A SHORT INTRODUCTION TO TWO-PHASE FLOWS Void fraction: Experimental techniques and simple models Herv´ e Lemonnier DTN/SE2T, 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
PHASE PRESENCE FUNCTION • Phase presence function definition: 1 si M ( r ) ∈ phase k, X k ( r , t ) � 0 si M ( r ) / ∈ phase k. • Phase indicator, FIP, fonction indicatrice de phase , χ k . • Space averaging: � 1 Conditional, < f > n � f k d V, D kn D kn � 1 | n � Plain, < | f> f d V. D n D n Void fraction: experimental techniques and simple models 1/38
AVERAGING OPERATORS (CT’D) • Time averaging: � k ( t ) � 1 X Conditional, f f ( τ )d τ, T k [ T k ] � Plain, f ( t ) � 1 f ( τ )d τ. T [ T ] • Commutativity of averaging operators: X R kn < f k > n = < | α k f k > | n . • Void fraction: average of the phase presence function • Void fraction, gas hold-up: taux de pr´ esence du gaz , taux de vide . Void fraction: experimental techniques and simple models 2/38
� VOID FRACTION ( α ) • Local time fraction (gas, void fraction) : α G ( r , t ) � X G = T G T . • Instantaneous space fraction. – Line fraction: � � L G = L G R G 1 ( t ) � < | X G > | 1 = L G + L L L – Area fraction: A G = A G R G 2 ( t ) � < | X G > | 2 = A G + A L A � � � � � � � – Volume fraction: V G = V G R G 3 ( t ) � < | X G > | 3 = V G + V L V Void fraction: experimental techniques and simple models 3/38
BASIC IDENTITIES • Commutativity ( f = 1): R kn = < | α k > | n = < | X k > | n . • Mean phase fraction. – Mean line-averaged, � � R G 1 = 1 R G 1 ( τ ) d τ = 1 α G d L T L [ T ] L – Mean area-averaged, � � R G 2 = 1 R G 2 ( τ ) d τ = 1 α G d A T A [ T ] A – Mean volume averaged, � � R G 3 = 1 R G 3 ( τ ) d τ = 1 α G d V T V [ T ] V • 7 precise definitions of the void fraction. The one to keep depends on the context (model). VF is always some average of X k . Void fraction: experimental techniques and simple models 4/38
� � � � OTHER RELATED DEFINITIONS • Instantaneous interfacial area: Γ 3 ( t ) � A i ( t ) V • Local interfacial area, is a time-averaged quantity: � 1 γ = | v i . n k | disc . ∈ [T] • Identity (commutativity of interaction terms), mean interfacial area: Γ 3 ≡ < | γ> | 3 • Γ 3 and γ can be measured. Void fraction: experimental techniques and simple models 5/38
EXPERIMENTAL TECHNIQUES FOR VOID FRACTION • Local void fraction. – Electrical probes – Optical probes • Line averaged void fraction. – Light attenuation (X or γ rays) • Area-averaged void fraction. – X-rays ou γ -rays, (one-shot) – Multi-beam densitometry – Neutrons diffusion (steel, steam-water, HP-HT) – Impedance probes • Volume averaged void fraction, – Quick closing valves, – Gravitational (hydrostatic) pressure drop, – Ultrasound attenuation (Bensler, 1990). • Medical imaging, CT, MRI. Void fraction: experimental techniques and simple models 6/38
� � LOCAL VOID FRACTION Electrical probes (different resistivity): Determination of the liquid phase indicator, X L ( r , t ). • Continuous phase (water), conducting, • Dispersed phase (air), non conducting • Threshold on current → X L → α L � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Void fraction: experimental techniques and simple models 7/38
� LOCAL VOID FRACTION Optical probes (different refraction index) : Determination of the gas indicator, X G ( r , t ). Principle: dish washer, rinsing liquid level indicator. � � � � � Water, freons, T < 110 o C � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Void fraction: experimental techniques and simple models 8/38
� � � � � � � � HOW TO SET THE THRESHOLD LEVEL? � � � � � • α = X L , depends on threshold: S 1 > S 2 ⇒ α L 1 < α L 2 . • Reference method: � � � � � � � � � � � ∆ p → R G 2 • It is recalled, < | α G > | 2 = R G 2 • Determine α G ( S ) on the section. � � � � � � � � � Find S such that, < | α G ( S ) > | 2 = R G 2 • Consistency check, not a calibration. � � � � Void fraction: experimental techniques and simple models 9/38
ELECTRICAL & OPTICAL PROBES Void fraction: experimental techniques and simple models 10/38
MULTIPLE TIP SENSORS • 2-sensor probe. Add assumptions: spherical bubbles, chord distribution → mean diameter, mean gas velocity. • 4-sensor probe. Determine interface orientation ( n k ), geometrical surface velocity, v i � n k • Local interfacial area, � 1 γ = | v i . n k | disc . ∈ [T] • Mean sauter diameter ( D 32 ), identity (bubbles), 6 α γ ≡ D SM Void fraction: experimental techniques and simple models 11/38
� � LIGHT (PHOTONS) ATTENUATION • X-rays or γ -rays. • Collimated beam, single spectral line • Beer-Lambert relation: [ µ ] = L − 1 � � � � d I = − µI d x, • Exponential absorption, µ absorption coefficient. • µ ρ : mass energy-absorption coefficient depends on f . � � Void fraction: experimental techniques and simple models 12/38
� � � PRACTICAL IMPLEMENTATION • X-Rays generator, γ source • Detection: photo multiplier (NaI, semi- conductors), counter. � � � � � � � � • Collimation: thick heavy metal block, � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � drilled, 0,5 mm � � • Collimated beam, single spectral line, • Integrate B-L on a finite length, � � − µ I = I 0 exp( − µL ) = I 0 exp ρ ρL • At low pressure, little absorption in the gas. Void fraction: experimental techniques and simple models 13/38
� � � MEASURING THE LINE PHASE FRACTION • D , diameter, e/ 2 wall thickness. � � � • Beer-Lambert equation: I = I 0 exp( − µ p e ) exp ( − µ L (1 − R G 1 ) D ) exp( − µ G R G 1 D ) � � • Definition of the gas line fraction: L G = L G R G 1 ( z, t ) � L G + L L D • Low pressure assumption: I G = I 0 exp( − µ p e ) I L = I 0 exp( − µ p e ) exp ( − µ L D ) � � I = I 0 exp( − µ p e ) exp ( − µ L (1 − R G 1 ) D ) R G 1 = ln I/I L Air-water flow, steam-vapor. ln I G /I L Void fraction: experimental techniques and simple models 14/38
UNCERTAINTY FACTORS � � � • Contrast → low energy � µ L � I G ≈ exp ρ L D I L ρ L • Statistical errors with counters → high energy � ∆ N 1 I ∝ N, ∝ N N � � � � � � � � � � � � • R L 1 fluctuations, I ∝ exp( R G 1 ) and exp f � = exp f , ∆ R G ≈ 0 , 20 (slug) , ∆ R G ≈ 0 , 05 (churn). I • Source stability: reference beam method, I → 0 . I ′ • Spectral hardening, direct calibration, I ( R L ), or use filters. Void fraction: experimental techniques and simple models 15/38
MEAN LINE GAS FRACTION After Bensler (1990, p. 60) • No water flow, J L = 0: R G 2 = 0,01, 0,04, 0,07, 0,10, 0,13, 0,16, 0,19. Void fraction: experimental techniques and simple models 16/38
MEAN LINE GAS FRACTION After Bensler (1990, p. 61) • Two-phase flow, J L = 2 m/s : R G 2 = 0,03, 0,061, 0,069, 0,089, 0,123. • Wall peaking, still an open problem for modeling... • Transition in profile shape, bubble clustering, slug flow. Void fraction: experimental techniques and simple models 17/38
� � � � � MEAN AREA FRACTION • Mean area fraction, R G 2 , � R � 1 R 2 − y 2 d y R G 2 = R G 1 ( y ) πR 2 � � − R • Computed tomography, axis-symmetric, R G 1 ( y ) ⇔ α G ( r ) R G 1 ( y, θ ) ⇔ α G ( X, Y ) • Instantaneous surface fraction, R G 2 ( t ) � � � � � � � � � � � � � � � � � � � � � � � � � � � � • Known limitations, Compton, diffusion ∆ R G 2 � 0 , 05 � � � � � � 0 < R G 2 < 0 , 8 Void fraction: experimental techniques and simple models 18/38
MEAN AREA FRACTION • Multibeam densitometer, R G 1 ( θ ) ⇔ α G ( r ) � � � � � � � � � • XCT, medical scanner � � � � � � R G 1 ( θ, φ ) ⇔ α G ( x, y ) � � � � � � � � � � � � � � • Neutron diffusion, 90˚ • Low attenuation in steel, diffusion on H nucleus • Neutron cinematography. Void fraction: experimental techniques and simple models 19/38
SUPER MOBY DICK After figures 63’r et 64’r from Jeandey et al. (1981). P 0 = 59 . 43 bar, T L = 269 . 2 o C, P sat = 54 . 31 bar. Void fraction: experimental techniques and simple models 20/38
THE ULTIMATE METHOD? • Nuclear magnetic resonance, NMR, MR imaging – Non intrusive method, – Magnetization (H, F), magnetic fields – Density (void fraction), velocity • Space and time resolution – 0D, 1D, 2D, etc. – Time averaged quantities, arbitrary space filters (LES). – Turbulent transport phenomena. • Routine for medicine, body (static) imaging, 1 mm 3 , arterial flow rate, still under development for flow imaging. Void fraction: experimental techniques and simple models 21/38
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