Carnot Efficiency

The Carnot efficiency is the thermal efficiency of a reversible heat engine operating between a hot reservoir at absolute temperature T_H and a cold reservoir at absolute temperature T_C (both in kelvin):

This result was derived by Sadi Carnot in 1824, before the formal development of thermodynamics, through reasoning about ideal reversible heat engines. It was later established rigorously within thermodynamics as a direct consequence of the second law and the definition of entropy.

Derivation from the second law

For any heat engine operating in a cycle, the first law requires W_\text{net} = Q_\text{in} - Q_\text{out}. The second law, expressed in terms of entropy, requires that the total entropy of the universe does not decrease. For a reversible engine exchanging heat only with two reservoirs at T_H and T_C, entropy balance gives:

which means Q_\text{out}/Q_\text{in} = T_C/T_H. Substituting into the thermal efficiency expression:

For any irreversible engine, entropy is generated internally, which requires Q_\text{out}/Q_\text{in} > T_C/T_H and therefore \eta < \eta_\text{Carnot}. This proves that the Carnot engine is the most efficient possible heat engine between two given reservoirs.

The Carnot cycle

The Carnot cycle is the idealised reversible cycle that achieves \eta_\text{Carnot}. It consists of four reversible processes: isothermal heat absorption from the hot reservoir at T_H, adiabatic (isentropic) expansion, isothermal heat rejection to the cold reservoir at T_C, and adiabatic (isentropic) compression back to the initial state. All heat transfer occurs reversibly — at infinitesimally small temperature differences — and there is no friction or other irreversibility. No real engine can implement the Carnot cycle because isothermal heat transfer requires infinite time (heat transfer rate goes to zero as the temperature difference goes to zero).

Implications for engineering

The Carnot efficiency shows that raising the hot reservoir temperature T_H or lowering the cold reservoir temperature T_C increases the maximum achievable efficiency. This drives power plant design toward ever-higher steam temperatures and pressures (supercritical and ultra-supercritical Rankine cycles) and higher turbine inlet temperatures in gas turbines (limited by blade material capability). It also quantifies the irreducible thermodynamic cost of heat rejection: for a given heat input, the minimum rejected heat is (1 - \eta_\text{Carnot}) Q_\text{in}; for a given work output, it is W(1/\eta_\text{Carnot}-1).

The gap between actual thermal efficiency and the Carnot efficiency — called the second-law efficiency or exergetic efficiency — measures how much of the thermodynamically available work is actually captured. A modern coal-fired steam plant may have a thermal efficiency of 45% operating between flame temperatures near 1800 K and an ambient sink at 300 K, giving a Carnot limit of 83% and a second-law efficiency of about 54%. The remaining potential is lost to irreversibilities in combustion, heat transfer, and turbomachinery.

Carnot efficiency for refrigeration

The Carnot efficiency concept extends to refrigeration and heat pump cycles. For a reversible refrigerator operating between T_C (cold space) and T_H (hot sink), the Carnot coefficient of performance is:

For a reversible heat pump:

These are the upper bounds on the performance of any real refrigerator or heat pump, and real devices fall below them for the same reasons that real engines fall below the Carnot thermal efficiency.