--- title: "Ideal Otto cycle diagrams and efficiency" subject: "Fundamental of Thermodynamics & Heat Transfer" chapter: 6 key: "semester_2/fundamental_of_thermodynamics_and_heat_transfer:otto-cycle-efficiency-compression-ratio" source: https://www.bctengineer.com/q/semester_2/fundamental_of_thermodynamics_and_heat_transfer/otto-cycle-efficiency-compression-ratio site: BCT Engineering --- # Ideal Otto cycle diagrams and efficiency Fundamental of Thermodynamics & Heat Transfer · Chapter 6: Thermodynamic Cycles - Asked in: 2068 Magh, 6 marks - Video: [Otto cycle (Basics, Working, Efficiency, pV diagram & Work done) Explained](https://www.youtube.com/watch?v=pG6KfB1Iv0s) — Engineering Funda - 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/otto-cycle-efficiency-compression-ratio - Whole chapter: https://www.bctengineer.com/md/semester_2/fundamental_of_thermodynamics_and_heat_transfer/chapter-6 ## Question Sketch an ideal Otto cycle on P-v and T-s diagrams. Also derive an expression for its efficiency in terms compression ratio. ## Short answer Otto cycle: isentropic compression (1-2), constant volume heat addition (2-3), isentropic expansion (3-4), constant volume heat rejection (4-1). Compression ratio r = v1/v2. Efficiency η = 1 − 1/r^(γ−1), depending only on r and γ. ## Answer The ideal Otto cycle is the air-standard cycle of the spark ignition engine. #### Diagrams ```text P-v diagram T-s diagram P T | 3 | 3 | |\ | /| | | \ | / | | | '. | / | | | '. | .' | | 2 '-. | 2 .' | | \ '-. | | 4 | '. '-. | | / | '-. '-. 4 | | .' | '-._ | | | .' | '-.._ | | 1.-' | '-- 1 | +-----------------------> v +--------------------> s v2 v1 1-2 isentropic compression 2-3 constant-v heat addition 3-4 isentropic expansion 4-1 constant-v heat rejection ``` #### Derivation Let r = v₁/v₂ be the compression ratio. Treat air as an ideal gas with constant cv and γ = cp/cv. Heat added at constant volume (2-3): q_in = cv (T₃ − T₂) Heat rejected at constant volume (4-1): q_out = cv (T₄ − T₁) η = 1 − q_out/q_in = 1 − (T₄ − T₁)/(T₃ − T₂) Isentropic compression 1-2 and expansion 3-4, with v₂ = v₃ and v₁ = v₄: T₂/T₁ = (v₁/v₂)^(γ−1) = r^(γ−1) T₃/T₄ = (v₄/v₃)^(γ−1) = r^(γ−1) Hence T₂/T₁ = T₃/T₄, which gives T₄/T₁ = T₃/T₂, so (T₄ − T₁)/(T₃ − T₂) = T₁/T₂ = 1/r^(γ−1) **η_Otto = 1 − 1/r^(γ−1)** The efficiency depends only on the compression ratio r and on γ; it rises as r increases. Example: r = 8, γ = 1.4: η = 1 − 1/8^0.4 = 56.5 %. In practice r is limited to about 10 by knocking. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. In the Otto cycle, heat is added and rejected at A. Constant entropy B. Constant volume C. Constant temperature D. Constant pressure Answer: B. Processes 2-3 and 4-1 are constant volume, so q = cv dT for both. (Key point: Heat transfer in Otto) 2. Which relations follow from the isentropic processes? A. T2/T1 = r^γ B. T2 = T3 and T1 = T4 C. T2/T1 = T3/T4 = r^(γ-1) D. T2/T1 = T4/T3 = r Answer: C. With v2 = v3 and v1 = v4, both ratios equal r^(γ-1). (Key point: Temperature ratios) 3. The ideal Otto cycle efficiency is A. 1 - 1/r^γ B. 1/r^(γ-1) C. 1 - r^(γ-1) D. 1 - 1/r^(γ-1) Answer: D. (T4 - T1)/(T3 - T2) = T1/T2 = 1/r^(γ-1), so η = 1 - 1/r^(γ-1). (Key point: Final efficiency) 4. What limits the compression ratio of an Otto engine to about 10? A. Cut-off ratio B. Low efficiency at high r C. Spark plug size D. Knocking Answer: D. Higher r raises efficiency, but in practice knocking limits r to around 10. 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