(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 determine the coefficient of performance (COP) for a heat pump, you need to know the temperatures of the hot and cold reservoirs. The COP is given by the equation:
COP = Qh / W
Where:
COP is the coefficient of performance
Qh is the heat absorbed from the hot reservoir
W is the work done on the system
(a) In this case, the hot reservoir temperature is 50.0°C, which is equivalent to 323.15K, and the cold reservoir temperature is -20.0°C, which is equivalent to 253.15K.
To calculate the COP, substitute the temperatures into the equation:
COP = 1 / (1 - Tc / Th)
Where:
Tc is the temperature of the cold reservoir (253.15K)
Th is the temperature of the hot reservoir (323.15K)
So, the COP is:
COP = 1 / (1 - 253.15 / 323.15) = 4.62
Therefore, the best coefficient of performance for this heat pump is 4.62.
(b) To calculate the amount of heat in kilocalories pumped into the warm environment when a certain amount of work is put into the heat pump, you need to convert the work from Joules to kilocalories.
Given the work input of 3.60 x 10^7 J (10.0 kW·h), you can convert it to kilocalories using the conversion factor: 1 cal = 4.184 J and 1 kcal = 1000 cal.
First, convert the work from Joules to kilocalories:
Work in kcal = (3.60 x 10^7 J) / (4.184 J/cal) / 1000 = 8600 kcal
Therefore, the heat pumped into the warm environment is 8600 kcal.
(c) To compare the cost of producing the same amount of heat by a heat pump and by burning natural gas, you need to consider the cost per unit of energy.
Given:
Cost of work input: 10¢/kW·h
Cost of natural gas: 88.0¢/therm (where 1 therm = 1.055 x 10^8 J)
First, calculate the cost of work input:
Cost of work input = (10¢/kW·h) x (10 kW·h) = $1 (since 1¢ = $0.01)
Next, calculate the cost of producing the same amount of heat by burning natural gas:
Energy input in therms = (3.60 x 10^7 J) / (1.055 x 10^8 J/therm) = 0.341 therm
Cost of natural gas = (0.341 therm) x ($0.88/therm) = $0.30 (rounded to the nearest cent)
Therefore, the cost of producing the same amount of heat by the heat pump is $1, while the cost of producing the same amount of heat by burning natural gas is $0.30.
To compare the costs:
Cost of heat pump / Cost of natural gas = $1 / $0.30 = 3.33
Therefore, the cost of producing the same amount of heat by the heat pump is approximately 3.33 times higher than the cost of using natural gas.