Illustrated Physics
physics, drawn
Concept

Carnot efficiency — where it appears

The largest fraction of the heat taken in that any engine can deliver as work, equal to one minus the ratio of the cold temperature to the hot. It is reached only by an engine that takes for ever, because a reversible heat flow needs a vanishing temperature difference to drive it.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

Two temperatures for the same sunlight. Sunlight described two ways, against how much it has been concentrated. The flat line is the temperature its spectrum belongs to — 5,762 K, the Sun's surface, which concentration does not change because a mirror does not alter a photon's energy. The rising curve is the temperature a blackbody would need in order to radiate the flux actually arriving: 394 K unconcentrated, 2213 K under a parabolic dish, and 5771 K at the geometric limit, where the two meet — computed here and checked against the Sun's own temperature, because a perfect concentrator reproduces the source's radiance and cannot exceed it. The gap between the two curves is the dilution, and dilution is entropy: the same energy spread over a hundred thousand times more directions occupies a hundred thousand times more modes. That is what a converter has to carry, and it is why the ceiling on solar conversion is not the Carnot efficiency between 5,762 K and 300 K.

The work a diluted beam will not do

Sunlight at the top of the atmosphere has the spectrum of a body at 5,762 kelvin and the energy flux of one at 394. The mismatch is not an accident of units: the light has been spread over a hundred thousand times more modes than it left in, and that dilution is entropy. Run it through a heat engine unconcentrated and five per cent of it is available as work.

optics · Etendue 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.

thermodynamics · Heat engines 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.

thermodynamics · Heat engines 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.

thermodynamics · Heat engines

Named alongside it

The objects these essays reach for when they reach for this one.

Available workHeat enginesThe second lawAtmospheric windowBlackbodyChemical potentialCoefficient of performanceConcentrationConductivityDetailed balanceDopingEmissivity

All concepts