An inventor proposed two new designs The first is a heat pum

An inventor proposed two new designs. The first is a heat pump that operates between two reservoirs of 8.0 degree C and 22.0 degree C, and delivers 15.0 kW of heat using 0.6 kW of electricity. Should you decide to develop this design? The second design is a refrigerator. It receives energy Q_c = 1000 kJ from a cold reservoir at T_c = 255 K and rejects energy Q_H =2000 kJ to a hot reservoir at T_H = 315 K. Should you decide to develop this design? Show your calculations.

Solution

We should check whether the given machines obeys second law of thermodynamics.

Heat pump

Ideal coefficient of performance = T_H/(T_H-T_C) = 295/(295-281) = 21.07

Coefficient of performance = Q_H/W

Maximum heat delivered Q_H = 21.07 x 0.6 = 12.64 kW

But it is said that given engine gives a maximum 15 kW which cannot be possible. So we should not develop this device which can\'t be realized.

Refrigerator

For a refrigerator maximum Coefficient of performance realized is the carnot refrigerator.

COP = T_C/(T_H-T_C) = 255/(315-255) = 4.25

Actual COP = Q_C/(Q_H-Q_C)

4.25 = 1000 /(Q_H-1000)

Q_H = 1235.29 kW

So the maximum energy the carnot refrigerator can reject is 1235.3 kW. But the given machine is said to reject 2000 kJ of heat which can\'t be realized. So we should not develop the machine.