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
[email protected], 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, oil 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: actes).
journals editing, conference organizations (proceedings,
– 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). Industrial occurrence and flow regimes
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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 to w a rd s tu rb in e
S T E A M G E N E R A T O R 7 0 b a r
• Sizing : SG: heat transfer and pressure drop. SGTR: critical flow at the safety valve. FSI: mechanical loading and vibrations.
s a fe ty v a lv e fro m c o n d e n se r
P R E S S U R IZ E R
R E A T O R V E S S E L 1 5 5 b a r P U M P
3 2 0 °C
S E C O N D A R Y C IR C U IT
2 9 0 °C
P R IM A R Y C IR C U IT
• 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 confinement 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 ≈ 355o C. Thermal power: 4250 MW. – Mass flow rate: 4928,6 kg/s per SG (4) – Core inlet temperature: 292,2o C – Core outlet temperature: 329,6o C • Secondary side, SG vapor pressure : 72,3 bar – Vapor temperature: 288˚ C – Feed water temperature: 229,5o C – 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 750
Linear approx. ρL, NIST
740 730
Density (kg/m3)
720 710 700 690 680 670 660 650 290
295
300
305 310 315 Temperature (°C)
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325
330
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MASS BALANCE OF THE PRIMARY CIRCUIT • Mass flow rate per loop (CL), ML ≈ 5023 kg/s • Inlet density: ρL1 (292o C, 155 bar) = 742, 41 kg/s. • Outlet density : ρL2 (330o C, 155 bar) = 651, 55 kg/s. Q1 =
ML = 6, 77m3 /s, ρ1
Q2 =
ML = 7, 71m3 /s ρ2
• Volumetric flow rates differ by 13%. • The volume of the primary circuit is 400 m3 ...
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WATER ENTHALPY AT 155 BAR 1550
Linear approx. h, NIST
1500
Enthalpy (kJ/kg)
1450
1400
1350
1300
1250 290
295
300
305 310 315 Temperature (°C)
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325
330
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PRIMARY SIDE HEAT BALANCE • Mass flow rate per loop (CL), ML ≈ 5023 kg/s. • Inlet enthalpy: hL1 (292o C, 155bar) = 1295 kJ/kg. • Outlet enthalpy: hL2 (330o C, 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 (292o C, 155bar) = 5, 2827 kJ/kg/K. P = ML CP ∆T ≈ 5023 × 201 103 = 1008 MW • Power differs by 10%. • Temperature drift: 31o C/hour.
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WATER ENTHALPY & INTERNAL ENERGY AT 155 BAR 1550
CP, NIST CV, NIST u, NIST h, NIST
1500
Heat capacity (kJ/kg/K)
6
1450 5 1400 4 1350 3
2 290
Enthalpy, Internal energy (kg/m3)
7
1300
295
300
305 310 315 Temperature (°C)
320
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1250 330
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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 3500
hL, EOS−NIST hV, EOS−NIST h, NIST Tables
3000
Enthalpy (kJ/kg)
2500 2000 1500 1000
500 0 0
50
100 150 200 250 Temperature (°C), Tsat = 99.61°C
300
350
Maximum liquid superheat, metastability limit >230 K Industrial occurrence and flow regimes
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WATER ENTHALPY AT 72 BAR 3500
hL, EOS−NIST hV, EOS−NIST h, NIST Tables
3000
Enthalpy (kJ/kg)
2500
2000
1500
1000
500 220
240
260 280 300 320 Temperature (°C), Tsat = 287.74°C
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360
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SECONDARY SIDE HEAT BALANCE • Mass flow rate per SG, ML ≈ 602 kg/s • Inlet enthalpy: hL1 (230o C, 72 bar) = 991.1 kJ/kg. • Outlet enthalpy: hV 2 (288o C, 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 100o C, 1 bar, ρV = 0, 5897 kg/m3 . • Perfect gas approximation: pV = RT , R = 8.316 J/mol/K, M = 18 g/mol. V =
RT = 3.103 10−2 m3 , p
ρ=
M = 0.5801 kg/m3 . V
• Steam at 288o C, 72 bar, ρV = 37.64 kg/m3 . • Perfect gas : pV = RT , R = 8.316 J/mol/K, M = 18 g/mol. RT V = = 6.481 10−4 m3 , p
M ρ= = 27.77 kg/m3 V
. • 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 X k
( H ,t)
1 t 0
M (H )
1 if x ∈ phase k Xk (r, t) = 0 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) D n
Space averaging operator (plain space average) Z 1 f dDn < | f> | n, Dn Dn • 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)
D
k n
Phase presence conditional average, Z 1 < fk >n , fk dDkn Dkn Dkn • 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
D
D k n
n
Dkn = Dn
Lk • n=1, line fraction, L1 + L2 Ak • n=2, surface fraction, A1 + A2 Vk • n=3, volume fraction, V1 + V2 Identity (proof left as an exercise) Rkn < f >kn = < | Xk f > | n
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w
FLOW RATE AND MASS FLOW RATE
• Instantaneous flow rate (m3 /s), wk = vk nz Z Qk (t) , wk dAk = Ak < wk >2
v k
k
k
Ak
n Z
• Instantaneous mass flow rate (kg/s) Z Mk (t) , ρk wk dAk = Ak < ρk wk >2 Ak
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TIME AVERAGING Time averaging on [T ], (plain) Z 1 t+T /2 f (t) , f (τ ) dτ T t−T /2
[T k] fk
Conditional time averaging, [Tk ] Z 1 X f k (t) , fk (τ ) dτ Tk [Tk ]
J t
t1 t-T /2
t2k T
t2k+ 1
Local time fraction, void fraction for gas/vapor
t2n t+ T /2
αk (r, t) ,
Tk = Xk (r, t) T
Identity, derives from the definitions, αk f
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X
= Xk f
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COMMUTATIVITY OF AVERAGING OPERATORS X
Rkn < fk >n = < | αk fk > | n Proof, from definitions: Rkn 1 T
Z
1 < fk >n = T Z
Z [T ]
(
Rkn Dkn
Z
fk dDkn
)
dt
Dkn (t)
Z
Z
1 1 1 dt Xk fk dt Xk fk dDn = dDn D D T n Dn n Dn [T ] [T ] ) Z ( Z 1 αk (r) X fk dt dDn = < | αk fk > | n Dn Dn Tk [Tk ]
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< | αk wX | 2 k > • Mean mass flow rate, Mk = ARk2 < ρk wk >2 = A< | αk ρk wX | 2 k >
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SUPERFICIAL VELOCITY • Mean volumetric flux, jk , Xk wk ≡ αk wX k • Mean mass flux, gk , Xk ρk wk ≡ αk ρk wX k • Superficial velocity (vitesse d´ebitante), Jk = < | jk > | 2=< |
αk wX | 2 k >
Qk = A
• Mixture superficial velocity, Q1 + Q2 J = J1 + J2 = A
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QUALITY • Mean mass flux, Gk , < | gk > | 2=< | αk ρk wX | 2= k >
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) 300
400, °C 380, °C 370, °C 350, °C 300, °C 200, °C Sat. ρL Sat. ρV
250
Pressure (bar)
200
150
100
50
0 0
100
200
300
400
500
600
700
800
900
1000
Density (kg/m3)
Critical pressure and temperature : ≈ 221 bar, 373,9o C. Industrial occurrence and flow regimes
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STEAM TABLES AND EOS (v, p) 300
T=400 °C T=380 °C T=370 °C T=350 °C T=300 °C T=200 °C Sat. Liq Sat. Vap
250
Pressure (bar)
200
150
100
50
0 0
0.005
0.01
0.015
0.02
0.025
0.03
Specific volume (m3/kg)
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STEAM TABLES AND EOS (ρ, p) 300
T=400 °C T=380 °C T=370 °C T=350 °C T=300 °C T=200 °C Sat. Liq Sat. Vap
250
Pressure (bar)
200
150
100
50
0 0
200
400
600
800
1000
Density (kg/m3)
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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) 6=< kk >2 ), • Energy balance, q, uniform heat flux distribution P = πqDz = M [h(z) − h1 ] = M [(xeq hV sat + (1 − xeq )hLsat ) − hL1 )] xeq
hL1 − hLsat πqDz + = hlv M hlv
• 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 properties, 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).
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TAITEL ET DUKLER (1980) MODEL
Flow regimes: • A-B: Bubbly (bulles) • A-D:Intermittent (poches, bouchons) • D-E: Churn (agit´e ) • E: Annular • B-C: Dispersed bubbles
Vertical ascending air-water flow, D = 50 mm, P = 1 bar. Industrial occurrence and flow regimes
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TAITEL ET DUKLER (1980) MODEL • Bubbly flow and intermittent (A): Bubble coalescence, zig-zag motion of bubbles. 3 1−α 2 JL = JG − (1 − α) U0∞ , α
αT = 0, 25,
U0∞ = 1, 53
g(ρL − ρG )σ ρ2L
14
• 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]0,5 νL0,08 1,12 J > 3, 0 0,10 0,48 (σ/ρL ) D • Intermittent and churn (D): churn flow ≡ development of slug flow. L J = 42, 6 √ + 0, 29 D gD
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TAITEL ET DUKLER (1980) MODEL • Annular (E): all liquid entrained by the gas, force balance, 1 2
JρG [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 1
– individual bubbles: UB = U0 inf (1 − α) 2 – In small diameter pipes, Taylor bubbles are slower than individual bubbles to coalescence towards slugs. 14 2 2 ρL dD 6 3, 78 (ρL − ρG )σ – Bubbly flow no longer exists.
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TAITEL ET DUKLER (1980) MODEL
Small diameter pipes: no bubbly flow.
Air water, D = 25 mm, P = 1 bar. Industrial occurrence and flow regimes
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APPLICATIONS : VertTD02
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
10
1 JL (m/s)
JL (m/s)
1
Bulles−Intermittent Bulles dispersées−Intermittent Bulles dispersées−Bulles Intermittent−Agité Intermittent−Annulaire
10
0.1
0.01
0.1
0.01
0.1
1 JG (m/s)
10
Air-water, 51 mm, 1 bar
100
0.1
1 JG (m/s)
10
100
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 liquid • Annular Modeling the transition based on mechanisms (Dukler & Taitel, 1986).
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FLOW PATTERN IN HORIZONTAL FLOW
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HORIZONTAL SLUG FLOW
α = 22% Industrial occurrence and flow regimes
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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) G
V L
V
V
_
G
2_ H A I I K H A _ _
• The Kelvin-Helmholtz instability, " # 12 (ρL − ρG ) cos βAG h VG > C2 , C2 ≈ 1 − dAL D ρG dh
9 A EC D J
S
S
G
A
S
A L
L
G
G R
• Base flow: smooth an horizontal interface SG SL Si Si τG −τL +τi + +(ρL −ρG )g sin β = 0 AG AL AL AG i
h
X 2 f (A, D, AL , PL , DL )−g(A, D, AG , PG , DG , Pi )−4Y −n 1 2 (dP/dz)LS 2 2 CL ρL JL ReLS . X = 1 −n = 2 (dP/dz)GS 2 CG ρG JG ReGS (ρL − ρG )g sin β Y = 41 2 C Re−m ρ J G G G G D2
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TAITEL AND DUKLER (1976) MODEL 1
Y positif −5 ,K> , K2 = sρG UL (ρL − ρG )Dg cos β νL ˜G sU ˜L U • Dispersed bubbles:
UL >
4AG g cos β Si fL
ρG 1− ρL
12
dp dz
12
8A˜G LS , T > , T = ˜ 2 (U ˜L D ˜ L )−n (ρL − ρG )g cos β S˜i U L 2
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TAITEL AND DUKLER (1976) MODEL
K=
2 ρG JG (ρL −ρG )Dg cos β
F = Curves
1 et 2
3
4
Coordinates
F, X
K, X
T, X
2 dp 4C ρJ −n = Re , dz D 2 S
T =
ρG ρG −ρL
12
12
DJL νL
JG
| |
dp dz LS
(ρL −ρG )g cos β
12
1 (dP/dz)LS 2 X = (dP/dz)GS
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. Industrial occurrence and flow regimes
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(Dg cos β) 2
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12
APPLICATIONS : HoriTD03 α=1° α=5°
10
10
1
1
JL (m/s)
JL (m/s)
D=12.5 mm D=50 mm D=300 mm
0.1
0.1
0.01
0.01 0.1
1
10 JG (m/s)
100
Air-Water, 50 mm, 1 bar, β = 0
0.1
1
10 JG (m/s)
100
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 Industrial occurrence and flow regimes
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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 Jk∗ ≈ Froude number. ∗ JG
1 2
=
J G ρG (gD(ρL − ρG ))
1 2
,
JL∗
1 2
J L ρL
=
1 2
,
(gD(ρL − ρG )) 12 3 m and C depend on NL = ρL gD µ(ρ2L −ρG ) ≡ Gr
∗ 12 JG
+
∗ 12 mJL
=C
L
m=1 NL > 1000 0, 88 < C < 1 (smooth inlet) , C = 0, 725 (sharp inlet)
m = 5, 6N −1/2 L NL < 1000 C = 0, 725
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FLOODING ET FLOW REVERSAL • Wallis model: no pipe length effect. ∗ – Some experiments show JG increase with the increase of L. Favors the wave instability mechanism.
– Many specific correlations. • Flow reversal ∗ ∗ – Wallis model, JG (FR) 6= JG (Flooding), hysteresis effect, pipe diameter effect. 1 2 J G ρG ∗ JG = 1 = 0, 5 (gd(ρL − ρG )) 2
– Puskina and Sorokin model, 1 2
Ku =
J G ρG (gσ(ρL − ρG ))
1 4
= 3, 2
• Control mechanisms: still an open problem. Industrial occurrence and flow regimes
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REFERENCES
Bankoff, S. G., & Chun Lee, Sang. 1986. Multiphase Science and Technolgy. Vol. 2. Hemisphere. Chap. 2-A critical review of the flooding literature, pages 95–180. Delhaye, J.-M. 2008. Thermohydraulique des r´eacteurs nucl´eaires. Collection g´enie atomique. EDP Sciences. Dukler, A. E., & Taitel, Y. 1986. Multiphase Science and Technolgy. Vol. 2. Hemisphere. Chap. 1-Flow pattern transitions in gas-liquid systems: measurement and modelling, pages 1–94.
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