Neutron Flux in a sub-critical reactor

[Oxlade]

Hi everyone,

I am supposed to derive the neutron flux equation provided for region A of my reactor. Just wondering if anyone can help me out since I stuck on the derivation for [1/r - 1/R2]; S/4πD aspect is very similar to a solving the constant for a point source spherical reactor

Here is my reactor diagram consisting of nested spheres (sorry, this is my first time posting):

Sphere 1 --> Lead target with radius of R0 (Target-region)
Sphere 2 --> Actinide Fuel surrounding lead target; has outer radius of R1 (A-region)
Sphere 3 --> Reflector surrounding the fuel with outer radius of R2 (E-region)

Neutron Flux vanished at R2 and the flux at A-region is given as:

Φ(r) = S/4πD [1/r - 1/R2]My attempts at deriving this equation always leaves me with exponential terms if I'm assuming I'm solving the diffusion equation for a finite spherical reactor with a point source. Can anyone help me?

 

[Astronuc]

Assuming symmetry, the current at the center of the Pb is zero. The current at the boundaries/interfaces on one side must equal that on the other side.

 

[raining Pete]

Hi there,

I'm not an expert in this topic, but I'll try my best to help. From what I understand, you are trying to derive the neutron flux equation for region A of your reactor. This equation is given as Φ(r) = S/4πD [1/r - 1/R2], where S is a constant, D is the diffusion coefficient, r is the distance from the center of the reactor, and R2 is the outer radius of the reflector.

To start, let's consider the diffusion equation for a finite spherical reactor with a point source:

∇²Φ(r) + B²Φ(r) = Sδ(r)

Where ∇² is the Laplacian operator, B² is the diffusion coefficient squared, Φ(r) is the neutron flux, and Sδ(r) is the point source term. To simplify things, let's assume that B² is constant and equal to D, and that the point source is located at the center of the reactor (r=0).

Integrating this equation over the volume of the reactor, we get:

∫∇²Φ(r) dV + ∫B²Φ(r) dV = ∫Sδ(r) dV

Using the divergence theorem, the first term on the left side becomes:

∫∇²Φ(r) dV = ∮ Φ(r) dA = 4πR²Φ(R)

Where R is the outer radius of the reactor. The second term on the left side is just D times the integral of Φ(r) over the volume of the reactor, which is just D times the total neutron flux.

Putting everything together, we get:

4πR²Φ(R) + DΦ = S

Solving for Φ, we get:

Φ(r) = S/4πD [1 - (R/r)²]

This equation is similar to the one given for region A, except that it is for the entire reactor and it includes a term for the distance from the center of the reactor. However, we can see that when r=R2, the neutron flux vanishes, which is consistent with the information given in the problem.

I hope this helps and that I didn't make any mistakes. Let me know if you have any questions or if anything is unclear. Good luck with your derivation!