(a) What is the best coefficient of performance for a heat pump that has a hot r

Question

(a) What is the best coefficient of performance for a heat pump that has a hot reservoir temperature of 60.0°C and a cold reservoir temperature of -20.0°C? (b) How much heat in kilocalories would it pump into the warm environment if 3.60 107 J of work (10.0 kW·h) is put into it? kcal (c) Assume the cost of this work input is 10¢/kW·h. Also assume that the cost of direct production of heat by burning natural gas is 87.0¢ per therm (a common unit of energy for natural gas), where a therm equals 1.055 108 J. Compare the cost of producing the same amount of heat by each method. (cost of heat pump / cost of natural gas)

Explanation / Answer

a)
Tc = - 20 oC = 253 K
Th = 60 oC = 333 K
COP = Th/(Th - Tc)
= 333/(333 - 253)
= 4.1625

b)
Work done, W = 3.6 x 107 J
Q/W = COP
Q is the heat supplied
Q = W x COP
= 4.1625 x .6 x 107 J
= 14.985 x 107 J
1 kcal = 4184 J
1 J = 1/4184 kcal
14.985 x 107 J = (14.985 x 107)/4184
= 3.58 x 104 kcal

c)
Cost of heat pump = 10 ¢/kW.h
Input work = 3.6 x 107 J = 10 kW.h
Total cost of input work by heat pump = 10 kW.h x 10 ¢/kW.h
= 100 ¢

Cost of natural gas = 87.0 ¢ per therm
= 87.0 ¢ per 1.055 x 108 J
When the input work of 3.6 x 107 J is supplied,
cost = (3.6 x 107 J ) x 87.0 ¢ per 1.055 x 108 J
= 29.687 ¢

Cost of heat pump / cost of natural gas = 100 ¢ / 29.687 ¢
= 3.368

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