A geothermal
heat pump absorbs 15
Btu/s of heat from the Earth 50 ft
below a house. This heat pump uses a 10 hP compressor.
a.) Calculate the COP of the heat pump.
|
|
b.) In the summer, the cycle is reversed to cool the house. Calculate the COP of the cycle when it is operated as an air-conditioner assuming the working fluid rejects 15 Btu/s to the Earth. |
|
|
|
|
|
Read : |
Here we must apply the
definition of COP for both refrigerators and heat pumps. |
|
|
|
|
|
|
|
|
Given: |
W |
10 |
hP |
|
QC |
15 |
Btu/s |
|
|
Find: |
a.) |
COPHP |
??? |
|
b.) |
COPR |
??? |
|
|
|
|
Diagram: |
|
|
|
|
|
|
|
|
|
|
|
|
|
Assumptions: |
1 - |
The heat pump and the refrigerator operate at steady-state. |
|
|
|
|
|
|
|
|
|
|
Equations
/ Data / Solve: |
|
|
Part a.) |
If the purpose is to cool the groundwater, then the device is a refrigerator. |
|
|
|
|
|
|
|
|
|
|
|
|
|
So, let's begin with
the definition of the coefficient of performance for a refrigerator. |
|
|
|
|
|
|
Eqn 1 |
|
|
|
We are given the
values of both QC and W,
so all we need to do is make the units consistent and then plug values into Eqn 1. |
|
|
|
Conversion Factors : |
1 hP = |
2545 |
Btu/h |
|
W |
25450 |
Btu/h |
|
|
QC |
54000 |
Btu/h |
|
|
|
|
|
|
|
COPR |
2.122 |
|
|
Part b.) |
If the purpose is to heat a
building, then the device is a heat pump. |
|
|
So, let's begin with
the definition of the coefficient of performance for a heat pump. |
|
|
|
|
|
Eqn 2 |
|
|
|
Next, we apply the 1st Law to the heat pump cycle, keeping in mind
that we traditionally do not use our sign convention when tie-fighter diagrams are used. The arrows on the diagrams indicate the direction that heat and work are moving. |
|
|
|
|
|
Eqn 3 |
|
|
|
Use Eqn 3 to eliminate QH from Eqn 2 to get : |
|
|
|
|
|
Eqn 4 |
|
|
|
Now, we can plug in
the numbers in consistent units that we used in part (a). |
|
|
|
COPHP |
3.122 |
|
|
|
Notice that : |
|
|
|
Eqn 5 |
|
|
|
This is always true for Carnot Cycles. |
|
|
Verify: |
No assumptions to verify that were
not given in the problem statement. |
|
|
|
|
|
|
|
|
|
|
Answers : |
COPR |
2.12 |
|
|
|
|
COPHP |
3.12 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|