Modeling and Measurement ofModeling and Measurement of ......Modeling and Measurement ofModeling and...

Modeling and Measurement ofModeling and Measurement ofModeling and Measurement of Interfacial Area Concentration in

T h Fl

Modeling and Measurement of Interfacial Area Concentration in

T h FlTwo-phase FlowTwo-phase Flow

Mamoru Ishii and Takashi Hibiki

Thermal-Hydraulics and Reactor Safety LaboratorySchool of Nuclear Engineering

P d U i itPurdue UniversityWest Lafayette, IN 47907

Thermal-Hydraulics and Reactor Safety Laboratory, School of Nuclear Engineering, Purdue University

Presentation OutlinePresentation OutlineIntroductionFormulation of Two-Fluid Model with Interfacial Area Transport Eq.p q

Two-Group Interfacial Area Transport Eq.Two-Group Momentum Eq.

Modeling of Sink and Source Terms in Interfacial Area Transport Eq.Modeling of Sink and Source Terms in Interfacial Area Transport Eq.Sink and Source Due to Bubble Breakup and CoalescenceSink and Source Terms Due to Phase ChangeSource Term Due to Wall NucleationSource Term Due to Wall Nucleation

Database to Evaluate Interfacial Area Transport Eq.Local Interfacial Area MeasurementDatabase for 8 X 8 Rod Bundle GeometryDatabase for 8 X 8 Rod Bundle Geometry

Benchmarking Interfacial Area Transport Eq.Benchmarking 1-D IATE in Adiabatic SystemsBenchmarking 1 D IATE in Condensation SystemsBenchmarking 1-D IATE in Condensation Systems

Future DirectionsConclusions


IntroductionIntroduction

The modeling philosophy of thermal-hydraulic system analysis codes treatsThe modeling philosophy of thermal-hydraulic system analysis codes treats interface structure using flow regimes and transition criteria that cannot dynamically represent the changes in interfacial structure (no time or length scale is incorporated into the transition criteria).

This leads to instantaneous changes in flow regime, which can not only induce non physical oscillations in system behavior but can also hamperinduce non-physical oscillations in system behavior but can also hamper code accuracy and robustness.

To better represent the effects of interfacial structure and regime transition, the use of a first order equation to characterize interfacial area transporthas been recommended (Ishii, 1975).has been recommended (Ishii, 1975).

ai : interfacial area concentrationvi : interfacial velocityΦ

( )ii i

aa

tΦ

∂+ ∇ ⋅ =

∂v

Φ : sink and source termst∂


Formulation of Interfacial Area Transport Equation (K t f ll i d I hii 1995)(Kocamustafaogullari and Ishii, 1995)

Boltzmann Transport Equation of Particles

( ) j ph

f dVf f S S

t V dt⎛ ⎞∂ ∂ ⎟⎜+∇ ⋅ + = +⎟⎜ ⎟⎜⎝ ⎠∂ ∂ ∑v

jt V dt⎝ ⎠∂ ∂

( ), , : distribution function, : volume, and : particle source and sink rates per unit

mixture volume due to particle interaction and phase changej phf V t V S Sx

mixture volume due to particle interaction and phase change

Interfacial Area Transport Equation

( ) ( ){ }23

i ii i g ph j ph

j

a aa

t tα

α η Φ Φα

∂ ∂⎛ ⎞⎟⎜+∇ ⋅ − +∇⋅ − = +⎟⎜ ⎟⎝ ⎠∂ ∂ ∑v vj

: rate of volume generated by nucleation source per unit mixture volume, and :

interfacial area source and sink rates per unit mixture volume due to particle interactionph j phη Φ Φ


p p

and phase change

Development of Two-group Interfacial AreaT t E ti (I hii d Hibiki 2005)Transport Equation (Ishii and Hibiki, 2005)

Fluid particle number density transport equation analogous to Interfacial Area Transport Eqtransport equation analogous to Boltzmann’s transport equation

Interfacial Area Transport Eq.

Two-group approachTwo group approach

Group 1Spherical/distorted bubble

Group 2Cap/slug/churn-turbulent

b bblgroup bubble group

4d,maxDgσΔρ

=Maximum Distorted Bubble Size Limit


, gΔρ

Two-group Interfacial Area Transport Equation(I hii d Ki 2004 I hii d Hibiki 2005)(Ishii and Kim, 2004; Ishii and Hibiki, 2005)

Two group Void Fraction Transport Equation

( ) ( ) ( ) 121gk g kgk g gk gk m

t

α ρα ρ Γ Δ

∂+∇⋅ = + −

∂v

Two-group Void Fraction Transport Equation

t∂

( ) ( )3 112 2 1 1 1 1 1

1

* *, , g crit

g j c g g ph cj Sm

Dm D D

t D

αΔ ρ η χ α η

⎡ ⎤⎧ ⎫∂⎪ ⎪⎪ ⎪⎢ ⎥= + +∇⋅ − ≡⎨ ⎬⎢ ⎥⎪ ⎪∂⎪ ⎪⎩ ⎭⎣ ⎦∑ vj⎣ ⎦

( ) ( ){ } ( )2 11 12 *a a α⎡ ⎤∂∂ ⎢ ⎥ ∑

Two-group Interfacial Area Transport Equation

( ) ( ){ } ( )2 11 11 1 1 1 1 1 1 1

1

23

*,

gi ii i c g g ph j ph

jg

a aa D

t t

αχ α η φ φ

α∂∂ ⎢ ⎥+∇ ⋅ = − +∇⋅ − + +⎢ ⎥∂ ∂⎣ ⎦

∑v v

( ) ( )22 22 gi ia aa

αα η

⎡ ⎤∂∂ ⎢ ⎥+ ∇ +∇v v( ) ( )

( ) ( )

2 22 2 2 2 2

2

2 111 1 1 1 2 2

3

*

gi ii i g g ph

g

gic g g ph j ph

at t

aD

t

α ηα

αχ α η φ φ

⎢ ⎥+ ∇ ⋅ = +∇⋅ −⎢ ⎥∂ ∂⎣ ⎦

⎡ ⎤∂⎢ ⎥+ +∇⋅ − + +⎢ ⎥∂ ∑

v v

v( ) ( )1 1 1 1 2 21

,c g g ph j phjg t

χ η φ φα ⎢ ⎥∂⎣ ⎦

∑


Two-group Momentum Equation(S t l 2003)(Sun et al., 2003)

Two group Momentum Equation

( ) ( ) ( )1 11 1 1 1 1 1 1 1 1

g g g Tg g g g g g g g g gp

tμ

α ρα ρ α α α ρ

∂ ⎡ ⎤+∇⋅ = − ∇ +∇⋅ + +⎢ ⎥⎣ ⎦∂

vv v gT T

Two-group Momentum Equation

( ) ( )

( )1 12 1 1 1 1g gi gi ig

tmΓ α

⎣ ⎦∂+ −Δ −∇ ⋅ +v MT

( )( ) ( ) ( )

( )

2 22 2 2 2 2 2 2 2 2

2 12 2 2 2 2

g g g Tg g g g g g g g g

i i i

pt

m

μα ρ

α ρ α α α ρ

Γ α

∂ ⎡ ⎤+∇⋅ = − ∇ +∇⋅ + +⎢ ⎥⎣ ⎦∂+ +Δ −∇ ⋅ +

vv v g

v M

T T

T( )2 12 2 2 2 2g gi gi igmΓ α+ +Δ ∇ +v MT

( )( ) ( ) ( )( )

11 1 1

g f f Tg f f f g f g f fp μ

α ρα ρ α α

⎡ ⎤∂ −⎢ ⎥⎣ ⎦ ⎡ ⎤ ⎡ ⎤+ ∇ ⋅ − = − − ∇ +∇⋅ − +⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦v

v v T T( ) ( ) ( )( )

( ) ( )

1 1 1

1 1

g f f f g f g f f

g f f fi if g fi

pt

α ρ α α

α ρ Γ α

+∇ ∇ +∇ +⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦∂+ − + + −∇ − ⋅

v v

g v M

T T

T


One-dimensional One-group Interfacial AreaT t E tiTransport Equation

Bubbly Flow Regime2-G Interfacial Area Transport Eq. → 1-G Interfacial Area Transport Eq.

One-dimensional Interfacial Area Transport Equation

( )a∂ ∂ ( )ii iz HE WE BB VT BC CDa

aa v

t zΦ Φ Φ Φ Φ Φ

∂ ∂+ = + + + − −

∂ ∂

HE: bulk liquid boiling, WE: bubble nucleation from active cavities,C C

( )dnWEWE DfN ,,ΦΦ =

BB: bubble breakup, VT: void transport, BC: bubble coalescence, CD: condensation

N : active nucleation site density f : bubble generation frequencyNn: active nucleation site density, f : bubble generation frequency,Dd: bubble departure size


Classification of Possible Interactions ofT B bbl (Hibiki d I hii 2000)Two-group Bubbles (Hibiki and Ishii, 2000)

Coalescence Breakup

Group 1(1+1 & 1)

Inter-group(1+2 & 2)

Inter-group(1+1 & 2)

Group 2(2+2 & 2)


Bubble Coalescence & Breakup Mechanism(K t f ll i d I hii 1995)(Kocamustafaogullari and Ishii, 1995)

Coalescence Mechanisms

Random Collision Wake Entrainment

Breakup MechanismsBreakup Mechanisms

Turbulent impact Shearing-off Surface instability


Modeling of Bubble Coalescence(Hibiki d I hii 2000)(Hibiki and Ishii, 2000)

Probable Coalescence Bubble random collision induced byProbable CoalescenceMechanism

Bubble random collision induced by turbulence in a liquid phase

Bubble Coalescence Rate

Bubble Collision Frequency

Coalescence Efficiency= x


(1) Turbulence is isotropic,(2) Bubble size lies in the inertial subrange.

Coalescence

Coalescence efficiency is an exponential function of time required for bubble

l i b li id fil thi iCoalescence Efficiency coalescence given by liquid-film-thinning

model and a contact time for two bubbles given by dimensional consideration.


Sink Term of Interfacial Area Concentration Due to B bbl C l (Hibiki d I hii 2000)Bubble Coalescence (Hibiki and Ishii, 2000)

Bubble Collision Coalescence

1 3 1 2 5 6 1 3K D⎛ ⎞⎛ ⎞


Coalescence Efficiency

( )

1 3

2 3,max

CC

b C

fD

γ αεα α

=− 1 2exp exp C f bC

CC

K Dt ρ ελ

τ σ

⎛ ⎞⎛ ⎞ ⎟⎜⎟⎜ ⎟⎜⎟= − = −⎜ ⎟⎜⎟ ⎟⎜ ⎟⎜ ⎜ ⎟⎜⎝ ⎠ ⎟⎝ ⎠

2 2⎛ ⎞ ⎛ ⎞

Rate of IAC Change

1 2 5 6 1 3

1 13 3C C C b C

i i

f na aα α

Φ φ λψ ψ

⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜⎟ ⎟= =⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠⎛ ⎞K =1 29

( )

1 2 5 6 1 31 32

5 3 1 2,max

exp C f bC

b C

K D

D

ρ εΓ α εα α σ

⎛ ⎞⎟⎜ ⎟⎜= − ⎟⎜ ⎟⎜ ⎟⎜ ⎟− ⎝ ⎠

KC=1.29


Modeling of Bubble Breakup(Hibiki d I hii 2000)(Hibiki and Ishii, 2000)

Probable Breakup Bubble eddy random collision inducedProbable BreakupMechanism

Bubble-eddy random collision induced by turbulence in a liquid phase

Bubble Breakup Rate

Bubble-Eddy Collision Frequency

Breakup Efficiency= x

Bubble-Eddy Collision Frequency

(1) Turbulence is isotropic,(2) Eddy size lies in the inertial subrange.(3) Eddy with size from cD to D can break

Breakup

(3) Eddy with size from cDb to Db can break up bubble with size of Db.

Breakup efficiency is an exponential function Breakup Efficiency

p y pof average energy of a single eddy and average energy required for bubble breakup


Source Term of Interfacial Area Concentration Due to B bbl B k (Hibiki d I hii 2000)Bubble Breakup (Hibiki and Ishii, 2000)

Bubble Eddy BreakupBubble-Eddy Collision Frequency

Breakup Efficiency

1 3E K⎛ ⎞⎛ ⎞

( )

1 3

2 3,max

BB

b B

fD

γ αεα α

=− 5 3 2 3exp expB B

B

f b

E Ke D

σλ

ρ ε

⎛ ⎞⎛ ⎞ ⎟⎜ ⎟⎟⎜ ⎜= − = − ⎟⎟⎜ ⎜⎟⎜ ⎟⎝ ⎠ ⎜ ⎟⎜ ⎟⎝ ⎠

Rate of IAC Change

2 21 1⎛ ⎞ ⎛ ⎞

K =1 59 1 3

1 13 3B B B e B

i i

f na aα α

Φ φ λψ ψ

⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜⎟ ⎟= =⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠⎛ ⎞KB=1.59 ( )

( )

1 3

5 3 5 3 2 31

,max

expB B

b B f b

K

D D

Γ α α ε σα α ρ ε

⎛ ⎞− ⎟⎜ ⎟⎜= − ⎟⎜ ⎟⎜ ⎟⎜ ⎟− ⎝ ⎠


Sink Term of Interfacial Area Concentration Due to B bbl C d ti (P k t l 2007)Bubble Condensation (Park et al., 2007)

Heat Transfer Control Region

Inertial-Control Region

( )2

4 1 BCD HC IC b c t Nuc Ja

c

Dn P N N ,

tΦ Φ Φ π α

⎧ ⎫⎪ ⎪⎪ ⎪= + = − +⎨ ⎬⎪ ⎪⎪ ⎪⎩ ⎭


c⎪ ⎪⎩ ⎭

Modeling of Wall Nucleation TermModeling of Wall Nucleation Term

W ll N l ti S T

2 : heated perimeter : cross sectional arean HN fD A

ξφ π ξ=

Wall Nucleation Source Term

, : heated perimeter, : cross-sectional areaWE d H cc

D AA

φ π ξ=

Key Models to Estimate Wall Nucleation Source Term

Active Nucleation Site Density, Nn

Key Models to Estimate Wall Nucleation Source Term

y, n

Bubble Departure Diameter, Dd

Bubble Departure Frequency, f


Active Nucleation Site DensityActive Nucleation Site Density

rr*

r

r1

r

Active nucleation site density images by Basu et al. (2002).


Active Nucleation Site Density (Hibiki and Ishii, 2003)Active Nucleation Site Density (Hibiki and Ishii, 2003)

Knowledge of Size and Cone Angle Distributions of Cavities Modelg g

( )2

21 exp exp 18n nN N f ,

Rθ λ

ρμ

+⎡ ⎤⎧ ⎫ ⎧ ⎫⎛ ⎞⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎟⎜ ⎢ ⎥= − − ⎟ −⎨ ⎬ ⎨ ⎬⎜ ⎟ ⎢ ⎥⎜ ⎟⎜⎪ ⎪ ⎪ ⎪⎝ ⎠ ⎪ ⎪⎪ ⎪ ⎩ ⎭⎣ ⎦8 cRμ⎝ ⎠ ⎪ ⎪⎪ ⎪ ⎩ ⎭⎩ ⎭ ⎣ ⎦

( ) ( )2 30 01064 0 48246 0 22712 0 05468 log *f ρ ρ ρ ρ ρ ρ+ + + + += + + ≡

5 2 -6=4.72 10 sites/m , =0.722 radian, =2.50 10 m, : contact angle,nN μ λ θ× ×

( ) ( ) -0.01064 0.48246 0 22712 0 05468 logf . . , ,ρ ρ ρ ρ ρ ρ= + − + ≡

*gρ Δρ ρ≡

( ){ }( ) ( ){ }

2 1

exp 1

g f f

c

fg g sat g sat

pR ,

i T T RTT

σ ρ ρ+=

− −{ } : gas temperature, : saturation temperature, : latent heat,

: gas constant based on a molecular weight. For example, the valueg sat fgT T i

R


of for water vapor is 462 J/(kg K)(=8.31 J/(mol K)/(18.0 10-3 kg/mol))

Active Nucleation Site Density (Hibiki and Ishii, 2003)Active Nucleation Site Density (Hibiki and Ishii, 2003)

107Data: Basu, Warrier & Dhir (2002)

θ=30o, Water-Copper θ=57o, Water-Copper

o otes/m

2 ]

10111012

Basu et al., θ=30o

Basu et al., θ=57o

Basu et al θ=80otes/m

2 ]

106

θ=75o-85o, Water-Zr-4 θ=90o, Water-Copper Hibiki & Ishii Kocamustafaogullari & Ishii (1983) Basu, Warrier & Dhir (2002)

en.,

Nn

[sit

107108109

1010 Basu et al., θ=80 Basu et al., θ=90o

Borishanskii et al., P=0.101 MPa Borishanskii et al., P=0.450 MPa Borishanskii et al., P=0.980 MPa Borishanskii et al., P=1.48 MPa Borishanskii et al., P=1.94 MPa

n.,

Nn,

cal

c. [s

it

105

θ=90o

θ=90o

θ=75o-80o

θ=57oNuc

l. Si

te D

e

104105106107

+50 %

Borishanskii et al., P=2.95 MPa Borishanskii et al., P=4.90 MPa Borishanskii et al., P=7.30 MPa Borishanskii et al., P=9.80 MPaBorishanskii et al P=14 7 MPaN

ucl.

Site

Den

100 101 102104θ=30o

Act

ive

N

Wall Superheat ΔT [K]

10210310

102 103 104 105 106 107 108 109101010111012

-50 %Borishanskii et al., P=14.7 MPa Borishanskii et al., P=19.8 MPa Zeng & Klausner, θ=5o

Benjamin & Balakrishnan, θ=38o

2

Act

ive

N

Wall Superheat, ΔTw [K] Active Nucl. Site Den., Nn, meas.

[sites/m2]


Bubble Departure DiameterBubble Departure Diameter

Time (ms): 0.0 0.2 0.4 0.6 1.0 1.2 Lift-offLift-off

Diameter (mm): 0.000 0.397 0.440 0.460 0.507 0.544 SlidingSliding

Time (ms): 1.4 1.6 1.8 2.0 2.2 2.4

CondensationVaporizationHeat Flux

CondensationVaporizationHeat Flux

i ( ) llDiameter (mm): 0.574 0.588 0.609 0.617 0.615 0.605 DepartureFlow DepartureFlow


Bubble Departure Diameter (Situ et al., 2008)Bubble Departure Diameter (Situ et al., 2008)

Balance of Forces on Bubble at Nucleation Site Model

FpFqs : surface tension force at -directionsxF x

y

: unsteady drag force (growth force)

at -directionduxF

x

FduxFslθi

x F

: shear lift force

: surface tension force at -direction

: unsteady drag force at

sl

sy

F

F y

F directiony

Fduy Fg

Fsx

Fsy

: unsteady drag force atduyF -direction

: pressure force

: gravity forcep

y

F

F : gravity force

: quasi-steady force g

qs

F

F


Bubble Departure Diameter (Situ et al., 2008)Bubble Departure Diameter (Situ et al., 2008)

∑ y sy duy p g qsF F F F F F .= + + + +∑

0F 0syF =Surface Tension Force

4 2444 fb α

U d D F 4443

fduy Jae i

bF N sin

αθ

π= −Unsteady Drag Force

4 ( ) 343p b f g bF F grπ ρ ρ+ = − −Pressure and Gravity Forces

1/12 12 0 796

6 3

/ nn

qs n

f f r b Reb

F.

v r Nπρ ν

−⎡ ⎤⎛ ⎞⎟⎜⎢ ⎥⎟= + +⎜ ⎟⎢ ⎥⎜ ⎟⎜⎝ ⎠⎢ ⎥⎣ ⎦Quasi-Steady Force


Bubble Departure Frequency (Situ et al., 2008)Bubble Departure Frequency (Situ et al., 2008)

Non-Dimensional Analysis Correlationy

0 8034 06 .fd qNBN . N .=

2dfDN ≡

Non-Dimensional Bubble Departure Frequency

fdf

N ,α

≡

Non-Dimensional Heat Flux Representing Nucleate Boiling Heat Transfer

qNB dqNB

f g fg

q DN ,

iα ρ

′′≡

f g fg

: bubble departure diameter,

: nucleate boiling heat flux calculated by using Chen's correlation (1966)b

qNB

D

q ′′


g y g ( )qNBq

Bubble Departure Frequency (Situ et al., 2008)Bubble Departure Frequency (Situ et al., 2008)

106

Basu's data Situ et al.'s data Thorncroft et al.'s upflow dataThorncroft et al 's downflow datae f [-

]

106

Basu's data Situ et al.'s data Thorncroft et al.'s upflow data Thorncroft et al.'s downflow dataN =10 7N0.634

d=fdD

2 d /αf [-

]

105

Thorncroft et al. s downflow data

lds N

umbe

r, R

103

Nfd=10.7NqNB

Prediction error = ±113%

Freq

uenc

y N

fd

104

Liqu

id R

enol

100

-D. D

epar

ture

F

+100%

-100%

0 10 20 30 40 50 60103

Jacob Number, Jaw [-]10-3 100 103 10610-3N

on-

Non-D. Nucl. Boil. Heat Flux, NqNB=q"NBDd/(αfρgifg) [-]


Local Interfacial Structure CharacterizationLocal Interfacial Structure Characterization

• Instrument: Multi-sensor Conductivity Probe• Measured Variables (Local)

1 1– Void fraction– Interfacial area concentration

1 1=

Δ ∑ti

j nij

aT v

– Interfacial velocity– Bubble number frequency and bubble chord length

T Di t ib ti f Th V i bl• Transverse Distribution of These Variables• Measurements for Two Bubble Groups Separately

– Group 1: Spherical and distorted small bubbles– Group 2: Taylor and churn-turbulent large bubbles


Multi-Sensor Conductivity ProbeMulti Sensor Conductivity Probe

~250 mm

Epoxy and conductive ink Stainless-steel tubing,OD: ~3 mm

Thermocouplewires

~50-70 mm

Epoxy

OD: 3 mmStainless-steel tubing

wires

Coated sensor body

< 50-micron

01Sensor tip (uncoated)~2.5 mm

2 3

~ 0.5-0.8 mm

Not scaled


Interfacial Area Measurement (Cont’d)Interfacial Area Measurement (Cont d)

impedanceΔ /Δt

t4

downstream sensor(sensor 2) sensor 1

vi=Δs/Δtt3

t2upstream sensor

(sensor 1)Δs

sensor 2

t2t1

timeΔt

Δs vb t1 t2 t3 t4


Conductivity Probe PortConductivity Probe Port


Benchmark with Image Analysis(50.8 mm ID Pipe Upward Flow: <jf > = 0.321 and <jg> = 0.179 m/s)

40 80

[%

]

imageexperiment

Frac

tion

AC 40

a i

[1/m]imageexperiment

α [

L sl = 10.5 cmL s = 49.8 cmj g =0.179m/sj =0 321m/s

Void

F IA

0

0.0 0.5 1.0r/R

j f =0.321m/s

Radial Position, r/R

0

0.0 0.5 1.0r/R

Radial Position, r/R


Database to Evaluate Interfacial Area Transport E ti Adi b ti T h FlEquation -Adiabatic Two-phase Flow-


Database to Evaluate Interfacial Area Transport E ti B ili T h FlEquation -Boiling Two-phase Flow-


Summary of Interfacial Area DatabaseSummary of Interfacial Area Database

• Test Section Geometry: Round pipe, Confined channel, Annulus and Rod BundleT t S ti Si 1 t 102• Test Section Size: 1 mm to 102 mm

• Flow Regime: Bubbly, Cap-bubbly, Slug and Churn-turbulent Flowsturbulent Flows,

• Flow Condition: <jg> up to 10 m/s, <jf> from -3.1 m/s to 5.0 m/s

• Thermal Condition: Adiabatic and Diabatic Flows• Gravity Condition: Normal and Micro Gravity Conditions

P C diti At h i P• Pressure Condition: Atmospheric Pressure


Database for 8 X 8 Rod Bundle GeometryDatabase for 8 X 8 Rod Bundle Geometry

dW

d

S

x

Subchannel CenterRod Gap CenterRod Gap Center



s]

101 Mishima and Ishii (1984) Bubbly to Cap Bubbly Cap-Bubbly to Cap-Turbulent Cap-Turbulent to Churn-TurbulentFlow Conditions for Local Data<j

f> [

m/s

B S

100

Flow Conditions for Local Data

9 14 16

4 10 13 5 6

17 19loci

ty,

< B-S

S-A

10-1

14

20

17 19

318215871

Liqu

id V

e

1211

perf

icia

l L

S-C

C-A

10-2 10-1 100 101 10210-2

Sup

Superficial Gas Velocity, <jg> [m/s]

S C


g

Local Data for 8 X 8 Rod Bundle Geometry at z/D=200( j 0 2 / d j 0 02 / )(<jf > = 0.2 m/s and <jg> = 0.02 m/s)

0.20G 1 C

300Group 1 at Centerm

-1]101

Mishima and Ishii (1984) Bubbly to Cap Bubbly [m

/s]

0.10

0.15

Group 1 at Center Group 1 at Gap

on,

<α>

[-

]

150

200

250Group 1 at Center Group 1 at Gap

Con

c.,

<a i>

[m

100

y p y Cap-Bubbly to Cap-Turbulent Cap-Turbulent to Churn-Turbulent Flow Conditions for Local Data

9 14 16

4 10 13 5 6

17 19

Vel

ocity

, <

j f>

B-S

S-A

0 00

0.05V

oid

Frac

tio

0

50

100

Inte

rfac

ial A

rea

C

10-2

10-120

1211

318215871

uper

ficia

l Liq

uid

S-C

C-A

0.0 0.2 0.4 0.6 0.8 1.00.00

Location from Center, x/W [-]0.0 0.2 0.4 0.6 0.8 1.0

0I

Location from Center, x/W [-]2.0

Group 1 at CenterG 1 t G/s

]

8

10Group 1 at CenterGroup 1 at Gapm

> [

mm

]

10-2 10-1 100 101 10210 2

Su


Wd

S

1.0

1.5 Group 1 at Gap

ocity

, <

v g> [

m/

4

6

8 Group 1 at Gap

Dia

met

er,

<D

Sm

x

S

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

Gas

Vel

o

0.0 0.2 0.4 0.6 0.8 1.00

2

Saut

er M

ean

D

Subchannel Center


Location from Center, x/W [-] Location from Center, x/W [-]Subchannel CenterRod Gap Center

1-D Data for 8 X 8 Rod Bundle Geometry( j 0 2 / d j 0 02 / )(<jf > = 0.2 m/s and <jg> = 0.02 m/s)

0.20 Group 1Spacer Grid Location 250

300 Group 1Spacer Grid Location[m

-1]

0.10

0.15

Sp ce G d oc o

actio

n,

<α>

[-]

150

200

250 p

ea C

onc.

, <

a i>

101

m/s

]

0 50 100 150 2000.00

0.05V

oid

Fra

0 50 100 150 2000

50

100

Inte

rfac

ial A

re

100

10 Mishima and Ishii (1984) Bubbly to Cap Bubbly Cap-Bubbly to Cap-Turbulent Cap-Turbulent to Churn-Turbulent Flow Conditions for Local Data

9 14 16

4 10 13 5 6

17 19loci

ty,

<j f>

[m

B-S

S-A

0 50 100 150 200Axial Position, z/DH [-]

0 50 100 150 200Axial Position, z/DH [-]

0 8

1.0 Group 1 Spacer Grid Location

m/s

]

4

5 Group 1 Spacer Grid Location

m>

[m

m]10-1

14 16

20

1211

17 19

318215871

rfic

ial L

iqui

d V

el

C-A

0.4

0.6

0.8

city

, <

<vg>>

[m

2

3

4

Dia

met

er,

<D

Sm

10-2 10-1 100 101 10210-2

Supe

r


S-C

0 50 100 150 2000.0

0.2

Gas

Vel

o

0 50 100 150 2000

1

Saut

er M

ean

D


Axial Position, z/DH [-] Axial Position, z/DH [-]


0.20G 1 t C t

300Group 1 at Centerm

-1]

101 Mishima and Ishii (1984) Bubbly to Cap BubblyCap-Bubbly to Cap-Turbulent>

[m

/s]

0.10

0.15

Group 1 at Center Group 1 at Gap

on,

<α>

[-

]

150

200

250Group 1 at Center Group 1 at Gap

Con

c.,

<a i>

[m

100

Cap Bubbly to Cap Turbulent Cap-Turbulent to Churn-Turbulent Flow Conditions for Local Data

9 14 16

4 10 13 5 6

17 19

d V

eloc

ity,

<j f>

B-S

S-A

0 00

0.05V

oid

Frac

tio

0

50

100

Inte

rfac

ial A

rea

C

2 1 0 1 210-2

10-120

1211

318215871

Supe

rfic

ial L

iqui

d

S-C

C-A

0.0 0.2 0.4 0.6 0.8 1.00.00

Location from Center, x/W [-]2.0

Group 1 at CenterGroup 1 at Gap/s

]

0.0 0.2 0.4 0.6 0.8 1.00I

Location from Center, x/W [-]

8

10Group 1 at CenterGroup 1 at Gap

m>

[m

m]

10-2 10-1 100 101 10210S


Wd

S

1.0

1.5p p

ocity

, <

v g> [

m/

4

6

8 p p

Dia

met

er,

<D

Sm

x

S

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

Gas

Vel

o

0.0 0.2 0.4 0.6 0.8 1.00

2

Saut

er M

ean

D

Subchannel Center




0.05Group 1

200Group 1m

-1]

0.03

0.04

Group 1 Spacer Grid Location

on,

<α>

[-

]

100

150


Con

c.,

<a i>

[m

101

m/s

]

0 00

0.01

0.02

Voi

d Fr

actio

0

50

Inte

rfac

ial A

rea

C

100


9 14 16

4 10 13 5 6

17 19loci

ty,

<j f>

[m

B-S

S-A

0 50 100 150 2000.00

Axial Position, z/DH [-]0 50 100 150 200

0I

Axial Position, z/DH [-]2.0


m/s

]

4


m>

[m

m]10-1

14 16

20

1211

17 19

318215871

rfic

ial L

iqui

d V

el

C-A

1.0

1.5

city

, <

<vg>>

[m

2

3

4

Dia

met

er,

<D

Sm

10-2 10-1 100 101 10210-2

Supe

r


S-C

0 50 100 150 2000.0

0.5

Gas

Vel

o

0 50 100 150 2000

1

Saut

er M

ean

D


Axial Position, z/DH [-] Axial Position, z/DH [-]


1.0G 1 t C t

600G 1 t C tm

-1]101

Mishima and Ishii (1984) Bubbly to Cap Bubbly [m

/s]

0.6

0.8Group 1 at Center Group 1 at Gap Group 2 at Center Group2 at Gap

on,

<α>

[-

]

300

400

500Group 1 at Center Group 2 at Gap Group 2 at Center Group2 at Gap

Con

c.,

<a i>

[m

100


9 14 16

4 10 13 5 6

17 19

Vel

ocity

, <

j f>

B-S

S-A

0 0

0.2

0.4

Voi

d Fr

actio

0

100

200

Inte

rfac

ial A

rea

C

10-2

10-120

1211

318215871

uper

ficia

l Liq

uid

S-C

C-A

0.0 0.2 0.4 0.6 0.8 1.00.0

Location from Center, x/W [-]0.0 0.2 0.4 0.6 0.8 1.0

0I

Location from Center, x/W [-]

2.5

3.0Group 1 at CenterGroup 1 at Gap

m/s

]

80

100 Group 1 at CenterGroup 1 at Gap

m>

[mm

]

10-2 10-1 100 101 10210 2

Su


Wd

S

1 0

1.5

2.0

p pGroup 2 at CenterGroup 2 at Gap

ocity

, <

v g> [

m

40

60

80 p p Group 2 at Center Group 2 at Gap

Dia

met

er,

<DSm

x

S

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

Gas

Vel

o

0.0 0.2 0.4 0.6 0.8 1.00

20Sa

uter

Mea

n

Subchannel Center




1.0Group 1

600

m-1

]101 Mishima and Ishii (1984) Bubbly to Cap Bubbly [m

/s]

0.6

0.8

Group 1 Group 2

on,

<α>

[-

]

300

400

500 Group 1 Group 2

Con

c.,

<a i>

[m

100


9 14 16

4 10 13 5 6

17 19

Vel

ocity

, <

j f>

B-S

S-A

0 0

0.2

0.4

Voi

d Fr

actio

0

100

200

Inte

rfac

ial A

rea

C

10-2

10-120

1211

318215871

uper

ficia

l Liq

uid

S-C

C-A

0.0 0.2 0.4 0.6 0.8 1.00.0

Location from Center, d/S [-]0.0 0.2 0.4 0.6 0.8 1.0

0I

Location from Center, d/S [-]

2.5

3.0 Group 1Group 2/s

]

80

100Group 1Group 2>

[m

m]

10-2 10-1 100 101 10210 2

Su


Wd

S

1 0

1.5

2.0

Group 2

ocity

, <

v g> [

m/

40

60

80 p

Dia

met

er,

<D

Sm

x

S

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

Gas

Vel

o

0.0 0.2 0.4 0.6 0.8 1.00

20Sa

uter

Mea

n D

Subchannel Center


Location from Center, d/S [-] Location from Center, d/S [-]Subchannel CenterRod Gap Center


0.6Group 1

500Group 1-1

]

101

m/s

]

0.3

0.4

0.5G oup Group 2 Spacer Grid Location

on,

<α>

[-]

300

400

Group 1 Group 2 Spacer Grid Location

Conc

., <

a i> [

m-

100


9 14 16

4 10 13 5 6

17 19loci

ty,

<j f>

[m

B-S

S-A

0 0

0.1

0.2V

oid

Frac

tio

0

100

200

Inte

rfac

ial A

rea

C

10-1

14 16

20

1211

17 19

318215871

rfic

ial L

iqui

d V

el

C-A

0 50 100 150 2000.0

Axial Position, z/DH [-]

0 50 100 150 2000I


2 0

2.5 Group 1 Group 2Spacer Grid Locationm

/s]

40

50 Group 1 Group 2Spacer Grid Locationm

> [

mm

]

10-2 10-1 100 101 10210-2

Supe

r


S-C

1.0

1.5

2.0 p

city

, <

<vg>>

[m

20

30

40 Spacer Grid Location

Dia

met

er,

<D

Sm

0 50 100 150 2000.0

0.5

Gas

Vel

o

0 50 100 150 2000

10

Saut

er M

ean


Axial Position, z/DH

[-] Axial Position, z/DH

[-]

1-D Data for 8 X 8 Rod Bundle Geometry1 D Data for 8 X 8 Rod Bundle Geometry

5G 1m

m]

101

m/s

] 3

4


met

er,

<D

Sm>

[m

100


9 14 16

4 10 13 5 6

17 19loci

ty,

<j f>

[m

B-S

S-A1

2

aute

r Mea

n D

iam

10-1

14 16

20

1211

17 19

318215871

rfic

ial L

iqui

d V

el

C-A4


m>

[m

m]

0 50 100 150 2000Sa


40

50 Group 1 Group 2Spacer Grid Locationm

> [

mm

]

10-2 10-1 100 101 10210-2

Supe

r


S-C

2

3

4

Dia

met

er,

<D

Sm

20

30

40 Spacer Grid Location

Dia

met

er,

<D

Sm

0 50 100 150 2000

1

Saut

er M

ean

D

0 50 100 150 2000

10

Saut

er M

ean


Axial Position, z/DH [-] Axial Position, z/DH

[-]

Benchmarking One-DimensionalI t f i l A T t E tiInterfacial Area Transport Equation

2 iaΦ Φ∂ ∂

+

One-Dimensional One-Group IATE under Steady Bubbly Flow Conditions

3i

i i TI RC gaa v v

z zΦ Φ α

α= − +

∂ ∂

Sink and Source Terms

1 2 5 6 1 32 1 3

exp RC f bRCK Dρ εΓ α ε

Φ⎛ ⎞⎟⎜ ⎟⎜= − ⎟⎜


( )5 3 1 2expRC

b RC ,maxDΦ

α α σ= ⎟⎜ ⎟⎜ ⎟⎜− ⎝ ⎠

( ) 1 31 KΓ α α ε ⎛ ⎞⎟⎜( )

( )1 3

5 3 5 3 2 3

1 expTI TITI

b TI ,max f b

K

D D

Γ α α ε σΦ

α α ρ ε

⎛ ⎞− ⎟⎜ ⎟⎜= − ⎟⎜ ⎟⎜ ⎟⎜− ⎝ ⎠


Benchmarking One-Dimensional Interfacial Area T t E ti (Hibiki d I hii 2002)Transport Equation (Hibiki and Ishii, 2002)

30

40 Experimental Data (Hibiki & Ishii, 1999) Total Void TransportB bbl R d C lli i

[m-1] 600

800 Experimental Data (Hibiki & Ishii, 1999) Total Void Transport[m

-1]

10

20 Bubble Random Collision Turbulent Impact

> - <

a i> z/D

=12

200

400Bubble Random Collision Turbulent Impact

> - <

a i> z/D

=12

-10

0

<j >=0 872 m/s <j >=0 0813 m/sCha

nge,

<

a i>

400

-200

0

<j >=3 49 m/s <j >=0 702 m/sCha

nge,

<

a i>0 20 40 60 80 100 120 140

-20

<jf> 0.872 m/s, <jg,0> 0.0813 m/sD=25.4 mmIA

C C

Axial Position, z/D [-]0 20 40 60 80 100 120 140

-600

-400 <jf> 3.49 m/s, <jg,0> 0.702 m/sD=25.4 mmIA

C C

Axial Position, z/D [-]


Benchmarking Interfacial Area Transport EquationS iti it A l i (Hibiki d I hii 2002)-Sensitivity Analysis- (Hibiki and Ishii, 2002)

1200

1400 Experimental Data

(Hibiki et al., 2001a)a i> [

m-1]

1200

1400 ΛD=0.50 ΛD=0.80Λ =1 00

<jf>=0.491 m/s<j >=0 190 m/s<a

i> [

m-1

]

800

1000

ΛD=0.50

entra

tion,

<

600

800

1000ΛD=1.00 ΛD=1.25 ΛD=1.50

<jg,0> 0.190 m/sD=50.8 mm

entra

tion,

<

200

400

600 ΛD=0.80

ΛD=1.00

ΛD=1.25<jf>=5.00 m/s<jg 0>=3.90 m/sA

rea

Con

ce

200

400

600

Are

a C

once

0 10 20 30 40 50 600

200 ΛD=1.50

g,0

D=50.8 mmIn

terf

acia

l

Axial Position, z/D [-]0 10 20 30 40 50 60

0

00 Experimental Data (Hibiki et al., 2001a)

Inte

rfac

ial

Axial Position, z/D [-]

0

0

b, ,setD

b,

D

DΛ ≡

0

0

: initial bubble diameter utilized for computation,

: initial bubble diameter observed in the experimentb, ,set

b,

D

D


0b, 0b,


One-Dimensional One-Group IATE under Steady Bubbly Flow Conditions

2 ii i TI RC CD

aa v vΦ Φ Φ α

∂ ∂= − − +

with Condensation

3i i TI RC CD gaa v v

z zΦ Φ Φ α

α+

∂ ∂


( )

1 2 5 6 1 32 1 3

5 3 1 2exp RC f bRCRC

b RC ,max

K D

D

ρ εΓ α εΦ

α α σ

⎛ ⎞⎟⎜ ⎟⎜= − ⎟⎜ ⎟⎜ ⎟⎜− ⎝ ⎠( )b RC ,max ⎝ ⎠

( )

( )1 3

5 3 5 3 2 3

1 expTI TITI

K

D D

Γ α α ε σΦ

α α ρ ε

⎛ ⎞− ⎟⎜ ⎟⎜= − ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠( )b TI ,max f bD Dα α ρ ε⎜ ⎟⎜− ⎝ ⎠

( )2

4 1 BCD HC IC b c t Nuc Ja

Dn P N NΦ Φ Φ π α

⎧ ⎫⎪ ⎪⎪ ⎪= + = − +⎨ ⎬⎪ ⎪


( )CD HC IC b c t Nuc Jact

+ +⎨ ⎬⎪ ⎪⎪ ⎪⎩ ⎭


0

50

[m

-1] p0=0.170 MPa, Tf0=102 oC, G=492 kg/m2s

0

50 p0=0.103 MPa, Tf0=98.0 oC, G=139 kg/m2s

[m

-1]

-100

-50

i> - <

a i> z/D

=1

150

-100

-50

i> - <

a i> z/D

=1

-200

-150 Experimental Data (Zeitoun, 1994) Total Heat Transfer Inertia Void TransportRandom CollisionC

hang

e,

<a i

-250

-200

-150 Experimental Data (Zeitoun, 1994) Predicted IAC Change Heat Transfer Inertia Void TransportRandom CollisionC

hang

e,

<a i

0 1 2 3 4 5 6-250

Random Collision Turbulent Impact

IAC

C

Axial Position, z/D [-]0 2 4 6 8

-300

Random Collision Turbulent Impact

IAC

CAxial Position, z/D [ - ]


Future DirectionF l ti f I t f i l A T t E tiFormulation of Interfacial Area Transport Equation

Formulation of Interfacial Area Transport EquationFormulation of Interfacial Area Transport Equation

1975 Ishii Basic concept of necessity of interfacial area transport equation

1995 Kocamustafaogullari and Ishii

Foundation of interfacial area transport equation

2003c Hibiki et al. Formulation of one-dimensional interfacial area transport equation in subcooled boiling flow

2003a Sun et al. Formulation of modified two-fluid model for two-gas momentum equations

2004 Ishii and Kim Formulation of two-group interfacial area transport equation

Future work • Extension of interfacial area transport equation to churn-turbulent-to-annular fl t itiflow transition

• Extension of interfacial area transport equation to annular and annular-mist flow regimes


Future DirectionD l t f M t T h iDevelopment of Measurement Techniques

Development of Measurement TechniquesDevelopment of Measurement Techniques1986 Kataoka et al. Mathematical foundation of interfacial area concentration to be measured by local

probe technique

1992 Revankar and Ishii Demonstration of double-sensor probe technique 1993 Revankar and Ishii Demonstration of multi-sensor probe technique1998 Hibiki et al. Development of improved double-sensor probe technique1998 Hibiki et al. Application of hot-film anemometry to liquid velocity measurement

1999 Wu and Ishii Monte Carlo simulation of double-sensor probe technique2000 Kim et al. Development of improved multi-sensor probe technique2004c Sun et al. Application of laser Doppler anemometer to liquid velocity measurement

Future work • Improvement of local probe technique to be applicable to highly three-dimensional flow

• Application of film thickness probe to measure annular flow characteristicsA li i f d l h i l i fl• Application of droplet measurement technique to measure annular-mist flow characteristics


Future DirectionD t b C t tiDatabase Construction

Database Construction

1998 Hibiki et al. Upward bubbly flow in vertical pipe (gas and liquid phases)

1999 Hibiki and Ishii Upward bubbly flow in vertical pipe (gas and liquid phases)p y p p (g q p )

2001 Bartel et al. Upward boiling bubbly flow in vertical annulus

2001a Hibiki et al. Upward bubbly flow in vertical pipe (gas and liquid phases)

2002 Sun et al. Upward bubbly flow in vertical large diameter pipe

2003a Hibiki et al. Downward bubbly flow in vertical pipe

2003b Hibiki et al. Upward bubbly flow in vertical annulus

2003 Kim et al. Upward bubbly flow in confined channel

2003b Sun et al. Upward cap-turbulent and transition to slug flows in vertical large diameter pipe

2003 Takamasa et al Bubbly flow in pipe under microgravity conditions2003 Takamasa et al. Bubbly flow in pipe under microgravity conditions

2004 Situ et al. Upward boiling bubbly flow in vertical annulus

2004a Sun et al. Upward cap-turbulent and churn-turbulent flows in confined channel

2005 Situ et al. Bubble lift-off and departure diameters

2007 Hazuku et al. Upward annular flow in vertical pipe

2007 Hibiki et al. Upward bubbly flow in vertical mini-channel

2008 Jeong et al. Upward cap-turbulent and churn-turbulent flows in vertical annulus

2008 Situ et al Bubble departure frequency2008 Situ et al. Bubble departure frequency

Future work • Development of extensive slug, churn-turbulent and annular flow data• Development of extensive data at elevated pressure• Development of extensive data in various flow channels (geometry, orientation and size)• Development of extensive wall nucleation data (active nucleation site density, bubble departure size and

frequency)• Development of extensive condensation and boiling data


• Development of extensive condensation and boiling data

Future DirectionSi k d S T M d liSink and Source Term Modeling

Sink and source term modelingSink and source term modeling1983 Kocamustafaogullari and

IshiiActive nucleation site density

1989 Riznic and Ishii Flashing source term1998 Wu et al. One-group model in pipe2000a Hibiki and Ishii One-group model in pipe2000b Hibiki and Ishii Two-group model in pipe2001b Hibiki et al. One-group model in small-diameter pipeg p p p2003a Fu and Ishii Two-group model in pipe2003 Hibiki and Ishii Active nucleation site density2004b Sun et al. Two-group model in confined channel2005 Situ et al Bubble lift off diameter2005 Situ et al. Bubble lift-off diameter2007 Park et al. Condensation sink term2008 Situ et al. Bubble departure diameter2008 Situ et al. Bubble departure frequency

Future work • Improvement of two-group model• Improvement of bubble departure diameter model• Improvement of bubble departure frequency model• Development of bulk boiling source model


Future DirectionI l t ti i t CFD C dImplementation into CFD Codes

Implementation into CFD codeImplementation into CFD codeFuture work • Implementation of interfacial area transport equation into CFD code

• Benchmarking CFD code against data showing fully 3-D behavior


Future DirectionM d li f I t f i l F d T b l M d lModeling of Interfacial Forces and Turbulence Models

• Lift Force Model • Zero-Equation Model

Interfacial Force Models Turbulence Models

• Tomiyama et al. (2002)• Hibiki and Ishii (2007)

• Wall Lubrication Force Model

• Sato et al. (1981) • One-Equation Model

• Kataoka and Serizawa (1995) • Antal et al. (1991)• Tomiyama (1998)

• Turbulence Dispersion Force Model

( )• Two-Equation Model

• Lopez de Bertodano et al. (1994)Turbulence Dispersion Force Model

• Lahey et al. (1993)• Burns et al. (2004)


ConclusionsConclusions

In relation to the modeling of the interfacial transfer terms in the two-fluid model, the concept of the interfacial area transport equation has been proposed to develop a constitutive relation for the interfacial area concentration. The changes in the two-phase flow structure can be g ppredicted mechanistically by introducing the interfacial area transport equation.(1) The basic concept of the interfacial area transport equation and its(1) The basic concept of the interfacial area transport equation and its

formulation have been briefly explained.(2) Available models of interfacial area sink and source terms and

existing databases have been reviewed.(3) Newly obtained data for 8 X 8 rod bundle geometry has been

presentedpresented.(4) The interfacial area transport equation has been benchmarked

using adiabatic bubbly flow and condensation bubbly flow data. (5) Future direction for this research has been also suggested.


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