The transport of neutrons through a medium of a physical system is commonly described by the mathematical description called transport theory. In large nuclear systems, fine-mesh technique such as the finite-difference method becomes extraordinarily expensive in solving the integral or integro-differential form of the transport equation. Therefore, in the last few decades several nodal (coarse-mesh) methods have been developed and implemented for the solutions of the neutron transport problems.
The objective of this thesis is to develop a hexagonal nodal transport method for analysis of non-rectangular assembly cores such as the fast breeder reactor core. For this purpose, the author developed a new nodal $S_N$ method called the Source Projection Analytic Nodal Discrete Ordinates Method (SPANDOM) which has unique features as the following.
SPANDOM does not invoke the transverse integration procedure but instead directly solves the two-dimensional discrete ordinates equation after the source term is projected and represented in high-order polynomials and/or exponential functions. The solution of the discrete ordinates transport equation is decomposed into its particular and homogeneous parts. They are then analytically solved with boundary condition.
With regard to the unit node for source representation in SPANDOM, three approaches have been developed for the hexagonal geometry: Triangle approach (SPANDOM-TA), Half-Hexagon approach (SPANDOM-HH), and Full-Hexagon approach (SPANDOM-FH).
In order to validate the accuracy and applicability of SPANDOM, the three approaches have been tested on two fast reactor benchmark problems and the numerical results are compared with those of the TWOHEX code. The results of comparison indicate that the present SPANDOM predicts accurately not only the effective multiplication factor but also the flux distributions in non-rectangular cores with hexagonal assemblies, even in the region where flux varies very rapidly. The region-averaged flux errors of SPANDOM-TA and SPANDOM-HH are relatively smaller than and those of SPANDOM-FH are comparable to the corresponding TWOHEX-6Δ results obtained using a spatial mesh of six triangles per hexagon. In $k_{eff}$ calculations, the accuracy of SPANDOM is much better than that of TWOHEX-6Δ for both `rods-in' and `rods-out' cases of two fast reactor benchmark problems. The good performance of SPANDOM is mainly attributed to the fact that SPANDOM represents the source distribution with high-order polynomial and/or exponential functions and then directly solves the 2-D discrete ordinates equation.
Most of the $S_N$ methods employ an acceleration technique to avoid excessive computing time and SPANDOM uses the asymptotic source extrapolation method to accelerate the convergence of solution. The results of comparison of computation times indicate that SPANDOM is faster than TWOHEX-6Δ calculation by factors of 1.1 to 2.8.
SPANDOM is not based on the transverse integration procedure and thus easily overcomes the difficulty in the approximation of transverse leakage in the existing nodal methods, which grants SPANDOM to have geometrical flexibility for various geometries such as triangle, half-hexagon, and hexagon.
Through the investigation of the effect of reentering models between SPANDOM and TWOHEX, it is noted that significant difference (several percents) in fluxes between $S_N$ methods can be induced by the reentering models used in the boundaries of the problem where the reentrant nodes exist.
It is also worth noting that SPANDOM gives continuous two-dimensional intranodal scalar flux distributions in a hexagon but that the other $S_N$ methods which can treat hexagonal nodes give only the node-averaged quantities.
From the results of benchmark calculations, it is concluded that SPANDOM has the sufficient accuracy for the nuclear design of reactor cores with hexagonal assemblies such as the fast breeder reactor.
