The laws of energy: conservation, entropy and efficiency: The limits to efficiency: Carnot and beyond

3.1 The Carnot limit caps every heat engine

A heat engine takes heat from a hot source, turns part of it into work and rejects the rest to a cold sink. Steam turbines, gas turbines, car and diesel engines, and the turbines in nuclear and geothermal plants are all heat engines. Carnot showed that even a perfect engine, with no friction and no leaks, cannot turn more than a certain fraction of the heat into work. That fraction is 1 minus the cold temperature divided by the hot temperature, with both measured in kelvin, the absolute scale that starts at minus 273 degrees Celsius.

The formula is worth working through. A modern coal plant raises steam to about 600 degrees Celsius, or 873 K, and rejects heat to cooling water at about 30 degrees, or 303 K. The Carnot limit is 1 minus 303 divided by 873, which is about 65%. Real plants reach 38 to 45% because heat must cross metal walls, the steam cannot be heated evenly at the top temperature, and turbines, pumps and fans all have losses.

Nuclear plants run cooler steam, typically under 300 degrees, because of the limits of reactor materials. Their Carnot limit is near 45% and their real efficiency about 33%, which is why statisticians count a nuclear kilowatt-hour as about three kilowatt-hours of heat.

The formula also explains why low-temperature heat is so hard to use. A geothermal well at 180 degrees, with cooling at 30, has a Carnot limit of about 33%, and real plants achieve 10 to 15%. Ocean thermal energy conversion, working between surface water at 27 degrees and deep water at 5, faces a limit near 7% and achieves perhaps 3%.

• Smil, Energy: A Beginner's Guide (2006) · via Smil Mastery Course, M1 'Entropy and the second law'
• Jaffe and Taylor, The Physics of Energy (2018) · via Smil Mastery Course, M1 'Entropy and the second law', 'Heat and how it moves'

3.2 Why engineers chase ever higher temperatures

Since the Carnot limit rises as the hot temperature rises, the main road to a more efficient heat engine is to run it hotter. Much of the history of power engineering is a story of materials that can survive higher temperatures and pressures.

Steam plants show the pattern. Early twentieth-century stations ran at a few hundred degrees and converted perhaps 10 to 20% of coal's energy into electricity. Subcritical plants of the 1960s to 1990s, like most of India's older fleet, run at about 540 degrees and reach 33 to 36%. Supercritical plants, now standard for new Indian and Chinese units, run steam above the critical pressure of water at roughly 540 to 570 degrees; ultra-supercritical plants push it to 600 degrees or more. Together they reach roughly 38 to 45%. Advanced ultra-supercritical designs aim at 700 degrees, but need nickel alloys costly enough to limit their use.

Gas turbines go further. Their combustion gases enter the first row of blades at around 1,500 degrees Celsius, hotter than the melting point of the blade metal itself. The blades survive only because they are single crystals of nickel superalloy, coated with ceramic and cooled by air blown through tiny internal passages. Smil regards the gas turbine as one of the great prime movers of the modern world, both in jet engines and in power stations.

A gas turbine alone converts about 35 to 42% of its fuel to electricity, but its exhaust leaves at 550 to 650 degrees, still hot enough to raise steam. Combined-cycle plants add a heat recovery boiler and a steam turbine to use that exhaust, stacking two engines in series. The result, about 60% and slightly more in the newest plants, is the highest efficiency of any fuel-burning power station.

• Smil, Energy: A Beginner's Guide (2006) · via Smil Mastery Course, M1 'Entropy and the second law'
• Smil, Energy in Nature and Society: General Energetics of Complex Systems (2008) · via Smil Mastery Course, M1 'Entropy and the second law'

3.3 Heat pumps run the engine backwards

Run a heat engine in reverse and you get a heat pump. Instead of using a flow of heat from hot to cold to produce work, it uses work, usually from an electric motor, to move heat from cold to hot. Refrigerators, air conditioners and heat pumps for buildings and industry are all the same machine pointed in different directions.

Because a heat pump moves heat rather than making it, it can deliver more heat than the electricity it consumes. The ratio of heat delivered to electricity used is the coefficient of performance (COP). A typical air-source heat pump heating a home has a COP of 3 to 4: one kWh of electricity moves two to three kWh of heat from the outside air, and the room receives the sum, three to four kWh. This does not break the first law, because the extra heat comes from outside. It does not break the second law either, because work is being spent to push heat uphill.

Carnot's reasoning sets the ceiling here too. The smaller the temperature lift, the higher the possible COP. Heating water to 35 degrees from air at 5 degrees has a theoretical COP of about 10; real machines manage 3 to 4. Lifting heat to 90 degrees for an industrial process gives a much lower COP, which is why industrial heat pumps work best where waste heat is already warm.

Air conditioners do the same in reverse. A good inverter AC removes three to five units of heat from a room for every unit of electricity, and dumps the heat plus the electricity outside. On the hottest afternoons, when the outdoor unit has to reject heat into air at 45 degrees, its COP falls just when cooling is needed most.

• Smil, Energy: A Beginner's Guide (2006) · via Smil Mastery Course, M1 'Entropy and the second law'
• Jaffe and Taylor, The Physics of Energy (2018) · via Smil Mastery Course, M1 'Entropy and the second law', 'Heat and how it moves'

See it in the table

Check your understanding

What is the Carnot limit for an engine running between 500 K and 300 K?

Why do gas turbine designers push inlet temperatures ever higher?

A heat pump with a COP of 3 uses 2 kWh of electricity. How much heat can it deliver?