---
title: "Brayton cycle; Atkinson and Stirling cycles"
subject: "Internal Combustion Engines"
chapter: 1
key: "semester_6/internal_combustion_engines:brayton-cycle-atkinson-stirling-notes"
source: https://www.bctengineer.com/q/semester_6/internal_combustion_engines/brayton-cycle-atkinson-stirling-notes
site: BCT Engineering
---

# Brayton cycle; Atkinson and Stirling cycles

Internal Combustion Engines · Chapter 1: Overview of thermodynamics of fuel-air cycles and real cycles

- Asked in: year not given, 4+4 marks
- Video: [Ideal BRAYTON CYCLE Explained in 11 Minutes!](https://www.youtube.com/watch?v=fZ80tkxWa-4) — Less Boring Lectures
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/internal_combustion_engines/brayton-cycle-atkinson-stirling-notes
- Whole chapter: https://www.bctengineer.com/md/semester_6/internal_combustion_engines/chapter-1

## Question

Derive the thermal efficiency of the air-standard Brayton cycle in terms of the pressure ratio and calculate it for a pressure ratio of 8. Write short notes on the Atkinson cycle and the Stirling cycle.

## Short answer

Brayton efficiency: η = 1 – 1/rp^((γ-1)/γ); for rp=8, η=44.8%. Atkinson cycle: expansion ratio > compression ratio, constant-pressure heat rejection, higher efficiency but lower power density; achieved by late intake valve closing. Stirling cycle: closed external-heat cycle with regenerator; ideal efficiency equals Carnot; any heat source; clean but large and slow.

## Answer

#### Brayton cycle efficiency
The Brayton (Joule) cycle is the ideal cycle of a gas turbine. It has isentropic compression 1-2 in a compressor, constant-pressure heat addition 2-3 in a combustor, isentropic expansion 3-4 in a turbine and constant-pressure heat rejection 4-1.

```text
 Air --> [Compressor] --> [Combustor] --> [Turbine] --> Exhaust
             1-2             2-3             3-4        4-1
              \_______________shaft__________/ --> load
```

$$
\begin{aligned}
Q_{in} &= c_p (T_3 - T_2), \qquad Q_{out} = c_p (T_4 - T_1) \\
\eta &= 1 - \frac{T_4 - T_1}{T_3 - T_2}
\end{aligned}
$$

Since $p_2 = p_3$ and $p_1 = p_4$, $T_2/T_1 = T_3/T_4 = r_p^{(\gamma-1)/\gamma}$. Hence $T_4/T_1 = T_3/T_2$ and

$$
\eta_{Brayton} = 1 - \frac{T_1}{T_2} = 1 - \frac{1}{r_p^{(\gamma-1)/\gamma}}
$$

For $r_p = 8$, $\gamma = 1.4$:

$$
\eta = 1 - 8^{-0.2857} = 1 - 0.552 = 0.448
$$

**Answer:** $\eta = 44.8\%$. It depends only on pressure ratio, not on the maximum temperature.

#### Atkinson cycle
- Same as the Otto cycle (isentropic compression, constant-volume heat addition) but the expansion goes on until the gas reaches atmospheric pressure. The expansion ratio is larger than the compression ratio.
- Heat is rejected at constant pressure.
- It extracts more work from the same fuel, so the efficiency is higher than Otto's.
- It is achieved with a long expansion stroke (original linkage design) or, in modern engines, by closing the inlet valve late (Miller/Atkinson-type timing). The cost is a lower power density, so it is used in hybrid cars.

#### Stirling cycle
- A closed cycle with an external heat source and a regenerator. Processes: isothermal compression, constant-volume heating (regenerator), isothermal expansion, constant-volume cooling (regenerator).
- With a perfect regenerator its efficiency equals the Carnot efficiency, $1 - T_L/T_H$.
- Heat is supplied from outside, so any fuel or solar heat can be used and combustion is clean and quiet.
- Drawbacks: large size, slow response and sealing problems, so it is rare in vehicles.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The Brayton cycle efficiency 1 - 1/r_p^((gamma-1)/gamma) depends on what?
   A. Pressure ratio only (and gamma), not the maximum temperature
   B. Maximum temperature only
   C. Compression ratio and cut-off ratio
   D. Heat added only
   Answer: A. T4/T1 = T3/T2 cancels the temperatures, leaving T1/T2 which depends only on r_p. (Key point: Brayton efficiency)

2. What is the Brayton efficiency at pressure ratio 8 and gamma = 1.4?
   A. 55.2%
   B. 44.8%
   C. 56.5%
   D. 68.5%
   Answer: B. 8^(-0.2857) = 0.552, so eta = 1 - 0.552 = 0.448. (Key point: Brayton numerical)

3. What distinguishes the Atkinson cycle from the Otto cycle?
   A. It uses constant-pressure heat addition
   B. It has isothermal processes
   C. The expansion ratio is larger than the compression ratio
   D. Its compression ratio is larger than the expansion ratio
   Answer: C. Expansion continues down to atmospheric pressure, extracting more work, at the cost of lower power density. (Key point: Atkinson cycle)

4. With a perfect regenerator, the Stirling cycle efficiency equals what?
   A. The Otto efficiency
   B. 1 - 1/r^(gamma-1)
   C. 50%
   D. The Carnot efficiency 1 - T_L/T_H
   Answer: D. Regeneration makes the constant-volume heat exchange internal, so only isothermal heat transfer remains, matching Carnot. (Key point: Stirling cycle)

## Related in Internal Combustion Engines

- [Otto cycle efficiency and mean effective pressure](https://www.bctengineer.com/question.md?key=semester_6%2Finternal_combustion_engines%3Aotto-cycle-efficiency-numerical) — Internal Combustion Engines, chapter 1
- [Efficiency ratio and specific fuel consumption](https://www.bctengineer.com/question.md?key=semester_6%2Finternal_combustion_engines%3Arelative-efficiency-sfc-numerical) — Internal Combustion Engines, chapter 8
- [Diesel cycle efficiency compared with Otto](https://www.bctengineer.com/question.md?key=semester_6%2Finternal_combustion_engines%3Adiesel-cycle-versus-otto-numerical) — Internal Combustion Engines, chapter 1
- [Fuel-air cycle and actual cycle losses](https://www.bctengineer.com/question.md?key=semester_6%2Finternal_combustion_engines%3Afuel-air-cycle-assumptions-losses) — Internal Combustion Engines, chapter 1
- [Volumetric efficiency from orifice air measurement](https://www.bctengineer.com/question.md?key=semester_6%2Finternal_combustion_engines%3Avolumetric-efficiency-orifice-air-consumption) — Internal Combustion Engines, chapter 8

## Related in other subjects

- [Brayton cycle efficiency and optimum pressure ratio](https://www.bctengineer.com/question.md?key=semester_7%2Fturbo_machines%3Abrayton-cycle-efficiency-optimum-pressure-ratio) — Turbo Machines, chapter 3
- [Ideal Brayton: efficiency and net power, pressure ratio 10](https://www.bctengineer.com/question.md?key=semester_2%2Ffundamental_of_thermodynamics_and_heat_transfer%3Abrayton-5m3s-pr10-1400k) — Fundamental of Thermodynamics & Heat Transfer, chapter 6
- [Brayton cycle pressure ratio 12: mass flow for 90 MW](https://www.bctengineer.com/question.md?key=semester_2%2Ffundamental_of_thermodynamics_and_heat_transfer%3Abrayton-pressure-ratio-12-90mw) — Fundamental of Thermodynamics & Heat Transfer, chapter 6
- [Air standard Brayton: state points, work, efficiency](https://www.bctengineer.com/question.md?key=semester_2%2Ffundamental_of_thermodynamics_and_heat_transfer%3Abrayton-0-18mpa-34c-max-2350) — Fundamental of Thermodynamics & Heat Transfer, chapter 6