Publications
120 results found
Gordon TL, Williams MMR, Eaton MD, et al., 2024, Itô-calculus based mathematical models for stochastic nuclear reactor kinetics and dynamics simulations of low neutron source nuclear power plant (NPP) start-up, Annals of Nuclear Energy, Vol: 202, ISSN: 0306-4549
This paper investigates the effect thermal feedback has on the stochastic nuclear reactor dynamics of lowneutron source nuclear power plant (NPP) start-ups. Stochastic mathematical and computational models arerequired to determine the probability of a stochastic power surge occurring during nuclear reactor start-up thatwould damage the nuclear fuel. The aim is to design the nuclear reactor, the nuclear fuel, and the operationalstart-up procedures in a manner that minimises the probability of a stochastic power surge occurring, whichdamages the nuclear fuel, to a prescribed level of probabilistic risk (10−8–10−5). Recently, the Pál-Bell equationshave been used for such low neutron source nuclear reactor start-up simulations. However, the stochasticnuclear reactor start-up models, based upon the Pál-Bell equations, cannot accommodate changes in themacroscopic neutron cross-sections arising from feedback processes. An alternative approach that could, inprinciple, include thermal feedback processes is the forward master equations. However, these are complex toimplement for multidimensional and multi-group stochastic nuclear reactor dynamics problems. In addition,time-dependent analog Monte Carlo models could be used but these are computationally prohibitive for mostnuclear reactor start-up simulations. This is due to the stringent requirements on the statistical accuracy of thesafety probability associated with stochastic power surges. Therefore, this paper uses an alternative Itô-calculusapproach to compute the stochastic properties required for low neutron source NPP start-up. The Itô-calculusapproach is an approximate mathematical method, compared to the more general Pál-Bell and Monte Carlomethods, for low neutron source nuclear reactor start-up and fast burst systems. Therefore, the implementationof the Itô calculus method is first validated against the Caliban fast burst nuclear reactor experimental waittime
Daniels JR, Williams MMR, Eaton MD, 2024, Transient analysis of the 1970 Windscale nuclear criticality incident, Progress in Nuclear Energy, Vol: 170, ISSN: 1878-4224
This paper describes a novel methodology for the analysis of transient nuclear criticality in layered aqueousemulsion-organic plutonium nitrate systems. The presented methodology includes point neutron kineticsequations coupled with phenomenological one-dimensional nuclear thermal hydraulics models, which describethe variation in mass, power, reactivity, temperature and voidage within the system. Equations to describethe mean neutron generation time and changes in reactivity due to changes in the system’s temperature andvoid fraction are formulated from fits of the results predicted by the criticality transient multiphysics model.Using these developed models, the nuclear criticality incident that occurred at the Windscale Works in 1970 isanalysed and presented. In this criticality transient, two immiscible fissile liquids (an aqueous solution and anorganic solvent) were disturbed such that an emulsion layer formed between them, inducing a transient nuclearcriticality excursion and yielding around 1015 fissions. It was determined that parameters such as emulsionglobule size and vessel outlet pipe radius had negligible impact on the total fission yield of the correspondingtransient when compared to variations in organic solvent volume and emulsion band thickness. Several possibleconfigurations were identified which could result in a transient similar to that of the Windscale Works in 1970.Notably, a system containing 39.75 L of organic solvent which underwent a transient with the formation ofan emulsion band that was a maximum of 5.4 cm thick and 54% aqueous solution by volume yielded 9 × 1014fissions, and thus is expected to closely resemble the 1970 Windscale nuclear criticality incident. However, theuncertainties associated with the original transient and the system parameters (e.g. thermophysical propertiesof the three layers) are such that the configuration is likely to vary from these exact quantities.
Daniels JR, Williams M, Eaton M, 2024, Steady-state analysis of the 1970 Windscale nuclear criticality incident, Progress in Nuclear Energy, Vol: 169, ISSN: 1878-4224
This paper presents several steady-state neutron transport models which are used to determine the behaviour of the nuclear criticality incident that occurred at the Windscale Works in 1970. The transfer vessel involved in this incident contained two immiscible fissile liquids (an aqueous phase and an organic phase) which had been disturbed such that an emulsion layer formed between them. Infinite domains of the aqueous, organic, and emulsion phases were modelled to enable the understanding of the effects of nuclear data library, neutron transport code and material on the calculated kinetics data, effective multiplication factor and macroscopic neutron cross sections. Six possible configurations of the 1970 Windscale criticality incident were simulated using three-dimensional models in MCNP, two-dimensional models in academic research code EVENT, and one-dimensional models in a collision probability (CP) code. Quantities of interest, such as reactivity, scalar neutron flux, neutron absorption and production rates, and kinetics data are presented. System reactivity, mean neutron generation time and neutron absorption and production rates are shown to increase with increasing emulsion thickness, whereas the total scalar neutron flux is shown to decrease. It is determined that the volume of organic phase present during the transient was likely to be around 39.0 L, as the emulsion thickness required to reach criticality (6.5 cm) was within the range cited in the literature. EVENT and the CP code showed deficiencies when calculating the scalar neutron flux through the plenum gas at the top of the transfer vessel, with the CP code overestimating the total scalar neutron flux in the system by as much as 19 %. The intrinsic neutron source, subcritical multiplication and subcritical power of the system were computed using Monte Carlo simulations. It was shown, by use of the Hansen criteria, that the subcritical system was likely to be in a strong neutron source regime (such t
Wilson SG, Eaton M, Kophazi J, 2024, Energy-dependent, self-adaptive mesh h(p)-refinement of a constraint-based continuous Bubnov-Galerkin isogeometric analysis spatial discretization of the multi-group neutron diffusion equation with dual-weighted residual error measures, Journal of Computational and Theoretical Transport, Pages: 1-64, ISSN: 2332-4309
Energy-dependent self-adaptive mesh refinement algorithms are developed for a continuous Bubnov-Galerkin spatial discretization of the multi-group neutron diffusion equation using NURBS-based isogeometric analysis (IGA). The spatially self-adaptive algorithms employ both mesh (h) and polynomial degree (p) refinement. Constraint-based equations are established across irregular interfaces with hanging-nodes; they are based upon master-slave relationships and the conservative interpolation between surface meshes. A similar Galerkin projection is employed in the conservative interpolation between volume meshes to evaluate group-to-group source terms over energy-dependent meshes; and to evaluate interpolation-based error measures. Enforcing continuity over an irregular mesh does introduce discretization errors. However, local mesh refinement allows for a better allocation of computational resources; and thus, more accuracy per degree of freedom. Two a posteriori interpolation-based error measures are proposed. The first heuristically minimizes local contributions to the discretization error, which becomes competitive for global quantities of interest (QoIs). However, for localized QoIs, over energy-dependent meshes, certain multi-group components may become under-resolved. The second employs duality arguments to minimize important error contributions, which consistently and reliably reduces the error in the QoI.
Daniels JR, Williams MMR, Eaton MD, 2024, Development and verification of a one-dimensional collision probability based neutron transport code to model axially heterogeneous cylindrical vessels containing aqueous and organic plutonium nitrate, Annals of Nuclear Energy, Vol: 196, ISSN: 0306-4549
This paper presents the development and verification of a collision probability (CP) code, capable of modelling neutron transport in one-dimensional slabs and axially heterogeneous cylinders with varying radii. The CP code is used to model layered systems of aqueous and organic plutonium nitrate, as process criticality accidents are more likely to occur in these systems compared to dry systems. The use of the CP code is desired as it offers a computationally inexpensive method for calculating neutron transport when compared to higher fidelity codes such as MCNP. For slab geometries, the CP code can be used effectively, given they contain at least 0.7 g cm−2 plutonium. The approximation employed by the CP code to model heterogeneous cylinders overestimated the rate of radial neutron leakage such that vessels with radii of 30.0 cm could not reliably calculate reactivity to within 1 $ of MCNP. Increasing the radii to 40.0 cm improved the accuracy of the CP code to within 1 $ of MCNP for systems containing at least 2.75 kg plutonium. The error in the CP code increased when used to model cylindrical geometries with dished ends and complete vessels with dished ends, such that systems with large dished ends and low plutonium content should be avoided. As a simple, neutronics based model, the CP code could be used as part of rough order of magnitude calculations for criticality transients, where high levels of accuracy are not required, given that potential errors in results have been previously identified.
Jones GS, Cooling CM, Williams MMR, et al., 2023, Transient nuclear criticality excursion analysis of highly dispersed particulate three-phase fluidised systems, Annals of Nuclear Energy, Vol: 180, ISSN: 0306-4549
The aim of this study was to perform sensitivity analysis, investigating how different fluidisation and sedimentation characteristics of three-phase wetted UO2 powder beds, may affect a transient nuclear criticality excursion initiated through the addition of water into a fissile powder bed. This type of postulated nuclear criticality accident scenario may occur in nuclear fuel fabrication facilities when a fire is fought through the use of water, supplied via an automatic or manual fire-suppression system. A similar scenario may also develop as a result of water leaks or flooding of the process area housing UO2 powder. The article introduces a model for gas-bubble induced fluidisation of a UO2 powder bed and examines how this phenomenon may affect the neutron kinetic response of a three-phase fluidised fissile powder system. Empirical analysis has shown that fissile suspensions form agglomerated structures when suspended in water, at agglomerate sizes that range from 18 to 40 . Simulation results indicate that both the critical gas velocity and rate of fluidisation may significantly affect transient nuclear criticality excursion dynamics. The re-distribution of fissile mass into a highly dispersed suspension generally reduces the reactivity of the system, however, depending on the H/U ratio, a positive reactivity may be added to the system. Low Péclet numbers in the suspension suggest that gas-bubble induced motion of the suspension causes a highly dispersive flow field. An oscillatory power response is predicted for low critical gas velocities where the reactivity of the system is predominantly governed by the re-distribution of fissile mass within the system. The frequency of these oscillations is greater for a higher hindered settling rate of powder particles. At a higher critical gas velocity, the transient nuclear criticality excursion is governed by the voidage reactivity feedback, making the response quite independent of fluidisation. In all cases, la
Jones G, Cooling CM, Williams MMR, et al., 2022, Wetting-induced volumetric collapse of UO2 powder beds and theconsequence on transient nuclear criticality excursions, Progress in Nuclear Energy, Vol: 154, ISSN: 0149-1970
Mathematical and computational models are proposed to simulate wetting-induced volumetric collapse of fissile powder beds. Slumping, nuclear thermal hydraulics, radiolytic gas, and steam production models are coupledwith point neutron kinetics to investigate transient nuclear criticality excursions in two 5-wt% enriched UO2 fissile powder beds with varying levels of wetting-induced volumetric collapse. The two beds are distinguishedby their mean powder particle size of 30 μm and 100 μm. For the UO2 powder beds modelled, the re-distribution of UO2 powder and moderator due to slumping introduced a negative reactivity into the system. This increasedthe amount of time taken for a delayed critical state to be reached once infiltration began, and also reduced the total fission energy generated over the course of the simulated transient. The total fission energy generatedranged from 42 MJ to 48 MJ 100 seconds after the initial nuclear criticality excursion was observed for the 30 μm sized UO2 powder bed. The fission energy of the larger sized powder bed (100 μm), varied from 42 MJ to 57 MJ. Larger discrepancies between the slumped and un-slumped initial peak power are predicted. Peak powers varied from 29.2 MW to 106 MW for the smaller-sized powder particles, whereas for larger particles, the peak powers varied from 255 MW to 501 MW.
Lampunio L, Duan Y, Eaton M, et al., 2022, Mean flow, turbulent structures, and SPOD analysis of thermal mixing in a T-junction with variation of the inlet flow profile, Energies, Vol: 15, ISSN: 1996-1073
This paper investigates the effects of different inlet flow profiles on thermal mixing within a T-junction using CFD simulations with the IDDES-SST turbulence model. The different combinations of inlet flow profiles are related to different stage in the flow entry region. The effects of the inlet flow profile on the mean and transient flow behaviour are assessed, while a spectral proper orthogonal decomposition and power spectral density analysis are performed to assess the underlying flow structures and the predominant frequency modes. It is found that the vortical structures associated with the horseshoe and hovering vortex systems consist of a single roll-up vortex for cases with uniformly distributed boundary conditions (BCs) at the branch inlet whereas a double roll-up vortex is observed for the other cases. The double roll-up vortex enhances the mixing locally due to the entrainment of fluid from the branch pipe in these vortical structures, which then results in a lower mean temperature distribution. The appearance of the secondary vortex pair and the nested vortices is delayed for cases with uniformly distributed BCs at the branch inlet, which again results in lower thermal mixing and consequently higher values of mean temperature when compared with the other cases. It is also found that the vorticity related to the counter-rotating vortex pair as well as to the second pair of vortices rotating in the opposite direction is higher for cases with uniformly distributed BCs at the branch inlet. Lastly, the combinations of inlet flow profiles lead to different coherent structures, and the dominant frequencies are of a Strouhal number of around 0.7 for uniformly distributed profiles at the branch inlet and in the range 0.4–0.5 for the other cases.
Wenman M, Grimes R, Tate J, et al., 2022, Parliament Committees. Delivering nuclear power: Written evidence from Imperial College London (NCL0026), Publisher: UK Parliment
Ferguson JA, Kópházi J, Eaton MD, 2022, NURBS enhanced virtual element methods for the spatial discretization of the multigroup neutron diffusion equation on curvilinear polygonal meshes, Journal of Computational and Theoretical Transport, Vol: 51, Pages: 145-204, ISSN: 2332-4309
The Continuous Galerkin Virtual Element Method (CG-VEM) is a recent innovation in spatial discretization methods that can solve partial differential equations (PDEs) using polygonal (2D) and polyhedral (3D) meshes. Recently, a new formulation of CG-VEM was introduced which can construct VEM spaces on polygons with curvilinear edges. This paper presents the application of the curved VEM to the multigroup neutron diffusion equation and demonstrates its benefits over the conventional straight-sided VEM for a number of benchmark verification test cases with curvilinear domains. These domains were constructed using a topological data-structure developed as part of this paper, based on the doubly-connected edge list, with curves and surfaces both represented using non-uniform rational B-splines (NURBS). This data-structure is used both to specify the geometry of the reactor and to represent the curvilinear polygonal mesh. We also present two separate methods of performing integrations on curvilinear polygons, one for homogeneous functions and one for non-homogeneous functions.
Jones GS, Winter GE, Cooling CM, et al., 2022, Mathematical and computational models for simulating transient nuclear criticality excursions within wetted fissile powder systems, Annals of Nuclear Energy, Vol: 169, Pages: 1-31, ISSN: 0306-4549
This paper describes a novel methodology for the analysis of nuclear criticality excursions in fissile powder beds under wetting conditions. These potentially hazardous powder, slurry and sludge systems maybe found in nuclear fuel manufacturing and fabrication facilities. A point neutron kinetics model was coupled with water infiltration, thermal–hydraulics and radiolysis models through the use of reactivity feedbacks. Good agreement in the water infiltration rate was found when comparing the water infiltrationmodel used in this paper to experiments conducted by the French Commissariat à l’Énergie Atomiqueet aux Énergies Alternatives (CEA). A case study was proposed whereby a sheet of fine water dropletsfrom a sprinkler system came into contact with an open-topped bed of low enriched UO2 powder.Simulations indicate that the mean powder particle size had a strong effect on the time required forthe water to percolate through the powder bed. Powder particle size was also predicted to have a moderate effect on the initial fission power spike. The fission energy released over the first 300 s of the nuclearcriticality transient ranged from 65:28 MJ to 97:98 MJ depending on mean powder particle size. This issimilar in magnitude to other simulated nuclear criticality excursions in powder beds. The model predictsthat the initial fission power spike would be limited by the production of radiolytic gas and to a lesserextent the effects of Doppler broadening and thermal expansion. As expected, boiling and the associatedsteam production, was found to be an important phenomenon in the reduction of the fission rate throughthe negative void reactivity effect of the steam.
Duan Y, North Ridao M, Eaton M, et al., 2022, Non-intrusive semi-analytical uncertainty quantification using Bayesian quadrature with application to CFD simulations, International Journal of Heat and Fluid Flow, Vol: 93, Pages: 1-17, ISSN: 0142-727X
To improve the safety, reliability, and performance of complex engineering systems, it is crucial to understand and quantify uncertainties. This paper presents a framework to non-intrusively and semi-analytically quantify the parametric uncertainty within CFD simulations using Bayesian quadrature (BQ). An in-house uncertainty quantification (UQ) code based upon this mathematical framework is developed. The code is then validated by applying it to quantify the uncertainty due to a varying parameter in a simple analytical test function. The mean and variance obtained using BQ are compared with those obtained from the analytical solution and stochastic simulation using the Latin hypercube sampling (LHS) method. The validation test case shows that BQ outperforms the LHS approach in terms of computational efficiency and accuracy. The UQ code is then utilised to characterise the uncertainty (due to the unknown inlet flow profile) of CFD predicted operating parameters of an industrial scale butterfly valve, as well as the uncertainties (due to the unknown high-wavenumber damping factor ) of a SAS-SST simulated bluff-body flow. It is found that the entry flow profile presents non-ignorable effects on the valve operating parameters. Meanwhile, the variance of the valve operating parameters changes with the valve opening. For the bluff-body flow, large variances of predicted flow properties exist in the region where the separate shear layer dominates because of varying. Moreover, the effect of is more significant on the turbulence quantities, as it acts on the generation of turbulent eddies directly.
Winter GE, Cooling CM, Eaton MD, 2022, A semi-empirical model of radiolytic gas bubble formation and evolution during criticality excursions in uranyl nitrate solutions for nuclear criticality safety assessment, Annals of Nuclear Energy, Vol: 165, Pages: 1-20, ISSN: 0306-4549
A novel mathematical and computational model for the formation and evolution of radiolytic gas in aqueous fissile solutions is presented. The model predicts the rate at which bubbles are formed and/or removed from the system using semi-empirical correlations calibrated by means of numerical simulation. The model is able to reliably predict the behaviour of aqueous fissile solutions, including transient effects due to the formation and removal of radiolytic gas. A further extension to the model enables its application to boiling systems.
Lampunio L, Duan Y, Eaton MD, 2021, The effect of inlet flow conditions upon thermal mixing and conjugate heat transfer within the wall of a T-Junction, Nuclear Engineering and Design: an international journal devoted to the thermal, mechanical, materials, and structural aspects of nuclear fission energy, Vol: 385, Pages: 1-20, ISSN: 0029-5493
This paper investigates the effects of different inlet velocity profiles on thermal mixing and conjugate heat transfer (CHT) within a T-junction. The flow domain is modelled using the Improved Delayed Detached Eddy Simulation (IDDES) turbulence model implemented within the commercial CFD software STAR-CCM+ 2020.1.1. The thermal analysis of the solid domain is also addressed within the CFD simulations. The OECD/NEA-Vattenfall experimental benchmark database is used to validate the CFD model. The effect of mesh sensitivity within the CFD simulations is also studied qualitatively and quantitatively. The influence of different inlet flow profiles on the CFD simulations is then assessed. Different combinations of inlet flow profiles, uniformly distributed and fully developed, are considered. Compared to the flow profile at the main pipe inlet, the flow profile at the branch pipe inlet presents a much more significant effect on the mean temperature distribution downstream of the T-junction. It is found that the flat flow profile at the branch inlet causes a higher temperature at the top wall, therefore a larger temperature gradient, and may lead to higher thermal stresses due to thermal stratification. The temperature distribution is more uniform for cases with the fully developed flow profile at the branch inlet. The variance of temperature is high at the sides of the pipe, regardless of the velocity profile used. The combination of flat flow profiles at both inlets causes the highest temperature variance. Moreover, the regions of maximum variance of temperature are located at different positions along the pipe section, depending on the combinations of inlet flow profiles.
Duan Y, Ahn JS, Eaton MD, et al., 2021, Quantification of the uncertainty within a SAS-SST simulation caused by the unknown high-wavenumber damping factor, Nuclear Engineering and Design, Vol: 381, Pages: 1-12, ISSN: 0029-5493
This paper aims to quantify the uncertainty in the SAS-SST simulation of a prism bluff-body flow due to varyingthe higher-wavenumber damping factor (Cs). Instead of performing the uncertainty quantification on the CFDsimulation directly, a surrogate modelling approach is adopted. The mesh sensitivity is first studied and thenumerical error due to the mesh is approximated accordingly. The Gaussian processes/Kriging method is used togenerate surrogate models for quantities of interest (QoIs). The suitability of the surrogate models is assessedusing the leave-one-out cross-validation tests (LOO-CV). The stochastic tests are then performed using the crossvalidated surrogate models to quantify the uncertainty of QoIs by varying Cs. Four prior probability densityfunctions (such as U(0, 1), N(0.5, 0.12), Beta(2, 2) and Beta(5, 1.5)) of Cs are considered.It is demonstrated in this study that the uncertainty of a predicted QoI due to varying Cs is regionallydependent. The flow statistics in the near wake of the prism body are subject to larger variance due to theuncertainty in Cs. The influence of Cs rapidly decays as the location moves downstream. The response of differentQoIs to the changing Cs varies greatly. Therefore, the calibration of Cs only using observations of one variablemay bias the results. Last but not least, it is important to consider different sources of uncertainties within thenumerical model when scrutinising a turbulence model, as ignoring the contributions to the total error may leadto biased conclusions.
Gordon T, Cooling CM, Williams MMR, et al., 2021, Numerical comparison of mathematical and computational models for the simulation of stochastic neutron kinetics problems, Annals of Nuclear Energy, Vol: 157, Pages: 1-27, ISSN: 0306-4549
This paper concerns numerical comparisons between five mathematical models capable of modelling the stochastic behaviour ofneutrons in low extraneous (extrinsic or fixed) neutron source applications. These models include analog Monte-Carlo (AMC),forward probability balance equations (FPB), generating function form of the forward probability balance equations (FGF), generatingfunction form of the backward probability balance equations (P´al-Bell), and an Itˆo calculus model using both an explicit andimplicit Euler-Maruyama discretization scheme. Results such as the survival probability, extinction probability, neutron populationmean and standard deviation, and neutron population cumulative distribution function have all been compared. The least computationallydemanding mathematical model has been found to be the use of the P´al-Bell equations which on average take four ordersof magnitude less time to compute than the other methods in this study. The accuracy of the AMC and FPB models have beenfound to be strongly linked to the computational e ciency of the models. The computational e ciency of the models decreasesignificantly as the maximum allowable neutron population is approached. The Itˆo calculus methods, utilising explicit and implicitEuler-Maruyama discretization schemes, have been found to be unsuitable for modelling very low neutron populations. However,improved results, using the Itˆo calculus methods, have been achieved for systems containing a greater number of neutrons.
Duan Y, Eaton MD, Bluck MJ, 2021, Fixed inducing points online Bayesian calibration for computer models with an application to a scale-resolving CFD simulation, Journal of Computational Physics, Vol: 434, Pages: 1-14, ISSN: 0021-9991
This paper proposes a novel fixed inducing points online Bayesian calibration (FIPO-BC) algorithm to efficiently learn the model parameters using a benchmark database. The standard Bayesian calibration (STD-BC) algorithm provides a statistical method to calibrate the parameters of computationally expensive models. However, the STD-BC algorithm does not scale well with regard to the number of data points and also it lacks an online learning capability. The proposed FIPO-BC algorithm greatly improves the computational efficiency of the algorithm and, in addition, enables online calibration to be performed by executing the calibration on a set of predefined inducing points.To demonstrate the procedure of the FIPO-BC algorithm, two tests are performed, finding the optimal value and exploring the posterior distribution of 1) the parameter in a simple function, and 2) the high-wave number damping factor in a scale-resolving turbulence model (scale adaptive simulation shear-stress transport model/SAS-SST). The results (such as the calibrated model parameter and its posterior distribution) of FIPO-BC with different inducing points are compared to those of STD-BC. It is found that FIPO-BC and STD-BC can provide very similar results, once the predefined set of inducing points in FIPO-BC is sufficiently fine. Given that fewer datapoints are needed in the proposed FIPO-BC algorithm, compared to the STD-BC algorithm, it will be a more computational efficient algorithm. In our demonstration test cases, the proposed FIPO-BC algorithm is at least ten times faster than the STD-BC algorithm. Meanwhile, the online feature of the FIPO-BC allows continuous updating of the calibration outputs and potentially reduces the workload on generating the database.
Latimer C, Kophazi J, Eaton M, et al., 2021, Spatial adaptivity of the SAAF and Weighted Least Squares (WLS) forms of the neutron transport equation using constraint based, locally refined, isogeometric analysis (IGA) with dual weighted residual (DWR) error measures, Journal of Computational Physics, Vol: 426, Pages: 1-31, ISSN: 0021-9991
This paper describes a methodology that enables NURBS (Non-Uniform Rational B-spline) based Isogeometric Analysis (IGA) to be locally refined. The methodology is applied to continuous Bubnov-Galerkin IGA spatial discretisations of second-order forms of the neutron transport equation. In particular this paper focuses on the self-adjoint angular flux (SAAF) and weighted least squares (WLS) equations. Local refinement is achieved by constraining degrees of freedom on interfaces between NURBS patches that have different levels of spatial refinement. In order to effectively utilise constraint based local refinement, adaptive mesh refinement (AMR) algorithms driven by a heuristic error measure or forward error indicator (FEI) and a dual weighted residual (DWR) or goal-based error measure (WEI) are derived. These utilise projection operators between different NURBS meshes to reduce the amount of computational effort required to calculate the error indicators. In order to apply the WEI to the SAAF and WLS second-order forms of the neutron transport equation the adjoint of these equations are required. The physical adjoint formulations are derived and the process of selecting source terms for the adjoint neutron transport equation in order to calculate the error in a given quantity of interest (QoI) is discussed. Several numerical verification benchmark test cases are utilised to investigate how the constraint based local refinement affects the numerical accuracy and the rate of convergence of the NURBS based IGA spatial discretisation. The nuclear reactor physics verification benchmark test cases show that both AMR algorithms are superior to uniform refinement with respect to accuracy per degree of freedom. Furthermore, it is demonstrated that for global QoI the FEI driven AMR and WEI driven AMR produce similar results. However, if local QoI are desired then WEI driven AMR algorithm is more computationally efficient and accurate per degree of freedom.
Ferguson J, Eaton M, Kophazi J, 2021, Virtual element methods for the spatial discretisation of the multigroup neutron diffusion equation on polygonal meshes with applications to nuclear reactor physics, Annals of Nuclear Energy, Vol: 151, Pages: 1-24, ISSN: 0306-4549
The Continuous Galerkin Virtual Element Method (CG-VEM) is a recent innovation in spatial discretisation methods that can solve partial differential equations (PDEs) using polygonal (2D) and polyhedral (3D) meshes. This paper presents the first application of VEM to the field of nuclear reactor physics, specifically to the steady-state, multigroup, neutron diffusion equation (NDE). In this paper the theoretical convergence rates of the CG VEM are verified using the Method of Manufactured Solutions (MMS) for a reaction-diffusion problem in the presence of both highly distorted and non-convex elements and also in the presence of discontinuous material data. Finally, numerical results for the 2D IAEA and the 2D C5G7 industrial nuclear reactor physics benchmarks are presented using both block-Cartesian and general polygonal meshes.
Winter GE, Cooling CM, Eaton MD, 2020, Linear energy transfer of fission fragments of 235U and nucleation of gas bubbles in aqueous solutions of uranyl nitrate, Annals of Nuclear Energy, Vol: 142, Pages: 1-19, ISSN: 0306-4549
Fission fragments emitted in a fissile solution create tiny gas bubbles, the size of which is determined by the linear energy transfer (LET) of the particles. The LET of fission fragments of 235U in aqueous solutions of uranyl nitrate has been determined, and using methods adapted from the literature, the size of gas bubbles generated along the tracks of these particles has been estimated, revealing important variations with respect to particle LET and solution properties. Empirical correlations are presented for the maximum radius of radiolytic gas bubbles in unsaturated solutions of uranyl nitrate as a function of solution temperature and concentration. These can be used to predict the critical concentration of dissolved hydrogen necessary for the appearance of gas voids during nuclear criticality transients. The findings are intended for use in a future model of nuclear criticality transients in aqueous fissile solutions for the purposes of nuclear criticality safety assessment.
Williams MMR, Eaton MD, 2020, A theory of low source startup based on the Pal-Bell equations (vol 102, pg 317, 2017), ANNALS OF NUCLEAR ENERGY, Vol: 140, ISSN: 0306-4549
Williams MMR, Eaton MD, 2020, Spatial effects in low neutron source startup and associated stochastic phenomena (vol 111, pg 616, 2018), ANNALS OF NUCLEAR ENERGY, Vol: 140, ISSN: 0306-4549
Latimer C, Kópházi J, Eaton MD, et al., 2020, A geometry conforming, isogeometric, weighted least squares (WLS) method for the neutron transport equation with discrete ordinate (SN) angular discretisation, Progress in Nuclear Energy, Vol: 121, Pages: 1-15, ISSN: 0149-1970
This paper presents the application of isogeometric analysis (IGA) to the spatial discretisation of the multi-group, source iteration compatible, weighted least squares (WLS) form of the neutron transport equation with a discrete ordinate (S) angular discretisation. The WLS equation is an elliptic, second-order form of the neutron transport equation that can be applied to neutron transport problems on computational domains where there are void regions present. However, the WLS equation only maintains conservation of neutrons in void regions in the fine mesh limit. The IGA spatial discretisation is based up non-uniform rational B-splines (NURBS) basis functions for both the test and trial functions. In addition a methodology for selecting the magnitude of the weighting function for void and near-void problems is presented. This methodology is based upon solving the first-order neutron transport equation over a coarse spatial mesh. The results of several nuclear reactor physics verification benchmark test cases are analysed. The results from these verification benchmarks demonstrate two key aspects. The first is that the magnitude of the error in the solution due to approximation of the geometry is greater than or equal to the magnitude of the error in the solution due to lack of conservation of neutrons. The second is the effect of the weighting factor on the solution which is investigated for a boiling water reactor (BWR) lattice that contains a burnable poison pincell. It is demonstrated that the smaller the area this weighting factor is active over the closer the WLS solution is to that produced by solving the self adjoint angular flux (SAAF) equation. Finally, the methodology for determining the magnitude of the weighting factor is shown to produce a suitable weighting factor for nuclear reactor physics problems containing void regions. The more refined the coarse solution of the first-order transport equation, the more suitable the weighting factor.
Kophazi J, Eaton M, McClarren R, et al., 2020, A geometry conforming isogeometric method for the self-adjoint angular flux (SAAF) form of the neutron transport equation with a discrete ordinate (SN) angular discretisation, Annals of Nuclear Energy, Vol: 136, Pages: 1-16, ISSN: 0306-4549
This paper presents the application of isogeometric analysis (IGA) to the spatial discretisationof the multi-group, self-adjoint angular flux (SAAF) form of the neutron transport equation witha discrete ordinate (SN) angular discretisation. The IGA spatial discretisation is based uponnon-uniform rational B-spline (NURBS) basis functions for both the test and trial functions. Inaddition a source iteration compatible maximum principle is used to derive the IGA spatiallydiscretised SAAF equation. It is demonstrated that this maximum principle is mathematicallyequivalent to the weak form of the SAAF equation. The rate of convergence of the IGA spatial discretisation of the SAAF equation is analysed using a method of manufactured solutions(MMS) verification test case. The results of several nuclear reactor physics verification benchmark test cases are analysed. This analysis demonstrates that for higher-order basis functions,and for the same number of degrees of freedom, the FE based spatial discretisation methods arenumerically less accurate than IGA methods. The difference in numerical accuracy between theIGA and FE methods is shown to be because of the higher-order continuity of NURBS basisfunctions within a NURBS patch as well as the preservation of both the volume and surfacearea throughout the solution domain within the IGA spatial discretisation. Finally, the numericalresults of applying the IGA SAAF method to the OECD/NEA, seven-group, two-dimensionalC5G7 quarter core nuclear reactor physics verification benchmark test case are presented. Theresults, from this verification benchmark test case, are shown to be in good agreement with solutions of the first-order form as well as the second-order even-parity form of the neutron transportequation for the same order of discrete ordinate (SN) angular approximation.
Jeffers RS, Kópházi J, Eaton MD, et al., 2020, Goal-based error estimation for the multi-dimensional diamond difference and box discrete ordinate (SN) methods, Journal of Computational and Theoretical Transport, Vol: 49, Pages: 51-87, ISSN: 2332-4309
Goal-based error estimation due to spatial discretization and adaptive mesh refinement (AMR) has previously been investigated for the one dimensional, diamond difference, discrete ordinate (1-D DD-SN) method for discretizing the Neutron Transport Equation (NTE). This paper investigates the challenges of extending goal-based error estimation to multi-dimensions with supporting evidence provided on 2-D fixed (extraneous) source and Keff eigenvalue (criticality) verification test cases. It was found that extending Hennart’s weighted residual view of the lowest order 1-D DD equations to multi-dimensions gave what has previously been called the box method. This paper shows how the box method can be extended to higher orders. The paper also shows an equivalence between the higher order box methods and the higher order DD methods derived by Hébert et al. Though, less information is retained in the final solution in the latter case. These extensions allow for the definition of dual weighted residual (DWR) error estimators in multi-dimensions for the DD and box methods. However, they are not applied to drive AMR in the multi-dimensional case due to the various challenges explained in this paper.
Duan Y, Ahn JS, Eaton MD, et al., 2020, QUANTIFICATION OF UNCERTAINTIES IN A SAS-SST SIMULATION CAUSED BY THE UNKNOWN HIGH-WAVE NUMBER DAMPING FACTOR USING SURROGATE MODELLING, Pages: 799-812
This paper aims to quantify the uncertainty in the SAS-SST simulation of a prism bluff-body flow due to varying the higher-wave number damping factor (Cs). Instead of running the uncertainty quantification on the CFD simulation directly, a surrogate modelling approach is adopted. The mesh sensitivity is first studied and the numerical error is approximated accordingly. The Kriging modelling method is used to create surrogate models for quantities of interest. The cross-validated Kriging models are constructed based on the CFD predictions. The stochastic tests are performed using the Kriging models, in order to obtain the mean and variance of the CFD prediction due to Cs varying uniformly in the range [0, 1]. In this study, the prediction of flow statistics near the prism block is subject to larger variance due to the varying Cs. The influence of the Cs rapidly decays as the location moves downstream of the prism block.
Ibekwe R, Cooling C, Trainer A, et al., 2020, Modelling the short-term and long-term behaviour of the Oklo natural nuclear reactor phenomenon, Progress in Nuclear Energy, Vol: 118, ISSN: 0149-1970
This paper presents a computational investigation of the short-term and long-term behaviour of the Oklo natural nuclear reactors, instances in the distant past in which natural uranium deposits developed self-sustaining nuclear chain reactions. For the first time, processes occurring on timescales of seconds (such as changing temperature, moderator availability and power) are coupled in a single simulation with processes occurring over timescales of thousands of years (such as changing enrichment, reactor geometry and isotopic composition). This simulation reproduces key features of the Oklo reactors found in the literature (the cyclic boiling and flow of water in and out of the reactor; the characteristic three-hour cycle time; the total energy released by the reaction), gives greater insight into their development and evolution, and demonstrates a non-cyclic, non-boiling regime of behaviour in the later stages of reactor operation that has not previously been described.
Duan Y, Cooling C, Ahn JS, et al., 2019, Using a Gaussian process regression inspired method to measure agreement between the experiment and CFD simulations, International Journal of Heat and Fluid Flow, Vol: 80, ISSN: 0142-727X
This paper presents a Gaussian process regression inspired method to measure the agreement between experiment and computational fluid dynamics (CFD) simulation. Because of misalignments between experimental and numerical outputs in spatial or parameter space, experimental data are not always suitable for quantitative assessing the numerical models. In this proposed method, the cross-validated Gaussian process regression (GPR) model, trained based on experimental measurements, is used to mimic the measurements at positions where there are no experimental data. The agreement between an experiment and the simulation is mimicked by the agreement between the simulation and GPR models. The statistically weighted square error is used to provide tangible information for the local agreement. The standardised Euclidean distance is used for assessing the overall agreement.The method is then used to assess the performance of four scale-resolving CFD methods, such as URANS k-ω-SST, SAS-SST, SAS-KE, and IDDES-SST, in simulating a prism bluff-body flow. The local statistically weighted square error together with standardised Euclidean distance provide additional insight, over and above the qualitative graphical comparisons. In this example scenario, the SAS-SST model marginally outperformed the IDDES-SST and better than the other two other, according to the distance to the validated GPR models.
Williams A, Williams M, Eaton M, 2019, Methods of studying the effect of rough surfaces on reactivity in two-dimensional reactor physics problems, Annals of Nuclear Energy, Vol: 130, Pages: 493-511, ISSN: 0306-4549
Rough and perturbed surfaces are examined in the field of reactor physics using homotopy, homogenisation and the Feinberg-Galanin method. The homotopy method allows a problem with a perturbed interface to be represented as an unperturbed problem with a modified boundary condition. Perturbation theory was applied to this method and the results were studied for the neutron transport equation, the neutron diffusion approximation and a hybrid ABH-diffusion approximation. The homogenisation of surface roughness in the fuelmoderator interface was also examined and compared to the homotopy approach. Finally, the effect of larger-scale geometric uncertainties was studied using the Feinberg-Galanin approach. The effectiveness of the methods is determined by examining the change in effective multiplication factor keff examined both directly and via the six-factor formula. Analytic solutions to approximations of perturbation theory differentials were capable ofaccurately predicting the change in keff when compared to numerical solutions. Examining a change in fuel pin radius of 20 µm, we find that for water moderated systems the overall change in eff k is close to 70 pcm and for graphite systems it is 18 pcm. This shows that water and graphite moderated cores are sensitive to small changes in geometry.
Perrier H, Denner F, Eaton MD, et al., 2019, On the numerical modelling of Corium spreading using Volume-of-Fluid methods, NUCLEAR ENGINEERING AND DESIGN, Vol: 345, Pages: 216-232, ISSN: 0029-5493
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