About

The Carnot efficiency defines the theoretical maximum fraction of heat convertible to work between two thermal reservoirs. No real engine exceeds this limit. The bound follows directly from the Second Law of Thermodynamics and depends only on the absolute temperatures of the hot reservoir T_H and cold reservoir T_C, measured in Kelvin. Miscalculating these values leads to oversized equipment, wasted fuel budgets, or failed feasibility studies for power plants, refrigeration systems, and industrial processes.

This calculator computes Carnot efficiency η, the coefficient of performance for both refrigeration (COP_ref) and heat pump (COP_hp) modes, and the maximum work extractable per unit of heat input. It accepts Kelvin, Celsius, or Fahrenheit. Note: the Carnot model assumes perfectly reversible processes with zero friction, no turbulence, and infinite heat-exchange time. Real-world efficiencies typically reach 40 - 65% of the Carnot limit due to irreversibilities.

Formulas

The Carnot efficiency represents the upper bound on thermal-to-mechanical energy conversion between two reservoirs at absolute temperatures T_H (hot) and T_C (cold).

η_Carnot = 1 − T_CT_H

For a Carnot refrigerator, the coefficient of performance measures cooling delivered per unit work input.

COP_ref = T_CT_H − T_C

For a Carnot heat pump, the COP measures heating delivered per unit work input.

COP_hp = T_HT_H − T_C

Maximum work output per unit of heat absorbed from the hot reservoir.

W_max = Q_H × η_Carnot

Temperature conversion to Kelvin (required for all formulas):

T_K = T_C + 273.15 T_K = (T_F − 32)1.8 + 273.15

Where η = Carnot efficiency (dimensionless, 0 - 1), T_H = absolute temperature of the hot reservoir (K), T_C = absolute temperature of the cold reservoir (K), Q_H = heat absorbed from the hot reservoir (J), W_max = maximum extractable work (J). Both T_H and T_C must be in Kelvin. T_H must strictly exceed T_C.

Reference Data

Heat Engine / System	T_H (K)	T_C (K)	Carnot η (%)	Typical Real η (%)
Nuclear PWR Steam Turbine	600	300	50.0	33 - 37
Coal-Fired Power Plant	838	300	64.2	33 - 40
Natural Gas Combined Cycle	1500	300	80.0	55 - 62
Diesel Engine (Truck)	1000	350	65.0	35 - 45
Gasoline Engine (Car)	1100	350	68.2	25 - 35
Gas Turbine (Jet Engine)	1800	300	83.3	35 - 40
Geothermal Binary Cycle	423	310	26.7	10 - 15
Ocean Thermal (OTEC)	298	277	7.0	2 - 3
Solar Parabolic Trough	663	300	54.8	15 - 22
Solar Tower Concentrated	900	300	66.7	20 - 28
Stirling Engine (Solar)	1000	310	69.0	30 - 40
Household Refrigerator	300	255	15.0	2 - 5 (COP)
Industrial Chiller	310	268	13.5	3 - 6 (COP)
Heat Pump (Residential)	318	273	14.2	3 - 5 (COP)
LNG Regasification Turbine	400	113	71.8	25 - 35
Supercritical CO₂ Cycle	823	305	62.9	45 - 50

Frequently Asked Questions

The Carnot efficiency derives from the ratio of absolute temperatures. Celsius and Fahrenheit scales have arbitrary zero points. Using them directly produces mathematically meaningless results. For example, 0°C is not "zero thermal energy" - it is 273.15 K. The Kelvin scale starts at absolute zero, where molecular motion ceases, making ratios physically meaningful.

When T_H = T_C, the efficiency becomes 0%. No temperature gradient exists, so no work can be extracted. This is consistent with the Second Law: heat flows spontaneously only from hot to cold. The COP values become undefined (division by zero) because no finite work can drive heat transfer between equal-temperature reservoirs.

Real engines achieve roughly 40 - 65% of the Carnot limit. Irreversibilities - friction in pistons, turbulence in fluid flow, finite heat-transfer rates, and combustion non-idealities - all degrade performance. The ratio of actual efficiency to Carnot efficiency is called the "second-law efficiency" or "exergetic efficiency." A combined-cycle gas turbine might reach 60% of its Carnot bound, while an automobile gasoline engine typically reaches only 35 - 50%.

Only if T_C = 0 K (absolute zero). The Third Law of Thermodynamics states that absolute zero is unattainable in a finite number of steps. Therefore, 100% Carnot efficiency is a theoretical impossibility. Even cryogenic systems operating near 4 K still have a non-zero T_C.

For a heat engine, η measures work output per unit heat input. For a refrigerator, COP_ref = (1 − η) ÷ η. For a heat pump, COP_hp = 1 ÷ η. Notably, COP_hp = COP_ref + 1 always holds. A small temperature difference yields very high COP values, which is why heat pumps are efficient when indoor-outdoor temperature differences are moderate.

Mathematically, yes - raising T_H while holding T_C constant increases η. Practically, material limits constrain this. Turbine blade alloys fail above roughly 1400 - 1600 K. Ceramic coatings push limits to approximately 1800 K. Beyond these thresholds, maintenance costs and failure rates negate the thermodynamic gains. Lowering T_C (e.g., using cold river water for condensers) is often more cost-effective.