multicomponent lbm method

8/17/2019 Multicomponent LBM Method

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Multicomponent Miscible

LB Models

Multi- Component

Multiphase

Miscible

Fluids/Diffusion(No Interaction)

Immiscible

Fluids

Single

ComponentMultiphase

Single Phase

(No Interaction)

N u m b e r o f

C o m p o n e n t s

Interaction

Strength

N a t u r e o

f

I n t e r a c t i o

n

ttracti!e

"epulsi!e

#o$%igh

Inherent

Parallelism


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Solute/Heat Transport

• Two approaches:

– Multicomponent with 0 or small interaction

parameter

– ‘Passive’ second component (namuro!


3/43

Diffusion Modeling with LB• "le#$% &P' )**+,/-oshino and namuro &nt. . um. Meth. 100+,

– Multicomponent 23

– Separate ‘passive’ distri4ution

• Shan and 5oolen &P' )**6,

– Multicomponent7 multiphase 23 (Phase separation possi4le!

– Separate ‘active’ distri4ution

– ‘8omplementar%’ densities

• 5i99usion coe99icient:

&' &*'

&

+

& −=

−= tslu D sτ

0

100

00

600

;00

)000

)100


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0

)00

100

+00

00

=00

600

>00

;00


5/43

"-D $est with wll• "inite s%stem:

• Standard ‘3ounce


6/43

2-D $est

• nstantaneous point source:

−+=

Dt

r

Dt

M C C

,ep

,

'

**

π

*

)0

))

)1

)+

)

)=

)6


7/43

ffective Diffusion

Tortuosit% (T A 2path/2! and percolation (15!


8/43

3%vit4 nd Bond

5umbe% 5o 3%vit4 3%vit4

Gravitational/capillary forces g gravitational acceleration

r characteristic pore radius

ρ diference in uid densities

interacial tension between uids

γ

ρ ∆=' gr

Bo

2x 3%vit4

Su%fce tension


9/43

ffective Diffusion Coefficients

0

0.)

0.1

0.+

0.

0.=

0 0.) 0.1 0.+ 0. 0.=

6olumet%ic 7i% Content

D 8 D o

Ma@well ();>+!

3uc#inBham ()*0!

Penman ()*0!

Marshall ()*=*!

MillinBton ()*=*!

CesselinB ()*61!

8urrie ()*6=!

C2'(Marshall!:

Moldrup et al (1000!

2x g

# g

g


10/43

. Solute is simulated b a second distribution f σ called the 0solute

component1 or σ -component. f σ corresponds to the fluid distribution function ecept $ith a

simpler e2uilibrium distribution

. Concentration3 // nalogous to fluid densit

. Diffusion coefficient3 // nalogous to !iscosit

Solute Transport (namuro method!

-oshino7 M. and T. namuro (100+! Int. J. Numer. Meth. Fluids 43, );+.


11/43

LBM Constant ConcentrationLBM Constant Concentration

σ σ σ σ σ σ σ σ σ

σ σ

ρ ρ ρ

ρ

444 56,787+7'7&7*7

7

www f f f f f f

f

a

a

++++++++=

=∑

56,

787+7'7&7*7 )(4

www

f f f f f f

++

+++++−= σ σ σ σ σ σ σ σ ρ

ρ

σ σ σ ρ ρ 44 9&

,,7 == w f σ σ σ ρ ρ 44 +

&667 == w f

σ σ σ ρ ρ 44 +&

557 == w f

9oshino' M nd $ 0nmu%o' 2##: 0nt 5ume% Meth ;luids /:' "


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8onstant 8oncentration 38 8ode

• // 8onstant concentration on north side.• 9or( iA0D iE2FD iGG!• • 9i A 9temp&2-, A ()./+6.!JrhoIsiBmaIprimeD• 9i&;, A ()./+6.!JrhoIsiBmaIprimeD• K• // 8onstant concentration on south side.• 9or( iA0D iE2FD iGG!•

• 9i A 9temp&0,&i,D• rhoIsiBmaIprime A 6.J( rhoIsiBmaI4ar• < ( 9i&0, G 9i&), G 9i&+,• G 9i&>, G 9i&, G 9i&;,!!D• 9i&1, A ()./ *.!JrhoIsiBmaIprimeD• 9i&=, A ()./+6.!JrhoIsiBmaIprimeD• 9i&6, A ()./+6.!JrhoIsiBmaIprimeD• K

• // 8onstant concentration on east side.• 9or( LA0D LE2-D LGG!• • 9L A 9temp&L,&2F, A ()./+6.!JrhoIsiBmaIprimeD• 9L&6, A ()./+6.!JrhoIsiBmaIprimeD• K• // 8onstant concentration on west side.• 9or( LA0D LE2-D LGG!• • 9L A 9temp&L,&0,D•

rhoIsiBmaIprime A 6.J( rhoIsiBmaI4ar• < ( 9L&0, G 9L&1, G 9L&,• G 9L&+, G 9L&6, G 9L&>,!!D• 9L&), A ()./ *.!JrhoIsiBmaIprimeD• 9L&=, A ()./+6.!JrhoIsiBmaIprimeD• 9L&;, A ()./+6.!JrhoIsiBmaIprimeD• K


13/43

LBM Constant Flux BoundariesLBM Constant Flux Boundaries

v D

DC vC C vC vC

C C D

++

=⇒=

+

−− &**

&

&

*

*

vC vC x

C D

x

=

+∂

∂−

+=

Maintaining true constant concentration dicult

Constant ux more common

Finite dierence approximation


14/43

LBM Fixed/Zero Diffusive FluxLBM Fixed/Zero Diffusive Flux

σ σ σ ρ ρ 44 9&

,,7 == w f σ σ σ ρ ρ 44 +

&667 == w f

σ σ σ ρ ρ 44 +&

557 == w f

∑ ⋅=a

aa D f j ne7σ

σ σ σ σ σ σ σ σ σ ρ ρ ρ 444*** 56,787+7'7&7*7 www f f f f f f j D +++−−+−+=

56,

787'74www

f f f j D

++

+++=

σ σ σ σ ρ

n


15/43

8onstant 5i99usive "lu@ 8ode

• // 8onstant di99usive 9lu@ 4oundar% onnorth side.• 9or( iA0D iE2FD iGG!• • 9i A 9temp&2-, A ()./+6.!JrhoIprimeD

• 9i&;, A ()./+6.!JrhoIprimeD• K• // 8onstant di99usive 9lu@ 4oundar% on

south side.• 9or( iA0D iE2FD iGG!• • 9i A 9temp&0,&i,D

• rhoIprime A 6.J( 0 G 9i&>, G 9i&, G 9i&;,!D• 9i&1, A ()./ *.!JrhoIprimeD• 9i&=, A ()./+6.!JrhoIprimeD• 9i&6, A ()./+6.!JrhoIprimeD• K

• // 8onstant di99usive 9lu@ 4oundar% on eastside.

• 9or( LA0D LE2-D LGG!• • 9L A 9temp&L,&2F, A ()./+6.!JrhoIprimeD• 9L&6, A ()./+6.!JrhoIprimeD

• K• // 8onstant di99usive 9lu@ 4oundar% on west side.• 9or( LA0D LE2-D LGG!• • 9L A 9temp&L,&0,D• rhoIprime A 6.J( 0 G 9i&+, G 9i&6, G 9i&>,!D• 9L&), A ()./ *.!JrhoIprimeD• 9L&=, A ()./+6.!JrhoIprimeD

• 9L&;, A ()./+6.!JrhoIprimeD• K

•


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Macroscopic overninB Nuations

• 5i99usion or heat eNuation

• 8onvection


17/43

Breakthrough Curves

• ==iston> ;low ? noiston> ;low ? nodispe%siondispe%sion

• Dispe%sed ;lowDispe%sed ;low

• Ret%ded8Ret%ded8

Dispe%sed ;lowDispe%sed ;low

Influent Solution3

Concentration C*

:ffluent Solution3

Concentration C


18/43


19/43

3rea#throuBh 8urve

0

10

:0

60

;0

)00

0 1 : 6 ; )0

$ime (4e%s)

C ( m g 8 l )

0nitil nd Bound%4

Conditions@

C(x'#) ! #

C(#'#AtA") ! "##

C(#'t") ! #

3ene%l Conditions@

! " m84e%

! #,

! #" m

1 0 m

q = 1m/y


20/43

Ta%lor 5ispersion


21/43

Ta%lor 5ispersion

mm D

W U D D'&*

''

+=

'

'

x

C D

x

C v

t

C

∂

∂+

∂

∂−=

∂

∂


22/43

Ta%lor/Oris 5ispersion

See also Stoc#man7 H.C.7 O lattice.


23/43

Pore olume


24/43


25/43

Pulse Source


26/43

Solute Transport in Porous Media


27/43

‘Pre-asympt tic’ transp rt

re-asympt tic’ transp rt

Test LBM against classicanalytical solutions to theC!" their #oundaryconditions" and concentrationdetection modes

$or% of van &enucthen et al

'()*( + ,--(. and reft and0u#er '()1*.

D

vL Br =

, Classical naltical Solutions

at + ;renner Numbers3

'

'

x

C D

x

C v

t

C

∂

∂+

∂

∂−=

∂

∂

m D

v Pe

=

(vn 3enuchten nd ie%eng' "*


28/43

ConcentrationConcentration Detecti netecti n

ModesModes 2esident

Core sectioning ow ow sampling

Flux

Column e!uent "prings #umping wells

Volume MassC r /=

FluxVolume

Flux Mass

C f =

r f C

x

C

v

DC +

∂

∂−=

C r - '

; C C f

- ' ; C

( < C r - &

; C )

vC x

C

DvC r f +∂

∂

−=


29/43

nalytical !olutionsnalytical !olutions

vn 3enuchten' M$h nd ie%eng ("*


30/43


31/43

Convection-Diffusion "#D $pen%Convection-Diffusion "#D $pen%

3 4 -5--66 lu ts7(

m 4 -5(66 lu, ts7(

L 4 ,8 lu

Br 4 vL/m 4 (


32/43

Convection-Diffusion "#D $pen%Convection-Diffusion "#D $pen%

m 4 -5---99 lu ts7(

Br 4 16-

-&

-+


33/43

&aylor-ris Dispersion&aylor-ris Dispersion

LB :trategy;

#eriodic$ gravitydriven ow

m

m

D

au D D

'&*

)'( ''+=

−=

'

&

+

&σ τ m D

'

ma'

a

u g

ν =

−=

'

&

+

&τ ν

D

uL Br

=Choose maimum !elocit

Poiseuille slit3 u < '/+ uma

Choose ;r

Molecular diffusion

coefficient@inematic !iscosit

Ara!itDispersion coefficient

Choose slit $idth

uma

g

u

DBr L =

L

a


34/43

&aylor-ris Dispersion&aylor-ris Dispersion

-&7 Dm < ***++++ lu'ts-&7 D < **'9

-&7 Dm < *&6 lu'ts-&7 D < *&6

-+7 Dm < ***++++ lu'ts-&

-&7 Dm < ****++++ lu'ts-&7 D < *'8

mm D

au

D D '&*

)'( ''

+= a L Pe /,,5


35/43

&ayl r/ris Dispersi n

ayl r/ris Dispersi n

Lttice gs ve%sion@ e%e-Reeves' S nd 1 Stoc&mn' "**+ Chem ng Sci ,2("*)@:2++-:2


36/43

Peclet 'um(er CorrelationsPeclet 'um(er Correlations

3%ube%t' D "**+ 0nt Mod h4s B' etude des deplcements

de fluides miscibles dns un milieu po%eux' Revue de lG0nstitut

;%ncis du et%ole

;%ied' nd M7 Comb%nous' "*+" 7dv 14d%oscience +'

"E*-2


37/43

Peclet 'um(er CorrelationsPeclet 'um(er Correlations


38/43

)i*+ ,eynolds 'um(er )i*+ ,eynolds 'um(er

:ingle cylinder" 2e < =($ned' ;luid Mech

"*,E (7lso Ftchi

Societ4 web pges)


39/43

Diffusi n Dead .nd P res and &ailin*

iffusi n Dead .nd P res and &ailin*


40/43


41/43

2e > )--


42/43

BuoyancyBuoyancy

!?uation of :tate

@ncorporation of #uoyancy as a velocitypertur#ation

( ) ( )*

*=*C*5

*

a

=-=C-Cf &/

ρ ρ τρ ρ

ρ ρ ∂∂∂∂

=

+++= ∑ geu aa

( ) ( )***)7( T T T

C C C

T C −∂∂

+−∂∂

+= ρ ρ

ρ ρ


43/43

3uo%anc%

ρ

ρ ρ τ σ ρ ++= ∑=geu

5

*

&

a

aa f

multicomponent lbm method

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