Spatial Expansion Functions Research Articles

Consistent spatial expansion functions on hexagonal surfaces for use in radiation transport methods

Some of the radiation transport solvers that are based on expansion function methods in hexagonal geometry generally use projection techniques that transform the spatial dependence of the angular flux in hexagonal geometry to Cartesian geometry by mapping and then expanding in terms of expansion functions (e.g., Legendre polynomials). However, this approach usually requires a relatively higher-order expansion in space for accurate representation of the spatial distribution of the flux. To overcome this limitation, a set of consistent spatial expansion functions have been developed in hexagonal geometry in this paper. These functions satisfy the orthogonality condition on hexagonal surfaces and guarantee the conservation of currents after expansion regardless of the truncation order. The latter is an important property for preservation of neutron balance.The new spatial expansion method has been implemented into the coarse mesh radiation transport (COMET) code and tested in both the single assembly and the full core Advanced High Temperature Reactor (AHTR) benchmark problems. The eigenvalues and assembly-averaged and stripe-wise fission density distributions predicted by the new method with 2nd order spatial expansion functions were found to exhibit similar accuracy as those computed by the current mapping method with 4th order spatial expansion functions. This indicates that the angular fluxes in AHTRs can be predominantly represented by a low order expansion if the new spatial expansion functions are used. As a result, to maintain the same accuracy, the computational efficiency of COMET is improved significantly (2–5 times) by using the new spatial expansion method.

Spatial expansion of the transport equation

The solution of the monoenergetic transport equation in slab geometry is expanded in terms of known spatial functions and unknown angular coefficients. Two general formalisms, a variational method and a moment-conservation method, are developed to obtain the necessary equations for the angular coefficients. The latter formalism is emphasized since it always yields neutron conservation. Further, it is shown that this formalism treats exactly any incident flux boundary condition, including the discontinuous vacuum boundary condition. For small systems with a highly peaked directional flux, a spatial expansion of the directional flux in integer powers of z (the spatial co-ordinate) is shown to yield extremely good results. Thus a polynomial expansion forms a complement to the widely used angular expansions, such as the P-N (spherical harmonic) method, which are most accurate for large systems with an almost isotropic directional flux. To emphasize the fact that the formalisms developed are applicable to any spatial expansion functions, the penetration (of a normal beam) problem is considered with exponential expansion functions. The analysis is shown to reduce to the exact transport result in all known limits.