Thermal Efficiency Calculator
Enter any two of heat input, work output, or thermal efficiency and the calculator solves for the third. Switch to Carnot mode to find the theoretical maximum efficiency from reservoir temperatures. You also get the heat rejected, the Carnot limit, and the second-law efficiency showing how close a real engine comes to the ideal.
What is thermal efficiency?
Thermal efficiency is the ratio of useful work output to the total heat energy supplied to a heat engine or thermodynamic cycle. It answers a simple question: of all the energy we put in, how much actually becomes useful work? Because the second law of thermodynamics forbids any engine from converting 100% of heat into work, some energy is always rejected to a cold sink - atmosphere, cooling water, a condenser. Thermal efficiency is expressed as a percentage and is the central performance metric for power plants, car engines, steam turbines, and any device that converts thermal energy to mechanical or electrical output.
The thermal efficiency formula
The fundamental equation is eta = Wout / Qin, where Wout is the net work produced per cycle and Qin is the heat supplied from the hot source. An equivalent form is eta = 1 - Qout/Qin, where Qout is the heat rejected to the cold sink. Energy balance requires Qin = Wout + Qout, so these two forms are mathematically identical. For a Carnot (reversible) engine operating between a hot reservoir at absolute temperature Th and a cold sink at Tc, the maximum possible efficiency is eta_Carnot = 1 - Tc/Th. No real engine can exceed this limit for a given pair of temperatures, regardless of the working fluid or design. Increasing Th or decreasing Tc are the two levers for raising the ceiling.
Second-law efficiency and real vs. ideal engines
Second-law efficiency compares a real engine to the Carnot ideal: eta_II = eta_actual / eta_Carnot. A value of 100% would mean the engine is fully reversible, which is impossible in practice due to friction, turbulence, heat leaks, and finite temperature differences. Modern combined-cycle gas turbines reach second-law efficiencies of around 70-80%, meaning they capture roughly three-quarters of the thermodynamic potential available between their reservoir temperatures. Simple reciprocating engines typically achieve 40-55% of the Carnot limit. The gap between first-law efficiency (absolute) and second-law efficiency (relative to the ideal) helps engineers identify where the biggest improvement opportunities lie.
How to use this calculator
Select a calculation mode from the dropdown. In "Solve for thermal efficiency" mode, enter the heat input and net work output in any energy unit (J, kJ, BTU, or cal). In the two reverse-solve modes you can back-calculate work output from a known efficiency and heat input, or calculate the required heat input from a known efficiency and work output. The Carnot mode calculates the theoretical maximum efficiency from reservoir temperatures in any temperature unit. Toggle "Compare to Carnot limit" in efficiency mode to see how your engine compares to the ideal for a given pair of reservoir temperatures - this gives the second-law efficiency and helps identify improvement potential.
Typical thermal efficiencies for common heat engines
| Engine / Cycle type | Typical efficiency range | Notes |
|---|---|---|
| Simple steam engine (historical) | 5 - 15% | Early Watt-era engines |
| Modern steam turbine (Rankine) | 35 - 45% | Coal/nuclear power plants |
| Combined-cycle gas turbine | 55 - 62% | Best available commercial technology |
| Gas turbine (Brayton cycle) | 30 - 40% | Jet engines and peaker plants |
| Gasoline engine (Otto cycle) | 25 - 35% | Typical passenger cars |
| Diesel engine | 35 - 45% | Trucks, ships, generators |
| Stirling engine | 30 - 40% | External combustion, quiet |
| Carnot engine (theoretical) | Up to ~80% | Impossible in practice; sets the ceiling |
Real-world engine types and their approximate thermal efficiency ranges. Actual values vary by design, fuel, and operating conditions.
Frequently asked questions
Can thermal efficiency ever be 100%?
No. The second law of thermodynamics states that any heat engine must reject some heat to a cold reservoir. Even the theoretically perfect Carnot engine reaches 100% only if the cold sink is at absolute zero (0 K), which is physically unattainable. Real engines are further limited by irreversibilities such as friction and heat losses, so practical efficiencies top out around 60-62% in the best commercial combined-cycle plants.
What is the difference between thermal efficiency and Carnot efficiency?
Thermal efficiency is the actual fraction of heat input converted to work in a real or idealised engine. Carnot efficiency is the theoretical maximum achievable by any engine operating between the same hot and cold reservoir temperatures - it depends only on those temperatures, not on design details. Real engines always fall below the Carnot limit. Their ratio is the second-law efficiency.
Does it matter which energy units I use?
No, as long as you use the same unit for both heat input and work output. The ratio Wout/Qin is dimensionless, so joules, kilojoules, BTU, and calories all give the same efficiency percentage when applied consistently. This calculator converts everything to joules internally and then converts results back to your chosen unit.
What does second-law efficiency tell me?
Second-law efficiency (also called exergetic efficiency) compares a real engine to the Carnot ideal for the same reservoir temperatures. A value of 60% means the engine captures 60% of the thermodynamic potential available between those reservoirs. It reveals how much room exists for engineering improvement without changing the reservoir temperatures, and it allows fair comparisons between engines operating under different conditions.
How do I increase the thermal efficiency of a heat engine?
There are two fundamental levers: raise the temperature of the hot source (Th) or lower the temperature of the cold sink (Tc). Both increase the Carnot ceiling. In practice, engineers also reduce irreversibilities - friction, turbulence, throttling losses, and heat leaks through insulation - to push the actual efficiency closer to that ceiling. Combined-cycle plants stack a gas turbine on top of a steam turbine, using the exhaust of one as the heat source for the other, which dramatically increases overall efficiency.
What is heat rejected (Qout) and why does it matter?
Heat rejected is the thermal energy discarded to the cold sink per cycle - the difference between heat input and useful work output. In power plants this waste heat warms cooling rivers, towers, or the sea. It matters because it represents lost potential, and in some combined systems (called cogeneration or combined heat-and-power plants) it is captured for space heating or industrial processes, raising the overall utilisation of the fuel even though the electrical efficiency remains the same.