electrochemical processes and porous media: mathematical ... · electrochemical processes and...

Weierstrass Institute forApplied Analysis and Stochastics

Electrochemical processes and porousmedia: mathematical and numericalmodeling

Jürgen FuhrmannAlfonso Caiazzo, Klaus Gärtner, Hartmut Langmach,Alexander Linke, Hong Zhao

Mohrenstrasse 39 · 10117 Berlin · Germany · Tel. +49 30 20372 0 · www.wias-berlin.deRICAM Workshop · Linz · 2011-10-05

Electrochemistry

Corrosion

Electroplating

Biological processes (heart beat etc.)

Batteries

Conversion of chemical energy stored in compounds within the cell intoelectrical energy

Different variants (low/high temperature, solid/liquid electrolyte . . . )

Fuel cells

Invented ≈ 1840 by Schönbein, Groves Conversion of chemical energy stored in hydrogen, methanol,

carbohydrates . . . into electrical energy Continuous supply of reactants, removal of products Different variants (low/high temperature, solid/liquid electrolyte . . . )

Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · (46)

Polymer electrolyte fuel cell - membrane electrode assembly (MEA)

Hydrogen fuel cell (H2-PEMFC)Direct methanol fuel cell (DMFC)

Anode channel

Porous layer

Reaction zone

Membrane

Cathode channel

Porous layer

Reaction zone

Carbon fiber + teflon

Carbon fiber + teflon + nafion + catalyst

Nafion (proton conducting polymer)



Void space

Polymer electrolyte

Porous matrix

Catalyst particles

CH3OH,H2OH2

2H2→ 4H+ +4e−2CH3OH +2H2O→ 2CO2 +12H+ +12e−

CO2

e−

e−

–

+

LoadH+

O2,N2

4H+ +4e−+O2→ 2H2O12H+ +12e−+3O2→ 6H2O

H2O,N2

Overall reaction: 4H2 +O2→ 2H2O+energyOverall reaction: 2CH3OH +3O2→ 4H2O+2CO2 +energy

Several MEA “sandwiches” combined into a stack.

Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · (46)


Hydrogen fuel cell (H2-PEMFC)Direct methanol fuel cell (DMFC)

Anode channel

Porous layer

Reaction zone

Membrane

Cathode channel

Porous layer

Reaction zone






Void space

Polymer electrolyte

Porous matrix

Catalyst particles

CH3OH,H2OH2

2H2→ 4H+ +4e−2CH3OH +2H2O→ 2CO2 +12H+ +12e−

CO2

e−

e−

–

+

LoadH+

O2,N2

4H+ +4e−+O2→ 2H2O12H+ +12e−+3O2→ 6H2O

H2O,N2



Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 3 (46)


Hydrogen fuel cell (H2-PEMFC)

Direct methanol fuel cell (DMFC)

Anode channel

Porous layer

Reaction zone

Membrane

Cathode channel

Porous layer

Reaction zone






Void space

Polymer electrolyte

Porous matrix

Catalyst particles

CH3OH,H2O

H2

2H2→ 4H+ +4e−

2CH3OH +2H2O→ 2CO2 +12H+ +12e−

CO2

e−

e−

–

+

LoadH+

O2,N2

4H+ +4e−+O2→ 2H2O12H+ +12e−+3O2→ 6H2O

H2O,N2







Anode channel

Porous layer

Reaction zone

Membrane

Cathode channel

Porous layer

Reaction zone






Void space

Polymer electrolyte

Porous matrix

Catalyst particles

CH3OH,H2O

H2

2H2→ 4H+ +4e−

2CH3OH +2H2O→ 2CO2 +12H+ +12e−

CO2

e−

e−

–

+

LoadH+

O2,N2

4H+ +4e−+O2→ 2H2O

12H+ +12e−+3O2→ 6H2O

H2O,N2







Anode channel

Porous layer

Reaction zone

Membrane

Cathode channel

Porous layer

Reaction zone






Void space

Polymer electrolyte

Porous matrix

Catalyst particles

CH3OH,H2O

H2

2H2→ 4H+ +4e−

2CH3OH +2H2O→ 2CO2 +12H+ +12e−

CO2

e−

e−

–

+

LoadH+

O2,N2

4H+ +4e−+O2→ 2H2O

12H+ +12e−+3O2→ 6H2O

H2O,N2

Overall reaction: 4H2 +O2→ 2H2O+energy

Overall reaction: 2CH3OH +3O2→ 4H2O+2CO2 +energy



Some physical effects

Electrolyte flow (free/porous media)

Transport + diffusion of dissolved species

Charge transport in electric field

Electrostatic potential distribution

(multistep) Reactions with electron transfer

Intercalation, heat transport, swelling . . .

Aging, ripening . . .

Simplifications: high velocity asymptotics, ideal mixing, lumped reactionsGeneral case:⇒ numerical modeling


J. Newman, K. Thomas-Aleya (2004):Electrochemical systems; A. Kulikovsky (2010): Analytical Fuel Cell Modeling

Charge transport

Charge transport is considered in electrodes and electrolytes.

Electrodes:

Solid (metal, carbon, semiconductor . . . ) Mostly electronic conductors: charge carriers are electrons (e−)

Electrolytes

Liquid or solid (aquatic solution, molten salt, polymer membranes) Ionic conductors: electrons are blocked, charge carriers are ions of different

type, e.g. protons (H+).

Mixed conductors show properties of both


Charge transport: Nernst Planck Poisson system

Transport of i-th dissolved species (i = 1 . . .n)due to diffusion, electromigration, advection indilute solution⇒ Nernst-Planck equation:

~Ni =−

electromigration︷︸︸︷ziuiFci∇φ −

diffusion︷︸︸︷Di∇ci +

advection︷︸︸︷ci~v

∂tci +∇ ·~Ni = ji

Distribution of charged species⇒ self-consistent electric field⇒ Poisson equation:

−∇ ·ε∇φ = Fn

∑i=1

zici

Variables

n number of speciesφ electrostatic potentialci species concentration~Ni molar fluxzi chargeui mobilityDi diffusion coefficientε electrostatic permeabilityF Faraday constantji Reaction~v Substrate velocity


Assumptions behind Nernst-Planck

Dilute solution theory:

Ignore interactions (collisions) between different dissolved species ci.Otherwise: Stefan-Maxwell terms, “concentrated solution theory”

Velocity field not influenced by moving ions.Otherwise: contribution to momentum balance

Fluid density not influenced by concentration changesOtherwise: variable density flow

Special case: semiconductor device equations.


Bulk electroneutrality

ε << F ⇒ electroneutrality in bulk (away from interfaces and boundaries)n

∑i=1

zici = 0

Sum up equations multiplied by zi:

∂t

(n

∑i=1

zici

)−∇ ·

(n

∑i=1

z2i uiFci∇φ +

n

∑i=1

ziDi∇ci +~vn

∑i=1

zici

)=

n

∑i=1

zi ji

Express e.g. c1:

z1D1c1 =−n

∑i=2

D1zici

No bulk reactions⇒

∇ ·

(n

∑i=1

Fz2i uici∇φ +

n

∑i=2

zi(Di−D1)∇ci

)= 0

Equal diffusion coefficients or small concentration gradients⇒Ohm’s law:

∇ ·κ∇φ = 0

(κ = F

n

∑i=1

z2i uici : conductivity

)Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 8 (46)

Nernst Planck Ohm system

Transport of i-th dissolved species (i = 2 . . .n)due to diffusion, electromigration, advection indilute solution⇒ Nernst-Planck equation:

~Ni =−

electromigration︷︸︸︷ziuiFci∇φ −

diffusion︷︸︸︷Di∇ci +

advection︷︸︸︷ci~v

∂tci +∇ ·~Ni = ji

Self-consistent electric field from Ohm’s Law:

∇ ·κ∇φ = 0

Variables

n number of speciesφ electrostatic potentialci species concentration~Ni molar fluxzi chargeui mobilityDi diffusion coefficientκ conductivityF Faraday constantji Reaction~v Fluid velocity


Special case: solid electrolyte/electrode

c1: mobile charge carriersc2: immobile charges in solid lattice (u2 = D2 = 0)~v = 0⇒ c2 = const

Electroneutrality,z1 =−z2⇒ c1 = c2

small ∇c2⇒ κ = z21u1Fc1

Electrodes (graphite, metal): c1 ↔ free e−

Polymer electrolytes in fuel cells: c1 ↔ free H+.


Inert electrode-electrolyte interface

--

--

-

--

--

--

--

--

-

-

-

-

--

--

--

--

--

--

--

-

-

-

-

++

+

++

+++

++

++

+

+

- +

- +

- +

- +

- +

- +

- +Electrode

Electrons

Fixed charges at lattice

Electrolyte

Free ions

Electrodes: electron concentration as c1 ⇒ “electron potential” φs

(Acidic) electrolytes: concentration of protons (H+) as c1 ⇒ “proton potential” φl

Local electroneutrality violated in boundary layer

Potential jump at interface

Electrons and protons attracting each other may accumulate on both sides ofthe interface creating a double layer

Simple model: double layer charge Qdl = Cdl(φs−φl)


Charge transfer reactions at Interfaces

--

--

-

--

--

--

--

--

-

-

-

-

--

--

--

--

--

--

--

-

-

-

-

++

+

++

+++

++

++

+

+

- +

- +

- +

- +

- +

- +

- +Electrode

Electrons

Fixed charges at lattice

Electrolyte

Free ions

Interface reaction with electron transfer

Reaction rates in mass action law depend on potential difference

n

∑i=1

aiAi n

∑i=1

biBi +ne−+nH+

r = k+enα(φs−φl )F

RT

n

∏i=1

[Ai]ai − k−e(1−α)(φs−φl )F

RT

n

∏i=1

[Bi]bi

(φs: electron potential in electrode, φl : proton potential in electrolyte)


Catalytic interface reactions

Most reactions need initial supply of energy in order to be started.

With a cataylist, this inital energy barrier may be much lower.

Adsorption to electrode surface modeled as reaction with free catalyst sites

(electrochemical) Reaction of adsorbates with lower activation energy

Desorption from electrode surface


Hydrogen Oxidation Reaction (HOR)

Anodic reaction in H2-PEMFCs – catalyst site, ·ad – adsorbed species.

H2 +2s 2Had , r1 = k+1

cH2

cre fH2

θ2− k−1 θ

2H

Had H+ + e−+ s, r2 = k+2 eαδφF/RT

θH − k−2 e(α−1)δφF/RT aH+ θ

cH2 : hydrogen concentrationθH : Share of catalyst sites occupied by hydrogenθ = 1−θH : Share of free catalyst sites

Typical way to include these processes into global model (for given δφ ).

~NH2 = D∇cH2 + cH2~v advection-diffusion in Ω

∂tcH2 +∇ ·~NH2 = 0 continuity in Ω

~NH2 ·~n+ r1 = 0 normal flux equals reaction on Γ⊂ ∂Ω

∂tθH − r1 + r2 = 0 evolution of catalyst coverage on Γ⊂ ∂Ω


Hydrogen Oxidation Reaction (HOR)

Anodic reaction in H2-PEMFCs – catalyst site, ·ad – adsorbed species.

H2 +2s 2Had , r1 = k+1

cH2

cre fH2

θ2− k−1 θ

2H

Had H+ + e−+ s, r2 = k+2 eαδφF/RT

θH − k−2 e(α−1)δφF/RT aH+ θ

cH2 : hydrogen concentrationθH : Share of catalyst sites occupied by hydrogenθ = 1−θH : Share of free catalyst sitesTypical way to include these processes into global model (for given δφ ).

~NH2 = D∇cH2 + cH2~v advection-diffusion in Ω

∂tcH2 +∇ ·~NH2 = 0 continuity in Ω

~NH2 ·~n+ r1 = 0 normal flux equals reaction on Γ⊂ ∂Ω

∂tθH − r1 + r2 = 0 evolution of catalyst coverage on Γ⊂ ∂Ω


Oxygen Reduction Reaction (ORR)

Multistep cathodic reaction in H2-PEMFC and DMFC

Oxygen reaction is split into several steps:

O2 + sk±1 O2,ad

O2,ad +H+ + e−k±2 HO2,ad

HO2,ad +H+ + e−k±3 H2O2,ad

H2O2,ad +2H+ +2e−k±4 2H2O+ s


Methanol Oxidation Reaction (MOR)

Anodic reaction in DMFC

CH3OH +2s1k±1 (CH2−OH)ad +Had

(CH2−OH)ad + s1k±2 (CH−OH)ad +Had

(CH−OH)ad + s1k±3 (C−OH)ad +Had

(C−OH)ad + s1k±4 (C−O)ad +Had

(C−O)ad +OHadk±5 (COOH)ad + s2

(COOH)ad +OHadk±6 CO2 +H2O+ s1 + s2

Hadk±7 H+ + s1 + e−

H2O+ s2k±8 OHad +H+ + e−


Open questions in catalysis modeling

Essentially, the way of modeling these reactions is guided by heuristics. Theproposed reaction chains follow one particuala hypothesis about the pathways.

Some missing effects:

Catalyst surface defects are preferred adsorption sites

Different crystal directions have different reaction speeds

Multisite adsorption processes

Restructuring of catalyst surface due to reaction

Elementary processes at electrodes may be more complex

Pt catalyst dissolution

Ripening of catalyst particles

More problems

“Sticky” intermediates (e.g. CO) block catalyst sites


Porous electrodes

Current proportional to interface area⇒ increase interface area per volume⇒ porous electrode

homogenization⇒ two coupled continua:

conducting matrix electrolyte in pore space

Solid-liquid interface is distributed in the volume

Surface reaction terms→ volume reaction terms

Fluid flow→ Darcy law

Nernst-Planck equation→ reactive transport in porousmedium

Newman/Tiedeman: Porous electrodes in batteries, double layer capacitors

Neu/Krassowska, Pennachio/Savare/Colli/Franzone: “Bidomain equation”(extracellular+intercellular space in cardiac tissue)

Moyne/Murad/Bennethum/Cushman: clays, concrete (coupled with swelling)


Potential distribution in porous electrodes

−∇ ·ksσ∇φs =−aF jF −∂tQdl

−∇ ·klκ∇φl =aF jF +∂tQdl

jF =i0F

eαF

RT (φs−φl)

− i0F

e(α−1) FRT (φs−φl)

Qdl =Cdla(φs−φl)

φs solid matrix potentialφl electrolyte potential

Cdl double layer capacityQdl double layer chargeks,kl pore space dependent factors

σ conductivity of pure solidκ conductivity of pure electrolytea interface area per volumei0 exchange current density

Model limitations/extensions:

Pore size should exceed thickness of boundary layers

Low concentration gradients


Proton Exchange Membrane

Nafion conductivity (Tohoku Univ.)

MEA by Gore Associates

e.g. Nafion c©DuPont: “Solid acid”: sulfonic acid groups fixed at

PTFE backbone

Dissociation in presence of liquid water⇒free H3O+ ions available for conduction

⇒ conductivity depends on water content

⇒ water management needs to guarantee itspresence

Problems associated with membrane

Reactant crossover

Swelling

free radical attacks from reactionintermediates


Two phase flow

Cathode

supply of O2 (gaseous, low solubility) remove H2O in gaseous and liquid form

Anode (DMFC)

supply of reactant dissolved in water removal of CO2 in dissolved and gaseous form

Liquid H2O is needed in order to maintain membrane conductivity

Phases need to move in opposite directions

Pores kept open for gas flow by admixture of teflon

Model by standard ansatz with modified capillary pressure/saturation curve


Mixed wettability: Bernoulli function based ansatz for s(pc)

0

0.2

0.4

0.6

0.8

1

-40 -30 -20 -10 0 10 20 30 40

Effe

ctiv

e Sa

tura

tion

se

Capillary Pressure pc

1:21:1

Bernoulli

Measured values for drying andwetting branches of se(pc) forquartz sand/teflon mixture withdifferent compositions (triangles);Least squares fit to Bernoullifunction based ansatz (lines)

Bernoulli function: B(x) = xex−1

pδc =α(pc− p 1

2)

se(pc) =−B′(

B(pδc )βw +B(−pδ

c )βn)

sw(pc) =sresw +(1− sres

g − sresw )se(pc)

Ustohal et. al. (1998): measured saturationcurves for quartz sand/teflon mixtures ofvarying composition

Divisek et. al. (2003): Bernoulli function basedparametrization fitting well to thesemeasurements

Measurements for fuel cell PTL ?

Hysteresis ?


Ustohal/Stauffer/Dracos, J. Cont. Hydrol. 1998; Divisek/Fuhrmann/Gärtner/Jung, JES 2003

Stefan-Maxwell: two phase flow with gas mixture

Stefan–Maxwell mean transport pore model for a mixture of ng gases empiricallycombined with two phase flow

∂t

(φ(1− sw)

pgiRT

)+∇ ·Ngi = Rg

chemi +Rg

evi

Ngi =−krg

ng

∑j=1

DSMi j (p1, . . . , png)∇pg j

pgi partial pressure of ith componentNgi molar flux of ith componentR gas constantT temperaturekrg(sw) rel. permeabilityDSM

i j Stefan Maxwell Diffusion matrixRg

evi evaporation reaction

Rgchemi chemical reactions


Stefan–Maxwell Gas Transport – Diffusion Matrix

Diffusion matrix DSMi j implicitely defined by

RT

(NgiDK

i+

ng

∑i=1

yiNg j− y jNgi

Dbmi j

)= ptot∇yi +yi

(Bi

DKi

+ng

∑i=1

Bi

Dmi j

y j

(1−

B j

Bi

))∇ptot.

ptot = ∑ pgi total gas pressureyi = pgi/ptot molar fractionsDK

i Knudsen diffusions coeff.Dm

i j binary diffusion coeff.λi mean free path gas i,r mean pore radiusKi = λi/r Knudsen numberBi effective permeability

Definition of capillarypressure: pc = ptot− pw


Further effects

Evaporation + Condensation

Volumetric reaction between dissolved and gas phase

Henry’s law gives equilibrium

Kinetic constant ?

Among other effects, the model contains reaction CO2|dissolved CO2|gas

Heat transport

Drag effect: force term in Darcy’s law form ion movement


Coupling between supply channels and porous electrodes

Challenge: two phase coupling

Droplets in H2 PEMFC CO2 bubbles in DMFC

Kandlikar et al., Mech Engr. 2009 N. Paust et al. IMTEK, Freiburg Univ.


Mathematical/Numerical Model of a DMFC MEA

1D/2D/3D- Numerical model of a DMFC MEA including all the processes describedabove in cooperation with FZ Jülich

MEA processes taken into account intemporal and spatial resolution.

11 coupled nonlinear partial differentialequations.

12 nonlinear algebraic equations perdiscretization node.

Resolution of catalytic reaction chains.

Two-phase flow model taking into accountmixed wettability.

0

0.5

1

0 500 1000 1500

Vol

tage

(V)

Current Density (A/m2)

Meas. 0.5 mol/lMeas. 1 mol/lMeas. 2 mol/l

Sim., 0.5 mol/lSim., 1 mol/lSim., 2 mol/l

Measured and calculatedpolarization curves at 60oC.


Divisek/Fuhrmann/Gärtner/Jung, J. Electrochem. Soc. (2003)

Influence of anodic saturation curve

0

0.2

0.4

0.6

0.8

1

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

Satu

rati

on (m

3 /m3 )

Capillary Pressure (bar)

p1/2=0.5 barp1/2=1.0 barp1/2=1.5 bar

0

0.5

1

0 500 1000 1500

Vol

tage

(V)


Meas. 0.5 mol/lp1/2=0.5 barp1/2=1.0 barp1/2=1.5 bar

Capillary pressure vs. saturation polarization curves

Best anode performance for more wettable material


J. Fuhrmann, K Gärtner, in: Device and Materials Modeling in PEM Fuel Cells, vol. 113 of Topics in Applied Physics, 2009

Influence of cathodic saturation curve

0

0.2

0.4

0.6

0.8

1

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

Satu

rati

on (m

3 /m3 )

Capillary Pressure (bar)

p1/2=0.5 barp1/2=1.0 barp1/2=1.5 bar

0

0.5

1

0 500 1000 1500

Vol

tage

(V)


measured, 0.5 mol/lmost wettable

medium wettableleast wettable

Capillary pressure vs. saturation polarization curves

Best cathode performance for less wettable material


J. Fuhrmann, K Gärtner, in: Device and Materials Modeling in PEM Fuel Cells, vol. 113 of Topics in Applied Physics, 2009

Flow cell modeling

Model simple xperimental devices to improve understanding of partial processes.

A+ s B+ s C + s Dissolved in fluidk±A k±B k±C Adsorption/Desorption

Aads

k±AB

Bads

k±BC

Cads Adsorbed at catalyst surface

H+ + e− H+ + e− Released ions

Thin-layer flow cell

cA inqin

Γin

cAout

cBout

cCout

Γout

Mass Spec

I + –

catalyst surface

Ω

Anode

Cathode

How does thefraction of thedesorbedintermediate Bdepend on thecatalystconcentration ?


Model multistep surface reaction in a flow cell

Rate expressions:

rA = k+A cAθf−k−A θA

rAB = k+ABθA−k−ABθB

rB = k+B cBθf− k−B θB

rBC = k+BCθB−k−BCθC

rC = k+C cCθf− k−C θC

Algebraic conditions for adsorbedspecies:

dθA/dt−rA− rAB = 0

dθB/dt−rB + rAB− rBC = 0

dθC/dt−rC + rBC = 0

θX – fraction of catalyst sites occupied by adsorbed species

θ f = 1−θA−θB−θC – fraction of free catalyst sites


Solute transport coupled with surface reaction

Stationary diffusion and transport of species X = A,B,C in velocity field~v:

∂cX

∂ t−div(DX gradcX −~vcX ) = 0

Boundary conditions:

cX = cX ,in on Γin Dirichlet

(DX gradcX − cX~v) ·~n =−cX~v ·~n on Γout Outflow

(DX gradcX − cX~v) ·~n = ce f fcat rX on Γcat Electrode reaction

∂cX

∂~n= 0 on Γ\ (Γin∪Γout∪Γcat) No flow

Outflow of species X :

Xout =∫

Γout

(DX gradcX − cX~v) ·~n ds

Relative amount of reaction intermediates:

Iout = Bout/(Bout +Cout)


H2O2 yield in catalytic oxygen reduction

0 0.2 0.4 0.6 0.8 1catalyst coverage

0

5

10

15

20

25

30

rela

tive

amou

nt o

f re

actio

n in

term

edia

te/%

O2(sol) H2O2(sol) H2O

O2(ads) H2O2(ads) H2O(ads)

Catalyst loading effect:H2O2 yield decreases with increasingcoverage of active Pt nanodisks on planarglassy carbon substrate (nanostructuredPt/GC model electrodes)

Top: Experimental, bottom: simulation.


Fuhrmann/Zhao/Langmach/Seidel/Jusys/Behm, Fuel Cells 2011

H2O2 yield in catalytic oxygen reduction

0 5 10 15 20 25 30

flow velocity /µL s-1

0

2

4

6

8

rela

tive

amou

nt o

f re

actio

n in

term

edia

te/%

O2(sol) H2O2(sol) H2O

O2(ads) H2O2(ads) H2O(ads)

H2O2 yield increases with increasingelectrolyte flow velocity for

two different densities of similarlysized Pt nanodisks

fully Pt covered electrode.

Top: Experimental, bottom: simulation.


Fuhrmann/Zhao/Langmach/Seidel/Jusys/Behm, Fuel Cells 2011

Choice of numerical methods

Challenges:

Nonlinear coupling

Dominant convection

Heterogeneous species distributions

Wishes:

Maximum principle (no unphysical oscillations)

Preservation of a priori bounds (positivity)

Convergence, Accuracy, Efficiency

Choices:

Voronoi finite volume method: stability but limited convergence ordere.g. Fuhrmann/Linke/Langmach, Appl. Num. Math. 2011

Stabilized FEM: high convergence order, but no preservation of a priori bounds.John/Knobloch, Comput. Methods Appl. Mech. Engrg. 2008

Discontinuous Galerkin: high flexibility wrt. grids, high convergence order, but nopreservation of a priori bounds, complicated discrete systems


Diffusion-Convection-Reaction system

N partial differential equations of convection/diffusion/reaction type in a one-, two orthree-dimensional domain with appropriate boundary conditions.

∂tb(~x,u)+∇ ·~j(~x,u)+ r(~x,u) = 0

u(~x, t) = (u1(~x, t) . . .uN(~x, t)): vector valued functionb(~x,u) = (b1(~x,u) . . .bN(~x,u)): vector of storage termsr(~x,u) = (r1(~x,u) . . .rN(~x,u)): vector of reaction terms~j(~x,u) = (~j1(~x,u) . . .~jN(~x,u)): vector of flux terms

~j(~x,u): sum of convective and diffusive fluxes.

Porous electrodes

Two phase flow

Density driven flow

Charge transport in semiconductor devices

Reaction-Diffusion systems

. . .


Two Point Flux Voronoi Finite Volume Method

~xK~xL

~xM

K L

Integral over space-time REV control volume

Newton/Leibniz, Gauss, quadrature

Approximation of inter-volume fluxes

0 =1

tn− tn−1

tn∫tn−1

∫K

(∂tb(~x,u)+∇ ·~j(~x,u)+ r(~x,u)

)dxdt

0 = |K|

(b(~xK ,un

K)−b(~xK ,un−1K )

tn− tn−1 + r(~xK ,unK)

)−∫

∂K

~j(~x,un) ·~nds

0 = |K|

(b(~xK ,un

K)−b(~xK ,un−1K )

tn− tn−1 + r(~xK ,unK)

)

− ∑L nb. of K

|∂K∩∂L||~xK −~xL|

g(~xK ,~xL,unK ,un

L)


Macneal, Quart. Math. Appl. 1953

Exponential fitting flux for linear problems

Scalar, linear convection diffusion flux in given velocity field v(~x):

j = D∇u+uv(~x)

Flux projection onto control volume normal~xK~xL:

vKL :=1

|∂K∩∂L|

∫∂K∩∂L

v(s) ·(~xK −~xL)ds

Upwind numerical flux:

g(uK ,uL) := D(

B(vKL

D

)uK −B

(−vKL

D

)uL

)

(B(ξ ) = ξ

e−ξ−1 : Bernoulli function)


Allen/Southwell, Quart. J. Mech. Appl. Math. 1955; Il’in, Mat. Zametki 1969; Scharfetter/Gummel, IEEE Trans. El. Dev. 1969

Generalization to nonlinear problems

Scalar, nonlinear convection diffusion flux in given velocity field v(~x):

j = D(u)∇u+F(u)v(~x)

Define g(uK ,uL) := G from the solution of a local Dirichlet problem for the projectionof the equation onto the edge~xK~xL:Let w(ξ ) := u(~xL +ξ (~xK −~xL)):

D(w)w′+F(w)vKL = Gw(0) = uL

w(1) = uK


Eymard/Fuhrmann/Gärtner, Numer. Math. 2006

Maximum principle

Stationary convection diffusion problem:

j = D∇u+uv(~x), ∇ · j = 0

Fundamental property of solution: “no unphysical oscillations”

Divergence free velocity field (∇ ·v = 0)⇒ continuous maximum principle:solution in a given point x is bounded by its values in a surroundig of x.

Discretely divergence free velocity projection vKL

∑L nb. of K

|∂L∩∂K||~xK −~xL|

vKL = 0

⇒ discrete maximum principle for discrete problem:solution in point xi is bounded by values in neigboring points x j.


Divergence free discrete fluxes

How to guarantee discrete divergence free condition when coupling to flowproblems?

From pointwise divergence free velocity field v(x) by exact calculation of

vKL =1

|K∩L|

∫K∩L

v(s) ·(~xK −~xL)ds

Analytical expressions (Hagen - Poiseuille, von Karman . . . ) Pointwisde divergence free finite elements

Finite volume scheme for flow problem leading to discrete divergence free fluxes

Darcy (porous media) flow: v = K∇p, vKL = K(pK − pL) Compatible finite volume solution for Navier-Stokes: work in progress


Divergence free finite elements

div(velocity space)⊂ pressure space

Lowest order Scott Vogelius elements:(Pd ,P−(d−1)) (d:space dimension)

Stable on macro triangulations

Maintain two independent discretizations for transport (FV) and for flow (FE)

(FV): For every triangle T , calculate contributions σKL;T = ∂K∩∂L∩T to∂K∩∂L

(FE): Calculate velocity projections vKL;T from continuous FE velocity field

(FV): Assemble vKL from vKL;T


Burman/Linke, App. Num. Math 2008; Fuhrmann/Linke/Langmach, Appl. Num. Math. 2011

Compatible finite volume method

After Nicolaides - extension of MAC scheme:

Choice of unknowns:

Pressures pK on simplex mesh nodes

Velocities qKL along simplex mesh edges ≡ normal to Voronoi box faces

Discrete operators divh,gradh, roth, vector calculus (up to boundary terms):

divh = gradTh

divh gradh = rotTh roth−gradh divh

rotTh gradh = 0

divh roth = 0

Discrete Stokes system: rotTh roth v−gradh p = 0

divh v = 0


Eymard/Fuhrmann/Linke, Proc. FVCA6, 2011

pdelib2/fvsys API

Problem description

Grid

Accumulation terms b(~xK ,uK)

Reaction terms r(~xK ,uK)

Fluxes between control volumes g(~xK ,~xL,uK ,uL)

Species sets varying between subdomains and boundary parts

Solution strategy

Adaptive time step selection

Nonlinear solver: Newton’s method

Linear solver: PARDISO (direct), AMG with point block smoothers

Implementation

OpenMP based parallelization

Portable to Linux/Unix/Mac/Windows

OpenGL online graphics, FLTK GUI, Lua scripting language


www.pdelib.org

Advantages of Voronoï Methods

Focus on conservation character of mathematical model

Straigthforward use of two point finite difference formulae, upwinding

Convergence results for many cases

Simplex based assembly loop as in FEM: data assembled from simplicialcontributions⇒ known techniques for parallelization etc. apply

Heterogeneities represented by mesh⇒ no averaging across interior boundaries

M-property + discrete maximum principle for Laplacian and properly upwindedconvection problems in 2/3D⇒ conservation of qualitative properties (positivity, dissipativity)⇒ physically meaningful a priori bounds

Implementation tolerant to Delaunay violations:Algebraic expression for “|∂K∩∂L|” may become negative, but still can becalculated


Challenges for Voronoï Methods

Simple scheme but considerable efforts in mesh generation.

No higher order version⇒ high accuracy solutions need very fine meshes.

Challenges

Reliable boundary conforming Delaunay meshing for general domains

Anisotropic problems −→ mesh alignment to anisotropy direction

Boundary + interior layer resolution −→ anisotropic meshes aligned with layers

Storage terms may be overestimated at heterogeneity boundaries−→ “double layer” meshing at boundary


Outlook

Current activities:

CO Electrooxidation on Pt surfaces (Experiments of K. Krischer, Munich)

Navier-Stokes/Porous media coupling with special emphasis on porouselectrodes (Poster of A. Caiazzo)

Corrosion modeling (with C.Chainais (Lille))

Nernst-Planck-Poisson-Navier-Stokes coupling using finite volume methods


electrochemical processes and porous media: mathematical ... · electrochemical processes and...

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