Two Carnot air conditioners, A and B, are removing heat from different rooms. Th

Question

Two Carnot air conditioners, A and B, are removing heat from different rooms. The outside temperature is the same for both rooms, 309.0 K. The room serviced by unit A is kept at a temperature of 294.0 K, while the room serviced by unit B is kept at 301.0 K. The heat removed from either room is 4330 J. For both units, find the magnitude of the work required and the magnitude of the heat deposited outside.

Explanation / Answer

The coefficient of performance for an heat pump in a cooling application is defined as amount of heat extracted from cold reservoir to work done to run the pump[1]: COP = Q_c/W => W = Q_c/COP For an ideal Carnot heat pump it takes its maximum value of: T_c /( T_h - T_c) For your two pumps COP_A = 294K/(309K - 294K) = 19.6 COP_A = 301K/(309K - 301K) = 37.625 (a) W_A = Q_c/COP_A = 4330J / 19.6 = 220.91 J (b) W_B = Q_c/COP_B = 4330J /37.625 = 115.08 J (c) Q_hA = W_A + Q_c = 220.91J + 4330J = 4550.91 J (c) Q_hB = W_B + Q_c = 115.08J + 4330J = 4445.08 J

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