Non-classical transport with angular-dependent path-length distributions. 2: Application to pebble bed reactor cores

We describe an analysis of neutron transport in the interior of model pebble bed reactor (PBR) cores, considering both crystal and random pebble arrangements. Monte Carlo codes were developed for (i) generating random realizations of the model PBR core, and (ii) performing neutron transport inside the crystal and random heterogeneous cores; numerical results are presented for two different choices of material parameters. These numerical results are used to investigate the anisotropic behavior of neutrons in each case and to assess the accuracy of estimates for the diffusion coefficients obtained with the diffusion approximations of different models: the atomic mix model, the Behrens correction, the Lieberoth correction, the generalized linear Boltzmann equation (GLBE), and the new GLBE with angular-dependent path-length distributions. This new theory utilizes a non-classical form of the Boltzmann equation in which the locations of the scattering centers in the system are correlated and the distance-to-collision is not exponentially distributed; this leads to an anisotropic diffusion equation. We show that the results predicted using the new GLBE theory are extremely accurate, correctly identifying the anisotropic diffusion in each case and greatly outperforming the other models for the case of random systems.Comment: 24 pages, 11 figures; Version 3: shortened title, corrected typo

Non-classical transport with angular-dependent path-length distributions. 1: Theory

This paper extends a recently introduced theory describing particle transport for random statistically homogeneous systems in which the distribution function p(s) for chord lengths between scattering centers is non-exponential. Here, we relax the previous assumption that p(s) does not depend on the direction of flight \Omega; this leads to an extended generalized linear Boltzmann equation that includes angular-dependent cross sections, and to an extended generalized diffusion equation that accounts for anisotropic behavior resulting from the statistics of the system.Comment: 22 pages; Version 3: shortened title; corrected typo

The Non-Classical Boltzmann Equation, and Diffusion-Based Approximations to the Boltzmann Equation

We show that several diffusion-based approximations (classical diffusion or SP1, SP2, SP3) to the linear Boltzmann equation can (for an infinite, homogeneous medium) be represented exactly by a non-classical transport equation. As a consequence, we indicate a method to solve diffusion-based approximations to the Boltzmann equation via Monte Carlo, with only statistical errors - no truncation errors.Comment: 16 pages, 3 figure

Advanced Variance Reduction for Global k-Eigenvalue Simulations in MCNP

The "criticality" or k-eigenvalue of a nuclear system determines whether the system is critical (k=1), or the extent to which it is subcritical (k1). Calculations of k are frequently performed at nuclear facilities to determine the criticality of nuclear reactor cores, spent nuclear fuel storage casks, and other fissile systems. These calculations can be expensive, and current Monte Carlo methods have certain well-known deficiencies. In this project, we have developed and tested a new "functional Monte Carlo" (FMC) method that overcomes several of these deficiencies. The current state-of-the-art Monte Carlo k-eigenvalue method estimates the fission source for a sequence of fission generations (cycles), during each of which M particles per cycle are processed. After a series of "inactive" cycles during which the fission source "converges," a series of "active" cycles are performed. For each active cycle, the eigenvalue and eigenfunction are estimated; after N >> 1 active cycles are performed, the results are averaged to obtain estimates of the eigenvalue and eigenfunction and their standard deviations. This method has several disadvantages: (i) the estimate of k depends on the number M of particles per cycle, (iii) for optically thick systems, the eigenfunction estimate may not converge due to undersampling of the fission source, and (iii) since the fission source in any cycle depends on the estimated fission source from the previous cycle (the fission sources in different cycles are correlated), the estimated variance in k is smaller than the real variance. For an acceptably large number M of particles per cycle, the estimate of k is nearly independent of M; this essentially takes care of item (i). Item (ii) can be addressed by taking M sufficiently large, but for optically thick systems a sufficiently large M can easily be unrealistic. Item (iii) cannot be accounted for by taking M or N sufficiently large; it is an inherent deficiency due to the correlations between fission source estimates. In the new FMC method, the eigenvalue problem (expressed in terms of the Boltzmann equation) is integrated over the energy and direction variables. Then these equations are multiplied by J special "tent" functions in space and integrated over the spatial variable. This yields J equations that are exactly satisfied by the eigenvalue k and J space-angle-energy moments of the eigenfunction. Multiplying and dividing by suitable integrals of the eigenfunction, one obtains J algebraic equations for k and the space-angle-energy moments of the eigenfunction, which contain nonlinear functionals that depend weakly on the eigenfunction. In the FMC method, information from the standard Monte Carlo solution for each active cycle is used to estimate the functionals, and at the end of each cycle the J equations for k and the space-angle-energy moments of the eigenfunction are solved. Finally, these results are averaged over N active cycles to obtain estimated means and standard deviations for k and the space-angle-energy moments of the eigenfunction. Our limited testing shows that for large single fissile systems such as a commercial reactor core, (i) the FMC estimate of the eigenvalue is at least one order of magnitude more accurate than estimates obtained from the standard Monte Carlo approach, (ii) the FMC estimate of the eigenfunction converges and is several orders of magnitude more accurate than the standard estimate, and (iii) the FMC estimate of the standard deviation in k is at least one order of magnitude closer to the correct standard deviation than the standard estimate. These advances occur because: (i) the Monte Carlo estimates of the nonlinear functionals are much more accurate than the direct Monte Carlo estimates of the eigenfunction, (ii) the system of discrete equations that determines the FMC estimates of k is robust, and (iii) the functionals are only very weakly correlated between different fission generations. The FMC method was developed only late in the project and has to date received limited testing. Current work, which is taking place after the conclusion of this project, involves further development and testing for more complex and realistic problems. We expect that the FMC method will become a practical tool for Monte Carlo criticality simulations

A theoretical derivation of the Condensed History Algorithm

Although the Condensed History Algorithm is a successful and widely-used Monte Carlo method for solving electron transport problems, it has been derived only by an ad-hoc process based on physical reasoning. In this paper we show that the Condensed History Algorithm can be justified as a Monte Carlo simulation of an operator-split procedure in which the streaming, angular scattering, and slowing-down operators are separated within each time step. Different versions of the operator-split procedure lead to (O([Delta]s) and O([Delta]s2) versions of the method, where [Delta]s is the path-length step. Our derivation also indicates that higher-order versions of the Condensed History Algorithm may be developed.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/29827/1/0000174.pd

An anatomically realistic lung model for Monte Carlo‐based dose calculations

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/134911/1/mp7284.pd

Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes II

In a recent article (Larsen, Morel, and Miller, J. Comput. Phys. 69, 283 (1987)), a theoretical method is described for assessing the accuracy of transport differencing schemes in highly scattering media with optically thick spatial meshes. In the present article, this method is extended to enable one to determine the accuracy of such schemes in the presence of numerically unresolved boundary layers. Numerical results are presented that demonstrate the validity and accuracy of our analysis.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/27867/1/0000280.pd

A new computational approach to nuclear aerosol problems

The nonlinear kinetic aerosol equation, describing the time evolution of an aerosol distribution within a well-stirred container, is formulated in a mathematically "conservative" form. A numerical method is then developed for which conservation of mass is automatically satisfied. This procedure simplifies the derivation of conservative numerical schemes by reducing the number of approximations that must be made. Comparisons between an exact solution of the kinetic aerosol equation and numerical approximations show the following: numerical solutions based on the conservative form of the kinetic equation are more accurate and are obtained more efficiently than numerical solutions based on the standard "nonconservative" form of the kinetic equation.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/28914/1/0000751.pd

Quantization of setup uncertainties in 3‐D dose calculations

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/135022/1/mp8756.pd