The Pennsylvania State University The Graduate School IMPROVEMENT OF COBRA-TF FOR MODELING OF PWR COLD- AND HOT-LEGS DURING REACTOR TRANSIENTS

Transcription

1 The Pennsylvania State University The Graduate School IMPROVEMENT OF COBRA-TF FOR MODELING OF PWR COLD- AND HOT-LEGS DURING REACTOR TRANSIENTS A Dissertation in Nuclear Engineering by Robert K. Salko c 2012 Robert K. Salko Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy December 2012

2 The dissertation of Robert K. Salko was reviewed and approved by the following: Maria N. Avramova Assistant Professor of Nuclear Engineering Dissertation Advisor, Chair of Committee Kostadin N. Ivanov Distinguished Professor of Nuclear Engineering Fan-Bill Cheung Professor of Mechanical and Nuclear Engineering Vincent Benitez Associate Professor of Music Emilian Popov Associate Researcher at Oak Ridge National Laboratory Arthur Motta Professor of Nuclear Engineering and Material Science and Engineering Chair of Nuclear Engineering Signatures are on file in the Graduate School.

3 Abstract COBRA-TF is a two-phase, three-field (liquid, vapor, droplets) thermal-hydraulic modeling tool that has been developed by the Pacific Northwest Laboratory under sponsorship of the NRC. The code was developed for Light Water Reactor analysis starting in the 1980s; however, its development has continued to this current time. COBRA-TF still finds wide-spread use throughout the nuclear engineering field, including nuclear-power vendors, academia, and research institutions. It has been proposed that extension of the COBRA-TF code-modeling region from vessel-only components to Pressurized Water Reactor (PWR) coolant-line regions can lead to improved Loss-of-Coolant Accident (LOCA) analysis. Improved modeling is anticipated due to COBRA-TF s capability to independently model the entrained-droplet flow-field behavior, which has been observed to impact delivery to the core region[1]. Because COBRA-TF was originally developed for verticallydominated, in-vessel, sub-channel flow, extension of the COBRA-TF modeling region to the horizontal-pipe geometries of the coolant-lines required several code modifications, including: Inclusion of the stratified flow regime into the COBRA-TF flow regime map, along with associated interfacial drag, wall drag and interfacial heat transfer correlations, Inclusion of a horizontal-stratification force between adjacent mesh cells having unequal levels of stratified flow, and Generation of a new code-input interface for the modeling of coolant-lines. The sheer number of COBRA-TF modifications that were required to complete this work turned this project into a code-development project as much as it was a study of thermal-hydraulics in reactor coolant-lines. The means for achieving iii

4 these tasks shifted along the way, ultimately leading the the development of a separate, nearly completely independent one-dimensional, two-phase-flow modeling code geared toward reactor coolant-line analysis. This developed code has been named CLAP, for Coolant-Line-Analysis Package. Versions were created that were both coupled to COBRA-TF and standalone, with the most recent version being a standalone code. This code performs a separate, simplified, 1-D solution of the conservation equations while making special considerations for coolant-line geometry and flow phenomena. The end of this project saw a functional code package that demonstrates a stable numerical solution and that has gone through a series of Validation and Verification tests using the Two-Phase Testing Facility (TPTF) experimental data[2]. The results indicate that CLAP is under-performing RELAP5-MOD3 in predicting the experimental void of the TPTF facility in some cases. There is no apparent pattern, however, to point to a consistent type of case that the code fails to predict properly (e.g., low-flow, high-flow, discharging to full vessel, or discharging to empty vessel). Pressure-profile predicitons are sometimes unrealistic, which indicates that there may be a problem with test-case boundary conditions or with the coupling of continuity and momentum equations in the solution algorithm. The code does predict the flow regime correctly for all cases with the stratification-force model off. Turning the stratification model on can cause the low-flow case void profiles to over-react to the force and the flow regime to transition out of stratified flow. The code would benefit from an increased amount of Validation & Verification testing. The development of CLAP was significant, as it is a cleanly written, logical representation of the reactor coolant-line geometry. It is stable and capable of modeling basic flow physics in the reactor coolant-line. Code development and debugging required the temporary removal of the energy equation and mass-transfer terms in governing equations. The reintroduction of these terms will allow future coupling to RELAP and re-coupling with COBRA-TF. Adding in more applicable entrainment and de-entrainment models would allow the capture of more advanced physics in the coolant-line that can be expected during Loss-of-Coolant Accident. One of the package s benefits is its ability to be used as a platform for future coolant-line model development and implementation, including capturing of the important de-entrainment behavior in reactor hot-legs (steam-binding effect) and flow convection in the upper-plenum region of the vessel. iv

5 Table of Contents List of Figures List of Tables Nomenclature Acronyms Acknowledgments ix xiv xv xxi xxiii Chapter 1 Introduction PWR Loss-of-Coolant Accident Computational Thermal-Hydraulic Modeling of PWR Coolant System Dissertation Overview Chapter 2 Overview of the COBRA-TF Thermal/Hydraulic Code Governing Equations Continuity Equations Momentum Equations Energy Equations Interfacial Area Transport Equation Closure of Governing Equations Flow Regime Map Wall Shear Phase Interface Shear Wall Heat Transfer v

6 2.2.5 Phase-Interface Heat Transfer Turbulent Mixing and Void Drift Entrainment Models Steam Tables Numerical Solution Chapter 3 Project History Phase I Phase II Phase III Chapter 4 Coolant-Line Governing-Equation Closure Coolant-Line Flow Regime Map Implementation Interfacial Drag Correlations Stratified Flow Regime Small-Bubble Flow Regime Slug Flow Regime Annular/Mist Flow Regime Mist Flow Regime Wall Drag Correlations Non-Stratified Flow Regimes Stratified Flow Regime Interfacial Heat Transfer Correlations Small-Bubble Flow Regime Slug Regime Dispersed Droplets Annular/Mist Regime Stratification Force Model Inclusion in Jacobian Chapter 5 CLAP Modeling Approach 68 Chapter 6 CLAPv1 Development CLAPv1 Solution Process CLAPv1 1-D Transport Equations vi

7 6.2.1 Continuity Equation Momentum Equation Energy Equation Interfacial Area Transport Equation Input and Output Deck Modifications Chapter 7 CLAPv2 Development CLAPv2 Solution Algorithms Separated Algorithm Fully Coupled Algorithm Governing Equations Separated Algorithm Momentum Equation Pressure-Correction Equation Volume Fraction Equations Fully Coupled Algorithm Momentum Equations Continuity Equations Geometric Constraint Chapter 8 Validation & Verification Experimental-Facility Description Experimental-Facility Model Code Simulations Void Fraction Numerical Algorithm Pressure Profile Convergence Chapter 9 Conclusions and Future Work Purpose Project Achievements Code-Development Notes PhD Contributions Evaluation of Achievements and Future Work vii

8 Appendix A CLAPv2 Input Deck 155 A.1 Geometry and Meshing Information A.2 Boundary Condition Information A.3 Convergence and Numerical Controls Appendix B TPTF-Modeling Results 160 B.1 Prediction of TPTF Void B.2 Void Sensitivity to Stratification Force B.3 Void Sensitivity to CLAPv2 Algorithms B.4 CLAPv2 Residuals for Coupled Algorithm B.5 CLAPv2 Residuals for Separated Algorithm Bibliography 181 viii

9 List of Figures 2.1 COolant Boiling in Rod Arryas - Two Fluid (COBRA-TF) normalwall flow regime map COBRA-TF hot-wall flow regime map Scalar-mesh cell and axial momentum-mesh cell configuration Scalar-mesh cell and transverse-momentum-mesh cell configuration COBRA-TF original solution scheme Possible section boundaries using the traditional COBRA-TF meshing capabilities COBRA-TF normal-wall flow regime map RELAP5-3D horizontal flow regime map Graphical representation of θ for calculation of u crit RELAP5-3D horizontal flow regime map Flowchart for the execution of p intfr Droplet drag coefficient as function of droplet Reynolds number Example of unequal liquid-level heights in adjacent scalar-mesh cells and resulting hydrostatic pressure forces Diagram of terms needed to define the stratification force Plot of partial stratification force term as function of α and corresponding curve-fit Partial stratification force term and its partial derivative with respect to α Universal coolant-line numbering scheme Representation of a staggered mesh generated for a coolant line COBRA-TF/CLAPv1 solution process Flowchart of the CLAPv2 separated-solution algorithm Flowchart of the CLAPv2 fully coupled solution algorithm Momentum-mesh cell and neighbors ix

10 7.4 Example of the Upwind Differencing Scheme (UDS) for determining the velocity to be used for momentum cell p east-face convection of momentum Diagram of scalar-mesh cell and its neighbors for the one-dimensional case Depiction of the TPTF test section COBRA-TF model for the Two-Phase-Flow Facility CLAPv1 models for the Two-Phase-Flow Testing Facility CLAPv2 models for the Two-Phase-Flow Testing Facility COBRA-TF, CLAPv1, CLAPv2, and RELAP5-MOD3 simulation results for TPTF Run COBRA-TF, CLAPv1, CLAPv2, and RELAP5-MOD3 simulation results for TPTF Run Fully coupled algorithm void-profile prediction for Run 722 with and without the stratification model Fully coupled algorithm void-profile prediction for Run 785 with and without the stratification model Fully coupled algorithm void-profile prediction for Run 749 with and without the stratification model COBRA-TF, CLAPv1, CLAPv2, and RELAP5-MOD3 simulation results for TPTF Run COBRA-TF, CLAPv1, CLAPv2, and RELAP5-MOD3 simulation results for TPTF Run Fully coupled algorithm void-profile prediction for Run 2514 with and without the stratification model Fully coupled algorithm void-profile prediction for Run 2541 with and without the stratification model Comparison of predicted and measured void for all 8 TPTF simulations using RELAP5-MOD3, CLAPv1, and CLAPv Run 728 void-profile comparison using the fully copuled and separated numerical algorithm Run 2541 void profiles using the separated numerical algorithm with and without the stratification force Fully coupled numerical-algorithm prediction of the pressure profile for Run 728 with and without the stratification force applied Fully coupled numerical-algorithm prediction of the pressure profile for Run 749 with and without the stratification force applied Fully coupled numerical-algorithm prediction of the pressure profile for Run 722 with and without the stratification force applied x

11 8.20 Fully coupled numerical-algorithm prediction of the pressure profile for Run 785 with and without the stratification force applied Fully coupled numerical-algorithm prediction of the pressure profile for Run 2508 with and without the stratification force applied Fully coupled numerical-algorithm prediction of the pressure profile for Run 2514 with and without the stratification force applied Fully coupled numerical-algorithm prediction of the pressure profile for Run 2535 with and without the stratification force applied Fully coupled numerical-algorithm prediction of the pressure profile for Run 2541 with and without the stratification force applied Normalized COBRA-TF mass flow rates throughout the test section for the TPTF simulations Ratio of momentum-cell total-mass flow rate to inlet total-mass flow rate for TPTF tests using CLAPv Normalized governing-equation residuals for Run 785 (fully coupled approach with stratification force) Normalized governing-equation residuals for Run 2541 (fully coupled approach with stratification force) Normalized governing-equation residuals for Run 728 (fully coupled approach with stratification force) Normalized governing-equation residuals for Run 2535 (fully coupled approach with stratification force) Ratio of inlet to outlet total-mass flow rate for TPTF cases using CLAPv Normalized residuals for Run 785 when using the segregated algorithm Normalized residuals for Run 2541 when using the segregated algorithm Normalized mass balance for TPTF runs using the separated algorithm B.1 COBRA-TF, CLAPv1, CLAPv2, and RELAP5-MOD3 simulation results for TPTF Run B.2 COBRA-TF, CLAPv1, CLAPv2, and RELAP5-MOD3 simulation results for TPTF Run B.3 COBRA-TF, CLAPv1, CLAPv2, and RELAP5-MOD3 simulation results for TPTF Run B.4 COBRA-TF, CLAPv1, CLAPv2, and RELAP5-MOD3 simulation results for TPTF Run B.5 COBRA-TF, CLAPv1, CLAPv2, and RELAP5-MOD3 simulation results for TPTF Run xi

12 B.6 COBRA-TF, CLAPv1, CLAPv2, and RELAP5-MOD3 simulation results for TPTF Run B.7 COBRA-TF, CLAPv1, CLAPv2, and RELAP5-MOD3 simulation results for TPTF Run B.8 COBRA-TF, CLAPv1, CLAPv2, and RELAP5-MOD3 simulation results for TPTF Run B.9 Fully coupled algorithm void-profile prediction for Run 722 with and without the stratification model B.10 Fully coupled algorithm void-profile prediction for Run 728 with and without the stratification model B.11 Fully coupled algorithm void-profile prediction for Run 749 with and without the stratification model B.12 Fully coupled algorithm void-profile prediction for Run 785 with and without the stratification model B.13 Fully coupled algorithm void-profile prediction for Run 2508 with and without the stratification model B.14 Fully coupled algorithm void-profile prediction for Run 2514 with and without the stratification model B.15 Fully coupled algorithm void-profile prediction for Run 2535 with and without the stratification model B.16 Fully coupled algorithm void-profile prediction for Run 2541 with and without the stratification model B.17 Run 722 void-profile comparison using the fully coupled and separated numerical algorithm B.18 Run 728 void-profile comparison using the fully coupled and separated numerical algorithm B.19 Run 749 void-profile comparison using the fully coupled and separated numerical algorithm B.20 Run 785 void-profile comparison using the fully coupled and separated numerical algorithm B.21 Run 2508 void-profile comparison using the fully coupled and separated numerical algorithm B.22 Run 2514 void-profile comparison using the fully coupled and separated numerical algorithm B.23 Run 2535 void-profile comparison using the fully coupled and separated numerical algorithm B.24 Run 2541 void-profile comparison using the fully coupled and separated numerical algorithm B.25 Normalized governing-equation residuals for Run 722 (fully coupled approach with stratification force) xii

13 B.26 Normalized governing-equation residuals for Run 728 (fully coupled approach with stratification force) B.27 Normalized governing-equation residuals for Run 749 (fully coupled approach with stratification force) B.28 Normalized governing-equation residuals for Run 785 (fully coupled approach with stratification force) B.29 Normalized governing-equation residuals for Run 2508 (fully coupled approach with stratification force) B.30 Normalized governing-equation residuals for Run 2514 (fully coupled approach with stratification force) B.31 Normalized governing-equation residuals for Run 2535 (fully coupled approach with stratification force) B.32 Normalized governing-equation residuals for Run 2541 (fully coupled approach with stratification force) B.33 Normalized residuals for Run 722 when using the segregated algorithm 177 B.34 Normalized residuals for Run 728 when using the segregated algorithm 177 B.35 Normalized residuals for Run 749 when using the segregated algorithm 178 B.36 Normalized residuals for Run 785 when using the segregated algorithm 178 B.37 Normalized residuals for Run 2508 when using the segregated algorithm B.38 Normalized residuals for Run 2514 when using the segregated algorithm B.39 Normalized residuals for Run 2535 when using the segregated algorithm B.40 Normalized residuals for Run 2541 when using the segregated algorithm xiii

14 List of Tables 4.1 Comparison of COBRA-TF and RELAP5-3D flow regime map void fraction boundaries Definition of the flow regime numbers used in CLAP Card 15.2 input variables Card 15.3 input variables TPTF test inlet conditions xiv

15 Nomenclature (α i α j ) Equilibrium void distribution for void drift α α est α i α j α v ρ mix Ḡ β c β p β tp β v X ɛ η γ Field volume fraction Estimated void resulting from point-jacobi form of the continuity equation Volume fraction of channel i Volume fraction of channel j Vapor void fraction Average mixture density between adjacent channels Average mass flux between adjacent sub-channels Continuity equation under-relaxation factor The pressure-correction under-relaxation factor Two-phase mixing coefficient Velocity under-relaxation factor Mesh-cell axial height Parameter for ensuring boundedness of void in the Carver method Fraction of phase-change mass flow rate taking place between vapor and droplets Parameter for ensuring boundedness of void in the Carver method xv

16 Γ Γ Ja kp Pr Re drop Re b Re i We Volumetric mass flow rate due to phase change Volumetric-evaporation rate Jacob number Coolant-line mesh-cell counter Prandtl number Droplet Reynolds number Bubble Reynolds number Phase-interface Reynolds number Webber number µ mb Small-bubble regime mixture viscosity µ v Vapor dynamic viscosity Φ Discrete governing-equation dependent variable Φ Last-iteration dependent-variable Φ C Φ k ρ ρ c ρ i ρ j ρ l ρ v τ i τ w θ Current-iteration dependent-variable obtained with direct solution Phase k two-phase-friction multiplier Phase density Density of the continuous phase Density of channel i Density of channel j Liquid density Vapor density Volumetric phase-interface shear force Volumetric wall-shear force Angle between vertical and stratified surface xvi

17 V a A drop A i,γ A i,film A i,ev A i,e A i,lb A i,lv A i,p A i,sb A i A l A p,sb A pw A s A v A x b c C HS C NS Field velocity vector Discrete governing-equation coefficient Surface area of a single droplet Volumetric rate of interfacial area change due to phase change Interfacial area of liquid film in annular/mist flow regime Droplet and vapor phase interface area Volumetric rate of interfacial area change due to entrainment/de-entrainment Interfacial area of a large bubble Continuous-liquid and vapor interfacial area Area (projected or interfacial) to be used in the interfacial drag force equation Interfacial area of small-bubble Interface surface area Liquid-phase cross-sectional area Projected area of small-bubble Surface area of heated surfaces in the mesh cell Conductor surface area in mesh cell Vapor-phase cross-sectional area Mesh-cell cross-sectional area (x normal) Discrete governing-equation coefficient Discrete governing-equation coefficient Horizontally-stratified flow regime constitutive relationship Non-stratified flow regime constitutive relationship C transition Constitutive relationship arrived at by logarithmic interpolation xvii

18 C D C p,l C p,v D D b d p,m D v dx line f f HM f i f w,k g G mix G ix G jx G k h h int,scl h int,scv h int,shl h int,shv h nb h c Coefficient of drag Liquid specific heat Vapor specific heat Pipe diameter Bubble diameter Governing-equation coefficient for non-k fields Hydraulic diameter of the vapor phase Coolant-line mesh-cell length Void fraction adjustment factor for ensuring geometric conservation Hughmark[3] friction factor Interfacial friction factor for continuous liquid Phase k wall-friction factor Gravity acceleration Mixture mass flux Axial mass flux of channel i Axial mass flux of channel j Phase k mass flux Enthalpy Sub-cooled liquid interface heat transfer coefficient Sub-cooled vapor interface heat transfer coefficient Super-heated liquid interface heat transfer coefficient Super-heated vapor interface heat transfer coefficient Nucleate-boiling heat transfer coefficient Convective-heat-transfer coefficient xviii

19 h f h g h i,shl h l h l h v k K form k f k i k k,w L M L en M D mass M T mass M L mom M T mom N drops N lb P p p p face q w Liquid saturation enthalpy Vapor saturation enthalpy Super-heated-liquid interfacial heat transfer coefficient Liquid enthalpy Stratified liquid-level height Vapor enthalpy Index of field Form loss coefficient Liquid thermal conductivity Drag force coefficient Phase k wall drag coefficient Volumetric mass transfer term Volumetric energy transfer due to phase change Mass transfer due to void drift Volumetric mass transfer from void drift and turbulent mixing Volumetric momentum source from phase change Volumetric momentum source from turbulent mixing and void drift Number of droplets in a mesh cell Number of large bubbles Pressure Pressure correction required to correct velocity Initial (guessed) pressure field Momentum at the momentum-cell face Volumetric wall heat transfer xix

20 R drop r lb S S S c S k S p T F T w u u u crit u rel u l u v V ij V lb V sb V v Droplet radius Large-bubble radius Discrete governing-equation source term Volumetric mass flow rate due to entrainment Portion of source term not including dependent variable Gap width between channels i and j Portion of source term including dependent variable Fluid temperature Conductor surface temperature Velocity correction required to conserve mass Initial (guessed) velocity resulting from initial pressure field Critical vapor velocity from Taitel & Dukler Relative liquid/vapor velocity Liquid phase velocity Vapor phase velocity Transverse velocity caused by turbulent mixing Large-bubble volume Volume of small bubbles Total volume of a scalar-mesh cell FSTRAT Constitutive-relationship parameter for logarithmic interpolation xx

21 Acronyms ASME American Society of Mechanical Engineers BOHL Beginning of Heated Length CHF Critical Heat Flux CLAP Coolant Line Analysis Package CFD Computational Fluid Dynamics COBRA-TF COolant Boiling in Rod Arryas - Two Fluid EOHL End of Heated Length ECC Emergency Core Cooling ECCS Emergency Core Cooling System EPRI Electric Power Research Institute GCBA Geometric Conservation-Based Algorithm HTC Heat Transfer Coefficient JAERI Japanese Atomic Energy Research Institute LHS Left-Hand Side LBLOCA Large-Break Loss-of-Coolant Accident LOCA Loss-of-Coolant Accident MCBA Mass Conservation-Based Algorithm xxi

22 SBLOCA Small-Break Loss-of-Coolant Accident LWR Light Water Reactor NRC Nuclear Regulatory Commission ONB Onset of Nucleate Boiling PSU Pennsylvania State University PWR Pressurized Water Reactor RBHT Rod Bundle Heat Transfer Facility RHS Right-Hand Side RMS Root-Mean Square SIMPLE Semi-Implicit Method for Pressure-Linked Equations TDMA Tri-Diagonal Matrix Algorithm T/H Thermal/Hydraulic TPTF Two-Phase Testing Facility T&D Taitel & Dukler UDS Upwind Differencing Scheme UPTF Upper Plenum Testing Facility V&V Verification and Validation xxii

23 Acknowledgments I would like to thank Mitsubishi Heavy Industries for providing the initial funding for this project. Also, I would like to thank Dr. Maria Avramova and Dr. Kostadin Ivanov for their assistance in obtaining the funding as well as additional funding to keep this project going. xxiii

24 Dedication To my wife and daughters my constant reminder to finish the dissertation. xxiv

25 Chapter 1 Introduction 1.1 PWR Loss-of-Coolant Accident The Loss-of-Coolant Accident (LOCA) is often the most limiting design-basis accident that can be experienced by a Light Water Reactor (LWR). Adequate understanding of reactor behavior and safety-system response during the transient will directly impact plant safety as well as the plant economics (e.g. increased power, improved fuel-cycle efficiency, relaxed operational regulations, etc.). The 1988 revision to the U.S. Nuclear Regulatory Commission (NRC) 10CFR50.46 requirements allowed for the use of best-estimate models in the calculation of LOCA transients. The revision was a result of improved understanding of the LOCA transient and associated Emergency Core Cooling System (ECCS) response. Great gains in plant safety and economics can be made by taking advantage of improvements in LOCA transient-behavior understanding and by continuously enhancing best-estimate methodologies used for reactor safety-system design and analysis. The coolant-system behavior following a LOCA will depend primarily on the break size that coolant is escaping from. A Large-Break Loss-of-Coolant Accident (LBLOCA), for example, will result in rapid depressurization of the reactor vessel and core uncovering (blowdown stage) until system pressure drops to the setpoint of the accumulators, triggering injection of borated coolant into the system (refill stage). After the lower plenum is refilled, the coolant rises in the vessel and recovers the core (reflood stage). The initial transient is fast, taking on the order of minutes[4].

26 2 The Small-Break Loss-of-Coolant Accident (SBLOCA), on the other hand, is characterized by a slow depressurization to the point where the saturation temperature is reached and a two-phase mixture begins to develop in the primary system. The reactor pumps are tripped on discovery of the break whereupon gravity causes phase separation and natural, two-phase circulation begins. The steam generators continue to provide cooling and the system pressure will plateau with respect to time of the transient. Generally, the core will be uncovered twice the first time driven by clearing of the loop seals and the second time driven by loss of coolant through the break. High-volume coolant injection will begin when system pressure has dropped below the accumulator pressure setpoint[4]. The key consideration during the LOCA transient is maintaining cooling to the reactor fuel rods and removal of core decay-heat to prevent fuel damage and radiation release. Because of its importance, extensive research has been done on the LOCA transient, which has led to a highly detailed mapping of anticipated transient events and to discovery of dominant phenomena that guide the transient behavior. For example, it has been observed in the full-scale tests of the Upper Plenum Testing Facility (UPTF)[1] in Germany that entrainment in the vessel and subsequent flow out of the broken cold-leg has significant effects on the downcomer liquid level during the refill stage of LBLOCA, which affects the core-reflood rate. Furthermore, the entrained-droplet field plays an important part in the reactor hot-leg during the refill/reflood stages since entrained droplets that reach the steam generator will vaporize, creating a back-pressure in the upper plenum that tends to resist core reflood (steam-binding effect). 1.2 Computational Thermal-Hydraulic Modeling of PWR Coolant System In keeping with the best-estimate methodology, it is necessary to utilize Thermal/Hydraulic (T/H) analysis codes containing a suitable array of validated models that can predict the dominant observed phenomena of the LOCA. An excellent choice for in-vessel T/H analysis is COBRA-TF, due, in part, to its two-fluid, three-

27 3 field (liquid, vapor, entrained droplets) modeling approach the importance of the unique entrainment-field behavior in the reactor downcomer has previously been described. However, system codes, such as RELAP and TRAC, are more suited to ex-vessel components of the primary coolant loop owing to their inclusion of special-component models (e.g. coolant pumps and accumulators). The advantages of both codes have previously been combined through coupling of the codes (see [5]) vessel simulation is performed by COBRA-TF while exvessel simulation is performed by the system code, with system properties being exchanged during the transient analysis. Coupling of RELAP5-3D and COBRA- TF has previously been performed for modelling of the full primary coolant loop[6, 7]. However, the traditional coupling location, where flow-field information passes between the two code simulation meshes, is the coolant-line/vessel junction point. It is instead proposed that the COBRA-TF modeling region be extended into the reactor cold- and hot-legs to capture the important, unique droplet behavior in those components. A coupling at the coolant-line/vessel junction would prohibit capturing droplet behavior in the lines due to the fact that RELAP5-3D does not consider the droplet field independently (its behavior is dependent on the continuous-liquid behavior). It is believed that by using a three-field approach in the coolant-lines, improved LOCA analysis will result (particularly due to capture of the aforementioned steam-binding effect). This proposition, however, raises several concerns since COBRA-TF was not designed for this type of modeling. Rather, COBRA- TF was designed for axial-dominated sub-channel flow typical of T/H behavior in reactor fuel bundles. The purpose of this project was to determine the requirements for performing vessel/coolant-line modeling using COBRA-TF and then to meet those requirements by further developing COBRA-TF. The end result of this work was the production of an independent add-on, dubbed Coolant Line Analysis Package (CLAP), which can be called by COBRA-TF to extend its modeling region into the reactor coolant-lines. This dissertation details the requirements of extending the modeling region into the coolant-lines, a history of the development work, models that were chosen for implementation into the code, the manner in which they were implemented, and, finally, Verification and Validation (V&V) results using the newly developed package for modeling of the Two-Phase Testing

28 4 Facility (TPTF) experimental data[2]. 1.3 Dissertation Overview This dissertation documents the efforts undertaken and accomplishments acheived throughout the pursuit of the project goal. The project started with an idea to extend COBRA-TF s modeling capabilities to the reactor coolant-line region. For this reason, the reader is first given an overview of the capabilities and mechanics of the COBRA-TF T/H code in Chapter 2. Throughout the duration of the project, several factors and decisions led to changes in the direction and focus of the work. To understand why certain decisions were made and how the project ended up where it is today, an overview of the project history is given in Chapter 3. An effort is made in the project-history chapter to keep the technical details to a minimum. After reading Chapter 3, the reader will understand that the project could more or less be broken up into three phases. After Chapter 3, the remainder of the dissertation is divided in a similar way. The first phase of work was focused on determining what would need to be done to make COBRA-TF applicable for coolant-line analysis. This involved a literary search and model selections. Chapter 4 discusses the flow regime map and constitutive models that were implemented into COBRA-TF for coolant-line analysis. The next two phases focused primarily on code development. Prior to starting this discussion, the reader is given a general overview of the modeling approach for reactor coolant-lines in Chapter 5 this modeling approach is common to the two CLAP versions developed in the following two phases. The second phase of work involved a complete restructuring of the necessary coolant-line models in order to better capture the unique geometry of the coolantline. This work involved separating the code modifications from COBRA-TF and led to the development of CLAP. Chapter 6 examines this code-development work as well as shortcomings that were encountered, which led to the need for a third phase. The code developed in Phase II was named CLAPv1. The third phase of work was entered to reduce the complexity of CLAP to more easily target coding errors. It involved almost completely de-coupling CLAP from the COBRA-TF solution and implementing a stand-alone numerical solution algorithm. Chapter

29 5 7 considers this new numerical solution algorithm and the revised form of the governing equations. The resulting code was named CLAPv2. At every stage of the project, the developed codes were tested against the ROSA TPTF experimental data by simulating the facility. All of the V&V results are summarized in Chapter 8. A final discussion of the project work and accomplishments made is given in Chapter 9.

30 Chapter 2 Overview of the COBRA-TF Thermal/Hydraulic Code The COolant Boiling in Rod Arryas - Two Fluid (COBRA-TF) computer code was developed originally in 1980 by the Pacific Northwest Laboratory under sponsorship of the NRC[8] in order to provide best-estimate calculation capability for LWR analysis. The original code version of COBRA-TF was implemented in the COBRA-TRAC code system[5] and further validated and refined as part of the FLECHT-SEASET 163-Rod Blocked Bundle Test and analysis program[9]. This version was transferred in the nineties to the Pennsylvania State University (PSU) to be used for the analysis of the Rod Bundle Heat Transfer Facility (RBHT) experimental facility data. This also served as the basis for establishing and further developing the PSU version of COBRA-TF, rebranded as CTF, and designed for stand-alone and coupled simulations of LWRs([10],[11]). This is also the code version that was used for this project. COBRA-TF uses the separated flow model along with independent modeling of three separate fields (liquid droplets, liquid film, and water vapor) for modeling two-phase flow. As the name implies, the code was developed for modeling of fuel-rod arrays; however, the code has found wider applications (e.g. [12]). The code contains both sub-channel and 3-D formulations of the momentum equations along with a wide array of T/H models important to LWR safety analysis (e.g. fuel-rod conduction, quench-front modeling, rod deformation, cladding-interaction models, heat transfer models for all boiling curve regions, non-condensable gas

31 7 model, etc.). The solution performed by the code is a time dependent one, which allows for capturing reactor behavior through important transient events like the LOCA. The remainder of this chapter aims to provide a breif overview of the fundamentals of COBRA-TF s two-phase flow modeling approach. Information is provided on the general governing equations employed for tracking mass, momentum, and energy across the fluid mesh. The numerical solution method and models used for providing closure to the governing equations are also relevant to the work in this dissertation. While the capabilties of COBRA-TF expand far beyond rudimentary numerical two-phase flow analysis, the majority of these additional capabilities are not mentioned in this dissertation since they are not directly relevant to the work presented here. However, a detailed treatise on the theory behind the COBRA-TF source code can be found in the CTF Theory Manual[13]. 2.1 Governing Equations Continuity Equations Three fields are modeled independently in COBRA-TF, leading to three sets of each of the governing equations. One exception is that the droplet and continuousliquid fields are assumed to be in thermodynamic equilibrium and, thus, share an energy equation. Three continuity equations appear in the form of Equation 2.1. ) t (α kρ k ) + div (α k ρ kvk = L k + Mmass T (2.1) Temporal and convection terms appear on the Left-Hand Side (LHS). The field source terms appear on the Right-Hand Side (RHS), which are evaporation/ condensation, represented by L k, and turbulent mixing/void drift, represented by Mmass. T All terms are defined on a per-unit-volume basis. The equations can be expanded out for each of the fields using Cartesian coordinates (3-D formulation). Equations 2.2, 2.3, and 2.4 show the expanded forms of the continuity equation for the vapor, liquid, and entrained droplet fields, respectively.

32 8 t (α vρ v ) + x (α vρ v u v ) + y (α vρ v v v ) + z (α vρ v w v ) = Γ (2.2) t (α lρ l ) + x (α lρ l u l ) + y (α lρ l v l ) + z (α lρ l w l ) = (1 η)γ S (2.3) t (α eρ e ) + x (α eρ e u e ) + y (α eρ e v e ) + z (α eρ e w e ) = ηγ + S (2.4) The evaporation/condensation is captured with the Γ term, which is the volumetric mass flow rate due to phase change. Phase change is viewed as taking place between the liquid fields (droplets and continuous-liquid) and the vapor field. The ratio of phase change happening between the two liquid fields is captured with the η symbol. The meaning of each term of the continuity equations is further described in Equation 2.5. [ ] [ ] [ ] Change of Mass x-direction y-direction + + with Time Mass Convection Mass Convection [ ] [ ] z-direction Phase Change + = ± Mass Convection Mass Transfer ([ ]) Entrainment/De-Entrainment ± Mass Transfer Momentum Equations (2.5) Momentum equations are formulated for each field and each direction of the flow. COBRA-TF is capable of a 3-D formulation of the momentum equations (9 momentum equations) or a sub-channel formulation of the momentum equations (6 momentum equations). The general formulation of the axial momentum equation is shown in Equation 2.6. ) ) (α k ρ kvk + div (α k ρ k u kvk = α k ρ k g α k P + τ w + τ i + Mmom L + Mmom T t (2.6)

33 9 Source terms include the body force due to gravity (axial momentum equations only), pressure gradient, wall shear, phase-interface shear, momentum source/sink due to phase change and momentum source/sink due to turbulent mixing and void drift. The general, Cartesian form of the COBRA-TF momentum equations have three different components for each of the three phases (9 equations in total). The three components account for momentum in the x-,y-, and z- directions. The expanded form of the z-component momentum equations are shown here for the three fields. t (α vρ v w v ) + x (α vρ v w v u v ) + y (α vρ v w v v v ) + z (α vρ v w v w v ) = α v P z τ wz,v τ iz,vl τ iz,ve + Γ w (2.7) t (α lρ l w l ) + x (α lρ l w l u l ) + y (α lρ l w l v l ) + z (α lρ l w l w l ) = α l P z τ wz,l + τ iz,vl + Γ w S w (2.8) t (α eρ e w e ) + x (α eρ e w e u e ) + y (α eρ e w e v e ) + z (α eρ e w e w e ) = α e P z τ wz,v + τ iz,vl τ iz,ve + Γ w + S w (2.9) The meaning of the terms in the momentum equations are as shown in Equation

34 10 [ ] [ ] Momentum Change z-direction Convection + with Time of x-direction Momentum [ ] [ ] z-direction Convection z-direction Convection + + = of y-direction Momentum of z-direction Momentum [ ] [ ] [ ] z-direction Wall Interfacial ± ± Pressure Force Shear Shear [ ] [ ] Momentum transfered Momentum transfered ± by Phase Change by entrainment (2.10) The convention is that the vapor field moves faster than the liquid fields, which is why the interfacial shear term is additive for the continuous-liquid and droplet fields and subtractive for the vapor field. Also, convention states that phase-change mass transfer is from the liquids into the vapor, accounting for the addition of Γ in the vapor equation and for the subtraction of Γ in the liquid equations. A tilde appears over the velocity terms multiplying by Γ and S because the velocity used will depend on the direction of the mass transfer. The convention is that the velocity of the donor field is used Energy Equations The general form of the energy equation is presented in Equation Since droplets and continuous-liquid are in thermal equilibrium, there are only two energy equations. ) t (α kρ k h k ) + div (α k ρ k h kvk = q wk P + α k t + M en L (2.11) An expanded, Cartesian form of the equations for vapor and liquid are shown in Equations 2.12 and t (α vρ v h v ) + x (α vρ v h v u v ) + y (α vρ v h v v v ) + z (α vρ v h v w v ) = ( ) αv qv T y ( αv q y v T z )z + Γ h g + Q P wv + α v t (2.12)

35 11 t ((α l + α e )ρ l h l ) + x (α lρ l h l u l ) + x (α eρ l h l u e ) + y (α lρ l h l v l ) + y (α eρ l h l v e ) + z (α lρ l h l w l ) + z (α eρ l h l w e ) = y ( αl q T l ) y z ( αl q T l )z + Γ h f + Q wl + (α l + α e ) P t (2.13) The terms of the energy equations are defined in Equation [ ] [ ] [ ] [ ] Change of Energy x-direction y-direction z-direction = with Time Convection Convection Convection [ ] Implicit Heat Turbulent ± Exchange Transfer + (Phase Change) [ ] [ ] Wall Heat Pressure + + (2.14) Transfer Work Turbulent mixing occurs only in the lateral directions (the x-direction is assumed in the axial direction) and it only affects the liquid phase Interfacial Area Transport Equation The interfacial area transport equation tracks the interfacial area of the droplet field throughout the flow field. Unlike the other governing equations, it is not solved inside the outer loop, but instead, it is solved after convergence of the outer loop is acheived for a timestep. The equation is written as: A i,d + div ( ) A i,d u e = A i,e + A i,γ (2.15) t Equation 2.15 simply means:

36 12 Rate of Interfacial area Efflux of Interfacial area change in interfacial area + interfacial area = generation by entrainment and + generation by concentration phase change concentration deposition (2.16) An expanded form of the interfacial transport equation is shown by Equation t A i,d + ( ) A ( ) x i,du e + A ( ) y i,dv e + A z i,dw e + = A i,e + A i,γ (2.17) 2.2 Closure of Governing Equations Prior to the application of a numerical solution to the sets of linearized, discrete governing equations, it is necessary to provide closure to them by giving values to the source terms. This includes calculating wall shear, phase-interface shear, wall heat transfer, phase-interface heat transfer, lateral exchange due to turbulent mixing and void drift, and mass transfer due to entrainment. discussed in this section. This is further Flow Regime Map The aforementioned closure relationships cannot be adequately captured for twophase flows without consideration for the behavior of the flow. This behavior can vary widely, causing orders-of-magnitude differences in heat, mass, and momentum transfer in the flow. To account for these differences, COBRA-TF includes flow regime maps for normal- and hot-wall conditions. The normal-wall flow regime map (where heated surface temperatures remain below the Critical Heat Flux (CHF) temperature) considers the small bubble, small-to-large bubble, churn/turbulent, and annular/mist flow regimes. The normal-wall flow regime map employed in COBRA-TF captures the flow regiems shown in Figure 2.1. When a heated surface in the analysis volume is greater than the CHF temperature, different flow regimes are possible. The COBRA-TF hot-wall flow regime map includes: inverted

37 13 annular, inverted slug, dispersed droplet, falling film, and top deluge flow regimes. The hot-wall flow regime map is shown in Figure 2.2. α< <α< <α<α crit α crit <α Small Bubble Small-to- Large Bubble Churn/ Turbulent Annular/ Mist Figure 2.1. COBRA-TF normal-wall flow regime map When the flow regime of a mesh cell is determined, consideration is made for the geometry of the two-phase interface and the interface surface area is calculated. This is necessary for the determination of interface heat and momentum transfer. An overview of the models for closing the governing equations is provided in the following sections. The overview is a general one at this point, but the specifics of certain models were important to this work and will be discussed in more detail in Chapter 4. The CTF manual may also be consulted for more detail on the following closure models[13].

38 14 Top Deluge Falling Film Quench Front Dispersed Droplets Inverted Slug Single Phase Liquid Figure 2.2. COBRA-TF hot-wall flow regime map Wall Shear The wall shear term, found in the momentum equations, includes wall friction and form loss contributions as shown in Equation τ w = dp ) + dp ) dx fric dx form (2.18) Two-phase frictional pressure drop is determined using the two-phase friction multiplier approach of Lockhart and Martinelli[14].

39 15 ) dp = f w,kg 2 k Φ 2 k (2.19) dx fric 2D h ρ k For normal-wall conditions, the liquid phase is considered in calculation of wall shear and Φ is evaluated as 1/α l. All wall-shear is carried by the liquid phase unless vapor void is very high. For hot-wall conditions, the vapor phase is instead used and Φ is 1/α v. All wall-shear is carried by the vapor phase in hot-wall conditions unless liquid void is very high. Form loss is included in the wall shear term and is carried by all three fields. It is captured by Equation ) dp k form = α k dx form 2 X ρ k u k u k (2.20) The wall shear is semi-implicit, meaning that it multiplies by the new iteration velocity when the momentum equations are being solved. Therefore, both Equations 2.19 and 2.20 are divided through by velocity when being calculated such that they are comprised of one old-iteration velocity and one new-iteration velocity Phase Interface Shear The drag-force equation is used to capture interface shear. τ i = ρu2 rel C DA i 2 XA x (2.21) The drag-coefficient term and the phase-interface-area term will both be dependent on the flow regime of the cell. The phase-interface shear, like the wall shear, has an explicit and implicit velocity.

40 Wall Heat Transfer Newton s law of cooling is used to characterize heat transfer between solid surfaces and the fluid. q A s wk = h c (T w T F ) A x X (2.22) The conductor surface temperature, T w, is obtained from the conduction equation solution for conductors. The convective-heat-transfer coefficient, h, is dependent on the heat transfer region. COBRA-TF recognizes: single-phase liquid or vapor convection, sub-cooled nucleate boiling, saturated nucleate boiling, transition boiling, inverted-annular-film boiling, dispersed-droplet-film boiling, and dispersed-droplet-deposition heat transfer Phase-Interface Heat Transfer The purpose of calculating interface heat transfer coefficients in COBRA-TF is to quantify the amount of mass transfer due to phase change. The net mass transfer is calculated by subtracting condensation terms (sub-cooled liquid and vapor) from evaporation terms (super-heated liquid and vapor terms). [ Γ h int,shl = h l h f + (h g h f )C [ p,l h int,scl h l h f + (h g h f )C p,l ] h int,shv h v h g (h g h f )C p,v h int,scv (h g h f )C p,l h v h g ] (2.23) An interfacial-heat-transfer coefficient must be calculated depending on the state of the phase (super-heated or sub-cooled) and depending on the flow regime of the mesh cell. The COBRA-TF manual[13] provides information on the interfacialheat-transfer coefficient models for specific flow regimes. When the volumetric evaporation rate, Γ, is calculated, it can be used to determine mass, momentum, and energy transfer between the three fields.