Illustrated Physics
physics, drawn
Thermodynamics

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.
16 min read 6 figures What stays the sameMade into an instrument

Assumes: The engine that pays back more than it takes · The work left in two buckets of water

The Carnot ceiling run backwards runs Carnot’s cycle backwards and finds that a machine can deliver three or four joules of heat for every joule of work it consumes. That is a heat pump, and the work is where the argument starts: a quantity of high-grade energy goes in, and a larger quantity of low-grade heat comes out somewhere useful.

Take the work away entirely. Allow the machine three reservoirs instead of two, at three temperatures, with heat free to cross each boundary — and no work crossing any boundary in either direction, no shaft, no piston, no wire.

Such a machine can refrigerate.

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.
Fig. 1 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 three cold temperatures and an ambient of 300 kelvin. The dots are what real machines achieve — between a fifth and a third of the ceiling.

The one inequality that fixes the ceiling

Label the three temperatures: TgT_g for the driving heat, T0T_0 for the ambient into which waste is rejected, and TcT_c for the cold space. Heat QgQ_g enters at the top, heat QcQ_c is lifted from the bottom, and heat Q0Q_0 leaves at ambient.

The first law with no work is one line:

Q0=Qg+Qc.Q_0 = Q_g + Q_c.

The second law is one more. The machine returns to its starting state each cycle, so its own entropy is unchanged, and the entropy carried in must not exceed the entropy carried out — entropy being a count that can grow and not shrink:

QgTg+QcTc≤Q0T0.\frac{Q_g}{T_g} + \frac{Q_c}{T_c} \le \frac{Q_0}{T_0}.

Substituting and rearranging gives the whole result:

QcQg≤Tg−T0Tg⋅TcT0−Tc.\frac{Q_c}{Q_g} \le \frac{T_g - T_0}{T_g}\cdot\frac{T_c}{T_0 - T_c}.

The right-hand side is a product of two things already established. The first factor is the Carnot efficiency of an engine between TgT_g and T0T_0. The second is the same ceiling inverted into a floor greater than one, for a refrigerator between T0T_0 and TcT_c.

Nothing in the derivation multiplied them together on purpose. They came out of one inequality, and they factorise because the machine is thermodynamically a Carnot engine driving a Carnot fridge, whether or not it contains anything resembling either.

Three flows of heat and no work anywhere. The heat crossing each boundary of a reversible three-reservoir refrigerator, per joule supplied to drive it, with the entropy each flow carries printed beside it. Heat enters at 450 kelvin and at 253, and the sum of the two leaves at 300. No work crosses any boundary in either direction, so the first law is simply that the heats add. The second law is the interesting one: the entropy arriving is 0.00931 joules per kelvin and the entropy leaving is 0.00931, and they are equal — which is exactly the condition that fixes how much heat could be lifted. A joule arriving at a high temperature carries little entropy and a joule arriving at a low one carries a lot, so the driving heat's entropy budget is what pays for the cold heat's.
Fig. 2 The three heat flows of a reversible machine of this kind, per joule supplied to drive it, with the entropy each carries. Heat in at 450 kelvin, heat in at 253, and the sum out at 300. The entropy arriving and the entropy leaving are equal, which is the condition that fixed how much could be lifted.

The entropy ledger is the clearer way to see what is being bought. A joule arriving at 450 kelvin carries 2.2 millijoules per kelvin of entropy; a joule arriving at 253 kelvin carries 4.0. So the driving heat is low-entropy energy, and what the machine sells is the right to bring in high-entropy energy from the cold space, paid for out of the driving heat’s entropy budget.

Work is simply the limiting case of zero entropy. A joule of work carries no entropy at all, which is what makes it the most valuable form energy comes in, and heat at a high temperature carries a little. The machine does not need work; it needs something with a low enough entropy per joule, and hot heat is that.

What the machine does instead of compressing

The thermodynamic argument says nothing about the mechanism, and the mechanism is worth a paragraph because it shows what replaced the compressor.

A compression fridge circulates a refrigerant: the compressor raises its pressure, it condenses in the hot coil giving up heat, it expands through a valve, and it evaporates in the cold coil taking heat in. The compressor is there to raise the pressure.

An absorption fridge raises the pressure without a compressor, by dissolving the refrigerant in a second liquid and boiling it back out somewhere else. Ammonia dissolves readily in water; a weak solution absorbs ammonia vapour from the low-pressure side, and heating the resulting strong solution drives the ammonia off at high pressure. The pump that circulates the liquid does exist and does negligible work, because pumping a liquid against a pressure difference costs far less than compressing a gas across the same difference — which is the practical content of a liquid being nearly incompressible.

The domestic version has no pump at all. A gas flame heats one leg of a sealed loop containing ammonia, water and hydrogen; the hydrogen equalises the total pressure everywhere while allowing the ammonia’s own partial pressure to differ between the legs; and circulation is driven by gravity and by the bubbles the flame makes. It contains no moving part whatever, it is silent, and it will run for decades.

Three flows of heat and no work anywhere. The heat crossing each boundary of a reversible three-reservoir refrigerator, per joule supplied to drive it, with the entropy each flow carries printed beside it. Heat enters at 380 kelvin and at 283, and the sum of the two leaves at 305. No work crosses any boundary in either direction, so the first law is simply that the heats add. The second law is the interesting one: the entropy arriving is 0.01160 joules per kelvin and the entropy leaving is 0.01160, and they are equal — which is exactly the condition that fixes how much heat could be lifted. A joule arriving at a high temperature carries little entropy and a joule arriving at a low one carries a lot, so the driving heat's entropy budget is what pays for the cold heat's.
Fig. 3 The same ledger for a machine driven by solar heat at 380 kelvin, cooling a space to 283 in a 305-kelvin ambient. The entropy still balances exactly, and the heat lifted per joule supplied has fallen to a fifth of the earlier case — because the driving heat is cooler and the ambient is warmer, and both factors move the wrong way at once.

The pump that is allowed to exist

There is a pump in most of these machines, and calling them work-free requires saying why it does not count.

The pump raises a liquid solution from the low-pressure side to the high-pressure side. The work it does per unit mass is the volume times the pressure difference, and a liquid’s volume per unit mass is small: pumping a kilogram of water across ten bar costs about a kilojoule. Compressing a kilogram of ammonia vapour across the same ten bar costs a few hundred kilojoules, because the gas has to be squeezed as well as moved and its volume per unit mass is three orders of magnitude larger.

So the pump’s work is a fraction of a per cent of the heat flows, and the machine is work-free to the accuracy of everything else in the analysis. That is not a fudge; it is the whole trick. The compressor has been replaced by dissolving the gas in a liquid, moving the liquid, and boiling the gas back out — and what makes the substitution profitable is that a liquid is nearly incompressible, so transporting it across a pressure difference is cheap.

The domestic version dispenses with the pump entirely by using a third gas at constant total pressure, so that nothing has to be moved across a pressure difference at all, and circulation is driven by heating one leg of a loop. It contains no moving part of any kind.

The comparison that is usually made wrongly

An absorption fridge is routinely described as inefficient. Its coefficient of performance is around 0.5 to 0.8 against 3 or 4 for a compression fridge, and the comparison is not a comparison.

The two numbers have different denominators. A compression fridge’s counts work going in; an absorption fridge’s counts heat. To compare them, the work has to be traced back to the heat that made it.

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 240-kelvin space it is 1.33. 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.
Fig. 4 The same ceiling over a wider range of driving temperature and for different cold spaces. The curves flatten: the engine factor tends to one as the driving temperature rises, so the whole expression tends to the fridge coefficient alone, and nothing above that is available at any temperature.

A power station converts heat to electricity at about 40 per cent — well below its own Carnot value, and for reasons that are not carelessness. A compression fridge with a coefficient of 3.5 running on that electricity lifts 3.5 joules per joule of work, which is 0.4×3.5=1.40.4 \times 3.5 = 1.4 joules per joule of heat burned. An absorption fridge lifts 0.7 per joule of heat burned.

So the compression chain is twice as good, not five times, and the factor of two is the real number. It buys silence, no moving parts, and the ability to run on heat nobody wanted — waste heat from an engine, steam from a process, sunlight on a collector — which for a machine whose fuel is free is the only comparison that matters. A machine that runs on something nobody wanted is competing against zero.

Solar cooling, which is the case the arithmetic was made for

The best argument for a machine that runs on heat is a place where heat is free and the demand for cooling rises and falls with it.

A flat-plate solar collector delivers water at 350 to 370 kelvin, and an evacuated-tube collector 380 to 420. Reading those off the ceiling at a 280-kelvin cold space: a flat plate at 360 kelvin gives a ceiling of 2.6 and an evacuated tube at 400 gives 3.5. Real single-effect machines achieve 0.6 to 0.8 of a coefficient, so a fifth to a quarter of the ceiling, and that is the number a system is designed around.

The alignment is what makes the idea attractive. Cooling demand in a hot climate peaks in the afternoon, which is when the collector delivers most; a compression chiller drawing that load from the grid does so at exactly the hour the grid is most stressed; and a thermal system can store its input in a hot tank far more cheaply than an electrical one can store electricity.

What has kept it from spreading is capital rather than physics. A solar absorption system needs the collector field, the chiller, the cooling tower and the storage, and its coefficient of 0.7 means the collector must gather 1.4 kilowatt-hours of heat for every kilowatt-hour of cooling. Photovoltaic panels feeding a conventional chiller with a coefficient of 3.5 need to gather roughly a third of a kilowatt-hour of electricity for the same cooling — and photovoltaic panels have become very cheap.

That comparison has moved a long way in twenty years and it moved on the cost of one component rather than on any thermodynamic argument. The ceiling in the first figure has not changed since the 1920s; what changed is what the alternative costs.

Where the machine cannot go

The region a heat-driven fridge can work in. Contours of how much heat a reversible three-reservoir machine lifts per joule supplied, in the plane of the driving temperature and the temperature of the cold space, at an ambient of 300 kelvin. Two edges bound the region and neither is negotiable: the driving heat must be above ambient, or there is no engine, and the cold space must be below it, or there is nothing to refrigerate. Approaching either edge sends the performance to zero. The contours run steeply, which is the practical content: deep refrigeration from modest heat is nearly impossible, and mild cooling from hot heat is easy. That is why absorption machines are used for air conditioning, where the cold space is fifteen degrees below ambient, and almost never for freezing.
Fig. 5 Contours of performance in the plane of the driving temperature and the cold space’s temperature, at 300 kelvin ambient. Two edges bound the region: the driving heat must be above ambient or there is no engine, and the cold space must be below it or there is nothing to refrigerate. Approaching either sends the performance to zero.

The contours run steeply downward toward low cold temperatures, and that is the practical shape of the subject. Mild cooling from hot heat is easy; deep refrigeration from modest heat is close to impossible.

Absorption chillers are therefore an air-conditioning technology. A cold space fifteen or twenty degrees below ambient, driven by steam at 400 to 450 kelvin, sits in the comfortable part of the plane. A freezer at 250 kelvin driven by the same steam sits where the contours crowd, and a cryogenic application is off the chart entirely.

The same figure says what a designer trades. Raising the driving temperature helps, with diminishing returns, and is limited by the chemistry — lithium bromide solutions crystallise, ammonia decomposes — rather than by thermodynamics. Lowering the ambient helps twice, through both factors, which is why these machines perform far better in a cool climate and why a cooling tower is worth its cost.

The same machine, reversed

Run the three flows the other way and the machine does something else useful, and the fact that it is the same device with the arrows turned round is worth seeing.

Supply heat at an intermediate temperature — waste heat from a process, at 360 kelvin say — and reject a part of it to ambient. The remainder can then be delivered at a temperature above where it came in, because rejecting some heat to a cold reservoir is again a supply of low entropy. That is a heat transformer, and it upgrades waste heat rather than producing cold.

The ceiling is the mirror of the refrigeration one. A machine taking QQ in at TmT_m, rejecting some at T0T_0 and delivering the rest at ThT_h can deliver at most a fraction

QhQm≤ThTm⋅Tm−T0Th−T0\frac{Q_h}{Q_m} \le \frac{T_h}{T_m}\cdot\frac{T_m - T_0}{T_h - T_0}

of what it took in — which for waste heat at 360 kelvin, ambient at 300 and delivery at 420 is 0.48. So about half the waste heat can be raised by sixty degrees and the other half is thrown away.

Whether that is worth doing is an economic question with a clear shape. Half of a free input at a higher temperature may be worth more than all of it at the temperature it arrived at, because what industrial processes need is heat above a threshold, and heat below the threshold is worth nothing at all. A heat transformer is a device for converting quantity into grade, at a fixed and computable exchange rate.

The region a heat-driven fridge can work in. Contours of how much heat a reversible three-reservoir machine lifts per joule supplied, in the plane of the driving temperature and the temperature of the cold space, at an ambient of 290 kelvin. Two edges bound the region and neither is negotiable: the driving heat must be above ambient, or there is no engine, and the cold space must be below it, or there is nothing to refrigerate. Approaching either edge sends the performance to zero. The contours run steeply, which is the practical content: deep refrigeration from modest heat is nearly impossible, and mild cooling from hot heat is easy. That is why absorption machines are used for air conditioning, where the cold space is fifteen degrees below ambient, and almost never for freezing.
Fig. 6 The operating region at a cooler ambient of 290 kelvin. Both edges have moved and the contours have opened out: a cooler surroundings helps through both factors of the product at once, which is why these machines are far more attractive in a temperate climate than in a hot one — and why the cooling tower that holds the rejection temperature down is worth its cost.

Every boundary in a real absorption chiller is irreversible

The ceiling assumes reversibility throughout, and nothing about the actual machine is reversible. Heat crosses every boundary through a finite temperature difference, the absorption and the boiling are irreversible mixing processes, and the solution is pumped through pipes. That is why real machines reach a fifth to a third of the figure drawn, and the gap is not a manufacturing defect but the price of running at a finite rate — the same trade a finite-time engine makes for an engine.

The reservoirs are assumed infinite. A real driving heat source cools as heat is taken from it and a real cold space warms, so the three temperatures move during operation, and the arithmetic of finite reservoirs applies here as much as to an engine.

And the mixing has been ignored. The working fluid is a solution whose composition changes round the cycle, so there is a chemical potential doing work in the accounting that a three-reservoir treatment does not contain. A proper analysis of an absorption cycle is a two-component phase diagram with a cycle drawn on it, and the three-reservoir ceiling is an upper bound on what any such cycle can do rather than a description of one.

Two other machines with no work in them

The three-reservoir idea is older and wider than refrigeration, and two other devices share the whole of the argument.

A thermoacoustic refrigerator has a loudspeaker, a stack of plates and a gas, and it moves heat along the stack because a parcel of gas oscillating in a sound wave — a second-order effect of exactly the kind an oscillating boundary layer produces — is compressed and displaced at the same time — so it picks heat up at one end of its excursion and drops it at the other. The version with no loudspeaker replaces it with a second stack across which a temperature difference is imposed by a flame: the flame drives a sound wave, and the sound wave drives the fridge. Heat in, cold out, one moving thing which is the gas.

And a gas-fired heat pump for a house is the same machine sized differently. Instead of burning gas to make heat at 340 kelvin for the radiators, it burns gas to drive an absorption cycle that lifts heat from outside, and delivers more heat to the house than the gas contains. A ceiling of 1.4 on the total heat delivered per unit burned is achievable, which is 40 per cent more than burning the gas directly — and the comparison with an electric heat pump depends entirely on how the electricity was made, exactly as in the section above.

What the three share is that nothing about them requires the intermediate step of making work. The engine is not there because an engine is needed; it is there because the analysis factorises that way, and a real machine does the two jobs with the same working fluid in the same vessel.

Two heats that are not interchangeable, and the fluid nothing draws

They cannot show that the cold heat and the driving heat are not interchangeable. Both are heat and both enter the machine, and the figures draw them as bars of the same kind. What distinguishes them is entirely the temperature they arrive at, which decides the entropy they carry, and a bar chart of joules has thrown that away — which is why the entropy is printed beside each bar rather than drawn.

Nor can they show the working fluid. The whole of the machine’s design is which pair of substances dissolve in one another with the right dependence on temperature over the right range, and that is a chemistry question the thermodynamics has no opinion about. Ammonia–water and lithium bromide–water are the two pairs in industrial use, and the reasons are corrosion, crystallisation and toxicity rather than anything on these axes.

And they cannot show what a double-effect machine does. The best chillers use the heat rejected from one stage to drive a second, which raises the coefficient above one — past the single-stage ceiling drawn here, because it is a different machine with more reservoirs. The ceiling in the figure is for three, and adding a fourth raises it.

Still open: whether the chemistry can be improved rather than the cycle

The thermodynamic ceiling has been known since the 1920s and the cycles that approach it are worked out. What limits real machines is the working pair: lithium bromide solutions crystallise if the concentration rises too far, which caps the driving temperature and forces the machine to run with a margin; ammonia is toxic and requires the whole circuit to be built to a pressure specification.

Searches for better pairs — ionic liquids, other salt solutions, organic absorbents — have run for decades and nothing has displaced the two incumbents. The requirement is a long conjunction: high solubility with a steep temperature dependence, low vapour pressure of the absorbent, no crystallisation, no corrosion, low toxicity, low cost. Candidates that meet five of those are common and ones that meet all of them are not, and whether the space has been searched or merely sampled is a fair question.

The habit worth carrying away is about what an engine actually needs. A machine that moves heat up a gradient needs a supply of low entropy, and work is only the purest form of that. Naming work in the requirement is a habit from machines that happened to have shafts in them; the requirement is entropy per joule, and hot heat satisfies it at a price that is exactly the Carnot efficiency between its own temperature and the surroundings.

Part 7 of 8

This essay is one argument about Heat engines. The others:

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 Part 6 — 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. 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 Part 8 — 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.

Where this goes next

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. Ways to arrange 10 coins. The number of distinct arrangements giving each number of heads, for 10 coins. Every individual arrangement is equally likely; the middle wins because there are more ways to reach it. Entropy is a count, and the arrow of time is arithmetic Nothing in mechanics prefers a direction. Entropy is not a force pushing things toward disorder — it is the observation that some outcomes have vastly more ways of happening than others. 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.

What links here

Essays that reach for this one mid-argument — the half of a link its own author cannot write down.

The objects named here

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

Available workCarnot efficiencyChemical potentialCoefficient of performanceEntropyExergyHeat enginesHeat flowRefrigerationReservoirReversibilityThe second law