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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
CFDOIL 2008, August 18-19 2
• 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
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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
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average size
increases
FB
defluidization
dominantaverage
size
decreases
FB expands
agglomeration and breakage
Polydisperse spray with evaporation
Lagrangian
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Eulerian
Polydisperse spray with evaporation
Lagrangian
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Eulerian
Spray evaporation and combustion
Gas-phase fuel Burnt gases
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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
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• 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
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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
CFDOIL 2008, August 18-19 10
• 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
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Define moments:
Elastic collisions conserve kinetic energy
Boltzmann equation: Hydrodynamic limitCollisions dominate � Maxwellian distribution
where
All higher-order moments depend only on
CFDOIL 2008, August 18-19 12
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)
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= fluid velocity (known from separate code)
Dilute flow: Collisions are negligible
Dense flow: Collisions are dominant
CFD models for dilute gas-particle flow
EulerianLagrangian
CFDOIL 2008, August 18-19 14
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)
CFDOIL 2008, August 18-19 15
• : 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
CFDOIL 2008, August 18-19 16
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
CFDOIL 2008, August 18-19 17
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
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Inverse problem solved on the fly in flow solver
weights abscissas
“Standard” Eulerian model for gas-solid flow
1-node quadratureMoment equations
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Pressure-less gas dynamics “sticky” particle limit
Trajectory crossing in 1D impinging flow
Number density (ρ) Mean velocity (v)
“Standard” 2-moment Eulerian model
CFDOIL 2008, August 18-19 20
Even low-order velocity statistics are incorrect!
Quadrature-based Eulerian model
2-node quadratureMoment equations
CFDOIL 2008, August 18-19 21
unclosed moment
Trajectory crossing in 1D impinging flow
Number density (ρ) Mean velocity (v)
2-node quadrature (4 moments)
CFDOIL 2008, August 18-19 22
All velocity statistics are predicted correctly!
3D “frozen” isotropic turbulence
Lagrangian 2-node quadrature
CFDOIL 2008, August 18-19 23
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
CFDOIL 2008, August 18-19 24
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
CFDOIL 2008, August 18-19 25
Crossing jets with collisions
1 % solids 10 % solids
CFDOIL 2008, August 18-19 26
Collision rate depends on volume fraction of solids
Particle-laden Taylor-Green flow
St<1/8π: Particles remain
in original
Initial conditions:
CFDOIL 2008, August 18-19 27
vortices
St>1/8π: Particles are
ejected from
vortices
Particle-laden Taylor-Green flow
St=0.3 > 1/8πNo collisions Strong collisions
CFDOIL 2008, August 18-19 28
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
CFDOIL 2008, August 18-19 29
• 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
CFDOIL 2008, August 18-19 30
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
CFDOIL 2008, August 18-19 31
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
CFDOIL 2008, August 18-19 32
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
CFDOIL 2008, August 18-19 33
• 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
CFDOIL 2008, August 18-19 34
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
CFDOIL 2008, August 18-19 35
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
CFDOIL 2008, August 18-19 36
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:
CFDOIL 2008, August 18-19 37
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
CFDOIL 2008, August 18-19 38
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= −
CFDOIL 2008, August 18-19 39
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
= −∆
= −∆
CFDOIL 2008, August 18-19 40
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∆
CFDOIL 2008, August 18-19 41
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
CFDOIL 2008, August 18-19 42
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
CFDOIL 2008, August 18-19 43
– 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