The Brayton Cycle: How a Gas Turbine Turns Pressure Into Power

A jet engine on the wing of an airliner and a gas turbine humming away in a power station look like very different machines, but thermodynamically they are the same animal. Both pull in air, compress it hard, burn fuel in the compressed stream, and let the hot gas expand through a turbine. The continuous, flowing nature of that process is what lets a turbine produce enormous power from a package far lighter than any piston engine of comparable output. The model that describes it is the Brayton cycle.

This article walks through the Brayton cycle, derives where its efficiency comes from, works a numerical example, and explains why the pressure ratio cannot simply be raised forever.

Why this calculation matters

The Brayton cycle is the reference model for gas turbines in aircraft propulsion, electrical power generation, ship propulsion, and mechanical drives for pumps and compressors. Whenever a designer asks how much useful work a turbine can extract from a given fuel flow, the ideal Brayton cycle gives the clean answer that the hardware then tries to approach.

It matters because the cycle exposes the central design knob of any gas turbine: the pressure ratio. That single number, set by the compressor, drives both the efficiency and the work output, and it interacts directly with the hottest temperature the turbine blades can survive. An engineer who understands the Brayton efficiency relation knows immediately how much is to be gained by adding compressor stages, and where that gain runs into the metallurgical wall of turbine inlet temperature. It also frames why gas turbines are often paired with a steam bottoming cycle: the Brayton exhaust is still hot enough to be worth a second cycle.

The core method

The ideal Brayton cycle is an air-standard model built from four steady-flow processes:

1 -> 2 : adiabatic (isentropic) compression in the compressor
2 -> 3 : constant-pressure heat addition in the combustor
3 -> 4 : adiabatic (isentropic) expansion through the turbine
4 -> 1 : constant-pressure heat rejection to the surroundings

Air is compressed, fuel is burned at roughly constant pressure to raise its temperature, the hot high-pressure gas expands through the turbine to produce work, and the spent gas is released. Part of the turbine's work drives the compressor; the remainder is the net output.

Applying the first law to the constant-pressure heat terms and using the isentropic pressure-temperature relations leads to a compact result for the thermal efficiency:

eta = 1 - 1 / r_p^((gamma - 1) / gamma)

Here eta is the thermal efficiency, r_p is the pressure ratio across the compressor (the compressor outlet pressure divided by the inlet pressure), and gamma is the ratio of specific heats of the working gas, about 1.4 for air.

The structure looks like the Otto cycle's efficiency formula, but the exponent is different. The Otto cycle uses a volume ratio raised to (gamma - 1); the Brayton cycle uses a pressure ratio raised to (gamma - 1)/gamma. The efficiency depends only on the pressure ratio and the gas — not on the size of the machine, not on how much fuel is burned, and not directly on the peak temperature. Raising r_p always raises the ideal efficiency, but with diminishing returns as the pressure ratio climbs.

A worked example

Take a gas turbine with a pressure ratio of r_p = 10, and treat the working fluid as air with gamma = 1.4.

Step 1 — form the exponent.

(gamma - 1) / gamma = (1.4 - 1) / 1.4 = 0.4 / 1.4 = 0.286

Step 2 — raise the pressure ratio to that power.

r_p^0.286 = 10^0.286 = 1.931

Step 3 — apply the efficiency formula.

eta = 1 - 1 / 1.931
eta = 1 - 0.518
eta = 0.482

So the ideal Brayton cycle for this turbine has a thermal efficiency of about 48 percent. That is the thermodynamic ceiling the pressure ratio sets. A real simple-cycle gas turbine lands lower — friction in the compressor and turbine, pressure losses in the combustor, and imperfect components all chip away at the figure.

The formula points clearly in one direction: push the pressure ratio up and the efficiency rises. Move from r_p = 10 to r_p = 20 and the ideal efficiency climbs to roughly 58 percent. So why not keep going? Because raising the pressure ratio raises the compressor outlet temperature, and the room left between that temperature and the fixed turbine inlet temperature limit is what produces net work. Past a certain pressure ratio, the net work per unit of air falls even though efficiency is still nudging upward, and the turbine becomes impractically large for the power it delivers. The best pressure ratio is a compromise between efficiency and specific work, anchored by how hot the turbine blades can run.

Common mistakes

Using a volume ratio instead of a pressure ratio. The Brayton cycle is driven by the compressor pressure ratio. Borrowing the Otto cycle's volume-ratio formula, or its exponent, gives the wrong efficiency. The (gamma - 1)/gamma exponent is the signature of a constant-pressure cycle.

Believing efficiency depends on turbine inlet temperature. In the ideal air-standard model it does not — efficiency is fixed by r_p and gamma alone. The turbine inlet temperature governs the net work and the practical pressure ratio, not the ideal efficiency itself. Both ideas are true at once, and confusing them is common.

Ignoring compressor work. A large fraction of the turbine's gross output is consumed by the compressor — often more than half. The net work is a difference between two big numbers, so small errors in either term swing the result hard.

Treating the ideal efficiency as the real efficiency. The 48 percent figure assumes perfect, lossless components. Real simple-cycle turbines fall short of it. Quote the ideal value as a bound, not a specification.

Forgetting that real gas properties shift. Combustion gas is hot and chemically different from cool air, so its gamma is lower than 1.4. Using 1.4 everywhere makes the prediction slightly optimistic; treat the air-standard result as an idealised reference.

Try the interactive NovaSolver calculator

Running the exponent once is straightforward, but feeling how the cycle trades efficiency against work as you move the pressure ratio is where the insight lives. The Brayton Cycle Simulator — Thermal Efficiency of a Gas Turbine on NovaSolver lets you set the pressure ratio, the specific heat ratio, the inlet temperature, and the turbine inlet temperature, then returns the thermal efficiency, the compressor outlet temperature, the turbine outlet temperature, and the net work, with a T-s diagram and efficiency curve that update as you adjust the inputs.

Related calculators

• Carnot Cycle Simulator — the temperature-limited efficiency ceiling that every real cycle, including the Brayton cycle, is measured against.
• Otto Cycle Simulator — Thermal Efficiency of a Spark-Ignition Engine — the petrol-engine cycle, useful for contrasting a volume-ratio cycle with a pressure-ratio one.
• Rankine & Refrigeration Cycle Simulator — the steam cycle often used as a bottoming cycle on the hot Brayton exhaust in combined-cycle plants.

You can browse the full set in the thermal engineering tools hub.

Closing note

The Brayton cycle reduces a gas turbine to one governing relationship: compress the air harder and you convert a larger share of the fuel's energy into work, following eta = 1 - 1/r_p^((gamma-1)/gamma). The pressure ratio is the lever, the turbine inlet temperature is the wall, and the best design lives in the tension between them. Run the formula for your own pressure ratio, remember that the ideal value is a ceiling rather than a guarantee, and the behaviour of jet engines and power-station turbines stops being mysterious and starts being predictable.