## Publications

21 results found

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.

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.

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.

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.

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.

Winter G, Cooling C, Williams M,
et al., 2018, Importance of parametric uncertainty in predicting probability distributions for burst wait-times in fissile systems, *Annals of Nuclear Energy*, Vol: 119, Pages: 117-128, ISSN: 0306-4549

A method of uncertainty quantification in the calculation of wait-time probability distributions in delayed supercritical systems is presented. The method is based on Monte Carlo uncertainty quantification and makes use of the computationally efficient gamma distribution method for prediction of the wait-time probability distribution. The range of accuracy of the gamma distribution method is examined and parameterised based on the rate and magnitude of the reactivity insertion, the strength of the intrinsic neutron source and the prompt neutron lifetime. The saddlepoint method for inverting the generating function and a Monte Carlo simulation are used as benchmarks against which the accuracy of the gamma distribution method is determined. Finally, uncertainty quantification is applied to models of the Y-12 accident and experiments of Authier et al. (2014) on the Caliban reactor.

Cooling CM, Williams MMR, Eaton MD, 2017, CALLISTO-SPK: A Stochastic Point Kinetics Code for Performing Low Source Nuclear Power Plant Start-up and Power Ascension Calculations, *Annals of Nuclear Energy*, Vol: 113, Pages: 319-331, ISSN: 0306-4549

This paper presents the theory and application of a code called CALLISTO which is used for performing NPP start-up and power ascension calculations. The CALLISTO code is designed to calculate various values relating to the neutron population of a nuclear system which contains a low number of neutrons. These variables include the moments of the PDF of the neutron population, the maturity time and the source multiplier. The code itself is based upon the mathematics presented in another paper and utilises representations of the neutron population which are independent of both space and angle but allows for the specification of an arbitrary number of energy groups.Five examples of the use of the code are presented. Comparison is performed against results found in the literature and the degree of agreement is discussed. In general the agreement is found to be good and, where it is not, plausible explanations for discrepancies are presented. The final two cases presented examine the effect of the number of neutron groups included and finds that, for the systems simulated, there is no significant difference in the key results of the code.

Cooling C, Adams G, Eaton M, 2016, Transitions from Stochastic to Point Kinetics Models in Fissile Solutions, *Transactions of the American Nuclear Society*, Vol: 115, Pages: 637-639, ISSN: 0003-018X

Adams G, Cooling C, Eaton M, 2016, Point Kinetics Modelling of Decay Heat and Xenon Effect, *Transactions of the American Nuclear Society*, Vol: 115, Pages: 633-636, ISSN: 0003-018X

Cooling CM, Williams MMR, Eaton MD, 2016, Coupled probabilistic and point kinetics modelling of fast pulses in nuclear systems, *Annals of Nuclear Energy*, Vol: 94, Pages: 655-671, ISSN: 1873-2100

This paper describes a probabilistic method of modelling point nuclear systemswith low numbers of neutrons including the effects of delayed neutron precursors andits coupling with standard point kinetics equations. This coupling allows the simulationof the non-deterministic progression of a system transitioning from subcritical tosupercritical and the resulting power peak. Through analysis of large numbers of realisationsvarious statistical parameters of such transients can be obtained. The methodof simulation presented here successfully replicates the survival and extinction probabilitiespredicted by the Backwards Master Equation and experimental and analyticresults from the literature regarding the Godiva reactor and extends the examinationof that reactor. In particular the effect of delayed neutrons on the simulated responseof Godiva is highlighted. With its implementation in a parallel computer code, themodel is able to simulate at a reasonable speed a range of systems where low neutronpopulations are important.

Major M, Cooling CM, Eaton MD, 2016, The Effect of A Changing Fuel Solution Composition on a Transient in a Fissile Solution, *Progress in Nuclear Energy*, Vol: 91, Pages: 17-25, ISSN: 0149-1970

This paper presents an extension to a point kinetics model of fissile solution undergoing atransient through the development and addition of correlations which describe neutronicsand thermal parameters and physical models. These correlations allow relevant parametersto be modelled as a function of time as the composition of the solution changes overtime due to the addition of material and the evaporation of water from the surface of thesolution. This allows the simulation of two scenarios. In the first scenario a critical systemeventually becomes subcritical through under-moderation as its water content evaporates.In the second scenario an under-moderated system becomes critical as water is added beforebecoming subcritical as it becomes over-moderated. The models and correlations usedin this paper are relatively idealised and are limited to a particular geometry and fissile solutioncomposition. However, the results produced appear physically plausible and demonstratethat simulation of these processes are important to the long term development oftransients in fissile solutions and provide a qualitative indication of the types of behaviourthat may result in such situations.

Cooling CM, Ayres DAF, Prinja AK,
et al., 2016, Uncertainty and global sensitivity analysis of neutron survival and extinction probabilities using polynomial chaos, *ANNALS OF NUCLEAR ENERGY*, Vol: 88, Pages: 158-173, ISSN: 0306-4549

Zamacinski T, Cooling CM, Eaton MD, 2014, A point kinetics model of the Y12 accident, *PROGRESS IN NUCLEAR ENERGY*, Vol: 77, Pages: 92-106, ISSN: 0149-1970

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- Citations: 2

Cooling CM, Williams MMR, Nygaard ET,
et al., 2014, An extension of the point kinetics model of MIPR to include the effects of pressure and a varying surface height, *ANNALS OF NUCLEAR ENERGY*, Vol: 72, Pages: 507-537, ISSN: 0306-4549

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- Citations: 5

Cooling CM, Williams MMR, Nygaard ET,
et al., 2014, A Point Kinetics Model of the Medical Isotope Production Reactor Including the Effects of Boiling, *NUCLEAR SCIENCE AND ENGINEERING*, Vol: 177, Pages: 233-259, ISSN: 0029-5639

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- Citations: 3

Cooling CM, Williams MMR, Nygaard ET,
et al., 2013, The application of polynomial chaos methods to a point kinetics model of MIPR:An Aqueous Homogeneous Reactor, *Nuclear Engineering and Design*, Pages: 126-152

This paper models a conceptual Medical Isotope Production Reactor (MIPR) using a point kinetics model which is used to explore power excursions in the event of a reactivity insertion. The effect of uncertainty of key parameters is modelled using intrusive polynomial chaos. It is found that the system is stableagainst reactivity insertions and power excursions are all bounded and tend towards a new equilibrium state due to the negative feedbacks inherent in Aqueous Homogeneous Reactors (AHRs). The Polynomial Chaos Expansion (PCE) method is found to be much more computationally efficient than that of Monte Carlo simulation in this application.

Buchan AG, Pain CC, Tollit TS,
et al., 2013, Simulated spatially dependent transient kinetics analysis of the Oak Ridge Y12 plant criticality excursion, *Progress in Nuclear Energy*, Vol: 63, Pages: 12-21

In June 1958 an accidental nuclear excursion occurred in the C-1 Wing of building 9212 in a process facility designed to recover enriched Uranium U(93) from various solid wastes. The accident was caused by the inadvertent flow of enriched uranyl nitrate into a 55 gallon drum which established a prompt critical nuclear excursion. Following the initial fission spike the nuclear system oscillated in power. The reaction was eventually terminated by the additional water which was flowing into the drum. The criticality excursion was estimated to have lasted approximately 20 min based upon nearby radiation measurement equipment with an estimated total fission yield of 1.3 × 1018 fissions of which the first fission spike contributed 6 × 1016 fissions.The traces from the radiation measurement devices indicated that most of the fissions occurred in the first 2.8 min, in which case the average power required for the observed fission yield was approximately 220 kW. After the first 2.8 min the system was postulated to have boiled causing a sharp decrease in density and reactivity of the system. This boiling probably reduced the power output from the system to a low level for the final 18 min of the excursion. This paper will aim to investigate the subsequent evolution of the Y12 excursion using the fundamentally based spatially dependent neutron/multiphase CFD kinetics simulation tool - FETCH. The reconstruction of the Y12 excursion using FETCH will follow the evolution of the excursion up until the uranyl nitrate starts to boil. The results of the FETCH simulation are presented and compared against the known measurements of the excursion from the radiation detection instruments located near the drum.

Cooling C, Williams MMR, Eaton M, et al., 2013, Point kinetics models of the medical isotope production reactor, Pages: 908-911, ISSN: 0003-018X

Buchan A, Eaton MD, Goddard AJH,
et al., 2012, Simulated transient dynamics and heat transfer characteristics of the water boiler nuclear reactor SUPO with cooling coil heat extraction, *Annals of nuclear energy*, Vol: 48, Pages: 68-83

The term “water boiler” reactor refers to a type of aqueous homogeneous reactor (AHR) that was designed, built and operated by Los Alamos in the 1940s. This was the first type of liquid fuelled reactor and the first to be fuelled with enriched Uranium. For security reasons the term “water boiler” was adopted and three versions were built: LOPO (for low power), HYPO (for high power) and SUPO (for super power) which were spherical shaped reactor vessels. The name was appropriate as the reactors appeared to boil although this was actually due to the release of radiolytic gas bubbles; although SUPO was operated during some studies close to the boiling point of uranyl nitrate. The final water boiler “SUPO” was operated almost daily as a neutron source from 1951 until its deactivation in 1974-23 years of safe, reliable operation. Many of the key neutron measurements needed in the design of the early atomic weapons were made using LOPO, HYPO and SUPO. More recently SUPO has been considered as a benchmark for quasi-steady-state operation of AHRs with internal cooling structures.This paper presents modelling and analysis of the coupled neutronic and fluid time dependent characteristics of the SUPO reactor. In particular the quasi-steady-state dynamics of SUPO have been investigated together with its heat transfer characteristics. In the simulations presented the SUPO reactor is modelled using the spatially dependent neutron/multiphase CFD simulation tool, FETCH, at a quasi-steady-state power of 25 kW. SUPO also possessed a cooling coil system that fed cooling water through the reactor for the extraction of the fission and decay heat. This cooling system, and the heat extraction, is modelled in the simulations using a new sub-modelling approach that is detailed here. The results from this simulation, such as gas fraction, gas generation rate, coolant rate and average temperature, are compared against the available experimental information.

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