Characteristics of Two-phase Flows in Vertical Pipe W. G. Sim/Hannam University, N. W. Mureithi/...

Characteristics of Two-phase Flows in Vertical Pipe

W. G. Sim/Hannam University, N. W. Mureithi/ Ecole Polytechnique, Montreal,

B.M. Bae/KAERI, M.J. Pettigrew/ Ecole Polytechnique, Montreal,

Characteristics of Two-phase Flows in Vertical Pipe

ABSTRACT: The characteristics of two-phase flow in a vertical pipe are investigated to gain a better understanding of vibration excitation mechanisms. An analytical model for two-phase flow in a pipe was developed by Sim et al. (2005), based on a power law for the distributions of flow parameters across the pipe diameter, such as gas velocity, liquid velocity and void fraction. An experimental study was undertaken to verify the model. The unsteady momentum flux impinging on a ‘turning tee’ (or a ‘circular plate’) has been measured at the exit of the pipe, using a force sensor. From the measured data, especially for slug flow, the predominant frequency and the RMS value of the unsteady momentum flux have been evaluated. It is found that the analytical method, given by Sim et al. for slug flow, can be used to predict the momentum flux.

W. G. Sim/Hannam University, N. W. Mureithi/ Ecole Polytechnique, Montreal, B.M. Bae/KAERI, M.J. Pettigrew/ Ecole Polytechnique, Montreal,

Hannam University , Ecole Polytechnique Montreal

Contents Introduction Drift Flux Model for Two-phase Flow in a Pipe

o Power Law for Distributions of Flow Parameters

o Average Values with Integral Analysis o Reynolds Transport Theorem Steady Momentum Flux, Unsteady Momentum Flux for Slug Flow

Experimental Investigations o Test Loops o Comparisons with Theory Conclusions


Introduction

Initial Motivation o Slender Structural Elements - Fretting Wear Damage

o Flow Mechanism of Two-phase Flow

Homogeneous Model - Only for Bubbly Flow

o Hydrodynamic Force – Momentum Flux

o Analytical Approach for Dynamic Response

Experimental Study, Reliable Prediction of Dynamic Response

Main Purpose o To investigate characteristics of two-phase flow in vertical pipe

an analytical model proposed based on a power law, experimental study undertaken

o To verify the analytical model, with experimental results

o To obtain information on the reaction force


Drift Flux Model for Two-phase Flow in a Pipe

Power Law for Distributions of Flow Parameters

Assumptions; - neglecting adherence or reflection of bubble at the surface of the wall Distributions of Flow Parameters

Void Fraction Velocity distribution for bubbly flow for slug flow for gas and liquid* Subscript “L” stands for local time average value

p

o

L

r

rr1

0

max

n

of

fL

r

rr

u

u1

0

max

m

og

gL

r

rr

u

u1

0

max


Average Values with Integral Analysis

Void Fraction

Velocity

Volumetric Quality

Flow Quality

Slip Ratio

max2

102 12)1(

)1(

1f

ff

r

fLL

o

f uC

Cdrrur

uo

max02 21

gg

r

gLL

o

g uCdrurr

uo

)21)(1(

222

1 2

max0

1

0max202 pp

pdr

r

rrr

rrdr

r

oo r p

oo

r

L

o

21

max

max1

1

11

1

f

g

f

g

f

g

f C

C

C

u

u

u

u

21

max

max1

1

11

1

f

g

f

g

f

g

f

g

f

g

f C

C

C

u

u

u

ux

21max

max 11

1 ff

g

f

g

g

f

f

g

CC

C

u

u

x

x

u

uS


Reynolds Transport Theorem

Momentum Equation

where

F R

D

V

f gf g A A

ggLgffLf

V V

ggLgffLf

Rgsp

dAUdAUdVUdt

ddVU

dt

d

FFFF

22)()(

PAFp

APF fris

g fV V

ffggg gdVgdVF

tifLfLfL erurutrU )()(),( '

tiLLL errtr )()(),( '

tiL erpptrP )(),( '


Steady Momentum Flux

o Momentum Flux by Liquid

o Momentum Flux by Gas

Momentum Multiplier

where mass flux

f gA A

ggLgffLfRs dAudAuF 22

2max210

2

02max

1

0max

2 )(12 ffmfmf

r n

of

p

of

A

ffLf AuCCdrr

rru

r

rrrdAu

o

f

2max

2ggmg

A

ggLg AuCdAug

m

A A

ggLgffLf

kgmAG

dAudAu

MM f g /32

22

AQQuuG ffggffgg /)()1(


Unsteady Momentum Flux for Slug Flow

Void Fraction;

Formulation;

- Frequency for Slug Flow by Heywood and Richardson(1979)

- Sequence of Momentum by liquid and Gas; Fourier series

C=0.0543 for vertical flow C =0.0434 for horizontal flow

Reduced Frequency;

l s

l t

F l

c

T o

F g

ot

s

t

st

T

c

l

l

l

ll

11

)21)(1(

22 2

max0

1

0max2 pp

pdr

r

rrr

r

orp

oo

02.1202.2

gD

j

DQQ

QC

L

uf

gf

f

t

gslug

02.12

2.002.2

1

g

j

jDC

j

DfS

f

slugN


Unsteady Momentum Flux for Slug Flow

By Liquid

By Gas

A

fLfoo

ol dAucT

tucT

tuTtF 2)22

()22

()0(

F l

c

T o

F g

1

2max )2cos())1(sin(

)1(2)1(

kslug

k

ffsfm tfkkk

uAK

1

)cos(k

kfkfo tAA

1

2max )2cos()sin(

)1(2)0(

kslug

k

ggsgmog tfkkk

uAKTtF

1

)cos(k

kgkgo tAA


RMS Value of Reaction Force by Slug FlowF R

D

V

F l

c

T o

F g

22 )(1)('maxmax fog

ofol

oRMS AF

T

cAF

T

cF

)1(2max ffsfm uAK


ExperimentalInvestigations

Test Loops

EPM HNU

EPM HNU

Test Cylinder Length (m) 1.52 1.01

Inner Diameter (mm)

20.8 30

Mixture Fine screen Multiple inlet holes (equally distributed )High contraction ratioControl volume for the reaction force Turning Tee Circular Plate (Diameter=280 mm)


Flow patterns (Taitel et al., 1980) selected for bubbly and slug flow and dynamic time traces of the dynamic reaction force(HNU)

0.0 0.5 1.0 1.5 2.0-0.4

-0.2

0.0

0.2

0.4

For

ce (

N)

0.0 0.5 1.0 1.5 2.0-0.6

-0.3

0.0

0.3

0.6

Fo

rce

(N

)

0.0 0.5 1.0 1.5 2.0-0.30

-0.15

0.00

0.15

0.30

Fo

rce

(N)

Time(s)

%20

%50

%75

smj /06.1

smj /04.1

smj /94.0


Typical force spectra given by EPM for

smj /2

%25 %50


Typical force spectra given by HNU

%20

%50

smj /71.0

smj /83.0

smj /59.0

smj /04.1

smj /85.0

smj /66.0


Steady parameters given by HNU ( )

%0 %20 %50

pmn ,7

)(fu

symbols),(MM

Blue; Green; Red;


Comparison of test results to analytical results ( __, o ; ) for

sec sec

EPM HNU

2,,7 pmn%50

0.5

0.0

0 5 10 15 20 25 30 35 400.0

2.0x10-9

4.0x10-9

6.0x10-9

8.0x10-9

1.0x10-8

1.2x10-8

1.4x10-8

1.6x10-8

1.8x10-8

2.0x10-8 0.5

0.25

0. 0

)/06.2( smj )/04.1( smj

)(

'

N

F R

nAnA


Comparison of test results to analytical results ( _ _ , ___ ; ) for (Blue) and (Red)

HNU EPM

2,,7 pmn%50

)(

'

N

F RMS

%75

0.0

0.5

1.0

1.5

0.0 0.2 0.4 0.6 0.8

0.0

0.5

1.0

1.5

2.0

0.00 0.50 1.00 1.50 2.00

fjfj


Reduced Frequency (Azzopardi and Baker, 2003)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.00 0.20 0.40 0.60 0.80

beta50%

beta75%0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.00 1.00 2.00 3.00 4.00

fj

fj

f

slug

j

DfS

S

S

EPM

HNU


Conclusions

An analytical model for two-phase flow in a pipe, based on a power law The integral forms easily incorporated into models for momentum flux

Reaction force exerted by the momentum flux at the exit of the pipe. – Two air-water loops were constructed.

– Momentum Flux (for bubbly flow; , slug flow; )

– Force spectra (for bubbly flow, slug flow)

– Reduced frequency (for slug flow)

Good agreement shown between the results

2,,7 pmn pmn ,7

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