Heat Engines
A heat engine converts heat into work by operating between a hot source and a cold sink. Learn thermal efficiency, the Carnot limit, heat rate, and how real engines — car engines, gas turbines, power plants — compare.
Every car engine, jet engine and thermal power station is a heat engine: it takes heat from a hot source (combustion, nuclear fission, concentrated sunlight), turns part of it into work, and rejects the rest to a cold sink (the atmosphere, a river).
The second law puts a hard ceiling — the Carnot efficiency — on how much heat can become work. Raising the hot temperature is the most powerful way to get closer to it.
An analogy
A water wheel: water falling from a high reservoir to a low one turns the wheel. The bigger the height difference (temperature difference), the more work you can extract.
Source, engine, sink
Heat in, work out, heat rejected. Typical efficiencies: car petrol engine 25–35 %, diesel 35–45 %, large marine diesel ≈ 50 %, coal plant 33–45 %, combined-cycle gas plant ≈ 60 %.
The Carnot limit
A reversible engine between T_H and T_L reaches 1 − T_L/T_H. With T_H = 800 K and T_L = 300 K, no engine can exceed 62.5 %. Real engines achieve perhaps 60–70 % of Carnot due to friction, heat transfer across finite ΔT and non-ideal processes.
Raising efficiency
Increase T_H (better materials, blade cooling, supercritical steam), lower T_L (colder cooling water, vacuum condensers), reduce irreversibilities, and recover waste heat (combined cycles, CHP).
Finite-time thermodynamics
Carnot efficiency comes with zero power output (infinitely slow). The Curzon–Ahlborn efficiency at maximum power, 1 − √(T_L/T_H), matches real power plants surprisingly well — a reminder that engineers optimise power and cost, not efficiency alone.
Formulas
- Thermal efficiency:
η = W_net / Q_H = 1 − Q_L/Q_H - Carnot efficiency:
η_Carnot = 1 − T_L / T_H - Heat rate:
HR = 3600 / η (kJ/kWh) - Fuel flow:
ṁ_fuel = Ẇ / (η · LHV)
Worked example: Carnot limit
Given: T_H = 800 K, T_L = 300 K
Find: Maximum efficiency.
- η = 1 − 300/800 = 0.625
Answer: 62.5 %
Worked example: Engine energy balance
Given: Fuel power 100 kW; Shaft output 40 kW
Find: Efficiency and heat rejected.
- η = 40/100 = 40 %
- Q_L = 100 − 40 = 60 kW (exhaust + coolant)
Answer: 40 %, 60 kW rejected
Common mistakes
- Wrong: Computing Carnot efficiency with °C. — Right: Use kelvin: 1 − 300/800, not 1 − 27/527.
- Wrong: Expecting real engines to reach Carnot. — Right: Irreversibilities and finite-rate processes keep them well below it.
Applications
- Combined-cycle power: Gas turbine exhaust (~600 °C) raises steam for a second cycle, reaching ~60 % efficiency.
- Combined heat & power: Using rejected heat for district heating raises total fuel utilisation to 80–90 %.
- Waste-heat recovery: Organic Rankine cycles and turbo-compounding extract work from exhaust heat.
Key takeaways
- η = W/Q_H = 1 − Q_L/Q_H.
- Carnot: 1 − T_L/T_H (kelvin) is the upper bound.
- Hotter sources and colder sinks raise efficiency.
- Combined cycles and CHP use rejected heat.
Going further
- Stirling engines: External-combustion engines with regenerators that can theoretically reach Carnot efficiency; used in solar dishes and submarines.
- Materials limit: Turbine inlet temperatures (~1600 °C in modern gas turbines) exceed blade melting points — enabled by single-crystal alloys, thermal barrier coatings and film cooling.
- Thermoelectrics: Solid-state heat engines (Seebeck effect) with no moving parts, used in RTGs for spacecraft, but with low efficiency (~5–8 %).