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 Symposyium du Département de Chimie Analytique,
Minérale et AppliquéeDavide Alemani University
of Geneva  Lattice Boltzmann (LB) andtime
splitting methodfor reaction-diffusion modelling

1. Reaction-diffusion in the environment.
2. The LB approach why and how
3. The time splitting method why and how
4. Some numerical results
5. Work in progress grid refinement

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The environmental problem
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Schematic representation of various chemical
species of a given element (M)
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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
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LB approach Why and how
Macroscopic Model
Mesoscopic Model (LB)
M L ML
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The LBGK model (1D)
LBGK Evolution Equations
Schematic Representation (1D)
Flux Computation
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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
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A detailed example of Time Splitting (RD)
coupled with LBGK approach
M L ML
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Flux at the electrode for a semilabile complex
Labile flux
Numerical flux
Inert flux
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Comparison between RD and NS with an exact
solution
Red circle values are taken from De Jong et al.,
JEC 1987, 234, 1
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Flux at the electrode for two complexes ML(1) and
ML(2) with very different time scale reaction
rates
ML(1)??ML(1) labile ML(2)??ML(2) inert
Labile flux
Inert flux
Numerical flux
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ML(1)??ML(1) labile ML(2)??ML(2) inert
Concentration profiles close to the electrode
Strong variation close to the electrode surface
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Error vs grid size and the equilibrium
constant
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Work in progress
A grid refinement approach for solving LBGK scheme
Problem to solve
M L(n) ?? ML(n)
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Work in progress
A grid refinement approach for solving LBGK scheme
Conservation of mass and flux at the grid
interface
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Thats all.Thanks to comeHoping to have been
clearHave a nice day