Solve Fick's Law Diffusion Problem with Mono-Energetic Neutron Sources

[afung]

The problem of the homework states:

Mono-energetic sources of neutrons emitting S neutrons/cm³ are distributed throughout an infinite moderator with diffusion coefficient, D (cm), and absorption cross section Ʃ_a (cm^-1)

Show that the neutron flux in the moderator is given by:

$\Phi$ = S/Ʃ_a

I know that the Diffusion equation is

dn/dt = S - Ʃ_a$\Phi$ + D$\nabla$²$\Phi$

For Steady-State the equation becomes

0= S - Ʃ_a$\Phi$ + D$\nabla$²$\Phi$

and $\nabla$J =0 if space independent thus giving the following equation as

dn/dt = S - Ʃ_a$\Phi$

So, my question to this problem is can you assume the following diffusion equation to be steady state and space independent to get this:

$\Phi$ = S/Ʃ_a ? or do I need to consider other variables?

 

[nilesh_pat]

Yes, you can assume the diffusion equation to be steady-state and space independent to get the result that the neutron flux in the moderator is given by \Phi = S/Ʃa. This can be derived by rearranging the steady-state diffusion equation as follows: 0 = S - Ʃa\Phi + D\nabla2\Phi Rearranging, we get: \Phi = (S + D\nabla2\Phi) / Ʃa Since \nablaJ = 0 if space independent, this simplifies to: \Phi = S / Ʃa