The dynamic Monte Carlo method for transient analysis of nuclear reactors

Neutrons have become a unique tool for non-destructive testing of materials but despite the increasing interest from science and industry, the number of accessible neutron sources has decreased in the last years due to the phasing out of nuclear research reactors. This creates a strong discrepancy between supply and demand for access to neutron facilities. Laser-Driven Neutron Sources (LDNS) have the potential to fill this gap but so far research on this topic is mostly focused on the neutron production and not on their utilization for material testing. This shortcoming is addressed by this thesis through the development of a setup designed to conduct Neutron Resonance Spectroscopy (NRS) on a laser-driven source. In this process, the question is answered how the involved components have to be optimized to enable this source type to be competitive with established neutron sources. This is accomplished via a meta-analysis on laser ion acceleration, a systematic review process of targetry solutions as well as Monte Carlo (MC) simulations for the neutron generation, moderation and transport. This is followed by an experimental verification of this technique through the identification of the isotopes 182W and 183W inside a tungsten sample. In addition, it was shown that the same setup was capable of determining the sample thickness of a 1mm indium plate behind a 2mm thick lead shield via a thermal neutron radiography. By using MC simulations for state of the art laser systems it was possible to predict the performance of LDNS to be comparable to existing sources while being orders of magnitude smaller in size. These results close the gap between LDNS as a theoretical concept and their application as a tool for material identification.

The Supercritical-Water-Cooled Reactor (SCWR) is the Generation IV reactor concept most closely related to current light water reactors (LWRs). The SCWR builds on the vast experience with today's LWRs and supercritical coal-fired power plants. Water at supercritical state is used as moderator and coolant, and reaches a temperature of about 500 °C at the core outlet, which enables a much higher thermal efficiency (∼ 44%) than possible with current-day LWRs. In the operating range of an SCWR, the supercrical water density can become as low as one seventh of the density of water at room temperature. In order to ensure a thermal neutron spectrum, large moderator regions are introduced into the fuel assembly designs, resulting in lattices with strong moderation heterogeneity. As such, the neutronics of these assemblies differs significantly from that of standard LWRs, and lies outside the validated domain of reactor physics codes. Previous comparisons between the deterministic code CASMO-4 and reference calculations carried out with the Monte Carlo code MCNP4C showed large differences in calculated pin-wise reaction rate distributions in perturbed SCWR lattices. The experimental validation of standard reactor physics codes for SCWR-representative neutronics conditions is thus clearly of key importance for the further development of this technology. The goal of this thesis is to provide an experimental database for such validation and to assess the level of performance of current-day codes for SCWR analysis. An SCWR-like fuel lattice, based on a Japanese assembly design proposal from 2001, has been investigated in the PROTEUS zero-power research reactor at the Paul Scherrer Institute. Measurements have been carried out on the unperturbed test lattice, as also on six other PROTEUS configurations corresponding to different types of perturbations of the reference lattice. The investigated perturbations include control rod related effects, moderator density changes, and the replacement of a fuel pin with gadolinium-poisoned fuel. For each experimental configuration, pin-wise distributions of the total fission rate (Ftot) and the 238U capture rate (C8), as well as of their ratio (C8/Ftot), have been obtained across the assembly and compared to computed values. The neutronics codes used for the calculations are the LWR assembly code CASMO-4E and the Monte Carlo code MCNPX. Additionally, the reactivity effects of removing individual pins from the unperturbed SCWR-like lattice have been measured and compared to predictions obtained using the two codes. The pin-wise reaction rate distributions predicted with MCNPX have been found to agree within two standard deviations with the measured values for the unperturbed, as well as all the perturbed, configurations. The 1σ uncertainty was in the order of 0.4%, 0.8%, and 2.2% for Ftot, C8, and C8/Ftot, respectively. MCNPX could thus be validated for the various SCWR-like conditions investigated and has, in turn, been used to validate CASMO-4E on simplified geometries. For the latter purpose, the unperturbed and perturbed SCWR-like lattices were modeled as reflected assemblies, and reaction rate distributions predicted with the two codes were compared. For the unperturbed lattice and for the configurations with local perturbations of the neutron absorption, the agreement between the codes was within ∼ 1% for all reaction rates. However, for lattices with locally perturbed moderation conditions, CASMO-4E was initially found to overestimate the reaction rates in the vicinity of the perturbations by up to 5%. The discrepancies were identified as resulting from the default leakage treatment in CASMO-4E, which applies a DB2 correction in a homogenized sense across the lattice. In this way, cases with a global leakage gradient were not being treated properly. Usage of the optional input card BZ2, which allows a region-wise leakage treatment, resolved this problem, and the codes then agreed mostly within 1% for all the configurations. The pin removal worth measurements in the unperturbed SCWR-like lattice have provided integral data complementary to the reaction rate distributions. MCNPX results were found to agree with experiment within the statistical uncertainty of typically ∼ 10% (1σ). The comparison of reflected-assembly results from MCNPX and CASMO-4E yielded significant differences in the calculated pin removal worths (up to 4.3σ). A decomposition analysis of these differences has indicated that the discrepancies, especially for the better moderated pin positions, are linked to the calculation of the additional moderator effect caused by the pin removal. Finally, the transferability of the code validation, carried out under the PROTEUS experimental conditions, to the most recently proposed SCWR assembly designs has been assessed. Since, under power-reactor conditions, the moderator and coolant water densities are considerably lower in the proposed SCWR assemblies than that in the PROTEUS test lattices, their neutron spectra are much harder. Assembly-averaged values of the integral parameters C8/Ftot and F8/Ftot have been found to be as much as 65% and 80% higher, respectively, than for the PROTEUS reference lattice. However, the agreement between CASMO-4E and MCNPX predictions for the proposed SCWR assembly designs is still very good, indicating that the accuracy of the deterministic calculations does not deteriorate markedly when considering these new designs under power-reactor conditions.

Dynamic Monte Carlo transient analysis for the Organization for Economic Co-operation and Development Nuclear Energy Agency (OECD/NEA) C5G7-TD benchmark

Neutron transport benchmark examples with web-based AGENT

Heat conduction [1] is usually modeled as a diffusion process embodied in heat conduction equation. The traditional numerical methods [2, 3] for heat conduction problems such as the finite difference or finite element are well developed. However, these methods are based on discretized mesh systems, thus they are inherently limited in the geometry treatment. This chapter describes the Monte Carlo method that is based on particle transport simulation to solve heat conduction problems. The Monte Carlo method is “meshless” and thus can treat problems with very complicated geometries. The method is applied to a pebble fuel to be used in very high temperature gas-cooled reactors (VHTGRs) [4], which is a next-generation nuclear reactor being developed. Typically, a single pebble contains ~10,000 particle fuels randomly dispersed in graphite– matrix. Each particle fuel is in turn comprised of a fuel kernel and four layers of coatings. Furthermore, a typical reactor would house several tens of thousands of pebbles in the core depending on the power rating of the reactor. See Fig. 1. Such a level of geometric complexity and material heterogeneity defies the conventional mesh–based computational methods for heat conduction analysis. Among transport methods, the Monte Carlo method, that is based on stochastic particle simulation, is widely used in neutron and radiation particle transport problems such as nuclear reactor design. The Monte Carlo method described in this chapter is based on the observation that heat conduction is a diffusion process whose governing equation is analogous to the neutron diffusion equation [5] under no absorption, no fission and one speed condition, which is a special form of the particle transport equation. While neutron diffusion approximates the neutron transport phenomena, conversely it is applicable to solve diffusion problems by transport methods under certain conditions. Based on this idea, a new Monte Carlo method has been recently developed [6-8] to solve heat conduction problems. The method employs the MCNP code [9] as a major computational engine. MCNP is a widely used Monte Carlo particle transport code with versatile geometrical capabilities. Monte Carlo techniques for heat conduction have been reported [10-13] in the past. But most of the earlier Monte Carlo methods for heat conduction are based on discretized mesh systems, thus they are limited in the capabilities of geometry treatment. Fraley et al[13] uses a “meshless” system like the method in this chapter but does not give proper treatment to the boundary conditions, nor considers the “diffusion-transport theory correspondence” to be described in Section 2.2 in this chapter. Thus, the method in this chapter is a transport theory treatment of the heat conduction equation with a methodical boundary correction. The

A neutron transport and thermal hydraulics coupling scheme to study xenon induced power oscillations in a nuclear reactor

This report describes the MCV (Monte Carlo - Vectorized)Monte Carlo neutron transport code [Brown, 1982, 1983; Brown and Mendelson, 1984a]. MCV is a module in the RACER system of codes that is used for Monte Carlo reactor physics analysis. The MCV module contains all of the neutron transport and statistical analysis functions of the system, while other modules perform various input-related functions such as geometry description, material assignment, output edit specification, etc. MCV is very closely related to the 05R neutron Monte Carlo code [Irving et al., 1965] developed at Oak Ridge National Laboratory. 05R evolved into the 05RR module of the STEMB system, which was the forerunner of the RACER system. Much of the overall logic and physics treatment of 05RR has been retained and, indeed, the original verification of MCV was achieved through comparison with STEMB results. MCV has been designed to be very computationally efficient [Brown, 1981, Brown and Martin, 1984b; Brown, 1986]. It was originally programmed to make use of vector-computing architectures such as those of the CDC Cyber- 205 and Cray X-MP. MCV was the first full-scale production Monte Carlo code to effectively utilize vector-processing capabilities. Subsequently, MCV was modified to utilize both distributed-memory [Sutton and Brown, 1994] and shared memory parallelism. The code has been compiled and run on platforms ranging from 32-bit UNIX workstations to clusters of 64-bit vector-parallel supercomputers. The computational efficiency of the code allows the analyst to perform calculations using many more neutron histories than is practical with most other Monte Carlo codes, thereby yielding results with smaller statistical uncertainties. MCV also utilizes variance reduction techniques such as survival biasing, splitting, and rouletting to permit additional reduction in uncertainties. While a general-purpose neutron Monte Carlo code, MCV is optimized for reactor physics calculations. It has the capability of performing iterated-source (criticality), multiplied-fixed-source, and fixed-source calculations. MCV uses a highly detailed continuous-energy (as opposed to multigroup) representation of neutron histories and cross section data. The spatial modeling is fully three-dimensional (3-D), and any geometrical region that can be described by quadric surfaces may be represented. The primary results are region-wise reaction rates, neutron production rates, slowing-down-densities, fluxes, leakages, and when appropriate the eigenvalue or multiplication factor. Region-wise nuclidic reaction rates are also computed, which may then be used by other modules in the system to determine time-dependent nuclide inventories so that RACER can perform depletion calculations. Furthermore, derived quantities such as ratios and sums of primary quantities and/or other derived quantities may also be calculated. MCV performs statistical analyses on output quantities, computing estimates of the 95% confidence intervals as well as indicators as to the reliability of these estimates. The remainder of this chapter provides an overview of the MCV algorithm. The following three chapters describe the MCV mathematical, physical, and statistical treatments in more detail. Specifically, Chapter 2 discusses topics related to tracking the histories including: geometry modeling, how histories are moved through the geometry, and variance reduction techniques related to the tracking process. Chapter 3 describes the nuclear data and physical models employed by MCV. Chapter 4 discusses the tallies, statistical analyses, and edits. Chapter 5 provides some guidance as to how to run the code, and Chapter 6 is a list of the code input options.

Self-avoiding walks on deterministic and random fractals: Numerical results

Authors

Monte Carlo (MC) simulations are often used in calculations of radiation transport to enable accurate prediction of radiation-dose, even though the computation is relatively time-consuming. In a typical MC simulation, significant computation time is allocated to following non-important events. To address this issue, variance reduction techniques (VRTs) have been suggested for reducing the statistical variance for the same computation time. Among the available MC simulation codes, electron gamma shower (National Research Council of Canada) (EGSnrc) is a general-purpose coupled electron-photon transport code that also features an even-handed, rich set of VRTs. The most well-known VRTs are the photon splitting, Russian roulette (RR), and photon cross-section enhancement (XCSE) techniques. The objective of this work was to determine the optimal combination of VRTs that increases the simulation speed and the efficiency of simulation, without compromising its accuracy. Selection of VRTs was performed using EGSnrc MC User codes, such as cavity and egs_chamber, for simulating various ion chamber geometries using 6 MV photon beams and 1.25 MeV 60Co photon beams. The results show that the combination of XCSE and RR yields the highest efficiency for ion-chamber dose calculations inside a 30 cm × 30 cm × 30 cm water phantom. Hence, properly selecting a different VRT without altering the underlying physics increases the efficiency of MC simulations for ion-chamber dose calculation.

To validate neutronics calculation for the blanket design of fusion-fission hybrid reactor, experiments for measuring reaction rates inside two simulating assemblies are performed. Two benchmark assemblies were developed for the neutronics experiments. A D-T fusion neutron source is placed at the center of the setup. One of them consists of three layers of depleted uranium shells and two layers of polyethylene shells, and these shells are arranged alternatively. The U-238 capture reaction rates are measured using depleted uranium foils and an HPGe gamma spectrometer. The fission reaction rates are measured using a fission chamber coated with depleted uranium. The other assembly consists of depleted uranium and LiH shells. The tritium production rates are measured using the lithium glass scintillation detector which is placed in the LiH region of the assembly. The measured reaction rates are compared with the calculated ones predicted using MCNP code, and C/E values are obtained.

Implementation of a dynamic Monte Carlo method for transients analysis with thermal-hydraulic feedbacks using MCNPX code

Preface. 1. Simulations and the Monte Carlo method. 1.1 Description of the method. 1.2 Convergence theorems. 1.3 Simulation of random variables. 1.4 Variance reduction techniques. 1.5 Exercises. 2. Markov chains. 2.1 Definitions and elementary properties. 2.2 Examples. 2.3 Strong Markov property. 2.4 Recurrent and transient states. 2.5 The irreducible and recurrent case. 2.6 The aperiodic case. 2.7 Reversible Markov chain. 2.8 Rate of convergence to equilibrium. 2.9 Statistics of Markov chains. 2.10 Exercises. 3. Stochastic algorithms. 3.1 Markov chain Monte Carlo. 3.2 Simulation of the invariant probability. 3.3 Rate of convergence towards the invariant probability. 3.4 Simulated annealing. 3.5 Exercises. 4. Markov chains and the genome. 4.1 Reading DNA. 4.2 The i.i.d. model. 4.3 The Markov model. 4.4 Hidden Markov models. 4.5 Hidden semi-Markov model. 4.6 Alignment of two sequences. 4.7 A multiple alignment algorithm. 4.8 Exercises. 5. Control and filtering of Markov chains. 5.1 Deterministic optimal control. 5.2 Control of Markov chains. 5.3 Linear quadratic optimal control. 5.4 Filtering of Markov chains. 5.5 The Kalman-Bucy filter. 5.6 Linear-quadratic control with partial observation. 5.7 Exercises. 6. The Poisson process. 6.1 Point processes and counting processes. 6.2 The Poisson process. 6.3 The Markov property. 6.4 Large time behaviour. 6.5 Exercises. 7. Jump Markov processes. 7.1 General facts. 7.2 Infinitesimal generator. 7.3 The strong Markov property. 7.4 Embedded Markov chain. 7.5 Recurrent and transient states. 7.6 The irreducible recurrent case. 7.7 Reversibility. 7.8 Markov models of evolution and phylogeny. 7.9 Application to discretized partial differential equations. 7.10 Simulated annealing. 7.11 Exercises. 8. Queues and networks. 8.1 M/M/1 queue. 8.2 M/M/1/K queue. 8.3 M/M/s queue. 8.4 M/M/s/s queue. 8.5 Repair shop. 8.6 Queues in series. 8.7 M/G/ queue. 8.8 M/G/1 queue. 8.9 Open Jackson network. 8.10 Closed Jackson network. 8.11 Telephone network. 8.12 Kelly networks. 8.13 Exercises. 9. Introduction to mathematical finance. 9.1 Fundamental concepts. 9.2 European options in the discrete model. 9.3 The Black-Scholes model and formula. 9.4 American options in the discrete model. 9.5 American options in the Black-Scholes model. 9.6 Interest rate and bonds. 9.7 Exercises. 10. Solutions to selected exercises. 10.1 Chapter 1. 10.2 Chapter 2. 10.3 Chapter 3. 10.4 Chapter 4. 10.5 Chapter 5. 10.6 Chapter 6. 10.7 Chapter 7. 10.8 Chapter 8. 10.9 Chapter 9. References Index.

Vibrations of fuel assemblies are an important issue in the safe operation of nuclear reactors, because they can challenge the integrity of the fuel with potential for radioactive releases. Reactor neutron noise-based techniques for monitoring vibrations are valuable for core diagnostic since they are not intrusive and make use of ordinary neutron flux measurements from ex-core and in-core detectors. The application of these techniques involves the solution of inverse problems that require numerical simulations capable of estimating the reactor neutron noise, given a model of the vibrations. For this purpose, several novel reactor neutron noise solvers have been developed in the CORTEX project using either Monte Carlo or deterministic methods, such as the discrete ordinates method, the method of characteristics, and the diffusion approximation. In the current work, these solvers have been scrutinized by computing the neutron noise induced by vibrations of one or multiple fuel pins in a simplified UOX fuel assembly benchmark, via proper variations of macroscopic neutron cross sections. The comparison of these neutron noise solutions obtained from the different methods shows novel insights into the simulation of neutron noise induced by mechanical vibrations, such as the challenges posed by the Monte Carlo method, the impact of the angular discretization on the application of the discrete ordinates method, and the accuracy of the diffusion approximation assessed via the higher-order neutron transport methods.

Deterministic and Monte Carlo methods are regularly employed to conduct lattice calculations. Monte Carlo methods can effectively model a large range of complex geometries and, compared to deterministic methods, they have the major advantage of reducing systematic errors and are computationally effective when integral quantities such as effective multiplication factor or reactivity are calculated. In contrast, deterministic methods do introduce discretization approximations but usually require shorter computation times than Monte Carlo methods when detailed flux and reaction-rate solutions are sought. This work compares the results of the deterministic code DRAGON to the Monte Carlo code Serpent in the calculation of the reactivity effects for a pressurized heavy water reactor (PHWR) lattice cell containing a 37-element, natural uranium fuel bundle with heavy water coolant and moderator. The reactivity effects are determined for changes to the coolant, moderator, and fuel temperatures and to the coolant and moderator densities for zero-burnup, mid-burnup [3750 MWd/t(U)] and discharge burnup [7500 MWd/t(U)] fuel. It is found that the overall trend in the reactivity effects calculated using DRAGON match those calculated using Serpent for the burnup cases considered. However, differences that exceed the amount attributable to statistical error have been found for some reactivity effects, particularly for perturbations to coolant and moderator density and fuel temperature.