Governing Equations

Neutron Transport

Gnat solves several governing equations. The first and chief among them is the multi-group neutron transport equation (NTE) with the fission sources omitted. The NTE has the following form:

where:

• is a position vector ().

• is the angular direction of travel ().

• is time ().

• is the neutron energy group.

• is the total number of groups.

• is the neutron angular flux for energy group ().

• is the neutron speed for a given energy group ().

• is the macroscopic neutron removal cross-section for group ().

• is the scattering cross-section and phase function from group into group , and from angle into angle
  ().

• is a group angular neutron source for group ().

The boundary conditions for Eq. (1) are the following:

where

• is the specular reflection direction ().

• is the incoming angular neutron flux for energy group ().

• is the specular reflection albedo (dimensionless).

• is the specularly reflected angular neutron flux for energy group ().

• is the boundary normal vector.

• is the boundary surface.

The unstabilized weak form of Eq. (1) is given by the following (with functional notation omitted for brevity):

The weak forms of Eq. (2) are given by the following (with functional notation omitted for brevity):

for the reflective boundary condition and

for the incoming flux boundary condition. The combined boundary condition has been decomposed into two boundary conditions for reflective boundaries and incoming flux bondaries such that . A special case of an incoming flux boundary condition where is the vacuum boundary condition.

The discrete ordiantes (S_n) method is used to discretize the angular variable in Eq. (3), Eq. (4), and Eq. (5). The scattering kernel is expanded in Legendre polynomials to yield a spherical harmonics representation for anisotropic scattering. These methods are discussed in the angular approach section.

Eq. (3) is known to be numerically unstable when solved using the finite element method, and therefore must be artificially stabilized. This is accomplished with two different options. The first is the self-adjoint angular flux (SAAF) method. The second is the upwinding method with discontinuous finite elements. Both of these approaches are discussed in the stabilization section.

Mass Transport and Activation

The second equation of state implemented by Gnat is the scalar transport equation for mobile isotopes:

where

• is the identifier for the current isotope.

• is the total number of isotopes.

• is the number density of isotope ().

• is the velocity of the fluid which isotope is immersed in ().

• is the decay constant of isotope ().

• is the microscopic absorbsion cross-section of isotope ().

• is the scalar (0th moment) of the neutron flux ().

• is the diffusion coefficient of isotope (), which may be a function of other field properties such as the fluid velocity.

• is the microscopic activation cross-section which results in the formation of isotope from isotope during neutron bombardment ().

• is the branching factor which results in the formation of isotope from isotope during a radioactive decay process.

The boundary conditions vary depending on the problem being modelled. The unstabilized weak form of Eq. (6) is given by the following (with functional notation omitted for brevity):

Similar to Eq. (3), Eq. (7) is known to be numerically unstable when solved with continuous finite elements. Stabilization of the isotope mass transport equation is accomplished with the streamline upwind/Petrov-Galerkin (SUPG) stabilization scheme. This approach is discussed in the the stabilization section.