Carnot Efficiency⚠ unverified
Physics / Thermodynamics · Maximum thermal efficiency of a heat engine between two reservoirs
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| T_hot | Th | K | 500.0 | Hot reservoir temperature |
| T_cold | Tc | K | 300.0 | Cold reservoir temperature |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| efficiency | η | — | Carnot efficiency |
The science & history
Understanding the Parameters
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$T_h$, $T_c$ — must be absolute (kelvin). Using degC without conversion is a common error. Require $T_h > T_c > 0$.
-
$\eta$ — fraction of heat input convertible to work at best. Example: $T_h=500\,\text{K}$, $T_c=300\,\text{K}$ → $\eta = 0.4$ (40%). Raising $T_h$ or lowering $T_c$ improves the limit.
Derivation (Approaching a Proof)
For a reversible cycle exchanging $Q_h$ at $T_h$ and rejecting $Q_c$ at $T_c$, Clausius equality gives $\oint dQ_{\mathrm{rev}}/T = 0$, so $Q_h/T_h = Q_c/T_c$. First-law net work $W = Q_h - Q_c$ (magnitudes for the engine), hence
$$\eta = \frac{W}{Q_h} = 1 - \frac{Q_c}{Q_h} = 1 - \frac{T_c}{T_h}.$$
Any irreversible engine between the same reservoirs has lower $\eta$ (Kelvin–Planck / second law).
History
Sadi Carnot (1824) reasoned about ideal heat engines; absolute temperature and the modern $\eta = 1 - T_c/T_h$ form followed Kelvin and Clausius in the mid-19th century.
Related Concepts: Thermal Efficiency, Otto Efficiency, Brayton Efficiency, Rankine Efficiency, Second Law Entropy, Exergy
Notes: Registry calculator carnot-efficiency (unverified). Ideal two-reservoir limit only.