Two-Phase Gas/Liquid Pipe Flow

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  AW  

2 1/ 2      WLW    LWL Vertical Pipe Flow Regime Map DEFINITIONS

Mass Flow Rate  m  , Volume Flow Rate (Q)

m mLGLLGG  m  Q  Q Mass Flux (G): m m m GGG L G   AAA LG

Volume Flux (J):

G GL GG JJJLG        m L G QQ LGV A m Volume Fraction Gas: ε Vol. Fraction Liquid: 1-ε

Phase Velocity: J J V,VL G LG1  

Slip Ratio (S): Mass Fraction Gas (Quality x):

V m S  G x  G VL mmGL

Mass Flow Ratio Gas/Liquid:

m x   GGS  m 1 - x 1 - LL

Density of Two-Phase Mixture:

m  G 1    L where x ε  x S 1 x ρGL / ρ is the volume fraction of gas in the mixture

Holdup (Volume Fraction Liquid):

S 1 x ρGL / ρ  φ 1  ε  x S 1 x ρGL / 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:

1 K  1  x / x ρ / ρ S   GL K 1 x  / x ρ / ρ  GL where

19 K 1 0.12 / Z 0.95 

1/ 6 1/ 8 1/ 4 Z NRe N Fr / 1 ε

HOMOGENEOUS GAS-LIQUID PIPE FLOW

Energy Balance (Eng’q Bernoulli Eqn)

2 f G 2 dx dz m G 2ν ρ g dP ρ DGL dL m dL m dL 2 dν 1 xG G dP where

νGL ν G  ν L 1 / ρ G  1 / ρ L For “frozen” flow (no phase change): dx  0 dL

dν If xG 2 G  1 flow is choked. dP For ideal gas:

1/ k ννGG 1   P1   ,     1 k / k P ρP  P  ρ kP   Ts1 For frozen ideal gas/liquid choked flow:

ρ kP Gcρ m m ε

For flashing flow (Clausius-Clapeyron eqn):

ν2 cT νG GL p  2 P T λGL Homogeneous Horizontal Flow

2 2fm G dL νGL dx ρ m dz ρD dP m 2 1 xG dνG / dP

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

c T T  x  p o s λGL Finite Difference Solution – solve for L

D2 ΔL - ΔP  G2 Δν  gΔZ / ν  ΣK   2 fit 4f νG m Dimensionless

4f ΔL 2 m ΣK  - Δη  G*2 Δε  gΔZ / P ν ε fit o o  *2 D εG where

Δη ΔP / Po * GG/Poρ o G ν o / P o

ε ν / νoo ρ / ρ  L= ΔL  i i

Using  Z   Lcos  , where θ is the pipe inclination angle with the vertical

for horizontal cos  = 0 for vertical up flow cos  = 1 for vertical down flow cos  = -1

*2 *2 4f ΔL 2 Δη G Δε / εG ΣK fit m  -  D 1 gDcosθ / 4 fPoo ν ε Procedure: Find G, Given L, Poe and P

. Select desired Δ P and determine  at each

pressure step from P o to P e

. Assume a value for G *

. Calculate ΔL at each pressure step. At choke point, ΔL  0

. Adjust until ΣΔL  L Ex: Flashing Water in Pipe

Given: G,Po ,T o , x o ,s o .ρ Go , ρ Lo 

Calculate: Pressure Drop over L

1. Assume a value for dL  ΔL ;

2. Est. f mm  fn  DG / μ  (Moody, Churchill, ~0.005) 3. ΔP G2  2 f ΔL / ρ D ρ fn ρ , ρ , x,S  1  m m , m L G 11 4. PP ΔP  x,T,ρ,ρ  Δx,ν   1 o 1 1s1G1L1s 1GL1 ρρ GL11 PPΔP 5.  ΔP2 2 1 2 Choked if ΔP 

Separated Pipe Flows

Each Phase Occupies a Specific Fraction of the Flow Area “Two-Phase Multiplier” for friction loss:

PP 2     ΦR    LL fm   fR Reference Single-Phase Flow:

R = L Total flow is liquid: G Lm m / A G

R = G Total flow is gas: GGm m / A G

R = L Gm Total flow is liquid in mixture:

GLm  1 x G

R = G Gm Total flow is gas in the mixture

GGm  xG Lockhart-Martinelli (1949)

PP  2     ΦLm    LL f   fLm or:

PP 2     ΦGm    LL f   fGm L-M Two-Phase Multipliers

C1 Φ12    Lm χχ2

22 ΦGm  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

PP    χ 2     LL     ffLm   Gm

2 2 P 2 fLm  1 x G  L fLm ρDL

22 P 2 fGm x G  L fGm ρDG

Friction Factors

f Lm is based on “liquid only” Reynolds No.

1 x GD N  ReLm μL f Gm is based on “gas-only” Reynolds No.

xGD N= ReGm mG Duckler et al. (1964) ρ Φ2  L α φ β φ Lm ρ and

2 ρG ΦGm  α φ β φ ρ where φ   1 ε  , ρ are “no slip” values and

 lnφ

α φ  1 2 1.281 0.478  lnφ  0.444  lnφ  34 0.094 lnφ  0.00843  lnφ

2 ρ φ2 ρ 1  φ βφ  L G  ρ φ ρ 1 φ mm The Reynolds Number is based on mixture properties:

DG N  βφ Rem   μm Sizing Relief Valves for Two-Phase Flow

m A  Gvalve

GKGvalve d ideal nozzzle

Assume Homogeneous Gas-Liquid Mixture in an Isentropic Nozzle

1/ 2 Pn dP GKρ2 dn ρ Po Discharge Coefficient ( K d )

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

If flow is choked (critical) use :

Kd ,gas

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

Kd ,liquid

TWO-PHASE DENSITY

ρ ερGL  1  ε ρ

Where ε is the volume fraction of gas:

x ε  x S 1 x ρ / ρ   GL x = mass fraction of gas phase (quality).

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

Flashing Flow Non-Equilibrium If L  10 cm Flashing is not complete if L 10 cm In this case, use L

x xo  x e  x o  10 L = nozzle length (cm)

x o = initial quality entering nozzle x e = local quality assuming equilibrium

If xo > 0.05, x = xe

Determine Quality, x e  fn( P )

• 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.

HDI – Homogeneous Direct Integration Exact Solution – Based on Numerical Finite Difference Equivalent of Nozzle Equation

1/ 2 1/ 2 Pn dP j n-1 P-P G ρ K -2 ρ K -4 j 1 j n n d n d   ρjo ρj 1 ρ j Po 

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)

Experimental Data

TABLE I VALVE SPECIFICATIONS (Lenzing, et al, 1997, 1998)

Valve KdG KdL Orifice Dia. Orifice Area (mm) B&R DN25/40 0.86 0.66 20 0.4869 (Bopp & Reuther Si63)

ARI DN25/40 0.81 0.59 22.5 0.6163 (Albert Richter 901/902)

1 x 2 “E” 0.962 0.729 13.5 0.2219 (Crosby JLT/JBS) Leser DN25/40 0.77 0.51 23 0.6440 (441) TABLE II FLOW CONDITIONS (Lenzing, et al, 1997, 1998)

Fluid Nom. Pressure Po Pb (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 Air-Water (Frozen) Flow

.Four Different Valves

.Three Different Pressures ARI DN25/40, Air/Water Calc 5 bar Data 5 bar 25000 Calc 8 bar

20000 Data 8 bar

) 2 15000

(kg/s m (kg/s 10000

dG K

5000

0 0.0001 0.001 0.01 0.1 1

xo LESER DN25/40, Air/Water Calc 5 bar Data 5 bar 25000 Calc 8 bar Data 8 bar

20000 Calc 10 bar ) 2 Data 10 bar

15000 (kg/s m

10000

dG K

5000

0 0.0001 0.001 0.01 0.1 1

xo B&R DN25/40, Air/Water Calc 5 bar Data 5 bar Calc 8 bar 35000 Data 8 bar 30000 Calc 10 bar Data 10 bar

) 25000 2

20000

(kg/s m 15000 dG

K 10000

5000

0 0.0001 0.001 0.01 0.1 1

xo Crosby 1x2 E, Air/Water Calc 5 bar

Data 5 bar 25000

20000

) 2

15000 (kg/s m

10000

dG K

5000

0 0.0001 0.001 0.01 0.1 1 xo 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 x=x+ x-x L/10 o e o 

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

If xo > 0.05, x = xe

Steam-Water Flashing (non-Equilibrium) Flow

.One Valve - Leser 25/40

. Four Different Pressures Leser DN25/40 Valve, Steam/Water, 5.4 bar

5000 Data 4500 HDI

4000 HNDI L=40mm ) 2 3500 3000

(kg/sm 2500

dG 2000 K 1500 1000 500 0 0.001 0.01 0.1 1

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

7000 Data

6000 HDI

HNDI L=40mm )

2 5000

4000 (kg/sm

dG 3000 K

2000

1000

0 0.001 0.01 0.1 1

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

8000 Data 7000 HDI HNDI L=40mm

6000

)

2 5000

4000

kg/sm

(

dG

K 3000

2000

1000

0 0.001 0.01 0.1 1

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

9000 Data 8000 HDI

7000 HNDI L=40mm ) 2 6000 5000

(kg/sm 4000 dG

K 3000 2000 1000 0 0.001 0.01 0.1 1

xo 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??