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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$%igh
Inherent
Parallelism
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Solute/Heat Transport
• Two approaches:
– Multicomponent with 0 or small interaction
parameter
– ‘Passive’ second component (namuro!
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Diffusion Modeling with LB• "le#$% &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
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"-D $est with wll• "inite s%stem:
• Standard ‘3ounce
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2-D $est
• nstantaneous point source:
−+=
Dt
r
Dt
M C C
,ep
,
'
**
π
*
)0
))
)1
)+
)
)=
)6
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ffective Diffusion
Tortuosit% (T A 2path/2! and percolation (15!
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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
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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
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. 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, );+.
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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
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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
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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
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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
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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
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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
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Ta%lor 5ispersion
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Ta%lor 5ispersion
mm D
W U D D'&*
''
+=
'
'
x
C D
x
C v
t
C
∂
∂+
∂
∂−=
∂
∂
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Ta%lor/Oris 5ispersion
See also Stoc#man7 H.C.7 O lattice.
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Pore olume
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Pulse Source
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Solute Transport in Porous Media
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‘Pre-asympt tic’ transp rt
re-asympt tic’ transp rt
Test LBM against classicanalytical solutions to theC!" their #oundaryconditions" and concentrationdetection modes
$or% of van &enucthen et al
'()*( + ,--(. and reft and0u#er '()1*.
D
vL Br =
, Classical naltical Solutions
at + ;renner Numbers3
'
'
x
C D
x
C v
t
C
∂
∂+
∂
∂−=
∂
∂
m D
v Pe
=
(vn 3enuchten nd ie%eng' "*
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ConcentrationConcentration Detecti netecti n
ModesModes 2esident
Core sectioning ow ow sampling
Flux
Column e!uent "prings #umping wells
Volume MassC r /=
FluxVolume
Flux Mass
C f =
r f C
x
C
v
DC +
∂
∂−=
C r - '
; C C f
- ' ; C
( < C r - &
; C )
vC x
C
DvC r f +∂
∂
−=
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nalytical !olutionsnalytical !olutions
vn 3enuchten' M$h nd ie%eng ("*
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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 (
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Convection-Diffusion "#D $pen%Convection-Diffusion "#D $pen%
m 4 -5---99 lu ts7(
Br 4 16-
-&
-+
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&aylor-ris Dispersion&aylor-ris Dispersion
LB :trategy;
#eriodic$ 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 $idth
uma
g
u
DBr L =
L
a
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&aylor-ris Dispersion&aylor-ris Dispersion
-&7 Dm < ***++++ lu'ts-&7 D < **'9
-&7 Dm < *&6 lu'ts-&7 D < *&6
-+7 Dm < ***++++ lu'ts-&
-&7 Dm < ****++++ lu'ts-&7 D < *'8
mm D
au
D D '&*
)'( ''
+= a L Pe /,,5
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&ayl r/ris Dispersi n
ayl r/ris Dispersi n
Lttice gs ve%sion@ e%e-Reeves' S nd 1 Stoc&mn' "**+ Chem ng Sci ,2("*)@:2++-:2
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Peclet 'um(er CorrelationsPeclet 'um(er Correlations
3%ube%t' D "**+ 0nt Mod h4s B' etude des deplcements
de fluides miscibles dns un milieu po%eux' Revue de lG0nstitut
;%ncis du et%ole
;%ied' nd M7 Comb%nous' "*+" 7dv 14d%oscience +'
"E*-2
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Peclet 'um(er CorrelationsPeclet 'um(er Correlations
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)i*+ ,eynolds 'um(er )i*+ ,eynolds 'um(er
:ingle cylinder" 2e < =($ned' ;luid Mech
"*,E (7lso Ftchi
Societ4 web pges)
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Diffusi n Dead .nd P res and &ailin*
iffusi n Dead .nd P res and &ailin*
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2e > )--
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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
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3uo%anc%
ρ
ρ ρ τ σ ρ ++= ∑=geu
5
*
&
a
aa f