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Symposyium du Département de Chimie Analytique, Minérale et Appliquée
Davide Alemani – University of Geneva Lattice Boltzmann (LB) and
time splitting methodfor reaction-diffusion modelling
1. Reaction-diffusion in the environment.2. The LB approach: why and how3. The time splitting method: why and how4. Some numerical results5. Work in progress: grid refinement
The environmental problem
Schematic representation of various chemical species of a given element (M)
Schematic representation of the physicochemical problem under investigation
Metal concentration:10-7 mol/m3 - 10-3 mol/m3
Diffusion coefficient:10-12 m2/s - 10-9 m2/s
Kinetic rate constants:10-6 s-1 - 109 s-1
ML
ML
LB approach: Why and how
bulk
electrode0
0
*MLL,M,XMLL,M,X
MLL
M
MLdLMa
2ML
2
MLML
2L
2
LL
2M
2
MM
cc
x
c
x
c
c
ckcckR
Rx
cD
t
c
Rx
cD
t
c
Rx
cD
t
c
Macroscopic Model
difc
fcd
ii
i
2,...,1
ML L, M,X2
1X,X
X,X
Mesoscopic Model (LB)
M,1fM,2f
2
21
2
XX,
2X
X
X,1X,1XX,2X,2
X,1X,1XX,1X,1
2)('
2)('
cx
tD
eqi
eq
eq
f
Rt
ffff
Rt
ffff
The LBGK model (1D)
1f
'1f'2f
2f
tt
t
x
xx
t
xv
t
xv
21
)(x M,2M,1
MMM ffDJ
LBGK Evolution Equations
Flux Computation
Schematic Representation (1D)
),(''),(
),('),(''
),(''''
),(),('
),(''
condition initial),(
),( )(
R
D
RD
ttxfttxf
ttxftxf
tttfTt
f
txftxf
tttfTt
f
txf
tttfTTt
f
Time splitting method: Why and How• Important when physical and chemical processes occur simultaneously
and rate constants vary over many order of magnitudes
• Enables to split a complex problem into two or more sub-problems more simply handled
NS
RD
DT
RT
),(
)),((2
),(),(
)),((2
),(),(
'
)],(),([),(),(
),(
),('),(
Reaction
implicitor Explicit
Diffusion
processstart
ReactionDiffusion
2,...,1
2,...,1
ttxf
ttxfRd
ttxfttxf
txfRd
ttxfttxf
f
txftxftxftttvxf
txf
ttxfftxf
i
djii
djii
i
ieqiiii
i
iii
ML
ML
M,1fM,2f
A detailed example of Time Splitting (RD) coupled with LBGK approach
Flux at the electrode for a semilabile complex
Labile flux: ak
Inert flux: 0a k
Numerical flux: 3a 10k
1*L Kc MLLM
d
a
k
k
Comparison between RD and NS with an exact solution
Red circle values are taken from:De Jong et al., JEC 1987, 234, 1
11MLL
7M
*L
3a 10101010 DDDKck
Flux at the electrode for two complexes ML(1) and ML(2) with very different time scale reaction rates
Labile flux:a,1k
a,2k
Inert flux: 0a,1 k 0a,2 k Numerical flux: 6a,1 10k 2
a,2 10k
M+L(1)ML(1) labile – M+L(2)ML(2) inert
Concentration profiles close to the electrode
Strong variation close to the electrode surface
M+L(1)ML(1) labile – M+L(2)ML(2) inert
Error vs grid size and the equilibrium constant x
d
*La'
k
ckK
10d
*La
k
ck
Work in progress
cxfx
cf intxA B
cfx int
fA fB
M + L(n) ML(n)
*L,,a
M
nnn ck
D
fc xgx
Problem to solve
A grid refinement approach for solving LBGK scheme
c ct cx
cxfx
intxA B
fluxlimlim
mass),(lim),(lim
intint
intint
x
c
x
c
txctxc
xxxx
xxxx
Conservation of mass and flux at the grid interface
ff ,2
cf ,1
Work in progress
A grid refinement approach for solving LBGK scheme
That’s all.Thanks to come
Hoping to have been clearHave a nice day
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