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Homework Help: Imaginary Ideal Gas Cycle Proof

  1. Nov 3, 2011 #1
    1. The problem statement, all variables and given/known data
    Given an imaginary ideal-gas cycle. Assuming constant heat capacities, show that the thermal efficiency is

    η = 1 - γ[((V1/V2)-1)/((P3/P2)-1)]

    Since i cant show you the cycle we are shown that

    l Qh l = which is absolute value of the heat at high temperature = Cv(T3-T2)
    l QL l = which is absolute value of the heat at low temperature = Cp(T1-T2)
    Cp/Cv = γ

    3. The attempt at a solution

    Ok so subing in these equations for thermal efficiency
    which is

    η = 1 - l QL l / l Qh l

    we get...

    η = 1 - γ(T1 - T2)/(T3 - T2)

    η = 1 - γ((T1/T3) - 1)

    This imaginary cycle only has a power stroke and we are assuming that its adiabatic...from this we concluded that

    T1V1^(γ-1) = T3V2^(γ-1)

    divide each equation we get

    V1^(γ-1)/P2^((1-γ)/y) = V2^(γ-1)/P3^((1-γ)/y)

    Now im not sure how to rearrange from here to make T1/T3 = (V1/V2)/(P3/P2)

    Any suggestions would be greatly appreciative.
    Last edited: Nov 3, 2011
  2. jcsd
  3. Nov 3, 2011 #2
    Since no one has replied I'm assuming some are confused as to what I'm talking about so here is the ideal gas cycle that I need to calculate the thermal efficiency from.
    Here is the link to the picture of the cycle
    Last edited: Nov 3, 2011
  4. Nov 3, 2011 #3

    I like Serena

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    Homework Helper

    Hi jrklx250s! :smile:

    Did you already try to apply the ideal gas law PV=RT?
  5. Nov 3, 2011 #4
    Hi Serena,

    Yes I believe so when i calculated the adiabatic processes for the power stroke... which i concluded that they were

    T1V1^(γ-1) = T3V2^(γ-1)

    And Since I need to make T1/T3 = (V1/V2)/(P3/P2)

    This means that T1 = (V1*P2)

    and T3 = (V2*P3)

    not sure how to conclude these from the two equations above. And I know its a simple alegbraic rearrangement that Im missing here.
  6. Nov 3, 2011 #5

    I like Serena

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    Careful. Let's start with (V1/V2)/(P3/P2).
    With some fraction manipulations this is equal to (P2*V1) / (P3*V2).

    Looking at the diagram you posted I can see that P1=P2 and that V1=V3.
    Furthermore you have that for instance P1*V1 = R*T1.

    Perhaps you can use that?
  7. Nov 3, 2011 #6
    Haha wow...thank you serena I was making this so much more complicated than it was.

    Yea of course you can just conclude that since

    so therefore...


    solving for both T's


    sub this in my previous equation and we get...
    η= 1 - ((V1/V2)-1)/((P3/P2)-1)

    Thank you.
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