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A SHORT INTRODUCTION TO TWO-PHASE FLOWS Industrial occurrence and flow regimes

Herv´ e Lemonnier DM2S/STFM/LIEFT, CEA/Grenoble, 38054 Grenoble Cedex 9 T´

• el. 04 38 78 45 40

herve.lemonnier@cea.fr, herve.lemonnier.sci.free.fr/TPF/TPF.htm ECP, 2011-2012

CLASSES CONTENTS (1/2)

• Introduction: CEA/Grenoble, scientific information
• Two-phase flow systems in industry and nature
• Flow regime
• Measuring techniques, composition (α)
• Simple models for void fraction prediction

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CLASSES CONTENTS (2/2)

• Balance equations
• 1D models, pipe flow
• Pressure drop and friction
• Heat transfer mechanisms in boiling
• Condensation of pure vapor
• Critical flow phenomenon

Recommended textbook: Delhaye (2008)

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THERMAL-HYDRAULICS

• Study of simultaneous flow and heat transfer, in French, thermohydraulique
• Phase: state of matter characterized by definite thermodynamic properties
• Two-phase: mixture of two phases (diphasique)
• Examples: air and water, oil and water (connate), water and steam,
• il and natural gas (multiphase), polyphasique).

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CEA/GRENOBLE RESEARCH CENTER

• CEA: Commissariat `

a l’Energie Atomique (15000 p)

• CEA/Grenoble: originates in 1956, founded by Louis N´

eel (4000p/2300 CEA)

• Heat transfer laboratories founded by Henri Mondin
• Nuclear energy directorate (5000 p, DEN)
• Department of nuclear technology (400 p, Cadarache, Grenoble, DTN)
• Department of reactors studies (400 p, Cadarache, Grenoble, DER)
• Labs of simulation in thermal-hydraulics (SSTH)
• Labs of experimental studies in thermal-hydraulics (SE2T)
• Thesis advising capabilities and referenced research groups for several Masters.

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LABS OF SIMULATION AND EXPERIMENTS IN THERMAL-HYDRAULICS

• Codes (logiciels) for safety studies, CATHARE.
• 3D Codes for two-phase boiling flows (Neptune).
• LES of single-phase flow and heat transfer (TRIO-U).
• Dedicated studies : safety and optimization of NR of various generations II, III

and IV, ship propulsion, cryogenic rocket engines.

• Analytic studies on boiling flows and critical heat flux (DEBORA)
• Thermal-hydraulic qualification of fuel bundles (OMEGA)
• Instrumentation development for single-phase and two-phase flows:

Can only be modeled a quantity which can be measured Applications→models and codes→experimental validation→instrumentation.

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SCIENTIFIC KNOWLEDGE AND INFORMATION

• How to solve a technical/scientic issue?
• Textbooks, books, journal papers: Library?.
• Scientific Societies:

journals editing, conference organizations (proceedings, actes). – La Soci´ et´ e fran¸ caise de l’´ energie nucl´ eaire – La Soci´ et´ e hydrotechnique de France – La Soci´ et´ e fran¸ caise de thermique – American nuclear society, thermal-hydraulics division (NURETH)

• Do you speak English? ...

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TWO-PHASE SYSTEMS IN INDUSTRY AND NATURE (1/4)

• Nuclear engineering: sizing, safety, decontamination (cleaning up)

– Loss of coolant accidents (LOCA-APRP). – Severe accidents w/o vessel retention. – Decontamination by using foam. – Nuclear waste reprocessing.

• Oil engineering, hot issue: two-phase production

– Transport. – Pumping. – Metering. – Oil refining (Chem. Engng).

• Oil engineering : safety

– Safety of installations. – LPG storage tanks and fire (BLEVE).

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TWO-PHASE SYSTEMS IN INDUSTRY AND NATURE (2/4)

• Chemical engineering

– Wastewater treatment (interfacial area and residence time). – Gas-liquid reactors (falling film, trickle bed, air-lift). – Mixing and separation. – Safety: homogeneous thermal runaway.

• Automotive industry

– Diesel fuel atomization. – Combustion in diesel engines. – Cavitation damage : power steering, fuel nozzles.

• Heat exchangers

– Condensers and evaporator/steam generator. – Boilers (critical heat flux, CHF), heaters.

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TWO-PHASE SYSTEMS IN INDUSTRY AND NATURE (3/4)

• Hydroelectricity and water distribution

– Water resources management: transients of pipings. – Priming of siphons.

• Space industry

– Cryogenic fuel storage (Vinci). – Thermal control of rocket engines (combustion chamber and nozzle). – Water hammer and pressure surges. – Cavitation in turbo-pumps. Instability (lateral loading) and damage.

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TWO-PHASE SYSTEMS IN INDUSTRY AND NATURE (4/4)

• Meteorology

– Storm formation, rain/hail, lightning. – Ocean and atmosphere exchanges, aerosols formation.

• Volcanology

– Critical flow of lava in wells. – Steam explosion. – Nu´ ees ardentes (Vesuvius, protection of Naples suburbs).

• Nivology

– Avalanches. – Snow maturation (three-phase / 2-component).

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NUCLEAR REACTORS, WATER COOLED

• Sizing :

SG: heat transfer and pressure drop. SGTR: critical flow at the safety valve. FSI: mechanical loading and vibrations.

• Safety :

LOCA, fuel cladding temperature, reference scenario

• Decontamination :

vessel, SG, minimizings wastes: foam

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NUCLEAR FUEL

• Fuel pellet.
• Rod ≈ 10 mm in diameter (first confine-

ment barrier).

• Fuel assembly 17 × 17.
• Control rods.
• Length ≈ 4 meters.
• Reactor core ≈ 4 m in diameter.
• Heat transfer: forced convection (7 m/s.)
• Thermal power 3000 ÷ 5000 MW.

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STEAM GENERATOR

• Tube-type, separates primary and secondary

circuits (second confining barrier).

• ≈ 5000 tubes, diameter 50 mm, height 10 m.
• Pressure: 155-70 bar.
• 3 or 4 SG and flow loops.
• Inverted U-Tubes.
• Secondary : two-phase flow.
• Issues : heat transfer and vibration damage.

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THE N4 PWR: SOME FIGURES

• Primary side pressure: 155 bar, Tsat ≈ 355oC. Thermal power: 4250 MW.

– Mass flow rate: 4928,6 kg/s per SG (4) – Core inlet temperature: 292,2oC – Core outlet temperature: 329,6oC

• Secondary side, SG vapor pressure : 72,3 bar

– Vapor temperature: 288˚ C – Feed water temperature: 229,5oC – Mass flow rate: 601,91 kg/s per SG (4)

• Assess the thermal balance of reactor and SG

Source: National Institute of Standards and Technology (NIST) (http://webbook.nist.gov/chemistry/fluid/)

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SOME BAD NEWS...

The following (low pressure) statements are rather wrong:

• The mass balance reads, Q1 = Q2, since water is incompressible,

at least weakly it is dilatable.

• The enthalpy is, h = CP T.
• For a liquid, CP ≈ CV , or h ≈ u.
• Steam is a perfect gas.

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WATER DENSITY AT 155 BAR

650 660 670 680 690 700 710 720 730 740 750 290 295 300 305 310 315 320 325 330 Density (kg/m3) Temperature (°C) Linear approx. ρL, NIST

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MASS BALANCE OF THE PRIMARY CIRCUIT

• Mass flow rate per loop (CL), ML ≈ 5023 kg/s
• Inlet density: ρL1(292oC, 155 bar) = 742, 41 kg/s.
• Outlet density : ρL2(330oC, 155 bar) = 651, 55 kg/s.

Q1 = ML ρ1 = 6, 77m3/s, Q2 = ML ρ2 = 7, 71m3/s

• Volumetric flow rates differ by 13%.
• The volume of the primary circuit is 400 m3...

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WATER ENTHALPY AT 155 BAR

1250 1300 1350 1400 1450 1500 1550 290 295 300 305 310 315 320 325 330 Enthalpy (kJ/kg) Temperature (°C) Linear approx. h, NIST

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PRIMARY SIDE HEAT BALANCE

• Mass flow rate per loop (CL), ML ≈ 5023 kg/s.
• Inlet enthalpy: hL1(292oC, 155bar) = 1295 kJ/kg.
• Outlet enthalpy: hL2(330oC, 155 bar) = 1517 kJ/kg.

P = ML∆h ≈ 5023 × 222 103 = 1115 MW

• 4-loop reactor power: 4460 MW.
• Linear approximation: h = CP T, CP (292oC, 155bar) = 5, 2827 kJ/kg/K.

P = MLCP ∆T ≈ 5023 × 201 103 = 1008 MW

• Power differs by 10%.
• Temperature drift: 31oC/hour.

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WATER ENTHALPY & INTERNAL ENERGY AT 155 BAR

2 3 4 5 6 7 290 295 300 305 310 315 320 325 330 1250 1300 1350 1400 1450 1500 1550 Heat capacity (kJ/kg/K) Enthalpy, Internal energy (kg/m3) Temperature (°C) CP, NIST CV, NIST u, NIST h, NIST

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PHASE CHANGE & METASTABLE STATES

• Phase rule: thermodynamic equilibrium of steam and vapor, temperature

and pressure are linked, saturation states v = n + 2 − ϕ = 1, p = psat(T), or T = Tsat(p)

• Phase coexistence pressure, resp. temperature
• Liquid only is thermodynamically stable above Tsat(p).
• Vapor only is thermodynamically stable below Tsat(p).
• Water must boil and comply with the second principle of thermodynamics
• Within the metastability T range, the fluid can be either two-phase or

single-phase.

• Metastable states usually not in tables, EOS is needed.

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WATER ENTHALPY AT 1 BAR

500 1000 1500 2000 2500 3000 3500 50 100 150 200 250 300 350 Enthalpy (kJ/kg) Temperature (°C), Tsat = 99.61°C hL, EOS−NIST hV, EOS−NIST h, NIST Tables

Maximum liquid superheat, metastability limit >230 K

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WATER ENTHALPY AT 72 BAR

500 1000 1500 2000 2500 3000 3500 220 240 260 280 300 320 340 360 Enthalpy (kJ/kg) Temperature (°C), Tsat = 287.74°C hL, EOS−NIST hV, EOS−NIST h, NIST Tables

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SECONDARY SIDE HEAT BALANCE

• Mass flow rate per SG, ML ≈ 602 kg/s
• Inlet enthalpy: hL1(230oC, 72 bar) = 991.1 kJ/kg.
• Outlet enthalpy: hV 2(288oC, 72 bar) = 2771 kJ/kg.

P = ML∆h ≈ 602 × 1780 103 = 1071 MW

• To be compared to 1115 MW on the primary side.

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STEAM IS A PERFECT GAS

• Steam density at 100oC, 1 bar, ρV = 0, 5897 kg/m3.
• Perfect gas approximation: pV = RT, R = 8.316 J/mol/K, M = 18 g/mol.

V = RT p = 3.103 10−2 m3, ρ = M V = 0.5801 kg/m3.

• Steam at 288oC, 72 bar, ρV = 37.64 kg/m3.
• Perfect gas : pV = RT, R = 8.316 J/mol/K, M = 18 g/mol.

V = RT p = 6.481 10−4 m3, ρ = M V = 27.77 kg/m3 .

• Perfect gas approximation under-estimate by 26%.

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TWO-PHASE FLOW VARIABLES

• Phase presence function
• Space averaging operators
• Instantaneous flow rates
• Time averaging operators
• Some mathematical properties
• Averaged flow rate and superficial velocity

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PHASE PRESENCE FUNCTION

• Xk(r, t) =

   1 if x ∈ phase k if x / ∈ phase k Measurable variable,

• Resistive probe (electrical impedance)
• Optical probe (refraction index)
• Thermal anemometry (heat transfer)

Subscripts : k = 1, 2, k = L, G, k = f, v etc.

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SPACE AVERAGING (1/3)

• Space averaging operator (plain space average)

< | f> | n 1 Dn

• Dn

f dDn

• n=1, line (chord in a pipe)
• n=2, surface (cross section)
• n=3, volume (some length of a pipe)

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SPACE AVERAGING (2/3)

• Phase presence conditional average,

< fk >n 1 Dkn

• Dkn

fk dDkn

• n=1, line
• n=2, surface (shown here)
• n=3, volume

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SPACE AVERAGING (3/3)

• Instantaneous phase fraction

Rkn(t) < | Xk(r, t)> | n = Dkn Dn

• n=1, line fraction,

Lk L1 + L2

• n=2, surface fraction,

Ak A1 + A2

• n=3, volume fraction,

Vk V1 + V2 Identity (proof left as an exercise) Rkn < f >kn= < | Xkf> | n

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FLOW RATE AND MASS FLOW RATE

• Instantaneous flow rate (m3/s), wk = vk nz

Qk(t)

• Ak

wk dAk = Ak < wk >2

• Instantaneous mass flow rate (kg/s)

Mk(t)

• Ak

ρkwk dAk = Ak < ρkwk >2

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TIME AVERAGING

• Time averaging on [T], (plain)

f(t) 1 T t+T/2

t−T/2

f(τ) dτ Conditional time averaging, [Tk] f

X k (t) 1

Tk

• [Tk]

fk(τ) dτ Local time fraction, void fraction for gas/vapor αk(r, t) Tk T = Xk(r, t) Identity, derives from the definitions, αkf

X = Xkf

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COMMUTATIVITY OF AVERAGING OPERATORS

Rkn < fk >n = < | αkfk

X>

| n Proof, from definitions: Rkn < fk >n = 1 T

• [T ]
• Rkn

Dkn

• Dkn(t)

fk dDkn

• dt

1 T

• [T ]

dt 1 Dn

• Dn

Xkfk dDn = 1 Dn

• Dn

dDn 1 T

• [T ]

Xkfk dt 1 Dn

• Dn
• αk(r)

Tk

• [Tk]

fk dt

• dDn = <

| αkfk

X>

| n Significant example: mean void fraction, fk = 1, salami theorem... Rkn = < | αk> | n

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MEAN FLOW RATES

• Mean volume flow rate,

Qk = ARk2 < wk >2 = A< | αkwX

k >

| 2

• Mean mass flow rate,

Mk = ARk2 < ρkwk >2 = A< | αkρkwX

k >

| 2

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SUPERFICIAL VELOCITY

• Mean volumetric flux,

jk Xkwk ≡ αkwX

k

• Mean mass flux,

gk Xkρkwk ≡ αkρkwX

k

• Superficial velocity (vitesse d´

ebitante), Jk = < | jk> | 2 = < | αkwX

k >

| 2 = Qk A

• Mixture superficial velocity,

J = J1 + J2 = Q1 + Q2 A

Industrial occurrence and flow regimes 35/61

QUALITY

• Mean mass flux,

Gk < | gk> | 2 = < | αkρkwX

k >

| 2 = Mk A

• Mixture mean mass flux,

G = G1 + G2 = M1 + M2 A

• Quality, (titre massique),

xk = Mk M , M = M1 + M2

• Volume quality,

βk = Qk Q , Q = Q1 + Q2

• Equilibrium (steam) quality...

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WATER THERMODYNAMIC DIAGRAM (ρ, p)

50 100 150 200 250 300 100 200 300 400 500 600 700 800 900 1000 Pressure (bar) Density (kg/m3) 400, °C 380, °C 370, °C 350, °C 300, °C 200, °C

• Sat. ρL
• Sat. ρV

Critical pressure and temperature : ≈ 221 bar, 373,9oC.

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STEAM TABLES AND EOS (v, p)

50 100 150 200 250 300 0.005 0.01 0.015 0.02 0.025 0.03 Pressure (bar) Specific volume (m3/kg) T=400 °C T=380 °C T=370 °C T=350 °C T=300 °C T=200 °C

• Sat. Liq
• Sat. Vap

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STEAM TABLES AND EOS (ρ, p)

50 100 150 200 250 300 200 400 600 800 1000 Pressure (bar) Density (kg/m3) T=400 °C T=380 °C T=370 °C T=350 °C T=300 °C T=200 °C

• Sat. Liq
• Sat. Vap

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EQUILIBRIUM QUALITY

• The thermodynamic equilibrium assumption,

TL = TV = Tsat(p), hk(Tk, p) = hk(Tsat(p), p) hksat(p)

• 1D model assumption, flat profiles (hk(r) =< kk >2),
• Energy balance, q, uniform heat flux distribution

P = πqDz = M[h(z) − h1] = M[(xeqhV sat + (1 − xeq)hLsat) − hL1)] xeq = hL1 − hLsat hlv + πqDz Mhlv

• Phase change enthalpy : hlv hV sat − hLsat
• Equilibrium quality varies linearly with position.
• h is the mean enthalpy (energy balance), non dimensional enthalpy,

xeq = h − hLsat hlv

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FLOW REGIMES

• Topological phase organisation in flows,

– Bubbles, bulles – Plugs and slugs, poches et bouchons – Liquid films, drops and droplets – No sharp transitions

• Modeling is the motivation for identification of flow regimes

– Single-phase flows: laminar-turbulent (NS or RANS) – Two-phase flows: structures of interfaces → model. – Shortcomings: fully developed flows, fuzzy transitions, hydrodynamic singularities.

• Control variables: flow rates, slope, direction, diameter, transport proper-

ties, inlet conditions etc.

• Examples: vertical and horizontal co-current flows.

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VERTICAL ASCENDING FLOWS

Flow regime transitions:

• Experiments, empirical,

flow rates, momentum fluxes.

• Transition modeling,

mechanisms, (Dukler & Taitel, 1986).

Industrial occurrence and flow regimes 42/61

TAITEL ET DUKLER (1980) MODEL

Vertical ascending air-water flow, D = 50 mm, P = 1 bar. Flow regimes:

• A-B: Bubbly (bulles)
• A-D:Intermittent (poches,

bouchons)

• D-E: Churn (agit´

e)

• E: Annular
• B-C: Dispersed bubbles

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TAITEL ET DUKLER (1980) MODEL

• Bubbly flow and intermittent (A): Bubble coalescence, zig-zag motion of bubbles.

JL = 1 − α α JG − (1 − α)

3 2 U0∞,

αT = 0, 25, U0∞ = 1, 53 g(ρL − ρG)σ ρ2

L

1

4

• Dispersed bubbles and bubbly : turbulent break up, small bubbles, rectilinear path (B),

dense packing (A) with αT = 0, 52 (D). 2[ρL/(ρL − ρG)g]0,5ν0,08

L

(σ/ρL)0,10D0,48 J1,12 3, 0

• Intermittent and churn (D): churn flow ≡ development of slug flow.

L D = 42, 6

• J

√gD + 0, 29

• Industrial occurrence and flow regimes

44/61

TAITEL ET DUKLER (1980) MODEL

• Annular (E): all liquid entrained by the gas, force balance,

Jρ

1 2

G

[σg(ρL − ρG)]

1 2 = 3, 1

See also flooding correlations & Ku.

• Small diameter pipes: bubbles (zig-zag) and occasionally Taylor bubbles.

– Taylor bubbles relative velocity: UT = 0, 35√gD – individual bubbles: UB = U0 inf(1 − α)

1 2

– In small diameter pipes, Taylor bubbles are slower than individual bubbles to coalescence towards slugs.

• ρ2

LdD2

(ρL − ρG)σ 1

4

3, 78 – Bubbly flow no longer exists.

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TAITEL ET DUKLER (1980) MODEL

Air water, D = 25 mm, P = 1 bar. Small diameter pipes: no bubbly flow.

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APPLICATIONS : VertTD02

0.01 0.1 1 10 0.1 1 10 100 JL (m/s) JG (m/s) Bulles−Intermittent Bulles dispersées−Intermittent Bulles dispersées−Bulles Intermittent−Agité L/D=50 Intermittent−Annulaire Intermittent−Agité L/D=100 Intermittent−Agité L/D=200 Intermittent−Agité L/D=500 0.01 0.1 1 10 0.1 1 10 100 JL (m/s) JG (m/s) Bulles−Intermittent Bulles dispersées−Intermittent Bulles dispersées−Bulles Intermittent−Agité Intermittent−Annulaire

Air-water, 51 mm, 1 bar Air-water, 25 mm, L/D = 100, 1 bar

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FLOW PATTERNS IN HORIZONTAL FLOW

Main flow regimes:

• Bubbly
• Plug
• Stratified, smooth or wavy
• Slug of gas and plugs of liq-

uid

• Annular

Modeling the transition based

• n mechanisms (Dukler & Tai-

tel, 1986).

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FLOW PATTERN IN HORIZONTAL FLOW

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HORIZONTAL SLUG FLOW

α = 22%

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DUKLER AND TAITEL (1976) MODEL

Transition criteria:

• Empirical, e.g.: Mandhane,

air-water, 25 mm, 1 bar.

• Mechanistic modeling of

transition (Dukler & Taitel, 1986).

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TAITEL AND DUKLER (1976) MODEL

• Stability of stratified flow, linear stability analysis

(back later on, see also exercises)

• The Kelvin-Helmholtz instability,

VG C2

• (ρL − ρG) cos βAG

ρG

dAL dh

1

2

, C2 ≈ 1 − h D

• Base flow: smooth an horizontal interface

τG SG AG −τL SL AL +τi Si AL + Si AG

• +(ρL−ρG)g sin β = 0

X2f(A, D, AL, PL, DL)−g(A, D, AG, PG, DG, Pi)−4Y = X2 =

1 2CLρLJ2 LRe−n LS 1 2CGρGJ2 GRe−n GS

=

• (dP/dz)LS

(dP/dz)GS

• .

Y = (ρL − ρG)g sin β

4 D 1 2ρGJ2 GCGRe−m G

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TAITEL AND DUKLER (1976) MODEL

0.2 0.4 0.6 0.8 1 0.001 0.01 0.1 1 10 100 1000 h/D X Y positif −5<Y<−3 Y<−10 Y=0 Industrial occurrence and flow regimes 53/61

TAITEL AND DUKLER (1976) MODEL

• Transition, stratified flow instability:

F 2

• 1

C2

2

˜ U 2

G d ˜ AG d˜ h

˜ AG

• 1,

˜ UG = A AG , F =

• ρG

ρG − ρL 1

2

JG (Dg cos β) 1

2

• Towards intermittent flow:

h D 0, 5

• Towards annular flow:

h D 0, 5

• Stratified smooth or wavy ?:

UG 4νL(ρL − ρG)g cos β sρGUL 1

2

, K 2 ˜ UG

• s ˜

UL , K2 =

• ρGJ2

G

(ρL − ρG)Dg cos β DJL νL

• Dispersed bubbles:

UL 4AG Si g cos β fL

• 1 − ρG

ρL 1

2

, T 2 8 ˜ AG ˜ Si ˜ U 2

L( ˜

UL ˜ DL)−n , T =  

• dp

dz

• LS

(ρL − ρG)g cos β  

1 2

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TAITEL AND DUKLER (1976) MODEL

Curves 1 et 2 3 4 Coordinates F, X K, X T, X K =

• ρGJ2

G

(ρL−ρG)Dg cos β

1

2

DJL νL

1

2

F =

• ρG

ρG−ρL

1

2

JG (Dg cos β)

1 2

T =

• | dp

dz|LS

(ρL−ρG)g cos β

1

2

X =

• (dP/dz)LS

(dP/dz)GS

• 1

2

• dp

dz

• S

= 4C D Re−n ρJ2 2 , Re = JD ν

• Laminar: C = 16, n = 1. Turbulent: C = 0, 046, n = 0, 2
• β: slope angle, β = 0, horizontal flow, β > 0, descending flows.

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APPLICATIONS : HoriTD03

0.01 0.1 1 10 0.1 1 10 100 JL (m/s) JG (m/s) D=12.5 mm D=50 mm D=300 mm 0.01 0.1 1 10 0.1 1 10 100 JL (m/s) JG (m/s) α=1° α=5°

Air-Water, 50 mm, 1 bar, β = 0 Air-water, 50 mm, 1 bar

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FLOODING AND FLOW REVERSAL

Flooding: transition from counter-current towards co-current up flow. Flow reversal: reverse transition

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EXPERIMENTAL DETERMINATION OF FLOODING

Modeling of flooding and flow reversal see also Bankoff & Chun Lee (1986).

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WALLIS MODEL

J∗

k ≈ Froude number.

J∗

G =

JGρ

1 2

G

(gD(ρL − ρG))

1 2 ,

J∗

L =

JLρ

1 2

L

(gD(ρL − ρG))

1 2 ,

J

∗ 1

2

G + mJ ∗ 1

2

L

= C m and C depend on NL =

• ρLgD3(ρL−ρG)

µ2

L

1

2 ≡ Gr

NL > 1000        m = 1 0, 88 < C < 1 (smooth inlet) C = 0, 725 (sharp inlet) , NL < 1000    m = 5, 6N −1/2

L

C = 0, 725

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FLOODING ET FLOW REVERSAL

• Wallis model: no pipe length effect.

– Some experiments show J∗

G increase with the increase of L. Favors the

wave instability mechanism. – Many specific correlations.

• Flow reversal

– Wallis model, J∗

G(FR) = J∗ G(Flooding), hysteresis effect, pipe diameter

effect. J∗

G =

JGρ

1 2

G

(gd(ρL − ρG))

1 2 = 0, 5

– Puskina and Sorokin model, Ku = JGρ

1 2

G

(gσ(ρL − ρG))

1 4 = 3, 2

• Control mechanisms: still an open problem.

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