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 Line Source
Let assume the neutron source (with strength S0) as an isotropic line source situated in an infinite cylindrical geometry. This line source is placed at r = 0. In order to solve the diffusion equation, we have to replace the Laplacian by its cylindrical form:
Since there is no dependence on angle Θ and z-coordinate, we can replace the 3D Laplacian by its one-dimensional form, and we can solve the problem only in radial direction. The source is assumed to be an isotropic source. Since the flux is a function of radius – r only, the diffusion equation (outside the source) can be written as (everywhere except r = 0):
This differential equation is called the Bessel’s equation and it is well known to mathematicians. In this case, the solutions to the Bessel’s equation are called the modified Bessel functions (or occasionally the hyperbolic Bessel functions) of the first and second kind, Iα(x) and Kα(x) respectively.
For r > 0, this differential equation has two possible solutions I0(r/L) and K0(r/L), the modified Bessel functions of order zero, which give a general solution:
To find constants A and C we use the identical procedure as in the case of infinite planar source. From finite flux condition (0≤ Φ(r) < ∞), that required only reasonable values for the flux, it can be derived, that A must be equal to zero. The term I0(r/L) 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) = C.K0(r/L)
where C is a constant that can be determined from source condition at r ➝0.
If S0 is the source strength, then the number of neutrons crossing a cylinder outwards in the positive r-direction must tend to S0 as r ➝0.
So that the solution may be written: