IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, 25-29 June 2007

1. IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, 25-29 June 2007 Governing Equations for Two-Phase N/C (T10) - Reyes 1

2. IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, 25-29 June 2007 Governing Equations for Two-Phase N/C (T10) - Reyes 2

3. IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, 25-29 June 2007 Governing Equations for Two-Phase N/C (T10) - Reyes 3 Lecture Objectives Describe the various models used to describe mass, momentum and energy transport processes in two-phase fluid flows related to natural circulation. Provide an overview of new models being considered for nuclear reactor safety computer codes.

4. IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, 25-29 June 2007 Governing Equations for Two-Phase N/C (T10) - Reyes 4 Outline Introduction Brief History of U.S. Nuclear Reactor Safety Computer Codes Two-Phase Flow Transport Equations One-Dimensional Two-Fluid Full Non-Equilibrium Transport Equations Two-Phase Mixture Transport Equations Two-Phase Drift Flux Transport Equations Two-Phase Flow Models for Reactor Analysis Advancements in Two-Phase Flow Modelling Conclusions

5. IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, 25-29 June 2007 Governing Equations for Two-Phase N/C (T10) - Reyes 5 Introduction The complexity of nuclear reactor geometry (e.g., multiple parallel paths and systems) coupled with transient two-phase fluid interactions make predictions of two-phase natural circulation behavior quite challenging A variety of methods have been used to model two-phase natural circulation in loops. Analytical Models (Solutions to Integration of transport equations around the loop). Systems codes (3,4,5 and 6 Equation Models)

6. IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, 25-29 June 2007 Governing Equations for Two-Phase N/C (T10) - Reyes 6 Introduction(Brief History) The FLASH computer code, developed by Westinghouse-Bettis, 1950�s. Simple"node and branch" approach to modeling suitable for some studies of single-phase flow in PWRs. Predecessor to the RELAP Series

7. IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, 25-29 June 2007 Governing Equations for Two-Phase N/C (T10) - Reyes 7 Introduction(Brief History) 1955 to 1975, Reactor Safety Research led to major advancements in boiling heat transfer and two-phase flow. Mid-1960s, Zuber�s development of the drift flux model. From the early 1970s to the present, the U.S. Nuclear Regulatory Commission supported the development of a number of computer codes to predict Loss-of-Coolant-Accident (LOCA) phenomenon. Idaho National Engineering Laboratory: (RELAP2, RELAP3, RELAP3B (BNL), RELAP4, RELAP5, TRAC-BF1) Los Alamos National Laboratory: (TRAC-PF1, TRAC-PD1) Brookhaven National Laboratory: (RAMONA-3B, THOR, RAMONA-3B, RAMONA-4B,HIPA-PWR and HIPA-BWR) In 1996, the NRC decided to produce the TRAC/RELAP Advanced Computational Engine or TRACE. (Combines the capabilities of RELAP5, TRAC-PWR, TRAC-BWR, and RAMONA. )

8. IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, 25-29 June 2007 Governing Equations for Two-Phase N/C (T10) - Reyes 8 Two-Phase Flow Transport Equations One-Dimensional, Two-Fluid, Full Non-Equilibrium One-Dimensional, Two-Phase Fluid Mixture One-Dimensional, Homogeneous Equilibrium Mixture(HEM) Transport Equations One-Dimensional, Two-Phase Drift Flux Transport Equations

9. IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, 25-29 June 2007 Governing Equations for Two-Phase N/C (T10) - Reyes 9 One-Dimensional, Two-Fluid, Full Non-Equilibrium (Uniform Density within each Phase,Constant Axial Cross-Sectional Area)

10. IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, 25-29 June 2007 Governing Equations for Two-Phase N/C (T10) - Reyes 10 One-Dimensional, Two-Fluid, Full Non-Equilibrium (Uniform Density within each Phase,Constant Axial Cross-Sectional Area)

11. IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, 25-29 June 2007 Governing Equations for Two-Phase N/C (T10) - Reyes 11 One-Dimensional, Two-Fluid, Full Non-Equilibrium (Uniform Density within each Phase,Constant Axial Cross-Sectional Area)(Neglecting Axial Heat Conduction and Axial Shear Effect) STAGNATION ENERGY: Thermodynamic internal energy and the kinetic energy of the fluid phase.

12. IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, 25-29 June 2007 Governing Equations for Two-Phase N/C (T10) - Reyes 12 One-Dimensional, Two-Phase Mixture Transport Equations (Uniform Density within each Phase,Constant Axial Cross-Sectional Area)

13. IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, 25-29 June 2007 Governing Equations for Two-Phase N/C (T10) - Reyes 13 One-Dimensional, Two-Phase Mixture Transport Equations (Uniform Density within each Phase,Constant Axial Cross-Sectional Area)

14. One-Dimensional, HEM Transport Equations (Uniform Density within each Phase,Constant Axial Cross-Sectional Area) Restrictions Imposed on Two-Phase Mixture Equations Thermal Equilibrium (Tl = Tv = TSAT), or Saturated Enthalpies (hl = hf and hv = hg)� Equal Phase Pressures (pl = pv = p) Equal Velocities (vl = vv = vm).

15. IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, 25-29 June 2007 Governing Equations for Two-Phase N/C (T10) - Reyes 15 One-Dimensional, Two-Phase Drift Flux Transport Equations (Uniform Density within each Phase,Constant Axial Cross-Sectional Area)

16. One-Dimensional, Two-Phase Drift Flux Transport Equations (Uniform Density within each Phase,Constant Axial Cross-Sectional Area)

17. Two-Phase Flow Models for Reactor Analysis

18. IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, 25-29 June 2007 Governing Equations for Two-Phase N/C (T10) - Reyes 18 Equivalent Approaches to Developing Model Balance Equations

19. Two-Phase Flow Models with Equal Phase Pressures (pv = pl = p)

20. Two-Phase Flow Models with Equal Phase Pressures (pv = pl = p)

21. IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, 25-29 June 2007 Governing Equations for Two-Phase N/C (T10) - Reyes 21 Two-Phase Flow Models with Equal Phase Pressures (pv = pl = p)

22. IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, 25-29 June 2007 Governing Equations for Two-Phase N/C (T10) - Reyes 22 Advancements in Two-Phase Flow Modeling(Interfacial Area Concentration Transport Model) Constitutive laws for interfacial transport are currently based on static flow regime maps. Efforts are underway to develop an interfacial area concentration transport model for dynamic flow regime modeling. Two-Group Interfacial Area Transport Model similar to Multi-Group neutron transport model. Group I consists of the spherical/distorted bubble group Group II consists of the cap/slug bubble group.

23. IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, 25-29 June 2007 Governing Equations for Two-Phase N/C (T10) - Reyes 23 Two-group bubble number density transport equations: Advancements in Two-Phase Flow Modeling(Interfacial Area Concentration Transport Model)

24. IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, 25-29 June 2007 Governing Equations for Two-Phase N/C (T10) - Reyes 24 Two-group Interfacial Area Transport Equations: Advancements in Two-Phase Flow Modeling(Interfacial Area Concentration Transport Model)

25. IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, 25-29 June 2007 Governing Equations for Two-Phase N/C (T10) - Reyes 25 The U.S. Nuclear Regulatory Commission (USNRC) is in the process of developing a modern code for reactor analysis. It is an evolutionary code that merges RAMONA, RELAP5, TRAC-PWR and TRAC-BWR into a single code. The reason for merging the codes, as opposed to starting new, is to maintain the sizable investment that exists in the development of input models for each of the codes. The consolidated code is called the TRAC/RELAP Advanced Computational Engine or TRACE. Advancements in Two-Phase Flow Modeling(TRACE Computer Code)

26. IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, 25-29 June 2007 Governing Equations for Two-Phase N/C (T10) - Reyes 26 TRACE is a component-oriented code designed to analyze reactor transients and accidents up to the point of fuel failure. It is a finite-volume, two-fluid, compressible flow code with 3-D capability. It can model heat structures and control systems that interact with the component models and the fluid solution. TRACE can be run in a coupled mode with the PARCS three dimensional reactor kinetics code. TRACE has been coupled to CONTAIN through its exterior communications interface (ECI) and can be coupled to detailed fuel models or CFD codes in the future using the ECI. TRACE has been coupled to as user-friendly front end, SNAP, that supports input model development and accepts existing RELAP5 and TRAC-P input models. Advancements in Two-Phase Flow Modeling(TRACE Computer Code)

27. IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, 25-29 June 2007 Governing Equations for Two-Phase N/C (T10) - Reyes 27 Advancements in Two-Phase Flow Modeling(TRACE Computer Code) � J. Staudenmeier, NRC

28. IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, 25-29 June 2007 Governing Equations for Two-Phase N/C (T10) - Reyes 28 Conservation Equations: (1) Mixture Mass (1) Vapor Mass (1) Liquid Momentum (1) Vapor Momentum (1) Mixture Energy (1) Vapor Energy Constitutive Equations: Equations of State Wall Drag Interfacial Drag Wall Heat Transfer Interfacial Heat Transfer Static Flow Regime Maps Advancements in Two-Phase Flow Modeling(TRACE Computer Code)

29. IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, 25-29 June 2007 Governing Equations for Two-Phase N/C (T10) - Reyes 29 Conclusions A Description of Two-Phase Flow Transport Equations has been provided: One-Dimensional, Two-Fluid, Full Non-Equilibrium One-Dimensional, Two-Phase Fluid Mixture One-Dimensional, Homogeneous Equilibrium Mixture(HEM) Transport Equations One-Dimensional, Two-Phase Drift Flux Transport Equations The 6, 5, 4, and 3 Equation Models have been discussed. A brief overview of new models being considered in the U.S. for nuclear reactor safety computer codes has been presented.