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A Implementing a weird-looking boundary condition (PDE/FDM)

  1. Apr 15, 2017 #1
    So I have this problem, taken from Kraus's heat transfer book.

    63ye6d.png

    So deriving the computational molecule, the conditions for (3.251a), (3.251b) is a bit of a no brainer. The issue I am having is about the boundaries for (3.251c) and (3.251d). This is actually the first time I have seen this kind of boundary condition.

    How do I deal with this? My hypothesis is to integrate the equations (lol) but the constants of integration stay around... I have no idea what to do at all. Bi and γ are constants. Thanks!
     
  2. jcsd
  3. Apr 15, 2017 #2
    Oh, to add.

    There's an analytical expression for the solution; and I am able to graph the resulting multivariate graph. I intend to do the numerical analysis as another way of solving it. The problem is, even if I implement the 'integrating the boundary condition' part or if I use the ghost point strategy, both of it does not work - the graph from the numerical solution is way too far from the analytical expression's graph.
     
  4. Apr 15, 2017 #3
    You're solving this by finite difference, correct?
     
  5. Apr 15, 2017 #4
    Yes sir. Finite difference.
     
  6. Apr 15, 2017 #5
    $$\theta (1+\Delta R, Z)-\theta (1-\Delta R, Z)=-2(\Delta R) Bi\ \theta (1,Z)$$
    so
    $$\frac{\partial ^2 \theta}{\partial R^2}=2\frac{(\theta (1-\Delta R, Z)-\theta (1,Z))-(\Delta R) Bi\ \theta (1,Z)}{(\Delta R)^2}$$
    or
    $$\frac{\partial ^2 \theta}{\partial R^2}=2\frac{(\theta (1-\Delta R, Z)-\theta (1,Z))}{(\Delta R)^2}-2\frac{Bi\ \theta(1,Z)}{\Delta R}$$
    or
    $$\frac{\partial ^2 \theta}{\partial R^2}=2\frac{(\theta (1-\Delta R, Z)-[1+(\Delta R) Bi]\theta (1,Z))}{(\Delta R)^2}$$
     
  7. Apr 15, 2017 #6
    Thanks for replying sir. Actually I did it already, but I seem to be getting erroneous results. Could you have my computational molecules for each boundary checked first?

    I intend to implement a solution similar to Gauss-Seidel iteration; thus I kept on factoring out the 'center' molecule.
     
    Last edited: Apr 15, 2017
  8. Apr 15, 2017 #7
    Sorry. I'll help with the finite differencing, but, as far as the debugging is concerned, you're on your own.
     
  9. Apr 15, 2017 #8
    Oh, it's alright sir.

    Apparently my computational molecules for the boundaries are correct; but the numerical solution is still given different values. I'm starting to think that the problem comes from the corners - problem is what should I do with the corners (0,1) and (1,1). What boundary should I invoke? Say for the upper left corner (0,1); is it 3.251b or 3.251d?
     
  10. Apr 15, 2017 #9
    Both
     
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