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Heat pump and COP; second law analysis

Short answer

A heat pump moves heat from cold to hot using work input; COP = QH/W. The second law limits maximum COP to TH/(TH−TL) (Carnot), requires work input per Clausius statement, and leads to second-law efficiency η_II = COP/COP_max. Entropy generation must be non‑negative.

Define heat pump and its COP. Explain how second law of thermodynamics can be applied to analyze the performance of the heat pump.

Heat pump

A heat pump is a cyclic device that transfers heat from a low-temperature source (outside air, ground, water) to a high-temperature space (a house in winter), using external work. It keeps the warm space warm.

   Warm space T_H  <----- Q_H
                           ^
                         [ HP ] <---- W (input)
                           ^
   Cold source T_L  -----> Q_L

Coefficient of performance

COP_HP = desired output/required input = Q_H/W = Q_H/(Q_H − Q_L)

It is always greater than 1, because Q_H = Q_L + W. It is related to the refrigerator COP by COP_HP = COP_R + 1.

Second law analysis

  1. Clausius statement. Heat cannot flow from cold to hot without work. So a heat pump cannot work without a work input (W > 0), and its COP is finite.
  2. Maximum COP. The second law says no heat pump can be better than a reversible one working between the same two temperatures. For a reversible (Carnot) heat pump, Q_H/Q_L = T_H/T_L, so

COP_HP,max = T_H/(T_H − T_L)

and any real heat pump has COP_HP ≤ T_H/(T_H − T_L). The temperatures are absolute. 3. Minimum work. For a heating load Q_H, W_min = Q_H (T_H − T_L)/T_H. 4. Effect of temperatures. The smaller the temperature lift (T_H − T_L), the higher the COP. A heat pump is more efficient in mild weather. 5. Second law efficiency. η_II = COP_actual/COP_max measures how close a real pump is to the ideal one. The gap comes from irreversibilities (friction, finite ΔT in the heat exchangers, throttling). 6. Entropy. For the heat pump together with its two reservoirs, the entropy generated per cycle is S_gen = Q_H/T_H − Q_L/T_L ≥ 0, and S_gen = 0 only for the reversible pump.

Example: house at 293 K, outside at 273 K: COP_max = 293/20 = 14.65. A real unit with COP 4 has η_II = 27 %.

✓ Answer checked by Claude Opus 5.5 (high effort)

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