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

 

[alphy]

Dear Student,

Thank you for reaching out for help with this assignment. I am happy to assist you in understanding the neutron flux profile in a spherical moderator.

First, let's define the terms in the given equation. The neutron diffusion equation describes the behavior of neutrons in a medium, where D is the diffusion coefficient, Ʃa is the absorption cross section, and S is the neutron source rate. In this case, we are considering an ideal moderator with a zero absorption cross section, which means that neutrons are not being absorbed by the medium. This simplifies the equation to:

D∇2\phi = -S

Next, we need to simplify this equation for a spherical moderator in spherical coordinates. In spherical coordinates, the Laplacian operator (∇2) is given by:

∇2 = (1/r2) ∂/∂r (r2 ∂/∂r) + (1/r2 sinθ) ∂/∂θ (sinθ ∂/∂θ) + (1/r2 sin2θ) ∂2/∂φ2

where r is the radial distance, θ is the polar angle, and φ is the azimuthal angle. Substituting this into the neutron diffusion equation, we get:

D(1/r2) ∂/∂r (r2 ∂\phi/∂r) + D(1/r2 sinθ) ∂/∂θ (sinθ ∂\phi/∂θ) + D(1/r2 sin2θ) ∂2\phi/∂φ2 = -S

We can further simplify this equation by noting that the moderator is spherically symmetric, meaning that the neutron flux profile \phi only depends on the radial distance r. This means that the second and third terms on the left side of the equation become zero. Therefore, our simplified equation becomes:

D(1/r2) ∂/∂r (r2 ∂\phi/∂r) = -S

Next, we need to determine the appropriate boundary conditions for this problem. Since the moderator is spherically symmetric, we can assume that the neutron flux is also symmetric about the origin. This means that the neutron flux \phi at r=0 is equal to the neutron flux at the extrapolated radius R. Therefore, our boundary conditions are:

\