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An Examination of Critical Heat Flux in a Nuclear Fuel Rod Using an Explicit Finite Difference Scheme

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An Examination of Critical Heat Flux in a Nuclear Fuel Rod Using an Explicit Finite Difference Scheme An Examination of Critical Heat Flux in a Nuclear Fuel Rod Using an Explicit Finite Difference Scheme By Alex Christensen A THESIS submitted to Oregon State University University Honors College in partial fulfillment of the requirements for the degree of Honors Baccalaureate of Science in Nuclear Engineering (Honors Scholar) Presented June 8, 2018 Commencement June 2018 AN ABSTRACT OF THE THESIS OF Alex Christensen for the degree of Honors Baccalaureate of Science in Nuclear Engineering presented on June 8, 2018. Title: An Examination of Critical Heat Flux in a Nuclear Fuel Rod Using an Explicit Finite Difference Scheme Abstract approved: Wade Marcum One of the most important components in the safety analysis of a nuclear reactor is its critical heat flux (CHF), as it can compromise the structural integrity of the clad and lead to the release of fission products into the primary coolant. Groeneveld et. al. has published a series of look-up tables used in the prediction of the onset of CHF. A forward-element finite difference method was written to model the temperature dis- tribution in a fuel rod and, using a series of look-up tables published by Groeneveld et. al, the CHF and critical heat flux ratio determined and compared against that previously published for a Westinghouse PWR. Two test cases were run using the finite difference method, one with a uniform volumetric heat generation distribution, the other a sinusoidal distribution. Included in the analysis of the sinusoidal dis- tribution are hot spot factors associated with typical LWRs. While the preliminary steady state test results appear accurate, future tests need to be performed using transient power distributions. Additionally, the assumption of no axial heat transfer significantly affects the critical heat flux ratio (CHFR) at nodes near the inlet and outlet of the core when using a sinusoidal heat generation profile. This assumption should be removed prior to the conduction of transient tests. Key Words: Critical Heat Flux, Departure From Nucleate Boiling, Finite Difference, Sub-channel Analysis, Thermal Hydraulics Corresponding e-mail address: [email protected] ©Copyright by Alex Christensen June 8, 2018 All Rights Reserved An Examination of Critical Heat Flux in a Nuclear Fuel Rod Using an Explicit Finite Difference Scheme By Alex Christensen A THESIS submitted to Oregon State University University Honors College in partial fulfillment of the requirements for the degree of Honors Baccalaureate of Science in Nuclear Engineering (Honors Scholar) Presented June 8, 2018 Commencement June 2018 Honors Baccalaureate of Science in Nuclear Engineering project of Alex Christensen presented on June 8, 2018 APPROVED: Wade Marcum, Mentor, representing the School of Nuclear Science and Engineering Dan LaBrier, Committee Member, representing the School of Nuclear Science and Engineering Guillaume Mignot, Committee Member, representing the School of Nuclear Science and Engineering Toni Doolen, Dean, Oregon State University Honors College I understand that my project will become part of the permanent collection of Oregon State University Honors College. My signature below authorizes release of my project to any reader upon request. Alex Christensen, Author Contents 1 Introduction 1 1.1 Motivation . .1 1.2 Objectives . .2 1.3 Overview of the Following Chapters . .3 2 Literature Review 4 2.1 The heat equation . .4 2.2 The use of numerical methods in heat transfer . .4 2.2.1 The explicit finite-difference method . .5 2.2.2 The implicit finite-difference method . .6 2.3 Critical Heat Flux . .7 3 Problem Description 11 4 Methodology 14 4.1 Calculations . 14 5 Results and Discussion 18 5.1 Steady State Test Cases . 18 5.1.1 Derivation of the Analytical Solution . 18 5.1.2 Comparison Between Analytical and Numerical Solutions . 20 5.2 Steady State CHF Tests . 34 6 Conclusion 39 7 List of Variables 43 8 Appendix 45 1 Introduction 1.1 Motivation The understanding of the temperature distribution in a nuclear fuel rod is paramount to the safety of nuclear energy. If a reactor's cladding exceeds its maximum allowable temperature, it may be at risk of becoming breached, allowing for the release of ra- dioactive contaminants into the primary cooling loop. Because of the inherent risks associated with nuclear energy, much effort is expended in the analysis of so-called design basis accidents. One type of design basis accident which must be taken into account is a control rod withdrawal accident. In this accident scenario, it is speculated that a control rod is suddenly ejected from the core. Assuming the reactor was previously at criticality (when the neutron population in any two consecutive fission generations is the same and power is unchanging), the massive positive reactivity insertion causes the core to become supercritical (where the neutron population and subsequent power level increase exponentially from generation to generation) [1]. Under many circumstances, the reactor core will actually become prompt critical, a unique case of supercriticality in which the prompt neutrons (those released directly from the fission reaction itself) alone are enough to bring the reactor critical. This phenomenon is often referred to as a reactor pulse. Under normal operating conditions, it is important for the amount of heat being gen- erated inside the reactor core to equal that which is removed by the passing coolant. Under these operating conditions, the temperature is held constant throughout the core as a function of time. These conditions are known as steady state. However, when the core becomes supercritical, this is no longer the case. Since the volumet- ric heat generation rate in the fuel is proportional to the total power of the core, as power rapidly increases, the temperature distribution of the fuel itself will begin increasing. As heat is not transferred instantaneously, it takes time for the heat to diffuse through the rest of the fuel rod (typically comprised of three regions: the fuel, gap, and clad), before being transferred to the coolant. Because of this latency, the system will remain in a transient state until power levels out and the temperature 1 has time to equalize. Due to the tight coupling between temperature and material properties, the rates at which temperatures change are highly non-linear. As power continues to increase, the amount of heat being transferred into the water will increase as well, resulting in ever increasing temperatures. Eventually, the water will begin to boil. As the thermodynamic quality (the ratio of vapor and liquid in the water as a function of mass) increases, a critical point is eventually reached when the vapor forms a film around the surface of the fuel element. The heat flux at which this occurs is known as the critical heat flux (CHF). Because steam is much less effective at transferring heat than liquid water, the amount of heat being transferred into the water will rapidly de- crease, causing the temperature of the cladding to increase dramatically in a fraction of a second [1]. This puts tremendous stress on the cladding, leading to its potential failure. Because of the importance in ensuring this does not occur, a common thermal limi- tation is placed on the heat flux between the clad and the coolant. One of the most widely utilized methods for predicting the onset of critical heat flux are a set of look- up tables published by Groeneveld et al. that interpolate on the input quantities mass flux, pressure, and equilibrium quality [2]. These tables are the result of a sta- tistical analysis of hundreds of independent databases. Most importantly, however, is that an assumption of thermal equilibrium was made in the creation of these tables. Little research has been conducted into the impact non-thermal equilibrium has on the value of CHF 1.2 Objectives The objective of this thesis to examine the prediction of critical heat flux in a standard Westinghouse pressurized water reactor. To do this, the current critical heat flux look- up tables published by Groeneveld et al. will be leveraged against the results obtained using an explicit finite difference scheme. 2 1.3 Overview of the Following Chapters • Literature Review: An overview of general background information on the topic, including heat transfer, finite difference methods, boiling, and critical heat flux. • Problem Description: This section contains a detailed description of the problem being modeled, including diagrams of a fuel rod and the binning layout used in the problem. • Methodology: This section contains the derivation of the explicit finite differ- ence method used, including the equations used for the node temperatures, the convection coefficient, and the methodology for updating water temperature as a function of height. • Results and Discussion: This section contains an analysis of the results ob- tained. It first describes a steady state test problem which was used to verify the finite difference codes ability to accurately model the parameters needed to determine critical heat flux. This includes the derivation of an analytical solu- tion for the temperature distribution in the fuel rod, which compared against the results of the numerical temperature distribution. This section also analyzes the CHF results obtained with varied parameters and compares them with those predicted by the Groeneveld look-up tables. • Conclusion: This section describes and attempts to justify the assumptions and simplifications used in this model and how they may have affected the results. It reiterates the importance of this work and the results obtained. Lastly, it examines several improvements that could be made to the model and outlines potential work to be done in the future. 3 2 Literature Review 2.1 The heat equation Derived from the energy transport equation, the heat equation is [4]: @T (~r; t) r · k(~r; T )rT (~r; t) +q _(~r; t) = ρ(~r; T )C(~r; T ) (1) @t where k is the thermal conductivity, C is the heat capacity, ρ is the density of the material, andq _ is the volumetric heat generation.
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