Reversed Carnot refrigerator: The coefficient of performance (COP) in terms of absolute temperatures T1 (low) and T2 (high) is
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ACOP = T2 / (T2 - T1)
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BCOP = T1 / (T2 - T1)
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CCOP = (T2 - T1) / T2
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DCOP = (T2 - T1) / T1
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ECOP = T1 / T2
Answer
Correct Answer: COP = T1 / (T2 - T1)
Explanation
Introduction / Context:The reversed Carnot cycle represents the upper theoretical limit of refrigeration performance operating between a cold reservoir at T1 and a hot reservoir at T2 (absolute temperatures). Its COP provides a benchmark for real systems.
Given Data / Assumptions:
- Ideal, reversible cycle between T1 (evaporator temperature) and T2 (condenser temperature).
- No pressure drops or non-idealities.
- Temperatures in Kelvin (or Rankine).
Concept / Approach:For a refrigerator, COP = desired effect / input = Q_L / W. For the Carnot refrigerator, Q_L / Q_H = T1 / T2 and W = Q_H − Q_L. Combine relations to obtain COP in terms of T1 and T2 only.
Step-by-Step Derivation:
1) From reversibility: Q_L / T1 = Q_H / T2.2) Therefore Q_H = Q_L * (T2 / T1).3) Work W = Q_H − Q_L = Q_L * (T2 / T1 − 1) = Q_L * (T2 − T1) / T1.4) COP = Q_L / W = T1 / (T2 − T1).Verification / Alternative check:Dimensionless check: temperatures cancel to a pure number; as T2 approaches T1, COP → ∞, consistent with the limiting reversible case.
Why Other Options Are Wrong:
- T2/(T2 − T1) is for a heat pump COP, not a refrigerator.
- (T2 − T1)/T2 or /T1 and T1/T2 are not the Carnot refrigerator COP.
Common Pitfalls:Confusing refrigerator COP with heat pump COP; always confirm whether the numerator is Q_L or Q_H.
Final Answer:COP = T1 / (T2 - T1)