A leaflet for an air-source heat pump says that

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · The laws of thermodynamics · The laws of thermodynamics

Thermal Physics, Ch Heat engines, refrigerators and the Carnot cycle

Problem

A leaflet for an air-source heat pump says that, with the house held at

T_h = 300\,\mathrm{K}

and the outside air at

T_c = 270\,\mathrm{K}

, the unit delivers

Q_h = 6.00\,\mathrm{kJ}

of heat into the house for every

W = 0.400\,\mathrm{kJ}

of electricity it consumes, and adds that "the textbook bound does not apply, because our refrigerant is a real vapour that boils and condenses, not an ideal gas, so for us

Q_h/Q_c

is not

T_h/T_c

". Treat the house interior and the outside air as reservoirs, and all heats and works as magnitudes. ((a)) Find the coefficient of performance

\mathrm{COP}_{\rm hp} = Q_h/W

the leaflet claims, and the largest value a heat pump between these two reservoirs can have. Is the claim possible? ((b)) Suppose the leaflet's pump X did exist, and let a reversible engine R working forwards between the same two reservoirs supply exactly the

0.400\,\mathrm{kJ}

per cycle that X consumes. Find the heat R takes from the house and the heat it rejects to the outside air, then add up what each reservoir gains over one cycle of the pair. How much heat would the pair move from the outside air to the house with no other effect, and which statement of the second law does that contradict? ((c)) State the value of

Q_h/Q_c

any reversible device between these reservoirs must have, and refute the leaflet's claim about the refrigerant.

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