Coarse-Grid Simulation of Riser Flows · 2020. 8. 13. · Original two-fluid model and constitutive...

1

Coarse-Grid Simulation of Riser Flows

Yesim Igci & S. SundaresanPrinceton University

NETL 2009 Workshop on Multiphase Flow ScienceMorgantown, WV

April 22, 2009

• DOE – UCR program• ExxonMobil Research & Engg. Co.

Art Andrews, Peter Loezos, Kapil Agrawal S. Pannala, M. Syamlal, T. O’Brien, S. Benyahia, R. BreaultNick Jones, Alvin Chen, Tim Healy, Rob Johnson, Rutton Patel

2

Roadmap for dilute gas-particle flows

Report on Workshop on Multiphase Flow Research, Morgantown, WV, June 6-7, 2006

Near-term (by 2009)• B3: Develop coarse-grained (filtered) two-fluid models

Mid-term (by 2012)•B3: Develop procedures to coarsen models for reactive flows

Connection to roadmap• Performed highly resolved simulations of a kinetic theory

based two-fluid model for gas-particle flow in various test domains

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

• Verified the fidelity of the filtered model. Validation remains to be done.

Characteristics of flows in turbulent fluidized beds & fast fluidized beds

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

• Identifiable macroscopic inhomogeneous structures

• (Kinetic theory based) two-fluid models are able to capture the above characteristics

But, they require high resolution and large computational times

• Do we really want to resolve all the fine structures?If we do not, then how should we modify the two-fluid model?

3

What we expect based on experimental data

Solution of discretized form of the kinetic theory based two-fluid model

What we get

30 m tall

76 cm channel width

75μm particles

2 cm grid

Gas vel = 6 m/sGas vel = 6 m/sSolids flux = 220 kg/m2.sSolids flux = 220 kg/m2.s

2-D simulations

Solids fraction field

Red – high

Blue -low

4

What we get in narrow channels with fine grids

Solution of discretized form of the kinetic theory based two-fluid model

What we get in wide channels and coarse grids

But in smaller channels and with finer grid resolution, kinetic theory based model does produce the right kind of spatial and temporal variations

5

Multiphase flow computations via two-fluid models

Density contour showing particle-

rich streamersIndividual

particles in gas

Reaction engineering need:Tools to probe macro-scale reactive flow features directly

All the constitutive models for the two-fluid models arefor nearly homogeneous

mixtures

Overview of the coarse-graining (filtering)

6

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-scale flow structures, without ignoring the possible consequence of the smaller scale structures.

7

8

First Step in development of Filtered Model

Density contour showing particle-

rich streamers

Individual particles in gas

Approach: Probe details of mesoscale structures and develop effective

coarse-grained equations

8Igci, et al., AIChE J., 54, 1431 (2008)

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.

9Igci, et al., AIChE J., 54, 1431 (2008)

The filtered drag coefficient, particle phase pressure and viscosity are now functions of particle volume fraction and filter size.

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 75 μμmm particles in airparticles in air

10

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

0 10 20 30 40 50

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Dimensionless horizontal location

Sca

led

filte

red

drag

coe

ffici

ent

Domain width: 102.80Domain width: 61.86Domain width: 41.12

The wall correction for the filtered drag coefficient is independent of channel width.

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

11

10 20 30 40 500.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Dimensionless distance

Sca

led

filter

ed d

rag

coef

ficie

nt

Filter size: 4.112Filter size: 2.056

The wall correction for the filtered drag coefficient is nearly independent of filter size.

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:

Dimensionless distance:

filter size

filter size2

1

t

gV FrΔ

Δ=

distance

distance2

1

t

gV FrΔ

Δ=

Wall Core

Grid size: 0.514

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

12

Verification of the filtered model

Compare macroscopic featuresCompare macroscopic features

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

1-D Linear Stability Analysis of filtered two-fluid model equations

13

Verification of the filtered model

Compare predictions of kinetic theory and filtered models

Height: 500 cm

Width: 30 cm




Free-slip boundary conditions

• Kinetic theory model: Grid size: 0.25 cm (0.514 dimensionless units)• Filtered model with closures corresponding to a filter size of 2 cm


14

Solution of discretized form of the filtered two-fluid model

15

0 100 200 300 400 5000

0.05

0.1

0.15

0.2

0.25

0.3

Elevation (cm)

Ave

rage

par

ticle

pha

se v

olum

e fra

ctio

n

Filtered model-- 1 cm gridFiltered model -- 0.5 cm grid

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

0 5 10 15 20 25 30

0.1

0.15

0.2

0.25

0.3

Distance (cm)P

artic

le p

hase

vol

ume

fract

ion

Filtered model -- 1cm gridFiltered mode -- 0.5 cm grid

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

Solution of discretized form of the filtered two-fluid model

16• Kinetic theory model: Grid size: 0.25 cm (0.514 dimensionless units)• Filtered model with closures corresponding to a filter size of 2 cm

Compare predictions of filtered models with different filter sizes

Height: 500 cm

Width: 50 cm




Free-slip boundary conditions

Verification of the filtered models

• Kinetic theory model: Grid size: 0.25 cm (0.514 dimensionless units)• Filtered model with closures corresponding to a filter sizes of 2 and 4 cm


17

Solution of discretized form of the filtered two-fluid models

18

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)

0 100 200 300 400 5000

0.05

0.1

0.15

0.2

0.25

Elevation (cm)

Ave

rage

par

ticle

pha

se v

olum

e fra

ctio

n

Filter size: 4cm; Grid size: 2cmFilter size: 4cm; Grid size: 1cmFilter size: 2cm; Grid size:1cm

0 10 20 30 40 50

0.1

0.15

0.2

0.25

0.3

Distance (cm)P

artic

le p

hase

vol

ume

fract

ion

Filter size: 4cm; Grid size: 2cmFilter size: 4cm; Grid size: 1cmFilter size: 2cm; Grid size: 1cm

Solution of discretized form of the filtered two-fluid models

19Filtered model with closures corresponding to filter sizes of 2 and 4 cm

10 20 30 40-140

-120

-100

-80

-60

-40

-20

0

20

40

Distance (cm)

Par

ticle

pha

se m

ass

flux

(g /

cm2

s)

Filter size: 4cm; Grid size: 2cmFilter size: 4cm; Grid size: 1cmFilter size: 2cm; Grid size: 1cm

Elevation = 3m

20

Computational times

• Assumption: Irrespective of the process device size, accurate integration of the kinetic theory based model will require grids as small as 0.25 cm (or smaller) for 75 μm particles.

• 2D simulations in the 30 cm x 500 cm channel:• 2 cm filtered model with 1 cm grid size ~ 40 times faster than

the kinetic theory model• Had it been done in 3D, the ratio would be ~ 100

• With a 2 cm filter size, the filter volume (3D) ~ 8 cm3 ; the grid volume (3D) ~ 1 cm3

• Large scale devices: grid volume (3D) ~ 100 - 1000 cm3 • The filtered model is expected to be ~ 105 – 106 times faster

than the original model!• Thus, the filtered model converts a virtually impossible

problem to a manageable problem!

21

Summary

• Performed highly resolved simulations of a kinetic theory based two-fluid model for gas-particle flow in various test domains

• Filtered the results to learn about constitutive relations for 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: future research project.

Extra slides

22

The filtered particle phase pressure is noticeably different in the core and the wall regions.

The wall correction for the filtered particle phase pressure is nearly independent of filter size.

, , with wall correction, , scaled *

( , r)

( )s filtered s

s filtered

s s

PP

P

φ

φ=

Filter size

*

, ,( ) ( , )s s s filtered periodic sP P Frφ φ Δ=

filter size

filter size2

1

t

gV FrΔ

Δ=

distance

distance2

1

t

gV FrΔ

Δ=

Wall Core10 20 30 40 50

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Sca

led

filter

ed p

artic

le p

hase

pre

ssur

e


23

The filtered particle phase viscosity is noticeably different in the core and the wall regions.

, , with wall correction, , scaled *

( , r)

( )s filtered s

s filtered

s s

μ φμ

μ φ=

Filter size

*

, ,( ) 1.15 ( , )s s s filtered periodic s Frμ φ μ φ Δ= ×

filter size

filter size2

1

t

gV FrΔ

Δ=

distance

distance2

1

t

gV FrΔ

Δ=

The wall correction for the filtered particle phase viscosity isnearly independent of filter size.

Wall Core10 20 30 40 50

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Sca

led

filter

ed p

artic

le p

hase

vis

cosi

ty


24

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+ =

25

Filtered continuity equations

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

( ) ( ) 0ts s

s s s

ρ φρ φ

∂+∇ ⋅ =

∂u ( ) ( ) 0

tf f

f f f

ρ φρ φ

∂+∇ ⋅ =

∂u

All filtered quantities are functions of

space and time

( ) ( )( ) ( )

0t

0t

s ss s s

f ff f f

ρ φρ φ

ρ φρ φ

∂+∇ ⋅ =

∂

∂+∇ ⋅ =

∂

u

u

26

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

⎛ ⎞⎜ ⎟

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

′ ′− ∇ +

14243u u u u u

f

σ

fluid-particleinteraction force

s sρ φ+1442443g

( ) ( )

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,

27

Filtered fluid phase momentum balance

( ) ( )stress due to

sub-filter scalefluctuations

effective sub-filter scaleflu

f f f f f f f f f f f f f

f f drag

pt

p

ρ φ ρ φ ρ φ φ

φ

⎛ ⎞⎜ ⎟

∂ ⎜ ⎟′ ′+∇ ⋅ = −∇ ⋅ − ∇⎜ ⎟∂⎜ ⎟⎜ ⎟⎝ ⎠

′ ′− ∇ −

14243u u u u u

f

id-particleinteraction force

f fρ φ+1442443g

( ) ( )

f f f f f f f f f drag f fptρ φ ρ φ φ ρ φ∂ +∇ ⋅ = − ∇ − +

∂u u u f g

Upon filtering,

28

Sub-filter scale correlations

stress due to stress due tosub-filter scale sub-filter scale

fluctuations fluctuations

effective sub-filter scalefluid-pa

s s s s f f f f

f f dragp

ρ φ ρ φ

φ

′ ′ ′ ′

′ ′− ∇ −

14243 14243u u u u

feffective sub-filter scale

rticle fluid-particleinteraction force interaction force

appearing in the fluid appearing in the particlephase equation phase equation

s f dragpφ

⎛ ⎞⎜⎜⎜

′ ′= − − ∇ +⎜⎜⎜⎜⎝ ⎠

1442443 1442443f

⎟⎟⎟⎟⎟⎟⎟

~ ,

, then

s f

s s f f

s s s s f f f f

If ρ φ ρ φ

ρ φ ρ φ

′ ′

>>

′ ′ ′ ′>>

u u

u u u u

So, only two terms to focus on

29

30

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.

2

1

t

gV FrΔ

Δ=

Scaled filter size

0 0.05 0.1 0.15 0.2 0.250

0.1

0.2

0.3

0.4

0.5

particle-phase volume fraction

dim

ensi

onle

ss fi

ltere

d dr

ag c

oeffi

cien

t

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

0.257

0.514

1.028

2.056

Domain size = 16 x 16

64 x 64 grids

16 x 16 x 16

0 0.05 0.1 0.15 0.2 0.250

0.1

0.2

0.3

0.4

0.5


dim

ensi

onle

ss fi

ltere

d dr

ag c

oeffi

cien

t

64 x 64 x 64

3-D

31

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8


dim

ensi

onle

ss fi

ltere

d dr

ag c

oeffi

cien

t

Filter Size = 4.112

resolution = 256x256resolution = 512x512resolution = 1024x1024

Dependence of the filtered drag coefficient on resolution (2-D)

Domain size

=132x132 (dim.less)

= 64 cm x 64 cm

Filter size

= 4 (dim.less)

= 2 cm

Corresponds to 1/8 cm

32

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.1

0.2

0.3

0.4

0.5

0.6


dim

ensi

onle

ss fi

ltere

d dr

ag c

oeffi

cien

t

Filter Size = 4.112

Domain size=32.896x32.896; resolution = 128x128Domain size=131.584x131.584; resolution=512x512

Filtered drag coefficient is independent of domain size (2-D)

Red: Domain size

= 32 x 32 (res: 128 x 128)

Green: Domain size = 128 x 128 (res: 512 x 512)

(dimensionless)

33

Filtered drag coefficient is independent of domain size (3-D)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.1

0.2

0.3

0.4

0.5


dim

ensi

onle

ss fi

ltere

d dr

ag c

oeffi

cien

tFilter Size = 2.056

Domain size=8.224³; resolution=32³Domain size=16.448³; resolution=64³

Blue: Domain size

= 8 x 8 x 8 (res: 32 x 32 x 32)

Green: Domain size = 16 x 16 x 16

(res: 64 x 64 x 64)

(dimensionless)

34

0 0.05 0.1 0.15 0.2 0.250

0.02

0.04

0.06

0.08

0.1

0.12

0.14


dim

ensi

onle

ss fi

ltere

d pa

rticl

e-ph

ase

pres

sure

0 0.05 0.1 0.15 0.2 0.250

0.02

0.04

0.06

0.08

0.1

0.12

0.14


dim

ensi

onle

ss fi

ltere

d pa

rticl

e-ph

ase

pres

sure

2

1

t

gV FrΔ

Δ=

,2

s eff

s t

pVρ

1 2 1cmFr Δ

= ⇒ Δ =

2-D

0.257

0.514

1.028

2.056

Domain size: 16 x 16

64 x 64 grids

16 x 16 x 16

64 x 64 x 64

3-D


Filtered particle phase pressure increases as filter size increases for both 2-D and 3-D

35

Filtered particle phase pressure increases as filter size increases for both 2-D and 3-D

• The effective pressure in 3-D is smaller than that in 2-D (smaller by 40%)

• The effective pressure increases nearly linearly with filter size

• Effective pressure is by and large due to the mesoscale velocityfluctuations once the filter size exceeds 1 cm for FCC particles.

• This means that it may not be necessary to consider a filtered granular energy equation, if we use a large enough filter size

{ , ,fromkinetictheory

2 s s s s s s eff s effpρ φ μ′ ′ + = +Iu u Sσ

36

0 0.05 0.1 0.15 0.2 0.250

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1


dim

ensi

onle

ss fi

ltere

d pa

rticl

e-ph

ase

visc

osity

0 0.05 0.1 0.15 0.2 0.250

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1


dim

ensi

onle

ss fi

ltere

d pa

rticl

e-ph

ase

visc

osity

2

1

t

gV FrΔ

Δ=

,3

s eff

s t

gV

μρ

1 2 1cmFr Δ

= ⇒ Δ =

2-D0.257

0.514

1.028

2.056

Domain size = 16 x 16

64 x 64 grids

16 x 16 x 16

64 x 64 x 64

3-D


Filtered particle phase viscosity increases as filter size increases for both 2-D and 3-D

37

Filtered particle phase viscosity increases as filter size increases for both 2-D and 3-D

• The effective viscosity in 3-D and 2-D are comparable

• The effective viscosity ~ (filter size)1.5–1.6

• Effective viscosity is by and large due to the mesoscalevelocity fluctuations once the filter size exceeds 1 cm for FCC particles.

• This means that it may not be necessary to consider a filtered granular energy equation

{ , ,fromkinetictheory

2 s s s s s s eff s effpρ φ μ′ ′ + = +Iu u Sσ

38

Coarse-Grid Simulation of Riser FlowsRoadmap for dilute gas-particle flowsCharacteristics of flows in turbulent fluidized beds & fast fluidized bedsWall correction to the filtered closuresThe filtered drag coefficient is noticeably different in the core and the wall regionsThe filtered drag coefficient is noticeably different in the core and the wall regionsVerification of the filtered modelVerification of the filtered modelVerification of the filtered modelsComputational timesSummaryExtra slidesThe filtered particle phase pressure is noticeably different in the core and the wall regions.The filtered particle phase viscosity is noticeably different in the core and the wall regions.Two-fluid model equationsFiltered continuity equationsFiltered particle phase momentum balanceFiltered fluid phase momentum balanceSub-filter scale correlationsSub-filter scale correlations

Coarse-Grid Simulation of Riser Flows · 2020. 8. 13. · Original two-fluid model and constitutive...

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Transcript of Coarse-Grid Simulation of Riser Flows · 2020. 8. 13. · Original two-fluid model and constitutive...