Gas Cycles: Otto, Diesel and Brayton

Air-standard cycles for petrol and diesel engines and gas turbines, compression ratio, efficiency and combined cycles.

On this page

The air-standard idea
The Otto cycle (petrol)
The Diesel cycle
The Brayton cycle (gas turbine)
Comparing the cycles
Combined cycles
Real engines
FAQ

The air-standard idea

Analyzing a real engine, with its combustion, valves and friction, is difficult. The air-standard method replaces it with a closed cycle of air (treated as an ideal gas with constant specific heats), heat added from outside and heat rejected to outside. It gives the right trends and the upper limit; real engines reach about half of the ideal efficiency. Throughout, k = 1.4, c_p = 1.005 and c_v = 0.718 kJ/kg·K.

The Otto cycle (petrol)

The spark-ignition engine: (1) isentropic compression of the mixture by the compression ratio r = V_max ÷ V_min; (2) heat addition at constant volume (the spark); (3) isentropic expansion (the power stroke); (4) heat rejection at constant volume (the exhaust valve opens). Its efficiency depends only on the compression ratio:

η_Otto = 1 − 1 ÷ r^k−1

Worked example: an Otto engine

Intake at 300 K, compression ratio 10, peak cycle temperature 2 000 K.

End of compression: T₂ = 300 × 10^0.4 = 753.6 K.
Heat added q_in = c_v (T₃ − T₂) = 0.718 × (2 000 − 753.6) = 894.9 kJ/kg.
End of expansion: T₄ = 2 000 ÷ 10^0.4 = 796.2 K; heat rejected q_out = 0.718 × (796.2 − 300) = 356.3 kJ/kg.
Net work = 538.7 kJ/kg; efficiency = 538.7 ÷ 894.9 = 60.2 percent, which matches 1 − 1 ÷ 10^0.4 = 60.2. The real engine would give 30 to 38 percent.
Power check for a 2 liter four-stroke at 6 000 rpm with a brake mean effective pressure of 1 000 kPa: P = bmep × V_d × rpm ÷ 120 = 100 kW.

The Diesel cycle

A compression-ignition engine compresses only air (so there is no knock limit) to a much higher ratio, then injects fuel that burns at roughly constant pressure while the piston moves. The ratio of volumes during heat addition is the cutoff ratio r_c.

η_Diesel = 1 − (1 ÷ r^k−1) × (r_c^k − 1) ÷ [k (r_c − 1)]

For r = 18 and r_c = 2, η = 63.2 percent ideal; real diesels reach 35 to 45 percent (large, slow marine diesels over 50 percent). A diesel also keeps its efficiency at part load because it throttles the fuel, not the air.

The Brayton cycle (gas turbine)

The gas turbine is a steady-flow machine: a compressor, a combustor (heat addition at constant pressure) and a turbine that drives the compressor and the load. It is the Brayton cycle, and its efficiency depends on the pressure ratio r_p:

η_Brayton = 1 − 1 ÷ r_p^(k−1)/k

Ideal efficiency of the Otto and Brayton cycles. Both rise with their ratio and flatten.

Worked example: a 10 MW gas turbine

Pressure ratio 12, compressor inlet 300 K, turbine inlet 1 400 K (1 127 °C).

Compressor exit: T₂ = 300 × 12^0.2857 = 610.2 K; compressor work = 1.005 × 310.2 = 311.7 kJ/kg.
Turbine exit: T₄ = 1 400 ÷ 12^0.2857 = 688.3 K; turbine work = 715.2 kJ/kg.
Net work = 715.2 − 311.7 = 403.5 kJ/kg; heat added = 793.8 kJ/kg; efficiency = 50.8 percent ideal.
The back-work ratio w_c ÷ w_t = 43.6 percent: more than 40 percent of the turbine output just drives the compressor (a steam turbine plant uses under 1 percent for its pump).
For 10 MW net, the air flow is 10 000 ÷ 403.5 = 24.8 kg/s.
With 87 percent compressor and 88 percent turbine efficiencies: T₂ = 657 K, T₄ = 774 K, net work 271.1 kJ/kg and efficiency 36.3 percent: the losses have cut the ideal value by about a third and left a hot exhaust.

Comparing the cycles

Cycle	Heat added at	Typical ratio	Ideal η	Real η	Typical use
Otto	constant volume	r = 9 to 12	59 to 63 %	28 to 38 %	cars, small engines
Diesel	constant pressure (approx.)	r = 16 to 22	60 to 68 %	35 to 50 %	trucks, ships, generators
Brayton	constant pressure	r_p = 10 to 40	50 to 65 %	30 to 42 %	jets, power stations
Rankine	constant pressure	see Steam Power Cycles: Rankine	40 to 55 %	33 to 45 %	power stations

Combined cycles

A gas turbine exhausts at 600 to 650 °C: too hot to waste. A combined cycle feeds that exhaust to a heat-recovery steam generator that runs a Rankine steam turbine (Steam Power Cycles: Rankine), so the high-temperature Brayton cycle sits on top of the low-temperature Rankine cycle. Modern plants reach 58 to 64 percent, the best of any fossil-fuel plant.

Worked example: adding a steam bottoming cycle

The real gas turbine above has an exhaust at 774 K; cooling it to 400 K releases 1.005 × (774 − 400) = 375.6 kJ/kg of heat.
A steam cycle converting 30 percent of it adds 112.7 kJ/kg, so the combined efficiency is (271.1 + 112.7) ÷ 747.2 = 51.4 percent, against 36.3 for the gas turbine alone.

Real engines

Real engines differ from the air-standard cycles in many ways: the working fluid changes composition and its specific heat rises with temperature, combustion takes time, and there are heat losses to the walls, friction, pumping losses (throttled petrol engines lose a lot at part load), valve timing and turbocharging. Turbochargers recover exhaust energy; Atkinson and Miller cycles use a longer expansion than compression to gain efficiency. Spark-ignition engines are limited by knock (self-ignition of the end gas), diesels by smoke and NOx.

Previous: Steam Power Cycles: Rankine. Next: Refrigeration and Heat Pumps.
Ideal Gases and Processes for isentropic relations.
Gears and Power Transmission for the drive line.
Thermodynamics and Steam: the whole guide.

FAQ

Why do diesels have a higher compression ratio than petrol engines?

Compression is air only, so there is nothing to knock. Fuel is injected at the end of compression.

Why is the gas turbine back-work ratio so high?

Compressing gas takes much more work than pumping a liquid. The turbine therefore needs a very high inlet temperature to produce a useful surplus.

What is bmep?

Brake mean effective pressure: the average pressure over the cycle that would give the shaft power. P = bmep × displacement × speed ÷ (2 × 60) for a four-stroke.

Why does efficiency matter less than fuel cost sometimes?

The best cycle depends on the load pattern, capital cost, emissions and fuel; a cheap peaking gas turbine may be better than a more efficient combined cycle that runs only a few hours a year.

References

Çengel, Y. A. and Boles, M. A., Thermodynamics: An Engineering Approach, McGraw-Hill (gas power cycles).
Heywood, J. B., Internal Combustion Engine Fundamentals, McGraw-Hill.
Boyce, M. P., Gas Turbine Engineering Handbook, Elsevier.

Last reviewed: October 2026. Ratios and efficiencies are typical.