Solutions of the Diffusion Equation – Non-multiplying Systems
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 non-multiplying system, i.e., in system where fissile isotopes are missing and therefore the fission cross-section is zero. The neutrons are emitted by external neutron source. We will assume that the system is uniform outside the source, i.e. D and Σa are constants.
Solution for the Infinite Planar Source
Let assume the neutron source (with strength S0) as an infinite plane source (in y-z plane geometry). In this geometry the flux varies so slowly in y and z allowing us to eliminate the y and z derivatives from ∇2. The flux is then a function of x only, and therefore the Laplacian and diffusion equation (outside the source) can be written as (everywhere except x = 0):
For x > 0, this diffusion equation has two possible solutions exp(x/L) and exp(-x/L), which give a general solution:
Φ(x) = Aexp(x/L) + Cexp(-x/L)
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.
From finite flux condition (0≤ Φ(x) < ∞), that required only reasonable values for the flux, it can be derived, that A must be equal to zero. The term exp(x/L) goes to ∞ as x ➝∞ and therefore cannot be part of a physically acceptable solution for x > 0. The physically acceptable solution for x > 0 must then be:
Φ(x) = Ce-x/L
where C is a constant that can be determined from source condition at x ➝0.
If S0 is the source strength per unit area of the plane, then the number of neutrons crossing outwards per unit area in the positive x-direction must tend to S0 /2 as x ➝0.