(a) What is the best coefficient of performance for a heat pump that has a hot reservoir temperature of 50.0°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 multiplied by 107 J of work (10.0 kW·h) is put into it?
(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 88.0¢ per therm (a common unit of energy for natural gas), where a therm equals 1.055 multiplied by 108 J. Compare the cost of producing the same amount of heat by each method. (cost of heat pump / cost of natural gas)
a. Tc=253.15k, Th=323.15
COP=1/(1-253.15/323.15)=4.62
To find the coefficient of performance (COP) for a heat pump, we need the temperatures of the hot reservoir (Th) and the cold reservoir (Tc).
(a) Given: Th = 50.0°C and Tc = -20.0°C.
To find the COP, we use the formula:
COP = 1 / (1 - Tc / Th)
Substituting the values:
COP = 1 / (1 - (-20.0°C) / 50.0°C)
COP = 1 / (1 + 0.4)
COP = 1 / 1.4
COP ≈ 0.714
Therefore, the COP for the given heat pump is approximately 0.714.
(b) To find how much heat the heat pump would pump into the warm environment when given a certain amount of work input, we need to use the formula:
Heat Pumped = Work Input / COP
Given: Work Input = 3.60 x 10^7 J (or 10.0 kW·h)
First, convert the work input from kW·h to Joules:
Work Input = 10.0 kW·h x 3.6 x 10^6 J/kW·h
Work Input = 3.60 x 10^7 J
Substituting the values:
Heat Pumped = 3.60 x 10^7 J / 0.714
Heat Pumped ≈ 5.04 x 10^7 J
To convert the heat pumped from Joules to kilocalories, we use the conversion factor:
1 kilocalorie = 4184 J
Heat Pumped = 5.04 x 10^7 J / 4184 J/kcal
Heat Pumped ≈ 12045 kcal
Therefore, the heat pump would pump approximately 12045 kilocalories of heat into the warm environment.
(c) To compare the cost of producing the same amount of heat by a heat pump and burning natural gas, we need their respective costs per unit of energy.
Given: Cost of work input = 10¢/kW·h
Cost of natural gas = 88.0¢ per therm (1.055 x 10^8 J)
First, calculate the cost of the work input for the heat pump:
Cost of work input = Work Input x Cost per unit of energy
Cost of work input = 3.60 x 10^7 J x 10¢/kW·h
Since 1 kW·h = 3.6 x 10^6 J:
Cost of work input = (3.60 x 10^7 J) / (3.6 x 10^6 J/kW·h) x 10¢
Cost of work input = 10¢ x 10
Cost of work input = 100¢
Next, calculate the cost of producing the same amount of heat by burning natural gas:
Cost of natural gas = Heat Pumped x Cost per unit of energy
Cost of natural gas = 12045 kcal x (88.0¢/1.055 x 10^8 J)
Since 1 therm = 1.055 x 10^8 J:
Cost of natural gas = 12045 kcal x (88.0¢/1.055 x 10^8 J) x (1.055 x 10^8 J/therm)
Cost of natural gas ≈ (12045 x 88.0) / 1.055
Finally, we can compare the costs:
Cost of heat pump / Cost of natural gas = Cost of work input / Cost of natural gas
Substituting the values:
Cost of heat pump / Cost of natural gas = 100¢ / ((12045 x 88.0) / 1.055)
Simplifying:
Cost of heat pump / Cost of natural gas ≈ (100 x 1.055) / (12045 x 88.0)
Cost of heat pump / Cost of natural gas ≈ 0.0957
Therefore, the cost of producing the same amount of heat by the heat pump is approximately 9.57% of the cost of producing heat by burning natural gas.