The engine that has to finish

A coal-fired station at West Thurrock ran between a boiler at 838 kelvin and cooling water at 298, so its Carnot ceiling was 64 per cent. It delivered 36. The usual gloss is that the difference is friction, turbine losses and the imperfection of engineering, and that a better plant would creep upward toward the ceiling.

It would not, and the reason has nothing to do with the quality of the engineering. The ceiling corresponds to a plant that produces no electricity at all.

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. — Fig. 1 Two efficiencies against the ratio of the reservoir temperatures. The upper curve is Carnot’s, reached only by an engine that runs infinitely slowly. The lower is the efficiency of an engine with finite thermal contact run for the most power. West Thurrock at 36 per cent, a CANDU reactor at 30 and Larderello’s geothermal plant at 16 are marked, and every one of them is closer to the lower curve.

What reversibility costs

The Carnot argument is exact and its exactness comes from the same place as its uselessness for design.

Reversibility costs time, and the Carnot loop is where the cost is hidden. Two isotherms and two adiabats enclose an area that is the work, and the ceiling that follows was fixed before any machine was described — but every step of that loop has to proceed through states arbitrarily close to equilibrium, which means arbitrarily slowly. The cycle is not slow because it is badly built. It is slow because it is reversible.

Every step of the cycle is reversible, and reversible means that the system is never more than infinitesimally out of equilibrium with what it is touching — the condition under which entropy is not created but merely moved. During the isothermal absorption, the working fluid must be at $T_h$ — because if it were below $T_h$ , the heat crossing the contact would be crossing a finite temperature difference, which generates entropy and is not reversible.

But heat crossing no temperature difference is heat crossing at no rate. The idealisation that makes the bound exact is the idealisation that makes the engine deliver nothing per hour.

Heat moves by a gradient: no difference, no flow. The equation that governs it has a rate in it, so transferring a given amount of heat across a vanishing temperature difference takes an unbounded time. That is the whole of the difficulty — the reversible cycle asks for exactly that transfer, and a machine that must finish cannot afford it.

What sets a thermal conductance at the molecular level is how far a carrier gets between collisions, and the consequence is economic rather than physical. A real heat exchanger has a finite conductance, that conductance costs money roughly in proportion to its area, and the temperature difference needed to push a given power through it is what the designer is buying. So the irreversibility is not a defect to be engineered away; it is a purchase, and the optimum is where the marginal cost of area equals the marginal value of efficiency.

The model that keeps the reversibility inside

Curzon and Ahlborn’s move, in 1975, was to keep Carnot’s engine and give it finite contact with the world.

The working fluid runs a reversible cycle between two internal temperatures $T_{hw}$ and $T_{cw}$ , and those are connected to the reservoirs through thermal conductances. Heat flows in at $\alpha(T_h - T_{hw})$ and out at $\beta(T_{cw} - T_c)$ . Everything irreversible has been placed in the two contacts and the engine itself is still perfect — which is why the model is called endoreversible.

The efficiency the fluid achieves is $1 - T_{cw}/T_{hw}$ , and the operator gets to choose it, by choosing how far the internal temperatures are pulled from the reservoirs. A small pull gives a high efficiency and a small heat flow. A large pull gives a large flow and a poor efficiency. There is a maximum in between.

The efficiency at which an engine delivers nothing. Power against efficiency for an engine with finite thermal contact at both ends, between reservoirs whose temperatures stand in the ratio 0.4. The curve starts at zero for the obvious reason — an engine of no efficiency does no work — and returns to zero at Carnot's 60 per cent for the reason worth the figure: reaching the ceiling requires the working fluid to sit at the reservoir temperatures, and then no heat crosses the contacts at all. In between it peaks, at η = 36.75 per cent — located by scanning the drawn curve — against 1 − √τ = 36.75 per cent, agreeing to 1.4e-5. So there is a choice to be made and it is not between good and bad engineering: an operator who wants the last few points of efficiency must accept a plant that produces almost nothing, and one who wants power must give up efficiency the second law would have allowed. The ceiling is real, and it is not on the frontier anybody is standing on. — Fig. 2 Power against efficiency for such an engine, between reservoirs in the ratio 0.4. Zero at zero efficiency for the obvious reason, and zero again at Carnot’s 60 per cent for the reason worth the figure. In between it peaks at 36.75 per cent, located by scanning the drawn curve, against 1 − √τ = 36.75 per cent.

The maximum is at

\eta^* = 1 - \sqrt{T_c/T_h},

which is remarkable for containing neither conductance. The two $\alpha$ and $\beta$ set how much power there is and not the efficiency at which it is greatest, so the result is as universal as Carnot’s within its own model — a function of the two temperatures and nothing else.

A Carnot cycle on pressure–volume axes. Two isothermal steps joined by two adiabatic ones, forming a closed loop. The gas expands 4.1-fold in reaching the cold reservoir at 0.70 of the hot one, and the area enclosed is the net work done over one cycle. — Fig. 3 The model itself, drawn as the loop it is: two isothermal steps joined by two adiabatic ones, with the gas expanding 4.1-fold on its way to a cold reservoir at 0.70 of the hot one. The area enclosed is the net work of one cycle. Every step is reversible by construction — the isothermal ones because the gas is always at the reservoir’s temperature, the adiabatic ones because nothing is exchanged — and that is exactly why the cycle takes for ever: a step at zero temperature difference transfers heat at zero rate.

Why the right-hand zero is the interesting one

The left-hand zero of that loop is trivial: an engine of no efficiency does no work.

The right-hand zero is the content. Approaching Carnot’s efficiency means letting the internal temperatures approach the reservoir temperatures, which means letting the temperature differences across the two contacts approach zero, which means letting the heat flow approach zero. The engine becomes perfect and stops.

So an operator has a genuine choice to make, and it is not a choice between good and bad engineering. Wanting the last few points of efficiency means accepting a plant that produces almost nothing; wanting power means giving up efficiency that the second law would have permitted. The ceiling is real and it is not the frontier anybody is standing on.

How well it describes real plant

The three plants in the opening figure were Curzon and Ahlborn’s own, and they are worth reading carefully because the agreement is good and not perfect.

Plant	$T_c$	$T_h$	Carnot	$1-\sqrt{T_c/T_h}$	Observed
West Thurrock, coal	298 K	838 K	0.64	0.40	0.36
CANDU, nuclear	298 K	573 K	0.48	0.28	0.30
Larderello, geothermal	353 K	523 K	0.33	0.18	0.16

The temperatures in that table are not the flame temperature and the ambient; they are the temperatures the working fluid is exchanging heat with, which for a steam plant means the boiler and the condenser. Getting that identification right is most of the work in applying the model to a real machine, and it is where the free energy rather than the heat becomes the quantity to keep track of.

Two of the three fall a little below the estimate and one is a little above, which is what a genuinely predictive model looks like as against a fitted one. The margins are a few percentage points, against sixteen to twenty-eight from the ceiling.

The one above the line is worth a note, because a model that is never exceeded is usually a bound and this is not one. A plant can beat the maximum-power efficiency, by running below maximum power, and a nuclear station whose fuel is cheap relative to its capital cost has a reason to.

The same trade, run backwards

The rung below this one ran the cycle in reverse and got a coefficient greater than one. That result inherits the same problem and the same resolution.

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. — Fig. 4 The ideal coefficient of performance of a heat pump against outside temperature, and a quarter of it. The ceiling on a pump is Carnot’s inverted, it is reached at zero heating rate, and the dashed fraction is roughly where a machine that actually heats a house sits.

A heat pump with infinite heat exchangers and infinite time would move heat at the ideal ratio and would take for ever to warm anything. A real one runs its refrigerant well below the outside air and well above the room, and every degree of that overshoot is efficiency spent to buy rate. The published seasonal performance figures for domestic units are around a quarter to a third of the ideal, and the endoreversible estimate for the same temperatures is in that range.

The number in a form worth remembering

Two limits, and the gap between them, on the temperatures that actually occur.

At a reservoir ratio of one half — a boiler at 600 K and cooling water at 300 — the ceiling is 50 per cent and the maximum-power efficiency is 29.3. At a ratio of a third the two are 66.7 and 42.3. At a ratio of nine tenths, which is a low-grade heat recovery job, they are 10 and 5.1.

The pattern is that the maximum-power efficiency is close to half the Carnot value over the whole range that matters, and exactly half in the limit of a small temperature difference. That is easy to see: for $T_c/T_h = 1-\epsilon$ , Carnot gives $\epsilon$ and one minus the square root gives $\epsilon/2$ . So a serviceable rule of thumb for any engine that has to produce something is half of Carnot, and the three plants above are all within a few points of it.

Entropy grows in proportion to the size of the system it belongs to, and that proportionality is what makes the subject usable. Entropy production in a heat exchanger is an extensive quantity, so doubling the plant doubles it — which means the efficiency at maximum power is a ratio, independent of scale, and the same formula covers a laboratory engine and a power station.

There is one further reading worth taking from the figure. Because the maximum is flat — the power falls only to second order either side of it — an operator can move several points away from the maximum-power efficiency and lose very little power. That flatness is why the observed plants scatter either side of the estimate rather than clustering on it, and it is why the model predicts a region honestly and a point only loosely.

The same loop in a solar cell

The power-against-efficiency loop is not a fact about heat engines. It appears wherever a device’s output depends on a variable the operator can set, with two extremes at which the output is zero, and the clearest non-thermal instance is a photovoltaic cell.

A solar cell has a current and a voltage, and the load decides the ratio between them. Short-circuit it and the current is maximal and the voltage is zero, so the power delivered is zero. Open-circuit it and the voltage is maximal and the current is zero, so the power delivered is zero again. The power is the product, and it has a maximum somewhere in between.

The parallel with this essay’s loop is exact in structure. The open-circuit voltage is the cell’s thermodynamic ceiling — the largest voltage the photon energy can support — reached only at zero output, in the same way that Carnot’s efficiency is reached only at zero power. The short circuit is the other zero. And the operating point that matters is neither.

That has a practical consequence that every installation deals with. A solar panel’s output depends steeply on where on that curve it is held, and the maximum shifts with illumination and with temperature — so a panel connected to a fixed load spends most of its day off the maximum. The electronics that sits between a panel and its battery therefore spends its time hunting for the maximum power point, adjusting the effective load continuously to hold the operating point there. A device whose entire job is to keep a system at the top of a curve like this one.

The quality of a cell is quoted, accordingly, not as an efficiency alone but as a fill factor — how close the maximum power comes to the product of the two extremes. That is a measure of how square the curve is, which is a measure of how much of the ceiling the device can deliver at a useful rate, and it is the photovoltaic version of the ratio this essay has been computing.

The device Carnot does not bound

The essay’s last caution deserves its own numbers, because the escape from Carnot’s ceiling is real and it does not escape the rate trade.

A fuel cell converts a fuel’s chemical free energy directly into electrical work, without ever making heat and then converting it. Its ceiling is therefore not Carnot’s but the ratio of the free energy released to the total energy released: for hydrogen at room temperature that is about 83 per cent, and it rises as the temperature falls, which is the opposite of a heat engine’s behaviour.

So a fuel cell is not bounded by Carnot at all, and the point is often made as though it settled something. It does not, because the same trade this essay is about applies to it in a different currency.

A fuel cell’s voltage falls as the current drawn from it rises — from the reaction’s own sluggishness at the electrodes, from the resistance of the electrolyte, and at high current from the difficulty of getting fuel to the reaction site fast enough. The efficiency is proportional to the voltage, so drawing more current means accepting a lower efficiency, and the power is again a product with a maximum in between.

The result is a polarisation curve with the same shape as everything else in this essay: an open-circuit voltage near the thermodynamic ceiling and no power, a limiting current with no voltage and no power, and a useful operating point between them. Real cells are run at around half to two thirds of their open-circuit voltage, which is to say at around half to two thirds of their ceiling — a familiar fraction.

Which is the honest summary of the fuel cell’s advantage. It is bounded by a different and higher ceiling, and it pays the same kind of price for producing anything, so the achievable efficiency is better than a heat engine’s by rather less than the ceilings suggest.

Who found it first

The result usually carries Curzon and Ahlborn’s names, from 1975, and it had been obtained twice before.

Novikov published it in 1957, in the context of nuclear power plant, and Chambadal published it the same year in a book on thermodynamics. Both derived the same expression by the same route, and neither had much effect: the result appeared as a calculation about a particular kind of installation rather than as a statement about engines in general, and it was not picked up.

What Curzon and Ahlborn added was the framing. They presented the model as a modification of Carnot’s own argument, gave the result as a bound of the same kind, and — decisively — put three real power stations beside it. The table in this essay is theirs, and it is what turned a calculation into a claim that could be argued about.

The pattern is worth naming because it is common. A result can be correct, published and available and still not exist as a piece of shared knowledge until somebody states what it is a result about. What was missing in 1957 was not the algebra but the sentence that the number is a companion to Carnot’s rather than a property of a reactor — and the comparison with measurement that made the sentence checkable.

Where the extra losses go

Three effects have been excluded from the model and are worth naming, because a real plant’s shortfall is the endoreversible loss plus these.

A steam plant does not run a Carnot cycle at all. It runs a Rankine cycle, whose heat absorption happens largely at constant temperature during boiling — the flat section of a heating curve — and whose remaining steps are nothing like adiabats. The finite-time argument still applies, because it depends only on heat having to cross a finite conductance, and not on which cycle is being run.

The cycle is not Carnot’s. Real cycles are shaped by what the working fluid can do, and the Rankine and Brayton cycles both absorb heat over a range of temperatures. That alone costs several points.

Friction and leakage. Turbine blades have boundary layers, seals leak, and the compressor in a gas turbine consumes a large fraction of what the turbine produces. These are the losses the word “inefficiency” usually refers to and they are the smaller part of the total.

Every part of a real plant has a time constant, and a cycle run faster than those constants allow does not merely lose efficiency — it stops executing the cycle it was designed for. Valves do not close, fluids do not reach equilibrium, and the p–V loop the plant traces is not the one on the drawing. That is the practical limit on running faster to get more power out.

And the cycle is run at a finite frequency. Even with perfect heat exchangers, a piston moving fast enough compresses the gas non-uniformly and the pressure at the piston face differs from the pressure in the bulk. That is a third source of irreversibility, it grows with speed, and it is what makes a real engine’s power peak and then fall as the speed rises.

The oldest version of the same argument

The trade is not modern. Carnot’s own memoir of 1824 contains it, in a passage usually skipped:

The ceiling was derived by imagining transfers made “without any difference of temperature”, and Carnot observed that a real engine must have such differences and that this is where its shortfall lies. What he did not have was a way to optimise the trade, because the optimisation needs a rate law for the heat transfer, and Fourier’s law was published the same year.

The oldest version of the argument is a boiler tube. A steam plant’s upper temperature is set by where water boils at a given pressure and by what the tubes will stand — not by anything thermodynamic — so supercritical plants exist because metallurgy advanced, and the Carnot ceiling has never been the binding constraint on a power station.

What the twentieth century added is the observation that the optimum is universal within the model — independent of the conductances — and that measured plant sits on it. The first is a piece of mathematics and the second is the surprise, because there is no reason a station designed by engineers with no knowledge of endoreversible thermodynamics should land on the maximum-power point of a model published in 1975. It lands there because fuel costs money and boilers cost money, and the economic optimum for those two costs is close to the physical one.

Where the model stops

The endoreversible model is a model, not a theorem. Its result is exact for its own assumptions — Newtonian heat transfer, all irreversibility in the contacts, a reversible interior — and changing those changes the answer. With radiative rather than conductive coupling the maximum-power efficiency is a different function, and with dissipation inside the engine as well it is different again.

“Maximum power” is not always the objective. An engine can be optimised for maximum work per unit of fuel, for minimum entropy production, for maximum profit given fuel and capital costs, or for maximum power, and these give four different operating points. The agreement with real plant above is partly because fuel and capital costs happen to push a station near the maximum-power point, and that is an economic fact rather than a physical one.

The conductances are treated as constants. A real heat exchanger’s conductance depends on the flow rate, the fouling and the temperature difference itself, and optimising the plant means sizing the exchangers, which changes $\alpha$ and $\beta$ and moves the whole curve.

And nothing here is about the fuel. The chemical energy in coal is a free energy, not a heat, and burning it to make heat before making work throws away a large fraction before the cycle begins. A fuel cell converts the same free energy directly and is not bounded by Carnot at all — which is the sharpest reminder that the ceiling applies to a particular architecture rather than to energy conversion in general.

What the pictures cannot show

The power-against-efficiency loop is drawn with power normalised to its own maximum, which conceals the fact that the maximum depends on the conductances and can be anything. Two plants with identical curves in these units differ by a factor of a thousand in megawatts.

Nor can the p–V diagram show what is different about the endoreversible engine. Its loop looks exactly like Carnot’s, because the fluid is running a reversible cycle; what has changed is the temperatures of the isotherms relative to the reservoirs, which are not on the diagram at all. Everything this essay is about happens between the diagram and the world.

Where this ladder goes next

Three rungs establish the same bound three times and mean something different by it each time: as a ceiling on work per unit of heat, as a ceiling on heat moved per unit of work, and now as a ceiling that costs infinite time to reach. The third reading is the one that makes the first two usable, because it says what the shortfall of a working machine is measuring.

The habit worth carrying away is a question to ask of any bound. What does the system have to be doing to achieve it? A limit derived from equilibrium arguments is achieved in equilibrium, and equilibrium means nothing is happening. The same structure recurs throughout this collection: the concentration limit is reached by a receiver in equilibrium with the Sun, the erasure bound is reached by an erasure taking infinite time, and the reversible limit of any process is the limit in which the process stops. A bound with a rate of zero attached is a real constraint and a poor target, and the useful question is always what the best achievable rate costs.

That closes this anchor. What remains about engines in this collection is not thermodynamic: the particular cycles — Otto, Diesel, Stirling, Rankine — are designs rather than principles, and what they are optimising is what a system at fixed temperature actually minimises under constraints that come from materials rather than from the second law.