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deebrwnfn11
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Homework Statement
Solve for the flux distribution using the 1D neutron diffusion equation in a finite sphere for a uniformly distributed source emitting S0 neutrons/cc-sec.
My problem right now is that I can't figure out the boundary conditions for this problem. We usually work with point sources in infinite domains when working with spherical geometry, so I am unfamiliar with setting up boundary conditions for finite spheres.
Homework Equations
The governing differential equation is:
[itex]
\frac{1}{r^2} \frac{d}{dr} r^2 \frac{d\phi}{dr} - \frac{1}{L^2}\phi(r) = \frac{-S_0}{D}
[/itex]
with a boundary condition:
[itex]
\phi(r=R) = 0
[/itex]
This problem is going to be turned into a computer code, so we were told not to use extrapolated boundary conditions. I just need help finding the boundary condition for r = 0
As I mentioned above, we've never worked with finite spheres before so I am not 100% certain that the solution involves sin and cos. Maybe sinh and cosh?
[itex]
\phi(r) = \frac{C_1}{r}sin(\frac{r}{L}) + \frac{C_2}{r}cos(\frac{r}{L}) + \frac{S_0L^2}{D}
[/itex]
The Attempt at a Solution
Physically, I know that the flux profile will be flat at the center of the sphere. I can't impose that the derivative equals zero because that would lead to a zero as the denominator in the coefficient terms. I've considered using a limit as r → 0, but that would cause the coefficient terms to shoot up to infinity. I don't think I can say that the neutron current at the center is equal to some value because there is a distributed source.
I am completely lost in finding this boundary condition. I think what I have written so far for the equations is right. If someone could point me in the right direction, or enlighten me of a mistake I have made I would be forever grateful.