A wellinsulated house requires 138 MJ/h to keep the indoor
temperature comfortable on a cold day. Under this load, the heat pump compressor uses 7.7 kW of electrical power.
a.) Determine the COP of the heat pump

b.) If the heat pump operates 125 hours in a winter month, what will the
homeowner spend on electric heat that month? Residential electricity costs
$0.11/kWh.

c.) How much would the
homeowner spend that month if she had electrical resistance heating
instead of a heat pump? 


Read : 
This is a
straightforward application of the definition of the COP of a heat pump. 










Given: 
Q_{H} 
138 
MJ/h 

Find: 
COP_{HP} 
??? 



W_{HP}
= 
7.7 
kW 







Price 
0.11 
$/kWh 







OpTime 
125 
h/month 
















Diagram: 











Assumptions: 
1  
The heat pump operates at steadystate. 


2  
There is no loss of efficiency when the heat pump is started up or shut
down by the thermostatic control system. 










Equations
/ Data / Solve: 


















Begin by writing the
definition for the COP of a heat pump : 


















Eqn 1 











We can use Eqn 1 to evaluate the COP_{HP}. Watch the units ! 











Q_{H} 
38.33 
kW 



COP_{HP} 
4.98 












First let's see how
much it would cost to deliver
Q_{H} to the home using an electrical
resistance heater. 











An electrical
resistance heater, at best, converts all of the electrical work supplied, W, into heat released into the home, Q_{H}. Therefore, in order to get 38.33 kW into your home, you must
buy 38.33 kW of electrical power. 

















W_{resist} 
38.33 
kW 











Now, we can determine
how much it would cost to
operate the resistance heater for 125 hr/month. 













Eqn 2 

















Cost_{resist} 
527.08 
$/month 











Now, we can apply Eqn 2 to determine the cost of
operating the heat pump for a month. 

















Cost_{HP} 
105.88 
$/month 











The difference between
these two costs is the monthly savings: 
Savings 
421.21 
$/month 










Verify: 
We cannot verify the steadystate assumption or the
assumption about the thermostatic control system based on the information
given in the problem statement. 










Answers : 
COP_{HP} 
5.0 

Savings 
421 
$/month 














Electrical
resistance heaters are not very popular, especially in cold climates. 











The thermal efficiency of a heat pump drops significantly as the outside temperature falls. 











When the outside temperature drops far enough that the COP_{HP} ~ 1, it becomes more practical to use the resistance heater ! 









