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Thermodynamics, maximum work

  1. Apr 22, 2012 #1

    fluidistic

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    Gold Member

    1. The problem statement, all variables and given/known data
    Each of three identical bodies has an equation of state U=NCT, with NC=2 cal/K. Their initial temperatures are 200 K, 250 K and 540 K. What is the maximum amount of work that can be extracted in a process in which these three bodies are brought to a final common temperature?


    2. Relevant equations
    Not sure.


    3. The attempt at a solution
    I know that the efficiency isn't constant during the entire process because the temperature of the bodies change.
    I've made a sketch with the following characteristics: An amount of heat [itex]dQ_1[/itex] leaves the body whose initial temperature is 540 K (let's call it body 1 with temperature [itex]T_1[/itex]). This amount of heat enters a machine that produces an amount of work [itex]dW_1[/itex] and transmit an amount of heat [itex]dQ_2[/itex] to the body (let's call it body 2 with temperature [itex]T_2[/itex]) that is initially at a temperature of 250 K. A similar process occur between the body 2 and 3.
    I think that the relation between [itex]dW_1[/itex], [itex]dQ_1[/itex] and [itex]dQ_2[/itex] is the following in case of maximum efficiency/maximum work produced:
    [itex]dW_1=-dQ_1 \left ( 1-\frac{T_2}{T_1} \right ) =-CdT_1\left ( 1-\frac{T_2}{T_1} \right )[/itex].
    Also [itex]dQ_2=dQ_1-dW_1 =CdT_1+CdT_1 \left ( 1-\frac{T_2}{T_1} \right )[/itex] and [itex]cdT_2=dQ_2-dQ_3=CdT_1 \left ( 2-\frac{T_2}{T_1} \right ) \Rightarrow dT_2=dT_1 \left ( 2-\frac{T_2}{T_1} \right )[/itex].
    Now I'm not sure what I've done is relevant nor do I know how to proceed further. I must get [itex]dW_1[/itex] and [itex]dW_2[/itex] (amount of work produced by a machine operating between body 2 and 3).
     
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