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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)
    T1P2^((1-γ)/y)=T3P3^((1-γ)/y)

    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.
    Thanks!
     
    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
    http://imageshack.us/photo/my-images/411/img1048u.jpg/
     
    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)
    T1P2^((1-γ)/y)=T3P3^((1-γ)/y)

    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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    Homework Helper

    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
    P1=P2
    V2=V3

    so therefore...
    P1V1=nRT1
    P3V3=nRT3

    P2V1=nRT1
    P3V2=nRT3

    solving for both T's

    T1=P2V1/nR
    T3=P3V2/nR

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

    Thank you.
     
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