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Sound Speed in a two phase mixture of gas and liquid

  1. Jan 27, 2009 #1
    1. The problem statement, all variables and given/known data

    A two phase mixture of gas and liquid (small air bubbles dispersed in water for example) may be treated as a continuum for the transmission of sound of long wavelengths. The liquid behaves as a heat reservoir, and pressure changes are approximately isothermal. Let rv be the ratio of gas volume to liquid volume and rm the ratio of gas mass to liquid mass

    rv = Vg / Vl

    rm = Mg / Ml

    If liquid compressibility is neglected, show that the sound speed is given approximately by:

    c^2 = [ (1 + rv)^2 * P ] / ( rv * rhol)

    Note: rhol = density of liquid = liquid mass / liquid volume

    2. Relevant equations

    Previous questions were to demonstrate these equations:

    rho = rhol * (1 + rm) / (1+ rv)

    P * rv = rhol * rm * R * T

    Note: rho density of the two phase mixture (ratio of mass of liquid and gas to volume of liquid and gas)
    rhol = density of liquid = liquid mass / liquid volume

    In the problem, pressure changes being isothermal, means the temperature won't vary when pressure is changed (as weel and the volume of the liquid as liquid compressibility is neglected)

    3. The attempt at a solution

    Apparently there is a simple physical way to solve this problem. Doing 2 drawing one with normal pressure and another with more pressure and the same liquid volume but only changes in the volume of gas.
    I tried to find the relation of the variation of pressure to the variation of volume as it will almost give me directly the solution as:
    c^2 = (d P) / (d rho)

    I also tried to find teh derivatives of the equations above from the equations in the part "relevant equations" but I was rapidly stuck (I may lack some mathematical abilities)
  2. jcsd
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