Governing Equations for TwoPhase NC T10 Reyes

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Department of Nuclear Engineering Radiation
Health Physics
GOVERNING EQUATIONS IN TWO-PHASE FLUIDNATURAL
CIRCULATION FLOWS(Lecture T10)
José N. Reyes, Jr. June 25 June 29,
2007 International Centre for Theoretical Physics
(ICTP) Trieste, Italy
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Course Roadmap
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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.

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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

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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)

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Introduction(Brief History)

• The FLASH computer code, developed by
  Westinghouse-Bettis, 1950s.
• Simple"node and branch" approach to modeling
  suitable for some studies of single-phase flow in
  PWRs.
• Predecessor to the RELAP Series

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Introduction(Brief History)

• 1955 to 1975, Reactor Safety Research led to
  major advancements in boiling heat transfer and
  two-phase flow. Mid-1960s, Zubers 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. )

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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

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One-Dimensional, Two-Fluid, Full Non-Equilibrium
(Uniform Density within each Phase,Constant
Axial Cross-Sectional Area)
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One-Dimensional, Two-Fluid, Full Non-Equilibrium
(Uniform Density within each Phase,Constant
Axial Cross-Sectional Area)
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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.

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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
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One-Dimensional, Two-Phase Mixture Transport
Equations (Uniform Density within each
Phase,Constant Axial Cross-Sectional Area)
Mixture Properties
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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
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One-Dimensional, Two-Phase Drift Flux Transport
Equations (Uniform Density within each
Phase,Constant Axial Cross-Sectional Area)
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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
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Two-Phase Flow Models for Reactor Analysis
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Equivalent Approaches to Developing Model Balance
Equations
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Two-Phase Flow Models with Equal Phase Pressures
(pv pl p)
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Two-Phase Flow Models with Equal Phase Pressures
(pv pl p)
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Two-Phase Flow Models with Equal Phase Pressures
(pv pl p)
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Advancements in Two-Phase Flow Modeling(Interfaci
al 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.

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Advancements in Two-Phase Flow Modeling(Interfaci
al 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.

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Advancements in Two-Phase Flow Modeling(Interfaci
al 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
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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.

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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.

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Advancements in Two-Phase Flow Modeling(TRACE
Computer Code) J. Staudenmeier, NRC
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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

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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.