Comparing Otto and Diesel cycles at the same compression ratio: Which statement is correct about their thermal efficiencies for a fixed compression ratio?

Introduction / Context:
The ideal Otto and Diesel cycles model spark-ignition and compression-ignition engines, respectively. For a fair comparison at the same compression ratio, the difference in the mode of heat addition leads to different efficiencies. Recognizing which cycle is more efficient under this condition is a staple concept in thermodynamics.

Given Data / Assumptions:

• Air-standard analysis with constant specific heats.
• Equal compression ratio r for both cycles.
• Otto: heat addition at constant volume; Diesel: heat addition at constant pressure with cut-off ratio ρ > 1.

Concept / Approach:
Otto cycle efficiency depends only on compression ratio: η_Otto = 1 − 1/r^(γ−1). Diesel efficiency also depends on cut-off ratio ρ and is always lower than Otto's at the same r because part of Diesel's heat addition occurs at larger specific volumes (lower average temperatures of heat addition). Therefore, for an equal r, η_Otto > η_Diesel.

Step-by-Step Solution:

Verification / Alternative check:
Plugging typical numbers (γ ≈ 1.4, r = 8–10) into standard formulas shows Otto’s η several points higher than Diesel’s unless ρ approaches 1 (which would trend toward Otto behavior).

Why Other Options Are Wrong:

• Same efficiency: False because Diesel's dependence on ρ reduces η.
• Diesel more efficient at same r: Opposite to theory.
• Efficiency independent of compression ratio: Both cycles increase η with r.

Common Pitfalls:
Confusing real-engine trends (Diesel engines often have higher r) with ideal-cycle comparisons at the same r.

Final Answer:
For a given compression ratio, Otto cycle is more efficient than Diesel cycle.