Formula & Calculator
Thermal Efficiency of a Heat Engine
Calculates the fraction of heat input to an engine that is converted into net useful work output.
Interpretation
Thermal efficiency of a heat engine is the net work output divided by the heat input from the hot reservoir. η = W_net / Q_in. It indicates how much of the heat is converted to useful work. Real engines have efficiencies below the Carnot limit.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| eta | Thermal efficiency (dimensionless) | |
| W_net | Net work output of the engine | kJ |
| Q_in | Total heat input to the engine | kJ |
What it means
The thermal efficiency of a heat engine is a measure of its performance, defined as the ratio of the net work output to the heat supplied from the high‑temperature reservoir: η = W_net / Q_in. By the first law, W_net = Q_in − Q_out, so η = 1 − (Q_out / Q_in). This shows that to maximise efficiency, heat rejection should be minimised. The second law imposes the maximum possible efficiency (Carnot). Real engines have lower efficiencies due to irreversibilities. Thermal efficiency is a key parameter in internal combustion engines, steam turbines, and gas turbines. It affects fuel economy and emissions. Improving thermal efficiency is a major goal in engine research, through higher compression ratios, better combustion, and waste heat recovery. The efficiency of power plants is often quoted as a percentage; for example, modern combined‑cycle plants achieve around 60%. Understanding thermal efficiency is essential for energy engineering and environmental policy.
Worked example
Thermal Efficiency – Two Examples
Real‑World| Parameter | Value |
|---|---|
| W_net | 100 kJ |
| Q_in | 300 kJ |
| Parameter | Value |
|---|---|
| W_net | 250 kJ |
| Q_in | 1000 kJ |
Common mistakes
- Net work Wnet: It is Qin − Qout, not just the work output from a single process.
- Heat input Qin: The heat added to the working fluid from the hot reservoir.
- Units: Both W and Q in Joules (or kJ).
- Efficiency < 1: Second law ensures η < 1 for any real engine.
- Carnot comparison: The thermal efficiency is always less than or equal to the Carnot efficiency.
Applications
Thermal efficiency of a heat engine is the ratio of net work output to heat input from the hot reservoir. It indicates how effectively a heat engine converts thermal energy into mechanical work. This parameter is used to compare different engine types (e.g., Otto, Diesel, gas turbines) and to assess the performance of power plants. Improving thermal efficiency leads to lower fuel consumption and reduced emissions. Engineers apply this formula in the design of combustion chambers, heat exchangers, and cycle configurations. It also guides the selection of operating pressures and temperatures. By optimising thermal efficiency, engineers can produce more power with less environmental impact.
- Design and optimisation of internal combustion engines
- Gas turbine and jet engine performance assessment
- Steam power plant cycle analysis
- Comparison of thermodynamic cycles
- Energy conversion system feasibility studies
Frequently Asked Questions
The thermal efficiency of a heat engine is the ratio of net work output to the heat input from the hot source: η = W_net / Q_in. It is the fraction of heat energy that is converted to useful work. It is always < 1 (or < 100%).
- W_net = net work delivered by the engine (J, kJ) – this is the difference between work done by the engine and work consumed (e.g., by pumps).
- Q_in = heat supplied to the working fluid (J, kJ) – usually from fuel combustion or a solar source.
- Using the total heat input (including heat rejected) – only the heat added to the cycle counts as Q_in, not the total fuel energy that may be lost as exhaust.
- Using the fuel's heating value instead of the heat actually absorbed – the fuel may have a heating value (LHV or HHV); you need the heat transferred to the working fluid.
- Confusing W_net with power – W_net is energy per cycle; if using power, ensure both sides are rates (e.g., kW).
- Applying the formula to a refrigerator – for a refrigerator, the coefficient of performance (COP) is used, not thermal efficiency.
The Carnot efficiency is the maximum possible efficiency (η_Carnot = 1 − T_c/T_h). The actual thermal efficiency is always less than or equal to the Carnot efficiency. The ratio η / η_Carnot is called the second‑law efficiency or effectiveness.
Thermal efficiency is based on the thermodynamic cycle (heat input to work). Brake thermal efficiency is the ratio of brake power (output shaft power) to the fuel energy input rate. Brake efficiency is lower than indicated thermal efficiency due to mechanical losses (friction, pumping).
You need the brake power (from dynamometer) and the fuel flow rate. The heat input rate is ṁ_fuel × LHV (lower heating value). Then η = P_brake / (ṁ_fuel × LHV). For a typical engine, this is around 25‑35%.
For the Otto cycle, η = 1 − 1/r^(γ−1). Higher compression ratio gives higher thermal efficiency. This is why modern engines use higher compression ratios (e.g., 10:1 to 12:1) to improve fuel economy.
Heat rejected (Q_out) is the heat that is not converted to work. By the First Law, W_net = Q_in − Q_out. Therefore, to improve efficiency, you must reduce Q_out (i.e., reject less heat). This is why heat engines operate with a high T_h and low T_c.
Thermal efficiency is specific to heat engines (heat → work). Energy efficiency is a broader term for any device converting energy from one form to another (e.g., electric motor: electrical → mechanical). For a heat pump, the coefficient of performance (COP) is used instead of efficiency.
Large power plants (combined cycle) can achieve thermal efficiencies of 50‑60%. Automobile engines are typically 25‑35% due to size constraints, part‑load operation, and emission control requirements. Efficiency is critical for power plants because fuel cost is the major operating expense.