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