1. International Centre for Theoretical Physics (ICTP)
Department of Nuclear Engineering & Radiation Health Physics
GOVERNING EQUATIONS IN TWO-PHASE FLUID NATURAL CIRCULATION FLOWS (Lecture T10)
José N. Reyes, Jr.
June 25 – June 29, 2007
International Centre for Theoretical Physics (ICTP)
Trieste, Italy
IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, June 2007

2. Course Roadmap
IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, June 2007

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

4. Outline Introduction Two-Phase Flow Transport Equations
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
IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, June 2007

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

6. Introduction (Brief History)
Natural Circulation Loop
“Node 1”
Mass & Energy Storage
“Node 2”
“Node 3”
Line 1 Resistance
Line 2 Resistance
“Node 7”
“Node 6”
“Node 5”
Line 6 Resistance
Line 5 Resistance
“Node 8”
“Node 4”
Line 3
Line 4
Line 7
Line 8
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
IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, June 2007

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. )
IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, June 2007

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

9. One-Dimensional, Two-Fluid, Full Non-Equilibrium (Uniform Density within each Phase,Constant Axial Cross-Sectional Area)
Phase “k” Mass Conservation:
Area Averaging:
IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, June 2007

10. One-Dimensional, Two-Fluid, Full Non-Equilibrium (Uniform Density within each Phase,Constant Axial Cross-Sectional Area)
Phase “k” Momentum Conservation:
IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, June 2007

11. Phase “k” Energy Conservation:
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)
Phase “k” Energy Conservation:
STAGNATION ENERGY: Thermodynamic internal energy and the kinetic energy of the fluid phase.
STAGNATION ENTAHLPY: Usual definition, however, it is expressed in terms of the stagnation energy.
IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, June 2007

12. One-Dimensional, Two-Phase Mixture Transport Equations (Uniform Density within each Phase,Constant Axial Cross-Sectional Area)
Mixture Mass Conservation:
Mixture Momentum Conservation:
Mixture Enthalpy Conservation:
IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, June 2007

13. One-Dimensional, Two-Phase Mixture Transport Equations (Uniform Density within each Phase,Constant Axial Cross-Sectional Area)
Mixture Properties:
IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, June 2007

14. Restrictions Imposed on Two-Phase Mixture Equations
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).
Mixture Mass Conservation:
Mixture Properties:
Mixture Momentum Conservation:
Mixture Energy Conservation:

15. One-Dimensional, Two-Phase Drift Flux Transport Equations (Uniform Density within each Phase,Constant Axial Cross-Sectional Area)
Relationship Between Drift Velocity and Relative Velocity:
Two-Phase Flow Regimes
Drift Velocity Equations
Churn-Turbulent Flow
Slug Flow
Annular Flow
IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, June 2007

16. One-Dimensional, Two-Phase Drift Flux Transport Equations (Uniform Density within each Phase,Constant Axial Cross-Sectional Area)
Mixture Mass Conservation:
Drift-Flux Momentum Conservation:
Drift-Flux Internal Energy Conservation:

17. Two-Phase Flow Models for Reactor Analysis
Two-Phase Fluid Models
Types of Constitutive Equations
(Flow Regime Dependent)
Wall Friction (phase or mixture) correlations
Wall Heat Transfer (phase or mixture) correlations
Interfacial Mass Transport Equation
Interfacial Momentum Transport Equation
Interfacial Energy Transport Equation
Thermodynamic Properties
Typical Two-Phase Fluid
Balance Equations
6-Equation Model
5-Equation Models
4-Equation Models
3-Equation Models
Numerics
Two-Fluid Non-Equilibrium
Balance Equations (6-Equations)
(2) Mass Conservation Equations
(2) Momentum Conservation Equations
(2) Energy Conservation Equations
Possible Restrictions
Equilibrium (Saturation)
Partial Equilibrium
Homogeneous
Slip ratio
Drift flux
Velocity
Temperature or Enthalpy
Two-Phase Fluid Model Calculated Parameters
6-Equation:
5-Equation:
4-Equation:
3-Equation:

18. Equivalent Approaches to Developing Model Balance Equations
1 Mixture Balance Equation +
1 Phase Balance Equation
2 Phase Balance Equation OR
IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, June 2007

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. Two-Phase Flow Models with Equal Phase Pressures (pv = pl = p)
IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, June 2007

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

23. Two-group bubble number density transport equations:
Advancements in Two-Phase Flow Modeling (Interfacial Area Concentration Transport Model)
Two-group bubble number density transport equations:
Group I
Group II
Sj is the net rate of change in the number density function due to the particle breakup and coalescence processes
Sph is the net rate of change in the number density function due to phase change
Sj,12 and Sj, 21 are the inter-group particle exchange terms.
IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, June 2007

24. Number Density Relation:
Advancements in Two-Phase Flow Modeling (Interfacial Area Concentration Transport Model)
Number Density Relation:
ai,k is the interfacial area concentration
 is the void fraction
k is the bubble shape factor.
Subscript “k” represents the bubble group.
Two-group Interfacial Area Transport Equations:
Group I:
Group II:
IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, June 2007

25. Advancements in Two-Phase Flow Modeling (TRACE Computer Code)
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.
IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, June 2007

26. Advancements in Two-Phase Flow Modeling (TRACE Computer Code)
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.
IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, June 2007

27. Advancements in Two-Phase Flow Modeling (TRACE Computer Code) – J
Advancements in Two-Phase Flow Modeling (TRACE Computer Code) – J. Staudenmeier, NRC
IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, June 2007

28. Advancements in Two-Phase Flow Modeling (TRACE Computer Code)
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
Additional Equations:
Non-condensable Gas
Dissolved Boron
Control Systems
Reactor Power
Calculated Parameters:
Vapor Void Fraction
Steam Pressure
Non-condensable Gas Pressure
Liquid Velocity and Temperature
Vapor Velocity and Temperature
Boron Concentration
Heat Structure Temperatures
IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, June 2007

29. Conclusions The 6, 5, 4, and 3 Equation Models have been discussed.
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.
IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, June 2007