Diffusion Equation – Point Source

Solutions of the Diffusion Equation – Non-multiplying Systems

As was previously discussed the diffusion theory is widely used in core design of the current Pressurized Water Reactors (PWRs) or Boiling Water Reactors (BWRs). This section is not about such calculations but provides illustrative insights, how can be the neutron flux distributed in any diffusion medium. In this section, we will solve diffusion equations:

in various geometries that satisfy the boundary conditions discussed in the previous section.

We will start with simple systems and increase complexity gradually. The most important assumption is that all neutrons are lumped into a single energy group. They are emitted and diffuse at thermal energy (0.025 eV).

In the first section, we will deal with neutron diffusion in a non-multiplying system, i.e., in a system where fissile isotopes are missing, the fission cross-section is zero. The neutrons are emitted by an external neutron source. We will assume that the system is uniform outside the source, i.e., D and Σ_a are constants.

Solution for the Point Source

Let us assume the neutron source (with strength S₀) as an isotropic point source situated in spherical geometry. This point source is placed at the origin of coordinates. To solve the diffusion equation, we have to replace the Laplacian with its spherical form:

We can replace the 3D Laplacian with its one-dimensional spherical form because there is no dependence on an angle (whether polar or azimuthal). The source is assumed to be an isotropic source (there is the spherical symmetry).  The flux is then a function of radius – r only, and therefore the diffusion equation (outside the source) can be written as (everywhere except r = 0):

If we make the substitution Φ(r) = 1/r ψ(r), the equation simplifies to:

For r > 0, this differential equation has two possible solutions exp(r/L) and exp(-r/L), which give a general solution:

Note that B is not usually used as a constant because B is reserved for a buckling parameter. To determine the coefficients A and C, we must apply boundary conditions.

To find constants A and C, we use the identical procedure as an infinite planar source. From finite flux condition (0≤ Φ(r) < ∞), which required only reasonable values for the flux, it can be derived that A must be equal to zero. The term exp(r/L)/r goes to ∞ as r ➝∞  and therefore cannot be part of a physically acceptable solution for r > 0. The physically acceptable solution for r > 0 must then be:

Φ(r) =  Ce^-r/L/r

where C is a constant that can be determined from source condition at x ➝0.

If S₀ is the source strength, then the number of neutrons crossing a sphere outwards in the positive r-direction must tend to S₀ as r ➝0.

So that the solution may be written:

 

See above:

Diffusion Theory