Bondarenko method for obtaining group cross sections in a multi-region collision probability model
Highlights
► A concise derivation for how to determine group cross sections using the Bondarenko method. ► A general extension to multi-region collision probability models. ► The method is demonstrated by comparison to Monte Carlo simulations.
Introduction
Hundreds of thousands of energy grid points are required to fully resolve the energy dependence of neutron interaction cross sections. Even at the petascale, a brute force resolution of this dependence in Monte Carlo and discrete ordinate techniques is far from practical. As a result, reactor physics codes typically use a ‘multigroup’ formulation where the neutron spectrum is represented by a few tens to hundreds of energy groups. Multigroup cross sections are then required that preserve the correct in-group reaction rates. However, producing these cross sections is not straightforward because the presence of in-group resonances affects the flux. Being able to produce the correct group-averaged cross sections then requires knowing the energy dependent flux within a group, which is often the quantity being sought.
Furthermore, the attempt to model a heterogeneous system requires additional considerations. The traditional method for treating heterogeneity involves applying an equivalence relation to the background cross section of the Bondarenko method (Gopalakrishnan and Ganesan, 1998; Joo et al., 2009; Kidman et al., 1972; Schneider et al., 2006a; Stamm'ler and Abbate, 1983). The desire to model next generation reactors with more complex geometries has led to the use of subgroup methods (Chiba, 2003; Cullen, 1974; Herbert, 1997; Huang et al., 2011). Self-shielding methods primarily differ in the accuracy with which they attempt to approximate the neutron flux within a group, and are well described in the reviews by Hwang (1982) and Herbert (2007).
The subgroup method is used when modeling next-generation reactors whose complex geometry precludes the use of the Bondarenko method. In this work we are concerned with modeling simple pin-cell geometries, in which case the Bondarenko method provides sufficient accuracy. The use of the Bondarenko method for heterogeneous systems requires the use of an effective escape cross section that describes the probability that a neutron may escape a resonance by leaving a region. There are a number of ways by which the escape cross section can be obtained. The methods vary in complexity, but most use the Wigner rational approximation and the mean chord length of a region in order to obtain an expression for a collision probability (either the first-flight escape probability or the fuel escape probability). Sometimes this approximation is adjusted by a Dancoff factor, Bell factor or by replacing the Wigner rational approximation with an N-term expression such as the one by Carlvik (Stamm'ler and Abbate, 1983; MacFarlane and Muir, 1994; Herbert and Marleau, 1991; Yamamoto, 2008).
In the present contribution we provide a review and simplified derivation for the escape cross section using a collision probability model for the transport of neutrons from one reactor region to another. To demonstrate the accuracy of the approach we use the Bondarenko method to generate multigroup reaction and kernel cross sections which we use with an in-house collision probability spectral solver to obtain a neutron spectrum. We compare the predictions of neutron spectrum and reaction rates for simulated fast and thermal spectrum reactors to a published benchmark (Rowlands et al., 1999) which is commonly used in the analysis of self-shielding methods (Herbert, 2005) as well as to results produced using MCNPX 2.7.0.
Section snippets
Group cross sections
We define a group structure with G groups that span a range of energies from EG to E0{eV}. The g-th energy group spans an energy range from Eg to Eg−1. Here we adopt the convention that the groups are ordered in descending energy so that the highest energy is E0 and the lowest energy is EG, and in general Eg < Eg−1. The width of an energy group is ΔEg = Eg−1 − Eg and is different for every group. The microscopic and macroscopic group cross sections for any interaction are respectively denoted
Accuracy of the Bondarenko method
Cross sections generated using Eq. (17) give excellent results compared to other computational approaches. To illustrate this, Eq. (17) was used to generate cross sections for an in-house multi-region collision probability code based on the fully benchmarked two-region VBUDS code (Schneider et al., 2006b, 2007). Infinite dilution and self-shielded cross sections were generated using NJOY (MacFarlane and Muir, 1994) for an infinite lattice light water reactor running fresh uranium dioxide fuel (
Conclusions
We have provided a review and simplified derivation for the implementation of the Bondarenko method for obtaining group cross sections in multi-region collision probability models. We have used the results in an in-house collision probability model to show how the group cross sections obtained in this way compare to those generated from infinite lattice Monte Carlo simulations of thermal and fast spectrum reactors as well as with the Rowlands benchmark for a thermal spectrum system. The results
Acknowledgments
We would like to thank the United States Nuclear Regulatory Commission for grant NRC-38-08-946 which helped to support this work.
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The authors contributed equally to this work.