department of nuclear engineering & radiation health physics iaea-ictp natural circulation...

IAEA-ICTP Natural Circulation Training Course, Trieste, Italy, 25-29 June 2007 Governing Equations for Two-Phase N/C (T10) - Reyes1

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




Course Roadmap

Opening Session

· INTRODUCTIONS

· ADMINISTRATION

· COURSE ROADMAP

Introduction

· GLOBAL NUCLEAR POWER

· ROLE OF N/C ADVANCED DESIGNS

· ADVANTAGES AND CHALLENGES

Local Transport Phenomena & Models

· LOCAL MASS, MOMENTUM AND ENERGY TRANSPORT PHENOMENA

· PREDICTIVE MODELS & CORRELATIONS

Integral System Phenomena & Models

· SYSTEMS MASS, MOMENTUM AND ENERGY TRANSPORT PHENOMENA

· N/C STABILITY AND NUMERICAL TECHNIQUES

· STABILITY ANALYSIS TOOLS

· PASSIVE SAFETY SYSTEM DESIGN

Natural Circulation Experiemnts

· INTEGRAL SYSTEMS TESTS

· SEPARATE EFFECTS TESTS

· TEST FACILITY SCALING METHODS

Reliability & Advanced Computational Methods

· PASSIVE SYSTEM RELIABILITY

· CFD FOR NATURAL CIRCULATION FLOWS



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.



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



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)



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

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 4Line 7

Line 8

izi

ii

i zngp

8

1

8

1



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. )



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



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

Phase “k” Mass Conservation:

Change Phase toDue Volumeper Unit RateTransfer Mass

AxisFlowAlongk""PhaseFluidofMassAveragedAreainChange

k""Phase Fluid of Mass Averaged Area

of Change of Rate Time

kkkkkk vzt

A

kk dAA

1Area Averaging:



Phase “k” Momentum Conservation:

FlowofDirectioninActingForcesGravityForceDraglInterfacia

FlowofAxisAlongGradientPressurePhaseFluid

Nto1StructuresonForcesDragPhaseFluidofSum

1

Change Phase todue Volumeper Unit RateTransfer Momentum

AxisFlowAlongk""PhaseFluidofMomentumAveragedAreainChange

2

k""Phase Fluid of Momentum Averaged Area

of Change of Rate Time

zkkzskkk

N

iizwk

zkskkkkkkk

ngnFpz

nF

nvvz

vt





(Neglecting Axial Heat Conduction and Axial Shear Effect)

Phase “k” Energy Conservation:

TransferHeatlInterfacia

sk

GravitytoDueWorkkPhase

kkk

N

i ikk

kk

okskkk

okkk

okk

QgvA

Pq

tp

hvhz

ut

""

Nto1StructurestoTransferHeatPhaseFluidofSum

1

FractionVoidinChangeswithAssociatedWorkPressurek""Phase

Change Phase todue Volumeper Unit RateTransfer Energy

AxisFlowAlongk""PhaseFluidofEnergyAveragedAreainChange

k""PhaseFluidofEnergyAveragedAreainChangeofRateTime

2

2k

kok

vuu

k

kk

ok

puh

0

• 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.



One-Dimensional, Two-Phase Mixture Transport Equations (Uniform Density within each Phase,Constant Axial Cross-Sectional Area)

Mixture Mass Conservation:

0

z

G

tmm

Mixture Momentum Conservation:

cos1

2

gz

pF

G

zt

Gm

mN

iwi

m

mm

Mixture Enthalpy Conservation:

z

pF

G

A

PqhG

zph

tm

N

iwi

m

mN

i i

iimmmmm

11



One-Dimensional, Two-Phase Mixture Transport Equations (Uniform Density within each Phase,Constant Axial Cross-Sectional Area)

1lvm

122

2

llvv

mm vv

G

1llvvm vvG

m

llvvm

hhh

1

m

lllvvvm G

vhvhh

1

m

llvvm

vvv

22

2 1

m

llvv

m G

vvv

332 1

1lvm ppp

Mixture Properties:

One-Dimensional, HEM Transport Equations (Uniform Density within each Phase,Constant Axial Cross-Sectional Area)

Mixture Momentum Conservation:

Mixture Energy Conservation:

• 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).


0

mmm v

zt

cos1

2 gz

pFv

zv

t m

N

iwimmmm

cos22 1

22

gvA

Pq

vhv

zp

vh

t mm

N

i i

ii

mmmm

mmm

Mixture Properties:

1lvm

m

llvvm

vvv

1

m

llvvm

hhh

1



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

41

241.1

l

glvj

gV

21

35.0

l

glvj

DgV

21

123

D

vV

v

ll

lvj

1vj

lvr

Vvvv

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

Drift-Flux Momentum Conservation:

Drift-Flux Internal Energy Conservation:


0

mmm v

zt

cos

1 1

2

gz

pF

V

zz

vv

t

vm

mN

iwi

m

vjlvmmm

mm

N

iwim

N

i i

ii

m

vjvlm

mm

m

vjlvvlmmmmm

FvA

Pq

V

zp

z

vp

Vuu

zvu

zu

t

11

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 FluidBalance Equations

·6-Equation Model·5-Equation Models·4-Equation Models·3-Equation Models

Numerics

Two-Fluid Non-EquilibriumBalance 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:

vlvl TTvvp ,,,,,

mvl vTTp ,,,,

vlvl TorTvvp ,,,,

vl vvp ,,, lvm TorTvp ,,,

mvp,,



Equivalent Approaches to Developing Model Balance Equations

1 Mixture Balance Equation

+1 Phase Balance

Equation

2 Phase Balance Equation

OR

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

6-Equation Model Conservation Equations Restrictions1 Constitutive Laws2 Calculated

Parameters Two-Fluid Non-Equilibrium (2) Mass Phase Balance (2) Momentum Phase Balance (2) Energy Phase Balance

None

(2) Phase wall friction (2) Phase heat flux friction (1) Interfacial mass (1) Interfacial momentum (1) Interfacial energy

vlv

l

TTv

vp

,,

,,

5-Equation Models

Two-Fluid Partial Non-Equilibrium (2) Mass Phase Balance (2) Momentum Phase Balance (1) Mixture Energy Balance SATv

SATl

TT

or

TT

(2) Phase wall friction (1) Mixture wall heat flux (1) Interfacial mass (1) Interfacial momentum

vl

vl

TorT

vvp ,,,

Two-Fluid Partial Non-Equilibrium (1) Mixture Mass Balance (2) Momentum Phase Balance (2) Energy Phase Balance

SATv

SATl

TT

or

TT

(2) Phase wall friction (2) Phase heat flux friction (1) Interfacial mass3 (1) Interfacial momentum (1) Interfacial energy

vl

vl

TorT

vvp ,,,

Slip or Drift Non-Equilibrium (2) Mass Phase Balance (1) Mixture Momentum Balance (2) Energy Phase Balance

Slip or Drift

Velocity

(1) Mixture wall friction (2) Phase heat flux friction (1) Interfacial mass (1) Interfacial energy (1) Slip velocity or Drift flux

mv

l

vT

Tp

,

,,

Homogeneous Non-Equilibrium (2) Mass Phase Balance (1) Mixture Momentum Balance (2) Energy Phase Balance

Equal

Velocity

mvl vvv

(1) Mixture wall friction (2) Phase heat flux friction (1) Interfacial mass3 (1) Interfacial energy

mv

l

vT

Tp

,

,,


4 -Equation Models

Conservation Equations Restrictions1 Constitutive Laws2 Calculated Parameters

Two-Fluid Equilibrium Model (1) Mixture Mass Balance (2) Momentum Phase Balance (1) Mixture Energy Balance

SATvl TTT (2) Phase wall friction (1) Mixture heat flux friction (1) Interfacial mass3 (1) Interfacial momentum

vl vvp ,,,

Drift Partial Non-Equilibrium (2) Mass Phase Balance (1) Mixture Momentum Balance (1) Mixture Energy Balance

Drift Velocity

SATlv TTorT

(1) Mixture wall friction (1) Mixture wall heat flux (1) Interfacial mass (1) Drift flux correlation

lv

m

TorT

vp ,,,

Slip Partial Non-Equilibrium (1) Mixture Mass Balance (1) Mixture Momentum Balance (2) Phase Energy Balance

Slip Ratio

SATlv TTorT

(1) Mixture wall friction (1) Mixture wall heat flux (1) Interfacial mass (1) Drift flux correlation

lv

m

TorT

vp ,,,

Homogeneous, Partial Non-Equilibrium: (1) Mixture Mass Balance (1) Mixture Momentum Balance (2) Phase Energy Balance

SATlv

mvl

TTorT

uuu

(1) Mixture wall friction (2) Phase wall heat flux (1) Interfacial mass3 (1) Interfacial energy

lv

m

TorT

vp ,,,




3-Equation Models

Homogeneous Equilibrium (HEM): (1) Mixture Mass Balance (1) Mixture Momentum Balance (1) Mixture Energy Balance

SATvl

mvl

TTT

uuu

(1) Mixture wall friction (1) Mixture wall heat flux

mup,,

Slip or Drift Equilibrium: (1) Mixture Mass Balance (1) Mixture Momentum Balance (1) Mixture Energy Balance

Slip or Drift Velocity

SATvl TTT

(1) Mixture wall friction (1) Mixture wall heat flux (1) Slip velocity or Drift flux

mup,,

1. Restrictions imposed on fluid phase velocities or temperatures (or enthalpies in lieu of temperatures). 2. Minimum number of constitutive laws. For example, for N structures in the flow, N structure heat

flux and N wall friction correlations may be required. 3. Interfacial mass transfer is required to determine interfacial momentum or interfacial energy transfer.



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.



• Two-group bubble number density transport equations:


jphphjjpm SSSSvn

t

n12,1,12,1,1,1

1

jphphjjpm SSSSvn

t

n21,2,21,2,2,2

2

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.



• Two-group Interfacial Area Transport Equations:


Group I:

Group II:

– ai,k is the interfacial area concentration

– is the void fraction

– k is the bubble shape factor.

– Subscript “k” represents the bubble group.

2

3,

k

kikk

an

• Number Density Relation:

1,11

1

1,12,1,12,1,

2

1,

1

11,1,

1,

3

2

3

1i

i

jphphjj

iii

i vt

aSSSS

ava

t

a

2,22

2

2,21,2,21,2,

2

2,

2

22,2,

2,

3

2

3

1i

i

jphphjj

iii

i vt

aSSSS

ava

t

a



• 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)



• 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) – J. Staudenmeier, NRC

SNAP

TRACE InputProcessing

ComputationalEngine

Other SupportApplications

3D NeutronKinetics

SNAP SystemModel Database

RELAP5ASCIIInput

TRAC-PASCIIInput

TRAC-BASCIIInput

Interprocess MessagePassing Service

Platform IndependentBinary File

SNAP

TRACE InputProcessing

ComputationalEngine

Other SupportApplications

3D NeutronKinetics

SNAP SystemModel Database

RELAP5ASCIIInput

TRAC-PASCIIInput

TRAC-BASCIIInput

Interprocess MessagePassing Service

Platform IndependentBinary File



• 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



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.

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