Brayton Cycle

The four-step idealisation

The Brayton cycle is the thermodynamic model of a gas turbine. Strip an aircraft jet engine, an industrial power-plant turbine, or a turbocharged piston engine of its rotating hardware and you find the same four-step heat-engine cycle underneath:

1. 1 → 2 · Isentropic compression. Atmospheric air enters the compressor at p_1, T_1 and is compressed to p_2, T_2. The compressor is an axial or centrifugal turbomachine driven by the turbine on a common shaft. Ideally adiabatic and reversible, so entropy is unchanged: T_2/T_1 = (p_2/p_1)^((γ−1)/γ) = r_p^((γ−1)/γ).
2. 2 → 3 · Constant-pressure heat addition. Compressed air enters the combustor, where fuel is sprayed in and burned in steady flow at approximately constant pressure. Heat q_in raises the temperature to the turbine-inlet temperature T_3 — the cycle's hottest point. In practice the combustor incurs a 2–5 % pressure drop; the textbook cycle treats it as exactly constant.
3. 3 → 4 · Isentropic expansion. Hot combustion gas expands through one or more turbine stages, dropping back to roughly atmospheric pressure. The turbine extracts work — first to drive the compressor on the same shaft, then to deliver net output (thrust in a jet, shaft power in a power-plant gas turbine).
4. 4 → 1 · Constant-pressure heat rejection. The hot, low-pressure exhaust dumps to atmosphere, cooling back to the ambient state. In an open cycle the working fluid is replaced rather than recooled in a heat exchanger — but thermodynamically it is the same constant-pressure heat-rejection leg.

Two isentropes and two isobars: a tilted rectangle on a T–s diagram, a banana-shaped loop on a P–V diagram. The enclosed area is the net work per unit mass.

Ideal efficiency and where it comes from

For a cold-air-standard analysis with constant heat capacities, the net work output per unit mass is

w_net = c_p (T_3 − T_4) − c_p (T_2 − T_1)

and the heat input is

q_in = c_p (T_3 − T_2)

Dividing — and applying the two isentropic relations T_2/T_1 = T_3/T_4 = r_p^((γ−1)/γ) — gives the famous closed-form expression for the ideal Brayton efficiency:

η_Brayton = 1 − T_1/T_2
          = 1 − (1/r_p)^((γ−1)/γ)
          = 1 − r_p^(−(γ−1)/γ)

For air (γ = 1.4), this becomes η = 1 − r_p^(−0.286). Plug in some numbers:

Pressure ratio r_p	Ideal η (γ = 1.4)	Real-engine application
4	33 %	Early turbojets (e.g. Jumo 004, 1944)
10	48 %	Small auxiliary power units, microturbines
20	57 %	Heavy-frame industrial gas turbines
30	62 %	Older high-bypass turbofans (CF6, RB211)
40	65 %	CFM56, V2500 commercial turbofans
50	68 %	CFM LEAP, PW1100G geared turbofan
60	70 %	GE9X (Boeing 777X)

These are ideal efficiencies — what a perfect cycle at that pressure ratio could in principle achieve. Real engines, after compressor and turbine losses, combustor pressure drop, and real-gas effects, deliver typically 25–40 % thermal efficiency as a standalone unit. Modern combined-cycle plants push the system efficiency past 60 % by recovering exhaust heat in a Rankine bottoming cycle.

Why specific work peaks at intermediate r_p

Efficiency rises monotonically with pressure ratio, but specific work — work delivered per kg of air — does not. Increasing r_p raises both the work the compressor consumes and the temperature limits of the turbine. Past a certain point the compressor swallows nearly the entire turbine output, leaving little net.

Differentiating w_net with respect to r_p with a fixed T_3/T_1 ratio gives an optimum pressure ratio:

r_p,opt = (T_3/T_1)^(γ/(2(γ−1)))

For T_3 = 1500 K, T_1 = 288 K, γ = 1.4: r_p,opt ≈ 14 for maximum specific work. Industrial gas turbines tend to sit near this optimum — they want power density. Aero engines push much higher r_p than the specific-work optimum, sacrificing some net work per kg of air to gain efficiency (fuel burn matters more than mass when you're hauling the engine 11 km up).

This is the classic engineering trade-off: efficiency monotone with r_p, specific work peaked at r_p,opt. Design choice depends on whether you are sized by fuel cost (push r_p high) or by power-per-weight (sit near r_p,opt).

Where the real cycle deviates from the ideal

Five effects pull the real engine away from the textbook curve:

• Compressor and turbine inefficiencies. Real adiabatic processes generate entropy. Polytropic efficiencies for state-of-the-art compressors and turbines are 88–93 %; isentropic efficiencies (comparing actual to ideal Δh across the whole machine) are slightly lower. The T–s diagram leans outward — the compression line tilts to higher T at the same exit pressure, the expansion line tilts to higher T at the same exit pressure.
• Combustor pressure drop. Practical combustors lose 2–5 % of inlet stagnation pressure to flame-stabilisation losses, dump diffuser losses, and dilution-jet mixing. The 2 → 3 leg is not exactly horizontal on a P–V plot.
• Real-gas effects. γ falls as T rises — from 1.4 for cold air to about 1.30 at 1500 K for combustion products. c_p rises. The closed-form efficiency that assumes constant γ over-predicts; engineers use either a corrected-γ approach or actual gas tables.
• Mass-flow asymmetry. Fuel adds 2–4 % to the gas mass between compressor exit and turbine inlet. The "single-stream" idealisation hides this.
• Bleed air. 10–20 % of compressor discharge is bled off for turbine-blade cooling, anti-ice, and cabin air; that air bypasses combustion and never produces full work. Modern designs route cooling air carefully to minimise the efficiency hit.

Three classical improvements: intercool, reheat, regenerate

Once you have the basic Brayton cycle, three modifications attack different losses.

• Intercooling. Split the compressor into low- and high-pressure spools and cool the air between them. Cooler gas requires less work to compress; the area on a T–s diagram representing compression work shrinks. Net effect: more net work per unit mass, but the average heat-addition temperature also falls, which can hurt cycle efficiency unless paired with regeneration.
• Reheat. Split the turbine and add a second combustor between the high- and low-pressure stages. The average heat-addition temperature rises, so does specific work. Used in the Alstom GT24/GT26 sequential-combustion turbines.
• Regeneration (recuperation). Hot turbine exhaust passes through a counter-flow heat exchanger that preheats the compressor discharge before it enters the combustor. Now part of the heat that would have been wasted in the exhaust is recovered. At low pressure ratios — where exhaust is much hotter than compressor discharge — regeneration adds dramatic efficiency, sometimes 15 percentage points. At high r_p the compressor discharge is already hot and the recuperator margin shrinks.

The intercooled-reheated-regenerated ("ICR") Brayton cycle approaches the theoretical Ericsson efficiency in the limit of infinitely many intercool/reheat stages. The Rolls-Royce WR-21 marine engine — fitted to the Royal Navy Type 45 destroyer — is a production ICR design, delivering > 40 % shaft efficiency from a single gas turbine.

The combined-cycle gas turbine

The single largest jump in real-world Brayton-engine efficiency comes from pairing it with a Rankine bottoming cycle. The exhaust of a 600 MW gas turbine still leaves the stack at 550–650 °C — far hotter than ambient — so an enormous amount of heat is being thrown away. A combined-cycle gas turbine (CCGT) plant routes that exhaust through a heat-recovery steam generator (HRSG), which raises high-pressure steam to drive a conventional steam turbine.

The thermodynamic logic is straightforward: the Brayton cycle handles the high-temperature top of the available heat-engine range (1500 K firing temperature → 600 °C exhaust); the Rankine steam plant picks up the bottom (600 °C → 30 °C condenser). Each cycle operates in its sweet spot.

Plant	Configuration	Net LHV η
Standalone GT (industrial)	Simple Brayton, r_p ≈ 20	32–40 %
Standalone GT (modern aero-derivative)	Simple Brayton, r_p ≈ 30+	40–43 %
CCGT (GE HA-class)	Brayton + 3-pressure Rankine	63–64 %
CCGT (Mitsubishi M501J)	Brayton + reheat Rankine	62 %+
Coal-fired ultra-super-critical	Rankine only	47 %

The roughly 25-percentage-point gap between simple-cycle GT and CCGT is the largest single lever in modern thermal-power-plant efficiency. CCGT is the dominant new build for natural-gas-fired electricity worldwide, including roughly a third of US electricity generation as of the mid-2020s.

Worked example: a CFM LEAP at cruise

Take the CFM International LEAP-1A turbofan at cruise altitude (11 km, T_1 ≈ 217 K, p_1 ≈ 23 kPa), idealising the gas-generator core as a simple Brayton cycle with r_p ≈ 50 and T_3 = 1700 K. Using γ = 1.4 throughout (acknowledged crude but instructive):

T_2/T_1 = r_p^((γ−1)/γ)
        = 50^0.286
        ≈ 3.18

T_2 ≈ 217 × 3.18 ≈ 690 K   (compressor discharge temperature)
T_4 = T_3 / 3.18 ≈ 535 K   (turbine exhaust temperature, ideal)

w_net  = c_p [(T_3 − T_4) − (T_2 − T_1)]
       = 1004 × [(1700 − 535) − (690 − 217)]
       = 1004 × [1165 − 473]
       = 1004 × 692
       ≈ 695 kJ/kg air

q_in   = c_p (T_3 − T_2) = 1004 × (1700 − 690) ≈ 1014 kJ/kg air

η_ideal = 1 − T_1/T_2 = 1 − 217/690 ≈ 68.5 %

The compressor work is roughly 41 % of the turbine work — typical for a high-r_p engine. After accounting for component efficiencies (compressor 88 %, turbine 91 %, combustor pressure drop 3 %, bleed 12 %) the actual gas-generator thermal efficiency drops to roughly 45–48 %. Couple that to a high-bypass fan and a propulsive efficiency of 75 %+ at cruise, and overall fuel-to-thrust-work efficiency lands near 35 % — the reason commercial aviation burns the fuel it does and not five times more.

Where in the cycle does what go wrong?

Leg	Ideal	Real-engine loss	Mitigation
1 → 2 compress	Reversible adiabatic	Entropy generation, surge margin loss	Variable stators, transonic blading, intercool
2 → 3 burn	Constant pressure	2–5 % pressure drop, NOx, soot, hot-streak	Lean premixed combustors, swirl, staging
3 → 4 expand	Reversible adiabatic	Entropy generation, blade-cooling penalty, tip leakage	Tip seals, film-cooled blades, single-crystal alloys
4 → 1 exhaust	Constant pressure to ambient	Heat dumped at high T — pure exergy loss	Regeneration; combined cycle

A short history

James Prescott Joule sketched the gas-cycle concept in 1851 — two adiabatic and two isobaric strokes of a perfect gas — to explore the relation between heat and mechanical work. The cycle therefore remains "the Joule cycle" in many European texts. George Brayton's 1872 patent of the "Ready Motor", a reciprocating constant-pressure engine that burned its fuel before the compression piston rather than after, made the cycle namable in American practice. Brayton's engine fizzled commercially — Otto's 1876 four-stroke spark-ignition engine outcompeted it — but the cycle survived because Frank Whittle (UK, 1937) and Hans von Ohain (Germany, 1939) built the first jet engines on a fundamentally Brayton template: continuous-flow compression, combustion, expansion. Every gas turbine since — aero or industrial — is a refinement on that 1939 architecture.

Where the Brayton cycle shows up

• Aircraft jet engines. Turbojets, turbofans, turboprops, turboshafts. The high-bypass commercial turbofan is the dominant air-transport prime mover.
• Industrial gas turbines. Power-plant electrical generation, mechanical drives for gas pipeline compressors, peaking plants. Often deployed as combined-cycle (CCGT).
• Marine gas turbines. Warships (LM2500, MT30, WR-21), high-speed ferries. Aero-derivative gas turbines dominate the warship-propulsion market.
• Auxiliary power units (APUs). Small gas turbines on aircraft and ground vehicles for on-board electrical and pneumatic power.
• Turbochargers. A topology-identical compressor-turbine pair driven by exhaust to boost piston-engine intake — informal Brayton.
• Closed Brayton cycles. Helium turbines for Generation-IV nuclear concepts; supercritical-CO₂ Brayton loops being prototyped (Sandia, GE, Echogen) for next-generation solar and nuclear power.

Common pitfalls

• Quoting "ideal Brayton efficiency" as if real engines reach it. The closed-form η = 1 − r_p^(−(γ−1)/γ) ignores every real-world loss. A modern aero engine at r_p = 50 ideally hits 68 %; the actual gas-generator efficiency is roughly 45–48 %.
• Forgetting that efficiency and specific work optimise at different r_p. Efficiency keeps rising with pressure ratio. Specific work peaks at r_p,opt and falls thereafter. An engine sized purely for efficiency may be impractically large per unit of output.
• Treating γ as 1.4 throughout. γ drops to roughly 1.30 at combustor outlet. Using cold-air γ for the whole cycle over-predicts efficiency by several percentage points.
• Conflating the gas-turbine core with the engine. A high-bypass turbofan's overall efficiency is the product of gas-generator thermal efficiency and propulsive efficiency — and propulsive efficiency depends on bypass ratio and flight Mach number, not the Brayton cycle alone.
• Calling a turbocharger a Brayton engine without flag. The components are identical, but the combustor is the host piston engine; thermodynamically it is best treated as an exhaust-driven recovery stage, not a standalone Brayton cycle.