Carnot Efficiency Formula
The Carnot efficiency formula η = 1 - Tc/Th gives the maximum possible efficiency of a heat engine.
Learn with examples.
The Formula
The Carnot efficiency sets the absolute upper limit on how efficient any heat engine can be. No real engine can exceed this theoretical maximum, which depends only on the temperatures of the hot and cold reservoirs.
This principle was established by French engineer Sadi Carnot in 1824. It is one of the cornerstones of the second law of thermodynamics. The formula shows that to maximize efficiency, you need either a very hot source or a very cold sink.
All temperatures must be expressed in kelvin (absolute temperature) for the formula to work correctly. An efficiency of 1 (100%) is only possible if the cold reservoir is at absolute zero, which is physically impossible. Real engines typically achieve 30-60% of their Carnot limit due to friction, heat losses, and irreversibilities.
Variables
| Symbol | Meaning |
|---|---|
| η | Carnot efficiency (dimensionless ratio, between 0 and 1) |
| Tc | Temperature of the cold reservoir (kelvin, K) |
| Th | Temperature of the hot reservoir (kelvin, K) |
Example 1
A steam engine operates between a boiler at 500°C and a condenser at 100°C. What is its maximum possible efficiency?
Convert to kelvin: Th = 500 + 273 = 773 K, Tc = 100 + 273 = 373 K
Apply the formula: η = 1 − Tc/Th = 1 − 373/773
η = 1 − 0.4826 = 0.5174
η ≈ 51.7% maximum efficiency
Example 2
A refrigerator maintains its interior at 5°C while exhausting heat to a room at 30°C. What is the maximum coefficient of performance (COP)?
Convert to kelvin: Tc = 5 + 273 = 278 K, Th = 30 + 273 = 303 K
The COP for a Carnot refrigerator is: COP = Tc / (Th − Tc)
COP = 278 / (303 − 278) = 278 / 25
COP = 11.12 (every 1 J of work moves 11.12 J of heat)
When to Use It
Use the Carnot efficiency formula to determine the theoretical maximum efficiency of any heat engine or refrigeration cycle.
- Evaluating steam turbines, internal combustion engines, and gas turbines
- Comparing real engine performance to the theoretical ideal
- Designing power plants to maximize thermal efficiency
- Understanding why low-temperature waste heat is hard to convert to work