Institute of Process Engineering, Chinese Academy of Sciences

Beijing, China. Email: nyang@home.ipe.ac.cn

2013-08-06

Ning Yang, Jinghai Li

6th NETL 2013 Workshop on Multiphase Flow Science, Morgantown

Stability-Constrained Multi-Fluid CFD Models for

Multiphase Systems

Complexity: Multiscale structure formation and evolution

Macro-scale: different flow regimes and regime transition Meso/Micro-scales: liquid vortices, bubble wakes, bubble swarms, bubble

deformation, bubble breakup/coalescence

Chen et al., AIChE J., 1994 Lin et al., AIChE J., 1996

Harteveld., 2005 Ruthiya et al., AIChE J., 2005

State of The Art: CFD simulation

• Model-dominated • Force models: drag, virtual mass, lift, turbulent dispersion force, etc. • Turbulence models: different choice for single-phase turbulence, bubble-induced

turbulence, dispersed or mixture model for two-phase turbulence, turbulence coupling

• Discretization schemes for convection term • Grid resolution • Boundary condition • fails to work for higher gas flow rate (Transition & heterogeneous regime) • NOT able to predict the regime transition

See Monahan et al., AIChE J., 2005

Requirements: CFD simulation

• Capture the dominant structures among the multi-scales

not only the macro-scale phase distribution

• Cover all the regimes

not only a single specific regime

• Predict the structure evolution (regime transition)

not only the evolution within each regime

Challenging problems

Macro-scale: regime transition, why ?

?

Meso/Micro-scales: how to describe the gas-liquid interaction ?

How can we incorporate the meso/micro effects into CFD simulation without the need to reconstruct the missing structure using DNS?

?

The EMMS model: Originally proposed for gas-solid fluidization

EMMS model

6 conservative equations Dilute

phase

Gas velocity Uc Solid velocity Upc Voidage ec Volume fraction f Cluster diameter dcl

Gas velocity Uf Solid velocity Upf Voidage ef

8 Variables

Dense phase

Energy-minimization multi-scale

Cluster-scale

Particle-scale: in dense-phase In dilute-phase

Energy Resolution

Structure Resolution

(Li & Kwauk, 1989)

Operating Conditions

Compromise between dominant

mechanisms

Stability Condition: Wst = min|ε= min Nst =min

Correlation between scales

The hierarchy in the EMMS model

Force balance equation for the dense phase

Force balance equation for the dilute phase

Force balance of interphase

g c f(1 )U fU f U= + −

p pc pf(1 )U fU f U= + −

( )2

g sfDf p g

p

34

UC g

dρ

ρ ρ= −

( )2

g scDc p g

p c

3 14 1

UC g

dρ ε ρ ρ

ε−

= −−

( )( )2

g siDi c p g

cl

34

UC g

dρ

ε ε ρ ρ= − −

Continuity of fluid

Continuity of solid

(1): Mass and force balances

(2): Correlation of meso-scale structure and meso-scale energy dissipation

Cluster diameter ( )p g p p max st,mf

cl pst st,mf

/ (1 )gU Nd d

N Nρ ρ ρ ε− − −

=−

( )p gfst g f

p

(1 ) min1

gN U f f U

ρ ρε εε ρ

−− = − − = −

(3): Stability condition

Yang, Progress in CFD, 2012; Chen et al., Chinese J Chem. Eng., 2012

8

Regime transition (choking)：The role of different constraints

A curved surface：solutions of mass and force balances

A curve：due to the correlation of meso-scale structure and meso-scale

energy dissipation

Two points：final solution subject to the stability condition

Yang, Progress in CFD, 2012; Chen et al., Chinese J Chem. Eng., 2012

0 5 10 15 20 25 30

0

20

40

60

80

100

120

Experimental

Simulation

Output solid flux (kg/m2s)Empirical correlations+CFD

Time (s)

Output particle flux

Simulation

Experiments

Two-Fluid Model

(Wen-Yu/Ergun drag laws) Two-Fluid Model

(EMMS drag model)

Yang et al., Ind. Eng. Chem. Res., 2004, 43, 5548-5561. Yang et al., Chem. Eng. J,, 2003

0 5 10 15 20 25 30

0

20

40

60

80

100

120

ExperimentalSimulation

Output solid flux (kg/m2s)EMMS+CFD

Time (s)

Clusters Homogeneous

Output particle flux

Simulation Experiments

Gas-Solid: Application of EMMS drag in CFD simulation of a CFB riser

The system can be self-adapted due to cluster formation!

Weinstein et al., Fluidization IV, 1983, 299-306; Li et al., Chem. Eng. Sci., 1998, 3367-3379

Axial distribution of solid concentration in a CFB riser

Yang et al., Ind. Eng. Chem. Res., 2004, 43, 5548-5561; Yang et al., CFB VIII, 2005, 291-298.

Two Fluid Model (TFM) Homogeneous

0

2

4

6

8

10

0.7 0.8 0.9 1.0

Hei

ght (

m)

I=20 kg I=15 kg

Voidage

TFM+EMMS Dense bottom, dilute top

Experiment: Dense bottom, dilute top

Ug=1.52m/s, Gs=14.3Kg/m2s

Extension to Gas-liquid system

DBS model

Operating Conditions

Compromise between dominant

mechanisms

Correlation between scales

3 conservation equations Large

bubbles

Gas velocity Ug,S Volume fraction fS Bubble diameter dS

6 Variables

Small bubbles

Dual-bubble-size model (DBS)

Large bubble

Small bubble

Gas velocity Ug,L Volume fraction fL Bubble diameter dL

Stability Condition: surf,S+L turb min.N N+ →

Liquid

Structure Resolution

Energy Resolution

(Yang et al, Chem. Eng. Sci, 2007)

12

Gas

Liquid

turbNsurfN

TN

Interface oscillation

Viscous dissipation

breakNInterface fracture

Path of energy transfer and dissipation (Zhao, 2006; Ge et al, CES,2007)

f

gU

Gas-liquid bubbly flow

Energy dissipated directly on micro-scale caused by relative motion between bubbles and liquid

Energy consumption due to meso-scale structure evolution

Scale-dependent energy resolution

+surf turbN N

breakN

totalNmin

max

(Zhao, PhD thesis, 2006; Ge et al, Chem. Eng. Sci.,2007)

Relationship of momentum transfer and energy dissipation

(Yang et al, Chem. Eng. Sci., 2010, 2011)

A new mechanism beyond transport equations

Large bubbles

Small bubbles

Liquid

Transport equations： Mass and momentum conservative equations (6 structure parameters), not closed!

Structure resolution

+surf turbN N min breakN max

Stability condition

+

Micro-scale dissipation A buffer for energy

dissipation Sustain the formation

and evolution of mesoscale structure

dissipated directly by relative motion of bubbles and liquid

Meso-scale dissipation

Energy resolution

Model assumption

(viscous dissipation ) Nturb≈ ε ( turbulent dissipation )

• Nbreak Ncoalescence (No net surface generated)

• Classical statistical theory of isotropic turbulence

•

b

min

0.52b

break b b BV f b BV b bb l b g0

( , ) ( , , ) d d ( , , )(1 )

d dN P d f c d f f df fλ

ω λ λ π σ λ ϕ ερ ρ

= ⋅ ⋅ ⋅ =− +∫ ∫

D,psurf T

D,b

1C

N NC

= −

(Luo & Svendsen, AIChE J., 1996)

Mathematical model (DBS):

2g,L2L l

L l DL L l3L L b

16 4 2 1

Uf Uf g C dd f f

πρ ρπ

= ⋅ ⋅ − ⋅ −

g,S g,L gU U U+ =

Variables: (fS, fL, dS, dL, UgS, UgL)

surf,S+L turb min.N N+ →

Small bubble:

Equations:

Large bubble:

Continuity:

Stability condition: subject to

the 6D space of structure parameter

18

Regime transition: why?

Camarasa et al., 1999, Chem. Eng. Proc., 38, 329-344; Ruthiya et al., 2005, AIChE Journal, 1951-1965

Yang, et al., Chem. Eng. Sci., 2010, 65, 517-526; Chen, et al., IECR, 2009, 48, 290-301

SBS≈DBS SBS≈DBS SBS≠DBS

Jump change of total gas hold-up

Jump change of global minimum of micro-scale energy dissipation within the space of structure parameters

Ug=0.128m/s

Ug=0.129m/s

Stability-constrained multi-fluid approach (SCMF)

DBS Model:

• Mass balance

• Force balance

• Closure (stability condition)

Two-fluid models:

• Mass conservation

• Momentum conservation

• Closure (Empirical)

vs.

Stability-constrained multi-fluid approach (SCMF)

• Mass conservation for two or multiple fluids

• Momentum conservation for two or multiple fluids

• Closure (stability condition)

20

Cd/db for gas and liquid phaseSCMF-A

Cd/db for dilute and dense phase

Structure parameters for dense phase

SCMF-B

22

SCMF-A Mass conservations:

Momentum conservations:

Assumption:

S L gu u u= = ( ) ( ) _ _ _

l ll l l liquid S liquid L liquid gasu

tε ρ

ε ρ∂

+∇ ⋅ = Γ +Γ = Γ∂

(1)

( ) ( ) _ _ _g g

g g g L liquid S liquid gas liquidut

ε ρε ρ

∂+∇ ⋅ = Γ +Γ = Γ

∂

(2)+(3):

( ) ( ) ( ), _Tl l l

l l l l l l eff l l l l l liquid gas

uu u P u u g

tε ρ

ε ρ ε µ ε ε ρ∂ +∇ ⋅ = − ∇ + ∇ + ∇ + + ∂

M (4)

(5)+(6):

( ) ( ) ( ), _

Tg g gg g g g g g eff g g g g g gas liquid

uu u P u u g

tε ρ

ε ρ ε µ ε ε ρ∂ +∇ ⋅ = − ∇ + ∇ + ∇ + + ∂

M

23

SCMF-B Mass conservations:

Momentum conservations:

Assumption: MS_L=ML_S=0

S l denseu u u= =

dense l Sε ε ε= +l S

dense l Sl S l S

ε ερ ρ ρε ε ε ε

= ++ +

(1)+(2): ( ) ( ) _ _ _

dense densedense dense dense liquid L S L dense Lu

tε ρ

ε ρ∂

+∇ ⋅ = Γ +Γ = Γ∂

( ) ( ) _ _ _L L

L L L L liquid L S L denseut

ε ρε ρ

∂+∇ ⋅ = Γ +Γ = Γ

∂ (3)

(4)+(5):

( ) ( ) ( )

( ) ( ){ } ( ), , _

dense dense densedense dense dense dense dense

Tl eff l S eff S S S dense dense dense L

uu u P

t

u u g

ε ρε ρ ε

µ ε µ ε ε ρ

∂+∇ ⋅ = − ∇

∂ + + ∇ + ∇ + +

M

( ) ( ) ( ), _TL L L

L L L L L L eff L L L L L L dense

uu u f P u u g

tε ρ

ε ρ µ ε ε ρ∂ +∇ ⋅ = − ∇ + ∇ + ∇ + + ∂

M (6)

24

Simulation case: Hills (1974)

Simulation case: Camarasa et al. (1999)

•Height: 2m; ID: 0.1m

•Initial liquid height: 1.35 m

• Perforated plate: 61 holes

• hole diameter: 1mm

• Number of meshes: 3.34 million

• k-ε mixture model

•First order upwind

• Time averaged data for H=1m

•Height: 1.3m; ID: 0.138m

•Initial liquid height: 0.9 m

• Perforated plate: 61 holes

• hole diameter: 2mm(0.4mm)

• Number of meshes: 5.31 million

• k-ε mixture model

•First order upwind

• Time averaged data for H=0.6m

SCMF-A vs. other drag models (Cd0, p, db)

2

2

422.5 5335 21640.5 , 0.128/

139.3 795 1500.3 , 0.128

g g gD b

g g g

U U UC d

U U U

− + ≤= − + >

0.687D0

16 48 8max{min[ (1 0.15Re ), ], }Re Re 3 4

EoCEo

= ++

D024 60.44Re 1 Re

C = + ++

D D0(1 ) pgC C ε= −

Tomiyama (1998)

White (1974)

DBS model (this work)

Effective drag coefficient:

Standard drag coefficient

SCMF-A

Correlations

27

Effect of correction factor on simulation CD0: Tomiyama; db=5mm; p: 0, 2, 4

D D0(1 ) pgC C ε= −

Ug=0.095m/s

Effect of correction factor on simulation CD0: White; db=5mm; p: 0, 2, 4

Ug=0.095m/s

D D0(1 ) pgC C ε= −

29

Simulation with the SCMF-A model (Gas holdup)

Ug=0.038m/s

Ug=0.095m/s

Ug=0.127m/s

30 Ug=0.038m/s Ug=0.095m/s Ug=0.127m/s

Simulation with the SCMF-A model

31

( )0.687

0

24 1 0.15Re Re 1000Re0.44 Re>1000

DC + ≤=

( )

( )( )

( )

0.75

267

2

24 1 0.1Re , viscous regimeRe

1 17.672 , distorted regime3 18.67

8 1 , churn3

gD b

g

g

fgC df

ερσ ε

ε

+

+∆ =

− turbulent flow regime

SCMF-B vs. other drag models:

Schiller-Naumann

Ishii-Zuber

( ) ( )1.51g gf ε ε= −

D D0(1 ) pgC C ε= − p=0

D D0(1 ) pgC C ε= −

2

2

337.90 6084.15 37066.03 , 0.101 m/s

142.22 661.62 897.53 , 0.101 m/sg g gD

b L g g g

U U UCd U U U

− + ≤ = − + >

( ), ,, , , , ,gs g L L L g S S SF U f d U f dε =

SCMF-B

32

Comparison between different models: Total gas holdup

Bubble column (Hills, 1974) Bubble column (Camarasa, 1999)

• Ishii-Zuber model over-estimates the total gas holdup at higher Ug

• Schiller-Naumann model under-predicts the total gas holdup at lower Ug

33

Simulation with the SCMF-B model (Total gas holdup)

The SCMF-B model can reproduce the plateau or shoulder of the gas holdup curve

Bubble column (Hills, 1974) Bubble column (Camarasa, 1999)

34

Comparison between SCMF-A, B and other models (Hills column)

Ug=3.8 cm/s Ug=9.5 cm/s Ug=12.7 cm/s

Radial profile of gas holdup

Ug=3.8 cm/s Ug=9.5 cm/s Ug=12.7 cm/s

Radial profile of liquid axial velocity

35

Comparison between SCMF-A, B and other models (Camarasa column)

Radial profile of gas holdup Radial profile of liquid axial velocity

36

SCMF-B

Ug=0.038m/s Ug=0.095m/s Ug=0.127m/s

37

Evolution from SCMF-A to SCMF-B

SCMF-A model1. Conservative equations for gas and liquid;2. Inlet flow rate Ug ;3. Drag coefficient (CD/db)T extracted from DBS model.

Step 11. Conservative equations for two “fluids”;2. Inlet flow rate Ug;3. Drag coefficient (CD/db)T extracted from DBS model.

Step 21. Conservative equations for two “fluids”;2. Inlet flow rate Ug,L;3. Drag coefficient (CD/db)T extracted from DBS model.

SCMF-B model1. Conservative equations for two “fluids”;2. Inlet flow rate Ug,L;3. Drag coefficient (CD/db)L extracted from DBS model.

Regime transition can be physically understood via the jump change of the minimum point of micro-scale energy dissipation in the 3D space of structure parameters.

Stability condition supply a new constraint for gas-liquid complex flow in addition to mass and momentum conservative equations

Conclusions and Prospects

Stability-constrained multi-fluid CFD approach

• may offer a closure for CFD and may be of significance to the fundamental of multiphase flow.

• are superior to traditional TFMs with empirical drag correlations: Without the need to adjust correction factors Suitable for low, intermediate and higher gas flow rates Can capture the plateau or shoulder of the gas holdup curve

• Each SCMF model has its strength and weakness by comparison. SCMF-A is better for lower and much higher flow rates. SCMF-B is better for prediction of overall gas holdup / at relatively higher gas flow rate / the wall region for lower gas velocity.

Recent publication

1. Yang, N* (2012) A multi-scale framework for CFD modeling of multi-phase complex systems based on the EMMS approach. Progress in Computational Fluid Dynamics, 12(2-3), 220-229.

2. Yang, N*, Wu, Z., Chen, J, Wang, Y., Li, J (2011) Multi-scale analysis of gas-liquid interaction and CFD simulation of gas-liquid flow in bubble columns. Chemical Engineering Science, 66(14), 3212-3222.

3. Yang, N*, Chen, J., Ge, W., Li, J. (2010). A conceptual model for analysing the stability condition and regime transition in bubble columns. Chemical Engineering Science. 65, 517-526, 2010. 4. Xiao, Q., Yang, N.*, Li, J., (2013). Stability-constrained multi-fluid CFD models for gas-liquid flow in bubble columns. Chemical Engineering Science. 100, 279-292. 5. Shu, S., Yang, N.*, (2013). Direct numerical simulation of bubble dynamics using phase-field model and lattice Boltzmann method. Industrial and Engineering Chemistry Research. DOI: 10.1021/ie303486y

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