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(1)UNIVERSITY OF SCIENCE AND TECHNOLOGY. Faculty of Energy and Fuels. Toward higher reliability of Monte Carlo modelling of nuclear reactors in time domain. Doctoral thesis Author: MSc Grzegorz Kępisty. Supervisor: A. Professor Jerzy Cetnar. 2017-05-11.

(2) Declaration of the author of this dissertation: Aware of legal responsibility for making untrue statements I hereby declare that I have written this dissertation myself and all the contents of the dissertation have been obtained by legal means.. date, signature of author. Declaration of the thesis supervisor: This dissertation is ready to be reviewed.. date, signature of supervisor. ii.

(3) Acknowledgements. I am grateful to my thesis advisor prof. Jerzy Cetnar for his time, patience and inspiring scientific discussions during these four years. I would like to thank colleague Przemysław Stanisz for all his support concerning technical issues, computer clusters environment and time spent on discussions. Without his help and experience some of my research would have been significantly delayed. I wish to express my gratitude to prof. Jerzy Janczyszyn and colleague Bartłomiej Symołon for their comments on editorial work. I am also grateful to the rest of Department of Nuclear Energy staff for kind atmosphere supporting standalone work. Last but not least, I would like to thank my family and Martyna for supporting me spiritually throughout these studies and my life in general.. Grzegorz Kępisty. iii.

(4) Abstract. The aim of this work is to indicate computational and modelling problems present during the application of the Monte Carlo method, and to find the solutions. Studies performed in the framework of the doctoral thesis are focused on the time domain of reactor simulations. In the articles, the factors of physical and numerical nature in considered models have been analysed. Many of those factors have not been qualitatively studied yet. The majority of the studies deal with equilibrium fuel cycle calculations (time scale of years). Several articles address the issue of computational cost in Monte Carlo simulations (time scale of days). One study concerns the application of the Monte Carlo method to the development of transient state studies (time scale of seconds). Qualitative assessment of multiple modelling errors and their impact on the final results is contribution of the performed research. The final output of the studies is better understanding of the simulation codes being used and a more complete definition of a time domain in the reactor research. The articles comprised in the doctoral thesis serve as a review of good (and bad) practices in the field of nuclear reactor Monte Carlo modelling. Many conclusions are valid also in case of deterministic tools.. iv.

(5) Streszczenie. Celem niniejszej pracy było wskazanie problemów obliczeniowych i modelowych obecnych przy użyciu metody Monte Carlo i znalezienie ich rozwiązań. Przeprowadzone badania były skupione wokół domeny czasu symulacji reaktorowych. Artykuły badały czynniki natury fizycznej lub numerycznej w rozpatrywanych modelach, z których duża część nie została wcześniej ilościowo przeanalizowana. Większość rozpatrywanych prac skupiała się na obliczeniach równowagowych cyklu paliwowego (skala rzędu lat). Część artykułów poruszała zagadnienie kosztu obliczeniowego w obliczeniach Monte Carlo (czas rzeczywisty rzędu dni). Jedna praca dotyczyła zastosowania metody Monte Carlo w rozwoju badań nad stanami nieustalonymi (skala rzędu sekund). Korzyścią z przeprowadzonych badań było ilościowe oszacowanie wpływu szeregu błędów modelu na jakość otrzymywanych wyników. Ukoronowaniem pracy jest lepsze zrozumienie używanych kodów symulacyjnych a także pełniejsze zdefiniowanie tytułowej domeny czasu w badaniach reaktorowych. Artykuły wchodzące w skład rozprawy stanowią przegląd dobrych (i złych) praktyk z zakresu modelowania Monte Carlo reaktorów jądrowych. Wiele wniosków jest słusznych również dla kodów deterministycznych.. v.

(6) Table of Contents 1.. About this dissertation .................................................................................................................... 2. 2.. The work context ............................................................................................................................. 4. 3.. Overview of the articles comprised in thesis .................................................................................. 5 3.1.. Article A1: Instabilities of Monte-Carlo burnup calculations for nuclear reactors (…) [1] ..... 5. 3.2.. Article A2: Burnup instabilities in the full-core HTR model simulation [2] ............................ 6. 3.3.. Article A3: Assessment of advanced step models for steady state Monte Carlo (…) [3] ........ 7. 3.4.. Article A4: Parametric studies of the PWR fuel assembly modelling with (…) [4] ................. 8. 3.5.. Article A5: Monte Carlo burnup in HTR system with various TRISO packing [5] .................... 9. 3.6.. Article A6: Underestimation of nuclear fuel burnup – theory, demonstration and (…) [6] . 10. 3.7.. Article A7: SFR mechanical scenarios and neutron transport transients with (…) [7] ......... 11. 3.8.. Article A8: Dominance ratio evolution in large thermal reactors [8] ................................... 12. 3.9.. Article A9: Local effects of fuel burnup in high temperature reactor [9]............................. 13. 3.10.. Article A10: On the discrepancies between FIMA and specific burnup [10] ..................... 14. 3.11.. Article A11: Statistical error propagation in HTR burnup model [11] ............................... 15. 4.. Summary and keynote................................................................................................................... 16. 5.. Bibliography................................................................................................................................... 19. 6.. Reprints ......................................................................................................................................... 22. 7.. Attachments (Statements of Contribution)................................................................................. 125.

(7) 1. About this dissertation The presented doctoral dissertation, in accordance with Article 13 paragraph 2 of the Act on Academic Degrees and Academic Titles (amended 18 March 2011), consists of a set of eleven thematically coherent articles. The articles have been published in scientific journals and nine of them appear on the list of journals with impact factor of the Ministry of Science and Higher Education. The articles present research problems concerning better utilization and understanding of the Monte Carlo method in the context of nuclear reactor models. The topic of fuel cycle calculations methodology is of special interest here. The results provide specific indications for numerical simulations in this field of science and illustrate the theses presented at the registration for the PhD conferment procedure.. The set consists of the following publications: 1) Instabilities of Monte-Carlo burnup calculations for nuclear reactors—Demonstration and dependence from time step model, Nuclear Engineering and Design 286 (2015) 49–59, [1] ; 2) Burnup instabilities in the full-core HTR model simulation, Annals of Nuclear Energy 85 (2015) 652–661, [2]; 3) Assessment of advanced step models for steady state Monte Carlo burnup calculations in application to prismatic HTGR, NUKLEONIKA 60:3 (2015) 523–529, [3]; 4) Parametric studies of the PWR fuel assembly modeling with Monte-Carlo method, Annals of Nuclear Energy 94 (2016) 189–207, [4]; 5) Monte Carlo burnup in HTR system with various TRISO packing, Annals of Nuclear Energy 92 (2016) 419–430, [5]; 6) Underestimation of nuclear fuel burnup – theory, demonstration and solution in numerical models, E3S Web of Conferences 10, 00022 (2016), SEED 2016, DOI: 10.1051/e3sconf/20161000022, [6]; 7) SFR mechanical scenarios and neutron transport transients with CAST3M code, Annals of Nuclear Energy 101 (2017) 226–236, [7]; 8) Dominance ratio evolution in large thermal reactors, Annals of Nuclear Energy 102 (2017) 85–90, [8]; 9) Local effects of fuel burnup in high temperature reactor, Logistyka 4 (2015) CD nr 3, 9114–9123, [9]; 10) On the discrepancies between FIMA and specific burnup, Progress in Nuclear Energy 98 (2017) 187-192, [10]; 11) Statistical error propagation in HTR burnup model, Annals of Nuclear Energy 105 (2017) 355–360, [11];. 2.

(8) In addition, the following articles have been published during the doctoral studies. These are not related or only partially related to the topic of the thesis:. 1) P. Stanisz, M. Oettingen, G. Kępisty, Depletion uncertainties propagation for replica calculations in the lead cooled fast reactor, Logistyka 4 (2015) CD nr 3, 9798–9808, ISSN 1231-5478; 2) G. Kępisty, M. Malicki, M. Orliński, On the physical explanation of lethargy use in nuclear reactor physics. Alternative energy binning technique, Dokonania Młodych Naukowców 5:4 (2014) 363–368, ISSN 2300-4436, Konferencja Młodych Naukowców nt. Wpływ młodych naukowców na osiągnięcia polskiej nauki - VI edycja : Gdańsk 25– 27.04.2014; 3) G. Kępisty, The directions of nuclear fuels technology development, In: Energia i paliwa 2015 (2016) 99–10, Kraków: Wydawnictwo Studenckiego Towarzystwa Naukowego 2016, ISBN: 978-83-932168-5-7; 4) G. Kępisty, Modeling of the fuel cladding burn up in pressurized water reactor (PWR), In: Zagadnienia aktualnie poruszane przez młodych naukowców 1:2 (2015) 416–420, Kraków : CREATIVETIME, 2015, e-ISBN: 978-83-63058-46-3; 5) M. Oettingen, G. Kępisty, Monte Carlo modelling of loosely coupled fissionable systems, Logistyka 4 (2014) CD nr 6, 4731–4739, ISSN 1231-5478; 6) P. Stanisz, G. Kępisty, M. Orliński, New trends using Monte Carlo methods for nuclear reactors, Dokonania Młodych Naukowców 5:4 (2014) 413–418, Konferencja Młodych Naukowców nt. Wpływ młodych naukowców na osiągnięcia polskiej nauki - VI edycja: Gdańsk 25–27.04.2014; 7) G. Kępisty, P. Stanisz, M. Oettingen, Scalability of the Monte Carlo continuous energy burnup code – MCB, In: CGW workshop'15: October 26–28, 2015 Kraków, Poland : proceedings (2015) 51–52, Academic Computer Centre CYFRONET AGH, ISBN: 97883-61433-14-9; 8) G. Kępisty, P. Stanisz, Stability and accuracy of step models for burnup calculations as an improvement of nuclear fuel cycle precision and logistics, Logistyka 4 (2014) CD nr 6, 4405–4413, ISSN 1231-5478; 9) I. Królikowski, G. Kępisty, M. Orliński, Thermal-hydraulic analysis of fuel block in high temperature reactor, Logistyka 4 (2015) CD nr 3, 9278–9285, ISSN 1231-5478; 10) G. Kępisty, Wodna magia, In: Poznaj nowoczesne technologie oraz zjawiska fizyczne : dzieci w świecie Internetu oraz niecodziennych tajemnic fizyki (2015) 20–25, Kraków: Dział Informacji i Promocji Akademii Górniczo-Hutniczej im. Stanisława Staszica w Krakowie, (AGH Junior), ISBN: 978-83-7464-782-3;. 3.

(9) 2. The work context Nuclear reactors belong to the most advanced systems in the area of contemporary techniques of electrical energy generation. Nuclear power is characterized by high reliability of energy supply, competitive energy prices, zero emission of carbon dioxide, long life time and independence from weather conditions [12-13]. The aforementioned advantages make nuclear power plants an option for the basis of energy mix in the developed countries. In the present situation of energy market and nuclear policy, nuclear technology is characterized by long construction time and high investment cost. Because of this, further reactor optimization, also at the level of computer modelling, has already been a standard for decades [14]. Simulation codes allow for numerical representation of physical systems – checking their parameters, investigating responses to various perturbations and performing design optimization. Nuclear reactor core physics belongs to the most complex computational tasks due to the structure of differential equations describing the system, the number of variables after discretization and several related numerical problems [15]. Consequently, research and development is a necessity both in terms of proper use and better understanding of presently existing tools and in terms of development of new methodologies. The framework of this thesis is focused on improving the reliability of Monte Carlo neutron transport calculations of nuclear reactors. This method is well-known and has been applied for tens of years in the field of neutron physics as a reference tool which lacks simplifications in terms of geometry mapping or particles interaction [16]. Numerous simulation codes are being developed in reactor science groups worldwide. Among the improvements we can distinguish new methods of variance reduction [17], computation acceleration [18], better parallelization [19] and transient modelling capabilities [20]. Monte Carlo simulations seem to have a stable position in upcoming research and development of nuclear reactors. This is under the condition that code users and developers will handle several challenges concerning computational cost, neutron source convergence or bias in predicted uncertainties [21]. The vast majority of research in the framework of this thesis concerns computations of fuel cycle, which is the basis for the prosperity of reactors in a wider scale. The improved methodology relates to the widely-understood time domain – better understanding and dealing with numerical instabilities. Another aspect is the influence of model scale and detail on time evolution of reactor state. An important element of this work is the MCB source code development [22]. A series of parametrical studies has contributed to the knowledge about good practices in using fuel transmutation codes. In particular, multiple assumptions and simplifications common in the area of core modelling have been verified, and their systematic impact on the results have been indicated and quantitatively assessed. The analysis of neutron source convergence evolution has been performed, which is particularly important, as it describes the stability of system both in the context of reactor work and computer simulations. The Monte Carlo method has also been applied to validate developed methodology dedicated to modelling transient states induced by core mechanical excitations. The obtained knowledge is not limited to a particular type of nuclear reactor or only to Monte Carlo codes.. 4.

(10) 3. Overview of the articles comprised in thesis 3.1.. Article A1: Instabilities of Monte-Carlo burnup calculations for nuclear reactors Demonstration and dependence from time step model [1]. Burnup simulations coupled to Monte Carlo neutron transport codes can exhibit computational instability when modelling nuclear systems. The origin of this problem is a strong coupling between neutron flux and nuclide field of fuel [23]. This issue occurs even in simplified, one-dimensional fuel assembly models from Pressurized Water Reactor (PWR), thus making equilibrium burnup calculations hardly feasible. The aim of this article is to investigate the presence and intensity of the aforementioned oscillations, depending on the time step length and dimensional size of the considered core models. Comparative calculations performed for one and two-dimensional model indicate a correlation between the length of active core and solutions susceptibility to instabilities. On the other hand, the intensity of the oscillation problem itself appears to be weakly sensitive to the time step length applied or the number of particles histories in neutron transport simulation. It results from the physical basis of the problem: oscillations are induced and maintained by the strongest known neutron absorber present in the system – 135 Xe [24], which is a fission product. In order to solve the problem, a literature review of available time step models and coupling schemes has been performed. Subsequently, Stochastic Implicit Euler method [25] has been chosen and implemented in the MCB source code due to its unconditional stability. The calculations made using this advanced methodology allowed to obtain: a) stable profile of neutron flux and power and b) physically correct isotopic evolution of fuel. The implemented coupling scheme contains additional correction considering the variation of neutron source intensity during the time step. An example of simulation with and without the solution is shown in Fig. 1 below (two-dimensional system of fuel assemblies from PWR reactor). Normalized power profile evolution (time steps). Normalized power profile evolution (time steps). 14. 1. 1. 12. 1,710E6. 10. 1,499E6 1,289E6 1,078E6. 8. 8,675E5 6,569E5. 6. 4,463E5 2,356E5 2,500E4. 0. 0. 4 2 2. Fig. 1. Power profile evolution over 16 subsequent time steps and various coupling schemes applied : staircase model (left) and Stochastic Implicit Euler (right) [1].. 5. 4.

(11) 3.2.. Article A2: Burnup instabilities in the full-core HTR model simulation [2]. The numerical instabilities studied in the previous article do not automatically prove their presence in other models of loosely-coupled nuclear systems. An example can be High Temperature Reactor (HTR), whose neutron physics processes significantly differ from the properties of Light Water Reactors. The aim of this article is to demonstrate the presence of computational oscillation in double-heterogeneous geometry of the whole core. Numerical results unambiguously indicate that a numerical system becomes unstable at a certain point of burnup procedure (about 1000-1200 days of irradiation under nominal power). It is an example of oscillation which is not driven by the level of xenon, but origins from gradual evolution of core properties (such as evolution of neutron source convergence speed, changes in fuel profile and others). The problem occurs in case of the simplest staircase coupling scheme being applied. What is more, an insufficient statistic in Monte Carlo neutron transport increases its intensity. The use of advanced schemes, such as bridge scheme or Stochastic Implicit Euler, allows to obtain stable and physically correct results. The examples of power density profile of various stability in subsequent time steps are presented in Figs. 2 and 3. Normalized power density profile in core at EOI - SIE, 100 d. Normalized power density profile in core - staircase 100 d. 25. Step 19. Step 20. Step 19. 25. Step 20. 20. 24. 20. 22. 1520. 20. 15. 18. 1015. Axial zone. Axial zone. 16 14 12 10 8. 10 5 10. 5 0. 6. 5. 0. 4. 0. 2 1. 2. 3. 4. 5. 6. 7. 8. 9. 10 1. 2. 3. Radial zone. 4. 5. 6. 7. 8. 9. 1. 10. 2 2. 3. 4 4. 5. 6. 6 7. 8. 8 9. 1. 10. 10 2. 3. 5. 6. 7. 8. 9. 10. Radial zone. Radial zone. Radial zone. 4. Figure 2. Power density profile in subsequent time steps for selected coupling schemes applied to HTR model [2].. 0. 20. Ax ia. 15. lz. on e. 10 5. 2. 4. 6. 8. 20 18 16 14 12 10 ep. 0. 20. Ax ial z. t es. Tim. 15 10. on e. 5 2. 4. 6. 8. 20 18 16 14 12 10 p. es. el (W/cm 3 ) nsity in fu. 2000 1000. Power de. 2000. 4000 3000. Power de. 4000. 6000 5000. nsity in fu. 6000. Power density profile in radial zone 5 - SIE method, 100 d. el (W/cm 3 ). Power denisty profile in radial zone 5 - staircase 100 d. te. Tim. Figure 3. Power density profile in the HTR fuel column in subsequent time steps and for selected coupling schemes [2].. 6. 0. 2.

(12) 3.3.. Article A3: Assessment of advanced step models for steady state Monte Carlo burnup calculations in application to prismatic HTGR [3]. Numerical models of reactor systems contain specific assumptions and simplifications allowing to obtain results of expected quality within limited computational cost. An example is the neglect of reactivity control presence in the core layout, which is of course prohibitive and unrealistic in the context of real core work. The purpose of this article is to assess how the presence of compensation rods impacts the stability of calculation and the reliability of the results obtained for the system of HTR. The primary conclusion of this research is the information that considering rods withdrawal in the core model efficiently improves the stability of fuel cycle simulation (Fig. 4).. Neutron flux profile (a.u.). 50 cm stepwise CR withdrawal. 40 30 20 10. m. e. ste. 9. 7. 5. p. 3. 80 0. 1. 70 0. 60 0. 50 0. 10 20 0 30 0 40 0 0. os. lp. ia Ax. n( itio. ). cm. 100. 200. 300. 400. 500. 600. 700. Irradia. Ti. 19 7 1 5 1 3 1 1 1. tion tim. e (day. 0 5d 105 d 205 d 305 d 405 d 505 d 605 d 705 d 805 d 905 d 1005 d 1105 d 1205 d 1305 d 1405 d 1505 d. ) Neutron Flux (a.u.). Neutron Flux evolution for CR withdrawal modeling. Neutron flux profile - staircase model (100 days step). Axial position (cm). Figure 4. Stability of neutron flux profile in the model with rods omitted (left) and the case containing them (right) [3].. The compensation rod included in the model improves the neutron source convergence (a peak of power follows uncovered fresh fuel). Besides more stable and representative power density profile, more precise simulation provides more reliable profile of spent fuel isotopic densities (shown in Fig. 5). Evolution of Pu239 mass profile - CR withdrawal modeling Profile of Pu239 mass (kg). Profile of Pu239 mass (kg). Evolution of Pu239 mass profile without CR modeling. 0,25 0,20 0,15 0,10 0,05. Irr. 0,10 0,05. (c. m). e. a Axi. ) ay. (d. ) ay. (d. 0,00 d 105 d 305 d 505 d 705 d 905 5d 110 5d 130 5d 150. tim. e. tim. 20 0. on siti l po. 0,15. n tio. n tio. ia ad. 10 0. 30 0. 8 70 00 0 6 50 00 0 40 0. 0,20. ia ad. Irr. 0,00 d 100 d 300 d 500 d 700 d 900 d 0 110 0d 130 0d 150. 0,25. 10 0. 20 0. 30 0. 8 70 00 0 6 50 00 0 40 0. n (c itio s o al p Axi. m). Figure 5. Isotopic mass of 239Pu in fuel zones of the system without control rods (left) and with reactivity control (right) [3].. 7.

(13) 3.4.. Article A4: Parametric studies of the PWR fuel assembly modelling with Monte-Carlo method [4]. Numerical modelling of nuclear systems requires the application of specific modelling practices affecting the reliability of results and providing acceptable computational simulation cost. The aim of this article is to analyze multiple factors occurring in models of the PWR fuel assembly in order to define their qualitative impact on predicted system characteristics in the time domain. Special attention is given to the significance of realistic reactivity control, applied density of coolant, time discretization in a model, spatial discretization of fuel zones, temperature discretization of fuel, application of various nuclear data, applied precision of Monte Carlo neutron transport and the presence of cladding-pellet gap inside the fuel rod. Basically, any neutron transport simulation of a core model requires clear decisions concerning the aforementioned factors, which surely do not exhaust the topic. Within the framework of the studies, numerous parametrical and comparative burnup calculations have been performed for the model of fuel assembly irradiated up to 43 GWd/tHM burnup. The analysis has been focused on variations of isotopic fuel evolution and the neutron multiplication factor. Some factors has appeared to be crucial (e.g. consideration of reactivity control or applied moderator density), whilst some of them have a negligible effect on global fuel transmutation (e.g. level of radial pellets discretization). Example differences in the outputs resulting from the use of boric acid as a reactivity control mechanism are shown in Fig. 6. The change of Conversion Ratio due to reactivity control. 0,65. 110. U235 Pu239 Pu241 Am243 Cm244. 100 90 80 70. Integral Conversion Ratio. Relative difference of isotope mass (%). Relative difference of isotope mass due to reactivity control 120. 60 50 40 30 20 10 0 0,0. 0,5. 1,0. 1,5. 2,0. 2,5. 3,0. 3,5. 4,0. Irradiation time (year). 0,60. 0,55. 0,50. No Boron With Boron. 0,45. 0,40. 0,5. 1,0. 1,5. 2,0. 2,5. 3,0. 3,5. 4,0. Time (y). Figure 6. Differences of isotopic densities (left) and discrepancies of the conversion ratio (right) due to consideration of reactivity control in the model of fuel assembly [4].. This is the first study which makes a precise analysis of the impact of so many factors on the reliability of computational results obtained. Knowledge about systematic and modelling uncertainties is one of the most important contributions of the considered research. Information about errors and uncertainties of observable results is probably the most important feature of any computer simulation, not only in case of nuclear systems. Conclusions complementary to those discussed here have been presented in article [26].. 8.

(14) 3.5.. Article A5: Monte Carlo burnup in HTR system with various TRISO packing [5]. Monte Carlo neutron transport allows for a reliable description of neutron physics processes in models of complex double-heterogeneous geometry of High Temperature Reactor. Spatial packing of fuel particles at the level of fuel compacts inside graphite blocks can be represented in various ways. The simplest method of cylinder filling using a cubic lattice does not provide most reliable results both in case of criticality calculations [27] and fuel cycle studies [28]. In this study five prismatic block models of HTR have been prepared, each having various fuel particles packing, in order to compare the results of burnup calculation. Besides classic arrangement of the cubic lattice truncated at the boundary of compacts, models oriented toward realistic particles packing at the boundary have been created. Other versions being investigated are: a model comprising a hexagonal lattice, a model based on stochastic displacements of fuel particles [29] and a case with positions generated in a random way. The scheme of the analyzed arrangements and the evolution of neutron multiplication factor discrepancies are shown in Fig. 7. Difference of Kinf in comparison to Cubic URAN 300. Kinf discrepancy (pcm). 200. In comparison to the results of Cubic URAN model. 100 0 -100. Truncated Cuboid Simple Cubic Simple Hexgonal Quasi Random. -200 -300 -400 -10. 0. 10. 20. 30. 40. 50. 60. 70. 80. Burnup (FIMA, %). Figure 7. Packing methods of fuel particles in five models considered (left) and the evolution of neutron multiplication factor differences (right) [5].. Numerical results show that the systems with more chaotic arrangements of fuel capsules exhibit a higher neutron flux in thermal and fast energy domain of spectrum (and lower at epithermal energies). This consequence of model details at the millimeter level results in discrepancies of fuel cycle at the scale of the whole core. Relative differences of isotopic concentrations reach a few percent while the neutron multiplication factor differs by 400 pcm. This is an example of modelling error propagation in Monte Carlo burnup calculation. The most practical conclusion of this work is a recommendation for avoiding fuel particle truncation at the edges of graphite cylinders. The optimal approach to the HTR geometry uses the function of stochastic displacements generation on-the-fly – during the execution of neutron transport simulation module.. 9.

(15) 3.6.. Article A6: Underestimation of nuclear fuel burnup – theory, demonstration and solution in numerical models [6]. For many years, the simplest staircase coupling scheme (aka. explicit Euler, beginningof-step approximation) has been applied in depletion codes based on Monte Carlo neutron transport calculations for evaluation of subsequent time steps [23]. This methodology is simple in implementation and fast in execution. Even though this model is only conditionally stable, it exhibits another deficiency. It systematically underestimates numerical fuel burnup in comparison to the burnup value declared by a code user (usually via irradiation history details) [4]. This article contains a theoretical description of the presented problem. The staircase scheme neglects a change of transmutation normalization coefficients within a time step. The bias of the obtained burnup value should be inversely proportional to the length of the time step in the model; thicker time mesh could be hypothetically applied. Of course, due to high computational cost of the Monte Carlo method it is not an acceptable solution. Advanced step models comprising the predictor-corrector methodology [30] help to account for a linear variation of the normalization in time intervals. As a first approximation, this should solve the discussed problem. Numerical tests performed with the SERPENT code [31] and the simple model of fuel assembly from the PWR system confirm the presence of the considered numerical problem and its predicted dependence from the step length. Numerical results are presented in Fig. 8. Fissions per Initial Metal Atom at End-of-Cycle. Numerical error of FIMA at End-of-Cycle. 6,4 1. 6,0. FIMA error (%). Output FIMA (%). 6,2. 5,8 5,6 5,4. euler predictor / staircase predictor-corrector. 5,2 5,0. 10. 100. euler predictor / staircase predictor-corrector Theoretical prediction. y db. 0,1. 0,01. ear. Lin. or avi beh. ory. the. te dic pre. 1E-3. 1000. 10. Step length (d). 100. 1000. Step length (d). Figure 8. Burnup of material in fuel assembly depending on the time step length applied in the model (left), predicted and observed errors for two considered coupling schemes (right) [6].. An important conclusion of the study is the observation that the considered error becomes smaller than numerical result precision in output files for steps shorter than 50 days (predictor-corrector in use). This provides a clear indication for users of transmutation codes. On the other hand, the obtained results allow to assess burnup underestimation even in case of the simplified methodology being applied.. 10.

(16) 3.7.. Article A7: SFR mechanical scenarios and neutron transport transients with CAST3M code [7]. Monte Carlo codes are treated as a reference method for deterministic programs, which are based on the neutron transport equation discretization [32-33]. The possibility of results comparison is a key issue for code developers and a major argument for the introduction of new methodologies into research and development of core multi-physics. This article describes research on transient states of the Sodium-cooled Fast Reactor (SFR) model using two deterministic codes. A new tool implemented in the code dealing with the neutron diffusion equation allows for direct accounting for geometry model deformations (CAST3M Neutron Transport Tool, CNTT). The key aspect of the study is validation of new model correctness using the Monte Carlo TRIPOLI-4 code [17]. The tests consist in preparing core bundle deformations using CAST3M tool and then adapting them to neutron transport models of CNTT and APOLLO3 codes [34]. Selected vibration modes of various amplitudes have been adapted to criticality studies in order to compare reactivity variations predicted by the considered tools. The results of validation calculations are presented in Fig. 9.. Figure 9. Reactivity changes in the core model: bundle compression (left) and core flowering (right) [7].. The results suggest a very good agreement of neutron transport response for all three applied tools and various system deformations. The new methodology implemented in the CAST3M code has been subsequently used to simulate several transient states of the same core system. Studies of this kind provide better understanding of neutron transport response of fast reactor configurations under mechanical excitation. The Monte Carlo method is helpful in developing other research tools in the field of nuclear reactor core physics. The presented studies show its usefulness for the research on transient states of reactor systems. This is a meaningful contribution to modelling of nuclear systems in the time domain according to the current state-of-art. The described research field is studied only in a few research groups world-wide.. 11.

(17) 3.8.. Article A8: Dominance ratio evolution in large thermal reactors [8]. The problem of neutron source convergence in Monte Carlo simulations is one of the most important challenges both for users and developers of computational codes [21]. The problem of this type exists even in case of deterministic codes [35] and is meaningful in spatially large nuclear reactor cores, such as those of pressurized water reactors (PWR). The neglect of this issue leads directly to the loss of results reliability (wrong neutron source). The speed of neutron source convergence is qualitatively measured by the so-called dominance ratio [36]. The convergence rate becomes problematically slow when its value approaches unity.. 1,000 0,999 0,998 0,997 0,996 0,995 0,994 0,993 0,992 0,991 0,990 0,989 0,988 0,987 0,986 0,985 0,984. 1,000 0,995. Dominance ratio - PWR core DR - PWR core - no boric acid 3 uncertainties 0. 50. 100. 150. 200. 250. 300. 350. 400. Dominance Ratio (DR). Dominance ratio. This article undertakes the topic of dominance ratio evolution as a function of fuel burnup for two reactor models: PWR (fresh fuel loading) and HTR (equilibrium loading). The studies have been performed using the MCB code and the method of source relaxation with spatial distribution of Dirac delta type [37]. The results clearly indicate that the dominance ratio tends to increase over fuel irradiation. On the other hand, its value is prone to the presence of localized strong neutron absorbers (e.g. compensation rods). The evolution of the dominance ratio as a function of fuel burnup (typical fuel cycles) is shown in Fig. 10.. 0,985 0,980 0,975 0,970 0,965 0,960 0,955 0,950 0,945 0,940. 3 uncertainties. 0,935 0,930. 450. DR HTR DR HTR - noCR. 0,990. -50. 0. 50. 100. 150. 200. 250. 300. 350. 400. 450. Irradiation time (EFPD). Irradiation time (EFPD). Figure 10. Dominance ratio evolution in the PWR system (left) and in the HTR system (right). The data series in red concerns the case without reactivity control in models [8].. The results are important for all studies in which Monte Carlo neutron transport tools are applied. They are of special significance in case of fuel cycle studies, because most of researchers apply a constant number of inactive neutron cycles at the start of the Monte Carlo run for all time steps. A number that is sufficient at the beginning of a fuel cycle may not provide the same source convergence quality at a higher burnup. An important conclusion is generally increasing computational cost of simulations due to required number of omitted neutron cycles. The results also contain the information concerning the operability of real nuclear reactors. The increase of the dominance ratio makes the system more prone to perturbations (less stable) and more difficult to control [38].. 12.

(18) 3.9.. Article A9: Local effects of fuel burnup in high temperature reactor [9]. Selection of optimal spatial discretization is one of the most basic problems in computer modelling of nuclear systems [39]. In case of double-heterogeneous geometry of High Temperature Reactor, it affects isotopic fuel evolution even at the level of a single graphite block. This article focuses on the comparison of transmutation procedure results for two models of HTR fuel prism differing in the number of burnable zones. The first one contains one fuel material, while the second one has 10 radial and 24 axial zones; altogether 240 separately burned materials. Although the discussed geometry with reflective boundary conditions is a kind of simplification, the observed effect has already occurred in the studies of whole core HTR model loaded with minor actinides [40]. The obtained results clearly indicate that fuel undergoes more intense burning at the vertical edges of a graphite block. It is visible in the fissile isotope density profile in Fig. 11 (initially flat) and in the outline of spatial burnup. The origin of these phenomena is a slight excess of graphite (about 2 cm) at the edges of the block, causing higher neutron thermalization in its proximity. On the other hand, the same is not visible in the radial direction of fuel zones.. 239Pu density profile at EOL. Ax ia. 20. lz. on. 15. e. 10. 5. 0. 1. 2. 3. 4. 5. 6. 7. 8. dia Ra. 9. 150 100. 20. Ax ia. e on lz. 15. lz. on. 10. es. 5 1. 2. 3. 4. 6. 5. 7. 8. z ial ad. 9. 50 0. Wd/t). nsity in fu. 250 200. Burnup (G. 350 300. Pu239 d e. 25. 0,50 0,45 0,40 0,35 0,30 0,25 0,20 0,15 0,10 0,05 0,00. el (g/cm3. ). Burnup profile for HTR fuel block at 800 EFPD. es. on. R. Figure 11. Fissile isotope density profile in the model with multiple fuel zones (left) and overall burnup profile (right) [9].. The aforementioned effect of the difference in burnup dynamics is not only of spatial nature. It appears that the effects of self-shielding in the model with multiple transmutation zones cause a relative increase of 239Pu total mass of 5% at the end of the irradiation period (1600 days). The analysis of time-space profiles suggests that axial discretization of fuel zones in HTR model is a key factor for obtaining reliable results in fuel cycle analyses. Moreover, it is inevitable for thermal-hydraulic studies in the context of hot-spot location prediction.. 13.

(19) 3.10. Article A10: On the discrepancies between FIMA and specific burnup [10] Fissions per Initial Metal Atom (FIMA, %) and burnup (GWd/tHM) are basic indicators of fuel consumption in the field of nuclear engineering [41]. The customary assumption is a linear proportionality between these two quantities, allowing for a comparison of results to those obtained by other authors or using other models. Physical analysis of both quantities shows that FIMA and burnup have various proportionality coefficient, depending on the considered content of the fuel. In particular, the coefficient depends on the ratio of the average energy released in nuclear fission to the average molar mass of heavy metal [42]. This ratio is different for uranium and plutonium isotopes, which implies a different value of FIMA for fixed energy amount extracted from the initial mass of fuels. What is more, the dependence between FIMA and burnup is not exactly linear even in case of a specified fuel cycle type. Thus, the considered effect is of purely physical nature and it can be observed both experimentally and computationally. Numerical tests have been performed using the SERPENT code. The fuels comprising single isotopes as well as realistic fuel vectors applied in industrial power plants have been analyzed. The results have confirmed the aforementioned discrepancies of FIMA values for short and very long irradiation periods. The example results are shown in Fig. 12 below. FIMA discrepancies (predictor-corrector model) 1,0 0,8. delta FIMA (%). 0,6. 233U-JEFF3.1 233U-ENDFB7 235U-JEFF3.1 235U-ENDFB7 239Pu-JEFF3.1 239Pu-ENDFB7 241Pu-JEFF3.1 241Pu-ENDFB7. 0,4 0,2 0,0 -0,2 -0,4 -0,6 -0,8 -1,0. 0. 100. 200. 300. 400. 500. 600. 700. 800. Burnup (GWd/tHM) Figure 12. Discrepancies of FIMA for mono-isotopic fuels in the model of pressurized water reactor assembly [10].. The discrepancy values obtained in the framework of the tests are comparable to experimental uncertainties of spent fuel measurement methods (relative precision of ~2%). Consequently, users of numerical tools must take into account the discussed effect as a potential source of systematic errors in comparative analyses.. 14.

(20) 3.11. Article A11: Statistical error propagation in HTR burnup model [11] Statistical uncertainty is an inevitable component of all tallies estimated during numerical experiments based on the Monte Carlo method. In particular, for Monte Carlo neutron transport the number of tracked particle histories is always limited by the increasing computational cost. Statistical uncertainties assessed by simulation codes can be underestimated several times due to a strong cross-correlation between neutron cycles in the power iteration method [43]. The problem becomes even more complex in case of fuel burnup calculations, because the uncertainties of nuclear reaction rates propagate on nuclide field in subsequent time steps. This can be shown using the independent replica method [44]. As a result, the isotopic densities in burnable cells also undergo some distribution around the average value when a random generator seed number is varied in subsequent samples of simulation. The aim of this article is to estimate statistical error propagation in the model of HTR system, with special emphasis on selected minor actinides nuclear reactions and on the evolution of nuclide field density variation. The studies of this type have not been performed yet for a whole-core double-heterogeneous geometry considered here. The most prominent results are presented in Fig. 13 below.. Space averaged values. 239Pu (n,f) 239Pu (n,g) 240Pu (n,f) 240Pu (n,g) 241Pu (n,f) 241Pu (n,g) Fission Heating. 13 12 11 10 9 8 7 6 5 4 3. up to 40-80 at last step. 2 1 0. 0. 200. 400. 600. 800. 1000. 1200. Irradiation time (EFPD). 1,000 300. Discarded cycles + 1 SD Dominance ratio + 3 uncert.. 250. 0,995 0,990. 200. 0,985. 150. 0,980. 100. 0,975. 50. 0,970. 0. 0. 200. 400. 600. 800. 1000. 1200. Dominance ratio. Real to apparent tally SD. 14. Number of discarded cycles test (MCNP). 15. 0,965. Irradiation time (EFPD). Figure 13. Statistical error propagation in the model of High Temperature Reactor (left) and analysis of neutron source convergence speed (right) [11].. The obtained results indicate that the simulation consists of two regimes of error propagation: the stable part (low fuel burnup, error underestimation of factor 2-4) and the unstable part (high burnup, uncertainties grow exponentially). The problem of the crosscorrelation between neutron cycles turns out to be more significant than in case of models considered by other authors. The problems of neutron source convergence are suggested by Shannon’s entropy a few time steps after the emergence of instability; however, the dominance ratio increase is well visible much earlier. This suggests that the automatic internal numerical test of the MCNP code [45] may be of limited sensitivity in case of certain models.. 15.

(21) 4. Summary and keynote Studies performed in the framework of the doctoral thesis are focused on the time domain of reactor simulations. The majority of the studies deal with equilibrium fuel cycle calculations (time scale of years). Several articles address the issue of computational cost in Monte Carlo simulations (time scale of days). One study concerns the application of the Monte Carlo method to the development of transient states studies (time scale of seconds). In the articles, the factors of physical and numerical nature in considered models have been analysed. Many of those factors have not been qualitatively studied yet. In each case, the effect of modelling factors on the results obtained the in time domain has been estimated. In such a way, a specific catalogue of good practices for users of Monte Carlo simulation codes has been created. Many conclusions are valid also in case of deterministic tools. Tab. 1 below presents a review of topics to be found in the series of articles composing the doctoral thesis. Table 1. Review of issues and problems considered in the series of articles comprised in the doctoral thesis.. Subject Modelling problems. A1 ✓. A2 ✓. A3 ✓. Numerical problems. ✓. ✓. ✓. Computational cost. ✓. ✓. A4 ✓. A5 ✓. A6. ✓ ✓. ✓. Optimization. ✓. Scale of millimeters. ✓. ✓. Scale of centimeters. ✓. ✓. Scale of meters. ✓. ✓. ✓. ✓. Equilibrium calculations Safety parameters. ✓. ✓. ✓. ✓. ✓ ✓. A8 ✓. A9 ✓. A10 A11 ✓ ✓. ✓. ✓. ✓. ✓. ✓. ✓. ✓. ✓. ✓. ✓. ✓. ✓. ✓. ✓. ✓. ✓. ✓. ✓. ✓. Transient states Good practices in simulations. A7 ✓. ✓. ✓. ✓. ✓. ✓. The aforementioned issues have been analyzed via numerical tests of various nuclear reactor models: from Pressurized Water Reactor (PWR), through High Temperature Reactor (HTR) to Sodium-cooled Fast Reactor (SFR). The results obtained are of general nature, even though most issues are at least partly dependent on neutron physics characteristics of a given core model. It seems that the biggest barrier for current applications of the Monte Carlo method in fuel cycle calculations are numerical oscillations in loosely coupled models of nuclear 16.

(22) systems (addressed in articles A1, A2, A3, A11). There are several techniques of reduction and elimination of this problem, e.g. the use of unconditionally stable coupling schemes, accounting for the presence and movement of control rods in a model or decreasing the time step length. In case of High Temperature Reactor model, the problem can be basically eliminated by accounting for the reactivity control (introduction of compensation rods), which brings asymmetry in the core. Unfortunately, in light water systems (e.g. PWR) the control is most often kept by the dilution of boric acid in the coolant. This makes the application of the predictor-corrector methodology necessary. The price for imposing stable burnup procedure is the increase of computational cost. In the context of whole core simulations, the burden may be unacceptable. Consequently, simplified geometry models become an alternative – those are free from numerical instabilities. Reactor models – both of whole core as well as fuel assembly or block – contain a series of assumptions, simplifications and approximations. The influence of those factors on the reliability of the obtained results has been quantitatively studied in article A4 for the PWR reactor and in A5 and A9 for the HTR system. Such elements as applied time and space discretization, accounting for reactivity control and nuclear data chosen for a model, allow to assess the quality of core physics representation. In addition, the obtained results help to make an estimation of eventual modelling error values (which in extreme cases can reach several tens of % in relative value). The assessment of the dominance ratio evolution as a function of reactor irradiation time at equilibrium state (presented in A8 and A11) provides valuable knowledge about changes of neutron source convergence speed. This information is meaningful both for computational techniques (increasing cost of reliable simulations) and for the safety of real nuclear systems. The problem is particularly important in case of large thermal reactors prevailing the industry of commercial reactors. Fuel consumption (measured in GWd/tHM burnup or %FIMA) is an indirect way of expressing irradiation time for reactor of constant power and overall fuel material utilization. Both quantities are commonly applied in case of measurements and simulations concerning fuel cycle and reactors economy. Consequently, it is useful to understand and distinguish between the burnup declared by a simulation code user and a numerical value obtained as an output of calculation (article A6). In general, these quantities do not have to be in accordance; a systematic phenomenon of numerical FIMA indicator underestimation is observed when the simplest coupling scheme is in use. On the other hand (as shown in the work A10) burnup and FIMA do not have to be strictly proportional to each other. The conversion coefficient between the considered quantities varies depending on the isotopic content of fresh fuel. Even in case of a fixed type of nuclear material and specific reactor model, the same amount of energy per mass unit can be generated from a various number of fissions (depending on the burnup). The results encourage careful comparative analyses and higher awareness in the field of fuel cycle research and development. The Monte Carlo neutron transport method indirectly affects the development of transient states modelling (as shown in A7). The example shows the application of criticality simulations to the estimation of reactivity variations induced by deformations of geometric core structure. The results are a reference for deterministic codes being developed, which are dedicated to studies on temporal system evolution at a time scale of hundreds of 17.

(23) milliseconds. The development of this class of programs is of major importance for better understanding of fast systems mechanics, also in the context of Generation 4 reactors [46]. This is an example of specific cooperation of various numerical tools – the Monte Carlo methodology supports deterministic codes and vice versa. Summing up, better understanding of reactor core evolution in the time domain is one of important targets in nuclear engineering. Due to the cost and complexity of these systems, intensive research and development using numerical tools is a necessity. The aim of this work is to indicate computational and modelling problems present during the application of the Monte Carlo method, and to find the solutions. Qualitative assessment of multiple modelling errors and their impact on the final results is another contribution of the performed research. The final output of the studies is better understanding of simulation codes being used and a more complete definition of a time domain in the reactor research (Fig. 14). The articles comprised in the doctoral thesis serve as a review of good (and bad) practices in the field of nuclear reactors Monte Carlo modelling.. Figure 14. Diagram presenting the articles which form the doctoral thesis and their location within various domains of time in modelling of nuclear reactors.. 18.

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(27) 6. Reprints This section contains reprints of articles A1-A11 constituting the doctoral thesis.. 22.

(28) 7. Attachments (Statements of Contribution) This section contains statements of my personal contribution to the considered articles. The documents give detailed information and are signed by co-authors.. 23 125.

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