TWENTE UNIVERSITY THE NETHERLANDS - TFE1 | Home · • APPLICATIONS OF FLUIDIZED SYSTEMS + heat exchange and drying + coating and granulation + gas purification via adsorption + chemical

MODELLING OF DENSE GAS-PARTICLE FLOWS

USING KINETIC THEORY OF GRANULAR FLOW

J.A.M. KUIPERS

TWENTE UNIVERSITYTHE NETHERLANDS

DENSE GAS-SOLID FLOWS“shifting sands” (Tanzania)

DENSE GAS-SOLID FLOWSclusters in co-current vertical gas-solid flows

INTRODUCTIONdense gas-particle flows in fluid bed family of contactors

1: bubbling bed2: turbulent bed3: circulating bed4: riser5: downer6: lateral staged bed7: vertical staged bed8: spouted bed9: floating bed10: twin bed

INTRODUCTION

• APPLICATIONS OF FLUIDIZED SYSTEMS

+ heat exchange and drying+ coating and granulation+ gas purification via adsorption+ chemical synthesis (acrylonitrile, maleic and phtalic anhydride)+ polymerization of lower olefines (propylene)+ Fischer-Tropsch synthesis+ Fluid Coking and Flexi-Coking+ combustion and incineration + Fluid Catalytic Cracking (FCC)

INTRODUCTIONFluid Catalytic Cracking (FCC) unit

ELEMENTARY PROPERTIES OF GAS-FLUIDIZED BEDSdense beds

FLOW REGIMES + KEY PHENOMENAmap of flow regimes in particle-laden flows

MULTI LEVEL MODELLING

• MULTI LEVEL MODELLING OF DENSE GAS-SOLID FLOWS

Van der Hoef et al., CES (2004)

DBM SIMULATIONindustrial size column

• SIMULATION CONDITIONS

Industrial scale column:• Dimensions: 4 m x 4 m x 8 m• Gas velocity: 2.5Umf=0.25 m/s

Emulsion phase properties:• Density: 400 kg/m3

• Viscosity: 0.1 Pa.s

Bubble properties:• Initial bubble size: 8 cm• Maximum bubble size: 80 cm• Typically ~ 5000 bubbles

DPM + KTGF SIMULATIONbubble formation: 15 cm bed

W=0.15 mdp=1.5 mm

ρs=2526 kg/m3

Ub=0.85 m/sUj=15.0 m/sNp=120000

DPM SIMULATIONspouted bed


/ 16.0 / 1.2sf mf bf mfu u u u= ↔ =

particle configuration particle velocity map


experimental simulated

/ 16.0 / 1.2sf mf bf mfu u u u= ↔ =

DNS (IBM)flow through cubic array of 64 particles at Rep=2.0

1003 Eulerian gridN=(dp/h)=20

Dimensionless drag

F=10.9 (computed)F=10.2 (analytical)

DNS (IBM)flow through random array of 1326 particles at Rep=120

DNS (IBM)flow through random array of 1326 particles at Rep=120

DNS (IBM)fluidization of 3600 discs

KINETIC THEORY BASED MODELS

• BASIC FEATURES

+ statistical mechanical description of particle-particle encounters

• ADVANTAGES

+ based on more fundamental description of particle-particle interactioncompared to classical two-fluid model

• DISADVANTAGES

+ incorporation of different particle properties (polydispersity) is quitedifficult and leads to (many) additional equations (CPU limitations)

KINETIC THEORY BASED MODELS

• LIMITATIONS AND PRESENT DIFFICULTIES

+ nearly spherical particles

+ not suited for dense gas-particle flows where (quasi-)static particlezones prevail (hoppers, fixed beds and moving beds)

+ incorporation of detailed particle-particle interaction models difficult

+ systems with broad distribution in physical properties (size, density)

+ systems with (rapid) changes in particle size (polymerization)

CONTINUUM MODEL BASED ON KINETIC THEORY

• DEFINITION OF PARTICLE VELOCITIES

+ instantaneous particle velocity:

+ ensemble averaged particle velocity:

+ fluctuating particle velocity:

• DISTRIBUTION OF FLUCTUATING VELOCITIES (KTG)

vcC −=

v

c

)2

exp()2

(2

2/3

kTmC

kTmnf −π

= Maxwell’s velocity distribution

Boltzmann constantnumber density


• TRANSPORT MECHANISMS FOR PARTICLE PROPERTY φ


• BOLTZMANN EQUATION IN TERMS OF

• BOLTZMANN EQUATION IN TERMS OF

• SUBSTANTIAL DERIVATIVE:

)()()()(tf

Fc

fcfF

rfc

tf e

∂∂

=•∂∂

+∂∂

•+∂∂

•+∂∂

)())(():()()(tf

FfCr

vCfC

rfC

Cf

DtvD

DtDf e

∂∂

=•∂∂

+∂∂

∂∂

−∂∂

•+∂∂

•−

)(r

vtDt

D∂∂

•+∂∂

=

c

vcC −=


• MAXWELL TRANSPORT EQUATION FOR PROPERTY φ

• ENSEMBLE AVERAGE OF PROPERTY φ

∫∫∫ φ>=φ< Cfd

=∫∫∫∂∂

φ>=φ<Δ=>∂φ∂

<∂∂

−>∂φ∂

<•−>∂φ∂

•<+>∂φ∂

•<+>φ

<

−•∂∂

>φ<+>φ<•∂∂

+>φ<

Cdtf

nC

Crv

CDtvD

CF

rC

DtDn

vr

nCnrDt

nD

e )()]:(

)([

)()()(

2

21)(

21

1

mCCCm

Cm

=•=φ

=φ

=φ

MICRO BALANCE EQUATIONS

• CONTINUITY EQUATIONS

• MOMENTUM EQUATIONS

0)()( =ρε⋅∇+ρε∂∂ ut ffff

0)()( =ρε⋅∇+ρε∂∂ vt ssss

gvupuuut fffffffff ρε+−β−τε⋅∇−∇ε−=ρε⋅∇+ρε∂∂ )()()()(

gvuppvvvt ssssssssss ρε+−β+∇−τε⋅∇−∇ε−=ρε⋅∇+ρε∂∂ )()()()(

MICRO BALANCE EQUATIONS

• GRANULAR TEMPERATURE EQUATION

• ADDITIONAL EQUATIONS AND CLOSURES

+ phase densities+ phase stress tensors and phase viscosities+ pseudo Fourier energy flux and solids pseudo conductivity + solids pressure+ interphase momentum exchange coefficient+ covariance between fluid and solid fluctuating velocities

γ−Θ−⋅β+Θ∇κε⋅∇+∇τε+−=Θρε⋅∇+Θρε∂∂ ]3[)()(:)()]()([

23

sfsssssssss ccvIpvt

for dense flows this termcan safely be neglected

CLOSURE OF MICRO BALANCE EQUATIONSinterphase momentum transfer coefficient

• Ergun equation (εf<0.8):

• Wen and Yu equation (εf>0.8):

• Drag coefficient:

vudd p

ff

p

f

f

f −ρ

ε−+μ

ε

ε−=β )1(75.1

)1(150

2

2

65.2

43 −ε−ρ

εε=β ff

p

sfd vu

dC

])(Re15.01[Re24 687.0

pp

dC += 44.0=dC

exponent depends onparticle Reynolds number Rep

Rep<1000 Rep>1000

f

pffp

dvuμ

ρε −=Re

DRAG CLOSUREErgun equation


• SUMMARIZING THE KEY FEATURES OF KTGF

+ particles interact through binary nonideal collisions (no friction)

+ non-ideal particle-wall collisions are accounted for

+ departure from Maxwellian (velocity) distribution function is small

+ one additional equation (“granular temperature equation”)

+ closures for solids viscosity, pressure and pseudo conductivity

+ random granular motion and no collective granular motion

NUMERICAL SOLUTION METHOD

• KEY FEATURES

+ explicit treatment of convection and diffusion terms

+ implicit treatment of porosity pressure in momentum equations+ staggered computational mesh+ high order schemes for convection for mass and momentum

stabilityconditions

1<Δ⎥⎥⎦

⎤

⎢⎢⎣

⎡

Δ+

Δ+

Δt

zu

y

u

xu zyx

1<Δ⎥⎥⎦

⎤

⎢⎢⎣

⎡

Δ+

Δ+

Δt

zv

y

v

xv zyx

]

)(1

)(1

)(1

1[21

222 zyx

tf

Δ+

Δ+

Δ

<Δν

]

)(1

)(1

)(1

1[21

222 zyx

ts

Δ+

Δ+

Δ

<Δν


• DEFINITION OF EULERIAN VARIABLES

scalar variablesx-velocity componenty-velocity componentz-velocity component

xy

z


• OVERALL COMPUTATIONAL STRATEGY PER CYCLE

compute convection and diffusion terms for g+s momentum equations

estimate g+s velocity distributions using old porosity + pressure fields

solve porosity and pressure Poisson equations using ICCG

solve granular temperature equation

compute g+s velocity distributions with new porosity + pressure fields

RESULTS OF KTGF MODELbubble formation at a jet using 3D model

RESULTS OF KTGF MODEL: MONODISPERSE SYSTEMSbubble formation: 30 cm bed

W=0.30 mdp=2.5 mm

ρs=2526 kg/m3

Ub=1.20 m/sUj=20.0 m/sNp=60000

RESULTS OF KTGF MODELeffect of restitution coefficient on bed dynamics

RESULTS OF KTGF MODELeffect of restitution coefficient on bed dynamics

RESULTS OF KTGF MODEL: BIDISPERSE SYSTEMSsegregation rate in bidisperse systems

RESULTS OF KTGF MODEL: BIDISPERSE SYSTEMSsegregation rate in bidisperse systems

RESULTS OF KTGF MODELeffect of lateral segregation on riser reactor performance

• RISER FLOW

+ particle diameter (FCC) 40 μm+ riser diameter 0.3 m+ superficial gas velocity 6.3 m/s+ solids mass flux 390 kg/(m2.s)

• FEATURES

+ two-fluid model incorporating kinetic theory of granular flow+ turbulence model: Prandtl mixing length model+ axi-symmetrical flow


• RISER FLOW

RADIAL PROFILE OF SOLIDS VOLUME FRACTION εs

0.00

0.04

0.08

0.12

0.16

0.20

0.0 0.5 1.0(r/R) [-]

[εs]


• RISER FLOW

RADIAL PROFILE OF AXIAL SOLIDS VELOCITY Vz

-3.0

1.0

5.0

9.0

13.0

17.0

0.0 0.5 1.0(r/R) [-]

vz [m.s-1]


• REACTION SCHEME:

• KINETICS:

13

12

11

2.0

15

25

−

−

−

=

=

=

sk

sk

sk

][[*]

[*][*][*]3

21

−⎯→⎯

+⎯→⎯+⎯→⎯+k

kk CBA

r k xr k x k xr k xr k

A s f A

B s f A B

C s f B

s s

= −

= + −

= +

= −

ε ρ

ε ρ

ε ρ

ε ρ

[*][*] ( )[*][*]*

1

1 2

2

3


• AXIAL PROFILE OF FLOW-AVERAGED FRACTIONS IN RISER

0.0

0.5

1.0

0.0 0.5 1.0(z/L) [-]

Xi [-]

A B

[*]C

NON-IDEAL

0.0

0.5

1.0

0.0 0.5 1.0(z/L) [-]

Xi [-]

A B

[*]C

IDEAL

CONCLUSIONS

• BUBBLING BED

+ significant effect of e on bubble dynamics

+ dissipation level to low (friction is not included in KTGF)

• CFB RISER

+ radial segregation in dense riser flow can be predicted

+ significant effect of radial segregation on riser reactor performance

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TWENTE UNIVERSITY THE NETHERLANDS - TFE1 | Home · • APPLICATIONS OF FLUIDIZED SYSTEMS + heat exchange and drying + coating and granulation + gas purification via adsorption + chemical