Heat Engine Efficiency Formula
Calculate the thermal efficiency of a heat engine: η = W/Q_h = 1 - Q_c/Q_h.
Includes Carnot limit and real-world examples.
The Formula
A heat engine converts heat into useful mechanical work. It absorbs heat Q_h from a hot reservoir, converts some of it into work W, and dumps the remainder Q_c into a cold reservoir. The thermal efficiency η (eta) measures what fraction of the absorbed heat becomes useful work.
Since energy is conserved: W = Q_h − Q_c, so the two forms of the efficiency formula are equivalent. No real heat engine can be 100% efficient — some heat must always be rejected to the cold reservoir. This is a consequence of the second law of thermodynamics.
The Carnot efficiency sets the absolute upper limit: η_Carnot = 1 − T_c/T_h (temperatures in Kelvin). Real engines (Otto, Diesel, Rankine, Brayton) are always less efficient than Carnot operating between the same temperatures. A modern coal power plant achieves about 35–40% efficiency. Combined-cycle gas turbines reach 55–60%. Car engines typically achieve 20–35%.
Improving efficiency means either raising the hot temperature or lowering the cold temperature. This is why steam turbines operate at the highest practical pressures and temperatures, and why waste heat recovery systems try to lower the exhaust temperature.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| η | Thermal efficiency (0 to 1) | Dimensionless |
| W | Net work output | Joules (J) |
| Q_h | Heat absorbed from hot reservoir | Joules (J) |
| Q_c | Heat rejected to cold reservoir | Joules (J) |
Example 1
A steam turbine absorbs 10,000 J of heat and produces 3,500 J of work.
η = W / Q_h = 3,500 / 10,000
η = 0.35 = 35% efficiency
Example 2
An engine absorbs 5,000 J and rejects 3,200 J. What is its efficiency?
η = 1 − Q_c/Q_h = 1 − 3,200/5,000 = 1 − 0.64
η = 0.36 = 36% efficiency
When to Use It
- Evaluating power plant performance
- Comparing heat engine designs
- Calculating fuel savings from efficiency improvements
- Understanding why 100% efficiency is thermodynamically impossible