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Can an analytical solution for PRKE be found with ramp reactivity ?

  1. Jul 14, 2006 #1
    Hi there,

    Before i state what my actual question is, let me give some background on some work i have done on this. I have been trying to solve the 0-D PRKE transient problem for a step and a ramp input numerically using couple of high precision methods.

    1) Using MATLAB's inbuilt ode23s solver.
    2) Using a custom written adaptive solver

    My problem is that i want to find the order of accuracy or maybe even just the exact error in my adaptive method in comparison to the actual true solution. Without the exact analytical solution, i would have to assume that the solution from MATLAB is exact and compute the error. But i am reluctant to do that because i want to test few embedded numerical methods with higher orders of accuracy than matlab's 2-3 method.

    So now, i did derive the analytical solution for PRKE with a step input (constant reactivity) and tested my method with the solution i computed analytically. It perfectly matches !

    But when the reactivity is a function of time (Ramp or say parabolic change), i end up with an integro-differential form which does require approximating the integral numerically, which in turn will introduce truncation errors in the calculation. Now is there any way to get around this ? OR have i completely missed out some other alternative ?

    I would appreciate any and all help that someone can provide. Thanks in advance !
     
  2. jcsd
  3. Jul 14, 2006 #2

    Astronuc

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    Staff: Mentor

    Please provide the differential equation for the PK model.
     
  4. Jul 14, 2006 #3
    Please see my attached document file for the equations and for the question in more detail. Please let me know if something is confusing !
     

    Attached Files:

  5. Jul 16, 2006 #4

    Astronuc

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    Staff: Mentor

    I think an analytical solution might be possible, and could be found using Laplace transforms.

    For a ramp, [itex]\rho(t)\,=\,\rho_o\,+\,\gamma\,t[/itex], but [itex]\rho_o[/itex] could be zero.

    The problem in the attachment shows one precursor (or an effective precursor) concentration. Is this correct?

    Numerically, one can use the Runge-Kutta method for solving first order differential equations.

    Interesting paper - "The reactor point-kinetics equations: semi-analytical methods versus numerical methods"
    http://sab.sscc.ru/imacs2005/papers/T2-I-72-0937.pdf
    See bottom of page 4 for discussion of Runge-Kutta method.
     
    Last edited: Jul 16, 2006
  6. Jul 16, 2006 #5
    Astronuc,

    Thanks for the reply.

    Yes. I have shown only one precursor group in the document because i just wanted to simplify the system. For the analytical solution though, i do not think it makes any difference but just couple more equations to solve to obtain the coefficients.

    I have been trying to use Laplace transforms for the PRKE but since my math with transforms is a bit rusty, it is taking me a while. But it is good to know that i am going in the right direction !

    And thanks for the great reference document. I am sure it will help me out further in my current research.

    I will try to derive the analytical solution and if not, i might have to go with the numerical solution with a very high accurate numerical scheme. I will let you know what i stumble onto along the way and post some interesting observations when i have my result.

    Thanks again !
     
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