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

  • #1
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11
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!
 

Answers and Replies

  • #2
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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
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Yes sir. Finite difference.
 
  • #5
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4,424
$$\theta (1+\Delta R, Z)-\theta (1-\Delta R, Z)=-2(\Delta R) Bi\ \theta (1,Z)$$
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
$$\frac{\partial ^2 \theta}{\partial R^2}=2\frac{(\theta (1-\Delta R, Z)-[1+(\Delta R) Bi]\theta (1,Z))}{(\Delta R)^2}$$
 
  • #6
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11
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:
  • #7
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4,424
Sorry. I'll help with the finite differencing, but, as far as the debugging is concerned, you're on your own.
 
  • #8
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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?
 
  • #9
20,569
4,424
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?
Both
 

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