 7 · thermodynamics
  8. One dimensionless group between an engine and Carnot. The efficiency of a thermoelectric couple against the temperature of its hot side, with the cold side at 300 kelvin, for 4 values of the figure of merit, and the Carnot ceiling drawn above them. The expression has exactly one material quantity in it — the dimensionless group formed from the Seebeck coefficient squared, the electrical conductivity, the temperature and the thermal conductivity — and everything else is the two temperatures. At 600 kelvin, a figure of merit of one gives 10.8 per cent against a Carnot ceiling of 50.0, and a figure of merit of four gives 22.6. The approach to the ceiling is slow: every doubling of the group buys less than the last, so the difference between a good material and a perfect one is smaller than the difference between a poor material and a good one.

    An engine with one number in it

    A thermoelectric couple has no moving part and no working fluid, and its efficiency is the Carnot value multiplied by a factor containing exactly one dimensionless group of material properties. Sixty years of effort have moved that group from about one to about two, and the reason it is hard is that its three ingredients are not independent: raising the conductivity ruins the coefficient it is squared against, and the only lever that is really free is the heat the lattice carries.

    part 8 · thermodynamics
  9. How far below the air a surface facing the sky falls. How much colder than the air a surface facing the night sky becomes, when insulated from below and with air at 5 °C, against how strongly the air can warm it, measured by the convection coefficient (3 to 5 W/m²K in still air, 20 or more in a breeze), for sky emissivities of 0.72, 0.82, 0.97 — clear and dry, clear and humid, and overcast. Under a clear dry sky in still air the surface sits 12.3 °C below the air; a breeze of 20 W/m²K halves that and more; under cloud the drop is 1.3 °C. The surface is radiating to something colder than the air: the upper atmosphere and, through the window near ten micrometres, space.

    The cold reservoir overhead

    On a clear, still night a car roof can grow frost while the air around it is several degrees above freezing. Nothing in the air is cold enough to freeze water. The roof is radiating its heat upward to a sky whose radiation back is that of something twenty degrees colder than the air — part of it the cold upper atmosphere, part of it space itself, seen through a window in the infrared where the air is transparent. The clear sky is a cold reservoir available everywhere, and it can be used: to make frost, to cool buildings below the air temperature even in sunshine, and in principle to run an engine at night.

    part 9 · thermodynamics
  10. A race between the flame and the end gas. The Livengood–Wu integral of the end gas — the fraction of its ignition delay used up — against time from the spark, for 95-octane fuel at compression ratios 9, 11 and 13; the dotted line marks when nine-tenths of the charge has burned. If the integral reaches one before then, the end gas ignites by itself while a sizeable part of the charge is still unburned, and knocks. At a ratio of 9 the flame wins easily: the integral ends at 0.90. At 11 it reaches one when 92 per cent of the charge has burned, just short of knock; at 13, when only 78 per cent has, and the rest detonates. The integral climbs fastest at the end, where the end gas is hottest and most compressed, which is why knock is heard late in the burn.

    The squeeze a petrol engine is not allowed

    Every textbook gives a petrol engine's ideal efficiency as one minus the compression ratio raised to a small negative power, and every reader can see that the efficiency keeps climbing as the ratio rises. Real petrol engines stop near ten or eleven. Nothing in the thermodynamics stops them. The fuel does: the charge being compressed is already fuel and air, and the burning gas behind the flame squeezes the rest until it ignites on its own, in a hammer-blow that wrecks pistons. An octane number buys nothing but time against that, and a diesel escapes it by compressing air and adding the fuel only when it wants ignition.

    part 10 · thermodynamics
  11. The extra work in letting the gas expand further. Pressure against volume, the volume in units of the combustion chamber and the pressure on a logarithmic axis, for two ideal engines that compress the charge by the same ratio of 10 and add the same heat: an Otto cycle that expands by 10 too and throws away gas still at 3.8 bar (dashed), and an over-expanded cycle that lets the burned gas expand 26.8 times, until it is back at the 1 bar it came in at (solid). The shaded region between the expansion line and the intake pressure, beyond the Otto engine's stroke, is where the extra work is done (on this logarithmic axis its area is not to scale): 136 kJ per kilogram of charge, raising the efficiency from 55.3 to 63.6 per cent. It is taken from gas that an Otto engine releases through its exhaust valve still pushing.

    The engine that expands further than it compresses

    A petrol engine's efficiency is usually quoted as a function of its compression ratio, and its compression ratio is held near ten because a higher one makes the fuel detonate. But the efficiency does not come from the compression. It comes from the expansion — from how far the burned gas is allowed to push the piston before it is thrown away — and an ordinary engine expands its gas exactly as far as it compressed it, simply because the same cylinder does both. Let the gas expand further than it was compressed and most of what a higher compression ratio would have given comes back with no knock at all. The price is paid in power, and it is why the most efficient petrol engines on the road are built this way.

    part 11 · thermodynamics
  12. Newcomen's dilemma. Two consequences of how cold a Newcomen engine's injection spray leaves its cylinder, against that temperature: the pressure difference the atmosphere then pushes the piston down with, one atmosphere less the vapour pressure of the water left in the cylinder; and the steam the next stroke wastes reheating the cylinder's iron skin, as a fraction of the most it wastes at 30 °C (8.2 cylinder-fulls). A cylinder cooled to 40 °C gives a push of 0.93 atmospheres and wastes 7.0 cylinder-fulls of steam a stroke; one left at 70 °C wastes half as much but pushes with only 0.69. Every choice was bad, because the one vessel had to be both hot, to receive the steam, and cold, to make the vacuum. Newcomen's engineers ran it in between.

    The cylinder that must not be cooled

    The first steam engines that did useful work, Thomas Newcomen's pumps of 1712, made a vacuum by spraying cold water into a cylinder full of steam. It worked, and on every stroke it cooled the cylinder, so that the next charge of steam condensed on the cold iron until the walls were hot again — several cylinders' worth of steam spent heating metal that the next spray would cool. James Watt saw in 1765 that the vessel was being asked to be hot and cold at once, and moved the cold to a separate condenser. The improvement was a factor of three or four in coal. A century later the reason had a name: heat passing from hot to cold across a large difference, doing nothing on the way, is work thrown away.

    part 12 · thermodynamics

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