Comparing refrigeration cycles: Among the listed cycles operating between the same temperature limits, which refrigeration cycle has the maximum possible coefficient of performance (COP)?

Introduction / Context:
The coefficient of performance (COP) measures refrigeration effectiveness: COP_R = Q_L / W_in. For any two thermal reservoirs, the second law sets an upper bound on COP. Identifying the theoretically best cycle helps benchmark real systems and diagnose inefficiencies.

Given Data / Assumptions:

• All cycles considered operate between the same cold (T_L) and hot (T_H) temperature limits.
• Ideal, reversible limits are admissible for comparison.
• No constraints on working fluid other than cycle definition.

Concept / Approach:
The Carnot refrigeration cycle is fully reversible and hence ideal. For given T_L and T_H, its COP is COP_Carnot = T_L / (T_H − T_L) (temperatures in Kelvin). No real cycle can exceed this bound. Practical vapor-compression cycles approach but never surpass it; air (Bell–Coleman) cycles are typically far below due to gas compression/expansion irreversibilities.

Step-by-Step Solution:
Define the reversible benchmark: COP_Carnot = T_L / (T_H − T_L).Any irreversibility reduces COP from the Carnot limit.Among the listed options, only the Carnot cycle is fully reversible by definition.Therefore, select 'Carnot cycle' as having the maximum COP.

Verification / Alternative check:
Compare with idealized vapor-compression: even with a reversible expander replacing the throttling valve, additional irreversibilities (heat transfer with finite ΔT, non-isentropic compression, real fluid effects) keep COP lower than the Carnot limit.

Why Other Options Are Wrong:
Air cycle: lower COP for the same temperature span. Vapor-compression (ordinary or with expander): better than throttling-only in the latter case but still below Carnot.

Common Pitfalls:

• Confusing 'best practical' with 'theoretical maximum'—the question asks the latter.
• Using Celsius instead of Kelvin in COP_Carnot formula.

Final Answer:
Carnot cycle