Polytropic exponent, thermodynamics question

  1. 1. The problem statement, all variables and given/known data

    Air is compressed to 6% of its initial volume in the cylinder of a diesel engine. In the process, its pressure increased from 1 bar to 41 bar.

    Calculate the polytropic exponent for this process and comment on the associated heat transfer.

    2. Relevant equations

    PV^λ = constant

    3. The attempt at a solution

    Pinital (Vinital)^λ = Pfinal(Vfinal)^λ

    Pinitial = 1 bar
    P final = 41 bar
    V initial = 100%
    V final = 94%

    Well I'm pretty much not sure if I'm going about solving this (probably simple) problem in the right way. Anyone care to help?

    Also do you have to change the units from bar to pascals?

    Thanks dudes and dudettes
     
  2. jcsd
  3. rude man

    rude man 5,511
    Homework Helper
    Gold Member

    Whew! "Polytropic exponent!" Ya live & ya learn!

    OK, so you know p1 and V1, giving you one equation with unknowns C and n:
    p1V1^n = C
    Then, what are p2(p1) and V2(V1)? Write the equation involving pp2 and V2 in therms of p1 and V1. That's a second equation with two unknowns C and n.

    Solve pre h.s. algebra.

    Now for part 2: if you assume an ideal gas, you can compute T1 and T2 (assume w.l.o.g. 1 mole of air), use the 1st law to express ΔU = (const.)ΔT, then work = ∫pdV and finally ΔQ = ΔU + W.
     
    Last edited: Nov 30, 2011
  4. I don't quite understand, do I know V1? :S

    No volume is given in the question but can you just use percentage values instead?

    So I'll have

    P1V1^n = C

    and

    P2V2^n = C

    ?

    Thanks by the way rude man!
     
  5. Q_Goest

    Q_Goest 2,966
    Science Advisor
    Homework Helper
    Gold Member

    Hi Studios,
    You have the right equation to solve for the polytropic exponent, but you need to better define Vi and Vf. If "Air is compressed to 6% of its initial volume..." then can you write a very short equation that relates Vi and Vf? Something like Vi = Vf C where C is a constant? Once you do that, you can find initial and final conditions so you can solve for the polytropic exponent.

    Do you know how the polytropic exponent varies depending on heat flux? For an adiabatic (no heat transfer) process, the process is isentropic, so what do you think the polytropic exponent would be equal to in that case? For the case where temperature is constant, PV = mRT = constant, right? In that case, what would be the polytropic exponent and how would you describe the heat transfer to the air during the process?
     
  6. rude man

    rude man 5,511
    Homework Helper
    Gold Member

    Sorry, my post was totally incomplete.

    Q Goest is giving you good leads.

    One thing that bothers me is I think we're all assuming an adiabatic process, which was not given but maybe was so defined in the relevant chapter of the textbook. I believe that assumption is generally made, so maybe I'm just being picky. For that matter we weren't given authority to assumean ideal gas either - but what can one accomplish without that?

    I suppose if n computes to cp/cv for air then we're entitled to assume isentropy.
     
    Last edited: Dec 1, 2011
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