Formula & Calculator
Carnot Efficiency
Calculates the maximum theoretically possible thermal efficiency of a heat engine operating between a hot and cold reservoir.
Interpretation
Carnot efficiency is the maximum possible efficiency of a heat engine operating between two reservoirs at temperatures T_hot and T_cold. It is η = 1 − (T_c / T_h). This sets the upper limit for real engine efficiencies.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| eta | Carnot (maximum) thermal efficiency (dimensionless) | |
| Tc | Absolute temperature of the cold reservoir | K |
| Th | Absolute temperature of the hot reservoir | K |
What it means
The Carnot efficiency is the theoretical maximum efficiency of any heat engine operating between two thermal reservoirs at temperatures T_hot (heat source) and T_cold (heat sink), both measured in kelvin. It is given by η_Carnot = 1 − (T_c / T_h). This result comes from the Carnot cycle, which consists of two isothermal and two adiabatic processes. The Carnot engine is reversible, and all reversible engines operating between the same reservoirs have the same efficiency. No real engine can exceed this efficiency due to the second law of thermodynamics. The Carnot efficiency increases as T_h increases or T_c decreases. In practice, engine efficiencies are lower due to irreversibilities (friction, heat losses, etc.). The concept is used to evaluate the quality of energy conversion and to guide design improvements. It also applies to refrigerators and heat pumps, where the coefficient of performance is bounded by the Carnot COP.
Worked example
Carnot Efficiency – Two Examples
Real‑World| Parameter | Value |
|---|---|
| Th | 600 K |
| Tc | 300 K |
| Parameter | Value |
|---|---|
| Th | 900 K |
| Tc | 300 K |
Common mistakes
- Temperature ratio: Use absolute temperatures (Kelvin) – not °C or °F.
- Hot and cold reservoirs: Th is the high‑temperature reservoir, Tc the low‑temperature one.
- Idealised efficiency: This is the maximum possible efficiency – real engines are always lower.
- Efficiency decimal vs. percentage: The formula gives a fraction; multiply by 100 for %.
- Reversible cycle: Carnot efficiency applies only to reversible (Carnot) cycles.
Applications
Carnot efficiency represents the maximum possible efficiency of a heat engine operating between two thermal reservoirs. Although idealised, it provides a theoretical benchmark against which real engines are compared. The formula is used to assess the potential of thermodynamic cycles and to identify areas for improvement in power generation and refrigeration. In practical engineering, it guides the selection of operating temperatures and working fluids. For example, increasing the hot‑side temperature or decreasing the cold‑side temperature improves efficiency, but material and economic constraints limit the extremes. The Carnot efficiency also educates engineers on the fundamental limits of energy conversion, reinforcing the importance of thermodynamics in sustainable energy systems.
- Performance benchmarking of real heat engines
- Optimisation of power plant operating conditions
- Design of high‑efficiency refrigeration cycles
- Thermodynamic education and concept illustration
- Feasibility studies for energy‑conversion systems
Frequently Asked Questions
The Carnot efficiency is the maximum possible efficiency of a heat engine operating between two thermal reservoirs at temperatures T_hot and T_cold. It is given by η_Carnot = 1 − (T_c / T_h). It represents the theoretical upper limit; no real engine can exceed it.
T_h is the absolute temperature of the hot reservoir (heat source) and T_c is the absolute temperature of the cold reservoir (heat sink). You must use an absolute temperature scale – Kelvin (K) or Rankine (°R). Using Celsius or Fahrenheit will give incorrect and possibly negative efficiencies.
- Using Celsius or Fahrenheit – the most frequent error; always convert to Kelvin or Rankine.
- Confusing the hot and cold reservoirs – T_c must be the lower temperature.
- Thinking it is an achievable efficiency – Carnot is an ideal limit; real engines have lower efficiencies due to irreversibilities.
- Applying it to refrigerators/heat pumps directly – for those, the coefficient of performance (COP) is used, though the Carnot COP also uses absolute temperatures.
Convert to Kelvin: T_h = 600 + 273 = 873 K, T_c = 40 + 273 = 313 K. η = 1 − (313/873) = 1 − 0.3585 = 0.6415 or 64.15%. Real power plants achieve 35‑45% due to irreversibilities.
For 100% efficiency, the ratio T_c/T_h would have to be zero. That requires either T_c = 0 K (absolute zero) or T_h = ∞, both of which are impossible in practice. Therefore, some heat must always be rejected to the cold reservoir.
For a refrigerator (cooling effect): COP_R = T_c / (T_h − T_c).
For a heat pump (heating effect): COP_HP = T_h / (T_h − T_c) = COP_R + 1. These are also maximum theoretical values.
As the difference (T_h − T_c) increases, the efficiency increases. This is why power plants operate with very high steam temperatures (to get higher efficiency) and reject heat at the lowest possible temperature (e.g., using cooling towers).
The Carnot engine establishes the upper bound on thermal efficiency for any engine operating between two fixed temperatures. It proves that efficiency depends only on the temperatures, not on the working fluid or engine design. This leads to the definition of thermodynamic temperature scale and the Second Law.
Yes, the Carnot cycle consists of two isothermal and two adiabatic processes, all of which are reversible. Reversibility means no entropy is generated; the engine operates without friction, heat loss, or other dissipative effects. This is why it achieves the maximum efficiency.
The Carnot efficiency is used as a benchmark to compare real engines. The actual efficiency is always lower, so the difference indicates the 'quality' of the design and the extent of irreversibilities. Engineers strive to approach the Carnot limit by reducing friction, improving heat transfer, and using advanced cycles (e.g., combined cycles).