Friday, April 14, 2006

HW #6 - P #1 - "Show That" Problem Using the K-P Statement of the 2nd Law - 6 pts

Using the Kelvin-Planck Statement of the Second Law of Thermodynamics, demonstrate the following corollaries.
a.) The COP of an irreversible heat pump cycle is always less than the COP of a reversible heat pump cycle when both cycles exchange heat with the same two reservoirs.
b.) All reversible heat pump cycles operating between the same two reservoirs have the same COP.

2 comments:

Anonymous said...

Ok, What exactly are we supposed to do on this one? Like the hint suggested, i drew two heat pumps absorbing from a single cold reservoir, and excreting in to the same hot reservoir. One of them has a longer work arrow going in, because it wastes some of this work irreversibly to make the same ammount of Qh. so simply by the definition of COP, Qh/W we can see that the irreversible cop is less. Where does the KP statement come in?

Dr. B said...

aiite 8:28 PM

Setup the 2 HP's so that they take in the same amount of heat from the cold reservoir, Qc. Assume the Irrev HP requires LESS work input than the rev HP because it has a higher COP. Then, when you reverse the reversible HP it becomes an HE that rejectes Qc to the cold reservoir. As a result, there is zero net heat transfer to or from the cold reservoir. Under these circumstance the cold reservoir can be combined with the HPirrev and the HE to form a new system that completely converts heat to work ! That violates the K-P statement of the 2nd Law. Conclude that it is not possible for an irrev HP to have a higher COP than a rev HP.

I know that is a lot to get in one paragraph, but that is how you always approach this type of problem. Assume that something that contradicts what you are trying to show. Then, show that this violates the 2nd Law. Conclude that the original statement cannot be false, so it must be true. Don't worry too much about the possibility of an equality.