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S. Sundaresan and Yesim IgciPrinceton University

Festschrift for Professor Dimitri Gidaspow's 75th Birthday – II

Wednesday, November 11, 2009: 3:15 PM

Bayou E (Gaylord Opryland Hotel)

(3:57-4:18 PM)

Supported by US DOE, ExxonMobil

Filtered Two-Fluid Model for Gas-Particle Suspensions

Characteristics of flows in turbulent fluidized beds & fast fluidized beds

• Persistent density and velocity fluctuationsWide range of length and time scales

• Identifiable macroscopic inhomogeneous structuresRadial variation of particle volume fraction and fluxes

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Two-fluid model equations

( ) ( )

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

∂+∇ ⋅ =

∂u

Solids

Fluid

Solids

Fluid

inertia solid phase stress interphaseinteraction

gravityeffective

buoyancy

( ) ( )

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+ =

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Typical closures used widely:Newtonian fluid model for gas-phase stressKinetic theory of granular materials for solid phase stress Inter-phase force – due to gas-particle drag

Gidaspow's 1994 book + many papers!

75 μm particles in air

64 x 64 128 x 128 256 x 256

Average particle volume fraction: 0.05

16x16x16 32x32x32 64x64x64

with MFIX(Multiphase Flow with Interphase eXchanges)2D Domain size : 64 cm x 64 cm

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Modeling challenge for gas-particle flows

Original two-fluid model and constitutive relations

Significant advances in the past three decades

Filtered two-fluid modelModified constitutive relations

forhydrodynamic termsspecies and energy

dispersioninterphase heat and mass

transfer rateseven modified reaction rate

expressions!

Develop models that allow us to focus on large-scaleflow structures, without ignoring the possible consequenceof the smaller scale structures.

5Jinghai Li: EMMS approach

Filter “data” generated through highly resolved simulations of two-fluid models

Snapshot of particle volume

fraction fields obtained in highly

resolved simulations of gas-

particle flows. Squares illustrate

regions (i.e. filters) of over which

averaging over the cells is

performed.

6Igci, et al., AIChE J., 54, 1431 (2008)

Filtered variables

0

0

0

s s s f f f s

s s s s s s s s s

f f f f f f f f f

φ φ φ φ φ φ φ

φ φ φ

φ φ φ

′ ′ ′= + = + =

′ ′= + = =

′ ′= + = =

u u u u u u

u u u u u u

All filtered quantities are functions of

space and time

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Filtered particle phase momentum balance

( ) ( )stress due to

sub-filter scalefluctuations

effective sub-filter scale

s s s s s s s s s s s s s f

s f drag

pt

p

ρ φ ρ φ ρ φ φ

φ

⎛ ⎞⎜ ⎟

∂ ⎜ ⎟′ ′+∇ ⋅ = −∇ ⋅ + − ∇⎜ ⎟∂⎜ ⎟⎜ ⎟⎝ ⎠

′ ′− ∇ +

u u u u u

f

σ

fluid-particleinteraction force

s sρ φ+ g

( ) ( )

s s s s s s s s s f drag s sptρ φ ρ φ φ ρ φ∂ +∇ ⋅ = −∇⋅ − ∇ + +

∂u u u f gσ

Upon filtering,

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Sub-filter scale correlations

( )

fromkinetictheory

, ,2

s eff s effs s s s s

s f f sd efg fra

p

p

μ

β

ρ φ

φ

′ ′ + = +

′ ′− ∇ + = −

Iu u S

f u u

σ

sφAll the effective quantities will depend on the choice of filter size, and so on; such filter size dependence is present in LES of turbulent flows as well.

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1

t

gV FrΔ

Δ=

Scaled filter size

Filtered drag coefficient decreases as filter size increases for both 2-D and 3-D

2

1

t

gV FrΔ

Δ=

eff t

s

Vg

βρ

Example: 75 μm; 1500 kg/m3; domain size = 8 cm

1 2 1cmFr Δ

= ⇒ Δ =

2-D

Domain size = 16 x 16

64 x 64 grids

16 x 16 x 16

64 x 64 x 64

3-D

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Wall correction to the filtered closures

2-D Kinetic theory based simulations

Height: 500 cm

Width:

The inlet gas superficial velocity: 93 cm/s.

The inlet particle phase superficial velocity is 2.38 cm/s.

The inlet particle phase volume fraction is 0.07.

Partial-slip BC for particle phase and free-slip BC for gas phase

20 cm, 30 cm,  50 cm

Grid size: 0.25 cm    (0.514 dimensionless units)

75 μm particles in air

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The filtered drag coefficient is noticeably different in the core and the wall regions

The wall correction for the filtered drag coefficient is independent of channel width and filter size (not shown).

The same conclusion applies for filtered particle phase pressure and viscosity.

, with wall correction, scaled *

( , r)

( )filtered s

filtered

filtered s

β φβ

β φ=

filter size

*

,( ) ( , )filtered s filtered periodic s Frβ φ β φ Δ=

Dimensionless filter size:

filter size

filter size2

1

t

gV FrΔ

Δ=

Dimensionless distance:

distance

distance2

1

t

gV FrΔ

Δ=

Filter size: 2.056

Grid size: 0.514

Wall CoreDimensionless distance

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Verification of the filtered model

Compare

Original two-fluid model

Filtered two-fluid model

Solve a test problem using the original two-fluid model equations

Solve the same test problem using the filtered two-fluid model equations

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Verification of the filtered model

Compare predictions of kinetic theory and filtered models

Height: 500 cm

Width: 30 cm

The inlet gas superficial velocity: 93 cm/s.

The inlet particle phase superficial velocity is 2.38 cm/s.

The inlet particle phase volume fraction is 0.07.

Free-slip boundary conditions

75 μm particles in air

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Mass inventory of particles in the riser

Increasing resolution

• Kinetic theory model: Grid size: 0.25‐0.50 cm   

The minimum grid size required for :

• 2cm‐filtered model: Grid size: 1.00 cm 

Grid size = Filter sizeor smaller

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Solution of discretized form of the filtered two-fluid model

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Filtered model with closures corresponding to a filter size of 2 cm

Nearly grid‐size independent time‐averaged profiles when grid size is less than or equal to one‐half of the filter size.

Elevation = 3m

Solution of discretized form of the filtered two-fluid models

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Filtered model with closures corresponding to filter sizes of 2 and 4 cm yield nearly the same time‐averaged results as long as grid size is less 

than equal to 0.5 (filter size)

Elevation = 3m

Compare predictions of filtered models with different filter sizes

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Summary

• Performed highly resolved simulations of a kinetic theorybased two-fluid model for gas-particle flow in various testdomains

• Filtered the results to learn about constitutive relationsfor the filtered two-fluid model

• Verified the fidelity of the filtered model

• Validation remains to be completed – in progress.• Filtered species and energy balance equations (and rate

of chemical reactions) remain to be developed: futureresearch project.

Dear Dimitri, Happy 75th!

Slide Number 1Characteristics of flows in turbulent fluidized beds & fast fluidized bedsTwo-fluid model equationsSlide Number 4Slide Number 5Slide Number 6Filtered variablesFiltered particle phase momentum balanceSub-filter scale correlationsSlide Number 10Wall correction to the filtered closuresThe filtered drag coefficient is noticeably different in the core and the wall regionsVerification of the filtered modelVerification of the filtered modelMass inventory of particles in the riserSlide Number 16Slide Number 17Summary