The subgroup decomposition method (SDM) has recently been developed as an improvement over the consistent generalized energy condensation theory for treatment of the energy variable in deterministic particle transport problems. By explicitly preserving reaction rates of the fine-group energy structure, the SDM directly couples a consistent coarse-group transport calculation with a set of fixed-source "decomposition sweeps" to provide a fine-group flux spectrum. This paper will outline the implementation of the SDM into the three-dimensional, discrete ordinates (SN) deterministic transport code TITAN. The new version of TITAN, TITAN-SDM, is tested using 1-D and 2-D benchmark problems based on the Japanese designed High Temperature Engineering Test Reactor (HTTR). In addition to accuracy, this study examines the efficiency of the SDM algorithm in a 3-D SN transport code. / Master of Science

Benchmark calculations are performed for neutron transport in a two material (binary) stochastic multiplying medium. Spatial, angular, and energy dependence are included. The problem considered is based on a fuel assembly of a common pressurized water nuclear reactor. The mean chord length through the assembly is determined and used as the planar geometry system length. According to assumed or calculated material distributions, this system length is populated with alternating fuel and moderator segments of random size. Neutron flux distributions are numerically computed using a discretized form of the Boltzmann transport equation employing diffusion synthetic acceleration. Average quantities (group fluxes and k-eigenvalue) and variances are calculated from an ensemble of realizations of the mixing statistics. The effects of varying two parameters in the fuel, two different boundary conditions, and three different sets of mixing statistics are assessed. A probability distribution function (PDF) of the k-eigenvalue is generated and compared with previous research. Atomic mix solutions are compared with these benchmark ensemble average flux and k-eigenvalue solutions. Mixing statistics with large standard deviations give the most widely varying ensemble solutions of the flux and k-eigenvalue. The shape of the k-eigenvalue PDF qualitatively agrees with previous work. Its overall shape is independent of variations in fuel cross-sections for the problems considered, but its width is impacted by these variations. Statistical distributions with smaller standard deviations alter the shape of this PDF toward a normal distribution. The atomic mix approximation yields large over-predictions of the ensemble average k-eigenvalue and under-predictions of the flux. Qualitatively correct flux shapes are obtained, however. These benchmark calculations indicate that a model which includes higher statistical moments of the mixing statistics is needed for accurate predictions of binary stochastic media k-eigenvalue problems. This is consistent with previous findings. / Graduation date: 2005

The subject of this thesis is the development of a nodal discretization of the low-order quasi-diffusion (QDLO) equations for global reactor core calculations. The advantage of quasi-diffusion (QD) is that it is able to capture transport effects at the surface between unlike fuel assemblies better than the diffusion approximation. We discretize QDLO equations with the advanced nodal methodology described by Palmtag (Pal 1997) for diffusion. The fast and thermal neutron fluxes are presented as 2-D, non-separable expansions of polynomial and hyperbolic functions. The fast flux expansion consists of polynomial functions, while the thermal flux is expanded in a combination of polynomial and hyperbolic functions. The advantage of using hyperbolic functions in the thermal flux expansion lies in the accuracy with which hyperbolic functions can represent the large gradients at the interface between unlike fuel assemblies. The hyperbolic expansion functions proposed in (Pal 1997) are the analytic solutions of the zero-source diffusion equation for the thermal flux. The specific form of the QDLO equations requires the derivation of new hyperbolic basis functions which are different from those proposed for the diffusion equation. We have developed a discretization of the QDLO equations with node-averaged cross-sections and Eddington tensor components, solving the 2-D equations using the weighted residual method (Ame 1992). These node-averaged data are assumed known from single assembly transport calculations. We wrote a code in "Mathematica" that solves k-eigenvalue problems and calculates neutron fluxes in 2-D Cartesian coordinates. Numerical test problems show that the model proposed here can reproduce the results of both the simple diffusion problems presented in (Pal 1997) and those with analytic solutions. While the QDLO calculations performed on one-node, zero-current, boundary condition diffusion problems and two-node, zero-current boundary condition problems with UO₂-UO₂ assemblies are in excellent agreement with the benchmark and analytic solutions, UO₂-MOX configurations show more important discrepancies that are due to the single-assembly homogenized cross-sections used in the calculations. The results of the multiple-node problems show similar discrepancies in power distribution with the results reported in (Pal 1997). Multiple-node k-eigenvalue problems exhibit larger discrepancies, but these can be diminished by using adjusted diffusion coefficients (Pal 1997). The results of several "transport" problems demonstrate the influence of Eddington functionals on homogenized flux, power distribution, and multiplication factor k. / Graduation date: 2003

Several computer codes based on one and two-group diffusion theory models were developed for SHUFFLE. The programs were developed to calculate power distributions in a two-dimensional quarter core geometry of a pressurized power reactor. The various coarse-mesh numerical computations for the power calculations yield the following: the Borresen's scheme applied to the modified one-group power calculation came up with an improved power distribution, the modified Borresen's method yielded a more accurate power calculations than the Borresen's scheme, the face dependent discontinuity factor method have a better prediction of the power distribution than the node averaged discontinuity factor method, Both the face dependent discontinuity factor method and the modified Borresen's methods for the two-group model have quite attractive features. / Graduation date: 1992

A condensed multigroup formulation is developed which maintains direct consistency with the continuous energy or fine-group structure, exhibiting the accuracy of the detailed energy spectrum within the coarse-group calculation. Two methods are then developed which seek to invert the condensation process turning the standard one-way condensation (from fine-group to coarse-group) into the first step of a two-way iterative process. The first method is based on the previously published Generalized Energy Condensation, which established a framework for obtaining the fine-group flux by preserving the flux energy spectrum in orthogonal energy expansion functions, but did not maintain a consistent coarse-group formulation. It is demonstrated that with a consistent extension of the GEC, a cross section recondensation scheme can be used to correct for the spectral core environment error. A more practical and efficient new method is also developed, termed the "Subgroup Decomposition (SGD) Method," which eliminates the need for expansion functions altogether, and allows the fine-group flux to be decomposed from a consistent coarse-group flux with minimal additional computation or memory requirements. In addition, a new whole-core BWR benchmark problem is generated based on operating reactor parameters in 2D and 3D, and a set of 1D benchmark problems is developed for a BWR, PWR, and VHTR core.

This work presents a whole-core benchmark problem based on a 2-loop pressurized water reactor with both UO₂and MOX fuel assemblies. The specification includes heterogeneity at both the assembly and core level. The geometry and material compositions are fully described and multi-group cross section libraries are provided in 2, 4, and 8 group formats. Simplifications made to the benchmark specification include a Cartesian boundary, to facilitate the use of transport codes that may have trouble with cylindrical boundaries, and control rod homogenization, to reduce the geometric complexity of the problem. These modifications were carefully chosen to preserve the physics of the problem and a justification of these modifications is given. Detailed Monte Carlo reference solutions including core eigenvalue, assembly averaged fission densities and selected fuel pin fission densities are presented for benchmarking diffusion and transport methods. Three different core configurations are presented in the paper namely all-rods-out, all-rods-in, and some-rods-in.

The generation of multigroup neutron cross sections is usually the first step in the solution of reactor physics problems. This typically includes generating condensed cross section sets, collapsing the scattering kernel, and within the context of diffusion theory, computing diffusion coefficients that capture transport effects as accurately possible. Although the calculation of multigroup parameters has historically been done via deterministic methods, it is natural to think of using the Monte Carlo method due to its geometric flexibility and robust computational capabilities such as continuous energy transport. For this reason, a stochastic cross section generation method has been implemented in the Mont Carlo code MCNP5 (Brown et al, 2003) that is capable of computing macroscopic material cross sections (including angular expansions of the scattering kernel) for transport or diffusion applications. This methodology includes the capability of tallying arbitrary-order Legendre expansions of the scattering kernel. Furthermore, several approximations of the diffusion coefficient have been developed and implemented. The accuracy of these stochastic diffusion coefficients within the multigroup framework is investigated by examining a series of simple reactor problems.

The following research contributes original concepts to the fields of deterministic neutron transport modeling and reactor power excursion simulation. A deterministic neutron transport code was created to assess the value of new methods of determining neutron current, fluence, and flux values through the use of view factor and average path length calculations. The neutron transport code is also capable of modeling the highly anisotropic neutron transport of deuterium-tritium fusion external source neutrons using diffusion theory with the aid of a modified first collision source term. The neutron transport code was benchmarked with MCNP, an industry standard stochastic neutron transport code. Deterministic neutron transport methods allow users to model large quantities of neutrons without simulating their interactions individually. Subsequently, deterministic methods allow users to more easily couple neutron transport simulations with other physics simulations. Heat transfer and thermoelastic mechanics physics simulation modules were each developed and benchmarked using COMSOL, a commercial heat transfer and mechanics simulation software. The physics simulation modules were then coupled and used to simulate reactor pulses in fast burst and externally driven nuclear assemblies. The coupled system of equations represents a new method of simulating reactor pulses that allows users to more fully characterize pulsed assemblies. Unlike older methods of reactor pulse simulation, the method presented in this research does not require data from the operational reactor in order to simulate its behavior. The ability to simulate the coupled neutron transport and thermo-mechanical feedback present in pulsed reactors prior their construction would significantly enhance the quality of pulsed reactor pre-construction safety analysis. Additionally, a graphical user interface is created to allow users to run simulations and visualize the results using the coupled physics simulation modules. / text

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In this dissertation, three different methods for solving the Boltzmann neutron transport equation (and its low-order approximations) are developed in general geometry and implemented in 1D slab geometry. The first method is for solving the fine-group diffusion equation by estimating the in-scattering and fission source terms with consistent coarse-group diffusion solutions iteratively. This is achieved by extending the subgroup decomposition method initially developed in neutron transport theory to diffusion theory. Additionally, a new stabilizing scheme for on-the-fly cross section re-condensation based on local fixed source calculations is developed in the subgroup decomposition framework. The method is derived in general geometry and tested in 1D benchmark problems characteristic of Boiling Water Reactors (BWR) and Gas Cooled Reactor (GCR). It is shown that the method reproduces the standard fine-group results with 3-4 times faster computational speed in the BWR test problem and 1.5 to 6 times faster computational speed in the GCR core. The second method is a hybrid diffusion transport method for accelerating multi-group eigenvalue transport problems. This method extends the subgroup decomposition method to efficiently couple a coarse-group high-order diffusion method with a set of fixed-source transport decomposition sweeps to obtain the fine-group transport solution. The advantages of this new high-order diffusion theory are its consistent transport closure, straight forward implementation and numerical stability. The method is analyzed for 1D BWR and High Temperature Test Reactor (HTTR) benchmark problems. It is shown that the method reproduces the fine-group transport solution with high accuracy while increasing the computationally efficiency up to 16 times in the BWR core and up to 3.3 times in the HTTR core compared to direct fine-group transport calculations. The third method is a new spatial homogenization method in transport theory that reproduces the heterogeneous solution by using conventional flux weighted homogenized cross sections. By introducing an additional source term via an “auxiliary cross section” the resulting homogeneous transport equation becomes consistent with the heterogeneous equation, enabling easy implementation into existing solution methods/codes. This new method utilizes on-the-fly re-homogenization, performed at the assembly level, to correct for core environment effects on the homogenized cross sections. The method is derived in general geometry and continuous energy, and implemented and tested in fine-group 1D slab geometries typical of BWR and GCR cores. The test problems include two single assembly and 4 core configurations. It is believed that the coupling of the two new methods, namely the hybrid method for treating the energy variable and the new spatial homogenization method in transport theory set the stage, as future work, for the development of a robust and practical method for highly efficient and accurate whole core transport calculations.