Heat pump vs. refrigerator relationship For the same device viewed as a heat pump and as a refrigerator, the coefficients of performance satisfy:
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A(C.O.P.)_HP = (C.O.P.)_R + 2
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B(C.O.P.)_HP = (C.O.P.)_R + 1
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C(C.O.P.)_HP = (C.O.P.)_R − 1
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D(C.O.P.)_HP = (C.O.P.)_R
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E(C.O.P.)_HP = 1 / (C.O.P.)_R
Answer
Correct Answer: (C.O.P.)_HP = (C.O.P.)_R + 1
Explanation
Introduction / Context:The same vapour-compression machine can be interpreted either as a refrigerator (cooling device) or as a heat pump (heating device). Their C.O.P.s are closely linked.
Given Data / Assumptions:
- Q_L = cooling effect, Q_H = heating effect, W = compressor work.
- (C.O.P.)_R = Q_L / W; (C.O.P.)_HP = Q_H / W.
- First law for the cycle: Q_H = Q_L + W.
Concept / Approach:Substitute the energy balance into the definitions to relate the performance ratios directly.
Step-by-Step Solution:Start: Q_H = Q_L + W.Divide by W: Q_H/W = Q_L/W + 1.Therefore, (C.O.P.)_HP = (C.O.P.)_R + 1.
Verification / Alternative check:Check a numerical example: if (C.O.P.)_R = 3, then (C.O.P.)_HP = 4. This matches the formula.
Why Other Options Are Wrong:Options A and C add or subtract the wrong constant; D ignores the energy balance; E is not generally true.
Common Pitfalls:Forgetting that a heat pump’s useful output includes both the cooling effect and the work input converted to heat at the hot side.
Final Answer:(C.O.P.)_HP = (C.O.P.)_R + 1