Air-Standard Cycles — Diesel vs. Otto at the Same Compression Ratio For the same compression ratio r, how does the air-standard efficiency of the Diesel cycle compare with that of the Otto cycle?

Introduction / Context:
Otto and Diesel cycles are idealized models for spark-ignition and compression-ignition engines. Comparing their efficiencies at equal compression ratio illuminates the effect of heat-addition mode (constant volume vs. constant pressure) on cycle performance.

Given Data / Assumptions:

• Air-standard analysis with ideal gas, constant specific heats.
• Same compression ratio r = V1/V2 for both cycles.
• Reversible, internally idealized processes (no friction, no real losses).

Concept / Approach:
For the Otto cycle, heat is added at constant volume, producing a higher peak temperature for the same r. For the Diesel cycle, a portion of heat is added at constant pressure (finite cutoff ratio), lowering the average temperature at which heat is added and reducing thermal efficiency relative to Otto when r is the same. Hence, eta_Diesel < eta_Otto for equal r (for any cutoff ratio > 1).

Step-by-Step Solution:
Start from standard efficiency relations: eta_Otto = 1 - 1/r^(k-1).Diesel efficiency depends on both r and cutoff ratio rc: eta_Diesel = 1 - (1/r^(k-1)) * ((rc^k - 1)/(k*(rc - 1))).For rc > 1, the multiplicative factor reduces eta_Diesel below eta_Otto at the same r.

Verification / Alternative check:
Plotting efficiencies vs. r for typical k = 1.4 shows the Otto curve above the Diesel curve for the same r, converging only as rc → 1 (which would approach Otto behavior).

Why Other Options Are Wrong:
“Greater than” or “equal to” would require zero cutoff (rc = 1) or different r; those conditions violate the stated comparison with typical Diesel heat addition.

Common Pitfalls:
Confusing real-engine compression ratios (Diesel engines often use higher r than SI engines) with the theoretical comparison at equal r. In practice, higher Diesel r can yield higher efficiency, but that is a different comparison.

Final Answer:
less than