Modeling and Simulation of the Transient Reactor Test Facility using Modern Neutron Transport Methods by Travis J. Labossi`ere-Hickman B.S., University of Tennessee, Knoxville (2016) Submitted to the Department of Nuclear Science & Engineering in partial fulfillment of the requirements for the degree of Master of Science in Nuclear Science & Engineering at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2019 ○c Massachusetts Institute of Technology 2019. All rights reserved.
Author...... Department of Nuclear Science & Engineering May 10, 2019 Certified by...... Benoit Forget Professor of Nuclear Science & Engineering Thesis Supervisor Certified by...... Kord Smith KEPCO Professor of the Practice of Nuclear Science & Engineering Thesis Supervisor Accepted by ...... Ju Li Battelle Energy Alliance Professor of Nuclear Science & Engineering and Professor of Materials Science & Engineering Chair, Department Committee on Graduate Students 2 Modeling and Simulation of the Transient Reactor Test Facility using Modern Neutron Transport Methods by Travis J. Labossi`ere-Hickman
Submitted to the Department of Nuclear Science & Engineering on May 10, 2019, in partial fulfillment of the requirements for the degree of Master of Science in Nuclear Science & Engineering
Abstract The Transient Reactor Test Facility (TREAT) has regained the interest of the nuclear engineering community in recent years. While TREAT’s design makes it uniquely suited to transient fuel testing, it also makes the reactor very challenging to model and simulate. In this thesis, we build a Monte Carlo model of TREAT’s Minimum Critical Mass core to examine the effects of fuel impurities, calculate a reference solution, and analyze a number of multigroup cross section generation approaches. Several method of characteristics (MOC) simulations employing these cross sections are then converged in space and angle, corrected for homogenization, and compared to the Monte Carlo reference solution. The thesis concludes with recommendations for future analysis of TREAT using MOC.
Thesis Supervisor: Benoit Forget Title: Professor of Nuclear Science & Engineering
Thesis Supervisor: Kord Smith Title: KEPCO Professor of the Practice of Nuclear Science & Engineering
3 4 Acknowledgments
This work would not have been possible without the support of the MIT CRPG. I would especially like to thank Will Boyd, Sterling Harper, and Guillaume Giudicelli for instructing me in the ways of OpenMC and OpenMOC; Carl Haugen for co- building the full TREAT reactor model; Zhaoyuan Liu for providing the Cumulative Migration Method data; and Professors Benoit Forget and Kord Smith for their advising and for putting up with me these past few years. I would like to thank the American Nuclear Society for providing me with the opportunities which led to discovering the CRPG and for finding my first job at Framatome. I also wish to express my gratitude to my brothers and sisters of the MIT Graduate Christian Fellowship for their edification and for reminding me what was really important. This material is based upon work supported under an Integrated University Pro- gram Graduate Fellowship. Any opinions, findings, conclusions or recommendations expressed in this publication are those of the author(s) and do not necessarily reflect the views of the Department of Energy Office of Nuclear Energy. This research made use of the resources of the High Performance Computing Center at Idaho National Laboratory, which is supported by the Office of Nuclear Energy of the U.S. Department of Energy and the Nuclear Science User Facilities under Contract No. DE-AC07-05ID14517.
5 6 Contents
1 Introduction 19
2 Background 21 2.1 Construction of TREAT...... 21 2.2 Core...... 21 2.2.1 Standard Fuel Assembly...... 23 2.2.2 Control Rod Fuel Assembly...... 24 2.2.3 Control rods...... 25 2.2.4 Dummy Assemblies...... 25 2.3 Fuel of TREAT...... 25 2.3.1 Boron Impurity...... 26 2.3.2 Hydrogen Impurity...... 27 2.3.3 Graphitization Fraction...... 27 2.4 Computational Methods...... 28 2.4.1 Monte Carlo...... 28 2.4.2 MGXS...... 28 2.4.3 MOC...... 29 2.5 Equivalence Methods...... 31 2.5.1 Cumulative Migration Method...... 31 2.5.2 Superhomog´en´eisation...... 33
3 TREAT Reference Solutions 35 3.1 Minimum Critical Mass Model...... 35
7 3.2 Fuel Composition...... 36 3.2.1 Base Model...... 36 3.2.2 Sensitivity to Boron...... 37 3.2.3 Sensitivity to Hydrogen...... 37 3.2.4 Sensitivity to Graphitization...... 38 3.3 Simplified OpenMC Models for OpenMOC Analysis...... 38 3.3.1 Single Standard Fuel Element...... 39 3.3.2 3×3 Colorset (Follower)...... 40 3.3.3 3×3 Colorset (Poison)...... 40 3.3.4 MCM2D (No Rods)...... 41 3.3.5 MCM2D (Follower)...... 41 3.3.6 MCM2D (Poison)...... 42 3.4 MGXS Generation: Energy Groups...... 42 3.5 MGXS Generation: Spatial Domains...... 44 3.5.1 Mesh Domain...... 44 3.5.2 Material Domain...... 47 3.5.3 Cell Domain...... 48 3.5.4 Universe Domain...... 49 3.5.5 Heterogeneous vs. Homogeneous Representation...... 49
4 Deterministic Results 51 4.1 Variables to Converge...... 51 4.2 Source Region Convergence...... 52 4.2.1 CMFD Convergence...... 53 4.2.2 1D Convergence Study...... 54 4.2.3 2D Colorset Convergence...... 56 4.2.4 2D Full-Core Convergence...... 64 4.3 Ray Spacing and Azimuthal Angle Convergence...... 74 4.3.1 Heterogeneous...... 74
8 4.3.2 Homogeneous...... 76 4.4 Polar Angle Convergence...... 76 4.5 Effects of Corrections for Homogeneous Models...... 77 4.5.1 Correcting with CMM...... 77 4.5.2 Correcting with SPH...... 86 4.6 Final Results...... 92 4.6.1 Single Fuel Element Results...... 92 4.6.2 3×3 Colorset (Follower) Results...... 93 4.6.3 3×3 Colorset (Poison) Results...... 94 4.6.4 MCM2D (No Rods) Results...... 95 4.6.5 MCM2D (Follower) Results...... 96 4.6.6 MCM2D (Poison) Results...... 97
5 Conclusions 99 5.1 Summary of Accomplishments...... 99 5.2 Recommendations for Future Analysts...... 100
A Enlarged Figures 103
B CMM Transport Correction Ratios 107
C Software Versions 109 C.1 OpenMC...... 109 C.1.1 Revision used for sensitivity analysis...... 109 C.1.2 Revision used for MGXS generation...... 109 C.2 OpenMOC...... 110
9 10 List of Figures
2.1 Top view of the TREAT core...... 22 2.2 Minimum Critical Mass core configuration...... 22 2.3 Axial cutaway of a standard fuel element...... 23 2.4 Axial cutaway of a control rod fuel element...... 24 2.5 Regions of a TREAT control rod...... 25 2.6 Axial cutaway of an aluminum-clad dummy element...... 26 2.7 Construction of a CSG cell from basic surfaces...... 29 2.8 Example of flat vs. linear source regions for a 1D gradient...... 30
3.1 Full-reactor MCM model at midcore...... 35 3.2 Legend for the subsequent materials plots...... 39 3.3 Single standard fuel element...... 39 3.4 3×3 control rod colorset (graphite follower inserted)...... 40
3.5 3×3 control rod colorset (B4C poison inserted)...... 40 3.6 MCM full core (no control rod elements)...... 41 3.7 MCM full core (graphite follower inserted)...... 41
3.8 MCM full core (B4C poison inserted)...... 42 3.9 Fine mesh over a single fuel element...... 44 3.10 1D XS plots with error bars...... 45 3.11 Fine mesh over a control rod (follower) fuel element...... 45 3.12 Mesh tally across the MCM2D (Poison) core...... 46 3.13 Thermal fission mesh tally across the MCM2D (Poison) core..... 47 3.14 Control rod (follower) fuel element, Material domain...... 47
11 3.15 Control rod (follower) fuel element, Cell domain...... 48 3.16 Control rod (follower) fuel element, Universe domain...... 49 3.17 Heterogeneous vs. Homogeneous representations of MCM2D Poison. 50
4.1 Azimuthal rings and sectors for a pincell in a box...... 52 4.2 Subdivision of a TREAT element...... 53 4.3 Geometry of toy problem...... 54 4.4 Flat source vs. linear source for toy problem...... 55 4.5 Converged LSR meshes for the single fuel element...... 57 4.6 Converged LSR meshes for the 3×3 colorset (follower)...... 59 4.7 Converged LSR meshes for the 3×3 colorset (poison)...... 62 4.8 Power distributions for the 3×3 colorset (poison) models...... 63 4.9 Converged LSR meshes for the MCFuel core...... 65 4.10 MCM2D (No Rods): 11 groups vs. 25 groups...... 66 4.11 MCM2D (Follower): 11 groups vs. 25 groups...... 68 4.12 Converged LSR meshes for the MCM2D (Follower) core...... 69 4.13 Power distributions for д = 0.508 cm vs. 0.254 cm...... 71 4.14 MCM2D (Poison): 11 groups vs. 25 groups...... 72 4.15 Converged LSR meshes for the MCM2D (Poison) core...... 73 4.16 CMM-corrected vs. uncorrected for MCM2D (No Rods)...... 80 4.17 CMM-corrected vs. uncorrected for MCM2D (Follower)...... 82 4.18 CMM-corrected vs. uncorrected for MCM2D (Poison)...... 84 4.19 Effect of CMM on the 11-group vs. 25-group power distribution... 85 4.20 SPH vs. uncorrected for 3×3 (Poison)...... 87 4.21 Effect of SPH+CMM on the 11-group vs. 25-group power distribution 88 4.22 SPH vs. uncorrected for MCM2D (Poison)...... 89 4.23 SPH+CMM vs. SPH only for MCM2D (Poison)...... 91 4.24 Final Single Fuel Element Plots...... 92 4.25 Final 3×3 Colorset (Follower) Plots...... 93 4.26 Final 3×3 Colorset (Poison) Plots...... 94
12 4.27 Final MCM2D (No Rods) Plots...... 95 4.28 Final MCM2D (Follower) Plots...... 96 4.29 Final MCM2D (Poison) Plots...... 97
A.1 MCM full core with the B4C poison inserted (enlarged)...... 103 A.2 Full reactor at midcore, radial view (enlarged)...... 104 A.3 Full reactor at midcore, axial view (enlarged)...... 105
13 14 List of Tables
3.1 Monte Carlo simulation parameters for eigenvalue calculations.... 36 3.2 Boron sensitivity...... 37 3.3 Hydrogen sensitivity...... 37 3.4 Graphitization sensitivity...... 38 3.5 11-group structure...... 43 3.6 25-group structure...... 43
4.1 FSR convergence for 1D problem...... 54 4.2 LSR convergence for 1D problem...... 55 4.3 Single fuel element LSR convergence, heterogeneous...... 56 4.4 Single fuel element LSR convergence, homogeneous...... 57 4.5 3×3 colorset (follower) LSR convergence, heterogeneous...... 58 4.6 3×3 colorset (follower) LSR convergence, homogeneous...... 58 4.7 3×3 colorset (poison) LSR convergence, heterogeneous...... 60 4.8 3×3 colorset (poison) LSR convergence, homogeneous...... 60
4.9 3×3 colorset (poison) convergence, where д퐶푅퐷 = 0.254 cm...... 61 4.10 MCM2D (No Rods) LSR convergence, heterogeneous...... 64 4.11 MCM2D (No Rods) convergence, homogeneous...... 65 4.12 MCM2D (Follower) LSR convergence, heterogeneous...... 67 4.13 MCM2D (Follower) convergence, homogeneous...... 67 4.14 MCM2D (Poison) LSR convergence, heterogeneous...... 70 4.15 MCM2D (Poison) convergence, homogeneous...... 70
4.16 MCM2D (Poison) convergence, where д퐶푅퐷 = 0.254 cm...... 71
15 4.17 ∆푘 (훿푎, 푁푎) for the heterogeneous 3×3 follower colorset...... 75
4.18 ∆푘 (훿푎, 푁푎) for the heterogeneous MCM2D (Poison) core...... 75
4.19 ∆푘 (훿푎, 푁푎) for the homogeneous MCM2D (Poison) core...... 76 4.20 Effect of CMM on× the3 3 (Poison) model, where д = 2.54 cm.... 78 4.21 Effect of CMM on the MCM2D (No Rods) model, withд = 1.016 cm 79 4.22 Effect of CMM on the MCM2D (Follower) model, withд = 1.016 cm 81 4.23 Effect of CMM on the MCM2D (Poison) model, withд = 1.016 cm. 81 4.24 Effect of SPH on× the3 3 (Poison) model, with д = 2.54 cm..... 86 4.25 Effect of SPH on the MCM2D (Poison) model, withд = 1.016 cm.. 88 4.26 Final Single Fuel Element Results...... 92 4.27 Final 3×3 Colorset (Follower) Results...... 93 4.28 Final 3×3 Colorset (Poison) Results...... 94 4.29 Final MCM2D (No Rods) Results...... 95 4.30 Final MCM2D (Follower) Results...... 96 4.31 Final MCM2D (Poison) Results...... 97
B.1 11-group CMM ratios...... 107 B.2 25-group CMM ratios...... 108
16 Acronyms and Abbreviations
ATF: Accident tolerant fuel BATMAN: Baseline Assessment of TREAT for Modeling and Analysis Needs BC: Boundary condition CMFD: Coarse mesh finite difference CMM: Cumulative migration method CRPG: (MIT) Computational Reactor Physics Group CSG: Constructive (or combinatorial) solid geometry FSR: Flat source region HEU: Highly enriched uranium IHM: Infinite homogeneous medium INL: Idaho National Laboratory LSR: Linear source region MC: Monte Carlo MCM: Minimum critical mass MGXS: Multigroup cross section(s) MOC: Method of Characteristics NRL: (MIT) Nuclear Reactor Laboratory PCM or pcm: per cent mille, i.e., 10−5 SPH: Superhomog´en´eisationor Superhomogenization SR: Source region TREAT: Transient reactor test facility XS: Cross section
17 18 Chapter 1
Introduction
The accident at the Fukushima Daiichi Nuclear Power Plant in 2011 sparked a renewed interest in accident tolerant fuel (ATF) in the nuclear engineering community. The need to perform transient testing of ATF led to the restart of TREAT, the transient reactor test facility, in 2017 [1]. TREAT’s unique design makes it well suited to performing destructive fuel testing, but it also makes TREAT difficult to simulate using conventional industry tools. One can model the neutronics of TREAT with few physical approximations using Monte Carlo methods. However, these are notorious for being computationally expensive. While stochastic Monte Carlo methods may be a good choice for a small number of simulations, they quickly become prohibitively expensive when many cases need to be run. Optimization problems, uncertainty quantification, multiphysics coupling, and transient simulations involving small time steps are just a few such examples. Clearly, a faster deterministic solution is needed. To this end, we wish to construct a best-estimate Monte Carlo model of the transient reactor test facility in the minimum critical mass (MCM) configuration. This model shall be run once each for a small number of variations, where it is used to calculate a reference eigenvalue, and a reference power distribution. We will also use the model to generate several libraries of multigroup cross sections (MGXS) over TREAT’s core in both heterogeneous and homogeneous representations. Using these cross sections, we then demonstrate an accurate deterministic solution of TREAT’s core using the Method of Characteristics.
19 20 Chapter 2
Background
2.1 Construction of TREAT
The purpose of the transient reactor test facility, as the name implies, is to perform transient testing of nuclear fuels. Fuel samples are placed in experiment vessels mimicking the environment inside a power reactor. From within the vessel, the fuel sample can undergo accident scenarios, destructive testing, and so on. Transient testing may entail very high peak powers, up to 18 GW in some tests. TREAT is designed to endure these rapid pulses and shut itself down safely. [2]
2.2 Core
TREAT’s core consists of a 19×19 array of elements (assemblies) surrounded by a permanent graphite reflector and a concrete biological shield (Fig. 2.1). Each element is 4 in.× 4 in. (10.16 cm × 10.16 cm) including the inter-assembly gap and features four vertical air channels, one at each corner. Each element is protected by zircaloy or aluminum cladding. Several notable TREAT core configurations are being studied by the nuclear engineering community. Our focus is on the Minimum Critical Mass (MCM) configuration, depicted in Fig. 2.2. MCM contains four different element species, which are described in the following
21 Figure 2.1: Top view of the TREAT core Source: Fig. 1.3 from [3].
Figure 2.2: Minimum Critical Mass core configuration Source: Fig. 1.3 from [4].
22 subsections of this chapter. For more detailed information, the reader is encouraged to review, “The Baseline Assessment of TREAT for Modeling and Analysis Needs” [3]. The document, colloquially referred to as “the BATMAN report,” contains best- estimate specifications for TREAT’s geometry, materials, and more.
2.2.1 Standard Fuel Assembly
Standard fuel elements (see [3], Section 2.1) contain a block of active fuel bounded on the top and bottom by graphite reflectors, or “plugs.” The active fuel (Section 2.3) is itself made of graphite with dispersed uranium, and clad with Zircaloy 3. Plugs are aluminum-clad. One such block is depicted in Fig. 2.3.
Figure 2.3: Axial cutaway of a standard fuel element Image source: Fig. 2.02 from [3].
23 2.2.2 Control Rod Fuel Assembly
Externally, control rod fuel elements ([3], Section 2.2) are nearly identical to regular fuel elements. Internally, they contain a zircaloy-clad tube into which control rods are inserted. See Fig. 2.4 below.
Figure 2.4: Axial cutaway of a control rod fuel element Source: Fig. 2.14 from [3].
24 2.2.3 Control rods
Control rods (see [3], Section 2.8) have several axial regions. The “poison” section
is made of steel-clad boron carbide (B4C). It is present in the core when TREAT is shut down. When rods are withdrawn, a ziconium-clad graphite “follower” section is present instead. Fig. 2.5 depicts TREAT Control Rod Number I, as used in the MCM core.
Figure 2.5: Regions of a TREAT control rod Figure adapted from Fig. 2.278, “Control Rod Number II,” in [3] to show the measurements for Rod Number I.
2.2.4 Dummy Assemblies
Surrounding the fuel are a number of “dummy” elements (see Sections 2.5, 2.6 of [3]) forming a radial reflector within the core. They are similar to the fuel elements, but are filled with graphite repurposed from Chicago Pile 2 (CP-2), which was itself built with graphite from CP-1. Several dummy elements adjacent to the fuel use zircaloy cladding over the height of the active fuel. The rest are aluminum-clad. One such element is shown in Fig. 2.6.
2.3 Fuel of TREAT
TREAT uses an unconventional fuel: highly enriched uranium (HEU) dispersed in a graphite matrix. There is a great deal of uncertainty concerning the exact composition of the fuel [3,4], which can vary widely depending on one’s assumptions. The chief
25 Figure 2.6: Axial cutaway of an aluminum-clad dummy element Source: Figure 2.149 from [3]. sources of uncertainty unique to the the fuel composition are the boron impurity concentration, hydrogen impurity concentration, and graphitization fraction.
2.3.1 Boron Impurity
The graphite matrix of the fuel contains an uncertain amount of boron. During fabrication, the graphite became contaminated with boron from the borated steel dividers used during the graphitization process [5]. Measured boron concentration varies considerably between measurements, but average weight fractions of anywhere
26 between 5.90 ppm and 7.60 ppm are commonly used. [3,4]
2.3.2 Hydrogen Impurity
The fuel may contain a significant hydrogen impurity. As per [3], Section 3.1:
The graphite fuel for TREAT contained approximately 1 wt.% hydrogen content prior to baking. The fuel blocks were then baked up to 950∘C to initiate graphitization and release volatile gases. To achieve complete graphitization and completely release most non-graphite materials, a baking temperature of approximately 3000∘C is necessary. At that point, the hydrogen content is < 50 ppm. . . The TREAT fuel fabrication process was simulated by B&W but only baked to 900∘C in February 2015 and then tested by NSL Analytical in April 2015 for impurity content. The measured hydrogen content was 0.097 wt.% (970 ppm).
The hydrogen impurity is commonly neglected in TREAT models [3,4,6], although it can have a strong impact when accounted for (Section 3.2.3).
2.3.3 Graphitization Fraction
Another consequence of the low temperature baking is incomplete graphitization of the fuel blocks. TREAT’s fuel was TREAT’s fuel was manufactured with carbon from graphite flour, thermax, and coal tar. During baking, the flour and a fraction ofthe coal tar graphitized, while the rest of the ingredients did not. It is thought that 59% of the total carbon in TREAT fuel should be considered graphite [3,7]. In neutronics calculations, this can be modeled by applying 푆(훼, 훽) tables for graphite to 59% of the fuel and treating the remaining 41% as free carbon.
27 2.4 Computational Methods
2.4.1 Monte Carlo
The neutron transport equation can be solved using Monte Carlo methods with relatively few approximations. Essentially, a large number of particles are spawned at possible fission sites in the reactor geometry and tracked until they are absorbed or leaked. The average behavior of particles over multiple batches and the standard deviation thereof is tallied for the reaction rates of interest to the analyst.
OpenMC
Our analysis was performed using OpenMC, the open source Monte Carlo neutral particle transport code developed by the MIT Computational Reactor Physics Group (CRPG) and Argonne National Laboratory [8]. OpenMC includes a rich Python API to facilitate the creation of models using constructive solid geometry (CSG), post-processing of tally data, and other useful tasks. Monte Carlo results presented in this thesis were generated with OpenMC 0.9.0 using continuous-energy ENDF/B-VII.1 cross sections. See See AppendixC for more details.
Constructive Solid Geometry
Constructive Solid Geometry (CSG) defines space as Regions relative to one or more quadratic Surfaces. A CSG Cell occupies some Region in space and may be filled by some material. A Cell may also be filled by a CSG Universe, which is a collection of other Cells. An example of how a CSG Cell might be built is shown in Fig. 2.7.
2.4.2 MGXS
The neutron transport equation can be simplified by discretizing the energy into discrete bins. For some arbitrary energy-dependent, spatially-dependent macroscopic
28 Figure 2.7: Construction of a CSG cell from basic surfaces The bottommost shapes, such as the blue sphere, are represented by CSG Surfaces. Together, the Surfaces are combined to create more complicated Regions, eventually forming the Cell at the top. Image source: [9]
cross section Σ(퐸, ⃗푟), a set of multigroup cross sections (MGXS) can be calculated by the integral:
퐸푔−1 ∫︀ ∫︀ Σ(퐸, ⃗푟)휑(퐸, ⃗푟)푑⃗푟푑퐸 퐸푔 푟∈푉 Σ푔,푖 = (2.1) 퐸푔−1 ∫︀ ∫︀ 휑(퐸, ⃗푟)푑⃗푟푑퐸 퐸푔 푟∈푉
This is commonly done by tallying reaction rates and scalar fluxes using a Monte Carlo transport code such as OpenMC.
The choice of spatial region 푉 used for MGXS set 푖 can greatly affect the accuracy of this approximation. Less error is introduced by defining a group cross section fora Cell with a single material than for a large segment of the geometry. The advantages and disadvantages of several spatial domains are discussed further in Section 3.4.
2.4.3 MOC
The method of characteristics simplifies the neutron transport equation by solving along discrete tracks (characteristics) through the geometry [10]. In MOC, neutron flux may be discretized over a number of energy groups, polar angles, azimuthal angles, and source regions. MOC is highly attractive for solving the transport
29 Figure 2.8: Example of flat vs. linear source regions for a 1D gradient Image source: [11] equation deterministically1 because it can preserve the spatial heterogeneities and angular dependence of a problem without complex meshing.
OpenMOC
OpenMOC is the open-source method of characteristics neutron transport solver developed by the MIT CRPG [12]. Like OpenMC, OpenMOC defines geometries using CSG, and their Python APIs contain compatability tools to make it trivial to transfer a geometry between the two codes. See AppendixC for version information.
Source regions
In MOC, each characteristic track crosses one or more source region (SR), which modifies the flux along that track. The simplest type of SR is the flat sourceregion (FSR), which uses a single value for the source term in that region. A linear source region (LSR) represents the source term with a linear function. Higher-order sources such as quadratic are also possible. OpenMOC provides both flat and linear source solvers. A comparison of theflux gradient for each type in a homogeneous medium is illustrated in Fig. 2.8. A vacuum
1Although stochastic MOC transport codes exist, for our purposes, MOC is a fully deterministic method.
30 boundary condition on the left-hand side causes a sharp gradient in the flux profile. As a general rule, more source regions or a higher order source are required to resolve steep gradients.
2.5 Equivalence Methods
2.5.1 Cumulative Migration Method
The assumption of isotropic MGXS can cause problems when there is a strong directional component to the neutron flux. In our case, the mean free path inthe 푧-direction is greater than that in 푥 or 푦 due to neutrons traveling through the air channels. Homogenizing over TREAT elements does not properly account for this preferential axial streaming. One of the interesting applications of the Cumulative Migration Method (CMM) is the calculation of directional diffusion coefficients [13]. In essence, the migration area of neutrons can be tracked in multiple directions and used to calculate a diffusion coefficient for each. While OpenMOC does not support directional cross sections,
one can use CMM diffusion coefficients to ‘correct’ the transport cross section Σ푡푟 for regions homogenized with air channels. We suspect that this method will show improved MOC results over models using the standard flux-limited definition of the transport cross section, Eq. 2.2:
Σ푡,푔,푖휑푔,푖 − Σ푠,푔→푔,푖휑푔,푖 Σ푡푟,푔,푖 = (2.2) 휑푔,푖 Consider a model of a single TREAT element with reflective boundary conditions on the 푥 and 푦 edges, forming an infinite lattice. Because all elements used in the MCM core are 푥-푦 symmetric, using either dimension to represent the transverse
∞ direction is equivalent; arbitrarly, we choose 푥. Let 퐷푔 represent the diffusion 퐶,∞ coefficient in energy group 푔 in the flux-limited approach, and let 퐷푔,푥 represent the diffusion coefficient for the 푥-direction using the Cumulative Migration Method. For each energy group, define a CMM correction ratio я as if it were a cross
31 section:
∞ 퐷푔 я푔 = 퐶,∞ (2.3) 퐷푔,푥 This allows us to pre-calculate a set of CMM corrections for a given species of TREAT element and use it to correct the diffusion coefficient or transport cross section
0 0 for the homogenized element in a core model. If 퐷푔,퐸 and Σ푡푟,푔,퐸 are calculated using their flux-limited definition for some artibrary element 퐸, they may be corrected:
(︃ 퐶,∞ )︃ 퐶 퐷푔,푥 0 1 0 퐷푔,퐸 = ∞ 퐷푔,퐸 = 퐷푔,퐸 (2.4) 퐷푔 я푔
and using Σ푡푟,푔 = 1/3퐷푔
(︂ ∞ )︂ 퐶 퐷푔 0 0 Σ푡푟,푔,퐸 = 퐶,∞ Σ푡푟,푔,퐸 = я푔Σ푡푟,푔,퐸 (2.5) 퐷푔,푥 Editing the transport cross section requires making a change to the within-group scatter cross section Σ푠,푔→푔. Define the absolute change in Σ푡푟,푔 as:
[︀ 퐶 0 ]︀ 0 ∆Σ푡푟,푔 = Σ푡푟,푔,퐸 − Σ푡푟,푔,퐸 = [я푔 − 1] Σ푡푟,푔,퐸 (2.6)
Thus, for a given correction ratio я푔 and flux-limited within-group scatter cross 0 section Σ푠,푔→푔,
퐶 0 0 0 Σ푠,푔→푔 = Σ푠,푔→푔 + [я푔 − 1] Σ푡푟,푔 = Σ푠,푔→푔 + ∆Σ푡푟,푔 (2.7)
This definition for the CMM correction factor, Eq. 2.3, assumes that the angular flux spectrum seen by the element in an infinite lattice is similar to thatseenby the element in a core. If there are significant changes to the flux—for instance, ifthe element is next to a control rod with little incoming thermal flux from that direction— this may be a poor assumption. In that case, CMM correction factors may need to be generated on the actual core geometry.
32 2.5.2 Superhomog´en´eisation
In most practical calculations, spatial homogenization and energy condensation lead to reaction rate errors. These errors tend to be especially pronounced in situations with significant spatial self-shielding. One common remedy for these errors isSuper- homog´en´eisation(SPH) [14, 15]. If one knows the reference solution, a set of equivalence factors 휇 can be defined such that the reaction rates are preserved in the approximate (multigroup) solution.
Let Σ푔,푖 and 휑푔,푖 be some macroscopic cross section and the scalar flux, respectively, in group 푔 on spatial domain 푖. These can be modified by the SPH factor for that group and domain 휇푔,푖 as follows:
ˆ ˆ 휑푔,푖 Σ푔,푖 = 휇푔,푖Σ푔,푖 and 휑푔,푖 = (2.8) 휇푔,푖 The factors can be solved for iteratively in multiple energy groups with one or more spatial domains. Let the superscript 푛 represent the iteration index, where 휑0 is the reference flux. The SPH factors for the 푛th iteration are calculated:
휑0 휇푛 = 푔,푖 (2.9) 푔,푖 ˆ푛 휑푔,푖 Now let Σ0 be the cross section initially calculated by the reference solution. One edits the cross sections for every reaction by multiplying it by the corresponding SPH factor.
ˆ 푛+1 푛 0 Σ푔,푖 = 휇푔,푖Σ푔,푖 (2.10)
The multigroup solver uses these modified cross sections Σˆ 푛+1 to find a new flux tensor 휑ˆ푛+1, and the process is repeated until 휇 converges. This can be done cheaply if the solver preserves the flux profile between iterations.
33 34 Chapter 3
TREAT Reference Solutions
3.1 Minimum Critical Mass Model
Using the specifications from Baseline Assessment of TREAT for Modeling and Analysis Needs, a detailed 3D model of the transient reactor test facility was created in OpenMC (Fig. 3.1). The reactor core was placed in the Minimum Critical Mass (MCM) configuration.
(a) Radial view (b) Axial view
Figure 3.1: Full-reactor MCM model at midcore Enlarged figures are available in AppendixA.
35 3.2 Fuel Composition
3.2.1 Base Model
As mentioned in Section 2.3, there is uncertainty as to the level of boron impurity, level of hydrogen impurity, and extent of graphitization of TREAT’s fuel. There are any number of possible combinations of impurities which make a given model critical. TREAT analysts can effectively “tune” the impurities within the bounds of
uncertainty until 푘eff is approximately 1. The MIT Nuclear Reactor Laboratory (NRL) provided us with a model where such an optimization had been done [16]. The nominal fuel material used 7.53 ppm boron impurity, no hydrogen, and few trace impurities. 푆(훼, 훽) tables for graphite were used for 59% of the fuel, consistent with the recommendations in Section 2.3.3.
This material definition was plugged into the 3D MCM model. The resulting 푘eff was 1.00604 ± 4 pcm, comparable to the value of 1.00413 ± 20 pcm reported for a
similar model in [4]. The 푘∞ of the material itself in a homogeneous medium was also calculated. The fuel composition from NRL was then perturbed in the infinite homogeneous medium (IHM) and in the 3D model to quantify the sensitivities to boron, hydrogen, and graphitization. Results were obtained using a branch of OpenMC 0.9.0 which allowed the use of partial 푆(훼, 훽) tables (Appendix C.1.1). ENDF/B-VII.1 was used for continuous- energy cross section data.1 The simulation parameters and nominal eigenvalues are reported in Table 3.1 below.
Particles Per Total Inactive Nominal Nominal Model Batch Batches Batches 푘 휎 (pcm) Inf. Hom. 106 100 10 1.76552 ±6 MCM 3D 107 100 35 1.00604 ±4
Table 3.1: Monte Carlo simulation parameters for eigenvalue calculations
1It should be noted that ENDF/B-VIII (now available) contains improved graphite data. The use of ENDF/B-VIII may have a noticeable impact upon the eigenvalues and sensitivities reported here.
36 3.2.2 Sensitivity to Boron
Table 3.2 records the eigenvalues observed when plugging the extreme values into our base Monte Carlo models. The final columns record the sensitivity to deviations from the nominal value of 7.53 ppm.
Model Boron 푘 휎 (pcm) ∆푘 (pcm) ∆푘/Boron IHM 5.90 ppm 1.79795 ±7 +3243 -1990 pcm/ppm 7.53 ppm 1.76552 ±6 ±0 – 7.60 ppm 1.76412 ±7 −140 -2000 pcm/ppm MCM 3D 5.90 ppm 1.02057 ±4 +1453 -891 pcm/ppm 7.53 ppm 1.00604 ±4 ±0 – 7.60 ppm 1.00542 ±4 −62 -886 pcm/ppm
Table 3.2: Boron sensitivity
3.2.3 Sensitivity to Hydrogen
The magnitude of the possible hydrogen impurity spans several orders of magnitude. We calculated the effects of logarithmically spaced levels of hydrogen in the fuelfrom the baseline of 0 ppm.
Model Hydrogen 푘 휎 (pcm) ∆푘 (pcm) ∆푘/Hydrogen IHM 0 ppm 1.76552 ±6 ±0 – 100 ppm 1.75748 ±5 −804 −8.0 pcm/ppm 1,000 ppm 1.68682 ±7 −7870 −7.9 pcm/ppm 10,000 ppm 1.18538 ±8 −58, 014 −5.8 pcm/ppm MCM 3D 0 ppm 1.00604 ±4 ±0 – 100 ppm 1.00904 ±4 +300 +3.0 pcm/ppm 1,000 ppm 1.02321 ±3 +1717 +1.7 pcm/ppm 10,000 ppm 0.87739 ±3 −12, 865 −1.3 pcm/ppm
Table 3.3: Hydrogen sensitivity
Two competing effects are in play. Parasitic absorption in hydrogen causesa sharp drop in 푘∞, which we observe in the IHM. However, hydrogen also has a higher slowing down power than that of graphite. In the finite 3D model, the addition of
37 a small amount of hydrogen raises 푘eff as more neutrons are thermalized before they can leak. Absorption once again dominates at high concentrations.
3.2.4 Sensitivity to Graphitization
The sensitivity of the eigenvalue with respect to 푆(훼, 훽) fraction is shown in Table 3.4 below. A value of 0% represents all of the graphite in the fuel as free carbon, and a value of 100% represents full graphitization. The final columns record the sensitivity relative to the baseline value of 59%.
Model 푆(훼, 훽) Graphite 푘 휎 (pcm) ∆푘 (pcm) ∆푘/Graphite IHM 0% 1.76702 ±6 +150 −2.5 pcm/% 59% 1.76552 ±6 ±0 – 100% 1.76354 ±6 −198 −4.8 pcm/% MCM 3D 0% 1.01553 ±4 +949 −16 pcm/% 59% 1.00604 ±4 ±0 – 100% 0.99164 ±4 −1440 −35 pcm/%
Table 3.4: Graphitization sensitivity
Neutrons tend to lose more energy when scattering with free carbon nuclei than with carbon bound in graphite. Neglecting 푆(훼, 훽) thermal scattering helps to moderate neutrons before they can be parasitically absorbed or leak from the core. The effect on 푘 is far more pronounced for the finite model due to the reduction in leakage.
3.3 Simplified OpenMC Models for OpenMOC Analysis
The 3D model with the permanent reflector and all of the ex-core features is needed for studying the eigenvalue of the system as a whole. From a cross section generation standpoint, however, it is less interesting. The physics we are interested in capturing takes place within the core. Furthermore, the metric for the MOC cross sections are not how well they predict the eigenvalue of real-life TREAT, but how precisely they
38 are able to reproduce the OpenMC solution. For these reasons, it was not necessary to analyze the full reactor model in OpenMOC. Several simplified models were prepared for 2D analysis. For the full-core models, the core liner, permanent reflector, and all other ex-core structure were removed, and vacuum boundary conditions were placed radially along the 19 × 19 core lattice. Smaller reflected models were considered as well. Each of the following models was simulated. OpenMC was used to calculate the MGXS and the reference eigenvalue, power distribution, and groupwise flux.
Legend
Fuel Graphite Air
Zircaloy, Boron Carbide Aluminum Stainless Steel
Figure 3.2: Legend for the subsequent materials plots
3.3.1 Single Standard Fuel Element
Figure 3.3: Single standard fuel element
The simplest model contains a single standard fuel assembly with reflective boundary conditions on each edge.
39 3.3.2 3×3 Colorset (Follower)
Figure 3.4: 3×3 control rod colorset (graphite follower inserted)
A 3×3 colorset2 with eight standard fuel elements and one control rod fuel element was modeled. The zircaloy-clad graphite follower portion of the control rod is placed in the control rod channel. Reflective boundary conditions are imposed on each edge.
3.3.3 3×3 Colorset (Poison)
Figure 3.5: 3×3 control rod colorset (B4C poison inserted)
Another 3×3 colorset was modeled. It is identical to that in Section 3.3.2, but with the steel-clad B4C poison segment of the the control rod inserted.
2Also referred to as a “minicore” or “supercell” in some texts
40 3.3.4 MCM2D (No Rods)
Figure 3.6: MCM full core (no control rod elements)
The first 2D full-core model considered was a simplified version of the MCM core,with all control rod fuel elements replaced by standard fuel elements. Vacuum boundary conditions are present on all four sides for every full-core model.
3.3.5 MCM2D (Follower)
Figure 3.7: MCM full core (graphite follower inserted)
41 A 2D full-core model of the MCM core with the graphite follower control rod segments present in the control rod fuel elements was considered. It is analogous to the core with all rods out.
3.3.6 MCM2D (Poison)
Figure 3.8: MCM full core (B4C poison inserted) An enlarged figure appears in AppendixA.
Another 2D MCM model contains the B4C poison control rod segments inserted in the core. It is analagous to the all rods in case. This model is otherwise identical to the core of Section 3.3.5.
3.4 MGXS Generation: Energy Groups
Multigroup cross sections for TREAT were generated in OpenMC according to two different energy group structures. When examining integrated reaction rates and eigenvalues in a single structure, it is possible that errors in different groups will fortuitously cancel out and cause a solution to look better than it is. The use of multiple distinct energy group structures is intended to highlight any such cancellation of error within one structure.
42 Group Energy, upper Group Energy, upper Number bound (eV) Number bound (eV) 1 2 E+7 1 2 E+7 2 3.3287 E+6 2 6.066 E+6 3 1.1562 E+5 3 3.679 E+6 4 3.4811 E+3 4 2.231 E+6 5 1.3270 E+3 5 1.353 E+6 6 8.1000 E+1 6 8.21 E+5 7 6.2500 E+0 7 5.00 E+5 8 2.0961 E−1 8 1.11 E+5 9 7.6497 E−2 9 9.118 E+3 10 4.7302 E−2 10 5.53 E+3 11 2.0010 E−2 11 1.4873 E+2 12 1.5968 E+1 Table 3.5: 11-group structure 13 9.877 E+0 14 4.000 E+0 15 1.855 E+0 16 1.150 E+0 17 1.097 E+0 18 1.020 E+0 19 9.72 E−1 20 6.25 E−1 21 3.5 E−1 22 2.8 E−1 23 1.4 E−1 24 5.8 E−2 25 3.0 E−2
Table 3.6: 25-group structure
The Idaho National Laboratory provided us with the 11-group structure displayed in Table 3.5. It was tailored to TREAT’s spectrum and has been shown to perform well on TREAT models at 300 K [6]. The CASMO 25-group structure [17] was also used for multigroup cross sections. The structure was intended for LWRs, which, like TREAT, are thermal systems. It has a similar number of thermal energy groups, but with different bounds and not customized to TREAT. Detailed spectral analysis of TREAT showing that 11 groups is sufficient has been done at INL [6]. We shall quantify the errors arising from the 11-group and 25-group discretizations versus the continuous-energy Monte Carlo.
43 3.5 MGXS Generation: Spatial Domains
As discussed in Section 2.4.2, the choice of spatial domain for MGXS generation can significantly affect the solution. In the next few pages, we will consider four different ways to represent cross sections spatially. Plots show cross sections inthe 11-group structure, but similar trends were observed at the same energies when using 25 groups. Group 10 out of 11 has been favored for the mesh plots because it contains the majority of the thermal neutron population.
3.5.1 Mesh Domain
Perhaps the most straightforward approach to tallying MGXS is the Mesh domain. A uniform Cartesian mesh is placed over the entire core, with each mesh cell receiving its own set of multigroup cross sections. This approach requires no prior knowledge of the spatial dependence of the cross sections, and depending on the mesh size, can preserve a degree of heterogeneity. An early batch of cross sections was calculated using a 120×120 mesh tally over a single standard fuel element (Figs. 3.9, 3.10). Within each energy group, cross sections in the element did not change except with material.
(a) Actual mesh (b) Mesh colored by Σ푡 (group 10) Figure 3.9: Fine mesh over a single fuel element. Mesh cells within a single region share the same value. Note that some mesh cells, such as the diagonal corners and the gap/clad region, overlap multiple materials.
44 0.004 0.4
0.003 0.3 ) ) 1 1 m m c c ( (
0.002 0.2
0.001 0.1
10 10 0.000 ±1 0.0 ±1
4 2 0 2 4 4 2 0 2 4 Radial distance (cm) Radial distance (cm)
(a) Σ푓 (group 10) (b) Σ푡 (group 10) Figure 3.10: 1D XS plots with error bars The cross section profile is completely flat across the fuel meat.
Similar trends were observed for the control rod fuel element, this one tallied using an 80×80 mesh. Apart from locations where the geometry did not align perfectly with the mesh, there was virtually no variation in the cross section within each component.
0
0.40 0.4 10 0.35 20 0.3 0.30
30 )
1 0.25 m c
( 40 0.2 0.20
50 0.15 0.1 60 0.10
10 70 0.05 0.0 ±1
4 2 0 2 4 Radial distance (cm) 0 10 20 30 40 50 60 70
(a) Σ푡 (group 10) with error bars (b) Mesh colored by Σ푡 (group 10) Figure 3.11: Fine mesh over a control rod (follower) fuel element.
A 190×190 mesh (10×10 per element) was laid across the MCM2D (ARI) core to observe the behavior of cross sections in a model where the neutron spectrum changes. Across the fuel, the cross sections were almost flat within uncertainty, except in the fastest groups (1-2 out of 11, or 1-3 out of 25) where reaction rates are low and leakage is high. There appears to be some small deviation within the B4C rods themselves. Because the behavior of the cross sections is very consistent throughout the core,
45 0.44 10 10 ±1 ±1 2.0 0.42
0.40 1.5 ) )
1 1 0.38 m m c c ( ( 1.0 0.36
0.5 0.34
0.32 0.0
100 75 50 25 0 25 50 75 100 100 75 50 25 0 25 50 75 100 Radial distance (cm) Radial distance (cm)
(a) Σ푡 (group 10) across control rods (b) Σ푡 (group 10) across standard fuel
0.32 0 2 ±1 0.30 25 2.0
0.28 50
1.5 0.26 75 ) 1 m c (
0.24 100 1.0
0.22 125
0.20 150 0.5
0.18 175 100 75 50 25 0 25 50 75 100 Radial distance (cm) 0 25 50 75 100 125 150 175
(c) Σ푡 (group 2) across control rods (d) Mesh colored by Σ푡 (group 10) Figure 3.12: Mesh tally across the MCM2D (Poison) core The control rod slices were taken from the first row of rodded elements, index 55 in (d). The standard fuel slice was taken in the row immediately below, at index 65. The total cross section appears to go to 0 at the center of the rods in (a) because so few thermal neutrons penetrated that far. it is not efficient to represent them using a mesh. The Mesh domain requires alarge number of spatial bins; to resolve the clad and air gaps in a different mesh cell than the fuel requires a 40×40 (0.254 cm × 0.254 cm) mesh per element, meaning 1600 spatial bins for a single element model or 577,600 for the whole core. The number of particles needed to obtain good statistics on such a fine mesh makes the Monte Carlo calculation far more expensive than it would otherwise need to be. Storing cross sections for each mesh cell also results in a large amount of data. On modern hardware, reading the cross sections from disk can take longer than running the entire MOC calculation. While the Mesh domain is useful for observing the behavior of cross sections over the core, it is a poor choice for practical calculations of TREAT.
46 0 10 ±1 0.0040 0.004 25
0.0035 50 0.003 0.0030 75 ) 1
m 0.0025 c ( 0.002 100
0.0020 125 0.001 0.0015
150 0.0010
0.000 175 0.0005 100 75 50 25 0 25 50 75 100 Radial distance (cm) 0 25 50 75 100 125 150 175
(a) Σ푓 (group 10) across control rods (b) Mesh colored by Σ푓 (group 10) Figure 3.13: Thermal fission mesh tally across the MCM2D (Poison) core
3.5.2 Material Domain
In a sense, the Material domain is conceptually the opposite of the Mesh domain. Reactions in each unique material in the OpenMC model contributes to the same tally, regardless of their location in the core. Spatially, the model is fully heterogeneous, as seen in Figure 3.14.
Figure 3.14: Control rod (follower) fuel element, Material domain. The control rod follower, rod channel, fuel block, cladding, and air gaps are represented explicitly as shown. Each color represents a unique material; for a legend, see Figure 3.2.
The Material domain requires very little storage. The control rod fuel element with the graphite follower pictured above is represented with a total of 4 bins. This approach also scales well; the most heterogeneous model, MCM2D Poison (Section 3.3.6), requires only 7 materials over the entire core. Slight inaccuracies may be introduced by lumping several components from
47 different parts of the core together; for instance, graphite deep into the radial reflector may see a different neutron spectrum than graphite in the zirc-clad dummy elements or in the control rod follower. This could be remedied manually by defining identical materials for components in different regions of the core, or automatically by using the Cell domain.
3.5.3 Cell Domain
In the Cell domain, each unique CSG Cell in the OpenMC model has its own set of cross sections. This is similar to the Material domain, but different components of the same material (or components in different types of assembly) will contribute to different MGXS tallies. This slightly increases the overall fidelity.
Figure 3.15: Control rod (follower) fuel element, Cell domain. The control rod follower, rod channel, fuel block, cladding, and air gaps are represented explicitly as shown. Each color represents a unique CSG Cell.
The Cell domain can lead to excessive discretization on some geometries. It is probably not necessary to represent the three zircaloy Cells of Figures 3.14 and 3.15 with separate cross sections, for example. Nevertheless, the additional overhead is trivial; 9 bins are needed to represent a control rod fuel element, or 21 for either MCM2D core where they are present.
48 3.5.4 Universe Domain
Each species of TREAT element is represented by a CSG Universe. By tallying cross sections over these Universes, elements are effectively homogenized. Cross sections are commonly generated in such a manner for use in diffusion codes.
Figure 3.16: Control rod (follower) fuel element, Universe domain. The control rod follower, rod channel, fuel block, cladding, and air gaps are all lumped together. Each color represents a unique CSG Universe.
Applying this approach sacrifices spatial detail. If one does not care aboutthe reaction rate distribution within the assembly, it can be a good approximation. Notwithstanding, there are some caveats. Smearing air channels with the rest of the element misrepresents the streaming term. In theory, this effect could be ameliorated by adjusting the transport cross section using CMM (Section 2.5.1). In models where
B4C control rods are present, homogenizing the rods with the rest of the assembly gives rise to significant spatial self-shielding errors. For those errors, SPH factors (Section 2.5.2) are needed.
3.5.5 Heterogeneous vs. Homogeneous Representation
We selected the Cell domain for the heterogeneous representation of TREAT’s core. While it requires slightly more storage than tallying MGXS based on materials, we considered it worth the tradeoff for a marginal boost in accuracy. Conversely, thehigh storage and computational requirements of a fine Mesh domain makes that approach unattractive.
49 MGXS generated over some homogenized Cartesian mesh are also desirable. Such cross sections would be compatible with a variety of mesh-based deterministic solvers, such as INL’s RattleSNake. To this end, we chose to generate Universe-based cross sections. An element-width coarse mesh tally would accomplish the same objective. However, the cross sections do not meaningfully vary for different elements of the same type (except for fast groups in the reflector, where reaction rates are very low). For the homogeneous model, the Mesh domain offers little additional accuracy in return for its increased computational overhead.
(a) Heterogeneous (b) Homogeneous
Figure 3.17: Heterogeneous vs. Homogeneous representations of MCM2D Poison
50 Chapter 4
Deterministic Results
All models described in Section 3.3 were simulated with OpenMC using the revision given in Appendix C.1.2 with full 푆(훼, 훽) tables for graphite used in the fuel. An additional 1D toy problem with 59% 푆(훼, 훽) (Section 4.2.2) used the slightly more recent revision in Appendix C.1.1. OpenMC calculated the eigenvalues, reaction rate distributions, and multigroup cross sections for the homogeneous and heterogeneous representations. Each model was simulated and converged in OpenMOC. Absolute bias in eigenvalue and relative bias in reaction rates were calculated as follows:
5 ∆푘 (pcm) = (푘푀푂퐶 − 푘푀퐶 ) × 10 (4.1)
(︂푅 − 푅 )︂ ∆푅 (%) = 푀푂퐶 푀퐶 × 102 (4.2) 푅푀퐶
4.1 Variables to Converge
An MOC solution needs to be converged in several variables:
∙ Azimuthal ray spacing: 훿푎 (cm).
∙ Number of azimuthal angles: 푁푎.
51 ∙ Number of polar angles: 푁푝.
∙ Source region mesh: ∆푥, ∆푦 (cm).
Several of these spatial discretization parameters are interdependent. The combination of 훿푎 and 푁푎 determines the total track coverage of the geometry. Additionally, the tracks must be fine enough to characterize all the source regions of interest.
Unless otherwise specified, cases presented in this chapter used 훿푎 = 0.1 cm,
푁푝 = 6, 푁푎 = 64 for heterogeneous problems, and 푁푎 = 32 for homogeneous problems.
4.2 Source Region Convergence
The source region size determines how coarsely the flux profile is captured in space. It is advantageous to enforce a minimum source region size to ensure that the problem is sufficiently discretized in space. Applying OpenMOC’s CMFD solver divides geometries along a regular Cartesian mesh, establishing a global maximum SR size of ∆푥 × ∆푦 across the problem. In PWR problems, the OpenMOC CMFD mesh typically aligns with the assembly lattice for core calculations, or with the pincell lattice for assembly calculations. Cylindrical pincells themselves are often subdivided into azimuthal rings and/or sectors, as pictured in Fig. 4.1.
Figure 4.1: Azimuthal rings and sectors for a pincell in a box Image source: Fig. 8 from [15].
TREAT has few cylindrical features other than the control rods. Instead of
52 using a ring-based approach, we subdivided elements along a uniform Cartesian mesh
(Fig. 4.2). Each element type 푖 then has a local maximum SR dimension of д푖.
Figure 4.2: Subdivision of a TREAT element Here, a standard fuel element is subdivided on an 8 × 8 mesh. The maximum SR size д = 1.27 cm.
Heterogeneous geometries contain a number of features with dimensions smaller than д, further increasing SR refinement. For homogeneous geometries, all source regions on an element are the maximum size.
4.2.1 CMFD Convergence
Coarse Mesh Finite Difference (CMFD) is a method commonly used to accelerate transport calculations using a low-order diffusion solver [18]. OpenMOC’s CMFD solver cuts up source regions along a Cartesian mesh, ensuring a minimum SR size and streamlining the tallying process. Instabilities can arise near strong sources or strong absorbers. OpenMOC’s CMFD solver frequently encountered instabilities with the default settings and an element- width mesh. Setting the number of 푘-nearest neighbor cells to 3 and reducing the
1 mesh width to /2 of the element pitch caused the solver to be stable in all cases. Convergence for the CMFD solver could then be sped up with a successive over- relaxation factor of 1.5. As a result, the minimum CMFD mesh was 2 × 2 per element, for a global maximum source region size (∆푥 × ∆푦) of 5.08 cm × 5.08 cm.
53 4.2.2 1D Convergence Study
TREAT’s core experiences a great deal of neutron leakage, potentially resulting in steep flux gradients. We set up a toy problem to demonstrate the effect ofSR convergence in a high leakage scenario.
Figure 4.3: Geometry of toy problem 푥- and 푦-axes in cm. Fuel is red; reflector graphite is gray. Reflective BCs are on the top, bottom, and left. A vacuum BC is on theright.
The problem simulates a 1D slab of fuel using 59% 푆(훼, 훽) tables for graphite. The fuel slab is 121.92 cm wide (60.96 cm wide with a reflective boundary condition), equal to the width of 12 TREAT elements. To the right of the fuel is a slab of CP-2 graphite of the same width, with a vacuum boundary condition imposed at its right edge. It is infinite in the 푦 and 푧 dimensions. The source mesh was gradually refined. Both energy groups’ eigenvalues and flux distributions were compared to a multigroup solution calculated in OpenMC (the partial 푆(훼, 훽) version, Appendix C.1.1) using the same cross sections. Results are presented in Tables 4.1 and 4.2.
N. Groups Mesh д (cm) 푘 ∆푘 (pcm) 11 24 × 2 5.080 1.49895 −9227 60 × 5 2.032 1.57156 −1965 120 × 10 1.016 1.58589 −533 240 × 20 0.508 1.58991 −130 480 × 40 0.254 1.59095 −26 25 24 × 2 5.080 1.49641 −9179 60 × 5 2.032 1.56859 −1961 120 × 10 1.016 1.58280 −540 240 × 20 0.508 1.58680 −140 480 × 40 0.254 1.58782 −38
Table 4.1: FSR convergence for 1D problem
54 N. Groups Mesh д (cm) 푘 ∆푘 (pcm) 11 24 × 2 5.080 1.59014 −107 60 × 5 2.032 1.59170 +49 120 × 10 1.016 1.59144 +23 240 × 20 0.508 1.59136 +15 480 × 40 0.254 1.59130 +9 25 24 × 2 5.080 1.58699 −121 60 × 5 2.032 1.58856 +36 120 × 10 1.016 1.58831 +11 240 × 20 0.508 1.58823 +3 480 × 40 0.254 1.58818 −2
Table 4.2: LSR convergence for 1D problem
LSR Flux 2 LSR Flux 2
1 1 -60 -40 -20 0 20 40 60 -60 -40 -20 0 20 40 60 0 0
FSR Flux 2 FSR Flux 2
1 1 -60 -40 -20 0 20 40 60 -60 -40 -20 0 20 40 60 0 0
10 4
FSR vs. LSR. Relative Error (%) 5 FSR vs. LSR. Relative Error (%) 2
0 0
5 2 -60 -40 -20 0 20 40 60 -60 -40 -20 0 20 40 60 Position (cm) 10 Position (cm) 4
(a) д = 5.080 cm (b) д = 2.032 cm
LSR Flux 2
1 -60 -40 -20 0 20 40 60 0
FSR Flux 2
1 -60 -40 -20 0 20 40 60 0
1.0
FSR vs. LSR. Relative Error (%) 0.5
0.0
0.5 -60 -40 -20 0 20 40 60 Position (cm) 1.0
(c) д = 0.254 cm
Figure 4.4: Flat source vs. linear source for toy problem Flux profiles are plotted for both SR types. While the difference in gradients maybe nearly invisible to the naked eye, plotting the relative difference reveals errors of −3 to +3% in the fuel and up to +10% in the reflector for д = 5.080 cm.
55 Flat source regions performed poorly relative to linear source regions. Biases with the finest FSR д of 0.254 cm were comparable to those of LSRs at1-2cm.Fig. 4.4 shows the flux gradient over the 1D problem for each source type. Based onthese results, we are motivated to discard flat source solutions and opt for linear source regions with a width of ≈ 1 cm or less for problems with steep gradients.
4.2.3 2D Colorset Convergence
The single element and colorset cases are somewhat more forgiving. Their reflective boundary conditions do not permit any leakage. We examined the convergence of the LSR mesh in heterogeneous and homogeneous representations of each model. All subdividing meshes were aligned with the 2 × 2 CMFD mesh to avoid creating partial-width SRs.
Single Standard Fuel Element
The single element model was introduced in Section 3.3.1. The OpenMC reference
푘∞ = 1.665609 ± 1.5 pcm.
N. Groups Mesh д (cm) 푘∞ ∆푘 (pcm) 11 2 × 2 5.080 1.665737 +13 4 × 4 2.540 1.665747 +14 10 × 10 1.016 1.665749 +14 20 × 20 0.508 1.665750 +14 40 × 40 0.254 1.665751 +14 25 2 × 2 5.080 1.665746 +14 4 × 4 2.540 1.665756 +15 10 × 10 1.016 1.665759 +15 20 × 20 0.508 1.665759 +15 40 × 40 0.254 1.665760 +15
Table 4.3: Single fuel element LSR convergence, heterogeneous
The single element was already converged with the minimum CMFD mesh of 2×2. The homogeneous version is effectively an infinite homogeneous medium. It wouldbe converged regardless of the source region size, but the CMFD solver requires a mesh.
56 N. Groups Mesh д (cm) 푘∞ ∆푘 (pcm) 11 2 × 2 5.080 1.665610 ±0 4 × 4 2.540 1.665610 ±0 10 × 10 1.016 1.665610 ±0 20 × 20 0.508 1.665610 ±0 40 × 40 0.254 1.665610 ±0 25 2 × 2 5.080 1.665610 ±0 4 × 4 2.540 1.665610 ±0 10 × 10 1.016 1.665610 ±0 20 × 20 0.508 1.665610 ±0 40 × 40 0.254 1.665610 ±0
Table 4.4: Single fuel element LSR convergence, homogeneous
(a) Heterogeneous:д = 5.080 cm (b) Homogeneous: д = 5.080 cm
Figure 4.5: Converged LSR meshes for the single fuel element
57 3×3 Colorset (Follower)
The 3×3 colorset with the zirconium-clad graphite follower inserted into one of the elements is described in Section 3.3.2. The OpenMC simluation reported
푘∞ = 1.623351 ± 1.9 pcm. Note that in the tables below, “Mesh” describes the local mesh along which each element is subdivided, not a global mesh over the 3×3 array.
N. Groups Mesh д (cm) 푘∞ ∆푘 (pcm) 11 2 × 2 5.080 1.623334 −2 4 × 4 2.540 1.623351 ±0 10 × 10 1.016 1.623355 ±0 20 × 20 0.508 1.623356 ±0 40 × 40 0.254 1.623356 ±0 25 2 × 2 5.080 1.623333 −2 4 × 4 2.540 1.623350 ±0 10 × 10 1.016 1.623354 ±0 20 × 20 0.508 1.623355 ±0 40 × 40 0.254 1.623355 ±0
Table 4.5: 3×3 colorset (follower) LSR convergence, heterogeneous
N. Groups Mesh д (cm) 푘∞ ∆푘 (pcm) 11 2 × 2 5.080 1.623338 −1 4 × 4 2.540 1.623341 −1 10 × 10 1.016 1.623341 −1 20 × 20 0.508 1.623341 −1 40 × 40 0.254 1.623341 −1 25 2 × 2 5.080 1.623333 −2 4 × 4 2.540 1.623335 −2 10 × 10 1.016 1.623335 −2 20 × 20 0.508 1.623335 −2 40 × 40 0.254 1.623335 −2
Table 4.6: 3×3 colorset (follower) LSR convergence, homogeneous
58 It is unsurprising that the case with the control rod follower converges similarly to that with just the fuel element. While some fuel graphite is displaced by the rod channel, it is replaced by zircaloy, air, and regular graphite. The resulting changes to the flux gradient and the bias are minimal.
(a) Heterogeneous: д = 5.080 cm (b) Homogeneous: д = 5.080 cm
Figure 4.6: Converged LSR meshes for the 3×3 colorset (follower)
59 3×3 Colorset (Poison)
The 3×3 colorset (Section 3.3.3) with the steel-clad B4C poison section inserted had an OpenMC reference eigenvalue of 푘∞ = 0.745309 ± 1.9 pcm.
The presence of the B4C absorbers adds additional angular dependence to heterogeneous problems (Section 4.3). The results presented in Table 4.7 were calculated using 256 azimuthal angles.
N. Groups Mesh д (cm) 푘∞ ∆푘 (pcm) 11 2 × 2 5.080 0.736608 −870 4 × 4 2.540 0.742372 −294 10 × 10 1.016 0.743689 −162 20 × 20 0.508 0.743936 −137 40 × 40 0.254 0.744084 −122 25 2 × 2 5.080 0.736470 −884 4 × 4 2.540 0.742299 −301 10 × 10 1.016 0.743577 −173 20 × 20 0.508 0.743826 −148 40 × 40 0.254 0.743979 −133
Table 4.7: 3×3 colorset (poison) LSR convergence, heterogeneous
N. Groups Mesh д (cm) 푘∞ ∆푘 (pcm) 11 2 × 2 5.080 0.703122 −4219 4 × 4 2.540 0.705618 −3969 10 × 10 1.016 0.705568 −3974 20 × 20 0.508 0.705518 −3979 40 × 40 0.254 0.705519 −3979 25 2 × 2 5.080 0.702436 −4287 4 × 4 2.540 0.704961 −4035 10 × 10 1.016 0.704911 −4040 20 × 20 0.508 0.704861 −4045 40 × 40 0.254 0.704861 −4045
Table 4.8: 3×3 colorset (poison) LSR convergence, homogeneous
Due to homogenization errors (primarily in spatial self-shielding), the homogeneous model converges to an answer several thousand pcm from the reference value; this will be addressed in Section 4.5. Nevertheless, it converges quickly, which
60 makes sense because there is no spatial detail to resolve within the elements. A 4 × 4 mesh per element (д = 2.540 cm) is sufficient. On the other hand, it is not clear from the eigenvalue alone if the heterogeneous solution is converged at the finest mesh of 40 × 40. Inspecting the power distribution confirmed that it is (Fig. 4.8). The additional source region refinement is required to resolve flux and reaction rates on the rodded element. Notwithstanding, itis unnecessary to refine the regular fuel elements to the same degree. Table 4.9 records eigenvalues for a series of cases where the standard fuel element SR mesh is refined whilst holding the control rod fuel element’s mesh at 40×40.A 4×4 mesh on the fuel elements (д퐹 푈퐸퐿 = 2.540 cm) in conjunction with the 40 × 40 mesh on the rodded element is effectively converged.
N. Groups Mesh д퐹 푈퐸퐿 (cm) 푘∞ ∆푘 (pcm) 11 2 × 2 5.080 0.742344 −297 4 × 4 2.540 0.744180 −113 10 × 10 1.016 0.744126 −118 20 × 20 0.508 0.744091 −122 40 × 40 0.254 0.744084 −122 25 2 × 2 5.080 0.742222 −309 4 × 4 2.540 0.744075 −123 10 × 10 1.016 0.744021 −129 20 × 20 0.508 0.743986 −132 40 × 40 0.254 0.743979 −133
Table 4.9: 3×3 colorset (poison) convergence, where д퐶푅퐷 = 0.254 cm
61 (a) Heterogeneous: д퐹 푈퐸퐿 = 2.540 cm, (b) Homogeneous: д = 2.540 cm д퐶푅퐷 = 0.254 cm Figure 4.7: Converged LSR meshes for the 3×3 colorset (poison)
62 63
Figure 4.8: Power distributions for the 3×3 colorset (poison) models A mesh of д퐶푅퐷 = 0.254 cm is sufficient to converge the problem, but this level of refinement is unnecessary for the standard fuel elements. Refining д퐹 푈퐸퐿 from 2.54 cm (4 × 4 per elment) to 0.254 cm (40 × 40 per element) changes the fission rate distribution by hundredths of a percent. Figure shows 25-group results. 4.2.4 2D Full-Core Convergence
The 2D full-core models include radial leakage. From the 1D convergence study in Section 4.2.2, we expect the maximum LSR size д to around 1 cm to resolve the flux gradient across the core in all the cases.
MCM2D (No Rods)
The MCM configuration with no control rod elements is described in Section 3.3.4.
The reference eigenvalue from the OpenMC model is 푘eff = 1.180659 ± 2.3 pcm.
The MCM (no rods) core appears to be converged in source region size by 0.508 cm (mesh of 20 × 20 per element) for the heterogeneous case and 1.016 cm (10 × 10) for the homogeneous. The eigenvalues for the 11-group models and homogeneous models converge to values with significant biases from the reference solution. The homogenization errors will be addressed in Section 4.5.
N. Groups Mesh д (cm) 푘eff ∆푘 (pcm) 11 2 × 2 5.080 1.189968 +931 4 × 4 2.540 1.191394 +1073 10 × 10 1.016 1.190743 +1008 20 × 20 0.508 1.190609 +995 40 × 40 0.254 1.190578 +995 25 2 × 2 5.080 1.182078 +142 4 × 4 2.540 1.183531 +287 10 × 10 1.016 1.182887 +223 20 × 20 0.508 1.182753 +209 40 × 40 0.254 1.182722 +206
Table 4.10: MCM2D (No Rods) LSR convergence, heterogeneous
64 N. Groups Mesh д (cm) 푘eff ∆푘 (pcm) 11 2 × 2 5.080 1.198948 +1829 4 × 4 2.540 1.199942 +1928 10 × 10 1.016 1.199156 +1850 20 × 20 0.508 1.199056 +1840 40 × 40 0.254 1.199158 +1850 25 2 × 2 5.080 1.191000 +1034 4 × 4 2.540 1.192026 +1137 10 × 10 1.016 1.191246 +1059 20 × 20 0.508 1.191147 +1049 40 × 40 0.254 1.191252 +1059
Table 4.11: MCM2D (No Rods) convergence, homogeneous
(a) Heterogeneous: д = 0.508 cm (b) Homogeneous: д = 1.016 cm
Figure 4.9: Converged LSR meshes for the MCFuel core
65 66
Figure 4.10: MCM2D (No Rods): 11 groups vs. 25 groups MCM2D (Follower)
The MCM model with the graphite follower portion of the control rods inserted is described in Section 3.3.5. The reference eigenvalue from OpenMC’s calculation is
푘eff = 1.162142 ± 3.2
N. Groups Mesh д (cm) 푘eff ∆푘 (pcm) 11 2 × 2 5.080 1.171575 +943 4 × 4 2.540 1.172946 +1080 10 × 10 1.016 1.172318 +1018 20 × 20 0.508 1.172189 +1005 40 × 40 0.254 1.172159 +1002 25 2 × 2 5.080 1.163717 +158 4 × 4 2.540 1.165113 +297 10 × 10 1.016 1.164492 +235 20 × 20 0.508 1.164363 +222 40 × 40 0.254 1.164332 +219
Table 4.12: MCM2D (Follower) LSR convergence, heterogeneous
N. Groups Mesh д (cm) 푘eff ∆푘 (pcm) 11 2 × 2 5.080 1.180478 +1834 4 × 4 2.540 1.181432 +1929 10 × 10 1.016 1.180676 +1853 20 × 20 0.508 1.180577 +1843 40 × 40 0.254 1.180686 +1854 25 2 × 2 5.080 1.185553 +1041 4 × 4 2.540 1.186501 +1140 10 × 10 1.016 1.185767 +1065 20 × 20 0.508 1.185672 +1055 40 × 40 0.254 1.185783 +1066
Table 4.13: MCM2D (Follower) convergence, homogeneous
As expected, the results and biases are similar to the MCFuel model. Despite the additional heterogeneous detail in the control rod fuel elements, a mesh of 20 × 20 per element (д = 0.508 cm) is still sufficient to converge the heterogeneous problem in SR size. The homogeneous case has converged at д = 1.016 cm once again.
67 68
Figure 4.11: MCM2D (Follower): 11 groups vs. 25 groups As with the MCFuel core, the use of 11 energy groups underpredicts power/overpredicts leakage near the reflector. The presence of the graphite followers does not appear to introduce additional spatial error. (a) Heterogeneous: д = 0.508 cm (b) Homogeneous: д = 1.016 cm
Figure 4.12: Converged LSR meshes for the MCM2D (Follower) core
69 MCM2D (Poison)
The MCM2D model with the the B4C poison portion of the control rods inserted
(Section 3.3.6) had a reference eigenvalue of 푘eff = 0.848922 ± 2.6 pcm as calculated by OpenMC. The heterogeneous cases were run using 푁푎 = 256 angles, instead of the usual 64 (Section 4.3). The homogeneous case still used 32 azimuthal angles.
N. Groups Mesh д (cm) 푘eff ∆푘 (pcm) 11 2 × 2 5.080 0.853055 +413 4 × 4 2.540 0.856727 +781 10 × 10 1.016 0.856729 +781 20 × 20 0.508 0.856724 +780 40 × 40 0.254 0.856785 +786 25 2 × 2 5.080 0.846864 −206 4 × 4 2.540 0.850561 +164 10 × 10 1.016 0.850551 +163 20 × 20 0.508 0.850546 +162 40 × 40 0.254 0.850609 +169
Table 4.14: MCM2D (Poison) LSR convergence, heterogeneous
N. Groups Mesh д (cm) 푘eff ∆푘 (pcm) 11 2 × 2 5.080 0.844078 −484 4 × 4 2.540 0.846720 −220 10 × 10 1.016 0.846112 −281 20 × 20 0.508 0.845991 −293 40 × 40 0.254 0.846074 −285 25 2 × 2 5.080 0.846864 −1122 4 × 4 2.540 0.850561 −856 10 × 10 1.016 0.850551 −913 20 × 20 0.508 0.850546 −925 40 × 40 0.254 0.850609 −916
Table 4.15: MCM2D (Poison) convergence, homogeneous
As with the 3×3 B4C model, the homogeneous solutions converge around д = 1.016 cm (10 × 10 mesh). It is not immediately clear from the eigenvalues where the heterogeneous solutions converged. The SR size of д = 2.540 cm looks promising, but we expect д to be closer to 0.5-1.0 cm. Inspecting the fission rate distributions
70 (a) 11 groups (b) 25 groups
Figure 4.13: Power distributions for д = 0.508 cm vs. 0.254 cm At a mesh of 10 × 10 per element, the error in normalized fission rate is well within ±0.1% for either energy group structure. of each д versus the finest (Fig. 4.13) result shows that the problem is well converged at д = 0.508 cm. As with the colorset model, we then fixed д퐶푅퐷 at 0.254 cm and refined the mesh of the rest of the problem (Table 4.16):
N. Groups Mesh д푂푇 퐻퐸푅 (cm) 푘∞ ∆푘 (pcm) 11 2 × 2 5.080 0.855411 +649 4 × 4 2.540 0.857419 +850 10 × 10 1.016 0.856903 +798 20 × 20 0.508 0.856792 +787 40 × 40 0.254 0.856785 +786 25 2 × 2 5.080 0.846864 +29 4 × 4 2.540 0.850561 +231 10 × 10 1.016 0.850551 +180 20 × 20 0.508 0.850546 +169 40 × 40 0.254 0.850609 +169
Table 4.16: MCM2D (Poison) convergence, where д퐶푅퐷 = 0.254 cm
With д퐶푅퐷 = 0.254 cm, the heterogeneous problem converges at a д of 0.508 cm (20 × 20 per element) for the fuel and reflector, consistent with the other full-core models. A noticeable bias persists in the 11-group results, both in the eigenvalue and the power distribution (Fig. 4.14).
71 OpenMOC fission Distribution OpenMOC fission Distribution OpenMC fission Distribution 11 groups 25 groups 0 0 0 1.8 1.8 1.8
5 1.6 5 1.6 5 1.6
10 1.4 10 1.4 10 1.4
1.2 1.2 1.2 15 15 15
1.0 1.0 1.0 20 20 20
0.8 0.8 0.8 25 25 25 0.6 0.6 0.6
30 30 30 0.4 0.4 0.4
35 35 35 0.2 0.2 0.2
0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 % Relative error of 11 groups % Relative error of 11 groups % Relative error of 25 groups vs 25 groups vs openmc vs openmc 0 1.00 0 1.00 0 1.00
0.75 0.75 0.75 5 5 5 72 0.50 0.50 0.50 10 10 10
0.25 0.25 0.25 15 15 15
0.00 0.00 0.00 20 20 20
0.25 0.25 0.25 25 25 25
0.50 0.50 0.50 30 30 30
0.75 0.75 0.75 35 35 35 1.00 1.00 1.00 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35
Figure 4.14: MCM2D (Poison): 11 groups vs. 25 groups As with the No Rods and Follower cases, the use of 11 energy groups underpredicts power/overpredicts leakage near the reflector.
The B4C rods themselves appear to introduce little additional error to the heterogeneous cases. (a) Heterogeneous: д퐶푅퐷 = 0.254 cm, (b) Homogeneous: д = 1.016 cm д푂푇 퐻퐸푅 = 0.508 Figure 4.15: Converged LSR meshes for the MCM2D (Poison) core
73 4.3 Ray Spacing and Azimuthal Angle Convergence
1 The azimuthal ray spacing 훿푎 specifies the desired linear distance between tracks.
The ray spacing must be fine enough resolve all SRs. Co-dependent with 훿푎 is the number of azimuthal angles, 푁푎. There must be enough angles to characterize the angular flux in the model. Together, 푁푎 and 훿푎 determine the number of tracks over a geometry of a given size.
Unless otherwise specified, all OpenMOC calculations presented so far have used
훿푎 = 0.1 cm and 푁푎 = 32 angles. Unlike a classical LWR, TREAT has few features where a ray spacing of finer than 0.1 cm would be interesting. The only “thin” structures are the air gaps and cladding, which are nearly transparent to neutrons; there is no IFBA or other such feature to resolve. Because TREAT’s fuel is nearly homogeneous (HEU dispersed in its own moderator), the angular dependencies and resonance self-shielding effects characteristic of an LWR lattice are less of a concern as well.
Now that we know the source region meshes required to converge our MOC models (Section 4.2), we shall examine ray and angle convergence on those meshes on several problems.
4.3.1 Heterogeneous
3×3 Follower
The heterogeneous 3×3 colorset with the graphite follower converged in LSR size with д = 5.280 cm. Table 4.17 records the eigenvalue bias (using the 25-group MGXS) as a function of the ray spacing and number of azimuthal angles on that mesh.
1OpenMOC’s track generator may adjust the ray spacing slightly to ensure cyclical tracks with the angles chosen.
74 훿푎
푁푎 1.00 cm 0.50 cm 0.10 cm 0.05 cm 8 angles −62 −40 +20 +12 16 angles −899 +203 −9 +1 32 angles +389 +125 −7 +1 64 angles +303 +194 ±0 +1 128 angles −21 +177 ±0 ±0
Table 4.17: ∆푘 (훿푎, 푁푎) for the heterogeneous 3×3 follower colorset
We selected the combination of 훿푎 = 0.1 cm and 푁푎 = 64 angles. However,
훿푎 = 0.05 cm and 푁푎 = 16 angles is also sufficient to converge the× 3 3 follower colorset.
MCM2D (Poison)
The most challenging model to converge is the MCM2D core with the B4C poison rods
inserted. The converged LSR mesh used д퐹 푈퐸퐿 and д푅퐸퐹 퐿 = 1.016 cm (corresponding
to 10 × 10 mesh per element) and д퐶푅퐷 = 0.254 cm (40 × 40 mesh per element).
훿푎 푁푎 1.00 cm 0.50 cm 0.10 cm 0.05 cm 8 angles +732 +71 +504 +500 16 angles −1046 +545 +248 +305 32 angles +699 +232 +250 +254 64 angles +304 +202 +218 +210 128 angles +415 +290 +198 +193 256 angles +297 +132 +180 +180
Table 4.18: ∆푘 (훿푎, 푁푎) for the heterogeneous MCM2D (Poison) core
The combination of 훿푎 = 0.1 cm and 푁푎 = 256 angles is fairly well converged.
75 4.3.2 Homogeneous
We expect the homogeneous representation to have little to offer in terms of angular dependency. The 25-group uncorrected MGXS library was used to generate the results in Table 4.19.
훿푎 푁푎 1.00 cm 0.50 cm 0.10 cm 0.05 cm 8 angles −840 −937 −932 −932 16 angles −918 −913 −919 −918 32 angles −893 −890 −913 −915 64 angles −886 −900 −913 −915 128 angles −897 −921 −913 −914 256 angles −903 −905 −913 −914
Table 4.19: ∆푘 (훿푎, 푁푎) for the homogeneous MCM2D (Poison) core
The values of 훿푎 = 0.1 cm and 푁푎 = 32 angles are sufficient to converge the most challenging homogeneous problem.
4.4 Polar Angle Convergence
For 2D problems, only a small number of polar angles are necessary to charaterize the angular flux. The default polar angle quadrature in OpenMOC is the Tabuchi- Yamamoto quadrature [19] with 3 levels (TY3). It represents a total of 6 angles when solving forward and backward on each track. Because the Tabuchi-Yamamoto quadratures only go up to TY3, it is de facto converged in polar angle.
76 4.5 Effects of Corrections for Homogeneous Models
Homogenizing elements is a source of error in the respective deterministic solutions. These errors can be substantial. As discussed in Section 2.4, however, methods exist for reducing the errors introduced with homogenization.
4.5.1 Correcting with CMM
The Cumulative Migration Method [13] provides us with a set of correction factors for the transport cross sections of homogenized TREAT elements, as described in Section 2.5.1. In theory, these corrections will reduce the error from homogenizing over air channels. CMM corrections for three element types–standard fuel, control rod fuel with follower, and control rod fuel with poison–were generated on an infinite lattice of each respective element (AppendixB)[20].
Single Standard Fuel Element
The homogeneous fuel element model is effectively an infinite homogeneous medium. CMM corrections are only applied to the transport and within-group scatter cross
sections, which do not affect 푘∞ in an IHM. Thus, use of the Cumulative Migration Method has no effect on this model.
3×3 Colorset (Follower)
Although the homogeneous 3×3 colorset with the control rod follower is not strictly an IHM, the control rod element is neutronically very similar to the standard fuel element. In fact, the CMM corrections are nearly identical for both element types. CMM has no effect on this model either.
77 3×3 Colorset (Poison)
CMM actually moves the OpenMOC eigenvalue further away from the reference value,
푘∞ = 0.745309 ± 1.9 pcm. Recall that the CMM correction factors for each type of element were generated in an infinite lattice of the respective element. Applying corrections generated with vastly different angular flux spectra is evidently nota valid application of the Cumulative Migration Method.
N. Groups CMM 푘∞ ∆푘 (pcm) 11 False 0.705618 −3969 True 0.704575 −4073 25 False 0.704961 −4035 True 0.704173 −4114
Table 4.20: Effect of CMM on the3×3 (Poison) model, where д = 2.54 cm
78 MCM2D (No Rods)
The angular flux in the fuel elements of the MCFuel core should be very similarto that of the medium in which the element’s CMM factors were generated. With the presence of radial leakage, corrections to the transport cross section should now have an effect on the solution.
The OpenMC 푘eff = 1.180659±2.3 pcm. CMM does remarkably well in preserving the eigenvalue, as seen in Table 4.21.
N. Groups CMM 푘eff ∆푘 (pcm) 11 False 1.199156 +1850 True 1.181488 +83 25 False 1.191246 1059 True 1.180799 +14
Table 4.21: Effect of CMM on the MCM2D (No Rods) model, withд = 1.016 cm
79 OpenMOC fission Distribution OpenMOC fission Distribution OpenMC fission Distribution mesh10_cmm mesh10_nocmm 0 0 0
1.3 1.3 1.3 5 5 5
1.2 1.2 1.2 10 10 10
1.1 1.1 1.1 15 15 15
1.0 1.0 1.0 20 20 20
0.9 0.9 0.9 25 25 25
0.8 0.8 0.8 30 30 30
0.7 0.7 0.7 35 35 35
0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 % Relative error of mesh10_cmm % Relative error of mesh10_cmm % Relative error of mesh10_nocmm vs mesh10_nocmm vs openmc vs openmc 0 1.00 0 1.00 0 1.00
0.75 0.75 0.75 5 5 5 80 0.50 0.50 0.50 10 10 10
0.25 0.25 0.25 15 15 15
0.00 0.00 0.00 20 20 20
0.25 0.25 0.25 25 25 25
0.50 0.50 0.50 30 30 30
0.75 0.75 0.75 35 35 35 1.00 1.00 1.00 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35
Figure 4.16: CMM-corrected vs. uncorrected for MCM2D (No Rods) Using CMM-corrected cross sections for this core results in a higher relative fission rate near the fuel/reflector interface, increasing the leakage from the core and lowering the eigenvalue. Figure shows 25-group results. MCM2D (Follower)
As previously mentioned, a control rod fuel element with the graphite follower portion inserted behaves similarly to a standard fuel element. Ergo, the results for the MCM core with the follower are similar to those of the MCFuel model. Eigenvalues are
much closer to the reference value of 푘eff = 1.162142 ± 3.2 pcm from OpenMC, and power distributions are improved as well.
N. Groups CMM 푘eff ∆푘 (pcm) 11 False 1.180676 +1853 True 1.163642 +150 25 False 1.172790 +1065 True 1.162701 +56
Table 4.22: Effect of CMM on the MCM2D (Follower) model, withд = 1.016 cm
MCM2D (Poison)
As with the 3×3 colorset with the B4C control rod inserted, the MCM2D model with all rods in is made less accurate with the application of CMM corrections. The desired eigenvalue of 푘eff = 0.848922 ± 2.6 pcm is severely underpredicted.
N. Groups CMM 푘eff ∆푘 (pcm) 11 False 0.846112 −281 True 0.831708 −1721 25 False 0.839766 −916 True 0.830952 −1797
Table 4.23: Effect of CMM on the MCM2D (Poison) model, withд = 1.016 cm
81 OpenMOC fission Distribution OpenMOC fission Distribution OpenMC fission Distribution mesh10_cmm mesh10_nocmm 0 1.4 0 1.4 0 1.4
5 1.3 5 1.3 5 1.3
10 1.2 10 1.2 10 1.2
1.1 1.1 1.1 15 15 15
1.0 1.0 1.0 20 20 20
0.9 0.9 0.9 25 25 25
0.8 0.8 0.8 30 30 30 0.7 0.7 0.7
35 35 35 0.6 0.6 0.6 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 % Relative error of mesh10_cmm % Relative error of mesh10_cmm % Relative error of mesh10_nocmm vs mesh10_nocmm vs openmc vs openmc 1.00 1.00 1.00 82 0 0 0
0.75 0.75 0.75 5 5 5
0.50 0.50 0.50 10 10 10
0.25 0.25 0.25 15 15 15
0.00 0.00 0.00 20 20 20
0.25 0.25 0.25 25 25 25
0.50 0.50 0.50 30 30 30
0.75 0.75 0.75 35 35 35 1.00 1.00 1.00 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35
Figure 4.17: CMM-corrected vs. uncorrected for MCM2D (Follower) The uncorrected model underpredicts leakage, resulting in too much power at the fuel/reflector interface. Using CMM-corrected cross sections more correctly calculates the leakage and improves the power distribution. Figure shows 25-group results. This page has been intentionally left blank.
83 OpenMOC fission Distribution OpenMOC fission Distribution OpenMC fission Distribution mesh10_cmm mesh10_nocmm 0 0 0
1.75 1.75 1.75 5 5 5
1.50 1.50 1.50 10 10 10
1.25 1.25 1.25 15 15 15
1.00 1.00 1.00 20 20 20
0.75 0.75 0.75 25 25 25
0.50 0.50 0.50 30 30 30
0.25 0.25 0.25 35 35 35
0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 % Relative error of mesh10_cmm % Relative error of mesh10_cmm % Relative error of mesh10_nocmm vs mesh10_nocmm vs openmc vs openmc 2.0 84 0 0 0
10 10 1.5 5 5 5
1.0 10 10 5 10 5
0.5 15 15 15
0.0 0 0 20 20 20
0.5 25 25 25 5 5 1.0 30 30 30
1.5 10 10 35 35 35 2.0 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35
Figure 4.18: CMM-corrected vs. uncorrected for MCM2D (Poison) CMM corrections increase the error on the rodded elements, although they slightly reduce error on the rest of the fuel. Note the change of scaale in the lower-left plot. Figure shows 25-group results. Interestingly, the CMM-corrected 11-group and 25-group results now agree with each other better. Previously, it was noted that 11-group cross sections tended to underpredict leakage, due to having less energy resolution in the fast groups. Fig. 4.19 reveals that the CMM-corrected 11-group and 25-group models have more consistent leakage. The improvement in accuracy of the neutron streaming appears to be competing with the bias caused by inapplicable CMM corrections. We expect that correction factors calculated on an appropriate domain in the full-core geometry will eliminate most of this bias.2
% Relative error of 11 groups % Relative error of 11 groups vs 25 groups vs 25 groups 0 1.00 0 1.00
0.75 0.75 5 5
0.50 0.50 10 10
0.25 0.25 15 15
0.00 0.00 20 20
0.25 0.25 25 25
0.50 0.50 30 30
0.75 0.75 35 35 1.00 1.00 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35
(a) Without CMM (b) With CMM
Figure 4.19: Effect of CMM on the 11-group vs. 25-group power distribution CMM correction factors greatly improve the agreement betweeen the results of the two group structures. Fission source discrepancies at the fuel/reflector interface are reduced by an order of magnitude.
2At publication, no tool exists to perform this sort of calculation. We look forward to seeing the results of such a calculation as CMM matures.
85 4.5.2 Correcting with SPH
Significant spatial self-shielding errors arise from the homogenization4 oftheB C control rod elements. These must be corrected with Superhomog´en´eisationfactors [14, 15] to improve the homogeneous MOC results, as mentioned in Section 2.5.2.
For each model with the B4C rods inserted, one set of SPH factors was calculated over all rodded elements. Regular fuel and reflector elements were excluded. SPH factors were converged to a tolerance of 10−5 (1 pcm) using a total of 12 iterations. In both cases, the final SPH factors were all < 1, lowering the effective cross sections in the control rod element.
3×3 colorset (Poison)
The 3×3 colorset results show a marked improvement with the use of SPH factors. Their effect on eigenvalue is basically the same regardless of whether CMM correction factors are also applied.
N. Groups SPH CMM 푘∞ ∆푘 (pcm) 11 False False 0.705618 −3969 True False 0.734222 −1109 True True 0.733768 −1154 25 False False 0.704961 −4035 True False 0.734580 −1073 True True 0.734022 −1129
Table 4.24: Effect of SPH on the3×3 (Poison) model, with д = 2.54 cm
86 87
Figure 4.20: SPH vs. uncorrected for 3×3 (Poison) Using SPH factors significantly reduces reaction rate errors in the element containing the control rod. Figure shows 25-group results. MCM2D (Poison)
SPH factors also reduce reaction rate errors on the full-core 2D MCM core with the
B4C poison portion of the control rods inserted.
N. Groups SPH CMM 푘eff ∆푘 (pcm) 11 False False 0.846112 −281 True False 0.857849 +893 True True 0.844089 −483 25 False False 0.839766 −916 True False 0.851075 +215 True True 0.843620 −530
Table 4.25: Effect of SPH on the MCM2D (Poison) model, withд = 1.016 cm
For both energy group structures, the use of SPH factors alone raises 푘eff by over 1100 pcm. Applying CMM corrections to all elements before the SPH iteration results in an eigenvalue with a negative bias for both energy group structures. As with the CMM-only case (Section 4.5.1), the use of CMM helps to correct the overprediction of fast neutron leakage when using 11-group MGXS (Fig. 4.23) and results in greater agreement between the two group structures (Fig. 4.21).
% Relative error of 11 groups % Relative error of 11 groups vs 25 groups vs 25 groups 0 1.00 0 1.00
0.75 0.75 5 5
0.50 0.50 10 10
0.25 0.25 15 15
0.00 0.00 20 20
0.25 0.25 25 25
0.50 0.50 30 30
0.75 0.75 35 35 1.00 1.00 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35
(a) Without CMM (b) With CMM
Figure 4.21: Effect of SPH+CMM on the 11-group vs. 25-group power distribution When used in conjunction with SPH factors, CMM corrections improve the agreement betweeen the 11-group and 25-group fission rate distributions even more than when used alone. Discrepancies at the fuel/reflector interface are reduced by two orders of magnitude.
88 OpenMOC fission Distribution OpenMOC fission Distribution OpenMC fission Distribution no SPH SPH (iter 12) 0 0 0
1.75 1.75 1.75 5 5 5
1.50 1.50 1.50 10 10 10
1.25 1.25 1.25 15 15 15
1.00 1.00 1.00 20 20 20
0.75 0.75 0.75 25 25 25
0.50 0.50 0.50 30 30 30
0.25 0.25 0.25 35 35 35
0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 % Relative error of no SPH % Relative error of no SPH % Relative error of SPH (iter 12) vs SPH (iter 12) vs openmc vs openmc 10.0 10.0 10.0 89 0 0 0
7.5 7.5 7.5 5 5 5
5.0 5.0 5.0 10 10 10
2.5 2.5 2.5 15 15 15
0.0 0.0 0.0 20 20 20
2.5 2.5 2.5 25 25 25
5.0 5.0 5.0 30 30 30
7.5 7.5 7.5 35 35 35 10.0 10.0 10.0 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35
Figure 4.22: SPH vs. uncorrected for MCM2D (Poison) Using SPH factors reduces reaction rate errors in elements containing control rods. Figure shows 25-group results. This page has been intentionally left blank.
90 91
Figure 4.23: SPH+CMM vs. SPH only for MCM2D (Poison) Figure shows 11-group results. 4.6 Final Results
This section aggregates the final converged deterministic results for each model. Plots depict the 25-group solutions.
4.6.1 Single Fuel Element Results
The standard fuel element case (Section 3.3.1) had a reference eigenvalue of 푘∞ =
1.665609±1.5 pcm from OpenMC. The MOC results shown used 훿푎 = 0.1 cm, 푁푎 = 64 for heterogeneous problem, and 푁푎 = 32 for the homogeneous problem.
MGXS д (cm) 푘∞ ∆푘 (pcm) 11-Group Het 5.080 1.665737 +13 25-Group Het 5.080 1.665746 +14 11-Group Hom 5.080 1.665610 ±0 25-Group Hom 5.080 1.665610 ±0
Table 4.26: Final Single Fuel Element Results
Geometry OpenMC Fission Distribution 1.010
1.005
1.000
0.995
0.990
% Relative error of heterogeneous % Relative error of homogeneous vs openmc vs openmc -5.08 -5.08 0.10
0.05
0.0 0.0 0.00
0.05
5.08 5.08 0.10
0.0 0.0 -5.08 5.08 -5.08 5.08
Figure 4.24: Final Single Fuel Element Plots
92 4.6.2 3×3 Colorset (Follower) Results
The 3×3 colorset with the zircaloy-clad graphite follower (Section 3.3.2) had a
reference eigenvalue of 푘∞ = 1.623351 ± 1.9 pcm from OpenMC. The final MOC
results used the same parameters as the single fuel element: 훿푎 = 0.1 cm, 푁푎 = 64
(heterogeneous), and 푁푎 = 32 (homogeneous).
MGXS д (cm) 푘∞ ∆푘 (pcm) 11-Group Het 5.080 1.623334 −2 25-Group Het 5.080 1.623333 −2 11-Group Hom 5.080 1.623338 −1 25-Group Hom 5.080 1.623333 −2
Table 4.27: Final 3×3 Colorset (Follower) Results
Geometry OpenMC Fission Distribution
1.2
1.1
1.0
0.9
0.8
% Relative error of heterogeneous % Relative error of homogeneous vs openmc vs openmc -15.24 -15.24 0.2
-10.16 -10.16 0.1 -5.08 -5.08
0.0 0.0 0.0
5.08 5.08 0.1 10.16 10.16
15.24 15.24 0.2
0.0 0.0 -5.08 5.08 -5.08 5.08 -15.24 -10.16 10.16 15.24 -15.24 -10.16 10.16 15.24
Figure 4.25: Final 3×3 Colorset (Follower) Plots
93 4.6.3 3×3 Colorset (Poison) Results
The reference eigenvalue for the 3×3 colorset with the B4C poison inserted (Section 3.3.3) was 푘 inf = 0.745309 ± 1.9 pcm according to OpenMC. The hetero- geneous cases used 푁푎 = 256 azimuthal angles, the homogeneous cases used 푁푎 = 32
azimuthal angles, and both used a ray spacing of 훿푎 = 0.1 cm.
MGXS д퐶푅퐷 (cm) д퐹 푈퐸퐿 (cm) 푘∞ ∆푘 (pcm) 11-Group Het 0.254 2.540 0.744180 −113 25-Group Het 0.254 2.540 0.744075 −123 11-Group Hom+SPH 2.540 2.540 0.734222 −1109 25-Group Hom+SPH 2.540 2.540 0.734580 −1073
Table 4.28: Final 3×3 Colorset (Poison) Results
Geometry OpenMC Fission Distribution
1.4
1.2
1.0
0.8
0.6
% Relative error of heterogeneous % Relative error of homogeneous vs openmc vs openmc -15.24 -15.24 2
-10.16 -10.16 1 -5.08 -5.08
0.0 0.0 0
5.08 5.08 1 10.16 10.16
15.24 15.24 2
0.0 0.0 -5.08 5.08 -5.08 5.08 -15.24 -10.16 10.16 15.24 -15.24 -10.16 10.16 15.24
Figure 4.26: Final 3×3 Colorset (Poison) Plots The homogeneous plot is shown with SPH.
94 4.6.4 MCM2D (No Rods) Results
According to OpenMC’s reference solution, 푘eff = 1.180659±2.3 pcm for the simplified minimum critical mass core where all control rod elements replaced by standard fuel elements (Section 3.3.4). The final MOC results presented below used 푁푎 = 64 angles for the heterogeneous representation, 푁푎 = 32 angles for the homogeneous representation, and 훿푎 = 0.1 cm for both.
MGXS д (cm) 푘eff ∆푘 (pcm) 11-Group Het 0.508 1.190609 +995 25-Group Het 0.508 1.182753 +209 11-Group Hom+CMM 1.016 1.181488 +83 25-Group Hom+CMM 1.016 1.180799 +14
Table 4.29: Final MCM2D (No Rods) Results
Geometry OpenMC Fission Distribution
1.2
1.0
0.8
% Relative error of heterogeneous % Relative error of homogeneous vs openmc vs openmc -96.52 -96.52 1.0
-64.35 -64.35 0.5 -32.17 -32.17
0.0 0.0 0.0
32.17 32.17 0.5 64.35 64.35
96.52 96.52 1.0
0.0 0.0 -96.52 -64.35 -32.17 32.17 64.35 96.52 -96.52 -64.35 -32.17 32.17 64.35 96.52
Figure 4.27: Final MCM2D (No Rods) Plots The homogeneous plot is shown with CMM.
95 4.6.5 MCM2D (Follower) Results
The MCM core with the graphite follower portion of the control rods present
(Section 3.3.5) had a reference eigenvalue of 푘eff = 1.162142 ± 3.2 pcm from the
Monte Carlo solution. As with the previous model, 푁푎 = 64 for the heterogeneous version, 푁푎 = 32 for the homogeneous version, and 훿푎 = 0.1 cm for both.
MGXS д (cm) 푘eff ∆푘 (pcm) 11-Group Het 0.508 1.172189 +1005 25-Group Het 0.508 1.164363 +222 11-Group Hom+CMM 1.016 1.163642 +150 25-Group Hom+CMM 1.016 1.162701 +56
Table 4.30: Final MCM2D (Follower) Results
Geometry OpenMC Fission Distribution 1.4
1.2
1.0
0.8
0.6
% Relative error of heterogeneous % Relative error of homogeneous vs openmc vs openmc -96.52 -96.52 1.0
-64.35 -64.35 0.5 -32.17 -32.17
0.0 0.0 0.0
32.17 32.17 0.5 64.35 64.35
96.52 96.52 1.0
0.0 0.0 -96.52 -64.35 -32.17 32.17 64.35 96.52 -96.52 -64.35 -32.17 32.17 64.35 96.52
Figure 4.28: Final MCM2D (Follower) Plots The homogeneous plot is shown with CMM.
96 4.6.6 MCM2D (Poison) Results
Finally, the MCM configuration with the B4C control rods inserted in the core
(Section 3.3.6) had an eigenvalue of 푘eff = 1.162142 ± 3.2 pcm as calculated by
OpenMC. The ray spacing used was 훿푎 = 0.1 cm in all the results, with 푁푎 = 256
azimuthal angles for the heterogeneous case and 푁푎 = 32 azimuthal angles for the homogeneous case.
MGXS д퐶푅퐷 (cm) д푂푇 퐻퐸푅 (cm) 푘eff ∆푘 (pcm) 11-Group Het 0.254 0.508 0.856792 +787 25-Group Het 0.254 0.508 0.850616 +169 11-Group Hom+CMM+SPH 1.016 1.016 0.844089 −483 25-Group Hom+CMM+SPH 1.016 1.016 0.843620 −530
Table 4.31: Final MCM2D (Poison) Results
Geometry OpenMC Fission Distribution
1.5
1.0
0.5
% Relative error of heterogeneous % Relative error of homogeneous vs openmc vs openmc -96.52 -96.52 4 -64.35 -64.35 2 -32.17 -32.17
0.0 0.0 0
32.17 32.17 2 64.35 64.35 4 96.52 96.52
0.0 0.0 -96.52 -64.35 -32.17 32.17 64.35 96.52 -96.52 -64.35 -32.17 32.17 64.35 96.52
Figure 4.29: Final MCM2D (Poison) Plots The homogeneous plot is shown with CMM and SPH.
97 98 Chapter 5
Conclusions
5.1 Summary of Accomplishments
Full TREAT Model in OpenMC
A detailed 3D OpenMC model of the Transient Reactor Test Facility in the Minimum Critical Mass core configuration was created (Section 3.1). It was used to quantify the sensitivities of the eigenvalue to boron impurities, hydrogen impurities, and partial graphitization of the fuel (Section 3.2). The final fuel composition made the MCM
3D model approximately critical, with 푘eff = 1.00604 ± 4 pcm.
MGXS Generation
Six 2D models simplified from the MCM core configuration were created tostudy multigroup cross section generation strategies for TREAT (Section 3.3). We examined several spatial domains for MGXS discretization (Section 3.5), selected appropriate domains for heterogeneous and homogeneous representations of TREAT’s core lattice (Section 3.5.5), and generated MGXS in two distinct energy group structures using OpenMC (Section 3.4).
99 Converged TREAT Models in OpenMOC
For each model, the various sets of cross sections were plugged into a deterministic simulation using OpenMOC. We studied the models’ convergence as a function of source region mesh (Section 4.2), ray spacing, and angle (Section 4.3). For the homogeneous models, we also quantified the effects of cross section corrections using the Cumulative Migration Method (Section 4.5.1) and Superhomog´en´eisation (Section 4.5.2). Finally, we present each model’s converged results for the two energy group structures in both the heterogeneous and homogeneous cases (Section 4.6).
5.2 Recommendations for Future Analysts
Fuel Impurities
The eigenvalue of TREAT is fairly sensitive to the assumptions made for the boron content, hydrogen content, and level of graphitization in the fuel. We ultimately used 7.53 ppm boron, no hydrogen, and carbon treated as 100% graphite for later analysis. Due to the sensitivity, we recommend caution; neglecting hydrogen in particular is a big assumption (Table 3.3).
Spatial Domains for MGXS
The spatial dependence of cross sections within the same material is little to none (Section 3.5.1). Therefore, Mesh-based MGXS are unnessary. We recommend the Cell domain for heterogeneous representations of TREAT. A homogeneous representation may be achieved by tallying cross sections on the Universe domain for each element type. To correct for homogenization errors, CMM correction factors generated on an infinite lattice have shown excellent performance in finite cores withoutB4C control rods (Section 4.5.1). However, they do not perform as well for models where the B4C rods are inserted. If CMM corrections can one day be calculated directly on the core of interest, we expect they will show a marked improvement for rodded cases as well.
100 Regardless of whether CMM is used, SPH factors appear to be necessary to obtain reasonable results on rodded models (Section 4.5.2).
Energy Groups for MGXS
For finite heterogeneous models, we recommend using more fast groups than appear in the 11-group structure given in Section 3.4. We observed that it tends to underpredict leakage and bias the eigenvalue of our 2D models. The CASMO 25-group structure, which has more fast groups and a similar number of thermal groups, consistently performed better. For finite homogeneous models, we have offer the same recommendation unless CMM correction factors are used. With CMM, the streaming term is corrected, and fewer energy groups are needed to capture the flux profile. We observed the 11-group results come much closer to agreement with the 25-group and reference solutions. For reflective models, the inclusion of more fast groups does little to improvethe results because leakage is no longer a concern.
Parameters for Converged MOC Solutions
We strongly recommend employing a linear source solver when simulating TREAT using MOC. Reflective models with no strong absorbers converged with the largest SR mesh attempted, 5.08 cm (equivalent to a 2 × 2 mesh per element). Finite models required an SR mesh of 1.016 cm (10×10 per element) to resolve the steep flux gradient caused by the vacuum boundary. Heterogeneous models with the B4C poison segment of the control rods inserted required a much finer mesh for the rodded element; we would recommend using an SR mesh of at most 0.254 cm (40 × 40) for it. The SR discretization dictates the ray spacing and number of angles needed to resolve the model. For homogeneous cases, we suggest a ray spacing of 0.1 cm coupled with a set of 32 azimuthal angles. For heterogeneous cases, rasing the number of angles to 64 is beneficial. When using a 0.254 cm mesh for heterogeneous models withB4C control rods, many more azimuthal angles are needed to converge the solution; we suggest somewhere on the order of 256.
101 102 Appendix A
Enlarged Figures
Figure A.1: MCM full core with the B4C poison inserted (enlarged)
103 Figure A.2: Full reactor at midcore, radial view (enlarged)
104 Figure A.3: Full reactor at midcore, axial view (enlarged)
105 106 Appendix B
CMM Transport Correction Ratios
Energy Upper я푔 я푔 я푔 Group Bound Standard Control Rod Control Rod Number (eV) Fuel Fuel (Follower) Fuel (Poison) 1 2 E+7 0.89687 0.91480 0.93361 2 3.3287 E+6 0.89368 0.89432 0.89767 3 1.1562 E+5 0.97517 0.96809 0.96836 4 3.4811 E+3 0.96955 0.96569 0.97622 5 1.3270 E+3 0.96814 0.96969 1.00920 6 8.1000 E+1 0.96990 0.96921 1.02087 7 6.2500 E+0 0.96962 0.96870 1.02088 8 2.0961 E−1 0.96872 0.96585 1.04133 9 7.6497 E−2 0.96579 0.96609 0.94992 10 4.7302 E−2 0.95624 0.95463 1.11551 11 2.0010 E−2 0.89298 0.90762 0.99304
Table B.1: 11-group CMM ratios
107 Energy Upper я푔 я푔 я푔 Group Bound Standard Control Rod Control Rod Number (eV) Fuel Fuel (Follower) Fuel (Poison) 1 2 E+7 0.93943 0.95776 0.97495 2 6.066 E+6 0.93566 0.94594 0.96034 3 3.679 E+6 0.98020 0.98255 0.99317 4 2.231 E+6 0.98349 0.98482 0.98650 5 1.353 E+6 1.00027 0.99651 1.00017 6 8.210 E+5 0.99630 0.99626 0.99495 7 5.000 E+5 0.97964 0.97860 0.97988 8 1.110 E+5 0.97667 0.96886 0.96879 9 9.118 E+3 0.97142 0.97036 0.97097 10 5.530 E+3 0.96989 0.96531 0.97330 11 1.487 E+2 0.96776 0.97021 1.00782 12 1.597 E+1 0.96895 0.96910 1.01800 13 9.877 E+0 0.97086 0.96973 1.01162 14 4.000 E+0 0.97279 0.97024 1.02561 15 1.855 E+0 0.96813 0.97045 1.03921 16 1.150 E+0 0.95796 0.96572 1.01481 17 1.097 E+0 0.97532 0.95363 1.07327 18 1.020 E+0 0.96434 0.96970 1.07152 19 9.720 E−1 0.96735 0.96578 1.04301 20 6.250 E−1 0.97122 0.97049 1.02041 21 3.500 E−1 0.96991 0.96778 1.01124 22 2.800 E−1 0.96529 0.96607 1.03232 23 1.400 E−1 0.96807 0.96480 1.01308 24 5.800 E−2 0.96117 0.96279 1.10034 25 3.000 E−2 0.92079 0.92569 1.04665
Table B.2: 25-group CMM ratios
108 Appendix C
Software Versions
C.1 OpenMC
| The OpenMC Monte Carlo Code Copyright | 2011-2017 Massachusetts Institute of Technology License | http://openmc.readthedocs.io/en/latest/license.html Version | 0.9.0
C.1.1 Revision used for sensitivity analysis
Git SHA1 | b6a91cc25a0a95f0fcfd92bc3eab155d830e2dcf
C.1.2 Revision used for MGXS generation
Git SHA1 | 7ba35310c8da200d896d418933d30237b3c1d414
109 C.2 OpenMOC
| The OpenMOC Method of Characteristics Code Copyright | 2012-2015 Massachusetts Institute of Technology License | https://mit-crpg.github.io/OpenMOC/license.html Git SHA1 | 7a2eee2ab9f3c98d167da62b6d4ad2f57494b8c3
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