Example 1
Calculate the ideal coefficient of performance (COP) for an airtoair heat pump used to maintain the temperature of a house at 70 °F when the outside temperature is 30 °F.Solution:
First, convert the Fahrenheit temperatures to Celsius temperatures using this formula:
$${T}_{hot}=\left(7032\right)\times \frac{5}{9}={21}^{o}C$$ $${T}_{cold}=\left(3032\right)\times \frac{5}{9}={1}^{o}C$$Next, convert the Celsius temperatures to Kelvin temperatures by adding 273.
$${T}_{hot}={21}^{o}\text{}C\text{}+\text{}273\text{}=\text{}294K$$$${T}_{cold}={1}^{o}C+273\text{}=\text{}272K$$
Finally, use the formula from the previous screen to solve for the COP.
$$COP=\left(\frac{{T}_{hot}}{{T}_{hot}{T}_{cold}}\right)$$
$$COP=\left(\frac{294K}{294K272K}\right)=\frac{294}{22}=13.3$$
The example above shows that for every watt of power we use (and pay for) to drive this ideal heat pump, 13.3 W is delivered to the interior of the house and 12.3 from the outside (we don’t pay for this). This seems to be a deal that one cannot refuse. However, the theoretical maximum is never achieved in the real world. In practice, a COP in the range of 2 to 6 is typical. Even with this range, it is an excellent choice, because for every watt of power that we use, we transfer 1 to 5 additional watts from outside.
Example 2
Compare the ideal coefficients of performance of the same heat pump installed in State College, PA and Ann Arbor, MI when the inside temperature of a house is maintained at 70°F at both locations and the outside temperatures on a given day were 40°F and 15°F at State College and Ann Arbor, respectively.
State College, PA  Ann Arbor, MI 

T_{hot}= 70 ºF = 21 ºC = 294 K  T_{hot}= 70 ºF = 21ºC = 294 K 
T_{cold} = 40 ºF = 4 ºC = 277 K  T_{cold} = 15 ºF = 9.4 ºC = 264 K 
$$COP=\frac{{T}_{hot}}{{T}_{hot}{T}_{cold}}\left(\frac{294}{294272}\right)$$

$$COP=\frac{{T}_{hot}}{{T}_{hot}{T}_{cold}}\left(\frac{294}{294264}\right)$$ 
=17.3  = 9.8 
During a heating season, the heat pump's efficiency increases on mild days and decreases on cold days.