--- title: "Ideal Brayton cycle: diagrams and efficiency" subject: "Fundamental of Thermodynamics & Heat Transfer" chapter: 6 key: "semester_2/fundamental_of_thermodynamics_and_heat_transfer:brayton-cycle-efficiency-derivation" source: https://www.bctengineer.com/q/semester_2/fundamental_of_thermodynamics_and_heat_transfer/brayton-cycle-efficiency-derivation site: BCT Engineering --- # Ideal Brayton cycle: diagrams and efficiency Fundamental of Thermodynamics & Heat Transfer · Chapter 6: Thermodynamic Cycles - Asked in: 2072 Asoj, 6 marks; 2071 Bhadra, 6 marks (asked 2 times) - 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_2/fundamental_of_thermodynamics_and_heat_transfer/brayton-cycle-efficiency-derivation - Whole chapter: https://www.bctengineer.com/md/semester_2/fundamental_of_thermodynamics_and_heat_transfer/chapter-6 ## Question Sketch an ideal Brayton cycle on P-v and T-s diagrams. Also derive an expression for its efficiency in terms of pressure ratio. ## Short answer The ideal Brayton cycle has two isentropic and two constant‑pressure processes. Efficiency derivation: η = 1 − q_out/q_in = 1 − (T₄−T₁)/(T₃−T₂). Using isentropic ratios T₂/T₁ = T₃/T₄ = r_p^((γ−1)/γ) yields η = 1 − 1/r_p^((γ−1)/γ). ## Answer The ideal Brayton cycle is the air-standard cycle of a gas turbine. It has two isentropic and two constant-pressure processes. #### Diagrams ```text P-v diagram T-s diagram P T | 2 ____________ 3 | 3 | \ \ | /| | \ \ | / | | '. '. | / | | '. '. | 2 _.-' | | '-._ '-._ | | 4 | '- 1 _________ 4 | | .-' | | | _.-' | | 1.-' +----------------------> v +--------------------> s 1-2 isentropic compression 2-3 constant-P heat addition 3-4 isentropic expansion 4-1 constant-P heat rejection ``` #### Derivation of efficiency Take air as an ideal gas with constant cp. Per kg of air: - Heat added (2-3, constant P): q_in = cp (T₃ − T₂) - Heat rejected (4-1, constant P): q_out = cp (T₄ − T₁) Thermal efficiency: η = w_net/q_in = 1 − q_out/q_in = 1 − (T₄ − T₁)/(T₃ − T₂) For the isentropic processes 1-2 and 3-4, with P₂ = P₃ and P₁ = P₄, the pressure ratio is r_p = P₂/P₁ = P₃/P₄: T₂/T₁ = (P₂/P₁)^((γ−1)/γ) = r_p^((γ−1)/γ) T₃/T₄ = (P₃/P₄)^((γ−1)/γ) = r_p^((γ−1)/γ) Hence T₂/T₁ = T₃/T₄, which gives T₃/T₂ = T₄/T₁. Then (T₄ − T₁)/(T₃ − T₂) = (T₁/T₂) × (T₄/T₁ − 1)/(T₃/T₂ − 1) = T₁/T₂ So η = 1 − T₁/T₂ **η_Brayton = 1 − 1/r_p^((γ−1)/γ)** The efficiency depends only on the pressure ratio and γ: it rises as r_p rises. For example, r_p = 10 and γ = 1.4 gives η = 1 − 1/1.931 = 48.2 %. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which processes make up the ideal Brayton cycle? A. Two constant-pressure and two constant-volume B. Two isothermal and two isentropic C. Two isentropic and two constant-pressure D. Two isentropic and two constant-volume Answer: C. Compression and expansion are isentropic; heat addition and rejection are at constant pressure. (Key point: Brayton processes) 2. Which gives heat rejected per kg? A. cp(T4 - T1) B. cp(T2 - T1) C. cp(T3 - T2) D. cv(T4 - T1) Answer: A. Heat rejection 4-1 is at constant pressure, so q_out = cp(T4 - T1). (Key point: Heat expressions) 3. Which relation lets the temperatures cancel in the derivation? A. T2 = T4 B. T2/T1 = T4/T3 only for reversible heat C. T2/T1 = T3/T4, since both isentropic processes have the same pressure ratio D. T1 = T3 Answer: C. Equal pressure ratios give T2/T1 = T3/T4, hence (T4 - T1)/(T3 - T2) = T1/T2. (Key point: Key step) 4. Final efficiency expression? A. 1 - 1/r_p^((g-1)/g) B. 1 - T3/T1 C. 1 - 1/r_p^g D. 1 - r_p^((g-1)/g) Answer: A. eta = 1 - T1/T2 = 1 - 1/r_p^((g-1)/g). (Key point: Final result) ## Related in Fundamental of Thermodynamics & Heat Transfer - [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 - [Ideal Otto cycle diagrams and efficiency](https://www.bctengineer.com/question.md?key=semester_2%2Ffundamental_of_thermodynamics_and_heat_transfer%3Aotto-cycle-efficiency-compression-ratio) — 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 - [Ideal Brayton cycle: net work per kg and efficiency](https://www.bctengineer.com/question.md?key=semester_2%2Ffundamental_of_thermodynamics_and_heat_transfer%3Abrayton-work-per-kg-efficiency) — Fundamental of Thermodynamics & Heat Transfer, chapter 6 - [Ideal diesel cycle: diagrams and efficiency](https://www.bctengineer.com/question.md?key=semester_2%2Ffundamental_of_thermodynamics_and_heat_transfer%3Adiesel-cycle-efficiency-derivation) — Fundamental of Thermodynamics & Heat Transfer, chapter 6 - [Ideal gas turbine 100 kW: pressure ratio, work, efficiency](https://www.bctengineer.com/question.md?key=semester_2%2Ffundamental_of_thermodynamics_and_heat_transfer%3Aideal-gas-turbine-100kw-pressure-ratio) — Fundamental of Thermodynamics & Heat Transfer, chapter 6 ## 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 cycle work and efficiency](https://www.bctengineer.com/question.md?key=semester_7%2Fturbo_machines%3Aideal-brayton-numerical) — Turbo Machines, chapter 3 - [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 - [Brayton cycle; Atkinson and Stirling cycles](https://www.bctengineer.com/question.md?key=semester_6%2Finternal_combustion_engines%3Abrayton-cycle-atkinson-stirling-notes) — Internal Combustion Engines, chapter 1