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Reflections on mathematical models and simulation of gas‐particle flows
Sankaran Sundaresan
Princeton University
Circulating Fluidized Beds – 10
May 2, 2011
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Outline
• Examples of flow characteristics• Modeling issues• Modeling approaches• Outlook
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With so many fine books and software products around,
what is there to say?
Why model? What to model?
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4
Side-by-SideFCC unit
Riser-Reactor
StrippingSteam
Reaction Products
Cyclone
Feed
RegeneratedCatalyst Standpipe
Air grid
Air
Regenerator
Why model? What to model?
What phenomena would we like to understand?
To baghouse
Lift-line air
fluidized bed
standpipe riser
A: aeration ports
AA
A
A
A
A
A
A
Slide valve
A
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Loop stability
0.48
0.49
0.50
0.51
0.52
0.53
0.54
0.55
0 20 40 60
Solid
s vo
lum
e fr
actio
n
Time (seconds)
Low aeration rate(Stable flow)
Srivastava et al. Powder Tech., 100, 173 (1998)
To baghouse
Lift-line air
fluidized bed
standpipe riser
A: aeration ports
AA
A
A
A
A
A
A
Slide valve
A
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Loop stabilityHigher aeration rate
(Unstable flow)
Srivastava et al. Powder Tech., 100, 173 (1998)
What is the true mechanism for the instability?How does the stability transition change with scale‐up?
To baghouse
Lift-line air
fluidized bed
standpipe riser
A: aeration ports
AA
A
A
A
A
A
A
Slide valve
A
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Flow characteristics in the riser
How does the flow pattern change with scale‐up? How fast is the radial dispersion? How effective is the contacting between gas and particles?
To baghouse
Lift-line air
fluidized bed
standpipe riser
A: aeration ports
AA
A
A
A
A
A
A
Slide valve
A
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Competing options
Choice depends on:• backmixing• contacting efficiency• attrition, erosion, etc.
How do these issues change with scale‐up?
How well can we control them?
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Jet streaming
How well can we predict them? How does such flow behavior
change with scale‐up?
Knowlton, et al., Powder Tech., 150, 72 (2005)
Attributed to gas compression in deep beds operating at low pressures
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Cyclone performance
Strongly swirling flow aids separation
Gas turbulence adversely affects separation
Competition between swirl and turbulence?
How do particle loading, design choices and throughputs affect this competition?
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Mechanism and effect of liquid injection on flow
Flow characteristics are affected by:• Agglomeration• Gas evolution
Adapted from Bruhns & Werther, 2005.
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Mechanism and effect of liquid injection on flow
Flow characteristics are affected by:• Agglomeration• Gas evolution
Adapted from Bruhns & Werther, 2005.
How well do we understand the local and global flows?
How will these flow structures change upon scale‐up?
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Why is it difficult to model and simulate?
Widely varying particle loading levels. As a result, different regimes of flow
Need to quantify the physical processes reasonably well in all these regimes
Flow is invariably unsteady with a wide range of length and time scales. Cannot resolve all of them.
Particle size distribution
Changing particle characteristics
Wet systems: agglomeration and breakup
To baghouse
Lift-line air
fluidized bed
standpipe riser
A: aeration ports
AA
A
A
A
A
A
A
Slide valve
A
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Why model? What models?
• Understand physical processes
• Develop simpler models
• Explore design alternatives
• Scale up and process retrofits
To baghouse
Lift-line air
fluidized bed
Standpipe Ris
erA:
Aeration ports
AA
A
A
A
A
A
A
Slide valve
A
Simulations:
• at the level of a few thousand particles
• at the device scale
• Euler (fluid) ‐ Euler (particles) models
• Euler (fluid) – Lagrange (particles)
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( ) ( )
f f f f f f f f f f ftρ φ ρ φ φ ρ φ∂
+∇⋅ = − ∇⋅ − +∂
u u u f gσ
( ) ( ) 0ts s
s s s
ρ φρ φ
∂+∇ ⋅ =
∂u
( ) ( ) 0tf f
f f f
ρ φρ φ
∂+∇ ⋅ =
∂uFluid
particle Phase stress
interphaseinteraction
gravity
effectivebuoyancy
( ) ( )
s s s s s s s s s f s stρ φ ρ φ φ ρ φ∂
+∇⋅ = −∇⋅ − ∇ ⋅ + +∂
u u u f gσ σ
1s fφ φ+ =
Solids
Solids
Fluid
Two‐fluid model equations
inertia
Gas‐fluidized beds, risers and standpipes: fluid phase stress ~ pressure only
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( ) ( )
f f f f f f f f f f fPtρ φ ρ φ φ ρ φ∂
+∇⋅ = − ∇ − +∂
u u u f g
( ) ( ) 0ts s
s s s
ρ φρ φ
∂+∇ ⋅ =
∂u
( ) ( ) 0tf f
f f f
ρ φρ φ
∂+∇ ⋅ =
∂uFluid
interphaseinteraction
gravity
effectivebuoyancy
( ) ( )
s s s s s s s s s f s sPtρ φ ρ φ φ ρ φ∂
+∇ ⋅ = −∇⋅ − ∇ + +∂
u u u f gσ
Solids
Solids
Fluid
Two‐fluid model equations
inertia
Gas‐fluidized beds, risers and standpipes: interphase interaction ~ drag force only
1s fφ φ+ =
particle Phase stress
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( ) ( )
f f f f f f f f f d f fPtρ φ ρ φ φ ρ φ∂
+∇⋅ = − ∇ − +∂
u u u f g
( ) ( ) 0ts s
s s s
ρ φρ φ
∂+∇ ⋅ =
∂u
( ) ( ) 0tf f
f f f
ρ φρ φ
∂+∇ ⋅ =
∂uFluid
interphaseinteraction
gravity
effectivebuoyancy
( ) ( )
s s s s s s s s s f d s sPtρ φ ρ φ φ ρ φ∂
+∇⋅ = −∇⋅ − ∇ + +∂
u u u f gσ
1s fφ φ+ =
Solids
Solids
Fluid
Two‐fluid model equations
inertia
Good “text‐book” drag force models are available in the literature for nearly homogeneous systems: e.g., Wen and Yu (1966)
Wen & Yu, Chem. Eng. Prog. Symp. Ser., 62, 100 (1966)
particle Phase stress
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( ) ( )
f f f f f f f f f d f fPtρ φ ρ φ φ ρ φ∂
+∇⋅ = − ∇ − +∂
u u u f g
( ) ( ) 0ts s
s s s
ρ φρ φ
∂+∇ ⋅ =
∂u
( ) ( ) 0tf f
f f f
ρ φρ φ
∂+∇ ⋅ =
∂uFluid
interphaseinteraction
gravity
effectivebuoyancy
( ) ( )
s s s s s s s s s f d s sPtρ φ ρ φ φ ρ φ∂
+∇⋅ = −∇⋅ − ∇ + +∂
u u u f gσ
Solids
Solids
Fluid
Two‐fluid model equations
inertia
• force chains at high particle loading
• binary collisions between particles
• particle streaming
• Important in hoppers, bins, standpipes
• Less so in fluidized beds and risers
• Even less in freeboard region & cyclones
1s fφ φ+ =
particle Phase stress
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2D Domain size : 64 cm x 64 cm
3D Domain size : 8 cm x 8 cm x 8 cm
Simulations performed with MFIX
256 x 256
64x64x64
Average particle volume fraction: 0.05
75 μm particles in air
Instability driven by: inertia, dependence of drag force on particle loading level, inelastic collisions
Fine structure
Stabilized by: weak particle phase stress
Weak stabilization – small length scale
Resolve or not resolve?
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Simple example: turbulent fluidized bed
Vg> Vt
Vg= 0; Vp = Vt
Sedimentation of a single particle
Vg= Vt; Vp = 0
Levitation of a single particle
Vg> Vt; Vp = ?
Vertical conveying of a single particle
Presence of other particles typically hinder!
Bed emptying time for turbulent fluidized beds are much longer than predicted by this model
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Simple example: turbulent fluidized bed
Bed expansion decreases as one improves resolution*
When one does not resolve all the flow structures, one must “correct” the drag force model
O’Brien & Syamlal, CFB‐4 (1993)Li & Kwauk (1994) – EMMS modelMcKeen & Pugsley (2003) – tune cluster size
*Parmentier et al., AICHE J. (2011); Igci et al., AICHE J. (2010)
Extent of correction depends on chosen resolution*,#
( )homod f sf β= −u u
Standard form
Modified form
( ) ( ) ( )( )homo
1d f sf c hβ φ= − − Δu u
Function of resolution
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Newly emerging drag force models
*Parmentier et al., AICHE J. (2011); Igci et al., AICHE J. (2010, 2011)
0 as 0
1 as c→ Δ→⎧⎨→ Δ→∞⎩
0 0.1 0.2 0.3 0.4 0.5 0.60
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Particle volume fraction
h 2D
( )h φ• Parmentier et al. (2011)
dynamically adjust c.
• Igci et al. add wall correction
Further refinements
( ) ( ) ( )( )homo
1d f sf c hβ φ= − − Δu u
• We typically use different grid resolutions when simulating pilot scale and commercial scale units
• The effective drag law is now different for the two cases!
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Newly emerging drag force models
*Li & Kwauk (1994)
0 as 0
1 as c→ Δ→⎧⎨→ Δ→∞⎩
0 0.1 0.2 0.3 0.4 0.5 0.60
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Particle volume fraction
h 2D
( )h φ
EMMS model*
( ) ( )( )homo
1d f s EMMSf hβ φ= − −u u
( ) ( ) ( )( )homo
1d f sf c hβ φ= − − Δu u
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Particle phase stress
particle phase stress
interphaseinteraction
gravity
effectivebuoyancy
( ) ( )
s s s s s s s s s f d s sPtρ φ ρ φ φ ρ φ∂
+∇⋅ = −∇⋅ − ∇ + +∂
u u u f gσSolids
inertia
Campbell, JFM, 465, 261 (2002); Tardos et al., Powder Technol., 131, 23 (2003): Lois & Carlson, Euro. Phys. Lett., 80, 58001 (2007).
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Discrete Element Method
• Newton’s equations
• Spring – dashpot contact model
• Open domain (LAMMPS*) + commercial
• Can include cohesion, liquid bridge, non‐spherical shape, size distribution
Cundall & Strack, Geotechnique, 29, 47 (1979); Zhu et al., CES, 63, 5728 (2008).
*LAMMPS code. http://lammps.sandia.govPlimpton, J. Comp. Phys., 117, 1 (1995)
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Quasi‐static
inertial
intermediate
Scaled Pressure
Quasi‐static intermediate
inertial
DEM simulations of simple shear flow
Scaled shear rate
Scaled Shear Stress
Chialvo et al. (2011).
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DEM simulations of simple shear flow
Quasi‐static
inertial
intermediate
inertial
Quasi‐static
intermediate
Rescaled shear rate
Rescaled Shear StressRescaled Pressure
Chialvo et al. (2011).
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Standpipe flow of FCC particles
0.20
0.30
0.40
0.50
0.60
0 1 2 3 4 5 6
External aeration rate (m3/hr)
Ave
rage
sol
ids
volu
me
frac
tion
Friction
0% 12% 24% 36% 48% 60%
Increasing external aeration
s
unstable
s s uSrivastava et al. Powder Tech.,
100, 173 (1998)
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Do such small levels of stress matter in fluidized beds and CFBs?
They influence the size of the small clusters and streamers
When not resolving all the flow structures, one must “correct”:
• the drag force model +
• effective stresses due to fluctuating
meso‐scale structures
DEM simulations of simple shear flowScaled Pressure Scaled Shear Stress
Chialvo et al. (2011).
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Chen et al., Application of Coarse Grained Drag Law in Computational Fluid Dynamics Simulations of Fluidized Beds, AIChE Annual Meeting (2008)
Time AveragedInstantaneous
Fully cylindrical cold flow model of Syncrude coker – 1/19th scale.Song et al., Powder Tech, 147, 126 (2004).
How to go from scaled down unit to full commercial scale?
• Traditional – keep the same grid size; not practical
• Remedy: Use larger grids with appropriately scaled constitutive laws
Coker model simulation
325,000 grids; 32 processors
1 computational day per second of real time
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Handling particle size distributionEuler‐Euler approachMultiple particle phasesMethod of moments
Generalize kinetic theoryGeneralize drag law
• Inherent size distribution
• Changing particle properties
• Cast the particle phase balance equations in a Lagrangian framework• Follow the motion of a few million test particles, referred to as parcels,
while treating the remaining (ghost) particles through mean field• Multi‐Phase – Particle In Cell method*
• Much easier to handle particle size distribution and changing particle properties; faster computations
*Originally derived directly from a probabilistic approach: D.M. Snider, J. Comp. Phys., 170, 523 (2001)
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{
Effective buoyancyparticlephase stress
weightall other fluid-particle interactions
p pp p s p f
s
pp p d
s
d vv v
dt
vv
ρφ
ρφ
= − ∇⋅ − ∇ ⋅
+ +
1424314243
123
u
g f
σ σ
Discrete Particle Model Approach
Snider, J. Comp. Phys., 170, 523 (2001); O’Rourke & Snider, CES, 65, 6014 (2010); O’Rourke et al., CES, 64, 1784 (2009).
• Particle phase stress term captures the effects of all collisions• No need to track collisions between parcels• Different parcels can have different underlying particle size, property, etc.• Parcels can be allowed to interact in the mean: mimic kinetic theory, transfer
liquids, etc.
What are the right drag law and effective stresses to use when all flow structures are not resolved?
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CPFD® Simulation of a Settler
Courtesy: Dale Snider & Ken Williams, CPFD Software, LLC.
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{
Effective buoyancyparticlephase stress
weightall other fluid-particle interactions
p p pp p s p f
s
pp p d
s
d vv v
dt
vv
ρφ
ρφ
= − ∇⋅ − ∇ ⋅
+ +
1424314243
123
u
g f
σ σ
Discrete Particle Model Approach
#Patankar & Joseph, IJMF, 27, 1659 (2001); #Benyahia & Galvin, IECR, 49, 10588 (2010); *Chu et al., CES, 66, 834 (2011)
• MP‐PIC: Particle phase stress term captures the effects of all collisions
• No need to track collisions between parcels
• What if we track collisions between parcels?#
• Pretty well in quasi‐static regime• Much less accurately in the inertial
regime
If parcel size = particle size:
Track all collisions
CFD – DEM*
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Discrete Particle Model with Collision Tracking
Cheng et al., PRL, 99, 188001 (2007); Radl et al. (2011)
0,pUr
Rsample
yjet
yx
particle reservoir(dp, φp)
Dtar
Pros and cons for tracking collisions between parcels?
Still evolving…
Compare scattering angles predicted with experimental data
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Verification vs. Validation
• Can the simulator correctly reproduce analytically obtainable results for some test problems?
• Can the independence of the predictions to simulator parameters be ascertained? Grid size, parcel size, etc.
• Model vs. model: CFD‐DEM, Euler‐Euler, MP‐PIC, etc.
• Comparison against experimental data
• Begin with a kinetic theory based Euler‐Euler model and filter to obtain a coarse‐grained Euler‐Euler model
• Show that both model yield the same solution
• Many early validation studies were based on 2D simulations
• Quantitative differences between 2D and 3D
• Often validation are with data in pilot scale units with some tuning
• How do we know that the same tuning will work for larger scale where we will have coarser grid resolution?
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Things to consider in validation
• Comparison against experimental data at a minimum of two different scales
• Flow regime maps – covering turbulent to fast fluidization
• Good data base for riser flows: • Vary gas flux while holding solids flux constant• Vary solids flux while holding gas flux constant
• Good database for standpipes are needed as well
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Outlook• More and more 3D simulations of the full CFB loop
• GPU based computing
• Standpipe flows: detailed experimental characterization and validation of simulations – more needed
• Both Euler‐Euler and Euler‐Lagrange will continue to develop; but, Euler‐Lagrange approach (Parcel based: with or without collision tracking) is very likely to gain more traction:• Ease of handling PSD and changing particle properties
• How to adapt the ideas on microscale modeling (such as kinetic theory) and coarse simulations developed for Euler‐Euler approach to the discrete methods – fertile topic
• More and more effort on wet systems
• More case studies on reacting flows
Carbon capture, chemical looping, methanol to olefin
+Old, but still very relevant problems