Air-standard Otto cycle – efficiency formula in terms of compression ratio For an air-standard Otto cycle, the thermal efficiency is given by which of the following expressions (r = compression ratio, γ = cp/cv)?

Introduction / Context:The Otto cycle models spark-ignition engines with heat addition at constant volume. Its air-standard efficiency depends only on compression ratio and the specific-heat ratio under ideal assumptions.

Given Data / Assumptions:

• Air-standard assumptions: ideal gas with constant specific heats, internally reversible compression/expansion.
• Compression ratio r = V1/V2.
• γ = cp/cv.

Concept / Approach:The well-known result for the Otto cycle is η_Otto = 1 − 1 / r^(γ − 1). It shows that increasing compression ratio raises theoretical efficiency, with diminishing returns governed by γ.

Step-by-Step Solution:Relate temperatures across isentropic compression: T2/T1 = r^(γ − 1).Relate temperatures across isentropic expansion similarly.Use η = 1 − (Q_out/Q_in) with constant-volume heat addition/removal proportional to temperature differences.Algebraic elimination yields η = 1 − 1 / r^(γ − 1).

Verification / Alternative check:Limits: as r → 1, efficiency → 0; as r increases, η approaches 1. Real engines deviate due to dissociation, heat losses, and incomplete combustion.

Why Other Options Are Wrong:

• “1 + r^(γ − 1)” and “r^(γ − 1) − 1” are dimensionless but not efficiency forms (and can exceed 1).
• “1 − r^(γ − 1)” becomes negative at typical r > 1.
• “1 − 1 / r^γ” uses the wrong exponent; the correct exponent is γ − 1.

Common Pitfalls:Mistyping the exponent or forgetting that efficiency must lie between 0 and 1.

Final Answer:1 − 1 / r^(γ − 1)