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Pressure distrubution inside accelerating vehicle

  1. Mar 24, 2012 #1
    1. The problem statement, all variables and given/known data
    We have an accelerating vehicle with constant a. Air inside it with mass m.
    It is kind of closed box of length L and front area S.
    First objetive: we want to derive the steady state pressure distribution inside the box.
    Second objetive: we want to derive the transient pressure distribution inside the box.
    2vdr6nl.jpg
    2. Relevant equations
    My first guess was to assimilate the problem to an atmosfere with gravitatory field of g=a and use the barometric formula for calculating the pressure distribution.
    Second try is to use simplified Navier eq. but I don´t know how exactly...

    3. The attempt at a solution
    First try with barometric formula:
    Force over the mass of air inside: Fa = m a
    Reaction pressure over the rear side: Pa=Fa/S
    Assuming barometric distribution inside with g=a (M molar mass of air)
    P(x)=C exp( -M a x / (RT) )
    with initial condition at rear side P(0)=C=Pa
    This gives a very weak variation with lenght (i.e for 60m it just changes 0,1Pa) so I don´t trust this solution.

    Second try:
    Assuming 1D problem, incompressible fluid and no viscosity:
    dV/dt = f - (1/ro) dP/dx
    If steady state: f = (1/ro) dP/dx

    update: checking dimensinally it seems that f is an acceleration so it should be a, and it is in opposite sense of x so:
    dP = -ro a dx

    integrating P(x)=-ro a x + C,

    Now I know that in x=0, I have Pa= m a / S, so P(x)= - ro a x + m a /S

    With 1m/s2 this gives a linear gradient of pressure about 1.2Pa/m length of the vehicle.
    ¿it makes sense this result?
    I would appreciate any comment or help!
     
    Last edited: Mar 24, 2012
  2. jcsd
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