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Thermodynamics: two stage-compressor

  1. Sep 17, 2011 #1
    1. The problem statement, all variables and given/known data

    Consider a two stage compressor. The working fluid is air. The first stage compressor takes the air at P_1 and T_1. It compresses the air to the pressure P_2. Between the two compressors, the air is cooled at constant pressure from T_2 to T_1 (I believe this is the other way around but this is how its written). It is then compressed by the second compressor to the pressure P_3

    a) draw the process in P-v coordinates
    b) Determine the work per unit mass to run the compressor, as a function of P_1, P_2, P_3, T_1, gamma and c_p (specific heat at constant pressure).
    c) Determine the pressure, P_2, which minimizes the compressor work.

    2. Relevant equations

    Delta(U) = Q - W (heat to system - work by system) (1)
    PV=mRT, where R=287 kg/(mol K) and m = 1kg (2)

    du = c_v*dT (3)
    dh = (dq when P is constant) = c_p*dT (4)

    3. The attempt at a solution

    My P-v diagram is as shown in the (crude) paint drawing

    Now to find the work I use the definition of pressure-volume work along the paths A-B and C-D to obtain:

    W_AB = R*T_1*LN(p_1/p_2)
    W_CD = R*T_2*LN(p_2/p_3)

    using specific heats and applying (3) and (4) to (1)
    W_BC = (T_2 - T_1)*(c_p - c_v) = R*(T_2 -T_1)

    Now I imagine that the answer is the sum of the above. Unfortunatelly though I have been unable to get rid of that T_2 and presumably in the process get gamma to appear somewhere. If I could achieve that I would simply calculate dW/dp_2 == 0 and hopefully get an answer. I would like some help in getting rid of T_2 and any hints or comments on my work thus far.
     

    Attached Files:

  2. jcsd
  3. Sep 18, 2011 #2
    B should be P2 T2
    C should be P2 T1
     
  4. Sep 18, 2011 #3
    Ok! But does this mean I should treat A-B and C-D as adiabatic processes or should I not make such a simplifying assumption?

    To say the process is adiabatic would mean that the compressor, compresses the gas fast enough for there to be any heat transfer....
     
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