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Thermodynamic Relation

  1. Sep 5, 2010 #1
    1. The problem statement, all variables and given/known data

    A gas enters a compressor and is compressed isentropically. Does the specific enthalpy (h) increase or decrease as the gas passes from inlet to exit?

    2. Relevant equations

    [tex]\left(\frac{\partial{h}}{\partial{p}} \right)_s= v\qquad(1)[/tex]

    3. The attempt at a solution

    Since the specific volume v is a positive number we know that pressure increases (since it is being compressed), then the enthalpy must also increase.


    This is the answer that was given in the book. I don't really like it. The left side of (1) is a differential change and hence the right hand side is a single value. When we extend this idea to a finite change, what happens to the right hand side?

    Does anyone see what I mean by "I don't like it?" We are looking at values of h and p at two different states 1 and 2. But what the heck is v supposed to do?

    I feel like to get the full story, we would need to integrate dh = v(p) dp. We know that v should decrease with an increase in p. Any thoughts?
     
  2. jcsd
  3. Sep 6, 2010 #2

    CompuChip

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    Science Advisor
    Homework Helper

    Personally I always find these thermodynamic problems tricky, but I suppose that you could take your equation (1) and integrate it from the initial to the final pressure:

    [tex]
    \int_{p_1}^{p_2} \left(\frac{\partial{h}}{\partial{p}} \right)_s \, \mathrm{d}p = \int_{p_1}^{p_2} v \, \mathrm{d}p
    [/tex]

    It follows (insert stuff about fundamental theorem of calculus here) that
    [tex]h_2 - h_1 = v (p_2 - p_1) [/tex]
    or, more compactly,
    [tex]\Delta h = v \Delta p[/tex]

    Then compression means that [itex]\Delta p > 0[/itex] (the final pressure is higher than the initial one) so [itex]\Delta h > 0[/itex] (the enthalpy increases).

    Feel better now?
     
  4. Sep 6, 2010 #3
    Hi CompuChip!

    I actually don't just yet since we know that v varies as well as h and p. It might not matter though seeing as it is always positive, but I still feel like it should be more thorough. It will depend on the difference of the product v2p2 - v1p1 I think. And thus it will depend on how v varies with p. I am just trying to see if there is something general we can say without knowing explicitly how v varies with p.
     
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