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Method of characteristics to solve the hyperbolic PDE- population balance equation

  1. Apr 12, 2012 #1
    1. The problem statement, all variables and given/known data

    The PDE: ∂n/∂t + G∂n/∂L=0

    The initial condition: n(0,L)=ns

    The boundary condition: n(t,0)=B/G

    The parameter B and G above are dependent upon process conditions and change at each time. They can be calculated with adequate experimental data.

    2. Relevant equations

    I know that the characteristic equations for this PDE are:

    ∂n/∂s=0 (1)
    ∂L/∂s=G (2)
    ∂t/∂s=1 (3)

    Solving the above:

    n=n0 (4)
    t=s (5)
    L=Gt+L0 (6)

    3. The attempt at a solution

    From what I can see, the above equations suggest that along a characteristic curve given by equation (6), the population density, n, at size L0, travels along the size axis with rate of growth, G.

    The initial condition, n(0,L)=ns describes the population of particles over a given size range. So the initial data I have is a number of values of population density, n, at a number of sizes.
    Does the solution suggest that each initial value of population density, n corresponding to an initial size, L0 will stay constant along equation (6) and correspond to a different size in the next time step???? I hope I am clear in what im saying here.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
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