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Homework Help: Laplace Transforms on Partial Differential Equations - Non-dimensionalization too

  1. Jul 26, 2007 #1
    Laplace Transforms on Partial Differential Equations - Non-dimensionalization too!!!

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

    The experiment described in the previous problem was analyzed from the point of view of long time [itex]\left(\frac{D_{AB}\,t}{L^2}\;>>\;1\right)[/itex]. We wish to reconsider this analysis in order to derive a suitable relationship for short time conditions [itex]\left(\frac{D_{AB}\,t}{L^2}\;<<\;1\right)[/itex]. For short time conditions, the effect of concentration-dependent diffusivity is minimized. In the analysis to follow, use dimensionless independent variables


    Apply Laplace transforms to the non-dimensional transport equation to show that

    [tex]\bar R\left( s \right)\; = \;\frac{1}{s}\; - \;\frac{1}{{s^{\frac{3}{2}} }}\;\left[ {\frac{{1\; - \;e^{ - 2\,\sqrt s } }}{{1\; + \;e^{ - 2\,\sqrt s } }}} \right][/tex]

    from the "fraction solute A remaining" equation


    2. Relevant equations

    PDEs, Non-dimensionalization, Laplace Transforms.

    A hint is given that we must obtain an expression for the Laplace transform of the composition [itex]x_A[/itex] that appears in the PDE below

    [tex]\frac{{\partial x_A }}{{\partial t}}\; = \;D_{AB} \,\frac{{\partial ^2 x_A }}{{\partial z^2 }}[/tex]

    with initial and boundary conditions




    3. The attempt at a solution



    Using the first equation to substitute into the PDE above

    [tex]\frac{{\partial x_A }}{{\partial t}}\; = \;\frac{\theta\,L^2}{t} \,\frac{{\partial ^2 x_A }}{{\partial z^2 }}[/tex]

    But what do I do about the squared partial derivative of z in the last term?

    I have been thinking about this problem for hours, I just don't know where to start in order to derive the equation that is requested. Any help would be greatly appreciated!
    Last edited: Jul 26, 2007
  2. jcsd
  3. Jul 26, 2007 #2


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    Well, how do [tex]\partial t[/tex] and [tex]\partial z[/tex] nondimensionalise?
  4. Jul 26, 2007 #3
    I don't know, how do I figure that out?
    Last edited: Jul 26, 2007
  5. Jul 26, 2007 #4


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    What's [tex]\partial z/\partial \zeta[/tex]?
    Last edited: Jul 26, 2007
  6. Jul 26, 2007 #5
    Let's focus on [tex]\partial z[/tex] first!

    How would I get that knowing that [tex]z\,=\,\zeta\,L[/tex]?

    Is [tex]\partial z\;=\;0[/tex]?
  7. Jul 26, 2007 #6


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    Sorry, the earlier post should've been [tex]\partial z/\partial \zeta[/tex].

    This is how you get [tex]\partial z[/tex]...
  8. Jul 26, 2007 #7


    And for the other "dimensionless independent variable"



    Now get an expression to change the variable from z to [itex]\zeta[/itex]



    Now substitute the two change of variable expressions above into the hint equation from the top




    now take a Laplace transform


    and set [itex]x(\theta\,=\,0)\;=\;x_0[/itex] to get a second order ODE


    Last edited: Jul 27, 2007
  9. Jul 26, 2007 #8
    Solving the ODE


    Now use the R equation?

    Last edited: Jul 26, 2007
  10. Dec 4, 2008 #9
    Just wondering whether we have to dimensionless everything you solve PDE. Is it possible to solve the PDE with the DAB in it.
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