Two Phase II

8/13/2019 Two Phase II

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Ref.: Brill & Beggs, Two Phase Flow in Pipes, 6thEdition, 1991.

Chapter 3.


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Two-Phase Flow CorrelationsVertical Upward Flow Pipeline (Duns & Ros)

1-Flow regimes boundaries: The flow regimes map is shown

in Figure 3-10. The flow regimes boundaries are defined as

a functions of the dimensionless quantities:Ngv,NLv,Nd,NL,

L1,L2,LsandLmwhere:- Ngv,NLv,NdandNLare the same as Hagedorn & Brown method.

-Ls= 50 + 36NLv and Lm= 75 + 84NLv0.75

- L1andL2are functions ofNdas shown in Figure 3-11.

Bubble Flow Limits: 0 NgvL1 +L2NLv

Slug Flow Limits: L1 +L2NLv Ngv Ls

Transition (Churn) Flow Limits: Ls < Ngv Lm


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2-Pressure gradient due to elevation change: The procedure

for calculating the pressure gradient due to elevation

change in each flow regimes is:

-Calculate the dimensionless slip velocity (S) based on the

appropriate correlation

-Calculate vsbased on the definition of S:

-CalculateHLbased on the definition of vs:

-Calculate the pressure gradient due to elevation change:

s

sLssmmsL

L

sL

L

sg

sv

vvvvvvH

H

v

H

vv

2

4)(

1

5.02

4 /)( LLs gSv

ggLLss

celevationHHwhereg

g

Z

P

d

d


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Correlations for calculating S in each flow regimes:

Bubble Flow:

F1,F2,F3andF4can be obtained from Figure 3-12.

Slug Flow:

F5,F6andF7 can be obtained from Figure 3-14.

Mist Flow: Duns and Ros assumed that with the high gas flowrates in the mist flow region the slip velocity was zero (s=n).

dLv

gv

Lv N

FFFwhere

N

NFNFFS 4

3

'

3

2

'

321 1

6'

62

7

'

6

982.0

5 029.0)1(

)1( FNFwhereNF

FNFS dLv

gv


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3-Pressure gradient due to friction:

Bubble Flow:

f1is obtained from Moody diagram ( ), f2is a correction

for the gas-liquid ratio, and is given in Figure 3-13, andf3is an

additional correction factor for both liquid viscosity and gas-liquid

ratio, and can be calculated as:

Slug Flow: The same as bubble flow regime.

321 /2d

d ffffwheredg

vvfZP tp

c

msLLtp

friction

sL

sg

v

vff

501 13

L

sLL dvN

Re


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Annular-Mist Flow: In this region, the friction term is based onthe gas phase only. Thus:

As the wave height on the pipe walls increase, the actual area

through which the gas can flow is decreased, since the diameter

open to gas is d.

After calculating the gas Reynolds number, , the two-

phase friction factor can be obtained from Moody diagram or rough

pipe equation:

2

22

,

2d

d

d

dvvddwhere

dg

vf

Z

Psgsg

c

sggtp

friction

g

sgg dvN

Re

05.0067.0

)/27.0(log4

14

73.1

2

10

d

for

dd

ftp


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Duns and Ros noted that the wall roughness for mist flow is affected

by the wall liquid film. Its value is greater than the pipe roughness

and less than 0.5, and can be calculated as follows (or Figure 3-15):

Where

Duns and Ros suggested that the prediction of friction loss could be

refined by using dinstead of d. In this case the determination of

roughness is iterative.

dv

NN

dNNfor

dvdNNfor

sgg

WeL

We

sgg

LWe

2

302.0

2

)(3713.0:005.0

0749.0:005.0

LL

L

L

sgg

we Nv

numberWeberN22

,)(


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4-Pressure gradient due to acceleration:

Bubble Flow: The acceleration term is negligible.

Slug Flow: The acceleration term is negligible.

Mist Flow:

Pg

vvEWhere

E

Z

P

Z

P

Z

P

or

Z

P

Pg

vv

Z

P

c

nsgm

k

k

fele

total

totalc

nsgm

acc

1

d

d

d

d

d

d

d

d

d

d


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Transition Flow: In the transition zone between slug and mistflow, Duns and Ros suggested linear interpolation between the flow

regime boundaries,LsandLm, to obtain the pressure gradient, as

follows:

Where

Increased accuracy was claimed if the gas density used in the mist

flow pressure gradient calculation was modified to :

MistSlugTransition Z

PB

Z

PA

Z

P

d

d

d

d

d

d

ALLLNB

LLNLA

sm

sgv

sm

gvm

1,

m

gvg

g

L

N '


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Two-Phase Flow CorrelationsVertical Upward Flow Pipeline (Orkiszewski)

Orkiszewski, after testing several correlations, selected the

Griffith and Wallis method for bubble flow and the Duns

and Ros method for annular-mist flow. For slug flow, he

proposed a new correlation.Bubble Flow

1-Limits: vsg / vm


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Whereftpis obtained from Moody diagram with liquid

Reynolds number:

4-Pressure gradient due to acceleration: is negligible in bubble

flow regimes.

Slug Flow

1-Limits: vsg / vm >LBandNgv< Ls

WhereLsandNgvare the same as Duns and Ros method.

dg

vf

Z

P

c

LLtp

friction 2d

d 2

L

LL dvN

Re


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2-Two-phase density:

The following procedure must be used for calculating vb:

1- Estimate a value for vb. A good guess is vb = 0.5 (g d)0.5

2- Based on the value of vb, calculate the

3- Calculate the new value of vbfrom the equations shown in the

next page, based onNRebandNReLwhere

4- Compare the values of vbobtained in steps one and three. If they

are not sufficiently close, use the values calculated in step three as

the next guess and go to step two.

Lbm

sggbsLL

svv

vvv

)(

L

bL dv

Nb

Re

L

mL dvN

L

Re


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Use the following equations for calculation of vb:

30001074.8546.0 ReRe6 bL NfordgNvb

80001074.835.0 ReRe6 bL NfordgNvb

5.0

5.0

2 59.135.0d

vwhereL

Lb

800030001074.8251.0 ReRe6

bL NfordgN


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The value of can be calculated from the following equations

depending upon the continuous liquid phase and mixture velocity.

Continuous

Liquid Phase

Value

of vmEquation of

Water < 10

Water >10

Oil 10

)log(428.0)log(232.0681.0)log(013.0

38.1 dv

d m

L

)log(888.0)log(162.0709.0)log(045.0

799.0 dv

d m

L

)log(113.0)log(167.0284.0)1log(0127.0415.1

dvd

mL

)log(63.0397.0)1log(01.0

)log(

)log(569.0161.0)1log(0274.0

571.1

371.1

d

d

vX

Xdd

Lm

L


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Data from literature indicate that a phase inversion from oil

continuous to water continuous occurs at a water cut of

approximately 75% in emulsion flow.

The value of is constrained by the following limits:

These constraints are supposed to eliminate pressure

discontinuities between equations for since the equation pairs

do not necessarily meet at vm=10 ft/sec.

L

s

bm

b

m

mm

vv

v

vForb

vvFora

1:10)

065.0:10)


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Whereftpis obtained from Moody diagram with mixture

Reynolds number:

4-Pressure gradient due to acceleration: is negligible in slug

flow regime.

Transition (Churn) Flow Limits: Ls < Ngv Lm

The same as Duns and Ros method.

bm

bsL

c

mLtp

friction vv

vv

dg

vf

Z

P

2d

d 2

L

mL dvN

Re


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Two-Phase Flow CorrelationsBeggs and Brill

Beggs and Brill method can be used for vertical, horizontal and

inclined two-phase flow pipelines.

1-Flow Regimes: The flow regime used in this method is acorrelating parameter and gives no information about the

actual flow regime unless the pipe is horizontal.

The flow regime map is shown in Figure 3-16. The flow

regimes boundaries are defined as a functions of the

following variables:

738.6

4

4516.1

3

4684.24

2

302.0

1

2

5.0,10.0

10252.9,316,

LL

LLm

Fr

LL

LLgd

vN


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Segregated Limits:

Transition Limits:

Intermittent Limits:

Distributed Limits:

2

1

and01.0or

and01.0

LN

LN

FrL

FrL

32and01.0 LNL FrL

43

13

and4.0or

and4.001.0

LNL

LNL

FrL

FrL

4

1

and4.0or

and4.0

LN

LN

FrL

FrL


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2-Liquid Holdup: In all flow regimes, except transition, liquidholdup can be calculated from the following equation:

WhereHL(0)is the liquid holdup which would exist at the sameconditions in a horizontal pipe. The values of parameters, a, band

care shown for each flow regimes in this Table:

For transition flow regimes, calculateHLas follows:

LLc

Fr

b

LLLL H

N

aHHH

)0()0()0()( :constraintwith,

Flow Pattern a b c

Segregated 0.98 0.4846 0.0868

Intermittent 0.845 0.5351 0.0173

Distributed 1.065 0.5824 0.0609

ABLL

NL

AHBHAH Fr

LLL

1,, 23

3

ent)(intermittd)(segregaten)(transitio


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The holdup correcting factor (), for the effect of pipe inclination

is given by:

Where is the actual angle of the pipe from horizontal. For

vertical upward flow, = 90o

and = 1 + 0.3 C. Cis:

The values of parameters, d, e, f andgare shown for each flow

regimes in this Table:

)8.1(sin333.0)8.1sin(1 3 C

.0n thatrestrictiowith,ln)1( CNNdC gFrfLveLL

Flow Pattern d' e f g

Segregated uphill 0.011 -3.768 3.539 -1.614

Intermittent uphill 2.96 0.305 -0.4473 0.0978

Distributed uphill No correction C= 0 , = 1

All patterns downhill 4.70 -0.3692 0.1244 -0.5056


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3-Pressure gradient due to friction factor:

fnis determined from the smooth pipe curve of the Moody

diagram, using the following Reynolds number:

The parameter Scan be calculated as follows:

For and for others:

S

ntp

c

mntp

f

eff

dg

vf

L

P

,

2d

d 2

42 )(ln01853.0)(ln8725.0ln182.30523.0

ln

yyy

yS

n

mn dvN

Re

)2.12.2ln(2.1/1 2 )( ySHy LL


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4-Pressure gradient due to acceleration: Although the

acceleration term is very small except for high velocity flow,

it should be included for increased accuracy.

sin,

1

d

d

d

d

d

d

d

d

d

d

s

celec

ssgm

k

k

fele

total

totalc

sgms

acc

g

g

dL

dP

Pg

vvEWhere

E

L

P

L

P

L

P

orL

P

Pg

vv

L

P


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Figure 3-10. Vertical two-phase flow regimes map (Duns & Ros).


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F4

F4

F3

F2


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F6

F5


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Figure 3-16 Beggs and Brill Horizontal flow regimes map

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