Two-Phase Flow and Heat Transfer in Highly Permeable Porous...

Oct 7, 2015

Transport in Porous Media1/36

M. Quintard

D.R. CNRS, Institut de Mécanique des Fluides, Allée Prof. C. Soula, 31400 Toulouse cedex – France

[email protected] http://mquintard.free.fr

Two-Phase Flow and Heat Two-Phase Flow and Heat Transfer in Highly Permeable Transfer in Highly Permeable

Porous MediaPorous MediaMichel Quintard

mailto:[email protected]

Transport in Porous Media 2/36M. Quintard

OutlineOutline

Background One-phase flow at high Re number Two-phase flow: quasi-static and dynamic models Hybrid Model: pore-scale / network-scale Macro-Scale Model with phase splitting for

structured porous media Boiling in porous media Conclusions


BackgroundBackground

Nuclear Engng, Nuclear Safety Chemical Engng: distillation, catalytic columns Flow (Oil,...) in gravel, blasted rocks, … High T geothermy


Upscaling One-Phase FlowUpscaling One-Phase Flow

Pore-scale equation

Upscaling


Closure: Stokes problemClosure: Stokes problem

PDEs for deviations

Re~0

(Whitaker, 1986)


Closure and Macro-Scale Closure and Macro-Scale EquationEquation

See Sanchez-Palencia (homog.), Whitaker, ...


Non Darcean regimesNon Darcean regimes

Heuristic: Forchheimer, Ergun, …

Upscaling? Darcy Weak Inertial Strong InertialWeak

Turbulent

~10 ~30 ~1000

~Re~Re3

~Re2

~Re2

~Re

Re

(passability)


Non Darcean regimesNon Darcean regimes

Laminar inertia effects → ~generalized Forchheimer equation

– Re → 0: Darcy– Re ~ 0: weak inertia, F.〈vβ ~ v〉 〈 β〉3 (Levy, Mei

& Auriault, ...)– Re > 0: strong inertia, F.〈vβ ~ v〉 〈 β〉2 (Whitaker,

1996; Lasseux et al., 2011;...)


Experimental Evidence: Darcy Experimental Evidence: Darcy regimeregime

4×4 mm prisms - Air flow

8.00E-09

8.50E-09

9.00E-09

9.50E-09

1.00E-08

1.05E-08

1.10E-08

1.15E-08

1.20E-08

1.25E-08

0 1 2 3 4 5

Re

µV/ (

-∂P

/ ∂z

-ρg

) (m

²)

PermeabilityDarcyNon Darcy

Clavier et al., 2014


Experimental Evidence: inertia Experimental Evidence: inertia regimeregime

Clavier et al., 2014

4×4 mm prisms - Water flow

y = 0.0002x2.3831y = 0.0064x1.1916

0.1%

1.0%

10.0%

100.0%

Re

(-∂P

/∂z

-ρg

-µU

/K)

/ (µ

U/K

)

Weak Inertial

Transition

1 10 100

Non-spherical particle beds - Air flow

0.0E+00

1.0E-04

2.0E-04

3.0E-04

4.0E-04

5.0E-04

6.0E-04

7.0E-04

0 500 1000 1500 2000

Re

ρU²

/ (-∂

P/∂z

-ρg

-µU

/K)

(m) c5x5

c5x8c8x12p4x4p6x6

Weak inertia Strong inertia


Turbulent flows in porous mediaTurbulent flows in porous media

Turbulence: time and spatial averaging (see book De Lemos, 2006; ...)– time and spatial averaging commute!– However: not necessarily the same result if sequential closure!?

• for one-phase flow: scheme “II” seems preferable –contrary to Antohe & Lage (1997), Getachew et al. (2000)–see discussion: Nakayama & Kuwahara (1999), Pedras and de Lemos (2001), etc...

• for multiphase flow?• Legitimacy of seq. Closure? → Simultaneous closure over [R3 R✕ t]? More

complex sequential closures (t → x → t → ...) depending on the hierarchy of scales?


Turbulent flows in porous media (continued)Turbulent flows in porous media (continued)

Localized Turbulence: i.e., nearly periodic (see Jin et al., 2015 for DNS results)– Spatial averaging of RANS models → generalized

Forchheimer equation, F not necessarily ~ v〈 β〉 or v〈 β〉2

Porous media turbulence models (i.e., modified k-ε, k-ω, etc...)?– Pedras & De Lemos, Nakayama & Kuwahara (1999), ...– note: useful for fluid/porous medium interface (D'Hueppe et

al., 2012)

Soulaine and Quintard, 2014

Example: structured packings


Two-Phase FlowTwo-Phase Flow

Pore-scale

lβ

L

β-phase

averaging volume V

l γ

γ-phase

σ-phase


Upscaling, quasi-static theoryUpscaling, quasi-static theory

Case of B.C. 4Whitaker, 1986; Auriault, 1987; Lasseux et al., 1996; ...

+ Re numbers + Dynamic Bond numberif ≈0

⇒ ?

Transfert de chaleur 15/36M. Quintard

Macro-Scale Models: quasi-Macro-Scale Models: quasi-staticstatic Heuristic (Muskat)

Upscalingimbibition

w = wetting phase

drainage

Pc

1- Sor 1Swi

Sw Sw

kr

1- Sor 1Swi

Phase interaction


Inertia EffectsInertia Effects

Ergun (Heuristic)

Schulenberg and Muler (1987) (Heuristic and ⛐)

Upscaling (Lasseux et al., 2008)


Importance of Cross-Terms, and Importance of Cross-Terms, and Non-Linear EffectsNon-Linear Effects

from Clavier et al. (2015)

see also Taherzadeh & Saidi (2015)

Case Vβ=0 :


Comparison various models: dPComparison various models: dP

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8

4 mm beads 〈 vl =0〉

Exp (Clavier et al.)

Clavier et al.

Lipinski

Reed

Hu&Theofanous

Schulenberg

Tung&Dhir

〈 vg (m/s)〉

Models without cross-terms


Comparison various models: Comparison various models: SaturationSaturation

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0 0.2 0.4 0.6 0.8

Exp (Clavier et al.)

Clavier et al.

Lipinski

Reed

Hu&Theofanous

Schulenberg

Tung&Dhir

4 mm beads 〈 vl =0〉

〈 vg (m/s)〉


More: Dynamic ModelsMore: Dynamic Models

impact of ∂S/∂t, Vα, av...:– Pseudo-functions in Pet. Engng– Quintard & Whitaker (1990, from large-scale

heterogeneity effects and multi-zone)– Hilfer (1998, multi-zone)– Panfilov & Panfilova (2005, meniscus)– Hassanizadeh and Gray (Irr. Therm., av as state

variable, 1993), also Kalaydjian (1987)...– ...


Examples of dynamic equationsExamples of dynamic equations

Quintard & Whitaker, 1990

...see also Petroleum Engng literatureon pseudo-functions!

ω

η

η

ω

0 0.5 1

Ωβ=0 Pa/m

Sβ

10-1

10-2

10-3

10-4

10-5

Ωβ=10-4Pa/m

Ωβ=10+4Pa/m


Mixed or Hybrid ModelsMixed or Hybrid Models Motivation: highly

permeable media, trickle beds

Example of challenging problem: jet dispersion

Trickle Bed (X-ray, IFP)

Horgue et al., 2013


Mixed or Hybrid ModelsMixed or Hybrid Models

Network model Dynamic rules (may come from local VOF

simulations)

Melli & Scriven, 1991; results from Horgue et al. (PhD CIFRE/IFP/IMFT), 2012

VOF

Dynamic Network Simulation


Application to tomographic Application to tomographic imagesimages


Macro-Scale Models with Phase Macro-Scale Models with Phase “Splitting”“Splitting” Example: Flow through Structured

Media

Upscaling with phase splitting (Soulaine et al., 2014): role of momentum exchange term

Mahr and Mewes (2007) Alekseenko (2008)

Model with liquid phase splitting


Macro-Scale Models with Phase Macro-Scale Models with Phase “Splitting”“Splitting” Example: Soulaine et al. Exp.

Comparison with Fourati et al. (2012) experiments

Model with liquid phase splitting

1. Identification on 1st stack2. Application to several stacks


Coupling with Heat Transfer: Coupling with Heat Transfer: boiling in porous mediaboiling in porous media

3-Temperature model:– decoupled 2-phase flow, quasi-steady →

Generalized Darcy-Forchheimer? (time-space ergodicity?)

– 3-T model, extension of 2-T model: Berthoud & Valette (1994); Petit et al. (1999); Duval et al. (2004);...based on quasi-steady approx.

– Heuristic: time averaging of averaged equations → porous media Nukiyama curves (see Sapin et al., 2014)Note: Highly open problem!

App. Nuclear safety


3T-Model3T-Model

withBoiling rate

Note: two-phase flow model (inertia+cross terms) +


Heat Exchange Terms: Impact of Heat Exchange Terms: Impact of phase configuration!phase configuration! Quasi-static two-phase flow theory

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.25

0.5

0.75

1

1.25

1.5

1.75

2

h×

10−

6(W

m−

3K

−1 ) Chang SLG

Chang SGLStaggered SLGStaggered SGL

⟨ Tl⟩l: exch. ⟨ Tl⟩l- T sat

Sl

s

g

l

R

εℓ = 0.28, εg = 0.42, Sℓ = 0.4Phase repartition:

VOF or Cahn-Hilliard Duval et al., 2004, ...


Pore-Scale ExperimentationsPore-Scale Experimentations

Sapin et al., 2014

Nucleate Boiling Film Boiling Intense Boiling

Pt wiring R0=100Ω

ceramic

ceramic coating

Remark Need time-averaging!

Slow motion!


Porous Media Nukiyama CurvesPorous Media Nukiyama Curves

0

5

10

15

20

25

-20 30 80 130 180

Wall fluxqps (W/cm²)

Tsat = Tp - Tsat (°C)

qchf

qmin

TLeidenfrost

Forced Convection

boiling crisis

Vapor film collapse

Impact on non-linear properties: kr, h, D, etc...?!


Critical FluxCritical Flux

0

5

10

15

20

25

30

-20 -10 0 10 20 30 40 50 60 70∆Tsat = Tp - Tsat (°C)

Qmp = 20mWQmp = 50 mWQmp = 100 mWQmp = 150 mW

q ps (

W/c

m2 )

Surrounding heating changes critical flux!...but the behavior up to CHF seems to be unaffected (contrary to modeling guess)?!


Macro-Scale Model?Macro-Scale Model?Bachrata et al., 2013 (model implemented in CATHARE Safety code): work confirms main required features (inertia, NLE,…)

→ ∃ Macro-scale behavior: confirmed by experiments but heuristic time averaging or time-space ergodicity

→ Need for adapted correlations for phase permeabilities and cross terms, inertia terms, heat transfer coefficients, etc...

Example for Solid-fluid exchange: IRSN Experiments:PRELUDE, SYLPHIDE, ...


Macro-Scale Model: reflooding Macro-Scale Model: reflooding simulationsimulation

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 3998000

99000

100000

101000

102000

103000

104000PRELUDE

mesh number

void fractiontotal heat flux [W]

wall T [°C]

gas vel. [m/s]

liq. vel. [m/s]pressure [Pa]

Qmax = 0.8 10+5

W/m2

Tmax = 414 °CVg,max= 3.9 m/sVl,max= 9.4 10

-3 m/s

dp = 2 mm


Example of Macro-Scale Example of Macro-Scale Simulations (Quenching)Simulations (Quenching)

15s

25s

15s

25s

Temperature field

550K

750K

steam

Water and steam velocities

4 m

1.2 m

QTregoures et al. (2003)


Conclusions (high permeable Conclusions (high permeable media)media) Various types of models Momentum equations:

– Generalized Forchheimer equation ~OK up to some high Re numbers, including localized turbulence

– Two-phase flow: need for cross-terms Boiling:

Need for: inertia, non-equilibrium, ... Problem: coupling and time averaging?

Effective properties Experimental procedures and interpretation? DNS?

Numerical aspects

Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Slide 31Slide 32Slide 33Slide 34Slide 35Slide 36

Two-Phase Flow and Heat Transfer in Highly Permeable Porous...

Documents

Transcript of Two-Phase Flow and Heat Transfer in Highly Permeable Porous...