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

Source page: https://www.materialinsights.org/learn/energy/laws-of-energy/p3/

[Understanding Energy](https://www.materialinsights.org/learn/energy/) / [Module 2: The laws of energy: conservation, entropy and efficiency](https://www.materialinsights.org/learn/energy/laws-of-energy/) Part 3 of 5

## The limits to efficiency: Carnot and beyond

If every heat engine must reject some heat, how much must it reject? In 1824 the French engineer Sadi Carnot answered with a formula that depends only on two temperatures. It explains why power stations rarely beat 45%, why combined-cycle gas plants reach 60%, why geothermal and ocean thermal plants are so inefficient, and why heat pumps can deliver more heat than the electricity they use.

### 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%.

Data Carnot limits and real efficiencies

| Engine | Hot side | Cold side | Carnot limit | Real efficiency |
|---|---|---|---|---|
| Ocean thermal plant | 27 C (300 K) | 5 C (278 K) | about 7% | about 3% |
| Geothermal power plant | 180 C (453 K) | 30 C (303 K) | about 33% | 10-15% |
| Nuclear fission reactor (PWR steam) | 285 C (558 K) | 30 C (303 K) | about 46% | about 33% |
| Supercritical coal plant | 600 C (873 K) | 30 C (303 K) | about 65% | 38-45% |
| Gas turbine and combined cycle | 1,500 C (1,773 K) | 30 C (303 K) | about 83% | about 60-63% |

Carnot limit = 1 - T_cold / T_hot with temperatures in kelvin (degrees Celsius + 273).
 Source: Smil, Energy: A Beginner's Guide (2006), via Smil Mastery Course, M1 'Entropy and the second law'  Case

#### Why Indian plants lose a few points in summer

The Carnot formula has two temperatures, and the cold one matters as well. When cooling water or air warms up in an Indian summer, the cold side of the engine rises and the attainable efficiency falls. Thermal plants and gas turbines typically lose some output and efficiency on hot days, just when air-conditioning demand peaks. Plants with sea-water cooling or dry cooling towers behave differently, but none escape the rule that a warmer sink means less work from the same heat.

- 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.

Process flowchart A combined-cycle gas plant: two engines in series Inputs **Natural gas**100 units of fuel energy Process steps 1 **Gas turbine**

burns gas at about 1,500 C; about 38 units of electricity
 2 **Heat recovery steam generator**

exhaust at about 600 C raises steam
 3 **Steam turbine**

about 22 more units of electricity from the recovered heat
 Outputs **Electricity**about 60 units **Low-grade heat to condenser and stack**about 40 units Source: Smil, Energy: A Beginner's Guide (2006), via Smil Mastery Course, M1 'Entropy and the second law'  In the data

#### What 1 percentage point is worth

Efficiency gains in big plants are worth a great deal. Raising a 1 GW coal unit running at a 70% capacity factor from 36% to 37% efficiency cuts its coal use by about 2.7%, roughly 100,000 tonnes of Indian coal a year, and its CO2 by about 150,000 tonnes. Across a fleet of over 200 GW, each percentage point is worth millions of tonnes of coal. This is why plant renovation and better operation are cheap climate measures even before any switch of fuel.

- 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.

Process flowchart A heat pump heating water Inputs **Electricity**1 kWh **Heat from outside air**about 2.5 kWh at 5 C Process steps 1 **Evaporator**

cold refrigerant absorbs heat from outside air
 2 **Compressor**

electric motor squeezes the vapour, raising its temperature
 3 **Condenser**

hot refrigerant gives up its heat to water at about 35-55 C
 Outputs **Useful heat**about 3.5 kWh (COP 3.5) Source: Smil, Energy: A Beginner's Guide (2006), via Smil Mastery Course, M1 'Entropy and the second law'  Common misreading

#### "300% efficient"

Heat pump marketing sometimes claims efficiencies of 300% or more, which sounds like it breaks the first law. It does not, because the machine is not converting electricity into heat; it is using electricity to move heat that already exists. The right measure is the coefficient of performance, and the right comparison is with the best alternative. A gas boiler at 95% delivers 0.95 units of heat per unit of gas. A heat pump with COP 3.5 delivers 3.5 units per unit of electricity, even if that electricity came from a plant only 40% efficient.

- 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
[H₂O Water](https://www.materialinsights.org/commodities/water/)[Ni Nickel](https://www.materialinsights.org/commodities/nickel/)

### Check your understanding

What is the Carnot limit for an engine running between 500 K and 300 K?
 1 - 300/500 = 0.4, so 40%.

Why do gas turbine designers push inlet temperatures ever higher?
 The Carnot limit rises with the hot-side temperature, so hotter turbines can convert more heat into work.

A heat pump with a COP of 3 uses 2 kWh of electricity. How much heat can it deliver?
 COP 3 means three units of heat delivered per unit of electricity, so 6 kWh.

[← Part 2: The second law: entropy and the quality of energy](https://www.materialinsights.org/learn/energy/laws-of-energy/p2/)[Mark read · Part 4: Heat and how it moves →](https://www.materialinsights.org/learn/energy/laws-of-energy/p4/)

## Other sections

- [Module overview](https://www.materialinsights.org/learn/energy/laws-of-energy/index.md)
- [The first law: energy is conserved](https://www.materialinsights.org/learn/energy/laws-of-energy/p1/index.md)
- [The second law: entropy and the quality of energy](https://www.materialinsights.org/learn/energy/laws-of-energy/p2/index.md)
- [Heat and how it moves](https://www.materialinsights.org/learn/energy/laws-of-energy/p4/index.md)
- [Exergy and the three stages of energy accounting](https://www.materialinsights.org/learn/energy/laws-of-energy/p5/index.md)
- [Key terms, quiz and sources](https://www.materialinsights.org/learn/energy/laws-of-energy/review/index.md)