Two Phase Pipeline Part II

Two Phase PipelinePart II

Ref.: Brill & Beggs, Two Phase Flow in Pipes, 6th Edition, 1991. Chapter 3.

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, Ls and Lm where:

- Ngv, NLv, Nd and NL are the same as Hagedorn & Brown method.

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

- L1 and L2 are functions of Nd as shown in Figure 3-11.

Bubble Flow Limits: 0 ≤ Ngv ≤ L1 + L2 NLv

Slug Flow Limits: L1 + L2 NLv ≤ Ngv ≤ Ls

Transition (Churn) Flow Limits: Ls < Ngv <Lm

Annular-Mist Flow Limits: Ngv > Lm


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 vs based on the definition of S:

- Calculate HL based on the definition of vs :

- Calculate the pressure gradient due to elevation change:

s

sLssmmsL

L

sL

L

sgs v

vvvvvvHHv

Hv

v2

4)(1

5.02

4 /)( LLs gSv

ggLLsscelevation

HHwheregg

ZP

dd


Correlations for calculating S in each flow regimes:Bubble Flow:

F1 , F2 , F3 and F4 can be obtained from Figure 3-12.

Slug Flow:

F5 , F6 and F7 can be obtained from Figure 3-14.

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

dLv

gvLv N

FFFwhereN

NFNFFS 4

3'

3

2'

321 1

6'

627

'6

982.0

5 029.0)1(

)1( FNFwhereNF

FNFS d

Lv

gv


3- Pressure gradient due to friction:

Bubble Flow:

f1 is obtained from Moody diagram ( ), f2 is a correction

for the gas-liquid ratio, and is given in Figure 3-13, and f3 is 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

tpc

msLLtp

friction

sL

sg

vv

ff50

1 13

L

sLL dvN

Re


Annular-Mist Flow: In this region, the friction term is based on the 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 – ε.

After calculating the gas Reynolds number, , the two-

phase friction factor can be obtained from Moody diagram or rough

pipe equation:

2

22

,2d

dddvvddwhere

dgvf

ZP

sgsgc

sggtp

friction

g

sgg dvN

Re

05.0067.0)/27.0(log4

1473.1

210

dfor

ddftp


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 d – ε instead of d. In this case the determination of roughness is iterative.

dvNN

dNNfor

dvdNNfor

sgg

WeLWe

sgg

LWe

2

302.0

2

)(3713.0:005.0

0749.0:005.0

LL

L

L

sggwe N

vnumberWeberN

22

,)(


4- Pressure gradient due to acceleration:

Bubble Flow: The acceleration term is negligible.

Slug Flow: The acceleration term is negligible.

Mist Flow:

Pgvv

EWhereE

ZP

ZP

ZP

orZP

Pgvv

ZP

c

nsgmk

k

fele

total

totalc

nsgm

acc

1dd

dd

dd

dd

dd


Transition Flow: In the transition zone between slug and mist flow, Duns and Ros suggested linear interpolation between the flow regime boundaries, Ls and Lm , 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 ZPB

ZPA

ZP

dd

dd

dd

ALLLN

BLL

NLA

sm

sgv

sm

gvm

1,

m

gvgg L

N '

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 < LB

2- Liquid Holdup:

Where the vs have a constant value of 0.8 ft/sec.

13.0and/2218.0071.1Where 2 BmB LdvL

ssgsm

s

mL vvvv

vvH /4)/1(15.01 2


3- Pressure gradient due to friction:

Where ftp is 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 > LB and Ngv < Ls

Where Ls and Ngv are the same as Duns and Ros method.

dgvf

ZP

c

LLtp

friction 2dd 2

L

LL dvN

Re

Two-Phase Flow CorrelationsVertical Upward Flow Pipeline (Orkiszewski)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 vb from the equations shown in the

next page, based on NReb and NReL where

4- Compare the values of vb obtained 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

sggbsLLs vv

vvv

)(

L

bL dvNb

Re

L

mL dvNL

Re


Use the following equations for calculation of vb:

30001074.8546.0 ReRe6

bLNfordgNvb

80001074.835.0 ReRe6

bLNfordgNvb

5.0

5.02 59.135.0

dvwhere

L

Lb

800030001074.8251.0 ReRe6

bLNfordgN


The value of δ can be calculated from the following equations depending upon the continuous liquid phase and mixture velocity.Continuous

Liquid PhaseValue of vm Equation of δ

Water < 10

Water >10

Oil <10

Oil >10

)log(428.0)log(232.0681.0)log(013.038.1 dv

d mL

)log(888.0)log(162.0709.0)log(045.0799.0 dv

d mL

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

d mL

)log(63.0397.0)1log(01.0)log(

)log(569.0161.0)1log(0274.0

571.1

371.1

dd

vX

Xdd

Lm

L


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 δ is constrained by the following limits:

These constraints are supposed to eliminate pressure discontinuities between equations for δ since the equation pairs do not necessarily meet at vm=10 ft/sec.

L

s

bm

bm

mm

vvvvForb

vvFora

1:10)

065.0:10)

Two-Phase Flow CorrelationsVertical Upward Flow Pipeline (Orkiszewski)3- Pressure gradient due to friction:

Where ftp is 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.

Annular-Mist Flow Limits: Ngv > Lm

The same as Duns and Ros method.

bm

bsL

c

mLtp

friction vvvv

dgvf

ZP

2dd 2

L

mL dvN

Re

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 a correlating 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:

738.64

4516.13

4684.242

302.01

2

5.0,10.0

10252.9,316,

LL

LLm

Fr

LL

LLgdvN


Segregated Limits:

Transition Limits:

Intermittent Limits:

Distributed Limits:

2

1

and 01.0or and 01.0

LNLN

FrL

FrL

32 and 01.0 LNL FrL

43

13

and 4.0or and 4.001.0

LNLLNL

FrL

FrL

4

1

and 4.0or and 4.0

LNLN

FrL

FrL


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

Where HL(0) is the liquid holdup which would exist at the same conditions in a horizontal pipe. The values of parameters, a, b and c are shown for each flow regimes in this Table:

For transition flow regimes, calculate HL as follows:

LLcFr

bL

LLL HNaHHH )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

NLAHBHAH FrLLL

1,,23

3ent)(intermittd)(segregaten)(transitio


The holdup correcting factor (ψ), for the effect of pipe inclination is given by:

Where φ is the actual angle of the pipe from horizontal. For vertical upward flow, φ = 90o and ψ = 1 + 0.3 C. C is:

The values of parameters, d’, e, f and g are shown for each flow regimes in this Table:

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

.0n that restrictio with ,ln)1( CNNdC gFr

fLv

eLL

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 , ψ = 1

All patterns downhill 4.70 -0.3692 0.1244 -0.5056


3- Pressure gradient due to friction factor:

fn is determined from the smooth pipe curve of the Moody

diagram, using the following Reynolds number:

The parameter S can be calculated as follows:

For and for others:

Sntp

c

mntp

f

effdg

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


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,

1dd

dd

dd

dd

dd

scelec

ssgmk

k

fele

total

totalc

sgms

acc

gg

dLdP

Pgvv

EWhere

ELP

LP

LP

orLP

Pgvv

LP

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

F4

F4

F3

F2

F6

F5

Figure 3-16. Beggs and Brill, Horizontal flow regimes map.

Two Phase Pipeline Part II

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