M.D. Eaton

The Inner-Element Subgrid Scale Finite Element Method for the Boltzmann Transport Equation

Abstract This paper presents a new multiscale radiation transport method based on a Galerkin finite element spatial discretization of the Boltzmann transport equation. The approach incorporates a discontinuous subgrid scale (SGS) solution within the continuous finite element representation of the spatial variables. While the conventional discontinuous Galerkin (DG) method provides accurate and numerically stable solutions that suppress unphysical oscillations, the number of unknowns is relatively high. The key advantage of the proposed SGS approach is that the solutions are represented within the continuous finite element space, and therefore, the number of unknowns compared with DG is relatively low. The applications of this method are explored using linear finite elements, and some of the advantages of this new discretization over standard Petrov-Galerkin methods are demonstrated. The numerical examples are chosen to be demanding steady-state mono-energetic radiation transport problems that are likely to form unphysical oscillations within numerical scalar flux solutions. The numerical examples also provide evidence that the SGS method has a thick diffusion limit. This method is designed to work under arbitrary angular discretizations, so solutions using both spherical harmonics and discrete ordinates are presented.

Space-dependent kinetics simulation of a gas-cooled fluidized bed nuclear reactor

In this paper we present numerical simulations of a conceptual helium-cooled fluidized bed thermal nuclear reactor. The simulations are performed using the coupled neutronics/multi-phase computational fluid dynamics code finite element transient criticality which is capable of modelling all the relevant non-linear feedback mechanisms. The conceptual reactor consists of an axi-symmetric bed surrounded by graphite moderator inside which 0.1 cm diameter TRISO-coated nuclear fuel particles are fluidized. Detailed spatial/temporal neutron flux and temperature profiles have been obtained providing valuable insight into the power distribution and fluid dynamics of this complex system. The numerical simulations show that the unique mixing ability of the fluidized bed gives rise, as expected, to uniform temperature and particle distribution. This uniformity enhances the heat transfer and therefore the power produced by the reactor.

Self-Adaptive Spherical Wavelets for Angular Discretizations of the Boltzmann Transport Equation

Abstract A new method for applying anisotropic resolution in the angular domain of the Boltzmann transport equation is presented. The method builds on our previous work in which two spherical wavelet bases were developed for representing the direction of neutral particle travel. The method proposed here enables these wavelet bases to vary their angular approximations so that fine resolution may be applied only to the areas of the unit sphere (representing the direction of particle travel) that are important. We develop an error measure that operates in conjunction with the wavelet bases to determine this importance. A procedure by which the angular resolution is gradually refined for steady-state problems is also given. The adaptive wavelets are applied to three test problems that demonstrate the ability of the wavelets to resolve complex fluxes with relatively few functions, and to achieve this a particular emphasis is placed on their ability to approximate particle streaming through ducts with voids. It is shown that the wavelets are capable of applying the appropriate resolution (as dictated by the error measure) to the directional component of the angular flux at all spatial positions. This method therefore offers a new and highly efficient adaptive angular approximation method.

Finite element-spherical harmonics solutions of the 3D Kobayashi benchmarks with ray-tracing void treatment

The finite element spherical-harmonics method is applied to the solution of the Kobayashi 3D benchmark problems. In particular, we evaluate the surface radiation exchange method based on raytracing which has been developed to circumvent the difficulty caused to the second-order, even-parity formulation by low-density regimes. This method brings several advantages which include obviating the need to explicitly solve the problem in voided regions and improving accuracy of the solution for a given order of angular approximation. Results produced by the computer code EVENT are presented for the six cases proposed. Comparisons with reference solutions show that the hybrid scheme can solve to reasonable accuracy most cases with relatively modest space-angle resolution. For the more difficult cases involving streaming in purely absorbing media, noticeable discrepancies were observed, but this indicated the need for a more judicious space-angle refinement (not attempted) rather than any deficiency of the hybrid scheme itself.

Discontinuous isogeometric analysis methods for the first-order form of the neutron transport equation with discrete ordinate (SN) angular discretisation

In this paper two discontinuous Galerkin isogeometric analysis methods are developed and applied to the first-order form of the neutron transport equation with a discrete ordinate ( S N ) angular discretisation. The discontinuous Galerkin projection approach was taken on both an element level and the patch level for a given Non-Uniform Rational B-Spline (NURBS) patch. This paper describes the detailed dispersion analysis that has been used to analyse the numerical stability of both of these schemes. The convergence of the schemes for both smooth and non-smooth solutions was also investigated using the method of manufactured solutions (MMS) for multidimensional problems and a 1D semi-analytical benchmark whose solution contains a strongly discontinuous first derivative. This paper also investigates the challenges posed by strongly curved boundaries at both the NURBS element and patch level with several algorithms developed to deal with such cases. Finally numerical results are presented both for a simple pincell test problem as well as the C5G7 quarter core MOX/UOX small Light Water Reactor (LWR) benchmark problem. These numerical results produced by the isogeometric analysis (IGA) methods are compared and contrasted against linear and quadratic discontinuous Galerkin finite element (DGFEM) S N based methods.

A model of heat transfer dynamics of coupled multiphase‐flow and neutron‐radiation

Purpose – To present dynamical analysis of axisymmetric and three‐dimensional (3D) simulations of a nuclear fluidized bed reactor. Also to determine the root cause of reactor power fluctuations.Design/methodology/approach – We have used a coupled neutron radiation (in full phase space) and high resolution multiphase gas‐solid Eulerian‐Eulerian model.Findings – The reactor can take over 5 min after start up to establish a quasi‐steady‐state and the mechanism for the long term oscillations of power have been established as a heat loss/generation mechanism. There is a clear need to parameterize the temperature of the reactor and, therefore, its power output for a given fissile mass or reactivity. The fission‐power fluctuates by an order of magnitude with a frequency of 0.5‐2 Hz. However, the thermal power output from gases is fairly steady.Research limitation/implications – The applications demonstrate that a simple surrogate of a complex model of a nuclear fluidised bed can have a predictive ability and has...

Finite Element Based Riemann Solvers for Time-Dependent and Steady-State Radiation Transport

Abstract A high-order, nonoscillatory scheme is described which solves the transient and steady-state Boltzmann transport equation on unstructured Finite Element (FE) meshes. Flux limiters are applied in the space and time domains resulting in a scheme which is both free from oscillations and globally high-order accurate in space and time. The method described is finite volume based and uses a consistent FE representation of the solution variables to obtain a high-order solution along the control volume boundaries. Careful inspection of the eigenstructure of the Riemann problem allows one to switch smoothly between a high-order and a low-order nonoscillatory solution.

An Investigation of Power Stabilization and Space-Dependent Dynamics of a Nuclear Fluidized-Bed Reactor

Abstract Previous work into the space-dependent kinetics of the conceptual nuclear fluidized bed has highlighted the sensitivity of fission power to particle movements within the bed. The work presented in this paper investigates a method of stabilizing the fission power by making it less sensitive to fuel particle movement. Steady-state neutronic calculations are performed to obtain a suitable design that is stable to radial and axial fuel particle movements in the bed. Detailed spatial/temporal simulations performed using the finite element transient criticality (FETCH) code investigate the dynamics of the new reactor design. A dual requirement of the design is that it has a moderate power output of ˜300 MW(thermal).

On an improved design of a fluidized bed nuclear reactor-I: Design modifications and steady-state features

Abstract This paper describes several modifications to the design of a fluidized bed nuclear reactor in order to improve its performance. The goal of these modifications is to achieve a higher power output, requiring an excess reactivity of 4% at maximum expansion of the bed. The modifications are also intended to obtain a larger safety margin when the reactor does not operate; a shutdown margin of 4% is required when the bed is in a packed state. The modifications include installing an embedded side absorber, changing the reactor cross-section area, and modifying the moderator-to-fuel ratio. The new design based on the modifications related to the aforementioned parameters achieves the desired shutdown margin and the excess reactivity. A model describing the coupling of neutronics and thermal/fluid dynamics is developed, and it is used to study the behavior of the reactor at steady conditions. The results show that the reactor can achieve a high output temperature of 1163 K and produce a thermal power of ~120 MW. Further, the results indicate that the power level of the reactor can be controlled easily by adjusting the flow of helium into the core without any further use of control rods or other active control mechanisms.

A Point Kinetics Model of the Medical Isotope Production Reactor Including the Effects of Boiling

Abstract Previously, a point kinetics model of the Medical Isotope Production Reactor has been presented, which included representations of instantaneous power, delayed neutron precursors, fuel solution temperature, radiolytic gas content, and coolant temperature. This model has been extended to include the effects of a vertically discretized temperature profile with a mixing of heat energy by eddies, boiling, and condensation and an extended model of bubble velocity and radius. It is found that the most striking change to the behavior of the system is caused by the effects of steam, which provides a strong negative feedback that tends to depress average powers in cases where the fuel solution temperature rises above the saturation temperature but can also lead to large, sharp power peaks through steam exiting the system (which can remove a large amount of negative reactivity in a short amount of time). The overall effect, however, does not lead to any unbounded power excursions. Possibilities for further extension of the model include the modeling of the composition of the plenum gas and the modeling of global pressure and its effects.