Neutron Flux Profile in a Spherical Moderator

[Israakaizzy]

Hello People
I need help with the following assignment:
It states:
Consider an ideal moderator with zero absorption cross section, Ʃ_a = 0, and a diffusion coefficient, D, which has a spherical shape with an extrapolated radius, R. If neutron sources emitting S neutrons/cm³sec are distributed uniformly throughout the moderator, the steady neutron diffusion equation is given by,
D∇²\phi -Ʃ_a\phi=-S

a) Simplify the above neutron diffusion equation for this moderator in spherical coordinates and state the appropriate boundary conditions.

By solving the simplified diffusion equation, obtain the neutron flux profile, \phi(r).

I know I need to divided the neutron diffusion equation and cancel out the absorption cross section and end up with something like:
∇²\phi = -S/D
and the particular solution would be something like S/Ʃ_a
but what's the general solution to:
D∇²\phi =0
in spherical coordinates?

 

[hmeier]

Israakaizzy,

I think it should be something like Asinh(λ.r)/r + Bcosh(λ.r)/r, applying the border conditions B=0.

hope it helps,

Hernán

 

[Israakaizzy]

ok
Just explain me what is \lambda equal to? Is it 1/L ?

 

[hmeier]

Israakaizzy,

You are right, it should be 1/L if Ʃa were different than 0.

I did the maths for the homogeneous part:
∇^{2}\phi=0

saying that:

\phi=\frac{\widehat{\phi}}{r}

The Lapplacian inspherical coordintates turns:

∇^{2}\phi=\frac{∂^{2}\widehat{\phi}}{∂r^{2}} + \frac{2}{r}\frac{∂\widehat{\phi}}{∂r}

proposing an exponential solution:

λ^{2}e^{λr} + \frac{2}{r}λe^{λr} = 0

So:

λ= -\frac{2}{r}

and finally:

\phi= \frac{A}{r} + B

Don´t forget that for the inhomogeneous part you have to use the Lapplacian in sphericals.

Regards,

Hernán