Refrigeration fundamentals: The relative coefficient of performance (relative C.O.P.) is defined as the ratio of the actual coefficient of performance to the theoretical (Carnot) coefficient of performance for the same temperature limits.

Introduction / Context:
The “relative coefficient of performance” is a handy performance index in refrigeration that benchmarks a real plant against the best possible ideal cycle working between the same temperature limits. It tells how close the real system comes to the Carnot refrigerator's performance.

Given Data / Assumptions:

• Actual C.O.P. refers to Q_L / W for the real system.
• Theoretical (Carnot) C.O.P. is the C.O.P. of an ideal reversible refrigerator operating between the same evaporator and condenser temperatures.
• Temperatures are steady and heat leaks are negligible.

Concept / Approach:
Performance comparison must be dimensionless and normalized to the best theoretical limit. Dividing the actual C.O.P. by the Carnot C.O.P. yields a number between 0 and 1 (typically), directly indicating effectiveness relative to the ideal.

Step-by-Step Solution:

Verification / Alternative check:
If a plant operates ideally, Actual C.O.P. equals Carnot C.O.P. and the relative C.O.P. becomes 1. Any irreversibilities will make it less than 1, which matches intuition.

Why Other Options Are Wrong:

• The reciprocal (Theoretical / Actual) can exceed 1 and does not measure closeness to the ideal correctly.
• The product Actual × Theoretical has incorrect units and no physical meaning.
• “None of these” is wrong because a standard definition exists.
• The difference divided by theoretical gives “percentage shortfall,” not relative C.O.P.

Common Pitfalls:
Using Celsius instead of kelvin in Carnot formulas; mixing up ideal heat pump C.O.P. with refrigerator C.O.P.; and comparing to the wrong theoretical temperature limits.

Final Answer:
Relative C.O.P. = Actual C.O.P. / Theoretical (Carnot) C.O.P.