Heat Engines & Efficiency — The Carnot Limit
Heat Engines
Every car engine, power plant, and jet turbine is a heat engine — a device that converts thermal energy into useful work. The efficiency of every engine in the universe is bounded by a simple fraction.
The maximum possible efficiency of any heat engine operating between a hot reservoir ($T_H$) and a cold reservoir ($T_C$):
$$\eta_{\text{max}} = 1 - \frac{T_C}{T_H}$$
Temperatures must be in Kelvin. This limit can never be exceeded — it's a law of nature.
This is a ratio subtracted from 1 — arithmetic and fractions. Yet this simple formula determines the fundamental limits of every engine ever built.
Why 100% Efficiency Is Impossible
Since $T_C > 0$ K (absolute zero is unreachable), the fraction $\frac{T_C}{T_H}$ is always positive, so $\eta_{\max} < 1$ always. To get close to 100%, you'd need $T_H \to \infty$ or $T_C \to 0$, both physically impossible.
A coal power plant operates with steam at 550°C (823 K) and exhausts to a cooling tower at 30°C (303 K). What is the maximum theoretical efficiency?
$$\eta_{\text{max}} = 1 - \frac{303}{823} = 1 - 0.368 = 0.632 = 63.2\%$$
Even the theoretical maximum is only 63%. Real plants achieve about 35–45%. No amount of engineering can break the Carnot limit — it's math built into the laws of physics.
Real Engine Efficiencies
| Engine Type | T_H (K) | T_C (K) | Carnot Limit | Actual Efficiency |
|---|---|---|---|---|
| Car engine | ~2,500 | ~300 | 88% | 25–35% |
| Coal plant | ~823 | ~303 | 63% | 35–45% |
| Nuclear plant | ~600 | ~300 | 50% | 33–37% |
| Jet engine | ~1,700 | ~250 | 85% | 35–45% |
The Carnot efficiency formula is a fraction subtracted from 1. This simple piece of algebra sets an absolute ceiling on every engine in the universe. Understanding ratios and proportions means understanding why we can never build a "perfect" engine.