1

University of Fukui

Two-phase Flow Fundamental

2

University of FukuiPhase change

Water is heated at 0.1MPa (Atomspheric pressure)

Increase of sensible heatρVCpΔT

Increase of steam ρVxLｌｖ

(Energy is stored as a form of latent heat)Temp

x: Steam quality

Heating Heating Heating Heating Heating Heating

Liquid Liquid LiquidLiquid

Vapor VaporVapor

3

University of Fukui

Ice・Water・Steam 3D chart

Ice

Ice & Vapor

Ice

Water and vapor

Ice Super-heatedsteam

Criticalpoint

Vapor

Ice

Liquid

Triplepoint

4

University of Fukui

Chart of water (T-v)

5

University of Fukui

T-S chart of steam

T (K)

S(kJ/kg K)

Critical point647.3K22.1MPa

LiquidVapor

6

University of Fukui

Boiling and condensation

Heated surface

satTwT

satwsat TTTSuperheat

−=Δ：

lT

lsatsub TTTSubcooling

−=Δ：

satT

Liquid filmCooling surface

Steam

satTvT

wT

satvsat TTTSuperheat

−=Δ：

wsatsub TTTSubcooling

−=Δ：

7

University of Fukui

Surface tension

Contact angle

Liquid(l)

Solid(s)

svσVapor(v)

lvσ

slσ

θσσσ coslvslsv +=

θ

Young’s equation

In case of large θ: bad wetness

8

University of Fukui

Vapor in superheated liquid

r

Superheated liquid

Vapor

rppp lv

σ2=−=Δ

( )vlsat

lvvl

sat TL

dTdp

ρρρρ−

=

Eq. of Clapeyron-Clasius

( )

rLT

rLTT

lvv

sat

lvvl

vlsatsat

ρσ

σρρ

ρρ

2

2

≈

−=Δ

9

University of Fukui

Boiling curve

(a) Start of boiling

(b) Nucleate boiling

(c) Nucleate boiling (d) Critical point

(e) Transition boiling

(f) Film boiling

10

University of Fukui

Flow regime in vertical or horizontal flow

気泡流Bubbly flow

スラグ流Slug flow

チャーン流（フロス流）Churn flow

環状噴霧流Annular -mist flow

噴霧流Mist flow

gg

Vertical flow

気泡流Bubbly flow

スラグ流Slug flow

波状流Wavy flow

環状流Annular flow

プラグ流Plug flow

層状流Stratified flow

g

Horizontal flow

11

University of FukuiFlow regime

0.01

0.1

1

0.1 1 10

BSSFFAAir/Water7 MPa

j l0 (m

/s)

jg0 (m/s)

Slug

Annular

Bubbly

Flow regime map for vertical flow

Froth

0.001

0.01

0.1

1

10

0.1 1 10 100

Present data

j (m/s)

j (m/s)

SlugPlug

Stratified Smooth StratifiedWavy

Annular

Bubbly

l

g

Flow regime for horizontal flow

12

University of Fukui

Quality x

gl

g

WWW

x+

=Wｇ: Vapor mass flow rate [kg/s]Wl: Liquid mass flow rate [kg/s]

Thermal equilibrium steam qualityisat: Enthalpy in aturated condition[J/ｋｇ]Llg: Latent heat of evaporation[kg/s]lgL

iix sat−=

lglg Lii

LTCpX satsubl

sub−

=Δ

=Δ ΔΤsub: Subcooling (K)Cpl: Specific heat [J/(kg・K)]

13

University of Fukui

Relationship between quality and void fraction

( ) lglggg

ggg

lllggg

ggg

lg

g

lg

g

uuu

uuu

GGG

WWW

x

αραραρ

αραραρ

−+=

+=

+=

+=

1

Gｇ: mass velocity of gas [kg/m2s]Gl: mass velocity of liquid [kg/m2s]

AA kk α=

AQ

j kk =

kkk uj α=kkkkkk juG ραρ ==

AGW kk =

onVoidfractik :α

k=l, g

ρ: density [kg/m3]

14

University of Fukui

Void-Quality correlation

( ) Sx

x

l

gg −

+=

11

1

ρρ

α

( )

50

11

1

1

.

g

l

xxe

xxe

eeS

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎠⎞

⎜⎝⎛ −

+

⎟⎠⎞

⎜⎝⎛ −

+

−+=ρρ

Correlation of Smithe=0.4

Slip ratio

Void fraction

15

University of FukuiCharacteristic of void

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

計測値相関式

ボイド

率, α

(-)

熱平衡蒸気クォリティ, x (-)

圧力 7MPa

二相域単相域

gl

g

GGG

x+

=

gl

g

QQQ+

=α

Pressure

MeasuredCorrelation

Single-phase Two-phase

Thermal equilibrium quality

Voi

d fra

ctio

n

16

University of Fukui

Example of void fraction of PWR fuel bundle

17

University of Fukui

Profile of void fraction and steam quality in the core

z=0

Exit

Inlet α=1

Void fraction

Heat flux

Steam quality

Steam quality in case of uniform heat flux

18

University of FukuiMoody’s chart

Reynolds number, Re

Hydraulicallysmooth

TurbulentTransition

λ＝64/Re

Equ

ival

ent

rela

tive

roug

hnes

s, ε

/D

Laminar

Fric

tion

fact

or, λ

Reynolds number, Re

Hydraulicallysmooth

TurbulentTransition

λ＝64/Re

Equ

ival

ent

rela

tive

roug

hnes

s, ε

/D

Laminar

Fric

tion

fact

or, λ

19

University of Fukui

Two-phase flow multiplier

222

211

2121

uP

uDilP

ρζ

ρλ

=Δ

=Δ

Pressure loss In single-phase flow

SPFTPF PP Δ=Δ 2φPressure loss In two-phase flow

Pressure loss by pipe wall friction

Pressure loss by local loss coefficient

20

University of FukuiFlow regime around a fuel pin

Single phase H.T.

Nucleate boilingSubcooled boiling

Saturated boiling

Forced convectiveevaporatin

Dryout

Pressure

Void fractionH.T.C.

Bubbly flow

Slug flowChurn flowAnnular flow

Annular mist flow

Mist flow

21

University of Fukui

Heat transfer coefficient for nucleate boiling

707035041007

.a

.

llvv

a.l

l

a plLqlPr.

khl

⎟⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛×= −

σνρ

⎟⎠⎞

⎜⎝⎛−=

2.6exp79.0 25.0 pqTsΔ

Jens-LottesΔTs: K, q: W/m2, p: MPa

Kutateladze

Rohsenow67067070 .

l

v.

llvv

a

sf

.l

l

a

Lql

CPr

khl

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

−

ρρ

νρ

la: Capillary Const.

22

University of FukuiCritical Heat Flux

( ) 41

21310/

v

vllvvc

gL.q ⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

ρρρσρ

Eq. of critical heat flux by Zuber