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Question About Mass and Mass Flow Rate Relations

  1. Nov 19, 2008 #1
    Mass is the product of density and volume.

    Mass flow rate is the product density, velocity, and cross-sectional area. (It's the derivative of mass with respect to time.)


    Bare with the syntax please...

    Looking at a sphere within a larger sphere, the volume of the difference is V=(4/3)*pi*(R^3-r^3) where R is a constant inner radius of the larger sphere. r is the radius of the smaller sphere, and it's not constant.

    Multiplying that by density gives us the mass of the region in between the spheres.


    Taking the derivative of that mass, however, with respect time, how would the equation look in relation to the description I gave above in the second line?

    This isn't homework as I'm not in school anymore. It's a personal question.

    It's a situation where the smaller sphere is increasing in radius as time is increased.
     
    Last edited: Nov 19, 2008
  2. jcsd
  3. Nov 20, 2008 #2
    Well, since the bigger sphere is kept constant, as least that's what I am assuming from the start....

    You leave the constants in front, including the R.

    So. dM/dt = -4*pi*r^2*density
     
  4. Nov 20, 2008 #3
    Out of curiosity, if the inner sphere is getting larger then is the mass exiting the annular space between the spheres or is the mass being compressed? Or does it matter...? dM/dt implies a rate of change of mass with respect to time. If mass is not exiting or entering the space between the spheres then dm/dt =0... Perhaps I've misunderstood the question.
     
  5. Nov 20, 2008 #4
    To first reply, yes, the outer sphere is constant.

    It matters. Adiabatic compression is taking place. Even though it matters, such effect will be taken into account once I have a rate equation in terms of mass.

    The effect will reflect on density.
     
  6. Nov 21, 2008 #5
    Problem statement: A spherical shell of radius r1, where: { r1 = f(t) ; i.e. r1 is a function of time} is enclosed inside a spherical shell of constant radius R. A fluid (e.g. gas, liquid) fills the region in between, with density rho. Assume f'(t) > 0 and compute the density rho, where: {rho = g(r,t) ; i.e. rho is a function of time and the radial distance from the origin}.

    Is this correct? I started to derive this but I want to clarify a few assumptions first. If we really want to do this, we should do it right.

    0.) What are the initial conditions? (initial density distributions, pressures, etc.)
    1.) What is the rate of motion of the inner sphere? How fast does it move, and does it have an acceleration?
    2.) Shall we assume the density in the spherical "annulus" is uniform? (I.e. the density is constant throughout the annulus.)
    3.) The fluid in between has constant transport properties; no thermodynamic change in state.

    Which assumptions would you like to consider? I don't think I could solve the problem without assumption (2). Also, I would need assumption (3), or else we would have to go into some thermodynamic analysis of the gas.

    Remark 1: Note the assumption of (2) implies we need not consider compression wave effects.
    Remark 2: Without assumption (3), the compression of the gas in the annulus causes a pressure gradient to develop inside the sphere pointing towards the center of the sphere; i.e. a force develops which tends to resist the encroachment of the inner sphere.

    This should give us a more clear understanding.
     
    Last edited: Nov 21, 2008
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