Advanced Reactive Multiphase Flowscfdoil.com.br/2008/pdf/keynotes/rodney_fox.pdf · • Kinetic...

Advanced Reactive Multiphase Flows

Dr. Rodney O. Fox

Herbert L. Stiles Professor

CFDOIL 2008, August 18-19 1

Chemical & Biological Engineering

Iowa State University

Ames, Iowa 50014, USA

• Multiphase flows can be characterized by

Polydisperse multiphase flows

• Particle position xp

• Particle velocity Up

• Particle internal coordinates:

• Particle size L


• Particle enthalpy hp

• Fluid velocity Uc

• Fluid pressure p

• Fluid composition cc

• Fluid enthalpy hc

Describe in terms of

a number density function

(NDF)

n(x,U,L) = #/volume

Fluidized-bed polymerization reactors

1-10m

Electrostatic/thermal agglomeration

Particle swarm Catalyst fragments

Challenges:Particulate processes, growth, aggregation and breakage

Mass & heat transfer to/from polymer particles

Catalyzed polymerization chemistry

All phenomena are highly coupled, have a strong influence on the fluid dynamics


Mixing, segregation

FB Reactor 100µµµµm-1cm

Particle swarm

Single particleInterface mass &heat transfer

10-100µµµµm

Catalyst fragments surrounded by polymers

Sub-particle1-100nm

Molecular phenomena, kinetics

Active Site

1-100Å

Agglomeration dominant

Breakage dominant

No agglomeration

Effect of particle size on fluidized-bed dynamics


average size

increases

FB

defluidization

dominantaverage

size

decreases

FB expands

agglomeration and breakage

Polydisperse spray with evaporation

Lagrangian


Eulerian

Polydisperse spray with evaporation

Lagrangian


Eulerian

Spray evaporation and combustion

Gas-phase fuel Burnt gases


Fully coupled Eulerian multi-fluid model for spray combustion

• Kinetic equations are used to model multiphase flows

made up of two discrete phases

� Continuous primary phase � fluid (e.g. liquid or gas)

� Disperse secondary phase � solid or liquid “particles”

(e.g. solid particles, liquid droplets)

Models for multiphase flows


• We seek to develop EulerianEulerian models that can describe the

evolution of the multiphase flow

� Particle size distribution (number density function)

� Nucleation, growth, evaporation, aggregation, etc.

� Finite Stokes number effects

Overview of multiphase CFD model

Multiphase CFD model

Momentum

equations

Mass & energy

equations

Chemical species

equations

Population

balance equation


Turbulence

theory

Mass & heat

transfer models

Detailed

chemistry

Aggregation,

breakage and growth

ISAT DQMOM

In-Situ Adaptive Tabulation: handles detailed chemistry

Direct Quadrature Method of Moments: handles PBE

1. Derive a transport equation for the NDF (real and phase

space � usually high dimensional!)

2. Develop models for physical processes in NDF transport

equation:

• Drag, coalescence, breakage, chemical reactions, etc.

• Models must be consistent with Lagrangian description

Modeling approach


• Models must be consistent with Lagrangian description

3. Choose a solution method for solving transport equation:

• Classical moment methods (with moment closures)

• Quadrature-based moment methods

• Direct method: discretize phase space

Classical example: Boltzmann equation

= number density of “particles” with velocity

Define moments:

accumulation + transport = collisions


Define moments:

Elastic collisions conserve kinetic energy

Boltzmann equation: Hydrodynamic limitCollisions dominate � Maxwellian distribution

where

All higher-order moments depend only on


All higher-order moments depend only on

Euler equations

Limiting case for Boltzmannflow solvers

(Knudsen number=0)

Kinetic representation of gas-particle flow

collisionsfluid drag

= fluid velocity (known from separate code)


= fluid velocity (known from separate code)

Dilute flow: Collisions are negligible

Dense flow: Collisions are dominant

CFD models for dilute gas-particle flow

EulerianLagrangian


Follow many, many particles Close and solve for

a few moments

(usually k = 0 and 1)Both methods should

predict the same

velocity statistics!

Dependence on particle Stokes number

Stokes drag

• : particles follow fluid (small d, viscous fluid)


• : particles ignore fluid (large d, heavy particles)

• : particle trajectories can cross (PTC)

is bimodal at PTC

Canonical example: Impinging flow

St > Stc Stc = 1/8π


Particles can cross the centerline

while the fluid cannot

Particle velocity at many points is

not unique!

• Number density function (NDF) has too many degrees of

freedom (internal variables) to discretize directly:

� Lagrangian approach � estimate NDF from statistical sample � estimates for physical quantities are “noisy”

� Eulerian approach � write transport equations for

moments of NDF � nonlinear terms are not closed

Quadrature-based Eulerian models


moments of NDF � nonlinear terms are not closed

• Quadrature methods replace moments by an equivalent

set of weights and abscissas:

• Nonlinear terms are closed with nα and vα

Order k moment:

Gaussian quadrature in 1D

• Product-Difference algorithm (McGraw 1997)

weights abscissas


Inverse problem solved on the fly in flow solver

weights abscissas

“Standard” Eulerian model for gas-solid flow

1-node quadratureMoment equations


Pressure-less gas dynamics “sticky” particle limit

Trajectory crossing in 1D impinging flow

Number density (ρ) Mean velocity (v)

“Standard” 2-moment Eulerian model


Even low-order velocity statistics are incorrect!

Quadrature-based Eulerian model

2-node quadratureMoment equations


unclosed moment

Trajectory crossing in 1D impinging flow

Number density (ρ) Mean velocity (v)

2-node quadrature (4 moments)


All velocity statistics are predicted correctly!

3D “frozen” isotropic turbulence

Lagrangian 2-node quadrature


Quadrature-based Eulerian model attains a steady state

Non-equilibrium gas-particle flows

1. Crossing particle jets (no collisions)

2. Particles reflecting off of a solid wall

PTC


2. Particles reflecting off of a solid wall

3. Particle-laden Taylor-Green flow with finite Stokes

PTC

Crossing jets in 2D

Inflow boundary conditions fix weights and abscissas


Crossing jets with collisions

1 % solids 10 % solids


Collision rate depends on volume fraction of solids

Particle-laden Taylor-Green flow

St<1/8π: Particles remain

in original

Initial conditions:


vortices

St>1/8π: Particles are

ejected from

vortices

Particle-laden Taylor-Green flow

St=0.3 > 1/8πNo collisions Strong collisions


Particle size distributions

• In many industrial fluidized-bed reactors, particle

mixing and segregation can play a very important role

– Segregation is used to remove product or separate different

solids

– Chemical reactions and mass/heat transfer depend on the

local particle size distribution (PSD)

• Detailed information about the PSD at different


• Detailed information about the PSD at different

operating conditions is crucial for design and scale up

• Multi-fluid model to describe particle mixing and

segregation phenomena

– Include size change due to chemical reactions

– Predict particle elutriation due to size/density differences

NDF for size and velocity

particle volume (mass)

particle velocity time

spatial location


Average number of particles with volume v and

velocity U located at spatial location x at time t

particle volume (mass) spatial location

Population balance equation for particle size

• Processes leading to continuous/discrete changes:

– Nucleation� produces new particles, coupled to local

solubility, and properties of continuous phase

– Growth � mass transfer to surface of existing particles,

coupled to local properties of continuous phase

– Restructuring� particle surface/volume changes due to

shear and/or physio-chemical processes


shear and/or physio-chemical processes

– Aggregation/Agglomeration� particle-particle

interactions, coupled to local shear rate, fluid/particle

properties

– Breakage� system dependent, but usually coupled to

local shear rate, fluid/particle properties

Many of these are coupled to local fluid flow

Multi-fluid model for dense gas-particle flow

• Mass balances

• Momentum balances

1

( ) ( )�

g g g g g gMt

αα

ε ρ ε ρ=

∂+∇⋅ = −

∂ ∑u

( ) ( )s s s s s gMt

α α α α α αε ρ ε ρ∂

+∇⋅ =∂

u g: Gas phaseg: Gas phase

ssαα: Solids phase : Solids phase αα=1,=1,…… NN

Mass transfer from gas

to each solid phase


1

( ) ( )�

g g g g g g g g g g gt

αα

ε ρ ε ρ ε ρ=

∂+∇⋅ = ∇⋅ + +

∂ ∑u u u f gσσσσ

1,

( ) ( )�

s s s s s s s s g s st

α α α α α α α α α βα α αβ β α

ε ρ ε ρ ε ρ= ≠

∂+∇⋅ = ∇ ⋅ + +

∂ ∑u u u - f f gσσσσ

Stress tensor Body forceMomentum

transfer between

phases

Kinetic theory of dense granular flow

• Two different methods to calculate

Plastic Regime Viscous Regime

Slow flow Rapid flow

Soil mechanics Kinetic theory of granular flow

sασ


• Granular temperature: indicator for random motion

23 1

2 3E Cα α α αΘ = Θ Θ =

Enduring contactsKinetic + collisions

Momentum transfer

• Momentum transfer between solid-solid phases

– The first term is derived from kinetic theory to account for collisions and sliding between particles

2 2

0

13 3

3(1 )( / 2 / 8) ( )*

2 ( )

f s s s s p p

s p s p

e C d d gF C P

d d

βαα β α β α ββα

α β

π π ε ρ ε ρ

π ρ ρ

+ + + −= +

+

u u

( )Fβα βα α β= − −f u u


collisions and sliding between particles

– The second term is a “hindrance effect”: force the two phases to behave like one phase when they are packed

– By changing Cf and C1, the segregation rate can be adjusted

*

25 * 10 *

0*

10 ( )

g g

g g g g

ifP

if

ε εε ε ε ε

>=

− ≤

dilute

dense

Size segregation in a fluidized bed

Average bed height Relative segregation rates

Small particle


Large particle

Gas void fraction

5 s 10 s 15s 30s 5 s 10s 15s 30s

Segregation: Dependence on gas flow rate

Mass fraction of large particles

1.10m / sg =u


1.25m / sg=u

Quadrature methods for continuous PSD

2 nodes:

1 2

1 1

2 2

: 1:1

670 , 0.0789

1245 , 0.5064

d m

d m

ω ω

µ ε

µ ε

=

= =

= =

1000 , 958 ,rms aved m d mµ µ= =

0.0 2.01.0

1 2 3

1 1

: : 1: 4 :1

460 , 8.5 3

958 , 0.3067

d m e

d m

ω ω ω

µ ε

µ ε

=

= = −

= =3 nodes:


2 2

3 3

958 , 0.3067

1456 , 0.2691

d m

d m

µ ε

µ ε

= =

= =

3 nodes:

4 nodes:

1 2 3 4

1 1

2 2

3 3

4 4

: : : 1: 9.9 :9.9 :1

287 , 5.70 4

745 , 0.09827

1171 , 0.3818

1629 , 0.1037

d m e

d m

d m

d m

ω ω ω ω

µ ε

µ ε

µ ε

µ ε

=

= = −

= =

= =

= =

0.0 2.01.0

0.0 2.01.0

Average particle size and standard deviation across the bed

Four locations:

bottom: y=0.5 cm

middle: y=2.5 cm, 4.5 cm


middle: y=2.5 cm, 4.5 cm

top: y=6.5 cm

Symbols: multi-fluid model

Lines: discrete particle

simulations

Mass transfer:

Heat transfer: (lumped model)

Mass and heat transfer models

62 ,

/

c b v

ps ps

M s sM w

ShD a

d d

c X M

κ

ρ

= =

=gT

sT

gCMC

diffusion diffusion ( )

gs s c v g M wM k a c c Mε= −

( )gs s f v g sH h a T Tε= −


Heat produced from polymerization:

convectionconvection

External boundary layerExternal boundary layer

pspspspsdddd

2222

Growing polymer particleGrowing polymer particle

λf: thermal conductivity

Db: monomer bulk diffusivity

( )

2

gs s f v g s

f

f

ps

H h a T T

�uh

d

ε

λ

= −

=

*

*

2

[ ]

[ ]3[ coth( ) 1],

2

rs s p M r

ps p

eA

H k c c H

d k c

D

εη

φ φη φ

φ

∆ =− ∆

−= =

Initiation

Propagation

Kinetic scheme for metallocene catalyst

*ikc c→

* * *

1( ) pk

n nP c M P ++ → 0

0

exp( / )

exp( / )

i s

d s

ii

d d

k k E RT

k k E RT

α

α

= −∆

= −∆


Decay* * 0( ) dk

n nP c P c→ +

c c : : potential catalyst active sitepotential catalyst active site

cc* * : : active catalyst siteactive catalyst site

cc00 : : dead sitedead site

PPnn** : : ““live” polymer chains of lengthlive” polymer chains of length n n

M: M: monomermonomer

PPnn : : “dead” polymer chains of length“dead” polymer chains of length nn

0

013.79( ) exp( / )

d s

g m

p s

d d

pp

Pk k E RT

α

α= −∆

Hot spots in the reactorNumber of particles Particle temperature rise

gε

sT∆


Conclusions

– CFD modeling of multiphase flows is complicated!

– A “mesoscopic” modeling approach based on a kinetic description is a useful starting point

– “Standard” Eulerian models do not work for large Stokes number or weakly collisional flows

– Quadrature-based Eulerian methods work for arbitrary St and Kn


St and Kn

– Extension to particle size distribution requires careful accounting for size-dependent flow dynamics

– Including chemical reactions is relatively straightforward but requires careful modeling of heat and mass transfer between phases

Acknowledgments

• Fluidized-bed polymerization reactor: – Michael Muhle, ExxonMobil

– Rong Fan, Ram Rokkam, ISU PhD students

• Spray combustion– Marc Massot, Ecole Centrale Paris, France

– Frederique Laurent, Ecole Centrale Paris, France

– Julien Reveillon, CORIA, France

– Venkat Raman, Aerospace Engineering, UT Austin

• Dilute gas-particle flows:– Olivier Desjardins, ME, U. of Colorado


– Olivier Desjardins, ME, U. of Colorado

– Philippe Villedieu, ONERA, France

– Olivier Simonin, IMFT, France

– Alberto Passalacqua, ISU postdoc

• Quadrature methods for particle size distribution:– Daniele Marchisio, Politecnico di Torino, Italy

– Denis Vigil, CBE, ISU

Funding: NSF, DOE, BASF, BP Chemical, Dow Chemical, Univation

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