A heat pump operates reversibly to maintain a house at 21 degree C during winter

Question

A heat pump operates reversibly to maintain a house at 21 degree C during winter. With the proper working fluid inside the serpentine tube of the heat pump, it is possible to transfer 30 kW of heat from the outside air at 32 degree F. What is the COP of the heat pump, gamma max? How much on a rate basis in kW is the heat pump capable of transferring heat to the house? What is the weekly electrical bill in Massachusetts if the heat pump runs an average of 12 hours per day? Use the average price for the Boston-Brockton area: www.bls.gov/rol/cpibosap.pdf What is the real bill if the actual COP is only 25% of the ideal COP in order to account for irreversibilities?

Explanation / Answer

Thot = 21oC = 294.15 K

Tcold = 32oF = 0oC = 273.15 K

COP = Thot/(Thot -Tcold) = 14

also

Qhot/Thot = Qcold/Tcold

put Qcold = 30kW and values of temperatures

Qhot = 32.306 kW

weekly usage of heater = 12*7 hours = 84 hours

and

work input to heat pump = 32.306-30 = 2.306 kW

energy consumed = (2.306*84) = 193.704 kWh

price = $0.151/kWh

so weekly electricity bill 0.151*193.704 = $29.25

new COP = 0.25*14 = 3.5

COP = Qhot/(Qhot-Qcold)

put COP = 3.5 and Qcold =30 kW

Qhot = 42 kW

Work input to pump = 42-30 = 12 kW

energy consumed = 12*12*7 = 1008 kWh

new bill = 1008*0.151 = $152.21

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