61
Two-Phase Gas/Liquid Pipe Flow Ron Darby PhD, PE Professor Emeritus, Chemical Engineering Texas A&M University

Two Phase Gas Liquid Pipe Flow

  • Upload
    ac2475

  • View
    38

  • Download
    8

Embed Size (px)

DESCRIPTION

Two PhaseGas Liquid Pipe Flow

Citation preview

Page 1: Two Phase Gas Liquid Pipe Flow

Two-Phase Gas/Liquid Pipe Flow

Ron Darby PhD, PE

Professor Emeritus, Chemical Engineering

Texas A&M University

Page 2: Two Phase Gas Liquid Pipe Flow

Types of Two-Phase Flow

»Solid-Gas

»Solid-Liquid

»Gas-Liquid

»Liquid-Liquid

Page 3: Two Phase Gas Liquid Pipe Flow

Gas-Liquid Flow Regimes

Homogeneous

Highly Mixed

“Pseudo Single-Phase”

High Reynolds Number

Dispersed –Many Possibilities

Horizontal Pipe Flow

Vertical Pipe Flow

Page 4: Two Phase Gas Liquid Pipe Flow

Horizontal Dispersed Flow Regimes

Page 5: Two Phase Gas Liquid Pipe Flow

Vertical Pipe Flow Regimes

Page 6: Two Phase Gas Liquid Pipe Flow

Horizontal Pipe Flow Regime Map

Page 7: Two Phase Gas Liquid Pipe Flow

1/ 2

G L

A W

1/ 22

W L W

L W L

Page 8: Two Phase Gas Liquid Pipe Flow

Vertical Pipe Flow Regime Map

Page 9: Two Phase Gas Liquid Pipe Flow

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

Page 10: Two Phase Gas Liquid Pipe Flow

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

Page 11: Two Phase Gas Liquid Pipe Flow

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

Page 12: Two Phase Gas Liquid Pipe Flow

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

m G L

1

G L

x S 1 x ρ / ρ

Page 13: Two Phase Gas Liquid Pipe Flow

Holdup (Volume Fraction Liquid):

G L

G L

S 1 x ρ / ρφ 1 ε

x S 1 x ρ / ρ

Page 14: Two Phase Gas Liquid Pipe Flow

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.

Page 15: Two Phase Gas Liquid Pipe Flow

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

Page 16: Two Phase Gas Liquid Pipe Flow

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

Page 17: Two Phase Gas Liquid Pipe Flow

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

Page 18: Two Phase Gas Liquid Pipe Flow

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

Page 19: Two Phase Gas Liquid Pipe Flow

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

Page 20: Two Phase Gas Liquid Pipe Flow

Finite Difference Solution – solve for

L

2

fit2

m

D 2ΔL - ΔP G Δν gΔZ / ν ΣK

4 f νG

Page 21: Two Phase Gas Liquid Pipe Flow

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

Page 22: Two Phase Gas Liquid Pipe Flow

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

Page 23: Two Phase Gas Liquid Pipe Flow

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

Page 24: Two Phase Gas Liquid Pipe Flow

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

Page 25: Two Phase Gas Liquid Pipe Flow

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

Page 26: Two Phase Gas Liquid Pipe Flow

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

Page 27: Two Phase Gas Liquid Pipe Flow

Lockhart-Martinelli (1949)

or:

2

Lm

f fLm

P PΦ

L L

2

Gm

f fGm

P PΦ

L L

Page 28: Two Phase Gas Liquid Pipe Flow

L-M Two-Phase Multipliers

2

Lm 2

C 1Φ 1

χ χ

2 2

GmΦ 1 Cχ χ

Page 29: Two Phase Gas Liquid Pipe Flow

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

Page 30: Two Phase Gas Liquid Pipe Flow

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

Page 31: Two Phase Gas Liquid Pipe Flow
Page 32: Two Phase Gas Liquid Pipe Flow

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

Page 33: Two Phase Gas Liquid Pipe Flow

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

Page 34: Two Phase Gas Liquid Pipe Flow

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

Page 35: Two Phase Gas Liquid Pipe Flow

The Reynolds Number is based on mixture properties:

mRe

m

DGN β φ

μ

Page 36: Two Phase Gas Liquid Pipe Flow

Sizing Relief Valves for Two-Phase Flow

valve

mA

G

valve d idea l nozzzleG K G

Page 37: Two Phase Gas Liquid Pipe Flow
Page 38: Two Phase Gas Liquid Pipe Flow

Assume Homogeneous Gas-Liquid Mixture in an Isentropic Nozzle

n

o

1/ 2P

d n

P

dPG K ρ 2

ρ

Page 39: Two Phase Gas Liquid Pipe Flow

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

Page 40: Two Phase Gas Liquid Pipe Flow

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

x S 1 x ρ / ρ

ε

Page 41: Two Phase Gas Liquid Pipe Flow

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

Page 42: Two Phase Gas Liquid Pipe Flow

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 )

Page 43: Two Phase Gas Liquid Pipe Flow
Page 44: Two Phase Gas Liquid Pipe Flow

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)

Page 45: Two Phase Gas Liquid Pipe Flow

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

Page 46: Two Phase Gas Liquid Pipe Flow

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

Page 47: Two Phase Gas Liquid Pipe Flow

Air-Water (Frozen) Flow

Four Different Valves

Three Different Pressures

Page 48: Two Phase Gas Liquid Pipe Flow

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

Page 49: Two Phase Gas Liquid Pipe Flow

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

Page 50: Two Phase Gas Liquid Pipe Flow

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

Page 51: Two Phase Gas Liquid Pipe Flow

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

Page 52: Two Phase Gas Liquid Pipe Flow

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

Page 53: Two Phase Gas Liquid Pipe Flow

Steam-Water Flashing (non-Equilibrium) Flow

One Valve - Leser 25/40

Four Different Pressures

Page 54: Two Phase Gas Liquid Pipe Flow

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

Page 55: Two Phase Gas Liquid Pipe Flow

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

Page 56: Two Phase Gas Liquid Pipe Flow

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

Page 57: Two Phase Gas Liquid Pipe Flow

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

Page 58: Two Phase Gas Liquid Pipe Flow

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.

Page 59: Two Phase Gas Liquid Pipe Flow

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)

Page 60: Two Phase Gas Liquid Pipe Flow

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

Page 61: Two Phase Gas Liquid Pipe Flow

Questions??