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The total work in a Carnot cycle - checking some work

  1. May 4, 2014 #1
    1. The problem statement, all variables and given/known data
    Let the radiant energy in a cylinder be carried through a Carnot cycle. The cycle consists of an isothermal expansion at temperature T, an infinitesimal adiabatic expansion where the temperature drops to T-dT, ad returning to the original state via an isothermal compression and adiabatic compression. What is the total work in the cycle? Assume P= u / 3 where u is a function of T only.


    2. Relevant equations

    To find the work in a Carnot cycle [itex] W= \int_{V_1}^{V_2} PdV[/itex], and a similar expression for points 2 to 3, 3 to 4 and 4 to 1.

    3. The attempt at a solution
    The expansion from V1 to V2 would therefore look like this:

    [itex]P = u/3 [/itex]
    so for the first leg of the cycle [itex]W = \int \frac{u}{3} dV[/itex] and since the first leg is isothermal that would mean P is constant. So [itex]W=\frac{1}{3}u(T)(V_2-V_1)[/itex]

    for the second leg of the cycle we can work from the total energy of the system. [itex]U = u(T)V[/itex]. And [itex]dU=u(T)dV + V\frac{du}{dT}dT[/itex]. Since the change of energy is equal to the work done, we can say [itex] W = u(T)dV + V\frac{du}{dT}dT[/itex] and if we add that to the first work expression and plug in the relevant values for V we have [itex] u(T)dV + (V_3-V_2)\frac{du}{dT}dT+ \frac{1}{3}u(T)(V_2-V_1)[/itex].


    Since the cycle's third and fourth leg look the same as the first two (except with opposite signs and T-dT in the temperature) we can add up all four and we have:

    [itex] -u(T)dV + (V_1-V_4)\frac{du}{dT}dT- \frac{1}{3}u(T)(V_4-V_3) + u(T)dV + (V_3-V_2)\frac{du}{dT}dT+ \frac{1}{3}u(T)(V_2-V_1).[/itex].

    The dT terms cancel out

    [itex] -u(T)dV + (V_1-V_4)du- \frac{1}{3}u(T)(V_4-V_3) + u(T)dV + (V_3-V_2)du+ \frac{1}{3}u(T)(V_2-V_1).[/itex].

    and the u(T)dV terms seem to as well. But I wanted to check if I was going about this the right way. The answer we are supposed to get has a du in the nominator...
     
  2. jcsd
  3. May 5, 2014 #2

    Andrew Mason

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    Can you use the efficiency of the Carnot engine operating between T and T-dT to determine the work produced as a function of dT and U?

    AM
     
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