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Two-Phase Gas/Liquid Pipe Flow

Ron Darby PhD, PE

Professor Emeritus, Chemical Engineering

Texas A&M University

Types of Two-Phase Flow

»Solid-Gas

»Solid-Liquid

»Gas-Liquid

»Liquid-Liquid

Gas-Liquid Flow Regimes

Homogeneous

Highly Mixed

“Pseudo Single-Phase”

High Reynolds Number

Dispersed –Many Possibilities

Horizontal Pipe Flow

Vertical Pipe Flow

Horizontal Dispersed Flow Regimes

Vertical Pipe Flow Regimes

Horizontal Pipe Flow Regime Map

1/ 2

G L

A W

1/ 22

W L W

L W L

Vertical Pipe Flow Regime Map

DEFINITIONS

Mass Flow Rate , Volume Flow Rate (Q)

Mass Flux (G):

L G L L G G

m m m Q Q

m

GL

L G

mmmG G G

A A A

Volume Flux (J):

Volume Fraction Gas: ε Vol. Fraction Liquid: 1-ε

Phase Velocity:

GL

L G

m L G

L G

m

GGGJ J J

Q QV

A

GL

L G

JJV , V

1

Slip Ratio (S): Mass Fraction Gas (Quality x): Mass Flow Ratio Gas/Liquid:

G G

L L

m xS

m 1 - x 1 -

G

L

VS

V

G

G L

mx

m m

Density of Two-Phase Mixture: where is the volume fraction of gas in the mixture

m G L

1

G L

xε

x S 1 x ρ / ρ

Holdup (Volume Fraction Liquid):

G L

G L

S 1 x ρ / ρφ 1 ε

x S 1 x ρ / ρ

Slip (S)

Occurs because the gas expands and speeds up relative to the liquid.

It depends upon fluid properties and flow conditions.

There are many “models” for slip (or holdup) in the literature.

Hughmark (1962) slip correlation Either horizontal or vertical flow:

where

G L

G L

1 K 1 x / x ρ / ρS

K 1 x / x ρ / ρ

19

0.95K 1 0.12 / Z

1/ 41/ 6 1/ 8

Re FrZ N N / 1 ε

HOMOGENEOUS GAS-LIQUID PIPE FLOW

Energy Balance (Eng’q Bernoulli Eqn)

where

2

2m

GL m

m

2 G

2 f G dx dzG ν ρ g

ρ D dL dLdP

dνdL1 xG

dP

GL G L G L1 / 1 / ν ν ν ρ ρ

For “frozen” flow (no phase change):

If flow is choked.

For ideal gas:

dx0

dL

2 Gdν

xG 1dP

1/ k

G G 1

1 k / k

T s 1

ν ν P1,

P ρP P ρ kP

For frozen ideal gas/liquid choked flow:

For flashing flow (Clausius-Clapeyron eqn):

2

GL pG

2

GLT

c T

P

νν

λ

m

m

ρ kPG cρ

ε

Homogeneous Horizontal Flow

Flashing Flow - Determine x from (adiabatic) energy balance (or thermo properties database):

2 m

GL m

m

2

G

2 fG dL ν dx ρ dz

ρ DdP

1 xG dν / dP

p o s

GL

c T Tx

λ

Finite Difference Solution – solve for

L

2

fit2

m

D 2ΔL - ΔP G Δν gΔZ / ν ΣK

4 f νG

Dimensionless

where

*2m

fit o o *2

4 f ΔL 2ΣK - Δη G Δε gΔZ / P ν ε

D εG

oΔη ΔP / P

*

o o o oG G / P ρ G ν / P

o oε ν / ν ρ / ρ

i

i

L= ΔL

Using , where θ is the pipe inclination angle with the vertical for horizontal = 0 for vertical up flow = 1 for vertical down flow = -1

cos

cos

Z Lcos

cos

*2 *2

fitm

o o

2 Δη G Δε / εG ΣK4 f ΔL-

D 1 gDcosθ / 4 fP ν ε

Procedure: Find G, Given

Select desired and determine at each

pressure step from to

Assume a value for

Calculate at each pressure step.

At choke point,

Adjust until

ΔL 0

PΔ o

Pe

P

*G

ΔL

ΣΔL L

o eL, P and P

Ex: Flashing Water in Pipe

Given:

Calculate: Pressure Drop over L

1. Assume a value for ;

2. Est. (Moody, Churchill, ~0.005)

3.

,

4.

5.

Choked if

dL ΔL

o o o o Go LoG,P ,T , x ,s .ρ , ρ

m mf fn DG / μ

2

1 m mΔP G 2 f ΔL / ρ D

1

1 1

1 o 1 1 s1 G1 L1 1 GLsG L

1 1P P ΔP x ,T , ρ , ρ Δx ,ν

ρ ρ

m L Gρ fn ρ , ρ , x,S

2 1 2P P ΔP

2ΔP

ΔP

Separated Pipe Flows

Each Phase Occupies a Specific Fraction of

the Flow Area

“Two-Phase Multiplier” for friction loss:

2

R

fm fR

P PΦ

L L

Reference Single-Phase Flow:

R = L Total flow is liquid:

R = G Total flow is gas:

R = Total flow is liquid in mixture:

R = Total flow is gas in the mixture

L mG m / A G

G mG m / A G

GmL

LmG 1 x G

GmG

GmG xG

Lockhart-Martinelli (1949)

or:

2

Lm

f fLm

P PΦ

L L

2

Gm

f fGm

P PΦ

L L

L-M Two-Phase Multipliers

2

Lm 2

C 1Φ 1

χ χ

2 2

GmΦ 1 Cχ χ

State Liquid Gas C tt turbulent turbulent 20 vt laminar turbulent 12 tv turbulent laminar 10 vv laminar laminar 5

L-M Correlating Parameter

Lm Gm

2

f f

P Pχ

L L

2 2

Lm

fLm L

2 f 1 x GP

L ρ D

2 2

Gm

fGm G

2 f x GP

L ρ D

Friction Factors

is based on “liquid only” Reynolds No.

is based on “gas-only” Reynolds No.

Lmf

LmRe

L

1 x GDN

μ

Gmf

GmRe

G

xGDN =

m

Duckler et al. (1964)

and

where

are “no slip” values

2 L

Lm

ρΦ α φ β φ

ρ

2 G

Gm

ρΦ α φ β φ

ρ

φ 1 ε , ρ

and

2

3 4

lnφα φ 1

1.281 0.478 lnφ 0.444 lnφ

0.094 lnφ 0.00843 lnφ

22

GL

m m

1 φρρ φβ φ

ρ φ ρ 1 φ

The Reynolds Number is based on mixture properties:

mRe

m

DGN β φ

μ

Sizing Relief Valves for Two-Phase Flow

valve

mA

G

valve d idea l nozzzleG K G

Assume Homogeneous Gas-Liquid Mixture in an Isentropic Nozzle

n

o

1/ 2P

d n

P

dPG K ρ 2

ρ

Discharge Coefficient ( )

Values given by manufacturer, or in the “Red Book”

If flow is choked (critical) use :

If flow is not choked (sub-critical) use :

d ,liquid

K

dK

d ,gasK

TWO-PHASE DENSITY

Where is the volume fraction of gas:

x = mass fraction of gas phase (quality).

S = slip ratio = vG / vL = (fn(x, ρL/ ρG, …etc)

G Lρ ερ 1 ε ρ

G L

xε

x S 1 x ρ / ρ

ε

Flashing Flow Non-Equilibrium If L 10 cm

Flashing is not complete if

In this case, use

L = nozzle length (cm)

= initial quality entering nozzle

= local quality assuming equilibrium

If xo > 0.05, x = xe

L 10 cm

o e o

Lx x x x

10

ox

ex

Determine Quality,

• The quality is determined as a function

of pressure by an energy balance on

the fluid along the flow path.

• The path is usually assumed to be

isentropic.

ex fn( P )

HDI – Homogeneous Direct Integration Exact Solution – Based on Numerical Finite

Difference Equivalent of Nozzle Equation

n

o

1/ 21 / 2P j n-1

j 1 j

n n d n d

j o j 1 jP

P - PdPG ρ K -2 ρ K -4

ρ ρ ρ

Required Information:

ρ vs P at constant s from Po to Pn in increments of

Pj to Pj+1 .

Can be generated from an EOS or from a

database (e.g. steam tables).

(If choked, Gn Gmax at Pn=Pc)

TABLE I

VALVE SPECIFICATIONS

(Lenzing, et al, 1997, 1998)

Valve KdG KdL Orifice Dia.

(mm)

Orifice Area

B&R DN25/40

(Bopp &

Reuther Si63)

0.86 0.66 20 0.4869

ARI DN25/40

(Albert Richter

901/902)

0.81 0.59 22.5 0.6163

1 x 2 “E”

(Crosby

JLT/JBS)

0.962 0.729 13.5 0.2219

Leser DN25/40

(441)

0.77 0.51 23 0.6440

Experimental Data

TABLE II

FLOW CONDITIONS

(Lenzing, et al, 1997, 1998)

Fluid Nom. Pressure

(bar)

(psia)

(psia)

Air/Water 5 72.495 14.644

Air/Water 8 115.993 14.644

Air/Water 10 144.991 14.644

Steam/Water 5.4 78.295 14.644

Steam/Water 6.8 98.594 14.644

Steam/Water 8 115.993 14.644

Steam/Water 10.6 153.690 14.644

oP bP

Air-Water (Frozen) Flow

Four Different Valves

Three Different Pressures

0

5000

10000

15000

20000

25000

0.0001 0.001 0.01 0.1 1

Kd

G(k

g/s

m2)

xo

ARI DN25/40, Air/Water Calc 5 bar

Data 5 bar

Calc 8 bar

Data 8 bar

0

5000

10000

15000

20000

25000

0.0001 0.001 0.01 0.1 1

Kd

G(k

g/s

m2)

xo

LESER DN25/40, Air/Water Calc 5 bar

Data 5 bar

Calc 8 bar

Data 8 bar

Calc 10 bar

Data 10 bar

0

5000

10000

15000

20000

25000

30000

35000

0.0001 0.001 0.01 0.1 1

Kd

G(k

g/s

m2)

xo

B&R DN25/40, Air/WaterCalc 5 bar

Data 5 bar

Calc 8 bar

Data 8 bar

Calc 10 bar

Data 10 bar

0

5000

10000

15000

20000

25000

0.0001 0.001 0.01 0.1 1

Kd

G(k

g/s

m2)

xo

Crosby 1x2 E, Air/Water Calc 5 bar

Data 5 bar

HNDI – Homogeneous Non-Equilibrium

Direct Integration

For flashing flows, equilibrium is not reached

until flow path length reaches 10 cm or more.

For L<10 cm, quality (x = gas mass

fraction) is lower than it would be at

equilibrium (xe).

For L < 10 cm, quality is estimated from

where xo is the initial (L = 0) quality (L in cm)

If xo > 0.05, x = xe

o e ox = x + x - x L / 10

Steam-Water Flashing (non-Equilibrium) Flow

One Valve - Leser 25/40

Four Different Pressures

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0.001 0.01 0.1 1

Kd

G(k

g/s

m2)

xo

Leser DN25/40 Valve, Steam/Water, 5.4 bar

Data

HDI

HNDI L=40mm

0

1000

2000

3000

4000

5000

6000

7000

0.001 0.01 0.1 1

Kd

G(k

g/s

m2)

xo

Leser Valve DN25/40, Steam/Water, 6.8 bar

Data

HDI

HNDI L=40mm

0

1000

2000

3000

4000

5000

6000

7000

8000

0.001 0.01 0.1 1

Kd

G (

kg

/sm

2)

xo

Leser Valve DN25/40, Steam/Water, 8 bar

Data

HDI

HNDI L=40mm

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0.001 0.01 0.1 1

Kd

G(k

g/s

m2)

xo

Leser Valve DN25/40, Steam/Water, 10.6 bar

Data

HDI

HNDI L=40mm

SUMMARY/MORAL

Two-Phase Flow is much more complex than single –phase flow, because of the wide variety of possible flow regimes, phase distributions, thermo/mechanical equilibrium/non-equilibrium, etc.

Correlations are complex and limited in scope.

Analysis requires good understanding of flow mechanism.

References

Baker, O., “Simultaneous Flow of Oil and Gas”, Oil & Gas J.,

53:185-195, 1954

Darby, R., “Fluid Mechanics for Chemical Engineers”, 2nd Ed.,

Ch. 15, Marcel Dekker, 2001

Darby, R., F.E. Self and V.H. Edwards, “Properly Size Pressure

Relief Valves for Two-Phase Gas/Liquid Flow”, Chemical

Engineering, 109, no. 6, pp 68-74, June, (2002)

Darby, R., “On Two-Phase Frozen and Flashing Flows in Safety

Relief Valves”, Journal of Loss Prevention in the Process

Industries, v. 17, pp 255-259, (2004)

Refs (cont’d)

Darby,R, F.E. Self and V.H. Edwards, “Methodology for Sizing Relief Valves for Two-Phase Gas/Liquid Flow”, Proceedings of the Process Plant Safety Symposium, 2001, AIChE National Meeting, Houston, TX, April 2001

Duckler, A.E, M. Wicks III and R.G. Cleveland, “Frictional Pressure Drop in Two-Phase Flow: A Comparison of Existing Correlations for Pressure Loss and Holdup, AIChE J., 10:38-43, 1964

Govier, G.W. and K. Aziz, “The Flow of Complex Mixtures in Pipes”, Van Nostrand Reinhold Co., 1972

Hughmark, G.A., “Holdup in Gas-Liquid Flow”, CEP, 58(4), 62-65, l962

Lockhart, R.W., and R.C. Martinelli, “Proposed Correlation of Data for Isothermal Two-Phase, Two-Component Flow in Pipes”, CEP, 45(1), 39-48, 1949

Questions??