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/18

Fluid-particle drag in polydisperse gas-solid suspensions

William Holloway, Xiaolong Yin, and Sankaran SundaresanNETL Workshop on Multiphase Flow Science

Morgantown, WV4/22/2009

8:40-9:00 am

1

/18

• Breakdown of Princeton Tasks

• Bidisperse drag formulation

• Simulation procedures

• Low Re results

• Moderate Re results

• Summary

Outline

2

/18

Connection to Roadmap

Princeton Tasks Roadmap

!"#$%&"' "( )*+,-'&.%/ 0+"1 2/%/.#3&4 5*(/ 6784 9::6

9:

2015

>2015

1. Fundamental aspects of stress and flow fields in dense

particulate systems.

2. Definition of material properties on relevant scales, along with

efficient ways to represent properties in models and establish

standards for material property measurements.

3. Given the practical need for continuum modeling capability,

identify the inherent limitations and how to proceed forward, e.g.,

hybrid models that connect with finer scale models (DNS, DEM,finite element, stochastic, etc.) for finer resolution.

4. Size-scaling and process control (particle / unit-op / processing

system) is critical to industrial applications.

2009

2012

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!"#$%& '('( ;,#.,/<-3 ,-=/+-(/ >"# .??#/%%-(< $/@ ,"'-3.+ .#/.% #/+.,/? ,"?/(%/ '&.%/ >+"1%A B+"3$ C -% '#-=.#-+@ 3"(3/#(/? 1-,& ,&/ -%%*/% -(

D.E+/ FAFG B #/>/#% ," -%%*/% -( D.E+/ FA9G H #/>/#% ," D.E+/ FAI .(? J#/>/#% ," D.E+/ FAKAA

! !"#$ %&'( ")* +&,-./ %01"'(/2

3

/18



Task 2.2:

LBM/DTIBM simulations of flow through assemblies of binary particle mixtures where the two types of particles have non-zero relative velocities.

!"#$%&"' "( )*+,-'&.%/ 0+"1 2/%/.#3&4 5*(/ 6784 9::6

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>2015











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!"#$%& '('( ;,#.,/<-3 ,-=/+-(/ >"# .??#/%%-(< $/@ ,"'-3.+ .#/.% #/+.,/? ,"?/(%/ '&.%/ >+"1%A B+"3$ C -% '#-=.#-+@ 3"(3/#(/? 1-,& ,&/ -%%*/% -(


! !"#$ %&'( ")* +&,-./ %01"'(/2

3

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Task 2.2:


Near-term:• Develop drag relations that can

handle particle size and density distributions; applicable over the entire range of solids volume fraction.

• Development of constitutive relations for continuum models from discrete models such as DEM or LBM.

!"#$%&"' "( )*+,-'&.%/ 0+"1 2/%/.#3&4 5*(/ 6784 9::6

9:

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>2015











2009

2012

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>2015











2009

2012

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2009

2012

A

B

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!"#$%& '('( ;,#.,/<-3 ,-=/+-(/ >"# .??#/%%-(< $/@ ,"'-3.+ .#/.% #/+.,/? ,"?/(%/ '&.%/ >+"1%A B+"3$ C -% '#-=.#-+@ 3"(3/#(/? 1-,& ,&/ -%%*/% -(


! !"#$ %&'( ")* +&,-./ %01"'(/2

3

/18



Task 2.2:


Near-term:• Develop drag relations that can

handle particle size and density distributions; applicable over the entire range of solids volume fraction.

• Development of constitutive relations for continuum models from discrete models such as DEM or LBM.

Mid-term:• Consider the effect of lubrication

forces in particle-particle interactions.

!"#$%&"' "( )*+,-'&.%/ 0+"1 2/%/.#3&4 5*(/ 6784 9::6

9:

2015

>2015











2009

2012

A

B

C

D

2015

>2015











2009

2012

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B

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2012

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!"#$%& '('( ;,#.,/<-3 ,-=/+-(/ >"# .??#/%%-(< $/@ ,"'-3.+ .#/.% #/+.,/? ,"?/(%/ '&.%/ >+"1%A B+"3$ C -% '#-=.#-+@ 3"(3/#(/? 1-,& ,&/ -%%*/% -(


! !"#$ %&'( ")* +&,-./ %01"'(/2

3

/18

Particle-particle drag

Total force experienced by particle type i

Pressure gradient

Fluid-particle drag force

fi = !!i"p + fDi ! fppdi

Fluid-particle drag vs. total force

4

/18

Bidisperse fluid-particle drag formulation:

Particle-particle drag

Total force experienced by particle type i

Pressure gradient

Fluid-particle drag force

fi = !!i"p + fDi ! fppdi

fD1 = !!11!U1 ! !12!U2

fD2 = !!21!U1 ! !22!U2

Average fluid-particle drag per unit volume of

suspension

Fluid-particle drag vs. total force

4

/18

Bidisperse drag formulation

fD1 = !!11!U1 ! !12!U2

fD2 = !!21!U1 ! !22!U2

Volume specific friction coefficient

λ - lubrication cutoff

r = λ

r

F

5

/18

Bidisperse drag formulation

fD1 = !!11!U1 ! !12!U2

fD2 = !!21!U1 ! !22!U2

Volume specific friction coefficient

low Re

moderate Re

λ - lubrication cutoff

r = λ

r

F

Fluctuating particle velocities found to be small contribution to the fluid-particle drag force (Wylie and Koch, JFM (2003), vol. 480, pp. 95-118) 5

!ij = !ij("i, "j , di, dj ,!Ui,!Uj , < u2i >, < u2

j >, #)!ij = !ij("i, "j , di, dj , #)

!ij = !ij("i, "j , di, dj ,!Ui,!Uj , #)

/18

Simulation Procedures

Numerical Method: Lattice Boltzmann

Fluid motion solved on a 3D cubic lattice with no slip boundary conditions

LBM References:Ladd and Verberg, J. Stat. Phys. (2001), vol. 104, pp. 1191-1251Yin and Sundaresan. Ind. Eng. Chem. Res. (2009), vol. 48, pp. 227-241 6

/18


• Generate initial configurations that satisfy binary hard sphere distribution.




/18



• Assign particles with velocities, but do not update particle positions. FROZEN SIMULATIONS (exact for Stokes flow, arguable for finite Re).




/18




• Apply pressure gradient to enforce a net zero flow rate of fluid.




/18




• Apply pressure gradient to enforce a net zero flow rate of fluid.

• Ensemble average multiple independent realizations.

• Solve for βij




/18

Low Re bidisperse systems

fD1 = !!11!U1 ! !12!U2

fD2 = !!21!U1 ! !22!U2

Yin and Sundaresan. Ind. Eng. Chem. Res. (2009), vol. 48, pp. 227-241 7

/18


fD1 = !!11!U1 ! !12!U2

fD2 = !!21!U1 ! !22!U2!12 = !21


/18


fD1 = !!11!U1 ! !12!U2

fD2 = !!21!U1 ! !22!U2!12 = !21

!11 + !12 = !1 = !fD1!fixed

!U

!21 + !22 = !2 = !fD2!fixed

!U

Recovery of fixed bed drag when ΔU1 = ΔU2


/18


fD1 = !!11!U1 ! !12!U2

fD2 = !!21!U1 ! !22!U2!12 = !21

!11 + !12 = !1 = !fD1!fixed

!U

!21 + !22 = !2 = !fD2!fixed

!U

!!11 !12

!21 !22

"=

!!1 ! !12 !12

!12 !2 ! !12

"



/18


fD1 = !!11!U1 ! !12!U2

fD2 = !!21!U1 ! !22!U2!12 = !21

!11 + !12 = !1 = !fD1!fixed

!U

!21 + !22 = !2 = !fD2!fixed

!U

!!11 !12

!21 !22

"=

!!1 ! !12 !12

!12 !2 ! !12

"


One free parameter

ΔU1 = ΔU2 (‘fixed bed’) simulations: Extract β1 and β2

ΔU1 ≠ ΔU2 (‘moving suspension’) simulations: Extract β12


/18

Low Re Bidisperse fixed beds

Definitions follow van der Hoef et al., JFM (2005), vol. 528, pp. 233-258

Sauter mean diameter

< d >=n!

i=1

nid3i

nid2i

Dimensionless size ratio

yi =di

< d >

!i =18µ"i(1! ")

d2i

F !Di"fixed(", yi)

Fixed bed friction coefficient

Dimensionless drag force on particle of type i in a bidisperse fixed bed

8

/18

Drag law for bidisperse fixed beds

Yin and Sundaresan, AIChE J., accepted (2009)van der Hoef et. al., JFM, (2005), vol. 528, pp. 233-254

Refinement of original correction proposed by

van der Hoef et al.

9

F !D"fixed =

10!

(1! !)2+ (1! !)2(1 + 1.5

!!)

F !Di"fixed =

11! !

+!

F !D"fixed !

11! !

"(ayi + (1! a)y2

i )

Drag in a bidisperse fixed

bed

Drag in a monodisperse

fixed bed

a = 1! 2.660! + 9.096!2 ! 11.338!3

/18

Drag law for bidisperse fixed beds

dP

dx=

18!µ!U

< d >2

!F !

D"fixed +1

1! !

!"I"III

"2II

! 1"" Integrating over a continuous size

distribution we can obtain the pressure drop through a polydisperse fixed bed

σI, σII, and σIII are first, second, and third order moments of a particle size distribution

Yin and Sundaresan, AIChE J., accepted (2009)van der Hoef et. al., JFM, (2005), vol. 528, pp. 233-254

Refinement of original correction proposed by

van der Hoef et al.

9

F !D"fixed =

10!

(1! !)2+ (1! !)2(1 + 1.5

!!)

F !Di"fixed =

11! !

+!

F !D"fixed !

11! !

"(ayi + (1! a)y2

i )

Drag in a bidisperse fixed

bed

Drag in a monodisperse

fixed bed

a = 1! 2.660! + 9.096!2 ! 11.338!3

/18

Low Re Bidisperse Fixed beds

Horizontal Axis:

Vertical Axis:F !

Di"fixed

F !D"fixed

yi

Average error: 3.9%Max error: 9.4%

10

/18

Low Re bidisperse suspensions

Particle-particle interaction proportional to the probability of mutual contact

!12

"1"2= !2#

!!1"1

!2"2

!1"1

+ !2"2

"Harmonic mean

Yin and Sundaresan, AIChE J., accepted (2009)

fD1 = !!1!U1 ! !12 (!U2 !!U1)fD2 = !!2!U2 ! !12 (!U1 !!U2)

11

/18

Low Re Bidisperse suspensions

0

100

200

300

400

500

0 20 40 60 80

phi=0.15 phi=0.20 phi=0.25 phi=0.30phi=0.35 phi=0.40 mono

0

100

200

300

400

0 20 40 60 80

phi=0.20 phi=0.30 phi=0.40 mono

0

100

200

300

400

0 20 40 60 80


0

100

200

300

0 20 40 60 80


001.01 =dλ 002.01 =dλ

005.01 =dλ 01.01 =dλ

! !!1"1

!!2"2

!!1"1

+ !!2"2

"Horizontal Axis:

Vertical Axis:

! !!12

"1"2

!!12 =

!12 < d >2

µ

!!1 =

!1 < d >2

µ!!

2 =!2 < d >2

µ

!

!"

d1

"= 1.313log10

!d1

"

"! 1.249

is a linear function of the harmonic mean of and!12!1

"1

!2

"2

12Yin and Sundaresan, AIChE J., accepted (2009)

/18

Simulations at finite Re

Frozen suspension at finite Re Moving suspension at finite Re

• Inertial lag prevents fluid from adapting to particle motion

instantaneously

13

/18

Simulations at finite Re

Frozen suspension at finite Re

• Fluid adapts instantaneously to particle motion

Moving suspension at finite Re

• Inertial lag prevents fluid from adapting to particle motion

instantaneously

13

/18

Simulation procedure at finite Re

Must extrapolate frozen simulations to infinite resolution to get an accurate measure of the drag force in a moving suspension.

Dim

ensi

onle

ss fl

uid-

part

icle

dra

g fo

rce

0 0.02 0.04 0.06−1200

−1150

−1100

−1050

−1000

−950

−900Re = 20 Ret = 2

Moving

Frozen

1r2h

van der Hoef et al., JFM (2005), vol. 528, pp. 233-258 (extrapolation procedure)

Average pore radius

rh =d(1! !)

6!

14

/18

Simulation procedure at finite Re

Must extrapolate frozen simulations to infinite resolution to get an accurate measure of the drag force in a moving suspension.

Dim

ensi

onle

ss fl

uid-

part

icle

dra

g fo

rce

0 0.02 0.04 0.06−1200

−1150

−1100

−1050

−1000

−950

−900Re = 20 Ret = 2

Moving

Frozen

1r2h

van der Hoef et al., JFM (2005), vol. 528, pp. 233-258 (extrapolation procedure)

less than 1 % difference

Average pore radius

rh =d(1! !)

6!

14

/18

Drag law for finite Re

fD1 = !!1!U1 ! !12 (!U2 !!U1)fD2 = !!2!U2 ! !12 (!U1 !!U2)

!12

"1"2= !2#

!!1"1

!2"2

!1"1

+ !2"2

"!

!"

d1

"= 1.313log10

!d1

"

"! 1.249

Fluid-particle drag relation for inertial, bidisperse, gas-solid

suspensions in relative motion

The fluid-particle drag in a bidisperse suspension in the high Stokes number limit

can be expressed in the following way:

1 11 1 12

2 21 1 22

D

D

2

2

f U U

f U U

! !

! !

" # $ # $

" # $ # $ (1)

where Dif is the fluid-particle drag force on particles of type i per unit volume

suspension, is the relative velocity between particles of type i and the fluid, and iU$

ij! represents the volume specific friction coefficient on particles of type i due to the

motion of particles of type j.

In the limit of a fixed bed, 1U U2$ " $ , the drag relationship given by Eqn. (1) to

be simplified in the following way:

% &

% &

1 1 12 1 12

2 21 1 2 21

D

D

2

2

f U U

f U U

! ! !

! ! !

" # # $ # $

" # $ # # $ (2)

where

% & *

2

18 1 i

i

i

Fd

' ' (! #

#" Di fixed . (3)

In Eqn. (3) i! is the fixed bed friction coefficient of a particle of type i per unit volume,

' is the total volume fraction, i' is the volume fraction of species i,( is the fluid

viscosity, is the diameter of a particle of type i, and id *

Di fixedF # is the dimensionless fluid-

particle drag force on particles of type i in a bidisperse fixed bed ( *

Di f# ixedF is made

dimensionless by normalizing by the Stokes drag % &i Ud $#')( 13 ).

15

/18


fD1 = !!1!U1 ! !12 (!U2 !!U1)fD2 = !!2!U2 ! !12 (!U1 !!U2)

!12

"1"2= !2#

!!1"1

!2"2

!1"1

+ !2"2

"!

!"

d1

"= 1.313log10

!d1

"

"! 1.249





1 11 1 12

2 21 1 22

D

D

2

2

f U U

f U U

! !

! !

" # $ # $

" # $ # $ (1)







% &

% &

1 1 12 1 12

2 21 1 2 21

D

D

2

2

f U U

f U U

! ! !

! ! !

" # # $ # $

" # $ # # $ (2)

where

% & *

2

18 1 i

i

i

Fd

' ' (! #

#" Di fixed . (3)






Di f# ixedF is made


F !Di"fixed =

11! !

+!

F !D"fixed !

11! !

"(ayi + (1! a)y2

i )

15

F !D"fixed =

!10!

(1! !)2+ (1! !)2(1 + 1.5

"!)

#

/18


fD1 = !!1!U1 ! !12 (!U2 !!U1)fD2 = !!2!U2 ! !12 (!U1 !!U2)

!12

"1"2= !2#

!!1"1

!2"2

!1"1

+ !2"2

"!

!"

d1

"= 1.313log10

!d1

"

"! 1.249





1 11 1 12

2 21 1 22

D

D

2

2

f U U

f U U

! !

! !

" # $ # $

" # $ # $ (1)







% &

% &

1 1 12 1 12

2 21 1 2 21

D

D

2

2

f U U

f U U

! ! !

! ! !

" # # $ # $

" # $ # # $ (2)

where

% & *

2

18 1 i

i

i

Fd

' ' (! #

#" Di fixed . (3)






Di f# ixedF is made


Beetstra, van der Hoef, and Kuipers., (2007) AIChE J., vol. 53, pp. 489-501

F !Di"fixed =

11! !

+!

F !D"fixed !

11! !

"(ayi + (1! a)y2

i )

15

F !D"fixed =

!10!

(1! !)2+ (1! !)2(1 + 1.5

"!)

#(1 + !BV K)

/18


fD1 = !!1!U1 ! !12 (!U2 !!U1)fD2 = !!2!U2 ! !12 (!U1 !!U2)

!12

"1"2= !2#

!!1"1

!2"2

!1"1

+ !2"2

"!

!"

d1

"= 1.313log10

!d1

"

"! 1.249





1 11 1 12

2 21 1 22

D

D

2

2

f U U

f U U

! !

! !

" # $ # $

" # $ # $ (1)







% &

% &

1 1 12 1 12

2 21 1 2 21

D

D

2

2

f U U

f U U

! ! !

! ! !

" # # $ # $

" # $ # # $ (2)

where

% & *

2

18 1 i

i

i

Fd

' ' (! #

#" Di fixed . (3)






Di f# ixedF is made



F !Di"fixed =

11! !

+!

F !D"fixed !

11! !

"(ayi + (1! a)y2

i )

!BV K =0.413Remix

240" + 24(1! ")4(1 + 1.5"

")(1! ")!1 + 3"(1! ") + 8.4Re!0.343

mix

1 + 103!Re!(1+4!)

2mix

15

F !D"fixed =

!10!

(1! !)2+ (1! !)2(1 + 1.5

"!)

#(1 + !BV K)

/18


fD1 = !!1!U1 ! !12 (!U2 !!U1)fD2 = !!2!U2 ! !12 (!U1 !!U2)

!12

"1"2= !2#

!!1"1

!2"2

!1"1

+ !2"2

"!

!"

d1

"= 1.313log10

!d1

"

"! 1.249





1 11 1 12

2 21 1 22

D

D

2

2

f U U

f U U

! !

! !

" # $ # $

" # $ # $ (1)







% &

% &

1 1 12 1 12

2 21 1 2 21

D

D

2

2

f U U

f U U

! ! !

! ! !

" # # $ # $

" # $ # # $ (2)

where

% & *

2

18 1 i

i

i

Fd

' ' (! #

#" Di fixed . (3)






Di f# ixedF is made



F !Di"fixed =

11! !

+!

F !D"fixed !

11! !

"(ayi + (1! a)y2

i )

Remix =| !Umix |< d > (1! !)

"

!BV K =0.413Remix

240" + 24(1! ")4(1 + 1.5"

")(1! ")!1 + 3"(1! ") + 8.4Re!0.343

mix

1 + 103!Re!(1+4!)

2mix

15

F !D"fixed =

!10!

(1! !)2+ (1! !)2(1 + 1.5

"!)

#(1 + !BV K)

/18


fD1 = !!1!U1 ! !12 (!U2 !!U1)fD2 = !!2!U2 ! !12 (!U1 !!U2)

!12

"1"2= !2#

!!1"1

!2"2

!1"1

+ !2"2

"!

!"

d1

"= 1.313log10

!d1

"

"! 1.249





1 11 1 12

2 21 1 22

D

D

2

2

f U U

f U U

! !

! !

" # $ # $

" # $ # $ (1)







% &

% &

1 1 12 1 12

2 21 1 2 21

D

D

2

2

f U U

f U U

! ! !

! ! !

" # # $ # $

" # $ # # $ (2)

where

% & *

2

18 1 i

i

i

Fd

' ' (! #

#" Di fixed . (3)






Di f# ixedF is made



F !Di"fixed =

11! !

+!

F !D"fixed !

11! !

"(ayi + (1! a)y2

i )

Remix =| !Umix |< d > (1! !)

"!Umix =

1!

n!

i=1

!i!Ui

!BV K =0.413Remix

240" + 24(1! ")4(1 + 1.5"

")(1! ")!1 + 3"(1! ") + 8.4Re!0.343

mix

1 + 103!Re!(1+4!)

2mix

15

F !D"fixed =

!10!

(1! !)2+ (1! !)2(1 + 1.5

"!)

#(1 + !BV K)

/18

Finite Re bidisperse suspension data

Holloway, Yin and Sundaresan, (in preparation)

Remix range: 0-40Φ1: Φ2 range: 1-3d1:d2 range: 1-2

Re1:Re2 range: -1:3Average error: 5%Max error: 25%

Horizontal axis: Simulated fDi* Vertical axis: Predicted fDi*

−400 0 400 800 1200−400

0

400

800

1200

! = 0.3

fD2*

fD1*

y=x

−1000 0 1000 2000 3000−1000

0

1000

2000

3000! = 0.4

fD2*

fD1*

y=x

−100 100 300 500 700 900−100

100

300

500

700

900! = 0.2

fD2*

fD1*

y=x

16

/18

Looking ahead

• Combine LBM results at moderate Re together with IBM results from Subramaniam group at higher Re.

• Perform freely evolving bidisperse simulations to investigate particle-particle collisional interactions in sedimenting systems.

17

/18

Summary

• Fluid-particle drag relation developed that accurately predicts fluid-particle drag in Stokesian suspensions with particle-particle relative motion and size differences.

• Drag relation extended to account for moderate fluid inertia in bidisperse suspension flows.

Acknowledgements

• US Department of Energy• ExxonMobil Corporation• ACS Petroleum Research Fund

18

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