DEVELOPMENT OF HIGH-FIDELITY MULTI- PHYSICS SYSTEM FOR LIGHT WATER REACTOR ANALYSIS

Transcription

1 The Pennsylvania State University The Graduate School College of Engineering DEVELOPMENT OF HIGH-FIDELITY MULTI- PHYSICS SYSTEM FOR LIGHT WATER REACTOR ANALYSIS A Dissertation in Nuclear Engineering by Jeffrey W. Magedanz 2013 Jeffrey W. Magedanz Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy December 2013

2 The dissertation of Jeffrey W. Magedanz was reviewed and approved* by the following: Maria Avramova Assistant Professor of Nuclear Engineering Dissertation Advisor Committee Co-Chair Kostadin Ivanov Distinguished Professor of Nuclear Engineering Committee Co-Chair Fan-Bill Cheung Professor of Mechanical and Nuclear Engineering Suzanne Shontz Adjunct Assistant Professor of Computer Science and Engineering Armin Seubert Gesellschaft für Anlagen- und Reaktorsicherheit (GRS) Special Member Arthur T. Motta Professor of Nuclear Engineering and Materials Science and Engineering Chair of Nuclear Engineering * Signatures are on file at the Graduate School. ii

3 Abstract Nuclear simulation codes are vital for optimizing the performance of reactors while ensuring that they remain within regulatory safety margins. These codes tend to specialize in one part of the physical phenomena of a reactor core, such as Reactor Physics, Thermal Hydraulics, or Fuel Performance. However, the overall behavior of the system depends on the synergistic interaction of all of these phenomena they cannot be entirely separated from each other. For example, the reactivity and thus rate of change of the power generation is affected by the fuel temperature and coolant density, and the power in turn deposits heat inside the fuel rods. Thus, in a code specializing in one field, parameters from the other fields have traditionally been determined by simplified models, or by boundary conditions. Given that such simplification is not always acceptable, especially for transient and accident analysis involving reactivity, coupled codes have become a common way of representing the feedback between fields the codes are combined in such a way that they exchange information. It is especially common to develop core simulators that consist of coupled neutron-kinetics (time-dependent Reactor Physics) and thermalhydraulics codes. The neutron-kinetics code calculates the neutron flux and the resulting power distribution in the core, which it passes to the thermal-hydraulics code, which in turn solves the heat transfer in the fuel rods, and the energy, momentum, and mass equations for each phase of the coolant, passing back to the neutron-kinetics code the feedback from the fuel and coolant conditions. The spatial exchange of feedback parameters currently is performed in a coarse-mesh framework based on assembly-wise neutronics model and channel-wise model in thermal-hydraulics. While current methods are adequate for many applications, there has been a tendency in recent years toward greater heterogeneity in reactor cores, due to the use of mixed-oxide (MOX) fuel, burnable absorbers, and longer cycles with consequently higher fuel burnup. The resulting asymmetry of the neutron flux and energy spectrum between regions with different compositions causes a need to account for the directional dependence of the neutron flux, instead of the traditional diffusion approximation. Furthermore, the presence of both MOX and high-burnup fuel in the core increases the complexity of the iii

4 heat conduction. The heat transfer properties of the fuel pellet change with irradiation, and the thermal and mechanical expansion of the pellet and cladding strongly affect the size of the gap between them, and its consequent thermal resistance. These operational tendencies require higher fidelity multi-physics modeling capabilities, and this need is addressed by the developments performed within this PhD research. The dissertation describes the development of a High-Fidelity Multi-Physics System for Light Water Reactor Analysis. It consists of three coupled codes CTF for Thermal Hydraulics, TORT-TD for Neutron Kinetics, and FRAPTRAN for Fuel Performance. It is meant to address these modeling challenges in three ways: (1) by resolving the state of the system at the level of each fuel pin, rather than homogenizing entire fuel assemblies, (2) by using the multi-group Discrete Ordinates method to account for the directional dependence of the neutron flux, and (3) by using a fuel-performance code, rather than a Thermal Hydraulics code's simplified fuel model, to account for the material behavior of the fuel and its feedback to the hydraulic and neutronic behavior of the system. While the first two are improvements, the third, the use of a fuel-performance code for feedback, constitutes an innovation in this PhD project. Also important to this work is the manner in which such coupling is written. While coupling involves combining codes into a single executable, they are usually still developed and maintained separately. It should thus be a design objective to minimize the changes to those codes, and keep the changes to each code free of dependence on the details of the other codes. This will ease the incorporation of new versions of the code into the coupling, as well as re-use of parts of the coupling to couple with different codes. In order to fulfill this objective, an interface for each code was created in the form of an object-oriented abstract data type. Object-oriented programming is an effective method for enforcing a separation between different parts of a program, and clarifying the communication between them. The interfaces enable the main program to control the codes in terms of high-level functionality. This differs from the established practice of a master/slave relationship, in which the slave code is incorporated into the master code as a set of subroutines. iv

5 While this PhD research continues previous work with a coupling between CTF and TORT-TD, it makes two major original contributions: (1) using a fuel-performance code, instead of a thermal-hydraulics code's simplified built-in models, to model the feedback from the fuel rods, and (2) the design of an object-oriented interface as an innovative method to interact with a coupled code in a high-level, easily-understandable manner. The resulting code system will serve as a tool to study the question of under what conditions, and to what extent, these higher-fidelity methods will provide benefits to reactor core analysis. v

6 Table of Contents List of Tables... ix List of Figures... x Acknowledgements... xiii 1 Introduction Nuclear Fuel Reactor Phenomena and Simulation Coupled Codes Purpose and Objectives of the Proposed PhD Research Outline of Dissertation Background Reactor Physics Neutron Transport Cross Sections Angular Discretization Reactor Kinetics Thermal Hydraulics Reactor types Conservation laws Flow regimes Boiling crisis and thermal hydraulic limits Thermal hydraulics codes Fuel Performance Fission product behavior Thermal conductivity Fission gas Thermal expansion, deformation, and mechanical behavior Gas gap Literature Review Coupled Codes Internal and external coupling Approaches Spatial coupling Numerics Fuel Temperature in Reactor Physics Sensitivity and uncertainty Modeling by Studsvik Scandpower Fuel Performance Simulation Limitations of traditional codes BISON Coupling of FRAPTRAN with GENFLO Coupling of AMPFuel with Denovo Description of Codes CTF Conservation equations vi

7 4.1.2 Meshing Flow regimes Heat conduction Surface heat flux Gap conductance Solution methods TORT-TD Original TORT Time-dependent version Original CTF/TORT-TD Coupling Coupling approach Spatial coupling Temporal coupling FRAPCON Thermal model Mechanical model Rod internal pressure and gas release Cladding corrosion FRAPTRAN Coolant conditions Thermal model Mechanical model Rod internal pressure and gas release Cladding corrosion Predictive Biases, Modifications, and Sensitivity in FRAPCON and FRAPTRAN 87 5 Description of Coupling Coupling Structure Code interfaces Main program Parallel FRAPTRAN rods Information Exchanged CTF-FRAPTRAN TORT-TD feedback Spatial Coupling FRAPTRAN rod surface Neutronic spatial coupling Temporal Coupling Steady-state mode Transient mode Code Modifications and Interaction with Interface CTF TORT-TD FRAPTRAN Summary Original CTF/TORT-TD Results PWR Models Description of Purdue Benchmark Steady-state model vii

8 6.1.3 Rod-ejection accident with assembly-lumped feedback Pinwise feedback BWR Assembly Model Model description Steady-state convergence Steady-state results Transient results Summary CTF/FRAPTRAN Reflood Results MT MT MT-6A CTF/TORT-TD/FRAPTRAN Model Descriptions FRAPCON FRAPTRAN CTF TORT-TD Steady-State Convergence CTF FRAPTRAN Rod Comparison Steady state Rod ejection accident Angular Discretization Comparison Steady state Rod ejection accident Summary Conclusions and Future Work References Appendix A Code Samples A.1. Steady-State Loop A.2. Feedback-Accessor Method A.3. Coupling Data Storage A.4. Feedback-Transfer Subroutine A.5. MPI Communication A.6. Changes to Original Source code Appendix B Sample Inputs B.1. CTF Input B.2. FRAPTRAN Input B.3. Coupling Input for 4x viii

9 List of Tables Table 2-1: Differences between PWR's and BWR's Table 6-1: Parameters for rod-ejection transient predicted by CTF/TORT-TD, from [22] Table 6-2: Difference due to pinwise feedback vs. assembly-lumped for HFP Table 6-3: REA power peak for pinwise feedback and assembly-lumped feedback Table 7-1: Calculation of single-phase vapor heat transfer coefficients from Dittus- Boelter correlation using conditions in coupled CTF Table 7-2: Calculation of film-boiling heat transfer coefficients from Groeneveld 5.9 correlation using conditions in standalone FRAPTRAN Table 8-1: As-manufactured cold fuel dimensions Table 8-2: Parameters input to FRAPCON for both fuel types Table 8-3: Nodalization of FRAPCON model for both fuel types Table 8-4: Parameters input to FRAPCON which differ between fuel types Table 8-5: Parameters input to FRAPTRAN for all pins Table 8-6: FRAPCON initialization time for each pin type Table 8-7: Nodalization in FRAPTRAN Table 8-8: Modeling options in CTF Table 8-9: CTF model boundary conditions Table 8-10: Sub-channel types Table 8-11: Sub-channel horizontal connection types Table 8-12: Axial model information Table 8-13: Fuel properties in CTF Table 8-14: TORT-TD model parameters Table 8-15: Axial nodalization Table 8-16: Cross section libraries used Table 8-17: Initial steady-state power Table 8-18: Results for multiplication factor Table 8-19: Runtime comparison for each code Table 8-20: Multiplication factor and rod worth (half-inserted) for different rod models Table 8-21: Runtime comparison for each code Table 8-22: Degrees of freedom in CTF pressure calculation and TORT-TD transport calculation Table 8-23: Multiplication factor for each angular discretization Table 8-24: Run time comparison Table 8-25: Multiplication factor and rod worth (half-inserted) for different angle orders Table 8-26: Runtime comparison for different angular discretizations ix

10 List of Figures Figure 2-1: Angles for Level Symmetric quadrature in one octant, from [7]. The numbers next to the points represent different groups of points with the same weighting factors. 17 Figure 3-1: Example of PWR cylindrical geometry fuel assembly mapping, from [1] Figure 4-1: Normal flow regimes, from [18] Figure 4-2: Hot wall flow regimes, from [18] Figure 4-3: Heat transfer regime selection logic, from [18] Figure 4-4: Organization of calls to TORT-TD within CTF Figure 4-5: FRAPCON iteration procedure Figure 5-1: Coupling interface structure Figure 5-2: Structure of communication between processes Figure 5-3: Illustration of FRAPTRAN rod connected to multiple CTF channels, and CTF channel connected to multiple FRAPTRAN rods Figure 5-4: Overlap-fraction-weighted mapping scheme Figure 5-5: Old coupling - each TORT-TD node on XY plane mapped to one CTF rod 97 Figure 5-6: New coupling - arbitrary mapping to CTF channels Figure 5-7: Linear interpolation at node centers Figure 5-8: Steady-state convergence procedure for alternating-steady-state mode in FRAPTRAN (left), and time-marching-convergence mode (right) Figure 5-9: Time step coupling scheme Figure 6-1: Configuration of quarter-core model, from [22] Figure 6-2: Core-average axial power distribution for Hot Full Power, from [22] Figure 6-3: Core-average axial power distribution for Hot Zero Power, from [22] Figure 6-4: 2x2 mini-core configuration, from [22] Figure 6-5: Total power during rod-ejection transient, from [22] Figure 6-6: Fuel temperature over time for rod-ejection transient, from [22] Figure 6-7: Moderator density over time for rod ejection transient, from [22] Figure 6-8: Fuel temperature axial distributions for assembly experiencing rod ejection, from [22] Figure 6-9: HFP relative fuel temperatures - pinwise feedback vs. assembly-lumped feedback Figure 6-10: HFP relative pin powers - pinwise feedback vs. assembly-lumped feedback Figure 6-11: Power spike during REA - pinwise feedback and assembly-lumped feedback Figure 6-12: Layout of ATRIUM-10 assembly, showing water rod, bypass, control blade (with absorbing part in purple), and partial-length fuel rods (in yellow), from [33] Figure 6-13: Subchannel type configuration, from [33] Figure 6-14: Layout of HELIOS model, from [33] Figure 6-15: Cross-section types and nodalization of TORT-TD model, from [33] Figure 6-16: Power distribution oscillation between steady-state iterations, from [33]. 119 Figure 6-17: Power distribution with convergence forcing method, from [33] Figure 6-18: Axial power distribution for controlled and uncontrolled cases, from [33]121 Figure 6-19: Axial coolant density distribution for controlled and uncontrolled cases, from [33] x

11 Figure 6-20: Radial power distribution at node 28 (109.1 cm), for uncontrolled (left) and controlled (right) cases, from [33] Figure 6-21: Radial power distribution at node 32 (129.4 cm), for uncontrolled (left) and controlled (right) cases, from [33] Figure 6-22: Transient inlet flow boundary condition, from [33] Figure 6-23: Total power over time for flow transient, from [33] Figure 7-1: Inner clad temperature at node 7 (elevation=78 in) and test level 13 (~72.5 in) Figure 7-2: Inner clad temperature at node 10 (elevation=114 in) and test level 17 (~118 in) Figure 7-3: Coolant temperatures at rod surface as used in FRAPTRAN Figure 7-4: Fuel-surface heat transfer coefficients as used in FRAPTRAN Figure 7-5: Axial temperature distributions early in the reflood (100 s) Figure 7-6: Axial temperature distributions later in the reflood (250 s) Figure 7-7: Inner clad temperature at node 7 (78 in) and average experimental clad temperature at 76.9 in Figure 7-8: Inner clad temperature at node 9 (102 in) and average experimental clad temperature at 97.9 in Figure 7-9: Inner clad temperature at node 6 (66 in) and experimental rod 2C at 61 in 132 Figure 7-10: Inner clad temperature at node 9 (102 in) and experimental rod 2B at 102 in Figure 7-11: Coolant temperature axial distribution at beginning of transient, as used in FRAPTRAN, for standalone and when coupled to CTF Figure 7-12: Rod surface heat transfer coefficients at each axial node at beginning of transient, as used in FRAPTRAN, for standalone and when coupled to CTF. Manuallycalculated results for he Dittus-Boelter and Groeneveld 5.9 correlations are shown for comparison Figure 7-13: Inner clad temperature compared between CTF and FRAPTRAN for 40 psi system pressure instead of 250 psi Figure 8-1: Subchannel and fuel rod configuration of 4x4 case Figure 8-2: Dependence of gap conductivity on burnup for axially-uniform power Figure 8-3: Time step size during rod ejection accident Figure 8-4: Arrangement of assembly types in TORT-TD model Figure 8-5: Control rod insertion during rod ejection accident Figure 8-6: Power during null transient for both convergence schemes Figure 8-7: Average moderator density during null transient for both convergence schemes Figure 8-8: Average fuel temperature during null transient for both convergence schemes Figure 8-9: Radial power distribution - FRAPTRAN rods Figure 8-10: Radial moderator density distribution - FRAPTRAN rods Figure 8-11: Radial fuel temperature distributions Figure 8-12: Relative radial power - CTF vs FRAPTRAN rods Figure 8-13: Temperature difference CTF vs. FRAPTRAN Figure 8-14: Axial power distribution Figure 8-15: Axial fuel temperature distribution xi

12 Figure 8-16: Axial relative power difference - CTF vs. FRAPTRAN rods Figure 8-17: Rod 6 (low-burnup MOX) fuel temperatures Figure 8-18: Rod 7 (high-burnup MOX) fuel surface temperature Figure 8-19: Rod 7 (high-burnup MOX) fuel centerline temperature Figure 8-20: Rod 7 (high-burnup MOX) average fuel temperature Figure 8-21: Rod 10 (low-burnup UO 2 ) fuel temperature Figure 8-22: Rod 11 (high-burnup UO 2 ) fuel surface temperature Figure 8-23: Rod 11 (high-burnup UO 2 ) fuel centerline temperature Figure 8-24: Rod 11 (high-burnup UO 2 ) average fuel temperature Figure 8-25: Rod 6 (low-burnup MOX) temperature drop in gap and fuel Figure 8-26: Rod 7 (high-burnup MOX) temperature drops Figure 8-27: Rod 10 (low-burnup UO 2 ) temperature drops Figure 8-28: Rod 11 (high-burnup UO 2 ) temperature drops Figure 8-29: Rod 6 (low-burnup MOX) gap conductivity Figure 8-30: Rod 7 (high-burnup MOX) gap conductivity Figure 8-31: Rod 10 (low-burnup UO 2 ) gap conductivity Figure 8-32: Rod 11 (high-burnup UO 2 ) gap conductivity Figure 8-33: Axial power distribution - HZP for all models Figure 8-34: Radial power distribution HZP for all cases Figure 8-35: Power spike during rod ejection Figure 8-36: Fuel temperature increase during power spike Figure 8-37: Axial power distribution at time = sec Figure 8-38: Axial fuel temperature distribution at time = sec Figure 8-39: Axial power relative difference at time = sec - CTF vs. FRAPTRAN rods Figure 8-40: Axial fuel temperature difference at time = sec - CTF vs. FRAPTRAN rods Figure 8-41: Radial power relative difference at time = sec - CTF vs. FRAPTRAN rods Figure 8-42: Radial fuel temperature difference at time = sec - CTF vs. FRAPTRAN rods Figure 8-43: Radial coolant temperature difference at time = sec - CTF vs. FRAPTRAN rods Figure 8-44: Radial power relative difference - diffusion vs. S Figure 8-45: Radial power relative difference S 2 vs. S Figure 8-46: Radial power relative difference S 4 vs. S Figure 8-47: Axial power relative difference from S 8 reference solution Figure 8-48: Axial power relative difference (HZP) S 4 vs. S Figure 8-49: Axial power relative difference (HZP) S 2 vs. S Figure 8-50: Axial power relative difference (HZP) diffusion vs. S Figure 8-51: Power spike for rod ejection accident Figure 8-52: Power increase for rod ejection accident with diffusion xii

13 Acknowledgements The author would like to extend his gratitude to all of those who made this PhD work possible. He would like to thank Distinguished Professor Kostadin Ivanov for is continual support and guidance, Assistant Professor Maria Avramova for her help with CTF, and Dr. Armin Seubert for his help with TORT-TD. The author would also like to thank Professor Arthur Motta, Professor Fan-Bill Cheung, and Adjunct Assistant Professor Suzanne Shontz, who reviewed and approved this dissertation. He would furthermore like to thank all his colleagues at the Reactor Dynamics and Fuel Management Group at the Pennsylvania State University, and at Gesellschaft für Anlagen- und Reaktorsicherheit in Garching, Germany, whose assistance and contributions were invaluable to this research. xiii

14 1 Introduction This research aims at the development of a system of software tools for the analysis of nuclear Light Water Reactors (LWRs). The need for more sophisticated reactor simulation, applied to the safety and efficient operation of the current fleet of nuclear reactors, has grown in recent years due to such factors as the use of mixed-oxide (MOX) fuel and burnable absorbers, as well as longer refueling cycles and higher burnups. These have impacts on the material behavior of the fuel rods, as well as the level of challenge in accurately modeling neutron transport in order to solve for the power distribution, by increasing the heterogeneity of the neutronic interaction properties (cross sections) of the fuel. 1.1 Nuclear Fuel Each LWR fuel assembly consists of an array of rods, each of which contains a stack of pellets made of uranium oxide (UO 2 ) or in some cases, mixed plutonium-uranium oxide (MOX). It is enclosed in a metal cladding, typically composed of a zirconium-based alloy such as Zircalloy-2 (used in BWRs) or Zircalloy-4 (used in PWRs). Between the cladding and pellet is a small gap which is filled with helium. The helium also fills plena at the two ends of the rod, where springs hold the fuel pellets in place. Because fuel rods are subject to burnup, radiation damage, high temperatures, and several phenomena that are influenced by these factors, such as deformation, fission gas release, and cladding oxidation, the modeling of their behavior is a very complex field. Fuel Performance is important for two reasons: Fuel failures, in which the cladding is breached and radioactive fission products may be released, must be avoided. Thus, it is necessary to be able to predict the circumstances in which it may occur. The thermal conductivity of the fuel and the pellet-clad gap may change, thus affecting the fuel temperature, which has an important feedback effect on the neutronic behavior of the reactor. 1

15 Fuel Performance codes are traditionally applied to the first of these. Part of the purpose of this research is to apply them to the second as current practice is to use much simpler models for the calculation of fuel temperature. The field of Fuel Performance will be described in greater detail in section Reactor Phenomena and Simulation Reactor simulation is highly multi-disciplinary, as it involves the interaction of a wide variety of physical phenomena, and their efficient numerical solution on a large scale. It may be split into roughly three fields: Neutronics (Reactor Physics) the solution of the neutron transport equation under steady-state and transient conditions, in order to find the power distribution Thermal-Hydraulics the solution of the coolant flow and heat transfer problem Fuel Performance the modeling of material behavior in the fuel, including swelling, stress, deformation, thermal expansion, fission gas release, and cladding corrosion It may be noted that there is some degree of overlap between Thermal Hydraulics and Fuel Performance. The simulations of the former must determine rod surface temperatures in order to calculate heat transfer to the coolant, and thus involve a heat conduction model within the fuel rods. Furthermore, there may be a model for the conductivity of the gap between the fuel and cladding, which can vary in thickness due to the expansion of both (see section 2.3.5). 2

16 The heat transfer solution is also vitally important to the Fuel Performance, not only in ensuring that the centerline temperature does not reach the melting point, but also because of a variety of effects, including thermal expansion and the impact of temperature on the release of gaseous fission products from the grains of fuel. 1.3 Coupled Codes As most reactor simulation codes currently specialize in one aspect of the reactor, it is common to couple them in order to account for feedback. Current coupled code systems for reactors simulation typically include two codes: Neutron Kinetics (NK) codes solve the time-dependent neutron transport problem. Thermal Hydraulics (TH) codes solve the fluid flow and heat transfer problem. Coupling may be internal or external [1]. With external coupling, the NK code contains a thermal hydraulic model, and the TH code contains all of the other relevant parts of the reactor system. Only a few variables, giving conditions at the inlet and outlet of the core, are passed between the two codes. Internal coupling involves modeling the core in the TH code. Many more variables, including the power, fuel temperature, and coolant conditions, must be passed between the two codes. This requires a great deal more code modification, but allows the TH code to determine feedback for the NK code, and thus allows greater accuracy. There are also two different approaches [1]: parallel processing coupling and serial integration coupling. Parallel coupling involves using a tool such as Parallel Virtual Machine (PVM) or Message Passing Interface (MPI) to pass information between the two codes, which are kept separate. With serial integration, on the other hand, the NK code is implemented as a routine in the TH code, and the two parts alternate. 1.4 Purpose and Objectives of the Proposed PhD Research Providing the technical basis for the long-term sustainability of the current Light-Water- Reactor (LWR)-based nuclear power generating capacity requires the creation of advanced simulation capabilities to determine safety, fuel failure and operability margins 3

17 of LWR cores. These developments are focused on the licensing and deployment of advanced fuel designs and extension of the life of current plants. Many of the related problems are being addressed, but important scientific questions remain and can be answered with further high-fidelity design and safety studies on actually existing and advanced LWR fuel designs. The phenomena of interest result from the synergistic interaction of various physical processes occurring at different spatial and time scales in LWR cores studied by different disciplines and need to be investigated in an integral fashion. Such an approach requires coupled multi-physics and multi-scale simulations performed in an efficient manner. The main point is that the aforementioned phenomena need to be understood mechanistically in order that their behavior can be predicted and safe economic operation can be ensured. LWRs design margins could be improved by integrated high-fidelity reactor core simulations, which involve modeling of neutronics, fluid and heat transfer (thermal-hydraulics), thermo-mechanics, and fuel behavior all with feedback effects. А code system capable of such simulations must take into account the coupling of different physical phenomena (multi-physics) at different spatial and time scales (multi-scale) during reactor operation and safety related transients. For example, the current tendency in the nuclear power industry towards high fuel burnup as well as the advanced fuels campaign mission for major further increases in fuel burnup is driven by the desire to maximize the utilization of resources and minimize the volume of radioactive waste. The challenges to achieve such extended burnups include developing a modeling capability and performing investigations to understand and predict key aspects of fuel and cladding performance. This dissertation describes the development and testing of a code system with advanced multi-physics capability for high fidelity predictions of existing and advanced LWR fuel. It combines the capabilities of four codes: CTF The Pennsylvania State University s (PSU) version of COBRA-TF, a transient subchannel thermal-hydraulics code. 4

18 TORT-TD a three-dimensional (3D) steady-state/transient neutron-kinetics code based on the discrete-ordinates method. FRAPCON-3 A steady-state Fuel Performance code used for calculating long term fuel behavior. FRAPTRAN The companion code of FRAPCON-3, used for short-term transient fuel behavior. The advanced capabilities of these codes will provide high-fidelity fuel assembly and core analyses at the pin and sub-pin levels (spatial scale) In regard to reactor core neutronics, current methodologies are mostly based on twogroup nodal (coarse-mesh) diffusion calculations using assembly-homogenized crosssections and assembly discontinuity factors that are obtained from unit assembly (reflective) lattice transport calculations. These methods are very efficient and accurate when applied to present-day LWR cores. But, as new, more complicated and highly heterogeneous reactor core designs are being introduced, the adequacy of the current nodal cross-section generation methodology is being challenged. The infinite lattice assumption used in the process of cross-section homogenization does not account for the actual inter-assembly interactions and is nowadays considered to be a major source of uncertainty in global core analysis. The assembly-homogenization error is considered by most reactor analysts as the most significant shortcoming of the standard methodology for core calculations. At present much international activity revolves around developing the so-called next generation core simulation methods that are based on explicit multigroup, heterogeneous, fine-mesh, transport-theory core calculations. In this regard a 3D multi-group discrete-ordinates (S N ) neutron-transport code that solves the steady-state and time-dependent multi-group Boltzmann transport equation with an arbitrary number of delayed neutron precursor groups has advantages compared to the Method of Characteristics (MOC) in terms of efficiency for direct 3D simulations especially for transient applications. In regard to reactor core thermal-hydraulics, the advances in fluid dynamics modeling provided by the Computational Fluid Dynamics (CFD) codes cannot be yet fully utilized 5

19 for 3D coupled reactor core transient simulations due to two major reasons: the extensive computational cost involved and the limitations in two-phase flow modeling. The subchannel codes, such as COBRA-TF, have advantages over CFD tools in terms of computational efficiency especially for 3D and transient applications. In regard to reactor core coupled neutronics/thermal-hydraulics, it has been demonstrated that there are many open issues about the accuracy/adequacy of the currently-used fuel rod models in sub-channel codes for coupled neutronics/thermal-hydraulic calculations. An example is the dynamic versus constant gap conductance modeling, which has proven to have significant effect on the calculated Doppler temperature and consequently on the multiplication factor. In the established benchmarks, usually, the gap conductivity for all the gaps is set to one value in all the rods (assemblies). However, in a real core, not all the assemblies are made of fresh fuel, and there are some once or twice burned assemblies. Those assemblies have lower gap conductivity and could be affecting the calculation done with the constant gap conductance model. Currently used sub-channels codes either perform traditional heat conduction calculations using pre-specified material properties or utilize greatly simplified fuel rod models. Therefore, in realistic calculations with feedback effects, Fuel Performance codes should be used for prediction of the fuel behavior during steady-state and transient conditions for high-fidelity calculations. The ability to achieve better Fuel Performance and burnup are directly related to the development of advanced tools for high-fidelity physics-based Fuel Performance modeling. Such tools aim to improve the understanding and prediction of nuclear fuel behavior. As it has been mentioned above, they include frameworks for investigating multi-physics feedback effects and coupling techniques. Higher-fidelity modeling may be needed, including of neutronic and thermal hydraulic behavior on smaller spatial scales (i.e. pin cells and subchannels), as well as greater angular resolution of the neutron flux. The development of physics-based predictive tools in order to prevent fuel failures, achieve higher burnups, and improve safety margins is highly dependent on such highfidelity modeling, as well as accurate coupling of Fuel Performance with neutronics and Thermal Hydraulics. Thus, it is important to answer the following questions: 6

20 1. What level of fidelity is necessary for thermal hydraulic and neutronic modeling for core-wide coupling with Fuel Performance models? 2. What are the requirements for the design of flexible coupling interfaces between Fuel Performance and other physical models in order to address challenges such as complex geometry and compatible resolution in coupling three-dimensional (3D) models (fuel/neutronics) with 1D or 2D models (flow)? 3. What is the impact of power and flow uncertainties on Fuel Performance simulations? The objective of this PhD research is to improve, develop, and validate the coupled code system based on CTF/TORT-TD. This involves not only comparing results to benchmarks, but also identifying characteristics of situations in which greater accuracy is gained from using a high-fidelity NK code such as TORT-TD, which uses Discrete Ordinates, as well as to study the effect of pin-by-pin thermal-hydraulic modeling (instead of lumped-assembly models). Furthermore, the Fuel Performance modeling capabilities of FRAPCON-3 and FRAPTRAN will be added to the system, and the results of the combined system will be compared to those obtained using CTF s less detailed models. 1.5 Outline of Dissertation This dissertation includes a total of nine chapters. Chapter 2 will give an overview of the background information in the field of Nuclear Engineering. This will include discussions of Reactor Physics, Thermal Hydraulics, and Fuel Performance. Chapter 3 reviews the literature regarding coupled codes, modeling of fuel temperature in Reactor Physics, and recent trends in Fuel Performance simulation and its coupling to other codes. Chapter 4 gives a description of the four codes listed above that are used in the High- Fidelity Multi-Physics System. 7

21 Chapter 5 describes the coupling. This includes discussions of the code structure, information exchanged, and the spatial and temporal coupling. Chapter 6 describes results that were obtained from a coupled version of CTF and TORT- TD that existed before this PhD work. Chapter 7 describes results that were obtained from several Loss-of-Coolant Accident (LOCA) simulations with CTF and FRAPTRAN, with a constant power distribution. These results do not have feedback from Neutron Kinetics. Chapter 8 describes results that were obtained from steady-state and transient calculations with CTF/TORT-TD/FRAPTRAN. It contrasts FRAPTRAN s rod modeling capability with simplified rods modeled in CTF, and also shows results for several different angular discretizations in TORT-TD. Chapter 9 gives the conclusions derived from this PhD research and proposes future work. 8

22 2 Background As previously noted, the simulation of nuclear reactors may be divided into three fields: Reactor Physics, Thermal Hydraulics, and Fuel Performance. Each of these, and their typical solution methods, will be detailed below. 2.1 Reactor Physics The neutron flux in a reactor is a vital quantity, as it determines reaction rates and thus both rates of fuel depletion and heat generation. This in turn makes it relevant to the efficient use of fuel, and to the operation of the reactor within safety limits. The solution of the neutron flux distribution within a reactor is the focus of the field of Reactor Physics. It requires analyzing the transport of neutrons by accounting for their birth, absorption, and scattering as a result of interactions with nuclei present within the reactor medium Neutron Transport The neutron population in a reactor is a function of time, location, energy (velocity), and angle. Its calculation must take into account streaming (the movement of neutrons along a free path of travel), the loss of neutrons due to absorption by nuclei, the effect of scattering from collisions with nuclei, and the birth of neutrons from fission or other sources. The transport equation [2] is r, E, Ωˆ, t t, E,, t, E, E,, t t Ωˆ r Ωˆ r r Ωˆ d Ω ˆ ' d E ' r, E ' E, Ω ˆ ' Ωˆ r, E ', Ω ˆ ', t s r, E, Ωˆ, t s, (2-1) where is the angular neutron flux, ˆΩ is the angle in question, v is the neutron velocity, r, E is the total neutron cross section for location r and energy E, t 9

23 ˆ ˆ s E E, Ω Ω is the neutron scattering cross section for energy E and angle Ωˆ multiplied by the share of neutrons that are scattered to energy E and angle ˆΩ, and is the neutron source, excluding the scattering source. The five terms in equation (2-1) respectively represent the rate of change of neutron flux at a point in space, the effect of neutron streaming, the loss of neutrons due to collisions, the source due to scattering, and the source due to emission of new neutrons. The equation is a variant of the Boltzmann transport equation, which describes the transport of molecules under the kinetic theory of gases [2]. However, the neutron transport equation is much easier to solve, being linear due to the fact that, unlike for a true gas, the neutrons may only collide with a fixed medium of nuclei. Equation (2-1) is known as the integral-differential form of the transport equation, and is based on a local neutron balance. However, a different form may be derived [3] by first re-writing it by combining the source and scattering terms, and writing the streaming term to emphasize the direction of neutron travel (time dependence will also be removed for simplicity. u ˆ, ˆ ˆ, ˆ ˆ, ˆ ˆ, ˆ t u u q u d du r Ω Ω r Ω Ω r Ω Ω r Ω Ω (2-2) u is the distance traveled in the direction of neutron travel. s r, E, Ωˆ, t q r uωω ˆ, ˆ is the neutron source in the given direction from scattering and fission combined. Equation (2-2) can then be integrated along the neutron path to yield [3], R ˆ, ˆ dr q R ˆ e rr Ω r Ω r Ω. 0 (2-3) This is known as the integral form of the equation. Instead of local balance at a particular point, it describes global balance along a particular direction. 10

24 2.1.2 Cross Sections Cross sections are a property of the medium through which neutrons travel, and describe the probability of interaction. There are two ways of describing cross sections either as microscopic or macroscopic. Microscopic cross sections apply to individual nuclei, and may be interpreted as the effective surface area that a neutron must hit in order to interact with an atom. Macroscopic cross sections are properties of materials, and may be calculated as the sum of the microscopic cross sections of all component nuclei, each multiplied by their nuclide density in the region. They may be interpreted as the probability of interaction per unit distance traveled by the neutron. Both cross section types are strongly dependent on neutron energy. As there are several ways that a neutron may interact with a nucleus, there are several types of cross sections representing the probabilities of each type of interaction. The most important of these are: Capture the neutron is absorbed and becomes a part of the nucleus. Fission the neutron is absorbed and causes the resulting unstable nucleus to fission. Scatter the neutron bounces off the nucleus into a new direction and new (usually lower) energy. Total the sum of capture, fission, and scatter the total probability of the neutron s current direction and energy being interrupted. In reactor analysis, the neutron flux is divided into discrete energy groups, and thus the cross sections must be averaged over these energy ranges. In order to preserve reaction rates, the shape of the neutron energy spectrum within the range must be known with some degree of accuracy. A group cross section is thus defined as [2]: g r g r, E r, g r, E de E de, (2-4) 11

25 where r is the spatial location, E is the neutron energy, r, E r, E is the cross section, and is the scalar flux, or total rate at which neutrons pass a point, per unit area, in all directions. Generation of cross sections is an important step in the process of reactor simulation. Full reactor cores cannot be modeled with their geometric complexity with enough energy groups to resolve all the details of the cross sections energy dependence, as this would have a prohibitive cost. Nevertheless, smaller numbers of energy groups cannot be used unless the spectrum is already known. Thus, calculations with larger numbers of energy groups are performed in simplified geometry, such as an individual pin in 1-D cylindrical geometry, or a 2-D fuel assembly. These calculations yield a spectrum which is used to evaluate the integral in (2-1), producing cross sections appropriate for smaller numbers of energy groups Angular Discretization There are a wide variety of numerical methods available for solution of the transport equation. These are based mainly on three discretization techniques [4]: finite differences or finite elements for differential terms, quadrature for integrals, and expansion methods. There are three formulations of the neutron transport equation: integro-differential, integral, and surface integral, of which the first two have been described above and are of most interest for reactor problems, whereas the last is of interest for problems in which only the angular fluxes entering and leaving a media are needed. Methods based on the integro-differential formulation are best for optically thick regions, and produces sparse matrices which are solved iteratively. The integral approach, on the other hand is best for optically thin regions, due to its focus on neutron streaming. It provides an exact 12

26 representation of geometry, but one limit on its applicability is that each geometry requires a specialized subroutine for the purpose of efficiency Spherical Harmonics The Spherical Harmonics method, known as P N, solves the integro-differential form of the transport equation by expanding the angular variable in terms of spherical harmonics as basis functions [3][4][5]. In 1D geometry, with a single angular variable, this reduces to Legendre polynomials. For example, in slab geometry, removing time and energy dependence, the flux can be given as 2l 1 x, l x Pl, 2 ll (2-5) where cos is the angle cosine with respect to the direction of geometry variation, x is the position, P l is the Legendre polynomial of order l, and l is the Legendre moment of order l. Another method for 1D geometry is the double- P N or DP N method [3], which splits the expansion for positive and negative direction cosines. The flux in this case is N 2l 1 l x Pl 2 1, 0 l0 x,. N 2l 1 l x Pl 2 1, 1 l0 (2-6) In general, PN 1 is compared to DP N 2 /2 as both have N flux moments. For most cases, especially optically-thick regions with large amounts of neutron diffusion, the PN 1 solution, with its higher Legendre order, is preferable. However, in regions with large variations in flux between the positive and negative directions, such as at boundaries 13

27 (especially vacuum boundaries), the DP N 2 solution is preferable due to its separate /2 treatment of the two directions. For higher-dimensional problems where there are two angular variables, the flux may be represented by an order-l set of spherical harmonic functions [5] as L l ˆ m m, ˆ l Yl r Ω r Ω, l0 ml (2-7) where m l is the flux moment of order l, m, and m Y l is the spherical harmonic function of order l, m, given by Y m l m cos m Pl cos, for m 0, m, sin m Pl cos, for m 0 (2-8) where m P l is the associated Legendre polynomial of order l, m. While the Spherical Harmonics method allows the transformation of the transport equation into a simple set of partial differential equations in terms of the flux moments, it is not accurate for low orders if there are high variations in the angular distribution of the flux [5]. Additionally, it accuracy increases only slightly with increasing orders, while the mathematical complexity increases quickly the number of terms for order N grows as N 1 2 [6]. Due to this complexity, the full P N method is rarely used outside of 1-D geometry. The Simplified P N, or SP N, method was created in order to reduce this complexity. It simplifies by starting from the P N equations in 1-D and changing all operators from 1-D to their 3-D equivalents, such as changing spatial derivatives to the gradient operator [6]. This results in a method that is easier to implement for higher orders in multiple dimensions, but does not converge to the exact solution in the limit as the order goes to infinity. 14

28 The SP N method results in a set of equations similar to the diffusion equation discussed below in As a result, it is often implemented in diffusion-based 3-D core solvers Discrete Ordinates The Discrete Ordinates [3][4][7] method tracks the neutron flux in a set of discrete directions, called S N, and is based on using Gaussian quadrature to integrate the total flux over all directions. The method is often referred to as S N. It is computationally efficient and has become the dominant method of solving the integro-differential form of the transport equation [3]. It will be covered in greater depth here than the other methods, as it is the method used by TORT-TD, the high-fidelity multi-physics system s Neutron Kinetics code. For each discrete direction n, the flux is defined as n, ˆ m r r Ω, (2-9) where Ω ˆ m is the angular unit vector. In order to integrate these angular fluxes to obtain the scalar flux, these directions are given weighting factors. In 1D geometry, the angles and weighting factors are chosen so that they exactly integrate an equivalent set of Legendre polynomials, either according to the P N or DP N method. In this way, an S N quadrature set yields a solution that is equivalent to a PN 1 or DP N 2 solution. /2 For quadrature based on PN 1, the angle cosines order Legendre polynomial: n are chosen as the zeros of the N th - N 0, 1,2,, P n N n (2-10) The weighting factors w n are then set according to 1 2 N wn Pl n l0, l 1,2,, N. n1 (2-11) 15

29 where l0 is the Kronecker delta, defined as 1 if the subscripts are equal, and 0 otherwise. For quadrature based on DP N, however, the angles are set by solving for the zeros of the order N /2 Legendre polynomial separately for both negative and positive direction cosines: N/2 N/2 The weighting factors are then given by P 2 1 0, n 1,2, N / 2 n P 2 1 0, n N / 2 1,, N (2-12) n N /2 n1 N wp nn/21 n l n l0 wp 2 1 n l n l (2-13) Given that quadratures based on PN 1 or DP N 2 yield solutions that are equivalent to /2 the corresponding S N solution, the same principle applies in that the former is better for optically thick, diffusion-dominated regions, whereas the latter is better near boundaries. For higher numbers of dimensions, a single direction cosine is insufficient to describe the angle, due to lack of symmetry with respect to the other angular variable. Generally, the angle Ω ˆ n is described in terms of These are constrained by n, n, and n, its cosines with respect to each axis n n n (2-14) A common choice for a full set of angles is Level Symmetric quadrature [3]. This method choses a set of directions that is symmetric with respect to 90 rotation in any axis. The order N of S N quadrature with this method describes the number of levels of angles from pole to pole, and thus N /2 is the number of levels in each octant. The result is set of NN 2 / 8 directions in each octant, arranged in a pattern as shown in Figure 2-1. An advantage of Level Symmetric quadrature is that if the number of dimensions is reduced such that there is only one angular variable, the same quadrature set may be used. 16

30 For a given direction cosine set, i, j, k, i, j, k 1,2,, N / 2, equation (2-14) becomes i j k (2-15) The cosines are also constrained by the following relations: i j1 k1 1 N /2 1 (2-16) (2-17) Given these constraints, the only degree of freedom is to choose 1. A lower value will lead to ordinates which are clustered near the poles, and a larger value will move them further away. Figure 2-1: Angles for Level Symmetric quadrature in one octant, from [7]. The numbers next to the points represent different groups of points with the same weighting factors. Given a Discrete Ordinates quadrature set, the flux is usually discretized by an approach known as the Diamond Difference method. The flux is a linear function from cell face to cell face, with the slope determined by the total cross section and by the total neutron source (scattering and fission) within the cell. The average flux within the cell is then the center value, linearly interpolated between the boundaries. For 1D geometry, for example the cell center flux for direction n is 1 i, n i1/2, n i1/2, n, 2 (2-18) where i 1/2, n and i 1/2, n are the cell boundary fluxes. 17

31 The Diamond Difference relation is then applied to the balance equation across a cell, which in Cartesian coordinates yields n i1/2, n i1/2, n ii i, n iqi, n, (2-19) where i is the width of node i, i is the total cross section in node i, and q in, is the total source for direction n in node i, including the scattering and fission source. The source is given by q ', S, i s n n i n i S N (2-20) where s n ' n is the scattering cross section from direction n to n, and S i is the combined fission and external source. Given that the source exists as an independent variable in equation (2-21), but depends on the fluxes, it must be solved iteratively. Such iterations on the source are called outer iterations [7], and pass through the energy groups from highest to lowest, sweeping across the mesh in the direction of neutron travel within each group. Any repetitive solution within an individual energy group, however, is called an inner iteration. Using equation (2-19), an algorithm may begin at one boundary of the problem domain, and sweep across in the direction of neutron travel. If the boundary condition is a vacuum boundary, no initial value for the incoming flux is necessary, and thus the system is easy to solve without iteration. For example, for 4 mesh cells and S 2 quadrature, the solution matrix has the form: 18

32 x q 1,1 1,1 x x 2,1 q 2,1 x x x 3,1 q3,1 x x x x 4,1 q4,1 x 1,2 q 1,2 x x 2,2 q2,2 x x x 3,2 q 3,2 x x x x 4,2 q 4,2 where the x s indicate nonzero matrix coefficients. The system of equations for such a problem with vacuum boundaries may thus be described by the equation where m 1 m Lψ q, (2-21) L is a lower-triangular coefficient matrix, m1 ψ is the flux solution vector for outer iteration m 1, and m q is the neutron source vector for outer iteration m. In the case of reflective boundaries, however, there is incoming flux at the boundaries of the problem, and the equation takes the form m 1 Lψ q m Uψ m, (2-22) where U is an upper-triangular matrix describing reflecting fluxes obtained from the previous iteration. 19

33 For more than one dimension, equation (2-21) or (2-22) is still applicable, and the sweeping procedure is similar. For each octant, the solution starts in the corresponding corner and sweeps in the direction of neutron travel. The cell center flux is then related to the incoming fluxes in multiple directions. For example, in 2D Cartesian geometry 2, 0 n, i1/2, j n, i, j n, i1/2, j n n, i1/2, j 2 n, i, j n, i1/2, j, n 0. 2, 0 n, i, j1/2 n, i, j n, i, j1/2 n 2, 0 n, i, j1/2 n, i, j n, i, j1/2 n (2-23) For problems in which there is a large amount of scattering within energy groups, the outer iterations may converge very slowly. Thus, acceleration methods become necessary. Two such methods are coarse mesh rebalance and synthetic acceleration. Coarse mesh rebalance takes advantage of the fact that the converged solution must obey the conservation of neutrons, and forces this on a modified solution at each iteration, in terms of large regions of the problem domain (i.e. a coarse mesh). This is done by multiplying by a constant, f m, for each region m, as follows: l fm l1 r, Ωˆ r, Ωˆ, r V, m (2-24) where l 1, ˆ l r Ω is the next iteration flux, r, Ω ˆ is the previous iteration unmodified flux, and V m is the region with index m. For each volume, the method is derived by integrating the balance equation over each region m. The exiting currents plus neutrons absorbed must then be equal to the source plus the entering current: dj dj dv dvs, r m' m' V V m, m ' m, m ' m m (2-25) 20

34 where mm, ' is the surface between volumes m and m, J is the incoming partial current, J is the outgoing partial current, r is the cross section for removal from the energy group (absorption plus scattering to other groups), is the scalar flux, and S is the group neutron source (fission plus scattering from other groups). In order to enforce the constraint in equation (2-25), (2-24) is applied as follows: V l l l dv r dj f m dj fm ' dvs m' m' V m m, m ' m, m ' m (2-26) This results in a system of equations relating each f m to the multiplying factors for the surrounding regions, f m '. By definition, the flux and partial currents are defined as l l d, ˆ Ω r Ω, (2-27) l ˆ l J dωω nˆ r, Ω ˆ, Ωn ˆ ˆ 0 l ˆ l J dω Ωnˆ r, Ω ˆ, Ωn ˆ ˆ 0 where ˆn is a unit vector normal to the surface. (2-28) (2-29) For the Discrete Ordinates method, equations (2-27) through (2-29) are calculated in terms of the directional fluxes using the quadrature formula. In the case of the partial currents, the dot product with the surface normal is the direction s cosine with the axis perpendicular to the surface. For 1D geometry: 21

35 J J 1 w l l i n n, i 2 n w l l, i1/2 n n n, i1/2 0 n w l l, i1/2 n n n, i1/2 0 (2-30) (2-31) (2-32) Thus, the equation that must be solved for f m in each region is A J f m1 l i1/2, i1/2 ii m1 V A J A J f m1 m im l l l i r, i i i1/2, i1/2 ii i1/2, i1/2 ii m A J f V S l i1/2, i1/2 ii m m im i i, (2-33) where A is the surface area between cells, and i 1/2 V i is the cell volume. The other quantities in (2-33) are as defined above. Another method of acceleration is the synthetic method, in which a lower-order solution is used to accelerate a higher-order solution. For transport methods, the diffusion approximation (described in section ) is used. To derive this method, the neutron balance equation may be written as where H S, (2-34) H Ωˆ s d Ω. (2-35) The low-order diffusion operator, on the other hand, is: 22

36 H D, L r (2-36) where D is the diffusion coefficient. If we re-write H as H H H L, and integrate (2-34) over angles, we get L d Ω H L H HL S. (2-37) Rearranging and noting that the diffusion operator H L is independent of direction, we get L Ω L. (2-38) H S d H H From this, the acceleration scheme is derived as l 1 l HL S d Ω H HL. (2-39) From this, the accelerated scalar flux, used for the updated source in the next iteration, can be written explicitly as. l1 l 1 l l H L s (2-40) Although solving the system of equations that result from (2-40) produces a larger matrix than (2-33), this matrix is linear (i.e. independent of iteration), whereas coarse-mesh rebalance produces a new matrix with each iteration. Also, the synthetic method generally yields better acceleration than coarse-mesh rebalance. However, for a very fine mesh, synthetic acceleration may become unstable, due to incompatibilities between the spatial discretization of the diffusion and Discrete Ordinates methods. 23

37 Method of Characteristics The Method of Characteristics (MOC) [4] is similar to Discrete Ordinates in that it contains a discrete set of directions. It involves solving for the flux along characteristic lines that run through the problem domain, and is an integral method. Equation (2-3) can be discretized across a region with a constant source as tl Q e 1 e tl t, (2-41) where is the angular flux entering the region, is the angular flux exiting the region along the same line, Q is the source from scattering and fission within the region, and L is the length of the track through the region. The Method of Characteristics can produce very accurate solutions due to its handling of neutron streaming, its ability to easily handle anisotropic scattering, and its ability to better resolve complex geometries. However, it is also very computationally expensive, due to the large number of neutron tracks that are required. Thus, it is more commonly used in pins and fuel assemblies to calculate cross sections Collision Probabilities The Collision Probabilities method [3][4] is a common integral method which removes angular dependence by integrating over all angle for neutrons which originate from a given region. It is based on calculating the probability that a neutron produced in one region will have its first collision in another given region. This yields a set of equations relating only the scalar flux in each region, without losing accuracy due to angular approximation. However, it is only first order accurate in space, and produces a dense matrix. Because of this, it is only used for small spatial domains. 24

38 Diffusion approximation Equation (2-1) can be integrated over all angle to yield the neutron continuity equation, which is the balance of neutrons for all directions [2], to yield 1 J t r, E r, E, t v t 0 s r,, r,, de E E E t s E t (2-42) where by definition,,,, ˆ, r E t E t d ˆ r Ω' Ω' and 4,,,,, JrEt ˆ E ˆ t d ˆ Ω r Ω Ω. 4 Although equation (2-42) eliminates the angular dependence of the flux, it complications remain in the neutron current J, which has no simple relationship with the scalar flux. However, a common approximation is to model the current using Fick s law for diffusion:, E, t D, E, t J r r (2-43) After a series of steps [2] that involve the assumption of linearly anisotropic scattering and apply the P 1 approximation, it can be shown that a reasonable formula for the diffusion coefficient is D r, E 1 1 3,, t r E r E 0 s, (2-44) where 0 is the average scattering angle cosine. The diffusion method is applicable for regions in which Fick s law is valid, i.e. they are much larger than the mean free path of the neutrons. It loses accuracy, however, near 25

39 boundaries, strong absorbers, or other strong heterogeneities, as these cause the flux to become much more angle-dependent. Generally, transport methods, i.e. methods which deal with the angular dependence of the flux, are used in simplified geometries to generate cross-sections. Diffusion is then used in core simulators such as PARCS or SIMULATE, due the core s larger size in comparison with the neutron mean-free-path, and due to diffusion s much greater efficiency in terms of computation time Reactor Kinetics The time dependent behavior of a reactor depends on the balance of neutrons, and its study is known as reactor kinetics [2]. If more neutrons are produced than are absorbed at a given moment in time, the reactor power will increase, whereas it will decrease if fewer are produced. Thus, the neutron production must be carefully balanced during normal operation. The neutron multiplication depends not only on the composition of the core, but also on the current conditions. Parameters such as fuel temperature and coolant density have a strong impact on cross sections, and thus provide feedback to the kinetics. A small fraction of the neutrons (about 0.7%) are not released spontaneously by fission, but rather come from the decay of fission products. These are vital to reactor kinetics, as without them, reactor control would be impossible. Prompt neutrons have a lifetime of about 10-4 seconds, which would cause far too small of a time constant for changes in reactor power. However, the effect of delayed neutrons is to increase this average lifetime to about 0.1 seconds, making control possible. The time-dependent neutron source with delayed neutrons can be described as 1, s s C, f f ss i i i1 n (2-45) 26

40 where s f, ss is the steady-state neutron source, is the fraction of delayed neutrons, n is the number of groups of delayed neutrons, is the decay constant of delayed neutron precursor i, and i Ci is the concentration of delayed neutron precursor i. The concentration of delayed neutron precursor i is in turn changed over time by C i t,, C r t r r t, i i f (2-46) where i is the fractional yield of neutrons from delayed precursor i, and f r is the average of the fission cross section over the entire spectrum. 2.2 Thermal Hydraulics The field of Thermal Hydraulics encompasses the heat transfer and fluid flow aspects of a reactor. It is vitally important for LWR safety for two main reasons: In LWR s, the coolant also serves as the moderator, and thus its temperature and density provide a great deal of feedback to the neutronic behavior of the reactor. Many design limits, such as the necessity of avoiding fuel melting, claddingcoolant reaction, and excessive cladding strain, are primarily related to the thermal-hydraulic behavior of the core. 27

41 2.2.1 Reactor types There are two types of Light Water Reactors: Pressurized Water Reactors (PWR s) have little or no boiling due to their higher operating temperature, and use heat exchangers called Steam Generators to create steam in a secondary loop. Boiling Water Reactors (BWR s) generate steam directly in the core and have no secondary loop. Modeling and computation for BWR s tend to be more challenging, due to the complex and important role of boiling in the core dynamics. A more detailed list of differences between the two reactor types is given in Table 2-1. Table 2-1: Differences between PWR's and BWR's PWR Higher core operating pressure (~15.5 MPa) Generate steam for turbines in steam generator heat exchanger Minimal boiling in core Coolant passes once through core, and then to steam generator Have cross flow between fuel assemblies Typically 17 x 17 fuel rod array Typical fuel rod diameter 0.94 cm Cladding typically composed of Zircaloy-4 Uses cylindrical control rods inserted into guide tubes within assemblies Uses boric acid to control reactivity BWR Lower core operating pressure (~7.2 MPa) Generate steam for turbines directly in core Core exit void fraction ~70%, quality ~15% Coolant which does not boil is re-circulated Have channel box surrounding each fuel assembly Originally 8 x 8 fuel rod array some new fuels have 9 x 9 or 10 x 10 Typical fuel rod diameter 1.25 cm Cladding currently always composed of Zircaloy-2 Uses cruciform control rods inserted between fuel assemblies Boric acid only used for emergencies Uses recirculation pump speed to control core void and thus reactivity feedback 28

42 2.2.2 Conservation laws The thermal hydraulic behavior is governed by three conservation laws [8] those of mass, momentum and energy. An adequate solution of this system of equations will yield the velocity, density, and temperature distributions of the coolant within the system. The conservation of mass within a volume of phase k is given by: d dt 0 dv v v n ds v v n ds k k k s k k s Vk Akj Aks (2-47) where V k A kj A ks k v k is the volume in question; is the surface of the region in question; is the surface between phase k and other phases; is the density of phase k; is the velocity of phase k; v s n is the interface velocity; is a surface normal. In (2-47), the three terms respectively represent the rate of change of total mass in the volume, the rate of mass transfer to surrounding regions, and the net rate of mass transfer to other phases. The momentum balance within V k is given by: d dt v dv v v v n ds p I n ds g dv k k k k s k k k Vk Sk Sk Vk (2-48) where Sk k is the surface of V ; is the stress tensor of the fluid; k 29

43 p k I g is the pressure; is the Kronecker delta tensor; is the body force acting on the fluid. In (2-48), the four terms respectively represent the rate of change of total momentum in the volume, the rate of momentum convection through the surface, the force due to surface stress, and the force due to gravity. The energy balance within V k is given by: d dt Vk u dv u v v n ds q p I v n ds k k k k k s k k k k Vk Sk Sk g v dv q dv k k k Vk (2-49) where u k is the internal energy of phase k; q k q k is the interface heat flux; is the volumetric heat deposition rate within the fluid. In (2-19), the five terms respectively represent the rate of change of total heat energy in the volume, the net rate of heat transfer due to convection through the surface, the rate of heat addition due to surface heating and viscous dissipation, the energy added due to gravity, and the heat energy deposited directly within the fluid. Due to the extremely computationally intensive nature of Computational Fluid Dynamics (CFD) simulations, which resolve the velocity field in detail, it is necessary to integrate these equations over a spatial scale for which it is practical to solve them for reactor problems. This may be on the scale of a single pin, an assembly, or a larger region of the core, but in any case, every cell is generally in contact with a wall. Thus, the momentum and temperature are primarily affected by wall friction and wall heat flux, respectively, rather than by diffusion within the fluid. 30

44 The computational cells include multiple phases, typically gas and liquid. The gas can include vapor and non-condensable gas. The liquid may be divided into separate phases representing continuous liquid and droplets. Heat and momentum transfer to and from walls and other phases must be governed by heat transfer and friction coefficients, respectively, and these are highly empirical in nature. They must represent aspects of the physics and the configuration of the phases which are not resolved explicitly within the computational mesh. These coefficients are dependent on flow regimes, which are generalizations of the form of the two phases occupying the same space Flow regimes A flow regime [8] is a pattern in the distribution of gas and liquid within an enclosed space, typically a one dimensional channel. The regime that occurs for a given flow depends on the channel geometry and orientation with respect to gravity, the pressure, and the gas and liquid flow rates. Some common regimes are: Bubbly flow small bubbles dispersed in a continuous liquid. Slug flow large irregularly shaped bubbles with liquid slugs in between, as well as some smaller bubbles. Churn flow a more chaotically shaped variant of slug flow. Annular flow a vapor core with a layer of liquid in contact with the walls. Stratified flow liquid in the lower portion and gas in the upper portion of a horizontally oriented channel. Inverse annular flow a so called hot wall flow regime, in which wall heating causes vapor to cover the walls despite a continuous liquid core in the channel. This tends to occur during a reflood. Each type of regime generally has its own set of correlations for determining frictional and heat transfer coefficients. The regime may be chosen by a simulation code from a flow regime map. Traditionally, the map s regions are defined in terms of flow rates for 31

45 single-dimensional channels. However, for three-dimensional flow conditions, volumetric parameters such as the void fraction may be used Boiling crisis and thermal hydraulic limits The cladding temperature limit is not a limiting condition for normal reactor operation, as the cladding must only be maintained at a certain temperature slightly above the boiling point of water. However, during transients when the cladding surface heat flux may increase, or the flow of coolant may decrease, excessive temperatures may be reached which may cause the cladding to melt or react with the coolant. Such high temperatures may be caused by two different phenomena that are classified as types of Critical Heat Flux (CHF) or boiling crisis: Departure from Nucleate Boiling (DNB) and dryout. Departure from Nucleate Boiling can also be described as low void fraction CHF, and is a concern mainly in PWR s. As the wall of a heated channel becomes hotter, and the rate of bubble generation (nucleation) on the surface increases, a point is reached in which the bubbles merge together and form a layer of vapor which does not allow the liquid in the center of the channel to reach the surface. This dramatically reduces the value of the heat transfer coefficient, causing a large rise in the temperature of the fuel and cladding. This condition is the same as the inverse annular flow previously mentioned. Dryout is a form of CHF that occurs with higher void fractions, and is a concern primarily in BWR s. In normal BWR conditions, bubbles form on the walls and then separate and move toward the center of the channel, forming the annular flow regime at the top of the channel. However, if too much of the water boils off, the liquid layer covering the walls becomes extremely thin and thus unstable causing the vapor film to completely evaporate in some places, resulting in poor heat transfer and high temperatures. Unlike DNB, which is dependent mostly on the local heat flux, dryout is dependent mostly on the total heat flux integrated along the length of the channel, as it only occurs at a given point if the upstream conditions have sufficiently reduced the liquid layer thickness. Both forms of Critical Heat Flux are sources of important safety limitations, as regulations require a sufficient safety margin from CHF conditions. The limit for a PWR 32

46 is in terms of the Minimum Departure from Nucleate Boiling Ratio (MDNBR), which is the ratio of the heat flux that would cause Departure from Nucleate Boiling to the current peak heat flux. The limit for a BWR, on the other hand, is in terms of the Minimum Critical Power Ratio (MCPR), the ratio of total channel power that would cause dryout to the current peak total channel power Thermal hydraulics codes Thermal-hydraulic codes that are specialized for reactor simulation include system codes and sub-channel codes. System codes are used primarily for representing a large portion of the reactor system, including the pressure vessel, coolant loops, and specialized components such as pumps, steam generators (for PWRs), and steam separators (for BWRs). They are applied to transients such as Loss Of Coolant Accidents (LOCA), and include RELAP5 and TRACE. They advanced codes have a two-fluid model for tracking the momentum and state of the gas and liquid components in two phase flow. Sub-channel codes are used primarily for the reactor core itself, with boundary conditions specified for the inlet and outlet. They are used to model an arbitrary arrangement of coolant channels in the core or a portion of the core, with no fixed coordinate system. Instead, each connection between adjacent channels must be individually specified. Lateral momentum is assumed to not be laterally convected, which simplifies the momentum equations. Thus, the flow is not treated in a fully three-dimensional manner. Sub-channel analysis may be described as modeling the flow as primarily axial with simple exchange of fluid between adjacent channels [9]. Sub-channel codes primarily focus on the scale of individual pin cells. This scale may be represented by either rod-centered or coolant-centered channels. With rod-centered channels, each rod is connected to only one channel. Coolant-centered channels connect to four rods, while each rod connects to four channels. This adds complication, and increases the number of required channels, but is considered to be more accurate, as in this case cross-flow between adjacent channels passes through a narrow gap between rods. Nevertheless, rod-centered channels can be more accurate for annular flow regimes 33

47 and for Critical Heat Flux around a single rod [9]. However, sub-channel codes have been validated primarily using the coolant-centered approach. Computational Fluid Dynamics (CFD) codes can also be applied to reactor coolant flow. Unlike the two types of codes mentioned above, which use empirical friction factors and channel average velocities, CFD codes resolve the details of the velocity field using a much finer mesh. While they are more accurate and less empirical, explicitly representing more of the physics, CFD simulations have a prohibitive cost in terms of processing time for the purposes of most reactor analysis. They are most useful as a supplementary tool applied to small parts of the reactor geometry. 2.3 Fuel Performance Fuel Performance studies the changes that fuel undergoes under reactor conditions, including deformation of both pellet and cladding [11], changes in thermal properties [8][10], and release of fission gas from the pellet to the gas plenum [8][11]. Although the oxide has a relatively low thermal conductivity and uranium density, it is chosen over other forms due to its high melting point, ability to weather the effects of burnup, and chemical compatibility with the cladding Fission product behavior As fuel is irradiated, fuel atoms (Uranium or Plutonium) are consumed and replaced by a wide variety of fission products [12], which are themselves unstable and undergo a chain of decay processes before yielding a stable atom. These atoms may or may not form oxides, and may affect the oxygen potential with high burnup, thus affecting the tendency of the fuel to react with the cladding, which consists of stainless steel in fast reactors. As there are two fission product atoms per fuel atom consumed, the fuel tends to swell, causing long-term changes in the fuel pellet dimensions. Furthermore, as will be discussed in later sections, the fission products may affect the thermal properties of the fuel, and they may be gases which can be released from the grains of solid material and into the gas gap and plenum. 34

48 While swelling and other effects of fission product buildup are much lower for light water reactors than for fast reactors, due to lower burnup, there is a great deal of uncertainty due to simplifying assumptions that are made to when modeling the fuel s extremely complex behavior. Each point in the fuel is generally regarded as a closed system, with the exception of gases such as Xenon. However, in the fuel, there is a great deal of migration of atoms, especially due to the influence of a strong temperature gradient. Cesium, which is very chemically active, can act as a gas in hotter regions, and condense in cooler regions, although it is usually assumed to be a solid. This behavior also applies to Rubidium, Iodine, and Tellurium. Some fission products can gather into particles which have a size of less than one micrometer. Also, some elements, such as molybdenum, can exist either as an oxide or a metal. There is thus a great deal of uncertainty about the chemical state of the fission products, and thus uncertainty in the prediction of thermal and mechanical property changes caused by composition changes in the fuel Thermal conductivity The thermal conductivity of oxide fuels is dependent on a number of variables, especially the temperature, of which it is a decreasing function at temperatures below 2000 C [8]. Porosity has an effect of lowering the thermal conductivity. It is defined as P volume of pores ( Vp ) 1 total volume ( V ) TD (2-50) where TD is the theoretical density. The porosity of fresh fuel, which is made by sintering UO 2 powder, can be controlled by means of the sintering conditions to be anywhere between 80% and 98% [12]. Higher porosity decreases thermal conductivity, and is thus undesirable. However, a certain degree of porosity is desired to allow space for fission gases to escape, limiting the swelling. Fuel pellets may also crack, and this likewise causes a decrease in the density and thermal conductivity [8]. 35

49 Deviation from stoichiometry, whether by increasing or decreasing the ratio of oxygen to uranium atoms from 2:1, lowers the thermal conductivity. LWR fuel is produced to be stoichiometric [12], and although the burnup removes uranium atoms, the presence of fission products reduces this effect, such that the fuel has been found to be stoichiometric even a high burnups [10]. However, the addition of fission products slightly decreases the conductivity [8]. This is the net result of dissolved fission products decreasing the conductivity, and precipitated fission products increasing it [10]. In MOX fuel, the conductivity is slightly lower than that of pure UO 2, and is becomes lower the greater the proportion of PuO 2 [8]. During irradiation, the fuel also experiences radiation damage, which decreases the thermal conductivity. In addition to gamma, alpha, and beta radiation, fission products tend to displace a large number of atoms before coming to a rest after a fission event, leaving a track of vacancies and interstitials [12]. However, radiation damage peaks the first few minutes of exposure [10], as high temperatures promote annealing of these defects Fission gas The behavior of fission gas is one of the most challenging phenomena to model [12], and causes two complementary effects: swelling due to the retention of the gases in the fuel, and release of the fission gases to the gas plenum, which places increased pressure on the cladding and can lead to failure. Fission gas includes Xenon and Krypton [12]. Xenon has a much larger yield, but is subject to decay and neutron absorption [2], and thus has a short-term impact, whereas Krypton is the primary contributor to the long-term gas release behavior [12]. In the coolest parts of the fuel pellet, near the periphery, fission gases are nearly immobile within the molecular structure, and occupy no more space than a solid fission product [12]. However, at hotter temperatures (between 1300 K and 1900 K), the gas precipitates into bubbles which, due to their much lower density, cause the fuel to swell, 36

50 and some of it is released. When very high temperatures (above 1900 K) occur, all gas is released relatively quickly, and swelling from gas is minimal. Gas bubbles form within the fuel through a nucleation process which can be either homogenous or heterogeneous [12]. Homogeneous nucleation refers to the formation of bubbles at random locations where gas molecules happen to accumulate, whereas heterogeneous nucleation is when some feature of the material structure, i.e. a defect, contributes to the accumulation of gas at a certain location. It is thought that this primarily occurs in parts of fission fragment tracks which have many vacancies. Bubbles may then migrate through the fuel, either by random diffusion, or under the influence of temperature or stress gradients Thermal expansion, deformation, and mechanical behavior Many phenomena can change the dimensions of the fuel pellet and cladding. These may change dimensions by differing amounts, increasing or decreasing the size of the gas gap. There are strong temperature gradients within a reactor, and temperatures may change quickly during a transient. Thus, thermal expansion is a strong factor in the fuel dimensions. The change in volume of the fuel due to a temperature change may be given by the coefficient of thermal expansion: 1 V V T p (2-51) However, since pressure plays a negligible role if it is not extremely large, the constant pressure requirement may be dropped [12] Gas gap The gas gap [8] between the fuel and cladding plays an important role in determining the fuel temperature. At the beginning of life, it is filled with hydrogen, but with burnup, Xenon and Krypton are added. Both the pellet and cladding may change dimensions, due to thermal expansion, swelling, and stress, affecting the gap size. Furthermore, when the pellet cracks, the fragments may be relocated outward, moving them towards or in 37

51 contact with the cladding. This significantly affects the temperature jump between the cladding and fuel. If the gap closes and the pellet is in contact with the cladding, the thermal resistance becomes much lower, and there is conduction at the points of contact, in addition to conduction through the gas. Furthermore, Pellet-Clad Mechanical Interaction (PCMI) may occur, in which the expansion of the fuel pushes against the cladding, becoming the driving force of the deformation of the latter. 38

52 3 Literature Review 3.1 Coupled Codes Thermal Hydraulics and Neutron Kinetics codes were originally developed for different purposes, and thus were not connected [14]. However, advances in computing power have made possible fully-coupled 3D simulations, which allow more in-depth analysis where previously point kinetics or 1D kinetics models were used. This avoids the necessity of excessive conservatism, such as by choosing an unfavorable power distribution Internal and external coupling Coupling between thermal hydraulic and kinetics codes may be either internal or external [1]. External coupling was introduced first, and uses the TH code to model the balance of plant the part of the reactor system outside of the core. The kinetics code, on the other hand, has its own built-in fuel rod and coolant model. Only the conditions at the core inlet and outlet are passed between codes. Thus, external coupling is simpler and requires less modification. Internal coupling, however, uses the TH code to model the fuel rods and coolant within the core itself. This requires more modification, as local power and thermal-hydraulic conditions are exchanged between codes for the entire core. However, it also allows better modeling of the core Thermal Hydraulics, and greater stability, as the fluid conditions are solved for the core and balance-of-plant regions together, whereas in external coupling, the two solutions are made separately and not converged Approaches Two codes may be coupled either through parallel processing, or by serial integration [1]. The latter requires more code changes, and involves combining the codes to run sequentially. This is typically done by converting the kinetics code into a subroutine run by the Thermal Hydraulics code, although the reverse may also be done. 39

53 Parallel processing requires fewer changes, and involves running the two codes in parallel using Parallel Virtual Machine (PVM) or Message Passing Interface (MPI). In this approach, the codes do not directly access each other s subroutines and variables, but communicate by messages sent to another process Spatial coupling Spatial coupling between codes may be either fixed, in which generally one fuel assembly in the kinetics code corresponds to one thermal-hydraulic channel, or flexible, in which the input specifies a mapping scheme [1]. Generally, the thermal-hydraulic model of a BWR consists of a set of 1D channels (as there is no cross flow), while that of a PWR has a 3D model in cylindrical geometry. Each BWR channel, or PWR radial node, may be linked to one or several fuel assemblies. An example of a PWR mapping scheme is shown in Figure 3-1. Figure 3-1: Example of PWR cylindrical geometry fuel assembly mapping, from [1] There are different approaches for combining assemblies to have lumped feedback from a single thermal-hydraulic channel. One approach is called power flattering assemblies 40

54 with similar power are combined, to minimize the blurring of feedback. Additionally, one must choose a thermal-hydraulic resolution by considering both computation time, as well as the need for resolving feedback given the type of problem to be run Numerics Coupling numerics [1] may be explicit, implicit, or semi-implicit. Explicit coupling means that old-time values of the exchanged variables are used, without converging them to the new-time solution. Semi-implicit coupling uses some old-time and some new-time variables, while implicit coupling completely converges the solutions in the different codes. Current time-marching methods run the codes sequentially, making a fully implicit solution impossible. Because of this, time step sizes are limited, as each code must not advance too far in time without updated feedback from the other code. Implicit solutions require extreme modifications to both codes, in order for the entire solution to be calculated at once. This can result in matrices that are extremely challenging to solve, and this is the subject of ongoing research [15]. 3.2 Fuel Temperature in Reactor Physics Sensitivity and uncertainty The average fuel temperature is a parameter used by Reactor Physics codes which has a great deal of uncertainty [16]. It is dependent on a large number of aspects of Fuel Performance, such as density, gap conductance, cladding creep and corrosion, and thermal expansion, and changes through the life of the fuel as it is irradiated. This complexity is rarely modeled in Reactor Physics for core design. It is dependent thus not only on the instantaneous local power, but also the power history [17]. Additionally, volume-average temperatures cannot be measured only estimated and measurement of temperatures at points, such as the fuel centerline, is inherently intrusive, affecting the behavior and temperatures of the fuel [16]. 41

55 When considering measured data, there is uncertainty in the fuel temperature s influence on reactivity it is very difficult to isolate the direct effect of the temperature itself [16], as the temperature also affects reactivity through other parameters. For example, the temperature history has a large impact on isotopics, resulting in a positive reactivity effect from higher historical temperatures [17]. Fuel Performance codes are usually developed by fuel vendors, and consequently based on different sets of proprietary data [16]. Because of this, there is a significant disparity between the results of different codes, on the order of 100 K, resulting in reactivity differences of about pcm. However, there is some non-proprietary data available, such as from the Halden experiments. Fuel Performance codes were also developed mainly for licensing purposes, which requires certain conservative modeling assumptions for phenomena such as cladding strain, corrosion, and rod internal pressure [16]. This means that they tend to be less applicable to best-estimate simulation. However, the codes FRAPCON and FRAPTRAN have recently undergone modifications to reduce biases and improve best-estimate capabilities, as will be discussed below in section Modeling by Studsvik Scandpower Work at Studsvik Scandpower [17] has been directed at developing a consistent fuel temperature model for all of its applications, including lattice physics, core loading, steady-state analysis, transient safety, and simulator training for operators. Studsvik Scandpower s Core Management System has been updated with the change of its Fuel Performance code from INTERPIN-CS to INTERPIN-3, which contains updated models based on Halden data. These modeling changes include less fuel densification with burnup, more fuel conductivity degradation due to burnup, and a decrease in the roll of gaseous convection in the fuel-clad gap conductance. These updates resulted in fuel temperatures that increase linearly with burnup, rather than remaining flat. The lattice physics code CASMO-4 uses fuel temperatures that are independent of burnup, identical in all pins, and radially flat within each pin. It was shown that this leads to a ~6% overprediction of the reactivity feedback due to fuel temperature, which was 42

56 deemed acceptable as it is less than the 10% uncertainty that is generally assumed for Doppler coefficients. The change from INTERPIN-CS to INTERPIN-3 resulted in an increase in the feedback from fuel temperature. This results in greater damping of peaks and oscillations in the power. Fuel temperature data from INTERPIN-3 thus produced a more accurate result in SIMULATE-3 (a 3D core simulator) for a power runback transient after a main coolant pump trip, whereas the model using INTERPIN-CS data over-predicted oscillations in the axial flux imbalance. Unlike SIMULATE-3, which uses INTERPIN-generated tables of fuel temperatures given power and burnup, the transient 3D code SIMULATE-3K has its own fuel pin model, which is far simpler than the INTERPIN model. However, it has been noted that this simpler model generally gives similar results for fuel temperature. The exception is for pins that experience significant fission gas release, which occurs with sustained high temperatures. 3.3 Fuel Performance Simulation Limitations of traditional codes In view of the strong interactions between the various aspects of Fuel Performance and the resulting mathematical problems, several assumptions are introduced for rendering the set of equations amenable to an accurate numerical solution after a reasonable computation time. The most important assumption deals with the rod geometry. Most Fuel Performance codes consider a cylindrical geometry and assume azimuthal symmetry, i.e., they consider cylindrical UO2 fuel pellets and metallic cladding separated by a helium filled gap. Furthermore, the radial temperature gradient is considered to be much larger in comparison with the axial temperature gradient. As a result, the majority of the codes the cylindrical fuel rod is represented in a so-called one-and-a-half dimension, whereby all transport processes are solved in one (radial) dimension and the axial segments are coupled via balance equations. During operation the ceramic fuel pellets undergo cracking. This has a strong impact on the fuel behavior and affects the 43

57 assumptions made. In addition to that, the various strategies adopted in the Fuel Performance codes to deal with cracking will be outlined. In order to cope with large local deformations, such as those occurring in the cladding for instance during loss of coolant accidents, some codes either provide a separate tool to analyze local deformations and stresses in two dimensions by means of finite element calculations, or a full two dimensional mechanical computation or a three dimensional finite element analysis is proposed are utilized. More recently, Fuel Performance codes are currently being developed either from scratch, or on the basis of commercial software for three-dimensional finite element method (FEM) simulations. The starting point for their development is to take full advantage of the improvements in hardware and software, as well as in numerical techniques since the conventional Fuel Performance codes - currently used by safety authorities have been developed. Nevertheless, the increase in the number of dimensions goes along with a substantial increase of the computational costs and offers only an added value for the analysis of certain accident situations with large deformations. Indeed, for normal operational conditions, onedimensional codes are at least as accurate in view of the numerous uncertainties, for instance those brought about by the stochastic cracking and friction processes. The uncertainties in fuel rod powers and geometry (sphericity, pellet cracking, etc.) cannot be avoided by those multi-dimensional codes and ultimately may prevent them from replacing more simplified codes of today for licensing purposes. In any case, they will provide a deeper understanding and an excellent supporting tool for designing advanced nuclear fuels. Apart from the main limitations caused by the geometrical assumptions, there are other limitations related to the empirical nature of material property models and some of the physical models, such as the relocation of cracked fuel segments. As a consequence of the use of empirical models, it is impossible to extrapolate the material properties and models beyond the range of operating conditions in which they have been fitted. Hence, when advanced materials are to be designed for innovative reactor types, new codes have to be developed for each combination. The required amount of experimental data is both time consuming and expensive. With the advent of advanced software and hardware 44

58 many attempts have been made to introduce more physics-based models that enable extrapolation to a large range of operating conditions, and to cover a large set of potential materials. This has led to the development of meso-scopic models, micro-scopic models and even simulations tools at the scale of the electronic structure. Each simulation tool requires specific experimental data for their development and validation. There are now attempts to couple the various simulations tools in what is commonly referred to as the multi-scale approach BISON A new Fuel Performance code, BISON [25][26], is being developed at Idaho National Laboratory (INL). It is capable of 2-D or 3-D calculations using finite elements, and is based on INL s Multiphysics Object-Oriented Simulation Environment (MOOSE). BISON uses burnup and temperature-dependent fuel properties, and calculates swelling from burnup and fission gas, as well as phenomena such as cladding creep and pressure in the rod plenum and pellet-clad gap. The models are implemented such that all unknowns in the fuel are solved concurrently, so that there is no need for iteration or predictor-corrector methods. Using the 3-D finite element method, BISON is capable of modeling separate fuel pellets, and the contact points between pellets and between pellet and clad. It is thus able to show how the gap closes first near the ends of the pellets, due to the shape of the swelling. Additionally, it can model a missing pellet surface which occurs when a piece of the side of a fuel pellet is not present. This causes high temperatures in the region with a large pellet-clad gap, increasing the thermal expansion and swelling due to fission gas, and leading to high clad stress. Such Pellet-Clad Mechanical Interaction (PCMI) is a frequent cause of fuel failure. Fuel Performance is very dependent on phenomena that occur at very small length scales smaller than the node size such as radiation damage. For this reason, BISON has been designed to be coupled with codes that model smaller scales, such as MARMOT, another MOOSE-based finite element code, which is aimed at microstructure evolution. 45

59 3.3.3 Coupling of FRAPTRAN with GENFLO Previous work [28] at VTT Energy in Finland has created a coupling between FRAPTRAN and the high-efficiency non-iterative Thermal Hydraulics code GENFLO. The purpose was to improve the treatment of flow regimes and heat transfer coefficients used in FRAPTRAN, as the built-in models are much simpler than those in GENFLO. Thus, FRAPTRAN is the master code, and calls GENFLO. At each time step, GENFLO first calculates the coolant temperature and heat transfer coefficients, and FRAPTRAN then uses this to make the conduction and Fuel Performance calculation. Note that this is different than COBRA-TF, in which time steps begin by calculating conduction and heat transfer coefficients based on coolant conditions in the previous time step [29]. The coupling was tested for two transients: a Large Break Loss Of Coolant Accident (LBLOCA) based on the Loviisa plant, a VVER-440, and an instability event in a BWR. For the first, a comparison is made to FRAPTRAN standalone. There were key differences in the prediction of temperature versus time, due the difference in heat transfer coefficients. This is due to the fact that FRAPTRAN s built-in models do not accurately deal with rapidly changing flow regimes. In particular, the coupled results for the LBLOCA show the influx of a small amount of coolant to the uncovered core is more effective at cooling the core than what is predicted by FRAPTRAN standalone. Although it is assumed that the coupled version gives more accurate results because the flow regime treatment is more detailed, the paper recognizes that more study of the GENFLO s models, such as DNB and CPR correlations, is warranted. Furthermore, no comparison to experimental data was made. GENFLO FRAPTRAN has a number of limitations: As in FRAPTRAN (see 4.5), only a single rod may be modeled at a time. Crossflow cannot be taken into account. There is no feedback to the Neutron Kinetics calculation. This is determined by separate codes, and the power distribution is set as a boundary condition. 46

60 There is no feedback to the overall reactor coolant system, and thus no feedback to the pressure, flow, and enthalpy boundary conditions. The purpose of the high-fidelity system developed in this work differs from GENFLO FRAPTRAN in that it aims to use the Fuel Performance code to increase the accuracy and capabilities of the Thermal Hydraulics code, rather than vice versa. With coupling to CTF and TORT-TD, the first two limitations will not apply. However, given that CTF is currently limited to vertical channels, and does not have the ability to model complex components such as pumps and steam separators, the system will still be limited to situations in which the inlet and outlet boundary conditions can be known a priori Coupling of AMPFuel with Denovo The Advanced Multi-Physics Nuclear Fuel Performance (AMPFuel) code is a 3-D finiteelements Fuel Performance code which was designed to evaluate temperatures and strains in all pins within a nuclear fuel assembly. Recent work at Oak Ridge National Laboratory has created a coupling between AMPFuel and the multi-group Discrete Ordinates code SCALE/Denovo, and demonstrated it with a full-assembly calculation [27]. This coupling is aimed at the long-term behavior of the fuel during depletion, rather than transient analysis. It is motivated by the fact that uncertainties in the power distribution can have a significant impact on uncertainties in the performance of the fuel. The demonstration problem modeled a 17-by-17 PWR fuel assembly for 10 depletion steps. The Denovo model contained 10 million cells, 128 angles, and 44 energy groups, for a total 56 billion degrees of freedom. The AMPFuel model contained 3.3 billion spatial elements, 8 Gauss points per element, and 4 unknowns per point in the thermomechanical calculation, for a total of 105 billion degrees of freedom. This problem was solved in a total of 10 hours of runtime on 40,000 cores. Of this runtime, 45 minutes were spent on Denovo s neutronics calculation, 15 minutes on heat transfer, and 9 hours on the mechanics calculation. The result of this high resolution calculation showed the effects of asymmetries in power within the rods. Near discontinuities such as control rods, significant differences may 47

61 occur in the power between the two sides of a rod, resulting in temperature differences. This asymmetry in the temperature may then cause asymmetry in the thermal strain, resulting in bowing of the fuel pins. 48

62 4 Description of Codes The multi-physics code system, developed in the course of this work, directly incorporates three codes into a single executable: TORT-TD for Neutron Kinetics, CTF for Thermal Hydraulics, and FRAPTRAN for Fuel Performance. FRAPCON, which is applied to long-term steady-state Fuel Performance, is left separate and not modified, but is relevant to the system because it generates the initial conditions used in FRAPTRAN. Each of these four codes will be described below. 4.1 CTF COBRA-TF (Coolant Boiling in Rod Arrays Two Fluid), a subchannel-based Thermal Hydraulics code, was developed at Pacific Northwest National Laboratory (PNNL) for the U.S. Nuclear Regulatory Commission (NRC) for best-estimate analysis of design basis accidents and anticipated transients in LWRs [22]. CTF is PSU s improved version, which has in recent years undergone significant improvement in both efficiency and modeling of two-phase flow. CTF s modeling advantages include a three-field representation of vapor, continuous liquid, and liquid droplets, the ability to model fully 3 dimensional heat conduction, and dynamic gap conduction between the fuel pellet and cladding. Before the coupling with TORT-TD, CTF was previously coupled with NEM, PSU s core simulator, which solves the diffusion equations using the Nodal Expansion Method. 49

63 4.1.1 Conservation equations For two-phase flow, CTF contains a three field representation of the fluid, accounting for continuous liquid, gas, and entrained droplets. For each dimension, it solves four continuity equations (gas is split into vapor and non-condensable components), three momentum equations, and two energy equations (the continuous liquid and droplets are assumed to be in thermal equilibrium). The momentum equation for phase k is t U U U k k k k k k k g P M M M d T k k k k k k k k, (4-1) where k k U k g is the volume fraction, is the density, is the average velocity, is the gravity vector, P k M k is the pressure, is the viscous stress tensor, is the net rate of momentum change due to phase change, d M k T M k is the net rate of momentum change due to drag from other phases, and is the net rate of momentum change due to turbulent mixing and void drift. As the fluid channels stretch from wall to wall, the stress term takes the form only of the friction from walls and other phases the details of the flow field are not modeled. A large number of empirical correlations are used to determine these friction factors, and these correlations are flow regime dependent. 50

64 The conservation of mass is given by t T U W k k k k k k k, (4-2) where k T W k is the net rate of phase change to phase k, and is the net rate of transfer to phase k from turbulent mixing and void drift. The conservation of energy is given by k k hk k k hku k t P Q h q Q t i T k k k k ik k k, (4-3) where h k Q k i h k is the specific enthalpy of phase k, is the average conduction vector, is the specific saturation enthalpy of phase k, q ik is the heat transfer per unit volume to phase k at the phase interface, and T Q k is rate of heat transfer due to turbulent mixing and void drift. Each conservation equation contains terms which relate state of the phase in a given cell to the other phases, and to the adjacent cells. These are referred to as closure terms, as they give closure to the system of differential equations, allowing them to be solved. These closure terms are dependent upon the flow regime a description of the form of the phases. A complex logical system is responsible for selecting these phases, as well as their related boiling and heat transfer regimes. The closure terms can be divided into two different categories: those that are applied within a given cell relating the phases, and those that define relationships between a cell 51

65 and the adjacent cells, or adjacent heat-conducting walls. The terms relating different cells, referred to as the macro-mesh cell closure terms, are: Wall shear and form loss Inter-cell drag Wall heat transfer Turbulent mixing and void drift The micro mesh cell closure terms, which describe the relationships between phases in a cell, are: Interfacial drag Interfacial heat transfer Entrainment and de-entrainment Top-down reflood entrainment/de-entrainment Meshing The conservation equations are solved in terms of three different meshes. The continuity mesh is the set of cells in which scalars such as pressure and temperature are solved. The momentum, however, is tracked on a staggered mesh of cells centered at the boundaries between continuity cells. Likewise, the horizontal movement of the fluid is tracked at cells centered on the interface between cells in adjacent channels. Whereas the axial momentum is always transferred from cell to cell, this is not always the case with the horizontal momentum, as in the sub-channel formulation, the horizontal connections do not take into account their orientation or position relative to other horizontal connections. However, if desired, an option can be chosen the carry this horizontal momentum. This requires the code input to include a detailed description of the relative positions of gaps that carry momentum. Flow regimes are evaluated at momentum cells. However, the closure terms defining heat and mass transfer are needed in the continuity cells. For this purpose, the terms are 52

66 calculated from their respective correlations at the momentum cells and the values used in the continuity cells are determined by linear interpolation. The mesh uses subchannel coordinates, which means that the connections between adjacent channels, and between rods and channels, are defined individually, as there is no global coordinate system. This allows a great deal of flexibility in the creation of meshes. Meshing is usually done in one of three ways: Pin-by-pin, in which each rod is separately modeled and each coolant channel is the size of a single pin cell. Whole assembly, in which each rod and coolant channel represents an entire fuel assembly. Graduate mesh refinement, in which size is smaller closer to an important part of the problem domain, such as in the vicinity of an ejected rod. Pin-by-pin meshes may use either coolant-centered or rod-centered channels, as mentioned above in section The coolant-centered approach is more common and has more validation experience. However, the rod-centered channel is more easily used for coupling to a pin-by-pin neutronics model, so that each neutronics cell belongs to only one coolant channel, rather than four Flow regimes The flow regime is the pattern of flow of liquid and vapor within a channel. These are divided into two categories: normal and hot wall. The hot wall flow regimes are those where the wall temperature is high enough to exceed the Critical Heat Flux Temperature at which a vapor blanket forms. The normal flow regimes are: Small bubble flow Small to large bubble flow (slug flow) Churn/turbulent flow 53

67 Annular/mist flow The hot wall flow regimes are: Inverted annular flow Dispersed droplet flow Falling film flow Top deluge flow A key parameter determined by these flow regimes is the interfacial area between phases. In many cases, it is determined by a set of correlations for the current regime. However, for some flow regimes, the interfacial area transport equation is used, linking the current cell with the adjacent cells. This equation is da id, dt A U A A, i, d e i, E i, (4-4) where A id, is the concentration of droplet interfacial area per unit volume, U e is the entrained droplet velocity, A ie, is the net rate of entrained droplet formation, and A i, is the net rate of interfacial area formation due to phase change. Small bubble flow is dominated by continuous liquid with a small void fraction less than 0.2 below the void fractions at which bubbles begin to coalesce. The interfacial area between phases for this regime is based on modeling the bubble diameter. The small to large bubble regime, also known as slug flow, is present when the void fraction is between 0.2 and 0.5. It has larger bubbles that have coalesced, but are they are not touching. The interfacial area is calculated based on bubble diameter. It is assumed that smaller bubbles have a void fraction of up to 0.2, whereas the component of the void fraction exceeding this is assumed to consist of larger bubbles. 54

68 Churn/turbulent flow is a chaotic regime that represents the transition zone between the small to large bubble regime, and the annular/mist regime. The void fraction, between 0.5 and 0.8, is high enough that the bubbles touch and combine, but not high enough for a stable liquid film on the walls. Instead of having its own correlations, the closure terms for interfacial drag and heat transfer are calculated for the small to large bubble flow, and for annular/mist flow, and interpolated based on the void fraction between 0.5 and 0.8. Annular/mist flow consists of a liquid film on the walls, with a vapor core which may contain entrained droplets. The interfacial area is calculated as a sum of the film and droplet surface areas, in which the film is assumed to be cylindrical in shape, and the droplet surface area is determined from the droplet size. Figure 4-1: Normal flow regimes, from [18] Inverted annular flow is the post-critical-heat-flux regime that occurs when the wall temperature exceeds the Critical Heat Flux temperature, and there is a hot-wall flow regime in the cell above, but the void fraction is low. It contains a vapor film covering the walls, with a sub-cooled liquid core. The interfacial area is determined by assuming a cylindrical shape. 55

69 Dispersed droplet flow is a regime dominated by continuous vapor, with all liquid in the form of droplets. In this case, the interfacial area of the droplets is influenced by the surrounding cells and determined by solving the interfacial area transport equation. Falling film flow is characterized by a vapor moving upward while the liquid film on the wall moves downward. The void fraction must be above 0.8. While it is a hot wall regime (i.e. post-critical-heat-flux), it requires a normal regime in the cell above. For this regime, the film interfacial area assumes a cylindrical shape, while the droplet interfacial area is determined from the transport equation. The top deluge regime, like falling film, is characterized by a quench front coming from above, but has a void fraction below 0.8. The interfacial area is calculated by assuming the liquid is in the form of large spherical slugs. 56

70 Figure 4-2: Hot wall flow regimes, from [18] Heat conduction The heat conduction equation for fuel rods and other solid structures is solved at a series of nodes located at the centers of regions which may consist of different materials. However, half-width regions are used to place node centers at inside and outside surfaces of unheated conductors, as well as at the fuel pellet surface and cladding inside and outside surfaces. The spatial discretization of the conduction problem is given by T CpV Ki i T i i Ti Ki i Ti Ti Q i Vi, t i, 1 1, 1 1 (4-5) where i is the density of node i, C pi, is the specific heat of node i, 57

71 V i is the volume of node i, Q i is the heat deposition rate of node i, and Kii, 1 is the conductivity between nodes i and i 1, given by the inverse of the resistance between the two nodes: K ii, 1 R 1 R i, i1 i1, i, (4-6) where Rii, 1 refers to the resistance of the side of node i which is closest to node i 1. Thermal expansion is not taken into account in the distance between nodes, and the V term is evaluated at the cold state to avoid loss of mass due to change in density. All surfaces not in contact with the fluid are assumed to be adiabatic, while all surfaces in contact with the fluid are given heat transfer coefficients Surface heat flux The coolant temperature from the previous time step is used to calculate heat fluxes from the surface, which are then an explicit source of heating for the current time step coolant solution. However, given that wall temperature and heat flux heavily influence each other, they may oscillate unless implicitly coupled. This is done non-iteratively by applying the derivative of the heat transfer coefficient with respect to the wall temperature [30]. The surface heat flux to each fluid phase is then H q H T T T T T T n l n n n l l S l S l S l TS, (4-7) where H is the heat-transfer coefficient, the subscripts S and l refer to the values for the surface and phase l, respectively, and the superscript n refers to the value from the previous time step. 58

72 CTF contains a library of many correlations for heat transfer coefficients, with a system of selection logic. These are based on the following heat transfer regimes: Single phase liquid (SPL) Single phase vapor (SPV) Sub-cooled nucleate boiling (SCB) Saturated nucleate boiling (NB) Transition boiling (TRAN) Inverted annular film (IAFB) Dispersed droplet film boiling (IADF) Dispersed droplet deposition heat transfer (DFFB) Figure 4-3: Heat transfer regime selection logic, from [18] During rewetting of fuel that had previously become in contact with vapor, large differences in boiling regime can exist within a small distance. Thus, the ordinary nodalization becomes incapable of sufficiently resolving the boiling conditions. To 59

73 correct this, CTF contains a quench front model, which creates new nodes within the area in which quenching occurs Gap conductance The gas gap between the fuel pellet and cladding may be assigned a constant heat transfer coefficient in the input, or the code s dynamic gap conductance model may be used. This is based on Fuel Performance codes GAPCON and FRAP, and computes the heat transfer as a total of three components contributions from radiation, conduction through the gas gap, and conduction through points of contact, if the cladding and fuel are in contact: Hgap Hrad Hgas Hsolid (4-8) The heat flux due to radiation is calculated by a straightforward version of the radiation equation [30]: 1 1 A 1, qr SB 1 T1 T2 1 A2 2 (4-9) where SB is the Stefan-Boltzmann constant, A 1 and A 2 are the fuel and cladding surface areas, respectively, and 1 and 2 are the fuel and cladding emissivities, respectively. The conduction through the gas gap, based on GAPCON-2 [30], is H gas kgas t g g g 1 2, (4-10) where k gas is the gas conductivity, t g is the gas gap width, and 60

74 g 1 and g 2 are the fuel pellet and cladding jump distances, which are factors that compensate for the nonlinearity of the temperature near the wall. However, for the fuel in contact with the cladding, the gas conduction coefficient becomes [30] H gas kgas exp 8.8*10 Pi R R g g 4.2* , (4-11) where P i is the contact pressure, and R 1 and R 2 are the surface roughnesses of the fuel pellet and cladding, respectively Additionally, when the fuel and cladding are in contact, the heat transfer coefficient for the contact points is [30] H solid 5k P R, m int R H 1 R2 M 1 n (4-12) where k m 2kk k 1 2 k 1 2, k 1 and k 2 are the thermal conductivities of the fuel and cladding, respectively, P H int M is the ratio of the interface pressure to the Meyer hardness, and R 1 1 is the ratio of fuel surface roughness to the distance between peaks. Both the gap width and interface pressure must be calculated by the deformation model, which takes into account both thermal and mechanical stress. Burnup-dependent changes in dimensions are not taken into account, so the input gap width must take these into account. In addition to the thermal and mechanical stresses, the FRACAS-I mechanics 61

75 model (see section 4.4.2) from FRAPCON-2 is used to compute outward relocation of fragments due to cracking, and its impact on the gap width and fuel pellet thermal conductivity Solution methods CTF uses a form of the SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) in order to solve the conservation equations. This algorithm is based on the following procedure: 1. Guess the initial pressure. 2. Solve the momentum equation to obtain the velocities. 3. Use the continuity equation to calculate the pressure correction. 4. Calculate the corrected pressure by adding the correction to the initial guess. 5. Calculate a corrected velocity field with the corrected pressure. 6. Solve remaining discretized equations (such as energy). 7. Treat corrected pressure as guess for new iteration, and repeat the above steps until convergence is obtained. A major difference in the implementation in CTF is that there is only one iteration step 7 is omitted. Instead, if convergence criteria are not met, a new iteration is attempted with the time step size reduced by half, in order to reduce the change in pressure between time steps. When the momentum equations are solved in step 2, the transverse momentum is solved first, calculated one gap at a time, followed by the axial momentum. When solving the momentum equations, the new-time mass flows are used only in the time-discretization term and the shear. For all other terms, an explicit, old-time value is used. The mass and energy equations are set up within each cell as f x 0, (4-13) 62

76 where x is the solution vector for the mass and energy, and f represents the system of conservation equations. This is solved using Newton/Raphson iteration, which takes the form where 0 f x f x Df x x x, x is the previous-step solution, and 0 (4-14) Df x is the Jacobian matrix, which contains the partial derivatives of the equations with respect to each variable on which they depend. The system of equations depends on the phasic volume fractions, enthalpy, and pressure in both the same cell and the adjacent cells. This means that the Jacobian matrix has more columns than rows. It takes the form Df x 0 C C C C C C C l l l l l l l h 1 h P P P v v v v l e J i 1 i n E E E E E E E v v v v v v v h 1 h P P P v v v v l v J i 1 i n, E E E E E E E l l l l l l l h 1 h P P P v v v v l v J i 1 i n C C C C C C C e e e e e e e h 1 h P P P v v v v l v J i 1 i n C C C C C C C v v v v v v v h 1 h P P P v v v v l v J i 1 i n (4-15) where v is the vapor volumetric fraction (void fraction), h v is the vapor specific enthalpy, h l is the liquid specific enthalpy, C l, E v and C e, and C v are the continuous liquid, entrained droplet, and vapor masses, E l are the vapor and liquid energies, 63

77 P J is the pressure in the current cell, and P P i1 i n are the pressures in the adjacent cells. The shape of the Jacobian in (4-15) shows that (4-14) cannot be solved by itself for each cell, as there are more unknowns than equations. Instead, the pressure equation must be solved separately over all cells. The procedure is: 1. Use Gaussian forward elimination, so that the last row only contains the pressure terms. 2. Solve the pressure equation over all cells. 3. Use the pressure solution for backward substitution and solve for all terms in the continuity and energy equations. The pressure matrix may be solved in one of three different ways. For small problems, direct inversion via Gaussian elimination may be used. However, the cost quickly becomes prohibitive for large problems, for which the Gauss-Seidel iterative method may be used. For this method, the domain is divided into several simultaneous solution groups of subchannels which have a strong mutual influence. However, for non-stationary problems, especially transients with large variations of the mass flow, Gauss-Seidel may converge extremely slowly. Thus, Krylov methods have recently been added from the SPARSKIT-2 library, which is free software available from the internet. Four different options are available in CTF: BSGS (Bi-Conjugate Gradient Method Stabilized) with ILUT (Incomplete LU factorization) GMRES (Generalized Minimum RESidual method) with no preconditioner BSGS (Bi-Conjugate Gradient Method Stabilized) GMRES (Generalized Minimum RESidual method) with ILUT 64

78 4.2 TORT-TD Original TORT The TORT code [19] was developed primarily by W. A. Rhoades at Oak Ridge National Laboratory (ORNL) as a three-dimensional successor to the two-dimensional Discrete Ordinates code DORT. It is a part of ORNL s DOORS package. It calculates the flux or fluence of neutrons or photons from sources in the problem domain or boundaries. The set of discrete angles may be biased in order to focus on a direction that is of interest, or it may be symmetric. TORT is primarily applied to deep penetration problems, but can also be applied to reactor eigenvalue calculations. However, it does not allow time dependence, and thus cannot be applied to reactor transients. A number of fix-up options were considered to deal with negative fluxes, including linear zero where the flux is extrapolated to zero at the location a negative flux is encountered; as well as step methods, where the value of the flux at a cell boundary is assumed to be equal to the upstream cell center; and nodal methods, in which the flux within a cell is represented by an expansion method. It was decided that the linear zero method had not been sufficiently validated for three dimensions, and that the best approach was a weighting factor method. The flux at a cell center is a linear combination of the cell edge fluxes, i N N N ic ic N ic, (4-16) where i is the weighting factor, which if equal to 0.5 yields the standard linear extrapolation, N is the cell center flux, Ni cis the upstream cell boundary flux, and Ni cis the downstream cell boundary flux. 65

79 The weighting factor is then determined by an equation of the form 1/ 2 T SV BN CN A N A N i 1 2 1/ 2 V B C 1/ 2 A Nic s jd ke n m1/2 m ic ic (4-17) in which for TORT, s m 1, and n is an arbitrarily chosen factor between 0 and 1, which is recommended to be at least 0.3, but it is not recommended for fine tuning as there is little sensitivity. Within each computational cell, the cross sections and neutron source are assumed to be constant. Although the cross sections may in reality vary somewhat within a cell, this level of precision is not required as long as the cell size is sufficiently small. In order to properly integrate the angular flux, the quadrature set is chosen such that W 1 m (4-18) and W m m W m m W m m 0. (4-19) Thus, all directions will be weighted equally. However, as mentioned above, it is possible to bias the quadrature set so that some directions are resolved with a greater number of ordinates Time-dependent version The multi-group Discrete Ordinates code TORT-TD (TORT-Time-Dependent) [20] was developed at Gesellschaft für Anlagen und Reaktorsicherheit (GRS) based on TORT, but has been modified to handle time-dependence and reactor kinetics phenomena such as delayed neutrons. Additionally, it has a different input format which is designed for reactor calculations. It is applied to the pin-by-pin modeling of LWR s and HTR s, using homogenized-pin-cell cross sections that are produced by a Lattice Physics code. Before being coupled with CTF, TORT-TD was coupled with the thermal-hydraulic system code ATHLET. 66

80 In addition to TORT-TD s S N solver, it contains a diffusion solver which may be used for comparison, or to obtain a quick solution. The time-dependent version was developed by creating an implicit time discretization which implements a source term that includes delayed neutron precursors, as in equation (2-45), as well as the precursor equation, (2-46). The implicit time discretization gives unconditional stability. The time step size is determined by the thermal-hydraulics code. TORT-TD s time steps work by iteratively solving a series of fixed-source problems. These time steps generally take less computation time than a steady-state calculation, but this depends on the number of iterations, which in turn depends significantly on how close is the initial estimate of the flux. Taking into account time difference, the cross sections are modified as tot 1 g ' r g r, v t g (4-20) where tot g r is the unmodified cross section for neutron group g, v g is the average neutron velocity in group g, and t is the time step size. The fission yield for group g is 6 d g ' g 1 gl l l l l1, (4-21) where g is the steady-state fission yield in group g, is the delayed neutron fraction, d gl is the neutron yield in group g from delayed group l, l is the decay constant for delayed group l, 67

81 l is the concentration of neutron precursors in group l, and l is the fraction of neutrons originating from delayed group l. The time-dependent external neutron source (i.e. other than prompt fission neutrons) is then ˆ 6 ex ˆ 1 1 ˆ d,,, q r Ω q r Ω r Ω C r, g g g gl l l l vgt t l1 (4-22) where ex q g is the unmodified external neutron source, g is the flux, and C l is the concentration of neutron precursors in group l. Given the modified source, flux, and cross section, the equation to be solved for the flux at the next time step is g ' 4 1 ˆ g 1 gg ', g, ˆ Ω g ' r r, Ω dωˆ r Ω' ˆ Ωˆ r Ω' ˆ. 1 r ' r q r Ωˆ ' v, g fg ' g g g ' (4-23) In order to accelerate the convergence of each time step, the initial guess for the fluxes for the new time step is given by extrapolation assuming an exponential form of the change in flux. The new flux is approximated by ˆ r, Ω,, 1 r Ωˆ r Ω ˆ, g g e g (4-24) where the inverse reactor period is g 1 r, Ωˆ ln t 1, ˆ g g r Ω r, Ω ˆ, (4-25) 68

82 Within TORT-TD, the cross sections are assumed to be a function of between three and six thermal-hydraulic parameters. For the coupling with CTF, three are used fuel temperature, coolant density, and boron concentration. The cross sections between the sampling points for all of these parameters are calculated using either cubic splines or linear interpolation. TORT-TD also increases the spatial accuracy of its solution by the use of Pin-cell Discontinuity Factors (PDF s) from Generalized Equivalence Theory [21]. This is analogous to the Assembly Discontinuity Factors (ADF s) that are used for homogenized-assembly calculations, and is a method of using a special factor to relate the cell boundary fluxes to the average fluxes. Using such PDF s, the fluxes in two adjacent cells are related as where f f, i1 i1/2 i i1/2 (4-26) i 1/2 and i 1/2 f and i 1 i i. are the fluxes on the left and right side of the boundary, and f are the discontinuity factors on the right side of cell i 1 and left side of cell Equation (4-26) is based on modeling the true flux at the interface as f f. i1/2 i1 i1/2 i i1/2 The cell center flux is related to the boundary fluxes as (4-27) 1 i 2 i1/2 i1/2 fi fi, (4-28) while the center flux may be determined by 1 ixi f i xi i 2fi 1 i 1/2 q i fi, (4-29) 69

83 where i is the total cross section, x i is the cell width, is the angle cosine in relation to the x direction, and q i is the neutron source. The discontinuity factors used above are generated by the Lattice Physics code used to produce the pin-cell-homogenized cross sections. The discontinuity factor for a given boundary at a given cell is simply the ratio of the flux at that boundary to the average flux. 4.3 Original CTF/TORT-TD Coupling A coupling previously existed between TORT-TD and CTF, which was structured somewhat differently than that created in this research. It is based on the earlier coupling of COBRA-TF with nodal-diffusion codes QUABOX/CUBOX and NEM, and of TORT- TD with system code ATHLET. Thus, the variable names and comments in the coupling modules and subroutines make references to these other codes. There are four sets of values that are exchanged between the two codes. The power deposited for the heat conduction solution in each fuel rod in COBRA-TF must be received from the neutronic solution in TORT-TD. Likewise, each pin cell in TORT-TD must receive the fuel temperature, coolant temperature, and coolant density from the associated COBRA-TF rod and channel Coupling approach The method of coupling is serial integration, in with TORT-TD is implemented as a subroutine that is called by COBRA-TF. After each successful time step in COBRA-TF, the subroutine trans calls the subroutine interface_ctf_tort, which decides whether to run a TORT-TD calculation, and calls additional subroutines to transfer values from COBRA-TF variables to TORT-TD, and vice-versa. 70

84 No CTF time step Transient? No Steady-state converged? Yes Yes Transfer CTF results to TORT-TD Run TORT- TD Transfer TORT-TD results to CTF Figure 4-4: Organization of calls to TORT-TD within CTF Spatial coupling In the radial plane, each pin-cell in TORT-TD is linked to exactly one hydraulic channel in CTF, and at most one fuel rod. However, a single hydraulic channel and/or single fuel rod may be linked to many neutronic pin cells. The CTF may resolve the thermal-hydraulic solution on the scale of individual pins, or on a larger scale such as assembly-by-assembly. However, each fuel pin must be contained within a single hydraulic channel. This is the so-called rod-centered modeling approach. Thus, the conventional method of using coolant centered sub-channels is not allowed by this version of the coupling, as each pin-cell would overlap with four different coolant channels. The axial nodalizations of the two codes need not be identical. The variables are mapped using the average of the overlapping axial levels, weighted by the overlap fraction. The value of a variable at position x, y, z in a TORT-TD node is calculated as a zfrac a x, y, z x, y, i tort i ctf i, (4-30) 71

85 where each i represents an axial level in CTF that overlaps with the TORT-TD level, zfrac i is the fraction of the TORT-TD node that overlaps with CTF node i, and x, y, i a ctf is the value in CTF. The value of the power in a CTF rod is calculated as a zfrac a i x, y, i ctf i tort x, y i, (4-31) where the set of all xy, represents all the pin cells linked to the given CTF rod, each i represents an axial level in TORT-TD that overlaps with the CTF level, is the power in axial level i in position xy,, and zfrac is the fraction of the CTF node that overlaps with TORT-TD node i Temporal coupling There are two modes of coupling with respect to time-step advancement: steady-state and transient. Only for the second of these is TORT-TD run for every time step in CTF. For steady-state, CTF is run alone, moving forward in time until convergence is achieved, using constant power. The final solution is then transferred to TORT-TD, which performs a steady-state calculation. The new power is then transferred back to CTF, which calculates a new steady-state heat transfer and coolant solution. The power used by CTF is the average of the power calculated by TORT-TD in the previous two steady-state iterations. This averaging prevents oscillations that may occur between two different power shapes that alternate between iterations [22]. The iterations are repeated until the change in all parameters passed between the codes is less than the convergence criterion of This is the so-called nested loop iteration or fixed point iteration. It is not an implicit coupling scheme but approximates one by adding a loop to the coupled marching scheme used in transient calculations. After this, the transient calculation is performed, if specified in the input. x, y, i a tort For transients, each time step begins with a CTF solution. After this, TORT-TD makes a calculation for the same time period. The next time steps begins with CTF using the i 72

86 power calculated from the previous time step in TORT-TD. There is thus no iteration between the two codes within a single time step. 4.4 FRAPCON FRAPCON-3 and its companion code FRAPTRAN were developed by the NRC to simulate Fuel Performance under steady-state and transient conditions, respectively. While FRAPCON-3 is primarily used for auditing vendors Fuel Performance codes, FRAPTRAN has been used to develop licensing limits for design-basis accidents [24]. Both codes model a single fuel rod per input, and determine thermal-hydraulic conditions using an enthalpy-rise model. Heat conduction is calculated by a finite difference method, and material properties are determined correlations from the MATPRO package. FRAPCON-3 [23] calculates temperature, pressure, and deformation of fuel rods. It produces time-dependent results, and takes into account changing boundary conditions (i.e. power and coolant conditions), but is assumes all changes are gradual enough that a steady-state can be assumed at each point in time. It models phenomena that include: Heat conduction (within rod and from rod to coolant) Fuel-cladding mechanical interaction Fission gas release Densification and swelling Cladding oxidation Cladding deformation The code has been validated for burnups up to 62 gigawatt-days per metric ton of heavy metal. It is limited to oxide fuels (Uranium oxide or MOX), and water coolant (light water or heavy-water), but can handle gadolinia content and zirconium diboride coatings. The thermal and mechanical solutions are linked due to the heat transfer resistance of the pellet-clad gap. The dimensions of the pellet, along with many mechanical properties, are highly dependent on temperature. In turn, the temperature is dependent on the mechanical behavior, as the expansion and contraction of the pellet and cladding affect the size and thus thermal resistance of the gap. Because of this, at each time step, the thermal and 73

87 mechanical solutions must be determined iteratively until the temperature difference across the gap is converged. Additionally, the gas release from the fuel must be calculated, which creates feedback to the cladding dimension through the internal rod pressure. Thus, for each time step, there are two layers of iteration. Begin time step Compute fuel and clad deformation Compute rod temperatures no T gap converged? yes Gas release and rod internal pressure Gas pressure converged? yes no Figure 4-5: FRAPCON iteration procedure In order to obtain the thermal solution, it is necessary to determine the bulk fluid temperature at each axial height, as well as the difference between the bulk fluid and clad surface temperatures, the temperature drop across the crud and oxide layers, cladding, and fuel-clad gap, as well as the temperature distribution within the pellet Thermal model The coolant temperature is given by a simple enthalpy-rise model, in which the bulk fluid temperature at axial location z is given by z D0 q z Tb z Tin dz, C o PGAf (4-32) 74

88 where T in q'' z C P G is the inlet temperature, is the heat flux at axial location z, is the coolant heat capacity, is the coolant mass flux, A f D 0 is the coolant flow area, and is the outer cladding diameter. The relationship between bulk fluid temperature and rod surface temperature is given by f / T z q z h f, (4-33) where the heat transfer coefficient h f is given by the Dittus-Boelter correlation: h f 0.023k Re De Pr , (4-34) where k is the thermal conductivity of the fluid, D e is the coolant channel heated diameter, Re is the Reynolds number, and Pr is the Prandtl number. The temperature drop for the oxide and crud layers is a simple conduction model that assumes the thickness is negligible compared to the radius: q z z T z k, (4-35) 75

89 where z k is the oxide or crud layer thickness, and is the oxide or crud layer thermal conductivity. The conduction across the cladding, however, takes into account the cylindrical geometry: T q z r ln / / 0 r r k c o i c (4-36) where ri and ro are the inner and outer radii of the cladding, respectively, and k c is the thermal conductivity of the cladding. The heat transfer between the cladding and pellet, however, is more complex, and is given by the sum of components due to radiation, conduction through the gas, and conduction across contact points. The temperature difference is T gap q z h, (4-37) where h h h h. radiation gas solid The heat transfer coefficient due to radiation is 2 2 hradiation F Tfs T ci Tfs T ci, (4-38) which is originates from manipulation of the radiation equation equation: q F T T 4 4 radiation fs ci, (4-39) where F 1/ ef rfs / rci 1/ ec 1, 76

90 is the Stefan-Boltzmann constant, e f and e c are the fuel and cladding emissivities, respectively, T fs and T ci are the fuel surface and inner cladding temperatures, respectively, and r fs and r ci are the fuel surface and inner cladding radii, respectively. The heat transfer coefficient due to conduction across the gas is h gas kgas x, (4-40) where k gas is the gas thermal conductivity, and x is the effective gap width. The heat transfer coefficient due to conduction at contact points is determined by one of three expressions, depending on the contact pressure: h solid KmPrel Rmult if Prel RE Km if Prel 910 RE KmPrel 6 if Prel 910 RE 6, (4-41) where P rel is the ratio of the contact pressure to cladding Meyer hardness, K 2 K K / K K is the geometric mean conductivity, and K f and K are the m f c f c conductivities of the fuel and cladding, 2 2 R R R is the average of the roughness of the fuel and cladding, R f and R, f c c c R mult 333.3Prel if Prel , and 2.9 if Prel

91 E 7 exp ln Rf The heat conduction and generation are governed by the heat equation:. S k T, x T x nds S x dv, V (4-42) where k T, x represents the thermal conductivity and S x represents the heat source. This equation is solved by the finite difference method in the following form: T s 1 1 s v v m Tm klmlm Tm Tm krmrm Pf P Qlm lm Qrm rm, (4-43) where Tm 1 klm T m Tm 1,, and are the temperatures at three adjacent mesh points, and krm are the average thermal conductivities in the mesh intervals on the negative and positive sides of m, respectively, s lm P f and s rm are the mesh interval widths, is the axial power factor for a given node, P Qlm is the average rod power, and Qrm are the radial power factors for the mesh intervals, and v lm and v rm are the volumes of the mesh intervals. Whereas the axial power profile is specified in the input, the radial power is calculated internally, using diffusion theory to calculate the flux and thus power within each mesh interval. The thermal conductivity of Uranium oxide fuel pellets at 95% of theoretical density is K 95 1 A a gad BT f Bu Bu g Bu h T E F exp 2 T T exp 0.04, (4-44) 78

92 where T is the temperature, Bu is the burnup, f Bu Bu Bu g Bu is the effect of fission products on the crystal matrix, is the effect of radiation defects, e QT / ht Q 6380 K, is the effect of temperature on the annealing of irradiation defects, A m-k/w, a , gad is the weight fraction of gadolinium, B 2.46E-4 m-k/w/k, E 3.5E9 W-K/m, and F K. From this, the conductivity for arbitrary density may be calculated as Kd K 95 d / d, (4-45) where d is the fraction of theoretical density Mechanical model The recommended model for the mechanical behavior of the pellet and cladding is known as FRACAS-I. It is also called the rigid pellet model, as it neglects the effects of stress on deformation of the fuel pellets. The gap between pellet and cladding may be open or closed. It is said to be closed with the pellet and cladding are in contact, due to fuel swelling and thermal expansion, relocation of fuel fragments due to cracking, and creep of the cladding under high coolant pressure. When this occurs, the pellet s expansion drives the expansion of the cladding, 79

93 both in the radial and axial directions, as there is assumed to be no slipping. However, the cladding has no influence on the fuel s expansion. The heat and stress deformation of the fuel is obtained from the following form of Hooke s law: 1 P 1 1 dt E, (4-46) where 1 is the total deformation, P 1 is the plastic strain, 1 is the stress, E is the modulus of elasticity, and is the coefficient of thermal expansion. The full form of (4-44) is applied to the cladding. However, for the fuel pellet, only the last term on the right is applied, taking into account thermal expansion. As the cladding is relatively thin, the stress does not vary across its thickness. In addition to the strain, the strength of the cladding is calculated in order to determine the amount of stress required to cause it to fail, leading to release of fission products to the coolant. The strength is calculated as m n K 3 p e 10, (4-47) where K is the strength coefficient, is the strain rate, m p e is the strain rate exponent, is the true strain at maximum load, and 80

94 n is the strain-hardening exponent. Within the fuel pellet, each radial ring is allowed to expand without interference from other rings. The equation is N s d RH r i 1 T T i i Tref i i i1, (4-48) where R H Ti T i T ref r i s i d i is the hot radius, is the coefficient of thermal expansion for ring i, is the temperature for ring i, is the reference temperature, is the width of ring i at the reference temperature, is the swelling strain, and is the densification strain. Axial expansion is similarly calculated. The total length change is determined by the axial ring with the greatest length, as the inner rings, which are hotter, expand more. In most cases, for dished pellets, this is the first ring outside the dish radius Rod internal pressure and gas release The internal pressure of the fuel rod is assumed to follow the ideal gas law, and is calculated by MR P N V V V V P V V V V T T T T T T T T g ch cr dsh por rf i P n1 g ch cr dsh por rf i, (4-49) where M is the total number of moles of gas, 81

95 R is the universal gas coefficient, n1 N V and T follows: is the axial level, and are the volumes and temperatures of each respective region, with subscripts as P g plenum gap ch central hole cr cracks dsh pellet dishing por open porosity rf i roughness interface The gas that is released from the fuel may be determined by one of three models, which may be chosen by the user: ANS-5.4, which is useful for determining the release of short-lived fission products. FRAPFGR, which was developed at PNNL, and is useful for initializing transient gas-release for FRAPTRAN. A version of the Forsberg and Massih model which has been modified by PNNL, which handles stable fission gas release better than the other two models, is recommended by PNNL, and is used by default. The ANS-5.4 model has been implemented according to the specifications. The Forsberg-Massih model is based on diffusion of gasses in a spherical grain, according to the equation dc D t C r t t r dt,, (4-50) 82

96 where C r, t t Dt is the concentration of gas, is the rate of gas production, is the diffusion coefficient, and 2 d 2 d r 2. dr r dr The Forsberg-Massih model has been modified to use a low-temperature, high burnup model whenever this would predict a greater release: where 5 F 710 BU C, (4-51) F is the gas release fraction, BU is the local burnup in GWD/MTU, and C 0.01 BU 40 /10 for burnup above 40, and 0 otherwise. The Forsberg-Massih model has also been modified to allow some of the released gas to re-enter the fuel. The FRAPFGR model has been designed to produce accurate results not only for the gas release, but for the concentrations within the fuel grains and on their boundaries, as this is needed for accurate transient initial conditions for FRAPTRAN. It takes into account grain restructuring and porosity due to high burnups on the rim of the fuel pellets. In addition to fission gas release, FRAPCON also takes into account the release of nitrogen originating from the time of fabrication, and the production and release of helium. 83

97 4.4.4 Cladding corrosion The increase in oxide layer thickness is calculated using a different correlation for each cladding type. All are strong functions of the temperature at the clad-oxide interface. Additionally, FRAPCON calculates the amount of hydrogen released by oxidation that is deposited in the cladding. For PWR conditions, constant pickup fractions are used. However, for BWR conditions, burnup-dependent correlations are used. 4.5 FRAPTRAN FRAPTRAN [24] analyzes rod behavior when reactor conditions are rapidly changing. It does not model long-term phenomena such as the effects of burnup. The conditions of the rod at the beginning of a transient can be read from FRAPCON-3 output, or entered by the user. FRAPTRAN can be applied to the analysis of design-basis accidents, as well as evaluating new fuel designs and deciding on regulatory limits. For each time step, the thermal, mechanical, and gas release problems are solved iteratively in the same manner as for FRAPCON as shown in Begin time step Compute fuel and clad deformation Compute rod temperatures no T gap converged? yes Gas release and rod internal pressure Gas pressure converged? yes no Figure 4-5. However, there is no calculation of burnup related effects such as densification and swelling, and there are additional models to take into account transient and high temperature phenomena. 84

98 4.5.1 Coolant conditions At the beginning of each time step, the coolant conditions are calculated using inlet mass flow, enthalpy, and pressure, which are given in the input. The void fraction is calculated from the quality by assuming a slip ratio equal to unity, i.e. the liquid and vapor move at the same speed. The quality is derived from the enthalpy, which is calculated by H H 1 G rq t z L, (4-52) where is the density, G is the coolant mass flux, H is the coolant specific enthalpy, L is the flow area divided by the heat transfer perimeter, is the surface heat flux, q is the heat generation rate, and r is the fraction of heat deposited directly to the coolant. Unlike FRAPCON, in which the heat leaving through the cladding surface is always equal to the heat deposited in the fuel due to steady state, FRAPTRAN must calculate the surface heat flux at each time step using heat transfer coefficients and the current coolant and cladding surface temperatures. Correlations are available for a wide variety of coolant conditions, including critical heat flux and reflood. However, it is explicitly stated in the manual that FRAPTRAN is not intended as a replacement for subchannel codes, which model such phenomena more accurately. Another option is to specify the coolant enthalpy at each node as a function of time. This can be generated by a Thermal Hydraulics code, and is considered preferable to the builtin models. 85

99 4.5.2 Thermal model The thermal solution is calculated in a similar way to FRAPCON, with the same correlations for the conductivity of fuel, gap, and other materials. However, a timedependent form of the conduction equation is used, which is T Cp dv ktd s qdv. t V s V (4-53) There are two options available for the heat deposition. The entire time-dependent heat generation, due to fission and decay of fission products, may be input. Alternately, the time-dependent fission heat generation rate may be specified, and the decay heat can be calculated by the code, using the ANS standard decay heat model. Two options are available for the temperature of the plenum gas. It may be assumed to simply be 10 F greater than the adjacent coolant temperature, or calculated with a more complex model which takes into account heat exchange with the surrounding materials Mechanical model The mechanical properties and dimensional changes of the fuel and cladding are calculated in the same way as FRAPCON, except that long-term phenomena such as fission product swelling are not modeled due to the smaller time scale. FRAPTRAN additionally takes into account ballooning, a localized expansion of the cladding. This model is activated when the effective plastic strain in the cladding is greater than the instability strain. The BALON2 subroutine calculates the shape and extent of the deformation until either the hoop stress or strain reaches an empirical limit, at which point a cladding failure is predicted to occur Rod internal pressure and gas release The gas composition at the beginning of the transient may be input or read from FRAPCON. The transient release of gas may be input or calculated. Calculation of transient gas release requires FRAPCON to have been run with the FRAPFGR model in order to set up the location of gas within the pellets. 86

100 4.5.5 Cladding corrosion The corrosion that occurs at normal coolant temperatures is sufficiently slow that it does not need to be taken into account in FRAPTRAN. Thus, the oxide layer thickness is only modified when high-temperature corrosion occurs. The rate of creation of oxide is calculated by the Cathcart/Pawel and Baker/Just models at temperatures above 1073K and 1000K, respectively. These take the following form: dk dt 1 K Aexp B / RT, (4-54) where K T is the oxide layer thickness, is the temperature, and A, B, and R are constants. 4.6 Predictive Biases, Modifications, and Sensitivity in FRAPCON and FRAPTRAN Recent research [31] examined the sensitivity and biases of FRAPCON 3.3 and FRAPTRAN 1.3 relative to the available data, and recommended changes that were applied to versions 3.4 and 1.4, respectively. Both codes, while intended for best estimate calculations, and having some conservative options, were considered by some to be intrinsically conservative in their models. In order to ensure that the codes yield best estimate solutions that are not biased relative to the available data, many test cases were run for normal operating conditions and design basis accidents. It was found that overall, the codes tended to give unbiased best estimate solutions that are within the scatter of the available data. However, several corrections were made in the new versions of both codes in order to eliminate a few existing biases: Hydrogen pickup was formerly over-predicted in Zircalloy-2. Corrosion was formerly under-predicted for ZirloTM. Corrosion was formerly over-predicted for M5TM. 87

101 Thermal expansion of UO2 was formerly under-predicted between 2500 K and 3000 K. Steam conductivity model did not predict data well and was changed to ASME model. Cladding irradiation creep was formerly predicted to occur too rapidly. Additionally, the study determined which models and parameters showed the greatest amount of sensitivity. Low to moderate sensitivity to fabrication uncertainty was found. However, there is a great deal of sensitivity to perturbations in the thermal conductivity of the fuel, in particular for the rod internal pressure, centerline temperature, and cladding hoop strain. The rod internal pressure is sensitive to thermal conductivity due to the effect of temperature on gas release. The diffusivity of fission gas within the fuel also has a strong effect on the gas release and rod pressure. 88

102 5 Description of Coupling The high-fidelity multi-physics system consists of a single Fortran-based program, which incorporates the source code of CTF, TORT-TD, and FRAPTRAN. The structure of this program will be described below. The CTF-TORT-FRAPTRAN program uses serial integration, integrating all three codes into a single executable, and running subroutines from each code sequentially. However, the approach is innovative and different than the traditional master/slave relationship (used in the original CTF/TORT-TD coupling described in 4.3), in which the slave code is integrated as a subroutine into the master code. Instead, a new main program was written, which controls and communicates with each code. Additionally, the implementation is not purely serial integration, as it uses parallelism in the coupling with FRAPTRAN, for reasons that will be described below in section Coupling Structure A key objective in the design of the coupling was to keep the codes as separate as possible. This will facilitate the incorporation of future versions of the codes into the coupling, and prevent changes in one code from affecting the coupling with the other codes. Furthermore, future developers should be able to concentrate on one code, or one part of the top-level program structure, without first having to understand the details of how all three codes work. In order to accomplish this objective, an object-oriented approach was used for the design of the coupling. In object-oriented programming, programs are designed in terms of abstract data types (called classes in some language) which group together a set of variables and functions, called methods, which act upon those variables. The variables are encapsulated that is, they are protected from access from parts of the program outside of the abstract data type. All communication with the rest of the program occurs by means of the data type s public methods, which is referred to as its interface. This regulation of the flow of information makes the code more easily understandable, and prevents changes in a data type from affecting the rest of the code, and vice-versa. 89

103 5.1.1 Code interfaces While the existing source codes of CTF, FRAPTRAN, and TORT-TD are not objectoriented, the coupling was written so that the main program may control them in an object-oriented manner. To accomplish this, an abstract data type was written for each code, and is referred to as the code s interface. The interface data type s methods are given exclusive access to the variables and subroutines of its respective code. Thus, the main program is not concerned with the details of each code, including variable and subroutine names. Instead, it calls the interface s methods to perform high-level actions such as: Get feedback variables that are to be sent to other codes Set feedback variables that were obtained from other codes Tell the code to perform an action, such as to initialize and read input, run a steady-state calculation, or advance in time Obtain information about the model, such as geometry Obtain information about the state of the calculation This layer of abstraction between the main program and the codes involved in the coupling has several advantages: 1. A developer may focus on only one code at a time, and may not have to understand the other codes. 2. If a code is changed, only the interface of that particular code will need to be changed. 3. The sequence of actions performed by the coupled system is more clear, as the main program is written in terms of high-level actions. 4. The interface data types, which do not make reference to any other codes, may be re-used for coupling with a different set of codes. Also, the main program may be re-used with minimal changes for coupling with additional codes, if those codes can be given interfaces which work similarly to the existing ones. 90

104 Each interface also has a data transfer module, with which is shares access with the original code s subroutines. This is so that information passed from the interfaces may be accessed from within the code, without the code having to directly access the interface. Such bi-directional access between the code and its interface would not be possible, due to a circular reference between source-code files. Main Program CTF Interface FRAPTRAN Interface TORT-TD Interface CTF Original Code FRAPTRAN Original Code TORT-TD Original Code Data Transfer Module Data Transfer Module Data Transfer Module Figure 5-1: Coupling interface structure Main program The main program executes actions as follows: 1. Initialize all codes: For each code interface, the initialize method is run, and the interface calls its code s subroutines for reading input, allocating memory, and preparing for the calculation. The coupling description input is read along with CTF s input. Additionally, the interfaces and main program allocate memory for the variables exchanged between codes, and the main program obtains and stores the nodalization information for each code. 91

105 2. Iteratively calculate steady-states for each code until convergence is obtained (as described in 5.4.1). 3. Advance in time in all codes until the end of CTF s time intervals is reached (as described in 5.4.2). During both the steady state and transient loops, conditions are passed between codes in four parts: 1. Before running FRAPTRAN, the rod surface hydraulic conditions are obtained from the CTF interface, mapped to FRAPTRAN s nodalization, and then given to the FRAPTRAN interface. 2. After running FRAPTRAN, rod hydraulic feedback is obtained from the FRAPTRAN interface, mapped to CTF s nodalization, and then given to the FRAPTRAN interface. 3. Before running TORT-TD, the hydraulic feedback for each coupling unit is obtained from the CTF interface, and if a rod is present, the fuel temperature feedback is obtained either from the CTF or FRAPTRAN interface, depending on which code models the rod. The feedback is then mapped to TORT-TD s nodalization and passed to the TORT interface. 4. After running TORT-TD, for each coupling unit, if a rod is present, the power is obtained from the TORT interface, mapped to the nodalization of the code modeling the rod, and passed to the corresponding code interface Parallel FRAPTRAN rods The FRAPTRAN code was designed to simulate a single fuel rod at a time, and thus would not allow multiple rods with different conditions without radical changes to nearly every file of its source code. Thus, the multiple-rod version of the code was implemented using parallelism several MPI processes are used, and each contains a separate copy of FRAPTRAN and all its variables. All rods must communicate with the main program, which is contained in one process which will be referred to as the main process, without significant changes to either the main program or FRAPTRAN interface. To accomplish this, an additional abstract data 92

106 type was created, serving as a wrapper for the FRAPTRAN interface. This is referred to as the outer interface. Instead of the main program directly accessing the normal FRAPTRAN interface, it accesses this outer interface, which imitates the normal interface from the point of view of the main program, but instead of communicating directly with FRAPTRAN, it sends information using MPI messages. These messages are received by the other processes, which only contain FRAPTRAN rods, and the information is passed on to the normal interface. Thus, the part of the program concerned with sending and receiving MPI messages is concentrated in a small part of the source code the outer interface and the subroutine which is run in the processes other than the main one. The scheme is illustrated in Figure 5-2. Process 1 Process 2 Process 3 Main Program Outer Interface, Rod 3 Outer Interface, Rod 2 Interface, Rod 1 Outer Interface, Rod 1 Interface Interface, Rod 2 Interface, Rod 3 FRAPTRAN Code, Rod 1 FRAPTRAN Code, Rod 2 FRAPTRAN Code, Rod Information Exchanged CTF-FRAPTRAN Figure 5-2: Structure of communication between processes FRAPTRAN is used as an alternative to CTF s less-detailed built-in fuel rod modeling capability. FRAPTRAN contains a complete model of the fuel rod up to the clad surface, while CTF s channels model the coolant conditions at the clad surface. Thus, the clad surface is the location at which information is exchanged. 93

107 The following parameters are calculated in CTF and passed to FRAPTRAN: 1. Liquid temperature 2. Vapor temperature 3. Liquid convection heat transfer coefficient 4. Vapor convection heat transfer coefficient 5. Derivative of liquid convection coefficient with respect to clad temperature 6. Coolant pressure These values serve as boundary conditions while FRAPTRAN makes a Fuel Performance calculation. FRAPTRAN in turn calculates the following parameters, which are passed to CTF: 1. Clad temperature 2. Change in rod cross-sectional area from nominal state 3. Change in rod outer perimeter from nominal state 4. Rate of heat transfer from clad to liquid 5. Rate of heat transfer from clad to vapor For each time step, the given amount of heating is applied to the coolant channels coupled to the rod, while the changes in rod area and perimeter are applied to the flow area and wetted perimeter. The clad temperature is used for determining flow regimes, friction factors, and the heat transfer coefficients to be sent back to FRAPTRAN TORT-TD feedback When a TORT-TD model is included, it is given the following variables for cross-section feedback: 1. Coolant temperature 2. Coolant density 3. Coolant boron concentration 4. Fuel temperature 94

108 Note that the coolant temperature and density are also referred to as the moderator temperature and density, because in a LWR, the coolant also serves as a moderator. The coolant temperature and density are determined from CTF s hydraulic solution. However, given that CTF does not currently include a boron tracking model, a constant value of ppm is passed. This may be replaced when the coupling is implemented in a version of CTF that includes boron tracking. Additionally, it should be noted that, although the coolant temperature is passed by the coupling, it is not currently used in TORT-TD only the other three variables (listed in 4.2.2) are used. The fuel rods may be modeled in CTF or FRAPTRAN, or a combination thereof. Thus, the fuel temperature is obtained from whichever code models the rod. Furthermore, the only variable transferred from TORT-TD, the power distribution, can be given to both CTF and FRAPTRAN. 5.3 Spatial Coupling FRAPTRAN rod surface Each FRAPTRAN rod is coupled to one or more CTF hydraulic channels (see Figure 5-3), each of which is assigned a weighting factor. As FRAPTRAN cannot take into account azimuthal variation of the fuel rod and its surface conditions, these weighting factors are used to calculate an average for each thermal-hydraulic variable at the clad surface. Similarly, the weighting factors are used to split the heat flux from the clad surface, and the area change due to clad expansion, among the CTF channels. 95

109 Figure 5-3: Illustration of FRAPTRAN rod connected to multiple CTF channels, and CTF channel connected to multiple FRAPTRAN rods The FRAPTRAN rod may connect to one or more axial sections in CTF, but must fill the whole length of these sections. The axial nodalizations need not be identical. If they differ, the variables will be mapped using an overlap-fraction-weighted-average scheme. That is, given codes A and B, a feedback variable given to a node in code B will be an average of the overlapping nodes from code A, each weighted by the fraction of node in code B that it occupies. This is shown in Figure 5-4. y 3 Code A nodes x 3 Code B nodes y 4 y 4 = x 1y 1 + x 2 y 2 + x 3 y 3 x 4 y 2 x 2 x 4 x 1 y 1 Figure 5-4: Overlap-fraction-weighted mapping scheme 96

110 5.3.2 Neutronic spatial coupling Feedback to and from TORT is computed in terms of neutronic mapping units, over which feedback is homogenized in the x-y plane. Each mapping unit consists of a set of one or more TORT axial node stacks at given x-y positions, as well as several CTF channels with weighting factors, and at most one rod, modeled either in CTF or FRAPTRAN. This is shown in Figure 5-6, and contrasts to the old spatial coupling shown in Figure 5-5. In summary, in the old coupling scheme the rod centered subchannel modeling approach is utilized while in the new coupling scheme the coolant centered subchannel model may be used. Figure 5-5: Old coupling - each TORT-TD node on XY plane mapped to one CTF rod Figure 5-6: New coupling - arbitrary mapping to CTF channels 97

111 The axial problem domain in CTF must be the same size as the active fuel region in TORT-TD. In the reflector region, the inlet and outlet values of the feedback variables are used. As with the CTF-FRAPTRAN coupling, the axial nodalizations may differ. The total power in all x-y nodes is mapped to the fuel rod by the same overlap-fraction weighting as shown in Figure 5-4. However, the cross-section-feedback variables from both fuel and coolant are interpolated linearly at the location of TORT-TD s cell centers, as shown in Figure 5-7. Power is mapped using the overlap-fraction-weighting method because it is a global quantity which must be conserved. The linear-interpolation method, however, does not conserve global quantities, but yields more accurate local values. Generally, TORT-TD is expected to have many more axial nodes (about twice as many) than CTF and FRAPTRAN. Thus, interpolating the feedback variables will ensure that they have a smooth distribution over TORT-TD s axial nodes. Figure 5-7: Linear interpolation at node centers 98

112 5.4 Temporal Coupling All three codes may have different time step sizes, although the step end times are synchronized. In general, a CTF time step coincides with one or more FRAPTRAN time steps, while a TORT-TD time step coincides with one or more CTF time steps Steady-state mode During steady state mode, successive iterations are made on the power distribution in TORT-TD. Within each of these, the thermal-hydraulic and fuel-performance calculations are converged in one of two ways. The first is to make alternating steadystate calculations in CTF and FRAPTRAN, as shown on the left side of Figure 5-8. The other is to march CTF and FRAPTRAN forward in time, as during the transient calculation, until CTF reaches a converged solution based on heat balance. 99

113 CTF steady state CTF time step FRAPTRAN steady state FRAPTRAN time step FRAPTRAN Converged? no CTF Converged? no yes yes TORT-TD steady state TORT-TD steady state TORT-TD Converged? no TORT-TD Converged? no yes yes Steady state reached Steady state reached Figure 5-8: Steady-state convergence procedure for alternating-steady-state mode in FRAPTRAN (left), and time-marching-convergence mode (right) Transient mode During the transient calculation, the three codes successively makes time steps, each using the feedback from the previously-calculated conditions in the other codes, without any iteration between them (i.e. the coupling is numerically explicit). The time step loop is based on CTF s steps, and the following sequence is used: 1. CTF sets its time step size and determines the the thermal-hydraulic conditions at the FRAPTRAN rod surfaces, based on the previous time step s conditions. 2. FRAPTRAN advances in time by the same amount of time as the CTF step size. If its maximum step size is smaller than that in CTF, then the interval is divided into the least possible number of equal steps that satisfy the limit. 100

114 3. CTF performs is hydraulic time step calculation, using the above-calculated heat flux from the FRAPTRAN rod. 4. TORT-TD makes a time step if the time since its last step is close to the value of the TORT-TD step size set in the input. Alternatively, TORT-TD may make a time step with every CTF step if specified in the input. Figure 5-9: Time step coupling scheme 5.5 Code Modifications and Interaction with Interface CTF The most important changes to CTF are as follows: 1. In setin, the subroutine which reads input, card groups 15 and 16 are read, which define the coupling to FRAPTRAN and TORT-TD, respectively. 2. In cobra_tf.f, program cobra_tf is changed to subroutine cobra_tf. 3. In prep3d.f, the forcing functions that control transient boundary conditions are changed to only be used for the first second in the case of a steady-state calculation, and only after the first second in the case of a transient calculation. This allows the inlet mass flow to be ramped at the beginning of the steady-state 101

115 calculation, without affecting the beginning of the transient. Likewise, it prevents the boundary conditions from changing during the steady-state calculation. 4. In temp.f, the linear power of a rod is set as the power received from TORT- TD, if applicable. 5. In heat.f, a section is added which calculates heat transfer coefficients and fluid temperatures for the surface of coupled FRAPTRAN rods, copying the same logic and formulas as CTF uses to determine the values for its own rods. 6. In intfr.f, the clad temperatures of coupled FRAPTRAN rods are taken into account when calculating the flow regime. 7. In the subroutine trans, the time step loop is removed, so that calling the subroutine cobra_tf only initializes the code and reads input. 8. The subroutine time_step was split into two parts, using an argument to decide whether to (1) set the time step size and do the heat transfer calculation, or (2) do the hydraulic calculation. This allows heat transfer coefficients to be calculated before FRAPTRAN is run, while the heat transfer from the FRAPTRAN rods is implemented in CTF s hydraulic calculation. Given that CTF time steps may fail and re-start with a different size, arguments were added to send this information to the interface so that it can re-start the time step. 9. In the subroutine post_step1, the steady-state convergence test is modified to take into account heat transferred to the coolant from FRAPTRAN rods. In addition to these changes in the CTF source code, the coupling makes changes in CTF by directly accessing the variables for wetted perimeter, flow area, and hydraulic diameter of the coolant channels, as well as the terms which define the total heat transferred to each hydraulic node at the given time step TORT-TD No changes were made to TORT-TD s source code the original coupling with CTF, based on the previous coupling with ATHLET, was maintained. All four cross section feedback variables are placed by the interface in the module ccn3d. Before a TORT-TD calculation is called, the subroutine ExchangeCobraArrays is called in order to move the values into TORT-TD s own variables. Likewise, ExchangeCobraArrays is called 102

116 again after a TORT-TD calculation to the resulting power into the ccn3d module, where the interface has access to it FRAPTRAN The most important changes to FRAPTRAN were as follows: 1. In fraptran.f, program fraptran is changed to subroutine fraptran_main. 2. In crank6.f, the loop over time steps is removed, so that when fraptran_main is called, the code is only initialized, and no time steps are run. Instead, comput, which makes steady-state or time advancement calculations, is called from the interface s run_steady_state and run_time_steps methods. 3. In iofiles.f, the expected input file is changed from FrapTran.inp to FrapTran##.inp, where ## is the rod index with two digits. 4. In power.f, if power is received from TORT-TD, the current linear power at each axial node is retrieved from the data transfer module and used to set the rate of heat generation in the rod. 5. In timstp.f the time step size is retrieved from the data transfer module, where it is determined according to the scheme described in In bcdinp.f, an option was added to set ctf= on in the input. While this is redundant, as it is the only possible choice for the coupled version, it makes it clear which FRAPTRAN input files are meant for the coupling. 7. In htrc.f, the coolant temperature and pressure, and fuel surface heat transfer coefficient, are obtained from the data transfer module in order to be used in the heat transfer calculation. In FRAPTRAN, the heat transfer between rod and coolant is calculated in terms of an average coolant temperature and average heat transfer coefficient, as opposed to CTF where there is a separate coefficient for each phase. Thus, the coefficients and temperatures were averaged in a way that conserves the heat flux from the clad surface: 103

117 (5-1) and, (5-2) where is the average heat transfer coefficient used in FRAPTRAN, is the liquid heat transfer coefficient received from CTF, ( ), is the vapor heat transfer coefficient received from CTF, is the liquid heat transfer coefficient derivative with respect to clad temperature, received from CTF, is the clad temperature from the most recent iteration in FRAPTRAN, is the clad temperature from FRAPTRAN that was used by CTF when it generated the heat transfer coefficients, is the average coolant temperature used by FRAPTRAN, is the liquid temperature received from CTF, and is the vapor temperature received from CTF. Similarly, FRAPTRAN calculates a single value for the rate of heat transfer from the rod to the coolant, whereas CTF uses separate values for the phases. Thus, the heat must be appropriately split between the two phases. Additionally, since several FRAPTRAN steps may be made at a time, within a single CTF step, the heat transfer are averaged over all steps in order to conserve the total amount of energy. The heat transfer rates are given by and (5-3), (5-4) 104

118 where i is the time step index, is the total rate of heat transfer, is the time step size, and are the rates of heat transfer to the liquid and vapor, respectively, and and are the fractions of the heat deposited in the liquid and vapor, respectively. These last two parameters are defined as and ( ) ( ) ( ) (5-5) 5.6 Summary ( ) ( ) ( ). (5-6) CTF, FRAPTRAN, and TORT-TD have been integrated into a single Fortran-based executable, in which communication with each code is accomplished by means of an object-oriented data type which is referred to as its interface. This layer of abstraction between the main program and each code provides both clarity and flexibility to the coupled system. Modifications to the original source codes are minimal, and can be placed into newer versions. They have no dependence on the other codes. The coupling primarily uses serial integration. However, parallelism with MPI is used to allow multiple copies of FRAPTRAN to simulate multiple fuel rods. 105

119 6 Original CTF/TORT-TD Results Before the coupling with FRAPTRAN, and before the object-oriented code-interfacebased coupling was created, the original CTF/TORT-TD coupling, described in 4.3, was tested for both a PWR and BWR model, demonstrating its capabilities for steady-state and transients, and for both single-phase and two-phase flow. 6.1 PWR Models Two different PWR models were tested [22] with CTF/TORT-TD a quarter-core model for steady state, and a 2x2 mini core for a rod-ejection transient. These were both based on the Purdue PWR MOX/UO2 Core Transient Benchmark. The 2x2 mini core model had versions with assembly-lumped feedback in CTF, and pinwise feedback from rod-centered sub-channels. This was used to show the difference made by pin-by-pin modeling Description of Purdue Benchmark The Purdue PWR MOX/UO2 Core Transient Benchmark [32] was meant to provide a well-defined problem with complete input data, for the purpose of evaluating core simulators. In particular, it is to be used to verify the accuracy of the powers of individual pins and assemblies. The quarter-core model that it describes is based on a 4-loop Westinghouse PWR, with reflecting boundary conditions to represent symmetry. It contains 17x17-pin fuel assemblies, some with uranium oxide and some with weapons-grade MOX. There are several different levels of burnup, representing fresh, once-burned, and twice-burned fuel. The core is surrounded by a 2.52 cm baffle, and a single row of reflector assemblies, beyond which are vacuum boundary conditions. The axial length is cm, or 12 ft. The benchmark contains three problems two steady states and one transient: 1. Hot Zero Power (HZP) steady state with power at 0.01% of nominal value, 560 K coolant temperature, and All Rods Inserted (ARI). 106

120 2. Hot Full Power (HFP) steady state with power at 100%, 560 K inlet temperature, All Rods Out (ARO). 3. Rod ejection transient the control rods in one assembly are quickly removed from the core, leading to a spike in the reactor power. 4. Figure 6-1: Configuration of quarter-core model, from [22] Steady-state model For the two steady states, the quarter core model described in the benchmark was implemented in CTF/TORT-TD and CTF/NEM. Cross sections were prepared using the lattice-physics code HELIOS. These cross sections are for individual homogenized pin cells, with 8 energy groups and P 1 Legendre expansion for scattering. The neutronic model used 6 delayed neutron groups, and S 4 level-symmetric quadrature. 107

121 The HFP results for the axial power distribution are shown in Figure 6-2, while HZP results are in Figure 6-3. It can be seen that there is less difference between the two codes for HZP, in which the axial fluid and moderator conditions are uniform, due to lack of heat generation. Thus, it appears that the discrepancy for HFP can be attributed to the differing thermal-hydraulic nodalizations for NEM/CTF, the CTF model had 26 nodes of equal length, whereas for TORT-TD/CTF, there were 19 nodes of differing lengths. Figure 6-2: Core-average axial power distribution for Hot Full Power, from [22] 108

122 Figure 6-3: Core-average axial power distribution for Hot Zero Power, from [22] Rod-ejection accident with assembly-lumped feedback For the Rod-Ejection Accident (REA), a 2x2-assembly mini-core model was created, instead of using the benchmark s quarter-core. This mini-core had reflective boundary conditions on two sides, and vacuum boundary conditions on two sides, representing a 4x4 core with 90-degree rotational symmetry, similarly to the quarter-core model. There were 50 axial nodes in both TORT-TD and CTF. However, while TORT-TD modeled each pin, CTF had one channel for each assembly. The configuration is shown in Figure 6-4. Figure 6-4: 2x2 mini-core configuration, from [22] 109

123 The transient was started from a Hot Zero Power state. The control rod group, located in the lower-right assembly, was moved from fully-inserted to fully-out between 1.0 and 1.1 seconds of transient time. The power rapidly increases due to the reactivity insertion, and then rapidly decreases due to the Doppler effect from the fuel temperature. Unlike the steady-state cases, in which NEM/CTF is compared to TORT-TD CTF, the rod-ejection transient was modeled in TORT-TD/ATHLET and TORT-TD/CTF, so it is the thermal-hydraulic code that is different rather than the neutron-kinetics code. As can be seen in Figure 6-5, the power rises higher with CTF. Additionally, Figure 6-6 shows that the fuel temperature decreases more quickly in CTF than ATHLET after the power peak, and Figure 6-7 shows that the moderator density drops lower. This is due to the codes differing fuel rod models and thermodynamic property tables. Figure 6-8 shows how the axial fuel temperature distribution changes for the assembly in which the rods are ejected. The temperature has significantly increased at the moment when the power reaches its peak, but it continues increasing and reaches its peak slightly afterward. Figure 6-5: Total power during rod-ejection transient, from [22] 110

124 Figure 6-6: Fuel temperature over time for rod-ejection transient, from [22] Figure 6-7: Moderator density over time for rod ejection transient, from [22] 111

125 Figure 6-8: Fuel temperature axial distributions for assembly experiencing rod ejection, from [22] Table 6-1: Parameters for rod-ejection transient predicted by CTF/TORT-TD, from [22] Pinwise feedback Peak power [MW] 4120 Peak time [ms] 78 Maximum core-average fuel 892 temperature [K] Time of maximum core-average 110 fuel temperature [ms] The same REA as above was run again using the same neutronic model, but with 1156 fuel-centered sub-channels in CTF, instead of 4 assembly-lumped channels. Additionally, a HFP calculation was made in order to show the difference in the steady state power distribution. For the HFP steady state case, Figure 6-9 shows the relative differences between the pinwise and assembly-lumped models for the fuel temperature radial distributions, while Figure 6-10 shows the same for power radial ditrsibutions. The assembly on the lower right has the lowest power, as it is adjacent to the vacuum boundary condition. Within each assembly, the pinwise-feedback model predicts lower fuel temperatures in the lower right corner, and thus higher powers, when compared to the assembly-lumped-feedback 112

126 model. This is because it is taken into account that the lower-powered pins within each assembly have lower fuel temperatures, and thus less of the resulting negative feedback. By smearing this feedback, the assembly-lumped-feedback model is conservative in that it exaggerates the power differences between pins. Figure 6-9: HFP relative fuel temperatures - pinwise feedback vs. assembly-lumped feedback 113

127 Figure 6-10: HFP relative pin powers - pinwise feedback vs. assembly-lumped feedback Table 6-2: Difference due to pinwise feedback vs. assembly-lumped for HFP Maximum Difference Minimum Difference Standard Deviation of Difference Power 1.1% -1.5% 0.6% Fuel Temperature 8.0% -8.7% 3.4% Figure 6-11 shows the power peaks during the REA for both models. The assemblylumped feedback model is conservative, predicting a peak that is 8% higher. The pinwise feedback model, on the other hand, by resolving the higher fuel temperatures in the pins with the highest powers, has a greater amount of negative feedback and thus a lower power peak. The results of this study show that the assembly-lumped feedback model, currently used in nuclear power industry and regulation, overestimates the REA consequences, thus providing smaller safety margins, which limit the operation flexibility of a nuclear power plant. 114

128 Figure 6-11: Power spike during REA - pinwise feedback and assembly-lumped feedback Table 6-3: REA power peak for pinwise feedback and assembly-lumped feedback Peak Power [MW] Peak Time [s] Assembly feedback Pinwise feedback BWR Assembly Model Testing of CTF/TORT-TD was performed for a BWR assembly model [33]. The purpose was to verify the coupling for two-phase flow, as well as for pin-by pin thermal-hydraulic modeling. A BWR, due to the vapor present at the top of the core, has much greater variation in the feedback due to moderator density, and thus variation in power, with lower power at the top. For this reason, the effects of feedback can be more clearly seen Model description The model used was an ATRIUM-10 fuel assembly from Sweden s Oskarshamn-2 reactor. It contains a 10x10 array of fuel pins, in which 8 rods are partial-length rods, and a 3x3 portion of the array was occupied by a water rod. This geometry is shown in Figure The fuel properties vary over 7 different axial sections, including the lower/upper reflectors, the low-enriched sections at the top and bottom, and the region above the partial-length rods. 115

129 The CTF model contains two axial sections above and below the top of the partiallength rods. However, the flow area and wetted perimeter above the rods is the same as below the geometry does not account for the absence of the rod. There are a total of 25 nodes, of which 2 are the upper and lower reflector regions, 15 are below the partiallength rods, and 8 above. Pressure-loss coefficients represent spacer grids at six different axial elevations. The model uses fuel-centered sub-channels, as the original version of CTF/TORT-TD allowed each TORT-TD node to only be coupled to a single CTF channel. There are five different sub-channel types due to the geometry, and their arrangement is shown in Figure Figure 6-12: Layout of ATRIUM-10 assembly, showing water rod, bypass, control blade (with absorbing part in purple), and partial-length fuel rods (in yellow), from [33] 116

130 Figure 6-13: Subchannel type configuration, from [33] Cross sections were prepared for TORT-TD using HELIOS, of which the model is shown in Figure These cross sections, for 8 energy groups and P 1 scattering expansion, are made at the pin level rather than assembly level. In general, there is a node for each pin, but some rows and columns were split into multiple nodes in order to place node boundaries at material boundaries in the control blade. The resulting cross section types are shown using color coding in Figure Axially, the cm-long active core region was split into 75 nodes in TORT-TD. The model also included bottom and top reflector regions 23 cm and 29.8 cm long, respectively. 117

131 Figure 6-14: Layout of HELIOS model, from [33] Figure 6-15: Cross-section types and nodalization of TORT-TD model, from [33] 118

132 POWER DENSITY [W/cm³] Steady-state convergence While attempting to obtain steady-state convergence, it was found that axial power distribution was alternating between two shapes, as shown in Figure 6-16, causing slowed convergence. This was due to the strong negative feedback that moderator density has on power, and vice-versa. In order to correct the problem, the power given to CTF s fuel rods during steady-state iterations was the average of that from the current and previous iteration in TORT-TD. With the feedback being averaged between the two modes, the calculation converges much better, as shown in Figure ITERATION 1 ITERATION 2 ITERATION 3 ITERATION AXIAL HEIGHT [cm] Figure 6-16: Power distribution oscillation between steady-state iterations, from [33] 119

133 POWER DENSITY [W/cm³] LAST ITERATION PENULTIMATE ITERATION AXIAL HEIGHT [cm] Figure 6-17: Power distribution with convergence forcing method, from [33] Steady-state results Two steady-state calculations were made, both with a total power of 5 MW in the assembly. One was without the control blade, and the other had the blade partially inserted to 100 cm above the bottom of the active region. The calculations yielded the expected results, with the partially-inserted rod moving the power distribution towards the top of the core, as shown in Figure This causes the moderator density to be higher at the bottom (Figure 6-19), as there is less production of vapor. The effect on the multiplication factor was also as expected. For the uncontrolled case, k = , whereas the blade insertion reduced it to k = Figure 6-20 shows the radial power distribution for both cases at an elevation which is occupied by the partially-inserted blade. As expected, the power here is much lower for the controlled case, and it is especially low near the blade. Also as expected, the power is higher near the water rod, as the pins there benefit from the greater moderation. Above the blade, it can be seen in Figure 6-21 that the power is greater for the controlled case, where the blade forces the power distribution upward. The radial power distribution for the controlled case here is in fact quite similar to that for the uncontrolled case at the 120

134 MODERATOR DENSITY [kg/m³] POWER DENSITY [W/cm³] lower elevation. Note, however, that the color scale is not the same for all graphs. Thus, the left side of Figure 6-20 and the right side of Figure 6-21 show similar relative power distributions, but different powers UNCONTROLLED CONTROLLED AXIAL HEIGHT [cm] Figure 6-18: Axial power distribution for controlled and uncontrolled cases, from [33] UNCONTROLLED CONTROLLED AXIAL HEIGHT [cm] Figure 6-19: Axial coolant density distribution for controlled and uncontrolled cases, from [33] 121

135 Figure 6-20: Radial power distribution at node 28 (109.1 cm), for uncontrolled (left) and controlled (right) cases, from [33] Figure 6-21: Radial power distribution at node 32 (129.4 cm), for uncontrolled (left) and controlled (right) cases, from [33] Transient results A transient simulation was made in which the inlet mass flow was reduced by 10% after one second of transient time. This produced the expected result as the amount of liquid decreases and the volume occupied by vapor increases, the negative feedback from the moderator density causes a reduction in power, as shown in Figure

136 ASSEMBLY POWER [MW] INLET MASSFLOW [kg/s] TIME [s] Figure 6-22: Transient inlet flow boundary condition, from [33] Summary TIME [s] Figure 6-23: Total power over time for flow transient, from [33] The coupling between CTF and TORT-TD has been verified for both PWR and BWR models. For the PWR case, the results for both steady-state and transient calculations have been compared to other codes, and all differences have been accounted for. For the BWR case, which includes two-phase flow and pin-by-pin modeling of the Thermal Hydraulics, the results are as expected according to physical reasoning. This shows that the coupling is functioning correctly, and that it can produce accurate results as a reasonable compromise between computational cost and modeling fidelity. 123

137 7 CTF/FRAPTRAN Reflood Results Three experimental LOCA cases were selected from FRAPTRAN s integral assessment [34] in order to test the coupling between CTF and FRAPTRAN. These are test cases used in FRAPTRAN s original validation, for which pre-existing input models are available. All three cases used were LOCA experiments from the National Research Universal (NRU) reactor, and include experimental data. These fuel rod failure cases primarily pertain to the domain of Fuel Performance codes, as they contain excessive cladding strain (ballooning) and rod failure, which thermal hydraulic codes are not designed to model. Thus, they primarily show how CTF improves FRAPTRAN s modeling fidelity, rather than vice versa. Results similar to those shown below for MT-1 and MT-6A have previously been published [35] at the NURETH-15 conference. 7.1 MT-1 The MT-1 (Material Test 1) experiment [36] was conducted to determine the response of a full length 3% enriched PWR fuel assembly to a LOCA. It includes cladding ballooning and rupture, as well as the resulting flow blockage, and is intended to reduce uncertainties in LWR licensing criteria. The test assembly contained 32 rods, each 12 feet in length, arranged in a 6-by-6 array with the corner rods removed. One was an instrument tube, and 20 were non-pressurized outer guard rods, which provided adiabatic boundary conditions to the 11 pressurized test rods. The experiment was designed to create a peak cladding temperature of 1172 K, with high temperatures sustained long enough to allow swelling and rupture. Fission driven by irradiation from the reactor was used to simulate the decay heat in fuel undergoing a LOCA. The test assembly was first brought to a steady state using steam cooling, then allowed to heat adiabatically in stagnant steam. After 32 seconds, a reflood begins with cooling water injected at m/s, causing a quench front to gradually move along the length of the fuel. The experiment resulted in the rupture of six of the 124

138 eleven pressurized rods, during the time period of 60 to 95 seconds after the beginning of the transient. A complete CTF model, including a fuel rod, was created based on the geometry and boundary conditions available from the FRAPTRAN input. Additionally, a coupled model was prepared in which the rod was modeled in FRAPTRAN, and the coolant in CTF. All of the models contained a single fuel rod in a single coolant channel, with 12 axial nodes. The results of the CTF, FRAPTRAN, and coupled CTF/FRAPTRAN models are shown below. Figure 7-1 shows that all three codes produce fairly similar results for the inner clad temperature at axial node 7, and closely match the data. However, both the CTF and coupled models show some oscillation. This is expected because of the coarseness of the nodalization as the liquid level crosses a node boundary, it causes an immediate cooling of the entire node. For this reason, a mesh refinement scheme is normally used in the vicinity of a quench front in order to reduce this effect, as described in However, the quench front model is not functional in the current version of CTF, nor has it been implemented in the coupling. Figure 7-2 shows more discrepancy between the codes, and between the simulation results and measured data, at axial node 10. When comparing with Figure 7-1, it is evident that the quench front advances much more quickly from node 7 to node 10 in the experimental data than in any of the models. This cannot be accounted for by limitations in the codes modeling capabilities. It is most likely that the boundary conditions differ the flow rate may be different than that at the beginning of the reflood. A more interesting result is that, while clad temperature in FRAPTRAN starts declining almost immediately at the beginning of the reflood, the trend in CTF and CTF/FRAPTRAN is a more rounded peak before the decline in temperature. Given that the same amount of heating is present in both models, and it may only leave the rod through the cladding surface into the coolant, this difference may only be accounted for by differences in the heat transfer coefficients and coolant temperatures. These values, as used in FRAPTRAN, are shown for the standalone and coupled models in Figure 7-3 and 125

139 Figure 7-4. In both codes, the heat transfer coefficients are very low until the quench front arrives. In FRAPTRAN, with its far simpler thermal-hydraulic model, the coefficient instantly changes to a constant, whereas in CTF, it spikes and then decreases. The difference in the clad temperatures before the arrival of the quench front seems to come from the coolant temperature, which rises much higher in CTF. In FRAPTRAN, the coolant temperature in all nodes instantly changes at the beginning of the reflood to that of the fluid being injected. This is an inaccuracy, as the coolant (steam) temperature should continue to increase until the rising liquid level reaches the node. Also of interest is that in Figure 7-4, for node 4, the heat transfer coefficient becomes very small after quenching, whereas for the other two nodes shown, 7 and 10, it decreases to much greater value which is comparable to that in FRAPTRAN. This is a significant difference between nodes, and between models, that merits explanation. It is due to CTF s more sophisticated treatment of heat transfer regimes, which differ along the length of the rod. The lower nodes are closer to the inlet, and the fluid reaching them is cooler, whereas at the top of the rod, the coolant has been heated to the boiling point. This can be seen in Figure 7-5 and Figure 7-6, which show how the axial temperature distribution changes over time. Generally, the vapor temperature is somewhat higher than the liquid temperature, but for the upper nodes that are below the quench front at 250 seconds, they are nearly equal. This means that large amounts of boiling can occur at the clad walls. Given that boiling is a much more efficient mode of heat transfer than simple convection, it is logical that nodes in such conditions should have much higher heat transfer coefficients. 126

140 Inner Clad Temperature [K] Inner Clad Temperature [K] TC-133BIR2 CTF FRAPTRAN Coupled Time [s] Figure 7-1: Inner clad temperature at node 7 (elevation=78 in) and test level 13 (~72.5 in) TC-174CIR1 CTF FRAPTRAN Coupled Time [s] Figure 7-2: Inner clad temperature at node 10 (elevation=114 in) and test level 17 (~118 in) 127

141 Surface Heat Transfer Coefficient [kw/m 2 C] Coolant Temperature [K] FRAPTRAN, all nodes Coupled, node 4 Coupled, node 7 Coupled, node Time [s] Figure 7-3: Coolant temperatures at rod surface as used in FRAPTRAN Time [s] FRAPTRAN, node 4 FRAPTRAN, node 7 FRAPTRAN, node 10 Coupled, node 4 Coupled, node 7 Coupled, node 10 Figure 7-4: Fuel-surface heat transfer coefficients as used in FRAPTRAN 128

142 Temperature [K] Temperature [K] Liquid Vapor Clad Axial Node Figure 7-5: Axial temperature distributions early in the reflood (100 s) Liquid Vapor Clad Axial Node Figure 7-6: Axial temperature distributions later in the reflood (250 s) 7.2 MT-4 MT-4 [37] had a similar configuration to MT-1, with 20 non-pressurized guard rods surrounding the 12 (instead of 11) test rods. The purpose of the experiment was to evaluate ballooning and rupture for adiabatic heating in the range of 1033 K to 1200 K. After a steady state with steam heating, the system was heated adiabatically until a reflood was begun 57 seconds later. Eleven of the test rods ruptured, doing so during the interval of seconds after the beginning of the adiabatic heating. 129

143 Inner Clad Temperature [K] Inner Clad Temperature [K] As with MT-1, models were prepared in FRAPTRAN, CTF, and CTF/FRAPTRAN. The results for the inner clad temperature are shown below. It is evident that the boundary conditions present in the experiment are significantly different than the documented ones used in the codes. Nevertheless, there appears to be good agreement between the codes, supporting the conclusion that the coupling has been implemented correctly MT-4, 76.9 in CTF FRAPTRAN Coupled Time [s] Figure 7-7: Inner clad temperature at node 7 (78 in) and average experimental clad temperature at 76.9 in Time [s] MT-4, 97.9 in CTF FRAPTRAN Coupled Figure 7-8: Inner clad temperature at node 9 (102 in) and average experimental clad temperature at 97.9 in 130

144 7.3 MT-6A Like MT-1 and MT-4, the MT-6A experiment [38] was intended to simulate a LOCA and the subsequent recovery. Adiabatic heating to 1200 K was intended to create the greatest possible cladding expansion, and this was followed by a reflood with the intent of determining how quickly the fuel may be cooled. As in MT-1, the reflood water was injected at a rate of m/s. Unlike both of the above cases, an error occurred in the control system which caused the system pressure to be 250 psi instead of 40 psi. This raised the saturation temperature from C to C, putting the injected coolant (36.8 C) much further beneath the boiling point. The test assembly contained 21 rods in a 5-by-5 array with the corner rods removed. All rods were pressurized and intended to expand and rupture. Adiabatic boundary conditions were created by an insulating shroud rather than by guard rods. Also, the fuel was designed to reduce temperature gradients in the cladding, allowing maximum expansion. The result was that all 21 rods failed between 58 and 64 seconds after the beginning of the transient, after experiencing large amounts of strain. The peak powers in the vicinity of the ballooning were lower than the peak powers elsewhere, demonstrating that the ballooning does not hinder heat transfer and may enhance it. The results for CTF, FRAPTRAN, and CTF/FRAPTRAN are compared below in Figure 7-9 and Figure As with MT-4, experimental data does not match well probably because the true heating was much greater than the predicted value. More important is that in FRAPTRAN, both elevations are quenched much earlier. Given that the most important difference between this case and the above two cases is the greater pressure, the CTF and FRAPTRAN models were re-run with a pressure of 40 psi, and the results are shown in Figure The result is nearly the opposite FRAPTRAN is quenched sooner than CTF, at least at the higher nodes. It appears that the differences between the methods of the two codes are highly sensitive to pressure and its effect on subcooling. 131

145 Inner Clad Temperature [K] Inner Clad Temperature [K] Time [s] 2C 61 in. CTF FRAPTRAN Coupled Figure 7-9: Inner clad temperature at node 6 (66 in) and experimental rod 2C at 61 in Time [s] 2B 102 in. CTF FRAPTRAN Coupled Figure 7-10: Inner clad temperature at node 9 (102 in) and experimental rod 2B at 102 in 132

146 Coolant Temperature [K] FRAPTRAN Coupled Axial Node Figure 7-11: Coolant temperature axial distribution at beginning of transient, as used in FRAPTRAN, for standalone and when coupled to CTF For FRAPTRAN, the initial temperature at both nodes is significantly higher than in the CTF and coupled cases. Given that the rod power is the same, this must be due to different coolant temperatures or heat transfer coefficients. Figure 7-11 shows that the coolant temperatures are nearly the same during the steam cooling at the beginning of the transient, whereas Figure 7-12 shows that the heat transfer coefficient received from CTF in the coupled case is significantly higher. This can be traced to the correlations used by the codes. For both CTF and FRAPTRAN, the single-phase vapor heat transfer coefficient is given by the Dittus-Boelter correlation [18][24], h spv k D Re Pr, e (7-1) where k is the vapor thermal conductivity, D e is the hydraulic diameter, Re is the Reynolds number: Re GD e, 133

147 c p Pr is the Prandtl number: Pr, k G is the mass flux, c p is the isobaric specific heat, and is the dynamic viscosity. Although in FRAPTRAN, the Dittus-Boelter correlation is designated as the means of calculating the single-phase vapor heat transfer coefficient, it does not always determine the value used [24]. Instead, the coefficient is calculated as the minimum of the Dittus- Boelter correlation and the correlation used for post-critical-heat-flux film boiling. This is done to provide a smooth transition between the two flow regimes. By default, the film boiling heat transfer coefficient is determined by the Groeneveld-5.9 correlation: h k, g FB ReHOM Prw Y De (7-2) where k g is the saturated vapor thermal conductivity, Re HOM is the homogeneous Reynolds number: Re HOM GxD e, g x is the static quality, is the void fraction, g is the saturated vapor thermal conductivity, Pr w is the Prandtl number evaluated at the wall temperature, f Y x g f is the saturated liquid density, and 0.4 g is the saturated vapor density., 134

148 For four axial nodes, the Dittus-Boelter and Groeneveld-5.9 correlations were manually evaluated in terms of the conditions in coupled-ctf and standalone FRAPTRAN, respectively. This is shown in Table 7-1 and Table 7-2, and the results are shown in Figure 7-12 alongside the values used by the codes. It is evident that CTF is using the Dittus-Boelter correlation, whereas FRAPTRAN is using the Groeneveld-5.9 correlation, which is the minimum of the two. It thus appears that there is a discontinuity between FRAPTRAN s correlations for film boiling and for single-phase vapor, as the former yields a much smaller coefficient than the latter for single-phase vapor conditions. The codes means of enforcing continuity affects not only the transition between flow regimes, but causes the film boiling correlation to be used even during a steady state with single-phase vapor. This results in a heat transfer coefficient that is approximately half of that calculated by the Dittus-Boelter correlation the standard correlation for single-phase flow. This supports the conclusion that FRAPTRAN s handling of fluid conditions is deficient and that a dedicated Thermal Hydraulics code should be used instead to generate heat transfer coefficients. Table 7-1: Calculation of single-phase vapor heat transfer coefficients from Dittus-Boelter correlation using conditions in coupled CTF Node Pressure [bar] Temperature [C] μ [Pa s] E E E E-05 c p [kg/{kg C}] k [W/{m C}] E E E E-02 D e E E E E-02 G [kg/{m 2 s}] E E E E+02 Re E E E E+04 Pr h spv [kw/{m 2 C}]

149 Rod Surface Heat Transfer Coefficient [kw/m 2 C] Table 7-2: Calculation of film-boiling heat transfer coefficients from Groeneveld 5.9 correlation using conditions in standalone FRAPTRAN Node Pressure [bar] Saturation Temperature [C] μ (saturation) [Pa s] E E E E-05 k (saturation) [W/{m C}] E E E E-02 Wall Temperature [C] μ (wall) [Pa s] E E E E-05 c p (wall) [kg/{kg C}] k (wall) [W/{m C}] E E E E-02 D e E E E E-02 G [kg/{m 2 s}] E E E E+02 Re (saturation) E E E E+04 Pr (wall) h FB [kw/{m 2 C}] FRAPTRAN Coupled Dittus-Boelter Groeneveld Axial Node Figure 7-12: Rod surface heat transfer coefficients at each axial node at beginning of transient, as used in FRAPTRAN, for standalone and when coupled to CTF. Manually-calculated results for he Dittus-Boelter and Groeneveld 5.9 correlations are shown for comparison 136

150 Inner Clad Temperature [K] CTF, node 6 CTF, node 9 FRAPTRAN, node 6 FRAPTRAN, node Time [s] Figure 7-13: Inner clad temperature compared between CTF and FRAPTRAN for 40 psi system pressure instead of 250 psi 137

151 8 CTF/TORT-TD/FRAPTRAN A set of problems was designed in order to test the coupling for the interaction of all 3 codes, as well as to show a contrast between the modeling of the fuel rods in CTF and in FRAPTRAN. The model created was of a set of 15 PWR pins in a 4-by-4 arrangement, with one control rod guide tube. There were four different fuel types located in the four corners low-burnup MOX, low-burnup UO 2, high-burnup MOX, and high-burnup UO 2. This arrangement, shown in Figure 8-1, creates strong flux gradients which challenge TORT-TD s high-fidelity angular treatment. It will also allow a contrast of CTF and FRAPTRAN s modeling of the fuel temperature feedback for two different fuel types and two different levels of burnup. Two different problems were run: a steady-state Hot Full Power (HFP) calculation, and a Rod Ejection Accident (REA), which starts from Hot Zero Power (HZP). Both of these were run with CTF/TORT-TD, with CTF modeling the fuel rods, and with CTF/TORT- TD/FRAPTRAN, with the rods modeled in FRAPTRAN. FRAPCON was used to initialize the fuel conditions for FRAPTRAN based on burnup. The HFP case is used to test the spatial distribution of parameters such as power and fuel temperature, and how this depends on the model. The REA shows how the system responds to feedback during a rapid transient in which there is a great deal of sensitivity to the fuel temperature. The CTF/FRAPTRAN rod feedback comparison will be conducted using the S 8 angular discretization in TORT-TD. Additional cases with CTF/TORT-TD were run with S 2 and S 4, as well as the diffusion approximation. This will show the dependence on the level of angular fidelity for the strong flux gradients in the problem, as well as a contrast in CPU time. 8.1 Model Descriptions The geometry and composition of the fuel pins was obtained from the Purdue MOX benchmark [32], while the cross sections used were those generated for the previous research described in section 6.1. The models created for each code will be described below. 138

152 The fuel pin types selected were a 4.25% enriched Uranium Oxide pin from an assembly with 4.2% average enrichment, and a MOX pin containing 5.0% Plutonium Oxide content, from an assembly with an average PuO 2 content of 4.3%. Each of these represents the most common pin type in its respective type of assembly. The basic dimensions of the fuel, applied to the models in each code, are shown in Table 8-1. Table 8-1: As-manufactured cold fuel dimensions Active Length [m] Pin pitch [m] Fuel pellet radius [m] Inner clad radius [m] Outer clad radius [m] Clad-pellet gap thickness [m] Clad thickness [m] Low burnup MOX High burnup MOX Low burnup UO High burnup UO Guide tube Figure 8-1: Subchannel and fuel rod configuration of 4x4 case FRAPCON Two models were prepared in FRAPCON one for each fuel type. Since the benchmark data does not include all of the geometry details required by the code, such as the plenum length and pellet dish size, these were set using values obtained from sample inputs. The only differences are in the fuel contents, shown in Table

153 Both fuel types were burned for a total of 1000 days at a constant, axially-uniform power, which was calculated as the average-pin nominal power from the benchmark. Although a real pin would be burned with higher powers in the middle, the axially-uniform power was used for two reasons: TORT-TD currently does not allow axial variation in the fuel burnup within a given axial section, although it does allow variation by using multiple axial sections. For simplicity, the model only uses one axial section for the active fuel region. Thus, with uniform burnup in FRAPCON, FRAPTRAN may use a burnup distribution that is consistent with that used in TORT-TD. With axially-uniform burnup, axial variation in the solution will show the differences due to the power and thus temperature distribution. With an axiallyvarying burnup, it would be difficult to separate the effects of local burnup from those of the current local temperature. Table 8-2: Parameters input to FRAPCON for both fuel types Cold plenum length [m] 0.22 Outer diameter of plenum spring [m] Diameter of the plenum spring wire [m] Number of turns in the plenum spring 20 Length of each pellet [m] Depth of pellet dish [m] Pellet dish shoulder width [m] Pellet surface roughness [m] 2.e-6 Expected density increase during operation [kg/m 3 ] 100. Clad type Zircaloy 4 Clad surface roughness [m] e-7 Initial gas pressure [Pa] 2.0e6 Coolant pressure [Pa] 1.55e7 Coolant inlet temperature [K] 560. Coolant mass flux [kg/{m 2 s}] Linear power [kw/m] (axially uniform) Time step interval size [days] 50 Burn time [days] 1000 Table 8-3: Nodalization of FRAPCON model for both fuel types Pellet radial nodes (equal cross-sectional area) 17 Axial nodes (equal length)

154 Gap Conductivity [kw/{m 2 K}] Table 8-4: Parameters input to FRAPCON which differ between fuel types UO 2 MOX U-235 enrichment [%] PuO 2 content [%] % of theoretical density After running FRAPCON, the dependence of the gap conductivity on burnup was examined, and it is shown in Figure 8-2. The parameter starts from a value of about 1.0e4 kw/m 2 K for fresh fuel, and quickly increases to a much larger value of about 1.0e5 once the fuel has expanded sufficiently to be in contact with the cladding. 1.20E E E E E+04 MOX UO2 2.00E E Burnup [MWD/kg] Figure 8-2: Dependence of gap conductivity on burnup for axially-uniform power FRAPTRAN The FRAPTRAN model required relatively less data, which was chosen for consistency with the FRAPCON model. Most of the data is obtained from the output of the FRAPCON model described above. The only differences between FRAPTRAN inputs were (1) the FRAPCON restart file used, depending of fuel type, and (2), the burn time from which the fuel properties are to be obtained, shown in Table

155 Table 8-5: Parameters input to FRAPTRAN for all pins Upper plenum volume [m 3 ] e-5 Number of coils in upper plenum spring 5 Uncompressed height of upper plenum spring [m] Uncompressed outer diameter of upper plenum spring [m] Diameter of upper plenum spring wire [m] Pellet dish depth [m] Pellet dish volume [m 3 ] 2.06e-7 As-fabricated fill gas temperature [K] Table 8-6: FRAPCON initialization time for each pin type Fuel FRAPCON Initialization Time [seconds] Type Low Burnup (0.15 MWD/kg) High Burnup (35.0 MWD/kg) UO e7 MOX e CTF Table 8-7: Nodalization in FRAPTRAN Pellet radial nodes (equal cross-sectional area) 15 Cladding nodes 2 Axial nodes (equal length) 20 The CTF model contains 25 sub-channels, arranged as shown in Figure 8-1. There are three sub-channel geometries, and two sub-channel horizontal-connection geometries, described in Table 8-10 and Table The modeling parameters chosen are based on the recommended and/or typical values for a PWR, and are described in Table 8-8. Given that this model has a very small number of nodes and is easy to solve, Gaussian elimination is used to solve the pressure equation. The coolant boundary conditions, shown in Table 8-9, were obtained from the Purdue benchmark data, with the flow rate based on the average flow over 4 pins. 142

156 Table 8-8: Modeling options in CTF Rod friction model Re Entrainment/deposition Original model Mixing / void drift Use-specified mixing coefficient Equilibrium drift factor phase turbulent mixing coefficient Solver Direct Gaussian elimination Maximum time step size [s] 0.01 (0.001 during rod ejection) Table 8-9: CTF model boundary conditions Inlet enthalpy [kw/kg] Inlet mass flow [kg/s] Outlet pressure [bar] 155. Table 8-10: Sub-channel types Area [m 2 ] Wetted Perimeter [m] Number of Sub-channels Central Side Corner Table 8-11: Sub-channel horizontal connection types Length [m] Width [m] Horizontal Loss Coefficient Number of Gaps Half Full Table 8-12: Axial model information Axial nodes 20 (equal size) Spacer grid loss coefficient Number of spacer grids 10 (at even-numbered nodes) For the cases in which CTF models the fuel rods, the dimensions shown in Table 8-1 were input, and the standard properties for fresh UO 2 were used. Two models were prepared: Model 1 has a gap conductivity of 1.0e4 kw/m 2 K, the typical value used in CTF. 143

157 Time Step Size [s] Model 2 has a gap conductivity of 1.0e4 kw/m 2 K in the low-burnup fuel, with a value of 1.0e5 kw/m 2 K set for the high-burnup fuel, based on the FRAPCON results in Figure 8-2. While CTF has a dynamic gap conductivity model as described in section 4.1.6, this was not used due to a memory allocation error that occurs in the code version that was used for the coupling. This error has since been corrected, so future work will include comparing this model to FRAPTRAN once the coupling has been incorporated into a newer version of the CTF source code. Table 8-13: Fuel properties in CTF Fuel properties Built-in UO 2 properties Clad properties Build-in zirconium-dioxide properties % theoretical fuel density Gap conductivity [kw/{m 2 K}] 1.0e4 (1.0e5 for high-burnup rods in model 2) During steady-state calculations, and for most of the REA, CTF s maximum time step size is set as 0.01 seconds. However, this is decreased to seconds for transient times between 0.99 and 1.2 seconds, in order to more accurately resolve the power spike that occurs during this time Time [s] Figure 8-3: Time step size during rod ejection accident 144

158 8.1.4 TORT-TD The TORT-TD model has 16 nodes on the x-y plane one for each fuel rod or guide tube. In the axial direction, it has a finer nodalization than CTF, with 40 axial nodes in the active core region. Unlike in CTF and FRAPTRAN, a reflector region is modeled above and below the active core. The primary model, which was used to compare CTF with FRAPTRAN, uses 8 neutron energy groups, and S 8 level-symmetric quadrature. However, as will be discussed below, models with S 2, S 4, and diffusion were also run for comparison. Table 8-14: TORT-TD model parameters Number of energy groups 8 Number of delayed neutron precursors 6 Legendre scattering order 1 Quadrature type Quadrature order Table 8-15: Axial nodalization Level-symmetric S 8 (main case) S 2, S 4, and diffusion also used Active core Length [m] Nodes 40 (equal size) Lower reflector Length [m] 0.3 Nodes 5 (equal size) Upper reflector Length [m] 0.3 Nodes 5 (equal size) The XY plane was divided into four fuel assemblies, each with four pins, as shown in Figure 8-4. There was one MOX assembly type, M43u, which was used twice for the high and low-burnup pins. Since the lower-left corner has a control rod guide tube, this was modeled as the assembly type U42c, which differs from the U42u type on the lower right. 145

159 M43u Burnup = 0.15 M43u Burnup = 35.0 U42c Burnup = 0.15 U42u Burnup = 35.0 Figure 8-4: Arrangement of assembly types in TORT-TD model The cross section libraries used were from those generated by HELIOS for the previous research discussed in section 6.1. Table 8-16 shows which libraries were used for which type of fuel pin. Table 8-16: Cross section libraries used Region UO 2 pin Control rod guide tube MOX pin reflector Cross section type WQ-U42-FF-unrodded, WQ-U42-FF-rodded WQ-U42-GT-unrodded, WQ-U42-GT-rodded WQ-M43-HF-unrodded WQ-U42-RF-unrodded The power input to TORT-TD, shown in Table 8-17, was determined by the pin-average nominal power for 15 fuel pins for the HFP steady-state case. The HZP at the beginning of the REA was taken as 0.01% of this value. Table 8-17: Initial steady-state power Power (HFP) [MW] Power (HZP) [MW] e-4 For the HFP steady state, the control rod is fully withdrawn. However, in the rod ejection accident, the rod begins inserted to 50% of the length of the core, and is withdrawn at a constant rate between 1.0 and 1.1 seconds, as shown in Figure

160 Control Rod Insertion [%] Normally, a rod ejection starts with a fully-inserted control rod. However, in this model, the ratio of fuel rods to ejected control rods is much lower than in a true PWR core. For this reason, 50% insertion was used to prevent the reactivity insertion from being excessive Time [s] Figure 8-5: Control rod insertion during rod ejection accident 8.2 Steady-State Convergence Originally, the coupling was written only with the alternating-steady-state convergence scheme for CTF and FRAPTRAN described in section However, this yielded a poorly converged solution, in which the conditions did not hold a steady state during a five second null transient. Because of this, the time-marching steady-state convergence scheme was created. The plots below show that the latter method produces a much better converged steady-state, in which the global conditions hold constant. In the following section, all results are obtained from models that use the time-marching steady-state convergence scheme. 147

161 Moderator Density [kg/m^3] Total Power [MW] Time-marching Alternating steady state Time [s] Figure 8-6: Power during null transient for both convergence schemes Time-marching Alternating steady state Time [s] Figure 8-7: Average moderator density during null transient for both convergence schemes 148

162 Fuel Temperature [K] Time-marching Alternating steady state Time [s] Figure 8-8: Average fuel temperature during null transient for both convergence schemes 8.3 CTF FRAPTRAN Rod Comparison Two cases, a steady state at HFP, and a REA starting from HZP, were run for three models two in which the fuel rods were modeled in CTF, and one in which they were modeled in FRAPTRAN. As mentioned above, in one CTF model all gap conductivities were set as 1.0e4 kw/m 2 K, and in the other, the high-depletion rods have gap conductivities of 1.0e5 kw/m 2 K Steady state The axially-averaged radial distributions of power, fuel temperature, and moderator density are shown below for the case in which the rods were modeled in FRAPTRAN. As can be seen in Figure 8-9, the power is higher in the MOX fuel than the UO2 fuel, and higher in the low-burnup fuel than the high-burnup fuel. Also worth noting is that the MOX pins with the highest power are those adjacent to the lower-powered pins. This is due to the small size of the problem domain, and the fact that high-energy neutrons travel much greater distances and are less-easily absorbed, while thermal neutrons travel shorter distances and are more easily absorbed. High energy neutrons that are born from fission are dispersed throughout the problem domain, so that high powers (and thus more neutron births) from adjacent pins have less of a positive impact on the power of a given pin. Instead, adjacent pins with a higher absorption cross section decrease the flux of 149

163 thermal neutrons in their vicinity, lowering the power of pins surrounded by high-crosssection pins. This effect is different from the behavior of a more realistic 17-by-17 fuel assembly, in which regions with a similar composition would be much larger in comparison with a neutron s diffusion length. In this case, being near a large number of high power pins should have a positive impact on the power of a given pin, due to a larger number of neutron births. Figure 8-9: Radial power distribution - FRAPTRAN rods Figure 8-10 shows the radial distribution of moderator density. As expected, the hottest coolant accumulates in the corner of the hottest fuel, while cooler coolant accumulates in the coolest corner. The coolest region is that around the control rod guide tube, in which there is no fuel rod generating power. 150

164 Figure 8-10: Radial moderator density distribution - FRAPTRAN rods The radial fuel temperature distributions for two cases are shown in Figure Unlike the power and moderator density distributions, there is a visible difference between the CTF rod and FRAPTRAN rod cases. In both cases, higher temperatures correspond to higher powers, but in the CTF case with a constant gap conductivity of 1.0e4, the fuel temperatures are higher than in FRAPTRAN. CTF Rods Case 1 FRAPTRAN Rods Figure 8-11: Radial fuel temperature distributions The plots below show the relative difference for the two CTF cases (where case 2 has a higher gap conductivity for the high-burnup fuel), using the solution with FRAPTRAN rods as the reference solution. For the first CTF case (a gap conductivity of 1.0e4 everywhere), all pins are hotter than in FRAPTRAN, with the largest errors in the highburnup pins. The resulting negative feedback pushes the power slightly toward the low- 151

165 burnup side. However, given the small size of the radial problem domain and the fact that neutrons cross it easily, this effect is very small, with maximum errors in the radial power of about 0.05%. For CTF case 2, however, the higher gap conductivity results in a lower rather than higher fuel temperature relative to FRAPTRAN for the high-burnup pins. Because of this, the power is somewhat less shifted to the left. Case 1 Case 2 Figure 8-12: Relative radial power - CTF vs FRAPTRAN rods Case 1 Case 2 Figure 8-13: Temperature difference CTF vs. FRAPTRAN The radially-averaged axial plots below show that in CTF, the axial power distribution is not dramatically different (Figure 8-14), but has some error, especially at the top and bottom, as shown in Figure This is due to the higher fuel temperature at the center (Figure 8-15), which displaces the power away from the center. In case 2 of CTF, the fact that the temperature is not as center-peaked mitigates this effect. 152

166 Average Fuel Temperature [C] Linear Power [kw/m] CTF 1 CTF 2 FRAPTRAN Axial Position [m] Figure 8-14: Axial power distribution CTF 1 CTF 2 FRAPTRAN Axial Position [m] Figure 8-15: Axial fuel temperature distribution 153

167 Relative Power Difference Axial Position [m] CTF 1 CTF 2 Figure 8-16: Axial relative power difference - CTF vs. FRAPTRAN rods Below, axial temperature distributions are shown for the four rods in the center 6, 7, 10, and 11, representing each type of fuel. The values for the fuel surface, centerline, and average temperature are shown. For rods 6 and 10 (low-burnup fuel), only case 1 is shown for CTF, as case 2 is not significantly different. However, for rods 7 and 11, CTF cases 1 and 2 contrast the results for constant gap conductivity of 1.0e4 and 1.0e5, respectively. In general, the fuel surface temperatures are higher in CTF case 1 relative to FRAPTRAN, whereas they match well for rods 7 and 11 for case 2, in which there is little temperature drop across the fuel-clad gap. In all cases, there is some discrepancy in the fuel centerline temperature, which is a function of both the temperature drop across the gap as well as across the fuel. This will be explored below. 154

168 Temperature [K] Fuel Temperature [K] Axial Position [m] CTF - Surface CTF - Center CTF - Average FRAPTRAN - Surface FRAPTRAN - Center FRAPTRAN - Average Figure 8-17: Rod 6 (low-burnup MOX) fuel temperatures CTF 1 CTF 2 FRAPTRAN Axial Position [m] Figure 8-18: Rod 7 (high-burnup MOX) fuel surface temperature 155

169 Temperature [K] Temperature [K] CTF 1 CTF 2 FRAPTRAN Axial Position [m] Figure 8-19: Rod 7 (high-burnup MOX) fuel centerline temperature CTF 1 CTF 2 FRAPTRAN Axial Position [m] Figure 8-20: Rod 7 (high-burnup MOX) average fuel temperature 156

170 Temperature [K] Fuel Temperature [K] Axial Position [m] CTF - Surface CTF - Center CTF - Average FRAPTRAN - Surface FRAPTRAN - Center FRAPTRAN - Average Figure 8-21: Rod 10 (low-burnup UO 2 ) fuel temperature Axial Position [m] CTF 1 CTF 2 FRAPTRAN Figure 8-22: Rod 11 (high-burnup UO 2 ) fuel surface temperature 157

171 Temperature [K] Temperature [K] CTF 1 CTF 2 FRAPTRAN Axial Position [m] Figure 8-23: Rod 11 (high-burnup UO 2 ) fuel centerline temperature CTF 1 CTF 2 FRAPTRAN Axial Position [m] Figure 8-24: Rod 11 (high-burnup UO 2 ) average fuel temperature In order to understand the reasons for the temperatures above, it is necessary to separate the temperature rises from the cladding to the fuel surface and from the fuel surface to the centerline. The former depends on the gap conductivity, of which the value calculated by FRAPTRAN is displayed in the output and can be seen directly. The centerline-surface temperature drop, however, is dependent on both the thermal conductivity of the fuel and the radial power distribution within the pellets. The current CTF model uses radiallyuniform power, whereas FRAPTRAN uses the radial distribution determined by FRAPCON, which uses the TUBRNP model [23]. The non-uniform radial power is 158

172 Temperature Drop [K] peaked at the surface of the pellet, thus reducing the temperature rise within the fuel. Also, CTF uses fresh UO 2 properties for all fuel types, not taking into account Plutonium content or burnup. From Figure 8-25 through Figure 8-28, it can be seen that CTF case 1 over-predicts the fuel-clad-gap temperature drop relative to FRAPTRAN for all rods, especially 7 and 11. However, in these rods, the gap temperature drops in CTF case 2 and FRAPTRAN are negligible due to the high gap conductivity. For both low-burnup rods (6 and 10), the fuel temperature drop is over-predicted in CTF relative to FRAPTRAN, which may be attributed in part to lack of accounting for radial power, and to a lower thermal conductivity. Since it is true both for MOX and UO 2, it is most likely that this difference is due to higher temperatures at the fuel surface caused by the lower gap conductivity, as the thermal conductivity is temperature-dependent. For the high-burnup rods (7 and 11), however, the fuel temperature drop is underpredicted in CTF more so in case 2, despite the surface temperature that matches well. This means that differing temperatures cannot explain the differences in the thermal conductivity, and thus the difference must be due to CTF not taking into account burnup, which reduces the thermal conductivity CTF - Gap CTF - Fuel FRAPTRAN - Gap FRAPTRAN - Fuel Axial Position [m] Figure 8-25: Rod 6 (low-burnup MOX) temperature drop in gap and fuel 159

173 Temperature Drop [K] Temperature Drop [K] Axial Position [m] CTF 1 - Gap CTF 1 - Fuel CTF 2 - Gap CTF 2 - Fuel FRAPTRAN - Gap FRAPTRAN - Fuel Figure 8-26: Rod 7 (high-burnup MOX) temperature drops Axial Position [m] CTF - Gap CTF - Fuel FRAPTRAN - Gap FRAPTRAN - Fuel Figure 8-27: Rod 10 (low-burnup UO 2 ) temperature drops 160

174 Temperature Drop [K] Axial Position [m] CTF 1 - Gap CTF 1 - Fuel CTF 2 - Gap CTF 2 - Fuel FRAPTRAN - Gap FRAPTRAN - Fuel Figure 8-28: Rod 11 (high-burnup UO 2 ) temperature drops Below, the fuel-clad gap conductivities calculated by FRAPTRAN are compared to the constants used in CTF. As can be seen, the conductivity is axial distribution of the gap conductivity is very non-uniform, being much higher in the center, reducing the temperature peak. For rods 7 and 11 (high-burnup), the value is much greater, as the gap has closed and most of the length of the fuel is in contact with the cladding, making the modeling of the gap conductivity less important. For low-burnup fuel, however, the gap conductivity is more important, and cannot be accounted for by the axially-constant value used in CTF. Additionally, since the burnup is axially-uniform, it is evident that the gap conductivity is not purely burnup-dependent. Rather, the thermal expansion of the hottest parts of the fuel is major factor in the contact between the fuel and cladding. 161

175 Gap conductivity [kw/{m 2 C}] Gap conductivity [kw/{m 2 C}] 3.50E E E E E E+04 CTF FRAPTRAN 5.00E E Axial Position [m] Figure 8-29: Rod 6 (low-burnup MOX) gap conductivity 1.20E E E E E E+04 CTF 1 CTF 2 FRAPTRAN 0.00E Axial Position [m] Figure 8-30: Rod 7 (high-burnup MOX) gap conductivity 162

176 Gap conductivity [kw/{m 2 C}] Gap conductivity [kw/{m 2 C}] 1.40E E E E E E+03 CTF FRAPTRAN 2.00E E Axial Position [m] Figure 8-31: Rod 10 (low-burnup UO 2 ) gap conductivity 1.20E E E E E E+04 CTF 1 CTF 2 FRAPTRAN 0.00E Axial Position [m] Figure 8-32: Rod 11 (high-burnup UO 2 ) gap conductivity 163

177 Table 8-18 shows the results for the multiplication factor for each case. CTF s overprediction of the fuel temperature leads to an under-prediction of k eff relative to FRAPTRAN, although this is significantly mitigated in case 2, which in which the temperature difference is not as great. Table 8-18: Results for multiplication factor CTF 1 CTF 2 FRAPTRAN k eff (unrodded) Difference from FRAPTRAN [pcm] Table 8-19 shows the runtimes for each code in each case. TORT-TD has much larger runtimes than the other codes. Additionally, it can be seen that the CTF and TORT-TD runtimes can vary for problems of identical size and problem time. This is because subtle differences in the calculation can influence the number of iterations required for a given time step or steady-state calculation. The FRAPTRAN runtimes are much smaller than in the other two codes, as the different rods are run in parallel, whereas CTF and TORT-TD use only one CPU. Table 8-19: Runtime comparison for each code CTF 1 CTF 2 FRAPTRAN Steady-state iterations Time steps TORT-TD runtime [s] CTF runtime [s] FRAPTRAN runtime [s] Total Per steady state Per time step Total Per steady state Per time step Total Per steady state Per time step

178 Linear Power [kw/m] Rod ejection accident Before running the REA transient, two different HZP steady states were compared one in which the rod was fully withdrawn, and one in which it was halfway inserted, as at the beginning of the REA. These two cases enabled the calculation of the worth of the control rod, which is shown in Table As expected, all the models produced identical results, as there is no feedback due to the negligible power. Table 8-20: Multiplication factor and rod worth (half-inserted) for different rod models CTF 1 CTF 2 FRAPTRAN k eff (unrodded) k eff (half-rodded) Half-rod reactivity worth (pcm) Figure 8-33 and Figure 8-34 show the axial and radial power distributions for both rodded and unrodded cases. As for the multiplication factor, the CTF and FRAPTRAN rod models produce identical results. The control rod insertion shifts the power away from the top of the rods, but does not significantly influence the radial power distribution. This is because almost all of the power is generated at the bottom, where the control rod does not influence the relative pin powers Rodded Unrodded Axial Position [m] Figure 8-33: Axial power distribution - HZP for all models 165

179 Total Power [MW] Rodded Unrodded Figure 8-34: Radial power distribution HZP for all cases Figure 8-35 shows the power spike that occurs about 1.12 seconds into the transient, while Figure 8-36 shows the increase in fuel temperature during this time, providing the negative feedback that stops and reverses the power increase. The sensitivity to temperature is extremely large such that extremely subtle differences in the temperature result in visible differences in the peak power. Compared to FRAPTRAN, CTF case 1 over-predicts the temperature, and thus under-predicts the power, whereas case 2 does the reverse CTF 1 CTF 2 FRAPTRAN Time [s] Figure 8-35: Power spike during rod ejection 166

180 Linear Power [kw/m] Fuel Temperature [K] CTF 1 CTF 2 FRAPTRAN Time [s] Figure 8-36: Fuel temperature increase during power spike Axial Position [m] CTF 1 CTF 2 FRAPTRAN Figure 8-37: Axial power distribution at time = sec 167

181 Axial Power Relative Difference Fuel Temperature [C] CTF 1 CTF 2 FRAPTRAN Axial Position [m] Figure 8-38: Axial fuel temperature distribution at time = sec Axial Position [m] CTF 1 CTF 2 Figure 8-39: Axial power relative difference at time = sec - CTF vs. FRAPTRAN rods 168

182 Temperature Difference [C] CTF 1 CTF Axial Position [m] Figure 8-40: Axial fuel temperature difference at time = sec - CTF vs. FRAPTRAN rods Figure 8-41 shows the differences in the peak radial power distribution between CTF and FRAPTRAN, while Figure 8-42 shows the temperature difference. For case 1, the lowburnup fuel temperatures are under-predicted by about 20 C, whereas the high-burnup temperatures are over-predicted by about 15 C. However, the higher gap conductivity for the high-burnup rods in case 2 reduces this difference as these rods more quickly transfer their heat to the coolant. This can be seen in Figure 8-43, which shows the coolant temperature differences between the cases with CTF and FRAPTRAN rods. For case 1, the coolant temperatures on the right side are under-predicted, as less heat has been transferred to the coolant, whereas this has been significantly reduced for case 2. CTF Rods Case 1 CTF Rods Case 2 Figure 8-41: Radial power relative difference at time = sec - CTF vs. FRAPTRAN rods 169

183 CTF Rods Case 1 CTF Rods Case 2 Figure 8-42: Radial fuel temperature difference at time = sec - CTF vs. FRAPTRAN rods CTF Rods Case 1 CTF Rods Case 2 Figure 8-43: Radial coolant temperature difference at time = sec - CTF vs. FRAPTRAN rods As can be seen in Table 8-21, TORT-TD dominates the runtime with its high-order angular discretization. FRAPTRAN has the lowest runtime, as all the rods run in parallel, whereas CTF and TORT-TD only use one core. 170

184 Table 8-21: Runtime comparison for each code CTF 1 CTF 2 FRAPTRAN Steady-state iterations Time steps TORT-TD runtime [s] CTF runtime [s] FRAPTRAN runtime [s] Total Per steady state Per time step Total Per steady state Per time step Total Per steady state Per time step Angular Discretization Comparison The model with the rods modeled in CTF was run with four different levels of modeling fidelity in TORT diffusion, S 2, S 4, and S 8, and a comparison was made using S 8 as the reference case. There is no comparison to the FRAPTRAN-rod case, and the CTF rods all have gap conductivities of 1.0e4 kw/m 2 K. Instead, the purpose here is to show the sensitivity to angular discretization in the power distribution and CPU time. Table 8-22 shows the number of degrees of freedom in the largest matrix calculation in the two codes. For CTF, this is the pressure calculation, with one degree of freedom for the pressure at each node center. For TORT-TD, the transport calculation contains a degree of freedom for each discrete angle for each energy group at each mesh cell. Table 8-22: Degrees of freedom in CTF pressure calculation and TORT-TD transport calculation CTF Nodes/degrees of freedom 500 TORT Nodes 800 Energy groups 8 Angles Degrees of freedom Diffusion S S S

185 8.4.1 Steady state The results below are for steady-state calculations that were followed by a 5-second null transient (a transient calculation with no changes in time). This null transient will only be discussed in terms of the CPU time spent on the time steps. The axially-averaged radial power relative differences from the reference case are shown below. There is a significant sensitivity to the angular discretization for lower orders with errors as high as 6% for diffusion, and 2.5% for S 2. However, for S 4, the results are similar to S 8. Diffusion s higher error is due to two factors. It does not take into account the neutron streaming the tendency of the neutrons to continue in their present direction and cross large parts of the problem domain but rather assumes that the neutrons change direction on a scale that is small compared to the regions at which the flux is resolved. Additionally, it does not take into account anisotropic scattering. The S N model in TORT- TD uses order-p 1 Legendre expansion for anisotropic scattering, whereas the diffusion approximation can only use P 0, which means the scattering is treated as isotropic. Figure 8-44: Radial power relative difference - diffusion vs. S 8 172

186 Figure 8-45: Radial power relative difference S 2 vs. S 8 Figure 8-46: Radial power relative difference S 4 vs. S 8 The axial power distribution is likewise very similar for S 4 and S 8, with moderate loss of accuracy for S 2. However, the error for diffusion is much greater, especially at the bottom of the core. 173