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

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

Fluidized-bed polymerization reactors

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

defluidization

dominantaverage

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

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)

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

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

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

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

– 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

( ) ( )�

g g g g g gMt

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

( ) ( )�

g g g g g g g g g g gt

ε ρ ε ρ ε ρ=

∂+∇⋅ = ∇⋅ + +

∂ ∑u u u f gσσσσ

( ) ( )�

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

• Granular temperature: indicator for random motion

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

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

f s s s s p p

s p s p

e C d d gF C P

βαα β α β α ββα

π π ε ρ ε ρ

π ρ ρ

+ + + −= +

( )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 *

10 ( )

g g g g

ε εε ε ε ε

− ≤

dilute

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:

670 , 0.0789

1245 , 0.5064

1000 , 958 ,rms aved m d mµ µ= =

0.0 2.01.0

: : 1: 4 :1

460 , 8.5 3

958 , 0.3067

ω ω ω

= = −

= =3 nodes:

958 , 0.3067

1456 , 0.2691

3 nodes:

4 nodes:

1 2 3 4

: : : 1: 9.9 :9.9 :1

287 , 5.70 4

745 , 0.09827

1171 , 0.3818

1629 , 0.1037

ω ω ω ω

= = −

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

top: y=6.5 cm

Symbols: multi-fluid model

Lines: discrete particle

simulations

Mass transfer:

Heat transfer: (lumped model)

Mass and heat transfer models

M s sM w

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

Growing polymer particleGrowing polymer particle

λf: thermal conductivity

Db: monomer bulk diffusivity

gs s f v g s

H h a T T

[ ]3[ coth( ) 1],

rs s p M r

H k c c H

φ φη φ

∆ =− ∆

−= =

Initiation

Propagation

Kinetic scheme for metallocene catalyst

*ikc c→

1( ) pk

n nP c M P ++ → 0

exp( / )

k k E RT

= −∆

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

013.79( ) exp( / )

Pk k E RT

α= −∆

Hot spots in the reactorNumber of particles Particle temperature rise

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

advanced reactive multiphase flowscfdoil.com.br/2008/pdf/keynotes/rodney_fox.pdf · • kinetic...

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