5.46 By supplying energy at an average rate of 24,000 kJ/h, a heat pump maintains the temperature of a dwelling at 20°C. If electricity costs 8.5 cents per kW · h, determine the minimum theoretical operating cost for each day of operation if the heat pump receives energy by heat transfer from (a) the outdoor air at - 7°C. (b) the ground at 5°C.

A) $18.36

B) $10.24

Explanation:

Heat supplied Q = 24000 kJ/h

J/s = watt

I hour = 3600 sec, therefore 24000 kJ/hr becomes 24000/3600 = 6.67 kW

Coefficient of performance of heating for heat pump is

COP = T2 / (T2 - T1)

T2 (hot temperature) = 20 °C

T1 (cold temperature) = - 7 °C

COP = 20 / (20 - (-7)) = 20/27 = 0.741

Also COP = Q/W

where W is the power input to the heat pump

0.741 = 6.67/W

W = 9 kW

Operating for one day is 24 hr

Therefore heat pump uses 9 x 24 = 216 kWh

Cost per day = ¢8.5 x 216 = ¢1836 = $18.36

For ground at 5°C

COP = 20 / (20-5) = 20/15 = 1.33

1.33 = 6.67/W

W = 5.02 kW

For 24 hr it becomes 24 x 5.02 = 120.48 kWh

Cost = 8.5 x 120.48 = ¢1024.08 = $10.24