The ultimate limit on heat engines
In 1824 the French engineer Sadi Carnot asked how much work a steam engine could possibly extract from heat. His answer, refined by Kelvin and Clausius, became a cornerstone of thermodynamics: no engine working between a hot reservoir at Th and a cold reservoir at Tc can beat
η = 1 − Tc ÷ Th
with both temperatures in kelvins. The rest of the heat must be rejected to the cold side, whatever the design.
Worked example
A steam cycle with steam at 1,000 °F (811 K) and cooling water at 80 °F (300 K), taking 100 kWh of heat from the boiler:
| Result | Value |
|---|---|
| Carnot efficiency | 63.0% |
| Maximum work | 63.0 kWh |
| Heat rejected, at best | 37.0 kWh |
| Maximum heat pump COP | 1.59 |
| Maximum refrigerator COP | 0.59 |
A real steam plant at these temperatures reaches about 40%: friction, pressure losses and the finite speed of heat transfer take a large share of the theoretical limit.
Heat pumps and refrigerators
Run backward, a heat engine becomes a heat pump: it uses work to move heat from cold to hot. The Carnot limit then sets the best possible coefficient of performance (COP):
- Heating: COP = Th ÷ (Th − Tc)
- Cooling: COP = Tc ÷ (Th − Tc)
The smaller the temperature lift, the higher the COP. A heat pump moving heat from 32 °F outdoors into a 95 °F floor-heating loop has a theoretical COP of about 8.8. Real heat pumps reach 3 to 4 in these conditions, and less when it is very cold outside or the water must be hotter. This is why heat pumps pair best with low-temperature emitters like radiant floors.
Real-world efficiencies
| System | Typical efficiency | Carnot limit |
|---|---|---|
| Car gasoline engine | 25–35% | above 80% |
| Coal steam power plant | 33–40% | about 60% |
| Combined-cycle gas plant | about 60% | about 80% |
| Geothermal plant (300 °F source) | 10–15% | about 30% |
Key points
- Always convert to kelvins: K = (°F − 32) ÷ 1.8 + 273.15, or °C + 273.15.
- Efficiency rises with a hotter source or a colder sink, which is why power plants push steam temperatures as high as materials allow.
- The limit applies to engines converting heat to work. Electric motors, fuel cells and batteries are not heat engines and are not bound by it.