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Two PhaseGas Liquid Pipe Flow
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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??