The Pennsylvania State University. The Graduate School. College of Engineering

Transcription

1 The Pennsylvania State University The Graduate School College of Engineering HIGH-FIDELITY MULTI-PHYSICS COUPLING FOR PREDICTION OF ANISOTROPIC POWER AND TEMPERATURE DISTRIBUTION IN FUEL ROD: IMPACT ON HYDRIDE DISTRIBUTION A Thesis in Nuclear Engineering by Ian J. Davis 2013 Ian J. Davis Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science August 2013

2 ii The thesis of Ian Davis was reviewed and approved* by the following: Maria Avramova Assistant Professor of Nuclear Engineering Thesis Advisor Kostadin Ivanov Distinguished Professor of Nuclear Engineering Arthur Motta Professor of Nuclear Engineering and Materials Science and Engineering Chair of Nuclear Engineering *Signatures are on file in the Graduate School

3 iii ABSTRACT Hydrogen absorbed into the nuclear fuel cladding during reactor exposure is distributed in highly inhomogeneous fashion as a result of temperature and stress gradients. To correctly describe and predict this hydrogen distribution there exists a need for multi-physics coupling to provide accurate heterogeneous temperature distributions in the cladding. The Department of Energy (DOE) recognized the need for better hydrogen modeling, and generously sponsored a project at the Pennsylvania State University (PSU) to investigate this need under the Nuclear Energy University Programs (NEUP). Furthermore, this study was conducted with the idea of a two-fold approach: 1. Combine accurate high-fidelity thermal-hydraulic models for heat transfer, reactor physics models for neutron flux, and thermal-mechanics models for fuel performance calculations to acquire detailed temperature and stress distributions in the fuel rod; 2. Analytically model and experimentally test the temperature and/or stress dependent hydrogen pick-up, diffusion, and precipitation in nuclear fuel cladding. The work and results specific to this thesis fall under the first part of the two-fold approach. Further detailing the first part, combination of the computational models is achieved by coupling a subchannel code to a neutronics code and a fuel performance code. The thermal-hydraulics code chosen is COBRA-TF (CTF), which is a subchannel code that was modernized and further developed at PSU. CTF s capabilities include modeling two-phase flow in transient or quasi-steady-state conditions for Light Water Reactor (LWR) design and safety analysis. The neutronics code used in this research is DeCART (supported by the University of Michigan (UM)), a code which uses the method of characteristics approach to calculate the three dimensional (3-D) neutron flux. The power distribution with respect to (x,y,z) coordinates calculated with DeCART was used to calculate the radial power fraction (ratio of power produced in one fuel pin to the power averaged over all the fuel pins in the array), and the axial and radial pin power distributions for input in CTF. This data was, in turn, used to generate the temperature

4 iv and stress distributions in the cladding. To do this, CTF was coupled to the fuel performance code BISON. BISON is a finite element code developed by Idaho National Laboratory (INL) to study all aspects of thermal and mechanical behavior of fuel rods in the reactor core. Outer clad surface temperature distributions from CTF and power distributions from DeCART are used to accurately model single fuel pins in BISON, which calculates temperature gradients in the cladding material.

5 v TABLE OF CONTENTS LIST OF FIGURES... vii LIST OF TABLES... x ACKNOWLEDGEMENTS... xi Chapter 1: Background and Introduction Background Importance of Zirconium in Reactor Core Design Corrosion of Zirconium Cladding Hydrogen Diffusion and Precipitation Clad Embrittlement and Failure Thermal-hydraulics Codes Neutronics Codes Fuel Performance Codes Introduction Research Objectives Thesis Outline Chapter 2: Utilized Computer Codes The Subchannel Thermal-Hydraulic Code CTF The Neutronics Code DeCART The Fuel Performance Code BISON Chapter 3: High-Fidelity Multi-Physics Coupling Exchange Parameters Coupling Scheme CTF to DeCART Coupling CTF to BISON Coupling DeCART to BISON Coupling Coupling Driver Hydrogen Model Balance equations description Implementation of the model in BISON Chapter 4: Test Cases Standalone BISON Benchmark with FRAPCON x4 PWR Sub-Assembly (fuel pins only) CTF Calculations DeCART Calculations... 60

6 vi 4.3-4x4 PWR Sub-Assembly (with guide tube) CTF-DeCART Coupled FRAPCON Integral Assessment Case: Rod Chapter 5: Results Standalone BISON Benchmark with FRAPCON x4 PWR Sub-Assembly (all fuel pins) x4 PWR Sub-Assembly (with guide tubes) Internal (Rod Position 6) CTF-DeCART Coupled FRAPCON Integral Assessment Case: Chapter 6: Conclusions and Future Work References Appendices Appendix A: Coupling Modules A-1: main.py A-2: _input.py A-3: _check_convergence.py A-4: _geometry.py A-5: _get_ctf_output.py A-6: _get_input_data.py A-7: _get_iso_conc.py A-8: _get_power_density.py A-9: _output.py A-10: _power_dist.py A-11: _run_ctf.py A-12: _run_decart.py A-13: _speed_up_convergence.py A-14: _temp_den_dist.py A-15: _write_ctf_input.py A-16: _write_decart_input.py

7 vii LIST OF FIGURES Figure 1-1: Example Mesh for a 10 Pellet Model in BISON [8]... 5 Figure 1-2: Hydride Effect on Impact Properties of Zircaloy-2 [3]... 8 Figure 2-1: Control Volume Defined in Cartesian Coordinates for CTF [21] Figure 2-2: Scalar Mesh Cell and Axial Momentum Mesh Cell Configuration [21] Figure 2-3: Scalar Mesh Cell and Transverse Momentum Mesh Cell Configuration [21] Figure 2-4: Angular Flux Indices in Region n [17] Figure 2-5: 2D-RZ Example Mesh from BISON Figure 2-6: Detailed View of the Inter-Pellet Gap in BISON Temperature Calculations Figure 3-1: External Coupling Diagram Between CTF, DeCART, and BISON Figure 3-2: DeCART Cell Notation for Each Flat Source Plane Figure 3-3: Flowchart for CTF-DeCART Coupling Figure 4-1: Rod Power History Figure 4-2: Rod Axial Power Shapes Figure 4-3: 4x4 PWR Sub-Assembly Layout Figure 4-4: Cross-Flow Pattern for a 4x4 Sub-Assembly Figure 4-5: Gap Connection Labels for a 4x4 Sub-Assembly Figure 4-6: DeCART 4x4 Sub-Assembly Diagram Figure 4-7: 17x17 Assembly Diagram from ORNL Figure 4-8: 4x4 Sub-Assembly Section with Internal Guide Tube Figure 5-1: Rod Average Fuel Temperature Comparison Between BISON and FRAPCON [43] Figure 5-2: Rod Average Cladding Temperature Comparison Between BISON and FRAPCON [43] Figure 5-3: Rod Rod Average Burnup Comparison Between BISON and FRAPCON [43]... 67

8 Figure 5-4: Rod Fission Gas Release Comparison Between BISON and FRAPCON Figure 5-5: Rod 7 Convergence of the Fuel Temperature at 0.0 MWd/kgU Figure 5-6: Rod 7 Convergence of the Moderator Temperature at 0.0 MWd/kgU Figure 5-7: Rod 7 Convergence of the Moderator Density at 0.0 MWd/kgU Figure 5-8: Rod 7 Converged Axial Power Distributions Figure 5-9: Rod 7 Converged Average Fuel Temperature Profiles Figure 5-10: Rod 7 Converged Moderator Temperature Profiles Figure 5-11: Rod 7 Converged Moderator Density Profiles Figure 5-12: Comparison of Coupled CTF-DeCART to the Standalone DeCART Axial Power Distribution for rod 7 at 0.0 MWd/kgU Figure 5-13: Comparison of Coupled CTF-DeCART to the Standalone DeCART Axial Power Distribution for rod 7 at 10.0 MWd/kgU Figure 5-14: Comparison of Coupled CTF-DeCART to the Standalone DeCART Axial Power Distribution for rod 7 at 20.0 MWd/kgU Figure 5-15: Comparison of Coupled CTF-DeCART to the Standalone DeCART Axial Power Distribution for rod 7 at 27.5 MWd/kgU Figure 5-16: Rod 7 Clad Outer Surface Temperature Distributions Figure 5-17: Close-up View of Mesh used in 360 Pellet Full Rod BISON Simulation with Coupled Boundary Conditions Figure 5-18: Clad Temperature Profile from BISON for Coupled Rod 7 at Years Figure 5-19: Clad Temperature Profile from BISON for Coupled Rod 7 at 1.93 Years Figure 5-20: Initial Hydrogen Concentration Along Fuel Rod Figure 5-21: Distribution of Hydrogen in Solid Solution at Days Figure 5-22: Distribution of Hydrogen in Solid Solution at Days Figure 5-23: Distribution of Hydrogen in Solid Solution at Days Figure 5-24: Radial Power Factors at BOC, MOC, and EOC for Internal Array Figure 5-25: Rod 2 Convergence of the Fuel Centerline Temperature at 0.0 MWd/kgU for Internal viii

9 ix Figure 5-26: Rod 2 Convergence of the Fuel Surface Temperature at 0.0 MWd/kgU for Internal Figure 5-27: Rod 2 Convergence of the Moderator Temperature at 0.0 MWd/kgU for Internal Figure 5-28: Rod 2 Convergence of the Moderator Density at 0.0 MWd/kgU for Internal Figure 5-29: Rod 2 Axial Power Shapes as a Function of Burnup for Internal Figure 5-30: Rod 2 and Rod 16 Converged Fuel Centerline Temperature Distributions for Internal Figure 5-31: Rod 2 and Rod 16 Converged Fuel Surface Temperature Distributions for Internal Figure 5-32: Rod 2 and Rod 16 Converged Moderator Temperature Distributions for Internal Figure 5-33: Rod 2 and Rod 16 Converged Moderator Density Distributions for Internal Figure 5-34: Rod 2 Clad Outer Surface Temperature Distributions for Internal Figure 5-35: Subchannel 7 and 19 Liquid Temperature Distributions for Internal Figure 5-36: Rod Axial Power Shapes (0-295 days) from the CTF-DeCART Coupling Figure 5-37: CTF-DeCART vs Actual Axial Power Shape (0-295 days) Figure 5-38: CTF-DeCART vs Actual Axial Power Shape ( days) Figure 5-39: CTF-DeCART vs Actual Axial Power Shape ( days) Figure 5-40: CTF-DeCART vs Actual Axial Power Shape ( days) Figure 5-41: CTF-DeCART vs Actual Axial Power Shape ( days)

10 x LIST OF TABLES Table 1-1: Mechanical Properties of Zr and ZrH 1.83 at Room Temperature... 7 Table 3-1: Hydrogen Model Constants Table 4-1: Rod Specifications from FRAPCON Table 4-2: Rod Power History Data from FRAPCON Table 4-3: Rod Axial Power Shape Data from FRAPCON Table 4-4: Rod Input Data for BISON from FRAPCON Table 4-5: Default Input Parameters in BISON Table 4-6: 4x4 PWR Sub-Assembly Specifications Table 4-7: CTF Input Parameters for a 4x4 PWR Sub-Assembly Table 4-8: DeCART Input Parameters for a 4x4 PWR Sub-Assembly Table 4-9: Guide Tube Specifications Table 5-1: Convergence of CTF-DeCART Coupled Calculations at 0.0 MWd/kgU for a 4x4 PWR Sub-Assembly with all Fuel Pins Table 5-2: Convergence of CTF-DeCART Coupled Calculations at 0.0 MWd/kgU for Internal... 90

11 xi ACKNOWLEDGEMENTS I would to thank my thesis advisor, Dr. Maria Avramova, for her support over the past two years to turn this project into a reality. Her guidance in helping me transition to graduate research and appreciate the diligence required in this field is most appreciated. I would also like to thank Dr. Kostadin Ivanov for his guidance in helping me mature academically as a researcher. His wealth of knowledge and depth of understanding in complex nuclear concepts has helped steer my work towards an attainable goal. I am indebted to my research partner, Olivier Courty, for accompanying me on this journey over the past two years. His drive and attention to detail in our collaborative studies has kept me grounded and focused on completing the task at hand. The success of the research thus far would not be the same without him. I would also like to thank Olivier s advisor, Dr. Arthur Motta, for his support during the past two years, and also for challenging me to break down problems faced in an attempt to truly understand the physics of what is happening. I need to give special thanks to the DOE for funding this project under the NEUP program (Project #: ). In particular, thank you to our technical point of contact, Brady Hanson, for his continued guidance in the direction of the project. His input and genuine interest in our work certainly aided in the strong line of communication between the DOE and PSU. I would also like to thank the group of BISON developers at the INL, especially Rich Williamson and Jason Hales, for their willingness to help us learn how to use and understand BISON. From the University of Michigan, I thank Dr. Tom Downar, and especially Dan Walter for his continued support in the use of DeCART. Lastly, to my parents, siblings, and friends, who also deserve recognition for their support and encouragement, thank you.

12 1 Chapter 1: Background and Introduction 1.1 Background The nuclear industry is always looking to improve the current capabilities of its commercial nuclear power reactors. One of the most significant challenges that nuclear power faces is fuel rod failure. Moreover, a single fuel rod failure can necessitate an unplanned reactor shutdown or even an emergency SCRAM, creating extra costs and possible safety concerns for utilities. To avoid such events, the US Nuclear Regulatory Commission (NRC) has placed a limit on the amount of exposure a single fuel rod can acquire, roughly 60 MWd/kgU in units of burnup. Much study has been done and continues to be performed with the hopes of extending fuel rod life to the limits that the NRC will allow Importance of Zirconium in Reactor Core Design A nuclear reactor core is characterized by very high pressures and temperatures, large thermal gradients, and intense nuclear radiation [1]. These extreme conditions lead to strict demands for the types of materials used in core components. Moreover, core components need to withstand large mechanical and thermal stresses, as well as large amounts of nuclear radiation, which can change the physical properties of the materials over time. While much is known about the material response to mechanical and thermal stress, the effects of intense nuclear radiation and environmental corrosion are not fully known. Cladding mechanical properties may change due to radiation damage; however, the fuel rod undergoes several annealing processes through its lifetime, which help to reverse some of the effects of radiation damage [2]. What is important to take away

13 2 is that both radiation damage and corrosion can lead to a loss of ductility, whether directly or indirectly, and can ultimately compromise the structural integrity of the cladding. Considering all of the detrimental effects of nuclear irradiation and the type of environment present inside a reactor core, Zirconium has proven to be a suitable metal for some structural components inside the core; most importantly, nuclear fuel rod cladding. Two of the most common Zirconium alloys used in the nuclear industry are Zircaloy-2 and Zircaloy-4. Both alloys contain trace amounts of tin, iron, chromium, and nickel in varying weight percentages [3]. Though, Zircaloy-4 was actually developed from Zircaloy-2 in the hopes to decrease the amount of hydrogen pick up in the core. The main difference between the two alloys is that Zircaloy-4 has reduced amounts nickel and iron. Furthermore, these alloys have relatively low neutron absorption crosssections, low corrosion rates, high melting temperatures (~1850 C), and strong heat transfer properties [3]. Though Zircaloy-4 had some improved material properties over Zircaloy-2, it is still limited in corrosion resistance for high burnup. Therefore, nuclear vendors continue to develop new alloys in the hopes of extending fuel rod life. Westinghouse has developed a Zirconium Low Oxidation (ZIRLO) alloy, which showed improved corrosion resistance in long-term irradiation testing [4]. AREVA has also developed an improved zirconium alloy, called M5, which also exhibited improved resistance to high burnup corrosion, especially during accident scenarios [5].

14 Corrosion of Zirconium Cladding During the lifetime of a nuclear fuel rod, the cladding undergoes a corrosion mechanism, given by Equation (1-1). Zr + 2H 2 O ZrO 2 + 2H 2 (1-1) The corrosion mechanism degrades the cladding material; the zirconium oxide that forms on the waterside of the fuel diminishes the heat transfer properties of the alloy. Along with the formation of the zirconium oxide, hydrogen is released by the corrosion reaction. The majority of this hydrogen remains in the coolant and is carried away; however, roughly 10-20% of the hydrogen produced in this reaction is absorbed into the cladding material. Noted in the following sections, significant amounts of hydrogen can cause issues in the structural competency of the cladding material [3] Hydrogen Diffusion and Precipitation Once absorbed into the cladding, hydrogen in solid solution diffuses by two primary mechanisms: diffusion by a concentration gradient (Fick s Law) and diffusion by a temperature gradient (Soret Effect). From an article in the Journal of nuclear materials, Sawatzky developed a semi-analytical equation to describe the diffusion of hydrogen in solid solution [6]. In the presence of concentration and temperature gradients, the diffusion of hydrogen is given by Equation (1-2). J = DN RT d(ln(n)) (RT + Q dx T dt dx ) (1-2)

15 4 J is the diffusion flux, D is the diffusion constant of hydrogen in zirconium, N is the concentration of hydrogen in solid solution, R is the gas constant, T is the temperature, and Q* is the heat of transport (empirical value according to the Soret effect) [6]. As more hydrogen diffuses into the cladding, a concentration gradient forms causing hydrogen disseminate inward from the outer edge of the clad. Recall that a temperature gradient also affects the hydrogen diffusion. Moreover, hydrogen tends to diffuse in the opposite direction of positive temperature gradients (i.e. towards colder regions). Temperatures and temperature gradients can vary in all three dimensions (r,θ,z) independent of each other. In the radial (r) direction, the fuel pellet inside the cladding is a heat source, while the water outside the cladding acts as a heat sink. The temperature drops roughly C across the width of the cladding [7]. In the azimuthal (θ) direction, geometric heterogeneity can play a role in altering the temperature gradients. For example, a fuel rod next to a guide tube encounters colder moderator temperatures on that side adjacent to the guide tube compared to the other three sides, which are next to other fuel pins. Refer to Figure 4-8 for a visual look at this type of scenario. Also, fuel rods located on the outer edges of the core may encounter temperature gradients from the core interior to the reflector region outside the fuel assemblies. In the axial (z) direction, coolant enters the core at about 287 C and exits at around 320 C [7]. The axial temperature gradient in the coolant directly affects the axial temperature gradient in the cladding. Also, the energy deposition from the fuel pellets is not uniform in the axial direction. As a simple approximation, the axial power distribution follows the cosine function with the peak energy deposition occurring halfway along the active fuel length. This also has a direct effect on the temperature profiles inside the cladding material. On a much smaller scale, current fuel pellet manufacturing practices often include dishes and chamfers in the pellet geometry to provide extra space for fission gas release, thermal expansion, and swelling. Refer to Figure 1-1, which comes from an article in the Journal of Nuclear Materials authored by the INL [8]. Temperature gradients form in the inter-pellet gaps solely due to the fact that there is less fuel material in this region of

16 5 the rod. Less fuel material means less fission, which means less energy deposition in the cladding. Considering both the variability in the concentration and temperature gradients, it is clear to notice the complex nature of hydrogen diffusion in cladding. Figure 1-1: Example Mesh for a 10 Pellet Model in BISON [8] Over time as more hydrogen ingresses into the cladding, the concentration of hydrogen in solid solution approaches the Terminal Solid Solubility for precipitation (TSSp) limit [9, 10]. Above this limit, hydrogen in solid solution precipitates into hydrides. It is important to note that the TSS follows an Arrhenius law, and is different for hydride precipitation versus dissolution. Specifically, for a given hydrogen concentration (wt.ppm) the temperature required to precipitate is lower than the temperature required to dissolve hydrides, creating a hysteresis effect. In an article

17 6 from the Journal of Nuclear Materials, Kearns describes an experiment using welded diffusion couples to estimate the TSS for zirconium alloys. Results from the experiment provide Equation (1-3), which gives the TSS for Zircaloy-2 and Zircaloy-4 below 550 C. It is noted that all hydrides dissolved above 550 C for Kearns experiments; samples were hydrided to relatively low nominal values of either 50 or 160 ppm hydrogen. TSS (wt. ppm) = exp ( 4302 T ) (1-3) T is the temperature in Kelvin. Kearn s study has been used as a basis for determining the effect of the hysteresis on the TSS equations. Several decades later, McMinn, referencing Kearn s work, published results from another experimental program to measure the dissolution and precipitation temperatures of hydrogen in Zircaloy [9]. Equation (1-4) and Equation (1-5) summarize his findings. C du (wt. ppm) = exp ( ) (1-4) T C pu (wt. ppm) = exp ( ) (1-5) T C du represents the TSS of dissolution and C pu represents the TSS of precipitation for hydrogen in zirconium. T is the temperature in Kelvin. With the use of Kearns and McMinn s equations, it is possible to estimate the steady-state amount of hydrides that have precipitated. However, the precipitation of solid state hydrogen into hydrides is not instantaneous; there are kinetics involved. The solid state hydrogen concentration reaches a supersaturation level before precipitation occurs. If precipitation is not instantaneous,

18 7 then a part of the hydrogen in supersaturation diffuses while the rest is precipitating. This affects the distribution of hydrides in the cladding. It should be noted that hydrides are immobile in the zirconium once precipitated. Marino attempted to model and numerically solve these precipitation kinetics in a two-dimensional computer program [11]. His model is shown in Equation (1-6). dc dt = J α2 (C C eq ) (1-6) dc/dt is the rate of change of the hydrogen in solid solution, J is the diffusion flux (Fick s law and Soret effect), C is the concentration of hydrogen in solid solution, C eq is the equilibrium concentration given by the TSS of precipitation, and α is the precipitation rate estimated by Marino [11] Clad Embrittlement and Failure With a brief understanding behind how hydrogen diffuses and precipitates in cladding, it is important to also look at how hydrides affect the material properties of zirconium alloys. Kimberly Colas discusses some of the material properties of hydrides in her dissertation [12]. Hydrides are notably brittle in tension, with a fracture toughness less than 2 MPa at room temperature and less than 4 MPa at 300 C. Colas references a study done by Xu et al, which compared some material properties of zirconium hydrides (ZrH 1.83 ) to pure zirconium (Zr), and presents the information shown in Table 1-1 [13]. Table 1-1: Mechanical Properties of Zr and ZrH 1.83 at Room Temperature Material Modulus (Gpa) Hardness (Gpa) K IC (MPa/m) Zr ~40 ZrH 1.83 ~

19 8 While these mechanical properties were measured well below normal operating temperatures, the effect of containing a far more brittle species in zirconium alloys is clear, especially if the amount is significant. Several studies have shown that the presence of hydride decreases the ductility of zirconium alloys on a larger scale. A reduction in ductility increases the probability of crack formation/propagation, as well as complete failure by fracture. The Oak Ridge National Laboratory (ORNL) released a document in 1962 discussing some of the detrimental effects of hydrides on Zircaloy-2, and why the nuclear industry decided to move towards Zircaloy- 4 for Pressurized Water Reactors (PWR) because of its reduced tendency to pick up hydrogen [3]. Figure 1-2 shows some of the data released by the ORNL highlighting the effect hydrides have in impact properties of Zircaloy-2. Figure 1-2: Hydride Effect on Impact Properties of Zircaloy-2 [3]

20 Clearly, it can be seen that an increased concentration of hydrides significantly reduces the impact energy needed to fracture the cladding Thermal-hydraulics Codes Thermal-hydraulics computer codes are responsible for predicting the thermal response of reactor coolant to the heat dissipated by fuel pellets. Furthermore, thermal-hydraulics codes include models for heat transfer, laminar and turbulent fluid flow, and two-phase flow, among others. In the nuclear industry, thermal-hydraulics codes are used to ensure a nuclear design meets the safety regulations of the NRC. These safety regulations cover normal operations conditions, as well as a multitude of accident scenarios. Several types of thermal-hydraulics codes exist, the most common being subchannel codes, computational fluid dynamics (CFD), and direct numerical simulation (DNS). While CFD and DNS provide significant more detail in their meshes, subchannel codes are more widely accepted as practical thermal-hydraulics codes. Section 2.1 contains more information about the thermal-hydraulics code that was used in this project Neutronics Codes In the simplest form, neutronics codes are used to study neutron flux behavior in the core. Based on material and geometry specifications, neutronics codes can predict local neutron flux for both normal operation and transient scenarios. Moreover, the most important aspect to neutronics codes are their use of cross-sections, which are generated or read in from libraries based on the type or material modeled (i.e. enriched UO 2, water, Zircaloy-4, etc.). The cross-sections are then used in the solutions of either the Diffusion Equation or the Transport Equation, resulting in flux distributions. Examples of both the Diffusion and Transport Equations can be found in Section 2.2.

21 10 Several types of neutronics codes exist today, separated into two main categories: probabilistic and deterministic. Probabilistic codes use the Monte Carlo methodology to provide a statistical analysis for modeling the interaction of nuclear particles with materials. Each particular type of interaction is modeled as an event with an associated probability of occurring. The events are grouped into probability density functions, and then sampled from to statistically describe what is occurring in the physical phenomenon. The most common Monte Carlo code used today is called MCNP [14]. The advantage to using Monte Carlo codes, like MCNP, is that there are no assumptions made, eliminating some errors. However, the Monte Carlo approach is extremely computationally expensive. Often times, without significant computing resources, models are limited to steady-state power calculations. Deterministic codes, on the other hand, solve the Diffusion and/or Transport equations to provide neutron flux distributions. There are numerous techniques for solving these equations, some of which are the finite difference approach, method of characteristics, and many other numerical methods through discrete ordinates methods or function expansions (i.e. Nodal Expansion Method (NEM)) [1]. The one thing that all deterministic codes have in common is the need to discretize space and cross-section groups, often using assumptions. These assumptions can introduce error into the calculation, but are necessary to solve the equations. Though deterministic codes require assumptions, they are significantly more practical in terms of computational expense compared to Monte Carlo methods. For that reason, deterministic codes are preferred in the use of core design for the nuclear industry. Some vendors have even developed their own deterministic neutronics codes for use in the private industry (i.e. PHOENIX/ANC from the Westinghouse Electric Company) [15]. Other common deterministic codes include DENOVO developed by ORNL, CASMO/SIMULATE from Studsvik [16], DeCART from the Korea Atomic Energy Research Institute (KAERI), Argonne National Laboratory (ANL), and the Purdue University with

22 continued support at the University of Michigan [17], and MPACT from the University of Michigan Fuel Performance Codes Fuel performance codes are used to predict the thermal and mechanical behavior of fuel rods under steady-state and transient scenarios. Generally, fuel performance codes are capable of modeling a single fuel pin at a time because of the detail needed to accurately model the effects of heat and radiation. Therefore, mesh sizes tend to be finer for fuel performance codes compared to some other neutronics and thermal-hydraulics codes. That being said, the mesh size can vary from under a hundred nodes to a few hundred thousand nodes, depending on the model. Also, fuel performance codes vary in their ability to model in one, two, or three dimensions. Two fuel performance codes are involved at some level in this project. One such NRC approved fuel performance code is FRAPCON-3.4: A Computer Code for the Calculation of Steady-State Thermal-Mechanical Behavior of Oxide Fuel Rods for High Burnup [18]. More specifically, FRAPCON measures the steady-state response of temperatures, pressures, and deformations in fuel rods for long-term irradiation in a reactor. To determine these responses, FRAPCON uses models for heat conduction, cladding elastic and plastic deformation, fuel-cladding mechanical interaction, fission gas release, rod internal pressure, and cladding oxidation. Similar to other fuel performance codes, FRAPCON requires power and bulk coolant boundary conditions. Though FRAPCON is an NRC approved fuel performance code, there are some limitations to the applications that it can model. For example, the current code can only model UO 2 fuel pellets and Zirconium alloy cladding materials cooled by light or heavy water reactor conditions. There is a validated burnup limit of 62 GWd/MTU; however, this is already beyond the current NRC limit

23 12 for burnup and FRAPCON developers are confident that the code would give reasonable predictions for burnups past 62 GWd/MTU. In terms of heat transfer and fission gas release models, only radial heat flow and release rates are calculated. Recommended time step sizes should remain between 0.1 days to 50 days. Regarding clad deformation, less than 5 percent strain is meaningfully captured by FRAPCON s models [18].

24 Introduction This study is funded by DOE under the Nuclear Engineering University Programs at the Pennsylvania State University. The official title of the sponsored project is Anisotropic azimuthal power and temperature distribution on fuel rod: Impact on hydride distribution. Dr. Arthur Motta is the principal investigator assigned to this project with collaboration from Dr. Kostadin Ivanov and Dr. Maria Avramova. The inclusion of three investigators for this project stems from the multiphysics nature of the project. On one side, the study of hydrogen pick-up, diffusion, and precipitation in cladding is heavily based on nuclear materials science. Conversely, the power and temperature distributions thought to drive the pick-up/diffusion/precipitation mechanisms are heavily based on neutronics, thermal-hydraulics, and thermal-mechanics. For this reason two graduate student research assistants are involved with the undertaking of this project: Ian Davis and Olivier Courty. Ian Davis works on the computational science side of the project under the supervision of Dr. Ivanov and Dr. Avramova. Olivier Courty works on the nuclear materials science side of the project under the supervision of Dr. Motta. Over the course of the project s first two years, several undergraduate students have been involved with these efforts: Tristalee Williams and James Kendrick aiding Ian Davis, and Daniel Nunez, Christopher Piotrowski and Kevin Cass aiding Olivier Courty. The primary goal of the NEUP project as a whole is to develop a practical and accurate means for studying the effect that power and temperature distributions have on the heterogeneous nature of hydrogen distributions inside the cladding. This thesis contains a significant part of the work that has been performed over the past two years to approach this goal. In the first year of the project much background information was gathered to determine the types of PWR models to be studied. Relevant information is presented in Section Once the appropriate models were chosen, computer codes were acquired and tested for applicability to the needs of the study.

25 14 Three separate codes were utilized in the studies that are provided in this thesis: thermal-hydraulics (CTF) [19], neutronics (DeCART) [17], and fuel performance (BISON) [8]. Also in the first year, an analytical model was developed to calculate hydrogen diffusion, an X-ray diffraction experiment was conducted to study hydride precipitation, and a second experiment was designed to study hydrogen diffusion. Refer to Olivier Courty s thesis for details on the experiments performed in the first year of the project [20]. The bulk of the work presented in this thesis occurred in the second year of the project. An external coupling was performed between the thermal-hydraulics and neutronics codes to generate reliable boundary conditions for the fuel performance code. Testing and some validation of this external coupling were also completed to justify its use. Concurrently, the analytical hydrogen model was directly implemented into the fuel performance code. Joint efforts between Ian Davis and Olivier Courty were taken to test the analytical hydrogen model in the fuel performance code, using the boundary conditions calculated in the external coupling. Those findings are summarized in this thesis. 1.4 Research Objectives Considering the computational science side of the project, the primary goal is to accurately predict the temperature distributions and gradients in the cladding of nuclear fuel rods. To attain this goal, three computer codes are coupled together, using the strengths of each code. First, the neutronics code DeCART and the thermal-hydraulics code are CTF externally coupled together using a driver to run the program. Two-way feedback is included in the neutronics to thermalhydraulics coupling. The neutronics code is one-way coupled to the third major code, the fuel performance code BISON; feedback flows from the neutronics code to the fuel performance code. Similarly, the thermal-hydraulics code is one-way coupled to the fuel performance code, with

26 15 feedback flowing from thermal-hydraulics to fuel performance. Regarding the full coupling of the thermal-hydraulics and fuel performance codes, this task is discussed in the Conclusions. Once the coupling procedures are completed, it needs to be tested. Several sub-assembly models are created, simulated, and analyzed to determine the usefulness of the coupling. Also, a corollary goal of this research is to benchmark the fuel performance code. Benchmarking this code was seen as a necessity due to the fact that the fuel performance code is still under development, and has not been fully validated yet. 1.5 Thesis Outline This section briefly describes the outline of the topics discussed in this thesis. For a detailed listing of the sections and page numbers, please refer to the Table of Contents. Chapter 1 describes the background and introduction of the thesis. In the background, the motivation for this work is explained, supported by information from literature on the topic. The introduction part of Chapter 1 goes into more detail about the proposal that was submitted and approved for this project. Here, information can be found about the funding agency, project title, research goals, etc. Chapter 2 discusses the details of computer codes used in this project. A significant amount of information about code models and some basic equations is included. Chapter 3 explains the coupling procedures that are used to provide two-way feedback between CTF and DeCART. Also, details about passing converged boundary conditions to BISON are provided. Furthermore, figures, equations, and flowcharts of the coupling provide some visual aid to help explain these procedures. Next, Chapter 4 describes all of the test cases that are modeled for benchmarking and/or coupling. Such cases include single rod models from FRAPCON, and sub-assembly arrays modeled after typical PWR specifications. Following. Chapter 4, Chapter 5 discusses the results of the benchmarking and coupling test cases. Relevant results include power distributions, temperature

27 16 distributions, convergence, fission gas release, etc. Finally, in Chapter 6 conclusions of this research are discussed. Also, plans for future work on this project are mentioned. References can be found after Chapter 6, and the Appendices contain coupling code written in Python.

28 17 Chapter 2: Utilized Computer Codes Computer codes are of great significance to the nuclear industry. The efficiency and reliability of modern nuclear codes have been paramount to the success of nuclear power over the last several decades. Nuclear reactors create such hostile conditions that often times, direct measurement of core parameters is not possible. Therefore, computer codes are used supplement direct measurement, especially during the design phase. Various types of nuclear computer codes include neutronics, thermal-hydraulics, fuel performance, lattice, and systems, among others. 2.1 The Subchannel Thermal-Hydraulic Code CTF CTF (COBRA-TF) is a two-fluid, three-field subchannel analysis code capable of modeling any vertical one-, two-, or three-dimensional component in the reactor vessel. CTF is a modernized and further improved version of COBRA-TF that was developed at Penn State University [21]. In CTF the fluid field is divided into a continuous liquid field, an entrained liquid drop field, and a vapor field. Like many thermal-hydraulic analysis codes, the equations of the flow field in CTF are solved using a staggered difference scheme in which the velocities are obtained at the mesh faces and the state variables are obtained at the cell center. CTF allows heat transfer surfaces and solid structures that interact significantly with the fluid to be modeled as rods and unheated conductors. CTF allows many parameters to be specified by the user or determined using benchmarked empirical correlations. In the last decade, CTF has been improved (including translation to FORTRAN 95), further developed and extensively validated for both PWR and Boiling Water Applications (BWR) applications. Attention has been given to improving code error checking and the input deck has been converted from fixed to free format. Then Krylov solver based numerical techniques have been implemented to enhance computational efficiency.

29 18 Improvements have been made to the turbulent mixing and direct heating models, and code quality assurance testing has been performed using an extensive validation and verification matrix. Finally, to improve code usability and enable easier future modifications and improvements, code documentation (including theory, programming, and user manuals) has been prepared. Nowadays it is one of the state-of-the-art codes for LWR thermal-hydraulic analyses. At the very core of the two-fluid model used in CTF are the mass, momentum, and energy conservation equations, which are written separately for each phase [21]. The generalized phasic mass conservation equation is given below in Equation (2-1). t (α kρ k ) + (α k ρ k V k ) = L k + M e T (2-1) The k subscript represents the fluid field under consideration; it can be l for the liquid film field, v for the vapor field, or e for the entrained droplet field. The first term on the Left- Hand Side (LHS) of the equation is the change of mass with respect to time and the second term is the advection of the field mass into or out of the volume. V is the field velocity. L k on the Right- Hand Side (RHS) of the equation represents the mass transfer into or out of phase k; modes of inter-phase mass transfer include evaporation/condensation and entrainment/de-entrainment. The last term in Equation (2-1), M T e, is the entrained droplet mass source term due to turbulent mixing. The next equation to define is the generalized phasic momentum conservation equation. t (α kρ k V k ) + x (α kρ k u k V k ) + y (α kρ k v k V k ) + z (α kρ k w k V k ) = α k ρ k g α k P + [α k (τ k ij + Tk ij )] + M k L + M k d + M k T (2-2) Similar to Equation (2-1), the first term on the LHS of Equation (2-2) is the change of volume momentum with time. The rest of the terms on the LHS represent the advection of

30 19 momentum. All terms on the LHS are multiplied by the velocity vector V k ; therefore, each term contains three components (u k i + v k j + w k k ), which results in three separate momentum equations for each of the three directions (in Cartesian coordinates). The terms on the RHS represent the gravitational force, pressure force, viscous and turbulent shear stress, momentum source/sink due to phase change and entrainment, interfacial drag forces, and momentum transfer due to turbulent mixing. It should be noted that the pressure force is independent of phase under the assumption that pressure is equal across all phases. Also, gravity is assumed to be the only body force. Though turbulent shear stress is present in Equation (2-2), it is not modeled in CTF. Rather, CTF uses a simple turbulent diffusion approximation to capture turbulent mixing. The final conservation equation to be defined is the generalized phasic energy conservation equation. t (α kρ k h k ) + (α k ρ k h k V k ) = [α k (Q k + q T k )] + Γ k h i P k + q wk + α k t (2-3) The first term on the LHS of Equation (2-3) is the change in phase energy with respect to time, and the second term is the advection of phase energy into or out of the cell. The terms on the RHS are k-phase conduction and turbulent heat flux, energy transfer due to phase-change, volumetric wall heat transfer, and the pressure work term. Assumptions with Equation (2-3) include no volumetric heat generation occurring in the fluid, radiative heat transfer only occurs between solid surfaces and the vapor/droplet fields, internal dissipation is negligible, and that pressure is uniform throughout the phases. With respect to how CTF interprets Equation (2-3), heat conduction T in the fluid is not modeled; Q k is equal to zero. Also, the q k term represents energy exchange by both turbulent mixing and void drift. In CTF, this term is only considered in the lateral and orthogonal directions. These directions are illustrated in Figure 2-1, which shows the control volume definition in CTF for Cartesian geometry.

31 20 x - vertical A X U axial (vertical) A Y y X A Z z Y V orthogonal Z W lateral (transverse) Figure 2-1: Control Volume Defined in Cartesian Coordinates for CTF [21] Another important aspect of CTF s framework is the computational mesh that is used to solve the conservation equations. With a defined mesh cell, the mass, momentum, and energy equations can be set up for each field in each of the mesh cells that together model the whole system. CTF utilizes two separate meshes, which are staggered from each other. The first mesh, referred to as the scalar mesh, is used to define all of the scalar variables: void fraction α, pressure P, enthalpy h, and the fluid properties. The second mesh, referred to as the momentum mesh, is used to define the fluid velocity field. The CTF Theory Manual claims that the choice to use a staggered mesh comes from issues with numerical stability and accuracy issues [21]. More information about these issues can be found in the work published by Patankar [22]. It should be noted that the momentum mesh is also split up into two individual meshes: transverse and axial. This is done because CTF solves for both the transverse and axial fluid velocities. See Figure 2-2 and Figure 2-3.

32 21 Figure 2-2: Scalar Mesh Cell and Axial Momentum Mesh Cell Configuration [21] Figure 2-3: Scalar Mesh Cell and Transverse Momentum Mesh Cell Configuration [21] To numerically solve the conservation equations, CTF uses a form of the Semi-Implicit Method for Pressure-Linked Equations (SIMPLE). The CTF Theory Manual again references Patankar s work, now regarding the steps of the SIMPLE algorithm [21, 22]. They are as follows: 1. Guess the pressure field, p*. 2. Solve the momentum equations to obtain fluid velocities, u*, v*, and w*.

33 22 3. Use the continuity equation to solve for the pressure field correction, p. 4. Calculate the corrected pressure field, p, by adding p to p*. 5. Calculate the corrected velocity field u, v, and w using the corrected pressure field. 6. Solve remaining discretized equations that influence the flow field (i.e. energy equation). 7. Treat the corrected pressure, p, as the new guessed pressure, p*, and repeat steps 1-6 until convergence is reached. CTF adapts this method to fit its own solution method. Regarding Step 1, a reference pressure is given to CTF by the user through input. CTF calculates the pressure field from the userdefined pressure, and use this as the initial guess. With an initial guess for the pressure field, CTF moves to solving the conservation equations at each time step in the modeled transient. Except for the first time step, each new time step will use the calculated pressure field from the previous step as a guess for the new time step pressure field. Regarding Step 2, CTF solves for the transverse momentum equations before the axial momentum equations. Regarding Step 3, CTF uses both the continuity and energy equations to solve for the pressure field correction, not just the continuity equation as stated in the algorithm. It is important to note that the pressure correction is done for every scalar cell in the mesh, resulting in a matrix of considerable magnitude to be solved. CTF offers matrix solution either by direct Gaussian elimination or by Krylov iterative methods. Steps 4 and 5 are straightforward in CTF s solution scheme. CTF s other remaining equations to be solved include the interfacial area transport equation for tracking the interfacial area of the droplet field, fuel rod heat transfer, and decay heat equations. Finally, CTF follows Step 7 by checking internal convergence criteria. If convergence is not met, then the current timestep is reduced in size and the first six steps are completed again. If convergence is met, then CTF is permitted to move to the next timestep, starting back at Step 1. The primary convergence criterion in the solution

34 23 scheme requires that the change in pressure for each cell between timestep iterations is not greater than a specified value. Though CTF is a transient code, there exists a pseudo-steady-state option. It is referred to as pseudo-steady-state because the time term still exists in all governing equations [21]. Moreover, CTF still solves the equations as a transient, but then compares a set of convergence criteria at each timestep. These convergence criteria include the amount of energy stored in the fluid, in the solid, and in the system, the global energy balance, and the global mass balance. 2.2 The Neutronics Code DeCART DeCART (Deterministic Core Analysis based on Ray Tracing) is a method of characteristics (MOC) neutronics code [17, 23]. DeCART was originally developed by KAERI as part of a United States Republic of Korea (US-ROK) collaborative National Energy Research Institute (I-NERI) project between KAERI, ANL, and Purdue University. DOE, the Electric Power Research Institute (EPRI), and Advanced Fuel Cycle Initiative (AFCI) have supported further development in the U.S. The University of Michigan is also a major supporter of DeCART and its development. Regarding the actual models in the code, DeCART is capable of modeling whole core simulations while calculating direct sub-pin level heterogeneous fluxes at power generating conditions of a PWR and BWR. Also, depletion and transient simulations are available. Input parameters include geometry, material composition, thermal operating conditions, and program execution control parameters. To solve the integral transport equation, DeCART uses the method of characteristics approach with discrete ray tracing [17]. Hursin, one of the authors of the DeCART Theory Manual, notes that the direct application of MOC to fully 3-D core models is extremely computationally expensive. For practical reasons, the choice was made to develop DeCART with MOC in the two

35 24 planar dimensions coupled to one dimensional (1-D) diffusion solutions in the axial direction. Specifically, nodal expansion methods (NEM) are used to obtain the lower order 1-D diffusion solution. The axial coupling is resolved using the 3-D coarse mesh finite difference (CMFD) solution. It should be noted that the justification for using a lower order 1-D solution in the axial direction comes from the fact that most of the heterogeneity in the core exists in the radial direction compared to the axial direction [17]. For the planar directions, the 2-D MOC approach is applied to the Boltzmann transport equation in steady-state, given in Equation (2-4). Ω φ(r, Ω, E) + Σ(r, E)φ(r, Ω, E) = q(r, Ω, E) (2-4) With the assumption of an isotropic fission source, the source term can be written as: q(r, Ω, E) = 1 4π χ(r, E) k de νσ f (r, E )ϕ(r, E ) 0 + de dω Σ s (r, Ω Ω, E E)φ(r, Ω, E ) 0 4π (2-5) In Equation (2-4) and Equation (2-5), φ(r, Ω, E) represents the angular flux per unit volume, per solid angle, and per energy. The scalar flux can be found by integrating the angular flux over the solid angle, shown in Equation (2-6). ϕ(r, E) = 4π φ(r, Ω, E)dΩ (2-6)

36 25 A derivation to find an expression of the angular flux can be found in DeCART Theory Manual [17]. Equation (2-7) shows this expression which will serve as the basic equation for determining angular flux with MOC. Note that this expression assumes a small enough region where the source term and materials are constant. φ (r r, Ω r, E) = φ (r r,0, Ω r, E) exp ( Σ (E)s ) + q (Ω r,e) [1 exp ( Σ (E)s )] (2-7) sin θ Σ(E) sin θ The LHS of Equation (2-7) represents the axially integrated angular flux for a computational plane of arbitrary thickness. To solve for the angular and scalar fluxes, DeCART divides the problem into N flat source regions. Recall that the assumptions used in the equations above require that the regions be small enough to consider constant material properties. Next, sets of parallel neutron rays are spaced equally across the model at varying angular orientations. The azimuthal angles are L equally spaced, and the polar angles are defined by a quadrature set which defines polar angles and their weights [17]. The multi-group formulation is used to separate the energy dependence into G energy groups. Using this discretization, the angular flux is shown in Equation (2-8). φ g E g l,m (r) = Δα φ (r, α l, θ m, E)dE (2-8) E g+1 α l is the azimuthal angle and θ m is the polar angle, which together make up the solid angle, Ω(α l, θ m ). g represents the energy group, and not shown is the notation, n, which represents the flat source region. The DeCART Theory Manual notes that the bar sign over the angular flux variable, which signifies the averaging in the axial direction, has been omitted for brevity, and also that Δα = 2π/N. Figure 2-4 below shows a visualization of the angular discretization scheme in DeCART.

37 26 Figure 2-4: Angular Flux Indices in Region n [17] With the average angular flux known, the solution to the 1-D diffusion equation using NEM needs to be defined. First, DeCART formulates radially integrated multi-group 1-D diffusion problem for all the cells in the problem. The derived 1-D multi-group diffusion equation for homogenous node j is given in Equation (2-9). D g j d2 ϕ g j (z) dz 2 j + Σ rg ϕ j g (z) = χ j g k eff νσ fg g j j ϕg (z) j + Σ sg g g g j ϕ g j (z) L rg (z) (2-9) The last term on the RHS of Equation (2-9) is further defined in Equation (2-10). j L rg (z) = 1 j h (J j,r xg(z) J j,l xg (z) + J j,r yg (z) J j,l yg (z)) r x,y (2-10)

38 27 Also, the spatially averaged flux is given in Equation (2-11). ϕ g j (z) = 1 A r r ϕ g j (r)dr (2-11) In DeCART s models, the solution to Equation (2-9) is achieved through source iteration. Source iteration means that the fission source is provided from the previous iteration step, while the scattering source is updated through the Gauss-Seidel manner in each groupwise solution process. At this point the diffusion equation is in a form that can be applied to NEM. Using NEM, a solution for the node averaged fluxes is determined [17]. It is important to note that the accuracy of the NEM solution improves when axial node size decreases. However, there is a trade off because smaller axial node sizes means more 2-D MOC planes are needed, which increases the computational cost of the solution. Thus, DeCART allows for the solution of the 1-D diffusion equation to occur with a finer mesh than is used for the 2-D MOC calculations. Finally, the 1-D NEM solution and 2-D MOC solution are coupled together through transverse leakage terms. For more information about how these transverse leakage terms are derived, please refer to the DeCART Theory Manual [17]. 2.3 The Fuel Performance Code BISON BISON is a nuclear fuel performance analysis tool that is currently under development at the Idaho National Laboratory in the United States [8, 24]. BISON was built using MOOSE (Multiphsics Object Oriented Simulation Environment) [25]. Basically, MOOSE is a multi-physics application framework designed to significantly reduce the expense and time required to develop new applications. BISON is a finite element code that can model a single fuel pin, individual fuel

39 28 pellet(s), or any single geometry element desired in two or three dimensions. Input parameters include the mesh file for geometry, operating conditions, thermal and mechanical boundary conditions, etc. BISON stores all output information to the mesh file, which can be analyzed in a visualization tool, such as Paraview [26]. Figure 2-5 shows an example 2-D mesh built with respect to the radial and axial directions. Due to symmetry, only half of the fuel rod needs to be modeled. Figure 2-6 depicts a close-up view of the clad temperature profile along the inter-pellet gap. The temperature scale is in Kelvin for this plot. Both Figure 2-5 and Figure 2-6 were created in Paraview. The purpose of showing these figures is to highlight level of detail included in a finite element type of code. The grid shown in Figure 2-5 represents the number of nodes per fuel pellet and clad region separately. Fuel Pellet Clad Fuel Centerline Outer Clad Wall Figure 2-5: 2D-RZ Example Mesh from BISON

40 29 Figure 2-6: Detailed View of the Inter-Pellet Gap in BISON Temperature Calculations As noted above, the BISON code is built on the MOOSE framework. Moreover, Williamson notes in an article published in the Journal of Nuclear Materials that MOOSE is a massively parallel, finite element-based framework to solve systems of coupled non-linear partial differential equations using the Jacobian-Free Newton Krylov (JFNK) method [8]. Regarding specific models in BISON, there are a number of governing equations to define the conservation of mass, momentum, and energy. First the conservation of energy is given in terms of heat conduction in Equation (2-12). ρc p T t + q E ff = 0 (2-12) T is the temperature; ρ is the density; C p is the specific heat; E f is the energy released by a single fission rate; F is the volumetric fission rate. The heat flux represented by q is shown in Equation (2-13).

41 30 q = k T (2-13) k is the material thermal conductivity. Next, the conservation of mass equation is shown in Equation (2-14). C t + J + λc S = 0 (2-14) C is the concentration; λ is the radioactive decay constant; S is the source rate for a given species of mass. The mass flux J is further defined in Equation (2-15). J = D C (2-15) D represents the diffusion coefficient. Another important definition for flux involves the diffusion of oxygen inside UO 2 fuel pellets, which describes BISON s model for fission product transport/release in the fuel pellets. Equation (2-16) shows this definition [27]. J = D ( C + CQ HRT 2 T) (2-16) D now represents the diffusivity; Q* is the heat of transport for oxygen; H is the thermodynamic factor of oxygen; R is the universal gas constant. Finally, the conservation of momentum is given by Equation (2-17). σ + ρf = 0 (2-17)

42 31 σ is the Cauchy stress tensor; f represents the body force per unit mass [8]. It should be noted that the main solution variable used in BISON is the displacement vector u. Furthermore, to determine the displacement vector for a particular node, the stress field needs to be known. BISON relates the displacement vector to the stress field through strain, using several kinematic and constitutive relations. Moreover, there are a number of material models, which can affect stress/strain. In UO 2 fuel BISON includes models for the thermal properties of both un-irradiated and irradiated fuel, swelling and densification, thermal and irradiation creep, pellet fracture, and fission gas production and release. BISON offers two models for the thermal properties of UO 2 : Fink-Lucuta and MATPRO [28-30]. Both models include temperature, porosity, and burnup dependent thermal conductivity data. Please refer to the Journal of Nuclear Materials article written by Williamson et al. for more detailed information about the exact equations/relations used in BISON [8]. The swelling model comes from empirical relations in MATPRO, while densification is determined using the empirical model found in ESCORE [30, 31]. The thermal and irradiation creep model for UO 2 also comes from MATPRO; specifically, it is the MATPRO FCREEP model [30]. To calculate pellet fracture, BISON models the phenomenon in two manners: relocation and smeared cracking. The fuel relocation model comes from a simple empirical relation found in ESCORE [31]. For the smeared cracking, Williamson references a publication by Rashid in 1974 [32]. In this model, cracking is determined when the material stress on a mesh cell exceeds some specified critical stress. Once this occurs, the stress on that particular mesh cell is reduced to zero, which also eliminates the material strength at the node. Increased cracking in fuel pellets makes the problem more difficult to solve numerically. Several fission gas release models are available in BISON, including the Forsberg-Massih model and a Simple Physic-Based Model, which captures the kinetics of both fission gas swelling and release in UO 2 [33, 34]. The Forsberg-Massih option only models fission gas release, using an empirical model for the swelling.

43 32 In the same way that fuel material models are implemented in BISON, there are material models for Zircaloy cladding. Thermal creep for Zircaloy is calculated in BISON using a model developed by Hayes and Kassner, which depends on some material constants, effective stress, activation energy, temperature, and the shear modulus [35]. The irradiation creep definition comes from an empirical model developed by Hoppe, which depends on fast neutron flux and stress [36]. Irradiation growth in the cladding is determined using an ESCORE empirical model [31]. Another important phenomenon to consider in fuel performance calculations is the heat conductance across the pellet-clad gap. BISON calculates the total conductance across the gap as a summation of the gas conductance, increased conductance due to solid-solid contact, and conductance due to radiant heat transfer. The gas conductance comes from the work done by Ross and Stoute; the increased conductance due to solid-solid contact is described in an empirical model from Olander, and the radiant conductance is derived from the Stephan-Boltzman law [8, 37, 38]. Again, please refer to the article in the Journal of Nuclear Materials written by Williamson et al. for detailed information about the equations from these models [8]. One of the most difficult phenomenon to model in a finite element fuel performance code is mechanical contact. For contact to occur between the pellet and clad, three criteria must be met: the penetration distance of one body into another must not be positive, the contact force opposing penetration must be positive in the normal direction, and either the penetration distance or contact force must be zero at all times. Similar to the work done by Heinstein and Laursen, BISON tracks node displacements, and prevents fuel pellet nodes from penetrating clad faces [39]. A clad face is defined by a set of clad nodes, with an associated area. Mechanical contact may provide added stress to the pellets and cladding, even causing clad failure.

44 33 Chapter 3: High-Fidelity Multi-Physics Coupling Code coupling between the neutronics, thermal-hydraulics, and fuel performance allows for the use of each of the code s strengths. Neutronics is particularly strong for cross-section generation and sub-sequent fission energy deposition calculations; thermal-hydraulics is strong in terms of the heat transfer and fluid mechanics calculations within the coolant; fuel performance is strong for determining the thermal and mechanical behavior of the fuel and clad material in the fuel rod. Furthermore, using these strengths allows the user to develop a better all-around model of the fuel rod. 3.1 Exchange Parameters Exchange parameters are those variables, which get passed from one code to another. To improve upon the thermal-hydraulics calculations in CTF, the code needs feedback from the neutronics code for power distribution in the fuel rods. Specifically, the axial power distribution (relative local power with respect to the axial direction), radial power distribution (relative local power with respect to the radial location inside the fuel pellet, and the radial power factor (the total relative power of one rod compared to the average power of the array. 3.2 Coupling Scheme CTF and DeCART are externally coupled together with bi-directional feedback. The coupling between DeCART and BISON is one-way with only feedback from DeCART to BISON. The coupling between CTF and BISON is one-way with only feedback from CTF to BISON. Figure 3-1 below shows the general coupling scheme.

45 34 Figure 3-1: External Coupling Diagram Between CTF, DeCART, and BISON CTF to DeCART Coupling In the axial direction (z), the spatial coupling between DeCART and CTF is one-to-one[17, 19]. It should be noted that DeCART can model the areas of the fuel rod below and above the active fuel height (i.e. the plenum). CTF does not model this region; CTF models from the bottom of the fuel rod to the top. In the radial direction (r), the spatial coupling between DeCART and CTF is one-to-one. The number of radial nodes used in DeCART and CTF can vary based on user preference. However, CTF only needs radial node input for the fuel region because this is the only heat producing region. Radial nodes may be added to the cladding, but they are not necessary for input data needed in DeCART. This should be kept in mind when including radial nodes in the DeCART input files. The spatial coupling in the azimuthal direction (θ) can vary between DeCART and CTF. DeCART has a number of options for azimuthal splicing of the cells. The choice of azimuthal dependence affects how the output is presented in DeCART. It should be noted that the CTF input cards do not have azimuthal dependence. The azimuthal dependence of fluid temperature

46 comes from the selection of subchannels surrounding the fuel rod. Therefore, if the DeCART model is split up into 4 or 8 azimuthal regions, data manipulation for the power distributions will need to be performed in order to be transferred correctly in the CTF input cards. Based on fuel specifications and all known general parameters, input files are made for both DeCART and CTF. At the beginning of calculations, the fuel, clad, and moderator temperatures and densities are be input as nominal values in the DeCART input file. First, DeCART is run. DeCART automatically makes a separate output file for the power density at every mesh cell and burnup step. These output files are manipulated into a form that CTF can understand. CTF can input radial power factors (ratio of the power in one rod to the average power of the array), the radial power distribution for each rod, and the axial power distribution for each rod. Each spliced 3-D cell has some associated volume with it. This associated volume is used to weight the volumetric power in each cell to be able to calculate the normalized power distribution for CTF. First, the power density is stored for every mesh cell on every flat source plane, represented as P i,j,k, where i is the rod number, j is the flat source plane number (axial plane number), and k is the cell number. Figure 3-2 shows an example of the cell notation for a flat source plane in DeCART. 35

47 36 Figure 3-2: DeCART Cell Notation for Each Flat Source Plane Assuming there was only one radial ring of power producing material (fuel), then cells would contain power density information. Since no power is produced in all other material regions (i.e. clad and moderator), those cells contain 0.0 s for power density. Next, each cell is multiplied by its associated volume to find the power in each cell. P i,j,k = P i,j,k V i,j,k (3-1) Now that the power is known for every cell in the problem, summate the power for every cell on a flat source plane. P i,j = P i,j,k k (3-2) Following the same process, the total power in each rod can be calculated by summing over the flat source planes for the respective rod.

48 37 P i = P i,j j (3-3) Total array power is given to DeCART as an input, but it can also be calculated by summing the total power of each rod. The calculated total array power is checked against the DeCART input value for array power to ensure that power is conserved during data manipulation. TP = P i i (3-4) The total number of fuel rods in the array shall be denoted as I; the total number of flat source planes for each fuel rod shall be denoted as J; the total number of cells in each flat source plane for each fuel rod shall be denoted at K. The average power density in all fuel rods of the array can be found using the following equation. P rod = TP IV i (3-5) V i is the total volume per rod; it is assumed all fuel rods in the array have the same volume of fuel material. Next, the normalized Radial Power Factor (RPF) for each rod can be determined by dividing the power density in each rod by the average power density per rod in the array. RPF i = P i 1 (3-6) V i P rod Since the volumes are identical for each rod, Equation (3-6) can also be written in another form.

49 38 RPF i = P ii TP (3-7) The next piece of information that CTF needs is the axial power distribution for each fuel rod. First, the average power density per flat source plane for each rod needs to be calculated. P i,plane = P i JV i,j (3-8) V i,j is the total volume for a flat source plane for a particular fuel rod; again, it is assumed that the flat source planes are equally spaced, meaning that V i,j is identical for any rod and flat source plane combination. Therefore, the normalized relative power in one flat source plane of a fuel rod is given by: RP axial i,j = P i,j 1 V = P i,jj (3-9) i,j P i,plane P i To calculate the radial power distribution on the sub-pin level, slightly different notation is needed. r will represent the radial region of interest with R total radial nodes, and l will now represent a cell contained in radial region r with L total cells in each radial region r. Starting with the power in each mesh cell, the power in each rod for each radial region on each flat source plane is represented by: P i,r,j = P i,r,j,l l (3-10) Thus, the power in each rod in each radial region summed over all flat source planes is given as follows:

50 39 P i,r = P i,r,j j (3-11) Similar to Equation (3-8), the average power density per radial region of each rod can be calculated using the equation shown below. P i,radial = P i RV i,r (3-12) The volume associated with a radial region of a fuel rod, V i,r, is assumed to be the same regardless of fuel rod and radial region combination. Therefore the normalized relative power for a radial region of a particular rod is given by: RP radial i,r = P i,r 1 V = P i,rr (3-13) i,r P i,radial P i Once the radial power factors and power distributions are calculated, CTF is run. Relevant results from CTF output include bulk coolant temperature and density, cladding inner and outer surface temperatures, fuel pellet surface temperature, and fuel pellet centerline temperature. These results are given with respect to the axial direction, and also split into four azimuthal regions. Since the CTF results are given in separate azimuthal sections, data manipulation is performed to be input into DeCART. First, the moderator, fuel surface, and fuel centerline temperatures are stored in separate arrays for every rod i, axial node j, and azimuthal node k. As an example, the moderator temperatures are represented by T mod i,j,k. In the current coupling scheme, temperature feedback is only given to DeCART with respect to the radial and axial directions. For that reason, the

51 moderator, fuel surface, and fuel centerline temperatures must be averaged about the azimuthal direction. An example of this can be seen in the equation below. 40 T mod i,j = 1 K T mod i,j,k k (3-14) K represents the total number of azimuthal sections in the CTF calculation; generally, there are 4 azimuthal sections. Note that this averaging is also done for the fuel surface and fuel centerline temperature arrays. Similarly, the moderator density is reported for each subchannel sc, and axial node j. Therefore, the moderator density is also averaged about a particular rod for every axial node. Finally, for simplicity only one fuel temperature is supplied to DeCART for every fuel rod at each axial level. The fuel surface temperatures and fuel centerline temperatures are combined, using a weighted average. T fuel i,j = (0.7)T surface cl i,j + (0.3)T i,j (3-15) The feedback from CTF and DeCART are passed to each other using an iterative scheme until a steady-state solution is converged upon. Convergence is measured by the relative change of some parameter from one coupled iteration to the next. Please find the definition of convergence for a local temperature calculation below: Convergence maximum ( T i+1 T i T i ) 10 3 (3-16) i represents the iteration number. For the purposes of this coupling, the convergence criterion was set at A convergence acceleration technique was also used to improve the efficiency of the

52 CTF-DeCART coupled calculations. Below shows an example of how the convergence acceleration would be used on a local temperature. 41 weighted T fuel,i = (1 actual ω)tfuel,i 1 + ωt fuel,i (3-17) Omega is generally 0.5 for PWRs. Once a steady-state solution has been reached, DeCART is permitted to deplete to the next time step. Isotopic concentrations are tracked during depletion for every rod i, flat source plane j, and cell k. The specifc isotopes of interest are U-235, U-238, Pu- 238, Pu-239, Pu-240, Pu-241, Xe-135, and Sm-149. DeCART reports isotopic concentrations in terms of number density [#/barn/cm]. Therefore, the same equations apply to determine the isotopic concentration for each axial plane of every pin, as were used to normalize the power. These isotopic concentrations are supplied to the DeCART input with the CTF feedback from the previously converged steady-state solution. Now the coupling is ready for the next steady-state calculation. Iterations in the new steady-state calculation keep the isotopic concentrations constant, while iterating on the temperature and density distributions to approach a new steady-state solution. This process continues until the final timestep is reached CTF to BISON Coupling Since BISON is a finite element code, the coupling between it and CTF cannot be one-toone. BISON requires a separate mesh file, which is significantly finer than any CTF input [19, 24]. However, BISON does not require boundary conditions to be placed on every single node. Boundary conditions can be taken with respect to the axial and radial directions in meters. Therefore, the coupling from CTF to BISON is pseudo-one-to-one. Also, BISON can only model one fuel pin at a time. Therefore, a 4x4 array of fuel pins modeled in a single CTF input will need

53 42 16 BISON input files to run. Current plans are to pick the hottest pin found in the CTF and DeCART calculations to be modeled in BISON. Because BISON is using finite element method, the code is very computationally expensive and modeling 16 two- or three-dimensional pins separately in BISON is not practical. Based on the fuel specifications and all known general parameters, input files are made for both CTF and BISON. Previous iterations between CTF and DeCART shall be carried out before this step. Results from CTF needed for BISON calculations include the outer surface temperature of the cladding. This temperature distribution is used as a boundary condition in BISON. Based on the BISON model, the cladding temperatures can be input as steady-state or time-dependent. Also, spatial dependence in the axial and azimuthal directions depends on the BISON model that is created (i.e. 2-D or 3-D) DeCART to BISON Coupling Similar to CTF and BISON, the spatial coupling between DeCART and BISON is not oneto-one. However, the input structure of BISON allows for flexibility when including neutronics data. It can be assumed that working input files have been generated for both DeCART and BISON. Similar to the way DeCART power data was manipulated for CTF, BISON utilizes normalized axial peaking factors for its single pin model [24]. It is important to note that the ability to add radial and azimuthal dependence on peaking factors exists. To explain how to do this, one must first understand how the peaking factors are applied in BISON. Generally, the axial peaking factors are supplied to BISON with a power history in W/m. The local power associated with some axial location of the fuel rod can be calculated by multiplying the linear heat rate in W/m (from the power history) with the axial peaking factor at that location. Thus, the power shape of the fuel rod in the axial direction is known. An alternative way to represent the power distribution in BISON is to

54 43 supply the fission rates. A fission rate distribution would be able to be calculated from the power distribution from DeCART, along with the energy per fission.

55 Coupling Driver A python driver was used to control all aspects of the CTF-DeCART coupling, and the writing of the output files for BISON s boundary conditions. The flowchart in Figure 3-3 shows the basic operating procedure for the coupling. Initialization using Python Driver Converge steady-state DeCART calculation Extract power density from DeCART and supply to CTF in the form of axial power profiles, radial power profiles, and relative pin powers Converge CTF (steady-state) with new power data Update DeCART input file (i.e. fuel and moderator temperatures, moderator densities, isotopic concentrations) Extract fuel temperature, moderator temperature, and moderator density First pass? YES NO Convergence of CTF exchange parameters? NO YES Convergence of DeCART exchange parameters? NO Write BISON boundary condition files YES YES Final time step? NO Finalize output files Update DeCART input with converged parameters END Converge DeCART depletion calculation Extract isotopic concentrations for U-235, U-238, Pu-238, Pu239, Pu- 240, Pu-241, Xe-135, and Sm-149 Figure 3-3: Flowchart for CTF-DeCART Coupling

56 Hydrogen Model A model describing the behavior of hydrogen in Zircaloy-4 was derived from the equation given in section The global model is described in section Then, the model was implemented into the fuel performance code BISON, as explained in section Details of this implementation are given in Olivier Courty s thesis [20] Balance equations description From the precipitation, dissolution, and diffusion models explained in section 1.1.4, the balance equation for hydrogen in solid solution and hydride concentration can be deduced. The variation of hydrogen in solid solution per unit of time is given by the sum of the net flux, the hydrogen created by the dissolution of hydride minus the hydrogen transformed into hydride due to precipitation. Based on the Sawatzky diffusion model given by Equation (1-2), the diffusion flux is equal to: J = D C ss DC ssq RT 2 T (3-18) Hydride precipitation occurs when the C ss surpasses the TSSp. Hydride dissolution occurs when the C ss becomes lower than the TSSd. The TSSp and TSSd values measured by McMinn [40] have been used for the current work. Recalling Equation (1-4) and Equation (1-5), TSSd (wt. ppm) = exp ( T (K) ) TSSp (wt. ppm) = exp ( [ T (K) ) (3-19)

57 According to Marino s Equation (1-6), the rate of precipitation (in wt.ppm/s) is given by: 46 R precipitation = α 2 (C ss TSSp) (3-20) The dissolution is assumed instantaneous by most authors and is assumed here. In order to simplify future calculations, a linear law for the dissolution is assumed, with a characteristic time very small compared to the precipitation characteristic time: R dissolution = β 2 (C ss TSSp) (3-21) Note: β α, β l2 D The diffusion coefficient is calculated using Kearns correlation [41]. D = A diff exp ( Q Diff RT ) (3-22) Four different cases have to be taken into account for the writing of the balance equations. In the first case, the concentration of hydrogen in solid solution is greater than the TSSp. Then, precipitation occurs according to the laws described above. Precipitation: if C > TSSp, { dc ss dt =. J α2 (C ss TSSp) } (3-23) dc p dt = α2 (C ss TSSp) In the second case, the concentration in solid solution is between the TSSp and the TSSd. This is the hysteresis area, where neither diffusion nor precipitation occurs.

58 Hysteresis: if TSSp C > TSSd, { dc ss = J dt } (3-24) dc p dt = 0 47 In the third case, the concentration in solid solution is below the TSSd. The hydrogen in the precipitated hydrides (C p ) dissolves so that the C ss matches the TSSd value. This is possible only if there are hydrides (C p > 0). Dissolution: if TSSd C and C p > 0 and J > 0, { dc ss dt = J + β2 (TSSd C ss ) } (3-25) dc p dt = β2 (TSSd C ss ) In the fourth and last case, the concentration in solid solution is below the TSSd, but there are no more hydrides to dissolve. In that case, the only change to hydrogen concentration comes from net diffusion flux. Diffusion only: if TSSd C and C p = 0, { dc ss =. J dt } (3-26) dc p dt = 0 The model constants have been taken from the literature and are summarized in Table 3-1.

59 48 Table 3-1: Hydrogen Model Constants Phenomenon Parameter Value Unit Source Comments Fick s law A Dff 7.90*10-7 m 2 /s [10] Longitudinal diffusion Q Diff 4.49*10 4 J/mol [10] Longitudinal diffusion Soret effect Q* 2.51*10 4 J/mol/K [42] Average value Precipitation A P 1.39*10 5 wt. ppm [40] Unirradiated Q P 3.45*10 4 J/mol [40] Unirradiated Dissolution A D 1.06*10 5 wt. ppm [40] Unirradiated Q D 3.60*10 4 J/mol [40] Unirradiated Precipitation A α 6.23*10 1 s 1/2 [42] kinetics Q α 4.12*10 4 J/mol [42] Implementation of the model in BISON The equations describing the behavior of the hydrogen in Zircaloy-4, introduced in Section 1.1.4, have been implemented in the fuel performance code BISON. The hydrogen concentration is governed by two balance equations. Since BISON is based on Galerkin Finite Element models, some derivations have to be applied to the equation. They are transformed into what is called their weak form. The calculation can be found in Olivier Courty s thesis [20]. Usually, the boundary conditions for the hydrogen are expressed in term of flux at the cladding surface. The flux is assumed to be zero for all the surfaces except the cladding/coolant interface. At this interface, the flux can be deduced from the oxidation kinetics values and the pickup values. These conditions have been used in all the hydrogen simulations. BISON is a based on the MOOSE framework and is a modular code. Once new equations, such as the hydrogen model, are implemented, they are solved internally with all the other equations (i.e. temperature and stress equations). At each timestep, the code calculates the value of each parameter, internally coupling the equations. However, the hydrogen model needs very small

60 49 timesteps (<250 s) due to the precipitation kinetics, which is a fast process. On the other hand, the temperature and stress calculation can be made at very large timesteps (~10 6 s). Calculating the solution for each hydrogen timestep in-between depletion timesteps is not practical. Moreover, there is no feedback effect implemented from the hydrogen concentration on any of the other parameters. Therefore, the calculation is usually made in two steps. First the temperature and stress values are calculated with large timesteps. Then, the hydrogen model is run with small timesteps, using interpolated values of the clad temperature from the first calculation. In this manner, the hydrogen model in BISON can run completely separately from the depletion calculations, reducing the overall computational expense of the simulation.

61 50 Chapter 4: Test Cases 4.1 Standalone BISON Benchmark with FRAPCON During Summer 2012, an undergraduate student, Tristalee Williams, took part in a research fellowship at Penn State. As part of the fellowship, Tristalee joined our research efforts for the summer and conducted a benchmark of BISON against the fuel performance code FRAPCON. A detailed report of the benchmark can be found here [43]. For this benchmark, a single rod case was chosen from FRAPCON s Integral Assessment document, which is a collection of experimental fuel rod cases that were modeled in FRAPCON to validate its models for the NRC s approval. The chosen case is titled 15309, and was a PWR fuel rod irradiated for 4.25 years in the Oconee nuclear power plant in South Carolina. Table 4-1 shows the rod specifications that were taken from the FRAPCON input for rod

62 51 Table 4-1: Rod Specifications from FRAPCON Type Value Units Reactor PWR Layout Single pin Fuel UO2 Enrichment 3.00% Fuel density g/cc % of theoretical density (10.96 g/cc) 95% Burnable poison None Clad Zircaloy-4 Clad density 6.55* g/cc Coolant H2O Fill gas Helium Fuel pellet radius cm Clad inner radius cm Clad outer radius cm Clad thickness 673 microns Pin pitch cm Active fuel height 358 cm Total power -- MW Average linear heat rate -- kw/m Core pressure Mpa Mass flow rate kg/s Inlet temperature C *Not given in FRAPCON data The total power and average linear heat rates are not shown in Table 4-1 because these values are not constant. The linear heat rates are given as a function of time, and combined with axial power shapes to compute the power produced by the fuel rod. Table 4-2 below shows the power history data for rod Figure 4-1 visualizes the power history data from Table 4-2.

63 52 Table 4-2: Rod Power History Data from FRAPCON Time [days] Linear Heat Rate [kw/m]

64 53 Figure 4-1: Rod Power History As stated, axial power shapes are provided to FRAPCON as a function of time. This data is presented in Table 4-3 and visualized in Figure 4-2. Table 4-3: Rod Axial Power Shape Data from FRAPCON Relative Power at various time steps days days days days Axial Location [m] days

65 54 Figure 4-2: Rod Axial Power Shapes Apart from the general specifications supplied by the FRAPCON input file, BISON needs some more rod information to fully build the model. Those parameters, which can be found in FRAPCON or calculated using FRAPCON parameters, are listed in Table 4-4. Remaining BISON input parameters not available in FRAPCON are default values; those are listed in Table 4-5. Table 4-4: Rod Input Data for BISON from FRAPCON Type Value Units Pellet height 1.78 cm # of pellets 201 Pellet dish height mm pellet end-dish shoulder width 1.27 mm m Fuel surface roughness 5.99E-07 m Clad surface roughness 5.00E-07 m Initial fill gas pressure 3.31 Mpa Plenum to fuel ratio Fuel to dish volume ratio* *ratio of actual volume to cylinder volume

66 55 Table 4-5: Default Input Parameters in BISON Type Value Units Energy per fission 3.20E-11 J/fission Young s modulus: UO E+11 Pa Zircaloy E+10 Pa Poisson s ratio: UO Zircaloy Thermal expansion coefficient: UO E-05 1/K Zircaloy E-06 1/K UO 2 grain radius 1.00E-05 m Clad thermal conductivity 16 W/mK Clad specific heat 330 kj/kgk 4.2 4x4 PWR Sub-Assembly (fuel pins only) For this test case, a 4x4 pin array comprising only fuel pins was modeled in CTF and DeCART. The fuel rods were modeled after typical PWR specifications; all general specifications can be seen in Table 4-6. Some specifications were taken from a study done at ORNL, using the Advanced Multi-Physics Nuclear Fuel Performance Code (AMPFuel) [44]. Other specifications were taken from typical PWR inputs built for use in SIMULATE-3, as part of the university version of the Studsvik Scandpower Code System (CMS) for research and education purposes [16]. Using DeCART s depletion calculations, the 4x4 array was burned up to 40.0 MWd/kgU (~1036 effective full power days).

67 56 Table 4-6: 4x4 PWR Sub-Assembly Specifications Type Value Units Reactor PWR Layout 4 x 4 Fuel UO2 Enrichment 3.45% Fuel density 10.4 g/cc % of theoretical density (10.96 g/cc) 95% Burnable poison None Clad Zircaloy-4 Clad density 6.55 g/cc Coolant H 2 O Fill gas Helium Fill gas density g/cc Fuel pellet radius cm Clad inner radius cm Clad outer radius cm Clad thickness 570 microns Pin pitch 1.26 cm Active fuel height cm Top reflector height cm Bottom reflector height cm Array power MW Average linear heat rate 18.5 kw/m Core pressure 15.5 Mpa Mass flow rate 4.86 kg/s Beginning of Cycle (BOC) boron loading 1000 ppm Inlet temperature 287 C Figure 4-3 shows the fuel pin layout for DeCART. As a first approximation, no guide tubes or burnable absorbers were included in the 4x4 model; however, such characteristics of PWR s are added once initial testing of the external coupling is complete. Also, in the same figure the dotted lines separate the 25 subchannels that were modeled around the fuel pins in CTF.

68 57 Figure 4-3: 4x4 PWR Sub-Assembly Layout CTF Calculations CTF inputs require geometry, state conditions, and boundary conditions to be specified. These input parameters are calculated based on pin and lattice dimensions along with core conditions that are given or assumed. Example calculations for these input parameters are shown in the following section. All necessary data is provided in Table 4-1. From the AMPFuel data, CTF parameters like the subchannel flow areas, wetted perimeters, gap distances between rods/boundaries, distance between the centers of subchannels, etc. Important calculated input parameters for CTF are shown in Table 4-7. Figure 4-4 shows the directional cross-flow pattern that was input for the 4x4 sub-assembly. Note the choice for directional flow is arbitrary; either direction can be chosen. For this model, all directional cross-flow is pointed toward the North or West directions. Figure 4-5 shows the subchannel gap connection labels used in the 4x4 subassembly CTF input.

69 58 Table 4-7: CTF Input Parameters for a 4x4 PWR Sub-Assembly Type Value Units Enthalpy of the fluid: inlet 1233 kj/kg outlet 1434 kj/kg Subchannel flow areas: center 8.77E-05 m 2 side 4.38E-05 m 2 corner 2.19E-05 m 2 Wetted Perimeter: center 2.99E-02 m side 1.50E-02 m corner 7.48E-03 m Gap thickness: center 3.08E-03 m side/corner 1.54E-03 m Distance between the center of two channels: center-to-center 1.26E-02 m side-to-side 1.26E-02 m side-to-center 9.45E-03 m side-to-corner 9.45E-03 m Pressure loss coefficient (vertical) of spacer

70 59 Figure 4-4: Cross-Flow Pattern for a 4x4 Sub-Assembly Figure 4-5: Gap Connection Labels for a 4x4 Sub-Assembly

71 DeCART Calculations Similar to CTF, DeCART inputs require geometry, state conditions, and boundary conditions to be specified. Among these parameters, DeCART also allows options for depletion. Table 4-8 shows the DeCART input parameters not already stated in Table 4-6 or Table 4-7. Table 4-8: DeCART Input Parameters for a 4x4 PWR Sub-Assembly Type Value Units Weight % in UO2: U % U % O % Weight % in Zircaloy-4 Natural Zr % Natural Sn 1.450% Natural Fe 0.145% Natural Cr 0.100% Natural Ni 0.007% Depletion step size 2.5 MWd/kgHM # of azimuthal divisions in fuel region 4 # of radial divisions in fuel region 3 # of axial divisions in fuel region 18 Albedo boundary conditions: top/bottom Vacuum north/south/east/west Reflective Included in the DeCART source package is a Java graphical user interface (GUI) called DeCARTograph [17]. Using this GUI, it is possible to visualize the model layout; the 4x4 PWR sub-assembly with only fuel pins diagram is depicted in Figure 4-6.

72 61 Figure 4-6: DeCART 4x4 Sub-Assembly Diagram 4.3 4x4 PWR Sub-Assembly (with guide tube) The next step in testing the capabilities of the CTF-DeCART coupling involves adding guide tubes to the model. A 4x4 sub-assembly model is chosen from a typical 17x17 PWR assembly that was modeled by the ORNL for the AMPFuel code [44]. The full assembly layout is depicted in Figure 4-7. Figure 4-8 shows the 4x4 section of this full assembly that was modeled and tested in the CTF-DeCART coupling.

73 Figure 4-7: 17x17 Assembly Diagram from ORNL 62

74 63 Figure 4-8: 4x4 Sub-Assembly Section with Internal Guide Tube All geometry parameters and state conditions will remain the same as the 4x4 PWR subassembly with all fuel pins. Refer to Table 4-6, Table 4-7, and Table 4-8 for these parameters. Table 4-9 shows the specifications for the guide tubes. Table 4-9: Guide Tube Specifications Type Value Units Guide tube inner radius cm Guide tube outer radius cm

75 CTF-DeCART Coupled FRAPCON Integral Assessment Case: Rod Similar to how Rod served as a benchmark for standalone BISON calculations, it was also be modeled in the CTF-DeCART external coupling. The purpose of this is to test how well the coupled codes predict the axial power profiles, which are already known from the FRAPCON input files. No new information is needed to run the coupled calculations. Please refer to Section 5.5 to observe this validation technique.

76 65 Chapter 5: Results 5.1 Standalone BISON Benchmark with FRAPCON The FRAPCON simulation of rod ran in 34 time steps of varying length, while the BISON simulation ran in 87 time steps of varying length [43]. Therefore, a linear interpolation is used to allow parameters/variables of interest (i.e. temperature, pressure, stress, etc.) to be compared between the two codes. Also, the percent error method, Equation (5-1), calculates the average difference between a calculated parameter in BISON and a calculated parameter in FRAPCON with respect to space and/or time. The FRAPCON value is set as the theoretical value, because it is an NRC approved nuclear performance code that has already been verified and validated. And the BISON value was set as the experimental value in the equation. %Error or Average Difference = BISON value Frapcon value Frapcon value 100 (5-1) The first variable compared is temperature. Figure 5-1 shows the comparison of average fuel temperature and Figure 5-2 shows the comparison of average cladding temperature between BISON and FRAPCON. The shapes of temperature predictions match well with each other and with the power history of the rod, shown in Figure 4-1. There is a 14% average difference between the BISON and FRAPCON calculated average fuel temperatures and only a 2% average difference between the calculated average clad temperatures. Also, notice that BISON over estimates the average fuel temperature while it under estimates the average cladding temperature. This phenomenon may be attributed to a difference in calculating fuel-clad gap conductance between the two codes, which may be based on calculating a different fuel-clad gap distance in each code.

77 T (⁰F) T (⁰F) Comparison of BISON and Frapcon Average Fuel Temperature BISON Frapcon Time (days) Figure 5-1: Rod Average Fuel Temperature Comparison Between BISON and FRAPCON [43] Comparison of BISON and Frapcon Average Cladding Temperature Time (days) BISON Frapcon Figure 5-2: Rod Average Cladding Temperature Comparison Between BISON and FRAPCON [43]

78 RAB (MWd/kgU) 67 The second variable compared in this benchmark was rod average burnup. Figure 5-3 shows the comparison of BISON and FRAPCON rod average burnup. The shape of the rod average burnup predictions match well together and the average difference between the two codes is 2%. 5.00E E E+01 Comparison of BISON and Frapcon Rod Average Burnup 2.00E E+01 BISON Frapcon 0.00E Time (days) Figure 5-3: Rod Rod Average Burnup Comparison Between BISON and FRAPCON [43] The final variable compared in this benchmark is fission gas release. Figure 5-4 shows the comparison of BISON and FRAPCON fission gas release. There is a major difference between the two codes. BISON is a newer fuel performance code that is still under development. Note that at the time of this benchmark, fission gas release was one of the models within the BISON code that was incomplete. INL has since introduced the Simple Physics-Based Model as an alternative option to the Forsberg-Massih model for fission gas release. An updated simulation with the new fission gas model may improve results. Fission gas release has a large impact on the pressure and stresses calculated within the fuel rod. It will also have an effect on gap conductance calculations, due to the makeup of the gas within the fuel rod void volumes and plenum.

79 % FGR Comparison of BISON and Frapcon Fission Gas Release Time (days) BISON Frapcon Figure 5-4: Rod Fission Gas Release Comparison Between BISON and FRAPCON It is evident that BISON compares well when calculating temperatures and rod average burnup of the fuel rod. BISON is also still under development and should continue to improve its models with time x4 PWR Sub-Assembly (all fuel pins) As discussed in Section 3.2 of this thesis, inputs for DeCART and CTF were built first. The necessary parameters were exchanged between the codes, and a 10-3 convergence criterion was placed on the relative change between the fuel temperatures, coolant temperatures, and coolant densities in successive iterations. For the initial testing of the offline/external coupling, the CTF-DeCART coupled simulation converged in less than 10 iterations at each depletion step. The convergence of CTF and DeCART at beginning of cycle (BOC), i.e. 0.0 MWd/kgU, can be seen in Table 5-1. Note that the maximum local change between iterations for exchange parameters is always in the fuel centerline

80 69 temperature for this particular model. Also, the convergence of the k-effective is shown in Table 5-1. The k-effective represents the ratio of the number of neutrons produced in one generation of fissions compared to the preceding generation. Once the iterations were completed, an input model was developed for the BISON calculations. For simplicity and due to BISON s limitations, a single pin (Pin 7) was chosen from the 4x4 array to be modeled in BISON. Pin 7 was picked arbitrarily because relative power among the other pins is close to identical. It should be noted that reflective boundary conditions were used. For a 4x4 sub-assembly with all identical fuel pins, the RPF between the rods is insignificant. The clad outer surface temperature profiles have been provided from CTF to BISON at different burnups. The axial power profiles have been provided from DeCART to BISON at different burnups. Table 5-1: Convergence of CTF-DeCART Coupled Calculations at 0.0 MWd/kgU for a 4x4 PWR Sub-Assembly with all Fuel Pins Iteration Converged Parameter Maximum Local Change b/w Iterations k-eff from DeCART 1 (first pass) moderator density moderator temperature fuel surface temperature fuel CL temperature *Maximum local change remains in the fuel centerline temperature for every iteration Referring to Table 5-1, it can be seen that the k-effective converges to a relative difference of 2.511E-06 and an absolute difference of 3 pcm (per cent milli) in the final iteration. A k-effective above 1 means that the model is super-critical; more neutrons are created in every generation than the one before it. Generally, commercial nuclear reactors keep the core average at a k-effective equal to /- a few pcm. However, considering that the current model is simply a 4x4

81 70 section cut-away of a PWR lattice, it is comprehensible to have a neutronics model with a k- effective above 1.0. Moreover, while the core average k-effective is equal to 1.0, their in-lies sections of the core above and below a k-effective equal to 1.0, essentially balancing each other. Figure 5-5 shows the convergence of the fuel temperature at the BOC depletion step, where burnup is equal to 0.0 MWd/kgU. Figure 5-6 shows the convergence of the moderator temperature at the BOC depletion step, where burnup is equal to 0.0 MWd/kgU. Figure 5-7 shows the convergence of the moderator density at the BOC depletion step, where burnup is equal to 0.0 MWd/kgU. Figure 5-5: Rod 7 Convergence of the Fuel Temperature at 0.0 MWd/kgU

82 71 Figure 5-6: Rod 7 Convergence of the Moderator Temperature at 0.0 MWd/kgU Figure 5-7: Rod 7 Convergence of the Moderator Density at 0.0 MWd/kgU

83 72 Referring to Figure 5-5, Figure 5-6, and Figure 5-7, it is evident that the exchange parameters in the moderator/coolant converge significantly faster than the fuel surface temperature of fuel centerline temperature. Also, take note of how the shape of the curves progresses in the three previous plots. Noted in Section 3.2.1, the first iteration captures CTF results, which include the DeCART feedback; however, the first DeCART simulation does not contain any temperature or density profiles. As a result, DeCART produces a perfectly cosine shaped axial power profile (see Figure 5-12 below), and this power shape is directly reflected in the fuel temperature profile. Progressing through the iterations, the effect of the temperature and density feedback is evident. Colder moderator temperatures, and higher densities, in the bottom region of the reactor cause the power to shift in this direction. More dense coolant leads to more neutron moderation, which leads to higher neutron flux, and ultimately, higher power output. Therefore, it would be expected to see the peak fuel temperature converge to region along the bottom half of the fuel rod. The effect is also seen in the moderator temperature and density profiles. Though the exit temperature is the same during each iteration, the moderator temperature profile increases more quickly over the length of the rod for the converged iteration. As expected, the moderator density follows the same pattern and decreases more quickly at the converged iteration. The effect of these converged temperature profiles on the axial power distribution is explained next. While monitoring the exchange parameters from iteration to iteration is important, it is also beneficial to look at the converged parameters as they evolve through depletion. Figure 5-8 shows the axial power distribution for rod 7 as a function of burnup. Figure 5-9 shows the converged average fuel temperature distribution for rod 7 as a function of burnup.

84 73 Figure 5-8: Rod 7 Converged Axial Power Distributions Regarding the BOC timestep from Figure 5-8, the effect of the moderator temperature profile is clearly shown. For the converged solution, the axial power distribution is slightly bottompeaked; peak axial power occurs at ~40% of the active fuel height. Due to reasons already explained above, this is expected. As depletion progresses, the fuel becomes more burned near the center of the rod compared to the ends of the rod. A greater level of burnup near the center of the rod suppresses the relative power in that region. Also, more fission products build up near the axial center of the rod, adding to the effect of the power suppression. Observe the axial power shape at end of cycle (EOC), 40.0 MWd/kgU; the axial power shape evolves from slightly bottom-peaked to flat, and finally to top-peaked. The evolution of axial power shape can be directly attributed to the axial offset anomaly. The NRC released a report prepared by ORNL in 2002, which discusses the effects of axial offset [45]. Specifically, Wagner attributes the axial offset seen in commercial PWRs to the difference in moderator density along the active fuel length; higher density moderator near the core inlet results in higher reactivity in that region, which increases the burnup at the bottom of the core faster than the top. At the same time, Wagner also notes that there are other of

85 74 factors that help to cause axial offset (i.e. control rod insertion, non-uniform operating history, etc.). Nevertheless, the effects of axial offset described in Wagner s report are clearly seen in the evolution of the axial power shape in Figure 5-8. Figure 5-9: Rod 7 Converged Average Fuel Temperature Profiles Note that the fuel temperature profiles in Figure 5-9 follow the axial power shapes in Figure 5-8 closely. This is expected and is a direct result of the coolant temperature profile. Explained earlier, there is a higher amount of reactivity present in the lower region of the core. The higher reactivity leads to more fission, releasing more heat in the fuel pellets. Thus, higher fuel temperatures are expected in regions of higher reactivity. Figure 5-10 below shows the converged moderator temperature profiles as a function of burnup and axial position. Figure 5-11 depicts the converged moderator density profiles for rod 7 as a function of burnup and axial position.

86 75 Figure 5-10: Rod 7 Converged Moderator Temperature Profiles Figure 5-11: Rod 7 Converged Moderator Density Profiles Referring to Figure 5-10 and Figure 5-11, the BOC profiles had been discussed in depth. In latter depletion steps, both the moderator temperature and density approach a more linear shape. The reason for this change in shape comes from the flattening of the axial power profile. If the axial power profile was completely uniform (peaking factor = 1.0 at every axial location), then a constant

87 76 heat flux would be deposited in the coolant. A constant heat flux along a 1-D direction leads to a linear increase in temperature. Except for the ends of the fuel rod, the axial power is relatively uniform. Observing the bottom and top of the profiles in Figure 5-10 and Figure 5-11, the curves are not linear as a result of the significant drop in relative power in these regions. From analysis of the exchange parameters and power shapes alone, one can justify the results that have been provided. Without experimental data to check the usefulness of the coupling for modeling 4x4 PWR sub-assembly sections with only fuel pins, a comparison was done between standalone DeCART calculations and coupled CTF-DeCART calculations. One of the primary goals of the CTF temperature and density feedback is to update the cross-sections during depletion. DeCART also offer an option to update the cross-sections at designated burnup values (i.e. every 2.5 MWd/kgU). The default cross-section update option occurs every 10 MWd/kgU in DeCART. This option was decreased to 2.5 MWd/kgU to observe the effect on power distribution over time. The details of this comparison are seen in Figure 5-12, Figure 5-13, Figure 5-14, and Figure Figure 5-12: Comparison of Coupled CTF-DeCART to the Standalone DeCART Axial Power Distribution for rod 7 at 0.0 MWd/kgU.

88 77 Though it cannot be seen in Figure 5-12, the axial power distribution calculated in a standalone DeCART calculation with extra cross-section updates directly overlaps the axial power distribution for the standalone DeCART calculation with default cross-section update. At BOC there is no added benefit for increasing the number of cross-section updates. Furthermore, without the temperature feedback to the cross-sections, the axial power distribution follows a perfect cosine shape. Figure 5-13: Comparison of Coupled CTF-DeCART to the Standalone DeCART Axial Power Distribution for rod 7 at 10.0 MWd/kgU.

89 78 Figure 5-14: Comparison of Coupled CTF-DeCART to the Standalone DeCART Axial Power Distribution for rod 7 at 20.0 MWd/kgU. Figure 5-15: Comparison of Coupled CTF-DeCART to the Standalone DeCART Axial Power Distribution for rod 7 at 27.5 MWd/kgU.

90 79 Through the progression of Figure 5-12 to Figure 5-15, both the coupled and standalone axial power distribution flattens out. Though, the effect is much more severe in the standalone DeCART calculations. The added cross-section updates do not have an effect on the latter depletion steps either. The cross-sections are determined in DeCART based on the inputted temperatures for various material regions. Without any sort of temperature distribution feedback given in the standalone DeCART models, the cross-sections will not change. For this reason, there is no change in the axial power profiles for DeCART models with an increased number of cross-section updates. Comparing the coupled calculations to the standalone DeCART model using the internal 1-D TH feedback model may provide a more interesting results. This is added to the list of future work. Also, the axial offset anomaly described in Wagner s report is not seen in the standalone DeCART calculations. While the power suppression is observed in the center region of the fuel, the power shape in the bottom half of the core is the same as in the top half of the core for any burnup in the standalone DeCART calculations. In most cases, it is known that more reactivity exists in the lower region of the core, and this is reflected when DeCART is coupled with CTF, receiving temperature and density feedback for cross-section generations. Figure 5-16 shows the converged clad outer surface temperatures from the 4x4 subassembly model with all fuel pins. These temperature profiles are passed to BISON, along with the converged axial power distributions, as boundary conditions for the fuel performance calculations.

91 80 Figure 5-16: Rod 7 Clad Outer Surface Temperature Distributions. Referring to Figure 5-16, the clad outer surface temperature profiles follow the same pattern as the power, temperature, and density. At BOC the peak location of the clad outer surface temperature occurs at a lower elevation than at EOC. Again this is an expected result of the axial power shape flattening out and eventually becoming top-peaked. With coupled boundary conditions, the BISON simulation is run on a custom finite element mesh. There are 6 radial cells per pellet, 3 radial cells for the clad width, and 16 axial mesh cells per pellet with the clad matching the axial mesh size. As a result, the total number of nodes in the model is well over 100,000. An example of part of the full rod mesh can be seen in Figure Note that due to symmetry of a 2-D model in the radial and axial direction, only half of the fuel pellet needed to be modeled. The BISON simulation was modeled to run for 4 years; however, the calculations could not converge after ~2 years. It was determined that pellet-clad contact occurred at ~1.5 years. It is well known that finite element codes struggle to converge to a solution for displacements once contact is perceived (See Section 2.3). Despite not converging to the final timestep, the effect of the coupled outer clad temperature profile is evident. Figure 5-18 depicts the

92 cladding temperature profile calculated by BISON early on in the simulation, and Figure 5-19 shows the cladding temperature profile calculated by BISON at the last converged time step. 81 Figure 5-17: Close-up View of Mesh used in 360 Pellet Full Rod BISON Simulation with Coupled Boundary Conditions.

93 82 Figure 5-18: Clad Temperature Profile from BISON for Coupled Rod 7 at Years Figure 5-19: Clad Temperature Profile from BISON for Coupled Rod 7 at 1.93 Years Referring to Figure 5-18 and Figure 5-19, each line of green data points represents cladding temperatures. The top-most line of green data points is the inner clad surface temperature, and the

94 83 bottom-most line is the clad outer surface temperature, specified by CTF. The important thing to take away from Figure 5-18 and Figure 5-19 is the level of detail provided by the BISON calculations. Note at the earlier timestep the difference in clad temperature in the radial direction. Close to the bottom of the fuel rod, the temperature differs only about 10 C in the radial direction; however, at the peak clad temperature location (~2 m in the axial direction) the temperature between the inner and outer clad surfaces differs by about C. Conversely, at the latter timstep note that the axial shape of the temperature profile remains fairly constant in the radial direction, except near the very bottom and very top of the rod. Also, the axial shapes of the cladding temperature profiles match almost exactly with the coupling outer cladding surface temperature supplied by CTF. With temperature distributions now available in BISON from coupled boundary conditions, the Hydrogen Model is imposed on the cladding of the fuel rod. Olivier Courty has run hydrogen diffusion/precipitation simulations using the temperature profiles calculated in BISON [20]. In one particular simulation, Olivier Courty applied the Hydrogen Model to the Rod 7 BISON simulation from the coupled 4x4 sub-assembly with all fuel pins. Recall in Section 3.5, the timesteps for the Hydrogen Model need to be significantly smaller than a BISON depletion calculation. Due to the computational expense of running a finite element code for a full 360 pellet fuel rod, it is not practical to run a hydrogen diffusion/precipitation model for several simulated years, when the time scale is on the order of hundreds of seconds. Therefore, Olivier Courty assumed an initial homogenous concentration of hydrogen in solid-solution in the cladding of the fuel rod. The concentration was set at 60 wt.ppm; see Figure Then, the Hydrogen Model was simulated for 1 day in the code, with the BISON calculated temperature profiles applied to the fuel rod. During that one day simulation, hydrogen diffused axially and radially inside the cladding. See Figure 5-21, Figure 5-22, and Figure 5-23 to observe the change in Hydrogen concentration in solid solution as a function of time and space.

95 84 Figure 5-20: Initial Hydrogen Concentration Along Fuel Rod Figure 5-21: Distribution of Hydrogen in Solid Solution at Days

96 85 Figure 5-22: Distribution of Hydrogen in Solid Solution at Days Figure 5-23: Distribution of Hydrogen in Solid Solution at Days

97 86 Immediately after the start of the simulation, the hydrogen in solid solution migrates towards the colder outer edge of the cladding. The top-most line of dots represents the hydrogen in solid solution at the nodes on the outer edge of the cladding. This is a direct result of the strong temperature gradient in the radial direction (30-40 C across ~500 microns) compared to the axial direction (30-40 C across ~3.66 m). Also recall that the difference in temperature in the radial direction is more significant at the axial middle of the rod, compared to the very bottom or very top of the rod. This difference in the magnitude of the radial temperature gradient causing the eggshape profile seen in the hydrogen concentration plots. A greater temperature gradient in the axial middle of the fuel rod leads to a more significant diffusion of hydrogen towards the colder outer clad nodes. It is important to point out the difference in scales between Figure 5-21, and Figure 5-22 and Figure Without realizing this information, the amount of diffused hydrogen appears to be unrealistically large at days (34.56 s). Nevertheless, as time progresses, Figure 5-22 and Figure 5-23 show the hydrogen responding to the temperature gradient in the axial direction as well. Diffusion by a concentration gradient also plays a role, but it is not overtly obvious with these results. It should be noted that the minimum TSSp for the cladding temperatures experienced in the simulation (~280 C in the coldest part of the clad) is estimated to be 77 wt.ppm from Equation (1-5). The concentration of hydrogen in solid solution is well below this limit, so no precipitation occurs.

98 x4 PWR Sub-Assembly (with guide tubes) As discussed in Chapter 3 of this thesis, inputs for DeCART and CTF were built first. The necessary parameters were exchanged between the codes, and a 10-3 convergence criterion was placed on the relative change between the fuel temperatures, coolant temperatures, and coolant densities in successive iterations. Steady-state convergence was met at each burnup step before depleting to the next time step. A sub-assembly section of a PWR 17x17 lattice was chosen to be modeled in the CTF-DeCART coupling. The results of those coupled simulations are summarized in this section of the thesis Internal (Rod Position 6) The Internal section contains a guide tube in pin position 6, according to the diagram in Figure 4-8. With 15 fuel pins, compared to all 16 fuel pins in the initial testing case, the total power of the array decreases to MW. For this model, coupled CTF-DeCART calculations were carried out to 37.5 MWd/kgU. The coupling was actually set up to deplete to 40.0 MWd/kgU; however, the Penn State High Performance Computing (HPC) Systems have a nominal time limit of 24 hours per submitted job. The coupled calculations converged to 37.5 MWd/kgU, but then were cut short for the last depletion step when the time limit was reached. The coupled CTF- DeCART calculations were ran on 2 nodes with 1 processer per node; computational time for this simulation ended at 24 hours. The first set of results to show is the radial power factors from their respective converged iterations at certain depletion steps. Figure 5-25 shows the radial power factors for all 16 pin locations in the form of bar graphs. The bar graph values are calculated in the coupling, using the equations described in Section Three time domains are represented: BOC,

99 middle of cycle (MOC), and EOC. The 2-D plane view images to the right of the bar graphs show the axially integrated pin powers for the respective burnups calculated internally in DeCART. 88

100 Figure 5-24: Radial Power Factors at BOC, MOC, and EOC for Internal Array 89