Reversed Carnot refrigerator: The coefficient of performance (COP) in terms of absolute temperatures T1 (low) and T2 (high) is

Difficulty: Easy

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)

Discussion & Comments

No comments yet. Be the first to comment!
Join Discussion